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$B_s^0 \rightarrow \mu^+ \mu^-$ in a flavor violating extension of MSSM

T0 review · 4 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read The $\mu\nu$SSM can accommodate the measured $B_s^0\to\mu^+\mu^-$ branching ratio while simultaneously matching $\bar{B}\to X_s\gamma$, and the combined fit narrows the model's viable parameter space.

desk verdict A solid but derivative B-physics constraint paper for the µνSSM; the main caveat is that the Wilson coefficients are borrowed without derivation, so the numerics aren't independently checkable. read the letter →

arxiv 2502.05466 v1 pith:G6ICGW26 submitted 2025-02-08 hep-ph

Kai-Kai Meng , Hai-Bin Zhang , Jin-Lei Yang This is my paper · ORCID
classification hep-ph PACS 12.60.Jv14.80.Da
keywords rareBdecaymunuSSMB_s^0->mu+X_sgammaflavorviolationsupersymmetryWilsoncoefficientsminimal
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 claims that the $\mu\nu$SSM -- a supersymmetric extension of the Standard Model in which the Higgs $\mu$ term is generated by right-handed sneutrino vacuum values -- can reproduce the measured branching ratio of the rare decay $B_s^0\to\mu^+\mu^-$ while simultaneously matching the measured rate of $\bar{B}\to X_s\gamma$. If true, this matters because rare $B$ decays are among the cleanest probes of new TeV-scale physics, and the result would identify which regions of the $\mu\nu$SSM parameter space remain viable. The authors further claim that the combined constraints narrow the parameter space, yielding specific allowed ranges for the trilinear stop coupling $A_t$, $\tan\beta$, and the charged-Higgs-mass-controlling parameters $\kappa$, $\upsilon_{\nu^c}$, and $A_\lambda$.

What carries the argument

The load-bearing object is the effective Hamiltonian for $b\to s\mu^+\mu^-$ written as a sum over the Wilson coefficients $C_{7,8,9,10,S,P}$ and their chirality-flipped partners, with the $\mu\nu$SSM contributions collected in eleven classes of one-loop diagrams (scalar, fermion, squark, charged-Higgs, $W$, and neutralino loops). The coefficients are evolved from the electroweak scale to $\mu\sim m_b$ with the leading-order anomalous-dimension matrix, and the branching ratio is assembled from the scalar, pseudoscalar, and axial-vector form factors $F_S^s$, $F_P^s$, and $F_A^s$ times the $B_s^0$ decay constant. What carries the numerical argument is the chain by which $\kappa$, $\upsilon_{\nu^c}$, and $A_\lambda$ raise $M_{H^\pm}$, which in turn suppresses both $\mathrm{Br}(B_s^0\to\mu^+\mu^-)$ and $\mathrm{Br}(\bar{B}\to X_s\gamma)$, while $A_t$ adjusts the stop-mediated contributions.

What would settle it

Recompute $C_{S,P,9,10}^{\mathrm{NP}}$ directly from the $\mu\nu$SSM mass eigenstates (eight neutral scalars, ten neutral fermions, six up-squarks) instead of importing the B-LSSM expressions, and check whether the predicted $\mathrm{Br}(B_s^0\to\mu^+\mu^-)$ still falls inside the experimental band at the quoted benchmark points; a shift larger than the current $1\sigma$ band would overturn the central claim.

Watch

Extended reading notes

Core claim

Within a minimal-flavor-violating scenario for the soft SUSY-breaking terms, the paper computes the Wilson coefficients for $b\to s\mu^+\mu^-$ from the eleven classes of one-loop diagrams that exotic $\mu\nu$SSM particles contribute, evolves them to the hadronic scale, and combines them with the SM contributions to obtain $\mathrm{Br}(B_s^0\to\mu^+\mu^-)$. The central finding is that the new physics contributions to both $B_s^0\to\mu^+\mu^-$ and $\bar{B}\to X_s\gamma$ are controlled mainly by $M_{H^\pm}$, $\tan\beta$, and $A_t$, with $\kappa$, $\upsilon_{\nu^c}$, and $A_\lambda$ entering through $M_{H^\pm}$. Scanning the relevant parameters under the condition that the SM-like Higgs mass sits near 125 GeV, the paper finds regions where both measured branching ratios are simultaneously accommodated, and it quotes the corresponding allowed ranges, e.g. $-8.5\,\mathrm{TeV}\lesssim A_t\lesssim -1.1\,\mathrm{TeV}$ for $\tan\beta=15$ from $B_s^0\to\mu^+\mu^-$.

