Документ взят из кэша поисковой машины. Адрес оригинального документа : http://theory.sinp.msu.ru/comphep_html/tutorial/node106.html
Дата изменения: Wed Aug 9 20:40:47 2000
Дата индексирования: Mon Oct 1 22:51:53 2012
Кодировка:

Lagrangian of Higgs field

In the framework of renormalizable field theory the general expression for the gauge invariant Higgs Lagrangian is:

$$L_{Higgs}= (\nabla_\mu \Phi)^+(\nabla^\mu \Phi) -\, \frac{\lambda}{2} (\Phi^+ \Phi - \frac{1}{2} \phi_0^2)^2 \;,$$

where $\lambda$ is a new coupling constant.

Some perturbations of the Higgs vacuum can be realized by means of the local gauge transformation which do not correspond to physical degrees of freedom. To separate physical and gauge degrees of freedom for a small perturbation of the Higgs vacuum we present $\Phi$ in the following form:

$$\Phi= \left(\begin{array}{c} i\,W^+_f \\ (\phi_0 + H -i\,Z_f)/\sqrt{2} \end{array}\right)\;.$$ (12)

Here $W^+_f$ and $Z_f$ are the unphysical Goldstone fields corresponding to variation of the Higgs vacuum caused by the $D(w^+,0,0,0)$ and $D(0,0,0,z)$ gauge transformations. The real field $H$ corresponds to a physical degree of freedom which is assoshated with the Higgs patricle.

The term $(-\, \frac{\lambda}{2}( \Phi^+ \Phi - \frac{1}{2} \phi_0^2)^2)$ contains the mass term for the $H$ field:

$$-\, \frac{\lambda}{2} (\phi_0)^2 \, H^2 \;,$$

and the following terms of self-interaction for $H$ and the Goldstone fields:

$$-\, \frac{\lambda}{2} \left( (W^+_f W^-_f + ( H^2 + Z_f^2)/2)^2 + 2\phi_0 H(W^+_f W^-_f + ( H^2 + Z_f^2)/2) \right)\;.$$ (13)

Usually $\lambda$ is expressed via the mass of $H$-boson $M_H$:

$$\lambda=\left(\frac{M_H}{\phi_0}\right)^2\;.$$

Let us remind that

$$\nabla_\mu \Phi = \partial_\mu \Phi - \hat D(W+_\mu,W-_\mu,A_\mu,Z_\mu) \Phi \;\;,$$

where due to (11)

$${\hat D(W+_\mu,W-_\mu,A_\mu,Z_\mu) \Phi=}$$

$$\frac{i\,g_2}{2} \left( \begin{array}{c} W^+_\mu (\phi_0 + H -i\,... ..._f - Z_\mu(\phi_0 + H -i\,Z_f)/\cos{\Theta_w} )/\sqrt{2} \end{array} \right)\;.$$

The term $(\hat D_\mu \Phi)^+(\hat D^\mu \Phi)$ gives us the mass terms for the $W$ and $Z$-bosons:

$$\left( \frac{g_2 \phi_0}{2} \right)^2 W^+_\mu {W^-}^\mu + \f... ...\frac{g_2 \phi_0}{2 \cos{\Theta_w} } \right)^2 Z_\mu Z^\mu \;,$$ (14)

and the following terms describing the interaction of a couple of vector bosons with Higgs and Goldstones:

$$\frac{g_2^2}{4}( W^+_\mu {W^-}^\mu ( H^2 +Z_f^2 +2\phi_0 H +2 W^+_f W^-_f)$$

$$+ 4 A_\mu A^\mu \sin^2{\Theta_w} W^+_f W^-_f$$

$$+ \frac{Z_\mu Z^\mu}{\cos^2{\Theta_w}}( (H^2 + Z_f^2)/2 + \phi_0 H + (1-2 \sin^2{\Theta_w})^2 W^+_f W^-_f)$$

$$+ 4 A_\mu Z^\mu \tan{\Theta_w}(1 -2 \sin^2{\Theta_w}) W^+_f W^-_f$$

$$-2i\tan{\Theta_w} W^+_\mu (\cos{\Theta_w} A^\mu - \sin{\Theta_w} Z^\mu) (\phi_0 + H -i\,Z_f) W^-_f$$

$$+ 2i\tan{\Theta_w} W^-_\mu (\cos{\Theta_w} A^\mu - \sin{\Theta_w} Z^\mu) (\phi_0 + H +i\,Z_f) W^+_f$$

$$)\;.$$ (15)

The constant $\phi_0$ now can be expressed in terms of the $W$-boson mass $M_W$:

$$\phi_0= 2\,M_W/g_2\;.$$

The $Z$-boson mass is related to the $W$-boson mass by means of constraint:

$$M_Z=M_W/\cos{\Theta_w}\;.$$

The term $( - (\hat D_\mu \Phi)\partial^\mu \Phi^+ - (\hat D_\mu \Phi)^+ \partial^\mu \Phi)$ gives us off-diagonal quadratic terms:

$$- g_2 \phi_0 \left( \frac{1}{2}( W^+_\mu \partial^\mu W^-_f ... ..._f ) +\frac{1}{\cos{\Theta_w}}Z_\mu \partial^\mu Z_f \right)$$ (16)

and the following terms of interaction:

$$\frac{g_2}{2}\left(\right. -H ( W^-_\mu \partial^\mu W^+_f + W^+_\mu \partial^\mu W^-_f ) + (\partial^\mu H)(W^-_\mu W^+_f +W^+_\mu W^-_f)$$

$$-i(Z_f ( W^-_\mu \partial^\mu W^+_f -W^+_\mu \partial^\mu W^-_f ) + (\partial^\mu Z_f) (W^+_\mu W^-_f - W^-_\mu W^+_f))$$

$$+i( 2\sin{\Theta_w} A_\mu + \cos{\Theta_w}(1-\tan^2{\Theta_w}) Z_\mu) ( W^-_f \partial^\mu W^+_f - W^+_f \partial^\mu W^-_f )$$

$$+ (\partial^\mu H) Z_\mu Z_f/\cos{\Theta_w} - (\partial^\mu Z_f) Z_\mu H/cos{\Theta_w}$$

$$\left. \right)\;.$$ (17)

The off-diagonal quadratic terms are canceled by the gauge fixing terms. See below.