Muons: Properties, Detection & Applications

• Muons are elementary charged leptons with a rest mass of 105.66 MeV/c², known for their weak decay and significant role in probing fundamental physics.
• They are produced predominantly through pion decay, enabling high-intensity beams critical for precision experiments and innovative imaging techniques.
• Their unique decay properties and time dilation effects support stringent tests of special relativity and explorations of physics beyond the Standard Model.

Muons are elementary charged leptons with two charge states, $\mu^+$ and $\mu^-$, a rest mass $M_\mu = 105.66\,\mathrm{MeV}/c^2$, and a rest-frame lifetime $\tau_0 = 2.1969811 \pm 0.0000022\ \mu\mathrm{s}$. They are much heavier than electrons, do not participate in the strong interaction, and live long enough to be transported, stored, stopped in matter, bound into muonium or muonic atoms, and used as probes in particle, nuclear, atomic, condensed-matter, imaging, and astrophysical contexts. Their distinctive role follows from the conjunction of large mass, weak decay, high penetration, and experimentally useful polarization and detectability (Gorringe et al., 2015, Maggiora, 2023, Durham, 2018).

1. Intrinsic properties and weak decay

Muon decay is the cleanest purely leptonic charged-current weak process in precision physics. The dominant ordinary channels are

$$\mu^- \rightarrow e^- + \overline{\nu}_e + \nu_\mu, \qquad \mu^+ \rightarrow e^+ + \nu_e + \overline{\nu}_\mu,$$

and, for muons at rest in vacuum, the survival law is exponential,

$N(t)=N_0 e^{-t/\tau}.$

Because the process is purely leptonic, it defines the overall weak scale through the Fermi constant. The review literature quotes

$$G_F = 1.166\,378\,7(6)\times 10^{-5}\ \mathrm{GeV}^{-2},$$

with $0.5$ ppm precision from the muon lifetime, and gives the radiatively corrected relation

$$G_F = \sqrt{\frac{192\pi^3}{\tau_\mu m_\mu^5}\, \frac{1}{1+\Delta q^{(0)}+\Delta q^{(1)}+\Delta q^{(2)}} }.$$

Muon decay also tests the chiral structure of the weak interaction through the Michel-parameter formalism, with Standard Model values $\rho=\delta=\tfrac34$, $\mu^-$0, and $\mu^-$1 (Gorringe et al., 2015, Maggiora, 2023).

Muon decay is experimentally unusual in being self-analyzing. Polarized muons are naturally produced in $\mu^-$2, and parity violation in the decay makes the angular and energy distribution of the decay positron or electron encode the muon spin. This is why the same unstable particle supports both precision lifetime measurements and spin-based observables such as storage-ring $\mu^-$3 precession. The long lifetime is also short enough that matter effects can become visible: in scintillator-based stopping experiments, $\mu^-$4 capture competes with decay, so the measured time spectrum can differ substantially from the free $\mu^-$5 lifetime unless capture on nuclei is modeled explicitly (Gorringe et al., 2015, Maggiora, 2023).

2. Production, beams, and muonium

Muon beams are usually produced indirectly through pion decay. At the Paul Scherrer Institut, the HIPA facility uses a $\mu^-$6 proton beam with current up to $\mu^-$7 and beam power $\mu^-$8, providing the world’s highest intensities of low-momentum muons. Present surface-muon rates are up to about $\mu^-$9, while the proposed HiMB line is intended to deliver on the order of $M_\mu = 105.66\,\mathrm{MeV}/c^2$0 surface muons, specifically $M_\mu = 105.66\,\mathrm{MeV}/c^2$1 below $M_\mu = 105.66\,\mathrm{MeV}/c^2$2 (Kirch, 2016). At the front-end design level, compact schemes based on an $M_\mu = 105.66\,\mathrm{MeV}/c^2$3 proton beam on a liquid-mercury target inside a $M_\mu = 105.66\,\mathrm{MeV}/c^2$4 solenoid combine capture, decay, chicane cleaning, RF bunching, phase rotation, and ionization cooling, with accepted yields of about $M_\mu = 105.66\,\mathrm{MeV}/c^2$5 muons per incident proton for each sign in the quoted neutrino-factory configuration (Stratakis et al., 2015).

