$\mathrm{O}(a)$ improvement of the flavour singlet scalar density in a setup with Wilson fermions

Fabian Joswig; Jochen Heitger; Pia Leonie Jones Petrak

REVIEW 2 major objections 5 minor 27 references

$\mathrm{O}(a)$ improvement of the flavour singlet scalar density in a setup with Wilson fermions

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper reports the first non-perturbative estimates of the O(a) improvement coefficient $g_S$ of the flavour singlet scalar density, finding values compatible with zero at the smallest couplings.

desk verdict First non-perturbative g_S estimates in the CLS range, clean Ward identity, but the b_g connection is still a plan and the omitted clover term leaves the O(a) ambiguity unquantified. read the letter →

arxiv 2502.08797 v1 pith:VJYZUZJY submitted 2025-02-12 hep-lat

Pia Leonie Jones Petrak , Jochen Heitger , Fabian Joswig This is my paper · ORCID

classification hep-lat PACS 11.15.Ha12.38.Gc

keywords O(a)improvementflavoursingletscalardensityg_SWardidentityWilsonfermionslatticeQCDb_gsigmaterms

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

The paper aims to determine, for the first time, the non-perturbative value of $g_S$, the O(a) improvement coefficient of the flavour singlet scalar density in lattice QCD with Wilson-clover fermions. It derives a lattice Ward identity in which $g_S$ is the only unknown and evaluates it on five ensembles with $N_f=3$ mass-degenerate quarks at couplings $g_0^2 \in [1.5,1.77]$, the range used in large-volume simulations. The resulting estimates decrease toward zero at small couplings and are compatible with zero within one standard deviation at the two smallest couplings. This supports the common practice of neglecting $g_S$ in meson and baryon $\sigma$-term calculations, and the paper also shows how the same identity must be extended with a clover-term insertion before it can deliver the gauge-coupling improvement coefficient $b_g$.

What carries the argument

The central object is the O(a)-improved flavour singlet scalar density $(S_I)^0$, with improvement pattern containing a power-divergent term proportional to $e_S$ and an O(a) term proportional to $g_S$ times the field-strength insertion $e\mathrm{Tr}[F_{\mu\nu}F_{\mu\nu}]$, discretised in the local, site-symmetric form $\{e\mathrm{Tr}[FF]\}_{S_g}$. The argument is carried by a Ward identity obtained by subtracting two axial Ward identities, one with the operator $S_0 O_{ext}$ and one with $1\cdot O_{ext}$; this subtraction cancels the divergent $e_S$ term and leaves $g_S$ as the only unknown once the renormalisation factors $Z$, $r_m$ and the improvement coefficient $c_A$ are supplied. The route to $b_g$ runs through $b_g = 2g_0^2 g_S$, which requires replacing $\{e\mathrm{Tr}[FF]\}_{S_g}$ by $\{e\mathrm{Tr}[FF]\}_{S_g} + a\,g_0^2\,(\partial c_{sw}/\partial g_0^2)\,(i/2)O_{clover}$ in the Ward identity.

What would settle it

Evaluate the extended Ward identity with the clover-term replacement of eq. (14) on the same five ensembles: if the resulting $g_S$ differs from the values in Fig. 1 by more than the quoted errors, the current estimates are only an intermediate quantity, and if the values remain zero-compatible after the clover term, the conclusion that $g_S$ is small in this coupling range is confirmed.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims: non-perturbative estimates for $g_S$ have been determined for the first time. The extraction uses the chiral Ward identity (10), solved on five ensembles at $g_0^2 \in [1.5,1.77]$ with nearly massless, O(a)-improved Wilson fermions and the tree-level Symanzik improved gauge action. The estimates, shown in Fig. 1 for two choices of the time interval, approach zero for small couplings and are compatible with zero within 1-$\sigma$ at the two smallest couplings, confirming that $g_S$ is small in this range. The paper further establishes that the relation $b_g = 2g_0^2 g_S$ holds only after an additional, clover-term-like contribution is included in the Ward identity, and it outlines the implementation of that contribution as the next step toward non-perturbative $b_g$.

Load-bearing premise

The result rests on the assumption that the field-strength term used to improve the singlet scalar density, built only from the gauge action, is the complete O(a) improvement, even though the paper itself notes that a clover-term piece must be added before $g_S$ can be translated into $b_g$; if that missing piece moves the numbers by more than the quoted errors, the values reported here are not the final $b_g$-relevant ones.

