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Calculation of the axial-vector coupling constant $g_A$ to two loops in covariant chiral perturbation theory

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Pith's one-line read The paper computes the leading two-loop corrections to the nucleon axial-vector coupling constant $g_A$ at chiral order $q^5$ in two covariant versions of two-flavor baryon chiral perturbation theory (EOMS and infrared regularization) and…

desk verdict Genuine first complete covariant two-loop g_A coefficients with plausible analytics, but the numerical 'moderate' claim rests on an unshown HB-to-EOMS LEC matching and on details deferred to a follow-up paper. read the letter →

arxiv 2505.05941 v2 pith:T24WFTVM submitted 2025-05-09 hep-ph hep-latnucl-exnucl-th

Véronique Bernard , Jambul Gegelia , Shayan Ghosh , Ulf-G. Mei{ss}ner This is my paper · ORCID
classification hep-phhep-latnucl-exnucl-th
keywords axial-vectorcouplingconstantg_Achiralperturbationtheorytwo-loopcorrectionscovariantbaryonChPTEOMSschemeinfraredregularizationpion-nucleonscattering
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes the complete leading two-loop contribution to the nucleon axial-vector coupling $g_A$ at chiral order $q^5$, i.e. the $M^4$ term in the chiral expansion, working in the two covariant renormalization schemes of baryon chiral perturbation theory: extended-on-mass-shell (EOMS) and infrared regularization (IR). The centerpiece is a closed expression for the scheme-independent coefficient $\alpha_4 = -\frac{7}{3} g_0 (1-g_0^2) + 16 g_0^5$, along with the scheme-dependent coefficients $\gamma_4$ and $\beta_4$ given in Eqs. (32) and (34). Evaluated at physical pion mass with two published sets of pion-nucleon scattering low-energy constants, the two-loop correction is $\Delta^{(4)} = 2.6\%$ for one set and $-17.4\%$ for the other, which the authors describe as 'rather moderate'. If correct, these coefficients are the first complete covariant two-loop $M^4$ results for $g_A$, and they give a controlled basis for chiral extrapolations of lattice QCD calculations of the axial coupling.

What carries the argument

The engine is the set of scalar two-loop master integrals $F_{\alpha\beta\gamma\delta\epsilon}(m_1,\ldots,m_5)$ with two integration momenta and up to five propagators, evaluated on shell ($p^2=m^2$) and expanded through order $M^5$. Each integral is split into a purely infrared part, a mixed part, and a regular part following the infrared-regularization decomposition; the regular parts are absorbed into low-energy constants, while the mixed and infrared parts satisfy the Ward identities. The wave-function renormalization factor $Z$ is built from the same integrals at one and two loops, and the requirement that the $\log(M/\mu)/\epsilon$ divergences cancel against one-loop insertions of the $d_i$ and $l_i$ LECs fixes $\alpha_4$ and provides internal checks of the calculation.

What would settle it

Compute the order-$q^6$ two-loop diagrams with $L_{\pi N}^{(2)}$ vertices and the $M^5$ corrections to $g_A$: if their combined contribution at the physical pion mass is comparable to or larger than the reported $|\Delta^{(4)}|$ (in particular for LEC set 2, where $\Delta^{(4)}=-17.4\%$), the claim that the leading two-loop corrections are moderate is falsified. A complementary check is a high-precision lattice value of $g_A$ at the physical pion mass that cannot be accommodated once these higher-order terms are included at their natural size.

