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Revisiting the Electron EDM in the NMSSM

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Pith's one-line read The electron's electric dipole moment in the NMSSM receives three overlooked two-loop contributions — charged-Higgs, W-boson, and Kite diagrams — that tighten the bound on the phase φ4 by about an order of magnitude while partly relaxing…

desk verdict Correct and important observation—NMSSM needs 2HDM-type two-loop EDMs—but the numerical bounds rest on scan points that were never validated for vacuum stability or collider constraints. read the letter →

arxiv 2507.06320 v3 pith:BONOZRQ5 submitted 2025-07-08 hep-ph hep-th

Kaifei Ning , Michael Ramsey-Musolf This is my paper
classification hep-phhep-th PACS 11.30.Er12.60.Jv13.40.Em
keywords electronelectricdipolemomentNMSSMCPviolationtwo-loopEDMdiagramsBarr-ZeechargedHiggselectroweakbaryogenesistwo-Higgs-doubletmodel
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 argues that the electron electric dipole moment (EDM) predicted by the Next-to-Minimal Supersymmetric Standard Model (NMSSM) receives three classes of two-loop contributions — charged-Higgs loops, $W$-boson loops, and a 'Kite' diagram, all of the type familiar from the two-Higgs-doublet model (2HDM) — that every earlier NMSSM EDM analysis overlooked because those analyses carried over Minimal Supersymmetric Standard Model (MSSM) results without accounting for tree-level CP violation in the extended Higgs sector. For the NMSSM-specific phases $\phi_3$, $\phi_4$, and $\phi_5$ these new diagrams dominate the electron EDM and reverse its sign relative to the previously known terms, tightening the constraint on $\phi_4$ by roughly an order of magnitude beyond what earlier studies concluded, while for $\phi_0'$ and $\phi_6$ the new terms partially cancel the known two-loop contributions and relax the bounds. The paper constructs a basis of 14 independent CP-violating phases and transfers the gauge-invariant 2HDM formulas to the NMSSM by a short set of substitution rules. A sympathetic reader would care because the revised limits reshape the parameter space in which NMSSM electroweak baryogenesis could explain the observed matter–antimatter asymmetry of the universe.

What carries the argument

The load-bearing object is the set of three two-loop diagram classes taken over from the two-Higgs-doublet model — the charged-Higgs loop $(d_E^f)^{H^\pm}$, the $W$-boson loop $(d_E^f)^{W^\pm}$, and the 'Kite' diagram $(d_E^f)^{\mathrm{Kite}}$ — each evaluated in the background-field gauge so each group is separately gauge invariant. They carry the argument because they are the only diagrams sensitive to tree-level CP violation in the NMSSM Higgs sector, and they are ported from the complex 2HDM calculation by the substitution $q_{i1}\to a_i$, $c_f\,\mathrm{Im}(q_{i2})\to g^P_{H_i\bar f f}$, and $\lambda_{iH^+H^-}\to\lambda_i$, with all phase dependence encoded in the neutral-scalar rotation matrix $O_{ij}$ and the tree-level Higgs couplings of Appendix A. The paper's organizing device is a basis of 14 independent CP-violating invariants — built by assigning spurion charges under the global $U(1)$ symmetries, including a new $U(1)_S$ — together with a table assigning each phase to the one-loop MSSM, two-loop MSSM, or 2HDM diagram groups; this classification is what lets the authors say which previously unconstrained NMSSM phases are actually probed by the electron EDM.

What would settle it

Recompute the electron EDM over the $\phi_4$–$\phi_5$ scan used for Fig. 6 with an independent implementation of the full one- and two-loop diagram set: the paper predicts that including the three new classes excludes points with $|\sin\phi_4|\gtrsim 0.05$ while $\phi_3$ and $\phi_5$ remain allowed across the full $\pm 0.1$ range, and that the exclusion disappears when the new diagrams are dropped. Separately, run the benchmark and scan points through a spectrum generator that accepts complex NMSSM parameters, checking for Landau poles, tachyonic scalars, and deeper minima: if many of the points used for the limits are not valid vacua once CP phases are switched on, the bounds in Figs. 4–6 would not correspond to physical NMSSM configurations.

