Probing Neutrinophilic Scalars in Muon Decays

Yongchao Zhang; Zhong Zhang

REVIEW

Probing Neutrinophilic Scalars in Muon Decays

Yongchao Zhang, Zhong Zhang

High Energy Physics - Phenomenology · Submitted 12 Sep 2026

Pith's verdict: Solid, thorough phenomenology on neutrinophilic scalars from muon decays; the main novelty is showing that GF constraints dominate for mφ > O(GeV).

T0 · AI review

How this review was made: An AI referee (deepseek/deepseek-v4-flash) read the paper on September 15, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 1 major objection · 7 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv ID2609.13964 (version 2)

Pith ID5OALF3VX, Pith's permanent record of this paper and its review

CitationYongchao Zhang, Zhong Zhang. "Probing Neutrinophilic Scalars in Muon Decays." arXiv:2609.13964 (2026). https://pith.science/paper/5OALF3VX

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

This paper does what it sets out to do: compute loop and four-body contributions to muon decay from a neutrinophilic scalar φ, combine them into an IR-safe inclusive width, and derive constraints from the difference between GF measured in muon decay and from electroweak fits or CKM unitarity. The main result — that for mφ ≳ 1 GeV the GF bound on |heμ| is the strongest existing limit — is new and clearly presented. The comparison with meson decays, EWPD, IceCube, CMB, and BBN is comprehensive and honestly labeled where approximations are made (flavor-universal assumptions for some bounds). The paper is well organized and the calculations are standard, using Package-X and the KLN theorem for IR cancellation.

The stress-test concern about the IR cancellation not being explicitly verified is minor. The paper states the cancellation and references the same mechanism in τ decay and charged meson decays; there is no reason to doubt it works here. A dedicated appendix showing the cancellation explicitly would strengthen the paper, but its absence is not a flaw — the logic is clear and the underlying physics is well established. The soft spot I see instead is that the central constraint relies on a 2σ difference between two GF values that could shift with future precision data. That is acknowledged by the authors, who show the preferred region as a band, and the conservative bound uses the 2σ level. This is appropriate for a limits paper.

I would send this to peer review. The paper is a competent piece of phenomenology that fills a gap in the literature. The authors know the existing bounds and are careful about overclaiming: they identify where their limits are approximate and where flavor-specific studies would be needed. The manuscript is clearly written and the figures are readable. I would recommend publication after a minor comment suggesting the IR cancellation be shown explicitly, e.g., by displaying the mφ-dependent terms in the sum of Eqs. (8) and (11).

Referee report

Minor objections (7)

[Abstract and Introduction] The abstract states the coupling |h_{eμ}| is constrained down to 'roughly 0.027' and the mass up to 'roughly 5.9 TeV'. These numbers appear in the text but the exact definition of the contour (e.g., 2σ C.L. limit) should be stated in the abstract.
[Eq. (7)] The approximation in Eq. (7) assumes m_ϕ ≪ m_W. However, the analysis later considers m_ϕ up to TeV scales. For m_ϕ ~ m_W or larger, this approximation breaks down. The authors should comment on the validity of the loop integral expression for m_ϕ > m_W and whether their results are reliable in that regime.
[Section V.A, paragraph below Eq. (14)] The text says 'the corresponding constraints are expected to be stronger' when using G^{EW}_F instead of G^{CKM}_F. It would be helpful to quantify this expectation or show the comparison in a plot.
[Fig. 5 caption and text] The BBN limit is stated as m_ϕ > 1.3 MeV from Ref. [106]. The shaded region in Fig. 5 seems to cover m_ϕ below ~1 MeV. Please clarify whether the BBN constraint excludes all m_ϕ below that value regardless of coupling, or if the region shown is for the specific coupling range. Also, the vertical band for BBN should be labeled as an excluded region (e.g., 'excluded by BBN') for clarity.
[p. 9, second paragraph] Typo: 'which are are presented as the gray shaded regions' – remove duplicate 'are'.
[Section V.C] The EWPD limits from Ref. [14] are described as 'bottom right panel in Fig. 1' for flavor-universal couplings. Since the coupling h_{eμ} is flavor-specific, the applicability of this constraint is approximate. A brief discussion of how the EWPD constraint might change for LFV couplings (e.g., via the PMNS matrix) would strengthen the comparison in Fig. 5.
[Section V.A, Eq. (15)] The right-hand side of Eq. (15) uses G_μ^F m_μ^5/(96π^3) times the difference in G_F values. The derivation of this expression from the relation between muon width and Fermi constant should be referenced or briefly explained.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

The Fermi constant from muon lifetime provides the strongest laboratory constraint on neutrinophilic scalars with masses above a GeV, excluding the lepton-flavor-violating coupling h_{eμ} down to 0.027 and scalar masses up to 5.9 TeV.

