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Chapter 11 - Weak Interaction, First observation of neutrino, Parity…
Chapter 11 - Weak Interaction
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.
Unitarity crisis: scattering cross-section blows up at high energy!
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): \( m_\nu < 0.8 eV \)
Weak force is also
flavor changing
(strange quark decay)
V-A structure
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
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
.
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.
Cowan-Reiens experiment (~1951)
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 \(\pi \pi \pi \)
Two final states have parity +1 and -1
Now we know they are \( K_s\) and \(K_L\)
P violation in weak interaction
Align spin of \(Co_{60}\) with B field.
During beta decay, electrons are emitted mostly
opposite to the spin direction
.
\( ^{60}Co \rightarrow ^{60}Ni^* + e^- + \bar{\nu}_e \)
Wu's experiment (1956)