Watch

Extended reading notes

Core claim

The central claim is that at order $q^5$ the full two-loop contribution to $g_A$ is captured by the coefficients $\alpha_4 = -\frac{7}{3}g_0(1-g_0^2)+16g_0^5$, with $\gamma_4$ and $\beta_4$ as given in Eqs. (32) and (34). The coefficient $\alpha_4$ is universal (the same in EOMS and IR up to analytic terms), its $g_0^5$ term agrees with the earlier heavy-baryon renormalization-group result after correcting a factor of two in $\tilde\alpha_4 = 2\alpha_2$, and the $\log^2 M$ and $\log M$ divergences cancel against one-loop insertions of the dimension-three low-energy constants. The regular pieces of the two-loop integrals are absorbed into the local counterterms $C$, so the difference between the two schemes appears only in $\beta_4$. With the two LEC sets from pion-nucleon scattering the resulting correction is moderate, $2.6\%$ for set 1 and $-17.4\%$ for set 2 at the physical pion mass, which the authors take as evidence that the chiral expansion of $g_A$ converges reasonably well at fourth order, though they stress that the order-$q^6$ two-loop diagrams and $M^5$ corrections remain to be worked out.

Load-bearing premise

The numerical conclusion that the two-loop corrections are moderate assumes that the omitted order-$q^6$ two-loop diagrams with $L_{\pi N}^{(2)}$ vertices and the $M^5$ corrections are small; the paper explicitly states that these pieces are still in progress and must be worked out before final conclusions.

Editorial extensions

If this is right

  • The scheme-independent coefficient $\alpha_4$ fixes the leading non-analytic two-loop correction to $g_A$ in any chiral extrapolation, so future lattice-QCD extrapolations of the axial coupling can be carried out with two-loop covariant control through order $M^4$.
  • With LEC set 1 the fourth-order correction is only $2.6\%$ at the physical pion mass, supporting the use of covariant baryon chiral perturbation theory for $g_A$ up to pion masses of about 300 MeV.
  • With LEC set 2 the same correction is $-17.4\%$, so the claimed 'moderate' size depends on which pion-nucleon scattering LEC set is used; the expansion is not uniformly small.
  • The order-$q^6$ two-loop diagrams with $L_{\pi N}^{(2)}$ vertices and the $M^5$ corrections are still needed before final conclusions; the paper explicitly leaves both to future work.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension would be to fit the new closed forms directly to $g_A$ lattice data as a function of pion mass, which could independently determine the combination of $\bar d_i$ LECs that enters $\gamma_4$ and $\beta_4$ and cross-check the pion-nucleon scattering values.
  • If the in-progress $q^6$ and $M^5$ corrections turn out to be as small as the $M^4$ term for LEC set 1, this would strengthen the case that the chiral expansion of $g_A$ is under control near the physical point; if they are large for set 2, the moderate-$M^4$ conclusion would need to be revised.
  • Because only $\beta_4$ differs between the EOMS and IR schemes (the difference being absorbed into the counterterm $C_0$), the scheme dependence of the full two-loop prediction could be used as a diagnostic for missing higher-order terms when comparing with precise lattice data.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper computes the leading two-loop (chiral order q^5) corrections to the nucleon axial-vector coupling g_A in two covariant formulations of two-flavor baryon chiral perturbation theory, EOMS and infrared regularization. The central analytic results are the coefficients alpha4 in Eq. (30), gamma4 in Eq. (32), and beta4 in Eq. (34), together with the claim that the two-loop M^4 contribution to g_A is moderate once existing pion-nucleon scattering LECs are inserted: Delta(4) = 2.6% for set 1 and -17.4% for set 2 at the physical pion mass. The derivation is supported by several internal checks, including cancellation of log/epsilon divergences, scale independence of the counterterm combination C, agreement with known two-loop nucleon self-energy results, and agreement with the g0^5 term of Ref. [14] after correcting a factor of two. The numerical part uses two sets of LECs from heavy-baryon analyses of pion-nucleon scattering and assumes the order-q^5 counterterm C0 to be of natural size.

