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Comparison of the EFT Hybrid and Three-Loop Fixed-Order Calculations of the Lightest MSSM Higgs Boson Mass

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that two independent high-precision calculations of the lightest MSSM Higgs mass agree to about 0.2–1 GeV for SUSY scales below 10 TeV, and that the measured Higgs mass excludes scales above $12.5\pm1.2$ TeV in the…

desk verdict Useful cross-check of a three-loop fixed-order MSSM Higgs mass calculation against FeynHiggs NNLL, but the unquantified DR/MS top-mass scheme mismatch leaves the claimed 0.2–1 GeV agreement without a clean interpretation. read the letter →

arxiv 1908.00693 v2 pith:RYKSFLBA submitted 2019-08-02 hep-ph

E. A. Reyes R. , A. R. Fazio This is my paper · ORCID
classification hep-ph
keywords MSSMHiggsmassthree-loopfixedorderNNLLresummationEFThybridcalculationheavySUSYlimitstopmixingscaleboundboson
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 two independent ways of computing the lightest Higgs boson mass in the Minimal Supersymmetric Standard Model (MSSM) — a direct three-loop diagrammatic calculation and a hybrid calculation that resums large logarithmic corrections — give consistent answers in the range where both should be trustworthy. In a simplified benchmark with a single supersymmetric mass scale and vanishing stop mixing, the two predictions agree to within about 0.2–1 GeV for SUSY scales below 10 TeV. The paper also turns the measured Higgs mass into a constraint on the model: in this benchmark, SUSY scales above $12.5\pm1.2$ TeV are excluded. The significance is that two methodologically different high-precision calculations cross-check each other, and that LHC data can already bound the scale of supersymmetry in this scenario.

What carries the argument

The machinery is the pole equation for the lightest CP-even Higgs mass, $p^2 - m_h^2 + \sum_{\ell=1}^3 \hat{\Pi}^{(\ell)}_{hh}=0$, evaluated in the heavy-SUSY limit where all supersymmetric masses are set to a common scale $M_\text{SUSY}$ and the low-energy effective theory is the Standard Model. The fixed-order side is assembled from a basis of 33 three-loop master integrals (the irreducible Feynman integrals obtained after integration by parts); the hybrid side adds a resummation shift $\Delta^\text{log}_{hh}$ that subtracts the logarithms already present in the fixed-order self-energies so they are not counted twice. That subtraction, together with the renormalization-scheme choices for the top quark, is what makes the comparison between the two approaches meaningful.

What would settle it

Rerun the comparison with the top-quark mass in the same renormalization scheme on both sides — for example, feed a $\overline{\text{DR}}$ top mass into the resummation or an $\overline{\text{MS}}$ top mass into the three-loop self-energy insertion — and check whether the difference stays inside the claimed 0.2–1 GeV band for $M_\text{SUSY}<10$ TeV. If the shift moves by several GeV, the agreement is largely a scheme artifact; if it stays within the band, the cross-check survives its weakest point.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central numerical discovery is a cross-check: the three-loop fixed-order prediction of $\mathcal{O}(\alpha_t\alpha_s^2)$ and the NNLL resummed prediction of the hybrid approach for $M_h$ agree to within about $0.2$ GeV in the interval $2.2 \lesssim M_\text{SUSY} \lesssim 7.4$ TeV and to at most about 1 GeV below 10 TeV, for a degenerate heavy-SUSY benchmark with $X_t=0$ and $\tan\beta=10$. The agreement is not limited to one parameter point: setting the gluino mass to 1.5 TeV or increasing the stop mixing parameter to $X_t/M_\text{SUSY}\approx 1.5$ keeps the difference small while changing the shape of the mass curve. Above 10 TeV the fixed-order result accumulates large logarithms of $M_\text{SUSY}/M_t$ and diverges from the resummed result, growing to tens of GeV by 40 TeV; the paper argues this region is experimentally irrelevant because the combined LHC mass measurement, $M_h^\text{exp}=125.09\pm0.24$ GeV, already excludes $M_\text{SUSY}>12.5\pm1.2$ TeV in the considered scenario.

Load-bearing premise

The load-bearing premise is that the unquantified conversion between the $\overline{\text{DR}}$ top-quark mass used in the three-loop insertion and the $\overline{\text{MS}}$ top-quark mass used in the hybrid resummation does not shift $M_h$ by as much as the claimed 0.2–1 GeV agreement; the paper itself notes that scheme conversion can produce large uncontrolled shifts.

