On the definition of fragmentation functions and the violation of sum rules

TL;DR

This paper examines the limitations of sum rules for quark fragmentation functions in QCD, highlighting the role of bound states at fractional momentum and proposing future research directions to better understand hadronization.

Contribution

It identifies a fundamental problem with the derivation of sum rules in QCD due to bound states in the final state, suggesting a revised understanding of fragmentation functions.

Findings

Bound states at fractional momentum z=0 affect sum rule validity.

Charge sum rules may be violated, momentum sum rules remain valid.

Proposes future research to connect experimental fragmentation functions with non-perturbative hadronization models.

Abstract

We point out a problem with the formulation and derivations of sum rules for quark fragmentation functions that impacts their validity in QCD, but which potentially points toward an improved understanding of final states in inclusive hard processes. Fragmentation functions give the distribution of final-state hadrons arising from a parton exiting a hard scattering, and the sum rules for momentum, electric charge, etc express conservation of these quantities. The problem arises from a mismatch between the quark quantum numbers of the initial quark and the fact that all observed final-state hadrons are confined bound states with color zero. We point that, in a confining theory like QCD, the Wilson line in the operator definition of a fragmentation function entails that the final state in a fragmentation function includes a bound state in the external field generated by the Wilson line. We justify this with the aid of general features of string hadronization. The anomalous bound states are restricted to fractional momentum $z=0$. They tend to invalidate sum rules like the one for charge conservation when applied to the fragmentation functions inferred from experimental data, but not the momentum sum rule. We propose to exploit our ideas in future studies as a way to relate the ffs extracted from inclusive cross sections to more detailed non-perturbative descriptions of final state hadronization. We also describe scenarios wherein the traditional sum rules might remain approximately valid with a reasonably high degree of accuracy.