Significance. If the analytic coefficients are correct, this is the first complete covariant two-loop M^4 calculation of g_A in EOMS and infrared regularization, and the result for alpha4 in Eq. (30) is a parameter-free prediction in terms of g0. The consistency checks cited in the paper strengthen confidence in the reduction. The numerical claim of moderate two-loop corrections is, however, less firmly established: it depends on uncalculated higher-order pieces, on an LEC scheme identification that is asserted rather than derived, and on a numerical input set whose reported central values are not accompanied by uncertainties. The analytic part is likely to be of lasting value; the convergence conclusion needs additional support or more careful wording.

major comments (4)
  1. [Sec. 4, Eqs. (32)-(35)] The quoted Delta(4) values are obtained by inserting the heavy-baryon bar d_i LECs of Ref. [36] directly into the EOMS expressions (32) and (34). This is a load-bearing scheme identification: Eq. (5) explicitly shows that at least g0 and d16 require finite scheme-dependent shifts between EOMS and other schemes, and analogous shifts generically affect the other d_i and l_i appearing in gamma4 and beta4. The statement in Sec. 4 that 'as we have done an expansion in M/m of the IR and EOMS expressions our results should in fact correspond to the one in the heavy baryon approach' is not a derivation. Because the percentages in Eq. (35) are evaluated at C0=0, any finite shift in gamma4 or beta4 changes the quoted numbers. Please either apply the explicit HB-to-EOMS conversion for all LECs used, or demonstrate that the conversion only renormalizes C0 and that the resulting variation is within the C0 = +/-15 GeV^-4 band already shown in Fig. 3.
  2. [Sec. 4, Eq. (35) and Fig. 3] The paper states that 'we find that g0 = 1.0 for set 1 and g0 = 1.3 for set 2, in order,' but it does not explain how g0 is determined. The quoted Delta(n) values imply that g0 is chosen so that the truncated series reproduces the physical g_A; this should be stated explicitly, since otherwise the smallness of Delta(4) at the physical point is partially an artifact of the fitting condition. More importantly, the convergence interpretation is not yet supported: for set 2, Delta(3) = 44.5% and Delta(4) = -17.4% are of comparable magnitude, and the paper itself notes in Secs. 2 and 5 that the order-q^6 two-loop diagrams and M^5 corrections are needed before final conclusions can be drawn. The abstract's claim that 'these corrections are rather moderate' should be restricted to the computed Delta(4) contribution, with the caveat that the omitted higher-order pieces have not been shown to be small.
  3. [Sec. 3, around Eqs. (12)-(16) and (30)-(34)] The manuscript defers all details of the 44-diagram two-loop calculation to Ref. [20], which is listed as 'in preparation.' Since Eqs. (30), (32), and (34) are the central results of the paper, the reader cannot independently verify the reduction, the master-integral decompositions, or the LEC contributions. The internal consistency checks are reassuring, but they are not a substitute for a complete derivation. Please include the full calculation, or at least the master-integral reduction and the contribution of each class of diagrams/LECs, either in the paper or in a supplementary file that is made available with the submission.
  4. [Sec. 4, Eq. (35)] The numerical percentages are quoted without any uncertainty estimate, despite the dependence on LEC fit errors, the unconstrained choice C0=0, and the scheme-matching issue raised above. In particular, for set 2 a shift of Delta(4) by a few percent would change the conclusion from 'moderate' to 'substantial.' Given that the full uncertainty analysis is deferred to Ref. [20], the quantitative claims in Eq. (35) should either be accompanied by an error estimate or be explicitly labeled as central values that are indicative only. At minimum, the corresponding Delta(4) values for C0 = +/-15 GeV^-4 should be listed, since Fig. 3 shows that the M^4 curves vary appreciably over that range.
minor comments (4)
  1. [Sec. 4] The value 'F_pi = 0.927 GeV' is a factor of 10 too large; the physical value is 0.0927 GeV. If the numerical evaluation actually used F = 0.0927 GeV (as the quoted results suggest), the text should be corrected to avoid making the calculation irreproducible.
  2. [Sec. 4] The sentence giving set 2 contains a typographical error: 'bar d_1 + bar d_2 = 3.39,GeV^-2' and 'bar d_13 = 27.7,GeV^-2' have misplaced commas. Also, 'g0 = 1.0 for set 1 and 1.3 for set 2, in order' should read 'respectively.'
  3. [Eq. (5)] The notation 'g0 -> g0 + delta g0|reg d16 -> d16 + delta d16|reg' is missing a separator; the two scheme shifts should be displayed as separate equations or separated by a comma for clarity.
  4. [Acknowledgements and author affiliations] There are minor proofreading issues, including the duplicated 'by by' in the acknowledgements and the misspelling 'Reserch' in the affiliation of the fourth author; these should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the two-loop M^4 coefficients are derived from the chiral Lagrangian and checked by scale independence, with LECs from independent pion-nucleon scattering analyses; the main caveats are acknowledged open higher-order terms.

