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Full two-loop QCD corrections to the Higgs mass in the MSSM with heavy superpartners

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Pith's one-line read This paper completes the two-loop QCD threshold corrections to the Higgs quartic coupling in the MSSM and finds the new terms shift the predicted Higgs mass by about 100 MeV.

desk verdict Solid, complete-looking two-loop QCD threshold corrections that close a known gap in the MSSM EFT Higgs-mass program, with honest scoping of the numerical impact. read the letter →

arxiv 1908.01670 v2 pith:RSWL2QDH submitted 2019-08-05 hep-ph

Emanuele Bagnaschi , Giuseppe Degrassi , Sebastian Pa{ss}ehr , Pietro Slavich This is my paper · ORCID
classification hep-ph
keywords MSSMHiggsmasstwo-loopthresholdcorrectionsquarticcouplingeffectivefieldtheoryQCDstrong-electroweaksupersymmetry
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

The paper fills the last missing strong-interaction piece in the two-loop matching of the Minimal Supersymmetric Standard Model (MSSM) onto the Standard Model: the threshold corrections to the quartic Higgs coupling that involve both the strong gauge coupling and the electroweak gauge or Yukawa couplings. Together with earlier results obtained in the gaugeless limit, this completes all two-loop QCD corrections at the SUSY scale. The paper also compares two independent full two-loop computations of the relation between the quartic coupling and the pole Higgs mass at the electroweak scale, finding agreement to about 20 MeV near the measured Higgs mass. In a representative heavy-superpartner scenario, the new corrections lower the predicted Higgs mass by about 100 MeV and raise the common SUSY scale needed for a 125 GeV Higgs by about 100 GeV, an effect comparable to the LHC's measurement accuracy.

What carries the argument

The central object is the two-loop mixed QCD-EW threshold correction $\Delta\lambda_{2\ell,\text{QCD-EW}}$ to the quartic Higgs coupling, which the paper splits into three pieces: two-loop one-particle-irreducible contributions from the effective potential, two-loop Higgs wave-function-renormalization contributions that require momentum-dependent self-energies, and a renormalization-scheme contribution that converts DR-renormalized MSSM couplings ($\lambda$, $g$, $g'$, and the top Yukawa) to the MS-renormalized SM couplings. In the degenerate superparticle limit these combine into the explicit formula in eq. (27). The machinery carries the argument because the momentum-dependent self-energy and scheme-conversion terms are the parts that the earlier gaugeless calculations could not see, and they are also the parts that turn the completed SUSY-scale matching into a genuine next-to-next-to-leading-logarithmic resummation step.

What would settle it

Repeat the matching in a split scenario with, for example, $m_{\tilde g}=10M_S$ and recompute the two-loop QCD-EW threshold correction including the gluino-mass-enhanced terms; if the predicted Higgs mass shifts by more than about 100 MeV relative to the degenerate-mass formula, or if the $\beta$-function check in eq. (28) fails, then the completeness and numerical-impact claims are confined to the degenerate-mass scenario.

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Extended reading notes

Core claim

The central claim is that the class of two-loop threshold corrections of $O(g_{t,b}^2 g_s^2)$, $O(g^4 g_s^2)$ and $O(g'^4 g_s^2)$ to the quartic Higgs coupling has now been computed, so that, combined with the earlier $O(g_t^4 g_s^2)$ and $O(g_b^4 g_s^2)$ results, the two-loop QCD corrections at the SUSY scale are complete. The calculation goes beyond the effective-potential method used for the earlier gaugeless corrections: it requires the momentum-dependent two-loop self-energies of the Higgs and Z bosons, plus scheme-conversion terms between dimensional reduction and dimensional regularization and between MSSM and SM couplings. In the degenerate-mass limit the combined result is given by eq. (27), and it passes the consistency check of reproducing the SM $\beta$-function term for the quartic coupling that is proportional to $\lambda g_s^2(g_t^2+g_b^2)$. The paper further maintains that, in the EFT approach, the electroweak-scale determination of the pole Higgs mass should be performed at the same perturbative order, and shows that two existing full two-loop determinations agree to about 20 MeV near 125 GeV, within the quoted theoretical uncertainty of about 150 MeV.

Load-bearing premise

The quantitative results assume that all superparticles, including the gluino (the gluon's supersymmetric partner), and the heavy Higgs doublet share one common mass scale, with only the Standard Model below it; if the gluino is much heavier than the squarks, additional two-loop terms appear that the quoted 100 MeV numbers do not include.

