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Application of the 3-Loop FlexibleEFTHiggs Method to the MSSM and the NMSSM

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Thomas Kwasnitza, Dominik Stöckinger, Alexander Voigt, Johannes Wünsche

High Energy Physics - Phenomenology · Submitted 27 Jun 2025 · 69 references

Pith's verdict: Solid first public N3LO+N3LL NMSSM Higgs mass tool with real cross-checks, but the ~0.7 GeV uncertainty claim rests on an unverified assumption about non-log-enhanced 2-loop terms.

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arXiv ID2506.22208 (version 4)
Pith IDXENO73KH, Pith's permanent record of this paper and its review
CitationAlexander Voigt, Dominik St\"ockinger, Johannes W\"unsche, Thomas Kwasnitza. "Application of the 3-Loop FlexibleEFTHiggs Method to the MSSM and the NMSSM." arXiv:2506.22208 (2025). https://pith.science/paper/XENO73KH
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The paper

The paper claims

This paper extends the three-loop FlexibleEFTHiggs hybrid calculation to the NMSSM, claiming approximately 0.7 GeV theory uncertainty for perturbative singlet couplings, and tests the MSSM version on non-degenerate supersymmetric spectra.

The reading

The paper aims to establish that a hybrid fixed-order/EFT calculation can predict the light CP-even Higgs pole mass in both the MSSM and the NMSSM with state-of-the-art precision, even when supersymmetric particle masses are highly non-degenerate. For the MSSM, it shows that the existing three-loop FlexibleEFTHiggs implementation is stable and reliable across a wide range of non-degenerate spectra, with typical total uncertainties of about 0.5 GeV. For the NMSSM, it presents a newly implemented three-loop matching calculation that includes full one-loop, MSSM-like two-loop, and dominant three-loop QCD corrections, reaching N3LO+N3LL accuracy. The central quantitative claim is that for the NMSSM couplings $\lambda$ and $\kappa$ below the perturbativity limit $\sqrt{\lambda^2+\kappa^2}<0.7$, the theory uncertainty remains about 0.7 GeV, making it one of the most precise NMSSM Higgs mass predictions available.

What carries the argument

The central object is the FlexibleEFTHiggs method, a hybrid calculation that matches the BSM model to the SM at a single scale $Q_m$ using the Higgs pole mass condition $(M_h^{\rm BSM})^2=(M_h^{\rm SM})^2$ to fix the SM quartic coupling $\hat\lambda_{\rm SM}(Q_m)$. The full-model parametrization means that the matching is expanded in BSM parameters rather than SM parameters, which resums QCD-enhanced trilinear-coupling terms and avoids an expansion in $v^2/M_S^2$. The method's uncertainty estimation relies on varying the matching scale $Q_m\in[Q_{\rm in}/2,2Q_{\rm in}]$, and the paper provides a structural analysis showing that this variation generates the missing logarithmic and non-logarithmic NMSSM-specific terms.

Extended reading notes

Core claim

The paper's central claim is that the newly implemented hybrid calculation in the NMSSM constitutes a code achieving state-of-the-art precision for the Higgs mass calculation in the NMSSM, with N3LO+N3LL accuracy in the relevant couplings. The method uses Higgs pole mass matching to determine the SM quartic coupling, and it is performed in the full-model parametrization, which resums QCD-enhanced terms involving the stop-mixing parameter and includes power-suppressed $v^2/M_S^2$ terms without expansion. The calculation includes all one-loop NMSSM contributions, MSSM-like two-loop corrections of $O(g_3^2(y_t^4+y_b^4)+(y_t^2+y_b^2)^3+(y_t^2+y_\tau^2)^3)$, and the dominant three-loop QCD contributions of $O(g_3^4y_t^4)$. The paper argues that matching-scale variation reliably simulates the missing NMSSM-specific higher-order terms, so that for $\lambda,\kappa$ satisfying $\sqrt{\lambda^2+\kappa^2}<0.7$ the total uncertainty is approximately 0.7 GeV, provided the unknown coefficients are of order one.

Load-bearing premise

The quoted 0.7 GeV uncertainty assumes that the unknown NMSSM-specific higher-order coefficients are of order one and that $\mu_{\rm eff}/M_S$, $A_\lambda/M_S$, and $A_\kappa/M_S$ are comparable, so that matching-scale variation simulates all missing terms; if those coefficients are numerically large, the true error could be substantially larger.

