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\begin{document}

\author{G.Quznetsov \\
%EndAName
quznets@geocities.com}
\date{29 November, 1998}
\title{Gauge Without Higgs}
\maketitle

\begin{abstract}
The gauge transformations of the standard electroweak model are
selfinconsistent. The removal of this inconsistency eliminates Higgs from
the gauge bosons theory.
\end{abstract}

Let us consider covariant derivation of the electroweak Lagrangian \cite{Kn}:

\[
\mathcal{D}_\mu =\partial _\mu -i\cdot g_1\cdot \frac Y2\cdot B_\mu -i\cdot
g_2\cdot \frac{\tau ^i}2\cdot W_\mu ^i\mbox{.} 
\]

Here: $g_1$, $g_2$, $Y$ are the real positive numbers, $B_\mu $, $W_\mu ^i$
are the real functions on $R^{3+1}$ and

\[
\tau ^1=\left( 
\begin{array}{cc}
0 & 1 \\ 
1 & 0
\end{array}
\right) \mbox{, }\tau ^2=\left( 
\begin{array}{cc}
0 & -i \\ 
i & 0
\end{array}
\right) \mbox{, }\tau ^3=\left( 
\begin{array}{cc}
1 & 0 \\ 
0 & -1
\end{array}
\right) \mbox{.} 
\]

Hence

\[
W_\mu =\tau ^i\cdot W_\mu ^i=\left[ 
\begin{array}{cc}
W_\mu ^3 & W_\mu ^1-i\cdot W_\mu ^2 \\ 
W_\mu ^1+i\cdot W_\mu ^2 & -W_\mu ^3
\end{array}
\right] \mbox{.} 
\]

Let electroweak transformation $U$ transforms $B_\mu $ and $W_\mu ^i$ as the
following \cite{Kn}:

\begin{equation}
B_\mu \rightarrow B_\mu ^{\prime }=B_\mu -\frac 1{g_1}\cdot \partial _\mu
\chi \mbox{,}  \label{t1}
\end{equation}

\begin{equation}
W_\mu \rightarrow W_\mu ^{\prime }=U\cdot W_\mu \cdot U^{-1}-\frac
i{g_2}\cdot \left( \partial _\mu U\right) \cdot U^{-1}\mbox{.}  \label{t2}
\end{equation}

Here $\chi $ is a real function on $R^{3+1}$ and $U$ is a $SU\left( 2\right) 
$ matrix function on $R^{3+1}$.

The real functions $w_\mu ^i$ exist for which:

\[
i\cdot \left( \partial _\mu U\right) \cdot U^{-1}=\tau ^i\cdot w_\mu ^i%
\mbox{.} 
\]

Since $U$ is everyone $SU\left( 2\right) $ transformation then for every $%
W_\mu $ a $SU\left( 2\right) $ transformation $U$ exists for which:

\[
\frac i{g_2}\cdot \left( \partial _\mu U\right) \cdot U^{-1}=U\cdot W_\mu
\cdot U^{-1}\mbox{.} 
\]

That is

\[
W_\mu \rightarrow W_\mu ^{\prime }=0\mbox{.} 
\]

And for every $B_\mu $ a real function $\chi $ exists for which:

\[
\frac 1{g_1}\cdot \partial _\mu \chi =B_\mu \mbox{,} 
\]

hence

\[
B_\mu \rightarrow B_\mu ^{\prime }=0\mbox{.} 
\]

Therefore in this case:

\[
\mathcal{D}_\mu \rightarrow \mathcal{D}_\mu ^{\prime }=\partial _\mu -i\cdot
g_1\cdot \frac Y2\cdot 0-i\cdot g_2\cdot \frac{\tau ^i}2\cdot 0\mbox{.} 
\]

Here the coefficients of the transforming components remained, only:

\[
\mathcal{D}_\mu \rightarrow \mathcal{D}_\mu ^{\prime }=\partial _\mu +0%
\mbox{.} 
\]

Hence the covariant derivation can not transformed by (\ref{t1}), (\ref{t2})
because zero can be transformed to zero, only.

Let us consider the Lagrangian \cite{Kn}:

\[
\begin{array}{c}
\mathcal{L}_{ferm}\left( SU\left( 1\right) \times SU\left( 2\right)
,leptons\right) = \\ 
=-\overline{e_R}\cdot i\cdot \gamma ^\mu \cdot \left( i\cdot g_1\cdot \frac{%
Y_R}2\cdot B_\mu \right) \cdot e_R- \\ 
-\left[ 
\begin{array}{cc}
\overline{\nu _L} & \overline{e_L}
\end{array}
\right] \cdot i\cdot \gamma ^\mu \cdot \left( i\cdot g_1\cdot \frac{Y_L}%
2\cdot B_\mu \right) \cdot \left[ 
\begin{array}{c}
\nu _L \\ 
e_L
\end{array}
\right] + \\ 
+\frac{g_2}2\cdot \left[ 
\begin{array}{cc}
\overline{\nu _L} & \overline{e_L}
\end{array}
\right] \cdot \gamma ^\mu \cdot \left[ 
\begin{array}{cc}
W_\mu ^0 & -\sqrt{2}\cdot W_\mu ^{+} \\ 
-\sqrt{2}\cdot W_\mu ^{-} & -W_\mu ^0
\end{array}
\right] \cdot \left[ 
\begin{array}{c}
\nu _L \\ 
e_L
\end{array}
\right] \mbox{.}
\end{array}
\]

