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REVIEW 3 major objections 4 minor 56 references

One-loop induced contributions to the rare decay of $A_0 \rightarrow h_0h_0\gamma$ in Two Higgs Doublet Models

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper reports the first analytic one-loop expressions for the rare decay A0 → h0 h0 γ in the CP-conserving two-Higgs-doublet model, with numerical checks and decay-rate predictions.

desk verdict First one-loop treatment of A0 -> h0 h0 gamma in 2HDM, but the printed amplitude violates Bose symmetry (missing k23 A* pole), so the numerical rates are unreliable as they stand. read the letter →

arxiv 2501.15239 v1 pith:RYFB73II submitted 2025-01-25 hep-ph

Dzung Tri Tran (Duy Tan Univ.) , L. T. Hue (VLU) , Thanh Huy Nguyen (HCMUS) , Vo Quoc Phong (HCMUS) , Khiem Hong Phan (Duy Tan Univ.) This is my paper · ORCID
classification hep-ph
keywords TwoHiggsDoubletModelraredecayCP-oddbosonone-loopamplitudePassarino-VeltmanfunctionsWardidentityultravioletfinitenessdifferentialwidth
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 aims to provide the first complete analytic expressions for the one-loop amplitude of the rare decay A0 → h0 h0 γ in the CP-conserving two-Higgs-doublet model, a model with two scalar doublets, a CP-odd Higgs A0, two CP-even Higgs bosons, and charged scalars. Because the process has no tree-level vertex, it begins at one loop and is sensitive to virtual charged-Higgs, W, Goldstone, ghost, and fermion contributions that other channels may not expose. The authors write the amplitude in scalar Passarino-Veltman integral functions, verify ultraviolet and infrared finiteness and the Ward identity numerically, and compute partial widths and differential widths at representative parameter points. If the expressions are correct, they give collider phenomenologists a ready-made prediction for a mode that could probe the extended scalar sector.

What carries the argument

The load-bearing object is the one-loop form factor F defined through the amplitude's Lorentz structure; once F is known, the squared amplitude and decay width follow by phase-space integration. F is built from Passarino-Veltman scalar integrals, the standard reduction basis into which one-loop diagrams are decomposed, including scalar A0, B0, C0, and D0 functions and their tensor coefficients. The derivation organizes F as a sum over resonant A0*, h0*/H*, and Z* poles multiplied by tree-level couplings, plus non-resonant box topologies classified by the number of charged-Higgs lines, which is what makes the analytic result testable and reusable.

What would settle it

Compute the fermion-box contribution for a single massive fermion at the benchmark point MA0 = 800 GeV, MH = 200 GeV, MH± = 250 GeV, tan β = 5, sβ−α = 0.95 using an independent Passarino-Veltman reduction; if the summed amplitude is not zero, the paper's cancellation claim fails. Alternatively, generate the full one-loop amplitude with a second independent diagram-generation and reduction chain and compare with the decay rate in Table 2 at MA0 = 800 GeV and tan β = 3; a disagreement beyond numerical integration error would falsify the completeness of the diagram set.

Watch

Extended reading notes

Core claim

The central claim is that the one-loop amplitude for A0 → h0 h0 γ factorizes into a single form factor F through the Lorentz structure iM = [F1 k1^μ + F2 k2^μ] εμ = F [k1^μ/(k1·k3) − k2^μ/(k2·k3)] εμ, with the Ward identity forcing F = (k1·k3)F1 = −(k2·k3)F2. The paper collects contributions from A*-pole, φ*-pole for φ = h0, H, Z*-pole, and box topologies, expresses each in Passarino-Veltman functions, and states that all fermion box diagrams and several fermion triangle classes vanish. It reports that the amplitude passes numerical ultraviolet- and infrared-finiteness tests and the Ward identity at the level of twelve or more digits, and it tabulates decay rates for THDM types I, II, X, and Y at benchmark points with A0 mass between 500 and 1200 GeV.

Load-bearing premise

The result depends on the assumption that the listed diagrams form a complete set and that all fermion box contributions vanish exactly, a cancellation stated without derivation; the ultraviolet and Ward-identity checks test consistency of the included terms, not whether any term is missing.

