The Quark Model — Question 3

Question 3

Beta decay changes one valence flavor through a charged current

In neutron beta decay the valence transition is $udd\to uud+e^-+\bar\nu_e$. The elementary charged-current step is $d\to u+W^-$ followed by $W^-\to e^-+\bar\nu_e$; the W is virtual.

1. Check electric charge and baryon number at the quark vertex.

2. Check charge and electron-family number at the lepton vertex and for the full decay.

Question 3 - Solution

Beta decay changes one valence flavor through a charged current

(a) Method and calculation

$$\boxed{-1/3=2/3-1,\quad B:\ 1/3=1/3+0}.$$$$udd\to uud:\quad\boxed{\Delta Q_{\mathrm{hadron}}=+e,\ \Delta B=0}.$$

(b) Interpretation and further calculation

$$W^-\to e^-+\bar\nu_e:\quad\boxed{-1=-1+0,\ L_e=1-1=0}.$$$$\boxed{n\to p+e^-+\bar\nu_e:\quad Q=0,\ B=1,\ L_e=0\ \text{on both sides}}.$$

Physical check

The neutron mass cannot produce a real W. A virtual propagator mediates the low-energy process without violating total energy conservation.