Editorial extensions

If this is right

  • For NMSSM parameter points with $\sqrt{\lambda^2+\kappa^2}<0.7$ and moderate $|x_t|$, the three-loop hybrid calculation yields a predicted Higgs mass with a total theory uncertainty of about 0.7 GeV.
  • In the MSSM, the three-loop FlexibleEFTHiggs calculation remains reliable for non-degenerate supersymmetric spectra, with uncertainties of order 0.5 GeV in most tested scenarios.
  • Scenarios with strong splittings among third-generation squarks or very large $X_b$ can exhibit uncertainties of several GeV, in which case the two-loop version is recommended.
  • The three-loop QCD corrections have only a small impact (about 0.1 GeV for degenerate spectra with $M_S\approx 3$ TeV), but can introduce discontinuities from hierarchy transitions during matching scale variation.
  • The full-model parametrization leads to a resummation of QCD-enhanced terms in the stop-mixing parameter, which is not present in pure EFT calculations.

Reading between the lines

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

  • The structural argument that matching-scale variation simulates missing NMSSM-specific terms is likely transferable to other singlet-extended supersymmetric models, since the omitted terms follow a similar coupling-structure template.
  • The paper's caveat that the reasoning fails for $\kappa\gg\lambda$ suggests that a dedicated scan varying the ratio $\kappa/\lambda$ at fixed $\sqrt{\lambda^2+\kappa^2}$ would reveal where the uncertainty estimate breaks down.
  • For scenarios with $M_3\gg M_S$, implementing the advocated MDR scheme could reduce the large matching-scale uncertainties observed in the MSSM part.
  • The observed convergence of the loop expansion supports the claimed precision, but including the missing two-loop electroweak corrections would provide a direct numerical check of the uncertainty estimate.
Original abstract

We perform an extensive analysis of the light CP-even Higgs boson pole mass in the MSSM and its dependencies on various parameters based on the 3-loop FlexibleEFTHiggs hybrid calculation which is implemented and publicly avaiable since recently in FlexibleSUSY. Our focus lies on the study of the robustness of the approach in scenarios of highly non-degenerate SUSY mass spectra. Also, we present an improved Higgs mass calculation in the NMSSM based on the same approach, which is published in the new version 2.9.0 of FlexibleSUSY as well. The calculation provides a treatment in the full-model parametrization, leading to an advantageous resummation of QCD-enhanced terms in the stop-mixing parameter and includes important 2-loop contributions as well as 3-loop QCD contributions in the MSSM limit. We assess the reliability of this new calculation by applying it to several distinct NMSSM scenarios. In this context, special attention is devoted to the estimation of NMSSM-specific theory uncertainty.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,054 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.31 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.21 · mentions code or data availability 0.78.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; .

We perform an extensive analysis of the light CP-even Higgs boson pole mass in the MSSM and its dependencies on various parameters based on the 3-loop FlexibleEFTHiggs hybrid calculation which is implemented and publicly avaiable since recently in FlexibleSUSY. Our focus lies on the study of the robustness of the approach in scenarios of highly non-degenerate SUSY mass spectra. Also, we present an improved Higgs mass calculation in the NMSSM based on the same approach, which is published in the new version 2.9.0 of FlexibleSUSY as well. The calculation provides a treatment in the full-model parametrization, leading to an advantageous resummation of QCD-enhanced terms in the stop-mixing parameter and includes important 2-loop contributions as well as 3-loop QCD contributions in the MSSM limit. We assess the reliability of this new calculation by applying it to several distinct NMSSM scenarios. In this context, special attention is devoted to the estimation of NMSSM-specific theory uncertainty.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We perform an extensive analysis of the light CP-even Higgs boson pole mass in the MSSM and its dependencies on various parameters based on the 3-loop FlexibleEFTHiggs hybrid calculation which is implemented and publicly avaiable since recently in FlexibleSUSY. contribution 0.98 0.94 0.11 asserted 1.00
s1 Our focus lies on the study of the robustness of the approach in scenarios of highly non-degenerate SUSY mass spectra. contribution 1.00 0.57 0.08 asserted 0.56
s2 Also, we present an improved Higgs mass calculation in the NMSSM based on the same approach, which is published in the new version 2.9.0 of FlexibleSUSY as well. contribution 0.99 0.83 0.11 asserted 1.00
s3 The calculation provides a treatment in the full-model parametrization, leading to an advantageous resummation of QCD-enhanced terms in the stop-mixing parameter and includes important 2-loop contributions as well as 3-loop QCD contributions in the MSSM limit. method 0.91 0.77 0.14 asserted 1.00
s4 We assess the reliability of this new calculation by applying it to several distinct NMSSM scenarios. method 0.62 0.94 0.06 asserted 0.98
s5 In this context, special attention is devoted to the estimation of NMSSM-specific theory uncertainty. contribution 0.83 0.86 0.08 asserted 0.71