A distinct line of development emphasizes beam quality rather than raw rate. The PSI muCool program aims at a phase-space reduction of $M_\mu = 105.66\,\mathrm{MeV}/c^2$6 with efficiency $M_\mu = 105.66\,\mathrm{MeV}/c^2$7 by stopping $M_\mu = 105.66\,\mathrm{MeV}/c^2$8 in cryogenic helium gas and compressing them with electric and magnetic fields plus gas-density gradients before extraction into vacuum. The core transport relation is

$M_\mu = 105.66\,\mathrm{MeV}/c^2$9

which rotates the drift direction as the collision frequency $\tau_0 = 2.1969811 \pm 0.0000022\ \mu\mathrm{s}$0 varies across the gas cell. Demonstrations reported transverse and longitudinal compression separately, including longitudinal compression from a $\tau_0 = 2.1969811 \pm 0.0000022\ \mu\mathrm{s}$1 swarm to below $\tau_0 = 2.1969811 \pm 0.0000022\ \mu\mathrm{s}$2 within $\tau_0 = 2.1969811 \pm 0.0000022\ \mu\mathrm{s}$3 (Belosevic et al., 2019).

Muonium, the bound state $\tau_0 = 2.1969811 \pm 0.0000022\ \mu\mathrm{s}$4, is central to this beam-quality program. PSI explicitly identifies improved Mu $\tau_0 = 2.1969811 \pm 0.0000022\ \mu\mathrm{s}$5-$\tau_0 = 2.1969811 \pm 0.0000022\ \mu\mathrm{s}$6 spectroscopy and possible tests of the free fall of Mu as downstream applications of slow, bright $\tau_0 = 2.1969811 \pm 0.0000022\ \mu\mathrm{s}$7 beams (Kirch, 2016). A further milestone was the demonstration of acceleration of thermal positive muons from thermal energy to $\tau_0 = 2.1969811 \pm 0.0000022\ \mu\mathrm{s}$8: muonium emitted from laser-ablated aerogel was ionized by resonant multiphoton excitation, electrostatically accelerated to $\tau_0 = 2.1969811 \pm 0.0000022\ \mu\mathrm{s}$9, and then accelerated in a radio-frequency quadrupole to $$\mu^- \rightarrow e^- + \overline{\nu}_e + \nu_\mu, \qquad \mu^+ \rightarrow e^+ + \nu_e + \overline{\nu}_\mu,$$0. The measured normalized transverse emittances were

$$\mu^- \rightarrow e^- + \overline{\nu}_e + \nu_\mu, \qquad \mu^+ \rightarrow e^+ + \nu_e + \overline{\nu}_\mu,$$1

horizontally and

$$\mu^- \rightarrow e^- + \overline{\nu}_e + \nu_\mu, \qquad \mu^+ \rightarrow e^+ + \nu_e + \overline{\nu}_\mu,$$2

vertically, corresponding to phase-space reduction factors of $$\mu^- \rightarrow e^- + \overline{\nu}_e + \nu_\mu, \qquad \mu^+ \rightarrow e^+ + \nu_e + \overline{\nu}_\mu,$$3 and $$\mu^- \rightarrow e^- + \overline{\nu}_e + \nu_\mu, \qquad \mu^+ \rightarrow e^+ + \nu_e + \overline{\nu}_\mu,$$4 relative to the incident surface beam (Aritome et al., 2024).