Editorial extensions

If this is right

  • If the amplitude is correct, the analytic result can be evaluated immediately at any allowed THDM parameter point through the cited one-loop integral packages, giving the first complete prediction for this rare mode.
  • At the benchmark points studied, the partial width rises from about 0.009 keV at A0 mass 500 GeV to about 19 keV at 1200 GeV, with only weak dependence on the four THDM Yukawa types.
  • The differential width in the Higgs-pair invariant mass develops a peak near the H mass of about 710 GeV, and the peak is suppressed for intermediate tan β between 4 and 6, which the paper attributes to cancellations between triangle and Z–γ mixing diagrams.
  • Because the fermion-type dependence enters only through Z*-pole fermion triangles, the mode is a relatively weak discriminator among THDM types I, II, X, and Y, but a genuine probe of the scalar and gauge sector.

Reading between the lines

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

  • If the same diagram-completeness logic holds, the identical decomposition should apply to sibling modes such as A0 → h0 H γ or H → h0 h0 γ; a natural next step is to test whether fermion boxes remain vanishing there, since the scalar masses and couplings differ.
  • The stated cancellation of fermion boxes could be checked symbolically with generic fermion mass and couplings; the paper gives no derivation, so an independent reduction would either confirm the claim or expose a missing finite contribution.
  • At the quoted rates, this decay will be challenging to observe directly at the LHC but could become relevant in high-luminosity or future collider settings; its main near-term value may be as a consistency constraint inside global THDM parameter fits.
  • Because the amplitude is expressed in standard one-loop integrals, the same analytic formulas could be reused for related processes after substituting different external masses and couplings.
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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. The paper presents analytic one-loop expressions for the rare decay A0 -> h0 h0 gamma in the CP-conserving two-Higgs-doublet model. The amplitude is decomposed into A0*-pole, phi*-pole, Z*-pole, and box-diagram contributions and expressed in terms of scalar Passarino-Veltman functions in the standard LoopTools/Collier convention. The authors report numerical checks of ultraviolet finiteness, infrared finiteness, and the Ward identity, and they use the resulting form factor to compute decay rates at several benchmark points and differential distributions in the h0h0 invariant mass.

Significance. If the stated formulas are complete and correct, this would be the first analytic treatment of a rare loop-induced 3-body Higgs decay in the 2HDM, and the explicit PV-function expressions would be a useful reference for phenomenology. The paper has real strengths: the analytic results are explicit, the numerical checks of UV finiteness (Table 3) and the Ward identity (Table 4) are meaningful consistency tests, and the benchmark decay rates and distributions give a first quantitative picture of the channel. However, the central claim depends on completeness of the diagram set, and the manuscript provides no independent cross-check and no derivation for the asserted vanishing of several fermion-loop classes. Since the internal checks are consistency conditions rather than tests of completeness, a missing finite contribution would not be detected by them.

