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REVIEW 3 major objections 3 minor 42 references

On the contribution from the light quarks to $H\to\gamma\gamma , \gamma Z$

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The light-quark contribution to H→γγ does not vanish: it starts linearly, not quadratically, in the light-quark masses, and shifts the predicted decay rate by about three percent.

desk verdict The qualitative SVV argument is right and refutes the 2025 claim, but the LMD numbers are an estimate and the 'exact' language oversells them. read the letter →

arxiv 2507.22551 v1 pith:JP4TD2HF submitted 2025-07-30 hep-ph

Marc Knecht , Baiyi Qian This is my paper
classification hep-ph PACS 12.38.-t14.80.Bn
keywords light-quarkcontributionsH→γγdecayH→γZchiralsymmetrybreakingscalar-vector-vectorcorrelatorlowest-mesondominancequarkcondensatelarge-N_cQCD
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 claims that the light $u,d,s$ quarks do contribute to the Higgs decay amplitudes $H\to\gamma\gamma$ and $H\to\gamma Z$, contrary to a recent claim that their contribution vanishes. The contribution is controlled by the QCD three-point function of a scalar quark density and two vector currents, which is an order parameter of chiral symmetry breaking: it vanishes at every order of perturbation theory, which is why the usual treatment sees a quadratic mass suppression, but it is non-zero non-perturbatively. Consequently the light-quark contribution starts linearly in the quark masses rather than quadratically, and for $H\to\gamma\gamma$ the summed light-quark couplings are enhanced by about three orders of magnitude, shifting the predicted decay rate by about three percent. The $H\to\gamma Z$ channel is barely affected. If the claim is right, the effect becomes visible as precision on the $H\to\gamma\gamma$ branching ratio approaches the percent level.

What carries the argument

The load-bearing object is the scalar-vector-vector three-point function $\Gamma^{abc}_{\mu\nu}(q_1,q_2)=\int d^4x\,d^4y\,e^{iq_1\cdot x}e^{iq_2\cdot y}\langle 0|T\{S^a(0)V^b_\mu(x)V^c_\nu(y)\}|0\rangle$, built from colour-singlet quark bilinears in three-flavour QCD, whose flavour structure matches the singlet and nonsinglet components of the Higgs and vector currents. In the chiral limit this correlator is an order parameter of chiral symmetry breaking, and the short-distance expansions (3.4) and (3.6) show a leading behaviour proportional to the quark condensate, which proves it is non-vanishing. To reach the physical decay point the paper uses the lowest-meson-dominance (LMD) ansatz $F_{\mathrm{LMD}}(q_1^2,q_2^2,q_3^2)=[a+b(q_1^2+q_2^2)+c\,q_3^2]/[(q_1^2-M_V^2)(q_2^2-M_V^2)(q_3^2-M_S^2)]$, a single-pole representation of the form factor. Matching to the operator-product expansion fixes $b=c=-\tfrac12\langle\bar q q\rangle_0$, and a sum rule over $e^+e^-$ widths fixes $a$; the resulting expression (4.5) interpolates from the deep-Euclidean regime to the physical kinematics $q_1^2=q_2^2=0$, $q_3^2=M_H^2$ (and $q_2^2=M_Z^2$ for the $Z$ channel), and it is this interpolation that produces the numerical enhancement.

What would settle it

A direct test would be a lattice QCD computation of the scalar-vector-vector three-point function with physical quark masses, including the flavour-singlet (disconnected) quark contractions, extrapolated to the Higgs kinematics $q_1^2=q_2^2=0$, $q_3^2=M_H^2$ and to $q_2^2=M_Z^2$ for the $Z$ channel: if the form factor came out zero at those points, or if the amplitude scaled quadratically with $m_q$ rather than linearly, the paper's central claim would be false. A complementary experimental check is a percent-level measurement of the $H\to\gamma\gamma$ rate: agreement with the standard perturbative prediction to better than the predicted 3% downward shift would rule out the LMD enhancement.

