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REVIEW 3 major objections 3 minor 123 references

Loop-corrected Trilinear Higgs Self-Couplings in the NMSSM with Inverse Seesaw Mechanism

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The inverse-seesaw neutrino sector shifts the loop-corrected SM-like Higgs self-coupling by up to 10.5% and the Higgs mass by up to 4.5%.

desk verdict Solid one-loop-plus-vanilla-two-loop calculation of the Higgs trilinear couplings in the NMSSM with inverse seesaw; the 10.5% headline effect is a one-loop ISS contribution, and the missing two-loop (s)neutrino terms are the real soft spot. read the letter →

arxiv 2506.02743 v1 pith:GFRMWNXJ submitted 2025-06-03 hep-ph

Thi Nhung Dao , Martin Gabelmann , Margarete Mühlleitner This is my paper
classification hep-ph
keywords NMSSMinverseseesawtrilinearHiggsself-couplingmasscorrectionsone-looptwo-loopHiggs-to-HiggsdecaysneutrinoYukawacouplings
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 establishes that in the NMSSM extended with an inverse seesaw mechanism (NMSSM-nuSS), the new neutrino and sneutrino states leave a measurable imprint on the Higgs potential, not just on neutrino masses. It presents the full one-loop corrections to all trilinear Higgs self-couplings, with full momentum dependence, plus the dominant top-Yukawa and strong-coupling two-loop corrections, all computed in the same renormalization scheme as the Higgs mass corrections. For parameter points that pass Higgs, neutrino, lepton-flavor-violation, and oblique-parameter constraints, switching on the seesaw sector changes the effective SM-like trilinear coupling by up to 10.5% and the SM-like Higgs mass by up to 4.5% relative to the NMSSM without it. The two shifts are strongly correlated, so precision Higgs measurements can constrain the neutrino sector, and any future extraction of the Higgs self-coupling from di-Higgs production must include these corrections.

What carries the argument

The load-bearing object is the effective SM-like trilinear coupling $\hat\lambda_{hhh}^{\rm eff}$, defined as the third derivative of the effective Higgs potential with respect to the neutral Higgs fields at zero external momenta, evaluated in the same mixed on-shell/$\overline{\rm DR}$ renormalization scheme used for the Higgs mass calculation. The inverse seesaw enters through a $9\times9$ neutrino mass matrix with block structure $(0, M_D, 0;\ M_D^T, 0, M_X;\ 0, M_X^T, \mu_X)$ and an 18-by-18 sneutrino mass matrix, and the new one-loop contributions come from triangle diagrams with (s)neutrinos in the loops. The two-loop part is the top/stop-sector contribution taken unchanged from the complex NMSSM without seesaw, which is what makes the mass and coupling corrections directly comparable and gives the reported correlation.

What would settle it

For the benchmark point P1, evaluate the dominant two-loop contributions involving the new sector, such as $O(\alpha_t y_\nu^2)$ or $O(y_\nu^4)$ corrections to the effective potential; if any of them shifts $\lambda_{hhh}^{\rm eff}$ or $M_h$ by more than about one percentage point, then the quoted numbers are not the complete higher-order prediction.

Watch

Extended reading notes

Core claim

The paper's claim is that the (s)neutrino sector contributes one-loop corrections to the effective SM-like trilinear Higgs self-coupling that are large enough to survive all applied constraints and are two to three times larger in relative terms than the corresponding corrections to the SM-like Higgs mass. The calculation is organized as an effective coupling at zero external momentum, built from the third derivative of the effective potential, and includes the full one-loop contributions from all sectors with complete momentum dependence, further reduced to the dominant one-loop (s)top and (s)neutrino contributions in the gaugeless limit, and then combined with the two-loop $O(\alpha_t\alpha_s)$ and $O(\alpha_t(\alpha_s+\alpha_t))$ corrections from the complex NMSSM. After requiring a 122-128 GeV SM-like Higgs, LEP/LHC mass bounds, Higgs search and signal-rate constraints, neutrino oscillation data, lepton-flavor-violating decay bounds, and oblique parameters $S,T,U$, the relative difference between the model with and without the seesaw sector reaches 10.5% for the effective SM-like trilinear coupling and 4.5% for the Higgs mass at the highest included order; at the benchmark point P1, elements of the neutrino Yukawa matrix such as $(y_\nu)_{21}=0.95$ drive the effect.

