Factorization and Non-Local 1/mb Corrections
in the Decay B̄ → Xsγ
Dissertation
zur Erlangung des Grades
“Doktor der Naturwissenschaften”
am Fachbereich Physik, Mathematik und Informatik
der Johannes Gutenberg-Universität
in Mainz
Michael Benzke
geboren in Essen
Mainz, 2011
Tag der Promotion: 10. Juni 2011
Abstract
In this thesis, a systematic analysis of the B̄ → Xsγ photon spectrum in the endpoint
region is presented. The endpoint region refers to a kinematic configuration of the final
state, in which the photon has a large energy mb − 2Eγ = O(ΛQCD), while the jet has a
large energy but small invariant mass. Using methods of soft-collinear effective theory and
heavy-quark effective theory, it is shown that the spectrum can be factorized into hard,
jet, and soft functions, each encoding the dynamics at a certain scale. The relevant scales
in the endpoint region are the heavy-quark mass mb, the hadronic energy scale ΛQCD and
√
an intermediate scale ΛQCDmb associated with the invariant mass of the jet. It is found
that the factorization formula contains two different types of contributions, distinguishable
by the space-time structure of the underlying diagrams. On the one hand, there are the
direct photon contributions which correspond to diagrams with the photon emitted directly
from the weak vertex. The resolved photon contributions on the other hand arise at
O(1/mb) whenever the photon couples to light partons. In this work, these contributions
will be explicitly defined in terms of convolutions of jet functions with subleading shape
functions. While the direct photon contributions can be expressed in terms of a local
operator product expansion, when the photon spectrum is integrated over a range larger
than the endpoint region, the resolved photon contributions always remain non-local. Thus,
they are responsible for a non-perturbative uncertainty on the partonic predictions. In
this thesis, the effect of these uncertainties is estimated in two different phenomenological
contexts. First, the hadronic uncertainties in the B̄ → Xsγ branching fraction, defined
with a cut Eγ > 1.6GeV are discussed. It is found, that the resolved photon contributions
give rise to an irreducible theory uncertainty of approximately 5%. As a second application
of the formalism, the influence of the long-distance effects on the direct CP asymmetry
will be considered. It will be shown that these effects are dominant in the Standard Model
and that a range of −0.6 < ASMCP < 2.8% is possible for the asymmetry, if resolved photon
contributions are taken into account.

Contents
1 Introduction 6
1.1 Motivation: Why we fight . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Structure of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Experimental Status of B̄ → Xsγ . . . . . . . . . . . . . . . . . . . . . . . 10
2 The Decay B̄ → Xsγ in Full QCD 14
2.1 The Weak Effective Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 Integrating out the Heavy Degrees of Freedom . . . . . . . . . . . . 14
2.1.2 The Wilson Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.3 The Renormalization Group . . . . . . . . . . . . . . . . . . . . . . 20
2.1.4 The Operator Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.1.5 The Wilson Coefficients at the Correct Scale . . . . . . . . . . . . . 22
2.2 The Differential Decay Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.1 Next-to-Leading Order Matrix Elements . . . . . . . . . . . . . . . 28
2.3 The Integrated Decay Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Conclusions of this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Heavy-Quark Effective Theory 34
3.1 Hadronic Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Heavy-Quark Effective Theory . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Leading Power Matching and OPE . . . . . . . . . . . . . . . . . . . . . . 41
3.3.1 The Light-Cone Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.2 Calculation Continued . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.3 The HQET Trace Formalism . . . . . . . . . . . . . . . . . . . . . . 43
3.3.4 The Leading Power and the Shape Function . . . . . . . . . . . . . 45
3.4 Higher Power Corrections to the Total Rate . . . . . . . . . . . . . . . . . 48
3.5 Conclusions of this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 Soft-Collinear Effective Theory 50
4.1 Kinematics and Power Counting . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Soft-Collinear Effective Theory . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.1 Gauge Transformations and Wilson Lines . . . . . . . . . . . . . . . 55
4.2.2 Expansion and Lagrangian . . . . . . . . . . . . . . . . . . . . . . . 56
2
4.3 Factorization at Leading Power . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.1 Tree Level Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.2 Loop Level Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.3 The Factorization Formula . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.4 Matching the Jet Function . . . . . . . . . . . . . . . . . . . . . . . 64
4.4 Renormalization Group Running . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Conclusions of this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Subleading Contributions in SCET 68
5.1 Matching onto SCET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.2 The Subleading Q7γ −Q7γ Contribution . . . . . . . . . . . . . . . . . . . 70
5.2.1 Tree Level Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2.2 Factorization and the Decay Rate . . . . . . . . . . . . . . . . . . . 71
5.2.3 The Subleading Jet Functions . . . . . . . . . . . . . . . . . . . . . 72
5.2.4 The Subleading Shape Functions . . . . . . . . . . . . . . . . . . . 74
5.2.5 Kinematic vs Hadronic Power Corrections . . . . . . . . . . . . . . 77
5.2.6 Strange Quark Mass Effects . . . . . . . . . . . . . . . . . . . . . . 79
5.3 Factorization at Subleading Power . . . . . . . . . . . . . . . . . . . . . . . 80
5.4 Operator Matching onto SCET . . . . . . . . . . . . . . . . . . . . . . . . 83
5.4.1 Photon Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4.2 Operator Matching of Q8g . . . . . . . . . . . . . . . . . . . . . . . 84
5.4.3 Operator Matching of Qq1 . . . . . . . . . . . . . . . . . . . . . . . . 86
5.5 Conclusions of this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6 Analysis of Subleading Contributions to the Photon Spectrum 88
6.1 Analysis of the Qq1 −Q7γ Contributions . . . . . . . . . . . . . . . . . . . . 88
6.1.1 Qq1 −Q7γ : Facts about the Soft Function . . . . . . . . . . . . . . . 93
6.2 Analysis of the Qq1 −Qq1 and Qq1 −Q8g Contributions . . . . . . . . . . . . 95
6.3 Analysis of the Q8g −Q8g Contribution . . . . . . . . . . . . . . . . . . . . 96
6.3.1 Q8g −Q8g: Details about the Factorization . . . . . . . . . . . . . . 100
6.4 Analysis of the Q7γ −Q8g Contribution . . . . . . . . . . . . . . . . . . . . 102
6.4.1 Q7γ −Q8g in the Vacuum Insertion Approximation . . . . . . . . . 105
6.5 Constraints from PT Invariance . . . . . . . . . . . . . . . . . . . . . . . . 106
6.6 Conclusions of this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 108
7 Phenomenology I:
Hadronic Uncertainties in the Integrated Decay Rate 110
7.1 Partially Integrated Decay Rate . . . . . . . . . . . . . . . . . . . . . . . . 110
7.2 Hadronic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.3 Analysis of the Qc1 −Q7γ Contribution . . . . . . . . . . . . . . . . . . . . 116
7.4 Analysis of the Q7γ −Q8g Contribution . . . . . . . . . . . . . . . . . . . . 118
7.5 Analysis of the Q8g −Q8g Contribution . . . . . . . . . . . . . . . . . . . . 120
7.6 Conclusions of this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3
8 Phenomenology II:
Direct CP Violation 124
8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.2 The CP Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
8.3 Phenomenology in the SM . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
8.4 Phenomenology beyond the SM . . . . . . . . . . . . . . . . . . . . . . . . 130
8.5 Comments on B̄ → Xdγ and B̄ → Xs+dγ . . . . . . . . . . . . . . . . . . . 131
8.6 Conclusions of this Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9 Conclusions 134
9.1 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
A Definitions and Numbers 138
A.1 The Penguin Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
A.2 Input Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
B Operator Matching onto SCET 141
B.1 Tree-Level Matching of Q8g . . . . . . . . . . . . . . . . . . . . . . . . . . 141
B.2 Tree-Level Matching of Qq1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
B.3 Loop-Level Matching of Qq1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
B.3.1 Matching with Aem gluon⊥ and As . . . . . . . . . . . . . . . . . . . . 145hc
B.3.2 Matching with Aem gluon⊥ and Ahc ⊥hc . . . . . . . . . . . . . . . . . . . . 145
4
5
Chapter 1
Introduction
1.1 Motivation: Why we fight
Today, the Standard Model of particle physics (SM) is able to explain most of the physics
at microscopic scales. Mathematically it corresponds to a gauge theory based on the local
symmetry group SU(3)c × SU(2)L × U(1)Y , which naturally decomposes into a strong
(SU(3)c) and an electroweak sector (SU(2)L × U(1)Y ).
Based on the symmetry group of the strong sector, the Lagrangian of quantum chromo-
dynamics (QCD) is derived, which describes the fundamental interactions between colored
particles, i.e., quarks and gluons. Due to the non-abelian nature of the strong interaction a
perturbative calculation of physical processes involving quarks and gluons is only possible
at energy scales above ∼ 1GeV, where the strong coupling constant is small.
The electroweak sector of the SM exhibits several interesting features that are not
present in QCD. First, all fermions are charged under the electroweak gauge group and
therefore participate in the weak interaction. Second, its symmetry group SU(2)L×U(1)Y
is spontaneously broken down to U(1)EM which generates the masses of the heavy vector
bosons W± and Z. In the SM, this symmetry breaking is realized by the introduction
of a new scalar doublet field with a non-vanishing vacuum expectation value (VEV). In
this realization the Higgs boson appears as a new physical particle, which has yet to be
discovered in experiments. Finally, there are Yukawa couplings between quarks of different
flavor. After diagonalizing the quark mass matrix, these give rise to flavor changing in-
teractions, whose strength is encoded in the Cabibbo-Kobayashi-Maskawa (CKM) mixing
matrix. Possible physical complex phases in this mixing matrix lead to violations of the
charge-parity (CP ) symmetry.
Modern particle physics strives to improve our understanding of natural processes at the
quantum level. Through the analysis of decays of quark-antiquark bound states (mesons)
it is possible to determine free parameters of the SM like the CKM matrix elements as
well to discover physics beyond the SM (BSM). One particularly interesting process of this
type is the decay of a B meson into a photon and a charmless hadronic final state with
6
non-zero strangeness:
B̄ → Xsγ .
The particle B̄ includes charged (B−) as well as neutral (B̄0) B mesons with a bottom
quantum number of -1. The corresponding process with a bottom quantum number of +1
can be deduced by CP conjugation. This decay is interesting for several reasons. First
it plays an important role in the determination of the CKM matrix element Vub via the
semi-leptonic decay B̄ → Xulν̄. To a good approximation the hadronic uncertainties of this
decay and B̄ → Xsγ are equal and therefore it is possible to eliminate this uncertainty in
the measurement of Vub through a measurement of the B̄ → Xsγ decay rate [1,2]. Second,
direct CP violation may be observed by comparing B̄ → Xsγ to its CP conjugate process.
Finally, the process is mediated by the flavor changing neutral current (FCNC) b → sγ.
In the SM such a process is only possible at loop level (see first diagram in Figure 1.1).
This opens a possible window into BSM physics, since new particles can contribute to
b W s b W̃ s
u, c, t ũ, c̃, t̃
γ γ
Figure 1.1: The b → sγ FCNC in the SM (left) and in the MSSM (right).
loop diagrams even if there is not enough energy to produce them directly. One example
would be the FCNC in the minimal supersymmetric standard model (MSSM) containing
a squark and a chargino (see second diagram in Figure 1.1). Furthermore, certain extra-
dimensional models can introduce FCNCs at the tree level, which would also cause a
measurable deviation from the SM prediction. For a recent review dealing with B̄ → Xsγ
in BSM models see [3].
The decay b → sγ itself is mediated by the weak interaction. However, theoretical
predictions are complicated by the fact that the quarks are not free but bound into hadrons
by the strong force. Since the QCD coupling constant is large at the typical hadronic
energy scales, a purely perturbative approach is inappropriate. Effective field theories
allow to separate the calculation into a high-energy (perturbative) and a low-energy (non-
perturbative) part. While the high-energy part can in principle be determined to any order
in perturbation theory, the effect of the low-energy part can be estimated by the use of
phenomenological models. Together with the uncertainties of the perturbative calculation
these hadronic uncertainties principally restrict the precision of any prediction concerning
the B̄ → Xsγ decay in the SM as well as BSM.
In the past considerable progress has been made in the ambitious endeavor of calculating
the decay rate of B̄ → Xsγ as precisely as possible. The starting point for all calculations is
an effective theory in which all fields with a mass above the typical energy scale of B decays
(the bottom-quark mass mb) are integrated out. At leading order the transition is then
7
mediated by a single local operator Q7, containing just a photon, a bottom- and a strange-
quark. Since the mass of the decaying b-quark is large compared to the energy scale of
the hadronic binding effects, it is possible to use a further effective theory that implements
an expansion in powers of the inverse heavy-quark mass 1/mb. The appropriate theory is
the heavy-quark effective theory (HQET, for a throughout review see e.g. [4]). As will be
shown, the first term in this expansion corresponds to the decay of a free quark, which can
be calculated perturbatively. Corrections to this result are suppressed by at least 1/mb.
Today, the perturbative calculation of the decay rate has been performed up to next-to-
next-to-leading order (NNLO) in renormalization-group improved perturbation theory [5].
However, experimental measurements only include B̄ → Xsγ decays with photon energies
above ∼ 1.7GeV. In this so called endpoint region the hadronic jet Xs has an energy of the
√
order of the large scalemb while its invariant mass is much smaller at the order of mbΛQCD
which itself is much larger than the hadronic energy scale ΛQCD. Due to this particular
setup, it is no longer possible to employ an expansion in terms of local operators (OPE).
Also, potentially large logarithms appear in the perturbative calculation, which need to be
resummed to make the perturbative expansion meaningful. This resummation is possible
within the framework of the soft collinear effective theory (SCET) [6,7]. Alternatively it is
possible to extrapolate the experimental data to a photon cut outside the endpoint region.
For a cut Eγ > 1.6GeV the perturbative calculation yields a branching fraction of [5]
B(B̄ → Xsγ) = (3.15± 0.23) · 10−4.
A dedicated analysis of cut-related effects and uncertainties gives the slightly lower value
B(B̄ → Xsγ) = (2.98 ± 0.26) · 10−4 [8]. The errors given above consist of higher order
perturbative uncertainties (3%), parametric dependencies (3%), a charm-quark mass in-
terpolation ambiguity (3%) and—most significantly—non-perturbative uncertainties (5%).
The size of these non-perturbative uncertainties was estimated by naive dimensional anal-
ysis. In order to systematically study these effects one has to go beyond the leading power
in the heavy-quark expansion. Some of these effects have already been considered [2,9,10],
but a dedicated analysis was missing.
The primary goal of this work is to use SCET for a systematic study of subleading
power1 corrections to the B̄ → Xsγ decay in the endpoint region, which has not been
performed before. It was shown in [11, 12] that the differential decay rate of B̄ → Xsγ
obeys a factorization formula at leading power in the SCET expansion
dΓ(B̄ → Xsγ) = H · J ⊗ S ,
where ⊗ denotes a convolution. Each of the components describe the physics at a certain
scale (hard, intermediate and soft). While the hard function H and the jet function J
are perturbatively calculable, the soft function S is related to a hadronic matrix element.
When integrating the differential rate over a range much larger than the endpoint region
1The term “subleading” always refers to higher order terms in the expansion parameter of the effective
theory, while “next-to-leading” refers to higher order terms of the perturbative expansion in the coupling.
8
(i.e. including photons with an energy such that mb − 2Eγ ≫ ΛQCD) the expression above
reduces to an expansion in terms of local operators [13]. In this thesis a new factorization
formula for the differential rate in the endpoint region, valid at any order in the 1/mb
expansion will be established. It will introduce matrix elements of non-local operators
that do not reduce to local ones in the integrated rate [14]. In the end, the effect of these
non-perturbative uncertainties on the integrated decay rate, as well as on the direct CP
violation will be estimated. The main results of this work have been published in [15, 16].
It will turn out, that the factorization resolves several issues of the partonic calculation,
such as large logarithms and dependencies on low-energy parameters like the strange quark
mass. The numerical analysis of the non-perturbative effects will show that the irreducible
uncertainty of the partonic prediction of the branching fraction amounts to approximately
5%. Furthermore, it will be found that these effects dominate the prediction of the SM
CP asymmetry.
1.2 Structure of the Thesis
This work can be roughly divided into two parts, with the borderline running between Sec-
tion 5.2.6 and 5.3. Before that line the theoretical background will be extensively reviewed,
in order to motivate the approach and put it into a conceptual context. Afterwards the
new findings of this work will be presented.
The first part starts with a brief review of the experimental situation in Section 1.3.
Chapter 2 will then introduce the fundamental concepts of the partonic calculation of the
B̄ → Xsγ branching fraction. This includes the derivation of an appropriate set of opera-
tors and a discussion of the influence of renormalization. During the presentation several
issues concerning the calculation will be noted. One of these issues are the corrections due
to the binding of the partons into hadronic states. In Chapter 3 the heavy-quark effec-
tive theory will be introduced, which offers a way to deal with hadrons that contain one
heavy quark. The soft-collinear effective theory, which offers a systematic way to resum
large logarithms and deal with the breakdown of the local OPE will be discussed in Chap-
ter 4. Following up, the subleading contributions in the SCET expansion are considered
in Chapter 5. The first part of this chapter will discuss the subleading contributions due
to an interference of the electromagnetic dipole operator with itself. The second part will
then set the stage for the systematic analysis of all subleading contributions, by discussing
the matching of QCD onto SCET. This analysis is explicitly performed in Chapter 6.
Chapters 7 and 8 will then apply the results of the analysis in two different phenomeno-
logical contexts, i.e., the irreducible non-perturbative uncertainty on the prediction of the
B̄ → Xsγ branching fraction and the direct CP asymmetry. Chapter 9 will sum up the
results and draw conclusions. Finally, three appendices contain information on the pen-
guin function, the set of parameters used in this work, and a complete list of the SCET
operators up to a suppression of 1/mb compared to the leading power.
9
1.3 Experimental Status of B̄ → Xsγ
In order to illustrate the motivation behind the complex theoretical apparatus developed
for describing rare inclusive B decays the experimental situation is briefly reviewed (for
a more detailed review see [17, 18]). The primary quantity of interest is the branching
fraction of B meson decays mediated by a flavor changing neutral current. One of the
most important processes of this type is the decay into a photon and a charmless hadronic
final state with non-zero strangeness. In contrast to the experimental point of view, which
prefers exclusive measurements for easier background elimination, it is more desirable from
a theoretical perspective to measure an inclusive process, since this makes the difficult
modeling of the hadronic final states unnecessary.
In the dedicated B factories BaBar [19] and Belle [20] the B mesons are produced
by operating an electron positron collider at the Υ(4S) resonance, which predominantly
decays into pairs of neutral (B0, B̄0) or charged (B+, B−) B mesons (in approximately
equal parts). These in turn primarily decay into charmed final states via the CKM favored
charged current process b → cW−. There are two ways to measure the much smaller
branching fraction into the charmless final state Xsγ, which is mediated by an FCNC. In
the so called “semi-inclusive” method several exclusive decays into a strange final state
and a photon are measured separately and added up. The “fully inclusive” method only
reconstructs the photon2. Since B̄ → Xsγ is dominated by the two-body decay b → sγ the
photon energy spectrum peaks at Eγ ≈ mb/2, while the deviations from a δ-function are
due to gluon bremsstrahlung and hadronic effects within the meson. This distinctive form
of the spectrum can be used to identify the photons from a B̄ → Xsγ decay.
However, there are several background processes that render the identification of the
B̄ → Xsγ photons difficult (see Figure 1.3). On the one hand there is the continuum back-
ground. One component of this continuum is due to photons radiated from the initial state
electrons before their annihilation e+e− → γq̄q (called “Continuum ISR” in Figure 1.3).
Another arises from the decay of inclusively produced π0/η, which subsequently decay via
π0/η → γγ, with one undetected photon (called “Continuum π0” in Figure 1.3). When
using the semi-inclusive method, this continuum background is reduced due to kinematical
constraints and the beam energy constraint, while for the inclusive methods event shapes
have to be used to identify signal events.
On the other hand there is the BB̄ background, which refers to processes in which the
B mesons are produced, but then decay via a process other than B̄ → Xsγ. The main
component of this background is due to B → Xπ0 or B → Xη followed by π0/η → γγ. In
order to get rid of this background some lower cutoff on the photon energy is imposed. It is
this cutoff, that leads to the breakdown of the local OPE and potentially gives rise to the
large logarithms in the perturbative calculation. Generally, the experimental uncertainties
become larger as the cutoff on the photon energy is lowered, as can be illustrated with the
2With high enough statistics a lepton tag (the lepton originates from the semi-leptonic decay of the
other B meson) can be used to reduce the background [22].
10
10 7 ToCoCo10 6 BBb→
10 5 
10 4
10 3
10 2 talntinuum ISRntinuusγ m π
0 10 0 0.5 1 1.5 2 2.5 3 3.5 4Eγ (GeV) 
Figure 1.2: Numbers of photons from various background processes and the signal for
b → sγ, taken from [21]. See text for the explanation of the different graphs.
latest fully inclusive data of the Belle collaboration [22].
B(B̄ → Xsγ)E >2GeV = (3.02± 0.10 −4γ stat ± 0.11syst) · 10
B(B̄ → Xsγ)E = (3.21± 0.11 ± 0.16 ) · 10−4γ>1.9GeV stat syst
B(B̄ → Xsγ)E >1.8GeV = (3.36± 0.13stat ± 0.25syst) · 10−4γ
B(B̄ → Xsγ)E >1.7GeV = (3.45± 0.15 −4γ stat ± 0.40syst) · 10
The first error is statistical, while the second one is systematic and includes uncertainties
due to background subtraction.
The shape of the photon spectrum and therefore the differential decay rate dΓ/dEγ is
of interest as well, as it contains information about the heavy-quark parameters, which are
related to the b-quark mass and its movement inside the hadron. Also, for a comparison of
experimental data with the perturbative calculation an even lower cutoff is necessary than
the ones achieved in measurements so far. For this purpose one assumes a certain model
for the shape of the photon spectrum and calculates extrapolation factors [23], which are
defined as the ratio between branching fractions with minimum photon energies above and
at 1.6GeV.
The current experimental world average for a photon energy of Eγ > 1.6GeV is given
as [24]
B(B̄ → Xsγ) = (3.55± 0.24± 0.09) · 10−4 ,
where the first error is the combined statistical and systematic, while the second error is
due to the extrapolation. As a final remark it is noted that inclusive experiments only
measure the sum of B̄ → Xsγ and B̄ → Xdγ. The B̄ → Xdγ part is subtracted by
calculating the ratio of the two decays in the SM.
11
# γ's / (50 MeV)/ pb-1
In order to proceed with precision physics at the intensity frontier in the future, two
super-B factories – Belle II and SuperB in Italy – have been proposed. Their design and
physics case can be read off from the latest progress reports of the two collaborations [25,26].
In conclusion, the experimental situation motivates a theoretical treatment that is able
to describe the photon spectrum of the B̄ → Xsγ decay in the endpoint region. The
following chapters will introduce the appropriate framework and will work out its impact
on the theoretical prediction.
12
13
Chapter 2
The Decay B̄ → Xsγ in Full QCD
In this chapter all the ingredients necessary to perform a partonic calculation of the B̄ →
Xsγ decay rate will be discussed. The phrase “partonic” refers to the interpretation of
hadrons as the sum of their constituent partons whose interactions can be perturbatively
calculated. Of course, this interpretation ignores effects of confinement, which are very
relevant and will be considered in the next chapter. The first ingredient is the weak
effective Lagrangian that is necessary to describe flavor-changing processes. It can be
derived by systematically integrating out the heavy degrees of freedom, which will be done
in Section 2.1. Along the way it will be demonstrated how the new effective terms in the
Lagrangian can be obtained from a local operator product expansion. In Section 2.1.3 it
will then be shown how the effective couplings of the new operators are determined at the
appropriate scale. Using these results the differential decay rate in the parton model is
calculated in Section 2.2.
2.1 The Weak Effective Lagrangian
The starting point for all calculations concerning radiative B decays is the weak effective
Lagrangian. The general procedure of deriving effective theories will be discussed in this
section. As an example the heavy gauge bosons W± will be explicitly integrated out. Since
there are no leptons in radiative B decays only the quark sector will be considered. The
presentation in this section will follow to some extent the comprehensive discussions in
in [27, 28].
2.1.1 Integrating out the Heavy Degrees of Freedom
On the elementary level the decay B̄ → Xsγ is mediated by the decay of a b-quark via the
weak interaction. In addition, the b-quark is bound into a hadron by the strong interaction.
The Standard Model Lagrangian necessary to describe such a process was introduced by
Glashow, Weinberg, and Salam. After spontaneous symmetry breaking (SSB) it is given
14
by [29]
LSM = Lquark + LCC + LW,kin + LZ + LField + LHiggs
∑
[ ]
Lquark = ūi(iD/ −mi )uiu + d̄i(iD/ −mid)di
i
[ ]
∑
L = √g i gū γµ(1− γ )V dj W+ + √ d̄iγµ − ∗ j − (2.1)CC 5 ij µ (1 γ5)Viju Wµ
2 2 2 2
i,j
L 1= − (∂ W+ − ∂ W+)(∂µW−ν − ∂νW−µ) +M2 W+ −µW,kin µ
2 ν
ν µ W µ W
where the sum runs over all quark families and ui and di represent the up- and down-
type quarks of the ith family and miu,d are the corresponding masses. The fields W
±
µ
are massive SU(2)L gauge bosons and g is the associated coupling constant. Since the
Lagrangian is given in the mass basis the charged currents (indicated by the subscript CC)
contain the CKM mixing matrix Vij, that gives rise to family changing interactions [30].
The expressions for LZ, which encodes the flavor-diagonal couplings of the Z-bosons and
its kinetic term, as well as those for LField and LHiggs, which contain the self-interaction
terms of the gauge fields and the Higgs sector introduced in order to achieve SSB, are not
explicitly given. Finally the covariant derivative is given by
iDµ = i∂ + eqeAµ + g A
a a
s µT , (2.2)
where e and gs are the electromagnetic and strong coupling constants, eq is the quark
charge in units of e (eu = 2/3, ed = −1/3), T a are the SU(3)c color generators and Aµ and
Aaµ are the photon and gluon gauge fields.
The typical energy scale of B decays is set by the mass mb of the heavy b-quark. Since
heavier fields can not be produced directly it is convenient to use an effective field theory
(EFT) obtained by removing the heavy modes as dynamical degrees of freedom. The
effective theory must nevertheless include the corrections due to virtual contributions of
these heavy particles. A consistent way to construct the EFT is to start with the generating
functional of the full SM and integrate out the heavy fields in the path integral formalism.
In the case of B decays these are the heavy gauge bosons fields W±µ and Zµ and the top
quark field t. The generating functional can be written as
∫ ∫
[ ]
Z[J ] = Dφ exp i d4x [LSM + J(x)φ(x)] . (2.3)
Here φ(x) represents all fields that can be external in the theory at hand and J(x) are the
∫
associated external sources. The symbol Dφ stands for the path integral over all fields
of the theory. The Feynman propagator of a field is then given by
DF (x− y) = 〈0|Tφ(x)φ(y)|0〉
( )( )
1 ∣− δ − δ ∣= i i Z0[J ]∣
Z[0] δJ(x) δJ(y) J=0 (2.4)
∫ ∫
Dφφ(x)φ(y) exp [i d4xLφ,kin]
= ∫ ∫D ,φ exp [i d4xLφ,kin]
15
where Z0 is the generating functional containing only the free field Lagrangian of φ, δ/(δJ)
is a functional derivative and T indicates a time-ordered product. The quantity in the
denominator of the last line of (2.4) is called a functional determinant and will always
cancel in the perturbative calculation of correlation functions. In order to integrate out
the W bosons from the SM generating functional the sources of W± are set to zero and
the exponential is split as follows
∫ ∫ ∫ ∫
[ ] [ ] [ ]
Z[J ] = Dφ exp i d4xLW,kin exp i d4xLCC exp i d4xLField
∫
[ ]
× exp i d4x [Lrem + J(x)φ(x)] .
The last term containing Lrem can be deduced from Equation (2.1) and is independent
of the W fields. If one is only interested in the effective Lagrangian for the quark fields,
the self-interaction term LField can be ignored as well. After expanding the exponential
containing the interaction terms in LCC, the path integral over the W bosons can be
explicitly performed by using (2.4). One finds
( )
∫ ∫ ∫
[ ]
D 1W+DW− exp i d4xL 1− d4xd4W,kin yLCC(x)LCC(y) + . . .
2
∫ ∫
[ ]
= DW+DW− exp i d4xLW,kin
( )
g2
∫
× 1− d4xd4y (J+µW+ + J−µW−) (J+νW+ −νµ µ x ν + J W−ν )y + . . .16
∫ ∫
[ ]
= DW+DW− exp i d4xLW,kin
( )
2 ∫
× 1− g d4xd4y (J+µ (x)J−ν (y) + J−µ (x)J+ν (y))DµνF (x− y) + . . . .16
Here the charged currents are defined as J+µ = ū
iγµ(1 − γ5)V jijd and J−µ = (J+ †µ ) . The
terms in the last line can be resummed into an exponential which introduces non-local
interaction terms in the new effective Lagrangian. These terms do not depend on the
W fields which have been integrated out. The remaining path integral over the kinetic
part of the W Lagrangian is identical to the functional determinant mentioned above and
drops out in the definition of correlation functions. After integrating out the W bosons
the effective Lagrangian for the quarks can therefore be written as
Leff = Lquark + Lint, eff + LZ + LHiggs, (2.5)
with
∫
L g
2
int, eff = i d
4y (J+µ (x)J
−
ν (y) + J
− + µν
16 µ
(x)Jν (y))DF (x− y).
16
In the unitary gauge one has
( )
∫
µν − d
4k −i µ ν
D (x y) = gµν − k k e−ik(x−y)F .(2π)4 k2 −M2 M2W W
Up to this point the Lagrangians of the full and effective theory are equivalent for any
process without external W bosons. At tree level, the momentum k of the propagator is
of the same order as the external momenta which are by definition much smaller than the
W mass MW , i.e., k
2 ≪ M2W . It is therefore possible to expand the propagator DµνF in
powers of 1/M2W . This expansion effectively rewrites the non-local interaction functional
as a series of local ones. One arrives at
∫
d4y J+µ (x)J
−(y)Dµνν F (x− y)
( )
∫ ∫ ∫
d4k i 1
= d4y J+(x) gµν + (k2gµν − kµkν) + . . . e−ik(x−y) d4µ p J−ν (p) e−ipy(2π)4 M2 M2W W
( )
i 1
= J+µ (x) g
µν + (∂2gµν − ∂µ∂ν) + . . . J−ν (x).M2 M2W W
Summing up, one might say that integrating out the W bosons at tree level introduced
a sum of effective local-interaction functionals of arbitrary dimension. The higher dimen-
sional terms are suppressed by appropriate powers of 1/M2W . In canonical quantum field
theory this corresponds to a sum of local operators. Beyond the tree level, these operators
will have coefficients that encode the difference between the full and effective theory due
to virtual effects of the particles which have been integrated out. Writing an EFT in terms
of local operators is called the operator product expansion [31]. It is important to note
that this local expansion was only possible due to the large mass scale MW . Later on,
EFTs will be introduced in which such a large scale is missing. These will also give rise to
non-local operators. The leading order term in the local expansion is given by
2
L(1) g − +µint, eff = − Jµ (x)J (x). (2.6)8M2W
This√is the well-known Fermi interaction of weak decays with the Fermi constant defined as
GF/ 2 = g
2/(8M2W ). The apparent weakness of the weak interaction is due to the 1/M
2
W
suppression of the leading contribution.
2.1.2 The Wilson Coefficients
In the above derivation of the effective Lagrangian, corrections due to the strong interaction
have been ignored. Since QCD is still part of the effective theory, the effective description
resembles the one of QCD in the infrared (IR). However, high-virtuality gluons enter loop
diagrams together with other fields that are integrated out. In this case the momentum
of the heavy particle can become large and a naive expansion of the heavy propagator in
17
powers of k2/M2W is no longer possible. This practical problem can be systematically dealt
with in the framework of the heavy-mass expansion (see [32] and references therein). In
the so-called matching procedure the high-virtuality gluon corrections are absorbed into the
coefficients of the local operators in the OPE. These can then be determined by comparing
the effects of the gluonic corrections in the effective and the full theory. Since the coefficients
encode the physics of gluons with a large virtuality, this calculation can be performed in
perturbation theory. As an example one might consider the four-fermion operator in (2.6).
In the case of B̄ → Xsγ one has to deal with
Qc := s̄ γµ1 i (1− γ5)ci c̄jγµ(1− γ5)bj, (2.7)
where indices i, j now represent color quantum numbers. The QCD corrections to a process
with four external quarks in the full and effective theory are shown in Figure 2.1.2. The first
Figure 2.1: NLO QCD corrections to the four-quark process in the full (first line) and
the effective theory (second line). Two crossed circles denote the insertion of a four-quark
operator.
diagram in each line gives rise to the renormalization of the quark currents. Since the quark
currents are conserved the divergence of these diagrams is canceled by the field-strength
renormalization of the quark fields. Furthermore, the finite terms of these diagrams are the
same in the full and effective theory. Therefore they do not contribute to the coefficient of
the effective theory operator. Calculating the last two diagrams in the effective theory and
in 4− 2ε dimensions to regulate the divergences yields (suppressing the factor G√F V ∗
2 cs
Vcb)
( )
α 1 µ2M si = −6i + ln − + constant s̄ γ
µ(1− γ a a
2 i 5
)Tijcj c̄kγµ(1− γ5)Tklbl, (2.8)4π ε p
where T aij are the SU(3)c generators and µ is the renormalization scale. Since the coeffi-
cients are by definition independent from the IR degrees of freedom, the external particles
can be treated as massless and it can be assumed that they all the same momentum.
The four-quark operator in (2.8) can be related to Qc1 by the following relation for the
generators:
( )
a 1 1TijT
a
kl = δilδkj − δijδkl . (2.9)2 Nc
18
where Nc is the number of colors. This obviously introduces a second operator with a
different color structure1
Qc2 := s̄iγ
µ(1− γ5)cj c̄jγµ(1− γ5)bi. (2.10)
This effect is called the mixing of two operators under QCD renormalization. In order
to renormalize all operators of the effective theory the diagrams in the second line of
Figure 2.1.2 have to be calculated with operator insertions of Qc1 and Q
c
2. Then a matrix-
valued operator renormalization constant has to be introduced to absorb all divergences:
Q reni = ZijQj . (2.11)
After renormalization the result of the effective theory calculation can be compared to the
full theory calculation of the diagrams in the first line of Figure 2.1.2 and the difference
must be accounted for by coefficients of the effective operators. Note that there is no
divergence in the full theory since the full W propagator renders the loop integral finite.
M2
The full theory diagrams therefore contain logarithms of the type ln W− 2 . Comparing thisp
to the logarithmic terms in (2.8) reveals that the scale µ takes the role of a factorization
scale, that divides the result into a high-energy part contained in the coefficients and a
low-energy part encoded in the matrix elements of the effective operators. At NLO the
coefficients of the operators Qc1 and Q
c
2 are given by
( )
3 α 2s M
C (µ) = 1 + ln W1 −
11
,
N 2c 4π µ 6
( ) (2.12)
− αs M
2
W − 11C2(µ) = 3 ln .
4π µ2 6
The coefficients of the effective theory operators are called Wilson coefficients [33].
The general procedure outlined above can be summarized as follows. In order to con-
struct an effective theory appropriate for energies below the electroweak scale MW the
fields above that scale have to be integrated out. This corresponds to writing down a sum
of all local operators allowed by the symmetries and quantum numbers of the problem and
suppressed by powers of 1/MW :
( )
G ∑L F 1eff = Lquark − √ V ∗CKMVCKM CiQi +O . (2.13)
2 M2
i W
The Wilson coefficients Ci can be interpreted as coupling constants of the new interactions
in the effective theory. But since the full theory is known they can also be calculated
from the SM. Since the coefficients encode the effects of the high-virtuality modes of the
full theory they can be calculated perturbatively, by comparing the results of the effective
and of the full theory at a large scale. Thereby a dependence on this factorization scale
1Note that the operators Qc c1 and Q2 often have reversed names in the literature.
19
is introduced into the coefficients. This scale dependence cancels if the matrix elements
of the operators are calculated at the same scale and in the same scheme. However, for a
certain low-energy process like a meson decay the matrix element of the effective operators
must be renormalized at the low scale. This can potentially introduce large logarithms in
the Wilson coefficients. For µ = 1GeV the O(αs) corrections to the Wilson coefficients are
M2
only suppressed by a factor of αs ln W2 ∼ 0.9 compared to the leading-order term resultingπ µ
in a bad convergence of the perturbative series. This problem can be solved by resumming
large logarithms by means of the renormalization group (RG) [34].
2.1.3 The Renormalization Group
In order to derive the renormalization group equations (RGE) one considers the dependence
of the Wilson coefficients on the scale µ. Since this scale was introduced by splitting the
calculation in a low- and a high-energy part, the product of the Wilson coefficients with
the operator matrix elements renormalized at µ is formally independent of the scale. Using
(2.11) one has
d ∑
Ci(µ)〈Qreni (µ)〉 = 0d lnµ
i
([ ] [ ] )
∑ d d
= Ci(µ) (Z
−1) −1ij(µ) 〈Qj〉+ Ci(µ) (Z )ij(µ) 〈Qj〉
d lnµ d lnµ
i,j
([ ] [ ] )
∑ d d
= C (µ) 〈Qren(µ)〉 − C (µ) (Z−1) (µ) Z (µ) 〈Qreni i i ik kj (µ)〉d lnµ d lnµ j
i,j,k
( )
∑ d ∑
= Ci(µ)− Cj(µ)γji 〈Qreni (µ)〉,d lnµ
i j
where the anomalous dimension matrix γ is defined as
[ ]
d
γji = Z
−1 Z . (2.14)
d lnµ ji
It only depends implicitly on the scale µ through the strong coupling constant αs. Under
the assumption that the operators of the expansion are linearly independent this yields the
RGE of the Wilson coefficients:
d
C~ (µ) = γT (αs(µ))C~ (µ). (2.15)
d lnµ
By solving this equation it is possible to relate the Wilson coefficients at a certain scale to
the coefficients at any other scale. This solves the problem of the large logarithms. The
coefficients are calculated at the electroweak scale where the logarithms are small and the
20
perturbative expansion is valid. Then they are “run down” to the low-energy scale by
using the solution of the RGE. The solution to (2.15) is given by
∫ αs(µ) γT
C~
(αs)
(µ) = Tα exp dαs C~ (µ0), (2.16)s
α (µ ) β(αs)s 0
where the exponential of a matrix is defined by its Taylor expansion and the ordering
symbol Tα sorts the matrices in order of increasing values of the parameter αs with largers
values standing to the left. The QCD β-function is defined as β(α dαss) = . For the cased lnµ
of B̄ → Xsγ the matching scale µ0 corresponds to the weak scale µW which is of the order
of the W or t-quark mass. The Wilson coefficients are subsequently “run” down to the
bottom-quark mass scale µb ∼ mb.
2.1.4 The Operator Basis
So far the effective theory only contains the current-current operators Qc1 and Q
c
2 that
arose by integrating out the W bosons from diagrams like the first one in Figure 2.2. They
form a complete set under QCD renormalization, i.e., no new operators are introduced by
QCD corrections. However, by integrating out all other modes above the electroweak scale
additional effective operators arise that are relevant for the B̄ → Xsγ decay. In fact, it
will be established in Section 2.2 that Q1,2 do not even give rise to a LO matrix element.
Still, due to RG running their contribution is very relevant (see Section 2.1.5). In the
full effective Lagrangian there appear six more operators. First there are the dimension-6
QCD penguin operators that originate from the second diagram in Figure 2.2. The external
b W s b W s
b s
W
t t
q g q γ
c c
Figure 2.2: SM diagrams relevant for the B̄ → Xsγ decay. The diagrams give rise to
current-current, penguin and electromagnetic dipole operators, respectively.
current coupling to the gluon can be decomposed into a V −A and a V +A structure. The
short-hand notation V − A represents a current with vector minus axial vector structure
q̄γµ(1−γ5)q while V +A correspondingly represents q̄γµ(1+γ5)q. Furthermore the current
can contain quarks of any flavor lighter than the electroweak scale (q = u, d, c, s, b). When
QCD corrections are included also different color structures are possible. In total, there
are four QCD penguin operators. More operators are introduced by the last diagram
in Figure 2.2. It contains the insertion of a mass operator into the b-quark line, which
gives rise to a chirality changing dimension 5 operator. This diagram goes by the name
of electromagnetic dipole or electromagnetic penguin operator. Finally, there is also a
21
chromomagnetic dipole operator where the photon in the last diagram is replaced by a
gluon. A complete set of linearly independent operators that again closes under QCD
renormalization is given by [35]
Qq1 = (q̄b)
q
V−A (s̄q)V−A , Q2 = (q̄ibj)V−A (s̄jqi)V−A ,
∑ ∑
Q3 = (s̄b)V−A q (q̄q)V−A , Q4 = (s̄ibj)V−A q (q̄jqi)V−A ,
∑ ∑ (2.17)
Q5 = (s̄b)V−A q (q̄q)V+A , Q6 = (s̄ibj)V−A q (q̄jqi)V+A ,
−emb
Q = s̄σ µν
−gmb
7γ µν(1 + γ5)F b , Q8g = s̄σµν(1 + γ5)G
µνb ,
8π2 8π2
where the sums run over all quark flavors lighter than the electroweak scale. The complete
effective Lagrangian, necessary to describe the decay b → sγ at leading order in 1/M2W is
given by
( )
∑ ∑
Leff = Lquark−
G√F λ C qq 1Q1+C2Qq2+ Ci Qi+C7γ Q7γ+C8g Q8g +h.c. , (2.18)
2
q=u,c i=3,...,6
where λq = V
∗
qbVqs and the Wilson coefficients Ci depend on the scale µ at which the
operators are renormalized. The unitarity of the CKM matrix λu + λc + λt = 0 has been
used to rewrite the coefficients of the penguin operators. Under QED renormalization
additional operators can mix with the operator basis given above, but due to the smallness
of the electromagnetic coupling α these effects turn out to be subleading and will be ignored
here. It is an advantage of the effective theory approach, that radiative FCNC decays can
be expressed in terms of matrix elements of the operators in (2.17) with Wilson coefficients
that are independent of the process. New physics can enter the calculation either through
the introduction of new operators and/or through additional contributions to the Wilson
coefficients of the operators in (2.17).