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Extended reading notes

Core claim

On the paper's own terms, previous NMSSM fermion-EDM calculations are incomplete: because the NMSSM, unlike the MSSM, has tree-level CP violation in its scalar Higgs sector, the three 2HDM-type two-loop diagram classes $(d_E^f)^{H^\pm}$, $(d_E^f)^{W^\pm}$, and $(d_E^f)^{\mathrm{Kite}}$ must be added to the familiar one-loop and Barr–Zee two-loop sets. Working in the general NMSSM with 14 independent CP-violating phases, the paper adapts the gauge-invariant complex-2HDM electron-EDM results through the substitutions $q_{i1}\to a_i$, $c_f\,\mathrm{Im}(q_{i2})\to g^P_{H_i\bar f f}$, and $\lambda_{iH^+H^-}\to\lambda_i$, and finds that for $\phi_3,\phi_4,\phi_5$ the new diagrams dominate, reversing the predicted EDM sign; in a numerical scan $\phi_4$ is limited to $|\sin\phi_4|\lesssim 0.05$ while $\phi_3$ and $\phi_5$ remain allowed across the full scanned range, so satisfying the bound $d_e\le 4.1\times 10^{-30}$ e cm demands a fine-tuned cancellation between $\phi_4$ and $\phi_5$. For $\phi_0'$ and $\phi_6$, conversely, the new contributions are smaller than and opposite in sign to the MSSM two-loop terms, so those phases stay weakly constrained. The upshot is that the electron EDM excludes or permits a qualitatively different set of NMSSM configurations than the standard formulas indicated, which is the paper's reason for revisiting electroweak-baryogenesis studies that relied on the older expressions.

Load-bearing premise

The load-bearing premise is that the parameter points at which the EDM limits are computed are genuine, stable NMSSM vacua, but the paper's spectrum check validates them only in the CP-conserving limit and at tree level, so the CP-violating points that actually set the phase bounds are not individually verified against false minima, Landau poles, or collider exclusions.

Editorial extensions

If this is right

  • The current electron-EDM bound $d_e\le 4.1\times 10^{-30}$ e cm restricts $\phi_4$ to $|\sin\phi_4|\lesssim 0.05$ in the scan, roughly an order of magnitude tighter than $\phi_3$ and $\phi_5$, which remain viable across nearly the full scanned range and evade the bound at the benchmark with $\sin\phi\lesssim 10^{-2}$.
  • Satisfying the bound requires fine-tuned interplay between $\phi_4$ and $\phi_5$; the allowed points in the $\phi_4$–$\phi_5$ plane cluster in the first and third quadrants, where the two phases generate opposite-sign EDM contributions that cancel.
  • For $\phi_0'$ and $\phi_6$ the new diagrams partially cancel the previously known two-loop contributions, so these NMSSM-specific phases remain weakly constrained and can take values of $O(10^{-1})$ without violating the bound.
  • The same three diagram classes are required in any MSSM extension with an enlarged Higgs sector and tree-level CP violation — the paper names the $\mu\nu$SSM — and the revised formulas supersede the standard NMSSM expressions used in earlier EDM and baryogenesis studies.

Reading between the lines

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

  • If the sign reversal the paper finds for $\phi_3,\phi_4,\phi_5$ holds, then earlier NMSSM electroweak-baryogenesis studies were tested against an incomplete EDM observable; re-deriving the baryogenesis transport equations with the revised CP-violating Higgs couplings could shift the viable parameter region rather than merely shrink it.
  • The partial cancellation for $\phi_0'$ and $\phi_6$ is a concrete accidental-suppression mechanism: an electron-EDM measurement reaching $\sim 10^{-31}$ e cm would distinguish a genuinely small underlying coupling from the cancellation the paper identifies.
  • The same diagrams feed quark chromo-EDMs and the Weinberg operator through the heavy internal loops already in the formulas, so a neutron- and atomic-EDM analysis built on these expressions could constrain the NMSSM phases differently than the electron alone.
  • Because the ordering of the bounds tracks ratios such as $|\kappa v_s/A_\lambda|\sim 0.1$ and $|\beta/\kappa v_s|\sim O(1)$, the pattern is checkable by design: parameter regions with larger $A_\lambda$ should show a weaker $\phi_4$ bound, and repeating the scan while varying that coupling would test the prediction.
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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