major comments (3)
  1. [Eq. (19) with Eq. (24)] The amplitude as written is not Bose-symmetric under exchange of the two identical h0 bosons. In the basis of Eq. (16), M = F [k1/(k1.k3) - k2/(k2.k3)].epsilon, the bracket changes sign under k1 <-> k2, so F must be antisymmetric. The first term in Eq. (19) contains only the k13 A0* pole, g_h0A0A0 F_A0^(Trig)(k13)/(k13 - M_A0^2), and Eq. (24) gives F_A0^(Trig) as a function of k13 only. The required k23 pole term with the same coupling is absent. This is a completeness error, not a matter of sign convention: the squared amplitude in Eq. (62) and the Dalitz-plot integral in Eq. (61) would differ from the correctly symmetrized rate. The fixed-phase-space checks in Tables 3 and 4 cannot detect this kinematic asymmetry. Please add the k23 partner term or explain explicitly why it vanishes.
  2. [Eqs. (20) and (21)] The printed scalar couplings contain a dimensionally inconsistent term. In Eq. (20), the bracket contains 8 M_W^2 s_W^2/(e^2 v^2 M^2), which has mass dimension -2 inside an expression that must have mass dimension 2; the same term appears in Eq. (21). Since M^2 is the soft-breaking scale with mass dimension 2, the numerator appears to be missing a factor of M^2 (or the denominator is wrong). This affects g_h0A0A0 in the A0* pole of Eq. (19) and g_h0h0h0 in the phi* pole, so the numerical rates in Table 2 depend on this quantity. The authors should correct the typo and re-run the numerical results.
  3. [Eq. (38) and preceding text] The assertion that all fermion box diagrams vanish, stated just before Eq. (38) as 'We confirm that the contributions from fermion f in the loop of all one-loop box diagrams are vanished in this case,' is not supported by any derivation or reference. The same applies to the asserted cancellations of fermion triangle contributions in the A0* and phi* pole groups. Because a missing finite fermion-box or fermion-triangle term would survive the UV and Ward identity checks, this is a load-bearing completeness assumption. Please provide an explicit argument, a symmetry-based proof, or a cross-checked numerical demonstration for each vanishing class.
minor comments (4)
  1. [Table 4] The table caption says the second and last columns show the two form factors F1 and F2, but the displayed header and entries appear to show (k1.k3)F1 - (k2.k3)F2 as a single complex quantity. Please clarify what is actually tabulated.
  2. [Eq. (21)] Equation (21) has an unbalanced closing parenthesis in the second term; this should be corrected for readability.
  3. [Section 3 and Eq. (26)] The text alternately refers to the package as 'Collider' and 'Collier'; the correct name of Ref. [26] is Collier. Please use one consistent name.
  4. [Table 2] The statement that 'the decay rates are proportional to MA0' is imprecise; the tabulated values grow faster than linearly between 500 and 1200 GeV. Please rephrase.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: the decay rates are derived from standard 2HDM Feynman rules and benchmark inputs, with no fitting of the target observable; self-citations supply couplings and phases but the one-loop calculation itself is self-contained.

  1. self citation load bearing [Section 3, Eq. (37) and the sentence after Eq. (39); Section 4 input setup]
    "Where the general coupling given in the above equations ... is taken as in Ref. [53]."

    The couplings g_{h0H±H∓} and g_{h0h0H±G∓} are imported from Refs. [17] and [53], whose author lists overlap with the present paper. These self-citations are load-bearing for the printed analytic formulas, but the cited quantities are standard 2HDM Feynman rules, not the target decay amplitude A0 -> h0h0 gamma. The central calculation is therefore not reduced to a prior computation of the same observable.

full rationale

The manuscript's central claim is the first analytic one-loop computation of the rare decay A0 -> h0h0 gamma in the CP-conserving THDM. The amplitude is assembled from standard Feynman rules and Passarino-Veltman reductions, and the phenomenological decay rates in Table 2 are evaluated at declared benchmark points with no parameter fitted to the computed width. The numerical checks (UV finiteness, Ward identity) are internal consistency tests, not predictions forced by inputs. Self-citations to Refs. [17], [52], and [53] supply couplings, renormalization schemes, and input conventions; these are standard ingredients rather than the target result, so the circularity score remains low. The skeptic's concern about a missing k23 A0*-pole term and the reader's concern about undemonstrated fermion-box cancellations are completeness and correctness risks, not circularity: an error of that kind would invalidate the result but would not make the derivation equivalent to its inputs. No step exhibits the required reduction of the predicted quantity to a fitted parameter or to a self-cited computation of the same process.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central analytic result rests on standard 2HDM Feynman rules, automated diagram generation, and previously published renormalization schemes. The main nonstandard assumption is the unproved vanishing of fermion-loop diagram classes. No new particles or forces are introduced, and the benchmark points are chosen inputs rather than fit results.

