On the Run from the Dark Side of the Muon

Abstract: 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.

• The paper develops a novel strategy exploiting muon PDFs to indirectly constrain BSM couplings via modifications of the perturbative DGLAP evolution.
• It analyzes a gauged Lμ-Lτ Z' boson case study, showing that its contributions can alter kinematic distributions by up to 1% in key observables.
• The study combines numerical resummation with a log-likelihood shape analysis to set exclusion limits that outperform traditional direct search methods.

Indirect Probes of New Physics via Muon PDFs at High-Energy Muon Colliders

Motivation and Context

This paper develops an analysis strategy for probing physics beyond the Standard Model (BSM) by exploiting modifications to parton distribution functions (PDFs) in high-energy muon beams, with implications for all hard processes at a future muon collider ("On the Run from the Dark Side of the Muon" (2602.16771)). Muon colliders are uniquely promising for BSM exploration due to their clean lepton environment and suppressed synchrotron losses at multi-TeV energies, allowing direct access to luminosity and center-of-mass regimes typically associated with hadron colliders.

The authors focus on the regime where, at energies much higher than the muon mass, collinear radiation of gauge bosons and leptons effectively endows the muon with a compositeness described by PDFs. BSM modifications to the DGLAP evolution equations can then leave indirect imprints in overall kinematic distributions—an avenue previously explored for dark photon searches in hadronic environments [McCullough et al., JHEP 08 (2022) 019]. Unlike traditional searches targeting direct production or off-shell effects, this approach enables indirect constraints on BSM couplings, with sensitivity below thresholds set by direct searches.

Formalism and BSM Case Study

The central premise is that, for factorization scales $Q \gg m_\mu$, PDFs for the muon and associated radiative partons (photon, $Z$, etc.) are calculable perturbatively, with boundary conditions anchored at $Q = m_\mu$. New physics alters the DGLAP evolution through additional splitting functions and couplings.

As a case study, the paper analyzes a gauged $L_\mu - L_\tau$ extension, introducing a $Z'$ vector boson with mass $M_{Z'}$ between 50–100 GeV and coupling $g'$. This $Z'$, at sufficiently high energies, becomes a partonic constituent, contributing to DGLAP evolution via splittings of type $f\to f+Z'$ and $Z'\to f+\bar{f}$ (for $f$ in $\{\mu^\pm, \tau^\pm, \nu_\mu, \nu_\tau\}$). The presence of the $Z'$ modifies both the muon and photon PDFs: the muon PDF is enhanced, while the photon PDF is depleted (the $Z'$ "steals" splitting probability), with induced changes propagating to kinematic distributions for all hard final states.

Perturbative accuracy is discussed, noting formal issues with massive partons' PDFs [Frixione et al., JHEP 12 (2023) 170] but emphasizing that in the regime $M_{Z'}^2 \ll Q^2$ collinear calculations remain reliable. The LePDF code [Garosi et al., JHEP 09 (2023) 107] is augmented to incorporate $Z'$ contributions, with proper inclusion of back-reaction effects in the numerically resummed DGLAP system.

Statistical Framework and Observable Design

The observable central to the analysis is the $\tau = x_1 x_2 = \hat{s}/s$ (where $x_{1,2}$ are beam momentum fractions and $\hat{s}$ is partonic center-of-mass energy), with the differential cross section $d\sigma/d\tau$ encoding the invariant mass spectrum. Final states are selected for maximal PDF sensitivity and practical reconstructibility: $\mu^+\mu^-$, $\gamma\gamma$, $\mu^\pm\gamma$, and $e^+e^-$. Cross sections are computed at tree level, convolved numerically over PDFs, and integrated over central detector rapidity cuts $|\eta| < 2.0, 2.5$.

PDF deviations induced by the $L_\mu - L_\tau$ $Z'$ reach up to $\mathcal{O}(1\%)$ in $d\sigma/d\tau$, determining the requisite systematic control for statistical sensitivity. Phase space is restricted to minimize nonperturbative and forward region corrections, and to ensure high detector efficiency.

A log-likelihood shape analysis is performed, combining all final states. Event yields are modeled as Poisson variables, $\tau$ distributions sampled, and systematics (notably luminosity uncertainty) included as nuisance parameters, profiled in the likelihood. Marginalization over normalization reveals that the shape of $d\sigma/d\tau$ is the dominant driver; deviation profiles are non-trivial and final state-dependent. Muon collider simulation studies are referenced for systematic uncertainties (Bartosik et al., 2022), and potential improvements (precision PDF matching, polarized PDFs, EW parton showers) are articulated.

Numerical Results and Comparison to Existing Constraints

Numerical exclusion contours are derived for the $L_\mu - L_\tau$ $Z'$ in $(g', M_{Z'})$ parameter space, with direct comparison to limits from ATLAS/CMS four-muon searches [CMS:2018yxg, ATLAS:2023vxg], neutrino trident production [Altmannshofer et al., Phys. Rev. D 89, 095033 (2014); CCFR:1991lpl], and direct muon collider projections (resonant $\tau^+\tau^-\gamma$ and monophoton channels) [Huang et al., Phys. Rev. D 103, 095005 (2021); Dasgupta et al., JHEP 12 (2023) 011].

For $M_{Z'}$ in the range $\sim$50–100 GeV, the indirect PDF-based approach provides the strongest projected sensitivity, surpassing both existing and prospective direct search strategies. The exclusion is driven by the distinctive shape deviations in $d\sigma/d\tau$ when the $Z'$ modifies the muon and photon PDFs.

Direct searches lose sensitivity for lower $Z'$ masses due to collimated decay products ($\tau^+\tau^-$ from light $Z'$ cannot be resolved), and systematic photon energy resolution limits in recoil searches. Further inclusion of additional final states or angular information in the statistical analysis could strengthen these projections.

Implications and Future Directions

This study demonstrates that indirect probes based on muon PDF modifications are a robust and theoretically controlled approach for BSM sensitivity at future high-energy muon colliders, particularly for scenarios with weak couplings and masses below direct production thresholds. Theoretical uncertainties in PDFs and parton showers can be systematically reduced via higher-order calculations and data-driven fits.

Practically, this framework extends to a wide class of BSM scenarios, including muon-philic scalars, axion-like particles, and heavy neutral leptons. The methodology—analyzing shape deviations in inclusive kinematic distributions driven by BSM-induced PDF evolution—can also be synergistically incorporated with forward physics, angular analyses, and more detailed detector simulations. Extensions to models with flavor-dependent couplings or additional dark sectors are straightforward.

A key theoretical implication is the inversion of electroweak precision paradigms: instead of probing heavy new physics via vacuum polarization corrections, high-energy lepton colliders can now stringently test weakly-coupled light states through their impact on parton evolution, opening new indirect BSM search avenues.

Conclusion

The paper systematically quantifies the indirect sensitivity to BSM effects via modifications to muon PDFs at future muon colliders, focusing on the $L_\mu - L_\tau$ gauge boson as a representative scenario. The indirect shape analysis outperforms direct searches for $Z'$ masses in the 50–100 GeV range, highlighting the power of high-energy precision lepton colliders in exploring weakly coupled, flavor-specific interactions. Further developments in theory, detector simulation, and statistical methodology will optimize and generalize this indirect search paradigm, enhancing the reach for a wide array of new physics models.