Load-bearing premise

The calculation assumes that the dominant two-loop corrections involving the new neutrino and sneutrino states are negligible, even though the scan uses neutrino Yukawa couplings of about one; if those missing two-loop terms are comparable to the one-loop neutrino contribution, the quoted 10.5% and 4.5% would change.

Editorial extensions

If this is right

  • At valid scan points like P1, the loop-corrected effective coupling $\lambda_{hhh}^{\rm eff}$ lies roughly in the range 195-229 GeV, so di-Higgs production rates computed with tree-level couplings would be off by several percent and should be recomputed with the corrected coupling.
  • The one-loop corrections to the heavy-Higgs decay $H_2\to H_1H_1$ are about 40% larger than the tree-level rate, and the (s)neutrino sector changes the branching ratios of non-SM-like Higgs decays by up to about 3.4%, which matters for Higgs search exclusions.
  • The strong correlation between the (s)neutrino-induced shifts in $M_h$ and $\lambda_{hhh}^{\rm eff}$ means a future measurement of the trilinear coupling can be combined with the measured Higgs mass to discriminate these seesaw scenarios from the NMSSM without the seesaw sector.
  • The new corrections are implemented in the public program NMSSMCALC-nuSS, making the predictions usable for further phenomenological scans and collider studies.

Reading between the lines

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

  • The paper's scan treats the neutrino parameters as real and diagonal except for $y_\nu$; allowing complex phases with the same one-loop machinery could shift the quoted $\Delta$ values and would also feed into electric dipole moments, which are not the dominant constraint here.
  • The 10.5% effect in $\lambda_{hhh}^{\rm eff}$ is comparable to the 3.5-8% accuracy projected for a 100 TeV collider, so such a machine could probe the inverse seesaw sector directly through di-Higgs production if the large-$y_\nu$ points survive updated LHC searches.
  • A simplified SM-plus-inverse-seesaw study cited in the paper found effects of +20% to +30% for $|Y_\nu|>3$; the supersymmetric case here yields smaller effects because fermion and sfermion contributions cancel, so scanning $y_\nu$ beyond the values used here would test whether the cancellation pattern changes the size or sign of $\Delta$.
  • Because the two-loop corrections involving the new sector are not computed, the most defensible reading of the 10.5% number is as an estimate of the one-loop (s)neutrino contribution on top of the NMSSM two-loop baseline rather than a complete next-to-next-to-leading-order prediction; a full two-loop calculation including $y_\nu$ would settle the residual uncertainty.
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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. This paper computes the loop-corrected trilinear Higgs self-couplings in the NMSSM extended by an inverse seesaw sector (NMSSM-nuSS). The one-loop contributions are computed with full momentum dependence and include all sectors; for the effective zero-momentum SM-like coupling, the dominant top/stop and (s)neutrino contributions are combined with the O(αtαs) and O(αt^2) two-loop corrections previously obtained for the vanilla CP-violating NMSSM. The results are implemented in the public code NMSSMCALC-nuSS and used to compute heavy-Higgs Higgs-to-Higgs decay widths and branching ratios. After a parameter scan subject to Higgs, neutrino, lepton-flavor-violation, and oblique-parameter constraints, the authors find that the inverse-seesaw sector changes the effective SM-like trilinear coupling by up to 10.5% and the SM-like Higgs mass by up to 4.5% relative to the NMSSM without the seesaw sector, with a strong correlation between the two corrections.

Significance. If the numerical result is robust, the paper establishes a quantitatively significant and correlated imprint of a low-scale seesaw sector on the Higgs potential, which is relevant for Higgs-pair production and future collider measurements of the trilinear Higgs coupling. The calculation is internally checked (UV finiteness, agreement with previous results in limits), the scan employs realistic experimental constraints, and the implementation in a public Fortran code makes the results usable for further phenomenological studies. The main caveat is that the claimed ISS effect is a one-loop-level estimate; the assessment below concerns that caveat.

