arxiv: 2604.13439 · v2 · submitted 2026-04-15 · ✦ hep-ph · hep-th

Recognition: unknown

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

Dustin Keller

Authors on Pith no claims yet

Pith reviewed 2026-05-10 13:50 UTC · model grok-4.3

classification ✦ hep-ph hep-th

keywords collinear factorizationQCDscheme invariancerepresentation theoremoperator product expansionparton distribution functionsinterface algebrarelative tensor product

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The pith

Scheme-invariant collinear factorization in perturbative QCD is represented by the relative tensor product of coefficient and hadronic modules over an interface algebra.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper addresses the long-standing issue that perturbative coefficients and non-perturbative correlators in QCD collinear factorization are defined only up to finite redefinitions from scheme choices and operator mixing. It encodes all admissible redefinitions in a single interface algebra object and treats the coefficient functions and hadronic matrix elements as modules over this algebra in a symmetric monoidal category. The central theorem states that the functor of balanced, scheme-invariant pairings is represented exactly by the relative tensor product of these modules, which is terminal among all quotients that preserve physical semantics. This construction turns the usual scheme ambiguity into a canonical, choice-independent carrier for observables. A general reader cares because the result supplies a universal language in which physical predictions are manifestly independent of arbitrary factorization conventions.

Core claim

The Core Representation Theorem identifies the universal scheme-invariant carrier: the functor of balanced (scheme-invariant) pairings is represented by the relative tensor product C⊗_A f, which is terminal among all quotients of the naive composite C⊗f that preserve scheme-invariant semantics. Standard physics inputs (symmetry constraints, locality from the OPE, and a stated accuracy truncation) canonically induce the interface algebra and the module structures without further choices, and a minimal closure principle completes any generating set of long-distance operators to an A-stable sector.

What carries the argument

The relative tensor product C ⊗_A f over the interface algebra A, which is the terminal quotient of the naive composite that preserves scheme-invariant semantics.

If this is right

Physical observables are obtained directly from the relative tensor product and remain unchanged under any scheme redefinition encoded by the algebra.
The minimal closure principle supplies a canonical way to enlarge any finite set of long-distance correlators to a complete A-stable sector.
Symmetry constraints, locality of the OPE, and truncation order determine the algebra and modules uniquely from standard QCD inputs.
Coefficients and correlators need never be chosen in a specific scheme; only their balanced pairing over the algebra is required for predictions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same categorical organization could be applied to other factorization theorems in QCD, such as transverse-momentum-dependent factorization, by constructing analogous interface algebras.
Numerical or symbolic implementations of the relative tensor product would allow automated, scheme-independent global fits of parton distributions.
The terminal property of the relative tensor product suggests a natural route to include power-suppressed corrections by enlarging the algebra while preserving the representation.

Load-bearing premise

The introduced interface algebra object fully encodes every admissible finite collinear counterterm and mixing kernel, and standard physics inputs induce the algebra and module structures without extra choices.

What would settle it

An explicit higher-order collinear counterterm or mixing kernel that cannot be realized as a morphism in the interface algebra, or a computed physical observable whose value changes when a redefinition outside the relative tensor product is applied.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Keller gives a categorical framing to scheme dependence in QCD factorization, with a representation theorem for the invariant carrier via relative tensor products over an interface algebra.

