Quantum Monte Carlo calculation of $\delta_{\rm NS}$ in $^{10}$C using an effective field theory approach

Abraham R. Flores; Emanuele Mereghetti; Garrett B. King; Joseph Carlson; Maria Piarulli; Robert B. Wiringa; Saori Pastore; Stefano Gandolfi

REVIEW 2 major objections 5 minor 5 cited by

Quantum Monte Carlo calculation of $\delta_{\rm NS}$ in $^{10}$C using an effective field theory approach

T0 review · 2 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper computes the nuclear-structure correction to 10C superallowed beta decay with quantum Monte Carlo wave functions and chiral EFT operators, and finds it agrees with both the traditional survey value and the recent no-core shell mo

desk verdict First QMC calculation of δ_NS in 10C, technically solid and transparent, but the 'good agreement' with prior results is carried by an arbitrary LEC band. read the letter →

arxiv 2509.07310 v2 pith:ICOKVPS5 submitted 2025-09-09 nucl-th

Garrett B. King , Joseph Carlson , Abraham R. Flores , Stefano Gandolfi , Emanuele Mereghetti , Saori Pastore , Maria Piarulli , Robert B. Wiringa This is my paper · ORCID

classification nucl-th

keywords superallowedbetadecaynuclear-structureradiativecorrectionsV_udextractionchiraleffectivefieldtheoryquantumMonteCarloGreen'sfunctioncarbon-10CKMunitarity

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 tries to establish that the nuclear-structure-dependent radiative correction to the superallowed beta decay of carbon-10 can be computed from first principles with quantum Monte Carlo wave functions and chiral effective field theory operators, and that the result agrees with the two existing determinations. If correct, it provides an independent, ab initio cross-check of the correction that feeds the most precise determination of the CKM matrix element V_ud, whose first-row unitarity is currently in tension with the Standard Model. The calculation splits the correction into an energy-independent piece carried by magnetic, tensor, spin-orbit, contact, and O(alpha^2) two-body operators, plus a smaller energy-dependent piece. Across four nuclear Hamiltonians, the QMC results give a long-range contribution of -(4.06 to 4.43) x 10^-3 and, once the unknown contact couplings are assigned an arbitrary range, a full delta_bar_NS compatible with the survey value -4.0(5) x 10^-3 and the NCSM value -4.22(32) x 10^-3. The largest remaining uncertainty is not the many-body method but two undetermined low-energy constants.

What carries the argument

The load-bearing object is the EFT decomposition of the radiative correction into an energy-independent piece delta_NS^(0) and an energy-dependent piece delta_NS^E, with the isospin structure separated into spectator-proton and spectator-neutron contributions. Each piece is a sum over nuclear matrix elements of two-body transition operators labelled Fermi, Gamow-Teller, tensor, and spin-orbit, defined through radial functions h(r) such as the ~1/r magnetic and spin-orbit pieces, the delta-function contact terms carrying the two unknown LECs g_V1^NN and g_V2^NN, and the logarithmic O(alpha^2) Fermi term. The many-body matrix elements and their radial densities C(r) are computed with variation

What would settle it

Measure the charge radius of 10B*(0+;1) and check the paper's claimed linear correlations with M_F^E and M_F^+, and/or compute the two contact couplings g_V1^NN and g_V2^NN on the lattice: if the true LECs land outside the +/- (1/m_N)(2F_pi)^-2 band, the central values and error bars of delta_bar_NS and V_ud shift by more than the quoted ranges.

Watch

Extended reading notes

Core claim

The paper's central claim is that the nuclear-structure-dependent radiative correction delta_bar_NS for 10C -> 10B* can be evaluated using chiral EFT two-body current operators and quantum Monte Carlo (VMC and GFMC) wave functions, and that the resulting correction is consistent with the standard survey value and the more recent dispersive no-core shell model calculation. Concretely, the GFMC magnetic-plus-spin-orbit part of delta_NS^(0) sits in the range -[4.06,4.43] x 10^-3, roughly 40 percent larger in magnitude than the delta_NS,B = -3.06(35) x 10^-3 used in the traditional survey, with the difference attributed to many-body correlations in the GFMC wave functions. The energy-dependent p

Load-bearing premise

Two short-range coupling constants in the nuclear operator are not known from experiment or QCD; the paper arbitrarily sets them to plus or minus one in dimensionless form, and this choice drives the largest uncertainty in the final correction to V_ud.

