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Дата изменения: Wed Aug 9 20:40:47 2000
Дата индексирования: Mon Oct 1 22:51:47 2012
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Vector bosons

Gauge theory of electroweak interactions is based on the $SU(2)\times U(1)$ group. So, we have a triplet of $SU(2)$ vector fields $W^{\alpha}_\mu(x)$ and a single vector field $B_\mu(x)$. The $SU(2)$ structure constants are presented by the absolutely antisymmetric tensor $\epsilon^{\alpha\beta\gamma}$. The Lagrangian of gauge fields is written down according to (1):

$$L_{GF} = -\, \frac{1}{4} {FW^{\alpha}}_{\mu \nu}(x) {FW_{\a... ...mu \nu}(x) -\, \frac{1}{4} FB_{\mu \nu}(x) FB^{\mu \nu}(x)\;,$$ (8)

where

$$\begin{eqnarray*} FW^{\alpha}_{\mu \nu}(x)& =& \partial_{\mu}W^{\alpha}_{\nu}(x)... ...}(x)& =& \partial_{\mu}B_{\nu}(x) - \partial_{\nu}B_{\mu}(x) \;; \end{eqnarray*}$$

$g_2$ is a coupling constant for the $SU(2)$ gauge interaction.

Infinitesimal local gauge transformations are defined as follows

$$(\hat D(w,b) W_{\mu})^{\alpha}(x) = g_2 \epsilon^{\alpha\beta \gamma} W^{\beta}_{\mu}(x) w^{\gamma}(x) + \partial_{\mu} w^{\alpha}(x) \;;$$

$$D(w,b) B_{\mu}(x) = \partial_{\mu} b(x) \;.$$ (9)

Let us express $W^1_\mu$ and $W^2_\mu$ in terms of a mutually conjugated couple $W^+_\mu$ and $W^-_\mu$

$$\begin{eqnarray*} W^1_\mu&=&(W^-_\mu+\,W^+_\mu)/\sqrt{2}\;; \\ W^2_\mu&=& (W^-_\mu-\,W^+_\mu)/(i \sqrt{2})\;. \end{eqnarray*}$$

Thus, the Lagrangian of self-interaction for the $SU(2)$ gauge fields in term of $W^+$ and $W^-$ has the form:

$$i\,g_2 \left(\partial_\mu W^3_\nu(W^+_\mu W^-_\nu - W^-_\mu W^+_\nu) +\partial_\mu W^+_\nu(W^-_\mu W^3_\nu - W^3_\mu W^-_\nu) \right.$$

$$\left. -\partial_\mu W^-_\nu(W^+_\mu W^3_\nu - W^3_\mu W^+_\nu) \right)$$

$$+g_2^2 \left( W^3_\mu W^+_\nu(W^-_\mu W^3_\nu - W^3_\mu W^-_\nu) +\frac{1}{2} W^+_\mu W^-_\nu (W^+_\mu W^-_\nu - W^-_\mu W^+_\nu) \right)\;.$$ (10)

All matter fields in the electroweak theory are either $SU(2)$ invariant singlets or belong to its fundamental representation. In the latter case they form doublets. Generators for these doublets are expressed via the Pauli $\sigma$-matrices

$$\hat \tau_{\alpha} = \hat \sigma_{\alpha}/2 \;.$$

Thus the infinitesimal local gauge transformations for doublets take a form:

$$D(w,b) \psi(x)= \frac{i\,g_2}{2}w^{\alpha}(x)\hat \sigma_{\alpha}\psi(x) + \frac{i\, g_1}{2} Y\, b(x) \psi(x) \;.$$

Here $g_1$ is the coupling constant of $U(1)$ gauge interaction. The constant $Y$ depends on a type of the doublet. It is called a hypercharge.

In the gauge theory of electroweak interaction the gauge fields interact with a scalar (Higgs) doublet which has a nonzero vacuum state. Without loss of generality one can put $Y=1$ for the Higgs doublet. By means of the gauge transformation the vacuum state of this field may be presented in the form:

$$\Phi_{\Omega}= \left( \begin{array}{c} 0 \\ \phi_0/\sqrt{2} \end{array} \right)\;,$$

where $\phi_0$ is a real constant.

As a result of spontaneous symmetry breaking the $W^{\alpha}_{\mu}$ and $B_{\mu}$ fields do not correspond to physical particles. Physical particles in this model are the photon ($A_\mu$), W-bosons ( $W^+_{\mu}\,, W^-_{\mu}$) and Z-boson ($Z_{\mu}$). The photon field $A_{\mu}$ is a combination of gauge fields responsible for the local gauge transformations which save the Higgs vacuum $\Phi_{\Omega}$:

$$A_{\mu} = B_{\mu} \cos{\Theta_w} + W^3_{\mu} \sin{\Theta_w}\;,$$

where the mixing angle $\Theta_w = \arctan(g_2/g_1)$. To complete $W^+$, $W^-$, and $A$ up to the orthonormal basis of gauge fields we introduce

$$Z_{\mu} = -B_{\mu} \sin{\Theta_w} + W^3_{\mu} \cos{\Theta_w}\;.$$

Let $w^+(x)$, $w^-(x)$, $a(x)$, and $z(x)$ be parameters of the local gauge transformation corresponding to the fields $W^+_\mu$, $W^-_\mu$, $A_\mu$, and $Z_\mu$. Then for a matter doublet with a hypercharge $Y$ the gauge transformation is given by the following expression:

$${\hat D(w^+,w^-,a,z) \Psi=}$$

$$\frac{i\,g_2}{2} \left[\sqrt{2} \left( \begin{array}{cc} 0 & w^+ ... ...a_w\,a\, \left( \begin{array}{cc} 1+Y & 0 \\ 0&Y-1 \end{array} \right) \right.$$

$$+ \cos\Theta_w \,z\, \left.\left( \begin{array}{cc} 1 -Y\tan^2\Theta_w & 0 \\ 0& -1 -Y\tan^2\Theta_w \end{array}\right) \right] \Psi\;.$$ (11)