\documentclass[aps,prd,preprint]{revtex4}
%\documentclass[aps,prl,twocolumn]{revtex4}
\usepackage{psfig}

\newcommand{\wt}{\widetilde}
\newcommand{\stl}{{\tilde{t}_1}}
\newcommand{\str}{{\tilde{t}_2}}
\newcommand{\st}{{\tilde{t}}}
\newcommand{\gi}{{\cal G}}
\begin{document}


\title{The Radiative Electroweak Symmetry Breaking in the MSSM at 2-loops} 
\date{\today}
\author{A. Dedes}
%\email{dedes@th.physik.uni-bonn.de}
\affiliation{Physikalisches Institut der Universitat Bonn, 
 Nussallee 12, 
 D-53115 Bonn, Germany}
\author{R.~Espinosa}
\affiliation{Madrid,  Madrid, Autonoma U., IFT, Spain}
\author{R.~J.~Zhang}
\affiliation{Princeton, Inst. Advanced Study , USA}

\begin{abstract}

We find the 2-loop tadpole corrections to the minimization
conditions of the effective potential in the MSSM.
We show that with the two-loop improvement, we almost
exactly cancel the renormalization group scale dependence of the 
vevs (or of the parameters $\mu$ and $B$).
Next, we  delineate the parameter space  where 
EWSB occurs. We find...



\end{abstract}

\maketitle

\section{The General Set-Up}

First of all we follow the notation of Ref.\cite{Espinosa:2000df}.
We ignore contributions from the electroweak gauge couplings 
and keep only terms in the Lagrangian which depend on the SU(3) gauge
coupling $g_3$ and the top Yukawa  coupling $Y_t$. {\bf (For large
$\tan\beta$ values we should also keep contributions from sbottoms
but at this point I do not know if we can straiforwardly include
this correction into your two loop effective potential. But I expect
that we can do it.)}. First we write the Higgs potential in terms
of the field $h_1$ and $h_2$ dependent masses,
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
V(h_1,h_2)=V_{vac}+V_0(h_1,h_2)+V_1(h_1,h_2)+V_2(h_1,h_2) \;,
\nonumber \\
\label{pot}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%
where $V_{vac}$ is the field independent vacuum self energy
(which is non-zero when SUSY is broken), $V_0$ is the tree level
potential
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray} 
V_0(h_1,h_2)=\frac{1}{2}(m_{H_1}^2+\mu^2)h_1^2
+\frac{1}{2}(m_{H_2}^2+\mu^2)h_2^2 + B_\mu h_1 h_2  \nonumber \\ \;,
\label{tree}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%
where all the parameters have been taken to be real, $V_1$ is
the 1-loop effective potential in the Landau gauge,
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray} 
16\pi^2 V_1(h_1,h_2)=3 \biggl [ H(m_{\stl}^2)+H(m_{\str}^2)-2 
H(m_t^2) \biggr ] + ...
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%
where ``...'' means other corrections proportional to the electroweak
gauge couplings, 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
m_{\stl}^2 &=& m_{\st}^2+m_t X_t \;\;,\;\; m_{\str}^2=m_{\st}^2-m_t X_t
\;\;,\;\; \nonumber \\[1mm] 
m_{\st}^2 & \simeq & M_S^2+m_t^2 \;\;,\;\; 
m_t X_t = \frac{h_t}{\sqrt{2}}(\mu h_1 + A_t h_2)\;\;,\;\; 
m_t^2=\frac{1}{2}h_t^2h_2^2 \;, \nonumber \\[1mm]
M_S &=& M_{\tilde{Q}}=M_{\tilde{U}} \;, 
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
and the function H is
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
H(m^2)=\frac{m^4}{2} \biggl (\ln\frac{m^2}{Q^2}-\frac{3}{2}\biggr ) \;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the above we have assumed that all supersymmetric particles
have roughly the same mass $M_S>>M_Z$. $V_2$ represents the 2-loop
contributions to the effective potential and this is given in the 
Appendix D of \cite{Espinosa:2000df}.

