On the Run from the Dark Side of the Muon

We present an analysis strategy for probing physics beyond the Standard Model via modifications to the parton distribution functions (PDFs) in a muon beam, which measurably alter the kinematics of all hard processes at a future muon collider. High-energy muon colliders represent an opportunity to probe new physics using precision measurements and novel search strategies. At sufficiently high energies, light particles act as ``constituents’’ of the muon described by PDFs. As a concrete case study, we apply this framework to an $L_μ - L_τ$ gauge boson and demonstrate that, for masses in the range of approximately 50–100 GeV, this indirect PDF-based approach outperforms traditional searches relying on direct gauge boson production. These results highlight muon PDF probes as a powerful and promising avenue for beyond the Standard Model physics searches at a future muon collider.

💡 Research Summary

The paper proposes a novel indirect search strategy for physics beyond the Standard Model (BSM) at a future high‑energy muon collider (MuC) by exploiting modifications to the muon’s parton distribution functions (PDFs). While muons are elementary particles, at energies far above their mass they radiate collinear gauge bosons and fermions, which can be treated as partonic constituents described by perturbatively calculable PDFs. The authors argue that new light particles, such as a gauge boson associated with a U(1){Lμ‑Lτ} symmetry (denoted Z′), will alter the DGLAP evolution equations governing these PDFs. Specifically, the splittings μ→μ+Z′ and Z′→f\bar{f} (with f = μ, τ, ν_μ, ν_τ) introduce a Z′ PDF that feeds back into the SM PDFs, enhancing the muon PDF and suppressing the photon PDF in the regime M{Z′}^2 ≪ Q^2.

To quantify the effect, the authors modify the public LePDF code to numerically solve the coupled DGLAP system including the Z′ contributions. They present the resulting PDFs at a representative factorization scale and illustrate percent‑level deviations from the SM for the muon and photon. These PDF changes translate into measurable distortions of the τ distribution, where τ ≡ \hat{s}/s = x_1 x_2 is the ratio of partonic to collider center‑of‑mass energy. The τ spectrum is directly observable as the invariant‑mass distribution of the final state.

Four clean final states are selected for analysis: μ⁺μ⁻, γγ, μγ, and e⁺e⁻. The choice is motivated by (i) high reconstruction efficiency, (ii) availability of analytic tree‑level amplitudes (since current generators do not support polarized PDFs), and (iii) dominance of initial‑state partons (μ, γ, μγ) that are most sensitive to PDF shifts. For each channel the authors compute the differential cross section dσ/dτ by convolving the partonic matrix elements with the modified PDFs, using Q = √\hat{s}/2 as the hard scale.

Statistical inference is performed with an extended log‑likelihood function for each channel, treating the observed τ values as Poisson‑distributed events. The total log‑likelihood is the sum over the four channels, allowing a combined fit that exploits the shape information of the τ spectra rather than overall normalizations. Systematic uncertainties are incorporated by introducing the integrated luminosity (assumed 10 ab⁻¹) as a nuisance parameter; its impact is found to be negligible, confirming that the analysis is driven by the spectral shape.

The resulting projected sensitivity shows that for Z′ masses between roughly 50 GeV and 100 GeV, the PDF‑based method can exclude gauge couplings g′ down to about 0.02 at 95 % confidence level. This surpasses existing constraints from ATLAS and CMS searches for pp → μ⁺μ⁻μ⁺μ⁻ (where Z′ is radiated off a final‑state muon) and from neutrino‑trident measurements (CCFR). Compared with direct searches at a 10 TeV muon collider in the ν\barνγ and τ⁺τ⁻γ channels, the indirect PDF approach provides the strongest limits in the 50–100 GeV window, while the τ⁺τ⁻γ channel dominates at higher masses.

The authors emphasize that this strategy inverts the traditional electroweak precision program: instead of measuring small deviations in well‑understood observables, one leverages the high‑energy partonic structure of the lepton beam to probe light, weakly‑coupled new physics. The method is systematically improvable: higher‑order perturbative calculations can reduce theory uncertainties in the PDFs and hard scattering, and dedicated event generators with electroweak parton showers and polarized PDFs will enable more realistic simulations. Detector considerations such as magnetic field strength for multi‑TeV muon momentum measurement and resolution effects must also be addressed in future work.

In summary, the paper demonstrates that muon‑PDF distortions constitute a powerful, complementary probe of BSM physics at future lepton colliders, offering sensitivity to regions of parameter space that are challenging for conventional direct searches.