Chapter 11 - Weak Interaction

Two final states have parity +1 and -1

Evidence of neutrino

Electron energy spectrum is continuous!

Beta decay: $n \rightarrow p + e^- + \bar{\nu}$

Pauli: "neutrino" is chargeless and massless

Fermi: 4-fermion theory; $G_F$ is the interaction strength.

KATRIN: measures neutrino mass through tritium ($^3H$ or T ) $ T_2 \rightarrow HeH^+ + e^- + \bar{\nu}_e $

Constrains effective $\nu_e$ mass: $ m_\nu^2 = \sum |U_{ei}|^2 m_i^2 $

Result (2022): $ m_\nu < 0.8 eV $

First observation of neutrino

Channel: inverse beta decay: $ \bar{\nu}_e + p \rightarrow n + e^+ $

Use water diluted with $ CdCl_2 $, where Cd has large neutron capture rate (second to Gadolinium) Gd)

Distinct signature: $ Cd + n \rightarrow Cd + \gamma $, and $e^+ + e^- \rightarrow \gamma \gamma $

Two light signals with a time separation are produced, detected by scintillators.

Parity violation

What is parity?

Spatial inversion: $ x \rightarrow -x $

Conserved in strong and EM interactions

Suspected violation: $\tau-\theta$ puzzle

Same mass, spin-0, but decay to either $\pi \pi$ or $\pi \pi \pi $

P violation in weak interaction

Two final states have parity +1 and -1

Now we know they are $ K_s$ and $K_L$

Align spin of $Co_{60}$ with B field.

During beta decay, electrons are emitted mostly opposite to the spin direction.

V-A structure

$ ^{60}Co \rightarrow ^{60}Ni^* + e^- + \bar{\nu}_e $

All possible Lorentz-Invariant interaction vertices: 5 bilinear forms

scalar $\bar{\psi} \phi $

pseudoscalar $ \bar{\psi} \gamma^5 \phi $

vector $ \bar{\psi} \gamma^\mu \phi $

axial vector $ \bar{\psi} \gamma^\mu \gamma^5 \phi $

tensor $ \bar{\psi} T^{\mu\nu} \phi $

From experiment: V－A. E.g. Michel parameter parametrized from {S, P, V, A, T} is $\rho \approx 0.75 $, agreeing with V-A prediction

vertex factor: $ \frac{-i g_w}{\sqrt{2}}\frac{1}{2}\gamma^\mu(1-\gamma^5) $

Mediator: W boson, spin-1. Propagator: $\frac{-i g_{\mu\nu}}{q^2 - m_w^2} $

Chiral structure. Chirality is the eigenvalue of $\gamma^5 $, which is -1 for V, A, +1 for S, P, T. Helicity (projection of spin on momentum direction) = chirality for massless particles.

$P_L = \frac{1}{2}(1-\gamma^5) $, $P_R = \frac{1}{2}(1+\gamma^5) $

Because $P_L$ is in the vertex factor, only LH particle and RH anti-particle chiral states participate! Parity is maximally violated!

After relating Fermi's theory to Feynman rules, we find $\frac{G_F}{\sqrt{2}} = \frac{g_w^2}{8m_w^2} $

From muon lifetime (2.2 $\mu s$) measurement, find $ G_F = 1.16 \times 10^{-5} GeV^{-2} $. Then $ \alpha_w = \frac{g_w^2}{4\pi} = \frac{1}{30} > \alpha_{EM} $

Helicity in Pion decay

Unitarity crisis: scattering cross-section blows up at high energy!

Weak force is also flavor changing (strange quark decay)

Pion decay chanel: $ \pi^- \rightarrow e^-\bar{\nu}_e $, and $ \pi^- \rightarrow \mu^-\bar{\nu}_\mu $

But decay rate to electron / muon is $ 10^{-4} $ !!

$ \bar{\nu} $ is always RH (both chirality and helicity); pion is spin-0, so lepton is also RH in helicity.

A RH helicity state can be decomposed: $ u_\uparrow = \frac{1}{2}(1+\frac{p}{E+m})u_R + \frac{1}{2}(1 - \frac{p}{E+m})u_L $

Therefore, the only relevant term to weak interaction is $ M \sim \frac{1}{2}(1 - \frac{p_l}{E_l + m_l}) $

If the lepton is highly relativistic (e.g. electron), decay rate is suppressed.

Cowan-Reiens experiment (~1951)

Wu's experiment (1956)