Editorial extensions

If this is right

Sigma-term determinations with Wilson fermions at these couplings can continue to set $g_S=0$ without introducing a bias larger than the present statistical uncertainty.
The clover-term extension of the Ward identity gives a concrete route to non-perturbative $b_g$ values in the coupling range $g_0^2 \in [1.5,1.77]$, where only one-loop perturbation theory is currently available.
Because the $g_S$ estimates are compatible with zero at the two finest lattices, the perturbative expectation of a small $g_S$ appears adequate near the continuum limit in this range.
The correlation functions and disconnected-diagram estimators developed here can be reused for other flavour-singlet Ward identities on the same ensembles.

Reading between the lines

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

If the missing clover-term contribution shifts $g_S$ by an amount comparable to or larger than the quoted errors, earlier sigma-term analyses that neglected $g_S$ would need a revised O(a) uncertainty estimate; the paper does not quantify that shift.
The same Ward identity could be evaluated at additional lattice spacings to test whether the trend toward zero continues monotonically; a non-monotonic behaviour would signal that the remaining O(a) ambiguity is not negligible.
The stochastic-estimator technology for the disconnected diagrams transfers to other flavour-singlet quantities, such as the topological-charge density or the singlet axial current, where analogous improvement coefficients appear.
A non-perturbative $b_g$ obtained through $b_g = 2g_0^2 g_S$ would also test the one-loop assumption currently used in the scale and mass settings of large-volume Wilson-fermion simulations.

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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

The one-line take: this is a genuine first non-perturbative estimate of g_S in the CLS coupling range, and the Ward identity trick to cancel the e_S divergence is clean, but the reported numbers are not yet the b_g-relevant coefficients, and the size of the O(a) ambiguity is not quantified.

What is actually new: figure 1 gives g_S at five couplings, g0^2 in [1.5,1.77], from three-flavour Wilson-clover ensembles on an LCP, using a new Ward identity that subtracts two singlet identities to eliminate the cubic divergence. The g_S values are small, consistent with zero at the two smallest couplings, matching the historical expectation that they can be neglected in sigma term calculations. Previous knowledge was one-loop perturbation theory and a b_g determination at lower couplings. The paper is honest that these numbers come with a caveat: relation (14) shows that to extract b_g via b_g=2 g0^2 g_S, an additional clover-term contribution must be included in the field-strength insertion, and the current analysis omits it. So the abstract's statement that the relation to b_g can be established is a plan, not something the present numbers deliver.

The strengths: the derivation of eq. (10) is careful; the inputs Z, r_m, c_A come from independent non-singlet determinations in their earlier papers, not from a fit to the same data; the error analysis uses the Gamma method and trivial-topology projection; the LCP tuning is good. The paper also clearly explains the additional diagrams needed for the clover term, which will allow a straightforward follow-up.

Soft spots: this is a proceedings paper, so no continuum extrapolation, correlation function definitions deferred to ref [11], and the data are not shipped. More substantively, the current g_S belongs to a specific discretisation of the field strength (gauge-action-based). The missing clover term is an O(a) ambiguity, and at a ~ 0.06-0.1 fm it could be comparable to the statistical errors and to the small values near the chiral point. Without a quantitative estimate of the shift, a reader cannot tell whether these g_S values are the ones needed for sigma terms (where any local symmetric discretisation should work) or only intermediate quantities. The paper's own discussion acknowledges this, so it is a conditional, not fatal, weakness.

Who it's for: lattice QCD practitioners computing sigma terms with Wilson fermions in the CLS range, and anyone working on b_g. It deserves serious refereeing as the first non-perturbative determination of g_S; I'd send it out, with the expectation that the authors either quantify the O(a) ambiguity or wait for the clover-term version.

