Group constraint relations for five-point amplitudes in gauge theories with SO(N) and Sp(2N) groups

TL;DR

This paper derives linear constraint relations among five-point amplitudes in SO(N) and Sp(2N) gauge theories up to four loops, revealing the number of independent amplitudes at each loop order.

Contribution

It introduces a group-theoretic method to derive loop-order constraints for five-point amplitudes in these gauge theories, extending previous results.

Findings

Identifies 6, 22, 34, 44, 50 constraints at loop orders 0 to 4.

Determines the number of independent amplitudes at each loop order.

Provides a systematic way to analyze color-ordered amplitudes in SO(N) and Sp(2N) theories.

Abstract

In this paper,linear constraint relations among loop-order five-point color-ordered amplitudes in $SO(N)$ and $Sp(2N)$ gauge theories are derived with the group-theoretic method. These constrains are derived up to four-loop order. It is found that in both theories, there are $n=6,22,34,44,50$ linear constraint relations at $L=0,1,2,3,4$ loop orders. Then the numbers of independent color-ordered five-point amplitudes are respectively $n_{ind.}=6,12,22,34,50$ at each loop order.