# Decay Rates: Pions u dbar Look at pion branching fractions (BF)

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Decay Rates: Pions u dbar Look at pion branching fractions (BF)
The Beta decay is the easiest. ~Same as neutron beta decay Q= 4.1 MeV. Assume FT=1600 s LogF=3.2 (from plot) F= 1600 gives partial width(-1) T=1600/F=1 sec or partial width = 1 sec-1 u dbar P461 - particles IV

Pi Decay to e-nu vs mu-nu
Depends on phase space and spin factors in pion rest frame pion has S=0 2 spin=1/2 combine to give S=0. Nominally can either be both right-handed or both left-handed But parity violated in weak interactions. If m=0 ---> all S=1/2 particles are LH and all S=1/2 antiparticles are RH neutrino mass = > LH electron and muon mass not = 0 and so can have some “wrong” helicity. But easier for muon as heavier mass L+ nu P461 - particles IV

Polarization of Spin 1/2 Particles
Obtain through Dirac equation and polarization operators. Polarization defined the degree of polarization then depends on velocity. The fraction in the “right” and “wrong” helicity states are: fraction “wrong” = 0 if m=0 and v=c for a given energy, electron has higher velocity than muon and so less likely to have “wrong” helicity P461 - particles IV

Pion Decay Kinematics 2 Body decay. Conserve energy and momentum
can then calculate the velocity of the electron or muon look at the fraction in the “wrong” helicity to get relative spin suppression of decay to electrons P461 - particles IV

Pion Decay Phase Space Lorentz invariant phase space plus energy and momentum conservation gives the 2-body phase space factor (partially a computational trick) as the electron is lighter, more phase space (3.3 times the muon) Branching Fraction ratio is spin suppression times phase space P461 - particles IV

Muon Decay Almost 100% of the time muons decay by
Q(muon decay) > Q(pion->muon decay) but there is significant spin suppression and so muon’s lifetime ~100 longer than pions spin 1/2 muon -> 1/2 mostly LH (e) plus /2 all LH( nu) plus 1/2 all RH (antinu) 3 body phase space and some areas of Dalitz plot suppressed as S=3/2 electron tends to follow muon direction and “remember” the muon polarization. Dirac equation plus a spin rotation matrix can give the angular distribution of the electron relative to the muon direction/polarization P461 - particles IV

Detecting Parity Violation in muon decay
Massless neutrinos are fully polarised, P=-1 for neutrino and P=+1 for antineutrino (defines helicity) Consider + + e+ decay. Since neutrinos are left-handed PH=-1, muons should also be polarised with polarisation P=-v/c (muons are non-relativistic, so both helicity states are allowed). If muons conserve polarisation when they come to rest, the electrons from muon decay should also be polarised and have an angular dependence: n m+ p+ Jn Jm p+  m+ + nm e+ n m+ Je Jn Jm m+  e+ + ne + nm P461 - particles IV

Parity violation in + + e+ decay
Experiment by Garwin, Lederman, Weinrich aimed to confirm parity violation through the measurements of I(q) for positrons. 85 MeV pion beam (+ ) from cyclotron. 10% of muons in the beam: need to be separated from pions. Pions were stopped in the carbon absorber (20 cm thick) Counters 1-2 were used to separate muons Muons were stopped in the carbon target below counter 2. P461 - particles IV

Parity violation in + + e+ decay
Positrons from muon decay were detected by a telescope 3-4, which required particles of range >8 g/cm2 (25 MeV positrons). Events: concidence between counters 1-2 (muon) plus coincidence between counters 3-4 (positron) delayed by ms. Goal: to measure I(q) for positrons. Conventional way: move detecting system (telescope 3-4) around carbon target measuring intensities at various q. But very complicated. More sophisticated method: precession of muon spin in magnetic field. Vertical magnetic field in a shielded box around the target. The intensity distribution in angle was carried around with the muon spin. P461 - particles IV

Results of the experiment by Garwin et al.
Changing the field (the magnetising current), they could change the rate (frequency) of the spin precession, which will be reflected in the angular distribution of the emitted positrons. Garwin et al. plotted the positron rate as a function of magnetising current (magnetic field) and compared it to the expected distribution: P461 - particles IV

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