read the letter

The one or two things to know are that this paper recasts the known redundancy between coefficients and PDFs in collinear factorization as a problem in symmetric monoidal categories, and it proves a Core Representation Theorem saying the scheme-invariant pairings are represented by the relative tensor product C ⊗_A f, which is terminal among suitable quotients. It also claims that standard inputs like symmetries, OPE locality, and truncation induce the interface algebra and module structures canonically, plus a closure principle for the operators. This is new as a formal structuring of presentation non-uniqueness, and it does well by giving a clean terminal object for what physicists already treat as invariant under finite scheme changes. The approach organizes the mixing kernels and counterterms in a way that could clarify higher-order work without case-by-case adjustments. The soft spot is the induction step for the interface algebra. The stress-test concern is fair to raise: if choosing generators for counterterms or defining how convolutions interact with truncation leaves any residual freedom, then distinct but equivalent algebras could produce non-isomorphic relative tensor products and weaken the universality claim. The abstract presents the induction as canonical from physics inputs, so the full derivation needs to show there is no leftover choice. No circularity or load-bearing fitting appears in what is described. This paper is for perturbative QCD theorists who handle factorization at higher orders and are willing to use categorical language. A reader already thinking about scheme ambiguities will see the value in the terminality result. It deserves a serious referee to check the explicit construction of the algebra and whether the terminal property survives the truncation. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a categorical formalism for scheme-invariant collinear factorization in QCD. It introduces an interface algebra object A encoding finite collinear counterterms and mixing kernels. Coefficient data C and hadronic data f are structured as right and left modules over A within a symmetric monoidal category. The central Core Representation Theorem asserts that the relative tensor product C ⊗_A f represents the functor of balanced scheme-invariant pairings and is terminal among quotients of the naive C ⊗ f that preserve scheme-invariant semantics. The paper claims that physics inputs canonically induce these structures and proves a closure principle for operator sets.

Significance. If the theorem and induction hold, this work provides a rigorous foundation for identifying universal scheme-invariant quantities in QCD factorization, potentially resolving ambiguities in PDF and coefficient function definitions. The categorical approach using relative tensor products and terminal objects is a novel contribution that could aid in systematic higher-order calculations. The paper includes a representation theorem and a closure principle, which are notable strengths. Its impact would be greater with explicit links to phenomenological applications.

major comments (2)

Core Representation Theorem: The assertion that C ⊗_A f is terminal among quotients preserving scheme-invariant semantics depends critically on A being canonically and uniquely induced by the physics inputs without residual freedom. If choices in generators for counterterms or the interaction of convolution with truncation accuracy are not fully constrained, distinct A's could produce non-isomorphic relative tensor products, undermining the universality claim. An explicit proof or construction showing isomorphism independence is required.
Section on canonical induction of the interface algebra: The claim that symmetry constraints, locality/OPE, and accuracy truncation canonically induce the interface algebra A and the module structures on C and f is load-bearing for the central result. The manuscript should provide the step-by-step induction to confirm that no additional choices are involved in defining the monoidal recomposition calculus.

minor comments (2)

The notation for the symmetric monoidal category and the relative tensor product should be introduced with more explicit definitions and examples from QCD to improve accessibility for the target audience.
Consider adding a diagram illustrating the module structures, the naive composite, and the quotient process to clarify the terminality argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The comments highlight important points regarding the canonical induction of the interface algebra and the universality of the Core Representation Theorem. We address each major comment below and have revised the manuscript accordingly to strengthen the presentation.

read point-by-point responses

Referee: Core Representation Theorem: The assertion that C ⊗_A f is terminal among quotients preserving scheme-invariant semantics depends critically on A being canonically and uniquely induced by the physics inputs without residual freedom. If choices in generators for counterterms or the interaction of convolution with truncation accuracy are not fully constrained, distinct A's could produce non-isomorphic relative tensor products, undermining the universality claim. An explicit proof or construction showing isomorphism independence is required.

Authors: We agree that the universality claim requires demonstrating that A is induced without residual freedom leading to non-isomorphic tensor products. The manuscript constructs A from the physics inputs, but to make the independence explicit we have added a new lemma (Lemma 4.3) and its proof in the revised Section 4. The proof shows that any two algebras A and A' induced by the same symmetry constraints, OPE locality, and truncation accuracy are canonically isomorphic via a unique algebra homomorphism that intertwines the module structures on C and f. Consequently, the relative tensor products are isomorphic, preserving the terminal property among scheme-invariant quotients. This addresses the potential for distinct generators or truncation interactions. revision: yes
Referee: Section on canonical induction of the interface algebra: The claim that symmetry constraints, locality/OPE, and accuracy truncation canonically induce the interface algebra A and the module structures on C and f is load-bearing for the central result. The manuscript should provide the step-by-step induction to confirm that no additional choices are involved in defining the monoidal recomposition calculus.

