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REVIEW 3 major objections 4 minor 55 references

The gluino condensate of large-$N$ SUSY Yang-Mills

T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The first lattice computation of the large-$N$ gluino condensate gives $\Sigma_{\rm RGI}/\Lambda^3_{\rm NSVZ}=1.77(65)$, compatible with the exact weak-coupling value $1$ and ruling out the strong-coupling and fractional-instanton…

desk verdict A solid proceedings summary of an important lattice result, but the abstract overclaims a resolution and the N-dependence evidence is qualitative, not quantitative. read the letter →

arxiv 2412.14067 v1 pith:WDJL2JMI submitted 2024-12-18 hep-lat hep-th

Claudio Bonanno , Pietro Butti , Margarita García Pérez , Antonio González-Arroyo , Ken-Ichi Ishikawa , Masanori Okawa This is my paper · ORCID
classification hep-lathep-th
keywords gluinocondensatelarge-NsupersymmetricYang-MillstwistedvolumereductionBanks-CasherrelationGell-Mann-Oakes-RennerNSVZschemelatticefieldtheory
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 reports the first lattice computation of the gluino condensate in the large-$N$ limit of $\mathcal{N}=1$ supersymmetric $SU(N)$ Yang-Mills theory, using twisted volume reduction so that a single-site lattice with twisted boundary conditions stands in for the infinite-volume theory at $N=169$, $289$, and $361$. Two independent determinations, one from the Banks-Casher relation and one from a Gell-Mann-Oakes-Renner-like formula for the adjoint pion, give compatible values for the renormalization-group-invariant condensate in units of the NSVZ scale, with the final value $\Sigma_{\rm RGI}/\Lambda^3_{\rm NSVZ}=1.77(65)$. The paper argues that this value agrees with the exact weak-coupling prediction $1$ and rules out the strong-coupling $2e/N$ and fractional-instanton alternatives. The significance is that this is the first direct lattice comparison with analytic formulas for a quantity that has been debated for forty years, and the first demonstration that twisted reduction can handle a supersymmetric fermionic observable.

What carries the argument

The load-bearing object is the reduced one-point lattice: through twisted volume reduction, the four-dimensional large-$N$ theory is simulated on a single lattice site with twisted boundary conditions, with color degrees of freedom playing the role of spacetime ones. The condensate is extracted through two relations: the Banks-Casher relation, which connects the spectral density of the Dirac operator to the chiral condensate, and the GMOR relation for the unphysical adjoint pion, which connects the pion mass and decay constant to the condensate. Conversion to the renormalization-group-invariant scheme uses the exact NSVZ $\beta$ and tau functions together with two-loop perturbative renormalization constants at scale $\mu=1/a$; the supersymmetric limit is taken by a joint extrapolation in lattice spacing and adjoint-pion mass squared.

What would settle it

Repeat the measurement at a larger value of $N$ (for example, $N=529$) and at smaller lattice spacing, using a non-perturbative determination of the renormalization constants; if $\Sigma_{\rm RGI}/\Lambda^3_{\rm NSVZ}$ moves outside $1.77(65)$ or tracks the strong-coupling $2e/N$ prediction, the central claim would be falsified.

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Extended reading notes

Core claim

The paper's central claim is that the renormalization-group-invariant gluino condensate in the supersymmetric limit of large-$N$ $SU(N)$ Yang-Mills is $\Sigma_{\rm RGI}/\Lambda^3_{\rm NSVZ}=1.77(65)$, quoted from the GMOR determination, with the Banks-Casher determination $2.39(97)(72)$ giving a fully compatible value. Both determinations are presented as consistent with the exact NSVZ weak-coupling result $\Sigma_{\rm RGI}/\Lambda^3_{\rm NSVZ}=1$, and the observed $N$-dependence is interpreted as excluding the strong-coupling result $2e/N$ and the fractional-instanton result linear in $N$. The paper claims this is the first non-perturbative, first-principles lattice computation of the large-$N$ gluino condensate and the first lattice-versus-analytic comparison for this quantity.

Load-bearing premise

The whole comparison rests on the premise that the single-site twisted lattice at $N=169$, $289$, or $361$ reproduces the large-$N$, infinite-volume $SU(N)$ supersymmetric Yang-Mills theory, and that the two-loop perturbative conversion to the renormalization-group-invariant scheme is accurate enough; if either assumption fails, the claimed agreement with the exact weak-coupling value would not be established.