Load-bearing premise

The whole calculation leans on the assumption that the B-LSSM loop formulas for the Wilson coefficients remain valid after simply renaming the particles, without being re-derived for the $\mu\nu$SSM's larger set of scalars, fermions, and squarks.

Editorial extensions

If this is right

  • If the paper is right, the $\mu\nu$SSM with minimal flavor violation is not ruled out by the two cleanest rare-$B$ measurements; explicit viable points exist, for instance $\kappa=0.5$, $\upsilon_{\nu^c}=2\,\mathrm{TeV}$, $A_\lambda=0.5\,\mathrm{TeV}$, $A_t=-3.6\,\mathrm{TeV}$, $\tan\beta=15$, with $m_h\approx125$ GeV.
  • The combined fit shrinks the allowed range of $A_t$ at large $\tan\beta$: for $\tan\beta=30$, $B_s^0\to\mu^+\mu^-$ allows $-4.5\lesssim A_t\lesssim 0.5$ TeV while $\bar{B}\to X_s\gamma$ requires $-7\lesssim A_t\lesssim 2.6$ TeV, so the intersection is narrower than either constraint alone.
  • Lower bounds follow for the $\mu\nu$SSM-specific parameters: the $B_s^0\to\mu^+\mu^-$ measurement demands $\kappa\gtrsim0.18$ for $\upsilon_{\nu^c}=2$ TeV (or $\kappa\gtrsim0.55$ for $\upsilon_{\nu^c}=1.5$ TeV) and $A_\lambda\gtrsim-0.14$ TeV for $\upsilon_{\nu^c}=2$ TeV.
  • Because $\kappa$, $\upsilon_{\nu^c}$, and $A_\lambda$ enter through $M_{H^\pm}$, the measurement effectively places a lower bound on the charged-Higgs mass in the $\mu\nu$SSM.

Reading between the lines

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

  • Editorial inference: the numerical bounds are computed with Wilson-coefficient formulas carried over from the B-LSSM by relabeling; inserting the full $\mu\nu$SSM spectrum (eight neutral scalars, ten neutral fermions, six up-squarks) from scratch could shift the quoted ranges.
  • Editorial inference: the route through $M_{H^\pm}$ predicts a correlation between the allowed parameter region and charged-Higgs mass, so direct charged-Higgs searches at the LHC provide a model-independent cross-check.
  • Editorial inference: the paper's note that the $\mu\nu$SSM contributes to $C_9$ implies the same viable regions should produce correlated shifts in other $b\to s$ observables, such as $B\to K^*\mu^+\mu^-$ angular distributions, which future data can test.
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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

4 major / 6 minor

Summary. The manuscript computes Br(B_s^0 → μ^+μ^-) in the μνSSM, an MSSM extension with three right-handed neutrino superfields whose VEVs generate the μ term, and combines it with Br(B̄ → X_sγ). The calculation follows the standard effective-Hamiltonian route: Wilson coefficients for b→sμ^+μ^- are taken, with the statement that they 'can be written as' the corresponding expressions, from the authors' earlier B-LSSM calculation of Ref. [67], evolved from the electroweak scale to μ ∼ m_b, and inserted into the standard B_s^0 → μ^+μ^- decay formula; the b→sγ result is imported from the authors' Ref. [19]. After imposing a diagonal (MFV) structure on the soft terms, most parameters are fixed (Table III), and the branching ratios are shown as functions of A_t, κ, υ_ν^c, and A_λ, with the 125 GeV Higgs-mass-motivated choices (A_t = −3.6 TeV, tan β = 15) as the benchmark. The curves in Figs. 2–4 cross the experimental band, and the paper concludes that the μνSSM can simultaneously accommodate both observables, narrowing the allowed ranges of these parameters. The analysis is a scan, not a fit; no other b→s observables are confronted.