2.1.5 The Wilson Coefficients at the Correct Scale
The effective Lagrangian derived in the previous section can now be used to calculate
the b → sγ decay rate. In order to do so one writes down all contributing diagrams
up to O(GF ), i.e., with one insertion of an effective operator and including QCD to the
desired order. Since the effective couplings in form of the Wilson coefficients are themselves
calculated by a perturbative expansion in αs it is necessary to introduce a systematic way to
define the order of the calculation. In the so called leading logarithmic (LL) approximation
the Wilson coefficients are determined by a matching at O(α0s) followed by running using
the O(α1s) correction to the anomalous dimension matrix. This running is necessary to
resum large logarithms of the ratio of the large matching scale and the much lower scale of
the b decay. The LL approximation resums logarithms of the form (αs ln
µW )n. The large
µb
scale µW can be equal to the mass of any of the integrated out particles while the low-energy
scale µb is the scale of the process at hand. To complete the leading-order calculation the
matrix elements must be calculated at the low-energy scale µb ∼ mb. In general these
22
matrix elements are non-perturbative quantities, but to a first approximation (that will be
justified later on in Chapter 3) it is possible to perform their calculation in a perturbative
expansion in powers of αs. For a leading-order calculation only tree-level matrix elements
need to be considered. All the necessary steps (matching, running, and calculation of the
matrix elements) have been performed first in [36]. However, there is one subtlety related
to the determination of the anomalous dimension matrix at O(αs). Determining the 6× 6
sub-matrix that induces the mixing between the four-quark operators Q1...6 requires the
calculation of the UV divergent part of the one-loop diagrams in the effective theory (e.g.,
the first diagram in Figure 2.3). The same is true for the mixing in the dipole sector. For
Figure 2.3: Diagrams relevant for the calculation of the LO anomalous dimension. Effective
operators are now denoted by one crossed circle. The last diagram actually vanishes in
certain renormalization schemes (see text).
example the second diagram in Figure 2.3 is responsible for the mixing of Q8g into Q7γ .
Finally the mixing of the four-quark operators Q1...6 into the dipole operators Q7γ,8g is
mediated by the third diagram of Figure 2.3. It is an interesting feature of the operator set
(2.17) that these diagrams yield a renormalization scheme dependent result [37]. Indeed,
this dependence cancels between the first order anomalous dimension and the one-loop
matrix elements. It is convenient to introduce renormalization scheme invariant Wilson
coefficients Ceffi through [38]
6 6
∑ ∑
Ceff7γ = C7γ + yiCi , C
eff
8g = C8g + ziCi , (2.19)
i=1 i=1
where yi, zi depend on the scheme. In the ’t Hooft-Veltman scheme all yi and zi are
zero, while in the naive dimensional regularization scheme y5 = −1/3, y6 = −1, and
z5 = 1, and all other yi and zi vanish. Since the diagram vanishes in any four-dimensional
renormalization scheme it is necessary to perform a two-loop calculation to obtain the
complete leading-order contribution to the anomalous dimension. The complete calculation
has first been performed in [37] and afterwards confirmed in [39]. In the LL approximation
23
the Wilson coefficients at the scale µ can be written as [40]
8
∑
Ci(µ) = k
aj
ijη (i=1. . . 6),
j=1
8
( )
∑
16 8 14 16
Ceff7γ (µ) = η
a
23C i7γ(MW ) + η 23 − η 23 C8g(MW ) + C1(MW ) hiη , (2.20)
3
i=1
8
∑
eff 14C8g (µ) = η 23C8g(MW ) + C1(MW ) h̄
ai
iη ,
i=1
where
C1(MW ) = 1,
3x3t − 2x2t −8x3 2t − 5xt + 7xtC7γ(MW ) = − ln xt + − ,4(xt 1)4 24(x 1)3t (2.21)
−3x2 3t −xt + 5x2t + 2xtC8g(MW ) = − ln xt + ,4(xt 1)4 8(x − 1)3t
with x 2 2t = mt/MW and η = αs(MW )/αs(µ). The calculable parameters ai, hi, h̄i, and
kij are given in Table XXVII of reference [40] (with reversed names of C1 and C2). A
numerical study of these Wilson coefficients at a scale µ ∼ mb demonstrates that Ceff7γ is
strongly enhanced by the QCD corrections associated with the operator Qq1 [41]. This
enhancement is also responsible for a strong µb dependence of the Wilson coefficients,
that is not compensated in the leading-order calculation [42]. Since the exact value of
the scale is arbitrary, the µb dependence gives rise to an uncertainty of the calculated
quantities. Varying the scale in the rangemb/2 < µb < 2mb yields a change of the branching
fraction by ±22% [44]. As has been shown in [38], this dependence can be significantly
reduced by performing an NLO calculation using the next-to-leading logarithmic (NLL)
approximation.
The NLL program requires the calculation of the Wilson coefficients through a matching
at O(α1s), while the anomalous dimension matrix is needed up to O(α2s). This corresponds
to the calculation of two-loop diagrams for the mixing between the four-quark operators
and for the mixing between the dipole operators. The mixing between the two sectors even
requires calculations at three-loop order. By using the NLL approximation logarithms of
the form αs(α ln
µW
s )
n are resummed in addition to the leading order resummation. The
µb
NLO matching of the four-quark operators was already performed in [45–47] while the
dipole operators were matched in [48, 49]. References [46, 47] also contained the complete
two-loop anomalous dimension matrix of the four-quark sector. The mixing between the
dipole operators was calculated in [50] and the mixing between four-quark and dipole
operators in [43]. Finally, the operator matrix elements must be considered at NLO [51]
which will be the topic of the next section. As expected, the complete NLL program
reduces the dependence of the branching fraction on scale variations notably [43]. Apart
24
from the dependence on the low-energy scale µb there are also uncertainties due to the
matching scale µW [44] as well as the renormalization scale of the quark masses. While
the former scale dependence is already reduced by the NLO calculation, the uncertainty
due to charm-quark mass scale µc is only introduced at that order. This is explained by
the already mentioned fact, that matrix elements containing long-distance charm-quark
loops (see the last diagram in Figure 2.3) vanish at the leading order. As has been shown
in [52], replacing the pole charm-quark mass by the MS mass at the scale µb can enhance
the prediction for the branching fraction by O(10%). It is therefore necessary to consider
the NNLO corrections, in order to reduce the dependence on the charm-quark mass scale.
The NNLO calculation (using the NNLL approximation) was performed over the course
of many years by a joint effort of several groups. The matching procedure required the
calculation of two-loop diagrams for the case of C1...6 [53] and three-loop diagrams for
C7γ,8g [54]. For the mixing between the four-quark [55] and between the dipole [56] opera-
tors three-loop calculations are necessary, while the mixing between four-quark and dipole
operators even demanded for a four-loop calculation [57]. Running the NNLO Wilson
coefficients down by using the solution to the RGE in NNLL approximation additionally
resums logarithms of the form α2s(α ln
µW )ns . In order to reduce the scale dependenceµb
through the charm-quark mass the calculation of three-loop matrix elements containing a
charm-quark loop must be performed. So far this has only been done in the so-called large-
β0 approximation [58]. Although this approximation does not reduce the scale dependence
on µc by itself, it serves as a starting point for an interpolation of the charm-quark mass
to physical values [59]. All the above results, together with the O(α2s) matrix elements for
O7γ [58, 60–62] were merged in [5] to produce the current final word on the perturbative
NNLO calculation of the B̄ → Xsγ branching fraction for a sufficiently low cut on the
photon energy. It was demonstrated that going beyond the NLO did indeed significantly
reduce the scale dependencies (see Figure 2.4). The left plot in Figure 2.4 illustrates how
3.8 3.8
3.6 3.6
3.4 3.4
3.2 3.2
2 4 6 8 10 2 4 6 8 10
Figure 2.4: Branching fraction (·10−4) plotted versus the variation of the low-energy scale
µb (left plot) and versus the charm-quark mass scale µc (right plot). Dotted, dashed, and
solid lines represent LO, NLO, and NNLO respectively. Plots taken from [5].
the variations of the branching fraction due to variations of the low-energy scale µb decrease
when going to higher orders in αs. The uncertainty due to the µc variations on the other
hand is only introduced at NLO and decreases at the NNLO (right plot). In [5] the total
uncertainty of the branching fraction due to scale variations was estimated to be 3% by
25
varying the scales in a range between one half and two times their central value.
This concludes the discussion of the effective Lagrangian required for the description
of low-energy weak decays: In addition to the usual QCD and QED Lagrangian for the
quark sector it contains the effective local operators Q1...8g. Their coefficients are calculated
from the full theory to a certain order in αs. In oder to resum large logarithms they are
subsequently run down to the low-energy scale of the process by solving the RGE. Today,
these Wilson coefficients are known at NNLO through a matching at O(α2s) and a running
in the NNLL approximation. However, up-to-date calculations of the B̄ → Xsγ branching
fraction [5] only make use of the NNLO result for Ceff7γ , while using the NLO result for
C1...6,8g. For a matching scale of µW = 160GeV and a low-energy scale of µb = 2.5GeV
the coefficients are explicitly given in Table 2.1. It can be seen that the current-current
C1 C2 C3 C4 C C C
eff eff
5 6 7γ C8g
1.123 -0.272 0.020 -0.048 0.010 -0.060 -0.326 -0.184
Table 2.1: Wilson coefficients at the scale µ = µb = 2.5GeV at NLO (for C1...6,8g) and
NNLO (for Ceff7γ ).
operators, together with the dipole operators are dominant. The next important step will
be the calculation of the B̄ → Xsγ decay rate in the framework of the weak effective theory.
This will be the topic of the next section.
2.2 The Differential Decay Rate
With the derivation of the weak effective Lagrangian complete, the differential decay rate
can now be calculated by the usual methods of quantum field theory. This section discusses
these methods and gives the LO and NLO order results in the parton model.
In the rest frame, the decay rate of an initial particle i into a final state f can in general
be written as [29]
∫ 3
→ 1 1 d pf 1Γ(i f) = |〈f |iT |i〉|2, (2.22)
2Mi V t (2π)2 2Ef
where 〈f | represents all final state particles and the phase space integral is taken over all
final state momenta. Since the initial state is assumed to have a definite momentum a dis-
crete normalization including the factor 1/V must be used. This factor cancels after using
“Fermi’s Trick” for the square of the momentum conservation δ-function. The transition
matrix T contains the dynamics of the scattering process.
When considering radiative B decays, the initial state is a pseudoscalar B meson. The
calculations are performed for the case of a neutral B̄0 with a bottom quantum number
of -1 that is composed of a b and d̄ quark. To a first approximation the light quark d̄
does not influence the decay of the heavy b-quark, which corresponds to the assumption
of the spectator model. In this case, the decay rate for the charged B− ∼ bū meson is
26
identical to the neutral one. Deviations from this equality will be explicitly mentioned
when applicable.
The final state consists of a hadronic part with a net strangeness of -1 and a photon.
Due to confinement, the final-state quarks are bound into hadrons. In general, it is not
possible to calculate the hadronic matrix element 〈Xhadrγ|iT |B̄0s 〉 from first principles.
The problem is associated with the QCD coupling constant αs, that becomes large for
energy scales at the typical hadronic binding energy, thereby invalidating the perturbative
expansion. However, the situation is simplified by considering the inclusive decay rate.
This corresponds to the sum over all final states f with strangeness -1 and containing a
photon. It will be shown in Section 3.1 that the sum over hadronic matrix elements is then
equal to the sum over free quark matrix elements up to corrections proportional to the
inverse heavy-quark mass. This equality requires the assumption of quark-hadron duality.
In the free quark approximation the inclusive decay rate can be written as
( ) ( )
∫ ∫
→ 1 1 d
3
∑ pf 1 |〈 part | 4 L | 〉|2 O 1Γ(B̄ Xsγ) = X γ iT exp i d x eff b + ,
2mb V t (2π)2 2E
s m
part f bXs
(2.23)
where T is the time ordering symbol from the definition of the transition matrix T =
∫
T exp i d4xL and the phase space integration is taken over all final state particle mo-
menta. At leading order in αs there is only one strange-quark (and the photon) in the final
state and the transition is only mediated by the operator Q7γ . The effect of the four-quark
operators Q1...6 is already included by using the effective Wilson coefficient C
eff
7γ introduced
in (2.19). Thus, it is only necessary to calculate the first diagram in Figure 2.5 that rep-
resents the tree-level matrix element 〈sγ|Q7γ|b〉. This calculation can be performed by
g(q′)
γ(q) γ(q) γ(q)
Q7γ
b(p)
Q ′
b(p) ′ b(p) ′
7γ s(p ) s(p ) Q7γ s(p )
Figure 2.5: Diagrams containing the operator Q7γ contributing at LO (first diagram) and
NLO (second and third diagram).
using Feynman rules to calculate the invariant matrix element M related to the transition
∑
matrix through 〈pf |iT |p 〉 = (2π)(4)δ4i (pi− pf )iMfi. For a vanishing strange-quark mass
this diagram is evaluated to
MLO i√GF eff embi 7 = C7γ (µ)λ ′t ū (p )/q /ε∗(1 + γ )u2 s 5 b(p), (2.24)2 4π
where ub and us are the quark spinors and the momenta are defined in Figure 2.5. Inserting
this result into (2.23) yields the differential decay rate in the free quark approximation at
leading order in αs
G2 m2
dΓLO (B̄ → X γ) = F bα |Ceff 2 3free s 7 (µ)λt| Eγ dEγ δ(mb − 2Eγ), (2.25)2π4
27
with the photon energy Eγ and the electromagnetic coupling constant α = e
2/(4π). As one
can see, the LO calculation yields only a δ-function contribution to the photon spectrum
located at the photon energy mb/2. Of course, this is to be expected from a simple two-
body decay. There are two types of corrections to the form of the spectrum. First, there
is a smearing of the peak originating from the fact, that the b-quark is not free but bound
into a hadron. This will be the topic of Chapter 3. Second, QCD corrections from gluon
bremsstrahlung diagrams introduce a non-trivial photon spectrum, as b → sγg constitutes
a three-body decay. This case will be considered now. To conclude the LO calculation, the
total LO decay rate is obtained by integrating over the photon energy which results in
2 5
ΓLO
GFmbα eff 2
free(B̄ → Xsγ) = |C (µ)λt| . (2.26)32π4 7γ
Since the Wilson coefficients are real in the SM, it is not necessary to take their absolute
value squared in the above equation. However, the Wilson coefficients will always be
assumed to be complex in this work. This simplifies the generalization of the analysis to
new-physics models.
2.2.1 Next-to-Leading Order Matrix Elements
The NLO corrections to the decay rate fall into two categories. On the one hand there are
the virtual corrections (e.g., the second diagram in Figure 2.5) that contain one additional
gluon loop. On the other hand the bremsstrahlung corrections (e.g., the last diagram in
Figure 2.5) introduce an additional, non-observable gluon into the final state. At NLO
also operators other than Q7γ contribute to the decay rate (for examples the diagrams in
Figure 2.6). As can be seen from Table 2.1 the Wilson coefficients of the QCD penguin
′
b(p) qQ s(p′) g(q ) q ′
γ(q) 1 γ(q) b(p) Q s(p )1
Q8g
b(p) ′ b(p) ′s(p ) γ(q) Q ′8g s(p ) g(q ) γ(q)
Figure 2.6: Matrix elements containing the operators Qq1 and Q8g. The first two diagrams
depict virtual, the last two bremsstrahlung contributions.
operators Q3...6 are numerically small, the corresponding diagrams will therefore be ignored.
The diagrams associated with the current-current operator Q2 vanish as they contain a
factor involving the trace of the color matrices T a, which is zero. This leaves the matrix
elements of the operators Qq1, Q7γ and Q8g to be calculated. In order to determine the
NLO differential decay rate the contributions of the processes b → sγ and b → sγg must
be added:
dΓNLO dΓvirt dΓbrems
= + . (2.27)
dEγ dEγ dEγ
The virtual part is determined by squaring the sum of all relevant b → sγ matrix elements
up to O(αs). This gives rise to interference terms containing different operators. The
28
interference will always contain one LO (O(α0s)) matrix element of Q7γ together with an
NLO (O(α1s)) matrix element ofQq1, Q7γ orQ8g. Just like the LO, the virtual NLO diagrams
describe a three-body process and therefore their contribution to the spectrum is given by
a δ-function. As will be shown below the virtual photon corrections to b → sg are relevant
as well.
The NLO bremsstrahlung contributions in dΓbrems/dEγ are due to interferences of all
b → sγg diagrams, which leads to an αs suppression of the squared matrix elements. These
diagrams are responsible for the form of the photon spectrum and were first calculated
in [63]. When the contribution of the interference between the matrix elements of the
operators Qi and Qj to the decay rate of b → sγg is denoted as Γbremsij the result can be
written as [64, 65]
dΓbrems dΓbrems dΓbrems dΓbrems dΓbrems dΓbrems brems
= 77 + 88 + 11 + 17 + 18
dΓ
+ 78 , (2.28)
dEγ dEγ dEγ dEγ dEγ dEγ dEγ
where all terms except the first two are completely finite and given by (for ms = 0)
2Eγ
dΓbrems 2 2 5
∫
∣ ∣
( )
m 2
11 GF |λt| αmb CFαs 1 | |2 2 b − z∣ ∣= C1 2eu dx (1 x) 1− F ,∣ ∣dE 32π4γ 4π mb 0 x
2Eγ
dΓbrems G2 | ∫λ |2αm5 C α 1 ( ( ))17 F t b F s m= Re (C Ceff∗ − b z1 )( 4)eu dx xRe 1− F ,
dEγ 32π4 4π m
7γ
b 0 x
dΓbrems Re (C Ceff∗)18 1 8g dΓ
brems
= e 17d , (2.29)
dE Re (C eff∗γ 1C7γ ) dEγ
dΓbrems 2 2 578 GF |λt| αmb CFαs 1= Re (Ceff∗ eff7γ C8g )(−4)e4 ddEγ 32π 4π mb
[ ]
( )2 ( ) ( )
× 2Eγ mb 2Eγ 2Eγ4 + + 8 1− ln 1− .
mb 2Eγ mb mb
Here CF = 4/3 is the Casimir operator of the fundamental representation of the SU(3)c,
eu and ed are the charges of up- and down-type quarks in units of e, z = m
2/m2c b and
everything about the penguin function F (x) is collected in Appendix A.1. Furthermore,
the scale dependence of the Wilson coefficients and the QCD coupling αs has been dropped
for brevity and the results were CP -averaged2
1 ( )
Γ = Γ + ΓCP . (2.30)
2
The contribution to the spectrum due to the interference of two matrix elements of Q7γ
diverges for Eγ → mb/2. This limit corresponds the emission of a soft bremsstrahlung
gluon which gives rise to an IR singularity in the fully integrated decay rate. Fortunately,
gauge invariance requires that this singularity cancels when the virtual O(αs) corrections
2Unless explicitly noted B̄ → Xsγ will therefore include B → Xs̄γ as well.
29
are added. Since the divergence is introduced by a phase-space integral this cancellation
takes place at the level of decay rates. In order to determine the differential decay rate
one has to first integrate the virtual and bremsstrahlung contributions, add them up and
subsequently differentiate the result with respect to the limits of integration [51]
m
∫ b ( )
2 dΓbrems dΓvirt
Γ(E0) = dEγ + ,
E dE dE0 γ γ (2.31)
dΓ(Eγ) −dΓ(E
∣
0) ∣
= .
∣
dEγ dE0 E0=Eγ
An alternative way is to subtract the divergence in the distribution sense by the introduc-
tion of the plus distribution. It is defined as
1 [ ]∫ ∫1 1 f(x)− f(1)
dx − f(x) = dx − , (2.32)0 1 x + 0 1 x
where f(x) is a smooth test function. The spectrum is then given just by the brems-
strahlung contribution with the divergent terms replaced by plus distributions that already
include the cancellation of the divergence due to the virtual contribution. Finally, since
the virtual contributions are proportional to a δ-distribution they add a constant term to
the integrated rate. For ms = 0 the NLO contribution to the differential rate due to the
the self interference of the operator Q7γ (virtual + bremsstrahlung) is given by
dΓNLO G277 F |λt|2αm5b CFαs= |Ceff 2
dx 32π4 4π 7γ
|
[
( ) [ − ] [ ]4π2 µ2× − − − ln(1 x) 15 + + 2 ln δ(1 x) 4 − − 73 m2 (2.33)b 1 x + 1− x +
]
− x+ 1 ln(1− x) + 7 + x− 2x2 ,
2
with the shorthand notation x = 2Eγ/mb. The µ dependence in the term proportional
to the δ-function is of ultraviolet (UV) origin and was introduced by the inclusion of the
field-strength renormalization in the LO contribution of Q7γ . The logarithm ln(1− x) on
the other hand is of IR origin and becomes large (but still integrable) for x → 1. There
is one more comment in order, concerning the m5b factor in the above expression. Two
of the mb’s originate from the normalization of Q7γ and should therefore be taken in the
same renormalization scheme as the Wilson coefficients [38], while the other three come
from on-shell external lines and therefore correspond to pole masses. For that reason the
replacement
m5b → m3b, polemb(µ)2, (2.34)
is implicitly assumed in (2.33). The final contribution to the NLO spectrum is due to the
interference of the operator matrix element of Q8g with itself. As in the case of Γ
brems
77 the
30
differential decay rate dΓbrems88 /dx diverges, now for Eγ → 0. This IR divergence cancels
when adding the virtual photon corrections to the process b → sg. This can once again be
considered by introducing plus distributions, but since the photon spectrum in the region
of low photon energies is of no interest here, this prescription can be immediately dropped.
This leads to [64]
dΓbrems88 G
2
F |λ |2αm5t= b CFαs |Ceff|28g e2dx 32π4 4π d
[ ]
× 1(4 ln(1− x)− 8− 4 ln r) + 2(x− 2) ln(1− x) + 8− x− 2x2 + 4(x− 2) ln r ,
x
(2.35)
where only the leading terms in the strange-quark mass in r = m2s/m
2
b has been kept. This
time it can not be set to zero since this would introduce another divergence. Note, that
the appearance of the low-energy scale ms in the above result is a genuine parton-model
effect, since QCD is non-perturbative at such scales. Finally, the b-quark masses in the
prefactor are assumed to have the same form as in (2.34).
This completes the discussion of the bremsstrahlung contribution to the differential
rate at NLO. In order to calculate the integrated decay rate the remaining virtual contri-
butions must be added and the integral over the photon energy has to be performed. The
virtual contributions relevant for divergence cancellation were already calculated in [63],
the remaining ones in [51].
2.3 The Integrated Decay Rate
Apart from the photon spectrum given by the differential decay rate, also the integrated
decay rate is of phenomenological interest. Its calculation with a lower cutoff on the photon
energy will be discussed in this section.
As was demonstrated in Chapter 1 the inclusive decay rate of B̄ → Xsγ is experi-
mentally only accessible for large photon energies. The branching fraction is therefore
calculated by integrating the differential decay rate only over the endpoint region
[ m m ]b b ΛQCD
Eγ ∈ (1− δ) , with δ ∼ . (2.36)
2 2 mb
Here Eγ is defined in the b-quark rest frame. Performing this integration over the Q7γ self
interference in (2.25) and (2.33) up to NLO yields (ms = 0)
[
G2 |λ |2αm5 ( 3t
Γ (δ) = F b |Ceff|2 CFαs 2δ77 7γ 1 + − 2 ln2 δ − 7 ln δ + δ(δ − 4) ln δ + 10δ + δ2 −32π4 4π 3
]
− − 4π
2 2 )
5 − µ2 ln ,
3 m2b
(2.37)
31
with the first term in the round brackets being a Sudakov double logarithm. This result
makes a possible problem of partonic calculations in the endpoint region transparent.
For small values of δ the logarithms become large and potentially need to be resummed
to make the perturbative expansion meaningful. For δ ∼ ΛQCD/mb the logarithm will
enhance the NLO correction by a factor of ∼ 2.3. As will be motivated in Section 5.2.5
√
the appropriate scale for αs is µ ∼ ΛQCDmb ∼ 1.5GeV which leads to αs ∼ 0.375.
Overall the NLO corrections will therefore not be significantly suppressed, motivating
a resummation of the logarithms. Just like in the weak effective theory introduced in
Section 2.1, the resummation can be performed by matching onto an appropriate effective
theory and solving the RG equations. This effective theory is the soft collinear effective
theory (SCET) which will be introduced in Chapter 4.
Without resorting to SCET the photon cutoff must be chosen sufficiently low to en-
sure the applicability of the perturbative expansion. As was discussed in Section 1.3, this
requires the extrapolation of the experimental prediction, introducing new systematic un-
certainties. As a compromise a photon cutoff of E0 = 1.6GeV is used in the literature.
This corresponds to δ = (mb − 2E0)/mb ≈ 0.3 and a logarithmic enhancement of the
NLO by ∼ 1.2. In [5] all results discussed in this chapter were combined to calculate the
branching fraction of B̄ → Xsγ with an E0 = 1.6GeV. Due to the scale dependences dis-
cussed in Section 2.1.5 a certain value for µb and µc had to be chosen. For central values
of µb = 2.5GeV and µc = 1.5GeV the partial (i.e. including a cut) branching fraction is
given by
B(B̄ → Xsγ)E >1.6GeV = (3.15± 0.23) · 10−4. (2.38)γ
Apart from the error due to higher order perturbative contributions that are estimated from
the scale variations, the error estimate in (2.38) also includes the uncertainty due to themc-
interpolation as well as parametric and non-perturbative uncertainties. In order to reduce
the dependence on the b-quark mass the branching fraction is defined by normalizing the
result for the integrated B̄ → Xsγ decay rate to that of the semi-leptonic decay rate [66].
Finally, in [8] the effects of the photon energy cut on the partially integrated decay rate
have been studied. This analysis yielded a lower theoretical prediction of
B(B̄ → Xsγ)E >1.6GeV = (2.98± 0.26) · 10−4. (2.39)γ
2.4 Conclusions of this Chapter
The parton model calculation of the B̄ → Xsγ branching fraction discussed in this chapter
required sophisticated perturbative calculations and yields an estimate that agrees with the
experimental result within their error bars. During the course of the presentation two major
topics were noted, whose discussion was postponed to later chapters. First, it is desirable
to resum the large logarithms appearing in (2.37). This can be achieved by using SCET,
which will be introduced in Chapter 4. Second, there are non-perturbative corrections to
the photon spectrum as well as to the integrated decay rate. The next chapter will discuss
32
how these non-perturbative effects influence the calculation by employing the heavy-quark
effective theory.
33
Chapter 3
Heavy-Quark Effective Theory
The major topic of this Chapter will be the analysis of the hadronic matrix elements that
appear in the calculation of decay rates involving hadrons. Due to confinement it is not
possible to calculate them in an analytic way. However, by considering an inclusive decay
rate only forward scattering matrix elements between heavy-quark states are of relevance.
All details and intricacies of this matter are investigated in Section 3.1. Afterwards a
systematic way to deal with matrix elements between meson states that consist of one
heavy quark plus light degrees of freedom is presented in Section 3.2. The heavy-quark
effective theory makes use of the fact that the heavy quark only interacts softly with the
light degrees of freedom and is therefore nearly on-shell. In Section 3.3 the operators
relevant for the leading order of the B̄ → Xsγ decay rate are matched onto HQET and the
resulting non-local matrix element expanded in terms of local operators. It turns out that
this expansion contains an infinite number of leading order terms that need to be resummed
into a shape function. During the course of these calculations a suitable light-cone basis
as well as the HQET trace formalism are introduced.
3.1 Hadronic Matrix Elements
One of the major “approximations” adopted in the partonic calculation of the inclusive de-
cay rate in Section 2.2 was to replace the sum over hadronic matrix elements 〈Xhadrs γ|iT |B̄〉
by a sum over free quark matrix elements 〈Xparts γ|iT |b〉. It is not immediately clear that
this is a valid approximation at all, since the hadrons in the initial and final states are
in fact complex entities made out of their valence quarks, sea quarks as well as gluons,
constituting a QCD bound state. The typical binding energy for B mesons is of order
ΛQCD, also called the confinement scale. Therefore the dynamics of this bound state can
not be calculated in perturbative QCD.
However, the situation is simplified for mesons that consist of a heavy and a light quark.
In this case the motion of the heavy quark is only slightly influenced by the light degrees of
freedom. It is then appropriate to calculate the decay of a free heavy quark and account for
its Fermi motion inside the hadron by convoluting the result with the momentum distri-
34
bution of the heavy quark [67]. Since the heavy quark is nearly at rest (in the rest system
of the meson) the momentum distribution peaks at zero momentum, corresponding to an
on-shell heavy quark. This argument motivates the free quark approximation of the pre-
ceding Chapter. A more rigorous formulation of this statement is possible in the language
of HQET and will be considered now. As it will turn out, the free quark approximation
is the first term in an expansion in powers of the inverse heavy-quark mass. Furthermore,
the momentum distribution function will be identified with a sum over an infinite set of
leading twist operators [68, 69].
So far only a method to deal with the hadronic initial state has been considered. The
final state of the matrix elements does not contain mesons with a heavy quark. Instead it
can be related to a partonic final state by considering the sum over all exclusive modes.
Starting from Equation (2.22) the decay rate can then be written as
∫
→ 1 1 d
3
∑ pf 1
Γ(B̄ X γ) = |〈Xhadrs γ|iT |B̄〉|2
2MB V t (2π)2 2E
s
f
Xhadrs
 
∫
1 1 3∑〈 | − † d pf 1= B̄ iT |Xhadrγ〉〈Xhadr s s γ| iT |B̄〉 (3.1)2MB V t (2π)2 2Ef
Xhadrs
1 1
= 〈B̄|T †T |B̄〉.
2MB V t
In the last step the completeness of the set of states in the large square brackets was used.
It is important to note that this set is only complete for transition matrices T that contain
operators with a photon field as well as the right quantum numbers to form a strange
hadron. The form of the last line of (3.1) suggests the use of the unitarity of the S-Matrix
S†S = 1 and therefore
T †T = 2ImT (3.2)
to further simplify the expression for the decay rate. This leads to the optical theorem
Γ(B̄ → 1 1Xsγ) = Im 〈B̄|T |B̄〉
2MB V t (3.3)
1
= ImM(B̄ → 1B̄) = DiscM(B̄ → B̄) ,
MB 2iMB
where M is the invariant matrix element defined above Equation (2.24). It has to contain
at least a photon and a strange-quark propagator in order to contribute to B̄ → Xsγ. Just
like in Section 2.2 the leading order contribution to the transition matrix can be depicted by
a diagram containing two operators Q7γ (see first diagram in Figure 3.1). When considering
these diagrams it should be kept in mind that the in- and outgoing b-quark lines are in
fact not contracted with the initial and final states which are still hadronic. But there are
more subtleties related to the use of the optical theorem.
First of all it is important to note that the discontinuity prescription in (3.3) implies
the sum of all possible cuts through each diagram. However, not all cuts contribute to
35
s s
Q s sb 7γ Q7γ b b Q1 Q7γ b b Q8g Q8g b
Figure 3.1: Diagrams of forwards matrix elements with cuts (depicted by dashed lines)
that contribute to B̄ → Xsγ.
B̄ → Xsγ as can be seen in the second diagram in Figure 3.1. Specifically, the cut
related to the dashed line on the left of the diagram corresponds to a decay b → sc̄c.
Therefore the the discontinuity must be replaced by a restricted discontinuity that only
sums over cuts that contain at least a photon and a strange-quark line. In such a case the
matrix element can not be represented by operator products with a single time-ordering
prescription (originating from the definition of the transition matrix T ). Instead the fields
on the in-side (i.e., the side with the ingoing b-quark) of the cut must be time-ordered
while the ones on the other side of the cut anti-time-ordered. This can be seen by going
back to Equation (3.1) and noting that it contains an adjoint transition matrix T † which
gives rise to an anti-time-ordered operator product. Naively eliminating T † by using the
unitarity relation (3.2) is therefore not allowed in the case of a restricted number of final
states. It is possible to systematically include the effects of not time-ordered operator
products in the path integral formalism by using the Schwinger-Keldysh approach [70]. A
concise recent discussion has been presented in [71]. The final result of this discussion can
be summarized by the simple rule that, for diagrams with a restricted set of allowed cuts,
the Feynman rules on the out-side of the cut must be complex conjugated. All this is of
course not relevant for the leading order contribution due to the operator Q7γ , as in that
case all cuts contribute to B̄ → Xsγ.
Another important subtlety linked to the use of the optical theorem is related to the
appearance of soft particles in the final state. This is illustrated in the third diagram of
Figure 3.1. In this example the soft s-quarks are part of the final state, yet they do not
correspond to a cut propagator, since cutting an internal line puts the particles on-shell.
However, soft particles interact strongly via the strong interaction which implies that they
are far off-shell. In the actual calculation soft fields therefore remain uncontracted in the
matrix elements.
Following the arguments presented in this section so far, it was possible to eliminate
the hadronic final states from the calculation by considering the inclusive rate. The term
quark-hadron duality refers to the assumption that this rate is the same for hadronic and
partonic final states. Summing up, the B̄ → Xsγ decay rate can be written in the following
way. For contributions with all cuts relevant to the rate the optical theorem can be used
36
to write
1 1
Γ(B̄ → Xsγ) = Disc〈B̄|T |B̄〉
2iMB V t
∫
1 1 i
= Disc〈B̄| d4x d4yT{L(x)L(y)}|B̄〉 (3.4)
2iMB V t 2
∫
1 i
= Disc〈B̄| d4xT{L(x)L(0)}|B̄〉 ,
2iMB 2
where translation invariance was used in the last step. Each of the Lagrangian insertions
must contain one of the weak effective operators (for the leading power in 1/MW ) presented
in (2.17) or the corresponding hermitian conjugate. Of course, additional Lagrangian inser-
tions containing QCD or QED vertices are also possible. If certain cuts do not contribute
to the rate only the restricted discontinuity may be considered which shall be written as
∫
1 i
Γ(B̄ → Xsγ) = Disc 4restr.〈B̄| d xL†(x)L(0)|B̄〉. (3.5)
2iMB 2
One Lagrangian insertion appears in hermitian conjugated form, since it originated from
an anti-time-ordered product of operators or, equivalently, from the adjoint transition
matrix in (3.1). The two Lagrangians can only be connected by a cut propagator. Again,
additional Lagrangian insertions are possible. They have to appear in time- or anti-time-
ordering depending on which side of the cut they are inserted.
The above discussion relates the inclusive decay rate B̄ → Xsγ to a forward scattering
matrix element between hadronic states. That means that even though one has achieved
the elimination of the hadronic final states from the calculation, there is still no way to
determine the decay rate from first principles. However, as was already mentioned at the
beginning of this section the particular nature of the B-hadron allows for simplifications.
The determination of matrix elements between B-meson states does not require the use of
full QCD. Instead an effective theory implementing the fact that the heavy quark of the
meson is nearly on-shell can be used, which allows for a systematic expansion in the inverse
heavy-quark mass and which features additional symmetries. Using this EFT it will still
not be possible to actually calculate the hadronic matrix elements, but the symmetries
allow one to express the matrix elements in terms of a limited set of parameters, that are
process independent and can in principle be determined from experiment. By considering
certain ratios of observables it is therefore possible to reduce the effect of non-perturbative
physics on a calculation. The appropriate EFT is HQET and will be introduced in the
next section. As a closing remark for this chapter on hadronic matrix elements, it should
be mentioned that even though a formal expansion in 1/mb is always possible, it does not
necessarily lead to a sum of local matrix elements. In fact, it is a major objective of this
work to analyze the space-time structure of the operators in the expansion.
37
3.2 Heavy-Quark Effective Theory
In the previous section the inclusive decay rate was related to a QCD forward matrix
element between B-meson states. Although it is not possible to find an exact solution for
the meson bound state in QCD, enough is known about the physical situation inside the
meson to use a simpler effective theory in place of full QCD. This theory is HQET (for a
throughout review see [4]). It is constructed to be equivalent to QCD for the case of the
special kinematical configuration present within the hadron: The B-meson consists of a
heavy b-quark (in case of the B̄) with a mass mb and a cloud of light quarks, anti-quarks
and gluons. These light degrees of freedom and the heavy quark form a bound state with
a typical confinement scale of O(ΛQCD) which corresponds to the typical hadronic length
scale of O(1 fm). The starting point for the construction of HQET is the statement that
the heavy-quark mass is much larger than the confinement scale.
mb ≫ ΛQCD (3.6)
There are two immediate consequences to this statement. First, the Compton wavelength
of the heavy quark is O(1/mb ∼ 0.01 fm) and therefore much smaller than the resolution
that a probe with the energy of the light degrees of freedom can achieve. It follows that in
the limit of an infinite heavy-quark mass the light partons have no information about the
flavor or the spin orientation of the heavy quark. They see the heavy quark only as a static
source for a color field. This gives rise to the spin-flavor symmetry of HQET which is a
new symmetry and not present in full QCD but only in the considered kinematic region.
As a consequence of this symmetry the decays of different mesons that contain a heavy and
a light quark can be related to each other. Equation (3.6) further implies that the heavy
quark within the meson interacts only with soft gluons and is therefore nearly on-shell. Its
momentum can thus be written as
pµb = m
µ µ
bv + k , (3.7)
where v is the velocity of the meson and k is the residual momentum. Since the meson is
a real particle v satisfies v2 = 1 which implies that the residual momentum puts the heavy
quark off-shell due to interactions with the light degrees of freedom. The b-quark being
close to the mass-shell requires all components of k to be of O(ΛQCD).
The basic idea of HQET is to integrate out a part of the heavy-quark field that repre-
sents fluctuations around the mass shell. In order to do this one proceeds in the same way
as described in Section 2.1 where the heavy gauge bosons were integrated out of the SM
Lagrangian. However, since the heavy quark is still a degree of freedom of HQET it first
has to be decomposed into an on- and off-shell part. After performing the path integral
over the off-shell part the new non-local operators can be expanded in powers of 1/mb.
Finally one has to determine the Wilson coefficients by matching full QCD onto the EFT.
The first step is to introduce energy projection operators P± that are defined by
1± v/
P± = , (3.8)
2
38
and that satisfy the usual projection relations P 2± = P± and P+P− = 0. Using the operator
P+ (P−) on an on-shell, i.e., k = 0 (anti-)particle spinor returns the spinor itself, while the
reverse projection yields zero.
P+ub(mbv) = ub(mbv) , P−ub(mbv) = 0 ,
(3.9)
P−vb(mbv) = vb(mbv) , P+vb(mbv) = 0 .
Consequently, if the particle is slightly off-shell the projection operators decompose the
particle spinor into large and small two-component spinors. In the rest frame of the meson
the large projection corresponds to the upper two components of the spinor ub. There-
fore the P+ projection of the heavy-quark spinor only annihilates a quark while the P−
projection annihilates an anti-quark.
In order to construct the HQET Lagrangian describing B̄ decays the b-quark field is
decomposed by using the relation P++P− = 1. Additionally the strongest x-dependence is
factored out. Thus, in momentum space the heavy-quark fields only depend on the residual
momentum and there is a separate field for every velocity. This is commonly indicated by
a label v on the heavy-quark field, but since only heavy quarks with a single velocity play
a role in this work, this label is dropped. This leads to
b(x) := e−imbv·xb̃(x) = e−imbv·x(P + P )b̃(x) := e−imbv·x+ − [h(x) +H(x)]. (3.10)
The small component field H which is non-zero only for off-shell quarks is going to be
integrated out. After this decomposition the QCD Lagrangian for the heavy quark can be
written as
LQCD = b̄(iD/ −mb)b
(3.11)
→ h̄ iv ·Dh− H̄(iv ·D + 2mb)H + h̄ iD~/H + H̄ iD~/ h ,
where the relations v/h = h and v/H = −H have been used and D~ µ is defined through
Dµ = vµv · D + D~ µ. Integrating out the small component field H in the path integral
formalism (entirely analogous1 to the procedure in Section 2.1) introduces a new term in
the action functional which is given by
∫
1
Seff = d
4x h̄(x) iD~/ iD~· / h(x). (3.12)iv D + 2mb
Since the strong x-dependence was factored out of the heavy-quark fields in (3.10), the
derivative acting on h is much smaller than 2mb and the above expression can be expanded
in terms of local action functionals. Up to first order in 1/mb this yields
( )
∫
1 1
S 4eff = d x h̄(x) iD~/ iD~/ h(x) +O . (3.13)
2m 2b mb
1In fact, there is one subtlety related to the appearance of the covariant derivative in the functional
determinant instead of a normal derivative. However, by choosing the axial gauge v · A = 0 the case
reduces to the one considered before [72].
39
Ignoring radiative corrections, which would introduce scale dependent Wilson coefficients
for each effective operator, the HQET Lagrangian for the heavy quarks may be written
as [73, 74]
( )
L 1 1HQET = h̄ iv ·Dh+ h̄(x) iD~/ iD~/ h(x) +O
2mb m2b
( ) (3.14)
· 1 ~ 2 1= h̄ iv Dh+ h̄(iD) h+ h̄ σ g Gµνµν s h+O
1
,
2mb 4m m2b b
where the field strength tensor G is defined by ig µνsG = [iD
µ, iDν ]. This Lagrangian is
equivalent to the one of QCD in the kinematic region of a slightly off-shell heavy quark.
It exhibits several interesting features. First, it is important to note that the covariant
derivatives only contain soft gluon fields, since hard fields would put the heavy quark
far off-shell, which is outside of the kinematic regime appropriate for HQET. Another
important observation is that the leading power term contains no Dirac matrices thereby
making the interaction of the heavy quark with (soft) gluons independent of its spin state.
Also there is no mass-dependence in the leading term, which implies that the Lagrangian
is the same for all heavy quarks. The spin-flavor symmetry mentioned at the beginning of
this section is therefore manifest in the HQET Lagrangian of (3.14).
At subleading power in 1/mb two new operators are introduced, given in the second
line of (3.14). The first corresponds to the kinetic energy due to the residual momentum,
while the second represents the “chromomagnetic” interaction of the heavy-quark spin with
the gluon field. The spin-flavor symmetry is therefore broken at subleading power. For
problems containing several heavy quarks at different velocity the full Lagrangian is given
by a sum of the Lagrangians like (3.14), each with the appropriate velocity. The heavy
quarks only interact by the exchange of soft gluons.
