粒子衰變

${\displaystyle c=\hbar =1}$

粒子壽命列表

μ子 / 反μ子 ${\displaystyle \mu ^{-}\,/\,\mu ^{+}}$ 105.6 ${\displaystyle 2.2\times 10^{-6}}$
τ子 / 反τ子 ${\displaystyle \tau ^{-}\,/\,\tau ^{+}}$ 1777 ${\displaystyle 2.9\times 10^{-13}}$

Z玻色子 ${\displaystyle Z^{0}\,}$ 91,000 ${\displaystyle 10^{-25}}$

生還概率

${\displaystyle P(t)=e^{-t/(\gamma \tau )}}$

${\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}}$為該粒子的勞侖茲因子

衰變率

${\displaystyle d\Gamma _{n}={\frac {(2\pi )^{4}}{2M}}\left|{\mathcal {M}}\right|^{2}d\Phi _{n}(P;p_{1},p_{2},\dots ,p_{n})}$

n為原衰變所生成的粒子數，
${\displaystyle {\mathcal {M}}\,}$為連接始態與終態的不變矩陣上的元，
${\displaystyle d\Phi _{n}\,}$ 為相空間的元，及
${\displaystyle p_{i}\,}$為粒子i四維動量

${\displaystyle d\Phi _{n}(P;p_{1},p_{2},\dots ,p_{n})=\delta ^{4}(P-\sum _{i=1}^{n}p_{i})\left(\prod _{i=1}^{n}{\frac {d^{3}{\vec {p}}_{i}}{(2\pi )^{3}2E_{i}}}\right)\,}$

${\displaystyle \delta ^{4}\,}$為四維的狄拉克δ函數

三體衰變

${\displaystyle d\Phi _{3}={\frac {1}{(2\pi )^{9}}}\delta ^{4}(P-p_{1}-p_{2}-p_{3}){\frac {d^{3}{\vec {p}}_{1}}{2E_{1}}}{\frac {d^{3}{\vec {p}}_{2}}{2E_{2}}}{\frac {d^{3}{\vec {p}}_{3}}{2E_{3}}}\,}$

四維動量

${\displaystyle p^{2}=E^{2}-({\vec {p}})^{2}=m^{2}\quad \quad \quad \quad (1)\,}$

${\displaystyle p^{2}=\left(p_{1}+p_{2}\right)^{2}=p_{1}^{2}+p_{2}^{2}+2p_{1}p_{2}=m_{1}^{2}+m_{2}^{2}+2(E_{1}E_{2}-{\vec {p}}_{1}\cdot {\vec {p}}_{2})\,}$

四維動量守恆

${\displaystyle p_{\mathrm {i} }=p_{\mathrm {f} }}$

在二體衰變中

${\displaystyle p_{M}=p_{1}+p_{2}}$

${\displaystyle p_{M}-p_{1}=p_{2}}$

${\displaystyle p_{M}^{2}+p_{1}^{2}-2p_{M}p_{1}=p_{2}^{2}}$

${\displaystyle M^{2}+m_{1}^{2}-2\left(E_{M}E_{1}-{\vec {p}}_{M}\cdot {\vec {p}}_{1}\right)=m_{2}^{2}.\quad \quad \quad \quad (2)\,}$

• ${\displaystyle {\vec {p}}_{M}=0\,}$，及
• ${\displaystyle E_{M}=M\,}$

${\displaystyle M^{2}+m_{1}^{2}-2ME_{1}=m_{2}^{2}.\,}$

${\displaystyle E_{1}={\frac {M^{2}+m_{1}^{2}-m_{2}^{2}}{2M}}.\quad \quad \quad \quad (3)\,}$

${\displaystyle E_{2}={\frac {M^{2}+m_{2}^{2}-m_{1}^{2}}{2M}}}$

${\displaystyle |{\vec {p}}_{1}|=|{\vec {p}}_{2}|={\frac {\sqrt {\left[M^{2}-\left(m_{1}+m_{2}\right)^{2}\right]\left[M^{2}-\left(m_{1}-m_{2}\right)^{2}\right]}}{2M}}.\,}$

${\displaystyle {\vec {p_{1}}}^{2}={\frac {(M^{2}+m_{1}^{2}-m_{2}^{2})^{2}-4m_{1}^{2}M^{2}}{4M^{2}}}\,}$
${\displaystyle {\vec {p_{1}}}^{2}={\frac {M^{4}+m_{1}^{4}+m_{2}^{4}-2m_{1}^{2}M^{2}-2m_{2}^{2}M^{2}-2m_{1}^{2}m_{2}^{2}}{4M^{2}}}\,}$
${\displaystyle {\vec {p_{1}}}^{2}={\frac {M^{4}-M^{2}(m_{1}+m_{2})^{2}-M^{2}(m_{1}-m_{2})^{2}+(m_{1}^{2}-m_{2}^{2})^{2}}{4M^{2}}}\,}$
${\displaystyle {\vec {p_{1}}}^{2}={\frac {M^{2}\left[M^{2}-(m_{1}-m_{2})^{2}\right]-(m_{1}+m_{2})^{2}\left[M^{2}-(m_{1}-m_{2})^{2}\right]}{4M^{2}}}\,}$
${\displaystyle |{\vec {p}}_{1}|={\frac {\sqrt {\left[M^{2}-\left(m_{1}+m_{2}\right)^{2}\right]\left[M^{2}-\left(m_{1}-m_{2}\right)^{2}\right]}}{2M}}.\,}$

${\displaystyle |{\vec {p}}_{2}|\,}$的推導也一樣。

二體衰變

...而在實驗室系中，母粒子大概以接近光速的速度移動，因此所發射的兩粒子，其角度會與質心系的不一樣。

從兩個不同的參考系

${\displaystyle \tan {\theta '}={\frac {\sin {\theta }}{\gamma \left(\beta /\beta '+\cos {\theta }\right)}}}$

衰變率

${\displaystyle |{\vec {p}}_{1}|=|{\vec {p_{2}}}|={\frac {[(M^{2}-(m_{1}+m_{2})^{2})(M^{2}-(m_{1}-m_{2})^{2})]^{1/2}}{2M}}}$

${\displaystyle d^{3}{\vec {p}}=|p|^{2}\,dpd\Omega =p^{2}\,d\phi \,d\left(\cos \theta \right)}$

${\displaystyle d\Gamma ={\frac {1}{32\pi ^{2}}}\left|{\mathcal {M}}\right|^{2}{\frac {|{\vec {p}}_{1}|}{M^{2}}}\,d\phi _{1}\,d\left(\cos \theta _{1}\right)}$