費米耦合常數

費米耦合常數記為${\displaystyle G_{F}}$，是描寫弱相互作用強度的一個物理常數：^[1]

${\displaystyle {\frac {G_{\text{F}}}{(\hbar c)^{3}}}={\frac {\sqrt {2}}{8}}{\frac {g^{2}}{m_{\text{W}}^{2}}}=1.16637(1)\times 10^{-5}\;{\textrm {GeV}}^{-2}\ .}$

這裏g 是弱相互作用的耦合常數, 和 m_W 是W玻色子的質量.

在標準模型中, 費米常數和 希格斯場真空期望值 關係是：${\displaystyle v=({\sqrt {2}}G_{\text{F}})^{\text{-1/2}}\simeq 246.22\;{\textrm {GeV}}}$^[2]

費米常數的測量

費米常數的測量是通過測量${\displaystyle \mu }$的壽命得到的，${\displaystyle \mu }$壽命公式：

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

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

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

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

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

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

參考資料

1. D. Chitwood et al., Phys.Rev.Lett. 99 (2007) 032001
2. Plehn, Tilman; Rauch, Michael. Phys. Rev. D. 2005, 72: 053008. Bibcode:2005PhRvD..72e3008P. arXiv:hep-ph/0507321 . doi:10.1103/PhysRevD.72.053008. 引文使用過時參數coauthors (幫助); 缺少或|title=為空 (幫助)