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arxiv: 2606.12959 · v1 · pith:Z4663MXXnew · submitted 2026-06-11 · ✦ hep-ph · nucl-th

Nucleon matrix elements of axial anomaly, axial currents and pseudoscalar currents in the QCD sum rule

Janardan Prasad Singh This is my paper

Pith reviewed 2026-06-27 06:45 UTC · model grok-4.3

classification ✦ hep-ph nucl-th
keywords QCD sum rulesnucleon matrix elementsaxial anomalypseudoscalar currentsaxial currentsparton distribution functionscoupling constantsquark gluon operators
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The pith

QCD sum rules express nucleon couplings to pseudoscalar and axial currents in terms of operator matrix elements and parton distribution moments.

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

The paper applies QCD sum rules to one-nucleon matrix elements of correlators built from pseudoscalar octet, isovector, isoscalar, axial anomaly, and axial currents. It derives expressions for the nucleon coupling constants to these currents using matrix elements of quark, gluon, and quark-gluon operators together with moments of parton distribution functions. Non-diagonal contributions from excited and continuum states are included on the phenomenological side. Two expressions for the pseudoscalar couplings are found, one depending only on parton moments, that agree numerically. The nucleon matrix element of the axial anomaly is highlighted because it is rarely studied.

Core claim

Using QCD sum rules the coupling constants of the nucleon with pseudoscalar octet, isovector and isoscalar currents, axial anomaly, and axial isovector and isoscalar currents have been expressed in terms of nucleon matrix elements of quark, gluon and quark-gluon composite operators and moments of parton distribution function. Contributions from non-diagonal matrix elements between the nucleon and excited or continuum states have been accounted for. For the pseudoscalar coupling constants two expressions have been obtained, one consisting of only moments of parton distribution function but yielding approximately same numerical result. The nucleon matrix element of axial anomaly has been analy

What carries the argument

One-nucleon matrix elements of current-current correlators analyzed via QCD sum rules, which equate the phenomenological side including couplings and excited states to the operator product expansion side with composite operator matrix elements and PDF moments.

If this is right

  • The pseudoscalar coupling constants admit an expression depending solely on parton distribution moments.
  • The axial anomaly nucleon matrix element can be isolated and computed this way.
  • Accounting for non-diagonal transitions improves the sum rule accuracy for these couplings.
  • These relations link nucleon structure at low energy to high-energy parton data.

Where Pith is reading between the lines

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

  • If the expressions hold, they could help resolve contributions to the nucleon spin from the axial anomaly.
  • Similar sum rule techniques might determine couplings for other hadrons or currents.
  • Lattice calculations of the relevant matrix elements could test these relations directly.
  • Extensions including higher twist operators could refine the numerical agreement between the two pseudoscalar expressions.

Load-bearing premise

The QCD sum rule approach with its operator product expansion and phenomenological parametrization of the spectral function accurately determines the couplings without large unaccounted contributions from higher states or other effects.

What would settle it

A lattice QCD computation of the nucleon matrix element of the axial anomaly that significantly differs from the value predicted by inserting measured PDF moments into the derived sum rule expressions.

Figures

Figures reproduced from arXiv: 2606.12959 by Janardan Prasad Singh.