full rationale

The central analytic results, Eqs. (30), (32), and (34), are obtained by evaluating the two-loop diagrams, wave-function renormalization, one-loop L^(3)_πN insertions, and counterterms from the chiral Lagrangian; the coefficients are checked by scale independence and by comparison with known IR/Z results, not by fitting g_A. The low-energy constants entering γ4 and β4 are taken from the independent pion-nucleon scattering analyses [36], which do not use g_A as input, so the claim that the leading two-loop corrections are moderate is not a self-prediction of g_A. The paper itself flags its main limitations (the q^6 two-loop diagrams with L^(2)_πN vertices and the M^5 corrections remain to be computed, Secs. 2 and 5, with details deferred to Ref. [20]), but these are acknowledged open calculations rather than circular inputs. The identification of HB bar-d_i with EOMS d_i^r at μ=M_π is a scheme-matching assumption that could affect the numerical percentages, and the value of g0 is asserted without an explicit fitting procedure; however, neither reduces any predicted coefficient to its own input by construction. No self-citation chain is invoked to forbid alternatives; Refs. [14], [18], [20], and [30] are comparisons or pointers, not load-bearing authority. Accordingly, no circular step can be exhibited, and the appropriate score is 0.

Assumptions & free parameters 9 free parameters · 6 assumptions · 0 invented entities

The derivation itself is parameter-free at the level of the analytic coefficients alpha4, gamma4, and beta4, but the numerical 'moderate corrections' conclusion imports a large set of LECs from prior pion-nucleon fits, assumes values for g0 and C0, assumes d15 = 0, and assumes the uncalculated higher orders are small. No new dynamical entities are introduced.