2 major / 6 minor

Summary. The manuscript revisits the electron EDM in the general (non-Z3) NMSSM with explicit CP violation. It constructs a basis of 14 rephasing-invariant CP-violating phases, identifies three classes of two-loop diagrams (charged-Higgs loop, W-boson loop, and Kite diagrams) that were not included in earlier NMSSM EDM calculations, and adapts the analytic 2HDM results of Ref. [46] to the NMSSM via a set of substitution rules. The numerical analysis, based on a benchmark point and a scan over the phases phi3, phi4, phi5, shows that the new diagrams dominate the electron EDM for these phases, tightening the bound on phi4 by roughly an order of magnitude relative to phi3 and phi5, while for phi'_0 and phi6 they partially cancel the previously known MSSM two-loop contributions. The paper concludes that the allowed NMSSM parameter space for electroweak baryogenesis is materially affected by these new contributions.

Significance. If the numerical results are correct, the paper provides a nontrivial correction to the NMSSM electron EDM literature and has direct implications for electroweak baryogenesis model-building. The identification of the missing diagrams is topologically well-motivated: the NMSSM scalar sector has tree-level CP violation, so the 2HDM-type diagrams must be present, and the adaptation from the type-II 2HDM is plausible. The paper also gives a clear basis of invariant phases. The main limitation is that the quantitative constraints are derived at parameter points whose vacuum stability and collider viability are verified only in the CP-conserving limit, so the specific phase limits in Figs. 5 and 6 should be treated with caution until that gap is closed. The paper is not circular: the EDM bound is an external constraint, and no parameter is tuned to reproduce it.

major comments (2)
  1. [Section 3, first paragraph and Figs. 4–6] The paper states that 'we confirm the viability of our benchmark points with NMSSMTools in the CP-conserving limit,' but the electron EDM constraints in Figs. 4–6 are evaluated at CP-violating values of the phases (e.g., sin(phi)=0.01 in Fig. 4 and the scan over sin(phi3,4,5) in Fig. 6). No check of Landau poles, false vacua, or collider exclusions is reported for those CP-violating points. Since the central quantitative conclusion is the revised bound |sin(phi4)| ≲ 0.05 and the claim that phi4 is an order of magnitude more constrained than phi3 or phi5, this gap is load-bearing: if a substantial fraction of the scan points have tachyonic scalars or deeper minima, the allowed regions shown in Fig. 6 would not correspond to physical NMSSM vacua. Please repeat the NMSSMTools checks (or an equivalent vacuum-stability and Landau-pole check) at the CP-violating points, at minimum for the benchmark and for a representative sample of the accepted and rejected scan points, and either restrict the scan to viable points or quantify how many points are affected.
  2. [Introduction and Section 2, with references to Refs. [30,36,37]] The paper's central novelty claim is that the three two-loop diagram classes (d_E^f)^{H±}, (d_E^f)^{W±}, and (d_E^f)^{Kite} 'have been overlooked in the literature.' However, the paper does not explicitly inventory which of these diagrams are present or absent in the most complete previous NMSSM EDM calculations, in particular Ref. [37], which is cited only in the context of cancellations in the electron EDM. To make the novelty claim verifiable, please add a short comparison (either a table or a few sentences) listing the diagram topologies included in Refs. [30,36,37] and any other relevant prior work, and state explicitly which of the three classes are new relative to each.
minor comments (6)
  1. [Section 2.1, paragraph after Eq. (3)] The counting of the tadpole conditions, especially the statement that ∂V/∂a_d and ∂V/∂a_u are degenerate, is presented in a single sentence; a short explanation of how the three imaginary-component tadpole equations reduce to two independent conditions and how φ7 and φ8 are determined would make the basis of 14 phases more transparent.
  2. [Section 3, paragraph on Fig. 5 and Eq. (8)] The phase convention in the numerical study is ambiguous: Eq. (8) sets φ'_0 = φ3 = φ5 = π, but the text says 'setting each phase to a non-zero value of sin(phi)=0.01.' Please state explicitly whether this means φ_i = arcsin(0.01) with cos(φ_i) > 0 or φ_i = π + δ with sin(δ) = 0.01 (with either sign), because the sign of cos(φ_i) affects the Higgs couplings and therefore the EDM.
  3. [Appendix A, Eqs. (A.3)–(A.5)] The derivation of the adapted couplings is compressed: in particular, the definition of the neutral-scalar rotation matrix O and the treatment of the VEV phases e^{iφ_u}, e^{iφ_d}, e^{iφ_s} in the Yukawa couplings are not spelled out. A brief statement of these conventions would make the substitution rule (A.5) checkable and would help readers apply the results in other bases.
  4. [Fig. 4 caption] The caption says 'The lengths of bars are shown in unit of dExp_e,' but the horizontal axis has values from -10 to 10; please state explicitly that the bar lengths are in units of the experimental bound dExp_e and clarify whether the 'Total' bar is the algebraic sum of the three displayed contributions.
  5. [Fig. 5 caption] The caption says 'The gray region denotes the current experimental bound,' but the plot shows signed values of d_e; please specify that the bound applies to |d_e| and give the numerical value (4.1 × 10^{-30} e cm).
  6. [Throughout] There are several typographical artifacts in the text, such as 'o ffers' in the abstract and 'e ffective' in Section 1; please run a spell-check and correct these and similar LaTeX/OCR artifacts.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the EDM comparison is an external check and the two-loop formulas are imported from independent published 2HDM calculations.