Here:

\[
\begin{array}{c}
W_\mu ^0=W_\mu ^3\mbox{,} \\ 
-\sqrt{2}\cdot W_\mu ^{+}=W_\mu ^1-i\cdot W_\mu ^2\mbox{,} \\ 
-\sqrt{2}\cdot W_\mu ^{-}=W_\mu ^1+i\cdot W_\mu ^2\mbox{.}
\end{array}
\]

Hence:

\[
\mathcal{L}_{ferm}\left( SU\left( 1\right) \times SU\left( 2\right)
,leptons\right) = 
\]

\[
=0.5\cdot \left[ 
\begin{array}{ccc}
\overline{\nu _L} & \overline{e_L} & \overline{e_R}
\end{array}
\right] \cdot \gamma ^\mu \cdot 
\]

\[
\cdot \left[ 
\begin{array}{ccc}
g_2\cdot W_\mu ^0+g_1\cdot Y_L\cdot B_\mu & -g_2\cdot \sqrt{2}\cdot W_\mu
^{+} & 0 \\ 
-g_2\cdot \sqrt{2}\cdot W_\mu ^{-} & -g_2\cdot W_\mu ^0+g_1\cdot Y_L\cdot
B_\mu & 0 \\ 
0 & 0 & g_1\cdot Y_R\cdot B_\mu
\end{array}
\right] \cdot \left[ 
\begin{array}{c}
\nu _L \\ 
e_L \\ 
e_R
\end{array}
\right] 
\]

Let \cite{Kn}:

\[
A_\mu =\frac{g_2\cdot B_\mu -g_1\cdot Y_L\cdot W_\mu ^0}{\sqrt{g_1^2+g_2^2}}%
\mbox{,} 
\]

\[
Z_\mu =\frac{g_1\cdot Y_L\cdot B_\mu +g_2\cdot W_\mu ^0}{\sqrt{g_1^2+g_2^2}}%
\mbox{.} 
\]

Hence:

\[
B_\mu =\frac{g_2\cdot A_\mu +g_1\cdot Y_L\cdot Z_\mu }{\sqrt{g_1^2+g_2^2}}%
\mbox{,} 
\]

\[
W_\mu ^0=\frac{-g_1\cdot Y_L\cdot A_\mu +g_2\cdot Z_\mu }{\sqrt{g_1^2+g_2^2}}%
\mbox{.} 
\]

Let $Y_L=1$, $Y_R=2$:

\[
\mathcal{L}_{ferm}\left( SU\left( 1\right) \times SU\left( 2\right)
,leptons\right) =0.5\cdot 
\]

\[
\ \ \cdot \left[ 
\begin{array}{ccc}
\overline{\nu _L} & \overline{e_L} & \overline{e_R}
\end{array}
\right] \cdot \gamma ^\mu \cdot \left( 
\begin{array}{c}
\left[ 
\begin{array}{ccc}
\frac{g_2^2}{\sqrt{g_1^2+g_2^2}}\cdot Z_\mu & -g_2\cdot \sqrt{2}\cdot W_\mu
^{+} & 0 \\ 
-g_2\cdot \sqrt{2}\cdot W_\mu ^{-} & -\frac{g_2^2}{\sqrt{g_1^2+g_2^2}}\cdot
Z_\mu & 0 \\ 
0 & 0 & 0
\end{array}
\right] + \\ 
+\frac{g_1^2}{\sqrt{g_1^2+g_2^2}}\cdot Z_\mu \cdot \left[ 
\begin{array}{ccc}
1 & 0 & 0 \\ 
0 & 1 & 0 \\ 
0 & 0 & 1
\end{array}
\right] + \\ 
+\frac{g_1^2}{\sqrt{g_1^2+g_2^2}}\cdot Z_\mu \cdot \left[ 
\begin{array}{ccc}
0 & 0 & 0 \\ 
0 & 0 & 0 \\ 
0 & 0 & 1
\end{array}
\right] + \\ 
+\frac{2\cdot g_1\cdot g_2}{\sqrt{g_1^2+g_2^2}}\cdot A_\mu \cdot \left[ 
\begin{array}{ccc}
0 & 0 & 0 \\ 
0 & 1 & 0 \\ 
0 & 0 & 1
\end{array}
\right]
\end{array}
\right) \ \cdot \left[ 
\begin{array}{c}
\nu _L \\ 
e_L \\ 
e_R
\end{array}
\right] \mbox{.} 
\]