This concludes the derivation of the HQET Lagrangian for heavy quarks matched at
tree level. But in order to calculate the forward scattering matrix elements of (3.4) it is
necessary to also include the the weak effective operators of (2.18). This can in principle
be done by using the Lagrangian (2.18) instead of the pure QCD Lagrangian as starting
point, then decomposing the b-quark fields as was done in (3.10) and finally integrating
out the small component field H. However, an equivalent approach is available that lends
itself to the use as a recipe. On the classical level, integrating out a particle corresponds
to solving its equation of motion and replacing the field in the Lagrangian by the solution.
In the case of the small component field H the solution is formally given by
H(x) = (iv ·D + 2m )−1iD~b / h(x)
iD~
( )
/ 1 (3.15)
= h(x) +O .
2mb m2b
The field H is introduced into the heavy Lagrangian only by the decomposition (3.10)
which can therefore immediately be rewritten as
[ ]
~ ( )
b(x) → e−imbv·x[h(x) +H(x)] → e−imbv·x iD/ O 1h(x) + h(x) + . (3.16)
2m m2b b
40
Using this replacement rule in the QCD Lagrangian (3.11) instantly yields the HQET
Lagrangian (3.14). The same replacement can be performed in the effective operators
Q1...8g. This yields an expansion of the operators in powers of 1/mb that can be written as
∞
∑
(j) (j)
Qi = Ci Qi , (3.17)
j=0
with j indicating the suppression of the operator by powers of 1/mb and C representing the
Wilson coefficients from the matching onto HQET. They can be calculated perturbatively
to the desired order in αs. However, at this point it is sufficient to perform the matching
onto HQET only at tree level.
As an example the leading power contribution to the decay rate in (3.4) will be con-
sidered in the next Section. Along the way two important discoveries will be made. On
the one hand it will become apparent that an infinite amount of local operators needs to
be resummed for the leading contribution in the endpoint region. This will be a further
motivation for the introduction of another EFT, namely SCET. On the other hand the
calculation will eventually lead to the proof of the statement, that the partonic calculation
yields just the leading power of the 1/mb expansion.
3.3 Leading Power Matching and OPE
With an appropriate EFT realizing a systematic expansion in terms of a small parameter
(1/mb) at hand it is now possible to consider the hadronic forward scattering matrix element
of (3.4) in more detail. The leading contribution is given by the insertion the operator Q7γ
and its adjoint which leads to
∫
1 G2
Γ(B̄ → Xsγ) = Disc〈B̄|i d4x F |C 2 †7γλt| T{Q
2iM 2 7γ
(x)Q7γ(0)}|B̄〉 , (3.18)
B
with the scale dependencies of the Wilson coefficient dropped for brevity and Q7γ defined
in (2.17). Matching Q7γ at tree level onto HQET yields
−emb µν −embQ7γ = s̄σµν(1 + γ5)F b = s̄σµν(1 + γ )∂µ5 Aνb
8π2 4π2
[ ]
− ~ ( )em (3.19)→ b e−imbv·xs̄σ (1 + γ )∂µ iD/ 1µν 5 Aν h+ h+O .
4π2 2m m2b b
Inserting the leading power matched expression for Q7γ into (3.18) and using Wick’s the-
orem to contract the photon and strange-quark fields yields after some algebra
→ 1 G
2 m2α
Γ(B̄ Xsγ) =
F b |C7γλt|2
2iM 4π3B
∫ 4
× 〈 | d q µ mbv/+ i∂/− /q g
νβ
Disc B̄ i h̄(0)σ q σ qαµν αβ (1 + γ4 5)h(0) |B̄〉,(2π) (mbv + i∂ − q)2 + iε q2 + iε
(3.20)
41
where the strange-quark mass is set to zero. The partial derivatives act on the heavy-
quark field and correspond to the residual momentum k in momentum space. Due to the
derivatives in the denominator the resulting operator is non-local making it desirable to
employ an OPE. This OPE will contain operators that are suppressed by powers of 1/mb
and therefore add to the subleading contributions from the expansion of (3.19). Before the
expansion is performed the partial derivative will be replaced by a covariant derivative.
This yields the propagator in the background field of the soft gluons in the B-meson. It
will also be convenient to introduce a light-cone 4-vector-basis and decompose all momenta
of the problem in this basis. All the necessary definitions will be collected in the following
section.
3.3.1 The Light-Cone Basis
First one introduces a light-like vector n̄ that points in the direction of the momentum of
the external photon q = Eγn̄. Another vector n is defined by the relation n + n̄ = 2v.
The decay products other than the photon will have a large momentum component in this
direction. It is then easy to prove that
n̄2 = 0 = n2, n̄ · v = 1 = n · v, n · n̄ = 2. (3.21)
In the B-meson rest frame an appropriate choice for the light-cone vectors would be nµ =
(1, 0, 0, 1) and n̄µ = (1, 0, 0,−1). All 4-vectors can be decomposed in this light-cone basis
nµ n̄µ
xµ = n̄ · x+ n · x+ xµ⊥. (3.22)2 2
Note that by definition the external momenta of the B-meson MBv, the photon q and the
jet PX = MBv − q all have vanishing perpendicular components. With this basis at one’s
disposal the OPE of the s-quark propagator may proceed.
3.3.2 Calculation Continued
In order to expand the HQET operator in Equation (3.20) the propagator is rewritten in
a geometric series
mbv/+ iD/ − /q mbv/+ iD/ − /q
=
(mbv + iD − q)2 + iε m 2b(mb − 2Eγ) +mb in ·D + (mb − 2Eγ)in̄ ·D + (iD) + iε
mbv/+ iD/ − ∞
[ ]
/q 2
i
∑
− in ·D= − −
in̄ ·D − (iD) .
mb(mb 2Eγ + iε) mb − 2Eγ + iε mb mb(mb − 2Eγ + iε)i=0
(3.23)
Taking the cut forced by the discontinuity prescription in (3.20) corresponds to the replace-
ment 1/(x+iε)n → −2πi(−1)nδ(n)(x). For Eγ outside the endpoint region (mb−2Eγ → mb)
the propagator can thus be expanded in a series of singular terms with higher orders sup-
pressed by powers of 1/mb. The first term in this expansion (i = 0) yields the expected
42
momentum conservation δ-function of a two-body decay. At leading power in 1/mb the
decay rate in (3.20) is therefore equal to
m
∫ b
→ 1 G
2
Fm
2 2
Γ(B̄ X γ) = b
α
s |C 2 37γλt| dEγ Eγ〈B̄|h̄(0)n̄/(1 + γ4 5)h(0)|B̄〉δ(mb − 2Eγ) ,MB 4π 0
(3.24)
where the limits of integration are set by the cuts through the leading power contribution.
Note that the axial vector part of the matrix element vanishes due to parity invariance.
Also the Dirac matrix n̄/ = n̄ ·vv/+~̄n/ can be replaced by 1 between two heavy-quark spinors.
So far the heavy-quark expansion was realized at two different points in the calculation.
First the dipole operator Q7γ was matched onto the leading power HQET operator, then
the strange-quark propagator was expanded in powers of 1/mb. In order to continue with
the systematic expansion the initial and final states must be written in an mb-independent
way. The states are mass-dependent because they are defined as eigenstates of the HQET
Lagrangian (3.14) that contains corrections of O(1/mb). It is more appropriate to define
the external states as eigenstates of only the leading term in (3.14) and include the power
corrections as perturbations [75]. From now on all states are considered to be eigenstates
of the leading power HQET Lagrangian and the necessary Lagrangian insertions of higher
power to compensate for this are implicitly assumed.
By utilizing the heavy-quark symmetries, the remaining matrix element can now be
related to a universal form factor that is especially simple in the case of forward scattering.
Along with other meson form factors it was introduced by Isgur and Wise in [76]. Even
though the case at hand is simple, it is useful for future purposes to derive a systematic
way of defining form factors from HQET matrix elements. This can be done by employing
the HQET trace formalism [77, 78] that is introduced in the next section. Afterwards the
calculation of the leading power contribution to B̄ → Xsγ is resumed.
3.3.3 The HQET Trace Formalism
Due to the complex interactions of the light degrees of freedom within the meson it is not
possible to analytically calculate hadronic matrix elements. Instead they are decomposed
into a minimal set of form factors which can then be measured or determined by numerical
calculations on a lattice [79,80]. Through the introduction of HQET it is possible to relate
many hadronic matrix elements at a certain order in the 1/mb expansion to a limited
number of form factors. The HQET trace formalism offers a systematic way to find these
relations.
An important simplification was achieved by using the eigenstates of the leading power
HQET Lagrangian as initial and final states. Ignoring the 1/mb corrections, these states
correspond to hadrons that contain a heavy quark with infinite mass and the light degrees
of freedom. These hadrons have a well defined behavior under the spin and Lorentz trans-
formations. Their ground state can be represented by a direct spinor product uhv̄l, where
uh is the heavy-quark spinor, satisfying v/uh(v) = uh(v) and v̄l represents the light degrees
43
of freedom which transform like an anti-quark with velocity v under Lorentz transforma-
tions. The light spinor satisfies v/v̄l(v) = −v̄l(v). Depending on the spin quantum numbers,
the object uhv̄l has the correct transformation properties to represent the ground state of
a pseudoscalar or a vector meson state. In this work only the pseudoscalar mesons are of
interest. In terms of Dirac matrices the states can be written in a covariant way by [77,78]
√
M − 1 + v/(v) = MB γ5. (3.25)
2
√
Note that the factor MB enters through the relativistic normalization of the states. In
the decay rate (3.24) this factor always cancels.
Even though nothing is known about the interactions of the light degrees of freedom,
the hadronic matrix element will have a certain structure. Consider the matrix element
〈B̄(v)|h̄Γh|B̄(v′)〉 , (3.26)
where the heavy-quark fields have a different velocity and Γ represents a general Dirac
structure. Due to Lorentz invariance this matrix element corresponds to a spinor product
with no open indices. At leading power in HQET the heavy quarks interact with gluons
without introducing a Dirac structure at the vertex. Therefore the spinor index corre-
sponding to the heavy quark from the initial and final state is directly connected to the
spinor indices of the Dirac structure Γ. The matrix element will thus contain
M ′ij(v)ΓjkMkl(v ). (3.27)
The open light quark spinor indices i and l can be contracted to any type of Dirac structure
that enters through the complex interactions of the light degrees of freedom. The matrix
element can therefore be written as
〈B̄(v)|h̄Γh|B̄(v′)〉 = tr [M(v)ΓM(v′)Ξ(v, v′)]. (3.28)
The Dirac matrix Ξ can in principle depend on the scale at which the operators are renor-
malized. Since the matching to HQET is only performed at tree level here, this dependence
has been dropped.
As an example the important case Γ = γµ will be considered. Since there are two
independent external vectors available, the most general form for Ξ is given by
Ξ(v, v′) = Ξ0(v, v
′) + v/Ξ ′1(v, v ) + v/
′ Ξ2(v, v
′) + v/v/′ Ξ (v, v′3 ). (3.29)
Contributions containing γ5 are not allowed due to parity invariance. Lorentz invariance
on the other hand implies that the scalar functions Ξi can only depend on the product
v · v′. Inserting (3.29) into (3.28) and performing the trace yields
〈B̄(v)|h̄γµh|B̄(v′)〉 = MB[Ξ0(v · v′)− Ξ (v · v′1 )− Ξ2(v · v′) + Ξ (v · v′)](vµ + v′µ3 ). (3.30)
44
The expression in square brackets is the Isgur-Wise function ξ(v · v′) [76]. It is important
to note that this function is the same for all heavy-light mesons due to the HQET flavor
symmetry. By using current conservation
∫
d3x h̄(x)γ0h(x)|B̄(v)〉 = 1 |B̄(v)〉 , (3.31)
the normalization of the Isgur-Wise function may be determined. In the forward scattering
case v = v′ one finds ξ(1) = 1. It follows that
〈B̄(v)|h̄γµh|B̄(v)〉 = 2MBvµ. (3.32)
Of course, it is immediately clear from Lorentz invariance that the forward scattering
matrix element in (3.32) is proportional to vµ, but the general procedure will be useful
several times later on. With the result (3.32) one is finally able to show that the decay
rate at leading power in the heavy-quark expansion corresponds to the decay rate for free
quarks. This will be done now.
3.3.4 The Leading Power and the Shape Function
The section started with the definition of the inclusive decay rate in terms of a forward scat-
tering matrix element. The operator of this matrix element was then matched onto HQET
at the leading power in 1/mb. This gave rise to a non-local HQET operator. Furthermore
the external hadron states were replaced by eigenstates of the leading power HQET La-
grangian. After expanding the operator in an OPE, the leading power local HQET matrix
element was considered and its form determined by the HQET trace formalism. Inserting
the result (3.32) into (3.24) yields
m
G2 m2
∫ b
→ F bα
2
Γ(B̄ X γ) = |C λ |2 dE 3s 4 7γ t γ Eγδ(mb − 2Eγ) , (3.33)2π 0
which is equivalent to the free quark result in (2.25), just as claimed at the end of Chapter 2.
This naturally leads to the question of at what order in 1/mb corrections do appear.
To answer this question higher orders in the matching (3.19) and in the expansion (3.23)
must be considered. In addition to the matching the higher power Lagrangian insertions
compensating for the use of the leading power eigenstates are relevant. This section will
deal with the higher orders of the expansion of the propagator in (3.23).
The expansion in (3.23) was based on the 1/mb suppression of the higher order terms
45
in the series. After calculating the discontinuity this expansion can be written as2
( )
mbv/+ iD/ − /q → −2πi /q iD/− v/− +(mbv + iD q)2 + iε mb mb mb
[
( ) · ( ) (× − 2Eγ in D ′ − 2Eγ 1 in ·
)2 ( )
D
δ 1 + δ 1 + δ′′ 1− 2Eγ + . . .
mb mb mb 2 mb mb
]
( ) ( )
2
− in̄ ·D − 2Eγ (iD) 2Eγδ 1 + · · ·+ δ′ 1− + . . . ,
m m 2b b mb mb
(3.34)
where the second line only contains powers of the first term in the square bracket of (3.23),
while the third line contains the first power of the second and third term in (3.23). Even
though all higher powers are formally suppressed by 1/mb it is possible to assign a certain
scaling to the δ-functions themselves. It follows, that in the endpoint region (2Eγ/mb → 1)
all terms in the second line of (3.34) are of the same order. The concept of scaling will be
carefully introduced in Section 4.1, but here it is sufficient to note that the leading singular
terms in the second line of (3.34) are all equally important. This phenomenon is called the
breakdown of the OPE. It indicates that an HQET based local OPE is not applicable in
the kinematic region of Eγ → mb/2. The ad-hoc solution to this problem is to resum the
leading singular terms (corresponding to the first term in the square brackets of (3.23)) into
a so-called shape function [81, 82]. This function only has support in the endpoint region
and is defined by its moment expansion under the energy integral. The resummation can
explicitly be performed by expanding the propagator (3.23) in a different way
mbv/+ iD/ − /q mbv/+ iD/ − /q
(mbv + iD −
=
q)2 + iε mb(mb − 2Eγ + in ·D + iε)
∞ [ ]i
∑
× − (mb − 2Eγ)in̄ ·D − (iD)
2
.
mb(mi=0 b − 2Eγ + in ·D + iε) mb(mb − 2Eγ + in ·D + iε)
(3.35)
Here the higher order terms in the expansion are truly suppressed and the leading power
contains the resummed terms of the second line of (3.34)
( )
mbv/+ iD/ − /q → − /q iD/
[ ]
− 2πi v/− + δ(mb − 2Eγ + in ·D) + . . . . (3.36)(mbv + iD q)2 + iε mb mb
By inserting this result into the decay rate (3.20) suggests a definition for the leading order
shape function through
− 〈B̄|h̄ δ(mb − 2Eγ + in ·D)h|B̄〉S(mb 2Eγ) = . (3.37)
2MB
2In order to eliminate derivatives of δ-functions the relation xδ′(x) = −δ(x) was used which holds for
non-singular test functions.
46
However, due to the derivative in the argument of the δ-function the shape function does
not correspond to a single, local HQET operator. Instead, it is a new non-perturbative
object, whose moments can be expressed in terms of HQET matrix elements. Performing
a moment expansion returns just the sum of terms in the second line of (3.34) with local
matrix elements as coefficients.
− 〈B̄|h̄ h|B̄〉 〈B̄|h̄ in ·Dh|B̄〉S(mb 2Eγ) = δ(mb − 2Eγ) + δ′(mb − 2Eγ)
2MB 2MB
〈 | · (3.38)1 B̄ h̄(in D)2h|B̄〉
+ δ′′(mb − 2Eγ) + . . .
2 2MB
The first moment was already determined in the leading power calculation above and is
equal to 1. The second moment is proportional to the HQET matrix element 〈B̄|h̄iDµh|B̄〉
which vanishes by applying the equation of motion iv · Dh = 0 derived from the leading
power Lagrangian [83]. The third moment requires the decomposition of the matrix element
〈B̄|h̄iDµiDνh|B̄〉 in terms of form factors. This can be achieved by employing the trace
formalism of Section 3.3.3
〈B̄|h̄ΓiDµiDνh|B̄〉 = tr [MΓM ((gµν − vµvν)Ξ + σµν0 Ξ1)]
[ ( ) ]
1 + v/ λ1 iλ2 1 + v/ (3.39)
=: MBtr Γ (g
µν − vµvν) + σµν ,
2 3 2 2
where the HQET parameters λ1 and λ2 are related to the matrix elements of the subleading
operators in the HQET Lagrangian (3.14). When inserting these operators into the formula
above one finds
〈B̄|h̄(iD~ )2h|B̄〉 = 2MBλ1 , (3.40)
〈 | 1B̄ h̄ σ Gµνµν h|B̄〉 = 6MBλ2 , (3.41)
2
which suggests the relation of λ1 to the kinetic energy of the heavy quark within the
meson and λ2 to the hyperfine chromomagnetic interactions. With these definitions the
third moment of the leading shape functions calculates to −λ1/3 (set Γ = nµnν). Of course,
the resummation can be extended to higher orders in 1/mb for the product of the second
or third term in the square brackets of (3.23) with any power of the first term.
The shape function can be systematically derived within the framework of SCET which
introduces an appropriate power counting. This will be the topic of Section 4.3.3. There,
also corrections of higher order in αs will be considered. At the leading order in αs and
1/mb the shape function just replaces the momentum-conservation δ-function in (3.33)
M
G2 2
∫ B
m α 2
Γ(B̄ → X F b 2 3sγ) = |C7γλt| dEγ E S(mb − 2Eγ) , (3.42)
2π4 γ0
where the upper limit of integration is now MB/2, since the effects of the residual momen-
tum k are taken into account. The shape function thus encodes the form of the photon
47
spectrum in the endpoint region. Since the deviation of the spectrum from the form of a
δ-function is at leading order in αs related to the residual momentum of the heavy quark,
the shape function can also be interpreted as the momentum distribution function of the
heavy quark within the meson. This is analogous to the parton distribution functions in
the description of proton collisions.
Since the shape function only has support over the endpoint region, it affects the
integrated rate only if the limits of integration are in that region (e.g. by the introduction
of a lower cut E0 ∼ mb/2 on the photon energy). For a much larger range (E0 → 0)
the shape function might be replaced by its moment expansion, with higher orders being
suppressed by powers of 1/mb. In that case there is no need to resum the terms in (3.34).
For the contributions considered here, the terms of order 1/mb then vanish due to the
equation of motion. However, later on subleading shape functions that are also relevant
outside the endpoint region will be introduced. These subtleties are not easily analyzed
in the framework of HQET, illustrating the demand for a more appropriate EFT (SCET).
In the remainder of this chapter the higher power contributions of the local OPE are
considered.
3.4 Higher Power Corrections to the Total Rate
In the last Section it was shown that outside the endpoint region the decay rate of B̄ → Xsγ
can be written as a local OPE in terms of HQET operators with the leading power term
corresponding to the free quark result. Corrections due to the expansion of the light quark
propagator did only appear at the order 1/m2b . Outside the endpoint region this correction
is given by the third moment of the leading order shape function that was found to be
proportional to the HQET parameter λ1. Additional contributions to the order 1/m
2
b are
introduced by matching Q7γ onto subleading HQET operators as prescribed in (3.19) and
by Lagrangian insertions. Instead of the matrix element 〈B̄| †T{Q7γ(x)Q7γ(0)}|B̄〉 the decay
rate in (3.18) will then contain matrix elements like
〈 | { †(1) (1)B̄ T Q7γ (x)Q7γ (0)}| 〉 〈 | {
†(2)
B̄ , B̄ T Q7γ (x)Q7γ(0)}|B̄〉 or
〈B̄|T{ †(1)Q (x)Q (0)L(1)7γ 7γ (y)}|B̄〉 etc.
(i)
where the superscript (i) denotes the power of 1/mb suppression and Q
(i)
7γ and L can be
read of from (3.19) and (3.14), respectively. Furthermore, the 1/mb term from the OPE
can be combined with any other 1/mb suppressed terms. The complete calculation was
performed in [84] and resulted in
[ ]
2 5
Γ(B̄ → G m α λ1 − 9λ2Xsγ) = F b |C7γλ 2t| 1 + . (3.43)
32π4 2m2b
This is the final justification for the use of the free quark approximation. If the total
inclusive rate is considered, non-perturbative corrections to the free quark result only
48
appear at order 1/m2b . Of course, the total rate is not accessible by experiment, as was
explained in Section 1.3. Thus, a dedicated analysis of the effects in the endpoint region
is of interest.
3.5 Conclusions of this Chapter
At the end of Chapter 2 the question of how non-perturbative effects influence the calcula-
tion of the B̄ → Xsγ decay rate was posed. These effects are encoded in hadronic matrix
elements. In this Chapter the heavy-quark effective theory was introduced, which made
it possible to expand the matrix elements of the inclusive decay rate in powers of 1/mb.
Each term in this expansion can be related to a limited set of HQET parameters. These
are still not analytically computable, but universal due to the heavy-quark symmetries. In
particular it was shown that the leading power contribution corresponds to the free quark
result used in Chapter 2 and that, in this framework, the corrections to the total rate
appear only at order 1/m2b . During the course of the calculation it was found that the
local OPE breaks down in the endpoint region. This was solved by the introduction of the
shape function. However, it was left open how radiative corrections affect this approach.
In order to deal with this kind of questions the shape functions must be introduced in a
more systematic way. This can be done in the framework of SCET, which will be the topic
of the next chapter.
49
Chapter 4
Soft-Collinear Effective Theory
It will be the aim of this chapter to introduce SCET in the context of B̄ → Xsγ in the
endpoint region and to demonstrate its advantages over the previous approaches. The
fundamental concepts of SCET are the systematic separation of the scales involved in the
problem and the implementation of an appropriate power counting. In Section 4.1 the
kinematics of B̄ → Xsγ in the endpoint region are analyzed, which leads to the identifica-
tion of the relevant scales and to the scaling of the fields. Afterwards, in Section 4.2, SCET
will be put forward as the proper EFT to deal with the challenges identified in the previous
chapters. This involves the derivation of the Lagrangian up to the required order in the
power counting. In order to calculate the differential B̄ → Xsγ decay rate, the leading
power SCET version of the effective heavy-to-light operator is introduced in Section 4.3
and its Wilson coefficient determined by matching QCD onto the SCET operator. This
first matching step separates physics at the hard scale from the lower scales. In a second
step, SCET is matched onto HQET thereby proving a complete separation of the different
scales for the decay rate. This factorization forms the basis of a resummation of the large
logarithms in Section 4.4.
4.1 Kinematics and Power Counting
The two preceding chapters introduced two basic concepts which are important for the
analysis of the B̄ → Xsγ decay rate. On the one hand a perturbative calculation was used
to determine the high-energy behavior of the particles involved. On the other hand an
expansion in powers of the inverse heavy-quark mass made it possible to systematically
deal with non-perturbative effects. However, both approaches encountered difficulties in
the endpoint region of large photon energy that were related to the smallness of the quantity
mb−2Eγ . While the perturbative calculation suffered from large logarithms for Eγ → mb/2,
the 1/mb expansion produced an infinite amount of leading order matrix elements. More
precisely, these difficulties are related to the appearance of several distinct energy scales.
First, there is the large energy scale set by the mass of the decaying meson MB. Second,
the size of the hadronic interactions introduce the scale ΛQCD. In the rest frame of the
50
meson the photon momentum q = Eγn̄ can be written as q = (Eγ, 0, 0,−Eγ), where Eγ
is of the same order of magnitude as MB in the endpoint region. The momentum of the
hadronic jet PX = MBv − q then corresponds to
− n̄ nPX = (MB 2Eγ) +MB , (4.1)
2 2
with a large component of order MB in the n-direction and a small component of order
ΛQCD in the n̄-direction. It will be convenient to introduce the notation
xµ = (n · x, n̄ · x, x⊥) = (x+, x−, x⊥) (4.2)
for any 4-vector decomposed in the light-cone basis of Section 3.3.1, where x⊥ denotes
both perpendicular components. Thus, the jet has a large energy EX = P
0
X = MB −Eγ ∼
√ √ √
O(MB) but a small invariant mass M 2X = PX = MB(MB − 2Eγ) ∼ O( ΛQCDMB).
This invariant mass sets an intermediate scale, which is larger than the hadronic scale ΛQCD
but smaller than the hard scale MB. Ratios of these scales are responsible for the large
logarithms in the perturbative calculation. Also, a power counting taking into account all
the different scales is necessary for a systematic expansion in a small parameter.
In order to implement the appropriate power counting, a small parameter λ is intro-
duced, that parameterizes the difference between the scales by
hard: MB ,
√
intermediate: Λ M = λ1/2QCD B MB , (4.3)
soft: ΛQCD = λMB .
The parameter λ is therefore of the order1 ΛQCD/MB ∼ 0.1. Instead of the meson mass
MB one might also choose the heavy-quark mass mb, both of which are the same at the
leading power in λ. By using the parameter λ, different momenta may be classified by their
scaling properties. A momentum component is said to scale like λ when its magnitude is
of order λmb. In the context of B̄ → Xsγ in the endpoint the following momentum regions
are of relevance.
hard (h): (p+, p−, p⊥) ∼ (1, 1, 1)
hard-collinear (hc): (p , p , p ) ∼ (λ, 1, λ1/2+ − ⊥ )
(4.4)
anti-hard-collinear (hc): (p+, p , p ) ∼ (1, λ, λ1/2− ⊥ )
soft (s): (p+, p−, p⊥) ∼ (λ, λ, λ)
Unfortunately the naming is not consistent in the literature. Especially in the earlier
works [7, 85] soft momenta were often referred to as ultrasoft, while hard-collinear were
called collinear. The different momentum modes will now be discussed.
In the case of B̄ → Xsγ hard momenta will appear in virtual corrections to the heavy-
to-light operators. As was discussed in Section 3.2 the large part of the heavy-quark
√
1Note, that in the literature λ is sometimes defined as ΛQCD/MB .
51
momentum mbv is extracted and only the soft residual momentum k remains dynamical.
In the endpoint region, the partons making up the final state jet can only have a large
momentum component in the primary direction n of the jet momentum (hence the name
“collinear”), but are allowed to have small components in the other directions. In the case
of a two-body decay of an on-shell heavy quark, the small components vanish. Their origin
is therefore dynamical. In order to guarantee the scaling of the invariant mass of the jet,
the small components scale as given in (4.4). Soft particles do not change the scaling of
either the invariant mass or the jet energy and can therefore also be part of the final state.
Furthermore, soft interactions are responsible for the binding of the initial state partons
into a hadron. The only particle in the final state with a large momentum component in
the n̄-direction may be the photon, but intermediate fields can be anti-hard-collinear as
well.
The basic idea of the soft collinear effective theory is to introduce separate fields for
each necessary momentum mode and then integrate out the modes at the perturbative
scales, namely hard fluctuations as well as fluctuations around the light-cone of the hard-
collinear particles. SCET therefore separates physics at the hard scale from the physics
at the intermediate or lower scale. As will be shown, this additional matching step (in
addition to the matching onto HQET) allows for the resummation of the large logarithms
that appear in the free parton calculation. Furthermore, since all scales are identified and
parameterized by λ, the power counting is definite and there will be no need to explicitly
resum an infinite amount of leading order operators as was the case in HQET.
A priori it is not clear that the momentum modes defined in (4.4) are sufficient to
describe B̄ → Xsγ. The requirement is, that the fields of SCET reproduce the correct
infrared behavior of the considered QCD processes. The decomposition of an QCD am-
plitude into the relevant momentum regions corresponds to an analysis using the method
of regions [86]. Depending on the problem, certain momentum modes do not contribute
due to the appearance of scaleless integrals. It turns out that the momentum modes of
(4.4) are indeed sufficient for inclusive decays. The effective theory containing only hard-
collinear and soft modes is called SCET-I. The inclusion of anti-hard-collinear modes leads
to SCET(hc,hc,s) which corresponds to the sum of two SCET-I with interchanged light-
cone directions n and n̄. The two sectors are connected when matching QCD diagrams
onto SCET operators. In the case of exclusive decays, a new momentum mode becomes
important. The collinear mode (c) describes the partons inside the hadron with an in-
variant mass of O(ΛQCD) and scales like (λ2, 1, λ) [87, 88]. After matching SCET(hc,c,s)
onto a theory containing only collinear and soft modes one arrives at SCET-II. In addition,
this theory contains soft-collinear messenger modes (i.e. modes not appearing as external
states) with a scaling of (λ2, λ, λ3/2) [89]. Since this work only deals with inclusive decays,
SCET-I is the relevant EFT and will henceforth just be referred to as SCET. The following
Section will review the derivation of the SCET Lagrangian for inclusive decays.
52
4.2 Soft-Collinear Effective Theory
The soft-collinear effective theory was first introduced in [6] to solve the problem of large
logarithms in the perturbative calculation of B̄ → Xsγ. This presentation will make use of
the position-space formalism presented in [85]. Since the theory is constructed to contain
only hard-collinear and soft momentum modes, the QCD fields are decomposed as
ψ(x) = ψhc(x) + q(x) , (4.5)
where the fields are defined by the scaling of their momenta
∂µψhc(x) ∼ (λ, 1, λ1/2)ψhc(x) and ∂µq(x) ∼ (λ, λ, λ)q(x). (4.6)
In analogy to the decomposition of the heavy-quark field in HQET, the hard-collinear fields
can be decomposed into a large and a small two-component spinor. This time, the small
component vanishes if the spinor describes a particle that is exactly on the light-cone. The
hard-collinear particles of the final state jet have a large momentum component in one
light-cone direction and small components in the other directions and are therefore nearly
on the light-cone. The appropriate projection operators are
n/n̄/ n̄/n/
Pn = and Pn̄ = , (4.7)
4 4
which satisfy Pnu(p+n̄) = 0 and Pnu(p−n) = u(p−n) for spinors of light-like particles and
correspondingly for Pn̄. As projection operators they furthermore satisfy Pn + Pn̄ = 1
which leads to the following decomposition for hard-collinear fields
ψhc(x) = Pnψhc(x) + Pn̄ψhc(x) =: ξ(x) + η(x) , (4.8)
with the decomposed fields satisfying n/ξ = 0 and n̄/η = 0. By calculating the propagator
it is possible to assign a certain scaling to the fields themselves.
∫ 4
〈 | { }| 〉 〈 | { }| 〉 d p n/ in̄ · p0 T ξ(x)ξ̄(y) 0 = Pn 0 T ψhc(x)ψ̄hc(y) 0 Pn̄ = e−ip(x−y) , (4.9)
(2π)4 2 p2 + iε
Since by definition p is a hard-collinear momentum the integrals over the various compo-
nents scales like dnp dn̄p dp1⊥ dp
2
⊥ ∼ λ2. The right hand side of (4.9) therefore scales like λ
which implies that ξ ∼ λ1/2. A similar consideration leads to the scaling properties of the
remaining fermion fields, where one has to keep in mind that the integration measure for
soft integrals scales like λ4. Summing up, one finds
ξ ∼ λ1/2, η ∼ λ and q ∼ λ3/2. (4.10)
The gluon fields are decomposed in an analogous way Aµ(x) = Aµhc(x) + A
µ
s (x). Their
scaling can be determined by contracting the gluon propagator in a general gauge with the
light-cone vectors n and n̄. One finds
Aµhc ∼ (λ, 1, λ1/2) and Aµs ∼ (λ, λ, λ) , (4.11)
53
which implies that the gluon fields scale just like their momenta. With these fields scalings
it is possible to assign a definite scaling to every operator made out of the effective theory
fields.
In order to describe the B̄ → Xsγ process in the language of SCET it is necessary to
determine the Lagrangian for the EFT fields. This is done by using the QCD Lagrangian
as a starting point and then decomposing the QCD fields into the SCET fields by using
(4.5) and (4.8). This leads to
L n̄/SCET = ξ̄ in ·Dξ + ξ̄ iD/ ⊥η + ξ̄gA/ hcq + η̄ iD/ ⊥ξ
2 (4.12)
n/
+ η̄ in̄ ·Dη + η̄gA/ hcq + q̄gA/ hcξ + q̄gA/ hcη + q̄ iD/ sq ,
2
where where g is the strong coupling and Ds = ∂ − igAs is the covariant derivative con-
taining only a soft gluon field. Vertices with only one hard-collinear field are forbidden
by momentum conservation. In a next step, the fluctuations around the light-cone η are
integrated out by solving the classical equation of motion in analogy to the procedure em-
ployed in the derivation of the HQET Lagrangian in Section 3.2. The solution is formally
given by
η = − 1 n̄/· (iD/ ⊥ξ + gA/ hcq). (4.13)in̄ D + iε 2
In contrast to the derivation of the HQET Lagrangian it is not possible to expand the
inverse covariant derivative in the case of SCET. This gives rise to non-local operators in the
Lagrangian and necessitates the inclusion of the +iε prescription to regularize the inverse
derivative. This regularization by +iε is arbitrary and drops out in physical quantities [85].
After integrating out the η-field the Lagrangian becomes
LSCET = Lξ + Lq + Lξq
L n̄/ · − 1 n̄/ξ = ξ̄ in Dξ ξ̄ iD/ ⊥ iD/ ⊥ξ
2 in̄ ·D 2
L 1 n̄/ (4.14)q = q̄ iD/ sq − q̄gA/ hc · gA/ hcqin̄ D 2
( ) ( )
L 1 n̄/ 1 n̄/ξq = ξ̄ gA/ hc − iD/ ⊥ · gA/ hc q + q̄ gA/ hc − gA/ hc · iD/ ⊥ ξ ,in̄ D 2 in̄ D 2
where the +iε prescription has been dropped for brevity. From the field scalings in (4.10)
and (4.11) it follows that the kinetic part of the hard-collinear Lagrangian Lξ scales like
∫
λ2. Since a position space integral d4x over an operator that contains at least one hard-
collinear field scales like λ−2 the action is of order λ0 and thereby contributes to the leading
power term. Similarly, the kinetic part of the soft Lagrangian scales like λ4 which is again
canceled in the action by the integral over soft fields. The interaction terms do not scale
homogeneously in general and need to be expanded. Specifically, the hard-collinear-soft
interaction terms only give rise to subleading contributions starting at O(λ1/2).
54
It should be noted that a Lagrangian derived by the steps above is formally equiva-
lent to QCD as long as the fields are not restricted to a certain momentum region. As
a consequence, hard fluctuations do not introduce new operators or change any of the
operator coefficients. This no-renormalization property has been discussed in [85]. The
situation changes with the introduction of the heavy-to-light operators of the weak effec-
tive Lagrangian, which constitute a source for the hard-collinear particles. In that case
it is necessary to match the QCD operators onto the SCET operators, order by order in
the power counting. These SCET operators of each order are determined by the replace-
ments (4.5) and (4.8) in the QCD operators. However, before explicitly performing the
matching, some comments concerning the behavior of the SCET Lagrangian under gauge
transformations are in order.
4.2.1 Gauge Transformations and Wilson Lines
With the introduction of separate hard-collinear and soft gluon fields, the Lagrangian is
no longer invariant under general SU(3)c gauge transformations. Instead separate hard-
collinear and soft gauge transformations are introduced that do not change the scaling of
the transformed fields. The transformations are defined by the scaling of their derivatives
∂µU ∼ (λ, 1, λ−1/2hc )U µhc and ∂ Us ∼ (λ, λ, λ)Us [12]. From this it is clear that the soft
fields do not transform under hard-collinear gauge transformations but transform in the
usual way under soft gauge transformations. The hard-collinear fields on the other hand
transform under soft as well as hard-collinear gauge transformations. The transformations
are given by
[
q →s s s i sUsq, ξ → U ξ, A → U A U †s s s s s + Us ∂, U †s ], Ahc → UsAhcU †g s ,
(4.15)
[
q →hc q, ξ →hc hc hc iUhcξ, As → A †s, Ahc → UhcAhcUhc + Uhc Ds, U
†
g hc
].
By definition, the kinetic terms of the Lagrangian (4.14) are manifestly invariant under
these transformation. However, the invariance is not obvious for the interaction terms.
This can be remedied by the introduction of Wilson lines. Of relevance are the Wilson line
W and the soft Wilson line S defined by
(
∫ 0 )
W (x) = P exp ig dt n̄ · A(x+ tn̄) , (4.16)
−∞
(
∫ 0 )
S(x) = P exp ig dt n̄ · As(x+ tn̄) , (4.17)
−∞
where the gluon field A contains hard-collinear as well as soft modes and P is the path
ordering symbol that sorts the gluon fields closer to the lower limit of the integration to
the right. By definition the covariant derivative acting only on the Wilson line vanishes
(n̄DW ) = 0 and (n̄DsS) = 0 which implies the important relations
n̄ ·DW = Wn̄ · ∂ and n̄ ·DsS = S n̄ · ∂ (4.18)
55
where the covariant derivative is also acting on everything on the right of the Wilson line.
Their behavior under gauge transformations is given by
W (x) →s s hc hcUs(x)W (x), S(x) → Us(x)S(x), W (x) → Uhc(x)W (x), S(x) → S(x),
(4.19)
where it is assumed that the gauge transformations do not change the fields at infinity,
U(x − n̄∞) = 1. With the help of these Wilson lines it is now possible to write down
the SCET Lagrangian (4.14) in a manifestly gauge invariant form [85]. However, the
Wilson line W contains the soft gluon field and therefore generates an infinite sum of terms
with increasing power in λ. In order to determine the SCET Lagrangian as a systematic
expansion in terms of λ the Wilson line must be expanded2. The leading power in that
expansion is just the hard-collinear Wilson line Whc
W (x) = Whc(x) +O(λ) ,
(
∫ 0 )
Whc(x) = P exp ig dt n̄ · Ahc(x+ tn̄) . (4.20)
−∞
But there are more terms in the Lagrangian (4.14) that do not scale homogeneously in λ
and that therefore need to be expanded.
4.2.2 Expansion and Lagrangian
First it is necessary to expand each vector field in terms of their components. In addition,
the soft fields in interaction terms with hard-collinear fields need to be multipole expanded
[90]. This is because the hard-collinear fields vary much faster in the light-cone direction n̄
and in the perpendicular direction than the soft field. It follows that the position argument
x scales like (1, λ−1, λ−1/2) and the soft field must be expanded in
q(x) = ex·∂q(0) = en·x n̄·∂/2+x⊥·∂⊥en̄·xn·∂/2q(0)
( )
· 1
(4.21)
= 1 + x⊥ ∂⊥ + n · ·
1
x n̄ ∂ + xµxν⊥ ⊥∂µ∂ν +O(λ3/2) q(x−),2 2
where x = n− n̄ · x and the derivative is taken before one sets x = x−. The complete2
expansion has been performed up to O(λ2) in [85] and the full Lagrangian is given there.
In this work only the leading power Lagrangian is of relevance as well as the first power of
the hard-collinear-soft interactions. They are given by
L(0) n̄/ 1 n̄/SCET = ξ̄ in ·Dξ + ξ̄ iD/ hc⊥ · iD/ hc⊥ ξ,2 in̄ Dhc 2 (4.22)
L(1)ξq = ξ̄ iD/ hc⊥Whcq + q̄W
†
hciD/ hc⊥ξ,
where it is implied that all soft fields are evaluated at the position x−. These terms are
invariant under hard-collinear and soft gauge transformations at this order in the power
2Note that the soft Wilson line is of order λ0 due to the scaling of the integration measure dt.
56
counting, i.e., gauge transformations leave the Lagrangian invariant up to higher power
terms. It is convenient to introduce gauge invariant building blocks through [87]
ξhc = W
†
hcξ, A
µ † µ µ † µ µ µ
hc = Whc(iDhcWhc), iDhc = WhciDhcWhc = i∂ +Ahc , (4.23)
where the brackets indicate that the derivative is only acting on the arguments within the
brackets. From the definition of the hard-collinear Wilson line follows that n̄ · Ahc = 0.
This object therefore scales like (λ, 0, λ−1/2). The scaling of the hard-collinear quark field
ξhc is not changed. With these definitions the vertex of a hard-collinear gluon with a
hard-collinear and soft quark can be written as
L(1)ξq = ξ̄hcA/ hcq + q̄A/ hcξhc , (4.24)
which is manifestly gauge invariant at this order.
The Lagrangian (4.14) describes the QCD dynamics of hard-collinear and soft particles.
This would be sufficient for the analysis of B̄ → Xsγ if the photon was only created at
the weak interaction vertices. Since by definition the jet only contains hard-collinear and
soft particles (otherwise the invariant mass would not be small) no interactions containing
anti-hard-collinear particles would be required. However, at subleading power anti-hard-
collinear particles can be created at the weak interaction vertex, which are subsequently
converted into a photon. For this conversion it is necessary to introduce the SCET La-
grangian for anti-hard-collinear particles. Its derivation proceeds in exactly the same way
as the derivation of the SCET(hc,s) Lagrangian presented above, except that every n and
n̄ are interchanged. The field ξ will always refer to the large spinor components with an
added label n or n̄ to indicate the large momentum component. From the definition of the
projection operators (4.7) then follows that n̄/ ξn̄ = 0. Also the gauge invariant building
blocks ξhc and Aµ are defined analogously to (4.23) by using Wilson lines along the nhc
direction
ξhc = W
† ξ , Aµ = W † (iDµ W ). (4.25)
hc n̄ hc hc hc hc
In order to implement the photon conversions, the covariant derivatives will also contain
the photon field D = ∂ − igA− eqeAem. Since the photon is a real particle that moves in
the n̄-direction only the perpendicular component of the photon field is non-zero and scales
like λ1/2. The complete SCET(hc,hc,s) Lagrangian is given by the sum of the SCET(hc,s)
and SCET(hc,s) Lagrangians. The mixing between the sectors will be considered when
matching the weak effective operators from QCD onto SCET.