Significance. The advertised result — that the μνSSM parameter space contains regions consistent, simultaneously, with the measured Br(B_s^0 → μ^+μ^-) and Br(B̄ → X_sγ) — would be a useful constraint statement for this model class, since these are among the most precise probes of flavor-changing neutral currents at the TeV scale. The paper deserves credit for using updated inputs (HFLAV averages, PDG 2024, FLAG decay constant), for presenting the full 11-class Wilson-coefficient structure in analytic form, and for emphasizing the conjunction of the two constraints rather than either one alone. The central assessment hinges on whether the Appendix A transfer from the B-LSSM to the μνSSM is legitimate: if it is, the numerical results are plausible and the paper is a solid scan-level phenomenological study; if it is not, the central claim is unsupported. As written, the missing vertex definitions, the unquantified theory uncertainty, and the asserted (not shown) 125 GeV Higgs-mass constraint prevent full verification.

major comments (4)
  1. [Appendix A (Eqs. A1–A13) and Section III] The statement in Appendix A that the b→sμ^+μ^- Wilson coefficients 'can be written as' the B-LSSM formulas of Ref. [67] is not supported by the manuscript. The sums in Eqs. (A1)–(A13) run over the μνSSM field content (8 neutral scalars S_l, 10 neutral fermions χ^0_l, 6 up-squarks U_i, 5 charged fermions χ_i), but the coupling factors C^L/R_{abc}, the scalar and fermion mass matrices, and the mixing entries encoded in the x_i are never defined for the μνSSM. The two models differ in gauge and matter content (B-LSSM has U(1)_{B-L} and a Z′, while the μνSSM has RPV couplings λ_i, κ_{ijk} and no Z′), so the C^L/R factors are not the same objects in the two models. Since Figs. 2–4 are the only numerical evidence for the central claim, a reader cannot reproduce the curves or verify that the μνSSM sneutrino and neutralino mixings are correctly included. The authors should provide the μνSSM vertex and mass-input definitions, or state explicitly how the B-LSSM formulas of Ref. [67] were translated and why the relabeling is complete.
  2. [Section IV (benchmark paragraph after Eq. (35))] The 125 GeV Higgs-mass constraint is load-bearing but unverifiable. The text asserts that A_t = −3.6 TeV with tan β = 15 is chosen 'to ensure the SM-like Higgs mass around 125 GeV,' but no Higgs-mass calculation, spectrum generator, or reference is given. With the parameters of Table III (λ = 0.05, κ = 0.5, υ_ν^c = 2 TeV, A_λ = 0.5 TeV, third-generation squark masses 2 TeV), the predicted m_h in the μνSSM, including the singlet and sneutrino sectors, is not obvious from inspection, especially because Figs. 2–4 scan a wide range of the very parameter (A_t) that most strongly affects m_h. The paper should report the computed m_h for the benchmark points used in Figs. 2–4, or at least show a m_h scan over the plotted ranges.
  3. [Section IV (gray band, Figs. 2–4)] The comparison band is constructed by linearly adding the SM theory error to the experimental 1σ error, while the μνSSM predictions are plotted as single curves with no theory uncertainty. The central claim of 'successfully accommodate' is therefore not quantified: the reader cannot tell whether a curve crosses the band by 1σ or by much more, nor what fraction of the scanned parameter space is excluded. Since the Wilson coefficients are computed at one-loop order and evolved with the leading-log approximation of Eqs. (16)–(19), a minimal theory-error estimate (for example, variation of the matching scale μ_EW, variation of the fixed parameters in Table III, or propagation of the f_{B_s}, m_b, and CKM input uncertainties through Eqs. (22)–(26)) should be shown for representative points.
  4. [Appendix A, Eq. (A13)] There is an internal inconsistency in the printed formulas: C^{(11)}_{S,NP} and C^{(11)}_{P,NP} are shown as identical expressions, whereas in every other S/P pair in Eqs. (A1)–(A7) the two coefficients differ through the sign combination (C^L + C^R) versus (−C^L + C^R) at the lepton–scalar vertex. As printed, the pseudoscalar coefficient has no parity-odd part, which would directly affect F^s_P in Eq. (24) and hence the branching ratio. Either a sign combination is missing in C^{(11)}_{P,NP} or the line is mislabeled; this needs correction and re-checking of the numerics.
minor comments (6)
  1. [Appendix A, Eq. (A4)] The second equality in Eq. (A4) is again labeled C^{(4)}_{S,NP}; by symmetry with the first line it should be C^{(4)}_{P,NP}.
  2. [Appendix A, Eq. (A14)] In the printed expressions for G_3 and G_4, the plus sign between the third and fourth terms is missing; as typeset, the terms involving x_4 appear to be multiplied inconsistently.
  3. [Abstract] The phrase 'Combined with the decay B̄ → X_sγ, the numerical results indicate...' is a dangling construction; the sentence should be rephrased, e.g., 'Together with the decay B̄ → X_sγ, the numerical results indicate...'.
  4. [Section IV (paragraph after Table III)] The statement that 'The parameter λ also have effects similar to those of the parameter υ_ν^c' is not demonstrated in any figure, because λ is fixed to 0.05 in Table III throughout the scans; either add a λ scan or rephrase the claim as an inference from Eqs. (34)–(35).
  5. [Table II] The table refers to 'Table I of Ref. [55]' for the remaining SM inputs; since the quoted branching ratios depend on those values, a self-contained list of the central values would improve reproducibility.
  6. [Figures 2–4] Several figure labels are garbled in the extracted text, for example 'Br(Bs0-> + _)' in Fig. 2; the axis labels should be checked to read Br(B_s^0 → μ^+μ^-) and Br(B̄ → X_sγ) consistently.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction is present: the B_s^0 -> mu+ mu- branching ratio is computed from scanned model parameters and compared to data after the fact, not fitted from the data. The main caveat is that the Wilson coefficients are imported from the authors' earlier B-LSSM calculation rather than re-derived for the mu neutrino SSM, which is a derivation-support concern rather than a circularity.