What kind of paper the classifier reports this is: method or tool 0.87 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v4
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Schematic illustration of the scales used in the refined FlexibleEFTHiggs method. The FlexibleEFTHiggs method is schematically illustrated in
Figure 2. Left panel: Dependence of Mh (and the stop pole masses Mt˜1,2 ) on the gluino mass parameter M3 with all other parameters fixed as in Eqs. (19). The kink in the red uncertainty band at M3 ≈ 8 TeV originates from the high-scale uncertainty determined according to Eq. (15). At that point, the maximal difference is no longer found at the upper end of the Qm variation interval but at the lower one. Right panel: Dependence of Mh on the stop mixing parameter xt for different values of M3. For M3 = 9 TeV, the uncertainty is drawn as light blue band. At(Qin) according to Eqs. (13)–(14). The…
Figure 3. Left panel: Mh as a function of the DR′ CP-odd Higgs mass mA. Right panel: Mh as a function of the µ-parameter. The parameters not varied are fixed to MS = 3 TeV, tan β = 20 and xt = −2.2. The green band (left panel) and the blue band (right panel) show the uncertainty ∆Mh. The brown band corresponds to the measured Higgs mass with its experimental uncertainty. 3.3.3 Dependence on (mq˜)33 and (mu˜)33 In
Figure 4. Left panel: Mh as a function of (mq˜)33 for different values of xt . Right panel: Mh as a function of (mu˜)33 for different values of xt . The parameters not varied are fixed to MS = 3 TeV and tan β = 12. All curves are obtained from a 2-loop matching except the black dashed ones which represent the 3-loop results only for xt = 2. Also for xt = 2, the uncertainty bands (corresponding to the 2-loop calculation) are drawn in red (left panel) and teal (right panel), respectively. The brown band corresponds to the measured Higgs mass with its experimental uncertainty. degenerate mass spe
Figure 5. Diagrams corresponding to contributions to λˆ SM(Qm) not covered by the calculation, where ϕ ∈ {ℜ(h 0 u ), ℜ(h 0 d )}. Figure a: non-log-enhanced 2-loop contribution of O
Figure 6. Comparison between the calculated Higgs mass values in the MSSM and the NMSSM in dependence on MS, obtained from the FlexibleEFTHiggs approach implemented in Flexib￾leSUSY. The left panel depicts the curves for xt = 0 and the right one for xt = √ 6. The red band represents the estimated uncertainty for the NMSSM calculation in the MSSM limit, while the blue band indicates the uncertainty for λ = κ = 0.2. −3 −2 −1 0 1 2 3 95 100 105 110 115 120 xt Mh / GeV MS = 1 TeV, tan β = 5 MSSM NMSSM (MSSM limit) NMSSM (λ = κ = 0.2) −3 −2 −1 0 1 2 3 105 110 115 120 xt Mh / GeV MS = 3 TeV, tan β…
Figure 7. Comparison between of the FlexibleEFTHiggs Higgs mass calculation in the MSSM and the NMSSM in dependence on xt , in analogy to
Figure 8. Comparison between the calculated Higgs mass values obtained from different ap￾proaches implemented in FlexibleSUSY. The dependence Mh(MS) is shown on the left side, whereas Mh(xt) is shown on the right side. The red band represents the estimated uncertainty for the 3-loop FlexibleEFTHiggs calculation, while the blue band indicates the uncertainty for the 3-loop FO calculation. Additionally, the NMSSMCalc result obtained with pole mass matching including full 1-loop matching corrections and 2-loop corrections of O(ˆg 2 3 yˆ 4 t + ˆy 6 t ) is plotted. NMSSMEFTHiggs, allows to switch…
Figure 9. Prediction of the Higgs pole mass Mh as calculated by the FlexibleEFTHiggs (FEFT) method at 1-, 2- and 3-loop precision in dependence on λ = κ for different scenrios. The red curve represents the FlexibleEFTHiggs 3ℓ result, for which the red band indicates the estimated uncertainty. The horizontal brownish band marks again the experimentally determined value of Mh with its 1σ-uncertainty. fine-tuning problem with regard to xt in the MSSM. The left plot in
Figure 10. On the left side, the differences between the Higgs masses calculated by using the FlexibleEFTHiggs approach with n-loop matching depending on λ and κ are shown. The dependence of the estimated theory uncertainty ∆Mh corresponding to missing higher order contributions for different values of tan β is depicted on the right side. Here, the vertical dashed line indicates the perturbativity limit, see Eq. (27). changes occur. While the uncertainty decreases with increasing λ and κ for tan β = 3, it increases for larger considered values of tan β. Beyond the perturbativity limit, the…

Figures reproduced from arXiv:2506.22208 by the authors.

References
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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Builds on 1 paper
Uses the method or tool of 11 papers
Compares with 8 papers
Cites as support 1 paper
Cites as background 11 papers

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