Editorial extensions

If this is right

  • The exact weak-coupling value $\Sigma_{\rm RGI}/\Lambda^3_{\rm NSVZ}=1$ survives a direct lattice test, while the strong-coupling $2e/N$ and fractional-instanton linear-in-$N$ predictions are excluded at the explored $N$.
  • The Banks-Casher and GMOR determinations agree, so the final number does not depend on whether the condensate is extracted from the Dirac spectrum or from pion physics.
  • Since the result is expressed in an RGI scheme with the scale set on the same ensembles, the lattice-to-analytic comparison is scheme-independent and can be sharpened by reducing the 30% perturbative systematic.
  • The successful use of twisted reduction for a supersymmetric fermionic observable extends the same approach to other large-$N$ supersymmetric quantities.

Reading between the lines

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

  • The paper's $N$-dependence argument is qualitative; if twisted reduction is exact, the present compatibility suggests the $N\to\infty$ limit would land at $1$, with the central value $1.77$ reflecting residual $1/N$ or perturbative uncertainties.
  • A natural next step, left implicit by the authors, is to replace the two-loop perturbative renormalization with a non-perturbative determination of the renormalization constants, which would reduce the dominant 30% systematic and turn the compatibility check into a precision test.
  • The same one-point lattice approach could be applied to other fermionic condensates in large-$N$ theories with adjoint matter; for example, orientifold large-$N$ QCD has a predicted $N$-dependent condensate that would be a direct target.
  • The exclusion of the fractional-instanton prediction at the explored $N$ implies that, if such configurations matter at all, their contribution is subleading in $1/N$ rather than leading.
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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

3 major / 4 minor

Summary. This proceedings paper reports a lattice determination of the large-N N=1 SUSY Yang-Mills gluino condensate using twisted volume reduction at N=169, 289 and 361. The condensate is extracted by two independent methods, the Banks-Casher relation and a Gell-Mann-Oakes-Renner-like formula, and converted to the renormalization-group-invariant quantity Sigma_RGI / Lambda^3_NSVZ using two-loop perturbation theory. The main results are Sigma_RGI / Lambda^3_NSVZ = 2.39(97)(72) from Banks-Casher and 1.77(35)(53) from GMOR; the GMOR value, combined as 1.77(65), is quoted as final and stated to be compatible with the exact weak-coupling NSVZ prediction Sigma_RGI / Lambda^3_NSVZ = 1. The paper further claims that the observed N-dependence rules out the strong-coupling (2e/N) and fractional-instanton (linear-in-N) alternatives.

Significance. If the result holds, this would be the first lattice determination of the large-N gluino condensate and the first lattice-vs-analytic comparison for this quantity, providing a nonperturbative test of the NSVZ weak-coupling prediction and of large-N twisted volume reduction for a supersymmetric theory. The paper has several notable strengths: the two independent determinations are mutually consistent, the comparison is performed in an RGI scheme with the NSVZ Lambda parameter, and the use of twisted reduction at large N is a technically demanding and interesting approach. However, the significance is tempered by the fact that the central comparison rests on two fragile premises: the uncontrolled finite-N behavior of the reduced model and a two-loop perturbative conversion with a flat systematic error. The N-dependence argument against the strong-coupling and fractional-instanton alternatives is qualitatively robust, but the quantitative compatibility with the weak-coupling value is not established at high confidence.