free parameters (5)
  • s_beta_minus_alpha = 0.95
    Chosen for the nearly-alignment scenario in Section 4; a benchmark input, not fitted to data.
  • tan_beta = 3 (Table 2), varied 2-8 in Figs. 5-6
    Chosen from the allowed parameter region discussed in Section 2; not fitted to the computed decay.
  • M_H = M_A0 - M_Z
    Benchmark relation used in Section 4 and Table 2; affects the numerical rates.
  • M_H+ = M_A0
    Benchmark relation used in Section 4 and Table 2; affects the numerical rates.
  • M^2 (Z2 soft-breaking scale) = M_H^2
    Set equal to M_H^2 in Section 4; part of the benchmark input set.
assumptions (6)
  • domain assumption The CP-conserving, softly broken Z2-symmetric 2HDM scalar potential and the four Yukawa types I, II, X, Y describe the model.
    Section 2 introduces the model and Feynman rules used for the whole calculation.
  • domain assumption The on-shell renormalization schemes from Refs. [49-52] are consistent and sufficient for this one-loop induced process.
    Invoked in Section 3 for fermion, gauge, and scalar sectors.
  • domain assumption The set of Feynman diagrams generated automatically by FeynArts is complete.
    Section 3 relies on automated diagram generation; no independent enumeration is presented.
  • standard math The Passarino-Veltman reduction and the numerical evaluation by LoopTools and Collier are correct.
    Every analytic expression in Section 3 is written in terms of these standard functions.
  • ad hoc to paper Fermion contributions to several triangle groups and to all box diagrams vanish.
    Stated without derivation before Eqs. (24) and (38); plausibly follows from a Furry-type charge-conjugation argument, but the paper does not show it.
  • domain assumption The Standard Model inputs, gauge boson widths, and Higgs widths are taken from Ref. [17].
    Section 4 says the SM inputs are the same as in Ref. [17], without listing them.

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Pith. "Pith review of One-loop induced contributions to the rare decay of $A_0 \rightarrow h_0h_0\gamma$ in Two Higgs Doublet Models." pith.science (2026). https://pith.science/paper/RYFB73II

@misc{pith2026250115239,
  author       = {Pith},
  title        = {Pith review of: One-loop induced contributions to the rare decay of $A_0 \rightarrow h_0h_0\gamma$ in Two Higgs Doublet Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RYFB73II}},
  note         = {Machine review of arXiv:2501.15239}
}
abstract

The analytic expressions for one-loop contributions to the rare decay process $A_0 \rightarrow h_0h_0\gamma$ within the CP-conserving of Two Higgs Doublet Models are first reported in this paper. Analytic results are presented in term of scalar one-loop Passarino-Veltman functions following the standard output of the packages~{\tt LoopTools} and {\tt Collier}. In this context, physical results for the computed process are easily generated by using one of these packages. The numerical checks are proposed to verify for the analytic results in this paper. The checks rely on the renormalization conditions that the decay amplitude must be the ultraviolet finiteness and infrared finiteness. The amplitude consisting of an external photon always obeys the Ward identity. This will be confirmed numerically in this article. In phenomenological results, the decay rates of $A_0 \rightarrow h_0h_0\gamma$ are evaluated at several points in the allowed regions of the parameter space. Furthermore, the differential decay widths with respect to the invariant mass of Higgs-pair in final states are studied.

Figures

Figures reproduced from arXiv: 2501.15239 by the authors.

Figure 1
Figure 1. One-loop triangle diagrams including the mixing of [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. One-loop triangle diagrams with φ-poles for φ ≡ h0, H, including the mixing of Z-γ. All particles like fermion f, vector bosons and scalar particles are considered to propagate in the loop diagrams. Within HF gauge, we have also Goldstone and Ghost particles exchanging in the loop. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. One-loop triangle diagrams with Z-pole including the mixing of Z-γ. There are two form factors relating to the contributions of Z-pole giving in the above equation like F (Trig) Z,13 and F (Trig) Z,23 . Different from the previous form factors, both factors for Z-pole have non-zero contributions from fermion f exchanging in the loop. We show the analytic results for one-loop form factors F (Trig) Z,ij as follows: F … view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: One-loop four-point Feynman diagrams contributing to th [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: The differential decay rates with respect to the invariant [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]
Figure 6
Figure 6. Figure 6: The differential decay rates with respect to the invariant [PITH_FULL_IMAGE:figures/full_fig_p026_6.png]

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