Although not required in this work, it is mentioned for completeness that the pure
gluonic part of QCD can be matched onto SCET in an analogous way [90]. The gluon fields
are split into their hard-collinear and soft components with each of the sectors looking the
same as in QCD. Interaction terms between the two sectors are restricted by momentum
conservation. In the next section the matching of the weak effective operators onto SCET
will be performed at leading power in λ. This will lead to an expression, in which the
physics of the different scales (4.3) are factorized.
57
4.3 Factorization at Leading Power
In order to demonstrate the factorization of the differential decay rate, the following steps
will be performed. First the effective heavy-to-light operator is matched onto the corre-
sponding SCET operator. Then the differential decay rate is calculated and the emerging
non-local operator decomposed into two parts, one of which can be calculated perturba-
tively the other corresponding to an HQET matrix element. Both matching steps can in
principle be performed at any order in αs at the appropriate scale.
4.3.1 Tree Level Matching
The leading power operator contributing to B̄ → Xsγ is derived from Q7γ given in (2.17).
In order to determine onto which SCET operator Q7γ matches exactly, the hard-collinear
strange-quark field is decomposed by using Equation (4.8) which at leading power corre-
sponds to the replacement s → ξn +O(λ). Furthermore the field-strength tensor needs to
be expanded, leading to
−emb µν − −emb n̄/Q7γ = s̄σµνF (1 γ5)b → ξ̄n i n · ∂A/ em⊥ (1− γ5)b+O(λ3) , (4.26)8π2 8π2 2
where mb is once again the running b-quark mass in the MS scheme (see Section 2.2.1). In
a second matching step the decay rate will be expressed in terms of a pure HQET matrix
element. Thus the b-quark field needs to be replaced by the heavy-quark field h defined in
(3.10).
Since the large component of a hard-collinear gluon field scales like λ0, an arbitrary
number of them can be added to any QCD diagram without changing the scaling of that
contribution. The effective theory can describe situations where the hard-collinear gluon
is emitted from another hard-collinear line. However, when the gluon is emitted from the
heavy-quark field, it puts the heavy quark far off-shell (see left diagram in Figure 4.1).
These modes are integrated out of the effective theory, which gives rise to an effective
s
ξ
Ahc
. . . →
. . . Ahc
h
b Ahc Ahc
Figure 4.1: Tree-level matching of an heavy-to-light operator. The resummation into
a Wilson line includes all gluon permutations in the QCD diagram. The double line
corresponds to a heavy-quark field.
operator that emits any number of hard-collinear gluons (right diagram in Figure 4.1).
The gluon emissions can be resummed into an eikonal form which just corresponds to the
58
hard-collinear Wilson line of (4.20). This Wilson line appears to the left of the heavy-quark
field, but since it commutes with the photon field, it can be absorbed into the hard-collinear
field ξn to yield the gauge invariant quantity ξhc. Therefore the sum of all hard-collinear
gluon emissions was necessary to generate a gauge invariant operator. At tree level in the
strong interaction the operator Q7γ therefore matches onto the SCET operator
(0) −emb n̄/
Q7γ (x) = e
−imbvxξ̄hc(x) i n · ∂A/ em⊥ (x)(1− γ5)h(x−) , (4.27)8π2 2
where the soft field h was multipole expanded as in (4.21) and the superscript (i) on
an operator indicates the suppression in powers of λ1/2 compared to the leading power.
When calculating amplitudes containing a heavy-quark field, it is necessary to add the
HQET Lagrangian (3.14) to the effective theory Lagrangian of SCET(hc,hc,s). Due to
momentum conservation the HQET Lagrangian only contains soft covariant derivatives
Ds. Also, effective operators connecting two or more collinear gluons to the heavy-quark
field are not allowed in an effective theory describing the decay of a B-meson into collinear
particles [85]. This concludes the tree level matching. Next the effect of virtual corrections
will be considered.
4.3.2 Loop Level Matching
Another way to describe the matching performed above is to say that at the tree level
(0)
the Wilson coefficient of the leading power SCET operator Q7γ is just C
eff
7γ . At the loop
(0)
level all QCD diagrams that match onto Q7γ (e.g. the first diagram in Figure 4.2) need to
be calculated and expanded in powers of λ. The difference to the corresponding effective
s ξ ξ ξ
→
, ,
b h h h
s hc
Figure 4.2: At loop level the QCD operator Q7γ (on the left) matches onto the SCET
diagrams on the right. The gluons in the second and third diagram on the left are soft and
hard-collinear, respectively.
theory diagrams (e.g. the other diagrams in Figure 4.2) is accounted for by the Wilson
(0)
coefficient. In the literature the matching of Q7γ onto Q7γ is associated with a Wilson
coefficient C9 [7] (not to be confused with the Wilson coefficients of the weak effective
59
Lagrangian). By evaluating the diagrams in Figure 4.2 this coefficient is calculated to
[
C · − αs(µ)CF mb mb n̄ · P mb9(n̄ P, µ) = 1 2 ln2 − 7 ln + 4 ln ln
4π µ µ mb µ
]
· · (n̄ P n̄ P n̄ · )P π2
+2 ln2 − 2 ln + 2Li2 1− + + 6 ,
mb mb mb 12
(4.28)
where P is the sum of all hard-collinear momenta and µ is the matching scale which must
be of the order of the hard scale mb to keep the logarithms small. The Wilson coefficient
depends on the hard-collinear momentum since its large component is of the same order
as the hard momenta. It was proven in [7] that gauge invariance requires the Wilson
coefficients to depend only on the sum of all hard-collinear momenta instead of each one
separately. If the light quark is the only hard-collinear particle in the process, n̄ ·P is equal
to mb at leading power. Since the Wilson coefficient depends on a momentum that still
appears in the matrix element, the terms of the effective Lagrangian need to be represented
by a convolution in position space
∫
L(0) GF (0)SCET, eff = √ λt Ceff7γ (µ) dt C̃9(t, µ)Q7γ (x+ tn̄) + . . . , (4.29)
2
where C̃9(t, µ) is the Fourier transform of C9(n̄ ·P, µ), Ceff7γ was defined in Section 2.1.5 and
(0)
the ellipses denote terms originating from other QCD operators that match onto Q7γ . Due
to these terms the complete matching coefficient is given by a sum of products of the weak
effective Wilson coefficients Ci and the Wilson coefficients of the matching of QCD onto
the SCET operators from [7]. This gives rise to the hard function Hγ which is given by [13]
[ ( )]
eff CFαs(µh) − 2 mb m
2
b
Hγ(µh) =C7γ (µh) 1 + 2 ln + 7 ln − 6−
π
4π µh µh 12
( )
eff CFαs(µh) −8 mb 11 − 2π
2 2πi
+ C8g (µh) ln + + (4.30)4π 3 µh 3 9 3
( )
CFαs(µh) 104 mb
+ C1(µh) ln + g(z) + εCKM[g(0)− g(z)] ,
4π 27 µh
where n̄ · P = mb was used, the matching scale was set to the hard scale µh ∼ mb and
g(z) is related to the penguin function (see Appendix A.1) with z = m2/m2c b . Furthermore
some small corrections were dropped for brevity and can be taken from Reference [13]. The
parameter εCKM contains a weak phase from the CKM matrix elements that will become
important in Chapter 8. Today, the hard function has been explicitly calculated at one
loop order and its two loop result has been deduced in [8]. It encodes the effects from hard
fluctuations which have been integrated out, while the effective theory reproduces the IR
behavior of QCD.
60
Matching QCD onto SCET was only the first step. In order to achieve a complete
separation of scales a second matching step (onto HQET) at a different scale is necessary.
The result of this first matching step can be given in terms of the leading power effective
Lagrangian mediating the decay B̄ → Xsγ
L(0) GF (0)SCET, eff = √ λt Hγ(µh)Q7γ (x) , (4.31)
2
(0)
where the operator Q7γ is renormalized at the scale µh. This Lagrangian, together with the
SCET (4.14) and HQET (3.14) Lagrangians is sufficient to describe B̄ → Xsγ at leading
power. It will now be used for the calculation of the decay rate.
4.3.3 The Factorization Formula
With the leading power SCET operator at hand, the decay rate for B̄ → Xsγ will be
calculated for the third time. Due to the systematic power counting of SCET the calcula-
tion will naturally lead to the non-local matrix element of Section 3.3.4. Furthermore, the
calculation will explicitly demonstrate how the decay rate factorizes into parts, that each
contain the physics at a certain scale. This will eventually lead to the resummation of the
large logarithms encountered in the partonic calculation. The calculation will once again
start from Equation (3.4). The leading power (and leading order in αs) contribution is
(0)
made up of two operators Q7γ and is illustrated on the left of Figure 4.3. Since cutting the
Figure 4.3: Leading power contribution to B̄ → Xsγ. The left diagram represents the
product of SCET operators, while the right diagram depicts the space-time structure of
the HQET operator (see text).
photon line does not affect the calculation of the hadronic matrix element, the diagrams
will always be drawn with the photon propagator already cut. Here and in the following
heavy-quark lines will always be indicated by a double line, while light quarks correspond
to single lines. After evaluating the vacuum matrix element of the time-ordered product
of photon fields and taking the residue of the photon propagator pole one arrives at
( )
dΓ 1 G2 m2F b(µ)α= |H (µ)λ |2E3γ t γ −
1
dEγ 2M 4π4B π
∫ (4.32)
× Im 〈B̄|i d4x ei(mv−q)xT{h̄(x−)γµ⊥(1− γ5)ξhc(x)ξ̄hc(0)γ⊥µ(1− γ5)h(0)}|B̄〉.
As there are no hard-collinear particles in the initial or final state, the time-ordered product
of the two hard-collinear fields can be perturbatively evaluated between vacuum states.
61
However, at higher orders in αs there can be soft gluon interactions between the heavy
and hard-collinear lines, making this separate calculation impossible. The solution to this
problem is to introduce “sterile” fields with the help of the soft Wilson line in (4.17). One
defines
(0) (0)µ
ξhc(x) = S
µ †
n(x−)ξhc (x), Ahc(x) = Sn(x−)Ahc (x)Sn(x−) , (4.33)
where the label n was added to the soft Wilson line in order to distinguish it from Sn̄ which
is used in the definition of anti-hard-collinear sterile fields. By using Equation (4.18) it
is easy to see that after the introduction of sterile fields the Lagrangian (4.14) no longer
mediates soft gluon emissions from hard-collinear particles. This decoupling transformation
is the basis for an all order factorization proof of the leading power B̄ → Xsγ decay rate.
The hard-collinear part of the calculation can be separated from the soft part and since
the typical hard-collinear scale is larger than ΛQCD it can be perturbatively calculated.
This factorization was already proven in [11], but the effective theory approach [12] makes
the procedure more transparent. Formally the hard-collinear physics are encoded in the
so-called jet function J that can be defined as
∫ 4
〈 | { (0) (0) d p n/Ω T ξ (x)ξ̄ (0)}|Ω〉 =: i n̄ · pJ (p2)e−ipxhc hc , (4.34)(2π)4 2
where Ω is the interacting vacuum and the Dirac structure is determined by the hard-
collinear propagator. Inserting this into (4.32) and evaluating the Dirac structures leads
to
( )
dΓ 1 G2Fm
2
b(µ)α 1= |H 2 3γ(µ)λt| E −
dEγ 2MB 2π4
γ π
[ ]
∫ ∫ (4.35)4
× d pIm d4x ei(mv−q−p)xn̄ · pJ (p2)〈B̄|T{[h̄Sn](x−)[S†h](0)}|B̄〉 ,
(2π)4 n
where once again parity invariance was used to eliminate the axial vector part. By using
the transformations
∫ ∫ ∫
(n ) ( n̄ · x )
[hS ] n̄ · x = dt δ − t [hS ](tn) = dω eiω n̄x dt e−iωtn n 2 [hSn](tn) , (4.36)
2 2 2π
it is possible to perform the x and p integrations in (4.35) which leads to
( )
dΓ 1 G2 2
= F
mb(µ)α | 1H (µ)λ |2 3
4 γ t
E
dE 2M 2π γ
−
γ B π
[ ]
∫ ∫ (4.37)
× J − dtIm dωm (m (m 2E + ω)) e−iωtb b b γ 〈B̄|[h̄Sn](tn)[S†h](0)|B̄〉 ,
2π n
where the time-ordering symbol was dropped since the HQET fields only create and destroy
quarks and their ordering is already dictated by the propagator. It is now useful to define
the leading power shape function by
∫
1 dt
S(ω) := e−iωt〈B̄|[h̄Sn](tn)[S†nh](0)|B̄〉. (4.38)2MB 2π
62
The product of the soft Wilson lines Sn(tn)S
†
n(0) = [tn, 0] is a straight line segment con-
necting the points tn and 0, with gauge fields closer to the point tn standing to the left
of those closer to 0. This finite Wilson line guarantees gauge invariance of the non-local
operator. The HQET operator is depicted on the right of Figure 4.3 with a dashed line
indicating the finite Wilson line. By taking the complex conjugate of the shape function
and using translational invariance, one can show that the shape function is real. Thus the
imaginary part prescription in (4.37) only acts on the jet function J which suggests the
definition of a new quantity J that will from now on be called jet function
2 − 1J(p ) = ImJ (p2). (4.39)
π
This leads to the fully factorized expression for the leading power differential B̄ → Xsγ
decay rate
2 ∫dΓ GFm
2
b(µ)α
Λ̄
= |Hγ(µ)λt|2E3γ dωmb J(mb(p+ + ω))S(ω) , (4.40)dEγ 2π4 −p+
where Λ̄ = MB −mb and p+ = mb − 2Eγ. The lower limit of integration is derived from
the start of the branch cut (or pole in case of a single particle jet) of the jet function in
the complex p2-plane. The upper limit on the other hand follows from the definition of
ω = n · k and
n · PB > n · pb ⇔ MB > mb + ω , (4.41)
where PB and pb are the B-meson and b-quark momenta, respectively. At leading power, Λ̄
can be set to zero. The shape function defined in (4.38) corresponds to the shape function
of Section 3.3.4 which can be shown explicitly by using Equation (4.18) and Sn(0)S
†
n(0) = 1
and by writing
∫ ∫
dt − ←−e iωt〈B̄| dt[h̄Sn](tn)[S†nh](0)|B̄〉 = e−iωt〈B̄|[h̄Sn](0)etn· ∂ [S†2π 2π nh](0)|B̄〉 (4.42)
= 〈 ←−B̄|h̄(0) δ(ω − n · D)h(0)|B̄〉 ,
which is equal to the earlier definition by translation invariance. However, this time the
shape function arose naturally at leading power in the expansion in λ and no resummation
of terms was required. Furthermore, the extension to higher orders in αs is well defined.
The problem posed in the conclusions of Chapter 3 is therefore solved.
The factorization formula (4.40) can also be used to solve the issue of the large log-
arithms encountered in the partonic calculation. They appeared because not all relevant
scales were separated and their ratios could generate large numbers. With the factoriza-
tion formula at hand, it is now possible to calculate the matching coefficients each at their
natural scale and subsequently evolve them to a common scale by using the renormaliza-
tion group equations. The matching of the hard functions at the hard scale was already
discussed above in Section 4.3.2. The matching of the jet function will be the next topic.
63
4.3.4 Matching the Jet Function
At tree level, the jet function J is related to the propagator of the hard-collinear particle.
Taking the imaginary part, the jet function J is calculated to
J(mb(p+ + ω)) = δ(mb(p+ + ω)). (4.43)
Inserting this into the decay rate (4.40) leads to the leading power and leading order
expression already derived in Section 3.3.4. At NLO the usual procedure is applied of
calculating full theory diagrams (in this case SCET) as well as effective theory diagrams
(HQET) and absorbing the differences into the Wilson coefficients, i.e., the jet function.
The relevant diagrams are given in Figure 4.4. Once again, HQET reproduces the IR
, , →
hc hc hc
Figure 4.4: NLO matching of the SCET decay rate onto HQET. The mirror of the second
diagram is not shown.
behavior of the full theory. This particularly includes the soft gluon corrections to the
SCET diagrams [1]. The NLO calculation has been performed in [1, 91] and leads to the
renormalized jet function
[ ]
CFαs(µi) ( )
mb J(mb(p+ + ω), µi) = δ(p+ + ω) 1 + 7− π2
4π
[µ2/m ] (4.44)[ ( )]
C i bFαs(µi) 1 mb(p+ + ω)
+ 4 ln − 3
4π p 2+ + ω µi ∗
that was matched at the intermediate scale µi. The star distributions are generalized
plus distributions (compare Section 2.2.1) and were introduced in [92]. Note, that if the
logarithm in (4.44) is evaluated at a soft scale µi ∼ p+ it relates to the problematic large
logarithms of Equation (2.33). It is an advantage of the factorization that the scale µi can
√
be set to the more appropriate value of O( ΛQCDmb). Just like the hard functions, the jet
functions are known at two-loop order [93]. After performing the two matching steps and
separating all the scales of the problem it is now possible to determine the RG running of
the quantities involved in the calculation. This will be the topic of the next Section.
4.4 Renormalization Group Running
Although somewhat outside the main line of this work, the procedure of resumming the
large logarithms will be briefly reviewed in order to further motivate the usage of the
effective theory. So far the decay rate has been factorized into three components, that
64
each encode the physics at a certain scale. In the first matching step from QCD to SCET
the hard function Hγ appeared as the Wilson coefficient. If the matching is performed at
the hard scale µh ∼ mb it contains no large logarithms and the one-loop result is given in
(4.30). The second matching step from SCET onto non-local HQET operators needs to
√
be performed at the intermediate scale µi ∼ ΛQCDmb, where the logarithms are again
small. This leads to the one-loop result in (4.44). The typical scale of the remaining
soft matrix element is of order ΛQCD. Thus, it can not be calculated perturbatively but
must be modeled [2] and the model parameters related to experimental quantities. As was
discussed in Section 3.3.4 the leading power shape function can be extracted from the form
of the photon spectrum. The modeling must be performed at the soft scale µ0 ∼ ΛQCD.
By using the RG equations the three results can be evolved to a common scale (compare
Section 2.1.3), which cancels the dependence of the decay rate on the hard and intermediate
scales. Explicitly one proceeds as follows. First the hard function is determined at the
hard scale and evolved down to the intermediate scale by solving
d
Hγ(n̄ · p, µ) = γJ(n̄ · p, µ)Hγ(n̄ · p, µ) , (4.45)
d lnµ
where γJ is the anomalous dimension of the heavy-to-light SCET current. Due to the simple
Dirac structure of the leading power SCET Lagrangian (4.14) the anomalous dimension is
the same for all type of Dirac structures in the current. Next one starts with a model of
the shape function at the soft scale and evolves it up to the intermediate scale by solving
the integro-differential evolution equation
∫
d
S(ω, µ) = − dω′ γS(ω, ω′, µ)S(ω′, µ). (4.46)
d lnµ
Since the shape function is related to a non-local operator it is renormalized by a convolu-
tion, instead of multiplicatively.
The anomalous dimension γJ can be determined from the singularities of the diagrams
on the right of Figure 4.2 (in addition to the diagrams necessary to calculate the field-
strength renormalization). At the leading order it can be written as [7, 94]
( )
γJ(n̄ · p, µ) = −
µ CFαs µ 5
Γ ′cusp(αs) ln · + γ (αs) = − ln − + . . . , (4.47)n̄ p π n̄ · p 4
where Γcusp is the universal cusp anomalous dimension governing the UV singularities of
Wilson lines with light-like segments [95]. Therefore the solution to the RG equation is
given by
∫ αs(µi)
[ ( · α ′ ) ]∫dα n̄ p
Hγ(n̄ · p, µi) = exp Γcusp(α) ln −
dα
′ + γ
′(α) Hγ(n̄ · p, µh)
α (µ ) β(α) µh α (µ ) β(α )s h s h
(4.48)
where β(α) is the QCD β-function. The cusp anomalous dimension has so far been derived
up to three-loop order [96], while a conjecture for the two-loop expression of γ′ has been
suggested in [13].
65
In order to determine the anomalous dimension γS of the shape function one has to
calculate the HQET amplitudes from the diagrams in Figure 4.5. Of course, a perturbative
s
s
Figure 4.5: Diagrams relevant for the calculation of the shape function anomalous dimen-
sion. A mirror diagram of the second graph is not shown.
calculation of the hadronic operators will not yield a realistic result for the functional form
of the shape function. However, the UV behavior, i.e., the µ dependence of the shape
function may be determined that way. The one-loop calculation of γS as well as a review
of earlier calculations has been presented in [1]. There, the result is given as
[( ) ( ′ ) ]
′ CFαs µγS(ω, ω , µ) = 2 ln − − 1 δ(ω −
′ − θ(ω − ω)ω ) 2 , (4.49)
π Λ̄ ω ω′ − ω +
which leads to the following solution of the RG equation
∫ 0 ′
V (µ ,µ ) 1 ′ S(ω , µ0)S(ω, µi) = e S i 0 dω , (4.50)
Γ(η) µη ′Λ̄−ω 0(ω − ω)1−η
where η is an integral over the cusp anomalous dimension and together with the evolu-
tion function VS is defined in [1]. Improving on this calculation, a two-loop analysis was
performed in [97].
Using Equations (4.48) and (4.50) all quantities in the factorized decay rate (4.40)
can be calculated at their appropriate scale and then run to a common scale, thereby
eliminating the scale dependencies on µi and µh. This running resums the large leading
logs that appeared in the partonic calculation. Thus, the problem stated in the conclusions
of Chapter 2 is solved.
4.5 Conclusions of this Chapter
This chapter analyzed the differential B̄ → Xsγ decay rate at leading power in the SCET
expansion. It was shown that the effective theory approach using SCET could be used to
write down a factorization theorem for the leading power decay rate. Schematically it can
be represented as
dΓ(B̄ → Xsγ) ∼ H J ⊗ S (4.51)
where H is the hard function, J the jet function and S the shape function each encod-
ing the physics at their respective scales. The symbol ⊗ denotes a convolution. This
approach solved the two major issues worked out in the previous chapters. On the one
66
hand, the factorization of the leading power decay rate allowed for the resummation of
the large logarithms appearing in the partonic calculation of Chapter 2. On the other
hand the systematic power counting naturally introduced the shape function of Chapter 3
and extended the definition to any order in αs. The framework presented in this chapter
points out two possible directions for improvements. First, the perturbative calculations
can always be pushed to some higher order. However, since these calculations have already
been performed at the NNLO [93,97], the next step is the extension of the analysis to the
next order in the power counting. This will be the topic of the next chapter.
67
Chapter 5
Subleading Contributions in SCET
Historically, the first SCET analysis beyond the leading power dealt with the 1/mb sup-
pressed contributions due to two operator insertions of the electromagnetic dipole operator
Q7γ . This chapter will on the one hand review the well-known results of this analysis and
on the other hand set the basis for an extension of the formalism to include any operator
insertions beyond the leading power. For this it is necessary to discuss the general proce-
dure of determining the relevant SCET operator basis in Section 5.1. As a first application
of this procedure the suppressed SCET operators deriving from Q7γ will be derived in
Section 5.2.1. Inserting these operators into the formula for the decay rate and matching
the result onto HQET leads to a factorization theorem at subleading power. At this order
the subleading jet and shape functions arise, which are discussed in Section 5.2.3 and 5.2.4.
Concluding the discussion of the Q7γ −Q7γ interference, the results will be related to the
QCD calculation.
When other operator interferences are considered, anti-hard-collinear fields play an
important role. It is due to their appearance that a distinction between “direct” and
“resolved” photon contributions can be made. These concept, along with the derivation of
a general factorization theorem will be discussed in Section 5.3. In the last section of this
chapter the subleading SCET operators, relevant for the resolved photon contributions will
be derived from the QCD effective operators. A complete list of the subleading operators,
some of which are required for the direct photon contributions, will be given in Appendix B.
5.1 Matching onto SCET
The previous chapter introduced the leading power heavy-to-light SCET operator (4.27)
that was shown to scale like λ5/2. It was found that at tree level only the QCD operator
Q7γ (see (2.17)) has a non-vanishing matching coefficient. At the loop level, there are also
contributions from Q8g and Q
q
1 as well as from the penguin operators Q3...6 which were
absorbed into the effective Wilson coefficient Ceff7γ (see (4.30)). This chapter will deal with
SCET operators beyond the leading power. The ultimate goal is to calculate the B̄ → Xsγ
differential decay rate at subleading power, i.e., including contributions suppressed by 1/mb
68
compared to the leading power. Since the decay rate contains at least two SCET operator
insertions, it is necessary to find the operators that are suppressed by λ1/2 (denoted with a
superscript (1)) as well as those suppressed by λ (denoted with a superscript (2)) compared
to the leading power.
Once again, the possible SCET operators can be determined by starting from the QCD
operators and replacing each field by its SCET counterpart. Afterwards the operator is
expanded in powers of λ. Further suppressed operators are introduced by considering
time ordered products of effective QCD operators and QCD Lagrangian insertions and
integrating out the internal lines not present in SCET. One example was already given
in Figure 4.1. There, the heavy quark emitted an arbitrary number of hard-collinear
gluons, that put the heavy quark far off-shell. These off-shell modes were integrated out
of the effective SCET+HQET Lagrangian, leading to a single leading power operator that
contained the large components of the hard-collinear gluons resummed into a Wilson line.
Beyond leading power the SCET operator may also contain the smaller components of
hard-collinear gluons. However, since they scale with a higher power of λ it would make no
sense to resum them into eikonal form. Thus, only a limited number of them are emitted
from the heavy-quark line, which is thereby put off-shell. It was already demonstrated
in Section 3.2 that integrating out this off-shell mode corresponds to solving the classical
equation of motion of the off-shell field. This time it is not the small component of a nearly
on-shell heavy quark that is integrated out, but the far off-shell component. Following
References [10, 85] one decomposes the QCD heavy-quark field into a nearly on-shell and
a far off-shell part
( )
b(x) = e−imv·x bonv (x) + b
off
v (x) , (5.1)
inserts this into the QCD Lagrangian and solves the equation of motion for boffv which
formally leads to
boffv = −
1 on
iD/ − − gA/ hcb . (5.2)m(1 v/) v
The nearly on-shell part bonv can be further decomposed into a large component spinor
h and a small component spinor H as was done in (3.10). By explicitly solving this
equation order by order in λ and introducing the gauge invariant building blocks (4.23) it
is possible to write down the necessary replacement, that derives the SCET operator from
the corresponding QCD operator. Writing down only the terms relevant for B̄ → Xsγ, one
finds
[
→ − n/ A iD/ s − n/n̄/ ·A − 1 1 1
]
b Whc 1 / hc⊥+ n hc · n·Ahc− · i∂/⊥A/ hc⊥+. . . , h (5.3)2mb 2mb 4mb in̄ ∂ mb in̄ ∂
where operators containing more than one hard-collinear gluon have been dropped, since
they are not required for the matching performed in this work. By the power counting
rules of Section 4.1 the first term on the right hand side is of order λ3/2, the second of order
λ2 while the rest is of order λ5/2. In general, it is also possible that the anti-hard-collinear
photon is emitted from the heavy-quark line. In that case the hard-collinear quark field is
replaced by the photon field, with the corresponding interchanges of n and n̄. As usual,
the +iε prescription regulating the inverse derivatives is not explicitly shown.
69
The expansion for a hard-collinear quark field can also be obtained by writing down
time ordered products of QCD operators with gluon emissions from the hard-collinear line
and integrating out the appropriate internal lines. For the case of hard-collinear gluon
emissions this leads to
[ ]
ψhc → −
1 n̄/
Wc 1 · iD/ ⊥ ξhc , (5.4)in̄ ∂ 2
where the first term on the right hand side is of order λ1/2 and the second of order λ. The
analogous relation applies to the anti-hard-collinear quark field. If a photon is emitted
from the light quark field, the perpendicular covariant derivative is replaced by a photon
field.
After the above expansions have been inserted into the heavy-to-light operator, all soft
fields must be multipole expanded as given in (4.21). Furthermore all vector fields can be
decomposed in the usual way (3.22). With this procedure it is an easy task to determine the
relevant operators up to λ7/2. The matching onto the SCET operators will be performed
at the leading order in the strong coupling constant g. Therefore the Wilson coefficients
of the QCD to SCET matching step are just products of the Wilson coefficients Ci of the
weak effective Lagrangian (see Section 2.1.5). In order to describe B̄ → Xsγ, each operator
must contain at least one hard-collinear particle which will be part of the jet, as well as one
anti-hard-collinear particle, either the photon or light partons that can be converted into
a photon and soft partons. Anti-hard-collinear particles can not be part of the final state
jet, since this would imply an invariant mass of O(mb) which is excluded by definition. In
principle the operator can contain any number of hard-collinear and soft particles that is
consistent with the desired order in the power counting. In the following, all the operators
relevant for B̄ → Xsγ will be considered. It was explained in Section 2.1.5 and 2.2.1 that
the operators Qq1, Q7γ and Q8g are the most relevant. The next section will deal with the
SCET operators descending from the QCD operator Q7γ .
5.2 The Subleading Q7γ −Q7γ Contribution
Before dealing with the subleading operators on more general grounds, the contribution due
to the electromagnetic dipole operator Q7γ will be considered in detail. Although already
introducing some new concepts (subleading jet and shape functions), it only represents a
certain type of contribution (direct). The detailed discussion of all possible types will be
postponed to the next section.
5.2.1 Tree Level Matching
By using the replacement rules of the last section it is possible to determine the SCET
operators that the QCD operator Q7γ matches onto at tree level. At higher orders in
αs other QCD operators may also match onto these SCET operators. Furthermore, new
operators can be introduced by radiative corrections. In order to determine the full set of
operators at a certain order in the power counting, a more systematic approach is necessary.
70
In [98,99] reparameterization invariance (redefinition of the light-cone vectors) of SCET as
well as the invariance under the SCET gauge transformations was used to perform this task.
At tree level, Q7γ matches onto the leading power operator already given in (4.27). Beyond
leading power it matches onto a set of operators already given in [85]. Suppressing a factor
− emb e−imb v·x2 and assuming an external photon with a momentum in the n-direction one4π
finds
(1) n̄/
Q7γA = ξ̄hc [in · ∂A/ em⊥ ] (1 + γ µ5) x⊥Dµh ,2
[ µ ]ν (5.5)
(2) n̄/
Q = ξ̄ [in · ∂A/ em n · x7γA hc ⊥ ] (1 + γ5) n̄ ·
x x iD/
Dh+ ⊥ ⊥DµDνh+ h ,
2 2 2 2mb
and
(1) [in · ∂A/ em]
Q7γB = ξ̄
⊥
hc gA/ hc⊥(1 + γ5)h ,
mb
n̄/ 1 [in · ∂A/ em(2)
Q = −ξ̄ [in · ∂A/ em ]7γB hc ⊥ ] (1 + γ ⊥5) · g n · Ahch+ ξ̄hc g n · Ahc(1− γ5)h2 in̄ ∂ mb
[in · ∂A/ em]
+ ξ̄ ⊥hc gA/ hc⊥(1 + γ5)xµ
m ⊥
Dµh
b
− n̄/ · em 1 (i∂/⊥ gA/ hc⊥)ξ̄hc [in ∂A/ ⊥ ] (1 + γ5)2 in̄ · h ,∂ mb
(5.6)
where the covariant derivative Dµ only contains soft fields. The operators are separated
into an “A” and “B” group depending on whether they contain hard-collinear gluon fields
or not. As usual, the soft fields depend on x− while the hard-collinear fields depend on x.
The +iε prescription of the inverse derivative is not explicitly given. Since every operator
deriving from Q7γ already contains the photon, there is no room for further anti-hard-
collinear particles. With these operators it is now possible to calculate the decay rate
using the effective theory.
5.2.2 Factorization and the Decay Rate
The subleading contribution to the B̄ → Xsγ decay rate due to insertions of the operator
Q7γ (henceforth called Q7γ − Q7γ contribution) can now be determined by inserting (5.5)
and (5.6) into the decay rate formula (3.4). In addition, insertions of the SCET or HQET
Lagrangians are possible. It turns out that the product of a leading power operator and
an operator suppressed by λ1/2 vanishes due to chirality and the fact that the heavy-to-
light operators contain only left handed particles [10]. Therefore the first correction to
the leading power decay rate is suppressed by λ ∼ 1/mb. If expressed in absolute powers
of λ the leading power operators are of order λ5/2, which implies that operators with a
scaling of λ3 and λ7/2 are needed to determine the subleading decay rate. Schematically
71
the following operator insertions contribute at subleading power
∫ ∫
(0) (2) (1) (1) (0) (0)
Q Q , Q Q , Q Q d4xL(2) (0) (1)i j i j i j , Qi Q d4j xL(1),
∫ ∫ ∫ (5.7)
(0) (2) 4 L(0) (0) (0)Q Q d x , Q Q d4xL(1) d4xL(1)i j i j ,
as well as the mirror contributions and with i = j = 7γ. Since all operators deriving from
Q7γ already contain a photon field, no further anti-hard-collinear fields are allowed. Thus
the matrix element in the decay rate (3.4) has the generic form
Disc〈B̄|h̄(x)[φs(xi) . . . φhc(yi) . . . ]h(0)|B̄〉 , (5.8)
where φs and φhc are soft and hard-collinear fields respectively. If they originate from an
operator their position argument will be either x (or x−) or 0. If they originate from a
Lagrangian insertion, they can be located at any point along the n light-cone between
the two heavy-quark fields. When matching onto HQET, the hard-collinear fields are
integrated out, while the soft fields remain in the matrix element. Thus there must be at
least two hard-collinear fields in the square brackets, whereas the number of soft fields is
only constrained by the power counting. So far the soft and hard-collinear fields can still
interact via the exchange of soft gluons. However, just as in the leading power case in
Section 4.3.3, the hard-collinear fields may be decoupled by the introduction of the soft
Wilson lines. The matrix element therefore factorizes into a hard-collinear and a soft part
〈B̄| (0)h̄(x−)[φs(xi−) . . . ]h(0)|B̄〉 ×Disc〈Ω|φhc (yi) . . . |Ω〉 , (5.9)
thereby proving a factorization formula for the subleading contribution due to two inser-
tions of Q7γ . The soft Wilson lines introduced by the decoupling transformation still reside
in the soft matrix element. In analogy to the procedure at leading power, a jet function
can be defined by the Fourier transform of the discontinuity of the vacuum matrix element
of the hard-collinear fields. At the leading power, the non-locality of the soft matrix ele-
ment leads to a convolution of the jet and soft function. Since the additional soft fields in
the subleading matrix element introduce even more non-localities one might expect multi-
dimensional convolution integrals. By using partial-fraction identities on the hard-collinear
propagators connecting the soft fields it was shown in [100] that there is in fact only one
convolution integral in the case of the tree-level Q7γ − Q7γ interference. However, as will
be demonstrated in Chapter 6 multi-dimensional convolution integrals do appear for other
operator combinations at the subleading power.
In general, the subleading contributions to the decay rate can be divided into two
types. One type contains operator insertions that are suppressed by the small components
of additional hard-collinear fields or their derivatives. The other type is suppressed by
additional soft fields. They will be considered in turn.
5.2.3 The Subleading Jet Functions
First the case of additional hard collinear fields will be analyzed. One example of such a
(1)
contribution is given by two insertions of the operator Q7γB of Equation (5.6) which leads
72
to an αs/mb suppressed contribution and is depicted on the left of Figure 5.1. Another
hc
hc
hc hc
hc
Figure 5.1: Subleading SCET diagrams containing hard-collinear loops. The short double
lines indicate possible cuts (recall that the photon lines are already drawn cut). The
diagram to the right represents the non-local HQET operator remaining after integrating
out the hard-collinear fields.
contribution of this type (depicted in the second diagram of Figure 5.1) is generated by
(0) (2)
the interference of the leading power Q7γ with the suppressed Q7γB plus an insertion of
the leading power SCET Lagrangian (4.14). Since there are no additional soft fields in
these examples, they both match onto an HQET matrix element that is of the same form
as the one at leading power (its space-time structure is depicted in the third diagram of
Figure 5.1). The jet function on the other hand will contain additional hard-collinear fields.
The factorization formula can therefore symbolically be written as
1 ∑
dΓ(B̄ → Xsγ) ∼ H ji ⊗ S , (5.10)
mb i
where H is the hard function (corresponding to |C7γ(µ)|2 in the case of tree level matching),
S is the leading power shape function introduced in Section 4.3.3, ji are the so-called
subleading jet functions and the symbol⊗ denotes a convolution. There are several different
subleading jet functions corresponding to the different terms in Equation (5.6), all of which
have been calculated in [101]. In general they are divergent quantities and need to be
renormalized at the intermediate scale µi. This scale dependence cancels with the scale
dependence of the subleading soft functions which can be determined by considering the
one-loop corrections. The sum of all renormalized subleading jet functions is given as
( )
2 CFαs(µi) p
2
j(p , µi) = 16 ln + 25− 16c 2RS θ(p ) , (5.11)
4π µ2i
where cRS is a constant depending on the renormalization scheme whose importance will
become apparent later on (Section 5.2.5). It vanishes in the MS scheme and is 1 in the
dimensional reduction scheme which corresponds to evaluating all Dirac structures in d = 4
dimensions, while loop integrals are evaluated with d = 4 − 2ε. The step function θ is
introduced by taking the imaginary part of the hard-collinear correlation function. This
result only applies if the SCET operators are matched at tree level. If the hard functions
are determined at NLO, additional subleading jet functions suppressed by αs(µh)αs(µi)/mb
are to be expected. In conclusion, the subleading Q7γ − Q7γ jet function contribution to
73
the decay rate is given by
dΓSJF 2 2
∫
77 GFmb(µ)α
Λ̄
= |Ceff7γ (µ)λt|2 3
1
Eγ dωmb j(mb(p+ + ω), µ)S(ω, µ). (5.12)dEγ 2π4 mb −p+
where j is given in (5.11). When integrating the differential decay rate over a range much
larger than the endpoint region, the leading shape function may be approximated by the
first term its moment expansion (3.38). It is then possible to perform the integration over
the terms of the subleading jet function by using
M
∫ B
2 1
dEγ 1 ≈ (mb − 2E0) ,
E0≪M 2B (5.13)
MB − ( − )∫ 2 mb(mb 2Eγ) ≈ 1 mb(mb 2E0)dEγ ln (mb − 2E0) ln − 1 ,
E µ
2 2 µ2
0≪MB i i
where the approximate expressions on the right are valid up to O(1/(mb − 2E0)2) (see
Section 3.3.4). It follows that the subleading jet function contributions are promoted to
leading power in the limit E0 → 0 which is consistent with the observation that power
corrections to the total rate due to Q7γ − Q7γ interference appear only at O(1/m2b). The
other subleading contributions due to insertions of Q7γ originate from additional soft fields
in the matrix elements. These will be considered now.
5.2.4 The Subleading Shape Functions
In contrast to the subleading jet functions, subleading contributions due to additional soft
fields (or derivatives) in the SCET operators already appear at tree level. The relevant
diagrams for the Q7γ−Q7γ interference are given in Figure 5.2. The first diagram in the top
s s s
s s s s s s
Figure 5.2: Examples for diagrams contributing at subleading power due to additional soft
fields or derivatives of soft fields (first diagram).
(1)
row might represent the product of two times the first term of Q7γA with the additional soft
derivative responsible for the suppression, or the interference of the leading power operator
74
(2)
with Q7γA. The second diagram of the top row contains the insertion of a power suppressed
SCET interaction term. Due to the additional gluons these contributions contain a factor
g2 = 4παs.
In References [9,10,100] it was proven that contributions of this type obey a factorization
formula that can be schematically represented as
1 ∑
dΓ(B̄ → Xsγ) ∼ H J ⊗ si , (5.14)
mb i
where J is the leading power jet function (4.43) and si are the subleading shape functions.
These shape functions correspond to non-local HQET matrix elements that contain ad-
ditional soft fields or derivatives compared to the leading power shape function. Using
the notation of [100] the first five diagrams in Figure 5.2 introduce the following non-local
HQET operators defining the subleading shape functions.
∫ ∫
dt −iωt 4 〈B̄|T{(h̄Sn)(tn)(S†s(ω) = m e d z nh)(0)L(z)}|B̄〉b
2π 2MB
∫ ∫
dt t− 〈B̄|(h̄Sn)(tn)
/nγ⊥n (S† µν †
− iωt 2 µ ν n
gGs Sn)(rn)(Snh)(0)|B̄〉
t(ω) = e dr
2π 0 2MB (5.15)
∫ ∫
− dt
t 〈B̄|(h̄Sn)(tn)(S†i(iD 2 †⊥) Sn)(rn)(S h)(0)|B̄〉
u(ω) = e−iωt dr n n
2π 0 2MB
∫ ∫
dt t− −iωt 〈B̄|(h̄S
/n
n)(tn) iσµν(S
†
ngG
µν †
s⊥Sn)(rn)(Snh)(0)|B̄〉v(ω) = e dr 2
2π 0 2MB
The soft Wilson lines were introduced by the decoupling transformation (4.33). They reflect
the space-time topology of the diagram after integrating out the hard-collinear fields and
they guarantee gauge invariance of the HQET operators. Note, that even though the
additional soft fields introduce new non-localities, the subleading shape functions depend
only on one convolution variable ω. As was already mentioned above, this property only
holds at the tree level of the Q7γ−Q7γ contribution. One can interpret t and v as non-local
generalizations of the B-meson matrix element of the subleading HQET chromomagnetic
operator (see (3.14)), and u as a non-local generalization of the matrix element of the kinetic
operator. The function s is related to the second diagram in Figure 5.2. In addition to
the shape functions given above, there is a four quark operator appearing at O(αs) which
is depicted in the last diagram of Figure 5.2. It is generated by the insertion of two hard-
collinear-soft interaction terms of the SCET Lagrangian (4.24) together with two leading
75
(0)
power Q7γ operators. This contribution defines the following shape functions
∫ ∫
dt t
∫ t
f (q)(ω) =− 2 e−iωtu dr du2π 0 r
× 〈B̄|[(h̄S )(tn)T
a
n ]k[T
a(S†nh)(0)]l[(q̄Sn)(rn)]ln/[(S
†
nq)(un)]k|B̄〉 ,
2MB
∫ ∫
dt t
∫ t (5.16)
f (q) −iωtv (ω) =− 2 e dr du2π 0 r
a
× 〈B̄|[(h̄Sn)(tn)T ]kn/γ5[T
a(S†nh)(0)]l[(q̄Sn)(rn)]ln/γ5[(S
†
nq)(un)]k|B̄〉 ,
2MB
where k and l are color indices and T a are the SU(3)c generators. These non-local operators
exhibit another interesting property. Since the hard-collinear fields in SCET carry large
momentum components, the particles created by these fields always propagate forward in
time. There is no interaction term in SCET that can turn a forward moving quark into a
backward moving anti-quark, since this would require a hard fluctuation. As a result, after
the convolution with the jet function, the quantum fields in the definition of the subleading
shape functions are ordered in the same way as they appear in the Feynman graphs [100].
