Small-instanton effects in an atlas of KSVZ axion models

Ning Chen; Saurabh K. Shukla

REVIEW 4 major objections 4 minor 77 references

Small-instanton effects in an atlas of KSVZ axion models

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Small-instanton effects from vector-like quarks can enhance the KSVZ axion mass to tens of MeV at fixed decay constant, breaking the standard QCD axion relation while leaving the axion-photon coupling unchanged.

desk verdict A systematic KSVZ instanton atlas with genuinely new UV-sensitivity benchmarks, whose headline mass numbers rest on the acknowledged but unquantified aligned-minima assumption. read the letter →

arxiv 2608.08602 v1 pith:7J64AWHE submitted 2026-08-09 hep-ph hep-th

Ning Chen , Saurabh K. Shukla This is my paper · ORCID

classification hep-phhep-th

keywords smallinstantonsKSVZaxionmodelsvector-likequarksmassaxion-photoncouplingstrongCPprobleminstantonzeromodesgaugeperturbativity

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 small-size instantons, the same topological fluctuations that generate the QCD axion potential, can be dramatically enhanced in KSVZ axion models by heavy vector-like quarks, and can then dominate the axion mass. The enhancement is controlled by how the instanton's fermion zero modes are saturated: each scalar-Yukawa loop used in place of two mass insertions shifts the instanton-size integral toward the ultraviolet, turning a suppressed correction into a power of $M_{\rm Pl}/\Lambda$ as large as $(M_{\rm Pl}/\Lambda)^{13}$ in the strongest multi-quark spectra. Scanning fourteen gauge-perturbative representations, the authors find that, if the QCD and small-instanton potentials are aligned, the physical axion mass can rise to 71.2 MeV at fixed decay constant (two copies of the octet-doublet representation $R_{13}$), while the axion-photon coupling is unchanged. If true, the standard QCD relation among mass, decay constant, and photon coupling no longer defines the full axion search band, and unswept regions of the $(m_a,g_{a\gamma\gamma})$ plane become legitimate targets.

What carries the argument

The machinery is the one-instanton vertex and the naive-dimensional-analysis (NDA) power counting for saturating its fermion zero modes. For every VQ set, the number of zero-mode pairs is $2|N|=2T(R_3)\dim(R_2)$, and a closure with $k$ scalar-Yukawa loops and $r$ mass insertions must satisfy $2|N|=2k+r$; the exponent $\Delta=b_3+r-4$ then decides whether the size integral is infrared-dominated ($\Delta>0$), logarithmic, or ultraviolet-dominated ($\Delta<0$). The fully Yukawa-closed term carries the largest power of $M_{\rm UV}/\Lambda$ but is weighted by $(y_Q^2/16\pi^2)^k$, and the one-instanton determinant prefactors $C_3(R_3,R_2)$ complete the susceptibility estimate. The atlas of fourteen perturbative VQ representations, with copy-number limits fixed by two-loop gauge running to the Planck scale, supplies these inputs for every spectrum.

What would settle it

A direct computation of the one-instanton determinant and the fermion zero-mode overlap integrals for the fully Yukawa-closed vertex of the two-copy $R_{13}$ spectrum at $\Lambda=5\times10^{11}$ GeV with $y_Q=1$ would settle the claim: if the exact result comes out orders of magnitude below the NDA estimate, the predicted $m_a\simeq71.2$ MeV collapses.

Watch

Extended reading notes

Core claim

The authors claim that small-instanton contributions to the axion potential in KSVZ models are governed by a counting rule: a vector-like quark (VQ) representation with $2|N|$ conjugate zero-mode pairs is closed by $k$ scalar-Yukawa loops and $r$ mass insertions with $2|N|=2k+r$, and the instanton-size integral is ultraviolet-dominated when $\Delta=b_3+r-4<0$. Because each Yukawa loop lowers $\Delta$ by two, maximally Yukawa-closed contractions give the largest UV enhancement: $(M_{\rm Pl}/\Lambda)^{31/3}$ for a single color-15 VQ, $(M_{\rm Pl}/\Lambda)^{13}$ for two copies of the octet doublet $R_{13}$, and the same power 13 for the distinct sets $S_3$-$S_7$. Under the stated assumption that the QCD and small-instanton potentials have equal periodicity and aligned minima, the total potential is $V(a)=-(\chi_{\rm QCD}+\chi_{\rm SI})\cos(\bar{\theta}+a/f_a)$, so the axion mass at fixed $f_a$ is enhanced: the largest values quoted are 3.43 eV for a single VQ, 71.2 MeV for two copies of $R_{13}$, and 65.3 MeV for the $N_{\rm DW}=22$ assignment of the distinct set $S_4$. The axion-photon coupling, by contrast, stays determined by $f_a$ and $E/N$, and the paper exhibits contours in the $(m_a,g_{a\gamma\gamma})$ plane that leave the conventional QCD-axion band.

