Gauge fixing and ghost terms for the t'Hooft-Feynman gauge

In the case of t'Hooft-Feynman gauge the gauge fixing terms are

$$-\frac{1}{2} (\partial^\mu A_\mu)^2 -\frac{1}{2} (\partial... ...Z\, Z_f)^2 -\vert \partial^\mu W^+_\mu + M_W\, W^+_f\vert^2\;.$$

The squared divergences of fields transform the quadratic part of Lagrangian for vector bosons to a diagonal form like (4). The squared Goldstone field terms give a mass to the Goldstone particle equal to the mass of the corresponding vector boson field. The off-diagonal quadratic terms, which follow from the gauge fixing Lagrangian, cancel the off-diagonal terms (17) up to complete divergency terms.

According to the general rule (3) the Faddeev-Popov Lagrangian is

$$\begin{eqnarray*} && -A_{\bar{c}} D(W^+c,W^-_c,A_c,Z_c)(\partial^\mu A_\mu) \\ ... ...c}} D(W^+c,W^-_c,A_c,Z_c)(\partial^\mu W^-_\mu + M_W\,W^-_f)\;. \end{eqnarray*}$$

Note that due to (9)

$$\begin{eqnarray*} \hat D W^+_{\mu}& =& i\,g_2\, ( W^+_\mu( \sin{\Theta_w} A_c+ \... ...s{\Theta_w} (W^+_\mu W^-_c - W^-_\mu W^+_c) +\partial_\mu Z_c\;, \end{eqnarray*}$$

and according to (14)

$$\begin{eqnarray*} D W^+_f &=& W^+_c M_W + \frac{g_2}{2} ( W^+_c ( H -i\,Z_f) +... ...ac{Z_c H} {\cos{\Theta_w}} - i(W^-_c W^+_f - W^+_c W^-_f) )\;. \end{eqnarray*}$$

After substitution of these derivatives to the Faddeev-Popov Lagrangian we see that it contains the quadratic part:

$$-A_{\bar{c}}\Box A_c -Z_{\bar{c}}\Box Z_c - W^-_{\bar{c}}(\Box W^+_c +M_W W^+_c) - W^+_{\bar{c}}(\Box W^-_c +M_W W^-_c)\;,$$

and the following vertices of interaction:

$$-g_2( i \,\sin{\Theta_w} (\partial^\mu A_{\bar{c}}) (W^+_\mu W^-_c - W^-_\mu W^+_c)$$

$$+ i\,\cos{\Theta_w} (\partial^\mu Z_{\bar{c}}) (W^+_\mu W^-_c - W^-_\mu W^+_c)$$

$$+ \frac{M_Z Z_{\bar{c}}}{2}\left(\frac{Z_c H}{\cos{\Theta_w}} - i(W^-_c W^+_f - W^+_c W^-_f) \right)$$

$$- i(\partial^\mu W^-_{\bar{c}}) ( W^+_\mu( \sin{\Theta_w} A_c+ \cos{\Theta_w} Z_c) -W^+_c(\sin{\Theta_w}A_\mu+\cos{\Theta_w} Z_\mu))$$

$$+ \frac{M_W W^-_{\bar{c}}}{2} ( W^+_c ( H -i\,Z_f) + i\,(2 \sin{\Theta_w} A_c + \cos{\Theta_w} (1-\tan^2{\Theta_w}) Z_c) \,W^+_f) ) )$$

$$+ i(\partial^\mu W^+_{\bar{c}}) ( W^-_\mu( \sin{\Theta_w} A_c+ \cos{\Theta_w} Z_c) -W^-_c(\sin{\Theta_w}A_\mu+\cos{\Theta_w} Z_\mu))$$

$$+ \frac{M_W W^+_{\bar{c}}}{2} ( W^-_c ( H + i\,Z_f) - i\,(2 \sin{\Theta_w} A_c + \cos{\Theta_w} (1-\tan^2{\Theta_w}) Z_c) \,W^-_f) ) )$$

$$)\;.$$ (18)