In the above formulas this corresponds to the requirement that t > u > r.
The subleading shape function presented in this section also contribute, in other com-
binations, to the semi-leptonic B̄ → Xulν̄ decay in the endpoint region. If these shape
functions were the only ones appearing in B̄ → Xsγ one might try to find weighted distri-
butions of B̄ → Xsγ and B̄ → Xulν̄ events, in which the subleading shape functions enter
in the same combinations [102]. However, it will be shown in Chapter 6 that there are
several additional unique shape functions that are relevant for B̄ → Xsγ. The tree level
subleading shape function contribution due to the Q7γ − Q7γ interference can be written
as
dΓSSF G2 2
∫ Λ̄
77 = F
mb(µ)α |Ceff 1(µ)λ |2E3 dωm J(m (p + ω), µ)
dE 2π4 7γ
t γ b b +
γ mb −p+
( [ ])
× −ωS(ω, µ) + s(ω, µ)− t(ω, µ) + u(ω, µ)− v(ω, µ)− πα f (s)s u (ω, µ) + f (s)v (ω, µ) ,
(5.17)
where J is the leading power jet function and the soft functions are given in (5.15) and
(5.16). Since the shape functions only have support over the endpoint region, integrating
the differential decay rate over a range much larger than ΛQCD justifies the replacement
M
∫ B ∫ Λ̄ m
2 b
− −iωt F ( , t)dEγ dω δ(mb 2Eγ + ω)e F (Eγ, t) → δ(t) 2 , (5.18)
E0≪M 2B 2Eγ−mb
where the exponential originates from the definition of the subleading shape functions
and Eγ was set to mb/2 which corresponds to the first order in the power counting. The
function F contains all quantities of Equation (5.17) that are not explicitly given in (5.18),
76
specifically the non-local matrix element. It follows that the subleading shape functions
of the Q7γ −Q7γ contribution reduce to local matrix elements in the total rate. However,
for t = 0 all subleading shape functions vanish as can be easily seen from their definitions.
This zero normalization of the subleading shape functions corresponds to the vanishing
1/mb corrections, discussed in Section 3.3.4. There, it was argued that the total rate was
well approximated by the partonic calculation because of these vanishing corrections. As
will be shown in Section 7.1, this does no longer hold when subleading shape functions of
other operator interferences are taken into account. However, before discussing this, the
relation of the QCD calculation to the effective theory calculation will be more carefully
analyzed.
5.2.5 Kinematic vs Hadronic Power Corrections
From the discussion of the two preceding sections it becomes apparent that the subleading
jet function and the subleading shape function contributions describe two fundamentally
different types of corrections. The subleading shape functions introduce new matrix ele-
ments and are therefore related to hadronic parameters. The subleading jet functions on
the other hand are perturbatively calculable and their importance increases when consid-
ering the total rate. They are therefore kinematically suppressed in the endpoint region,
whereas the shape functions parameterize hadronic corrections through higher powers in
the heavy-quark expansion. The two types of suppression imply, that the decay rate of
the Q7γ −Q7γ contribution integrated over a large enough range can be expressed in terms
of local operators organized in a double expansion in powers of ∆/mb and ΛQCD/∆ [13],
where ∆ is related to the photon energy cut ∆ = mb − 2E0. With a cut in the endpoint
region, the factor ΛQCD/∆ is of O(1) which requires a resummation of the corresponding
terms into a shape function (see Section 3.3.4).
Before the establishment of the systematic factorization (5.10) the kinematically sup-
pressed contributions were deduced from the partonic calculation close to the endpoint.
In that region the emission of hard-collinear and soft bremsstrahlung gluons is possible.
Hard gluons, on the other hand are forbidden, since they would lead to an invariant mass
of the final state of order mb. Thus, the partonic calculation already contains all con-
tributions due to the subleading jet functions, which originate from hard-collinear gluon
bremsstrahlung, as was discussed in Section 5.2.3. However, since the different regions
were not properly factorized, the partonic result will also contain contributions from the
soft region. If this soft contribution appears at the same order in the power counting as
the hard-collinear one, it must be absorbed into the subleading shape functions. Such an
absorption is indeed necessary for the Q7γ − Q7γ contribution, as will be shown in this
section.
The general procedure of deriving the kinematic power corrections from the parton
model was suggested in [13]. The starting point is the partonic results in (2.29), (2.33) and
(2.35). The appropriate scale for the calculation is µi since one considers hard-collinear
gluon emissions. The Wilson coefficients on the other hand are evaluated at the hard
scale, as usual. Although the anomalous dimensions of the subleading SCET and HQET
77
operators are generally unknown, by using the solution to the RG equation derived at the
leading power to run between the two scales, it is possible to resum the Sudakov double
logarithms of the partonic result [13]. Finally, it is necessary to include the effects of the
residual b-quark momentum in the n̄ direction as it is of the same order as the hard-collinear
momentum in that direction. This corresponds to the replacement
n · p = mb − 2Eγ → n · p+ n · k = mb − 2Eγ + ω. (5.19)
The result must then be convoluted with the leading power shape function and expanded
around Eγ = mb/2. The first term in this expansion contains the subleading jet function
contributions. For the case of he Q7γ −Q7γ interference this leads to1
part Λ̄ ( )∫dΓ77 G
2
Fm
2
= b
(µ)α | eff |2 3 1 CFαs(µ) mbC7γ (µ)λt Eγ dω 16 ln − 15 S(ω, µ).dE 4γ 2π mb 4π −p p + ω+ +
(5.20)
Comparing this to the subleading jet function expression in (5.11) one finds that the
ln(mb) term agrees in both expressions, while the p+-dependent terms and the constant
accompanying the logarithms differ. This difference is due to the effects of the soft region in
the partonic result. In order to prove that the effective theory approach properly factorizes
the partonic result, it is necessary to calculate the partonic one-loop contributions to the
subleading shape functions (5.15) as well as the tree-level matrix elements of the soft
operators in (5.16). Since the perpendicular component of the heavy-quark velocity can be
freely chosen it has been set to zero in this work. As a result the matrix elements of the
soft operators corresponding to the subleading shape functions t and v vanish at one-loop
order, since they only contain gluons with transverse polarization. The tree-level matrix
elements of the soft operators corresponding to fu and fv also vanish, since they contain
scaleless integrals over the n̄-components of the light quark momenta. Finally, the matrix
element of the soft operator corresponding to s is non-zero only when the heavy quarks are
off-shell with a non-zero n-component of the residual momentum. As a result, it does not
contribute if the residual momentum is set to zero in the partonic calculation. Therefore,
the only remaining quantities left to be calculated from (5.17) are ωS and u. The relevant
diagrams are depicted in Figure 4.5. The shape function ωS can be extracted from the
calculation performed in [1]. Setting v · k = 0 and using dimensional regularization one
finds for the unrenormalized shape function
( )
CFαs(µ) 4 µ
2
ω Sbare(ω) = θ(−ω) − − 4 ln − + 4 , (5.21)4π ǭ ( ω)2
where 1/ǭ = 1/ǫ − γE + ln 4π regularizes the UV pole. The one-loop correction to the
subleading shape function u is calculated to [101]
( )
− CFαs(µ) 12 µ
2
ubare(ω) = θ( ω) + 12 ln − − 20 . (5.22)4π ǭ ( ω)2
1Note, that a part of the subleading contributions of the Q7γ −Q7γ interference is already included in
the kinematic factor E3γ = (mb + ω)
3/8 of the leading power in (4.40) (see [13] for details).
78
Subtracting the pole in the MS scheme, one obtains from (5.17) for the subleading shape
function part of the differential decay rate
( )
dΓSSF ∣ G2 m2(µ)α 277 ∣ = F b |Ceff 1 CFαs(µ) µ7γ (µ)λ |2E3t γ 16 ln − 24 , (5.23)∣dEγ part 2π4 mb 4π p2+
where the subscript “part” indicates that the naive partonic expressions were used instead
of the non-perturbative subleading shape functions. In order to compare the effective with
the full theory result, the shape functions in (5.11) and (5.20) are replaced by δ-functions,
which corresponds to the on-shell heavy quark of the partonic calculation. The sum of
the partonic expression for the subleading jet function and the subleading shape functions
(5.23) is then given by
( )
dΓSJF ∣77 dΓ
SSF ∣ 2 2
∣ 77 G m (µ)α 1 C∣ F b | eff 2 3 Fαs(µ) mb+ = C
∣ ∣
dE dE 2π4 7γ
(µ)λt| Eγ 16 ln + 1− 16cRS ,
γ part γ part mb 4π p+
(5.24)
which coincides with the partonic expression in (5.20) if the renormalization scheme depen-
dent constant is set to one. This choice is in fact the correct one, since (5.17) was derived
in [100] by evaluating all traces of Dirac structures in d = 4 dimensions. Thus, the sub-
leading jet function should be renormalized using the same scheme which just corresponds
to setting cRS = 1. Equation (5.24) then proves that the subleading contribution derived
from the partonic expression contains contributions from the hard-collinear as well as the
soft region. The logarithm in (5.20) originates from a combination of hard-collinear and
soft scales
mb mbp+ µ
2
ln = ln + ln (5.25)
p 2+ µ p2+
that are properly factorized in the effective theory approach.
The calculation presented above is equivalent to a method of regions analysis of the
one-loop QCD forward scattering amplitudes contributing to B̄ → Xsγ [101]. The hard-
collinear and soft regions exactly reproduce the partonic expressions for the subleading
jet and shape functions, respectively. No further regions are necessary to reproduce the
full QCD result. Hard gluon loops only play a role in the matching of the effective QCD
operators onto SCET, but do not give rise to subleading contributions, as they only depend
on scales of order mb [101]. Before proceeding with operator interferences other than
Q7γ −Q7γ a final comment is in order concerning strange-quark mass effects.
5.2.6 Strange Quark Mass Effects
Throughout this thesis the strange-quark mass was set to zero, which turns out to be an
excellent approximation numerically. However, taking ms to be non-zero gives rise to a
subleading jet function proportional to m2/p2s hc, where phc is the momentum of the hard-
collinear jet containing the strange-quark [103]. If one adopts the scaling ms ∼ O(ΛQCD),
this function scales as ΛQCD/mb in the endpoint region and contributes to F77. From [103]
79
follows that the extra contribution is
∫ Λ̄
Fms77 (Eγ, µ) = −H̃77 dωmb jm(ω + p+, µ)S(ω, µ) , (5.26)
−p+
where it was defined jm =
1 Im Jmm . Both H̃77 = 1+O(α ms) and Jm can be found in [103].π b b
At the lowest order in αs,
m2
jm(ω + p+, µ) =
s δ′(ω + p+) +O(αs) , (5.27)
mb
and indeed jm is suppressed by m
2
s/(mbΛQCD) compared to the leading-order jet function
J(p2, µ). One could argue that for ms ≈ 100MeV and ΛQCD ∼ 500MeV the parameter ms
should scale as a higher power of the SCET expansion parameter λ, e.g. m ∼ λ2s . This
would imply that jm can be neglected at order ΛQCD/mb. Contributions of jm to coefficient
functions Fij other than F77 are further suppressed by hard functions H̃ij = O(αs). They
will not be considered in the following, since they are bound to be tiny.
This concludes the discussion of the subleading contribution due to two insertions of
the operator Q7γ . The next section will deal with the new effects that are introduced by
the insertions of the other effective operators Qq1 and Q8g.
5.3 Factorization at Subleading Power
In the middle of the 2000s, the effective theory treatment of B̄ → Xsγ included the proof
of the leading power factorization [12], the two loop calculation of the leading power soft
and jet function [93, 97], the proof of tree-level factorization at subleading power for the
Q7γ −Q7γ contribution [9,10,100,106,107] as well as the the contribution of other operator
insertions at one-loop order in the hard function and through kinematical power corrections
[13]. Furthermore, hints at the importance of the non-perturbative corrections existed in
the form of a soft scale dependence of the partonic result for the Q8g − Q8g interference
[64, 104] and a non-perturbative correction to the total rate from the interference of Qc1
with Q7γ [108–111]. However, a detailed analysis of the subleading hadronic corrections
was missing.
The distinguishing feature of the Q7γ − Q7γ contributions is the fact that the photon
is always emitted directly from the weak-interaction vertex. In principle, the operators
Qq1 and Q8g can also match onto SCET operators with the photon directly emitted from
the vertex. One possibility was discussed in Section 4.3.2 where QCD was matched to
the leading power SCET operator at the one-loop level. In the case of Qq1 and Q8g it is
also possible, that the photon couples to the light partons. Processes of this kind will be
called resolved photon contributions to be contrasted with the direct photon contributions
where the photon couples to the weak vertex. It is clear that all subleading contributions
considered so far were direct ones. The resolved contributions can be interpreted as a probe
of the hadronic substructure of the photon. There is no analogue to these contributions in
80
the semi-leptonic decays, because the lepton-neutrino pair can only couple to light partons
via W -boson exchange.
In order for a light parton to emit the anti-hard-collinear photon it must be anti-hard-
collinear itself. The conversion of an hard-collinear particle into an anti-hard-collinear one
would require a hard fluctuation which is integrated out of SCET. Therefore an extension of
the subleading factorization formulas (5.10) and (5.14) which only contained hard-collinear
particles, is required. The proof of factorization proceeds in analogy to the direct case.
First the QCD operators are matched at the hard scale onto SCET(hc,hc,s). In the case of
Q7γ the resulting SCET operator was not allowed to include anti-hard-collinear particles,
since there was no way to convert them into hard-collinear or soft ones, and they were not
allowed to appear in the final state jet. Now, however, anti-hard-collinear particles may
be converted into a photon by an insertion of the SCET Lagrangian. The QCD operators
therefore also match onto SCET operators including anti-hard-collinear modes. A general
matrix element, appearing in the calculation of the differential decay rate then looks like
Discrestr.〈B̄|h̄(x)[φs(xi) . . . φhc(yi) . . . φhc(zi) . . . φ′ (z′i) . . . ]h(0)|B̄〉. (5.28)hc
Note that it is now necessary to consider only the restricted cut, as explained in Section 3.1.
For the same reason, two different types of anti-hard-collinear fields φhc and φ
′ have been
hc
introduced. Each type only appears on one side of the cut, either connected to the initial
or final state B-meson via the weak vertex. In other words, an anti-hard-collinear line must
never be cut, since this would change the scaling of the invariant mass of the final state jet.
In the case of B̄ → Xsγ the matrix element must contain at least two hard-collinear fields
to generate the final state jet, either a pair of strange-quarks or of other light particles. In
the latter case, the strange-quarks must appear as soft fields. Since each field scales with
a certain power of λ, the maximum number of fields of each type is determined by the
desired order in the power counting. If there are no anti-hard-collinear fields in the matrix
element, the result will correspond to a direct contribution, otherwise to a resolved one.
Due to the presence of the anti-hard-collinear fields the multipole expansion of the
soft fields becomes more subtle. The correct form of the expansion must then be deter-
mined on a case-by-case basis for each operator, rather than derived from a simple set of
rules. It will be shown that the heavy-quark fields must always be expanded about x−,
while the other soft fields can depend on either x− or x+, or both. After the decoupling
transformation (4.33) the matrix element decomposes into
〈B̄| (0)h̄(x)[φs(xi) . . . ]h(0)|B̄〉 ×Disc〈Ω|φhc (yi) . . . |Ω〉 × 〈Ω|
(0) | 〉〈 | ′(0)φ (zi) . . . Ω Ω φ (z′i) . . . |Ω〉,hc hc
(5.29)
where the soft matrix element once again contains the soft Wilson lines, introduced by
the decoupling transformation. Now they do not only connect the fields along the n-light-
cone but also the fields that are displaced in the n̄ direction by the anti-hard-collinear
propagators. The soft matrix element is therefore gauge invariant.
The second matching step, from SCET to HQET is performed by integrating out the
(anti-)hard-collinear fields. This can be done perturbatively since the corresponding scales
belong to the short-distance regime. The Wilson coefficients of this matching are just the
81
vacuum expectation values of the time ordered product of the (anti-)hard-collinear fields.
It is now possible to define general jet and shape functions through the Fourier transform
of the matrix elements
[ ] [ ]
(n) ∼ 〈 | (0) | 〉 (n)Ji Disc Ω φhc (yj) . . . Ω , Si ∼ 〈B̄|h̄(x−)[φs(xi∓) . . . ]h(0)|B̄〉 , (5.30)
F.T. F.T.
where the superscript (n) indicates the suppression in powers of 1/mb and “F.T.” designates
a Fourier transform. The subleading shape functions of (5.15) fit into this scheme and
(1)
correspond to Si . In addition to these objects resolved photon contributions give rise to
a new type of jet function
[ ]
(n) ∼ 〈 | (0)J̄i Ω [φ (zk) . . . ]|Ω〉 , (5.31)hc F.T.
which corresponds to an uncut anti-hard-collinear propagator dressed by Wilson lines. As
was discussed in Section 4.3.3, the vacuum correlation function of the photon field factorizes
trivially and is not considered in this discussion.
The above discussion suggests the validity of a general factorization formula for the
differential B̄ → Xsγ decay rate in the endpoint region to all orders in 1/mb
∞
∑ ∑
→ 1 (n) (n) ⊗ (n)dΓ(B̄ Xsγ) = Hi Jmn i Si
n=0 b i
∞ [ ]
∑ 1 ∑ ∑(n) (n) ⊗ (n) ⊗ (n) (n) (n) ⊗ (n) ⊗ (n) ⊗ (n)+ H J S J̄ + H J S J̄ J̄
mn i i i i i i i i i
n=1 b i i
(5.32)
which generalizes and includes the previous formulas (4.51), (5.10) and (5.14). The first line
contains the direct, the second line the resolved photon contributions. One can distinguish
between single and double resolved contributions depending on whether there are anti-
hard-collinear propagators on one or on both sides of the cut. Note that the notation
is symbolic. Objects denoted by the same symbol in the various terms refer, in general,
to different quantities. The symbol ⊗ denotes a convolution over all soft momentum
variables shared by the jet functions and shape function. A graphical representation of the
factorization formula is shown in Figure 5.3. The resolved contributions exhibit another
interesting property, that will explicitly be demonstrated in Section 7.1: It was discussed
in Sections 5.2.3 and 5.2.4 that the subleading Q7γ − Q7γ (direct) contributions are only
relevant in the endpoint region and vanish when the total rate is considered. This is no
longer true for the resolved contributions.
When talking about factorization this always refers to the the kind of proof presented
in this section. It implies that it is always possible to decompose the differential B̄ →
Xsγ decay rate in the endpoint region into the parts considered above. But it does not
imply, that this decomposition allows for a consistent resummation of large logarithms
and that the renormalization procedure discussed in Section 4.4 will work at all orders in
82
J J J
H J H H J H H J H
S S S
B B B B B B
Figure 5.3: Graphical illustration of the three terms in the factorization formula (5.32).
The dashed lines represent soft interactions that are factored off the remaining building
blocks by the decoupling transformation and the heavy-quark expansion.
perturbation theory. These issues will be analyzed in more detail in Section 6.3.1, when an
explicit example is available. The next step in the systematic analysis of the subleading
corrections is to match the QCD operators Qq1 and Q8g onto the corresponding SCET
operators.
5.4 Operator Matching onto SCET
It was motivated in Section 2.1.5 that the only QCD operators not severely suppressed
by their Wilson coefficients are Qq1 and Q8g. The matching of these operators proceeds
in analogy to the Q7γ matching by making the replacements discussed in Section 5.1.
As was explained in Section 5.2.2, only operators with a scaling of λ5/2, λ3 and λ7/2
are of relevance. For simplicity, the matching will only be performed at tree-level for
hard quantum corrections. This procedure will generate a large number of possible SCET
operators, all of which are collected in Appendix B. However, only a limited number
of them are relevant for the calculation of the subleading resolved contributions. Their
matching will be explicitly presented in this section.
The second matching step, from SCET onto HQET, will be performed at the one-loop
level for (anti-)hard-collinear quantum fluctuations associated with the leading power shape
function in (4.38). These correspond to direct contributions. Finally, the Wilson coeffi-
cients of the new subleading shape functions will be computed at tree-level, but including
the 4παs contributions resulting from tree-level (anti-)hard-collinear gluon exchange.
In order to determine which of the several possible subleading SCET operators are of
relevance, the basic requirements will once again be collected from the previous sections.
First, it is required that the final state only contains one anti-hard-collinear particle, namely
the photon. All other particles in the final state must either be hard-collinear or soft. There
must be at least one hard-collinear particle in the final state in order to have a jet with an
√
invariant mass at the required scale ΛQCDmb. Furthermore, there can be an arbitrary
number of anti-hard-collinear fields in the operators as long as they are converted to the
final state photon plus soft particles by SCET Lagrangian insertions. The number of
83
these soft and hard-collinear particles is only restricted by the desired overall scaling of
the operator. Before matching the operators themselves, the necessary conversions will be
collected.
5.4.1 Photon Conversions
As can be seen from the discussion in Section 4.2.2 (with hc replaced by hc) the conversion
of an anti-hard-collinear field into a photon and a soft field is suppressed by at least
λ1/2. The interaction of the photon with an anti-hard-collinear field on the other hand is
unsuppressed in (4.14). This leads to the possible conversions illustrated in Figure 5.4.
The last diagrams of each line, which involve three gluon fields, are not needed for the
hc hc
hc
hc s hc s hc s
hc s s
hc s hc
s
Figure 5.4: O(λ1/2) (top row) and O(λ) (bottom row) conversions of anti-hard-collinear
particles into a photon accompanied by soft particles. Only some representative diagrams
are shown.
tree-level analysis of this work. The scaling of the conversions does not change by adding
any number of anti-hard-collinear gluons to the ingoing side. The conversions at order λ1/2
can thus be written as
ξ → Aemhc ⊥ + q , ξ + ξ̄ → Aemhc hc ⊥ + As , Ahc +A → Aemhc ⊥ + As , (5.33)
while the ones at order λ are
A → Aemhc ⊥ + q + q̄ , A emhc → A⊥ + As + As . (5.34)
With these conversion rules at hand it is possible to decide which SCET operators con-
tribute to the subleading resolved photon contributions.
5.4.2 Operator Matching of Q8g
When matching the chromomagnetic dipole operator Q8g of (2.17), one can discard the
non-abelian part of the field strength tensor Gµν , since only the first order in g needs to
84
be considered. A priori, it is possible to match the s-quark onto either a hard-collinear or
anti-hard-collinear quark and the gluon onto a hard-collinear or anti-hard-collinear gluon.
Matching the s-quark onto a soft particle would only give rise to a contribution beyond
the subleading power. Also, the gluon and the s-quark can not both be anti-hard-collinear,
since the necessary conversions would again lead to a suppression beyond the subleading
power. However, for the resolved contribution one of them needs to be anti-hard collinear,
which implies two possible SCET operators that descend from Q8g. They are illustrated
in Figure 5.5. The first one (first row) is suppressed by λ compared to the leading power
hc
s
hc + O(λ) hc s
O(λ5/2) hc
hc
hc + O(λ1/2) s
O(λ5/2) hc
Figure 5.5: SCET Operators descending from Q8g that give rise to resolved photon contri-
butions plus the necessary conversions by Lagrangian insertions.
and can be written as follows. Here and in the rest of this section the factor −gmb e−imb v·x
4π2
will be suppressed, leading to
(2) n̄/
Q (x) = ξ̄ (x) [in · ∂A/ (x)] (1 + γ )h(x ) , followed by O(λ) conversion
8g,hc gluon hc 2 hc⊥
5 −
(5.35)
which contributes to the differential decay rate at subleading power in combination with
the leading power operator (4.27). The superscript (i) now denotes the total suppression
in powers of λ1/2 of the complete set of fields on one side of the cut. In the present
case it consists of the operator scaling of O(λ5/2) plus the O(λ) scaling of the Lagrangian
insertion. In the above expression the gluon field strength tensor has been expanded after
the introduction of the gauge invariant building block (4.23)
σ µν µ ν µ ν †µνigG = σµν [iD , iD ] = σµνWhc[iD , iD ]Whc hc hc hc hc
[ n̄/
=Whc 2 [in · ∂A/2 hc⊥] (5.36)
( )
A − · A n̄/ n/ − n/ n̄/
]
+2 [i∂/⊥ / hc⊥ i∂⊥ hc⊥] + [in · ∂ n̄ · A ] W † ,2 2 2 2 hc hc
where the terms in the second and third line are of order λ1/2 and λ, respectively.
85
If the s-quark is matched onto an anti-hard-collinear field it needs to be converted into
the photon and a soft quark. This conversion costs λ1/2, so the lowest power insertion
possible is of order λ3. This leads to (see second row of Figure 5.5)
(1) n/
Q (x) = ξ̄ 1/2
8g, hc quark hc
(x) [in̄ · ∂A/ hc⊥(x)] (1 + γ5)h(x−) , followed by O(λ ) conversion.
2
(5.37)
Two insertions of this operator give rise to a double resolved contribution. The subleading
contributions due to insertions of a SCET operator descending from Q8g will be discussed
in Sections 6.3 and 6.4.
q
5.4.3 Operator Matching of Q1
In order to determine the resolved photon contributions that include the current-current
operator Qq1, it has to be matched onto a SCET operator containing at least one anti-hard-
collinear field. The remaining quark fields in Qq1 are then matched onto the heavy-quark
field and two hard-collinear fields, since a soft quark would lead to a scaling larger than
λ7/2. However, such an operator can not interfere with the leading power SCET operator
which only contains one hard-collinear field. It is therefore necessary to match Qq1 onto a
SCET operator containing two anti-hard-collinear fields and convert them into a soft gluon
and the photon via a penguin loop, by using the appropriate conversion rule in (5.33). The
situation is depicted in Figure 5.6. Note that the conversion into a single gauge boson
hc hc s
hc + O(λ1/2)
hc
O(λ3) hc hc
Figure 5.6: SCET Operator descending from Qq1 that gives rise to resolved photon contri-
butions plus the necessary conversion by Lagrangian insertions.
vanishes in the ’t Hooft-Veltmann scheme, as was discussed in Section 2.1.5. The anti-
hard-collinear fields can be either a charm- or up-quark, while the hard-collinear field must
be the strange-quark. This leads to an O(λ7/2) SCET operator insertion, namely
q (2)
Q1 = e
−imbv·xξ̄ µ 1/2hcγ (1−γ5)h(x−) ξ̄hc(x)γµ(1−γ5)ξhc(x) , followed by O(λ ) conversion.
(5.38)
In the interference with the leading power SCET operator this gives rise to a single-resolved
photon contribution. This contribution will be discussed in detail in Section 6.1.
86
5.5 Conclusions of this Chapter
In this chapter a factorization theorem for the subleading contribution to the differential
B̄ → Xsγ decay rate in the endpoint region has been presented in (5.32). The concepts of
subleading jet and shape functions were generally introduced in (5.30). In addition a new
type of jet function (5.31) appeared in the calculation of resolved photon contributions. It
was related to an uncut anti-hard-collinear propagator dressed by Wilson lines. After this
general considerations about the subleading contributions to B̄ → Xsγ, the remainder of
this work will deal with each contribution in detail, using the SCET operators of Section 5.4
as a starting point. For this, it is convenient to write the CP -averaged B̄ → Xsγ differential
decay rate in the endpoint region as
[
dΓ G2 α|V V ∗|2 ∫ Λ̄ ( )
= F
tb ts m2(µ)E3 |Hγ(µ)|2 dωmb J mb(ω + p+), µ S(ω, µ)
dEγ 2π4
b γ
−p+
] (5.39)
1 ∑ [ ∗ ]+ Re Ci (µ)Cj(µ) Fij(Eγ, µ) + . . . ,mb i≤j
where the first and second line represent the leading and subleading power contribution,
respectively, while the ellipses denote terms of even higher order in the power counting
and the real part prescription originates from the CP -averaging. The leading power has
been discussed in detail in Chapter 4 (compare Equation (4.40)). At the subleading power,
the Q7γ − Q7γ contribution has been discussed in this chapter. It was found that the
(a)
function F77 decomposes into a subleading jet function part F77 and a subleading shape
(b)
function part F77 . The former is given in (5.12), the latter in (5.17). The other Fij will
be determined in the next chapter. In principle, all possible combinations of the operators
Qq1, Q7γ and Q8g contribute to B̄ → Xsγ. However, only three of them, corresponding to
F17, F78 and F88 actually contribute at O(1/mb), as will also be shown in the next chapter.
87
Chapter 6
Analysis of Subleading Contributions
to the Photon Spectrum
In this chapter all the relevant subleading contributions to the differential B̄ → Xsγ
decay rate in the endpoint region will be analyzed in detail. As was discussed in the
previous chapter, this includes the interference of operators descending from Qq1 with those
descending from Q7γ in Section 6.1, the interference of operators descending from Q8g
with themselves in Section 6.3, as well as the interference of operators descending from
Q8g with those descending from Q7γ in Section 6.4. The results of the calculation will
exemplify the factorization formula (5.32), introduced in the last chapter. However, there
are some subtleties concerning the factorization of the Q8g −Q8g contribution. These will
be discussed in Section 6.3.1. Finally, some properties of the soft functions will be discussed
in their respective sections and in Section 6.5.
q
6.1 Analysis of the Q1 −Q7γ Contributions
In order to determine the direct photon contribution for the interference of the operators
Qq1 and Q7γ one has to evaluate the diagrams in Figure 5.1. The first diagram in that
Figure contains two insertions of O(λ3) SCET operators, which can be read off from (5.6)
and (B.13). There is more than one possibility for insertions that generate the second
diagram in Figure 5.1. The vertex on the right can either be the leading power SCET
operator (4.27) or an O(λ3) operator from Q7γA. In the former case the vertex on the left
corresponds to an O(λ7/2) operator from (B.13), else to an O(λ3) operator. In either case
an additional leading power SCET Lagrangian (4.14) insertion is necessary, to connect the
hard-collinear gluon to the hard-collinear strange-quark. The calculation yields
( )
∫
(a) CFαs(µ) 2
Λ̄
F17 (Eγ, µ) = − dω S(ω, µ) , (6.1)4π 3 −p+
which involves a convolution of the leading shape function (4.38) with a jet function con-
sisting of the cut of a hard-collinear loop. This result coincides with the kinematical power
88
corrections (see Section 5.2.5) derived from the partonic result in (2.29) by expanding
around Eγ = mb/2. To obtain this result the scaling m
2
c =O(ΛQCDmb) for the charm-
quark mass was adopted, meaning that the ratio m2c/mb remains a constant of order ΛQCD
in the heavy-quark limit.
Much more interesting is the single resolved photon contribution arising from this oper-
ator pair. As shown in the left graph of Figure 6.1, it is obtained by combining the O(λ3)
SCET four-quark operator in (5.38), which contains two anti-hard-collinear quark fields
in addition to a hard-collinear strange-quark and a heavy quark, with the leading-power
contribution (4.27) descending from Q7γ . According to the rules (5.33), the conversion
hc
hc
s
Figure 6.1: Diagrams arising from the matching of the Qq1 −Q7γ contribution onto SCET
(left) and HQET (right). As a resolved photon contribution, the diagram contains non-
localities in the light-cone direction n, denoted by the vertical dashed line in the second
diagram.
of the two anti-hard-collinear fields into a photon and a soft gluon costs a factor of λ1/2,
giving a total suppression with respect to the leading term of λ ∼ ΛQCD/mb. When the
(anti-)hard-collinear fields are integrated out in the second matching step, one obtains the
HQET diagram shown in the right graph of Figure 6.1. Here and below, horizontal (ver-
tical) dotted lines represent Wilson lines along the n (n̄) direction. The HQET matrix
element corresponding to the graph on the right in Figure 6.1 contains a soft gluon field
in addition to the two heavy quarks. In the language of the factorization formula (5.32)
this corresponds to a single resolved contribution with a subleading shape function. For
the contribution from the operator Qc1 one obtains
∫ Λ̄
(b) 2
F17,c(Eγ, µ) = (1− δu) dω δ(ω + p+)3 −∞
∞ (6.2)[ ( − )]∫ 2× dω1 m iεRe 1− F c g17(ω, ω1, µ) ,
−∞ ω1 + iε 2Eγ ω1
where the CKM-suppressed parameter δu is defined by
Re [λ∗ ∗qC1(µ)(−λt)C7γ(µ)]
δq = |λ |2Re [C∗ , (6.3)t 1(µ)C7γ(µ)]
89
while the subleading shape function g17 is defined by
∫ ∫
dr
g (ω, ω , µ) = e−iω r
dt
1 e−iωt17 1
2π 2π
( ) ( † ) ⊥ ( † ) ( )〈B̄| h̄S (tn) n̄/(1 + γ ) S S (0) iγ n̄ S gGαβ× n 5 n n̄ α β n̄ s Sn̄ (rn̄) S
†
n̄h (0)|B̄〉
.
2MB
(6.4)
The penguin function F (x) is defined in Appendix A.1. For 0 < x < 1/4 this function
develops an imaginary part. Note that a power counting for the charm-quark mass is
adopted such that m2c = O(mbΛQCD). The argument of the penguin function in the
convolution (6.2) then counts as O(1). Another comment concerning the charm-quark
mass is in order. In the analysis of the B̄ → Xsγ decay rate and photon spectrum,
it is customary to adopt for the charm-quark mass a running mass defined at a hard-
√
collinear scale µhc ∼ mbΛQCD ∼ mc [52]. For instance, the default value adopted in [5]
is mc = mc(1.5GeV). This scale choice is indeed appropriate for charm-quark mass effects
residing in the jet functions entering the factorization formula (5.32). On the other hand,
charm-quark mass effects also enter some of the hard functions in the factorization formula,
for instance via the coefficient Hγ(µ) in (4.30) [13], or via phase-space functions such as
those shown in (2.29). In this case the charm-penguin loops are probed at virtualities of
order mb, and it is therefore appropriate to use a running mass mc = mc(µh) evaluated at a
hard scale µh ∼ mb. This can have important numerical effects, enhancing the theoretical
prediction for the total decay rate by up to 3%.
The structure of the soft Wilson lines in (6.4), which are directed either along n or n̄,
follows when the decoupling transformation is applied to the (anti-)hard-collinear fields in
SCET to remove their soft interactions from the effective Lagrangian and absorb them into
eikonal factors. The Wilson lines reflect the space-time topology of the HQET diagrams
shown on the right-hand side in Figure 6.1. The two weak vertices shall be labeled by
coordinates 0 (left) and x = tn + x+ + x⊥ (right), and the vertex of the soft gluon by
y = rn̄ + y− + y⊥. The multipole expansion of the effective-theory fields implies that
x+,⊥ and y−,⊥ can be set to zero at this order. Gauge invariance then requires that the
fields h̄(tn) and Gs(rn̄) are joined by a Wilson line, and the rules of SCET determine that
this line consists of two segments: a straight line [tn, 0] along the light-like direction n
followed by a straight line [0, rn̄] along the light-like direction n̄. The fields Gs(rn̄) and
h(0) are joined by a straight Wilson line [rn̄, 0] along the light-like direction n̄. Using
that [tn, 0] = Sn(tn)S
†
n(0) etc., one recovers the structure of the Wilson lines in the non-
local operator in (6.4). It is noted for completeness that soft functions closely related to
the functions g17 in (6.4) and g11 in (6.23) were introduced, in a context not related to
B̄ → Xsγ decay, in [112].
There is more to the space-time structure of the soft operator that is worth pointing
out. Since hard-collinear fields in SCET carry large momentum components, the particles
created by these fields always move forward in time. As a result, after convolution with
the jet functions, the quantum fields in the definition of the subleading shape functions are
90
ordered in the same way as they appear in Feynman graphs [100]. The operators considered
in this work contain fields that propagate along the two light-like directions n and n̄, as
indicated by the dotted lines in the right graph in Figure 6.1. If one assigns coordinate 0
to the first of the weak vertices in the figure, the gluon is emitted at space-time point rn̄
with r > 0. This is ensured by the iε prescription in the jet function in (6.2), since
∫
dω e−iω
i
1r
1 = 2π θ(r) . (6.5)
ω1 + iε
The gluon thus lives at a later time than the field h(0) (recall that n0 = n̄0 = +1), and
indeed it appears to the left of that field.
Another comment is in order concerning the structure of the result (6.2). From the
diagrams shown in Figure 6.1 one derives one half times the expression (6.2) without the
real-part prescription in front of the integral and in expression (6.3). The mirror diagrams
not shown in the figure, in which the two weak vertices are interchanged, give an analogous
contribution with the complex conjugate Wilson coefficients and CKM matrix elements,
the complex conjugate penguin function F ∗(r),1 the propagator factor 1/(ω1− iε), and the
soft function
∫ ∫
′ dr dtg17(ω, ω1, µ) = e
iω1r e−iωt (6.6)
2π 2π
( ) ( ) ( ) ( )
×〈B̄| h̄Sn̄ (tn) (−iγ
⊥n̄ ) S† gGαβα β n̄ s Sn̄ (tn+ rn̄) S
† †
n̄Sn (tn) n̄/(1 + γ5) Snh (0)|B̄〉 ,
2MB
which is related to the original one by complex conjugation: g′17(ω, ω1, µ) = [g17(ω, ω1, µ)]
∗.
To show this, one uses translational invariance to shift all position arguments by −tn and
then changes the sign of the integration variable t. The sum of the diagrams in Figure 6.1
plus their mirror graphs thus gives a real result, and after averaging over CP -conjugate
decay modes one obtains (6.2).
The real-part symbols in (6.3) and (6.2) refer to different kinds of complex param-
eters. The various products of Wilson coefficients and CKM factors carry, in general,
CP -violating weak phases. The convolution of the jet and soft functions, on the other
hand, can carry CP -even, strong-rescattering phases, which in principle can result either
from anti-hard-collinear loops (i.e., the jet functions J̄i) or from the soft matrix elements
themselves. (However, in Section 6.5 it will be argued that the soft functions are real.)
When both types of phases are present, a non-zero direct CP asymmetry arises. The
subleading power corrections provide new mechanisms for generating such an asymmetry.
This will be discussed in Chapter 8.
The range of support of the soft functions in HQET can be derived in analogy to
Section 4.3.3 by noting that the light-cone projections n ·pi and n̄ ·pi of all parton momenta
in the B meson must be non-negative, and that the total momentum of all partons in the
B meson is MBv. Since in HQET the momentum of a heavy quark is decomposed as
1This is because the fields in the charm-quark loop of the mirror graphs are anti-time-ordered.
91
pb = mbv + k, where k is the residual momentum, it follows that
∑ ∑
n · pi + n · k = Λ̄ , n̄ · pi + n̄ · k = Λ̄ , (6.7)
i=6 b i 6=b
where n · k > −mb and n̄ · k > −mb. In the heavy-quark limit mb → ∞ it follows that
−∞ < n · k ≤ Λ̄ and 0 ≤ n · pi < ∞ (for i =6 b), and similarly for n̄ · k and n̄ · pi.
In the special case of the soft function g17 in (6.4), the variable ω corresponds to the
residual-momentum component n · k of the initial-state heavy quark, while ω1 can either
correspond to the component n̄·pg of a gluon in the final-state B meson or to the component
−n̄ · pg of a gluon in the initial-state B meson. It thus follows that −∞ < ω ≤ Λ̄ and
−∞ < ω1 < ∞. This implies that the penguin-loop function F (x) is sampled over both
positive and negative values of its argument.
In principle, each of the six QCD penguin four-quark operators in the effective weak
Hamiltonian can give rise to a similar contribution, either via loops of massless quarks
(q = u, d, s) or via a charm-quark loop. (b-quark loops lead to further power suppression.)
Of these options only Qc1 has both a large Wilson coefficient C1 ∼ 1 and a large CKM
factor V ∗cbVcs = O(λ2), so it will give rise to the dominant effects. However, for later use
the case of the CKM-suppressed operator Qu1 will also be considered. Using that F (0) = 0,
it follows that the resolved photon contribution resulting from this operator is given by
∫ Λ̄ ∫ ∞
(b) 2 dω1
F17,u(Eγ, µ) = δu Re dω δ(ω + p+) g17(ω, ω1, µ) . (6.8)3 −∞ −∞ ω1 + iε
Note that the soft function is the same as in (6.2).
Next the convergence properties of the convolution integrals in (6.2) and (6.8) will be
investigated. In the UV region, for ω1 ≫ ΛQCD, the first integral approaches the form
of the second one, since mass effects become negligible. It follows that the convolution
over ω1 converges as long as the soft function g17 vanishes for ω1 → ±∞. In general, the
asymptotic behavior of the soft functions for large values of the ωi variables can be analyzed
using short-distance methods [1]. This shows, for instance, that the leading shape function
behaves as S(ω, µ) ∼ 1/ω modulo logarithms for ω → −∞. For the present case, naive
dimensional analysis suggests the behavior g17(ω, ω1, µ) ∝ ω1 for large ω1 but fixed ω, in
which case the convolution integral would diverge linearly. To obtain such a contribution,
however, would require a non-zero matrix element of the soft operator between two on-shell
b-quarks. But this matrix element vanishes by Lorentz invariance. A non-zero contribution
is only obtained if, in addition to the heavy quarks, one adds a soft external gluon. This
costs two orders in power counting, so that the asymptotic fall-off is at least as strong as
g17(ω, ω1, µ) ∝ 1/ω1 for ω1 → ±∞. It follows that the convolution integrals (6.2) and (6.8)
are UV convergent.
