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

Heidi Rzehak; Nick Murphy

REVIEW 1 major objections 4 minor 120 references

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

T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper predicts that, in the MSSM with heavy superpartners and complex phases, the lightest Higgs mass swings by almost 20 GeV when the trilinear phase varies, while the CP-odd admixture falls below 0.5% for charged Higgs masses above…

desk verdict A careful, honest NLL EFT calculation of the complex MSSM Higgs sector that is new in combination, but the headline 20 GeV phase effect sits on large one-loop thresholds with no two-loop control. read the letter →

arxiv 1909.00726 v2 pith:AZY2UKTU submitted 2019-09-02 hep-ph

Nick Murphy , Heidi Rzehak This is my paper · ORCID

classification hep-ph

keywords MSSMHiggsmassCPviolationcomplexphasestwo-Higgs-doubletmodeleffectivefieldtheorytwo-loopRGEsnext-to-leadinglogarithmsCP-oddadmixture

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 sets out to predict the Higgs sector of the Minimal Supersymmetric Standard Model (MSSM) in the regime where all superpartners are heavy, the two Higgs doublets are light, and the MSSM parameters can be complex. Its central result is that the phase of the common trilinear coupling, $\phi_A$, is not a minor detail: at $\tan\beta = 5$ and $|A| = |\mu| = 3M_s$, varying $\phi_A$ moves the predicted lightest-Higgs mass by almost 20 GeV, and even at $\tan\beta = 20$ the shift reaches about 5 GeV. The calculation carries this through a one-loop matching of the MSSM onto a type-III two-Higgs-doublet effective theory at the SUSY scale, followed by two-loop renormalization-group running that keeps all complex phases, so that next-to-leading logarithms of the heavy scale are resummed. A second finding is that the CP-odd admixture of the lightest Higgs boson falls below half a percent once the charged Higgs mass exceeds roughly 260 GeV in the studied scenario. The paper matters because it delimits how much CP violation from the heavy SUSY sector can survive in the 125 GeV Higgs boson and how strongly the Higgs-mass prediction depends on the unknown phases.

What carries the argument

The load-bearing object is the effective-field-theory chain from the MSSM to a low-energy type-III two-Higgs-doublet model, i.e. a two-Higgs-doublet model in which both doublets couple to up- and down-type fermions. At the SUSY scale $M_s$, one-loop matching produces complex quartic couplings $\lambda_5,\lambda_6,\lambda_7$ and 'wrong'-Yukawa couplings $h'_t,h'_b$ from box, triangle, and wave-function-renormalization diagrams; the field redefinition in Eq. (30) ensures canonically normalized kinetic terms. The two-loop RGEs for the 2HDM carry all phases, and the 'top-down' iterative solution fixes the high-scale MSSM couplings so that SM couplings match their measured values at $M_t$. The pole masses are then computed by one of three methods: (a) one-loop-corrected 2HDM mass matrix at $M_{H^\pm}$, (b) the same at $m_t$, (c) matching to the SM and evaluating a SM pole mass for large $M_{H^\pm}$. The default option (d) adds $\gamma_{\mathrm{resum}}$ to the (1,1), (1,2), and (2,2) entries of the Higgs-basis mass matrix before rotating back, resumming $\ln(M_{H^\pm}/m_t)$ while retaining the full 2HDM mixing information; this is the mechanism that produces the quoted interpolated masses and the CP-odd admixtures.

What would settle it

Compute the missing two-loop threshold corrections for the complex MSSM in the same scheme and re-evaluate $m_h$ as a function of $\phi_A$ at $\tan\beta = 5$ and $|A| = |\mu| = 3M_s$; if the almost-20 GeV swing collapses to below the few-GeV level, the central claim would be refuted, and on the experimental side an HL-LHC measurement of the tau-Yukawa CP angle $\phi_\tau$ would decide whether the predicted admixture near $\phi_\tau \simeq 4^\circ$ for $M_{H^\pm} \sim 500$ GeV is present or excluded.

