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NA64μ, a muon-beam missing-energy experiment, sets the first direct bounds on two unbounded four-lepton SMEFT operators and three νSMEFT operators, and can break a flat direction in muon global fits.

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2026-08-03 22:09 UTC pith:XCST3X3C

load-bearing objection Useful first recast of NA64mu data to muonic four-fermion operators, with a real caveat: the numerical bounds depend on an unvalidated transfer of the dark-photon acceptance function. the 2 major comments →

arxiv 2511.11801 v2 pith:XCST3X3C submitted 2025-11-14 hep-ph

First bounds on effective muon interactions using the NA64μ experiment at CERN

Paolo Crivelli , Josu Hernandez-Garcia , Jacobo Lopez-Pavon , Victor Martin Lozano , Laura Molina Bueno This is my paper
classification hep-ph
keywords SMEFTνSMEFTNA64μmissing-energy searchesfour-fermion operatorsmuon–nucleus scatteringheavy neutral leptonsWeizsäcker–Williams approximation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates that the NA64μ experiment — a 160 GeV muon beam impinging on a lead target, looking for invisible particles via missing energy and momentum — provides the first direct experimental constraints on a class of four-fermion effective interactions involving muons that were previously unbounded. These include two neutral-current four-lepton operators of the Standard Model Effective Field Theory (SMEFT), which no other experiment has constrained directly, and three operators of νSMEFT, the extension of SMEFT that adds fermions that are singlets of the Standard Model gauge group, such as heavy neutral leptons and dark-sector states. Using the published 1.98×10¹⁰ muons-on-target dataset with no candidate events, the authors compute the signal rate with the Weizsäcker–Williams photon-flux approximation and derive 90% CL bounds of about 10⁻² GeV⁻² on the Wilson coefficients of these operators, with projected sensitivities of about 10⁻⁴ GeV⁻² — new-physics scales near 100 GeV — for the planned 10¹⁴-MOT upgrade. If the reinterpretation holds, the results break the four-muon flat direction of neutrino-trident global fits, making the individual four-muon operators separately measurable for the first time, and they bridge the gap between today's low-energy searches and a future muon collider.

Core claim

Reinterpreting the first NA64μ dataset (1.98×10¹⁰ muons on target, zero candidates) as an EFT search, the paper computes the μN → μNχ₁χ₂ cross section in the Weizsäcker–Williams approximation and convolves it with the acceptance κ(k²) via Eq. (22). Requiring at most 2.44 signal events at 90% CL, it derives the first direct bounds on the NC-SMEFT operators [OLL]μμαα and [OLℓ]ααμμ — |[c]|/v² ≤ 1.0×10⁻² GeV⁻² today, 1.4×10⁻⁴ GeV⁻² at the upgrade — and on the νSMEFT operators Oμμ, Oμ_LN and Oμμ_LNLℓ, at (0.5–1.5)×10⁻² GeV⁻² today and ~10⁻⁴ GeV⁻² projected, for m_N ≲ 1.5–3 GeV. With the trident bound on [ĉLL]μμμμ, the individual four-muon coefficients become separately bounded, breaking the flat

What carries the argument

The load-bearing device is the factorized cross section of Eq. (14): dσ(μN → μNχ₁χ₂) = dσ^{2→3}_{μν} |c/Λ²|² (dk²/2π) ξ_{μν}, where ξ_{μν} is the integrated two-body current of the invisible neutral-fermion pair — computable in closed form — and dσ^{2→3}_{μν} is the muon–nucleus bremsstrahlung process in the Weizsäcker–Williams approximation (nucleus scattering treated as scattering off a flux of quasi-real photons). The experiment enters through κ(k): the geometric acceptance as a function of the invisible-pair invariant mass squared, k² = (k₁ + k₂)², taken from the NA64μ dark-photon Monte Carlo and assumed to describe the EFT signal. The signal count, Eq. (22), is NS = NMOT (ρ_N/m_N) Lᵉᶠᶠ_

Load-bearing premise

The detector acceptance κ(k) — computed by Monte Carlo for an on-shell dark-photon mediator — is assumed, without dedicated simulation or assigned systematic uncertainty, to describe the EFT signal and to depend only on the invisible-pair invariant mass k²; the Table I bounds stand or fall on that transfer.