Load-bearing premise

The load-bearing premise is that the QCD and small-instanton potentials have the same periodicity and their minima coincide, so their strengths simply add; a relative phase would change the mass-shift formula and generically produce a CP-violating minimum.

Editorial extensions

If this is right

Small-instanton contributions can become the dominant source of the axion mass: once $\chi_{\rm SI}\gtrsim\chi_{\rm QCD}$, the mass shift follows $\Delta m_a/m_{a,\rm QCD}\simeq\sqrt{\chi_{\rm SI}/\chi_{\rm QCD}}$ and the axion can weigh tens of MeV with $f_a$ fixed.
At fixed $f_a$ and anomaly ratio $E/N$, $g_{a\gamma\gamma}$ is unchanged, so the enhanced mass moves a model horizontally in the $(m_a,g_{a\gamma\gamma})$ plane: two copies of $R_{13}$ reach $m_a=71.2$ MeV and the distinct set $S_4$ reaches 65.3 MeV.
The strongest enhancements are $(M_{\rm Pl}/\Lambda)^{31/3}$ for a single VQ and $(M_{\rm Pl}/\Lambda)^{13}$ for identical copies and distinct VQ sets, so fully Yukawa-closed spectra are the ones that deviate most from the QCD relation.
Several predicted contours lie outside the conventional QCD-axion band but inside currently unexcluded regions; for example, the set $S_2=\{R_{11},R_{13}\}$ with $(E/N,N_{\rm DW})=(34/33,22)$ gives $|g_{a\gamma\gamma}|\simeq3.16\times10^{-14}$ GeV$^{-1}$ at $m_a\simeq81.5$ eV.
The atlas also identifies one-VQ spectra that stay close to the QCD relation (e.g., $R_{11}$, $R_{12}$, $R_{13}$) alongside those that leave it ($R_{14}$, $R_{15}$, and multi-copy cases), so the framework classifies models by the size of their expected deviation.

Reading between the lines

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

A natural next step, not taken in the paper, is to quantify the aligned-minima assumption: if the two potentials carry a relative phase, the same machinery predicts a displaced minimum and an induced electric dipole moment roughly proportional to $\chi_{\rm SI}/\chi_{\rm QCD}$, which neutron-EDM searches could bound.
The counting rule transfers directly to other axion constructions: adding more PQ-charged scalars or larger gauge representations would generate extra Yukawa closures, so similar atlases could be drawn for other axion models with even larger UV exponents.
For axion dark matter, an enhanced mass at fixed $f_a$ changes the relation between decay constant and relic abundance, so a spectrum that seems excluded in the standard band could reappear as a heavier axion with the same photon coupling in higher-frequency haloscope searches.
The NDA estimates are estimates of the leading size; computing the exact fermion zero-mode overlap integrals for a fully Yukawa-closed vertex would convert the quoted masses into sharper numbers, and could lower them significantly.

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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 paper is a systematic extension of the instanton-NDA program to an atlas of KSVZ spectra: single VQ, identical copies, distinct sets, with two-loop perturbativity cuts. What's genuinely new is the catalog itself—the endpoint exponents, the (M_Pl/Λ)^13 maximal scalings for identical copies and distinct sets, and the concrete (m_a, g_aγγ) trajectories. The internal arithmetic checks out: I re-derived the endpoint exponents for R13^(2) and R1^(25) and they match. The authors are honest about the framework coming from Csáki et al. and about what they did not compute.

The soft spot is the aligned-minima assumption. Eq. (4.1) adds QCD and small-instanton susceptibilities as if the two cosine potentials share a minimum. The stress-test is right that this is not derived: phases in the m_Q and y_Q insertions of Eq. (2.18) can rotate the small-instanton term relative to the QCD term, giving V = -χ_QCD cos θ - χ_SI cos(θ - δ). For the headline benchmarks (two copies of R13, set S4), where χ_SI/χ_QCD is huge, a generic δ displaces the minimum by O(1), so the quoted masses—71.2 MeV and 65.3 MeV—and the strong-CP-free conclusion are conditional on δ=0. The authors flag this in Secs. I, IV, and V, so it is acknowledged rather than hidden. But the abstract and summary present the numbers without the caveat, which will mislead casual readers. A referee should ask for a phase scan or at least an explicit statement that δ is a free parameter that can ruin the CP-conserving benchmark.

