
\documentclass[12pt]{article}
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\begin{document}

\author{G.Quznetsov \\
%EndAName
quznets@geocities.com}
\date{October 23, 1998}
\title{The Massive Gauge Bosons without the Higgs Mechanism}
\maketitle

\begin{abstract}
The simplest massive gauge bosons Lagrangian, invariant for SU(2)
transformations without the Higgs mechanism, is constructed.
\end{abstract}

\tableofcontents

I use the following notation:

the Pauli matrices:

\[
\tau _1=\left[ 
\begin{array}{cc}
0 & 1 \\ 
1 & 0
\end{array}
\right] \mbox{, }\tau _2=\left[ 
\begin{array}{cc}
0 & -i \\ 
i & 0
\end{array}
\right] \mbox{, }\tau _3=\left[ 
\begin{array}{cc}
1 & 0 \\ 
0 & -1
\end{array}
\right] \mbox{,} 
\]

the identity matrix:

\[
E=\left[ 
\begin{array}{cc}
1 & 0 \\ 
0 & 1
\end{array}
\right] \mbox{.} 
\]

\section{The Projection Matrices}

Let $U$ be any $SU\left( 2\right) $ transformation:

\[
U=\left[ 
\begin{array}{cc}
\cos \left( \varepsilon \right) +i\cdot n_3\cdot \sin \left( \varepsilon
\right) & \left( i\cdot n_1+n_2\right) \cdot \sin \left( \varepsilon \right)
\\ 
\left( i\cdot n_1-n_2\right) \cdot \sin \left( \varepsilon \right) & \cos
\left( \varepsilon \right) -i\cdot n_3\cdot \sin \left( \varepsilon \right)
\end{array}
\right] \mbox{,} 
\]

$\varepsilon $, $n_1$, $n_2$, $n_3$ are a real functions on $R^{3+1}$ and:

\[
n_1^2+n_2^2+n_3^2=1\mbox{.} 
\]

Let $a^{\left( +\right) }$ and $a^{\left( -\right) }$ be the normalized
eigenvectors of $U$:

\begin{eqnarray*}
U\cdot a^{\left( +\right) } &=&\exp \left( i\cdot \lambda \right) \cdot
a^{\left( +\right) }\mbox{,} \\
U\cdot a^{\left( -\right) } &=&\exp \left( -i\cdot \lambda \right) \cdot
a^{\left( -\right) }\mbox{,} \\
\left| a^{\left( +\right) }\right| &=&1=\left| a^{\left( -\right) }\right| %
\mbox{,} \\
a^{\left( +\right) \dagger }\cdot a^{\left( -\right) } &=&0=a^{\left(
-\right) \dagger }\cdot a^{\left( +\right) }\mbox{.}
\end{eqnarray*}

In this case let:

\[
Q=\left[ 
\begin{array}{cc}
a_1^{\left( +\right) } & a_1^{\left( -\right) } \\ 
a_2^{\left( +\right) } & a_2^{\left( -\right) }
\end{array}
\right] \mbox{.} 
\]

\[
X^{\left( +\right) }=Q\cdot \left[ 
\begin{array}{cc}
1 & 0 \\ 
0 & 0
\end{array}
\right] \cdot Q^{\dagger }\mbox{,} 
\]

\[
X^{\left( -\right) }=Q\cdot \left[ 
\begin{array}{cc}
0 & 0 \\ 
0 & 1
\end{array}
\right] \cdot Q^{\dagger }\mbox{.} 
\]

In this case:

\[
X^{\left( +\right) }\cdot X^{\left( -\right) }=\left[ 
\begin{array}{cc}
0 & 0 \\ 
0 & 0
\end{array}
\right] =X^{\left( +\right) }\cdot X^{\left( -\right) }\mbox{,} 
\]

\begin{eqnarray*}
X^{\left( +\right) }\cdot X^{\left( +\right) } &=&X^{\left( +\right) }%
\mbox{,} \\
X^{\left( -\right) }\cdot X^{\left( -\right) } &=&X^{\left( -\right) }%
\mbox{,}
\end{eqnarray*}

\[
X^{\left( +\right) \dagger }=X^{\left( +\right) }\mbox{, }X^{\left( -\right)
\dagger }=X^{\left( -\right) } 
\]

and

\[
X^{\left( +\right) }+X^{\left( -\right) }=\left[ 
\begin{array}{cc}
1 & 0 \\ 
0 & 1
\end{array}
\right] \mbox{.} 
\]