Editorial extensions

If this is right

The long-range magnetic-plus-spin-orbit part of delta_NS^(0) comes out at -[4.06,4.43] x 10^-3 in GFMC, about 40 percent larger in magnitude than the survey's delta_NS,B, so the difference is attributed to many-body correlations rather than to physics missing from the EFT.
With the arbitrary contact-LEC range included, EFT+QMC, the traditional survey analysis, and the NCSM dispersion calculation all agree on delta_bar_NS within errors, removing a reason to suspect a large nuclear-structure error in V_ud extraction.
The 10C-only extraction of V_ud from EFT+QMC falls between 0.97336 and 0.97355, compatible with the survey-based (0.97318) and NCSM-based (0.97317) values within the experimental error; the theory error from the unknown LECs is larger than the spread from the four Hamiltonians.
Because the spin-independent matrix elements M_F^E and M_F^+ correlate strongly with the charge radius of 10B*, a measured radius of the 0+ daughter would directly reduce the model dependence of the energy-dependent correction.
The corrected spin-orbit operator and the non-negligible L^CM term change the operator set of Ref. [48]; these changes matter most in systems where the LS and L^CM contributions do not cancel.

Reading between the lines

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

If the two contact LECs were determined from lattice QCD or from modeled two-nucleon weak amplitudes, the dominant uncertainty in this approach would drop, likely making EFT+QMC competitive with or more precise than the dispersive NCSM evaluation.
The strong empirical correlation between radius and the spin-independent matrix elements suggests a cheap experimental route: measuring the charge radius of the short-lived 10B* 0+ state would pin down the model dependence of the largest energy-dependent term without waiting for a full QCD calculation of the LECs.
The sensitivity of the spin-dependent matrix elements to OPE correlations, and the nodal densities that drive GFMC changes, warn that calculations in heavier superallowed emitters need wave functions that reproduce spin-isospin correlations, not just energies and radii.
Clarifying how pion-range contributions depend on the definition of the isospin limit, as the paper notes, is a prerequisite for consistently separating delta_C from delta_bar_NS at higher chiral orders; until that is settled, the current delta_C input is taken from the survey analysis rather than computed in the same framework.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First thing to know: this is the first QMC calculation of the nuclear-structure correction δ_NS in 10C, and it is a careful, transparent piece of work. The catch is that the abstract's 'good agreement' with previous evaluations is bought with an arbitrary band for two low-energy constants, not with the calculation itself.

The genuinely new content is real: VMC and GFMC matrix elements for δ_NS with four Hamiltonians, including the corrected spin-orbit operator (sign and center-of-mass term) and an explicit check that the Fermi matrix element comes out at 7.07. The operator definitions in Appendix A are complete enough to reproduce, and the paper honestly reports that the dominant uncertainty is the undetermined LECs g_V1^NN and g_V2^NN. The density plots and the VMC/GFMC comparison are useful diagnostics. This is a solid methodological step, not a precision extraction.

The soft spots are in how the result is framed. The LEC-free content says one thing; after adding the ±1 naturalness range it says another. The magnetic-plus-spin-orbit piece alone is about 3σ above the Hardy-Towner δ_NS,B, and the full LEC-free central value sits ~0.5–0.9×10^-3 (roughly 2σ) above the NCSM result. The ±(0.5–0.8)×10^-3 band from g̃=±1 converts this into 'agreement within error.' That is underdetermination, not prediction, and the abstract oversells it. There is also a tension inside the paper: the two-nucleon weak amplitudes from AV18 and NV2-Ia agree to better than 2%, which suggests small contact LECs, while the ±1 assignment permits contact terms ~10% of the magnetic ones. The paper never resolves which regime is physical.

So for whom? This is for people working on V_ud, on chiral EFT currents, and on ab initio nuclear structure. It is a useful benchmark and a start toward fitting the LECs from multiple isotopes or two-nucleon data. I would send it to a serious referee. The referee should ask for a less loaded abstract and a more direct treatment of the LEC range — for example, a figure showing δ_NS as a function of the LEC values, or constraints from two-nucleon amplitudes.