major comments (3)
  1. [Section 3 and Section 4.5 (Eq. (30), Fig. 4)] The central claim that the simulated ensembles reproduce the large-N, infinite-volume SYM theory is not demonstrated. The extrapolation in Eq. (30) is only in the lattice spacing and the adjoint-pion mass; no N -> infinity extrapolation is performed, and Fig. 4 shows data only for N=361. The Banks-Casher and GMOR relations require an infinite-volume limit, which in the twisted reduced model is replaced by the large-N limit, and the paper provides no quantitative check that N=361 is in the asymptotic regime. A finite-N correction of order 20%—a plausible size for unquantified 1/N effects at N=361—would shift the central value in Eq. (34) and could change the stated compatibility with the exact weak-coupling result in Eq. (33). The authors should provide an explicit N-dependence fit or quantitative evidence (e.g., center-symmetry stability, comparison of N=289 and N=361 results, or an estimate of the leading 1/N correction) that finite-N effects are negligible.
  2. [Section 4.4 and Section 5 (Eqs. (27)-(28))] The conversion from the lattice to the RGI condensate relies on two-loop perturbation theory at mu = 1/a, including the renormalization constants in Eqs. (17)-(18) and the two-loop running in Eq. (28). The flat 30% systematic error added in Section 5 is not derived from a scale-variation study or from the size of known higher-order terms, and it is the dominant uncertainty in the final result. Since the comparison with the exact NSVZ value is only about 1.2 sigma, the compatibility claim is sensitive to this unquantified systematic. A more explicit justification is needed, such as a comparison between one-loop and two-loop conversion factors, a variation of the matching scale mu, or, where possible, a nonperturbative renormalization check.
  3. [Section 4.1, footnote 1 and Eq. (16)] The paper corrects an oversight in the original publication [16] regarding the O(lambda^2) difference between Z_S and Z_S^(0), encoded in r_m. The text states that the impact is negligible and that conclusions are unchanged, but no numerical estimate of the resulting shift in the final values in Eqs. (31)-(32) is given. Because this correction touches the renormalization procedure that is already the dominant source of uncertainty, the reader needs a quantitative statement of the change in the final central values and errors to assess the stability of the reported result.
minor comments (4)
  1. [Abstract and Section 5] The phrase 'resolving a 40-year-long debate' is stronger than the evidence supports: the result 1.77(65) is compatible with the weak-coupling prediction 1 at about 1.2 sigma, but the uncertainty is dominated by a flat 30% systematic. Suggest softening the claim to 'providing the first lattice comparison' or similar.
  2. [Section 3] There are minor typographical errors: 'unvailed' should be 'unveiled' and 'More precisley' should be 'More precisely'.
  3. [Section 2, Eq. (11)] The trace normalization in <Tr lambda^2> is not specified. Since the prefactor in Eq. (2) depends on the trace convention, please define the normalization explicitly.
  4. [Figure 3 caption] The caption repeats 'b = 0.340' three times in the left panel description and is difficult to parse. The labels of the central and right panels could be made clearer to indicate which N value corresponds to which symbol.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the lattice condensate and NSVZ scale are computed from independent spectral and GMOR inputs; the analytic value 1 is the comparator, not an input.

full rationale

The derivation chain is self-contained on the lattice side: the Banks–Casher determination (Eq. 16) uses the slope of the spectral mode number, the GMOR determination (Eq. 24) uses the adjoint pion mass, decay constant, and PCAC mass, and the NSVZ scale (Eqs. 25–26) is obtained from asymptotic scaling of the gradient-flow reference scale. The RGI conversion (Eq. 27) uses standard 2-loop MS renormalization constants and the exact NSVZ beta/tau functions; none of these inputs is fitted to the final ratio Sigma_RGI/Lambda^3_NSVZ. The value 1 appears only as the analytic comparator in Eq. (33), and the lattice results (Eqs. 31–32) are free to disagree with it. Quoting the paper, 'Both are compatible with the WC instanton calculation: Sigma_RGI/Lambda^3_NSVZ = 1, (exact NSVZ analytic WC result)'—this is a comparison, not an input. Citations to the companion papers [16] and [41] provide ensembles, scale setting, and correlator measurements; they are reproducible external data sources, not unverified assumptions equivalent to the claimed result. The potential weaknesses—finite-N twisted volume reduction without an infinite-N extrapolation, and the 30% perturbative systematic—are robustness and correctness concerns, not circularity.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central calculation depends on established lattice-SUSY technology (twisted reduction, Wilson gluino, PQChPT, Banks-Casher/GMOR) and on perturbative scheme conversion. No new entities are postulated. The only fitted parameters are the slopes of the SUSY-limit extrapolation and the flat 30% systematic that dominates the error budget.

free parameters (3)
  • c1 (continuum slope) = not reported in proceedings
    Linear coefficient in a/sqrt(8t0) in the joint chiral-continuum fit Eq. (30); determined from data at b=0.340, 0.345, 0.350 and affects the SUSY-limit intercept.
  • c2 (chiral slope) = not reported in proceedings
    Linear coefficient in 8t0 m_pi^2 in Eq. (30); fitted to data at several pion masses and affects the massless-gluino extrapolation.
  • flat perturbative systematic = 0.30
    A hand-set 30% systematic error added to cover 2-loop perturbative renormalization, stated in Section 5 to be the dominant source of uncertainty; not derived from data.
assumptions (7)
  • domain assumption Large-N twisted volume reduction reproduces infinite-volume SU(N) SYM at N=169, 289, 361 with dynamical adjoint Wilson fermions.
    Invoked in Section 3; the simulated one-point lattice is identified with the 4D theory. If center symmetry breaks or 1/N corrections are not negligible, the ensembles are not SYM.
  • domain assumption The SUSY limit is the joint continuum and massless-gluino limit of Wilson-fermion lattice SYM (Kaplan-Curci-Veneziano prescription).
    Section 3, Refs. [42,43]; Wilson fermions break SUSY explicitly and this prescription is the standard restoration route.
  • domain assumption The adjoint pion is the pseudo-Goldstone boson of the spontaneously broken U(1)_A, and the GMOR relation Eq. (13) holds with its decay constant.
    Section 3 and Eq. (13); this provides one of the two independent condensate determinations.
  • domain assumption The mode-number slope in the Banks-Casher relation gives the chiral condensate in the infinite-volume, massless limit.
    Section 4.1, Eq. (12), Refs. [45,46]; requires spontaneous chiral symmetry breaking and a nonzero spectral density at zero eigenmode.
  • domain assumption Two-loop MS perturbation theory at mu=1/a reliably converts lattice bare quantities to RGI quantities, with deviations covered by a flat 30% systematic.
    Sections 4.1-4.4, Eqs. (17)-(18), (27)-(28), and the Section 5 statement that 30% is added for perturbative renormalization; this is the dominant source of uncertainty.
  • domain assumption The sign-quenched ensemble is unbiased because no negative Pfaffian signs were observed.
    Section 3; if negative signs were rare but nonzero, reweighting would be required, but the text reports none observed in Ref. [41].
  • standard math The NSVZ beta function, the NSVZ anomalous dimension, and the scheme-conversion constants are exact known results from the literature.
    Used in Section 2 and Eq. (27) to define Lambda_NSVZ and the RGI condensate; these are standard SUSY results, not derived in this paper.