Hence, a real functions $u$, $v$, $b$ on $R^{3+1}$ exist, for which:

\[
X^{\left( +\right) }=\left[ 
\begin{array}{cc}
b & u+i\cdot v \\ 
u-i\cdot v & 1-b
\end{array}
\right] 
\]

and

\[
X^{\left( -\right) }=\left[ 
\begin{array}{cc}
1-b & -u-i\cdot v \\ 
-u+i\cdot v & b
\end{array}
\right] \mbox{.} 
\]

Hence:

\[
\partial _\mu b=-2\cdot \frac{u\cdot \partial _\mu u+v\cdot \partial _\mu v}{%
\sqrt{1-4\cdot \left( u^2+v^2\right) }}\mbox{,} 
\]

\[
\partial _\mu X^{\left( +\right) }=\left[ 
\begin{array}{cc}
\partial _\mu b & \partial _\mu u+i\cdot \partial _\mu v \\ 
\partial _\mu u-i\cdot \partial _\mu v & -\partial _\mu b
\end{array}
\right] \mbox{,} 
\]

\[
\partial _\mu X^{\left( -\right) }=\left[ 
\begin{array}{cc}
-\partial _\mu b & -\partial _\mu u-i\cdot \partial _\mu v \\ 
-\partial _\mu u+i\cdot \partial _\mu v & \partial _\mu b
\end{array}
\right] \mbox{,} 
\]

\[
\partial _\mu X^{\left( +\right) \dagger }=\partial _\mu X^{\left( +\right) }%
\mbox{, }\partial _\mu X^{\left( -\right) \dagger }=\partial _\mu X^{\left(
-\right) }\mbox{,} 
\]

\[
X^{\left( +\right) }\cdot \left( \partial _\mu X^{\left( +\right) }\right)
\cdot X^{\left( +\right) }=\left[ 
\begin{array}{cc}
0 & 0 \\ 
0 & 0
\end{array}
\right] =X^{\left( -\right) }\cdot \left( \partial _\mu X^{\left( -\right)
}\right) \cdot X^{\left( -\right) }\mbox{,} 
\]

\begin{eqnarray*}
X^{\left( -\right) }\cdot \left( \partial _\mu X^{\left( +\right) }\right)
\cdot X^{\left( +\right) } &=&\left( \partial _\mu X^{\left( +\right)
}\right) \cdot X^{\left( +\right) }\mbox{,} \\
X^{\left( +\right) }\cdot \left( \partial _\mu X^{\left( -\right) }\right)
\cdot X^{\left( -\right) } &=&\left( \partial _\mu X^{\left( -\right)
}\right) \cdot X^{\left( -\right) }
\end{eqnarray*}

and

\[
\partial _\mu X^{\left( +\right) }=-\partial _\mu X^{\left( -\right) }%
\mbox{.} 
\]

Let if

\[
\psi =k_1\cdot a^{\left( +\right) }+k_2\cdot a^{\left( -\right) } 
\]

then

\[
\begin{array}{c}
\psi ^{\left( +\right) }=k_1\cdot a^{\left( +\right) }\mbox{,} \\ 
\psi ^{\left( -\right) }=k_2\cdot a^{\left( -\right) }\mbox{.}
\end{array}
\]

In this case:

\begin{eqnarray*}
X^{\left( +\right) }\cdot \psi &=&\psi ^{\left( +\right) }\mbox{,} \\
X^{\left( -\right) }\cdot \psi &=&\psi ^{\left( -\right) }\mbox{.}
\end{eqnarray*}

$U$ can be expressed by the projection matrices as the following:

\[
U=\exp \left( i\cdot \lambda \right) \cdot X^{\left( +\right) }+\exp \left(
-i\cdot \lambda \right) \cdot X^{\left( -\right) }\mbox{.} 
\]

Hence,

\[
U^{-1}=\exp \left( -i\cdot \lambda \right) \cdot X^{\left( +\right) }+\exp
\left( i\cdot \lambda \right) \cdot X^{\left( -\right) } 
\]

and

\begin{eqnarray*}
\partial _\mu U &=&i\cdot \left( \partial _\mu \lambda \right) \cdot \exp
\left( i\cdot \lambda \right) \cdot X^{\left( +\right) }-i\cdot \left(
\partial _\mu \lambda \right) \cdot \exp \left( -i\cdot \lambda \right)
\cdot X^{\left( -\right) }+ \\
&&+\exp \left( i\cdot \lambda \right) \cdot \left( \partial _\mu X^{\left(
+\right) }\right) +\exp \left( -i\cdot \lambda \right) \cdot \left( \partial
_\mu X^{\left( -\right) }\right) \mbox{.}
\end{eqnarray*}

