Past exam of the particle physics course of the University of Oxford / 2015 / 7 / c

In ${\tau }^{-}\to {\ell }^{-}{\bar{\nu }}_{\ell }{\nu }_{\tau }$, charge and missing four-momentum show an invisible neutral component beyond the visible charged lepton. The distributions are consistent with three-body V-A interaction decay; hadronic modes such as ${\tau }^{-}\to {\pi }^{-}{\nu }_{\tau }$ instead have two-body missing-mass kinematics. Production and decay accounting assigns $L_{\tau }({\tau }^{-})=L_{\tau }({\nu }_{\tau })=1$ and their antiparticles minus one. Searches for wrong-flavour and neutrinoless channels constrain violations. These observations establish a useful approximately conserved tau-family number in ordinary interactions, rather than proving an exact law from missing energy alone.

To establish that the invisible particle is distinct from ${\nu }_e$ and ${\nu }_{\mu }$, combine decay evidence with the absence of unsuppressed tau production by conventional electron- and muon-flavour beams, and direct tau neutrino charged current interactions. DONUT observed the short displaced tau lepton decay in nuclear emulsion after a neutrino interaction. ^[ref] Neutrino oscillation means individual family numbers are not exact conserved quantities during propagation, even though they are an excellent approximation for these local decay assignments.

Write the leptonic partial decay width asHere $G_F$ is defined from muon decay, so the coupling denominator fixes the convention. Measure ${\Gamma }_{\ell }=B_{\ell }/{\tau }_{\tau }$. The ratio $B_{\mu }/B_e$ tests $|g_{\mu }/g_e{|}^2$ after the known phase space and radiative corrections. Comparing $B_e/{\tau }_{\tau }$ with the muon lifetime and the fifth power of the mass ratio tests $|g_{\tau }/g_{\mu }{|}^2$. Hadronic comparisons such as tau-to-pion versus pion-to-muon decay supply additional universality tests with their own hadronic and electromagnetic corrections.

Use a high-multiplicity hadronic mode with only one invisible neutrino, and reconstruct the total visible four-momentum $p_h$. In the tau lepton rest frame,Fit the endpoint of the hadronic invariant mass distribution, preferably together with its laboratory energy boundary at known beam energy. The two-dimensional boundary uses more information than the single largest observed mass. Detector smearing, unseen neutral products, backgrounds, beam radiation and hadronic spectral modeling must be included. Absence of a displaced endpoint generally sets an upper bound rather than a positive neutrino mass measurement. Strictly the flavour state contains several mass eigenstates; an unresolved endpoint constrains the corresponding weighted mass spectrum.