major comments (3)
  1. [Sec. 3, Eq. (33)] The two-loop contributions Δλ^{(2,αsαt)} and Δλ^{(2,α_t^2)} are imported unchanged from the CP-violating NMSSM without the inverse seesaw sector and therefore contain no dependence on yν, λX, MX, Aν or BμX. Since the quoted 10.5% effect (Sec. 4.1, Fig. 4) is generated entirely by the one-loop (s)neutrino diagrams, the paper's central number is a one-loop estimate of the ISS contribution dressed with common two-loop top/stop corrections, not a complete higher-order prediction. At the benchmark point P1, (yν)21 = 0.95, so mixed two-loop terms of O(yν^2 αt) and O(yν^4) are not a priori negligible; the authors should either compute or estimate them, or provide a conservative uncertainty band (e.g. from scale variation). Without this, the abstract's 'up to 10.5%' is conditional on an unquantified missing contribution.
  2. [Sec. 3.1, after Eq. (56)] The neutral-Higgs–sneutrino couplings g_{h_i \tilde n_j \tilde n_k} that enter the central one-loop expression Eq. (54) are not written out; the text says they are 'very lengthy' and will be provided 'upon request'. This makes the derivation impossible to verify from the paper alone and blocks reproduction of the numerical results. Please include the full expressions in an appendix or supplementary material.
  3. [Sec. 4.1, Figs. 2 and 3, with Eq. (58)] The one-parameter variations around P1 show SM-like Higgs masses up to 143 GeV (Fig. 2) and relative corrections Δ up to about 40%, far above the 10.5% quoted for the scan. The caption labels green triangles as points satisfying all constraints, but points with Mh = 143 GeV cannot satisfy the 122–128 GeV window of Eq. (58). The text should explicitly distinguish points that pass the Higgs-mass/Higgs-signal constraints from those that do not, and should state that the headline 10.5% applies only to the scan points that satisfy Eq. (58). As written, the reader cannot tell whether the extreme Δ values are included in the claim.
minor comments (3)
  1. [Abstract and Sec. 5] The abstract quotes 'up to 10%' while Sec. 4.1 and the conclusions quote 10.5%; please unify the number.
  2. [Sec. 3.2] The effective couplings used for the numerical Δ are computed in the gaugeless limit and include only the (s)top and (s)neutrino one-loop contributions; the 'full one-loop' statement in the abstract would be clearer if it referred to the momentum-dependent calculation of Sec. 3.1 and the effective couplings were described as 'dominant one-loop' contributions.
  3. [Sec. 4.1] There is a typo 'In particuler' in the discussion of Fig. 3, and in Sec. 5 the phrase 'can reach reach 4.5%' contains a duplicated word.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the 10.5% (s)neutrino effect is a scan output, and the imported two-loop terms are independent of the ISS sector.

full rationale

The central claim is derived via Eq. 33 as tree-level plus one-loop NMSSM-nuSS corrections plus two-loop O(alpha_t alpha_s) and O(alpha_t^2) corrections imported from Refs. [76,77]. The one-loop ISS contribution is computed with new explicit diagrams in Eq. 54, carrying dependence on y_nu, lambda_X, M_X, A_nu and the sneutrino mixing matrices; it is not defined in terms of the final 10.5% or 4.5% numbers. The two-loop pieces were previously calculated for the CP-violating NMSSM without the inverse seesaw sector, contain none of the ISS parameters, and are embedded only in the sectors that are identical between the two models; they therefore cannot artificially generate the claimed ISS-induced shift. The quantity Delta in Eq. 64 is a ratio of two separate code outputs (NMSSMCALC-nuSS versus NMSSMCALC) at the same loop order, and the quoted maxima are outputs of a parameter scan subject to external Higgs, neutrino, lepton-flavor-violation, and oblique-parameter constraints; no parameter is fitted to the target 10.5% or 4.5% values. Self-citations to Refs. [51,52,76,77] are present and load-bearing as sources of the model setup, of the mass-calculation framework, and of the vanilla-NMSSM two-loop results, but those cited results do not assume or contain the claimed (s)neutrino-induced coupling shift, so they constitute independent support under their stated approximations. The paper is also transparent in Sec. 4.1 that the (s)neutrino sector enters only at one-loop order and that the two-loop corrections do not depend on neutrino parameters; this is a completeness caveat about omitted ISS-dependent two-loop contributions, not a circular step. No fitted-input-renamed-as-prediction, self-definitional, uniqueness-import, ansatz-by-citation, or known-result-renaming pattern is present.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim depends on the model Lagrangian (Eqs. 1-3), the inverse seesaw hierarchy, and the renormalization scheme. No parameter is fitted to the claimed 10% result; the scan ranges and renormalization scale are chosen by hand and affect the maxima. No new entities are introduced here: the six singlet superfields are inherited from Refs. [51,52].

