PX156 - J2 - decay channels

• particles can decay to different final states with different characteristic decay rates, ${\mathrm{\Gamma }}_i$
• each different final state is called a 'channel'
• each ${\mathrm{\Gamma }}_i:$ 'partial decay rate'
• the total decay rate:

• to understand which decay channels dominate in a particle's decay scheme, the branching fraction is used

branching fraction

• branching fraction: for a given channel, $i$, its branching fraction is:

• AKA branching ratio

• eg: what is $BF(Z^0\to \nu \overline{\nu })?$
  • assuming $\mathrm{\Gamma }(X^0\to l\overline{l})=\mathrm{\Gamma }(Z^0\to q\overline{q})=\mathrm{\Gamma }(Z^0\to \nu \overline{\nu })$
  $$BF(Z^0\to \nu \overline{\nu })=\frac{{\mathrm{\Gamma }}_i(Z^0\to \nu \overline{\nu })}{{\mathrm{\Gamma }}_{tot}}$$
  • numerator: since there are 3 different flavours of neutrinos:
  $${\mathrm{\Gamma }}_i(Z^0\to \nu \overline{\nu })=3\mathrm{\Gamma }(Z^0\to {\nu }_e{\overline{\nu }}_e)$$
  • denominator:
  $${\mathrm{\Gamma }}_{tot}=\sum _{i=1}^N{\mathrm{\Gamma }}_i=N\mathrm{\Gamma }(Z^0\to {\nu }_e{\overline{\nu }}_e)$$
  • what is $N$?
    • $\mathrm{\Gamma }(Z^0\to l\overline{l}):3:l=e,\mu ,\tau$
    • $\mathrm{\Gamma }(Z^0\to \nu \overline{\nu }):3:{\nu }_x,x=e,\mu ,\tau$
    • $\mathrm{\Gamma }(Z^0\to q\overline{q}):$
      • $m(Z^0)<<2m_{top}⟹$ ignore top
      • no flavour changing neutral currents
      • $q=u,d,s,c,b$
      • quark channels: $5\times$number of colours $=15$
  $$\begin{array}{rl}BF(Z^0\to \nu \overline{\nu })& =\frac{3\mathrm{\Gamma }(Z^0\to {\nu }_e{\overline{\nu }}_e)}{N\mathrm{\Gamma }(Z^0\to {\nu }_e{\overline{\nu }}_e)}\\ & =\frac{3}{N}\\ & =\frac{3}{3+3+15}\\ & =\frac{1}{7}\end{array}$$