Watch

Extended reading notes

Core claim

In the paper's own terms, the claim being established is an extension of existing heavy-SUSY Higgs predictions to include CP violation: after one-loop matching of the complex MSSM to a type-III two-Higgs-doublet model and two-loop phase-keeping RGE evolution, the phase $\phi_A$ of the common trilinear coupling is a controlling parameter for the spectrum. For $\tan\beta = 5$ and $|A| = |\mu| = 3M_s$, $m_h$ changes by almost 20 GeV as $\phi_A$ runs from $0$ to $360^\circ$; at $\tan\beta = 20$ the variation shrinks to about 5 GeV, and the sign of the effect depends on the ratio $|A|/M_s$. The gluino phase $\phi_{M_3}$ moves $m_h$ by at most about 1.3 GeV, while the heavy neutral Higgs masses shift by roughly 3.5 GeV ($H_2$) and 1.5 GeV ($H_3$) with $\phi_A$, and their CP-odd content is exchanged between the two states. The CP-odd component of the lightest Higgs is computed from the mixing matrix in the rotated basis; it is maximal near $\phi_A \approx 120^\circ$ and drops below 0.5% already for $M_{H^\pm} \simeq 260$ GeV in the benchmark scenario. The paper also claims that the interpolating pole-mass approximation, which adds the resummed correction $\gamma_{\mathrm{resum}} = v^2(m_t)\lambda_{\rm SM}(m_t) - v^2(M_{H^\pm})\lambda_{\rm SM}(M_{H^\pm})$ to the Higgs-basis mass matrix, agrees well with the pure 2HDM determination at small $M_{H^\pm}$ and with the decoupled-SM determination at large $M_{H^\pm}$, making it a reliable default.

Load-bearing premise

The load-bearing premise is that the omitted two-loop threshold corrections—which are not yet available for complex MSSM parameters in the dimensional-reduction scheme—are small enough not to change the quoted GeV-level phase dependence; the paper does not estimate their size.

Editorial extensions

If this is right

At $\tan\beta = 5$ with $|A| = |\mu| = 3M_s$, the phase $\phi_A$ changes the predicted lightest-Higgs mass by almost 20 GeV, so fixed-phase scans miss a controlling parameter of the heavy-SUSY Higgs sector.
The same phase dependence shrinks to about 5 GeV at $\tan\beta = 20$, and its sign depends on $|A|/M_s$, meaning that the CP-phase correction cannot be absorbed into a universal shift.
The gluino phase $\phi_{M_3}$ shifts $m_h$ by at most about 1.3 GeV in the considered scenarios, so it is a subleading source of CP-driven mass uncertainty.
For the heavy neutral Higgs bosons, $\phi_A$ moves $m_{H_2}$ by about 3.5 GeV and $m_{H_3}$ by about 1.5 GeV, and the two states exchange their CP-odd character as $\phi_A$ varies, with only weak dependence on $M_{H^\pm}$.
The CP-odd component of the lightest Higgs falls below 0.5% once $M_{H^\pm} \gtrsim 260$ GeV in the benchmark scenario, so for the experimentally preferred $M_{H^\pm} \geq 500$ GeV the predicted admixture is just below the expected LHC reach of roughly $\phi_\tau \simeq 4^\circ$.

Reading between the lines

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

Editorial inference: the missing two-loop thresholds are the main caveat; if they are approximately phase-independent, the quoted differences—like the 20 GeV swing—survive, but if they are phase-sensitive they could either enhance or erase the effect.
Editorial inference: the low-scale phases of $\lambda_5,\lambda_6,\lambda_7$ generated by complex RGE running feed directly into electric-dipole-moment predictions, so the same calculation provides a route to map which high-scale CP phases are experimentally allowed; the paper explicitly leaves this EDM check for future work.
Editorial inference: combining the fast drop of the CP-odd component with LHC bounds on low $M_{H^\pm}$ and high $\tan\beta$ suggests that a measurably CP-violating 125 GeV Higgs in this heavy-SUSY setup would require a low charged-Higgs mass or a smaller $M_s$ than the 30 TeV benchmark; an HL-LHC $\tau\tau$ measurement could test this directly.
Editorial inference: a natural testable extension is to recompute the benchmark with the missing two-loop thresholds and $\tan\beta$ resummation included, or to port the same matching-and-running machinery to the next-to-minimal supersymmetric standard model, and check whether the phase swing remains of order several 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.