3. Proper time, lifetime, and acceleration

Muons are a canonical test of relativistic time dilation because atmospheric production heights are far larger than the distance $$\mu^- \rightarrow e^- + \overline{\nu}_e + \nu_\mu, \qquad \mu^+ \rightarrow e^+ + \nu_e + \overline{\nu}_\mu,$$5 implied by the rest-frame lifetime. The standard relation

$$\mu^- \rightarrow e^- + \overline{\nu}_e + \nu_\mu, \qquad \mu^+ \rightarrow e^+ + \nu_e + \overline{\nu}_\mu,$$6

explains their survival to sea level, and the same kinematics enters accelerator-based lifetime tests. A pedagogical estimate quoted for a $$\mu^- \rightarrow e^- + \overline{\nu}_e + \nu_\mu, \qquad \mu^+ \rightarrow e^+ + \nu_e + \overline{\nu}_\mu,$$7 muon gives $$\mu^- \rightarrow e^- + \overline{\nu}_e + \nu_\mu, \qquad \mu^+ \rightarrow e^+ + \nu_e + \overline{\nu}_\mu,$$8 and a substantial survival fraction from upper-atmosphere production to ground (Maggiora, 2023).

The storage-ring case isolates a subtler issue: whether very large transverse acceleration changes the decay clock beyond the usual Lorentz factor. In the CERN and Brookhaven storage-ring experiments, $$\mu^- \rightarrow e^- + \overline{\nu}_e + \nu_\mu, \qquad \mu^+ \rightarrow e^+ + \nu_e + \overline{\nu}_\mu,$$9 muons circulated in a ring of diameter $N(t)=N_0 e^{-t/\tau}.$0 with $N(t)=N_0 e^{-t/\tau}.$1, giving

$N(t)=N_0 e^{-t/\tau}.$2

The measured lifetime agreed with the special-relativistic prediction to about $N(t)=N_0 e^{-t/\tau}.$3, and the lifetime of stored muons matched the lifetime of muons moving inertially in a straight line at the same speed. Farley’s analysis framed this as a test of an “effective gravity” inferred from circular acceleration and argued that a naive gravitational-potential interpretation would predict a large, unobserved effect; the conservative standard-relativistic reading is that the decay law is governed by proper time along the worldline and, in uniform circular motion in flat spacetime, depends on speed rather than centripetal acceleration as an independent variable (Farley, 2015).

The same point was extended in Farley’s discussion to an ESR storage-ring lithium-ion experiment, where $N(t)=N_0 e^{-t/\tau}.$4 and $N(t)=N_0 e^{-t/\tau}.$5, and spectroscopy verified ordinary time dilation to $N(t)=N_0 e^{-t/\tau}.$6 parts per million. Under the same storage-ring potential argument, an optical shift of about $N(t)=N_0 e^{-t/\tau}.$7 would have been expected, but no such shift was observed. This suggests that large circular acceleration in flat spacetime does not introduce an additional clock effect beyond the usual Lorentz factor, while the broader claim that “gravity does not change the scale of time at all” remains the controversial part of the argument rather than the storage-ring lifetime observation itself (Farley, 2015).

4. Detection and identification techniques

Muon detection spans very different regimes, from delayed-coincidence lifetime apparatus to high-rate collider identification. A compact example is the A.M.E.L.I.E. scintillator system, which uses a single $N(t)=N_0 e^{-t/\tau}.$8 plastic scintillator and a $N(t)=N_0 e^{-t/\tau}.$9 SiPM to detect a stopping muon and the later decay electron or positron in the same volume. Its timing logic opens a $$G_F = 1.166\,378\,7(6)\times 10^{-5}\ \mathrm{GeV}^{-2},$$0 non-retriggerable window after a first high-threshold pulse and interprets a second high-threshold pulse as the decay stop signal, with effective timing resolution set to $$G_F = 1.166\,378\,7(6)\times 10^{-5}\ \mathrm{GeV}^{-2},$$1 (Maggiora, 2023).

