Roman N. Lee

LiteRed 1.4: a powerful tool for reduction of multiloop integrals

We review the Mathematica package LiteRed , version 1.4.

Analytic Results for Massless Three-Loop Form Factors

We evaluate, exactly in d, the master integrals contributing to massless threeloop QCD form factors. The calculation is based on a combination of a method recently suggested by one of the authors (R.L.) with other techniques: sector decomposition implemented in FIESTA, the method of Mellin-Barnes representation, and the PSLQ algorithm. Using our results for the master integrals we obtain analytical expressions for two missing constants in the ϵ-expansion of the two most complicated master integrals and present the form factors in a completely analytic form.

Reducing differential equations for multiloop master integrals

A bstractWe present an algorithm of the reduction of the differential equations for master integrals the Fuchsian form with the right-hand side matrix linearly depending on dimensional regularization parameter ϵ. We consider linear transformations of the functions column which are rational in the variable and in ϵ. Apart from some degenerate cases described below, the algorithm allows one to obtain the required transformation or to ascertain irreducibility to the form required. Degenerate cases are quite anticipated and likely to correspond to irreducible systems.

Master integrals for four-loop massless propagators up to weight twelve

Abstract We evaluate a Laurent expansion in dimensional regularization parameter ϵ = ( 4 − d ) / 2 of all the master integrals for four-loop massless propagators up to weight twelve, using a recently developed method of one of the present coauthors (R.L.) and extending thereby results by Baikov and Chetyrkin obtained at weight seven. We observe only multiple zeta values in our results. Therefore, we conclude that all the four-loop massless propagator integrals, with any integer powers of numerators and propagators, have only multiple zeta values in their epsilon expansions up to weight twelve.

Group structure of the integration-by-part identities and its application to the reduction of multiloop integrals

The excessiveness of integration-by-part (IBP) identities is discussed. The Lie-algebraic structure of the IBP identities is used to reduce the number of the IBP equations to be considered. It is shown that Lorentz-invariance (LI) identities do not bring any information additional to that contained in the IBP identities, and therefore, can be discarded.

Critical points and number of master integrals

A bstractWe consider the question about the number of master integrals for a multiloop Feynman diagram. We show that, for a given set of denominators, this number is totally determined by the critical points of the polynomials entering either of the two representations: the parametric representation and the Baikov representation. In particular, for the parametric representation the corresponding polynomial is just the sum of Symanzik polynomials. The relevant topological invariant is the sum of the Milnor numbers of the proper critical points. We present a Mathematica package Mint to automatize the counting of the master integrals for the typical case, when all critical points are isolated.

Analytic epsilon expansion of three-loop on-shell master integrals up to four-loop transcendentality weight

We evaluate analytically higher terms of the ϵ-expansion of the three-loop master integrals corresponding to three-loop quark and gluon form factors and to the three-loop master integrals contributing to the electron g − 2 in QED up to the transcendentality weight typical to four-loop calculations, i.e. eight and seven, respectively. The calculation is based on a combination of a method recently suggested by one of the authors (R.L.) with other techniques: sector decomposition implemented in FIESTA, the method of Mellin-Barnes representation, and the PSLQ algorithm.We evaluate analytically higher terms of the epsilon-expansion of the three-loop master integrals corresponding to three-loop quark and gluon form factors and to the three-loop master integrals contributing to the electron g-2 in QED up to the transcendentality weight typical to four-loop calculations, i.e. eight and seven, respectively. The calculation is based on a combination of a method recently suggested by one of the authors (R.L.) with other techniques: sector decomposition implemented in FIESTA, the method of Mellin--Barnes representation, and the PSLQ algorithm.

Four-loop corrections with two closed fermion loops to fermion self energies and the lepton anomalous magnetic moment

A bstractWe compute the eighth-order fermionic corrections involving two and three closed massless fermion loops to the anomalous magnetic moment of the muon. The required four-loop on-shell integrals are classified and explicit analytical results for the master integrals are presented. As further applications we compute the corresponding four-loop QCD corrections to the mass and wave function renormalization constants for a massive quark in the on-shell scheme.

Dimensional recurrence relations: an easy way to evaluate higher orders of expansion in ε

Applications of a method recently suggested by one of the authors (R.L.) are presented. This method is based on the use of dimensional recurrence relations and analytic properties of Feynman integrals as functions of the parameter of dimensional regularization, d . The method was used to obtain analytical expressions for two missing constants in the e -expansion of the most complicated master integrals contributing to the three-loop massless quark and gluon form factors and thereby present the form factors in a completely analytic form. To illustrate its power we present, at transcendentality weight seven, the next order of the e -expansion of one of the corresponding most complicated master integrals. As a further application, we present three previously unknown terms of the expansion in e of the three-loop non-planar massless propagator diagram. Only multiple ζ values at integer points are present in our result.

Calculating multiloop integrals using dimensional recurrence relation and D-analyticity

We review the method of the calculation of multiloop integrals recently suggested in Ref.[R. Lee, Nuclear Physics B 830 (2010) 474, 0911.0252 ]. A simple method of derivation of the dimensional recurrence relation suitable for automatization is given. Some new analytic results are given.