Let us denote:

\[
V_\mu ^{\left( W\right) }=\left[ 
\begin{array}{ccc}
\frac{g_2^2}{\sqrt{g_1^2+g_2^2}}\cdot Z_\mu  & -g_2\cdot \sqrt{2}\cdot W_\mu
^{+} & 0 \\ 
-g_2\cdot \sqrt{2}\cdot W_\mu ^{-} & -\frac{g_2^2}{\sqrt{g_1^2+g_2^2}}\cdot
Z_\mu  & 0 \\ 
0 & 0 & 0
\end{array}
\right] \mbox{,}
\]

\[
V_\mu ^{\left( b\right) }=\frac{g_1^2}{\sqrt{g_1^2+g_2^2}}\cdot Z_\mu \cdot
\left[ 
\begin{array}{ccc}
1 & 0 & 0 \\ 
0 & 1 & 0 \\ 
0 & 0 & 1
\end{array}
\right] \mbox{,} 
\]

\[
V_\mu ^{\left( Z\right) }=\frac{g_1^2}{\sqrt{g_1^2+g_2^2}}\cdot Z_\mu \cdot
\left[ 
\begin{array}{ccc}
0 & 0 & 0 \\ 
0 & 0 & 0 \\ 
0 & 0 & 1
\end{array}
\right] \mbox{,} 
\]

\[
V_\mu ^{\left( A\right) }=\frac{2\cdot g_1\cdot g_2}{\sqrt{g_1^2+g_2^2}}%
\cdot A_\mu \cdot \left[ 
\begin{array}{ccc}
0 & 0 & 0 \\ 
0 & 1 & 0 \\ 
0 & 0 & 1
\end{array}
\right] \mbox{.} 
\]

In this case:

\[
\mathcal{D}_\mu =\partial _\mu -\frac i2\cdot V_\mu ^{\left( W\right)
}-\frac i2\cdot V_\mu ^{\left( b\right) }-\frac i2\cdot V_\mu ^{\left(
Z\right) }-\frac i2\cdot V_\mu ^{\left( A\right) }\mbox{.} 
\]

$V_\mu ^{\left( W\right) }$ can not transformed as:

\[
V_\mu ^{\left( W\right) }\rightarrow V_\mu ^{\left( W\right) \prime }=U\cdot
V_\mu ^{\left( W\right) }\cdot U^{-1}-i\cdot \left( \partial _\mu U\right)
\cdot U^{-1} 
\]

since a $SU\left( 2\right) $ matrix $U$ exists for which:

\[
i\cdot \left( \partial _\mu U\right) \cdot U^{-1}=U\cdot V_\mu ^{\left(
W\right) }\cdot U^{-1}\mbox{.} 
\]

And $V_\mu ^{\left( b\right) }$ can not transformed as:

\[
V_\mu ^{\left( b\right) }\rightarrow V_\mu ^{\left( b\right) \prime }=V_\mu
^{\left( b\right) }-\partial _\mu \chi 
\]

since a real function $\chi $ exists for which:

\[
\partial _\mu \chi =V_\mu ^{\left( b\right) }\mbox{.} 
\]

Therefore:

\[
V_\mu ^{\left( W\right) }\rightarrow V_\mu ^{\left( W\right) \prime }=U\cdot
V_\mu ^{\left( W\right) }\cdot U^{-1}\mbox{,} 
\]

\[
V_\mu ^{\left( b\right) }\rightarrow V_\mu ^{\left( b\right) \prime }=U\cdot
V_\mu ^{\left( b\right) }\cdot U^{-1}\mbox{,} 
\]

\begin{eqnarray*}
\left( V_\mu ^{\left( Z\right) }+V_\mu ^{\left( A\right) }\right)
&\rightarrow &\left( V_\mu ^{\left( Z\right) }+V_\mu ^{\left( A\right)
}\right) ^{\prime }= \\
&=&U\cdot \left( V_\mu ^{\left( Z\right) }+V_\mu ^{\left( A\right) }\right)
\cdot U^{-1}-i\cdot \left( \partial _\mu U\right) \cdot U^{-1}-\partial _\mu
\chi \mbox{.}
\end{eqnarray*}

\begin{thebibliography}{9}
\bibitem{Kn}  for instance, Gordon Kane, 
\mbox{\it{ Modern Elementary Particle
Physics}.}

(Addison-Wesley Publishing Company, Inc., 1987)
\end{thebibliography}

\end{document}