Editorial extensions

If this is right

  • The heavy-MSSM prediction for $m_h$ no longer has any missing two-loop strong-interaction matching terms, so the remaining SUSY-scale uncertainty is set by uncomputed electroweak-only two-loop pieces and by higher orders.
  • In the degenerate scenario, the new corrections move the predicted Higgs mass down by about 100 MeV and move the common SUSY scale that yields $m_h = 125.09$ GeV up by about 100 GeV.
  • The electroweak-scale comparison shows that the two full two-loop calculations of the quartic-to-pole-mass relation agree to about 20 MeV near the observed mass, so that side of the EFT chain is not the dominant uncertainty.
  • This is a necessary step toward a full NNLL resummation of the logarithmically enhanced corrections to the MSSM Higgs mass.
  • If superpartners are discovered in the TeV range, the Higgs mass becomes a sharper precision constraint on stop masses and mixing, with theoretical and experimental uncertainties of comparable size.

Reading between the lines

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

  • The same computational pieces, momentum-dependent two-loop self-energies plus DR-MS scheme conversions, could be reused to compute the remaining electroweak-only two-loop matching corrections, which would complete the full two-loop matching rather than just its QCD part.
  • The numerical direction of the shift, toward a slightly heavier SUSY scale for the same Higgs mass, suggests that if the MSSM is realized with stops near the current LHC reach, the mixed QCD-EW corrections slightly tighten the region of parameter space compatible with 125 GeV.
  • A two-step EFT that decouples the gluino before the squarks would provide a direct test of the degenerate-scale assumption; in split-SUSY spectra the inferred value of $M_S$ could move by more than the quoted 100 GeV.
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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

0 major / 3 minor

Summary. This paper computes the two-loop threshold corrections to the quartic Higgs coupling in the MSSM-to-SM EFT matching that involve both the strong and electroweak gauge couplings (the mixed QCD-EW corrections). Section 2 decomposes the correction into 1PI, wave-function-renormalization, and renormalization-scheme pieces, and gives a complete combined result in the degenerate superpartner-mass limit in Eq. (27). Section 3 compares two independent full two-loop computations of the SM relation between the pole Higgs mass and the MS-renormalized quartic coupling, refs. [85] and [86] (the code mr), finding agreement at the level of about 20 MeV near m_h = 125 GeV. Section 4 quantifies the numerical impact in a common-scale SUSY scenario: roughly -100 MeV on the predicted Higgs mass and about +100 GeV on the inferred stop mass scale. Section 5 discusses limitations, including the case where the gluino is much heavier than the squarks and the scenario would require a separate decoupling step.

Significance. If correct, the result closes the last missing two-loop strong-interaction contribution to the quartic-coupling matching in the single-scale MSSM EFT, thereby completing the two-loop QCD part of the threshold correction. The paper's main strengths are the clean decomposition of the calculation, the explicit degenerate-mass formula in Eq. (27), the nontrivial cross-checks against the independent self-energy results of refs. [31-33], the cancellation of IR divergences between Eqs. (14)-(17), the DRED-DREG shift, and the RGE-consistency check in Eq. (28). The reported numerical impact is modest and honestly assessed as subdominant to the gaugeless two-loop corrections, while being comparable to current LHC Higgs-mass precision. The stated limitations—the common-scale numerical scenario, the neglect of first-two-generation Yukawa couplings, and the gluino-mass-hierarchy caveat in Sec. 5—are explicit and do not undermine the algebra of the correction, though they bound the domain of the completeness claim.

minor comments (3)
  1. [Sec. 2 and Sec. 5] The full generic-mass results are described as available on request, but they are not included in the manuscript or as an ancillary file; I request that they be provided as supplementary material so that the completeness claim for arbitrary superpartner masses can be independently checked and reused.
  2. [Abstract and Sec. 5] The phrase 'completes the calculation of the two-loop QCD corrections to the quartic coupling' should be explicitly qualified by the single-scale assumption under which the calculation is performed, given the paper's own discussion that a much heavier gluino would require an additional threshold-decoupling step.
  3. [Eq. (14)] The IR-divergent logarithm ln(-p^2/Q^2) should be accompanied by a standard +i0 prescription or an explicit statement of the convention used, since the expression as printed is ambiguous for p^2 < 0.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new two-loop corrections are computed from scratch and checked against independent external results; reliance on prior work is normal, not a reduction.

full rationale

The central mixed QCD-EW two-loop threshold correction is not derived from the quantity it is used to predict. The paper computes the 1PI contribution from the two-loop squark effective potential (Eqs. (3)-(12)), the wave-function-renormalization contribution from an asymptotic expansion of two-loop Higgs self-energies generated with FeynArts/TRACER (Eqs. (13)-(17)), and the scheme/threshold pieces from DRED-DREG differences and gauge/Yukawa shifts (Eqs. (18)-(26)). Each piece is cross-checked against independently published results: the self-energies of refs. [31-33], the SM quartic beta function of ref. [103], and the EW-scale relation of refs. [85,86] via the mr code. The combined result in Eq. (27) is the sum of these pieces, not an ansatz or a fitted quantity. Reliance on the authors' earlier refs. [65,72] for the one-loop and gaugeless two-loop matching is a normal use of previously published parameter-free results; the present paper does not define its new quantities in terms of the target prediction, nor does it fit any parameter to the Higgs mass and then call the result a prediction. Stated limitations, such as the common SUSY scale in the numerical scenario, the neglect of first-two-generation Yukawa couplings, and the possible gluino-mass hierarchy, bound the numerical applications but do not create circularity. No circular step can be exhibited from the paper's own equations or citations.