The behavior of the soft functions in the IR region can not be derived from a perturba-
tive analysis. In the present case, however, it suffices to make the reasonable assumption
that g17(ω, ω1, µ) is non-singular at ω1 = 0. Using the expansion
− − 1 − 1 11 F (x) = − − . . . (6.9)
12x 90x2 560x3
92
valid for large x, one finds that for small ω1 the convolution integral (6.2) arising from the
charm-quark loop behaves as
[ ( )]
∫ ∫
dω1 − m
2
1 F c
− iε Eγ
g17(ω, ω1, µ) ≈ − dω1 g17(ω, ω1, µ) . (6.10)
ω1 + iε 2E ω 6m2γ 1 c
ω1≈0 ω1≈0
For the convolution integral (6.8) arising from the up-quark loop, one finds instead
∫ ∫
dω1 dω1
g17(ω, ω1, µ) = P g17(ω, ω1, µ)− iπ g17(ω, 0, µ) , (6.11)
ω1 + iε ω1
where the symbol P denotes the Cauchy principal value of the integral. It follows that the
convolution integrals indeed exist as long as the subleading shape function is non-singular
at ω1 = 0. Note that for the case of the up-quark loop it is important that the integral
over ω1 runs over both positive and negative values. Previous authors have already pointed
out that the up-quark loop contribution to the B̄ → Xsγ decay rate, while strongly CKM
suppressed, is described by an uncalculable long-distance contribution [108,110,111]. The
relation (6.11) provides a rigorous field-theoretic definition of this contribution in terms of
a well-defined, non-local soft matrix element.
For phenomenological purposes it is useful to define a new function
[ ( )]
∫
2 ∞ dω 21 mq − iε
f17,q(ω, µ) = 1− F g17(ω, ω1, µ) . (6.12)
3 −∞ ω1 + iε (mb + ω)ω1
The final expression for the Qq1 −Q7γ contribution can then be written as
( )
∫
CFαs(µ)
Λ̄
∑
F17(Eγ, µ) = −
2
dω S(ω, µ) + δq Re f17,q(−p+, µ) . (6.13)
4π 3 −p+ q=c,u
Note that the argument of the penguin function entering f17,q(−p+, µ) is m2q/[(mb −
p+)ω1] = m
2
q/(2Eγ ω1), as it should be. It was already mentioned at the beginning of
this section, that the direct part in (6.13) corresponds just to the result derived by ex-
panding the parton model expression in (2.29) around the endpoint (for m2c ∼ ΛQCDmb).
The resolved part on the other hand is a new hadronic effect. Even though there is no way
to explicitly calculate these non-perturbative contributions, some information about them
is obtainable.
Qq6.1.1 1 −Q7γ: Facts about the Soft Function
One fact that is known about this non-perturbative contribution is that the definition of
the soft function g17 in (6.4) implies the normalization condition
∫ Λ̄ ∫ ∞ 〈B̄| h̄ n̄/ iγ⊥n̄ αββ gG h |B̄〉
dω dω1 g17(ω, ω1, µ) =
α s = 2λ2 , (6.14)
−∞ −∞ 2MB
93
where (3.39) was used to evaluate the matrix element of the local quark-gluon operator in
terms of the hadronic parameter λ2. Moreover, the HQET trace formalism of Section 3.3.3
implies that the soft function can be written as
[ ⊥ ]g α17(ω, ω1, µ) = tr M n̄/(1 + γ5) iγα MΞ (v, n̄, ω, ω1, µ)
[ ]
1 + v/ 1 + v/ ( )
= MBtr n̄/(1 + γ ) iγ
⊥
5 α iγ
α⊥Ξ1(ω, ω
α⊥
1, µ) + iγ n̄/Ξ2(ω, ω1, µ)
2 2
= 4MBΞ2(ω, ω1, µ)
(6.15)
where the most general decomposition of Ξα was used. It follows from this argument that
the factor (1 + γ5) in (6.4) can be replaced by 1, since the part of the trace involving γ5
vanishes. It is then easy to see that
∫ Λ̄ ∫ Λ̄ [ ]∗
dω g17(ω, ω1, µ) = dω g17(ω,−ω1, µ) . (6.16)
−∞ −∞
One can constrain the function g17(ω, ω1, µ) further by looking at its first moments
with respect to ω and ω1. These can be related to linear combinations of HQET matrix
elements with three covariant derivatives. Such matrix elements can be expressed in terms
of two hadronic parameters, ρ1 and ρ2, via [113]
〈B̄| ( )h̄Γ α δ βαδβ iD iD iD h |B̄〉 1 1 + v/ [ ρ ]1 ρ2 1 + v/
= tr Γαδβ (g
αβ − vαvβ) vδ + iσαβvδ .
2MB 2 2 3 2 2
(6.17)
Thus, one finds
∫ Λ̄ ∫ ∞ 〈B̄| h̄ n̄/γ⊥α in ·D [iDα⊥, in̄ ·D]h |B̄〉dω ω dω1 g17(ω, ω1, µ) = = −ρ2 ,
−∞ −∞ 2MB (6.18)
∫ Λ̄ ∫ ∞ 〈B̄| h̄ n̄/γ⊥α [[iDα⊥, in̄ ·D], in̄ ·D]h |B̄〉dω dω1 ω1 g17(ω, ω1, µ) = = 0 ,
−∞ −∞ 2MB
where the HQET parameter ρ2 is related to the parameter ρ
3
LS introduced in [114] via
ρ3LS = 3ρ2. The vanishing of the first moment with respect to ω1 of g17 is not a coincidence.
As will be shown in Section 6.5, g17 is in fact a real function. Relation (6.16) then implies
that all the odd moments in ω1 vanish.
As a final comment it is added that even in the limit where the charm-quark is treated
as a heavy quark, mc = O(mb), the penguin contribution to the photon spectrum must still
be described by a subleading shape function. In this limit the argument of the penguin
function in (6.2) is of order mb/ΛQCD. Expanding then the function [1 − F (x)] to first
order in 1/x leads to2
( ) ( )
∫
→ −mb + ω dt
⊥ † αβ
f (ω, µ) e−iωt
〈B̄| h̄Sn (tn) n̄/(1 + γ5) iγα n̄β Sn gGs h (0)|B̄〉
17,c .
18m2c 2π 2MB
(6.19)
2Only the first term in this expansion gives rise to a UV-convergent convolution integral.
94
Integrating this expression over ω, and dropping higher power corrections, one obtains a
contribution to the total decay rate proportional to
∫ mb mbλ2
dp+ f17,c(−p+, µ) → − . (6.20)2
−Λ̄ 9mc
This non-perturbative correction to the total rate was already considered in [108–111] (the
correct sign was obtained in the last reference). This gave an early hint at the weaknesses
of the local OPE approach on B̄ → Xsγ. With the above derivation its origin can be
related to the effect of a resolved photon contribution.
q q q
6.2 Analysis of the Q1 −Q1 and Q1 −Q8g Contributions
The power-counting rules described in Appendix B show that for these two cases there do
not exist operators arising at order 1/mb in the heavy-quark expansion that contain soft
fields other than the two heavy quarks. In particular, the diagrams shown in Figure 6.2
contribute to the B̄ → Xsγ photon spectrum only at order 1/m2b . This is an important
hc
hc hc
s s s s
Figure 6.2: Examples of SCET diagrams giving rise to resolved photon contributions sup-
pressed by at least two powers of of 1/mb. The left graph arises from the pairing Q
q
1 −Qq1,
while the right one contributes to the Qq1 −Q8g term.
finding. Since for the case of two charm-quark loops the first diagram in the figure is
proportional to the large Wilson coefficient |C 21| ∼ 1, it has sometimes been mentioned as
a potentially dangerous source of power corrections. It follows from the present analysis
that this contribution scales as (ΛQCD/m
2
b) relative to the leading term. It is therefore
expected to be a small correction (see below).
It thus remains to calculate the leading power corrections to the direct photon term
in the factorization formula, by analyzing graphs of the type shown in Figure 5.1. Note
that here only the first diagram on the left in this figure can contribute. After a straight-
forward calculation (summing over q = c, u), one finds once again contributions involving
a convolution of the leading shape function with a jet function consisting of the cut of a
hard-collinear loop. The results are the same in the two cases and given by
∫ Λ̄
(a) 2Eγ (a) CFαs(µ) 2
F11 (Eγ, µ) = F18 (Eγ, µ) = dω S(ω, µ) . (6.21)mb 4π 9 −p+
95
This is the obvious generalization of the parton-model results in (2.29). Note that the
prefactor mb/2Eγ in the result for F18 follows from the SCET calculation, and therefore it
is presented here. In the endpoint region this factor equals 1 up to power corrections, and
it could therefore be omitted.
In order to substantiate the statement about the smallness of the O(1/m2b) double
resolved photon contribution represented by the first diagram in Figure 6.2, this graph is
evaluated explicitly. The result is
∣ ∣2 ∫ Λ̄ ∫ ∞ ∫
(b) 1 λ 2
∞
∣ c ∣
F ∣ ∣11 (Eγ, µ) = − dω δ(ω + p+) dω1 dω2m ∣ ∣b λt 9 −∞ −∞ −∞
[ (
× 1 − m
2
c −
)] [ ( )]
iε 1 2
1 F 1− F ∗ mc − iε g11(ω, ω1, ω2, µ) ,
ω1 + iε 2Eγ ω1 ω2 − iε 2Eγ ω2
(6.22)
where the 1/mb prefactor indicates the additional power suppression. The soft function is
defined as
∫ ∫ ∫
dt dr
g (ω, ω , ω , µ) = e−iωt e−iω r
du
1
11 1 2 e
iω2u g α βµν n̄ n̄
2π 2π 2π
( ) ( ) ( ) ( ) ( ) ( )
×〈B̄| h̄S (tn) S
† gGνβS (tn+ un̄)Γ S†n̄ n̄ s n̄ n̄Sn (tn) S
†
nSn̄ (0) S
† µα †
n̄ gGs Sn̄ (rn̄) Sn̄h (0)|B̄〉 ,
2MB
(6.23)
where Γ = n̄/(1−γ5). This function satisfies g11(ω, ω1, ω2, µ) = [g11(ω, ω2, ω1, µ)]∗, which im-
(b)
plies that F11 is real. In order to obtain an estimate of the magnitude of this contribution,
the penguin functions are expanded to first order using (6.9). This yields
( )2 ∣ ∣2 ∫ ∫
1 2E ∞ ∞(b) ≈ − γ
∣λc ∣ mb
F11 (Eγ, µ) ∣ ∣ dω1 dω2 g11(−p+, ω1, ω2, µ) , (6.24)648 m ∣b λ ∣t m4c −∞ −∞
where the remaining double integral over the soft function scales like Λ3QCD. For any
reasonable value of this quantity, the prefactor 1/648 and the additional 1/mb suppression
render this contribution negligible. For instance, if one models the double integral by
Λ411 S(−p+, µ) with some hadronic scale Λ11 ∼ ΛQCD, the contribution of this term relative
to the leading direct photon contribution in (4.40) is approximately given by
∣ ∣2 ( )4 ( )4
1 ∣− C1(µ)
∣ Λ11
∣ ∣ ≈ − Λ112 · 10−4 . (6.25)
648 ∣C ∣7γ(µ) mc 0.5GeV
6.3 Analysis of the Q8g −Q8g Contribution
As it will turn out, this contribution is more subtle than the remaining ones, so its calcula-
tion will be presented in more detail. The direct contribution is generated by two insertions
96
(1)
of the operator Q8g, hc quark given in (B.2). Since there is no leading power operator con-
taining a photon and descending from Q8g (at tree level in the matching) only the first
diagram of Figure 5.1 is relevant for the direct photon contribution. The same diagram is
given more suggestively in the first line of Figure 6.3. The second line of Figure 6.3 shows
hc
hc
hc hc
hc
s s
Figure 6.3: Diagrams arising from the matching of the Q8g −Q8g contribution onto SCET
(left) and HQET (right). Dashed lines denote non-localities obtained after (anti-)hard-
collinear fields have been integrated out.
the resolved contribution which is generated by two insertions of the operator (5.37) and
its associated conversion. The operator in (5.35) does not contribute to the Q8g − Q8g
interference, since it is already suppressed by λ compared to the leading power and there
is no leading power operator descending from Q8g to combine it with. Also, there is no
contribution due to a matching of Q8g onto an operator containing any soft fields at order
1/mb in the heavy-quark expansion. The right panel of Figure 6.3 shows the HQET dia-
grams obtained after the (anti-)hard-collinear fields have been integrated out in the second
matching step.
The direct photon contribution involves a convolution of the leading shape function in
(4.38) with a subleading jet function consisting of the cut of a hard-collinear loop. In the
present case the jet function is divergent and needs to be regularized. Using dimensional
regularization and subtracting its 1/ǭ pole in the MS scheme yields
( )2 ( )∫
(a) CFαs(µ) m
Λ̄
b 2 mb(ω + p+) 1 4
F88 (Eγ, µ) = dω ln + − cRS S(ω, µ) . (6.26)4π 2E 2γ −p 9 µ 9 9+
The scheme-dependent constant cRS vanishes in the MS scheme, cMS = 0. If, on the other
hand, the dimensional reduction scheme is adopted, in which the Dirac algebra is performed
97
in 4 rather than d = 4− 2ǫ dimensions, then cDR = 1. It will be shown later that the final
answer for F88 is scheme independent.
The double resolved photon contribution gives rise to a more complicated structure, as
the resulting soft matrix element contains four quark fields located at different space-time
points. One finds
( )2∫ Λ̄ ∫ ∞ ∫
(b) 8 mb dω
∞
1 dω2
F88 (Eγ, µ) = παs(µ) dω δ(ω+p ) g
cut
+ 88 (ω, ω1, ω2, µ),9 2Eγ −∞ −∞ ω1 + iε −∞ ω2 − iε
(6.27)
where the subleading shape function has been defined by
∫ ∫ ∫ ∫
∑
gcut
dr du dt
88 (ω, ω1, ω2, µ) = e
−iω1r eiω2u e−iωt
2π 2π 2π
Xs
( ) ( ) ( ) ( ) ( ) ( )
× 〈B̄| h̄Sn (tn)T
A S†S (tn) Γ S†n n̄ n̄ n̄s (tn+ un̄)|X †s〉 〈Xs| s̄Sn̄ (rn̄)Γn̄ Sn̄Sn (0)TA S†nh (0)|B̄〉
2MB
∫ ∫ ∫
dr
= e−iω r
du
1 eiω2u
dt
e−iωt
2π 2π 2π
( ) ( ) ( ) ( ) ( ) ( )
× 〈B̄| h̄Sn (tn)T
A S†S †n n̄ (tn) Γn̄ Sn̄s (tn+ un̄) s̄Sn̄ (rn̄)Γ
†
n̄ Sn̄Sn (0)T
A S†nh (0)|B̄〉
.
2MB
(6.28)
As before, the Wilson lines render the soft matrix element gauge invariant. The sum over
soft intermediate states Xs with strangeness S = −1 in the first equation arises since in
this particular case the hard-collinear jet does not contain the strange-quark. Note that
only color-octet partonic states contribute to the sum, not physical hadronic ones. As was
already argued in Section 3.1 a complete sum can be performed, since the operators ensure
the correct quantum number for the intermediate states. This gives rise to the second
equation, in which the two strange-quark fields are not time ordered but appear in the
order shown in the formula. In the definition above
n̄/n/ n/n̄/
Γn̄ = (1 + γ5) , Γn̄ = (1− γ5) (6.29)
4 4
are projectors onto two-component light-quark spinors. In deriving the result (6.27) the
Dirac structure has been simplified using the identity [115]
α n/n̄/ µ n̄/n/ 2 n/n̄/ n̄/n/γ⊥ γ⊥ ⊗ γ⊥µ γ⊥α = (d− 2) ⊗ , (6.30)4 4 4 4
where d is the number of space-time dimensions.
According to the discussion of Section 6.1, it follows that gcut88 has support for −∞ <
ω ≤ Λ̄ and −∞ < ω1,2 < ∞. Note the difference in the sign of the iε terms in the
two anti-hard-collinear propagators in (6.27), which is due to the fact that the anti-hard-
collinear fields connected to the right weak vertex in the diagrams are anti-timed-ordered.
Consequently, after convolution with the jet functions the position variables r and u in
98
(6.28) are restricted to positive values, such that the fields in the soft 〈Xs| . . . |B̄〉 matrix
elements are time ordered, while those in the 〈B̄| . . . |Xs〉 matrix elements are anti-time-
ordered, as it should be. Finally, the second equality in (6.28) implies the relation
[ ]∗
gcut cut88 (ω, ω1, ω2, µ) = g88 (ω, ω2, ω1, µ) , (6.31)
and since the convolution in (6.27) is symmetric in ω1 and ω2 up to complex conjugation,
it follows that the final result is real. Contrary to (6.14) there is no useful normalization
condition for the soft function gcut88 .
For phenomenological purposes, it will be convenient to define a new, real function
∫
2 ∞
∫
dω ∞1 dω2
f (ω, µ) = gcut88 − 88 (ω, ω1, ω2, µ) , (6.32)9 −∞ ω1 + iε −∞ ω2 iε
in terms of which the second contribution to the photon spectrum is simply
( )2
(b) mb
F88 (Eγ, µ) = 4παs(µ) f88(−p+, µ) . (6.33)2Eγ
Note that the poles at ω1 = 0 and ω2 = 0 are regularized by the iε prescriptions, in analogy
with (6.11).
One comment is in order concerning the structure of this result in light of the general
factorization formula (5.32). The present case is the only example of a double resolved
photon contribution at order 1/mb. The jet function for the hard-collinear gluon is given
by the cut of the gluon propagator, which up to trivial prefactors yields J(p2) = δ(p2) with
p2 = mb(ω + p+), in analogy with the tree-level expression for the quark jet function of
(4.43). The jet functions for the two anti-hard-collinear quark propagators are, up to a
trivial numerator factor, given by J̄(p2) = 1/(p2 + iε), where p2 = 2Eγ ω1,2 in the present
case. Hence, the triple convolution can be recast in the form (omitting scale dependences
for brevity)
∫ ∫ ∫
dω1 dω2
dω δ(p+ + ω)
cut
− g88 (ω, ω1, ω2)ω1 + iε ω2 iε
∫ ∫ ∫
( ) [ ]∗
= H mb dω J mb(p+ + ω) 2Eγ dω1 J̄(2Eγ ω1) 2Eγ dω2 J̄(2E ω ) g
cut
γ 2 88 (ω, ω1, ω2),
(6.34)
with the hard matching coefficient corresponding at tree level to the product of Wilson
coefficients |C8g|2 in (5.39). This result is in agreement with the factorization formula
(5.32). Even though all seems well with the above formula, the presentation has hidden an
important subtlety. The next section will deal with the renormalization properties of the
individual quantities in the factorization formula.
99
6.3.1 Q8g −Q8g: Details about the Factorization
A priori, each of the quantities in Equation (6.34) is scale dependent. However, the scale
dependence should cancel (up to terms of order α2s) in the sum of the two contributions
(6.26) and (6.27). This is in correspondence to the cancellation of divergences between the
hard-collinear and soft region in a method of regions analysis of Q8g −Q8g in full QCD. In
order for this cancellation to happen the convolution (6.32) must contain a µ-dependent
term at zeroth order in the strong coupling. This fact is incompatible with a multiplicative
renormalization of SCET operators as given in Equation (4.46). The situation can be
contrasted to the Q7γ −Q7γ case in Section 5.2.5, whose matrix element turned out to be
less singular. The resolution of this puzzle is that the additional convolution integrals over
the soft function themselves are not convergent. In order to demonstrate this, one calculates
the asymptotic behavior of the soft function for large values ω1,2 ≫ ΛQCD, corresponding
to highly energetic light quarks. This behavior can be extracted using short-distance
methods [1]. At leading order in perturbation theory, one just has to replace the light-
quark fields in the definition (6.28) by a cut propagator and perform some phase-space
integrations. Working in d = 4− 2ǫ dimensions leads to
∣ 1−ǫ ∫ Λ̄
gcut
CF θ(ω∣ 1)ω −ǫ
88 (ω, ω1, ω
1 ′ ′ ′
2, µ) =∣ 2−ǫ − δ(ω1−ω2) dω S(ω , µ) (ω − ω) +. . . .ω1,2≫ΛQCD (4π) Γ(1 ǫ) ω
(6.35)
Corrections to this result are suppressed by powers of αs or ΛQCD/ω1,2. The limit ǫ → 0
is smooth and gives rise to a dependence gcut88 ∝ ω1 δ(ω1 − ω2). It is then obvious that the
double convolution integral in (6.32) is logarithmically divergent in the UV region.3 When
the convolution is understood in the usual sense as an integral over renormalized functions,
then this divergence is not regularized.
On the other hand, the explicit expression (6.35) shows that the convolution integral
would be regularized by the dimensional regulator if the limit ǫ → 0 was taken after the con-
volutions have been evaluated. In that case one obtains a 1/ǫ pole from the UV-divergent
convolution integral, which needs to be subtracted in the MS scheme. Thus the following
procedure will be applied: a hard cutoff ΛUV is introduced and the convolution integral
is split up in a low-momentum region defined by ω1, ω2 < ΛUV and a high-momentum
region defined by the complement. In the high-momentum region it is possible to replace
the soft function by the perturbative expression (6.35) up to higher-order terms in αs and
power-suppressed contributions. Then the high-momentum contribution to the double con-
volution integral is evaluated before taking the limit ǫ → 0. In doing so, it is important
to reinstate a factor (1− ǫ)2 from the Dirac algebra, see (6.30), and a factor µ2ǫ from the
conversion of the bare coupling constant g2 into the renormalized coupling 4παs(µ). This
3Note that according to (6.35) there are no UV divergences from the region of large negative values of
ω1,2. In this region the convolution integrals are cut off by non-perturbative dynamics.
100
leads to
[
∫ ∞ ]∫2 − 2 2ǫ dω
∞
1 dω2
f88(ω, µ) = (1 ǫ) µ − g
cut,bare
88 (ω, ω1, ω2)9 −∞ ω1 + iε −∞ ω2 iε MS subtracted
∫ ∫
2 ΛUV dω ΛUV1 dω2
= cut− g88 (ω, ω1, ω2, µ) (6.36)9 −∞ ω1 + iε −∞ ω2 iε
∫
C Λ̄
( ′ )
− F ′ ′ ΛUV(ω − ω)dω S(ω , µ) ln + 2− 2cRS ,
72π2 µ2ω
where cRS is the same scheme-dependent constant as in (6.26). This expression is indepen-
dent of the auxiliary scale ΛUV, which for consistency should be taken to be several times
ΛQCD, so that perturbation theory can be trusted. In the above result the dependence on
the factorization scale of dimensional regularization is explicit, and it is now evident that
the sum of the two contributions (6.26) and (6.27) is both scale and scheme independent.
Indeed, one finds
( )2 ( )∫
C Λ̄Fαs(µ) mb 2 mb 1
F88(Eγ, µ) = ln − dω S(ω, µ)
4π 2Eγ 9 ΛUV 3 −p+ (6.37)
( )2 ∫ Λ ∫8 m UV dω ΛUVb 1 dω2
+ πα (µ) gcuts − 88 (−p+, ω1, ω2, µ) .9 2Eγ −∞ ω1 + iε −∞ ω2 iε
The large logarithm ln(mb/ΛUV) in the first term results from the ratio of the hard-collinear
scale mb(ω + p+) in (6.26) and the soft (yet perturbative) scale ΛUV(ω + p+) contained
in the function f88(−p+, µ) in (6.33). Resumming these large logarithms would require
solving evolution equations in the effective theory. In the case of UV-divergent convolution
integrals, the derivation of such equations is an open problem.
The problem of divergent convolution integrals in SCET has been encountered pre-
viously in the context of heavy-to-light form factors [88, 115–117] and power-suppressed
contributions to hadronic B-meson decays [35,118]. It is to some extent still an open ques-
tion whether these integrals indicate a failure of factorization, or whether they can be cured
by a generalization of the theoretical framework of SCET (an attempt in this direction was
initiated in [119]). An important difference is that in all previous cases these divergences
were of IR origin. In the current case, however, the convolution integrals diverge in the
UV. Such divergences appear to be rather generic in the description of higher-order power
corrections, because the resulting convolution integrals contain higher powers of soft mo-
mentum variables. Still, the presence of this effect casts doubt on the factorization proof
in Section 5.3. It is not obvious that the treatment discussed in this section will work
at higher orders in perturbation theory and allow for a consistent resummation of large
logarithms. In that sense the derivation of factorization is incomplete.
Note that the result (6.37) is insensitive to the mass of the strange-quark, as it should
be. This is in contrast with the parton-model expression derived in [64] and shown in (2.29).
The IR regulator ms introduced in the parton-model calculation is replaced in real QCD
by a subleading shape function, i.e., by a hadronic matrix element of a non-local operator.
101
One recovers the parton-model expression for F88 if one calculates the soft matrix element
in perturbation theory, i.e., if one assumes the validity of (6.35) also at small values of ω1,2
and introducesms as an IR regulator, which replaces ω
−ǫ(ω′−ω)−ǫ1 → [ω1(ω′−ω)−m2]−ǫs in
this formula. Of course, such a treatment can not be justified due to the non-perturbative
nature of QCD at low energies.
It was argued in [104] that the IR-sensitive terms in the Q8g − Q8g contribution to
the B̄ → Xsγ photon spectrum can be absorbed into photon fragmentation functions of a
strange-quark or gluon. While no formal proof of this assertion was given in that paper,
it is likely to be true in the kinematic region away from the endpoint, where the splitting
processes s → γ + s and g → γ + g can be treated using the collinear approximation.
In the language of SCET this means that the partons after the splitting are still anti-
hard-collinear fields, and hence the photon energy can not be near the endpoint. In the
endpoint region, on the other hand, these partons are soft, and they do not factorize from
the remaining soft matrix element. Hence, in this region the non-perturbative physics
is encoded in a complicated subleading four-quark shape function rather than a simpler
fragmentation function.
6.4 Analysis of the Q7γ −Q8g Contribution
Evaluating the diagrams in Figure 5.1 for this operator pair, yields for the direct photon
contribution
∫
(a) CFαs(µ) mb 10
Λ̄
F78 (Eγ, µ) = dω S(ω, µ) , (6.38)4π 2Eγ 3 −p+
which generalizes the parton-model result (2.29) in the endpoint region. The case of the
Q7γ−Q8g interference term is special in that, even though the parton-model expression does
not indicate any problematic feature that would call for non-trivial soft contributions, there
actually do exist some O(1/mb) effects that are described by subleading shape functions.
Moreover, these effects remain non-local even for the total decay rate [14] (see also [105]).
In order to study the resolved photon contributions, either one of the two SCET op-
erators in (5.35) and (5.37) arising from the matching relation for Q8g is combined with
the leading-order operator in (4.27) descending from Q7γ . In both cases, the conversion
of the anti-hard-collinear fields gives rise to one or more soft quark fields. The relevant
SCET diagrams are depicted in the left panels in Figure 6.4, while the corresponding soft
graphs resulting after the second matching step are shown in the right panels. In the first
case, the second soft quark is generated by an insertion of a subleading term of the SCET
Lagrangian. Evaluating the first contribution in detail yields
∫ ∫ ∫
(b) 16 m
Λ̄ ∞
b dω
∞
1 dω2
F78 (Eγ, µ) = παs(µ) Re dω δ(ω + p+)3 2Eγ −∞ −∞ ω1 + iε −∞ ω2 − iε (6.39)
[ ]
× ḡ78(ω, ω cut1, ω2, µ)− ḡ78 (ω, ω1, ω2, µ) .
The soft function ḡ78 arises when the hard-collinear strange-quark line in the left diagram
in the first row of Figure 6.4 is cut, while the function ḡcut78 originates from the cut through
102
hc
hc hc
s s
hc
s s
Figure 6.4: Diagrams arising from the matching of the Q7γ −Q8g contribution onto SCET
(left) and HQET (right). Dashed lines denote non-localities obtained after (anti-)hard-
collinear fields have been integrated out.
the hard-collinear gluon line. In this latter case the soft strange-quark line must also be
cut. Specifically, the corresponding subleading shape functions are defined as
∫ ∫ ∫
dr −iω r du iω u dtḡ (ω, ω , ω , µ) = e 1 e 2 e−iωt78 1 2
2π 2π 2π
( ) ( ) ( ) ( ) ( )
× 〈B̄|
(6.40)
h̄S A †n (tn)T Γn Sns (un) s̄Sn̄ (rn̄) Γ S
† A †
n̄ n̄Sn (0)T Snh (0)|B̄〉 ,
2MB
and
∫ ∫ ∫ ∫
dr du dt ∑
ḡcut(ω, ω , ω , µ) = e−iω1r iω2u −iωt78 1 2 e e2π 2π 2π
Xs
( ) ( ) ( ) ( ) ( )
×〈B̄| h̄Sn (tn)T
A Γ †n Sns ((t+ u)n)|Xs〉 〈Xs| s̄Sn̄ (rn̄) Γn̄ S†n̄Sn (0)TA S†nh (0)|B̄〉
2MB
∫ ∫ ∫
dr −iω r du dt= e 1 eiω2u e−iωt (6.41)
2π 2π 2π
( ) ( ) ( ) ( ) ( )
×〈B̄| h̄Sn (tn)T
A Γ †n Sns ((t+ u)n) s̄Sn̄ (rn̄) Γn̄ S
†
n̄Sn (0)T
A S†nh (0)|B̄〉 ,
2MB
where Γn̄ was introduced in (6.29), and Γn is defined in the same way as Γn̄ but with n and
n̄ interchanged. One half of the contribution shown in (6.39), but without the real part
103
prescription, arises from the original diagrams, while the mirror diagrams not shown in
the figure give the complex conjugate of the above expressions. The two results combined
give a real result, as indicated in (6.39). Note that there is no need to insert a time-
ordering symbol in front of the light-quark fields in the definition of ḡ78 in (6.40), since
after convolution with the jet functions the integration variables r and u are restricted to
take positive values, and hence the light-quark fields have a space-like separation. In the
second equation in (6.42), on the other hand, the fields are not time ordered because the
non-local four-fermion operator arises upon performing a sum over intermediate states, as
shown in the first equation.
Consider next the contribution shown in the second row of Figure 6.4. In this case the
soft light-quark pair can carry any flavor. This yields
∫ Λ̄ ∫ ∞ ∫ ∞
(c) mb 1
F78 (Eγ, µ) = 4παs(µ) Re dω δ(ω + p+) dω1 dω2 (6.42)2Eγ −∞ −∞ −∞ ω1 − ω2 + iε
[( ) ( ) ]
× 1 1 (1) 1 1 (5)+ − g78 (ω, ω1, ω2, µ)− − − g78 (ω, ω1, ω2, µ) ,ω1 + iε ω2 iε ω1 + iε ω2 iε
where the subleading shape functions have been defined by
∫ ∫ ∫
(1) dr du dt
g (ω, ω , ω , µ) = e−iω1r eiω2u e−iωt78 1 2 2π 2π 2π
( ) ( ) ( ) ( ) ( )
× 〈
∑
B̄| h̄Sn (tn) S†S An n̄ (0)T n̄/(1 + γ ) S†5 n̄h (0)T q e q̄S A †q n̄ (rn̄) n̄/ T Sn̄q (un̄)|B̄〉 ,
2MB
∫ ∫ ∫
(5) dr
g (ω, ω , ω , µ) = e−iω
du dt
1r
78 1 2 e
iω2u e−iωt
2π 2π 2π
( ) ( ) ( ) ( ) ( )
× 〈
∑
B̄| h̄Sn (tn) S†nSn̄ (0)TA n̄/(1 + γ5) S†n̄h (0)T q eq q̄Sn̄ (rn̄) n̄/γ A †5 T Sn̄q (un̄)|B̄〉 ,
2MB
(6.43)
where the sum extends over light quark flavors (q = u, d, s), and eq denote the quark electric
charges in units of e. Again, one half of the contribution shown in (6.42), but without the
real part prescription, arises from the original diagrams, while the mirror diagrams not
shown in the figure give the complex conjugate of the above expressions.
In these definitions the light-quark fields are time-ordered, as indicated by the T sym-
bols. That this is the appropriate ordering can be seen as follows. After convolution with
the jet functions, for the terms containing the propagator 1/(ω1 + iε) in the second line of
(6.42) the integration variables r and u are restricted to the range r > u > 0. These terms
correspond to the Feynman graph shown on the left in the second row of Figure 6.4, in
which q̄(rn̄) should appear to the left of q(un̄). For the terms containing the propagator
1/(ω2 − iε) the integration variables are restricted to the range u > r > 0. These terms
correspond to the analogous Feynman graph with the opposite direction of the fermion
arrow on the light-quark line, for which q̄(rn̄) should appear to the right of q(un̄). Hence,
the proper ordering is indeed the ordering according to (light-cone) time. On the other
104
hand, arguments along the lines discussed in [120] suggest that the time-ordering prescrip-
tion is, in fact, not required for forward matrix elements and fields at light-like separation.
In what follows it will be assumed that the T symbol can be dropped in (6.43).
Very little is known about the complicated four-quark shape-functions defined in (6.40),
(6.42), and (6.43). By the general arguments presented in Section 6.1 it follows that the
(1,5)
soft functions ḡ78 and g78 have support for −∞ < ω ≤ Λ̄ and −∞ < ω1,2 < ∞. However,
in the case of ḡcut78 it is necessary that ω1,2 > 0. Note also the symmetry property
∫ Λ̄ [ ]∗ ∫ Λ̄
(1,5) (1,5)
dω g78 (ω, ω1, ω2, µ) = dω g78 (ω, ω2, ω1, µ) , (6.44)
−∞ −∞
which follows from the definitions of the soft functions in (6.43). While nothing specific
could be said about the subleading shape function in the Q8g −Q8g interference, the situ-
ation is slightly improved in the Q7γ −Q8g case.
6.4.1 Q7γ −Q8g in the Vacuum Insertion Approximation
(1,5)
The fact that in the case of g78 the operators involve light quarks of all flavors offers a
strategy for modeling their matrix elements between B-meson states. Unlike the case of the
four-quark operators encountered for gcut, ḡ , and ḡcut88 78 78 , here it is possible to (very roughly)
estimate the matrix element by inserting the vacuum intermediate state between the two
light-quark fields. The “vacuum-insertion approximation” (VIA) is used extensively in the
study of local four-quark operator matrix elements, and a priori there is no reason why it
should work less accurately for non-local operators. Following [14] leads to
∫ Λ̄ ( )∣ 2
(1,5) ∣
dω g78 (ω, ω1, ω2, µ) = −
F (µ)
e 1− 1 φBspec +(−ω1, µ)φB∣ 8 N2 +(−ω2, µ) , (6.45)−∞ VIA c
where espec denotes the charge of the spectator quark inside the B meson, i.e., espec =
2/3 for B±, and espec = −1/3 for B0 and B̄0. The quantity F√(µ) is the HQET matrix
element corresponding to the asymptotic value of the product fB MB in the heavy-quark
limit [121]. Finally, φB+(ω, µ) is the leading light-cone distribution amplitude of the B
meson [122]. It is a real function with support for ω > 0, which vanishes at ω = 0 and
asymptotically falls off like 1/ω modulo logarithms [123]. Useful forms for this function
have been derived based on QCD sum rules [122,124,125], the relativistic quark model [126],
and model-independent moment relations obtained using the operator-product expansion
[123, 127]. The support of φB+(ω, µ) implies that only negative values of ω1 and ω2 give
rise to non-zero contributions in (6.45), which is in accordance with the fact that q̄ (q)
describes an anti-quark in the initial (final) state.
105
To conclude this analysis, the following phenomenological functions are defined
∫ ∞ ∫ ∞
(I) 4 dω1 dω2 [ ]
f78 (ω, µ) =
cut
3 −∞ ω1 + iε −∞ ω2 −
ḡ78(ω, ω1, ω2, µ)− ḡ78 (ω, ω1, ω2, µ) ,iε
∫ ∞ ∫ ∞
(II) 1
f78 (ω, µ) = dω1 dω2 (6.46)
−∞ −∞ ω1 − ω2 + iε
[( ) ( ) ]
× 1 1 (1) 1 1 (5)+ − g78 (ω, ω1, ω2, µ)− − − g78 (ω, ω1, ω2, µ) .ω1 + iε ω2 iε ω1 + iε ω2 iε
In the VIA, one finds
{ }
[ ]
Λ̄ ( )∫ 2 ∫∣ F (µ) 1 1 ∞ φB
2
(II) ∣
dω f (ω, µ) = −e 1− + 2πi dω +(ω, µ)78 ∣ spec ,2
−∞ VIA 8 N
2
c λB(µ) 0 ω
( )
∫ Λ̄ ∣ 2
(II) ∣
dωRe f78 (ω, µ) = −
F (µ) − 1 1espec 1 , (6.47)∣ 2 2
−∞ VIA 8 Nc λB(µ)
∫∞
where λ = dω φBB +(ω, µ)/ω denotes the first inverse moment of the B-meson light-cone0
(I,II)
distribution amplitude [35]. In terms of the functions f78 , the direct and resolved photon
contributions to the B̄ → Xsγ photon spectrum can be summarized as
∫
CFαs(µ) m 10
Λ̄
b
F78(Eγ, µ) = dω S(ω, µ)
4π 2Eγ 3 −p+ (6.48)
m [ ]b (I) (II)
+ 4παs(µ) Re f
2E 78
(−p+, µ) + f78 (−p+, µ) .
γ
This concludes the analysis of each of the subleading contributions to the B̄ → Xsγ
differential decay rate in the endpoint region. Some information about the new subleading
shape functions appearing in this analysis was already extracted in Section 6.1.1 and this
section. The next section will consider the properties of the subleading shape functions
under PT transformations.
6.5 Constraints from PT Invariance
In the expressions presented in the previous sections there are four potential sources of
complex phases: weak (CP -violating) phases from the CKM matrix elements and the
Wilson coefficients, and strong (CP -conserving) phases from the new jet functions J̄i and
the various subleading shape functions. The CKM phases are non-zero only for theQq1−Q7γ
contributions (with q = c, u), where they are suppressed by two powers of the Cabbibo
angle. The Wilson coefficients are real in the Standard Model, although they can be
complex in many of its extensions (see e.g. [128]).
A unique property of the resolved photon contribution is that the new jet functions J̄i
are given in terms of full propagators (dressed by Wilson lines) and not by cut propagators.
106
As a result, these functions are in general complex and give rise to strong phases. Since
√
the relevant scale of the jet functions is 2Eγ ΛQCD, which is perturbative in the endpoint
region, these strong phases are calculable in perturbation theory. The other potential
source of strong phases are the soft functions, whose phases are of a non-perturbative
nature. When studying the effects of the resolved photon contributions on the rate and
CP asymmetry in B → Xsγ decay, it is important to obtain some handle on these non-
perturbative phases. In order to do so, the invariance of strong-interaction matrix elements
under parity (P ) and time reversal (T ) is employed.
Under the combined transformation PT , a spinor field ψ(x) transforms as PTψ(x)PT =
ΛPT ψ(−x), where in the Weyl representation of the Dirac matrices ΛPT = −γ0γ1γ3 up to
an irrelevant phase factor. The soft Wilson line Sn(x) in (4.38) transforms into Sn(−x).4
Finite-length Wilson lines, as they appear in the definitions of the soft functions, transform
as PT [tn, 0]PT = PT Sn(tn)S
†
n(0)PT = Sn(−tn)S†n(0) = [−tn, 0]. Finally, the external
B-meson states transform as PT |B̄(v)〉 = −|B̄(v)〉. Also, because the time-reversal trans-
formation is anti-linear, matrix elements get complex conjugated under application of PT .
Consider now the definition of the soft function g17 in (6.4). Using the fact that the
position-space strong-interaction matrix element is PT invariant leads to
∫ ∫
dr dt
g (ω, ω , µ) = e−iω1r e−iωt17 1 (6.49)
2π 2π
( ) ( ) (
〈B̄| h̄S (−tn) n̄/(1− γ ) S†S (0) iγ⊥n̄ S†
) ( )
× n 5 n n̄ α β n̄ gG
αβ
s Sn̄ (−rn̄) S†h (0)|B̄〉∗n̄ ,
2MB
where it was used that Λ†PT n̄/ iγ
⊥ Λ = n̄/ iγ⊥α PT α and Λ
† ⊥ ⊥
PT n̄/γ5 iγα ΛPT = −n̄/γ5 iγα . However,
it was already argued after (6.15) that the term containing γ5 vanishes. Hence, PT trans-
forms the position-space matrix element into the complex conjugate of the same matrix
element with all position arguments xi replaced by −xi. By taking the complex conjugate
of relation (6.50) and reversing the sign of the integration variables r and t, it then follows
that g17(ω, ω1, µ) is a real function.
(1,5)
An analogous argument can be presented for the soft functions ḡ78 in (6.40) and g78
in (6.43), where it is important however that it is possible to avoid the time-ordering
prescription for the soft light-quark fields. The HQET trace formalism can be used to
show that in the definitions of these matrix elements only even numbers of γ5 matrices can
give rise to non-vanishing contributions, and then the Dirac structures are in all cases even
under PT . In contrast to (6.15) the shape function ḡ78 corresponds to a four-quark matrix
element. Therefore there are two possibilities for additional Dirac structures at higher
order. They can appear either between the two light quarks or between a light quark and
4Strictly speaking the lower limit of integration is also changed from −∞ to +∞, but this provides an
equally valid definition of the same object.
107
an external state. This leads to
[ ]
1 + v/ 1 + v/
ḡ78(ω, ω1, ω2, µ) = MBtr ΓA Ξ1(v, n̄) ΓB Ξ2(v, n̄)
2 2
[ ] [ ] (6.50)
1 + v/ 1 + v/
+MBtr γ5 ΓA Ξ3(v, n̄) tr Ξ4(v, n̄) ΓB γ5 ,
2 2
where ΓA = Γn and ΓB = Γn̄, and for brevity the dependence of the coefficient functions
Ξi on ω, ω1, ω2, and µ has been suppressed. A similar expression, but with different
(1,5)
matrices ΓA and ΓB, holds for g78 after a Fierz transformation. The most general Lorentz-
invariant decompositions of the functions Ξi involve products of up to four v/, n̄/, and γ
α
⊥
matrices, where all transverse indices must be contracted. No γ5 matrices appear in this
decomposition. Note also that by using the relation n + n̄ = 2v n/ can be eliminated in
favor of v/ and n̄/. With only two independent external vectors v and n̄, however, it is
impossible to saturate the four indices of an ǫαβγδ symbol, and hence only even numbers of
γ5 matrices in the product structure ΓA⊗ΓB can give rise to non-zero traces. For the case
of ḡ78 considered above, it follows that one can replace 16 Γn⊗Γn̄ → n̄/n/⊗n̄/n/−n̄/n/γ5⊗n̄/n/γ5.