Authors: We concur that a fully explicit step-by-step induction is essential to confirm canonicity. The original manuscript outlines the induction from the three inputs but does not break it down sequentially. In the revision we have expanded Section 3.2 with a detailed inductive construction: (i) symmetry constraints fix the generators of A as an algebra object; (ii) locality and OPE determine the relations and the left/right module actions on f and C respectively; (iii) the accuracy truncation completes the structure by imposing a finite filtration that is stable under the monoidal recomposition. At each step we verify that the physical requirements leave no free choices, so the monoidal calculus is uniquely determined. This makes the load-bearing claim fully rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under stated assumptions

full rationale

The Core Representation Theorem is presented as a categorical result in a symmetric monoidal category where the interface algebra A encodes admissible counterterms and mixing kernels, with C and f as modules, and the relative tensor product C⊗_A f shown terminal among quotients preserving scheme-invariant semantics. The paper states that standard physics inputs (symmetry constraints, locality/OPE, accuracy truncation) canonically induce A and the module structures, followed by a minimal closure principle. No equations or steps reduce the claimed universal carrier to a fitted parameter, self-referential definition, or load-bearing self-citation; the induction is asserted from external QCD principles without internal redefinition of the target result. The structure is therefore independent of the theorem's conclusion and does not match any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 2 invented entities

The central claim rests on the existence and properties of an interface algebra that captures all finite scheme redefinitions, together with the assumption that QCD locality, OPE, and symmetry constraints induce the module structures without extra data. No numerical free parameters are introduced. Invented entities are the interface algebra and the relative tensor product construction.

axioms (3)

standard math Symmetric monoidal category axioms and module category structures over an algebra object
Invoked to organize coefficient and hadronic data as right/left modules and to define the relative tensor product.
domain assumption Collinear factorization and leading-twist OPE hold up to controlled power corrections in perturbative QCD
Stated in the abstract as the physical setting in which the scheme non-uniqueness arises.
domain assumption Standard physics inputs (symmetry constraints, locality/OPE, accuracy truncation) canonically induce the interface algebra and module structures
Claimed in the abstract as the mechanism that determines the algebra without additional choices.

invented entities (2)

Interface algebra object no independent evidence
purpose: To encode admissible finite collinear counterterms and mixing kernels that induce scheme redefinitions
New algebraic object introduced to formalize presentation non-uniqueness; no independent evidence provided beyond the construction itself.
Relative tensor product C ⊗_A f no independent evidence
purpose: To represent the universal scheme-invariant carrier of balanced pairings
Constructed as the terminal quotient preserving scheme-invariant semantics; defined within the paper's categorical framework.

pith-pipeline@v0.9.0 · 5547 in / 1805 out tokens · 23942 ms · 2026-05-10T13:50:44.781293+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 4 canonical work pages

[1]

Collins.Foundations of Perturbative QCD

John C. Collins.Foundations of Perturbative QCD. Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology. Cambridge University Press, 2011

2011
[2]

Collins, Davison E

John C. Collins, Davison E. Soper, and George Sterman. Factorization of hard processes in qcd. In A. H. Mueller, editor,Perturbative QCD, volume 5 ofAdvanced Series on Directions in High Energy Physics, pages 1–91. World Scientific, Singapore, 1989

1989
[3]

E. C. Aschenauer, R. S. Thorne, and R. Yoshida. Structure functions.Phys. Rev. D, 110:030001, 2024. doi:10.1103/PhysRevD.110.030001; review PDF: https://pdg.lbl.gov/ 2024/reviews/rpp2024-rev-structure-functions.pdf. 21

work page doi:10.1103/physrevd.110.030001 2024
[4]

Kenneth G. Wilson. Non-lagrangian models of current algebra.Phys. Rev., 179:1499–1512, 1969

1969
[5]

Kenneth G. Wilson. Operator-product expansions and anomalous dimensions in the thirring model.Phys. Rev. D, 2:1473–1478, 1970

1970
[6]

Wilson and W

Kenneth G. Wilson and W. Zimmermann. Operator product expansions and composite field operators in the general framework of quantum field theory.Commun. Math. Phys., 24:87–106, 1972

1972
[7]

Smits and Mark Kotanchek

Guido F. Smits and Mark Kotanchek. Pareto-front exploitation in symbolic regression. In Una-May O’Reilly, Tina Yu, Rick L. Riolo, and Bill Worzel, editors,Genetic Programming Theory and Practice II, pages 283–299. Springer, 2005

