Techniques for solving two-loop massive Feynman integrals

Abstract

The Large Hadron Collider (LHC) is dedicated to the task of performing high energy scattering collisions. The success of the physics program at the LHC also demands high precision of theoretical predictions. We need to calculate multi-loop scattering amplitudes in perturbative quantum field theory to obtain precise theoretical predictions. Feynman integrals are the building blocks for scattering amplitudes and therefore the calculation of scattering amplitudes at higher orders in perturbation theory also requires a deep understanding of multi-loop Feynman integrals. Now is the age when we need to evaluate two-loop corrections to processes with massive particles, like the top quark and the electroweak bosons. The calculation of two-loop Feynman integrals with masses, using the available tools, often turns out to be a difficult task. The algebraic structure of the Feynman integral has proven to be a great help in computing these integrals. In practice, we use the method of differential equations to solve a particular Feynman integral. We can exploit the properties of dimensionally regulated integrals to find a basis of integrals which we call the master integrals. Solving these master integrals correspond to solving the Feynman integral. The use of a particular canonical form' makes the solution of the differential equation for the master integrals simpler and lets us immediately write them down in terms of iterated integrals at all orders in the dimensional regularization parameter. In the case of mostly massless processes, the integrals are known to evaluate to a special class of functions known as the multiple polylogarithms. This is not true starting from two loops. The simplest single scale example for this case is given by the very famous sunrise integral which is known to contain an elliptic curve and needs elliptic generalizations of multiple polylogarithms in order to write down the solution.

In this work, we present two examples of Feynman integrals which depend on multiple scales. The first one is the planar double box integral with a closed top loop, which is required for the top-pair production. This integral enters the next-to-next-to-leading order (NNLO) contribution for the process $pprightarrow t bar{t}$ and was a bottleneck for a long time. The system of differential equations for the double box integral is governed by three different elliptic curves, which originate from different sub-topologies. In order to solve the differential equation satisfied by the master integrals in this case, we use the factorization properties of the Picard--Fuchs operator associated with the elliptic' topologies to bring down the system of differential equation to the one coupled in blocks of sizes $2 times 2$ at worst, at order $epsilon^0$. We also use the linear form for the differential equation in order to solve the system as iterated integrals in the dimensional regularization parameter conveniently. The other example presented is that of the two-loop master integrals relevant to mixed QCD-EW corrections to the decay $Hrightarrow b bar{b}$ through a $Htbar{t}$ coupling. This has been done keeping full dependence on the heavy particle masses $(m_t,; m_H; text{and}; m_W)$, but neglecting the $b$-quark mass. In this case, the system of differential equations for the master integrals can be brought to the canonical form and the master integrals can be expressed entirely in terms of multiple polylogarithms.