The Use of Schoonschip and Form in Perturbative Lattice Calculations

TL;DR

This paper presents the development of algebraic computation codes using Schoonschip and Form for lattice perturbation theory, addressing the challenges of lattice-specific gamma matrix manipulations, and applies them to compute renormalization constants in lattice QCD.

Contribution

The authors developed specialized routines for lattice gamma matrices within Schoonschip and Form, enabling automated algebraic calculations in lattice perturbation theory.

Findings

Successfully computed 1-loop renormalization constants for deep inelastic scattering operators.

Addressed and solved the issue of gamma matrix handling on the lattice.

Demonstrated the applicability of the codes to complex lattice QCD calculations.

Abstract

Using the formal languages Schoonschip and Form, we have developed general codes that are able to carry out all the algebraic manipulations needed to perform analytic lattice calculations, starting from the elementary building blocks (propagators and vertices) of each Feynman diagram. The main difficulty resides in the fact that, although there are many built in instructions to deal with Dirac gamma-matrices, Schoonschip and Form have been conceived having in mind a continuum theory, which is invariant with respect to the Lorentz group. On the lattice, on the contrary, a field theory is only invariant with respect to the hypercubic group, contained in the (euclidean) Lorentz group and not every pair of equal indices should be summed over. Being impossible to directly use the `gammatrics' of Schoonschip and Form as they are, special routines have been developed to correctly treat gamma matrices on the lattice, while using as much as possible of the built in Schoonschip and Form commands. We have used our codes to compute, in 1-loop perturbation theory in lattice QCD, the renormalization constants and mixing coefficients of the operators that enter in the determination of the first two moments of deep inelastic scattering structure functions.