full rationale

The paper's derivation chain is self-contained with respect to circularity. The central numerical quantity is the electron EDM, which is compared against the experimental bound d_e <= 4.1e-30 e cm from Ref. [16]; no parameter is fitted to reproduce that bound. The new 2HDM-type diagrams (d_E^f)^{H±}, (d_E^f)^{W±}, and (d_E^f)^{Kite} are obtained by adapting Ref. [46]'s gauge-invariant two-loop calculation via the substitutions of Eq. (A.5), an external published calculation, not by a self-citation chain or by defining the phases in terms of the EDM. The phases phi3, phi4, and phi5 are defined independently in Eq. (5), and their relative constraints (e.g., phi4 stronger than phi3 by |kappa v_s/A_lambda| ~ O(0.1)) follow from the quoted coupling expressions in Eq. (A.4), not from fitting. The only self-referential element, Ref. [56] ('in preparation'), is a forward pointer to future work and carries no load in the derivation. The limitation that benchmark points are checked with NMSSMTools only in the CP-conserving limit while EDM constraints are evaluated at CP-violating points is a physical-validity and correctness concern about the scanned parameter space, not a circularity: it does not make any predicted quantity equal to an input by construction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central calculation rests on imported 2HDM two-loop integrals, a tree-level Higgs potential, heavy-sfermion suppression, and an unverified extension of CP-conserving vacuum validation to the CP-violating parameter space. No new particles or fields are introduced. The benchmark and scan parameters are hand-chosen inputs, not fitted outputs, but they control the quantitative size of the reported phase constraints.