Watch

Extended reading notes

Core claim

The central claim is that the light-quark contribution to the $H\to\gamma\gamma$ and $H\to\gamma Z$ amplitudes does not vanish and is linear in the light-quark masses $m_u$, $m_d$, $m_s$, not quadratic as in the usual perturbative treatment. The object that carries the argument is the $\langle SVV\rangle$ three-point function, the correlator of a colour-singlet scalar quark density with two vector currents, which in the chiral limit (all quark masses set to zero) is an order parameter of spontaneous chiral symmetry breaking: the operator-product expansion gives a leading short-distance behaviour proportional to the quark condensate $\langle\bar q q\rangle_0$, so the correlator cannot be identically zero even though it vanishes at every order of perturbation theory. The decay amplitudes factor a single power of the light-quark mass times this non-vanishing hadronic function, and that is the origin of the linear-mass behaviour. Evaluated in the lowest-meson-dominance approximation to the $N_c\to\infty$ limit of QCD, with parameters fixed by the short-distance constraints, a sum rule, and lattice inputs, the summed light-quark effective couplings for $H\to\gamma\gamma$ come out larger by about three orders of magnitude than the one-loop perturbative values, giving $\Gamma(H\to\gamma\gamma)_{\mathrm{LMD}}=0.97\,\Gamma(H\to\gamma\gamma)_{\mathrm{pQCD}}$, while the $H\to\gamma Z$ contribution is barely modified.

Load-bearing premise

The result assumes that the lowest-meson-dominance form factor, whose parameters are fixed by the short-distance expansion and a sum rule, faithfully represents the true non-perturbative correlator at the physical decay point — two on-shell photons (or a photon and a $Z$) with the Higgs on shell; if the true form factor vanished or behaved very differently at those kinematics, the predicted enhancement, and possibly the linear-mass conclusion itself, would fail.

Editorial extensions

If this is right

  • If the claim holds, the light-quark contribution to $H\to\gamma\gamma$ is roughly a thousand times larger than the one-loop perturbative estimate, moving the predicted rate to $\Gamma(H\to\gamma\gamma)_{\mathrm{LMD}}=0.97\,\Gamma(H\to\gamma\gamma)_{\mathrm{pQCD}}$; this shift becomes visible once branching-ratio precision reaches the percent level.
  • The $H\to\gamma Z$ amplitude gains no comparable enhancement: the light-quark couplings grow only modestly and the rate prediction is essentially unchanged, so this channel will not discriminate the effect.
  • The non-perturbative treatment removes the renormalization-scheme and scale ambiguity of the perturbative light-quark contribution, because the final couplings depend only on ratios of quark masses and on mass-times-condensate products, which are renormalization-group invariant.
  • Within the large-$N_c$/LMD description the light-quark contribution is purely real; generating imaginary parts would require subleading $1/N_c$ effects from multi-particle intermediate states, which the paper leaves for future work.

Reading between the lines

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

  • A lattice QCD evaluation of the $\langle SVV\rangle$ form factor at the exact Higgs kinematics, or at small momenta with controlled extrapolation, would settle the dispute directly; the paper uses lattice inputs for masses and condensates but does not propose this computation, and varying the light-quark mass on the lattice would test the predicted linear scaling.
  • Because the enhancement is driven by the external scale $M_H$ (the $q_3^2$ term in the LMD numerator), the same linear-mass mechanism should apply to heavier CP-even scalars in extensions of the standard model: their light-quark-induced $\gamma\gamma$ corrections would not decouple like a perturbative $m_q^2$ term.
  • The same reasoning applies to any amplitude built from a scalar density and two vector currents in QCD, so other precision observables sensitive to such matrix elements may hide similar non-perturbative linear-quark-mass effects that purely perturbative estimates miss.
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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 / 3 minor

Summary. The paper studies the contribution of the light quarks (u, d, s) to the Standard Model amplitudes for H→γγ and H→γZ, using non-perturbative QCD. The authors argue that the relevant object is the ⟨SVV⟩ three-point function, which they show via short-distance OPE (Eqs. (3.4), (3.6)) does not vanish identically in the chiral limit. They then evaluate this correlator in the lowest-meson-dominance (LMD) approximation to the large-Nc limit, obtaining a form factor (Eq. (4.5)) whose parameters are fixed by the OPE, a sum rule, and external inputs. The resulting light-quark contribution is linear in the quark masses, not quadratic, and for H→γγ is larger in magnitude by about three orders than the lowest-order perturbative evaluation, changing the total H→γγ rate by about 3% (Eq. (5.9)). For H→γZ the effect is small. The paper claims that the non-vanishing and linear-mass behavior are exact consequences of QCD short-distance properties.

