费米耦合常数

费米常数的测量是通过测量${\displaystyle \mu }$的寿命得到的，${\displaystyle \mu }$寿命公式：

${\displaystyle {\frac {\hbar }{\tau _{\mu }}}={\frac {G_{F}^{2}m_{\mu }^{5}}{192\pi ^{2}}}F(\rho ){\biggl [}1+H_{1}(\rho ){\frac {{\widehat {\alpha }}(m_{\mu })}{\pi }}+H_{2}(\rho ){\frac {{\widehat {\alpha }}^{2}(m_{\mu })}{\pi ^{2}}}{\biggl ]},}$

这里${\displaystyle \rho ={\frac {m_{e}^{2}}{m_{\mu }^{2}}}}$和

${\displaystyle F(\rho )=1-8\rho +8\rho ^{3}-\rho ^{4}-12\rho ^{2}\ln \rho =0.99981295,}$

${\displaystyle H_{1}(\rho )={\frac {25}{8}}-{\frac {\pi ^{2}}{2}}-{\Big (}9+4\pi ^{2}+12\ln \rho {\Big )}\rho +16\pi ^{2}\rho ^{3/2}+{\mathcal {O}}(\rho ^{2})=-1.80793,}$

${\displaystyle H_{2}(\rho )={\frac {156815}{5184}}-{\frac {518}{81}}\pi ^{2}-{\frac {895}{36}}\zeta (3)+{\frac {67}{720}}\pi ^{4}+{\frac {53}{3}}\pi \ln 2-(0.42\pm 0.002)_{\text{had}}-{\frac {5}{4}}\pi ^{2}{\sqrt {\rho }}+{\mathcal {O}}(\rho )=6.64,}$

${\displaystyle {\widehat {\alpha }}(m_{\mu })^{-1}=\alpha ^{-1}+{\frac {1}{3\pi }}\ln \rho +{\mathcal {O}}(\alpha )=135.901}$