free parameters (3)
  • Benchmark point BP hyperparameters (Eq. 6-7) = tan(beta)=2.11, lambda=0.65, kappa=0.2, v_s=494 GeV, alpha=6.34e4 GeV^2, beta=200 GeV, M2=2M1=200 GeV, M3=2.5 TeV…
    Hand-picked representative NMSSM point used for all numerical EDM constraints in Section 3. The quantitative bounds depend on this choice and are not fitted to data.
  • Scan ranges for phi3, phi4, phi5 and Higgs-sector parameters (Eq. 9) = sin(phi3,4,5) in [-0.1, 0.1]; lambda in [0.5, 0.7], kappa in [0.1, 0.3], v_s in [200, 500] GeV, sqrt(alpha) in [200…
    Sampled ranges chosen by hand. The shape of the allowed regions in Fig. 6 depends on these ranges, not on a data-driven prior.
  • Phase benchmark settings (Eq. 8) = Phi1,2,3,4,5 = phi0,1,2,4,6,8,9 = 0; phi0' = phi3 = phi5 = phi7 = pi; single-phase scans set sin(phi) = 0.01
    Central values and scan amplitude are illustrative inputs that set the scale of the EDM bars in Figs. 4 and 5. The relative ranking of phase constraints is tied to these settings.
assumptions (6)
  • domain assumption The two-loop EDM integrals from the complex 2HDM calculation of Ref. [46] are correct and can be adapted to the NMSSM via the substitution rules (A.5).
    The paper does not re-derive the integrals; it imports them from Ref. [46] and maps NMSSM couplings onto 2HDM quantities. The substitution requires the NMSSM Higgs sector to match the type-II 2HDM structure.
  • domain assumption The tree-level Higgs potential and tadpole equations are sufficient to determine the CP-violating Higgs spectrum and couplings used in the EDM evaluation.
    Section 3 states that the analysis treats the Higgs potential at tree level. Radiative corrections to masses and mixings in the CP-violating case are not included.
  • domain assumption NMSSMTools validation in the CP-conserving limit implies that the CP-violating benchmark and scan points are physically viable, with no Landau poles, false vacua, or collider exclusions.
    The paper only checks viability in the CP-conserving limit. The assumption that non-zero CP phases do not invalidate these points is not tested.
  • domain assumption Sfermion loops inside the Kite and charged-Higgs diagrams are negligibly suppressed by heavy sparticle masses.
    Section 2.2 states this suppression and neglects sfermion contributions, but no numerical demonstration of the suppression size is provided.
  • standard math The global U(3)^5, U(1)_PQ, U(1)_R, and U(1)_S symmetries can be used to reduce the CP-violating phases to a basis of 14 independent physical phases.
    This is a standard field-redefinition argument following Refs. [42, 43], used to define the phase basis in Section 2.1.
  • standard math Perturbative QFT and gauge invariance ensure that the one-loop MSSM, two-loop MSSM, and 2HDM-type diagram groups are separately gauge invariant.
    Invoked in Section 2.2 to justify summing diagram groups; the gauge-invariance cancellations are stated rather than demonstrated in the paper.

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Cite this review

Pith. "Pith review of Revisiting the Electron EDM in the NMSSM." pith.science (2026). https://pith.science/paper/BONOZRQ5

@misc{pith2026250706320,
  author       = {Pith},
  title        = {Pith review of: Revisiting the Electron EDM in the NMSSM},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BONOZRQ5}},
  note         = {Machine review of arXiv:2507.06320}
}
read the original abstract

The Next-to-Minimal Supersymmetric Standard Model (NMSSM) with explicit CP violation offers a promising framework for explaining the observed baryon asymmetry while remaining consistent with stringent electric dipole moment (EDM) bounds. In this work, we identify a previously overlooked set of two-loop diagrams that contribute to fermion EDMs in the NMSSM. Our analysis of the electron EDM shows that, for certain NMSSM specific CP-violating phases, these diagrams partially cancel existing contributions, thereby relaxing the associated constraints. In other phases, the diagrams dominate and tighten the bounds, which in turn necessitates more finely tuned parameters to reconcile successful baryogenesis with current EDM limits.

Figures

Figures reproduced from arXiv: 2507.06320 by the authors.

Figure 1
Figure 1. One-loop diagrams contributing to fermion EDM in the MSSM; left: [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Two-loop diagrams contributing to fermion EDM in the MSSM; [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Two-loop diagrams contributing to fermion EDMs in the background [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: Magnitudes of electron EDM (in unit of e cm) induced by MSSM [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: Parameters points that pass the limit of electron EDM. Red and green [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

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