GRACE at ONE-LOOP: Automatic calculation of 1-loop diagrams in the electroweak theory with gauge parameter independence checks

TL;DR

GRACE-at-one-loop presents a comprehensive, automated framework for calculating one-loop electroweak corrections in the Standard Model, from model-file generation to cross sections and event generation. The core innovations include a robust non-linear gauge fixing scheme that enables rigorous gauge-parameter independence checks, a novel tensor-integral reduction approach, and a complete renormalisation procedure with on-shell conditions tailored for unstable particles. The paper validates the system with extensive numerical tests, UV/IR finiteness, and gauge-parameter checks across a wide set of two-to-two processes, demonstrating remarkable agreement with existing results. These developments significantly advance automated, high-precision predictions for multi-leg processes and pave the way for reliable one-loop computations in future collider phenomenology.

Abstract

We describe the main building blocks of a generic automated package for the calculation of Feynman diagrams. These blocks include the generation and creation of a model file, the graph generation, the symbolic calculation at an intermediate level of the Dirac and tensor algebra, implementation of the loop integrals, the generation of the matrix elements or helicity amplitudes, methods for the phase space integrations and eventually the event generation. The report focuses on the fully automated systems for the calculation of physical processes based on the experience in developing GRACE-loop. As such, a detailed description of the renormalisation procedure in the Standard Model is given emphasizing the central role played by the non-linear gauge fixing conditions for the construction of such automated codes. The need for such gauges is better appreciated when it comes to devising efficient and powerful algorithms for the reduction of the tensorial structures of the loop integrals. A new technique for these reduction algorithms is described. Explicit formulae for all two-point functions in a generalised non-linear gauge are given, together with the complete set of counterterms. We also show how infrared divergences are dealt with in the system. We give a comprehensive presentation of some systematic test-runs which have been performed at the one-loop level for a wide variety of two-to-two processes to show the validity of the gauge check. These cover fermion-fermion scattering, gauge boson scattering into fermions, gauge bosons and Higgs bosons scattering processes. Comparisons with existing results on some one-loop computation in the Standard Model show excellent agreement. We also briefly recount some recent development concerning the calculation of mutli-leg one-loop corrections.

Figures (20)

• Figure 1: GRACE System Flow.
• Figure 2: An example of a model file in GRACE.
• Figure 3: Particle description in LanHEP
• Figure 4: Entering the Higgs interaction in LanHEP
• Figure 5: Introducing wave function counterterms in LanHEP