Group-theoretic relations for amplitudes in gauge theories with orthogonal and symplectic groups

TL;DR

This paper extends a group-theoretic method to derive linear constraints on four-point gluon amplitudes in SO(N) and Sp(2N) gauge theories, revealing loop-order dependent relations and independent amplitude counts.

Contribution

It applies a pure group-theoretic iterative method to SO(N) and Sp(2N) gauge theories, deriving new linear amplitude constraints up to four loops.

Findings

Identifies specific numbers of constraint relations at each loop order.

Determines the count of independent four-point amplitudes at each loop order.

Extends previous SU(N) results to orthogonal and symplectic gauge groups.

Abstract

It is important to find nontrivial constraint relations for color-ordered amplitudes in gauge theories. In the past several years, a pure group-theoretic iterative method has been proposed to derive linear constraints on color-ordered amplitudes in SU(N) gauge theories. In this paper, we use the same method to derive linear constraints on four-point gluon amplitudes in SO(N) and Sp(2N) gauge theories. These constraints are derived up to four-loop order. It is found that there are $n=1,6,10,13,16$ constraint relations at $L=0,1,2,3,4$ loop orders in both SO(N) and Sp(2N) cases. Correspondingly, the numbers of independent four-point color-ordered amplitudes are $2,3, 5, 8, 11$ at $L=0,1,2,3,4$ loop orders in both theories.