Differential equations for two-loop four-point functions

Abstract

At variance with fully inclusive quantities, which have been computed already at the two-or three-loop level, most exclusive observables are still known only at one-loop, as further progress was hampered so far by the greater computational problems encountered in the study of multi-leg amplitudes beyond one loop. We show in this paper how the use of tools already employed in inclusive calculations can be suitably extended to the computation of loop integrals appearing in the virtual corrections to exclusive observables, namely two-loop four-point functions with massless propagators and up to one off-shell leg. We find that multi-leg integrals, in addition to integration-by-parts identities, obey also identities resulting from Lorentz-invariance. The combined set of these identities can be used to reduce the large number of integrals appearing in an actual calculation to a small number of master integrals. We then write down explicitly the differential equations in the external invariants fulfilled by these master integrals, and point out that the equations can be used as an efficient method of evaluating the master integrals themselves. We outline strategies for the solution of the differential equations, and demonstrate the application of the method on several examples.

Two-loop QCD corrections to the heavy quark form factors: the vector contributions

“…As explained in [14,15,16], the form factors coming from the computation of the trace operation introduced in the previous Section are expressed in terms of several hundreds of scalar integrals. It is possible to express all these integrals as a combination of only 17 independent scalar integrals, called the Master Integrals (MIs) of the problem, by means of the so-called Laporta algorithm [17], using integration-by-parts identities [18], Lorentz invariance [19] and general symmetry relations. This reduction algorithm is performed exactly in D = (4 − 2ǫ) dimensions [20].…”

Master integrals for the 2-loop QCD virtual corrections to the forward–backward asymmetry

“…2. method of [11], but also the Lorentz invariance identities [12] and the symmetry relations, that can occur in particular mass configurations. The identities can then be solved by standard techniques (Gauss substitution rule), whose implementation is, however, algebraically very demanding.…”

QED vertex form factors at two loops

“…As the spinor traces are in D-dimensional space-time, in all the above formulas Eqs. (7)(8)(9)(10)(11), all the r.h.s., strictly speaking, should be multiplied by the overall constant (1/4) Tr1, where Tr1 is the trace of the unit Dirac matrix in D-continuous dimensions. The overall constant is in fact undetermined for arbitrary D, except for its limiting value at D = 4, which is 1.…”