Referee Report

2 major / 5 minor

Summary. This paper presents the first QMC calculation of the nuclear-structure-dependent radiative correction \bar{\delta}_{NS} for the superallowed 10C→10B* decay, using the EFT formalism of Refs. [48,49]. The authors evaluate the relevant two-body operator matrix elements (magnetic, tensor, spin-orbit, contact, and O(α^2) Fermi) with VMC and GFMC wave functions generated from four Hamiltonians: AV18+UX, AV18+IL7, NV2+3-Ia, and NV2+3-Ia*. They benchmark the Fermi matrix element (M_F = 7.07), analyze the transition densities, and study the role of OPE correlations. The GFMC long-range magnetic-plus-spin-orbit contribution is -[4.06,4.43]×10^{-3} (Eq. (34)); the O(α^2) two-body contribution is -[0.21,0.40]×10^{-3} (Eq. (37)); and the energy-dependent term is +[0.97,1.17]×10^{-3} (Eq. (39)). Adding an arbitrary dimensional-analysis range for the unknown contact LECs (Eq. (A14) with \tilde{g}=±1) makes the total \bar{\delta}_{NS} compatible with the Hardy-Towner and NCSM values, and leads to the V_ud extractions in Eqs. (46)–(49).

Significance. If the underlying EFT derivation is accepted, this is a useful independent many-body evaluation of \bar{\delta}_{NS}. The paper is careful in its operator definitions, provides a transparent decomposition of uncertainties, corrects a sign in the spin-orbit operator, restores the L_CM term, and includes a simple Fermi-matrix-element sanity check. The radial-density analysis and the correlation study in Sec. V are valuable diagnostics. The main limitation is structural rather than mathematical: the two contact LECs are undetermined, and the quoted agreement with previous evaluations is obtained by adding an arbitrary LEC band to the calculated central values. The authors are transparent about this, but the abstract's "good agreement" language goes beyond what the LEC-free calculation supports. The work is nonetheless a meaningful first step in benchmarking QMC against existing shell-model and dispersive results, provided the claims are recalibrated.

major comments (2)

[§VI, Table V, Abstract] The claim of "good agreement" is not supported by the LEC-independent part of the calculation. Summing the GFMC entries in Table V (mag+LS, O(α²), and δ_E^NS) without the contact-LEC term gives δ_NS central values of approximately −3.3 to −3.7×10^{-3} across the four interactions. These lie 0.55–0.93×10^{-3} above the NCSM value −4.22(32)×10^{-3} (Eq. (36)), i.e., about 2σ, and 0.3–0.7×10^{-3} from the Hardy–Towner value −4.0(5)×10^{-3} (Eq. (35)). The ±(0.48–0.77)×10^{-3} band from Eq. (A14) with \tilde{g}=±1 is an arbitrary dimensional-analysis prior; it only widens the error bars and does not move the central values toward the comparison targets. The abstract and the conclusion ("very good agreement" with Ref. [50]) should be revised to state that the results are compatible only after including this prior, and the LEC-free comparison should be displayed separately.
[Appendix A, Eq. (A18)] The O(α^2) three-body transition operator is not evaluated. Because the O(α^2) two-body contribution is already −[0.21,0.40]×10^{-3} (Eq. (37)), an uncomputed three-body term of comparable size would enter at the quoted precision. The manuscript should either compute/estimate this contribution or explicitly identify it as a missing piece in the uncertainty budget. As written, "which we have not evaluated" leaves a known omission that is not reflected in the error bars.

minor comments (5)

