標準模型」タグアーカイブ

素粒子標準模型:おまけ:折りたたまれている項の展開

ここでほとんどの項はこのまま解析出来るが、\(~\left(
D_\mu W_\nu^+-D_\nu W_\mu^+
\right)
\left(
D^{\dagger\mu}W^{-\nu}-D^{\dagger\nu}W^{-\mu}
\right)~\)に関しては、多くの項が折り畳まれているので分解しておく。
\begin{eqnarray}
D_\mu W^+_\nu-D_\nu W^+_\mu
=
\partial_\mu W^+_\nu-\partial_\nu W^+_\mu
-ie\cot\theta_w(Z_\mu W^+_\nu-Z_\nu W^+_\mu)
-ie(A_\mu W^+_\nu-A_\nu W^+_\mu)
\end{eqnarray}
より
\begin{eqnarray}
&&\left(
D_\mu W_\nu^+-D_\nu W_\mu^+
\right)
\left(
D^{\dagger\mu}W^{-\nu}-D^{\dagger\nu}W^{-\mu}
\right)\\
&=&
(\partial_\mu W^+_\nu-\partial_\nu W^+_\mu)
(\partial^\mu W^{-\nu}-\partial^\nu W^{-\mu})\nonumber\\
&&+e^2\cot^2\theta_w(Z_\mu W^+_\nu-Z_\nu W^+_\mu)(Z^\mu W^{-\nu}-Z^\nu W^{-\mu})
+
e^2(A_\mu W^+_\nu-A_\nu W^+_\mu)(A^\mu W^{-\nu}-A^\nu W^{-\mu})\nonumber\\
&&
+e^2\cot\theta_w(A_\mu W^+_\nu-A_\nu W^+_\mu)(Z^\mu W^{-\nu}-Z^\nu W^{-\mu})
+
e^2\cot\theta_w
(Z_\mu W^+_\nu-Z_\nu W^+_\mu)(A^\mu W^{-\nu}-A^\nu W^{-\mu})\nonumber\\
&&
+
ie(\partial_\mu W^+_\nu-\partial_\nu W^+_\mu)
((\cot\theta_w Z^\mu+A^\mu)W^{-\nu}-(\cot\theta_w Z^\nu+A^\nu)W^{-\mu})\nonumber\\
&&-
ie(\partial_\mu W^-_\nu-\partial_\nu W^-_\mu)
((\cot\theta_w Z^\mu+A^\mu)W^{+\nu}-(\cot\theta_w Z^\nu+A^\nu)W^{+\mu})\\
\nonumber\\
&=&
(\partial_\mu W^+_\nu-\partial_\nu W^+_\mu)
(\partial^\mu W^{-\nu}-\partial^\nu W^{-\mu})\nonumber\\
&&+2e^2\cot^2\theta_w
(Z_\mu Z^\mu W^+_\nu W^{-\nu}-Z_\mu W^{+\mu}Z_\nu W^{-\nu})
+
2e^2(A_\mu A^\mu W^+_\nu W^{-\nu}-A_\mu W^{+\mu}A_\nu W^{-\nu})\nonumber\\
&&
+2e^2\cot\theta_w
(A_\mu Z^\mu W_\nu^+ W^{-\nu}-A_\mu W^{-\mu}Z_\nu W^{+\nu}+h.c)
\nonumber\\
&&
+
2ie
(
(\cot\theta_w Z^\mu+A^\mu)W^{-\nu}\partial_\mu W^+_\nu

(\cot\theta_w Z^\mu+A^\mu)W^{-\nu}\partial_\nu W^+_\mu
)+h.c
\end{eqnarray}
以上より、ラグランジアンの中に入っている時の係数を考慮して、以下の結果を得る。
\begin{eqnarray}
&&-\frac{1}{2}\left(
D_\mu W_\nu^+-D_\nu W_\mu^+
\right)
\left(
D^{\dagger\mu}W^{-\nu}-D^{\dagger\nu}W^{-\mu}
\right)\\
&=&
-\frac{1}{2}(\partial_\mu W^+_\nu-\partial_\nu W^+_\mu)
(\partial^\mu W^{-\nu}-\partial^\nu W^{-\mu})\nonumber\\
&&-e^2\cot^2\theta_w
(Z_\mu Z^\mu W^+_\nu W^{-\nu}-Z_\mu W^{+\mu}Z_\nu W^{-\nu})
-e^2(A_\mu A^\mu W^+_\nu W^{-\nu}-A_\mu W^{+\mu}A_\nu W^{-\nu})\nonumber\\
&&
-e^2\cot\theta_w
(A_\mu Z^\mu W_\nu^+ W^{-\nu}-A_\mu W^{-\mu}Z_\nu W^{+\nu}+h.c)
\nonumber\\
&&
-ie
(
(\cot\theta_w Z^\mu+A^\mu)W^{-\nu}\partial_\mu W^+_\nu

