Momentum Sum Rule Is Violated in The Operator Product Expansion in QCD At The High Energy Colliders

TL;DR

This paper demonstrates that the momentum sum rule in the operator product expansion (OPE) in QCD is violated at high energy colliders due to the incorrect assumption about the gauge invariant energy-momentum tensor operator.

Contribution

It reveals that the commonly assumed relation for the gauge invariant energy-momentum tensor operator does not hold, leading to a violation of the momentum sum rule in QCD OPE.

Findings

The momentum sum rule is violated in QCD OPE at high energies.

The relation $<P| { m f T}^{++}(0)|P>=2(P^+)^2$ holds only for the hadron's energy-momentum tensor.

The violation arises from the gauge invariance and the operator definitions within QCD.

Abstract

To prove the momentum sum rule in the operator product expansion (OPE) in QCD at high energy colliders it is assumed that $<P| {\hat T}^{++}(0)|P>=2(P^+)^2$ where $|P>$ is the momentum eigenstate of the hadron $H$ with momentum $P^\mu$ and ${\hat T}^{++}(0)$ is the $++$ component of the gauge invariant color singlet energy-momentum tensor density operator ${\hat T}^{\mu \nu}(0)$ of all the quarks plus antiquarks plus gluons inside the hadron $H$. However, in this paper, we show that this relation $<P| {\hat T}^{++}(0)|P>=2(P^+)^2$ is correct if ${\hat T}^{\mu \nu}(0)$ is the energy-momentum tensor density operator of the hadron but this relation $<P| {\hat T}^{++}(0)|P>=2(P^+)^2$ is not correct if ${\hat T}^{\mu \nu}(0)$ is the gauge invariant color singlet energy-momentum tensor density operator of all the quarks plus antiquarks plus gluons inside the hadron. Hence we find that the momentum sum rule is violated in the operator product expansion (OPE) in QCD at high energy colliders.