Editorial extensions

If this is right

  • Below about 10 TeV in this benchmark, the two high-precision predictions for $M_h$ can be used interchangeably at the 1 GeV level.
  • Above 10 TeV, the fixed-order three-loop result accumulates large logarithms and should not be used; the resummed hybrid result is the reliable one there.
  • In the heavy-SUSY scenario with $\tan\beta\gtrsim 10$, the measured Higgs mass excludes $M_\text{SUSY}>12.5\pm1.2$ TeV.
  • Increasing $|X_t/M_\text{SUSY}|$ up to about 1.5 reduces the difference between the two calculations by roughly a factor of seven compared with $X_t=0$.
  • The point $X_t/M_\text{SUSY}=2.4$ is a maximum of $M_h$ at every $M_\text{SUSY}$, which makes it the reference point for the minimal required SUSY scale.

Reading between the lines

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

  • Editorial inference: a scheme-uniform rerun of the top-quark mass is the cheapest decisive check of whether the 0.2–1 GeV agreement is physical or an artifact of the $\overline{\text{MS}}$/$\overline{\text{DR}}$ conversion.
  • Editorial inference: the same comparison could be extended beyond the degenerate benchmark to non-universal spectra where $M_A$ differs from the sfermion scale, since the fixed-order three-loop calculation is claimed to be valid across the whole MSSM parameter space.
  • Editorial inference: a future combined Run-2 measurement with a smaller uncertainty would, if the bound holds, push the excluded SUSY scale downward and sharpen the tension with naturalness.
  • Editorial inference: checking negative values of $X_t/M_\text{SUSY}$ would test whether the improved agreement near +1.5 is symmetric or tied to the sign of the stop mixing angle.
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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 / 5 minor

Summary. The manuscript compares the authors' recent three-loop fixed-order O(alpha_t alpha_s^2) calculation of the lightest MSSM Higgs-boson mass with the NNLL EFT-hybrid prediction implemented in FeynHiggs 2.14.3. In the heavy-SUSY single-scale benchmark with tan(beta)=10, A_f=0 and zero stop mixing, the two predictions are stated to agree to within about 0.2-1 GeV for MSUSY below roughly 10 TeV. The paper also reports that the agreement improves when a gluino threshold or a non-zero stop mixing is introduced, and it uses the combined CMS/ATLAS measurement M_h^exp = 125.09 +/- 0.24 GeV to derive an upper bound on the SUSY scale, concluding that the region MSUSY > 12.5 +/- 1.2 TeV is excluded in the considered scenario.

Significance. If accepted, the comparison is a useful cross-check between a diagrammatic three-loop fixed-order calculation and a publicly available resummed hybrid code in a region not covered by the earlier H3m benchmark, and the derived upper bound on MSUSY is a concrete phenomenological statement. The paper is commendably explicit about the FeynHiggs flag settings and about the region where large fixed-order logarithms spoil the result. However, the central numerical claims rest on an unquantified top-quark scheme mismatch and on an uncertainty estimate that applies to only one of the two curves, so the agreement and the exclusion bound require qualification before they can be taken at face value.