[Eq. (7) and Eq. (A4)] The tensor operator definition in Eq. (7) contains a typo: S^(jk)(r̂) = 3 r̂·σ^(j) r̂·σ^(k) − σ^(i)·σ^(j) should read − σ^(j)·σ^(k), as in Eq. (A4).
[Sec. V.A] The text says the L_CM term is "actually non-negligible" in relation to Ref. [48], but later states that the LS matrix element is negligible because of a cancellation between L and L_CM. Please clarify that the individual contributions are non-negligible while their sum is small.
[Eq. (A13)] The regulator is written as R_S = 0.8 fm^{-1}, but the Gaussian in Eq. (A13) requires R_S to have dimensions of length. This is presumably a typo for R_S = 0.8 fm; since the contact matrix elements M_CT depend on R_S, this should be corrected.
[Eqs. (33), (46)–(49)] The error labeled σ_{g_NN_V} in Eqs. (46)–(49) is the width of an assumed LEC range, not a determined uncertainty. The manuscript should explicitly state this is a prior and that the central values would shift if the LECs were found to lie outside the assumed ±1 range.
[Sec. VI, Eq. (44)] In the sentence following Eq. (44), "the first error" and "the second error" are clear, but the notation (56)gV(87)µ could be made more explicit by saying "the first error is from g_V and the second from the scale µ."

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central δ_NS calculation is not fitted to the 10C decay data; the EFT operators come from independent prior derivations, and the arbitrary LEC band is an honest uncertainty, not a self-referential fit.

full rationale

The paper's derivation chain is: (i) adopt EFT two-body operators from Refs. [48,49] (Sec. II); (ii) compute their nuclear matrix elements with VMC/GFMC for four Hamiltonians (Secs. III–V); (iii) combine them with external inputs (g_V, phase-space factors, t, δ_C) to form δ_NS and extract V_ud; (iv) compare with Hardy-Towner [45] and NCSM [50]. No step uses the 10C beta-decay data or the comparison values as an input to fix the calculation. The two LECs g_V1^NN and g_V2^NN are explicitly not fitted: Eq. (A14) sets dimensionless couplings to ±1, and Table V labels this as 'arbitrary values.' The resulting error band widens the uncertainty until overlap with Refs. [45,50] occurs, so the claimed 'good agreement' is an overlap of an a priori LEC range with external results, not a prediction forced by construction. That is an underdetermination/correctness concern, not a circular reduction. The self-citations to Refs. [48,49] (authors overlap with Gandolfi and Mereghetti) provide the operator formalism, but those are published, independent derivations already applied to 14O, and the present paper corrects a sign error in the spin-orbit operator and adds the L_CM term, demonstrating that the cited result is not being used as an unexamined black box. The V_ud comparison shares δ_C and t with Ref. [45], but the paper states this assumption explicitly and it does not feed back into the δ_NS result. No equation in the paper is equivalent to its inputs by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central quantities are matrix elements of operators derived in prior work by overlapping authors; the genuinely new inputs are the QMC wave functions and the four Hamiltonians, which are from previous publications. No new particles or forces are introduced. The main ledger entries are the two undetermined contact LECs and the regulator choices, all of which are explicitly acknowledged in the text.

free parameters (3)

g_V1^NN, g_V2^NN (contact LECs) = Not fitted; set to +/-(1/m_N)(2F_pi)^-2, i.e., g_tilde = +/-1
Enter the short-range contact operators in Eq. (A10); the paper uses the dimensional-analysis estimate (Eq. A14) and the resulting range dominates the error budget of delta_NS and V_ud (Table V).
R_S (regulator for contact delta functions) = 0.8 fm^-1 (as printed; presumably fm)
Gaussian regulator width in Eq. (A13) chosen by hand; for AV18 it is admitted to be inconsistent with the potential's short-range dynamics (Appendix A).
Lambda (cutoff for O(alpha^2) log operator) = R_A^-1 = (1.2 A^1/3)^-1
Chosen in Eq. (A15); the paper states the Lambda dependence cancels with the treatment of the Fermi function, so it is a scheme choice rather than a fitted value.

