Muon (${\mu }^{-}$)

The muon is a spin-one-half charged lepton, the second-generation counterpart of the electron. A negative muon has electric charge $-e$, while the antimuon has charge $+e$. It has no quark substructure in the Standard Model and participates in electromagnetic and weak interactions.

A muon is unstable and decays weakly; many muons reaching ground level come from cosmic-ray cascades in the atmosphere. Their penetrating tracks make them useful in detector calibration. A simple lifetime measurement stops muons in a detector and measures the delay to a charged decay product, forming a distribution of decay times. Capture of negative muons by nuclei can change the observed disappearance rate, so material and muon charge matter when interpreting that distribution. Muon decay supplies a precise low-energy weak-interaction test.

The rest mass of the muon, approximately $105.66\mathrm{MeV}$ in natural units. Keeping it in phase space is necessary in precision lepton universality tests.

Ordinary negative-muon decay is ${\mu }^{-}\to e^{-}+{\bar{\nu }}_e+{\nu }_{\mu }$; the positive-muon channel has the conjugate particles. The process proceeds through a weak charged current. At energies much smaller than the W mass, a four-fermion interaction with the Fermi constant describes it.

At leading order, neglecting the electron mass and radiative corrections, the width is $\Gamma =G_F^2{m}_{\mu }^5/(192{\pi }^3)$ in natural units. The fifth power reflects the interaction dimension together with three-body phase space. The electron has a continuous energy distribution because two neutrinos share the missing energy and momentum. Precision extraction of the Fermi constant uses the measured lifetime and includes electron-mass and radiative corrections. A stopping experiment measures disappearance or decay timing, so nuclear capture must be distinguished from free decay when relevant.

The proper mean lifetime of a free muon, approximately $2.2\mu \mathrm{s}$. Independent particle decays give survival probability $\exp (-\Delta \tau /{\tau }_{\mu })$. In a frame where the Lorentz factor is constant, $\Delta \tau =\Delta t/\gamma$; interactions of a stopped negative muon in matter can add capture channels.

A vertically incident particle annihilating with a stationary target into two equal-energy muons gives each half the incident longitudinal momentum, while their transverse momenta are opposite. At height $H$ above a locally flat surface, each proper flight time is $2H{m}_{\mu }{c}^2/[c(p_{\mathrm{in}}c)]$, since $\gamma v_z=p_z/m_{\mu }$. Independent exponential particle decays then give the displayed joint survival probability. Atmospheric energy loss, scattering and curvature are omitted; replacing both trajectories by vertical paths would miss the two-body angular correction.

For an unpolarized parent charged lepton undergoing a three-body leptonic V-A interaction decay, neglecting daughter masses and radiative corrections, the normalized charged-daughter energy distribution in the parent rest frame is $f(x)=2x^2(3-2x)$, where $x=2E_{\ell }^{\ast }/m_{\mathrm{parent}}$ lies in $[0,1]$. Direct integration gives ${\int }_0^{1}f(x)dx=1$ and ${\int }_0^{1}xf(x)dx=7/10$. A specified parent spin orientation also produces angular terms, so this one-variable law alone does not describe polarized samples.

For a negative muon with rest-frame spin vector $\mathbf{s}$, the leading massless-electron distribution is $d\Gamma /(dxd\Omega )=G_F^2{m}_{\mu }^5{x}^2[(3-2x)+(1-2x)\hat{\mathbf{k}}\cdot \mathbf{s}]/(384{\pi }^4)$, where $x=2E_e/m_{\mu }\in [0,1]$. Its final-spin sum is $64G_F^2(k\cdot q^{\prime })(q\cdot (p-m_{\mu }s))$. The spin vector is axial whereas the electron direction is polar, so their scalar product reverses under parity. Its occurrence demonstrates parity violation by the left-chiral weak charged current.

The antimuon is the antiparticle of the muon. It has electric charge $+1$ and lepton number $-1$.