Muon PDF Probes in Collider Physics

• Muon PDF probes are analyses that treat muons as sources of perturbatively calculable partons and decay spectra, enabling precise modeling of their role in high-energy collisions.
• They employ advanced PDF evolution via generalized DGLAP equations to disentangle contributions from electroweak, QCD, and interference effects, crucial for both Standard Model precision and BSM investigations.
• Practical applications include probing muon-decay Michel parameters, vector boson fusion, and mixed Z/γ dynamics, which help refine measurements and search for signals of new physics.

Muon PDF probes are analyses in which the muon is treated as a source of perturbatively calculable partons, or, in a distinct analogy, as a parent particle whose decay spectra are parameterized and constrained through secondary scattering observables. In the collider usage, the relevant objects are electroweak parton distribution functions $f_{i/\mu}(x,Q)$ for muons, neutrinos, gauge bosons, quarks, gluons, and interference states such as the mixed $Z/\gamma$ PDF; in the decay-spectrum usage, the Michel parameters $P_x$ and $w_x$ compactly encode how the muon’s energy is distributed among decay products and can be probed through coherent elastic neutrino–nucleus scattering (CE$\nu$NS) (Capdevilla et al., 2024, Marzocca et al., 2024, Asadi et al., 18 Feb 2026, Bresó-Pla et al., 25 Feb 2025).

1. Partonic description of the high-energy muon

At energies far above the muon mass, collinear radiation of gauge bosons and fermion pairs from the incoming muon becomes enhanced by logarithms of the ratio of the hard scale to the particle masses. Emissions such as

$$\mu \to \mu \gamma,\quad \mu\to\mu Z,\quad \mu\to \nu_\mu W,\quad \gamma\to f\bar f$$

justify a PDF description of the muon at multi-TeV colliders. The inclusive cross section for a final state $F$ is written as

$$\sigma_{\mu^+\mu^-\to F} = \sum_{i,j}\int_0^1 dx_1\,dx_2\; f_{i/\mu^+}(x_1,Q)\, f_{j/\mu^-}(x_2,Q)\, \hat\sigma_{ij\to F}(\hat s, Q,\mu_R),$$

with $\hat s=x_1x_2s$. In this framework, the “initial state” is no longer a pure on-shell muon, but a muon plus a cloud of collinear electroweak and QCD partons (Asadi et al., 18 Feb 2026).

The evolution of these PDFs is governed by generalized DGLAP equations. In the notation used for muon-collider PDF evolution,

$$Q^2\frac{d f_B(x,Q)}{d Q^2} = P_B^v(Q)\, f_B(x,Q) + \sum_{A,C} \frac{\alpha_{ABC}}{2\pi}\, \widetilde{P}_{BA}^C \otimes f_A + \frac{v^2}{16\pi^2Q^2} \sum_{A,C} \widetilde{U}^C_{BA}\otimes f_A.$$

The perturbative boundary condition for a muon beam is

$Z/\gamma$0

This is a central distinction from proton PDFs: the muon beam has no nonperturbative input comparable to hadronic structure functions (Asadi et al., 18 Feb 2026).

The partonic content generated by the evolution includes the valence muon, radiatively induced charged leptons and neutrinos, the electroweak gauge bosons $Z/\gamma$1, $Z/\gamma$2, and $Z/\gamma$3, quarks and gluons generated via $Z/\gamma$4, and interference PDFs. In the Standard Model there are two mixed channels singled out by quantum numbers: transverse photon–$Z/\gamma$5 interference and longitudinal $Z/\gamma$6–Higgs interference. In the broken phase, the transverse $Z/\gamma$7 sector must therefore be treated as a matrix of PDFs,

$Z/\gamma$8

rather than as independent scalar distributions (Marzocca et al., 2024).