At collider scale, one representative strategy avoids conventional iron flux return. The “Muon Identification without Iron” design for the 4th Concept detector combines a dual-readout DREAM-type calorimeter with a dual-solenoid spectrometer. The central dual-readout signature is

$$G_F = 1.166\,378\,7(6)\times 10^{-5}\ \mathrm{GeV}^{-2},$$2

because the muon’s ionization contribution appears in scintillation $$G_F = 1.166\,378\,7(6)\times 10^{-5}\ \mathrm{GeV}^{-2},$$3, while direct Cerenkov light from the muon is essentially absent in the fibers; radiative electromagnetic losses contribute equally to $$G_F = 1.166\,378\,7(6)\times 10^{-5}\ \mathrm{GeV}^{-2},$$4 and $$G_F = 1.166\,378\,7(6)\times 10^{-5}\ \mathrm{GeV}^{-2},$$5 and cancel in $$G_F = 1.166\,378\,7(6)\times 10^{-5}\ \mathrm{GeV}^{-2},$$6. For isolated tracks, the paper quotes pion rejection against muons from about $$G_F = 1.166\,378\,7(6)\times 10^{-5}\ \mathrm{GeV}^{-2},$$7 at $$G_F = 1.166\,378\,7(6)\times 10^{-5}\ \mathrm{GeV}^{-2},$$8 to $$G_F = 1.166\,378\,7(6)\times 10^{-5}\ \mathrm{GeV}^{-2},$$9 at $0.5$0, with a mean bending field of about $0.5$1 and an integral bending power of about $0.5$2 in the external muon system (0709.0768).

Muon identification in collider data also extends beyond a single scalar isolation variable. In CMS Open Data at $0.5$3, weakly supervised Classification Without Labels was used to distinguish prompt muons from non-prompt muons using low-level particle-flow constituents within $0.5$4 of the muon. A Particle Flow Network reached $0.5$5 and a prompt efficiency of $0.5$6 at $0.5$7 background efficiency, compared with $0.5$8 and $0.5$9 for the best cone-based isolation network. A single data-selected energy flow polynomial nearly closed the gap, reaching $$G_F = \sqrt{\frac{192\pi^3}{\tau_\mu m_\mu^5}\, \frac{1}{1+\Delta q^{(0)}+\Delta q^{(1)}+\Delta q^{(2)}} }.$$0 and $$G_F = \sqrt{\frac{192\pi^3}{\tau_\mu m_\mu^5}\, \frac{1}{1+\Delta q^{(0)}+\Delta q^{(1)}+\Delta q^{(2)}} }.$$1 under the same metric. This suggests that conventional isolation compresses away nontrivial angular and compositional information in the local radiation pattern (Witkowski et al., 2023).

5. Atmospheric, cosmic-ray, and astrophysical roles

Cosmic-ray interactions in the upper atmosphere continuously produce muons through pion decay, creating a natural flux that dominates charged cosmic radiation at sea level in the simplified classroom description and supplies a passive probe for many experiments. One review quotes a sea-level flux of approximately

$$G_F = \sqrt{\frac{192\pi^3}{\tau_\mu m_\mu^5}\, \frac{1}{1+\Delta q^{(0)}+\Delta q^{(1)}+\Delta q^{(2)}} }.$$2

with angular distribution roughly proportional to $$G_F = \sqrt{\frac{192\pi^3}{\tau_\mu m_\mu^5}\, \frac{1}{1+\Delta q^{(0)}+\Delta q^{(1)}+\Delta q^{(2)}} }.$$3 and average energy around $$G_F = \sqrt{\frac{192\pi^3}{\tau_\mu m_\mu^5}\, \frac{1}{1+\Delta q^{(0)}+\Delta q^{(1)}+\Delta q^{(2)}} }.$$4 (Durham, 2018). A didactic treatment gives an average vertical intensity $$G_F = \sqrt{\frac{192\pi^3}{\tau_\mu m_\mu^5}\, \frac{1}{1+\Delta q^{(0)}+\Delta q^{(1)}+\Delta q^{(2)}} }.$$5, corresponding to $$G_F = \sqrt{\frac{192\pi^3}{\tau_\mu m_\mu^5}\, \frac{1}{1+\Delta q^{(0)}+\Delta q^{(1)}+\Delta q^{(2)}} }.$$6, summarized as the rule of thumb of about $$G_F = \sqrt{\frac{192\pi^3}{\tau_\mu m_\mu^5}\, \frac{1}{1+\Delta q^{(0)}+\Delta q^{(1)}+\Delta q^{(2)}} }.$$7 (Maggiora, 2023).