The other limitations are milder and standard for NDA: no error bars on order-one coefficients, a single-threshold approximation (Λ=m_σ) with M_Q varied independently, and fixed copy counts at Λ=5×10^11 GeV. None of these undercuts the qualitative message—small instantons can dominate the axion curvature in plausible KSVZ spectra, modifying the standard mass–coupling relation. The paper is a useful map of where that happens, with concrete targets for searches outside the QCD band.

Who this is for: axion model-builders and experimental colleagues who want to know where heavy KSVZ spectra might land. It deserves a serious referee. I would want the phase treatment strengthened before publication, but the core atlas and the qualitative conclusion should survive.

Recommendation: send to peer review; ask for a quantitative discussion of misaligned phases and a more careful statement of the benchmark conditions.

Referee Report

4 major / 4 minor

Summary. The paper computes small-instanton contributions to the axion potential in KSVZ models with vector-like quarks, using the instanton-NDA power counting of Ref. [48]. It classifies the allowed zero-mode closures for a single VQ, for multiple identical copies, and for sets of distinct VQ representations, and it imposes one- and two-loop gauge perturbativity up to the Planck scale. The endpoint exponent Delta = b3 + r - 4 determines whether each contribution is UV or IR dominated, and the authors identify the strongest UV enhancements as (M_Pl/Lambda)^{31/3} for a single VQ and (M_Pl/Lambda)^{13} for identical copies and distinct VQ sets. Assuming equal periodicity and aligned minima of the QCD and small-instanton potentials, the paper then derives the enhanced axion mass and the resulting axion-photon coupling, reporting benchmark values such as m_a = 71.2 MeV for two copies of R13 and m_a = 65.3 MeV for the S4 set, and displaying contours in the (m_a, g_aγγ) plane against current experimental bounds.

Significance. The paper provides a systematic and internally coherent atlas of small-instanton effects in KSVZ axion models. The zero-mode counting, the endpoint classification, and the perturbativity filter are presented explicitly, and the group-theoretic determinants are collected in an appendix, which makes the calculation reproducible from the text. The identification of the strongest UV scalings, (M_Pl/Lambda)^{31/3} and (M_Pl/Lambda)^{13}, is a useful result that is likely robust in ordering even if the numerical prefactors are NDA estimates. If the aligned-potential premise holds, the modified m_a-g_aγγ relation opens new search regions beyond the conventional QCD-axion band, which is of genuine phenomenological interest. The main weaknesses are that the headline masses and couplings depend on an acknowledged but unquantified phase-alignment assumption, and that the numerical benchmarks are order-one NDA estimates presented without an error budget. The paper is therefore best read as a model-space survey with indicative predictions rather than as a precision calculation.

major comments (4)