major comments (3)
  1. [Section 4, top-quark mass scheme (runningMT=1 vs runningMT=3)] The comparison is made between two calculations that use different top-quark mass schemes. The FeynHiggs NNLL result is obtained with runningMT=1 (SM MS-bar top mass at NNLO), while the three-loop fixed-order insertion via FHAddSelf uses runningMT=3 (DR-bar top mass), at the common renormalization scale mu_r = MSUSY. No conversion between m_t^MS(mu_r) and m_t^DR(mu_r) is provided, and Section 1 itself lists scheme conversion of input parameters as a source of potentially large shifts from uncontrolled higher-order terms. For MSUSY at the TeV scale the MS-DR difference in m_t is of order several GeV and can shift M_h by an amount comparable to the claimed 0.2-1 GeV agreement. Please quantify this scheme shift, or perform the comparison with a single top-quark mass scheme.
  2. [Section 4 and lower panels of Figs. 2-4] The blue band shown in the figures is the FHUncertainties estimate for the FeynHiggs NNLL prediction, as stated in the text, but the red three-loop fixed-order curve is shown without any uncertainty estimate. The statement that the 0.2 GeV difference is 'within the theoretical uncertainty (blue band)' is therefore not valid: that band does not apply to the fixed-order calculation, and no combined or fixed-order-specific uncertainty is given. Without an uncertainty estimate for the red curve, the agreement claim of 0.2-1 GeV has no well-defined significance.
  3. [Section 4, Figs. 5-6 and the exclusion bound] The derivation of the bound MSUSY > 12.5 +/- 1.2 TeV does not state explicitly which M_h prediction is used for the contours in Figs. 5 and 6. This distinction matters because above 10 TeV the fixed-order and NNLL curves differ by up to tens of GeV. If the contours are based on the NNLL curve, the sentence saying that the region where the three-loop results 'blow up' is excluded by the measured Higgs mass is misleading, since an unreliable fixed-order prediction cannot by itself exclude a region; if the contours are based on the fixed-order curve, the bound is not trustworthy above 10 TeV. Please state the curve used and define the exclusion using the appropriate prediction and its uncertainty.
minor comments (5)
  1. [Section 4, paragraph after Fig. 6] The text says 'MSUSY must be at most 12.5 +/- 1.2 GeV'; the unit should be TeV.
  2. [Section 4, flag settings] For reproducibility, the values of the input parameters not controlled by the listed FeynHiggs flags (in particular the top-quark mass, alpha_s, and any FeynHiggs defaults used for the MSSM parameters) should be listed explicitly.
  3. [Figure 4 caption] The caption refers to 'the NNLO results of FeynHiggs' while the text and other captions describe the same curves as NNLL; the terminology should be made consistent.
  4. [References] References [14] and [35] are duplicates of the same ATLAS/CMS combined measurement and one should be removed.
  5. [Section 4, exclusion bound for non-zero mixing] The quoted 12.5 +/- 1.2 TeV uncertainty is associated with the zero-mixing curve only; for other values of X_t/MSUSY no uncertainty is shown, and the text should state that the bound with its uncertainty applies to the X_t=0 case.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the comparison is between an independent three-loop calculation and the external FeynHiggs code, and the exclusion bound uses external experimental data without fitting.

full rationale

The paper's central comparison is a benchmark of the authors' three-loop fixed-order O(alpha_t alpha_s^2) result against FeynHiggs 2.14, an independent publicly available code. The three-loop curve comes from the authors' earlier paper [34], but that self-citation is not circular: the calculation is a diagrammatic fixed-order computation, it was previously checked against H3m for low SUSY scales, and in this paper it is compared against a separate numerical implementation rather than being derived from FeynHiggs. No equation in the paper reduces one prediction to the other by construction. The NNLL FeynHiggs result is not fitted to the CMS/ATLAS mass; it is an independent prediction from the code, and the brown band is the measured value. The exclusion bound MSUSY > 12.5 +/- 1.2 TeV is obtained by intersecting the FeynHiggs NNLL prediction for Mh(MSUSY) with Mh_exp = 125.09 +/- 0.24 GeV, which is a standard theory-versus-experiment exclusion rather than a parameter fit. The different top-quark mass schemes used for the two curves (runningMT=1 vs runningMT=3) are a possible source of systematic uncertainty, as the paper itself notes in the introduction that scheme conversion can produce large shifts, but this affects the interpretation of the agreement and is not circularity. There is no fitted parameter renamed as a prediction, no uniqueness theorem imported from the authors' prior work, and no ansatz smuggled in via citation. The paper is self-contained as a numerical comparison and an exclusion study, so the circularity score is 0.

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

The comparison rests on the benchmark scenario and on the trustworthiness of the two calculations. No new parameters are fitted: the SUSY masses, tan(beta), Xt, and the gluino mass are chosen inputs. The main unstated assumption is that the DR/MS top-mass scheme conversion does not introduce a shift comparable to the claimed agreement.