What would settle it

A dedicated Monte Carlo of the contact-operator process μN → μNχ₁χ₂ (continuous invariant-mass pair), run through the NA64μ detector response to obtain an EFT version of κ(k), and a recomputation of the 90% CL limits of Table I using it in Eq. (22) in place of the on-shell dark-photon acceptance: if the resulting bounds shift substantially from 1.0×10⁻² GeV⁻² (current) and 1.4×10⁻⁴ GeV⁻² (upgrade), the paper's numerical claims are not robust.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Two previously unbounded neutral-current SMEFT operators — [OLL]μμαα and [OLℓ]ααμμ, with α = μ or τ — now carry 90% CL bounds |[c]|/v² ≤ 1.0×10⁻² GeV⁻² from the published 1.98×10¹⁰-MOT run, projected to reach 1.4×10⁻⁴ GeV⁻² (Λ ≳ 83 GeV) with the 10¹⁴-MOT upgrade.
  • The four-muon flat direction of neutrino-trident global fits (Eq. 7) can be broken by combining NA64μ with the existing trident constraint, bounding the individual coefficients at |[cLL]μμμμ|/v² ≲ 5.0×10⁻⁵ and |[cLℓ]μμμμ|/v² ≲ 1.4×10⁻⁴ GeV⁻², so the two muon operators become separately measurable.
  • Three νSMEFT operators that couple muons to SM-singlet fermions N — candidate heavy neutral leptons or dark-sector states — are bounded for m_N ≲ 1.5–3 GeV, with current limits around (0.5–1.5)×10⁻² GeV⁻² and projected reach ~10⁻⁴ GeV⁻² (Λ ≳ 100 GeV).
  • The low-energy NA64μ probe covers the non-resonant, contact-interaction part of the parameter space that a future muon collider would access through on-shell production, making the two experimental programs complementary, as the paper notes in its outlook.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The bounds inherit the transfer of the dark-photon acceptance κ(k) to the EFT signal (Section V, after Eq. 22): a contact operator produces an invisible pair with a continuous invariant-mass spectrum and different angular and momentum distributions than an on-shell Z′. The paper asserts this transfer is valid but does not quantify its uncertainty; a dedicated EFT Monte Carlo of the acceptance coul
  • The same factorized cross-section machinery applies directly to NA64's electron running, which would probe the electron analogs of these operators with the same formulas; as the paper itself notes, LEP monophoton limits already dominate the electron case, so the muon channel is where this method adds genuinely new reach.
  • The νSMEFT reach dies off above m_N ≈ 1.5–3 GeV purely through phase-space closing (Eq. 21), so NA64μ constrains only light singlet fermions; combining these bounds with beam-dump or HNL-decay searches that cover higher masses would map the full (m_N, coefficient) plane for each operator.
  • For the νSMEFT operators the analysis is one-operator-at-a-time; if several were active simultaneously, their interference terms (the Cn coefficients of Eq. 20) could relax or sharpen the limits, so a simultaneous-fit version of the analysis would be a natural follow-up.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper reinterprets the NA64μ missing-energy search at CERN (1.98×10^10 muons on target, zero observed events) to set 90% CL limits on dimension-6 four-fermion operators that are otherwise poorly constrained: two SMEFT four-lepton operators [O_LL]^{μμαα} and [O_Lℓ]^{ααμμ}, and three νSMEFT operators involving a light singlet fermion N. The authors compute elastic μ-nucleus scattering cross sections for μN→μNχχ in the Weizsäcker-Williams approximation, convolve them with the NA64 acceptance function κ(k) taken from the dark-photon analysis of Ref. [12], and obtain current bounds and future sensitivities for 3.5×10^11 and 10^14 MOT. They report first direct bounds on some of these operators and argue that NA64μ can break the four-muon flat direction of Eq. (7).