2. Neutrino and gauge-boson content as collider probes

The most distinctive electroweak constituent of the muon is the muon-neutrino PDF generated by the charged-current splitting

$Z/\gamma$9

This induces a distribution $P_x$0 that is strongly biased toward large $P_x$1, because the emission is dominantly soft in the gauge boson and the neutrino tends to take most of the muon’s energy. At both 3 TeV and 10 TeV muon colliders, the $P_x$2 and $P_x$3 luminosities are the largest at large $P_x$4, while gauge-boson–gauge-boson luminosities are subdominant in that regime (Capdevilla et al., 2024).

This structure makes several Standard Model processes into direct probes of the neutrino content of the muon. In

$P_x$5

the dominant signal channel is

$P_x$6

while the main background is vector boson fusion through $P_x$7 and $P_x$8. For

$P_x$9

the corresponding neutrino-initiated subprocess is

$w_x$0

In both classes of observables, high-$w_x$1 regions become increasingly dominated by the $w_x$2 luminosity, so differential distributions in $w_x$3, rapidity, and $w_x$4 provide an experimental handle on $w_x$5 itself (Capdevilla et al., 2024).

A related use of “muon PDF probes” appears in multi-TeV vector boson fusion. At a muon collider, collinear radiation of $w_x$6 and $w_x$7 bosons off energetic muons makes the beams act like sources of effective electroweak gauge bosons, so that heavy-state production can be organized schematically as

$w_x$8

Within a model-independent heavy-scalar framework with couplings $w_x$9 and $\nu$0, observing

$\nu$1

implies $\nu$2 and $\nu$3, so that the basis-independent invariant

$\nu$4

is nonzero. Under the stated assumptions, this is sufficient to establish CP violation in the scalar sector. In this sense, the effective “PDF of the muon into $\nu$5” becomes the production-side lever arm for a bosonic CP-violation test (Li et al., 26 Nov 2025).

3. The mixed $\nu$6 PDF and interference structure

The mixed $\nu$7 PDF is an interference object rather than a probability density in the ordinary sense. It arises because, after integrating over the azimuth of the collinear emission, interference between different helicity states vanishes, but interference between distinct species survives when they share the same unbroken quantum numbers and can enter the same splitting and hard amplitude. In the Standard Model, this selects exactly the transverse $\nu$8–$\nu$9 sector and the longitudinal $$\mu \to \mu \gamma,\quad \mu\to\mu Z,\quad \mu\to \nu_\mu W,\quad \gamma\to f\bar f$$0–Higgs sector (Marzocca et al., 2024).

At leading order, the effective-vector-boson approximation for $$\mu \to \mu \gamma,\quad \mu\to\mu Z,\quad \mu\to \nu_\mu W,\quad \gamma\to f\bar f$$1 is accidentally suppressed by the small vector coupling combination

$$\mu \to \mu \gamma,\quad \mu\to\mu Z,\quad \mu\to \nu_\mu W,\quad \gamma\to f\bar f$$2

This suppression is not present in the leading-logarithm resummed result. Once electroweak evolution polarizes the muon PDFs and includes higher-order splittings, the mixed $$\mu \to \mu \gamma,\quad \mu\to\mu Z,\quad \mu\to \nu_\mu W,\quad \gamma\to f\bar f$$3 PDF becomes comparable in size to other electroweak gauge-boson PDFs. Extending the analytic approximation to $$\mu \to \mu \gamma,\quad \mu\to\mu Z,\quad \mu\to \nu_\mu W,\quad \gamma\to f\bar f$$4 lifts the accidental cancellation through double gauge-boson emission and Sudakov-enhanced terms of the form $$\mu \to \mu \gamma,\quad \mu\to\mu Z,\quad \mu\to \nu_\mu W,\quad \gamma\to f\bar f$$5, bringing the analytic treatment into good agreement with the full LL-resummed numerical evolution (Marzocca et al., 2024).