2005
[8]

Distilling free-form natural laws from experimental data

Michael Schmidt and Hod Lipson. Distilling free-form natural laws from experimental data. Science, 324(5923):81–85, 2009

2009
[9]

William La Cava, Bogdan Burlacu, Marco Virgolin, Michael Kommenda, Patryk Orzechowski, Felipe Olivetti de Fran¸ ca, Yuge Jin, and Jason H. Moore. Contemporary symbolic regression methods and their relative performance.Adv. Neural Inf. Process. Syst., 34:1–16, 2021

2021
[10]

Brunton, Joshua L

Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems.Proc. Natl. Acad. Sci. USA, 113(15):3932–3937, 2016

2016
[11]

Ai feynman: A physics-inspired method for symbolic regression.Sci

Silviu-Marian Udrescu and Max Tegmark. Ai feynman: A physics-inspired method for symbolic regression.Sci. Adv., 6(16):eaay2631, 2020

2020
[12]

Interpretable machine learning for science with pysr and symbolicregression.jl, 2023

Miles Cranmer. Interpretable machine learning for science with pysr and symbolicregression.jl, 2023

2023
[13]

2024 Interpretable scientific discovery with symbolic regression: a review

Nour Makke and Sanjay Chawla. Interpretable scientific discovery with symbolic regression: A review.Artif. Intell. Rev., 57(2), 2024. doi:10.1007/s10462-023-10622-0; arXiv:2211.10873

work page doi:10.1007/s10462-023-10622-0 2024
[14]

Bl¨ umlein, V

J. Bl¨ umlein, V. Ravindran, and W. L. van Neerven. On the Drell–Levy–Yan relation to O(α2 s).Nucl. Phys. B, 586:349–381, 2000

2000
[15]

Bl¨ umlein and M

J. Bl¨ umlein and M. Saragnese. TheN 3LO scheme-invariant QCD evolution of the non-singlet structure functionsF NS 2 (x, Q2) andg NS 1 (x, Q2).Phys. Lett. B, 820:136589, 2021

2021
[16]

Springer, 1991

William Fulton and Joe Harris.Representation Theory: A First Course, volume 129 of Graduate Texts in Mathematics. Springer, 1991

1991
[17]

Springer, Cham, 2015

Thomas Becher, Alessandro Broggio, and Andrea Ferroglia.Introduction to Soft-Collinear Effective Theory, volume 896 ofLecture Notes in Physics. Springer, Cham, 2015

2015
[18]

C. W. Bauer and M. Neubert. Heavy-quark and soft-collinear effective theory. inReview of Particle Physics(Particle Data Group), 2024. Review chapter PDF available from PDG

2024
[19]

Lin, Mapping parton distributions of hadrons with lattice QCD, Prog

Huey-Wen Lin. Mapping parton distributions of hadrons with lattice qcd.Prog. Part. Nucl. Phys., 144:104177, 2025. arXiv:2506.05025. 22

work page arXiv 2025
[20]

A hybrid renormalization scheme for quasi light-front correlations in large-momentum effective theory.Nucl

Xiangdong Ji, Yu Liu, Andreas Sch¨ afer, Wei Wang, Yi-Bo Yang, Jian-Hui Zhang, and Yong Zhao. A hybrid renormalization scheme for quasi light-front correlations in large-momentum effective theory.Nucl. Phys. B, 964:115311, 2021

2021
[21]

M. Beneke. Renormalons.Phys. Rept., 317:1–142, 1999. arXiv:hep-ph/9807443, doi:10.1016/S0370-1573(98)00130-6

work page doi:10.1016/s0370-1573(98)00130-6 1999
[22]

Hopf algebras, renormalization and noncommutative geometry.Commun

Alain Connes and Dirk Kreimer. Hopf algebras, renormalization and noncommutative geometry.Commun. Math. Phys., 199:203–242, 1998

1998
[23]

Factorization algebra, 2023

Kevin Costello and Owen Gwilliam. Factorization algebra, 2023

2023
[24]

Williams

Minghao Wang and Brian R. Williams. On the renormalization and quantization of topological-holomorphic field theories, 2024. 23

2024