\section{The Gauge Bosons}

Let $W_\mu ^{\left( 1\right) }$, $W_\mu ^{\left( 2\right) }$, $W_\mu
^{\left( 3\right) }$ be any real functions on $R^{3+1}$and let:

\[
W_\mu =W_\mu ^{\left( 1\right) }\cdot \tau _1+W_\mu ^{\left( 2\right) }\cdot
\tau _2+W_\mu ^{\left( 3\right) }\cdot \tau _3\mbox{,} 
\]

\[
D_\mu =\partial _\mu -i\cdot g\cdot W_\mu \mbox{.} 
\]

Let for the $U$ transformation:

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

Let us denote:

\[
V_\mu ^{\left( +\right) }=W_\mu \cdot X^{\left( +\right) }\mbox{ and }V_\mu
^{\left( -\right) }=W_\mu \cdot X^{\left( -\right) }\mbox{.} 
\]

In this case:

\[
V_\mu ^{\left( +\right) }+V_\mu ^{\left( -\right) }=W_\mu 
\]

and if for the $U$ transformation:

\[
\begin{array}{c}
V_\mu ^{\left( +\right) }\rightarrow V_\mu ^{\left( +\right) \prime }= \\ 
=\left( \exp \left( -2\cdot i\cdot \lambda \right) \cdot E+\left( 1-\exp
\left( -2\cdot i\cdot \lambda \right) \right) \cdot X^{\left( +\right)
}\right) \cdot V_\mu ^{\left( +\right) }+ \\ 
+\frac 1g\cdot \left( \partial _\mu \lambda \right) \cdot X^{\left( +\right)
}-\frac ig\cdot \left( 1-\exp \left( -2\cdot i\cdot \lambda \right) \right)
\cdot \left( \partial _\mu X^{\left( +\right) }\right) \cdot X^{\left(
+\right) }\mbox{,}
\end{array}
\]

and

\[
\begin{array}{c}
V_\mu ^{\left( -\right) }\rightarrow V_\mu ^{\left( -\right) \prime }= \\ 
=\left( \exp \left( 2\cdot i\cdot \lambda \right) \cdot E+\left( 1-\exp
\left( 2\cdot i\cdot \lambda \right) \right) \cdot X^{\left( -\right)
}\right) \cdot V_\mu ^{\left( -\right) }- \\ 
-\frac 1g\cdot \left( \partial _\mu \lambda \right) \cdot X^{\left( -\right)
}-\frac ig\cdot \left( 1-\exp \left( 2\cdot i\cdot \lambda \right) \right)
\cdot \left( \partial _\mu X^{\left( -\right) }\right) \cdot X^{\left(
-\right) }
\end{array}
\]

then

\[
V_\mu ^{\left( +\right) \prime }+V_\mu ^{\left( -\right) \prime }=W_\mu
^{\prime }\mbox{.} 
\]

Let us denote:

\[
W_\mu ^{\left( +\right) }=X^{\left( -\right) }\cdot W_\mu \cdot X^{\left(
+\right) }\mbox{ and }W_\mu ^{\left( -\right) }=X^{\left( +\right) }\cdot
W_\mu \cdot X^{\left( -\right) }\mbox{.} 
\]

Hence:

\[
W_\mu ^{\left( +\right) }=X^{\left( -\right) }\cdot V_\mu ^{\left( +\right) }%
\mbox{ and }W_\mu ^{\left( -\right) }=X^{\left( +\right) }\cdot V_\mu
^{\left( -\right) }\mbox{.} 
\]

Therefore, for the $U$ transformation:

\[
\begin{array}{c}
W_\mu ^{\left( +\right) }\rightarrow W_\mu ^{\left( +\right) \prime }= \\ 
=\exp \left( -2\cdot i\cdot \lambda \right) \cdot W_\mu ^{\left( +\right)
}-\frac ig\cdot \left( 1-\exp \left( -2\cdot i\cdot \lambda \right) \right)
\cdot \left( \partial _\mu X^{\left( +\right) }\right) \cdot X^{\left(
+\right) }\mbox{,}
\end{array}
\]

\[
\begin{array}{c}
W_\mu ^{\left( -\right) }\rightarrow W_\mu ^{\left( -\right) \prime }= \\ 
\ =\exp \left( 2\cdot i\cdot \lambda \right) \cdot W_\mu ^{\left( -\right)
}-\frac ig\cdot \left( 1-\exp \left( 2\cdot i\cdot \lambda \right) \right)
\cdot \left( \partial _\mu X^{\left( -\right) }\right) \cdot X^{\left(
-\right) }\mbox{.}
\end{array}
\]