assumptions (4)
  • domain assumption All SUSY masses are degenerate at a single scale MSUSY and the decoupling limit applies (eq. 6).
    The entire comparison and bound are computed in this heavy-SUSY benchmark; results need not hold for non-degenerate spectra.
  • domain assumption The O(alpha_t alpha_s^2) fixed-order calculation of [34] is correct and uses a valid mixed OS/DR renormalization with zero external momentum and gaugeless limit.
    The paper inserts this self-cited three-loop result via FHAddSelf and uses it as one side of the comparison; no independent check is provided in this paper.
  • domain assumption FeynHiggs 2.14.3 NNLL resummation is a reliable reference at high MSUSY and its uncertainty band of about 0.6 GeV covers the true missing higher-order uncertainty.
    The paper uses FeynHiggs as the benchmark and takes its FHUncertainties band as the error estimate for the agreement.
  • ad hoc to paper The scheme conversion between the DR top quark mass (runningMT=3) and the MS top quark mass (runningMT=1) does not produce shifts large enough to affect the agreement claim.
    The two settings are used simultaneously without quantifying the conversion, despite the paper citing scheme conversions as a known source of large shifts.

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Cite this review

Pith. "Pith review of Comparison of the EFT Hybrid and Three-Loop Fixed-Order Calculations of the Lightest MSSM Higgs Boson Mass." pith.science (2026). https://pith.science/paper/RYKSFLBA

@misc{pith2026190800693,
  author       = {Pith},
  title        = {Pith review of: Comparison of the EFT Hybrid and Three-Loop Fixed-Order Calculations of the Lightest MSSM Higgs Boson Mass},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RYKSFLBA}},
  note         = {Machine review of arXiv:1908.00693}
}
abstract

The lightest Higgs boson mass of the Minimal Supersymmetric Standard Model has been recently computed diagrammatically at the three-loop order in the whole supersymmetric parameters space of the SUSY-QCD sector. The code FeynHiggs combines one- and two-loop fixed-order with the effective-field-theory calculations for the same Higgs mass. The two numerical predictions agree considering the scenario of only one SUSY-scale and vanishing stop mixing parameter below 10 TeV. The agreement is improved by introducing an additional supersymmetric scale and a non-zero stop mixing. Additionally, the combined CMS/ATLAS Higgs mass value was used to derive an upper bound on the needed SUSY scale. In the considered scenario, values above the scale $12.5\pm1.2~\rm{TeV}$ are excluded.

Figures

Figures reproduced from arXiv: 1908.00693 by the authors.

Figure 1
Figure 1. Basis of three-loop Master Integrals. The dashed line represents a massless propagator. The thin solid line is the propagator with a mass at the electroweak scale Mt and the thick solid line depicts the propagator involving the SUSY scale MSUSY . 3 EFT Hybrid Calculation of Mh When there is a large mass hierarchy between the electroweak scale and the scale of the SUSY parti￾cles, the fixed-order computations of the … view at source ↗
Figure 2
Figure 2. Comparison of the Mh predictions of Feyn￾Higgs with the three-loop fixed-order computation of Mh at O(αtα 2 s) in the heavy SUSY limit. The dot-dashed and the dashed lines are the fixed-order results of Feyn￾Higgs at one and two -loop level respectively. The blue dotted line contains the NNLL resummation of the large logarithms in FeynHiggs. The blue band corresponds to the uncertainty in the NNLL prediction taken f… view at source ↗
Figure 3
Figure 3. Numerical comparison of the Mh predictions in a scenario where Mg˜ = 1.5 TeV and Xt/MSUSY = 0. These plots follow the same conventions as in the Fig￾ure 2. Up: Evolution of Mh as a function of MSUSY . Down: Differences between the three-loop fixed-order and the FeynHiggs predictions. merically small and can be safely neglected, as was shown in [69]. We have considered a gluino mass of Mg˜ = 1.5 TeV. The inclusion of… view at source ↗
Figures from the paper (3 more)
Figure 2
Figure 2. Figure 2: Finally, we have studied the dependence of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png]
Figure 5
Figure 5. Figure 5: Dependence of Mh on MSUSY and Xt in the heavy SUSY limit. We have used tanβ = 10. The gray lines represent the values of MSUSY and Xt which pro￾duce the same Higgs boson mass. The predicted value of Mh increases monotonically with MSUSY . aries for the region of rMSSM …
Figure 6
Figure 6. Figure 6: Region of rMSSM parameters in the heavy SUSY limit which is compatible with the central value and the combined uncertainty of the CMS/ATLAS Higgs boson mass, Mexp h = 125.09 ± 0.24 GeV. Up: Gray lines represent the points (MSUSY , tanβ) compatible with a 125.09 GeV Hig…

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Forward citations

Cited by 2 Pith papers

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Reviewed August 14, 2026 · model on record in the stance chip above.