\documentclass{beamer} % BB Mar 20 2012.

\usetheme{fsuheplat}

%AB: color definitions
\definecolor{abtitlecolor}{rgb}{0.0,0.255,0.494}
\definecolor{absecondarycolor}{rgb}{0.0,0.416,0.804}
\definecolor{abprimarycolor}{rgb}{1.0,0.686,0.0}

%AB: redefinition of beamer colors
\setbeamercolor{title}{fg=abtitlecolor}
\setbeamercolor{frametitle}{fg=abtitlecolor}
\setbeamercolor{palette tertiary}{fg=white,bg=abtitlecolor}
\setbeamercolor{palette secondary}{fg=white,bg=absecondarycolor}
\setbeamercolor{palette primary}{fg=black,bg=abprimarycolor}
\setbeamercolor{structure}{fg=abtitlecolor}

\setbeamerfont{section in toc}{series=\bfseries}
\def\b{\color{blue}}
\def\r{\color{red}}
\def\s{\color{black}}

%AB: remove navigation icons
\beamertemplatenavigationsymbolsempty

\title[Vector boson mass]{\textbf{Vector boson mass generation \\
       without new particles}}

\author[Bernd Berg]{Bernd A. Berg}

\institute[]{Florida State University}

\date[APS April 2012 Meeting]{APS April Meeting, Atlanta, April 1, 2012
    \\~~~\\ References: arXiv 0909.3340; PRD 82, 1145070, 2010; \\
                        \b arXiv 1109.5861.}
% , 0911.3692. }  \date{\today}


\begin{document}

\frame{\titlepage}

\frame{\frametitle{\r Introduction } \b

Explicit breaking of gauge symmetry through massive vector bosons 
can be avoided by the \r Higgs mechanism, \b which leads to perturbative
renormalizability. Nevertheless, introduction of the Higgs particle
into a theory of matter fields interacting through vector bosons 
remains quite adhoc.
This provides an opening for \r new physics \b solutions of which 
supersymmetry and string theory are most popular. 
\medskip

Though new physics 
should emerge on the way to the Planck scale, there appears no strong 
reason to expect experimental signals on the LHC energy scale.  \r 
Occam's razor \b suggests to stay with fermions and vector bosons.
\medskip

Notably, the original arguments for the Higgs rely on perturbation
theory. Our non-perturbative understanding of QFT developed later.
Here it is suggested to address the problem of vector boson mass
generation from scratch within the \r non-perturbative LGT framework.
}

\frame{\frametitle{\r SU(2) deconfining phase transition} \b

For massive vector bosons we need SU(2) in a \r deconfined \b phase.

Order parameter is the expectation value of the Polyakov loop: Single 
peak confined, double peak deconfined. A transition can be achieved 
by increasing the physical temperature, but in a fundamental theory 
we have to stay at zero temperature. A coupling of parallel SU(2) 
matrices \s \vskip -20pt
\begin{eqnarray} \nonumber % \label{Sadd} 
  {\rm Re}\,{\rm Tr}\,\left[V_{\nu}(x+\hat{\mu}a)\,V^*_{\nu}(x)\right]
\end{eqnarray} \b
is well-suited to align Polyakov loops, but breaks SU(2) gauge 
invariance. To rescue local U(1)$\,\otimes\,$SU(2) invariance of 
matter \\ fields a concept of \r extended gauge invariance \b is
introduced: \s
\begin{eqnarray} \nonumber % \label{aGU}
  U'_{\mu}(na) &\to& G(na) U'_{\mu}(na) G^{-1}(na+\hat{\mu}a)\,,
  \\ \nonumber % \label{aGV}
  V'_{\mu}(na) &\to& G(na) V'_{\mu}(na) G^{-1}(na+\hat{\mu}a)\,,
\end{eqnarray} \b
with \s $G\in$ U(1)$\,\otimes\,$SU(2),
$U'\in$ U(1)$\,\otimes\,$SU(2)$_g$,
$V'\in$ U(1)$_g\,\otimes\,$SU(2).
}


\frame{\frametitle{\r Action } \s
\begin{eqnarray} \nonumber % \label{S}
  S &=& \frac{\beta_a}{2} \sum_p {\rm Re}\,{\rm Tr}\,U'_p
     + \frac{\beta_b}{2} \sum_p {\rm Tr}\,V'_p 
     + \frac{\lambda}{2}\sum_{\mu\nu} S_{\mu\nu}^{\rm add}\,,\\
  S^{\rm add}_{\mu\nu}\ &=& {\rm Re}\,{\rm Tr}\, \nonumber
   \left[U'_{\mu}(x)\,V'_{\nu}(x+\hat{\mu}a)\,U'^{\dagger}_{\mu}
   (x+\hat{\nu}a)\,V'^{\dagger}_{\nu}(x)\right]\,.              
\end{eqnarray} \b
\b is invariant under these transformations and the partition function 
is \s
\begin{eqnarray} \nonumber % \label{S}
  Z\ =\ \int\prod_n\prod_{\mu=1}^4dU'_{\mu}(na)\,dV'_{\mu}(na)\,e^S\,.
\end{eqnarray} \r
Numerical Results: \b
Properties as function of $\s\lambda$ have been investigated by MC
calculations. After fixing $\s\beta_a$ in the Coulomb phase of U(1) and
$\s\beta_b$ in the scaling region of confined SU(2), one finds for small
$\s\lambda$ the same results as for \s $\lambda=0$: \b A massless U(1) 
photon and confined SU(2) with a glueball spectrum. Increasing \s 
$\lambda$, \b a strong first order phase transitions takes place.
\vskip 20pt
}