In extensive air showers, muons are both composition tracers and diagnostics of hadronic interaction models. The Pierre Auger Observatory measures the muonic shower component most cleanly in inclined hybrid events, using the relative muon number $$G_F = \sqrt{\frac{192\pi^3}{\tau_\mu m_\mu^5}\, \frac{1}{1+\Delta q^{(0)}+\Delta q^{(1)}+\Delta q^{(2)}} }.$$8 and its energy scaling. Auger and other experiments find a persistent “muon deficit” in simulations: the observed number of muons is about $$G_F = \sqrt{\frac{192\pi^3}{\tau_\mu m_\mu^5}\, \frac{1}{1+\Delta q^{(0)}+\Delta q^{(1)}+\Delta q^{(2)}} }.$$9 to $\rho=\delta=\tfrac34$0 larger than predicted by post-LHC models, and the global discrepancy across experiments is described as about the $\rho=\delta=\tfrac34$1 level. Yet the measured relative fluctuations of the muon number are consistent with expectations derived from the composition inferred from $\rho=\delta=\tfrac34$2, suggesting that the modeling failure is more selective than a gross misdescription of first-interaction fluctuations (Góra, 2022).

Other detectors probe complementary sectors of atmospheric muon phenomenology. ALICE, $\rho=\delta=\tfrac34$3 underground with about $\rho=\delta=\tfrac34$4 of rock overburden and a $\rho=\delta=\tfrac34$5 vertical muon threshold, measured atmospheric muon bundles and found five high-multiplicity events with $\rho=\delta=\tfrac34$6 in $\rho=\delta=\tfrac34$7 live days, corresponding to $\rho=\delta=\tfrac34$8; the observed high-multiplicity rate was successfully described by a heavy primary composition using newer QGSJET II-04 modeling (Sitta, 2017). HAWC, at $\rho=\delta=\tfrac34$9 altitude, observed near-horizontal muons mainly above $\mu^-$00 zenith with a mean rate $\mu^-$01, while nearby volcanoes provided several km water equivalent of directional overburden and hence multi-TeV energy thresholds (Barber et al., 2017). In water or ice, still higher-energy muons become intrinsically stochastic: detailed Monte Carlo studies from $\mu^-$02 to $\mu^-$03 show large range straggling, with average range increasing from $\mu^-$04 at $\mu^-$05 to $\mu^-$06 at $\mu^-$07, while bremsstrahlung dominates the shortest-range tail and direct pair production characterizes the longest-range trajectories (Okumura et al., 2011).

Muon physics also enters extreme-density astrophysics. In the first $\mu^-$08 after binary neutron star merger, post-processed merger remnants with $\mu^-$09–$\mu^-$10 and $\mu^-$11–$\mu^-$12 were found to contain non-negligible muon populations. Depending on the baryonic equation of state, the net muon fraction is between $\mu^-$13 and $\mu^-$14 of the net electron fraction. Muons alter the trapped-neutrino flavor hierarchy so that, deep inside the remnant, muon antineutrinos are the most abundant trapped species, followed by electron antineutrinos, and they modify the remnant pressure by up to $\mu^-$15 relative to calculations neglecting them (Loffredo et al., 2022).

6. Imaging and applied uses

Because muons are heavy charged leptons with suppressed radiative losses, they are exceptionally penetrating probes of matter. In safeguards and arms-control contexts, cosmic muons act as a passive radiographic beam with no artificial source. The basic imaging mechanisms are transmission and multiple Coulomb scattering, with sensitivity to path-integrated density and to inverse radiation length. A dry-cask demonstration at Idaho National Laboratory used two $\mu^-$16 muon trackers on an MC-10 cask with 24 basket slots, 18 of which contained spent fuel, over about 9 weeks; missing multiple assemblies in the two leftmost columns were detected at more than $\mu^-$17, while a single missing assembly in one column produced a $\mu^-$18 deviation, corresponding to $\mu^-$19 confidence that the column was not fully populated (Durham, 2018).