Referee Report

2 major / 5 minor

Summary. The paper reports a first non-perturbative determination of the O(a) improvement coefficient g_S of the flavour singlet scalar density in Nf=3 lattice QCD with Wilson-clover fermions and tree-level Symanzik-improved gauge action. The authors derive a lattice Ward identity (eq. (10)) from two subtracted chiral Ward identities, using Schrödinger functional boundary conditions and previously determined renormalisation/improvement parameters Z, r_m, and c_A. The identity is solved on five CLS ensembles with g0^2 in [1.5,1.77], yielding g_S estimates that approach zero at small coupling and are compatible with zero at the two smallest couplings. The paper also discusses the relation b_g = 2 g0^2 g_S, pointing out that an additional clover-term contribution (eq. (14)) is required for this relation and that the current results cannot yet be used to extract b_g.

Significance. If the results hold, they provide the first direct non-perturbative estimates of g_S in the coupling range used in large-volume CLS simulations, which is relevant for O(a) improvement of meson and baryon sigma terms. The derivation is careful and the numerical analysis uses standard tools (Gamma-method errors, trivial-topology projection, LCP ensembles with nearly massless quarks). The paper is transparent about the missing clover term and clearly states that the b_g extraction is future work. The main value is a proof-of-principle and a set of finite-lattice-spacing estimates that can guide sigma-term calculations, though continuum extrapolation and the clover-term correction remain to be completed.

major comments (2)

[Sections 4 and 5, eq. (14)] The reported g_S is defined without the clover-term contribution that eq. (14) shows is necessary for the b_g relation. The paper itself calls this "merely an O(a) ambiguity" (Section 4). At the lattice spacings used (a ~ 0.06-0.1 fm), an O(a) ambiguity can be comparable to the quoted statistical errors and to the small extracted values near the chiral point, so the numbers in Fig. 1 may not be the g_S needed for sigma-term improvement if a different discretisation of the field strength is ultimately adopted. A quantitative estimate of the difference between the two definitions (e.g., a perturbative estimate or an evaluation of the clover correlation functions on one ensemble) should be provided, or the abstract should make clear that the presented g_S is an intermediate, definition-dependent quantity and that the b_g extraction is not yet realised.
[Introduction and Section 5] The claim that this is the first non-perturbative determination of g_S needs qualification with respect to Ref. [7], which determines b_g non-perturbatively in the neighbouring coupling range g0^2 in [0.4,1.5]. Since b_g = 2 g0^2 g_S holds when the clover term of eq. (14) is included, Ref. [7] effectively provides a non-perturbative g_S in a different scheme/coupling range. The paper should explicitly state why the present determination is novel (direct Ward identity, different definition, different coupling range) or soften the "first time" wording to avoid an overstated novelty claim.

minor comments (5)

[Abstract] The phrase "a relation to b_g ... can also be established, allowing for its non-perturbative extraction as well" is forward-looking; the abstract should make clear that the extraction of b_g is not performed in this paper and that the presented g_S cannot yet be used for that purpose.
[Section 4, Fig. 1] The figure caption notes that the perturbative prediction is not directly comparable to the data, but the main text would benefit from repeating this caveat when the data are discussed, to prevent the reader from inferring a discrepancy.
[Footnote 1 and Section 4] The sign convention g_S = -d_S with respect to Ref. [7] is only given in a footnote; it should also appear in the main text where g_S is first introduced, because the comparison with perturbative expressions and with future work depends on this sign.
[Table 1] The column heading "Nsep [MDU]" is ambiguous; the text explains it, but the table would be clearer with a heading such as "Nsep [MDU]" followed by a footnote, or an explicit statement that Nsep is the separation in molecular dynamics units.
[Section 4, eqs. (11)-(12)] The definitions of the connected correlation functions are deferred to Ref. [11]; since this is a proceedings paper, a brief restatement or at least a clear mapping of the notation (e.g., "con", "disc") to the definitions in the appendix of [11] would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: g_S is solved from a Ward identity whose inputs come from independent non-singlet determinations; the omitted clover term is a scheme limitation, not a circular reduction.

full rationale

The central claim — a first non-perturbative estimate of g_S — is obtained by solving the Ward identity (10) for g_S as the only unknown. The inputs Z, r_m and c_A are taken from refs. [11] and [12], which are published non-perturbative determinations based on flavour non-singlet Ward identities and a separate axial-current improvement condition; neither assumes the value of g_S, so these self-citations are independent support rather than circular inputs. The perturbative expression (3) is used only as an order-of-magnitude reference, and the paper explicitly states that the non-perturbative g_S results cannot be directly compared with it because of the missing clover-term contribution in eq. (14). That omission is a scheme/completeness limitation: the reported g_S is defined with the gauge-action-based discretisation {eTr[FF]}_{Sg}, and the paper transparently says that an additional clover-term-like discretisation is needed before b_g can be extracted via b_g = 2 g0^2 g_S. This is not a circular reduction — the Ward identity is a genuine constraint relating measured correlation functions to g_S, with no fitted parameter being relabelled as a prediction. The caveat in Section 5 that the present g_S cannot yet be used for b_g is an honest limitation, not a sign that the derivation reduces to its own inputs. Hence the circularity score is 0.