free parameters (2)
  • Renormalization scale μ_R = not specified (enters Eqs. (42)-(44) and (57))
    Observed one-loop corrections depend on the renormalization scale at which logarithms are evaluated; the paper defines μ_R but does not state its numerical value.
  • Scan bounds for (MX)ii, yν, Aν and other ISS parameters = see Table 1
    The headline maxima (10.5% for the coupling, 4.5% for the mass) are maxima over these hand-chosen ranges; different ranges would change the quoted maxima.
assumptions (5)
  • domain assumption The NMSSM-nuSS superpotential Eq. (1) and soft terms Eq. (3) define the theory.
    All loop corrections are computed in this framework; the result depends on its particle content and interactions.
  • domain assumption Inverse seesaw hierarchy |m_μX| << |m_MD| << |m_MX| is satisfied.
    Used to set up the neutrino sector and justify Eq. (20); violation would change the loop contributions.
  • ad hoc to paper Two-loop vanilla NMSSM corrections are valid in NMSSM-nuSS without (s)neutrino two-loop terms.
    This is the central assumption; no two-loop corrections involving (s)neutrinos are computed, even though yν is O(1).
  • domain assumption Effective THCs at zero external momentum in the gaugeless limit are a good approximation.
    Used in Sec. 3.2 to obtain Eq. (39); acknowledged approximation, could be unreliable for light pseudoscalar states.
  • domain assumption Mixed OS-DR renormalization scheme from [51] applies consistently to the THCs.
    Counterterms in Eqs. (36)-(38) rely on the scheme; scheme choice affects finite parts.

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Cite this review

Pith. "Pith review of Loop-corrected Trilinear Higgs Self-Couplings in the NMSSM with Inverse Seesaw Mechanism." pith.science (2026). https://pith.science/paper/GFRMWNXJ

@misc{pith2026250602743,
  author       = {Pith},
  title        = {Pith review of: Loop-corrected Trilinear Higgs Self-Couplings in the NMSSM with Inverse Seesaw Mechanism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GFRMWNXJ}},
  note         = {Machine review of arXiv:2506.02743}
}
abstract

The higher-order corrections for the SM-like Higgs boson mass and the trilinear Higgs self-couplings in the Next-to-Minimal Supersymmetric extension of the Standard Model (NMSSM) with Inverse Seesaw Mechanism are significant and highly correlated. We present here the full one-loop corrections to the trilinear Higgs self-couplings supplemented by the dominant top-Yukawa and strong coupling induced two-loop corrections from our previous calculations in the complex NMSSM. These corrections are performed consistently with the corresponding Higgs boson mass corrections. We discuss in detail the new effects from the extended neutrino and sneutrino sectors on both the trilinear Higgs self-couplings and the SM-like Higgs boson mass. When compared to the case of the NMSSM without Inverse Seesaw Mechanism, the new effects can be up to 10\% for the effective SM-like trilinear Higgs self-couplings, and up to 4.5\% for the SM-like Higgs boson mass for valid parameter points, i.e. points satisfying the Higgs data, the neutrino data, the constraints from the charged lepton flavor-violating decays, and the new physics constraints from the oblique parameters $S, T, U$. The new corrections are also included in the Higgs-to-Higgs decays for the heavy Higgs states and implemented in the new version of the Fortran code NMSSMCALC-nuSS.

Figures

Figures reproduced from arXiv: 2506.02743 by the authors.

Figure 1
Figure 1. Upper left panel: the effective trilinear coupling of the SM-like Higgs boson at one-loop [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Similar to Fig. 1 but ( [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Dependence of the SM-like triple Higgs self-coupling, the Higgs mass and the relative [PITH_FULL_IMAGE:figures/full_fig_p031_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Scatter plots for the effective SM-like triple Higgs self-couplings (upper and lower left) [PITH_FULL_IMAGE:figures/full_fig_p032_4.png]
Figure 5
Figure 5. Figure 5: Upper panels: The decay widths at tree-level (green), one-loop (red), two-loop [PITH_FULL_IMAGE:figures/full_fig_p033_5.png]

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