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

The derivation is a parameter-free two-loop computation; the MSSM inputs (MS, Xt, tan(beta), Ab = A(tau) = At) and SM inputs (mt, alpha_s, GF, mZ, mW, mb) are external. The validity of the claimed completeness rests on the modeled scenario: a single decoupling scale for all MSSM states, the SM as the low-energy EFT in the unbroken phase, and the neglect of first-two-generation Yukawa couplings. The paper flags the heavy-gluino caveat in Sec. 5. The asserted non-degradation from mixing a full two-loop EW side with a partial two-loop SUSY side (Sec. 3) is a methodological assumption. No invented entities are introduced.

assumptions (4)
  • domain assumption All MSSM superparticles and the heavy Higgs doublet decouple at a common scale MS, leaving the SM as the EFT below; O(v^2/MS^2) terms are negligible.
    This defines the scenario in which the corrections are claimed to complete the two-loop QCD matching and underpins the numerics of Sec. 4. The paper flags in Sec. 5 that a much heavier gluino would introduce mass-ratio-enhanced corrections.
  • domain assumption Yukawa couplings of the first two generations are neglected in the squark contributions.
    Sec. 2.1 states: 'our calculation neglects the Yukawa couplings of the first two generations'. Standard in the MSSM, but it is part of the completeness claim.
  • standard math The asymptotic expansion of the two-loop self-energies in the heavy superparticle masses is valid in the limit v to 0 (unbroken EW phase) and p^2 to 0.
    Used in Sec. 2.2 for the WFR contributions; the paper cross-checks the results against the explicit analytic formulae of refs. [31-33].
  • ad hoc to paper The relation between the pole Higgs mass and lambda(QEW) can be used at full two-loop EW accuracy even though the SUSY-scale side is only partially two-loop.
    Sec. 3 asserts this inclusion 'does not degrade' the overall accuracy because the two sides are separately free of log-enhanced terms. This is a methodological assumption, not a proven bound.

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Cite this review

Pith. "Pith review of Full two-loop QCD corrections to the Higgs mass in the MSSM with heavy superpartners." pith.science (2026). https://pith.science/paper/RSWL2QDH

@misc{pith2026190801670,
  author       = {Pith},
  title        = {Pith review of: Full two-loop QCD corrections to the Higgs mass in the MSSM with heavy superpartners},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RSWL2QDH}},
  note         = {Machine review of arXiv:1908.01670}
}
read the original abstract

We improve the determination of the Higgs-boson mass in the MSSM with heavy superpartners, by computing the two-loop threshold corrections to the quartic Higgs coupling that involve both the strong and the electroweak gauge couplings. Combined with earlier results, this completes the calculation of the two-loop QCD corrections to the quartic coupling at the SUSY scale. We also compare different computations of the relation between the quartic coupling and the pole mass of the Higgs boson at the EW scale. We find that the numerical impact of the new corrections on the prediction for the Higgs mass is modest, but comparable to the accuracy of the Higgs-mass measurement at the LHC.

Figures

Figures reproduced from arXiv: 1908.01670 by the authors.

Figure 1
Figure 1. Difference (in MeV) between the Higgs mass given as input to the code mr and the Higgs mass obtained by inserting in eq. (34) the value of λ(mt) computed by mr, as a function of the input Higgs mass. that the calculations of refs. [85] and [86] differ substantially in what concerns the renormalization of the Higgs vev and the corresponding treatment of the tadpole contributions (they also differ in the treatment of … view at source ↗
Figure 2
Figure 2. Difference (in MeV) between the predictions for the Higgs mass obtained with and without the inclusion of the mixed QCD–EW corrections to the quartic Higgs coupling, as a function of a common SUSY scale MS, for Xt = √ 6 MS (lower, blue lines) or Xt = 2 MS (upper, red lines), and Ab = Aτ = At. In each set of lines the solid one is obtained with tan β = 20 and the dashed one with tan β = 5. The star on each line marks… view at source ↗
Figure 3
Figure 3. Values of the common SUSY scale MS and of the stop mixing term Xt that lead to mh = 125.09 GeV, for tan β = 20 and Ab = Aτ = At. The dotted line is obtained including only the one-loop threshold corrections to the Higgs quartic coupling, the dashed line includes the two-loop corrections in the gaugeless limit, and the solid line includes also the effect of the mixed QCD–EW corrections. includes also the effect of th… view at source ↗

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Higgs-Boson Masses and Mixings in the MSSM with CP Violation and Heavy SUSY Particles

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