(1) (5)
Both of these product structures are even under PT . In the case of g78 and g78 , it follows
similarly that n̄/(1 + γ5)⊗ n̄/ → n̄/⊗ n̄/ and n̄/(1 + γ5)⊗ n̄/γ5 → n̄/γ5 ⊗ n̄/γ5. Once again, these
(1) (5)
product structures are even under PT . Therefore the functions ḡ78, g78 , and g78 are all
real.
Finally the functions ḡcut88 in (6.28) and ḡ
cut
78 in (6.42) will be considered. They are
defined in terms of sums over intermediate states |Xs〉, which without loss of generality
can be chosen to be eigenstates of PT with eigenvalues ±1. After summing over the
polarizations of the intermediate states and integrating over their momenta, one finds that
each term in the sum over states can be written as a product of two traces, in analogy to
the second term in (6.50). The same arguments as above then show that ḡcut and ḡcut88 78 are
real.
In conclusion, all of the subleading shape functions are real. The strong phases men-
tioned in the introduction to this section thus arise only from the new jet functions J̄i.
Given that the soft functions are real, it now follows from (6.16), (6.31), and (6.44) that
∫ ∫ (1,5)
dω g17(ω, ω1) is an even function of ω1, and that g
cut
88 (ω, ω1, ω2) and dω g78 (ω, ω1, ω2)
are symmetric under the exchange of ω1 and ω2.
6.6 Conclusions of this Chapter
This chapter presented a systematic and throughout analysis of the 1/mb corrections to
the B̄ → Xsγ photon spectrum in the endpoint region. The results can be summarized by
108
giving the complete list of the quantities Fij appearing in (5.39). They are
( )
∫
C α (µ) Λ̄F s mb(ω + p+)
F77(Eγ, µ) = dω 16 ln + 9 S(ω, µ) + F
SSF
77 (Eγ, µ) ,4π −p µ
2
+
Λ̄ ( )∫CFαs(µ) 2 mb(ω + p+) 1
F88(Eγ, µ) = dω ln − S(ω, µ) + 4παs(µ) f88(−p+, µ) ,
4π 2−p 9 µ 3+
∫
CFαs(µ) 10
Λ̄ [ ]
(I) (II)
F78(Eγ, µ) = dω S(ω, µ) + 4παs(µ) Re f (−p+, µ) + f (−p+, µ) ,
4π 3 78 78−p+
( )
∫
CFαs(µ) 2
Λ̄
∑
F17(Eγ, µ) = − dω S(ω, µ) + δq Re f17,q(−p+, µ) ,
4π 3 −p+ q=c,u
∫
CFαs(µ) 2
Λ̄
F11(Eγ, µ) = F18(Eγ, µ) = dω S(ω, µ) ,
4π 9 −p+
(6.51)
with the leading shape function defined in (4.38), δq in (6.3), and F
SSF
77 in (5.17) while
the the phenomenological functions fij are convolutions of subleading shape functions
and defined in the previous sections of this chapter. These results provide important
informations for phenomenological applications. The following chapters will analyze them
in detail. In Chapter 7 the effect of the new subleading shape functions on the partially
integrated decay rate will be considered. Since these non-perturbative quantities can not
be calculated explicitly, they constitute a hadronic uncertainty in the determination of
the integrated decay rate. This is also true for an integration range much larger than
the endpoint region. In contrast to the subleading shape functions of the Q7γ − Q7γ
contribution, the new subleading shape functions do not reduce to local matrix elements
if integrated over a large range. The reason for this lies in the non-local structure of
the HQET matrix elements in the n direction. This non-locality is due to the anti-hard-
collinear propagators and is not “integrated out” by considering the total rate.
The other phenomenological application is the determination of the hadronic uncer-
tainty in the direct CP asymmetry, observed in the decays of B- and B̄-mesons. This will
be considered in Chapter 8.
109
Chapter 7
Phenomenology I:
Hadronic Uncertainties in the
Integrated Decay Rate
It has been argued in Chapter 1 that the branching fraction of B̄ → Xsγ contains useful
information concerning the determination of SM parameters, as well as for restricting the
parameter space of new physics models. For practical reasons explained in Section 1.3,
this branching fraction only includes photons with an energy above a certain cut. The
corresponding theoretical prediction, based on the parton model, was given in (2.38). In
this calculation the uncertainty due to non-perturbative effects was estimated to be about
5% [5].
The last chapter systematically identified the non-perturbative corrections to the B̄ →
Xsγ photon spectrum in the endpoint region at O(1/mb). By integrating these results over
the photon energy, it is now possible to estimate the size of the non-perturbative corrections
to the branching fraction. As it turns out, some of these corrections fall outside the realm of
the local OPE and need to be parameterized by integrals over subleading shape functions.
In Section 7.1 the partially integrated decay rate with a lower cutoff will be defined and
related to the results of the previous chapter. Section 7.2 will then define a set of hadronic
quantities that parameterize the uncertainties due to the new subleading contributions.
Subsequently, each of the relevant contributions will be considered in Sections 7.3 to 7.5.
Finally, the results will be summed up in Section 7.6 and an improved estimate of the
hadronic uncertainties in the partonic calculation of the branching fraction will be given.
7.1 Partially Integrated Decay Rate
For phenomenological purposes, it is most interesting to study the partial B̄ → Xsγ decay
rate
∫ MB/2 dΓ
Γ(E0) := dEγ , (7.1)
E dE0 γ
110
obtained by integrating the photon spectrum over a region E0 < Eγ < MB/2. Provided
that ∆ := mb − 2E0 is much larger than ΛQCD, the direct photon contributions to this
integrated rate can be calculated in terms of local operator matrix elements [13] using a
combined expansion in powers of ∆/mb and ΛQCD/∆. The latter expansion corresponds to
the moment expansion of the leading shape function, given in (3.38). In the limit E0 → 0
one obtains the total decay rate, and ∆ = mb; however, as was discussed in Section 1.3,
the rates measured experimentally are obtained with values of E0 larger than 1.7GeV, so
that ∆ < 1.25GeV.
An important feature of the resolved photon contributions studied in this work is that
they do not reduce to local operator matrix element in the limit ∆ ≫ ΛQCD. Rather, the
corresponding contributions to the integrated decay rate must still be described in terms
of matrix elements of non-local operators. This implies that the corresponding theoretical
uncertainties do not reduce significantly as the cutoff E0 is taken out of the endpoint region.
This will now be illustrated by deriving expressions for the first-order power corrections to
the integrated decay rate
[
G2Fα|VtbV ∗ 2Γ(E ) = ts| m20 b(µ)m3 |H 2b γ(µ)| [1 +O(αs)]32π4
] (7.2)
1 ∑ [
+ Re C∗
]
i (µ)Cj(µ) F̄ij(∆, µ) + . . . ,mb i≤j
valid for ∆ ≫ ΛQCD. Here mb denotes the pole mass of the b-quark. The dots represent
terms of order 1/m2b and higher, which are ignored. The integrated coefficient functions
are obtained as
∫ ∆
F̄ij(∆, µ) = dp+ Fij(Eγ, µ) , (7.3)
−Λ̄
where p+ = mb − 2Eγ . As will be explained below, with the exception of gcut88 the non-
perturbative soft functions have support for values ω = O(Λ 1QCD). In the limit ∆ ≫ ΛQCD,
the ω integrals in the definitions of the subleading shape function can then be performed
over the entire range from −∞ to Λ̄, and this leads to simplifications. However, the
integrals over the remaining ωi variables can not be simplified.
For the direct photon contributions, the integrals (5.13) will again be needed
∫ ∆ ∫ Λ̄
dp+ dω S(ω, µ) ≈ ∆ ,
−Λ̄ −p+ (7.4)
∆ Λ̄ ( )∫ ∫ mb(ω + p+) ≈ mb∆dp+ dω ln S(ω, µ) ∆ ln − 1 .2
−Λ̄ −p µ µ
2
+
It follows that the direct photon terms contribute to (7.2) at order ∆/mb in power counting
and can be computed using a local operator-product expansion. In the formal limit E0 → 0
these terms are promoted to O(1) contributions.
1Radiative tails of these functions, which can exhibit power behavior and extend to larger ω values will
be ignored. These effects only contribute at higher orders in αs.
111
Next the subleading shape-function contributions to the integrated decay rate will be
discussed. For the operator pair Q7γ − Q7γ , the subleading shape-function contributions
also reduce to matrix elements of local operators, as discussed in detail in [9, 10, 100, 107].
From (5.24) it then follows that
CFαs(µ)
( m )b
F̄77(∆, µ) = ∆ 16 ln + 1 . (7.5)
4π ∆
Note that, at order 1/mb, the non-zero strange-quark mass effect discussed in Section 5.2.6
integrates to zero in the partially integrated decay rate (at tree level in αs), as long as
∆ ≫ ΛQCD. Similarly, for the operator pairs Qq1 − Qq1 and Qq1 − Q8g, only direct photon
contributions contribute at order 1/mb, and one obtains
CFαs(µ) 2
F̄11(∆, µ) = F̄18(∆, µ) = ∆ . (7.6)
4π 9
The remaining contributions, all of which contain resolved photon terms, are more
interesting. For the operator pairs Qq1 −Q7γ , one obtains
( ) ∞ [ ( )]∫C α 2F s(µ) 2 2 dω1 mc − iεF̄17(∆, µ) = − ∆+ (1−δu) Re 1− F h17(ω1, µ) ,
4π 3 3 −∞ ω1 + iε mb ω1
(7.7)
where
∫ Λ̄ ∫ Λ̄
h17(ω1, µ) = dω g17(ω, ω1, µ) ≈ dω g17(ω, ω1, µ)
−∆ −∞ (7.8)
( ) ( ) ( )
∫
dr 〈B̄| h̄Sn̄ (0) n̄/ iγ⊥n̄ †β Sn̄ gGαβSn̄ (rn̄) S†n̄h (0)|B̄〉
= e−iω1r α s .
2π 2MB
The integral over p+ in (7.3) eliminates the δ(ω+ p+) distribution in (6.2), and integrating
the soft function g17(ω, ω1, µ) in (6.4) over ω then eliminates the t-integral and sets t = 0,
so that part of the non-localities of the operator are eliminated. However, the gluon field
is still smeared out on the n̄ light-cone. Note that there is no contribution from the up-
quark penguin loop to the integrated rate. As noted in Section 6.5, the integral over ω of
(b)
g17(ω, ω1, µ) is symmetric in ω1, so that the integral over F17,u in (6.8) vanishes.
2
The case of the Q7γ −Q8g interference leads to
∫ ∞ ∫CFαs(µ) 10 dω
∞
1 dω2 (5)
F̄78(∆, µ) = ∆ + 4παs(µ) Re − h78 (ω1, ω2, µ) , (7.9)4π 3 −∞ ω1 + iε −∞ ω2 iε
2It has been pointed out in the past that up-quark penguin loops might give rise to an O(ΛQCD/mb)
uncertainty in the integrated rate for B̄ → Xdγ decay [111], where unlike in B̄ → Xsγ they are not CKM
suppressed. Applying the present analysis to B̄ → Xdγ shows that this contribution actually vanishes,
removing that source of uncertainty in the integrated decay rate. Note that the same is not true for the
CP asymmetry in B̄ → Xdγ decay, where the corresponding contribution is proportional to h17(0), which
is non-zero in general.
112
where in analogy with (7.8) it has been defined
∫ ∫
(5) dr
h (ω , ω , µ) = e−iω1r
du iω2u
78 1 2 e (7.10)2π 2π
( ) ( ) ( ) ( )
×〈B̄|
∑
h̄S A †n̄ (0)T n̄/γ5 Sn̄h (0) q eq q̄Sn̄ (rn̄) n̄/γ5 T
A S†n̄q (un̄)|B̄〉
.
2MB
(1)
Note that the contribution from g78 vanishes, since the integral over ω of this function is
symmetric under the exchange of ω1 and ω2. Likewise, the contributions from the functions
ḡ78 and ḡ
cut
78 to the integrated decay rate cancel each other. This follows from the fact that
the two non-local operators in (6.40) and (6.42) coincide for t = 0. In the VIA one finds
( )
∣ C α (µ) 10 2
∣ F s
F̄78(∆, µ) = ∆−
παs(µ)
espec 1−
1 F (µ)
. (7.11)
∣
VIA 4π 3 2 N2c λ
2
B(µ)
The second term coincides with the result derived first in [14].
Finally, for the case of the operator pair Q8g −Q8g, it follows from (6.37)
( )
CFαs(µ) 2 mb 1
F̄88(∆, µ) = ln − ∆
4π 9 ΛUV 3
Λ Λ (7.12)∫ ∫8 UV dω UV1 dω2
+ πα cuts(µ) − h88 (∆, ω1, ω2, µ) ,9 −∞ ω1 + iε −∞ ω2 iε
where
∫ Λ̄
hcut88 (∆, ω1, ω2, µ) = dω g
cut
88 (ω, ω1, ω2, µ) . (7.13)
−∆
Naively, one would expect that for ∆ ≫ ΛQCD this function becomes independent of ∆
and reduces to the expression
∫ ∫
? dr du
hcut(ω , ω , µ) = e−iω1r iω2u88 1 2 e (7.14)2π 2π
( ) ( ) ( ) ( ) ( ) ( )
×〈B̄| h̄Sn (0)T
A S†nS (0) Γ S
† † A †
n̄ n̄ n̄s (un̄) s̄Sn̄ (rn̄)Γn̄ Sn̄Sn (0)T Snh (0)|B̄〉 ,
2MB
in which case the second term in (7.12) would be strictly positive. However, in the present
case the limit t → 0 in (6.28) is singular, since then the separation between the two light-
quark fields s and s̄ becomes light-like. As a result, the integral over ω in (7.13) diverges
linearly as ∆ is raised to infinity, and hence it must be evaluated at large but finite ∆.
In other words, unlike for the other soft functions, the support of the function gcut88 is not
restricted to values ω = O(ΛQCD) but extends to large negative values of ω. This is in
accordance with the asymptotic behavior derived in (6.35).
The convolutions of the soft functions with anti-hard-collinear jet functions in the re-
sults given above can not be expressed in terms of local operator matrix elements, but
rather define unknown hadronic parameters of order ΛQCD. These are the sources of gen-
uine, first-order power corrections to the integrated decay rate, which are not reduced by
lowering the cutoff E0 on the photon energy. In the following sections the size of these
hadronic effects will be estimated.
113
7.2 Hadronic Uncertainties
The results of the previous sections can be used to quantify the effect of the resolved photon
terms on the B̄ → Xsγ photon spectrum. In this section the decay rate integrated over a
sufficiently wide energy range will be considered.
In order to estimate the irreducible theoretical uncertainty from these new non-local
effects on the integrated decay rate, it is useful to define the function
F Γ(E0)− Γ(E0)|OPEE(∆) = | (7.15)Γ(E0) OPE
where E0 is the lower cutoff on the photon energy, and ∆ = mb − 2E0. This definition is
such that the true decay rate Γ(E0) is obtained from the theoretical expression Γ(E0)|OPE
obtained using a local operator product expansion by multiplying it with [1 + FE(∆)].
Note that Γ(E0)|OPE refers to the formula used in previous calculations of the B̄ → Xsγ
rate, see e.g. [5]. The function FE(∆) corresponds to the relative theoretical error made in
these calculations due to the neglect of non-local power corrections from resolved photon
contributions. Note that the non-perturbative effect from (6.20) is already included in the
OPE result.
At order 1/mb this yields
{
[ ]
1 ( )FE(∆) = F̄77(∆, µ)−
CFαs(µ) mb
∆ 16 ln + 1
mb 4π ∆
[ ]
C1(µ) CFαs(µ) 2 mbλ2
+ F̄17(∆, µ) + ∆ +
C7γ(µ) 4π 3 9m2c
[ ] (7.16)
C8g(µ) CFαs(µ) 10
+ F̄78(∆, µ)− ∆
C7γ(µ) 4π 3
}
( )2 [ ( )]
C8g(µ) CFαs(µ) 2 mb∆ 5
+ F̄88(∆, µ)− ∆ ln − + . . . ,
C7γ(µ) 4π 9 m2s 9
where it was assumed that the Wilson coefficients are real (like in the Standard Model)
and effects proportional to Vub have been neglected. Note that the terms in the first line
on the right-hand side vanish due to the relation (7.5). The various other contributions
can be expressed in terms of suitably defined hadronic parameters of order ΛQCD, using
the expressions for the quantities F̄ij(∆, µ) derived in the previous section under the as-
sumption that ∆ ≫ ΛQCD. Making explicit the dependence on the Wilson coefficients and
factors of the strong coupling g2 = 4παs leads to
C 2 specF 1(µ) Λ17(m(∆) = c/mb, µ) C8g(µ) Λ (µ)E + 4παs(µ) 78
C7γ(µ) mb C7γ(µ) mb
(7.17)
( )2 [ ]
C8g(µ) Λ88(∆, µ)
+ 4παs(µ) −
CFαs(µ) ∆ ∆
ln + . . . ,
C7γ(µ) mb 9π mb ms
114
where
2 ∫(m ) ∞
[ ( ) ]
c dω
2
1
Λ17 , µ = ecRe 1−
mc − iε mb ω1F + h
2 17
(ω1, µ) ,
mb −∞ ω1 mb ω1 12mc
∫ ∞ ∫ ∞
Λspec
dω1 dω2 (5)
78 (µ) = Re h (ω1, ω2, µ) ,
−∞ ω1 + iε −∞ ω2 − iε 78
[ Λ ( )]∫ ∫UV dω ΛUV1 dω2 CF ΛUV
Λ 2 cut88(∆, µ) = es − 2h88 (∆, ω1, ω2, µ)− ∆ ln − 1 .2−∞ ω1 + iε −∞ ω2 iε 8π ∆
(7.18)
In the case of Λ17 and Λ88 the appropriate powers of the quark electric charges have been
factored out. Because of the sum over light-quark flavors in (7.11), the parameter Λspec78
receives contributions proportional to any one of the light-quark charges. The resulting
hard breaking of isospin symmetry implies that its value will be different for charged and
neutral B mesons, even in the limit of exact isospin symmetry of the strong interaction. It
will be show in Section 7.4 that, in certain approximation schemes, Λspec78 is proportional
to the electric charge of the spectator quark in the B meson.
Note that the parameters m2c/mb and ∆ entering the arguments of Λ17 and Λ88 count
as O(ΛQCD). The dependence on the strange-quark mass in (7.17) arises only because the
function FE(∆) is defined as the deviation from the partonic rate Γpart(E0). The true decay
rate Γ(E0) in (7.15) is independent of ms. Note also that the result for Λ88 is formally
independent of the UV cutoff ΛUV, and that it is the only hadronic parameter in (7.17)
that depends on the quantity ∆. In the formal limit where the cut on the photon energy
is removed, ∆ → mb, the linear growth (modulo logarithms) of the parameter Λ88 with ∆
implies that the corresponding contribution to FE(∆) is promoted from a power-suppressed
to a leading-order effect. Indeed, it is well known that in this limit there exists a leading-
power, non-perturbative Q8g −Q8g contribution related to the photon fragmentation off a
strange-quark or gluon [104]. For practical applications this observation is irrelevant. It
will be argued in Section 7.5 that, for realistic values of E0 outside the endpoint region, the
dependence of Λ88 on ∆ is very weak, and therefore the function FE(∆) is almost equal to
a constant.
Without further information about the soft functions, the Λij parameters are expected
to be of order ΛQCD apart from the electric charges factored out in (7.18). This would lead
to very large effects of up to 30% on the decay rate. Fortunately, it is possible to constrain
the values of Λ17 and Λ
spec
78 by means of simple considerations, as will now be discussed.
The input parameters used for the estimates in the following discussion are collected in
Appendix A.2. The accuracy of the calculations is such that the scale dependence of the
subleading soft functions and the corresponding hadronic parameters is irrelevant. Even
though their µ dependence is indicated in the formulae given above, to properly control
this dependence would require to extend the calculations to the next order in the expansion
in powers of αs(µ).
115
7.3 Analysis of the Qc1 −Q7γ Contribution
In order to obtain a reasonable estimate for the parameter Λ17, first everything known about
the function h17(ω1, µ) defined below (7.8) will be collected. As proven in Section 6.5, this
function must be real, and the symmetry relation (6.16) then implies that it is an even
function of ω1. It follows that all odd moments of h17 vanish. Moreover, from (6.14) the
normalization of h17 is fixed to 2λ2. About the higher even moments nothing definite is
known, but one can expect them to be proportional to an appropriate power of ΛQCD
times a not too large numerical factor. Finally, as a soft function, h17 should not have any
significant structures, such as peaks or zeros, outside the hadronic energy range.
The first functions that come to mind are an exponential and a Gaussian,
λ2 |ω1| 2λ ω
2
2 1
h17(ω1, µ) = e
−
σ , or h17(ω1, µ) = √ e− 2σ2 , (7.19)
σ 2πσ
for which all even moments are finite. As long as σ ≪ 4m2c/mb ≈ 1.1GeV, which with the
power counting adopted in this thesis is formally of order ΛQCD, then for all relevant ω1
values the argument of the penguin function F (x) entering the definition of Λ17 in (7.18)
is much larger than 1/4, which is the radius of convergence for the Taylor expansion given
in (6.9). It is then a good approximation to expand the penguin function [1 − F (x)] to
O(1/x3). The first term in this expansion corresponds to the non-perturbative correction
identified in [108], which was already included in the partonic result and subtracted in
(7.18). It therefore does not contribute to FE(∆). The next term gives rise to an odd
moment of h17 and thus vanishes. The third term in the expansion contributes the amount
3
expanded − ec mΛ = b17 λ2 〈ω21〉 (7.20)280 m6c
to Λ17. Here 〈ω21〉 denotes the (normalized) variance of the function h17(ω1, µ), which
equals 2σ2 for the exponential form and σ2 for the Gaussian. For a typical hadronic
scale σ = 0.5GeV this gives Λexpanded17 = −6.9MeV and −3.4MeV, respectively. Here and
below the input parameters collected in Appendix A.2 have been used. The corresponding
contributions to the decay rate are very small, below 0.5% in magnitude.
It is an interesting observation that, due to a weaker numerical suppression, certain
1/mb corrections to Λ17 can give a contribution of comparable size. They arise from the
fact that the first moment of the function g17(ω, ω1, µ) with respect to ω does not vanish,
see (6.18). In order to calculate the resulting power-suppressed term, one replaces the first
relation in (7.18) by
∫ ∫
(m2 ) Λ̄ ∞c dω1Λ17 , µ = ecRe dω
mb −∞ −∞ ω1
{ } (7.21)
( )3 [ ( )]
m + ω m2× b − c − iε mb ω11 F + g
2 17
(ω, ω1, µ) ,
mb (mb + ω)ω1 12mc
116
where the factor (mb+ω )3 appears because of the prefactor E3γ in (5.39). Expanding nowmb
the penguin function to first order yields
{ }
2 ∫ Λ̄ ∫(m ) ∞
( )4
dω1
Λ c17 , µ = ecRe dω −
ω mb ω1 mb ω1
1 + + + . . . g17(ω, ω1, µ) ,
m 2 2b −∞ −∞ ω1 mb 12mc 12mc
(7.22)
where the dots represent higher-order terms in the expansion of the penguin function,
which in particular give rise to the contribution (7.20). The expression shown above yields
a 1/mb-suppressed contribution to the parameter Λ17, which is denoted by δΛ17. It is
proportional to the normalized first moment of the function g17 with respect to ω, which
according to (6.14) and (6.18) is given by 〈ω〉 = −ρ3LS/(6λ2) ≈ 0.24GeV. This leads to
2ρ3
δΛ = LS17 ≈ −(9.8± 5.2)MeV , (7.23)
27m2c
which is formally a power correction proportional to Λ2QCD/mb to the result in (7.20).
Here ρ3LS = 3ρ2 corresponds to the spin-orbit term of the HQET Lagrangian introduced
in (3.14).
In practice, it turns out that (7.20) provides a reasonable approximation only as long
as σ < 0.3GeV. Performing the convolution integral in (7.18) exactly, one finds that for
both model functions in (7.19) the resulting value of |Λ17| is maximized for certain values
of σ, which depend on the functional form of h17. Using the input parameters collected in
Appendix A.2, one obtains (Λexp17 )max = −4.6MeV for σ = 0.51GeV with the exponential
model, and (ΛGauss17 )max = −8.1MeV for σ = 0.77GeV with the Gaussian model. Note that
the maximum values are smaller in magnitude than those one would derive from (7.20)
with these values of σ.
The above estimates do not provide a conservative bound on the size of the hadronic
parameter Λ17. A significantly larger effect can be obtained if the soft function g17(ω, ω1, µ)
exhibits a tail outside the region |ω1| ≪ 4m2c/mb. In analogy with the leading-order shape
function, it is to be expected that the function g17 exhibits a radiative tail proportional to
1/ω1 for large ω1. But even at the non-perturbative level, it is conceivable that a significant
contribution to the integral results from the region of larger ω1 values. Consider, as an
example, the model
2λ ω2 − Λ2 ω22 1 − 1h 217(ω1, µ) = √ e 2σ− , (7.24)2πσ σ2 Λ2
which for Λ and σ of order ΛQCD satisfies all requirements one would reasonably impose
on the soft function. The solid curve in Figure 7.1 shows this function evaluated with σ =
0.5GeV and Λ = 0.425GeV. It features regions of positive and negative values and hence
is less constrained at larger ω1 by the fact that the normalization is fixed to 2λ2. Having
values of either sign is not problematic, because there is no probabilistic interpretation
of the subleading soft functions. The long-dashed line in the figure shows the weight
function under the convolution integral in the definition of Λ17 in (7.18), including the
charge factor ec. With the above parameter choices for the soft function, it follows that
117
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
ω1 [GeV]
Figure 7.1: Model function h17(ω1, µ) from (7.24) in units of GeV, with σ = 0.5GeV and
Λ = 0.425GeV (solid line); weight function under the convolution integral in the definition
of Λ17 in (7.18) in units of GeV
−1 (long-dashed line); weight function including 1/mb
corrections, obtained by the substitution ω → 〈ω〉 in (7.21) (short-dashed line). See text
for explanations.
Λ17 = −42MeV. By using another set of values, a correction with the opposite sign and of
the same magnitude can be obtained. For example, taking σ = 0.5GeV and Λ = 0.575GeV
yields Λ17 = 27MeV. If the 1/mb corrections as shown in (7.21) are included, using (mb +
ω) → (mb+〈ω〉) = (mb−ρ3LS/6λ2), one finds−62MeV and 21MeV, respectively. Of course,
these are just illustrative values, and one could obtain even larger negative or positive values
by reducing the separation between σ and Λ, which however will also increase the value of
the soft function at ω1 = 0. Nevertheless, based on these considerations, it seems that
−60MeV < Λ17 < 25MeV (7.25)
is a reasonably conservative range, which will be adopted for the analysis below. While
this allows for a value significantly larger in magnitude than the naive estimate (7.20), it
nevertheless strongly suggests that Λ17 is considerably smaller in magnitude than ΛQCD.
Note that the effect of a value of Λ17 near the extreme values indicated above would
be of the same magnitude as the effect of the leading-order, non-perturbative correction
from [108], which corresponds to −m 2bλ2/(9mc) ≈ −48MeV.
7.4 Analysis of the Q7γ −Q8g Contribution
It is instructive to analyze this contribution using the language of flavor symmetry of the
strong interaction. Due to the weighting by the quark electric charges, the relevant four-
quark operator in (6.43) is a pure SU(3) octet, which can be decomposed into two parts
118
corresponding to isospin I = 0, 1. The Wigner-Eckart theorem implies that
( )
Λspec
1 (8) ± 1 (8) 1 (8) (8) (8)78 = ΛI=0 Λ6 2 I=1 = Λ6 I=0 − ΛI=1 + espec ΛI=1 , (7.26)
where the upper (lower) sign in the first equation refers to charged (neutral) B mesons, and
as before espec denotes the electric charge of the spectator quark in units of e. In the limit
(8) (8)
of unbroken SU(3) flavor symmetry, it follows that ΛI=0 = ΛI=1, since both parameters
arise from the matrix element of the same SU(3) octet operator. Hence, in this limit one
obtains
∣
Λspec
(8)
∣
78 = espec ΛI=1 . (7.27)SU(3)
Interestingly, the VIA discussed in Section 6.4.1 also predicts that Λspec78 is proportional
to espec [14], and this fact can be used to obtain a model estimate of the relevant SU(3)
reduced matrix element. From (6.47) it follows
( )
2
∣ ∣
(8) (8) 1 F (µ)
Λ ∣I=1 = Λ
∣
I=0 = − 1− ∈ [−386MeV,−35MeV] , (7.28)VIA VIA N2c 8λ2B(µ)
where in the last step the parameter ranges discussed in Appendix A.2 have been used.
According to (7.26), the isospin-averaged decay rate [Γ(B̄0 → Xsγ)+Γ(B− → Xsγ)]/2
(8)
depends only on ΛI=0, while the isospin difference [Γ(B̄
0 → Xsγ) − Γ(B− → Xsγ)] is
(8)
proportional to ΛI=1. While a priori these two non-perturbative parameters are unrelated,
they coincide both in the SU(3) flavor-symmetry limit and in the VIA as has just been
shown. It was first pointed out in [129] that, in the limit of exact SU(3) flavor symmetry,
the isospin-averaged decay rate can be related to the isospin asymmetry,
Γ(B̄0 → Xsγ)− Γ(B− → Xsγ)
∆0− =
Γ(B̄0 → , (7.29)Xsγ) + Γ(B− → Xsγ)
without employing the VIA. This asymmetry has been measured by the BaBar Collab-
oration using two different experimental methods. For the “sum-over-exclusive-modes
method” with Eγ > 1.9GeV, they find ∆0− = (−0.6± 5.8± 0.9± 2.4)% [130], where the
errors are statistical, systematic, and due to the production ratio B̄0/B−, respectively. For
the “recoil method” with Eγ > 2.2GeV, they obtain instead ∆0− = (−6± 15± 7)% [131],
where the errors are statistical and systematic, respectively. The naive average of these
two results is ∆0− = (−1.3± 5.9)%. At the order of the present calculation, the parameter
(8)
ΛI=1 is related to ∆0− via
∣
(8) −C7γ(µ) mbΛ ∣I=1 = ∆ ≈ (59± 268)MeV , (7.30)exp 0−C8g(µ) 2παs(µ)
where in the last step the average experimental result with its large uncertainty given
above has been used. This value is consistent with the prediction (7.28) obtained in the
VIA within errors, even though the central value has the opposite sign.
119
(8)
Allowing for SU(3) flavor-symmetry breaking at the level of 30%, i.e. ΛI=0 = (1 ±
(8)
0.3) ΛI=1, one finally obtains
Λspec
(8)
78 = (espec ± 0.05) ΛI=1 ≈ −4.5GeV (espec ± 0.05)∆0− , (7.31)
which is meant as a range, not an error bar. This formula implies that, within the quoted
uncertainty, the isospin asymmetry also determines the flavor-averaged value of Λspec78 . For
the corresponding contribution to the flavor-averaged value of the function FE(∆), it fol-
lows that
∣
Favg ∆0−E (∆)∣ = −(1± 0.3) , (7.32)78 3
which adds SU(3)-breaking effects to the estimate derived in [129]. Note that this relation
is independent of the values of the Wilson coefficients and other theoretical parameters.
Due to the current large experimental uncertainties in the measurement of the isospin
asymmetry, it is difficult to give a reliable estimate for Λspec78 . Based on (7.28) and (7.30),
(8)
the parameter ΛI=1 is expected to be negative (assuming that the VIA is sufficiently reliable
to predict the sign correctly), but since the experimental value allows for the entire range in
(7.28) at the level of two standard deviations, it is not possible to restrict that range further
at present. A future, more accurate measurement of ∆0− could improve the situation.
7.5 Analysis of the Q8g −Q8g Contribution
Unfortunately, very little is known about the soft function hcut88 entering the definition of
the hadronic parameter Λ88 in (7.18). Its asymptotic behavior for large values of ω1 and ω2
can be derived from (6.35), and it ensures that Λ88 is independent of the UV cutoff ΛUV.
Note that the second term in the definition of Λ88, which contains the logarithm of ΛUV/∆,
is bound to give a very small contribution to Λ88, because (CF e
2
s∆)/(8π
2) < 3MeV is very
small for realistic values E0 ≥ 1.6GeV. Thus, it is expected that the hadronic parameter
Λ88 receives its dominant contributions from values ω1,2 = O(ΛQCD), for which no useful
constraints on the soft function hcut88 exist. For the same reason, it is expected that the
linear growth of Λ88 for large ∆ is a numerically irrelevant effect. It then follows that the
function hcut88 (∆, ω1, ω2, µ) is approximately equal to the function h
cut
88 (ω1, ω2, µ) shown in
(7.15), even though this relation is not strictly valid. As mentioned earlier in the paragraph
following that equation, this form would imply that the contribution to Λ88 resulting from
the double integral in (7.18) were strictly positive.
In summary, the hadronic parameter Λ88(∆, µ) is expected to be, to a good approx-
imation, independent of ∆ and given by a positive, non-perturbative constant of order
e2s ΛQCD:
Λ88(∆, µ) ≈ e2s Λ(µ) , Λ(µ) > 0 . (7.33)
This expectation is supported by using different models for the soft function hcut88 , for
example by writing it as a product of two functions f1(ω1) f2(ω2) and using various models
such as exponentials or Gaussians. A particularly simple example is provided by functions
120
hcut88 (ω1, ω2, µ) that are symmetric in both ωi variables and have support for ωi = O(ΛQCD).
In this case the third relation in (7.18) implies Λ(µ) ≈ 2π2hcut88 (0, 0, µ), and the value at
the origin scales like hcut88 (0, 0, µ) ∼ ΛQCD. For the numerical analysis, the rather generous
range 0 < Λ(µ) < 1GeV will be considered. Even for the largest value, the suppression by
the charge factor e2s = 1/9 in (7.33) implies that the effect of this term on the decay rate
is very small.
7.6 Conclusions of this Chapter
With the estimates of the hadronic parameters Λij at hand it is now possible to study the
implications of the analysis for the function FE(∆) in (7.17). Using the parameter values
collected in Appendix A.2, one obtains from (7.25) and (7.33) the contributions
∣
F ∣E ∈ [−1.7,+4.0]% ,17
∣ (7.34)
F ∣E ∈ [−0.3,+1.9]% .88
The value of FE|88 depends slightly on ∆ and is obtained using ∆ = 1.45GeV, correspond-
ing to a cut at E0 = 1.6GeV. For the case of FE|78, the charge-averaged contribution will
be considered and the estimates will be given using both methods, the VIA and the isospin
asymmetry. In the latter case, the extremal values are obtained by assuming 30% SU(3)
violation and taking the 95% confidence level experimental range. This yields
∣
F VIA∣E ∈ [−2.8,−0.3]% ,78
∣
F exp
(7.35)
∣
E ∈ [−4.4,+5.6]% (95% CL) .78
In order to obtain a conservative estimate of the combined theoretical analysis, a Bay-
sean approach will be adopted and various contributions added up using the scanning
method. In this way, the final result turns out to be
−4.8% < F (∆) < +5.6% (VIA for ΛspecE 78 ) , (7.36)
where the theoretical estimate for FE|78 has been used. When the experimental estimate
is used instead, the range is expanded to
−6.4% < FE(∆) < +11.5% (Λspec78 from ∆0−) . (7.37)
It is emphasized that the estimates in this sections should be considered as ranges , within
which the actual values of FE are expected to lie, without making a statement about the
most likely values within these ranges.
If in the future a more precise value of the isospin asymmetry can be measured, this
could be used to reduce the uncertainty range somewhat. If, for example, it is assumed
that the true isospin asymmetry lies in the center of the interval predicted by the VIA,
∆0− = +4.6%, then in the absence of experimental uncertainties one would derive F |expE 78 ∈
121
[−2.0,−1.1]%, where the remaining uncertainty stems from the unknown effects of SU(3)
breaking. In this “ideal” case, the combined result would be
−4.0% < FE(∆) < +4.8% (ideal case) . (7.38)
There seems to be no possibility to reduce this uncertainty in the foreseeable future, given
that no theoretical tools exist to constrain the non-local matrix elements defining the soft
functions entering the various resolved photon contributions studied in this work. Thus,
the range in (7.38) can be considered as the irreducible theoretical uncertainty affecting
any theoretical prediction of the B̄ → Xsγ branching fraction.
122
123
Chapter 8
Phenomenology II:
Direct CP Violation
This chapter will discuss the effects of the resolved photon contributions of Chapter 6
on the direct CP asymmetry. Since these effects are not covered within the local OPE
approach, they constitute a new non-perturbative uncertainty on the partonic calculations
of the past. After a short introduction on CP violation in Section 8.1, the relevant formulae
are worked out in Section 8.2 and the non-perturbative effects of the resolved contributions
are parameterized. Sections 8.3 and 8.4 will then present a numerical analysis of the new
effects in the SM and beyond. Finally the CP asymmetry in B̄ → Xsγ is contrasted to
the one in B̄ → Xdγ in Section 8.5.
8.1 Preliminaries
A certain process is said to violate the CP symmetry directly, if the rate of this process
differs from the rate of the CP -conjugate process. In the case of B̄ → Xsγ this can be
written as
∆Γ = Γ(B̄ → Xsγ)− Γ(B → Xs̄γ) 6= 0 . (8.1)
Such a difference is possible when at least two partial amplitudes contributing to the
process carry different weak and strong phases. The weak phase is associated with the
coupling constants in the Lagrangian, while a non-zero strong phase is generated by Feyn-
man diagrams with internal lines, that are able to go on-shell. In the SM a weak phase is
introduced through the complex parameters of the CKM matrix.
The partonic calculation of the B̄ → Xsγ amplitude introduces different strong phases
at O(αs), which give rise to CP violation. The calculation was performed in [128] under
the assumption that the rate can be expressed in terms of a local OPE if integrated over
a large enough range. However, it was discussed in the previous chapters that there exist
contributions that do not fit into the local OPE approach, even in the total rate. These
contributions could be expressed in terms of convolutions of subleading shape functions
with jet functions J and J̄ (see Section 5.3). It was noted, that even though the subleading
124
shape functions are real, the corresponding jet functions can introduce a strong phase. This
gives rise to a non-perturbative CP violating contribution.
An appropriate quantity to parameterize CP violating effects is the CP asymmetry
ACP defined by
Γ(B̄ → Xsγ)− Γ(B → Xs̄γ)
ACP = → → . (8.2)Γ(B̄ Xsγ) + Γ(B Xs̄γ)
Just like the CP averaged decay rate discussed in the previous chapters, the CP asymmetry
can be used as a probe for new physics. In particular, new physics models can generate
complex Wilson coefficients, thereby introducing new phases for CP violation. On the
other hand, while the experimental error of the CP averaged rate has been reduced below
the 10% level (see Section 1.3), the CP asymmetry is much less certain. The current world
average is [24]
ACP = −(1.2± 2.8)% . (8.3)
The theoretical analysis, based on the parton model predicts for the CP asymmetry in the
SM [132]
ASMCP = (0.44
+0.15
−0.10 ± 0.03+0.19−0.09)% , (8.4)
where the errors refer to uncertainties associated with the quark-mass ratio mc/mb, CKM
parameters, and higher-order perturbative corrections. It follows from this prediction, that
any measured value for the CP asymmetry outside the range 0 < ACP < 1% would be a
clear signal of new physics. Such models have been considered in [13,17,133–141].
The CP asymmetry defined above contains the partially integrated decay rate. As
usual, the integration is defined with a lower cut E0 on the photon energy (see Section 7.1).
The experimental result given in (8.2) is an averaged value from measurements using values
for E0 between 1.9 and 2.2GeV. The theoretical prediction used E0 = 1.6GeV. In this case,
however, it was demonstrated in [128] that the dependence of ACP on the cut is relatively
small (a few percent).
The effective theory analysis of B̄ → Xsγ presented in the previous chapters estab-
lished the following contributions to the differential decay rate (compare Equation (5.39)).
The leading power contribution is at leading order in αs irrelevant for an analysis of CP
violation, since it contains neither weak nor strong phase differences (recall that the lead-
ing shape function is CP invariant). At NLO the leading power contains both, weak and
strong phase differences, encoded in the hard function |H |2γ , see (4.30). The strong phases
are due to hard virtual corrections as depicted in Figure 2.6. When integrated over a
range much larger than the endpoint region, the leading shape function can be expanded
in powers of ΛQCD/∆ with ∆ = mb − 2E0, as explained in Section 3.3.4. The first term
in that expansion corresponds to the partonic calculation of the hard virtual loops. At
subleading power there are direct and resolved contributions. It has been shown explicitly
in Chapter 6, that the direct contributions correspond to QCD bremsstrahlung diagrams,
expanded around the endpoint. If integrated over a large enough range the direct contri-
butions are promoted to leading power. These can also give rise to CP violating phases.
Finally, the resolved contributions can potentially contribute to CP violation. However,
some of them vanish in the integrated rate (see Section 5.2.4). Thus, if the cut on the
125
photon energy is assumed to be sufficiently low, i.e., such that ∆ = few×ΛQCD, the direct
contributions (at any power) can be calculated in terms of a local operator matrix elements
using a combined expansion in ∆/mb and ΛQCD/∆. The partonic result corresponds to
the leading power in ΛQCD/∆ and includes all powers in ∆/mb. However, the resolved
contributions can never be expressed in terms of local matrix elements. Their effects have
to be added to the partonic result. In the following sections, the CP asymmetry including
direct as well as resolved contributions will be given, assuming a sufficiently low cut. While
the direct contributions will depend explicitly on the cut, the cut effect for the resolved
contributions can be ignored as explained in Section 7.1.