The reading

This paper studies how a hypothetical scalar particle that couples primarily to neutrinos (a neutrinophilic scalar) would affect muon decays. The most sensitive processes are a 1-loop correction to the standard muon decay μ→eνν and the associated four-body decays μ→eννφ, both induced by a lepton-flavor-violating coupling h_{eμ} to Majorana neutrinos. Infrared divergences in these processes cancel when the widths are summed, yielding a finite inclusive decay width that depends logarithmically on the scalar mass. By comparing the precisely measured Fermi constant G_F from muon decay with independent determinations from electroweak fits and CKM unitarity, the coupling |h_{eμ}| is constrained below 0.027 and the scalar mass m_φ below 5.9 TeV. For m_φ ≳ GeV, these bounds are stronger than any existing laboratory, astrophysical, or cosmological limit, making muon decay the most sensitive probe of this class of models.

What carries the argument

The central mechanism is the cancellation of infrared divergences between the virtual 1-loop correction to μ→eνν and the tree-level four-body decays μ→eννφ, enforced by the Kinoshita-Lee-Nauenberg theorem. This cancellation produces a physical inclusive width proportional to |h_{eμ}|^2 multiplied by a logarithm log($m_φ^{2}$/$m_W^{2}$), which becomes sensitive to high scalar masses through the running of the effective coupling. The key object is the lepton-flavor-violating coupling h_{eμ} of the scalar to Majorana neutrinos in the effective Lagrangian ℒ_M = -½ h_{αβ} φ ν^c_α ν_β + H.c. The comparison of three independent determinations of the Fermi constant (from muon decay, from electroweak fits, and from CKM unitarity) provides the experimental lever arm.

Extended reading notes

Core claim

The paper establishes that the 1-loop correction of a neutrinophilic scalar φ to the Standard Model decay μ→eνν, combined with the real emission channels μ→eννφ, leads to an infrared-safe inclusive decay width whose expression contains a logarithmic dependence on m_φ. Using the 2σ tension between G_F from muon decay and from the global electroweak fit (which is larger by about 0.07%), the authors derive the constraint |h_{eμ}| < 0.027 for scalar masses above a few MeV, and m_φ ≲ 5.9 TeV if |h_{eμ}| is of order one. For m_φ ≳ 1 GeV, this G_F constraint is more stringent than all other existing limits from charged meson decays, electroweak precision data, IceCube neutrino observations, cosmic microwave background, and big bang nucleosynthesis. In contrast, the three-body decay μ→eφφ, being proportional to the fourth power of the coupling, yields only a weak constraint g_eff ≲ 0.4.

Load-bearing premise

The analysis assumes that the scalar φ interacts only with neutrinos via the effective Lagrangians in Eq. (1a,b) and that no other new physics contributes to the muon decay width; if φ has additional couplings (e.g., to charged leptons or quarks) or if other BSM sources modify the muon width, the constraints on h_{eμ} could be altered.

Editorial extensions

If this is right

The G_F measurement from muon decay now gives the strongest laboratory bound on the LFV coupling h_{eμ} for scalar masses above roughly 1 GeV, improving on previous EWPD limits by a factor of about 5 at m_φ = 1 GeV.
Scalar masses up to 5.9 TeV can be probed indirectly through radiative effects on muon decay, far exceeding the reach of direct production at colliders for this coupling strength.
The ∼2σ tension between G_F from muon decay and from the global electroweak fit can be alleviated by the combined contribution of the φ-loop and four-body decays, with a preferred parameter space shown as a green band in Fig. 5.
Exotic muon decays such as μ→eφφ are much more weakly constrained (g_eff ≲ 0.4) and are subdominant to other bounds, confirming that the main discovery window is the inclusive h_{eμ} channel.
Future high-precision measurements of the muon lifetime and Michel parameters at experiments like Mu3e can further tighten the constraint on |h_{eμ}| and potentially detect the four-body final states.

Reading between the lines

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

If the current 2σ tension in G_F determinations persists in future data, the h_{eμ} coupling region preferred by the tension (green band in Fig. 5) becomes a concrete target for direct searches of μ→eννφ at Mu3e or MEG II, which could identify the four-body final state through the electron energy spectrum.
The analysis assumes no other new physics modifies the muon decay width. If additional BSM sources exist (e.g., a charged scalar also coupling to leptons), the constraints on h_{eμ} could be relaxed or strengthened depending on the sign of interference with the SM.
This work highlights that muon decays, historically used to define G_F, are themselves powerful probes of lepton-flavor-violating neutrino interactions—a role often overshadowed by meson decays or collider searches. The inclusive approach (virtual + real emission) is essential to obtain finite predictions.
Future high-intensity muon facilities (Mu3e, Mu2e, MACE) can improve the sensitivity to h_{eμ} by one to two orders of magnitude, both via refined G_F determination and by directly searching for the invisible scalar in four-body decays.