free parameters (9)
  • g0 (chiral-limit axial coupling) = set1: 1.0, set2: 1.3
    Needed to plot g_A(M). The paper states these values but does not state whether they are obtained by matching the physical g_A or from fits, so the quoted Delta(4) percentages depend on this choice.
  • C0(m) (order-q^5 counterterm combination) = scanned in [-15,15] GeV^-4
    Not determined in the paper; assumed natural size about 1/Lambda_chi^4 with Lambda_chi = 0.6 GeV. The upper and lower M^4 curves depend on this scan.
  • LEC set 1 (c2, c3, c4) = c2=3.51, c3=-6.63, c4=4.01 GeV^-1
    Taken from combined piN -> piN and piN -> pi pi N fits in Ref. [36]; enters alpha3 and the one-loop insertions.
  • LEC set 1 (d-bar values) = d1+d2=4.37, d10=-0.8, d11=-15.6, d12=5.9, d13=13.6, d14=-7.43, d16=0.4, d18=-0.8 GeV^-2
    Dimension-three LECs from Ref. [36]; enter the Delta(4) coefficients gamma4 and beta4.
  • LEC set 2 (c2, c3, c4) = c2=4.89, c3=-7.26, c4=4.74 GeV^-1
    Second LEC set from Ref. [36], based on a different power counting; drives the larger Delta(3) and Delta(4) values.
  • LEC set 2 (d-bar values) = d1+d2=3.39, d10=10.9, d11=-30.9, d12=-10.9, d13=27.7, d14=-7.36, d16=-3.0, d18=-0.8 GeV^-2
    Alternative dimension-three LEC set from Ref. [36]; enters the numerical M^4 correction.
  • l3(m) and l4(m) = l3=1.4e-3, l4=3.7e-3
    Mesonic LECs used to determine F and M in the chiral limit; enter gamma4 and beta4.
  • m (chiral-limit nucleon mass) = 0.87 GeV
    Input from Ref. [35]; sets the scale for mixed and regular terms in the EOMS expressions.
  • d15 = 0 (assumed)
    Only the combination d14 - d15 is known from fits; d15 is set to zero without a stated uncertainty, which affects LEC combinations in gamma4 and beta4.
assumptions (6)
  • domain assumption The chiral Lagrangian of Refs. [15,19,18] contains all operators needed at orders q^3 and q^4, and the LECs d_i and l_i renormalize as in Eqs. (18)-(20).
    The whole calculation is built on this operator basis and renormalization scheme, introduced in Secs. 2 and 3.
  • domain assumption In the EOMS and IR frameworks, loop integrals split into infrared, mixed, and regular parts, and the regular parts can be absorbed into LECs without changing physical predictions.
    This is the defining property of the renormalization schemes, stated in Sec. 3 around Eq. (13).
  • domain assumption An expansion in M/m is valid, so the covariant IR and EOMS results reduce to the heavy-baryon expressions.
    Sec. 4 states: 'As we have done an expansion in M/m of the IR and EOMS expressions our results should in fact correspond to the one in the heavy baryon approach.'
  • ad hoc to paper The uncalculated order-q^6 two-loop diagrams and M^5 corrections are small enough to leave the qualitative conclusion unchanged.
    Secs. 2 and 5 explicitly leave these contributions to future work; the numerical claim of moderate corrections relies on this assumption.
  • ad hoc to paper The order-q^5 counterterm combination C0 is of natural size, C0 ~ 1/Lambda_chi^4 with Lambda_chi = 0.6 GeV.
    Assumed in Sec. 4 to set the width of the M^4 band; no independent determination is provided.
  • domain assumption The external LEC fits in Ref. [36] are reliable and do not introduce a hidden scheme mismatch when inserted into the EOMS/IR expressions.
    The numerical estimates in Sec. 4 depend on these published values and on the d15 = 0 assumption.

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Pith. "Pith review of Calculation of the axial-vector coupling constant $g_A$ to two loops in covariant chiral perturbation theory." pith.science (2026). https://pith.science/paper/T24WFTVM

@misc{pith2026250505941,
  author       = {Pith},
  title        = {Pith review of: Calculation of the axial-vector coupling constant $g_A$ to two loops in covariant chiral perturbation theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T24WFTVM}},
  note         = {Machine review of arXiv:2505.05941}
}
abstract

We present a calculation of the leading two-loop corrections to the axial-vector coupling constant $g_A$ in two covariant versions of two-flavor baryon chiral perturbation theory. Taking the low-energy constants from a combined analysis of elastic and inelastic pion-nucleon scattering, we find that these corrections are rather moderate.

Figures

Figures reproduced from arXiv: 2505.05941 by the authors.

Figure 1
Figure 1. Two-loop diagrams contributing to the nucleon self-energy. The same topologies contribute also to [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Two-loop diagrams contributing to gA. The wavy line represents the axial current, the dashed ones the pion and the solid lines the nucleon. where one has Z1−loop = − 9 32π 2F2 M2 g 2 0  16π 2λ + log M µ + 1 3 − πM 2m − ϵ  log2 M µ + 2 3 log M µ + 1 24 (6 + π 2 ) − π m M  log M µ − 1 6 (1 + 6 log 2) + 3 16π 2F2 M2 g 2 0  16π 2λ + log m µ − 1 2 − ϵ  log2 m µ − log m µ + 1 4  5 + π 2 6  (9) with λ = 1 16π 2… view at source ↗
Figure 3
Figure 3. gA as a function of M for set 1 (left panel) and set 2 (right panel) of the LECs given in the text. The blue dash-dotted, the black dashed and the red solid lines represent the results up to M2 , M3 and M4 , respectively. The upper M4 curve corresponds to C0(m) = 15 GeV−4 , whereas the lower M4 curve represents the case C0(m) = −15 GeV−4 . The pink circle denotes the physical value of gA. third and fourth order for … view at source ↗

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Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.