\frame{\frametitle{\r } \b \vskip -10pt
Action parts at $\s\beta_a=1.1$ in the Coulomb phase and $\s\beta_b=2.3$ 
in the SU(2) scaling region:
\begin{figure} \begin{center}
\includegraphics[width=0.7\textwidth]{fig/actions} 
\end{center} \end{figure} 
Plaquette expectation values on a $\s 12^4$ lattice as function 
of $\s \lambda$ (ordered o and disordered d starts).
\vskip 10pt
}


\frame{\frametitle{\r Polyakov loop histograms} \b

\begin{figure} \begin{center}
\includegraphics[width=0.7\textwidth]{fig/SU2ploop} 
\end{center} \end{figure} 

SU(2) Polyakov loop histogram $\s H$ at $\s \lambda=0.5$ (left) 
and $\s \lambda=1$ (right). 
\smallskip

\r Photon: \b Massless on both sides of the transition.
\medskip

}


\frame{\frametitle{\r Massive vector boson in deconfined phase} \b
\begin{figure} \begin{center}
\includegraphics[width=0.7\textwidth]{fig/fsamw} 
\end{center} \end{figure} \vskip -10pt

$\beta_a=\lambda$, $\beta_b=2\lambda$ fits of $a\,m_W(\lambda,N)$ 
for $a\,m_W(\lambda)=a\,m_W(\lambda,\infty)$. \bigskip

\centerline{\r Continuum Limit?} \vskip 10pt
}


\frame{\frametitle{\r Classical } \b
With matter fields \s
\begin{eqnarray} \nonumber % \label{L}
  &~& L =
    \overline{\psi}\left(i\gamma_{\mu}D^a_{\mu}-m\right)\psi 
  + \overline{\psi}\left(i\gamma_{\mu}D^b_{\mu}-m\right)\psi\,,
  \\ \nonumber &~& 
  - \frac{1}{2}{\rm Tr}\left(F^a_{\mu\nu}F^a_{\mu\nu}\right) 
  - \frac{1}{2}\left(F^b_{\mu\nu}F^b_{\mu\nu}\right)
  -\frac{\lambda}{4}\ {\rm Tr}\left(F^{\rm add}_{\mu\nu}
  F^{\rm add}_{\mu\nu}\right)\,.
\end{eqnarray} \b
Here \s $D^a_{\mu}=\partial_{\mu}+ig_aA'_{\mu}$ and $D^b_{\mu}=\partial_{
\mu}+ig_b B'_{\mu}$ \b are gauge covariant derivatives and the additional 
field tensor is \s
\begin{eqnarray} \nonumber \label{Fadd}
  F^{\rm add}_{\mu\nu} &=& g_b\partial_{\mu}B'_{\nu} -
  g_a\partial_{\nu}A'_{\mu} + i\,g_ag_b\left[A'_{\mu},B'_{\nu}\right]\,.
\end{eqnarray} \b
In gauge \s $B'_{\mu}\to B_{\mu}$, $A'_{\mu}\to A_{\mu}$, \b one gets \s
\begin{eqnarray} \nonumber % \label{Ladd}
  L^{\rm add}\ =\ 
  -\frac{\lambda\,g_a^2}{16}\,\left(\partial_{\mu}a_{\nu}\right)^2\
  -\frac{\lambda\,g_b^2}{16}\,\left(\partial_{\mu}b^i_{\nu}\right)^2\,.
\end{eqnarray} \b
These pieces are found in U(1) and SU(2) effective Lagrangians usually 
obtained by integrating over gauge transformations with a Gaussian 
weighting function, but here without the Faddeev-Popov ghost fields.
\r Transitions to longitudinal modes, divergent self-energy.
\vskip 10pt
}


\frame{\frametitle{\r Non-perturbative quantum continuum limit} \b 
\vskip -20pt
\centerline{ $a m_w\ \to\ a m_{\min}\ >0\ ?$}
\begin{figure} \begin{center}
\includegraphics[width=0.7\textwidth]{fig/amw} 
\end{center} \end{figure} \vskip -10pt
\centerline{\r $a m_w\ \to\ 0$ preferred by MC data!}

}

\frame{\frametitle{\r Summary and conclusions} \b

The presented MC calculations indicate divergence of the correlation 
length \s $\xi/a\to\infty$, $\xi=m_W^{-1}$ for $\lambda\to\infty$, \b
contradicting perturbation theory and supporting a quantum continuum 
limit. Of course, we cannot exclude that for larger systems and 
coupling constants the behavior may turn around and support \s $m_W(
\lambda)\to m_{\rm min}>0$. \b While this is correct, the perturbative 
approach is in essence vulnerable to similar criticism. There is no 
proof that perturbation theory describes the true nature of a QFT. 
\bigskip

Besides moving on to new horizons, we should perhaps keep an open 
mind for the third logical possibility that a deeper understanding 
of conventional QFT could unveil new models with massive vector
bosons. The unexpected numerical results provide no answers, but 
indicate that it is worthwhile to continue this investigation.
}

\end{document}