This interference PDF has direct phenomenological consequences. In high-energy Compton scattering,

$$\mu \to \mu \gamma,\quad \mu\to\mu Z,\quad \mu\to \nu_\mu W,\quad \gamma\to f\bar f$$6

the ratio

$$\mu \to \mu \gamma,\quad \mu\to\mu Z,\quad \mu\to \nu_\mu W,\quad \gamma\to f\bar f$$7

is negative and ranges from $$\mu \to \mu \gamma,\quad \mu\to\mu Z,\quad \mu\to \nu_\mu W,\quad \gamma\to f\bar f$$8 up to tens of percent at large transverse momentum and backward muon rapidity. For a 10 TeV muon collider with $$\mu \to \mu \gamma,\quad \mu\to\mu Z,\quad \mu\to \nu_\mu W,\quad \gamma\to f\bar f$$9, the effect exceeds $F$0 in several $F$1 bins and exceeds $F$2 in the bin $F$3 TeV and $F$4. The corresponding $F$5 background is small in the most sensitive region (Marzocca et al., 2024).

The same interference structure affects Higgs observables. In associated $F$6 production, the inclusion of the $F$7 PDF shifts the 10 TeV Standard Model prediction from $F$8 fb to $F$9 fb in the $$\sigma_{\mu^+\mu^-\to F} = \sum_{i,j}\int_0^1 dx_1\,dx_2\; f_{i/\mu^+}(x_1,Q)\, f_{j/\mu^-}(x_2,Q)\, \hat\sigma_{ij\to F}(\hat s, Q,\mu_R),$$0 limit, while at 3 TeV the change is at the $$\sigma_{\mu^+\mu^-\to F} = \sum_{i,j}\int_0^1 dx_1\,dx_2\; f_{i/\mu^+}(x_1,Q)\, f_{j/\mu^-}(x_2,Q)\, \hat\sigma_{ij\to F}(\hat s, Q,\mu_R),$$1 level. This is already comparable to the $$\sigma_{\mu^+\mu^-\to F} = \sum_{i,j}\int_0^1 dx_1\,dx_2\; f_{i/\mu^+}(x_1,Q)\, f_{j/\mu^-}(x_2,Q)\, \hat\sigma_{ij\to F}(\hat s, Q,\mu_R),$$2 precision anticipated for a high-luminosity $$\sigma_{\mu^+\mu^-\to F} = \sum_{i,j}\int_0^1 dx_1\,dx_2\; f_{i/\mu^+}(x_1,Q)\, f_{j/\mu^-}(x_2,Q)\, \hat\sigma_{ij\to F}(\hat s, Q,\mu_R),$$3 measurement, so the mixed PDF must be included consistently in any percent-level extraction of $$\sigma_{\mu^+\mu^-\to F} = \sum_{i,j}\int_0^1 dx_1\,dx_2\; f_{i/\mu^+}(x_1,Q)\, f_{j/\mu^-}(x_2,Q)\, \hat\sigma_{ij\to F}(\hat s, Q,\mu_R),$$4 and $$\sigma_{\mu^+\mu^-\to F} = \sum_{i,j}\int_0^1 dx_1\,dx_2\; f_{i/\mu^+}(x_1,Q)\, f_{j/\mu^-}(x_2,Q)\, \hat\sigma_{ij\to F}(\hat s, Q,\mu_R),$$5 (Marzocca et al., 2024).

4. Beyond-the-Standard-Model distortions of muon PDFs

Muon PDF probes are also a BSM search strategy. In an $$\sigma_{\mu^+\mu^-\to F} = \sum_{i,j}\int_0^1 dx_1\,dx_2\; f_{i/\mu^+}(x_1,Q)\, f_{j/\mu^-}(x_2,Q)\, \hat\sigma_{ij\to F}(\hat s, Q,\mu_R),$$6 model with a new gauge boson $$\sigma_{\mu^+\mu^-\to F} = \sum_{i,j}\int_0^1 dx_1\,dx_2\; f_{i/\mu^+}(x_1,Q)\, f_{j/\mu^-}(x_2,Q)\, \hat\sigma_{ij\to F}(\hat s, Q,\mu_R),$$7, the Lagrangian

