# Hidden Analytic Relations for Two-Loop Higgs Amplitudes in QCD

## TL;DR

This paper reveals hidden analytic relations between two-loop Higgs amplitudes in QCD, extending the maximal transcendentality principle to include quark external states and connecting QCD results with ${
m extbf{N}=4}$ SYM theory.

## Contribution

It generalizes the maximal transcendentality principle to Higgs amplitudes with quark states in QCD and uncovers new relations with ${
m extbf{N}=4}$ SYM theory.

## Key findings

- Maximally transcendental parts of $H
ightarrow q ar{q} g$ and $H
ightarrow 3g$ amplitudes are equivalent.
- The correspondence extends to two-loop form factors of more general operators.
- Relations between Higgs amplitudes and minimal density form factors in ${
m extbf{N}=4}$ SYM are observed.

## Abstract

We compute the Higgs plus two-quark and one-gluon amplitudes ($H \rightarrow q \bar{q} g$) and Higgs plus three-gluon amplitudes ($H \rightarrow 3g$) in the Higgs effective theory with a general class of operators. By changing the quadratic Casimir $C_F$ to $C_A$, the maximally transcendental parts of the $H \rightarrow q \bar{q} g$ amplitudes turn out to be equivalent to that of the $H \rightarrow 3g$ amplitudes, which also coincide with the counterparts in ${\cal N}=4$ SYM. This generalizes the so-called maximal transcendentality principle to the Higgs amplitudes with external quark states, thus to the full QCD theory. We further verify that the correspondence applies also to two-loop form factors of more general operators, in both QCD and scalar-YM theory. Another interesting relation is also observed between the planar $H \rightarrow q \bar{q} g$ amplitudes and the minimal density form factors in ${\cal N}=4$ SYM.