素粒子標準模型:レプトンセクター:レプトンセクターのまとめ

\begin{eqnarray}
\mathcal{L}_{lepton}
&=&
\sum_{I=1,2,3}
\left[
i\bar{L}_{Ii}\bar{\sigma}^\mu(D\mu L_I)_{i} + i{\ell}_{I_R}^\dagger\sigma^\mu D_\mu {\ell}_{I_R} \right] -\sum_{I,J=1,2,3}y^l_{IJ}\left(
\bar{L}_{Ii}H_i\ell_{J_R}+c.c\right)\\
&\Downarrow&SSB&ユニタリ変換\nonumber\\
&=&\sum_{I=1,2,3}
\left[\bar{\psi}_{\nu_I}i\Slash{\partial}P_L\psi_{\nu_I}
+
\bar{\psi}_{\ell_I}\left(i\Slash{\partial}-m_{\ell_I}\right)\psi_{\ell_I}
-\frac{m_{\ell_I}}{v}h\bar{\psi}_{\ell_I}\psi_{\ell_I}
\right.\nonumber\\
&&~~~~~~~~~~~~~~-e\bar{\psi}_{\ell_I}\gamma^\mu\psi_{\ell_I}A_\mu + \frac{e}{\sqrt{2}\sin\theta_w} \bar{\psi}_{\nu_I}\gamma^\mu P_L\psi_{\ell_I}W^+_\mu + \frac{e}{\sqrt{2}\sin\theta_w} \bar{\psi}_{\ell_I}\gamma^\mu P_L\psi_{\nu_I}W^-_\mu\nonumber\\
&&~~~~~~~~~~~~~~
\left.+
\left(
\frac{e}{2\sin\theta_w\cos
\theta_w}
\left(
\bar{\psi}_{\nu_I}\gamma^\mu P_L\psi_{\nu_I}-
\bar{\psi}_{\ell_I}\gamma^\mu P_L\psi_{\ell_I}
\right)
+
e\tan\theta_w
\bar{\psi}_{\ell_I}\gamma^\mu\psi_{\ell_I}
\right)Z_\mu\right]\nonumber\\
\nonumber\\
&=&
\sum_{I=1,2,3}
\left[\bar{\psi}_{\nu_I}i\Slash{\partial}P_L\psi_{\nu_I}
+
\bar{\psi}_{\ell_I}\left(i\Slash{\partial}-m_{\ell_I}\right)\psi_{\ell_I}
-\frac{m_{\ell_I}}{v}h\bar{\psi}_{\ell_I}\psi_{\ell_I}\right.\nonumber\\
&&~~~~~~~~~~~~~~\left.+
J^\mu_{\ell_I,em}A_\mu+J^\mu_{\ell_I,Z}Z_\mu
+
J^{\mu}_{\ell_I,+}W^-_\mu+J^{\mu}_{\ell_I,-}W^+_\mu\right]
\end{eqnarray}
\begin{eqnarray}
J^{\mu}_{\ell_I,+}&\equiv&
\frac{e}{\sqrt{2}\sin\theta_w} \bar{\psi}_{\ell_I}\gamma^\mu P_L\psi_{\nu_I}\\
J^{\mu}_{\ell_I,-}&\equiv&\frac{e}{\sqrt{2}\sin\theta_w}\bar{\psi}_{\nu_I}\gamma^\mu P_L\psi_{\ell_I}\\
J^\mu_{\ell_I,em}&\equiv&-e\bar{\psi}_{\ell_I}\gamma^\mu\psi_{\ell_I}\\
J^\mu_{\ell_I,Z}&\equiv&\frac{e}{2\sin\theta_w\cos
\theta_w}
\left(
\bar{\psi}_{\nu_I}\gamma^\mu P_L\psi_{\nu_I}
-\bar{\psi}{\ell_I}\gamma^\mu P_L\psi{\ell_I}
\right)
+
e\tan\theta_w
\bar{\psi}_{\ell_I}\gamma^\mu\psi_{\ell_I}\nonumber\\
&=&
\frac{e}{2\sin\theta_w\cos
\theta_w}
\left(
\bar{\psi}_{\nu_I}\gamma^\mu P_L\psi_{\nu_I}
-\bar{\psi}_{\ell_I}\gamma^\mu P_L\psi_{\ell_I}
\right)
\tan\theta_w J^\mu_{\ell_I,em}
\end{eqnarray}

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