# Violation of the Kluberg-Stern--Zuber theorem in SCET

## TL;DR

This paper demonstrates that the classical Kluberg-Stern--Zuber theorem does not hold in SCET, showing non-zero mixing of operators vanishing by equations of motion, which impacts the calculation of anomalous dimensions in QCD scattering amplitudes.

## Contribution

It reveals the breakdown of the Kluberg-Stern--Zuber theorem in SCET and identifies the necessary field choices for correct infrared divergence reproduction in QCD.

## Key findings

- Non-vanishing mixing of eom operators in SCET.
- Identification of preferred fields for infrared divergence matching.
- Divergence of convolution integrals at subleading power.

## Abstract

A classic result, originally due to Kluberg-Stern and Zuber, states that operators that vanish by the classical equation of motion (eom) do not mix into "physical" operators. Here we show that and explain why this result does not hold in soft-collinear effective theory (SCET) for the renormalization of power-suppressed operators. We calculate the non-vanishing mixing of eom operators for the simplest case of $N$-jet operators with a single collinear field in every direction. The result implies that---for the computation of the anomalous dimension but not for on-shell matrix elements---there exists a preferred set of fields that must be used to reproduce the infrared singularities of QCD scattering amplitudes. We identify these fields and explain their relation to the gauge-invariant SCET Lagrangian. Further checks reveal another generic property of SCET beyond leading power, which will be relevant to resummation at the next-to-leading logarithmic level, the divergence of convolution integrals with the hard matching coefficients. We propose an operator solution that allows to consistently renormalize such divergences.