$$\sigma_{\mu^+\mu^-\to F} = \sum_{i,j}\int_0^1 dx_1\,dx_2\; f_{i/\mu^+}(x_1,Q)\, f_{j/\mu^-}(x_2,Q)\, \hat\sigma_{ij\to F}(\hat s, Q,\mu_R),$$8

adds new parton species and mixed states to the DGLAP system. Below the electroweak scale, for $$\sigma_{\mu^+\mu^-\to F} = \sum_{i,j}\int_0^1 dx_1\,dx_2\; f_{i/\mu^+}(x_1,Q)\, f_{j/\mu^-}(x_2,Q)\, \hat\sigma_{ij\to F}(\hat s, Q,\mu_R),$$9, the evolution includes $\hat s=x_1x_2s$0, $\hat s=x_1x_2s$1, $\hat s=x_1x_2s$2, $\hat s=x_1x_2s$3, $\hat s=x_1x_2s$4, and $\hat s=x_1x_2s$5; above the electroweak scale it tracks $\hat s=x_1x_2s$6, $\hat s=x_1x_2s$7, $\hat s=x_1x_2s$8, $\hat s=x_1x_2s$9, $$Q^2\frac{d f_B(x,Q)}{d Q^2} = P_B^v(Q)\, f_B(x,Q) + \sum_{A,C} \frac{\alpha_{ABC}}{2\pi}\, \widetilde{P}_{BA}^C \otimes f_A + \frac{v^2}{16\pi^2Q^2} \sum_{A,C} \widetilde{U}^C_{BA}\otimes f_A.$$0, $$Q^2\frac{d f_B(x,Q)}{d Q^2} = P_B^v(Q)\, f_B(x,Q) + \sum_{A,C} \frac{\alpha_{ABC}}{2\pi}\, \widetilde{P}_{BA}^C \otimes f_A + \frac{v^2}{16\pi^2Q^2} \sum_{A,C} \widetilde{U}^C_{BA}\otimes f_A.$$1, and $$Q^2\frac{d f_B(x,Q)}{d Q^2} = P_B^v(Q)\, f_B(x,Q) + \sum_{A,C} \frac{\alpha_{ABC}}{2\pi}\, \widetilde{P}_{BA}^C \otimes f_A + \frac{v^2}{16\pi^2Q^2} \sum_{A,C} \widetilde{U}^C_{BA}\otimes f_A.$$2. Numerically, the main effects are an enhancement of the muon PDF at $$Q^2\frac{d f_B(x,Q)}{d Q^2} = P_B^v(Q)\, f_B(x,Q) + \sum_{A,C} \frac{\alpha_{ABC}}{2\pi}\, \widetilde{P}_{BA}^C \otimes f_A + \frac{v^2}{16\pi^2Q^2} \sum_{A,C} \widetilde{U}^C_{BA}\otimes f_A.$$3 and a suppression of the photon PDF, both at the percent level for the benchmark $$Q^2\frac{d f_B(x,Q)}{d Q^2} = P_B^v(Q)\, f_B(x,Q) + \sum_{A,C} \frac{\alpha_{ABC}}{2\pi}\, \widetilde{P}_{BA}^C \otimes f_A + \frac{v^2}{16\pi^2Q^2} \sum_{A,C} \widetilde{U}^C_{BA}\otimes f_A.$$4, $$Q^2\frac{d f_B(x,Q)}{d Q^2} = P_B^v(Q)\, f_B(x,Q) + \sum_{A,C} \frac{\alpha_{ABC}}{2\pi}\, \widetilde{P}_{BA}^C \otimes f_A + \frac{v^2}{16\pi^2Q^2} \sum_{A,C} \widetilde{U}^C_{BA}\otimes f_A.$$5 (Asadi et al., 18 Feb 2026).