8.2 The CP Asymmetry
The direct photon contribution to the CP asymmetry is given by the expressions for the
CP asymmetry from [128] and the Qc1 −Qu1 interference term from [142]
∣
dir Γ(B̄ → Xsγ)− Γ(B → Xs̄γ) ∣∣ACP (δ) = Γ(B̄ → Xsγ) + Γ(B →
∣
Xs̄γ) ∣
Eγ>(1−δ)mb/2
{
αs 40 8z
[ ]
= | | Im[C
∗
1C7γ ]− v(z) + b(z, δ) Im[(1 + ǫ ∗2 s)C1C7γ]C7γ 81 9
− 4 8zIm[C C∗8g 7γ] + b(z, δ) Im[(1 + ǫs)C ∗1C ]9 27 8g
}
16z
+ b̃(z, δ) |C1|2 Im[ǫs] +O(α2s) , (8.5)27
where z = m2c/m
2
b , δ = 1 − 2E0/mb, b(z, δ) = g(z, 1) − g(z, 1 − δ), and b̃(z, δ) = g̃(z, 1) −
g̃(z, 1− δ), with
( ) ( ) ( )
π2 π2 28 4
v(z) = 5 + ln z + ln2 z − + ln2 z − z + − ln z z2 +O(z3),
3 3 9 3
{ }
√ √ √
( )
y y 3y(y − 2z) 4z
g(z, y) = θ(y − 4z) (y2 − 4yz + 6z2) ln + − 1 − 1−
4z 4z 4 y
{ [ ]
√ √
( )2 √ √( )2
− − y y − y y − π
2
g̃(z, y) = θ(y 4z) y 2L2 + 1 + 2 ln + 1 +
4z 4z 4z 4z 6
√ √
[ ] ( )
4z 6z2 (y)− − − − y y2 + y 4z + 2 ln ln + − 1
y y z 4z 4z
}
√
( )
4 + 3y − 6z − 4z+ 1 . (8.6)
4 y
126
The parameter ǫs is the only source of CP violation in the SM and is defined as in [128]
to be
λ V ∗u ubVus λ
2(iη̄ − ρ̄)
ǫs = =
6
∗ = +O(λ ), (8.7)λt VtbVts 1− λ2(1− ρ̄+ iη̄)
with the Wolfenstein parameters λ = sin θC ≈ 0.225, ρ̄ = 0.144 and η̄ = 0.342. For brevity
the scale dependencies of the Wilson coefficients Ci and the strong coupling αs have been
dropped. The numerically most important contributions arise from the interference of
charm- and up-quark penguin graphs with virtual or real gluon emission with the leading
electromagnetic dipole amplitude.
The CP asymmetry due to resolved contributions can be easily derived from the results
in Chapter 6. Keeping in mind that the earlier results were CP averaged, it is obvious that
the dual real part prescriptions in (5.39) and in the definition of the F̄ij need to be replaced
by imaginary part prescriptions times −2. By decomposing the uncut propagators in J̄
into a Cauchy principle value and a δ-function, some of the integrals in the definition of
the F̄ij can be explicitly performed. This leads to
∣
∣
Ares
Γ(B̄ → Xsγ)− Γ(B → Xs̄γ) ∣
CP = ∣Γ(B̄ → Xsγ) + Γ(B → Xs̄γ) ∣
δ≈1
π { }[ ∗ ] [ ] [ ]= | | Im (1 + ǫs)C1C7γ Λ̃
c
17 − Im ǫsC ∗ u1C7γ Λ̃17 + Im C8gC∗7γ 4πα Λ̃B̄m C 2 s 78 ,b 7γ
(8.8)
where (once again omitting the scale dependence of the Wilson coefficients and the Λ̃ij
parameters)
u 2Λ̃17 = h17(0) ,3
( )
∫
c 2
∞ dω m2
Λ̃ c17 = f h17(ω) ,3 4m2 (8.9)c ω mbω
mb
∫ ∞ [ ]
B̄ dω (1) (1)Λ̃78 =2 h78 (ω, ω)− h78 (ω, 0) ,
−∞ ω
and where the hij are defined in Section 7.1 and f is the imaginary part of the penguin
function (see Appendix A.1)
( )
√ √
1 1
f(x) = 4x ln + − 1 . (8.10)
4x 4x
(5)
Note that the contribution from g78 vanishes, since the integral over ω of this function
is symmetric under the exchange of ω1 and ω2. Likewise, the contributions from the
functions ḡ78 and ḡ
cut
78 to the CP asymmetry cancel each other. Note also that the soft
(1)
function h78 (ω1, ω2, µ) is symmetric under the exchange of ω1 and ω2. Compared to the
127
direct photon contributions which are αs suppressed, the resolved photon contributions are
1/mb suppressed.
The total CP asymmetry is given by the sum of (8.5) and (8.8)
{( )
tot π 40 α 8z α
[ ]
s s Λ̃
c
ACP (δ) = | | − v(z) + b(z, δ) +
17 Im [C1C
∗
7γ]C 27γ 81 π 9 π mb
( )
Λ̃u c [ ]− 17 − Λ̃17 8z αs+ v(z) + b(z, δ) Im [ǫ C ∗s 1C ]
m 9 π 7γb
( )
− 4 αs − Λ̃
B̄
4πα (µ) 78s Im [C C
∗
8g 7γ]9 π mb
}
8z αs 16z αs
+ b(z, δ) Im [(1 + ǫ )C ∗ 2s 1C8g] + b̃(z, δ) |C1| Im [ǫs] + . . . ,27 π 27 π
(8.11)
where the ellipses represent terms of order α2s, αs/mb, 1/m
2
b and higher. This formula is
model independent and is valid also for extensions of the SM, in which the dipole coefficients
C7γ and C8g receive new CP violating contributions. The phenomenological implication of
this formula will be explored in the following sections.
8.3 Phenomenology in the SM
In order to properly estimate the importance of each term in (8.11), a few approximations
will be made. However, the numerical calculations will always be performed with the
full formula. If the charm-quark mass is assumed to scale like m2c =O(ΛQCDmb), the
above expression may be expanded in powers of z, δ =O(ΛQCD/mb). Then contributions
proportional to b(z, δ) and b̃(z, δ) scale as (Λ 2QCD/mb) and can be neglected to a good
approximation, while the term proportional to v(z) contributes at first order in ΛQCD/mb
and can be simplified by expanding v(z) to zeroth order in z. This yields the approximation
{([ ] )
c
Atot
π 40 40 Λc αs Λ̃
CP (δ) ≈ 17 ∗| − + Im [C1C ]C |27γ 81 9 m 7γb π mb
( ) ( ) }
Λ̃u − Λ̃c 40 Λ α 4α Λ̃B̄− 17 17 c s s+ Im [ǫsC1C∗7γ]− − 4παs(µ) 78 Im [C ∗8gC7γ ] ,mb 9 mb π 9π mb
(8.12)
which is independent of the cutoff in δ. The scale parameter Λc is just like the parameters
Λ̃ij of O(ΛQCD) and defined by
( )
m2c − 2 m
2
b 4 mb π
Λc := 1 ln + ln
2 − , ≈ 0.38GeV, (8.13)
mb 5 mc 5 mc 15
128
where the parameters given in Appendix A.2 have been used. As can be seen from (8.12)
the resolved photon contributions yield numerically significant corrections to the direct
terms since αs/π ∼ ΛQCD/mb. In the second term, which is the only one present in the
SM, the resolved contribution is likely more important due to the double suppression of
the direct term.
In order to obtain ranges for the non-perturbative parameters Λ̃ij the same methods
are employed as for the integrated rate in Chapter 7. Note, that the parameter Λ̃B̄78 in
(8.8) depends on the flavor of the spectator quark inside the B̄-meson. It has been shown
in Section 7.4 that Λ̃B̄78 = especΛ̃78, where espec denotes the electric charge of the spectator
quark in units of e. Evaluating the hadronic matrix element of the corresponding non-local
four-quark operator in the VIA yields
2 ∫
B̄ ≈ 2fBM
∞ B
B φ (ω)
Λ̃78 espec dω
+ . (8.14)
9 0 ω
Therefore the size of the hadronic parameter
[ ]
∫ 2
1 ∞ φB(ω1, µ)
:= dω +1 . (8.15)
νB(µ)2 0 ω1
with the light-cone distribution amplitude φB+ needs to be estimated. Although it was not
discussed in the literature in the past, one can use the analysis of [123] to estimate its
size. The results are summarized in Table 8.1. The other parameters in (8.15) are given
µ [GeV] 1.0 1.5 2.0 2.5
ν−2B [GeV
−2] 1.21± 0.26 0.91± 0.16 0.77± 0.12 0.68± 0.10
Table 8.1: Values of ν−2B using the analysis of [123]. See text for a discussion of the
appropriate scale.
in Appendix A.2. In order to be conservative, νB will be taken to be between 0.5GeV and
1.5GeV, which covers the range in Table 8.1 as well as the value that can be obtained from
a QCD sum rules analysis similar to that of [125]. This leads to a range
17MeV < Λ̃78 < 190MeV . (8.16)
The parameters Λ̃q17 can be estimated by employing the same models as studied in Sec-
tion 7.3. Note that Λ̃u17 only depends on the value of the soft function h17 at the origin.
This leads to the following ranges
−330MeV < Λ̃u17 < 525MeV ,
(8.17)
−9MeV < Λ̃c17 < 11MeV .
One observes that Λ̃u17 and Λ̃78 are of order ΛQCD, while the charm-loop contribution Λ̃
c
17 is
much smaller. The slight preference for positive values of Λ̃q17 is due to the normalization
129
constraint (6.14). Note that in the formal limit mc → mu = 0 the values of Λ̃c17 and
Λ̃u17 coincide. However, a strong GIM violation is to be expected, owing to the fact that
the integral in the second relation in (8.9) starts at 4m2c/mb ≈ 1.1GeV, at which the
soft function es expected to take already rather small values, since it is governed by non-
perturbative dynamics and must vanish for ω → 0.
With the above results at hand, the SM CP asymmetry can be predicted and the effect
of the non-perturbative corrections estimated. The factorization scale is chosen to be µ =
2GeV (for which αs(µ) = 0.307, and C1(µ) = 1.204, C7γ(µ) = −0.381, C8g(µ) = −0.175
at leading order), since the strong phases required for a non-zero CP asymmetry arise
either from GIM violations related to charm-quark loops (for which µ ∼ 2mc), or from cut
√
hard-collinear propagators (for which µ ∼ ΛQCDmb). This leads to
( )
Λ̃u − Λ̃c
ASM ≈ 1.15× 17 17CP + 0.71 %. (8.18)300MeV
Although the Λ̃c17 is much smaller than Λ̃
u
17, it is still given in (8.18) to make explicit the fact,
that the CP asymmetry vanishes in the formal limit mc = mu due to the GIM mechanism.
The numerically most important contribution arises from the up-quark penguin graph
with emission of a soft gluon at a light-like distance from the heavy quarks. Uncertainties
associated with the input parameters are not shown, since they are much smaller than the
non-perturbative uncertainty. The central value 0.71% for the direct photon term is larger
than the one obtained in [132] since smaller values for µ and mc have been used here.
It is observed that the resolved photon term is parametrically larger than the direct
photon term, which contains an additional αs suppression. Using the model estimates for
Λ̃ij given above leads to the range
−0.6% < ASMCP < 2.8% , (8.19)
which covers most of the experimentally allowed range. Only a value of the asymmetry
below −2% could be interpreted as a sign of new physics, as in this case Λ̃u17 < −700MeV
would be much larger than the model expectations. The case of new physics will be
explored in more detail in the next section.
8.4 Phenomenology beyond the SM
In order to illustrate the impact of the resolved photon terms, a class of new physics models
is considered in which the dominant non-standard effects are encoded in the values of the
dipole operators which will be parameterized in the form
C CSM7γ 7γ iθ C8g C
SM
7 8g= r e , = r eiθ87 . (8.20)
C CSM C CSM
8
1 1 1 1
130
Using that arg(−ǫs) ≈ −γ in the SM, one obtains for the default parameters
( )
e Λ̃ r
A specCP [%] = 6.27 + 11.98 espec ×
78 8
sin(θ7 − θ8)
100MeV r7
( ) ( )
Λ̃c 1 Λ̃u − Λ̃c 1
+ 10.12 + 2.14× 17 sin θ7 + 0.74 + 1.26× 17 17 sin(γ + θ7)
10MeV r7 300MeV r7
r8 0.037 r8
+0.18 sin(θ8) + sin γ − 0.004 sin(γ + θ ) .
r2 r2 2
8
7 7 r7
(8.21)
The flavor-averaged CP asymmetry (AeuCP +A
ed
CP )/2 can be obtained by replacing espec →
1/6. Then the resolved photon contributions are subdominant except for the third term,
which is already present in the SM. In principle, very large asymmetries are possible from
the first and second terms [128], which however are already ruled out by the data.
A non-trivial feature of this analysis is that the resolved photon contributions induce a
flavor dependent term in the CP asymmetry already at order ΛQCD/mb. In the SM such
effects are suppressed, compared with (8.18), by at least one additional factor of ΛQCD/mb
and are thus bound to be negligible. This suggests the measurement of the CP asymmetry
difference
e − e ≈ 2 Λ̃78 C8g ≈ × Λ̃78 r∆A := A u 8CP CP A dCP 4π αs Im 12% sin(θ8 − θ7) (8.22)mb C7γ 100MeV r7
as a sensitive probe for flavor physics beyond the SM. Even though it will be difficult to
determine the value of the hadronic parameter Λ̃78 with any reasonable accuracy, the differ-
ence ∆ACP can easily reach the level of 10% in magnitude if either the electromagnetic or
the chromomagnetic dipole coefficients (or both) receive a sizable CP violating new-physics
phase. It is important in this context that the Wilson coefficient of the chromomagnetic
operator can be much enhanced with regard to the SM value, so that r8/r7 ∼ a few is
possible [128]. Before concluding, it is interesting to consider the the case of B̄ → Xdγ
and compare it to B̄ → Xsγ.
8.5 Comments on B̄ → Xdγ and B̄ → Xs+dγ
All of the expressions in the previous sections also apply to the CP asymmetry in B̄ → Xdγ,
where one only needs to change ǫs to ǫd, defined as
V ∗ubV
ǫ = ud
ρ̄− iη̄
d
V V ∗
=
tb td 1−
= O(1) . (8.23)
ρ̄+ iη̄
As a result, the CP asymmetry for the B̄ → Xdγ decay in the SM differs by a factor
Im ǫd/Im ǫs ≈ −22 from the B̄ → Xsγ case.
131
In the endpoint region, there are two other potential sources of difference between
b → sγ and b → dγ that however do not contribute to the integrated CP asymmetry at
order 1/mb. The first is the four-quark shape functions which are sensitive to the flavor of
q in b → qγ [9, 10, 100]. However, as shown in Section 7.1 these effects integrate to zero
in the partially inclusive CP asymmetry at order ΛQCD/mb. The second potential source
arises if one adopts the scaling ms ∼ O(ΛQCD). This gives rise to a new subleading jet
function which is suppressed by ΛQCD/mb in the endpoint region (see Section 5.2.6). At
order ΛQCD/mb, it only contributes to the interference Q7γ with itself and not to the CP
asymmetry. The interference of Q7γ with other operators is suppressed by two powers of
ΛQCD/mb in the endpoint region. Outside of the endpoint region these strange-quark mass
effects are even further suppressed.
Another observable of interest is the CP asymmetry for B̄ → Xs+dγ. Considering only
the direct photon contributions, the sum of the SM CP asymmetries of B̄ → Xdγ and
B̄ → Xsγ vanishes in the U-spin limit (ms = md), as was first pointed out in [143]. This
can be seen explicitly by applying unitarity of the CKM matrix
∆ΓSM(B̄ → Xsγ) + ∆ΓSM(B̄ → Xdγ) ∝ Im [V V ∗(V ∗ V + V ∗ub tb us ts udVtd)] = 0. (8.24)
From the discussion in the previous section it follows that the resolved photon contributions
do not violate this relation at order ΛQCD/mb. It is not inconceivable, though, that beyond
this order there could be resolved photon contributions that might affect this relation, but
these are beyond the scope of this work.
8.6 Conclusions of this Chapter
It was demonstrated in Chapter 6 that resolved photon contributions give rise to new,
calculable CP violating strong phases related to (anti-)hard-collinear jet functions, which
are convoluted with real, non-perturbative shape functions. These phases are responsible
for a sizable hadronic uncertainty in the partonic calculation of the CP asymmetry. In
this chapter it was shown, that the dominant part of these uncertainties is related to the
interference of the electromagnetic dipole amplitude with the amplitude for an up-quark
penguin transition accompanied by soft gluon emission. It was parameterized in the form
of Λ̃u17 in (8.9) and a possible range of values for this quantity was estimated. It turned out,
that the long-distance contribution encoded in Λ̃u17 is parametrically leading in the SM and
will be difficult to disentangle from possible new physics effects. A measurement of the
CP asymmetry difference (8.22) seems more promising as a new physics probe. Finally it
was found that the untagged CP asymmetry for the B̄ → Xs+dγ decay vanishes in the SM
(up to U-spin breaking corrections [128, 143, 144]) even after the resolved terms are taken
into account.
132
133
Chapter 9
Conclusions
9.1 Summary and Outlook
This work explored the origin and implications of non-perturbative contributions to the
decay rate of B̄ → Xsγ. The charmless, radiative and inclusive decay B̄ → Xsγ is relevant
as a test of the SM as well as a probe for physics beyond the SM. For example, it is used
in combination with the semi-leptonic decay B̄ → Xulν̄ to determine the CKM matrix
element Vub. On the other hand it is able to provide stringent bounds on several new
physics models [3].
The partonic calculation of the B̄ → Xsγ decay rate considers the decay of a free,
on-shell b-quark and has been performed practically completely up to NNLO [5] (see Chap-
ter 2). This calculation predicts a branching fraction of B(B̄ → Xsγ) = (3.15±0.23) ·10−4
for all B̄ → Xsγ decays containing a photon with an energy above a cut E0 > 1.6GeV.
This cut on the photon energy is necessary in order to eliminate the BB̄ background from
the experimental measurements (see Chapter 1).
Of course, there are several effects that are not considered in the partonic prediction.
The most relevant one stems from non-perturbative corrections, due to the large QCD
coupling constant at low (hadronic) energies. In the framework of HQET it can be shown
that the partonic calculation corresponds to the leading term in an expansion in powers
of 1/mb and that the non-perturbative corrections only appear at higher orders in the
expansion (see Chapter 3). However, this only applies to situations where the cut on the
photon energy is chosen sufficiently low (so that mb−2E0 → mb). The cut of E0 > 1.6GeV
was chosen to satisfy this requirement. It was already known for some time that the
expansion can no longer be expressed in terms of local operators when considering the
endpoint region (Eγ → mb/2). In that case the partonic result has to be convoluted with
a shape function, that is defined by the Fourier transform of a non-local HQET matrix
element, and that can be interpreted as the momentum distribution of the heavy quark
within the meson [81] (see Chapter 4).
Later on it was discovered, that there are contributions to the integrated B̄ → Xsγ
decay rate at O(1/mb) which do not fit into the framework of a local OPE even if the cut
134
is taken outside the endpoint region [14]. It was one of the major objectives of this work
to systematically analyze all 1/mb suppressed contributions of this type at tree level in
perturbation theory. The appropriate approach for such an analysis makes use of SCET
[6,7] (see Chapter 4). By employing SCET it was possible to proof a factorization formula
for the differential B̄ → Xsγ decay rate in the endpoint region which is valid at any order
in the 1/mb expansion (see Chapter 5). This formula can be written as
∞
∑
→ 1
∑
(n) (n)
dΓ(B̄ Xsγ) = Hi Ji ⊗
(n)
S
mn i
n=0 b i
∞ [ ]
∑ 1 ∑ ∑(n) (n) ⊗ (n)+ Hi Ji Si ⊗
(n) (n) (n) (n) (n)
J̄i + Hi Ji ⊗ Si ⊗ J̄i ⊗
(n)
J̄i ,mn
n=1 b i i
(9.1)
(n)
where Hi are hard functions appearing in the matching of QCD onto SCET at the hard
(n) (n)
scale µh ∼ mb, Ji and J̄i are the jet functions corresponding to the matching coefficient
√ (n)
from SCET onto HQET at µi ∼ ΛQCDmb, and Si represents the shape functions, which
encode the non-perturbative dynamics. The explicit formula is given in (5.39). On the
one hand this factorization resolves some of the issues of the partonic calculation, like
large logarithms or the dependence on low-energy parameters in (2.35). On the other
hand it provides new insights concerning the possible space-time structure of the relevant
(n)
contributions. While the usual jet functions Ji are defined as cut propagators of hard-
(n)
collinear particles, the new type of jet function J̄i corresponds to full propagators dressed
by Wilson lines. They appear in the so-called resolved photon contributions, in which the
photon is not emitted directly from the vertex but from one of the light partons. The
(n)
contributions that do not contain a J̄i are called direct photon contributions and they
can be expressed as a double expansion in powers of ΛQCD/∆ and ∆/mb when considering
the integrated rate with a cut far outside the endpoint region.
In contrast, the resolved contributions are new and not covered in the local OPE ap-
proach, yet they have important implications for phenomenology. They significantly con-
tribute to the uncertainty in the prediction of the branching fraction as well as in the
determination of the direct CP asymmetry. The second major topic of this work was to
estimate the possible size of these effects. Due to their non-perturbative nature, they can
not be calculated analytically. Unfortunately it is also not possible to determine their
values on the lattice, since they contain non-local matrix elements separated by light-like
distances.
A careful analysis revealed that the phenomenologically relevant resolved contributions
at O(1/mb) are due to interferences of weak effective operators Qq1 − Q7γ , Q7γ − Q8g and
Q8g −Q8g (see Chapter 6). Models for the shape functions and a relation to the measured
isospin asymmetry have been used to estimate the influence of these contributions on the
integrated rate with a cut outside the endpoint region (see Chapter 7). It turned out that a
simple dimensional estimate of ΛQCD/mb (possibly enhanced by Wilson coefficients) tends
to overestimate the effect. In the Qq1−Q7γ case the normalization (6.14) restricted the size,
135
while the contributions due to Q7γ −Q8g in (6.45) and Q8g −Q8g in (6.32) were suppressed
by charge factors. Nevertheless, it was found that the resolved photon contributions are
responsible for an irreducible uncertainty of at least
−4.0% < FE(∆) < +4.8% , (9.2)
where the Q7γ −Q8g contribution was extracted from the measured isospin asymmetry in
the hypothetical limit of no experimental uncertainties. This result therefore represents
the “ideal” case and it is hard to conceive a way to further reduce this uncertainty.
The effect of the resolved contributions on the direct CP asymmetry can be determined
by considering the imaginary parts of the convolutions of the jet and shape functions. All
(n)
shape functions are real (see Chapter 6), yet the new jet functions J̄i carry a strong
phase. This phase generates new long-distance contributions to the CP asymmetry (see
Chapter 8), which turn out to be the dominant effects in the SM, since they are enhanced
by a factor 1/αs compared to the SM in (8.18). The largest of these effects is due to the
interference of the electromagnetic dipole amplitude with an up-quark penguin amplitude
accompanied by a soft gluon emission. It was found that the SM prediction for the direct
CP asymmetry should be given as a range
−0.6% < ASMCP < 2.8% , (9.3)
compared to the ∼ 0.5% prediction of the parton model [132]. Since such a large range
can possibly hide new physics effects, a measurement of the asymmetry rate difference in
(8.22) was suggested.
In summary, the analysis presented in this thesis established the factorization formula
(9.1) for the photon spectrum in the endpoint region, and provided estimates for the ef-
fects of the resolved photon contributions on the branching fraction (9.2) and the direct
CP asymmetry (9.3). In the light of pending advances on the intensity frontier [25,26], the
results will have important implications for the determination of SM parameters, as well
as for the search for new physics, which might be hidden by the non-perturbative effects
analyzed in this thesis. Even though these effects can only be estimated at present, the
systematic approach based on the effective theories SCET and HQET has been shown to
provide a suitable framework for their identification and their exact field theoretic defini-
tion.
136
137
Appendix A
Definitions and Numbers
A.1 The Penguin Function
This appendix will introduce the penguin function, appearing in calculations where two
lines of a four-quark operator are contracted to a penguin loop. Feynman diagrams, such
as the last one in Figure 2.6 generally are proportional to a function [63]
2G(t) + t
· ′ (A.1)(q q ) t
where q and q′ are the momenta of the gauge bosons emitted from the loop, t = 2q · q′/m2c
(for a charm-quark loop) and G(t) is defined by
∫ 1 dy ( )
G(t) = ln 1− t(1− y)− iε . (A.2)
0 y
It is now convenient to introduce the penguin function F (x) through
( )
1
F (x) = −2xG (A.3)
x
which implies
( )
1
F (x) = 4x arctan2 √ (A.4)
4x− 1
which is real for x < 0 and x > 1/4 and whose real and imaginary parts for 0 < x < 1/4
are also given by
( )
√ √
1 1
ReF (x) = π2x− 4x ln2 + − 1 (A.5)
4x 4x
( )
√ √
1 1
ImF (x) = 4πx ln + − 1 . (A.6)
4x 4x
138
3.0 3.0
2.5 2.5
2.0 2.0
1.5 1.5
1.0 1.0
0.5 0.5
0.0 0.0
- 0.2 0.0 0.2 0.4 0.6 0.8 - 5 0 5 10
Figure A.1: The penguin function F (x) (on the left) and F (1/x) (on the right). The full
and dashed lines represent the real and imaginary part, respectively. Also illustrated are
the expansions for large x (on the left) and small x (on the right).
Due to the definition (A.3) the penguin function of the inverse argument as well as the
penguin function itself are plotted in Figure A.1. The expansion in (6.9) converges for |x| >
1/4 and is also illustrated in the Figure. Furthermore, it can be seen that F (x) vanishes
for x → 0 which justifies the approximation F (z/x) → 0 in (2.29) for m2c ∼ ΛQCDmb.
139
A.2 Input Parameters
In this Appendix all input parameter values used in the various numerical calculations are
collected. As the default choice for the factorization scale µ entering the Wilson coefficients,
the strong coupling constant, and the various hadronic quantities, the hard-collinear scale
√
µ = 1.5GeV is taken, which is indeed a scale of order mbΛQCD. This is an appropriate
scale choice, given that RG evolution effects are neglected.
The b-quark mass enters the expressions either via the photon energy (Eγ ≈ mb/2 near
the peak of the spectrum) or as the heavy-quark expansion parameter. It is therefore
appropriate to adopt a low-scale subtracted heavy-quark mass, such as the mass defined in
the shape-function scheme [1]. Specifically, mb = 4.65GeV will be used. The charm-quark
mass enters as a running mass in charm-penguin diagrams with a soft gluon emission,
which are characterized by a hard-collinear virtuality. Therefore mc = mc(µ) defined in
the MS scheme will be used, with µ = 1.5GeV fixed as described above. This corresponds
to the choice adopted in [5], and following these authors mc(µ) = 1.131GeV is assumed.
Finally, for the strange-quark mass ms = mb/50 is taken, which is the value commonly
adopted in the literature on B̄ → Xsγ decay.
Furthermore input values for some HQET matrix elements are needed. The parameters
λ and ρ32 LS are extracted from a global fit to B̄ → Xcl ν̄ experimental data by the Heavy
Flavor Averaging Group (HFAG) [24]. Unfortunately, in many cases these and other
parameters are extracted from a combined fit to B̄ → Xcl ν̄ and B̄ → Xsγ, an approach
that was criticized in [145]. Only recently HFAG has started quoting also values obtained
using only semi-leptonic data. The most recent results are λ2 = (0.12 ± 0.02)GeV2 and
ρ3LS = (−0.17 ± 0.09)GeV3 [146]. For simplicity, the central values for these quantities
will always be used. For the first inverse moment of the B-meson light-cone distribution
amplitude, the range 250MeV < λB < 750MeV is taken, which covers predictions obtained
using QCD sum rules and other methods [122–127]. Finally, to the level of accuracy√of the
calculations in this work, the parameter F can be extracted from the relation F = fB MB,
and using fB = (193± 10)MeV [147] one obtains 0.177GeV3 < F 2 < 0.217GeV3.
140
Appendix B
Operator Matching onto SCET
This appendix will complete the list of SCET operators relevant for B̄ → Xsγ and de-
scending from Q q8g and Q1 up to a suppression of 1/mb compared to the leading power
operator (4.27). The operators descending from Q7γ were already given in (5.5) and (5.6).
In order to determine the SCET operators the methods introduced in Section 5.1 will be
applied.
B.1 Tree-Level Matching of Q8g
As was already discussed in Section 5.4, there are three relevant possibilities to match
the gluon and the light quark onto SCET fields. Either they are both hard-collinear
or one is hard-collinear and the other anti-hard-collinear. Soft fields are excluded by the
power counting. The leading operators for the combination of hard-collinear and anti-hard-
collinear fields were already given in Section 5.4 but will be repeated here for completeness.
For the case of Q8g containing an anti-hard-collinear gluon and a hard-collinear s-quark,
the conversion of the anti-hard-collinear gluon into a photon is O(λ) in power counting. In
this case, Q8g is matched onto a single operator (suppressing the a factor −gmb e−imb v·x2 ):4π
(2) n̄/
Q = ξ̄hc [in · ∂A/ hc⊥] (1 + γ5)h , followed by O(λ) conversion. (B.1)8g, hc gluon 2
If the s-quark is matched onto a hard-collinear field, the anti-hard-collinear photon
needs to be emitted from either the b or s-quark lines. This is illustrated in Figure B.1. The
operator Q8g would then match onto SCET operators that contain both a hard-collinear
gluon and an anti-hard-collinear photon suppressed by mb if the photon is emitted from
the b-quark line, and by n̄ · ∂ if the photon is emitted from the s-quark line. Furthermore
the expansion of Gµν given in (5.36) and the multipole expansion of the heavy-quark field
141
SCET
+ −→
Figure B.1: Graphical illustration of the tree-level matching procedure for the operator
Q8g for the case when the s-quark is matched onto a hard-collinear field.
need to be considered. This leads to
(1) 1
Q8g, hc quark = ξ̄hc ←−edeA/ em⊥ [in̄ · ∂A/ hc⊥] (1 + γ5)h ,
in̄ · ∂
(2) 1
Q8g, hc quark = ξ̄hc ←−edeA/ em⊥ [in̄ · ∂A/ hc⊥] (1 + γ )x
µ
5
· ⊥
Dµh
in̄ ←−∂
− i ∂/ ⊥ n̄/ξ̄ emhc ←− [in̄ · ∂A/ hc⊥]edeA/· 2m ⊥
(1 + γ5)h
in̄ ∂ b
1 em n̄/+ ξ̄hc ←−edeA/ ⊥ [i∂/⊥A/ hc⊥ − i∂⊥ · Ahc⊥] (1 + γ5)h (B.2)
in̄ · ∂ 2
− n̄/ξ̄ emhc [i∂/⊥A/ hc⊥ − i∂⊥ · Ahc⊥] edeA/ ⊥ (1 + γ5)h2mb
[ ]
1 n̄/ i
+ ξ̄hc ←−edeA/ em⊥ n̄ · ∂ n · Ahc (1 + γ5)h
in̄ · ∂ 2 2
[ ]
n̄/ i
+ ξ̄hc n̄ · ∂ n · A emhc edeA/ ⊥ (1 + γ5)h .2mb 2
If the s-quark is matched onto an anti-hard-collinear field, it can only be converted to
an anti-hard-collinear photon and a soft s-quark. The conversion costs λ1/2, so the lowest-
order operator possible is O(λ3). Considering all the possible structures for Gµν and the
multipole expansion, one finds
(1) n/
Q = ξ̄ [in̄ · ∂A/ ] (1 + γ )h , followed by O(λ1/2) conversion,
8g, hc quark hc hc⊥ 52
(2) n/
Q = ξ̄ µ 1/2
8g, hc quark hc
[in̄ · ∂A/ hc⊥] (1 + γ5)x
2 ⊥
Dµh , followed by O(λ ) conversion, (B.3)
+ ξ̄hc [i∂/⊥A/ hc⊥ − i∂⊥ · Ahc⊥] (1 + γ5)h , followed by O(λ1/2) conversion,
[ ]
n/ i
+ ξ̄hc n̄ · ∂ n · Ahc (1 + γ5)h , followed by O(λ1/2) conversion.2 2
142
q
B.2 Tree-Level Matching of Q1
The tree-level matching of Qq1 onto a SCET operator containing two anti-hard-collinear
and one hard-collinear field was already given in Section 5.4. There, it was argued, that
all other possibilities for the matching of the light quark fields do not contribute to sub-
leading B̄ → Xsγ since there is no operator they can interfere with to generate a 1/mb
suppressed contribution to the differential decay rate. However, at order 1/m2b there are
more possibilities to match Qq1. To simplify the notation the operator will be written as as
Qq1 = s̄Γ1q q̄ Γ2b, where Γ1 ⊗ Γ2 = γµ(1− γ5)⊗ γµ(1− γ5). At tree level, the light quarks
can only be matched onto hard-collinear or anti-hard-collinear fields. A matching of any of
the light quarks onto even one soft field would lead to a suppression of O(λ4). As a result,
Qq1 is matched at O(λ3), before taking into account any conversions. When there is more
than one anti-hard-collinear quark field, no conversion is allowed at tree-level. Hence, the
following cases remain (suppressing the e−imb v·x factor).
s̄ = ξ̄hc, q = ξhc, q̄ = ξ̄hc: The anti-hard-collinear photon can either be emitted from
the heavy or one of the hard-collinear quark lines. This leads to four possible O(λ7/2)
operators:
q (2) n̄/
Q =− ξ̄ Γ e eA/ emh ξ̄ Γ ξ + ξ̄ e eA/ em n̄/ 11 hc 1 d ⊥ hc 2 hc hc d ⊥ ←−Γ1h ξ̄hcΓ2ξhc2mb 2 in̄ · ∂ (B.4)
n̄/ 1 n̄/ 1
+ ξ̄hcΓ1h ξ̄hc edeA/
em
⊥ ←−Γ2µξ emhc − ξ̄hcΓ1h ξ̄hcΓ2 edeA/2 · 2 ⊥ · −→
ξhc .
in̄ ∂ in̄ ∂
s̄ = ξ̄hc, q = ξ , q̄ = ξ̄hc and s̄ = ξ̄hc, q = ξhc, q̄ = ξ̄ : In this case the anti-hard-hc hc
collinear quark needs to be converted to a photon via ξ emhc → A⊥ + q and ξ̄hc → Aem⊥ + q̄,
which is of O(λ1/2). This yields
q (2)
Q = ξ̄ Γ h ξ̄ Γ ξ , followed by O(λ1/21 hc 1 hc 2 hc ) conversion, (B.5)
+ ξ̄hcΓ1h ξ̄
1/2
hcΓ2ξhc , followed by O(λ ) conversion.
s̄ = ξ̄ , q = ξhc, q̄ = ξ̄hc: Supplemented with SCET Lagrangian for ξ̄hc → Aem⊥ + q̄,hc
only one O(λ7/2) operator is possible:
q (2)
Q 1/21 = ξ̄hcΓ1h ξ̄hcΓ2ξhc , followed by O(λ ) conversion. (B.6)
q
B.3 Loop-Level Matching of Q1
Loop-level matching is only relevant for Qq1, when the two up-type quarks are contracted
and a number of gauge bosons are emitted from the internal lines. The contribution of
three gauge bosons would lead to further power or loop suppression, so only one or two
143
bosons need to be considered. The one gauge boson loops are already included in the
effective Wilson coefficients Ceffi defined in Section 2.1.5.
A more involved contribution arises for the loops with two external bosons, which is
the main focus of this section. These can only be one photon and one gluon. Two external
gluons would lead to further power or loop suppression. As usual, the b-quark is matched
onto the heavy-quark field h. Since there is already a photon in the operator, the s-quark
can not be anti-hard-collinear. On the other hand, a soft s-quark would lead to power
suppression, so it must be hard-collinear. There are two possibilities to match the gluon
emitted from the loop: it can either be hard-collinear or it can be soft. If the gluon is
hard-collinear, the loop momentum is hard for momentum conservation reasons. If the
gluon is soft, the loop momentum is anti-hard-collinear which just corresponds to the case
considered in Section 6.1. For the first case, a photon and a hard-collinear gluon, it is
necessary to calculate the loop diagram in QCD in order to perform the matching. For
the second case, a photon and and a soft gluon, one would need the tree-level matching of
Qq1 onto the operator of (5.38). The conversion of the two anti-hard collinear quarks to a
photon and a soft gluon would be calculated in SCET. Alternatively, one can calculate the
process in QCD and use the fact that the two calculations are equivalent, since only one
momentum region, anti-hard-collinear, contributes in this case. The result can be read off
from the two gluon calculation in [148], where one of the gluons is replaced by a photon.
Using the notation of [148] the amplitude is given by
A e eq g= (T a)mns̄ ∗µ ∗a νmΓ2AµνΓ1bnǫ (q1)ǫ (q2)δijδkl , (B.7)
4π 4π 1 2
where
[ ]
mq qµqν
Aµν = [(4r − 1)F (r)− 4r] gµν − + [F (r)− 1] iǫραµν (qρ1 − qρ2) γαγ52r mq
(B.8)
2r − F (r)+ [F (r) 1] [qµiǫ ρ σ ρ σ α 5 ρ σ 52 ρσανq1q2 − qνiǫρσαµq1q2 ] γ γ − iǫρσµνq1q2γ .mq mq
Here mq and eq are the mass and charge of the quark in the loop, q1, ǫ1 (q2, ǫ2) are the
momentum and polarization of the photon (gluon), r = m2q/q
2 − iǫ, q2 = (q1 + q2)2, and
F (r) is the penguin function defined in Appendix A.1. Alternatively, one could write in a
gauge-invariant notation
A ǫ∗µ(q )ǫ∗a ν
1 F (r)
µν 1 1 2 (q2) = [(1− 4r)F (r) + 4r] F µνGa + G̃a F ρµiγ52m µν 2m ρµq q (B.9)
− 2
( )
+ [1 F (r)] Ga F̃ µβ + F aµβ α
2 µα µα
G̃ iq γβγ5 .
q
Here the convention
F̃ µν −1= ǫµναβFαβ (ǫ0123 = −1) (B.10)
2
was used in addition to the fact that for an external gluon q ∗a2 · ǫ2 = 0. In the following
sections the cases of an anti-hard-collinear and a hard loop momentum will be considered
144
in turn. As already mentioned above, the former case is equivalent to the calculation of the
SCET diagram in Section 6.1. The latter case on the other hand is necessary to determine
the direct photon contributions involving insertions of the operator Qq1.
B.3.1 Matching with Aem gluon⊥ and Ahc s
The diagram with an anti-hard-collinear photon and a soft gluon emitted from the internal
line already contributes at O(λ7/2). Since Agluons scales homogeneously ∼ (λ, λ, λ), it is
natural to use the gauge invariant form for the resulting operators, rather than decomposing
the gauge fields into their light-cone components. Furthermore, only the axial part of Aµν
in (B.9) yields a non-zero result, as the Γi in (B.7) are the usual V − A Dirac structures,
when matching Qq1. Therefore one only needs
( )
A ǫ∗µ
2
µν 1 (q1)ǫ
∗a ν
2 (q2) = G
a F̃ µβ + F G̃aµβ iqαµα µα γβγ5[1− F (r)]q2
(B.11)
( )
≈ 2 Ga µβµαF̃ iqαγβγ5[1− F (r)] ,q2
which follows from the fact, that qαFµα vanishes at the lowest order in λ.
The loop can consist of any up-type quark that is not integrated out in the weak effective
Lagrangian. When matching onto SCET the u quark should be taken to be massless, so
F (m2/q2u ) can be replaced by 0 (see Appendix A.1). The charm-quark mass scales like
m2c ∼ mbΛQCD, which is of the same order as q2. As a result, F (m2 2c/q ) should not be
expanded for c-quarks.
Thus, in position space the Qq1 operators are matched onto
( )
u (2) eeu
Q a β −imbv·x1 (x) = ξ̄2 hc(x)T γ (1− γ5)h(x)e4π
∫
d4q d4× 1 q2 ei(q1+q )x 12 i(qα + qα)gGa (q ) ǫµβρσ2 Fρσ(q1) ,
(2π)4 (2π)4 (q + q )2 + iǫ 1 2 µα1 2
∫
( ee ) d4c (2) c q1 d
4q2
Q (x) = ξ̄ (x)T aγβ(1− γ )h(y)e−imbv·x ei(q1+q2)x1 hc 5 (B.12)4π2 (2π)4 (2π)4
[ ( )]
× 1 − m
2
1 F c − iǫ i(qα1 + qα)gGa µβρσ(q + q )2 + iǫ (q + q )2 2 µα(q2) ǫ Fρσ(q1) ,1 2 1 2
where the dependence of the momentum-space Fρσ and G
a
µα on q1 and q2 is shown explicitly.
gluon
B.3.2 Matching with Aem⊥ and Ahc ⊥hc
In this case q2 is hard, i.e. q2 ∼ m2b . Therefore F (m2 2q/q ) can be expanded around zero for
charm-quarks as well as for up-quarks. The first order correction resulting from this ex-
pansion gives a power suppressed contribution, so F (m2q/q
2) can be set to zero. Depending
145
on the polarization of the gluon field Qq1 either matches onto a O(λ3) or an O(λ7/2) opera-
tor. Furthermore corrections from the multipole expansion of the heavy-quark need to be
included. In total one finds that Qq1 is matched onto (suppressing a factor
eeqg −imb v·x
2 e )4π
q (1) n̄/
Q = ξ̄ ǫ ν1 hc ⊥µν [Ahc⊥ n · ∂Aem µ⊥ ] (1− γ5)h ,2
q (2)
Q1 = ξ̄hc ǫ⊥µν [A
em
⊥ · ∂⊥Aν µhc⊥] γ⊥(1− γ5)h
1
+ ξ̄ em ν µhc ǫ⊥µν [A⊥ n̄ · ∂ n · Ahc] γ⊥(1− γ5)h2 (B.13)←−
− i ∂/ ⊥ξ̄hc ←− ǫ [Aem µ⊥µν ⊥ n̄ · ∂Aνhc⊥] (1− γ5)h
in̄ · ∂
n̄/
+ ξ̄ ǫ [Aν em µ µhc ⊥µν hc⊥n · ∂A⊥ ] (1− γ5)x⊥Dµh ,2
where ǫ 1 α β⊥µν = ǫ2 αβµνn̄ n .
146
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