Significance. If the numerical limits are robust, the paper delivers a genuinely new set of constraints: several four-lepton and νSMEFT operators listed in Table I are currently unbounded, so any direct measurement is valuable. The analytic cross-section formulas in Appendix A are explicit and the statistical treatment (NS≤2.44, Feldman-Cousins) is transparent. The paper does not fit any parameter to produce the limits; it uses a published null event count and fixed cross-section expressions, which is a strength. The main weakness is the transfer of the dark-photon acceptance κ(k) to EFT contact-operator signals; the central quantitative claims depend on this unvalidated step.

major comments (2)
  1. [Section V, Eq. (22) and Fig. 4] The acceptance function κ(k) is taken from the NA64 dark-photon Monte Carlo of Ref. [12], where it is an efficiency for an on-shell Z′ produced via bremsstrahlung, including the cuts on the scattered muon momentum (p<80 GeV) and veto activity. The paper argues that because NA64 is only sensitive to missing energy/momentum, the acceptance is a function of k²=(k1+k2)². This does not follow: for a contact operator the final state is a continuous-invariant-mass fermion pair, and for fixed k² the outgoing muon momentum and angle depend on the energy fraction x and on the operator structure. The acceptance should be weighted with the EFT differential distribution in x and k²; using the on-shell Z′ acceptance as a universal κ(k²) can bias the event rate in Eq. (22) in either direction. Since every bound in Table I and Eqs. (23)–(24) is derived from Eq. (22), a dedicated simulation of the EFT si
  2. [Section V, Eq. (20)–(22) and Appendix A] The acceptance transfer problem is compounded for the νSMEFT operators with finite HNL mass m_N. The functions β(k,m_N) and the C_n coefficients in Appendix A produce different angular and momentum distributions of the final-state muon as m_N varies. The dark-photon κ(k) was computed for a massless invisible decay product; extrapolating it to m_N up to ~2 GeV (as in Fig. 6) is an assumption that should be validated. At minimum, the authors should show that κ(k) is insensitive to the χχ angular distribution, or provide a quantitative estimate of the resulting uncertainty on the m_N-dependent bounds.
minor comments (4)
  1. [Section V, Eq. (23)] The bound in Eq. (23) contains a term −0.1 |[cLL][cLℓ]/v⁴|. The sign and coefficient of the interference term are not derived in the text; please state whether this is a conservative choice and how the 0.1 was obtained.
  2. [Fig. 4] The y-axis label "(k)" should be κ(k); also the legend could state more clearly that the magenta curve is the published 2022 acceptance, while the yellow and purple curves are expected projections.
  3. [Appendix A] The subsection titles "O_αLN and O_ff" use notation that is not introduced in the main text (O_ff appears as O_μμ in Eqs. (9)–(10)). Please align the notation with Eq. (9)–(11).
  4. [Table I] For the 1.98×10^10 MOT row, "Current NA64μ sensitivity" is in fact a 90% CL observed bound (magenta shaded region in Fig. 5). Consider renaming to "Current NA64μ bound" to distinguish it from the expected sensitivity of the 2023–2024 dataset.