Significance. If the quantitative results held, the paper would overturn the usual neglect of light-quark contributions in Higgs decays and correct a recent claim (ref. [9]) that these contributions vanish at the non-perturbative level. The identification of the ⟨SVV⟩ correlator as the correct object and the demonstration that it is not identically zero are solid and important. The paper is careful to use external data (e+e- widths, lattice quark condensates, PDG/FLAG masses) rather than fitting to the target amplitudes, so there is no circularity. The predicted few-percent effect on H→γγ is potentially relevant for future precision measurements at HL-LHC and FCC-ee. However, the central quantitative claim relies on the LMD model, whose validity at the physical kinematics is the main weakness; the exactness claimed in the abstract and Section 6 goes beyond what the OPE alone establishes.

major comments (3)
  1. [Abstract and Section 6] The statements that the light-quark contribution is linear in the quark masses at the physical point are presented as exact ('These statements are exact since they rest only on the identification of the ⟨SVV⟩ correlators ... and on their short-distance properties in QCD'). This overreaches the evidence. The OPE results (3.4) and (3.6) constrain the correlator only in deep-Euclidean or mixed deep-Euclidean regimes; they do not determine the form factor F at q1^2=q2^2=0, q3^2=M_H^2 (or q2^2=M_Z^2), where two of the momenta are on shell and q3 is timelike. A correlator that is nonzero in the Euclidean region could in principle vanish at these specific physical kinematics. The linear-in-mass behavior of the physical amplitudes therefore rests on the LMD ansatz of Eq. (4.2), not on an exact QCD statement. Please revise the abstract and Section 6 to clearly distinguish the exact non-vanishing of the SVV correlator from the model-dependent evaluation at the physical point.
  2. [Section 4, Eqs. (4.2)-(4.5)] The LMD form factor retains only the lowest vector and scalar poles. In the large-Nc limit, radial excitations in each channel are not parametrically suppressed (each meson contributes at order 1 in 1/Nc). Their contribution to FLMD at q3^2=M_H^2, where the scalar pole is evaluated far from its mass, is not estimated. Since the numerical enhancement in Table 1 and Eq. (5.9) is dominated by the OPE-determined b and c terms evaluated at the physical point, the quantitative results should be framed as an estimate within a specific model, with an estimate of the truncation uncertainty or a justification for why higher resonances are negligible at these kinematics. As written, the paper's quantitative predictions (e.g., the three-order enhancement and Γ_LMD=0.97Γ_pQCD) are not established beyond the LMD ansatz.
  3. [Section 5, Table 1] The comparison between the LMD and pQCD columns mixes amplitudes with different analytic structure: the LMD results are purely real because of the pole approximation, while the pQCD results are complex because of physical thresholds. For the u and d quarks the imaginary part in pQCD is comparable to or larger than the real part. The statement of an 'increase ... by at least three orders of magnitude' should specify whether it refers to |C| or Re C, and the interpretation of the enhancement should account for the fact that the two evaluations belong to different approximations (large-Nc poles versus perturbative cuts). Please clarify this point and discuss whether a direct comparison of absolute values is meaningful.
minor comments (3)
  1. [Section 5, Eq. (5.1) and footnote] The quoted value c~ = 4.6(0.8)·10^-3 appears inconsistent with the footnote stating that the central value is '0.52 instead of 0.46'; please clarify the units and whether the table uses the updated value.
  2. [Throughout] There are several typographical errors: 'runnung' before Eq. (5.3), 'regine' in Section 2, and 'an the one hand' in Section 6. The title also renders 'H→γγ, γZ' with a missing space in 'toH → γγ, γZ' in the header.
  3. [Section 3, Eq. (3.4)] The OPE formula (3.4) would benefit from an explicit statement of the kinematic conditions (all q_i^2 large and Euclidean) and the order of the neglected terms; the current text says 'deep-Euclidean' but does not specify how large the momenta must be for the leading term to dominate.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central non-vanishing and linear-mass claims rest on OPE and chiral-symmetry arguments, and the LMD parameters are fixed by external inputs rather than by the target H decay amplitudes.