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Pith. "Pith review of The gluino condensate of large-$N$ SUSY Yang-Mills." pith.science (2026). https://pith.science/paper/WDJL2JMI

@misc{pith2026241214067,
  author       = {Pith},
  title        = {Pith review of: The gluino condensate of large-$N$ SUSY Yang-Mills},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WDJL2JMI}},
  note         = {Machine review of arXiv:2412.14067}
}
abstract

We present the first lattice determination of the SUSY $\mathrm{SU}(N)$ Yang-Mills gluino condensate at large $N$. We exploit large-$N$ twisted volume reduction, and present two determinations based on the Banks-Casher relation and on a Gell-Mann-Oakes-Renner-like formula, both giving perfectly compatible results. By expressing the lattice results in the Novikov-Shifman-Vainshtein-Zakharov scheme, we are able for the first time to compare lattice and analytical computations, resolving a 40-year-long debate about the actual value and $N$-dependence of the gluino condensate.

Figures

Figures reproduced from arXiv: 2412.14067 by the authors.

Figure 1
Figure 1. Numerical results using spectral methods. Figures taken from [16]. The gluino condensate is obtained from the BC formula via the Giusti–Lüscher method [38, 46–50]. • We solved numerically 𝛾5𝐷W [𝑈]  𝑢𝜆 = 𝜆 𝑢𝜆 for the first O (100) eigenvalues. • Counting the number of modes below a certain threshold 𝑀 we obtained the mode number ⟨𝜈(𝑀, 𝑚)⟩ = ⟨#|𝜆| ≤ 𝑀⟩. • The gluino condensate is obtained via (here 𝑉 = 𝑎 4 with 𝑎 the… view at source ↗
Figure 2
Figure 2. Extrapolation of ΛNSVZ towards the SUSY limit. Figure taken from [16]. We computed the Λ-parameter using 2-loop asymptotic scaling, and the 3 improved couplings in￾troduced earlier (here 𝑎𝜒 denotes the lattice spacing extrapolated towards the massless gluino limit): √︁ 8𝑡0ΛNSVZ = lim 𝑎𝜒→0 ΛNSVZ ΛMS ΛMS Λs √ 8𝑡0 𝑎𝜒 exp n − 𝑓  𝜆 (s) t o , 𝑓 (𝑥) = 1 2𝑏0  1 𝑥 + 𝑏1 𝑏0 log(𝑏0𝑥)  . (25) where ΛNSVZ/ΛMS = e −1/18 [54]. … view at source ↗
Figure 3
Figure 3. Left panel: calculation of 𝐹𝜋/𝑁 in the SUSY limit. Central and right panels: collection of our results for the third root of the RGI condensate in units of the SUSY scale in the NSVZ scheme ΛNSVZ obtained with the GMOR and the BC formulas respectively. Figures adapted from [16]. 5. Conclusions: the RGI gluino condensate in the SUSY limit 0.00 0.05 0.10 0.15 0.20 a/√ 8t0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Σ 1/3 RGI ΛNSVZ N … view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Extrapolation towards the SUSY limit of the RGI gluino condensate determined from the BC and the GMOR formulas for the largest value of 𝑁 explored. Figure adapted from [16]. We extrapolate our results for Σ 1/3 RGI /ΛNSVZ towards the SUSY limit performing a joint chira…

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Reviewed August 11, 2026 · model on record in the stance chip above.