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\begin{document}


\noindent
\hspace*{-36mm}
\includegraphics[scale=1.01,clip]{./hintergrund}


\vspace*{-1140mm}

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\vspace{1cm}

%% Titel
\vspace*{10mm}
\begin{center}
\fcolorbox{white}{white}
{
  \begin{minipage}[b]{600mm}
    \begin{center}
      \vspace*{10mm}
       \Huge{\sf
	\textcolor{cyan}{\bf Condensation in two flavor scalar electrodynamics with non-degenerate quark masses}}\\[7mm]
	\Large{\bf{Alexander Schmidt} \sf{, Philippe de Forcrand, Christof Gattringer} }
        \vspace{-1cm}
    \end{center}
   \vspace*{1cm}
  \end{minipage}
}
\end{center}


\vspace{3cm}

%%%%%%%%%%%%%%%%%%%%%%% 2 columns %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{multicols}{2}


%%%%%%%%%%%%%%%%%%%%%%% MOTIVATION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\large \centering{\textcolor{cyan}{\LARGE\sf Motivation}}

\vspace{1.0cm}

\begin{minipage}[b]{350mm}

  We study two-flavor scalar electrodynamics with two non-degenerate quark masses to find out about the characteristics of the condensation of this system induced by a finite chemical potential. 

  \vspace{-24pt}
\end{minipage}
\vspace{2.0cm}


%%%%%%%%%%%%%%%%%%%%%%% ACTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\large \centering{\textcolor{cyan}{\LARGE\sf The action}}

\vspace{1.0cm}

\begin{minipage}[b]{350mm}

  In the conventional notation the lattice action is given by (the lattice constant is set to $a=1$)

  \vspace{1cm}

  \begin{flushleft}
    \small
    {\color{cyan}Gauge field $U_{\vec{n},\mu}$} \quad 
    {\color{magenta}1st flavor Higgs field $\phi_{\vec{n}}^1$} \quad
    {\color{ForestGreen}2nd flavor Higgs field $\phi_{\vec{n}}^2$} \quad \quad
    {\color{gray}$U_{\vec{n},\mu} \in U(1)$, $\phi_{\vec{n}} \in \mathbb{C}$}
  \end{flushleft}
  \begin{eqnarray}
    S \hspace{0.1cm} & = & S_G[U] + S_H[U,\phi] \label{latac} \\ \nonumber \\
    S_G & = & -  \beta \sum_{\vec{n}} \sum_{\mu < \nu}  Re \; {\color{cyan}U_{\vec{n},\mu} \, U_{\vec{n} + \hat{\mu}, \nu} \, U_{\vec{n} + \hat{\nu}, \mu}^\star \, U_{\vec{n},\nu}^\star} 
    \\
    S_{H} & = & \sum_{\vec{n}}\! \Bigg[ \kappa^1 \mid \!\! {\color{magenta}\phi^1_{\vec{n}}} \!\! \mid^2
    + \lambda^1 \mid \!\! {\color{magenta}\phi^1_{\vec{n}}} \!\! \mid^4  
    + \kappa^2 \mid \!\! {\color{ForestGreen}\phi^2_{\vec{n}}} \!\! \mid^2
    + \lambda^2 \mid \!\! {\color{ForestGreen}\phi^2_{\vec{n}}} \!\! \mid^4 \Bigg ] \ \\
    &-& \sum_{\vec{n}}\! \Bigg[ \sum_{\mu}\! \Bigg( e^{\delta_{\mu 4} \mu^1}{\color{magenta}{\phi^1_{\vec{n}}}^\star} \, {\color{cyan}U_{\vec{n},\mu}} \,  {\color{magenta}\phi^1_{\vec{n}+\widehat{\mu}}}   
    + e^{-\delta_{\mu 4} \mu^1} {\color{magenta}{\phi^1_{\vec{n}}}^\star} \, {\color{cyan}U_{\vec{n} - \widehat{\mu},\mu}^\star} \,  {\color{magenta}\phi^1_{\vec{n}-\widehat{\mu}}} \Bigg) \Bigg] \nonumber \\
    &-& \sum_{\vec{n}}\! \Bigg[ \sum_{\mu}\! \Bigg( e^{\delta_{\mu 4} \mu^2}{\color{ForestGreen}{\phi^2_{\vec{n}}}^\star} \, {\color{cyan}U_{\vec{n},\mu}^\star} \,  {\color{ForestGreen}\phi^2_{\vec{n}+\widehat{\mu}}}   
    + e^{-\delta_{\mu 4} \mu^2} {\color{ForestGreen}{\phi^2_{\vec{n}}}^\star} \, {\color{cyan}U_{\vec{n} - \widehat{\mu},\mu}} \,  {\color{ForestGreen}\phi^2_{\vec{n}-\widehat{\mu}}} \Bigg) \Bigg]
    \nonumber
  \end{eqnarray}
  

  \vspace{0.2cm}

  with $\beta$ the inverse gauge coupling, $\kappa^i$ the mass parameters and $\lambda^i$ the Higgs couplings.

  \vspace{-24pt}
\end{minipage}
\vspace{2.0cm}


%%%%%%%%%%%%%%%%%%%%%%% FLUX ACTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\large \centering{\textcolor{cyan}{\LARGE\sf Flux representation of the action}}

\vspace{1.0cm}

\begin{minipage}[b]{350mm}

  {\textcolor{cyan}{\Large\sf The basic idea}} 
  is to expand the partition sum and perform the summation over the original degrees of freedom.