Figure 1
Figure 1. Figure 1: Plots of expressions of M 2s2 Π (88) 0 , Π(88) 4 and a zero-line for comparision. From combination of Π(88) 0 and Π(88) 41 one gets : (χ8) 2 + 2M2X i (χ8i) 2 e −(Mi−M) 2/sh M 2M3 i − 2M(Mi − M) M2 i χ ′ 8i χ8i − (Mi − M) 2 sM2 i i = − 7 2 M4 s 2 (A u 4 + A d 4 + 4A s 4 ) + 149 24 M6 s 3 (A u 6 + A d 6 + 4A s 6 ) ≡ Π (88) 4 (29) 9 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Two plots, each one of which gives (χ8) 2 . It is amazing that χ8 and χ8i depend only A q n’s after combining the two sum rules given by Eqs. (24) and (25). The coefficient of (⃗q2 ) does not give any useful information. In a similar way we can proceed for χgχ8. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Plots of expressions of M 2s2 Π (08) 0 , Π(08) 4 and a zero-line for comparision. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Two plots, each one of which gives χgχ8. In Figs. (1, 3), we have shown that (M/(2s 2 ))|Π (88) 0 | << |Π (88) 4 | and (M/(2s 2 ))|Π 08) 0 | << |Π (08) 4 |. In principle, Π(88) 41 or Π(88) 4 should be fitted as a + (b + c/s)e d/s with the requirement that a > 0 and c, d < 0, whereas there is a priori no requirement on b. However, such functions can be parameterized in terms of just 3 parameters and the fou… view at source ↗
Figure 5
Figure 5. Figure 5: Plots of OPE expressions of -2M ∂ ∂⃗q2 B [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Plots of OPE expression of -2M ∂ ∂⃗q2 B [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Plots of OPE expression of M ∂ ∂⃗q2 B [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Plots of OPE expression of − M 3 ∂ ∂⃗q2 B [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Two plots, each one of which gives (χ − P ) 2 . From fits of Πu−d P41 (Fig. 9a) and Πu−d P4 (Fig. 9b), where fittings are done over the same interval of s, we get (χ − P ) 2=(0.259, 0.210) while from fits of Πu+d P41 (Fig. 10a) and Πu+d P4 (Fig. 10b), where fittings 19 [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Two plots, each one of which gives χ − P (χ + P + χg/4). ⟨P, S|J − µ5 |P + q, S′ ⟩ = ¯u(P, S)[g 3 A(q 2 )γ ν γ5 + h 3 A(q 2 )qµγ5]u(P + q, S′ ) ⟨P, S|J + µ5 |P + q, S′ ⟩ = ¯u(P, S)[g u+d A (q 2 )γµγ5 + h u+d A (q 2 )qµγ5]u(P + q, S′ ) (54) On taking divergence of the above equations and taking the limit q 2 → 0, one gets : χ − P = g 3 A/2, (55) χ + P + χg/4 = g u+d A /2 (56) Similarly, for s-quark alone (… view at source ↗
read the original abstract

We have analyzed one-nucleon matrix elements of current-current correlators; the currents consist of pseudoscalar octet, isovector and isoscalar currents, axial anomaly, and axial isovector and isoscalar currents. Using QCD sum rules, the coupling constants of nucleon with each of these currents have been expressed in terms of nucleon matrix elements of quark, gluon and quark-gluon composite operators and moments of parton distribution function. On the phenomenological side, contribution from the non-diagonal matrix elements of operators between nucleon and its excited states or continuum states have also been accounted for. For the pseudoscalar coupling constants of the nucleon two expressions have been obtained in which one of them consists of only moments of parton distribution function but yielding approximately same numerical result as the other one. Of particular interest is the nucleon matrix element of axial anomaly which has been largely ignored in the current literature.

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.

Referee Report

0 major / 0 minor

Summary. The manuscript analyzes one-nucleon matrix elements of current-current correlators involving pseudoscalar octet, isovector and isoscalar currents, the axial anomaly, and axial isovector and isoscalar currents. Using QCD sum rules, the coupling constants of the nucleon with each current are expressed in terms of nucleon matrix elements of quark, gluon and quark-gluon composite operators and moments of parton distribution functions. Non-diagonal matrix elements between the nucleon and excited or continuum states are included on the phenomenological side. Two independent expressions are obtained for the pseudoscalar couplings, one of which depends only on PDF moments and yields approximately the same numerical result as the other. The nucleon matrix element of the axial anomaly is isolated as a point of particular interest.

Significance. If the central derivations hold, the work supplies explicit relations for these nucleon couplings within the QCD sum-rule framework, including a non-trivial consistency check between two expressions for the pseudoscalar case. Isolating the axial-anomaly matrix element addresses a quantity that has received limited attention. The inclusion of non-diagonal contributions on the phenomenological side is a standard but necessary refinement that strengthens the approach.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation to accept. We appreciate the recognition given to the derivation of the coupling constants via QCD sum rules, the consistency check between the two expressions for the pseudoscalar couplings, and the isolation of the nucleon matrix element of the axial anomaly.

Circularity Check

0 steps flagged

Derivation self-contained within QCD sum rule framework

full rationale

The paper applies standard QCD sum rules to one-nucleon matrix elements of current-current correlators involving pseudoscalar, axial, and anomaly currents. Couplings are expressed via OPE matching to nucleon matrix elements of composite operators and PDF moments, with explicit inclusion of non-diagonal nucleon-to-excited/continuum contributions on the phenomenological side. Two independent expressions for pseudoscalar couplings are obtained (one reducing to PDF moments), and the axial-anomaly matrix element is isolated. No quoted equations show a result defined in terms of itself, a fitted parameter renamed as a prediction, or load-bearing self-citation chains; the relations follow directly from the sum-rule matching procedure without reduction to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit list of fitted parameters, background axioms, or new postulated entities; the approach relies on standard QCD sum rule machinery whose details are not visible.

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