[Sec. IV A, Eq. (4.1); Secs. I, V] The central numerical claims rest on the assumption that the QCD and small-instanton potentials have equal periodicity and aligned minima, so that V(a) = -(χ_QCD + χ_SI) cos(θ̄ + a/f_a). The text acknowledges in Secs. I and V that relative phases require a model-dependent treatment, but this is exactly the premise on which Eq. (4.5) and the benchmark masses 71.2 MeV and 65.3 MeV depend. In the one-instanton amplitude of Eq. (2.18), the mass and Yukawa factors can carry complex phases; summing instanton and anti-instanton amplitudes generically gives -χ_QCD cos θ - χ_SI cos(θ - δ) for a relative phase δ, not Eq. (4.1). For the spectra with χ_SI/χ_QCD ≫ 1, an O(1) δ displaces the minimum by an O(1) amount, induces strong CP violation, and changes the mass-shift formula. The qualitative statement that small instantons can dominate the curvature is robust, but the quantitative atlas and the claim that the standard relation is simply modified without a CP problem are conditional on δ = 0. The authors should either derive or quantify this alignment, for example by scanning the phase and showing which conclusions survive, or clearly mark all m_a benchmarks and g_aγγ contours as applying only in the strictly aligned limit.
[Sec. II, Eq. (2.20); Sec. IV A] The instanton-size integral is evaluated with a single threshold Λ = m_σ over the whole range 1/M_UV ≤ ρ ≤ 1/Λ, with b3 fixed. In the mass plots, M_Q is scanned over many orders of magnitude, including values well above Λ. Once M_Q exceeds Λ, the vector-like quark is no longer active above its own mass, so b3 and the RG-invariant scale Λ_G should be matched at M_Q as well; using a single threshold changes both the endpoint exponent Δ and the prefactor. As written, the contours in Figs. 3 and 4 for M_Q ≫ Λ are not derived from the stated matching procedure. The authors should either restrict the scan to M_Q ≤ Λ or extend Eq. (2.20) to piecewise intervals with matching at both M_Q and m_σ.
[Sec. III, Eq. (3.4); Sec. IV A] There is an internal inconsistency in the relation between M_Q, y_Q, and Λ. From Eq. (3.4), m_σ = sqrt(2 λ_Φ) v and M_Q = y_Q v / sqrt(2). With λ_Φ = 1 and the identification Λ = m_σ, one has v = Λ and hence y_Q = sqrt(2) M_Q/Λ, whereas Sec. IV A states that 'Eq. (3.4) gives y_Q = 2 M_Q/Λ' and Fig. 5 uses M_Q = y_Q Λ/2. The factor sqrt(2) propagates into the quoted benchmark masses and couplings. The paper should state the exact convention used and recompute the affected numbers, or explain why the relation differs from Eq. (3.4).
[Sec. IV A, Figs. 3-6] The numerical predictions are NDA estimates with no assessment of the order-one uncertainties in the determinant prefactors C3, the loop factors (2π/α_s)^6, and the identification of Λ with m_σ. The paper quotes m_a = 71.2 MeV and m_a = 65.3 MeV as if they were sharp values. Because χ_SI appears with powers such as (M_UV/Λ)^13, an O(1) change in Λ changes the enhancement by a large factor, and the quoted benchmark masses are correspondingly fragile. The abstract and Sec. V should present these benchmarks as order-of-magnitude indicators, or include an explicit error budget, before they are used to define search targets.

minor comments (4)

[Abstract] The sentence 'while the axion-photon of g_aγγ remains controlled by f_a and the anomaly ratio of E/N' is grammatically incomplete and should read 'while the axion-photon coupling g_aγγ remains controlled by f_a and the anomaly ratio E/N'.
[Sec. II, after Eq. (2.16)] The phrase 'the scalar field is an active propagating propagating degree of freedom' contains a duplicated word; it should be 'an active propagating degree of freedom'.
[Sec. IV A, first paragraph] The authors explicitly state that the maximum multiplicities and distinct-VQ sets are fixed at Λ = 5×10^11 GeV and then reused for other values of Λ in the mass and coupling plots. This is an acknowledged limitation, but its quantitative impact on the displayed contours is not estimated; a sentence quantifying the expected shift of the perturbativity bounds would help.
[Figs. 3 and 4] The figures use colored symbol legends that may be hard to distinguish in monochrome print; a labeled legend or distinct line styles for the individual R_i spectra would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No material circularity: chi_SI is computed from an independent instanton-NDA integral, and the aligned-potential assumption is an explicit premise, not a derived output.

full rationale

The derivation chain is: choose KSVZ VQ representations and copy numbers (Tab. I) from gauge-perturbativity; compute chi_SI from the instanton-size integral Eq. (2.18) with zero-mode closure Eq. (2.19) and endpoint classification Eq. (2.22); then translate the combined susceptibility into m_a via Eq. (4.2) and into g_aγγ via the standard anomaly formula Eq. (4.8). At no point is a 'prediction' used to fix chi_SI, the copy numbers, y_Q, M_Q, or Lambda. The parameters y_Q=1, Lambda=5e11 GeV, M_Q=Lambda/2 are stated benchmark inputs, and the quoted masses (71.2 MeV, 65.3 MeV) are outputs of those inputs, not fits. The only load-bearing external framework is the instanton-NDA prescription of Ref. [48], which is cited as an independent calculation, and the standard instanton/lattice inputs Refs. [36,37,38,60]; none of these is authored by the present authors except a non-load-bearing GUT reference [23] in a general list [19-25]. The aligned-minima assumption behind Eq. (4.1) is explicitly stated in the abstract, Sec. I, and Sec. IV, and its failure is acknowledged in Sec. V ('relative phases between the QCD and ultraviolet potentials require a model-dependent treatment'); that makes the quoted numbers conditional, but it is a stated model assumption rather than a circular definition or a fitted input renamed as a prediction. Accordingly, no specific circular step can be exhibited, and the paper is self-contained against external benchmarks for what it claims.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The atlas is built on the instanton-NDA framework of Ref. [48] and on several stated modeling choices: aligned potentials, a single matching scale Lambda = m_sigma, and perturbativity up to M_Pl. Model parameters (Yukawa, VQ mass, matching scale, PQ charges) are scanned rather than fitted, so the predictions are parametric consequences of the assumed spectra.