Desk Editor's Note

read the letter

You should know this is a careful, honest EFT calculation, not a breakthrough. It computes the MSSM Higgs spectrum with heavy SUSY and complex phases by matching to a type-III 2HDM at one loop, running with two-loop complex RGEs, and resumming NLLs. That specific combination is new relative to Refs. [64,65,92], and the numerical study of phase dependence and CP-odd admixtures is useful. The paper is also unusually candid about its limitations—Section 4 explicitly says two-loop thresholds are unknown for complex parameters in DR, and the omission of tanβ resummation is stated.

The technical core looks sound. The one-loop matching formulas are given in detail and agree with Ref. [92] up to gauge terms; the RGEs are checked against Ref. [106]; the interpolation method (d) is cross-checked against the 2HDM and SM limits. The three pole-mass options are sensible, and the authors explain when each is applicable. The qualitative results—phase dependence of several GeV at low tanβ, and a CP-odd admixture that drops below 0.5% for MH+ above about 260 GeV in the benchmark—are likely robust.

The soft spot is the quantitative headline. The 20 GeV phase dependence shown in Fig. 5b is calculated at |A|=|µ|=3Ms, where the one-loop box corrections to the quartic couplings are comparable to their tree-level values. The CP-violating couplings λ5-λ7 are loop-induced and receive exactly these large A-dependent terms, so the phase sensitivity is driven by a threshold correction that is not parametrically small. With two-loop thresholds omitted and no uncertainty estimate given, a shift of several GeV in mh is plausible. The authors do not overclaim—they say "sizeable" and "couple of GeV"—but the specific numbers in Fig. 5b should be treated as uncontrolled until the missing pieces are estimated.

The remaining issues are minor. The non-iteration of the 125.15 GeV input in footnote 6 is acceptable. The tanβ≤20 restriction is conservative. No code or external benchmark against existing public tools is provided, which makes independent verification harder.

This paper deserves a serious referee and probably a conditional accept. I would ask for an uncertainty estimate, a benchmark at a less extreme |A|/Ms, and a comparison with FeynHiggs or FlexibleSUSY in the real-parameter limit. It will be useful for people working on MSSM Higgs EFT and CP violation in the Higgs sector.

Referee Report

1 major / 4 minor

Summary. This paper presents an effective-field-theory calculation of the Higgs-boson spectrum in the MSSM with heavy superpartners and complex parameters. The MSSM is matched to a type-III 2HDM at one-loop order at the scale Ms, the 2HDM parameters are evolved down with two-loop RGEs that include complex phases, and the pole masses are computed with several methods: using the 2HDM at MH+, using the 2HDM at mt, matching to the SM for large MH+, and an interpolation method (d) that resums the dominant logarithms while retaining the full 2HDM mass matrix. The central numerical findings are that the mass of the lightest neutral Higgs boson can vary by up to about 20 GeV as the common phase of At and Ab is varied in the extreme benchmark |A|=|µ|=3Ms, tanβ=5, and that the CP-odd admixture of the lightest Higgs boson drops below 0.5% for MH+≈260 GeV in their scenario. The paper also studies the dependence on the gluino phase and the masses and mixings of the heavy Higgs bosons.

Significance. If the quantitative results hold, this work is a valuable step toward next-to-leading-log resummed predictions for the CP-violating MSSM with heavy sfermions. It provides the first complete one-loop matching of the complex MSSM to a type-III 2HDM including field redefinitions, two-loop RGEs with complex phases, and a transparent comparison of different pole-mass schemes. The interpolation method (d) is cross-checked against the pure 2HDM and SM limits for real parameters, and the authors are explicit about the missing two-loop thresholds and the restriction tanβ≤20 without tanβ resummation. The qualitative conclusion that CP phases can shift the lightest Higgs mass by several GeV and can control the size of the CP-odd admixture is plausible and interesting. However, the main numerical claim is made in a benchmark where the one-loop matching is not a small correction, and the absence of any uncertainty estimate limits the quantitative reach of the paper.

major comments (1)

