How to compute one-loop Feynman diagrams in lattice QCD with Wilson fermions

Abstract

We describe an algebraic algorithm which allows to express every one-loop lattice integral with gluon or Wilsonfermion propagators in terms of a small number of basic constants which can be computed with arbitrary high precision. Although the presentation is restricted to four dimensions the technique can be generalized to every space dimension. We also give a method to express the lattice free propagator for Wilson fermions in coordinate space as a linear function of its values in eight points near the origin. This is an essential step in order to apply the recent methods of Lüscher and Weisz to higher-loop integrals with fermions.In [1] we presented a general technique which allows to express every one-loop bosonic integral at zero external momentum in terms of two unknown basic quantities which could be computed numerically with high precision. This method allows the complete evaluation of every diagram with gluon propagators.We have recently generalized [2] this technique to deal with integrals with both gluonic and fermionic propagators. For the fermions we use the Wilson action [3]. Notice that our method depends only on the structure of the propagator and thus it can be applied in calculations with the standard Wilson action as well as with the improved clover action [4]. We show that every integral at zero external momentum can be expressed in terms of a small number of basic quantities (nine for purely fermionic integrals, fifteen for integrals with bosonic and fermionic propagators). The advantage of this procedure is twofold: first of all every Feynman diagram can be computed in a completely symbolic way making it easier to perform checks and verify cancellations; moreover the basic constants can be easily computed with high precision and thus the numerical error on the final result can be reduced at will. Although the presentation is restricted to * Speaker at the conference.four dimensions the technique can be generalized to every space dimension. Definewhere n i are positive integers, p and q are arbitrary integers (not necessarily positive),is the denominator appearing in the propagator for Wilson fermions 2 . In the following when one2 To be precise, D F (k, m f ) is the denominator in the propagator for Wilson fermions only for m f = 0. For m f = 0 the correct denominator would bêHowever in our discussion m f will only play the role of an infrared regulator and thus it does not need to be the true fermion mass. The definition (3) is easier to handle than (4).

Two-loop critical mass for Wilson fermions

“…[5] for the pure gauge theory and extended in Ref. [6,7] to include Wilson fermions. This algorithm allows to express every one-loop integral with gluon and Wilson-fermion propagators in terms of a small number of basic constants that can be computed with arbitrary precision [6].…”

High-accuracy two-loop computation of the critical mass for Wilson fermions

“…Our result was expressed in terms of three purely bosonic constants Z 0 , Z 1 and F 0 and of 12 numerical constants that appear in the presence of Wilson fermions. The numerical values of these constant are obtained by using a powerful recursive method that gives very precise results [5,6]. This algorithm generalizes the method we introduced for purely bosonic integrals in [7].…”

High-precision computation of two-loop Feynman diagrams with Wilson fermions

“…At the Lattice conference of last year we presented [1] an algebraic algorithm that allows to express every one-loop lattice integral with gluon or Wilson-fermion propagator in terms of a small number of basic constants which can be computed with arbitrary high precision [2]. This was a generalization of what we did previously with purely bosonic integrals [3] and it was also an essential step in order to apply the coordinate-space method by Lüscher and Weisz [4] to higher-loop integrals with fermions.…”