full rationale

The paper's chain is: define the mu nu SSM (Eqs. 4-5); construct the b -> s mu+ mu- effective Hamiltonian (Eqs. 11-15); quote one-loop Wilson coefficient expressions in Appendix A; then scan SUSY parameters and compare the resulting Br(B_s^0 -> mu+ mu-) and Br(Bbar -> X_s gamma) with the experimental bands in Figs. 2-4. No parameter is fit to these two branching ratios: the parameters At, tan beta, kappa, nu_nu^c and A_lambda are varied and the allowed regions are read off only after the comparison. The branching-ratio formula (Eqs. 22-26) is standard and does not contain the measured value as an input. The one load-bearing self-citation is in Appendix A, where the Wilson coefficients for the mu nu SSM are asserted to 'can be written as' the formulas from the authors' own B-LSSM paper [67], without displaying the mu nu SSM vertex factors C^L,R or deriving them for the enlarged scalar/fermion sectors. This is a missing-derivation or support issue for the numerical claim, not a circularity: the quoted loop functions, masses and mixing entries are inputs, not outputs, and the paper does not re-label a fitted parameter as a prediction. Therefore no step reduces by construction and the central numerical claim has independent content, although the Appendix A transfer from B-LSSM to mu nu SSM should be made explicit and checked.

Assumptions & free parameters 10 free parameters · 5 assumptions · 0 invented entities

The calculation rests on the µνSSM model plus a set of hand-picked and scanned parameters. No new particles or forces are introduced by this paper. The main assumptions are the MFV reduction, the zero neutrino-Yukawa approximation, and the reuse of Wilson coefficient formulas from the authors' earlier B-LSSM work.

