Singlet and non-singlet three-loop massive form factors

Abstract

We consider Quantum Chromodynamics with external vector, axial-vector, scalar and pseudo-scalar currents and compute three-loop corrections to the corresponding vertex function taking into account massive quarks. We consider all non-singlet contributions as well as those singlet contributions where the external current couples to a massive quark loop. We apply a semi-numerical method which is based on expansions around singular and regular kinematical points. They are matched at intermediate values of the squared partonic center-of-mass energy s which allows to cover the whole kinematic range for negative and positive values of s. Our method permits a systematic increase of the precision by varying the expansion depth and the choice of the intermediate matching points. In our current set-up we have at least seven significant digits for the finite contribution of all form factors. We present our results as a combination of series expansions and interpolation functions which allows for a straightforward use in practical applications.

Towards gg → HH at next-to-next-to-leading order: Light-fermionic three-loop corrections

“…The final step is to use the "expand and match" approach [17,18] to obtain "semianalytic" results which cover the whole s range. Note that this approach properly takes into account the threshold effects at the point s = 4m 2 t .…”

“…However, after an expansion in the Mandelstam variable t (see Refs. [5,6,16]) and the application of the "expand and match" [17,18] method to compute the master integrals, one obtains semi-analytic results which cover a large part of the phase space. Such a result allows the study of the renormalizations scheme dependence at three-loop order.…”

Analytic approximations of $2\to 2$ processes with massive internal particles

“…These integrals are obtained with the help of differential equations using the "expand and match" approach developed in Refs. [22,23]. The boundary conditions are obtained from the large-m t limit, in which the integrals are simple and can be computed analytically.…”