(\cot\theta_w Z^\mu+A^\mu)W^{-\nu}\partial_\nu W^+_\mu
)+h.c
\end{eqnarray}

素粒子標準模型:標準模型のまとめ

今までの結果を纏めると以下のラグランジアンに帰着する。
\begin{eqnarray}
\mathcal{L}_{SM}
&=&
\frac{1}{2}\partial_\mu h\partial^\mu h
+
\frac{M^2_Z}{2v^2}Z_\mu Z^\mu\left(2vh+h^2\right)
+
\frac{M_W^2}{v^2}W_\mu^+W^{-\mu}(2vh+h^2)
-\frac{1}{2}m_h^2h^2-\frac{1}{4}\lambda vh^3-\frac{1}{16}\lambda h^4
\nonumber\\
&&
-\frac{1}{2}\left(
D_\mu W_\nu^+-D_\nu W_\mu^+
\right)
\left(
D^{\dagger\mu}W^{-\nu}-D^{\dagger\nu}W^{-\mu}
\right)
-\frac{1}{4}Z_{\mu\nu}Z^{\mu\nu}
-\frac{1}{4}
A_{\mu\nu}A^{\mu\nu}\nonumber\\
&&
+
ig_2(
c_wZ_{\mu\nu}
+
s_wA_{\mu\nu}
)
W^{+\mu}W^{-\nu}
+
\frac{1}{2}g^2_2\left(
W_\mu^-W^{-\mu}
W_\nu^+W^{+\nu}-
W_\mu^-W^{+\mu}
W_\nu^+W^{-\nu}
\right)\nonumber\\
&&
+
\frac{1}{2}M_Z^2Z_\mu Z^\mu
+
M^2_WW_\mu^+W^{-\mu}-\frac{1}{4}G^a_{\mu\nu}G^{a\mu\nu}\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\\
&&+
\sum_{I,J=1,2,3}\left[
\bar{\psi}^F_{u_I}((i\Slash{\nabla}-m_{u_I})\psi_{u_I})^F
+
\bar{\psi}^F_{d_I}((i\Slash{\nabla}-m_{d_I})\psi_{d_I})^F
-\frac{m_{u_I}}{v}h\bar{\psi}^F_{u_I}\psi^F_{u_I}
-\frac{m_{d_I}}{v}h\bar{\psi}^F_{d_I}\psi^F_{d_I}
\right.\nonumber\\
&&~~~~~~~~~~~~~~~~~~~
+
\left(
\frac{2e}{3}\bar{\psi}^F_{u_I}\gamma^\mu\psi_{u_I}^F-
\frac{e}{3}\bar{\psi}^F_{d_I}\gamma^\mu\psi_{d_I}^F
\right)A_\mu
+\frac{e}{\sqrt{2}\sin\theta_w}\left[
U_{IJ}\bar{\psi}_{u_I}^F\gamma^\mu P_L\psi^F_{d_J}W^+_\mu + U^\dagger_{IJ}\bar{\psi}_{d_I}^F\gamma^\mu P_L\psi^F_{u_J}W^-_\mu \right]
\nonumber\\
&&~~~~~~~~~~~~~~~~~~~\left.
+\left\{
\frac{e}{2\sin\theta_w\cos\theta_w}\left(
\bar{\psi}^F_{u_I}\gamma^\mu P_L\psi^F_{u_I}
-\bar{\psi}^F_{d_I}\gamma^\mu P_L\psi^F_{d_I}
\right)
-\frac{2e\tan\theta_w}{3}
\bar{\psi}^F_{u_I}\gamma^\mu{\psi}^F_{u_I}
+\frac{e\tan\theta_w}{3}
\bar{\psi}^F_{d_I}\gamma^\mu{\psi}^F_{d_I}
\right\}Z_\mu\right]\nonumber\\
\end{eqnarray}