The same paper describes treaty-verification-oriented muon-induced neutron imaging. Negative muons that stop in fissile material can emit neutrons after capture, and neutron-tagged muon tracks can be backprojected by laminography. A Los Alamos demonstration with three $\mu^-$20 uranium cubes, a single tracker $\mu^-$21 above the cubes, $\mu^-$22 of steel shielding, and stilbene neutron detectors localized the sources in a 24-hour measurement. The muonic-atom lifetimes quoted for uranium,

$\mu^-$23

set the characteristic capture timescale used for neutron tagging (Durham, 2018).

Muon imaging is also being extended to cultural heritage. A study of the “low-size limit of muography” examined wooden statues enlarged to $\mu^-$24 and $\mu^-$25, with hidden air or bronze cylinders, using CRY-generated atmospheric muons and Geant4 transport. In the optimistic ideal-detector setup, $\mu^-$26 million muons, corresponding to roughly $\mu^-$27 hours, were sufficient for scattering-based detection of a bronze inclusion in the smaller object, while absorption-based reconstruction on the larger statue used a 2-hour equivalent exposure and selected only $\mu^-$28 muons to enhance contrast (Moussawi et al., 2023). In gamma-ray astronomy, muons also become useful analysis objects rather than nuisances: for imaging atmospheric Cherenkov telescope arrays, analytic modeling of Cherenkov light from individual atmospheric muons suggests that, above $\mu^-$29, arrays containing at least one very large telescope could achieve hadron rejection levels up to $\mu^-$30 while retaining high gamma-ray efficiency, by exploiting the relative muon richness of hadronic showers (Olivera-Nieto et al., 2021).

7. Precision frontiers and emerging directions

Modern precision muon physics spans flavor, dipole moments, atomic spectroscopy, and weak interactions with nuclei. The contemporary review literature treats three CLFV channels as flagship probes: $\mu^-$31 The limits quoted are

$\mu^-$32

$\mu^-$33

and

$\mu^-$34

while the Standard Model rate for $\mu^-$35 with neutrino mixing is below $\mu^-$36. Any observed CLFV signal would therefore be unequivocal evidence for new physics (Gorringe et al., 2015).

The muon anomalous magnetic moment remains equally central. Using

$\mu^-$37

the review quotes an experimental value

$\mu^-$38

to be compared with representative Standard Model evaluations

$\mu^-$39

corresponding to discrepancies of $\mu^-$40 and $\mu^-$41, respectively. The storage-ring observable is the anomalous precession frequency

$\mu^-$42

measured from the oscillatory time spectrum of decay positrons (Gorringe et al., 2015).

Bound-state systems extend this precision frontier. Muonium hyperfine spectroscopy yielded

$\mu^-$43

together with $\mu^-$44 and $\mu^-$45 at $\mu^-$46 ppb, while muonic hydrogen spectroscopy by CREMA produced

$\mu^-$47

the result that crystallized the proton-radius puzzle (Gorringe et al., 2015). Muon capture adds hadronic weak structure: MuCap extracted the singlet capture rate

$\mu^-$48

implying an induced pseudoscalar coupling consistent with chiral perturbation theory (Gorringe et al., 2015).

An emerging direction is relativistic quantum information with muons. In tree-level $\mu^-$49 scattering, the outgoing muon-electron spin system can be treated as a two-qubit mixed state reconstructed from the polarized scattering amplitudes. The proposal for “quantum state tomography with muons” studies beam energies $\mu^-$50, $\mu^-$51, and $\mu^-$52; at $\mu^-$53, with a muon flux $\mu^-$54, a $\mu^-$55 aluminum target stack, and a one-day run, the expected number of entangled events is $\mu^-$56. This suggests that muons may become experimentally usable carriers of Bell-inequality and tomography observables over a wide relativistic range unavailable to most other massive elementary particles (Gao et al., 2024).