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

No new particles or entities are introduced. The only quantity determined is the standard improvement coefficient g_S. The key input that carries risk is the definition of the field-strength discretisation in the improvement pattern, not an invented entity.

assumptions (6)

domain assumption Wilson-clover fermions and tree-level Symanzik improved gauge action describe QCD at these couplings.
This is the discretisation used throughout; it is the physical model of the theory being studied.
domain assumption Schrödinger functional boundary conditions with one-loop boundary improvement coefficients c_t are sufficient for O(a) improvement of the boundary.
Stated in Section 3; relies on the standard ALPHA framework for mass-independent renormalisation and improvement.
domain assumption Ward identities hold in a fixed topological sector, so projection to the trivial sector is valid.
Invoked in Section 3 with reference [19]; this is a standard practice to avoid critical slowing down of topological charge.
domain assumption O(am) terms can be neglected at the simulated quark masses.
PCAC masses in Table 1 are small (e.g. 0.000011(14) at the finest lattice), so the O(am) term in eq. (9f) is dropped and the Ward identity is valid up to O(a^2).
ad hoc to paper The improvement pattern of the singlet scalar density, eq. (8), with the specific discretisation {eTr[FF]}_{Sg}, is complete up to O(a^2).
This is the load-bearing assumption identified in the review; the paper itself shows that an additional clover term is needed for the b_g relation, eq. (14).
standard math The renormalisation parameters Z, r_m from [11] and c_A from [12] are used as external inputs.
These are prior non-perturbative determinations from the same collaboration, not derived in this paper.

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Cite this review

Pith. "Pith review of $\mathrm{O}(a)$ improvement of the flavour singlet scalar density in a setup with Wilson fermions." pith.science (2026). https://pith.science/paper/VJYZUZJY

@misc{pith2026250208797,
  author       = {Pith},
  title        = {Pith review of: $\mathrmO(a)$ improvement of the flavour singlet scalar density in a setup with Wilson fermions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VJYZUZJY}},
  note         = {Machine review of arXiv:2502.08797}
}

abstract

We report on our Ward identity determination of the $\mathrm{O}(a)$ improvement coefficient for the flavour singlet scalar density, namely $g_\mathrm{S}$, from three-flavour lattice QCD with Wilson-clover fermions and the tree-level Symanzik improved gauge action. We employ five couplings, $g_0^2 \in [1.5,1.77]$, that cover the range used in large-volume CLS simulations. While $g_\mathrm{S}$ itself is for instance relevant for the $\mathrm{O}(a)$ improvement of meson and baryon sigma terms, a relation to $b_\mathrm{g}$, the $\mathrm{O}(a)$ improvement parameter of the gauge coupling, can also be established, allowing for its non-perturbative extraction as well. With Wilson fermions, $b_\mathrm{g}$ is in principle required for full $\mathrm{O}(a)$ improvement at non-vanishing sea quark masses. We outline our procedure for extracting $b_\mathrm{g}$.

Figures

Figures reproduced from arXiv: 2502.08797 by the authors.

**Figure 1.** 𝑔S results obtained via eq. (10) for 𝑁f = 3 using our 𝑍 and 𝑟m interpolation formulas from [11] to construct the 𝑍𝑟m values at the given couplings needed. The perturbative prediction from eq. (3) (dashed red line) cannot be directly compared to the 𝑔S data points, see text below. The time interval [𝑡1, 𝑡2] (in eq. (10)) is set to [𝑇/4, 3𝑇/4] (labelled by T/4) or [𝑇/3, 2𝑇/3] (labelled by T/3) as shown by different co… view at source ↗

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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Heitger, F

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work page Pith review arXiv 1997