assumptions (5)

domain assumption The two-body chiral EFT operators of Refs [48,49] form the complete set of O(alpha epsilon_chi) and O(alpha Q/M_pi) contributions to delta_NS.
Used throughout Section II; the present paper corrects a sign and adds the L_CM term, but assumes no other missing operators at this order.
domain assumption The mixed-estimate GFMC formula (Eq. 21) gives accurate off-diagonal matrix elements for operators that do not commute with the Hamiltonian.
Section III; Eq. (19) is explicitly stated to be approximate for non-commuting operators.
ad hoc to paper The three-body O(alpha^2) transition operator in Eq. (A18) is negligible.
Appendix A.1 states 'which we have not evaluated'; its omission is assumed to be numerically irrelevant without a quantitative estimate.
domain assumption The isospin-breaking correction delta_C from Ref [45] can be combined with the EFT-calculated delta_NS.
Section VI, used in Eqs. (41)-(45); the paper concedes delta_C should ideally be computed with the same many-body method and interactions.
domain assumption Setting the isospin limit with M_pi = M_pi0 in the two-body weak currents and potentials.
Appendix A.3; the paper notes the pion-range contributions depend on this definition and could change under a different isospin limit.

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Pith. "Pith review of Quantum Monte Carlo calculation of $\delta_{\rm NS}$ in $^{10}$C using an effective field theory approach." pith.science (2026). https://pith.science/paper/ICOKVPS5

@misc{pith2026250907310,
  author       = {Pith},
  title        = {Pith review of: Quantum Monte Carlo calculation of $\delta_\rm NS$ in $^10$C using an effective field theory approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ICOKVPS5}},
  note         = {Machine review of arXiv:2509.07310}
}

abstract

We compute radiative corrections to the superallowed $\beta$ decay of $^{10}{\rm C}$ in an effective field theory approach using nuclear matrix elements obtained from quantum Monte Carlo calculations. These corrections are an important ingredient in the extraction of the Cabibbo-Kobayashi-Masakawa quark mixing matrix element $V_{ud}$, and the role of this work is to illuminate the uncertainties arising from nuclear structure. Our results provide good agreement with both the traditional extraction of $V_{ud}$, as well as with a more recent evaluation performed using the no-core shell model and a dispersion formalism. The dominant uncertainty in this approach is the presence of two unknown low-energy constants that enter into the relevant nuclear matrix elements. Future determinations of these low-energy constants -- either from QCD or modeling them with two nucleon amplitudes -- would improve the precision of the extraction in this formalism.

Figures

Figures reproduced from arXiv: 2509.07310 by the authors.

**Figure 2.** FIG. 2: Normalized operator densities according to Eq. (24) computed with VMC using the [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** FIG. 3: Comparison of the matrix elements obtained with the Ia interaction (filled symbols) vs Ia* [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4: (a) [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reviewed August 4, 2026 · model on record in the stance chip above.

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Operators contributing to the energy-independent correctionδ (0) NS 28

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inner radiative corrections

Pion-range contributions to δE NS 30 References31 I. INTRODUCTION In the Standard Model (SM) of particle physics, charged-current interactions are mediated by the Cabibbo–Kobayashi–Maskawa (CKM) quark mixing matrix [1, 2], which is predicted to be unitary. By independently measuring the individual CKM elements, the unitarity of the CKM matrix can be teste...

2020

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Operators contributing to the energy-independent correctionδ (0) NS AtO(α), ¯δNS receives contributions from magnetic, contact and spin-orbit operators. The radial functions are hmag GT,p(r) = 4hmag T,p (r) = gA 3mN 1 +κ p r ,(A8) hmag GT,n(r) = 4hmag T,n (r) = gA 3mN κn r ,(A9) hCT GT,p(r) =− 4π 3 (gNN V1 +g NN V2 )δRS (r), hCT GT,n(r) =− 4π 3 (gNN V1 −g...

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HereM π =M π0, andZ π is determined from the pion mass splitting M 2 π± −M 2 π0 = 2e2F 2 π Zπ,(A20) implyingZ π ∼0.8

Operators contributing to the energy-dependent correction δE NS The energy-dependent correction receives the following contributions hE F,p(r) =− r 2RA , hE F,n(r) = 0, hEπ GT,p(r) =−h Eπ GT,n(r) = g2 AZπ 3 e−Mπr 72MπRA 12 + 12Mπr−M 2 πr2 , hmeπ GT,p(r) =−h meπ GT,n(r) = g2 AZπ 3 e−Mπr 72MπRA 15−21M πr+M 2 πr2 , hEπ T,p(r) =−h Eπ T,n(r) = g2 AZπ 3 e−Mπr 7...

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