full rationale

The derivation chain is: (i) identify the light-quark contribution with the <SVV> three-point function; (ii) use the OPE to show the correlator is nonzero in the chiral limit; (iii) approximate the form factor with the LMD ansatz whose parameters are fixed by short-distance constraints, a sum rule, and external hadronic inputs; (iv) evaluate numerically. The non-vanishing and linear-in-mass claims do not reduce to a fitted parameter: the quark-mass factor is factored out explicitly in eqs. (2.5) and (2.6), and the form factors are evaluated in the chiral limit, so linearity follows if the form factor does not vanish, which is supported by the OPE results (3.4) and (3.6) with the quark condensate as the leading term. The LMD parameters are not fitted to H->gamma-gamma or H->gamma-Z amplitudes: c_tilde comes from e+e- partial widths (eq. 5.1), m_q<q_bar q>_0 from lattice QCD (eq. 5.2), quark masses from PDG/FLAG (eq. 5.3), and b and c from the OPE constraints (4.3). The self-cited sum rule (4.4) fixes only parameter a, and the paper itself notes that a is numerically subdominant: the numerator of FLMD is dominated by the term proportional to (q^2+M_H^2), larger by three to four orders of magnitude than the term proportional to c_tilde. Thus no load-bearing self-citation chain forces the result. The LMD ansatz is openly an approximation, and Section 6 explicitly qualifies the quantitative conclusions ('should certainly be taken cum grano salis'), so the model dependence is acknowledged rather than disguised as a derivation. The claim in Section 6 that the non-vanishing/linear statements are 'exact' may overstate the support for the on-shell value, but that is a correctness or model-extrapolation concern, not circularity. No equation was found in which a fitted input is renamed a prediction or in which the target quantity is defined into the input.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The quantitative results rest on the LMD model with parameters fixed by external data: two resonance masses chosen by hand, a sum-rule constant from e+e- data, and quark masses and condensates from PDG, FLAG, and lattice QCD. The central qualitative conclusion, non-vanishing and linear in mass, relies on the assumption that the SVV correlator does not vanish at the physical point, an extrapolation from the OPE region. No invented entities are introduced.

free parameters (3)
  • M_V, lowest vector meson mass = 770 MeV
    Chosen by hand as the rho meson mass to define the LMD form factor in eq (4.5); appears in the denominators of FLMD and affects the numerical size.
  • M_S, lowest scalar meson mass = 1400 MeV
    Chosen by hand as the lightest scalar state that survives in the large-Nc limit; the paper notes the result is not very sensitive to this value as long as it is about 1 GeV.
  • Tilde c, sum-rule constant = 4.6(0.8) x 10^-3
    Taken from ref [30], determined from e+e- partial widths of vector resonances; fixes the parameter a in FLMD through eq (4.4).
assumptions (4)
  • domain assumption Chiral limit: the three-point functions are evaluated with mu = md = ms = 0 after factoring out the quark masses
    Section 2, after eq (2.4): the authors focus on effects linear in light-quark masses and evaluate the correlators in the chiral limit. This is standard in chiral perturbation theory but an approximation for physical quarks.
  • domain assumption Spontaneous chiral symmetry breaking occurs, so the quark condensate <qqbar>_0 is nonzero
    Required for the SVV correlator to be a non-vanishing order parameter in the OPE, eq (3.4). The nonzero condensate is supported by lattice QCD and is a standard property of QCD.
  • ad hoc to paper The deep-Euclidean OPE results (3.4) and (3.6) control the form factor at physical kinematics
    The paper extends the short-distance OPE behavior to the physical point via the LMD ansatz. This is an assumption not proven from OPE alone, since the value at q1^2 = 0, q2^2 = 0, q3^2 = M_H^2 could in principle vanish even if the correlator is nonzero at short distances.
  • domain assumption Large-Nc limit with lowest-meson dominance (single-pole ansatz) describes the SVV form factor
    Section 4, eq (4.2): the form factor is approximated by the lightest vector and scalar resonance poles. This is an approximation to QCD with Nc = 3, and the authors acknowledge it is approximate.

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Pith. "Pith review of On the contribution from the light quarks to $H\to\gamma\gamma , \gamma Z$." pith.science (2026). https://pith.science/paper/JP4TD2HF

@misc{pith2026250722551,
  author       = {Pith},
  title        = {Pith review of: On the contribution from the light quarks to $H\to\gamma\gamma , \gamma Z$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JP4TD2HF}},
  note         = {Machine review of arXiv:2507.22551}
}
abstract

This Letter addresses the contribution from the light $u , d , s$ quarks to the amplitudes for the decay modes of the Brout-Englert-Higgs scalar boson $H$ into two photons ($H\to\gamma\gamma$) or to a photon and a neutral weak gauge boson ($H\to\gamma Z$), taking into account the non-perturbative aspects of QCD. Contrary to a recent claim, the contribution from the light quarks does not vanish. Rather, it is shown that, in contrast to the perturbative evaluations usually considered in the literature, this contribution to the amplitudes starts with a term that is linear, and not quadratic, in the masses of the light quarks, thus pointing toward a sizeable enhancement of their contribution to $H \to \gamma \gamma$.

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