The associated collider observable is the invariant-mass fraction

$$Q^2\frac{d f_B(x,Q)}{d Q^2} = P_B^v(Q)\, f_B(x,Q) + \sum_{A,C} \frac{\alpha_{ABC}}{2\pi}\, \widetilde{P}_{BA}^C \otimes f_A + \frac{v^2}{16\pi^2Q^2} \sum_{A,C} \widetilde{U}^C_{BA}\otimes f_A.$$6

through the distributions $$Q^2\frac{d f_B(x,Q)}{d Q^2} = P_B^v(Q)\, f_B(x,Q) + \sum_{A,C} \frac{\alpha_{ABC}}{2\pi}\, \widetilde{P}_{BA}^C \otimes f_A + \frac{v^2}{16\pi^2Q^2} \sum_{A,C} \widetilde{U}^C_{BA}\otimes f_A.$$7 in the final states $$Q^2\frac{d f_B(x,Q)}{d Q^2} = P_B^v(Q)\, f_B(x,Q) + \sum_{A,C} \frac{\alpha_{ABC}}{2\pi}\, \widetilde{P}_{BA}^C \otimes f_A + \frac{v^2}{16\pi^2Q^2} \sum_{A,C} \widetilde{U}^C_{BA}\otimes f_A.$$8. A reference setup uses $$Q^2\frac{d f_B(x,Q)}{d Q^2} = P_B^v(Q)\, f_B(x,Q) + \sum_{A,C} \frac{\alpha_{ABC}}{2\pi}\, \widetilde{P}_{BA}^C \otimes f_A + \frac{v^2}{16\pi^2Q^2} \sum_{A,C} \widetilde{U}^C_{BA}\otimes f_A.$$9 TeV, $Z/\gamma$00, central cuts $Z/\gamma$01 or $Z/\gamma$02, and $Z/\gamma$03. Within that framework, the profile-likelihood analysis is almost insensitive to the luminosity uncertainty because the sensitivity is dominated by the shape of $Z/\gamma$04, not by overall normalization. In the mass window $Z/\gamma$05–$Z/\gamma$06 GeV, the PDF-based reach lies significantly below the direct-search curves from $Z/\gamma$07 and $Z/\gamma$08, so it probes smaller $Z/\gamma$09 couplings than those associated-production channels at the same collider (Asadi et al., 18 Feb 2026).

The neutrino PDF also modifies new-physics production channels directly. For a heavy scalar $Z/\gamma$10 doublet $Z/\gamma$11, the charged-current mode

$Z/\gamma$12

is present only because of the muon’s $Z/\gamma$13 content. The inclusive LePDF-based prediction agrees with the fixed-order simulation of

$Z/\gamma$14

within scale uncertainties, and the kinematic distributions show the expected ISR pattern: the radiated $Z/\gamma$15 is mostly forward and low-$Z/\gamma$16, while the heavy scalars are central and high-$Z/\gamma$17 (Capdevilla et al., 2024).

A further example is resonant single production of an axion-like particle $Z/\gamma$18 through VBF. In a 10 TeV collider, the $Z/\gamma$19-induced interference contribution typically modifies the total cross section by $Z/\gamma$20 for $Z/\gamma$21 TeV, with the sign and magnitude depending on the combinations $Z/\gamma$22, $Z/\gamma$23, and $Z/\gamma$24. This makes the mixed-PDF sector relevant not only for Standard Model precision studies but also for coupling extraction in BSM resonance searches (Marzocca et al., 2024).

5. Muon-decay spectra as “muon PDFs” in the Michel-parameter sense

A distinct use of the phrase “Muon PDF probes” appears in stopped-muon decay. In this context, the energy distributions of $Z/\gamma$25, $Z/\gamma$26, and especially $Z/\gamma$27 are treated as objects analogous to PDFs: they encode how the muon’s energy is distributed among decay products and are parameterized by neutrino Michel parameters $Z/\gamma$28 and $Z/\gamma$29. These parameters are quadratic functions of the effective muon-decay couplings $Z/\gamma$30, so deviations from the Standard Model values $Z/\gamma$31 and $Z/\gamma$32 diagnose non-$Z/\gamma$33 Lorentz structures, right-handed currents, or right-handed neutrinos in muon decay (Bresó-Pla et al., 25 Feb 2025).