[Section 4, Eqs. (15)–(21), Fig. 5(b)] The central quantitative claim—that the lightest Higgs mass varies by almost 20 GeV with the phase ϕA—is not controlled by the published calculation. For the benchmark |A|=|µ|=3Ms with h_t(Ms)≈0.8, the one-loop box contributions to the quartics, e.g., Δλ_1^(4) = −κ/2 h_t^4 |µ̂|^4 ≈ −0.1, are comparable to the tree-level values λ_i ≈ 0.1–0.3 in Eq. (13). The CP-violating couplings λ5, λ6, λ7 are entirely loop-induced and receive exactly these A-dependent terms, so the phase sensitivity in Fig. 5(b) is dominated by threshold corrections that are not parametrically small. Section 4 states that two-loop thresholds are unknown for complex parameters in the DR scheme, and no uncertainty estimate is given anywhere. Since two-loop terms are suppressed by only one additional loop factor, the missing O(h_t^4 g_s^2) and O(|A|^6 h_t^6) pieces can plausibly shift m_h by more than 1–2 GeV and can modify the phase dependence itself, not merely the overall mass. The qualitative conclusion that phases can shift m_h by several GeV is likely robust and is a worthwhile result, but the specific magnitude near 20 GeV and the masses close to 105 GeV in Fig. 5(b) are not quantitatively supported. I recommend adding an estimate of the missing higher-order uncertainty, for example by renormalization-scale variation or by using the O(h_t^4 g_s^2) two-loop thresholds of Ref. [92] (converted to DR) as a proxy, and softening the quantitative statements in the abstract and Section 7 accordingly.

minor comments (4)

[Section 4] There is a typo in the sentence describing the loop-function evaluation: "evalued" should be "evaluated".
[Eq. (56)] The symbol "γreum" is a typo for "γresum".
[Section 6.5] Please clarify that the notation P^2_{i3} in the CP-odd percentage definition denotes |P_{i3}|², since the mixing matrix is in general complex.
[Abstract and Section 6.4] The abstract quotes the phase dependence as "on the order of a couple of GeV," while Fig. 5(b) shows almost 20 GeV for the extreme benchmark |A|=|µ|=3Ms. Please state both the typical and maximal values, and explicitly connect the maximal value to the benchmark at the boundary of the calculation’s validity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Higgs masses are computed from MSSM inputs via one-loop matching and two-loop RGEs; the phase dependence is an un-fitted output.

full rationale

The central derivation is self-contained: the MSSM soft parameters and phases specified at M_s enter the one-loop matching conditions (Eqs. (15)-(43)) to define 2HDM couplings, which are evolved with two-loop complex RGEs and then converted to pole masses through Eqs. (49), (50), and (57). No parameter is fitted to the target m_h; the numerical minimization in step 2(c) adjusts high-scale gauge and Yukawa couplings to reproduce the low-scale SM gauge and Yukawa observables, not the Higgs mass. The headline phase dependence of m_h follows directly from the loop-induced complex couplings, especially lambda5-lambda7 in Eqs. (19)-(21) and (28)-(29), and their running, and is therefore an independent prediction rather than a renamed input. The only self-referential element is footnote 6, which acknowledges that the SM MS-bar inputs were extracted using M_SM_h = 125.15 GeV without iterating the calculation; this is an external measured value used as input, not the output redefined as an input, and the paper correctly notes that iteration would be the more sophisticated procedure. Missing two-loop threshold corrections are a stated accuracy limitation (Section 4) and a correctness risk, but do not amount to circularity.

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

The model parameters (Ms, tanβ, MH+, A, μ, phases) are inputs scanned by hand, not fitted to the target Higgs mass. No new particle or force is introduced. The only quantity fixed by the measured Higgs mass is the conversion of SM inputs at Mt (MhSM=125.15 GeV), which introduces a mild circularity as acknowledged in footnote 6.

free parameters (6)

|A|/Ms = 3 (also 2 in some scans)
Chosen to maximize the threshold corrections and CP-violating effects, providing 'an estimate of the largest effects' (Section 6.1).
|μ|/Ms = 3 (also 1 in some scans)
Set equal to |A|/Ms in the default scenario; reducing it to 1 shifts mh by up to about 6 GeV (Section 6.4).
ϕA (common phase of At and Ab) = 2.1 rad ≈ 120°
Chosen because it 'maximizes roughly the size of the CP-odd component of the lightest neutral Higgs boson' (Section 6.1).
tanβ = 5 (also 10, 15, 20)
Restricted to ≤20 because tanβ resummation is not included; phase dependence is larger for smaller tanβ.
MH+ = 500 GeV (also 200-1000 GeV)
Chosen in the experimentally preferred region; CP-odd admixture drops rapidly with increasing MH+.
Ms = 3-30 TeV
SUSY mass scale is scanned as an input; heavier Ms reduces the phase dependence of the couplings.

