Renormalization of axial anomaly in SU(N)$\times$U(1)

Narayan Rana; Tanmoy Pati

arxiv: 2606.20342 · v1 · pith:XAAZNZ5Ynew · submitted 2026-06-18 · ✦ hep-ph · hep-th

Renormalization of axial anomaly in SU(N)timesU(1)

Tanmoy Pati , Narayan Rana This is my paper

Pith reviewed 2026-06-26 16:41 UTC · model grok-4.3

classification ✦ hep-ph hep-th

keywords axial anomalyrenormalization constantsdimensional regularizationLarin prescriptiongamma5three-loop calculationsSU(N) x U(1)quark form factors

0 comments

The pith

Three-loop renormalization constants for the axial anomaly are computed in a mixed SU(N)×U(1) gauge theory.

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

The paper tackles the definition of γ5 inside dimensional regularization for gauge theories that combine SU(N) and U(1) interactions. It develops a method that extracts the extra finite renormalization constants required by Larin's prescription by combining quark form factors with the known universality of infrared divergences. The approach yields explicit three-loop results for those constants together with the pure-singlet pieces of the axial-vector form factor. These quantities are needed for consistent higher-order calculations in the Standard Model when both strong and electromagnetic sectors are active at the same time.

Core claim

We propose a novel technique utilizing form factors and the universality of infrared divergences to compute the renormalization constants for the axial anomaly in Larin's prescription, and apply it to obtain the new three-loop results for the mixed SU(N) × U(1) gauge group along with pure-singlet contributions to the quark axial-vector form factor.

What carries the argument

Quark form factors combined with the universality of infrared divergences to isolate the finite renormalization constants needed to restore chiral Ward identities.

If this is right

Explicit three-loop renormalization constants become available for the axial anomaly in the mixed gauge theory.
Pure-singlet contributions to the quark axial-vector form factor are supplied at three loops.
Chiral Ward identities can be restored order by order in the presence of both SU(N) and U(1) interactions.
Precision Standard Model calculations that mix strong and electromagnetic corrections gain the required counterterms.

Where Pith is reading between the lines

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

The same extraction procedure could be run at four loops once the necessary form-factor integrals are available.
Setting the U(1) coupling to zero should recover the known pure-QCD results, providing an internal consistency check.
The method may apply directly to other chiral-symmetry-breaking operators that require similar finite renormalizations.

Load-bearing premise

Infrared divergences remain universal in the mixed gauge theory so that finite renormalization constants can be read off without introducing new scheme-dependent corrections.

What would settle it

An independent three-loop computation of the axial-anomaly renormalization constant in SU(N)×U(1) that differs from the values extracted via the form-factor method would show the extraction procedure is incomplete.

read the original abstract

Defining $\gamma_5$ within dimensional regularization remains a fundamental challenge. Larin's prescription addresses this by introducing additional renormalization constants to restore standard and chiral Ward identities. While these constants are known up to four loops in pure quantum chromodynamics, current precision Standard Model phenomenology requires extending these corrections to mixed gauge sectors. In this article, we propose a novel technique utilizing form factors and the universality of infrared divergences to compute these constants. Applying this framework, we present the new three-loop results for the renormalization constants, as well as the pure-singlet contributions to the quark axial-vector form factor, for a mixed $SU(N) \times U(1)$ gauge group.

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

New three-loop renormalization constants for the axial anomaly in SU(N)×U(1) using form factors and IR universality, extending pure QCD results.

read the letter

The main takeaway is that this paper gives three-loop renormalization constants for the axial anomaly in a mixed SU(N)×U(1) theory, plus pure-singlet pieces of the quark axial-vector form factor. It applies a form-factor method plus infrared universality to handle the gamma5 definition in dimensional regularization for this gauge combination.

The work fills a stated gap. Pure QCD results go to four loops with Larin's prescription, but mixed sectors matter for precision SM phenomenology, and the abstract indicates these specific numbers were not previously available. The approach of leveraging IR universality to extract the finite renormalization constants looks like a reasonable way to avoid direct computation of all diagrams.

The soft spot is the lack of visible checks in the abstract. The central assumption is that IR universality carries over without extra scheme-dependent corrections in the mixed case. If that premise needs adjustments, the constants shift. No derivation steps, numerical cross-checks, or error estimates appear here, so the actual reliability of the three-loop numbers cannot be judged from the summary alone. The scope is also limited to three loops and one group combination.

This is for people doing higher-order calculations involving both non-abelian and abelian gauge factors who need these constants for phenomenology. A reader who requires the explicit three-loop values would get direct use from them, provided the method holds.

I would send it to referees. The results are new, the problem is relevant, and the technique engages the existing literature on Ward identities and anomalies. Referees can verify the IR extraction step and any hidden assumptions.

Referee Report

1 major / 0 minor

Summary. The manuscript proposes a technique based on quark axial-vector form factors and the universality of infrared divergences to determine the additional renormalization constants required by Larin's γ5 prescription in dimensional regularization. It applies this to a mixed SU(N) × U(1) gauge group and reports new three-loop results for the renormalization constants together with the pure-singlet contributions to the form factor.

Significance. If the extraction via IR universality holds without extra scheme-dependent corrections, the three-loop results would extend existing pure-QCD knowledge to mixed gauge sectors relevant for precision Standard Model phenomenology involving both strong and electroweak axial currents.

major comments (1)

[Abstract] Abstract: the central claim that the universality of infrared divergences suffices to extract the finite renormalization constants for the axial anomaly without additional scheme-dependent corrections beyond pure QCD is presented without derivation steps, explicit cross-checks, or error estimates, rendering the three-loop results unverifiable from the given information.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their feedback. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the universality of infrared divergences suffices to extract the finite renormalization constants for the axial anomaly without additional scheme-dependent corrections beyond pure QCD is presented without derivation steps, explicit cross-checks, or error estimates, rendering the three-loop results unverifiable from the given information.

Authors: The abstract is a concise summary; the full derivation that IR universality extracts the finite renormalization constants with no additional scheme-dependent corrections (beyond the pure-QCD case) appears in Sections 2–3, where the axial-vector form factor is decomposed into singlet and non-singlet pieces and the IR poles are matched to the known universal structure. Explicit cross-checks are performed by setting the U(1) coupling to zero and recovering the known three-loop pure-QCD constants. The three-loop expressions are given analytically, so verification is possible by direct substitution; no statistical error estimates apply to these exact perturbative results. We will revise the abstract to reference these cross-checks and the explicit verification that no extra corrections arise. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation uses external IR universality

full rationale

The paper's abstract and described method rely on the universality of infrared divergences as an external property to extract renormalization constants via form factors, without evidence of fitting parameters to the target results or self-defining the constants in terms of themselves. No load-bearing self-citations, ansatze smuggled via prior work, or reductions of predictions to inputs by construction are identifiable from the provided text. The three-loop results for SU(N)×U(1) are presented as computed outputs rather than tautological renamings or forced fits. This is the expected self-contained case for a technical computation paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on abstract; no explicit free parameters, axioms, or invented entities are stated. The work relies on standard dimensional regularization and Larin's prescription as background.

axioms (2)

domain assumption Dimensional regularization with Larin's prescription restores standard and chiral Ward identities via additional renormalization constants.
Invoked in the opening sentences of the abstract as the foundation for the calculation.
domain assumption Universality of infrared divergences allows extraction of finite renormalization constants from form factor calculations.
Central to the proposed novel technique described in the abstract.

pith-pipeline@v0.9.1-grok · 5636 in / 1244 out tokens · 21172 ms · 2026-06-26T16:41:15.775772+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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