Standard Model and Beyond · Problem 7.10

The Standard Electroweak Theory

Physics — Particle Physics · neutron-decay

Problem

Calculate (a) the neutron lifetime, (b) the e−e^- polarization, and (c) the electron asymmetry with respect to the neutron spin direction in terms of gVg_V and gAg_A defined in (7.121). Work in the approximation that the proton momentum is negligible compared to its mass. Hint: write out the amplitude explicitly using the Pauli-Dirac representation for the spinors and γ\gamma matrices. Use the helicity basis for the leptons. (d) Repeat the lifetime calculation, this time using trace techniques, summing and averaging over all spins, and calculate the eνe\nu angular correlation coefficient αeν\alpha_{e\nu}. The latter is defined by the angular distribution 1+αeνβep^e⋅p^ν1 + \alpha_{e\nu}\beta_e \hat{p}_e \cdot \hat{p}_\nu, where p⃗ν\vec{p}_\nu can be measured using p⃗ν≡−(p⃗p+p⃗e)\vec{p}_\nu \equiv -(\vec{p}_p + \vec{p}_e).
Intuitive explanation

Neutron beta decay is a weak interaction where a neutron transforms into a proton, electron, and antineutrino—a process governed by vector and axial-vector coupling constants gVg_V and gAg_A. The key is to compute the decay amplitude using spinor representations and helicity states, then extract physical observables (lifetime, polarization, correlations) by squaring the amplitude and integrating over phase space. Part (d) shifts to trace methods to efficiently sum over unmeasured spins, revealing how the electron and antineutrino directions correlate through the weak interaction's parity-violating structure.

Intuitive explanation

Neutron beta decay is a weak interaction where a neutron transforms into a proton, electron, and antineutrino—a process governed by vector and axial-vector coupling constants gVg_V and gAg_A. The key is to compute the decay amplitude using spinor representations and helicity states, then extract physical observables (lifetime, polarization, correlations) by squaring the amplitude and integrating over phase space. Part (d) shifts to trace methods to efficiently sum over unmeasured spins, revealing how the electron and antineutrino directions correlate through the weak interaction's parity-violating structure.

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