free parameters (5)

y_Q = scanned; set to 1 in g_aγγ plots
VQ Yukawa coupling; controls the scalar-Yukawa closures and the VQ mass M_Q = y_Q Lambda / 2.
M_Q = scanned in mass plots
VQ mass; enters through mass insertions (M_Q rho)^r and the assumed threshold structure.
Lambda = 5e11 GeV for copy bounds; varied 5e10 to 5e16 GeV in coupling plots
Matching scale identified with the radial-mode mass; sets the instanton-size integration range.
lambda_Phi = 1
Scalar quartic coupling; fixes m_sigma = sqrt(2) v and is chosen to set the matching condition.
DeltaX_a = +/-1 scanned for distinct VQ sets
PQ charge assignments; scanned to extremize E/N and N_DW in the axion-photon analysis.

assumptions (4)

domain assumption QCD and small-instanton potentials have equal periodicity and aligned minima.
Used in Eq. (4.1) to add susceptibilities; if misaligned, the mass shift formula and CP properties change. Stated in Sec. I and Sec. IV.
domain assumption The instanton-NDA power-counting rules of Ref. [48] correctly estimate small-instanton amplitudes for these models.
The whole susceptibility calculation uses these rules; order-one uncertainties are not quantified.
domain assumption A single matching scale Lambda = m_sigma describes the change in active field content, with b3 constant over the integration range.
Identified in Sec. III; non-degenerate VQ thresholds are neglected, as acknowledged in Sec. V. Enters Eqs. (2.18)-(2.22).
domain assumption Gauge couplings remain perturbative, alpha_i < 1, up to M_Pl; spectra failing this are discarded.
Selection criterion for the atlas; if relaxed, different or larger UV powers could appear.

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Pith. "Pith review of Small-instanton effects in an atlas of KSVZ axion models." pith.science (2026). https://pith.science/paper/7J64AWHE

@misc{pith2026260808602,
  author       = {Pith},
  title        = {Pith review of: Small-instanton effects in an atlas of KSVZ axion models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7J64AWHE}},
  note         = {Machine review of arXiv:2608.08602}
}

abstract

We investigate small-instanton contributions to the axion potential across a range of KSVZ models containing vector-like quarks~(VQs), using naive dimensional analysis. We consider scenarios containing a single VQ, multiple identical copies, and sets of distinct VQs, requiring in each case that the gauge couplings remain perturbative up to the Planck scale under the two-loop gauge running. The associated fermion zero-mode content varies between these cases, requiring different combinations of mass insertions and scalar--Yukawa contractions for its saturation. Increasing the copies of VQs can render the instanton-size integral dominated by instantons of the smallest size, corresponding to the scale near the ultraviolet~(UV) cut-off. The resulting contribution then becomes sensitive to the UV completion and the induced potential can compete with, or dominate over, the ordinary QCD contribution. Assuming that the QCD and small-instanton potentials are aligned, we determine the resulting axion-mass shift and its consequences for the axion--photon coupling. When the small-instanton induced susceptibility becomes comparable to or larger than the QCD susceptibility, the physical axion mass of $m_a$ is enhanced at fixed decay constant~$f_a$, while the axion-photon of $g_{a\gamma\gamma}$ coupling remains controlled by $f_a$ and the anomaly ratio of $E/N$. The standard QCD relation among $m_a$, $f_a$ and $g_{a\gamma\gamma}$ is consequently modified, opening new regions of the $(m_a,g_{a\gamma\gamma})$ plane for axion searches.

Figures

Figures reproduced from arXiv: 2608.08602 by the authors.

**Figure 2.** FIG. 2. The ’t Hooft vertex of SU(3) [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** FIG. 3. Axion-mass shift as a function of the VQ mass at (a) Λ = 5 [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Axion-mass shift for the distinct-VQ spectra, for the same values of Λ as in Fig. [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]

**Figure 5.** Figure 5: FIG. 5. The plot shows the variation of absolute axion–photon coupling as a function of the physical [PITH_FULL_IMAGE:figures/full_fig_p024_5.png]

**Figure 6.** Figure 6: FIG. 6. Axion–photon predictions for the distinct-VQ sets [PITH_FULL_IMAGE:figures/full_fig_p026_6.png]

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