free parameters (10)
  • lambda = 0.05 (fixed)
    Chosen from parameter space analysis of Ref. [34]; affects neutral scalar sector and Higgs mass.
  • kappa = 0.1 to 1 (scanned); 0.5 in benchmark plots
    Varies charged Higgs mass via Eq. (34); a central parameter for the B decay rates.
  • tan beta = 15 or 30 in plots; 15 chosen for final analysis
    Scanned because it strongly affects both branching ratios; tied to Higgs mass constraint.
  • upsilon_nu^c = 1.5 to 2.5 TeV; 2 TeV in benchmark
    Unique µνSSM parameter, the right-handed sneutrino VEV, scanned and benchmarked.
  • A_lambda = 0.5 TeV in benchmark; scanned from about -0.2 to 1 TeV in figures
    Affects charged Higgs mass through Eq. (35); constrained by B_s -> mu+ mu- data.
  • A_kappa = -300 GeV
    Set by tachyon avoidance condition Eq. (33); minimal influence on the branching ratios.
  • At (Au3) = -3.6 TeV (chosen for 125 GeV Higgs); scans over -5 to 5 TeV
    Stop mixing parameter; the main driver of the SUSY corrections to both decays.
  • M2 = 1 TeV
    Gaugino mass parameter set using GUT relations and LHC gluino bounds.
  • Third-generation squark masses = 2 TeV (m_Q3, m_uc3, m_dc3)
    Set at 2 TeV to enhance the At effect; first two generations at 3 TeV from LHC constraints.
  • Slepton masses (m_ec) = 1 TeV
    Negligible impact on the observables; chosen for simplicity.
assumptions (5)
  • domain assumption The µνSSM superpotential in Eq. (4) defines the theory under study.
    The entire calculation assumes this particular supersymmetric model and its particle content.
  • ad hoc to paper Minimal flavor violation ansatz in Eq. (28) reduces the soft parameters to a diagonal set.
    This simplification is chosen for tractability rather than forced by data; the central claim depends on this assumption.
  • domain assumption The approximations Y_nu_i = 0 and upsilon_nu_i = 0.
    Justified by neutrino oscillation constraints from Ref. [58]; these simplify the calculation significantly.
  • standard math The effective Hamiltonian framework for b -> s transitions, Eqs. (11)-(13).
    Standard low-energy effective theory from literature; the paper adopts it without modification.
  • domain assumption Tachyon avoidance condition in Eq. (33), leading to A_kappa = -300 GeV.
    The model requires non-tachyonic sneutrino masses; the constraint is used to fix A_kappa.

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Cite this review

Pith. "Pith review of $B_s^0 \rightarrow \mu^+ \mu^-$ in a flavor violating extension of MSSM." pith.science (2026). https://pith.science/paper/G6ICGW26

@misc{pith2026250205466,
  author       = {Pith},
  title        = {Pith review of: $B_s^0 \rightarrow \mu^+ \mu^-$ in a flavor violating extension of MSSM},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G6ICGW26}},
  note         = {Machine review of arXiv:2502.05466}
}
abstract

$B$ meson rare decays play a crucial role in exploring new physics beyond the standard model. In this study, we explore the rare decay process $B_s^0 \rightarrow \mu^+ \mu^-$ in a flavor violating extension of the Minimal Supersymmetric Standard Model (MSSM), namely the $\mu$-from-$\nu$ SSM ($\mu\nu$SSM). Combined with the decay $\bar{B}\rightarrow X_s\gamma$, the numerical results indicate that the $\mu\nu$SSM can successfully accommodate the experimental data for $B_s^0 \rightarrow \mu^+ \mu^-$ and additionally narrow down the parameter space.

Figures

Figures reproduced from arXiv: 2502.05466 by the authors.

Figure 1
Figure 1. FIG. 1: The Feynman diagrams contributing to [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: When [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. , we can also observe that both Br(B0 s → µ +µ −) and Br(B¯ → Xsγ) decrease as υν c increases. From Eqs.(34, 35), it can be seen that, similar to κ, an increase in υν c will also lead to an increase in MH± . Therefore, both υν c and κ can influence the theoretical predictions for Br(B0 s → µ +µ −) and Br(B¯ → Xsγ) by affecting MH± . The parameter λ also have effects similar to those of the parameter υν c . From Eqs.… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: When tan [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

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

  1. $B \to X_{\mathrm{s}} l^{+} l^{-}$ in the $\mu$ from $\nu$ Supersymmetric Standard Model

    hep-ph 2026-07 conditional novelty 4.0 of 10

    In the μνSSM, B→X_s l+l− is dominated by charged-Higgs C7 in the low-q² region and C9/C10 Z-penguins in the high-q² region, with the forward-backward asymmetry set by C7C10 and C9C10 interference.

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