  \vspace{0.5cm}

  {\textcolor{cyan}{\Large\sf As an example}} 
  we look at a single nearest neighbour term
  \begin{eqnarray}
    Z \; \propto \; e^{\phi_x^\star \, U_{x,\nu} \,\phi_{x+\widehat{\nu}}}
    \; = \; \sum_{k_{x,\mu}}    \frac{1}{ (k_{x,\mu})!} \; 
    \bigg[ \, \phi_x^\star \, U_{x,\nu} \,\phi_{x+\widehat{\nu}} \bigg]^{\, k_{x,\mu}} \quad .
    \nonumber
  \end{eqnarray}

  Performing the summation over $\phi^i$ our partition sum no longer depends on the fields $\phi^i$
  \begin{eqnarray*}
    Z \; = \; \sum_{\{\phi\}} \sum_{\{U\}} \; e^{-S_G(U)-S_H(U,\phi)} &=& \sum_{\{\phi\}} \sum_{\{U\}} \; e^{-S_G(U)} \sum_{\{k,l\}} F(U,\phi,k,l) \\
    &=& \sum_{\{k,l\}} \sum_{\{U\}} \; e^{-S_G(U)} \underbrace{\sum_{\{\phi\}} F(U,\phi,k,l)}_{\textnormal{perform this summation}} \quad .
  \end{eqnarray*}

  {\textcolor{cyan}{\Large\sf Finally}}
  we end up with a real and positive partition sum plus constraints for the dual degrees of freedom
  \begin{eqnarray*}
    Z \; = \; \sum_{\{k,l\}} \sum_{\{p\}} FB(k,l,p) = \hspace{-0.5cm} \sum_{\{p, k^1, l^1, k^2, l^2\}} \hspace{-0.5cm} {\cal W}(p,k,l) \, {\cal C}_B(p,k^1,k^2) \, {\cal C}_F(k^i) \quad .
  \end{eqnarray*}

  \vspace{0.2cm}

  %\begin{center}
    %\includegraphics[height=13cm]{dofs.pdf}
  %\end{center}

  \vspace{-24pt}
\end{minipage}
\vspace{2.0cm}


%%%%%%%%%%%%%%%%%%%%%%% PHASE DIAGRAM %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\large \centering{\textcolor{cyan}{\LARGE\sf Phase diagram} \cite{PhysRevLett.111.141601}}

\vspace{1.0cm}

\begin{minipage}[b]{350mm}

  \begin{center}
    \includegraphics[height=22cm]{phasediagram.pdf}
  \end{center}

  \vspace{-24pt}
\end{minipage}
\vspace{2.0cm}

%%%%%%%%%%%%%%%%%%%%%%% MASS CORRELATORS %%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\large \centering{\textcolor{cyan}{\LARGE\sf Mass correlators in the confined phase (preliminary)}}

\vspace{1.0cm}

\begin{minipage}[b]{350mm}
  
  The masses of the bound states $U_1$ and $U_2$ are split because we set the effective masses of the two flavours to different values.

  \vspace{-0.5cm}

  \begin{center}
    \includegraphics[height=14.5cm]{mass.pdf}
  \end{center}

  \vspace{-24pt}
\end{minipage}
\vspace{2.0cm}

%%%%%%%%%%%%%%%%%%%%%%% CONDENSATION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\large \centering{\textcolor{cyan}{\LARGE\sf Condensation (preliminary)}}

\vspace{1.0cm}

\begin{minipage}[b]{350mm}

  We here show different observables as function of $\mu$. The dotted lines show the masses $U_1$ and $U_1$ determined from the plots above.

  \begin{center}
    \includegraphics[height=35.8cm]{finmu_840.pdf}
  \end{center}

  \vspace{-24pt}
\end{minipage}
\vspace{2.0cm}

%%%%%%%%%%%%%%%%%%%%%%% SUMMARY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\large \centering{\textcolor{cyan}{\LARGE\sf Summary}}

\vspace{1.0cm}

\begin{minipage}[b]{350mm}

  Although we studied the condensation of the system with two non-degenerate quark masses, we do not see two seperate condensation points, as we would have expected in first place. At the moment we are doing further simulations to better understand the finite mu transition of the system and the consequences of having two different quark masses. 
  
  \vspace{-24pt}
\end{minipage}
\vspace{2.0cm}

%%%%%%%%%%%%%%%%%%%%%%%%%% Acknowledgments %%%%%%%%%%%%%%%%%%%%%%%%%%%%

\hrule
\vspace{1.0cm}
\large \centering{\textcolor{cyan}{\Large\sf Acknowledgments}}

\vspace{1.0cm}

\begin{minipage}[b]{350mm}
This work was supported by the Austrian Science Fund, FWF, through the Doctoral
Program on {\it Hadrons in Vacuum, Nuclei, and Stars} (FWF DK W1203-N16).
\end{minipage}

%%%%%%%%%%%%%%%%%%%%%%%%%% References %%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\hrule
\vspace{1.7cm}
\large \centering{\textcolor{cyan}{\Large\sf References}}

\vspace{-1.0cm}

\begin{minipage}[b]{350mm}
    %\begin{multicols}{2}

      % \hrulefill
      \vspace{-8cm}
      \footnotesize
      \bibliographystyle{plain}
      \bibliography{bib}
      \vspace{-3cm}

    %\end{multicols}\vspace{-24pt}
  \end{minipage}

\end{multicols}
\end{document}