Circularity Check

0 steps flagged

No significant circularity: the NA64mu bounds are derived from an external null event count and independent global-fit inputs; the dark-photon acceptance transfer is a physics assumption, not a circular reduction.

full rationale

The derivation chain is: (i) identify SMEFT/νSMEFT operators that are unbounded per independent global fits [8,9]; (ii) compute the EFT production cross section analytically in Eqs. (13)-(21) and Appendix A; (iii) combine with the published NA64mu null event count (N_S ≤ 2.44 at 90% CL) to set bounds via Eq. (22). No parameter is fitted to the data to produce the quoted limits; the null result and acceptance κ(k) are external experimental inputs, and the global-fit constraints used to claim operators are unbounded come from unrelated groups. The only potentially questionable step is importing the acceptance κ(k^2) computed for an on-shell dark photon mediator in Ref. [12] into the EFT signal. The paper justifies this by saying NA64 is only sensitive to missing energy/momentum. Whether this transfer is quantitatively justified is a physics/robustness question, not a circularity: the acceptance was computed externally and is not constructed from the EFT bounds it is used to derive. If the transfer is wrong, the numerical bounds could shift, but they would not be made true by definition. Likewise, Eq. (7) is not a prediction but an input from independent trident/global-fit literature. Therefore the central claim (first direct constraints on these operators) has independent content and no step reduces to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The central calculation assumes the WW approximation and NA64's dark-photon acceptance; no free parameters are fitted. The acceptance transfer is the most fragile input, and the nuSMEFT results also assume HNL invisibility and single-operator dominance.

axioms (5)
  • domain assumption Weizsacker-Williams equivalent photon approximation for mu N -> mu N chi1 chi2, Eq. (17)
    The 2->3 process is factorized into a photon flux and an on-shell subprocess. Standard in NA64 analyses but an approximation; its accuracy for contact operators is not quantified in this paper.
  • ad hoc to paper NA64 acceptance kappa(k) from dark-photon MC applies to EFT signals and depends only on k^2=(k1+k2)^2
    Stated after Eq. (22) and used in Eq. (22); no dedicated simulation for nu nu or N N final states is provided, and this is the most fragile input to the quoted bounds.
  • domain assumption Zero expected background and null observation: NS <= 2.44 at 90% CL
    Based on the NA64 2022 dataset with no candidate events; the recast ignores the published background estimate, which if non-zero would only strengthen the limits.
  • domain assumption Heavy neutral leptons N are effectively invisible in the nuSMEFT signal
    The signal is defined as missing energy carried by neutral particles; if N decays inside the detector, events would be vetoed. Lifetime and coupling dependence are not discussed.
  • domain assumption Dimension-6 truncation and neglect of d=5 operators and SM interference
    Stated in Sections II and IV; standard EFT practice, but the validity of the EFT at the quoted Lambda values near 10 GeV is not scrutinized.

pith-pipeline@v1.3.0-alltime-deepseek · 12078 in / 15018 out tokens · 146774 ms · 2026-08-03T22:09:50.193882+00:00 · methodology

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read the original abstract

We analyze how NA64$\mu$ can contribute to the global SMEFT program demonstrating that it can probe two effective four lepton operators completely unbounded so far and break one of the current flat directions. Furthermore, we also study an extension of SMEFT that includes fermion singlets of the SM gauge group in the low energy field content. This effective field theory, usually dubbed $\nu$SMEFT, is well motivated by the observation of light neutrino masses and leptonic mixing. We find that NA64$\mu$ can constrain three unbounded four fermion operators of the $\nu$SMEFT. We derive the current leading bounds on these operators and compute the future sensitivity. Our results fill the gap between the current experimental program and a possible future muon collider able to probe this type of New Physics.

Figures

Figures reproduced from arXiv: 2511.11801 by Jacobo Lopez-Pavon, Josu Hernandez-Garcia, Laura Molina Bueno, Paolo Crivelli, Victor Martin Lozano.

Figure 1
Figure 1. Figure 1: FIG. 1. Feynman diagrams for the two to four neutrino and HNL production processes at NA64 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic illustration of the NA64 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Feynman diagrams contributing to neutral fermion production in the EFT at NA64 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Geometrical acceptance on the signal efficiency as a function of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. 90% CL NA64 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. 90% CL NA64 [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

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Forward citations

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