At the Spallation Neutron Source, the delayed $Z/\gamma$34 and $Z/\gamma$35 fluxes from

$Z/\gamma$36

feed the CE$Z/\gamma$37NS signal measured by COHERENT. After convolution with the CE$Z/\gamma$38NS cross section, the delayed event rate can be written in terms of effective rescaling factors $Z/\gamma$39 and $Z/\gamma$40, with

$Z/\gamma$41

This makes the recoil-energy and timing distributions of CE$Z/\gamma$42NS into probes of the muon-decay spectrum itself (Bresó-Pla et al., 25 Feb 2025).

Using CsI and LAr data from COHERENT, the experimentally best-constrained linear combination is

$Z/\gamma$43

This is the first direct experimental constraint involving the muon antineutrino Michel parameters $Z/\gamma$44 and $Z/\gamma$45, and only the second constraint on the electron-neutrino Michel parameters $Z/\gamma$46 and $Z/\gamma$47. The quoted one-at-a-time $Z/\gamma$48 ranges are

$Z/\gamma$49

Within this usage, CE$Z/\gamma$50NS acts as a precision production-side measurement of the muon’s decay “PDFs,” rather than of electroweak initial-state radiation (Bresó-Pla et al., 25 Feb 2025).

6. Experimental foundations, uncertainties, and limitations

Muon PDF probes depend on both beam preparation and theoretical control. A compact front-end muon source optimized for a Neutrino Factory or muon-collider program begins with an 8 GeV proton beam on a liquid mercury jet inside a 20 T solenoid, followed by a 20 T $Z/\gamma$51 2 T taper over 6.00 m, a 2 T decay channel, a bent-solenoid chicane, RF bunching and phase rotation, a matching section, and an ionization-cooling channel. The quoted performance is about $Z/\gamma$52 and $Z/\gamma$53 per incident 8 GeV proton within the reference acceptance, a train of $Z/\gamma$54 well-defined bunches centered around $Z/\gamma$55, and a transverse normalized rms emittance reduced by a factor of $Z/\gamma$56 to $Z/\gamma$57. The chicane and absorber reduce downstream energy deposition by more than an order of magnitude compared to a scheme without them. This source is explicitly described as directly relevant to using muons as high-energy probes once further acceleration is provided (Stratakis et al., 2015).

Theoretical systematics remain substantial. For the $Z/\gamma$58 PDF, factorization-scale variations $Z/\gamma$59 and $Z/\gamma$60 give relative changes of $Z/\gamma$61 around $Z/\gamma$62 GeV and $Z/\gamma$63 around $Z/\gamma$64 TeV, whereas QED-like pieces are at the $Z/\gamma$65 and $Z/\gamma$66 level, respectively. For mixed $Z/\gamma$67 effects, the scale spread is at the few-percent level in benchmark Compton and $Z/\gamma$68 studies. In the $Z/\gamma$69 case study, varying the hard scale by a factor of two changes $Z/\gamma$70 at the $Z/\gamma$71 level, larger than the percent-level deviations induced by the benchmark $Z/\gamma$72 couplings. These numbers indicate that higher-order electroweak evolution, matched fixed-order matrix elements, and improved treatment of massive and mixed PDFs are required for precision applications (Capdevilla et al., 2024, Marzocca et al., 2024, Asadi et al., 18 Feb 2026).

Experimental limitations are equally specific. Muon-collider detector concepts include forward shielding to suppress beam-induced background, so only objects with

$Z/\gamma$73

are reconstructed in some analyses. Many vector-boson-fusion muons and neutrinos therefore escape detection and appear as missing energy. Sensitivity estimates in several studies are purely statistical, with systematics such as background normalization, b-tagging efficiencies, and detector calibrations not yet included. This suggests that muon PDF probes are already structurally well defined, but that their ultimate reach will depend on coordinated advances in beam delivery, electroweak PDF theory, event-generator implementations, and detector realism (Li et al., 26 Nov 2025, Capdevilla et al., 2024, Asadi et al., 18 Feb 2026).