Let us denote:

\[
w_\mu ^{\left( +\right) }=X^{\left( +\right) }\cdot W_\mu \cdot X^{\left(
+\right) }\mbox{ and }w_\mu ^{\left( -\right) }=X^{\left( -\right) }\cdot
W_\mu \cdot X^{\left( -\right) }\mbox{.} 
\]

Hence:

\[
w_\mu ^{\left( +\right) }=X^{\left( +\right) }\cdot V_\mu ^{\left( +\right) }%
\mbox{ and }w_\mu ^{\left( -\right) }=X^{\left( -\right) }\cdot V_\mu
^{\left( -\right) }\mbox{.} 
\]

Therefore, for the $U$ transformation:

\[
w_\mu ^{\left( +\right) }\rightarrow w_\mu ^{\left( +\right) \prime }=w_\mu
^{\left( +\right) }+\frac 1g\cdot \left( \partial _\mu \lambda \right) \cdot
X^{\left( +\right) } 
\]

and

\[
w_\mu ^{\left( -\right) }\rightarrow w_\mu ^{\left( -\right) \prime }=w_\mu
^{\left( -\right) }-\frac 1g\cdot \left( \partial _\mu \lambda \right) \cdot
X^{\left( -\right) }\mbox{.} 
\]

In this case:

\begin{eqnarray*}
\left( W_\mu ^{\left( +\right) \dagger }-W_\mu ^{\left( -\right) }\right)
\cdot \left( W_{\left( +\right) }^\mu -W_{\left( -\right) }^{\mu \dagger
}\right) &=&\left[ 
\begin{array}{cc}
0 & 0 \\ 
0 & 0
\end{array}
\right] \mbox{,} \\
\left( W_\mu ^{\left( +\right) \prime \dagger }-W_\mu ^{\left( -\right)
\prime }\right) \cdot \left( W_{\left( +\right) }^{\mu \prime }-W_{\left(
-\right) }^{\mu \prime \dagger }\right) &=&\left[ 
\begin{array}{cc}
0 & 0 \\ 
0 & 0
\end{array}
\right] \mbox{.}
\end{eqnarray*}

Let us denote:

\[
\left( \tau _2\cdot M\cdot \tau _2\right) ^{*}=\overline{M}\mbox{.} 
\]

In this case:

\begin{eqnarray*}
\left( w_\mu ^{\left( +\right) }+\overline{w_\mu ^{\left( -\right) }}\right)
\cdot \left( w_{\left( +\right) }^\mu +\overline{w_{\left( -\right) }^\mu }%
\right) &=&\left[ 
\begin{array}{cc}
0 & 0 \\ 
0 & 0
\end{array}
\right] \mbox{,} \\
\left( w_\mu ^{\left( +\right) \prime }+\overline{w_\mu ^{\left( -\right) }}%
^{\prime }\right) \cdot \left( w_{\left( +\right) }^{\mu \prime }+\overline{%
w_{\left( -\right) }^\mu }^{\prime }\right) &=&\left[ 
\begin{array}{cc}
0 & 0 \\ 
0 & 0
\end{array}
\right] \mbox{.}
\end{eqnarray*}

All are zero.

\section{The Violation of the Symmetry}

Let a constant phase breeze $\exp \left( i\cdot \varphi \right) $ breathes
in this world.

\[
W_\mu \rightarrow \exp \left( i\cdot \varphi \right) \cdot W_\mu \mbox{.} 
\]