assumptions (5)

domain assumption The two-loop RGEs for a general 2HDM with complex parameters, taken from Refs. [99-103], correctly describe the running of all couplings.
The paper does not list the full RGE functions; it states it checked agreement with Ref. [106] and relies on this external framework for the NLL resummation.
domain assumption One-loop threshold corrections at Ms, computed with all soft masses equal to Ms and neglecting O(g_a g_b^y) terms with a+b=4, are sufficient; two-loop thresholds are not included.
Section 4: 'We do not include two-loop corrections to the thresholds, since they are still unknown for complex parameters.' The central mass values depend on this omission.
domain assumption The SM couplings at Mt extracted using MhSM = 125.15 GeV can be used without iterating the conversion.
Footnote 6 explicitly says a more sophisticated approach would iterate the conversion depending on the predicted Higgs mass, but a 'one time' conversion is considered sufficient.
ad hoc to paper The phases of M1 and M2 can be chosen to minimize EDM constraints and do not affect the Higgs mass prediction.
Section 6.1 states this assumption; these phases do not enter the calculation, so their EDM implications are deferred to future work.
domain assumption For the log resummation in option (d), the 2HDM can be treated as a CP-even type-II 2HDM with only top-Yukawa couplings.
Section 5, option (d): CP-violating and wrong-type Yukawa couplings are loop-suppressed and neglected in the resummation, which is valid for small tanβ.

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Cite this review

Pith. "Pith review of Higgs-Boson Masses and Mixings in the MSSM with CP Violation and Heavy SUSY Particles." pith.science (2026). https://pith.science/paper/AZY2UKTU

@misc{pith2026190900726,
  author       = {Pith},
  title        = {Pith review of: Higgs-Boson Masses and Mixings in the MSSM with CP Violation and Heavy SUSY Particles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AZY2UKTU}},
  note         = {Machine review of arXiv:1909.00726}
}

read the original abstract

We calculate the Higgs-boson mass spectrum and the corresponding mixing of the Higgs states in the Minimal Supersymmetric Standard Model (MSSM). We assume a mass-hierarchy with heavy SUSY particles and light Higgs bosons. To investigate this scenario, we employ an effective-field-theory approach with a low-energy Two-Higgs-Doublet Model (2HDM) where both Higgs doublets couple to up- as well as down-type fermions. We perform a one-loop matching of the MSSM to the 2HDM and evolve the parameters to the low energy scale by exploiting two-loop renormalization group equations, taking the complex parameters into account. For the calculation of the pole mass, we compare three different options: one suitable for large charged Higgs masses, one for low charged Higgs masses, and one approximation that interpolates between these scenarios. The phase dependence of the mass of the lightest neutral Higgs boson can be sizeable, i.e. on the order of a couple of GeV depending on the scenario. In addition, we discuss the CP composition of the neutral Higgs bosons.

Figures

Figures reproduced from arXiv: 1909.00726 by the authors.

**Figure 2.** Pictorial description of the mass calculation. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** The quartic couplings’ dependence on Ms for the default scenario: tan β = 5, ϕA = 2.1 ≈ 120◦ , ϕµ = ϕM3 = 0, |µ| = |A| = 3Ms, MH+ = 500 GeV. The red curves are the result of employing complex RGEs, and the blue real RGEs. Dashed lines are the couplings at the high scale Ms, and solid lines are couplings at the scale MH+ . 16 [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: The mass of the light Higgs boson is shown in dependence on the scale [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

**Figure 5.** Figure 5: The upper row presents the mass of the lightest Higgs boson depending on the phase [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

**Figure 6.** Figure 6: Dependence of the mass of the lightest Higgs boson on the phase [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

**Figure 7.** Figure 7: The other parameters of the scenario are tan [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

**Figure 8.** Figure 8: The dependence of the size of the CP-odd component on [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]

**Figure 9.** Figure 9: The dependence of the mass of the second lightest and the heaviest Higgs boson on the [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]

**Figure 10.** Figure 10: The dependence of the CP-odd component in percent for the next-to lightest Higgs [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]

Discussion (0). Continue with ORCID to comment.

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