A Core Representation Theorem for Scheme-Invariant Collinear Factorization in QCD

TL;DR

This paper introduces a universal scheme-invariant framework for collinear factorization in QCD, formalizing the redundancy in defining perturbative and non-perturbative components.

Contribution

It formalizes presentation non-uniqueness in QCD factorization using category theory and introduces an interface algebra object that captures scheme-invariant structures.

Findings

Identifies the universal scheme-invariant carrier as a relative tensor product.

Organizes coefficient and hadronic data as modules over an interface algebra.

Proves a minimal closure principle for long-distance operators.

Abstract

Collinear factorization and the leading-twist operator product expansion (OPE) in perturbative QCD express suitably inclusive observables in scale-separated kinematics as composites of perturbative short-distance coefficients with universal long-distance non-perturbative correlators such as parton distribution functions (PDFs), up to controlled power corrections. A persistent structural feature is \emph{presentation non-uniqueness}: coefficients and correlators are not individually physical, but are defined only up to finite factorization-scheme redefinitions induced by collinear subtractions and renormalized-operator mixing. We formalize this redundancy categorically by introducing an \emph{interface algebra object} encoding admissible finite collinear counterterms/mixing kernels and by organizing coefficient data and hadronic data as right/left modules over this algebra in a symmetric monoidal category encoding the chosen recomposition calculus. Our main result, the \emph{Core Representation Theorem}, identifies the universal scheme-invariant carrier: the functor of balanced (scheme-invariant) pairings is represented by the relative tensor product $C\otimes_A f$, which is terminal among all quotients of the naive composite $C\otimes f$ that preserve scheme-invariant semantics. Finally, we show how standard physics inputs (symmetry constraints, locality/OPE, and a stated accuracy truncation) canonically induce the interface algebra and module structures, and we prove a minimal closure principle for completing a generating set of long-distance operators/correlators to an $A$-stable sector.