Original abstract

The non-standard self-interactions of neutrinos could be induced by (light) scalars that couple primarily to neutrinos. In this work, we investigate the effects of neutrinophilic scalars $\phi$ on muon decays at the tree, 1-loop and 2-loop levels, considering both Dirac and Majorana neutrinos. The most phenomenologically interesting processes are the 1-loop decay $\mu \to e \phi \phi$ and $\phi$-induced decay $\mu \to e \gamma$ at the 2-loop order from the same coupling $g_{\rm eff}$, and the 1-loop corrections of $\phi$ to the SM muon decay and the four-body muon decays $\mu^- \to e^- \nu_\mu \nu_\mu \phi,\, e^- \bar\nu_e \bar\nu_e \phi$, both induced by the coupling $h_{e\mu}$ of $\phi$ to Majorana neutrinos. The current experimental bounds on $\mu \to e \gamma$ provide the most stringent limits on the coupling $g_{\rm eff}$, up to roughly 0.064, for the scalar mass $m_\phi \gtrsim 8.7$ MeV. The precise $G_F$ measurements constrain the coupling $|h_{e\mu}|$ down to roughly $0.027$, with the scalar mass $m_\phi$ constrained up to roughly $5.9$ TeV. For $m_\phi \gtrsim {\cal O}({\rm GeV})$, the $G_F$ measurements are more stringent than other existing laboratory, astrophysical and cosmological constraints. Some other $\phi$-induced processes are also possible, which are, however, heavily suppressed by the tiny neutrino masses and/or the loop factor.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,905 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.61 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.88 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The non-standard self-interactions of neutrinos could be induced by (light) scalars that couple primarily to neutrinos. In this work, we investigate the effects of neutrinophilic scalars $\phi$ on muon decays at the tree, 1-loop and 2-loop levels, considering both Dirac and Majorana neutrinos. The most phenomenologically interesting processes are the 1-loop corrections of $\phi$ to the SM muon decay and the four-body muon decays $\mu^- \to e^- \nu_\mu \nu_\mu \phi,\, e^- \bar\nu_e \bar\nu_e \phi$, both induced by the coupling $h_{e\mu}$ of $\phi$ to Majorana neutrinos. The infrared divergences in these processes cancel with each other, yielding an infrared-safe inclusive decay width. The precise $G_F$ measurements constrain the coupling $|h_{e\mu}|$ down to roughly $0.027$, with the scalar mass $m_\phi$ constrained up to roughly $5.9$ TeV. For $m_\phi \gtrsim {\cal O}({\rm GeV})$, the $G_F$ measurements are more stringent than other existing laboratory, astrophysical and cosmological constraints. Some other $\phi$-induced processes are also possible, which are, however, heavily suppressed by the small couplings, tiny neutrino masses and/or the loop factor.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The non-standard self-interactions of neutrinos could be induced by (light) scalars that couple primarily to neutrinos.	background 1.00	0.05	0.03	hedged possibility 1.00
s1	In this work, we investigate the effects of neutrinophilic scalars $\phi$ on muon decays at the tree, 1-loop and 2-loop levels, considering both Dirac and Majorana neutrinos.	contribution 1.00	0.90	0.05	asserted 0.87
s2	The most phenomenologically interesting processes are the 1-loop corrections of $\phi$ to the SM muon decay and the four-body muon decays $\mu^- \to e^- \nu_\mu \nu_\mu \phi,\, e^- \bar\nu_e \bar\nu_e \phi$, both induced by the coupling $h_{e\mu}$ of $\phi$ to Majorana neutrinos.	contribution 0.86	0.77	0.13	asserted 0.98
s3	The infrared divergences in these processes cancel with each other, yielding an infrared-safe inclusive decay width.	result 0.90	0.82	0.05	asserted 1.00
s4	The precise $G_F$ measurements constrain the coupling $\|h_{e\mu}\|$ down to roughly $0.027$, with the scalar mass $m_\phi$ constrained up to roughly $5.9$ TeV.	result 0.99	0.61	0.62	asserted 0.99
s5	For $m_\phi \gtrsim {\cal O}({\rm GeV})$, the $G_F$ measurements are more stringent than other existing laboratory, astrophysical and cosmological constraints.	interpretation 0.55	0.62	0.28	asserted 1.00
s6	Some other $\phi$-induced processes are also possible, which are, however, heavily suppressed by the small couplings, tiny neutrino masses and/or the loop factor.	result 0.50	0.51	0.04	hedged possibility 0.67

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.63.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Feynman diagrams for the 1-loop three-body decay

**Figure 2.** FIG. 2. Feynman diagrams for the 1-loop decay

**Figure 3.** FIG. 3. Feynman diagrams for the four-body muon decays

**Figure 4.** that the GF constraint and the preferred regions associated with the mild tension between G µ F and GCKM F are excluded by other laboratory, astrophysical and cosmological limits in this figure. Such a conclusion applies also to the corresponding case with Majorana neutrinos, as mentioned in Section III. B. Limits on |heµ| The constraint on the scalar mass mϕ and the coupling |heµ| from the precise measurements of G µ F and GEW F at 2σ C.L. in Eq. (15) is shown as the red shaded region in

**Figure 6.** FIG. 6. Feynman diagram for the 1-loop two-body decay

**Figure 7.** FIG. 7. Feynman diagrams for muonium–antimuonium oscil

**Figure 8.** FIG. 8. Feynman diagrams for the 2-loop decay

Showing 7 of the paper's 8 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2609.13964 by the authors.

References

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.