In this case:

\begin{eqnarray*}
\left( W_\mu ^{\left( +\right) \dagger }-W_\mu ^{\left( -\right) }\right)
\cdot \left( W_{\left( +\right) }^\mu -W_{\left( -\right) }^{\mu \dagger
}\right) &\neq &\left[ 
\begin{array}{cc}
0 & 0 \\ 
0 & 0
\end{array}
\right] \mbox{,} \\
\left( W_\mu ^{\left( +\right) \prime \dagger }-W_\mu ^{\left( -\right)
\prime }\right) \cdot \left( W_{\left( +\right) }^{\mu \prime }-W_{\left(
-\right) }^{\mu \prime \dagger }\right) &\neq &\left[ 
\begin{array}{cc}
0 & 0 \\ 
0 & 0
\end{array}
\right] \mbox{,}
\end{eqnarray*}

but

\[
\left( W_\mu ^{\left( +\right) \prime \dagger }-W_\mu ^{\left( -\right)
\prime }\right) \cdot \left( W_{\left( +\right) }^{\mu \prime }-W_{\left(
-\right) }^{\mu \prime \dagger }\right) =\left( W_\mu ^{\left( +\right)
\dagger }-W_\mu ^{\left( -\right) }\right) \cdot \left( W_{\left( +\right)
}^\mu -W_{\left( -\right) }^{\mu \dagger }\right) \mbox{.} 
\]

Therefore,

\[
\left( W_\mu ^{\left( +\right) \dagger }-W_\mu ^{\left( -\right) }\right)
\cdot \left( W_{\left( +\right) }^\mu -W_{\left( -\right) }^{\mu \dagger
}\right) 
\]

is invariant for this gauge transformation.

And

\begin{eqnarray*}
\left( w_\mu ^{\left( +\right) }+\overline{w_\mu ^{\left( -\right) }}\right)
\cdot \left( w_{\left( +\right) }^\mu +\overline{w_{\left( -\right) }^\mu }%
\right) &\neq &\left[ 
\begin{array}{cc}
0 & 0 \\ 
0 & 0
\end{array}
\right] \mbox{,} \\
\left( w_\mu ^{\left( +\right) \prime }+\overline{w_\mu ^{\left( -\right) }}%
^{\prime }\right) \cdot \left( w_{\left( +\right) }^{\mu \prime }+\overline{%
w_{\left( -\right) }^\mu }^{\prime }\right) &\neq &\left[ 
\begin{array}{cc}
0 & 0 \\ 
0 & 0
\end{array}
\right] \mbox{,}
\end{eqnarray*}

but

\[
\left( w_\mu ^{\left( +\right) \prime }+\overline{w_\mu ^{\left( -\right) }}%
^{\prime }\right) \cdot \left( w_{\left( +\right) }^{\mu \prime }+\overline{%
w_{\left( -\right) }^\mu }^{\prime }\right) =\left( w_\mu ^{\left( +\right)
}+\overline{w_\mu ^{\left( -\right) }}\right) \cdot \left( w_{\left(
+\right) }^\mu +\overline{w_{\left( -\right) }^\mu }\right) \mbox{.} 
\]

Therefore,

\[
\left( w_\mu ^{\left( +\right) }+\overline{w_\mu ^{\left( -\right) }}\right)
\cdot \left( w_{\left( +\right) }^\mu +\overline{w_{\left( -\right) }^\mu }%
\right) 
\]

is invariant for this gauge transformation, too.

\section{Lagrangian}

Let, as usual,:

\[
W_{\mu \nu }=\partial _\mu W_\nu -\partial _\nu W_\mu +i\cdot g\cdot \left[
W_\nu ,W_\mu \right] \mbox{.} 
\]

Since

\[
W_\mu =W_\mu ^{\left( +\right) }+W_\mu ^{\left( -\right) }+w_\mu ^{\left(
+\right) }+w_\mu ^{\left( -\right) } 
\]

then

\[
\mathcal{L}_{GB}=-\frac 14\cdot W_{\mu \nu }\cdot W^{\mu \nu }+ 
\]

\[
+\frac 12\cdot m_w^2\cdot \left( w_\mu ^{\left( +\right) }+\overline{w_\mu
^{\left( -\right) }}\right) \cdot \left( w_{\left( +\right) }^\mu +\overline{%
w_{\left( -\right) }^\mu }\right) + 
\]

\[
+\frac 12\cdot m_W^2\cdot \left( W_\mu ^{\left( +\right) \dagger }-W_\mu
^{\left( -\right) }\right) \cdot \left( W_{\left( +\right) }^\mu -W_{\left(
-\right) }^{\mu \dagger }\right) 
\]

is the Lagrangian for the massive gauge bosons $w$, $W^{\left( +\right) }$
and $W^{\left( -\right) }$, invariant for $SU\left( 2\right) $
transformation.

\end{document}