Inspection of the extrema of the two variables function
$V(h_1,h_2)$ leads to the following cases:



 {\it i) Radiative Electroweak Symmetry Breaking :} The $SU(2)\times
U(1)$
symmetry is broken at some scale $Q<Q_R$ where $Q_R$ is called
REWSB-scale. The potential (\ref{pot}) should satisfy the condition
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\Biggl [V_{h_1h_1}(Q_R)V_{h_2h_2}(Q_R)-V_{h_1h_2}^2(Q_R) \Biggr ]_{h_1=h_2=0}
=0  \;,
\label{rewsb}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%
where $V_{h_ih_j}=\frac{\partial^2 V}{\partial h_i \partial h_j}$.

 {\it ii) Stability of the potential :} There must be a scale $Q=Q_I$
such that the potential $V(h_1,h_2)$ is bounded from below,
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
V(h_1=h_2\rightarrow \infty ; Q_I ) >0 \;.
\label{stable}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%

{\it iii) Global EW minimum :} Neccesary and sufficient
conditions for a global minimum arround the EW vacuum expectation
values $v_1, v_2$ are,
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\Biggl [V_{h_1}=V_{h_2}\Biggr ]_{h_1=v_1, h_2=v_2} &=& 0 \;, 
\label{min1} \\[2mm]
\Biggl [V_{h_1h_1}\;\;\;,\;\;\; 
V_{h_2h_2}\Biggr ]_{h_1=v_1, h_2=v_2} &>& 0 \;, \label{min2}\\[2mm]
\Biggl [V_{h_1h_1}V_{h_2h_2}-V_{h_1h_2}^2 \Biggr ]_{h_1=v_1,
  h_2=v_2} &>& 0 \;, \label{min3} \\[2mm]
\Biggl [V(h_1,h_2)  \Biggr ]_{h_1=v_1, h_2=v_2} &<& 
 \Biggl [V(h_1,h_2,\Phi_i)  \Biggr ]_{h_1=v_1, h_2=v_2, \Phi_i=x_i} \;,
\label{global}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%% 
where $V_{h_i}=\frac{\partial V}{\partial h_i}$ and $\Phi_i$ are other
scalar fields of the MSSM with v.e.v's $x_i$. In this letter we 
{\it assume} the validity of (\ref{global}). Eqs
(\ref{min2},\ref{min3}) imply $\det({\cal M}_h^2)>0$. 

Ones  eqs.(\ref{min1}-\ref{global}) are satisfied then also
eq.(\ref{rewsb})  should be
satisfied.   Then the only thing that  is left is to calculate the
tadpole corrections {\it i.e., eqs.((\ref{min1},\ref{min2})} at 2 loop
level.  After checking  the scale invariance
of the VEVs we constraint 
the running parameters of the MSSM under the requirement that 
the 2-minimization conditions, (\ref{min1},\ref{min2})
are satisfied or not.







\section{1-loop corrections}
At one loop it is rather trivial to prove the scale 
invariance of the potential. Thus, with $\Omega_\Lambda = 
-V(h_1=h_2=0)=-\frac{3}{16\pi^2} M_S^4\ln\frac{M_S^2}{Q^2}$ and the
1-loop RGE's 
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\frac{\partial m_1^2}{\partial t} &=& \frac{3 h_t^2}{16\pi^2}  \mu^2 \\
\frac{\partial m_2^2}{\partial t} &=& \frac{3 h_t^2}{16\pi^2} \biggl [
m_2^2+2 M_S^2 + A_t^2 \biggr ] \\
\frac{\partial B_\mu}{\partial t} &=& \frac{3 h_t^2}{16\pi^2}\biggl [
\mu A_t +\frac{1}{2} B_\mu \biggr ] \\
\frac{\partial h_2^2}{\partial t} &=& -\frac{3 h_t^2}{16\pi^2} h_2^2 \;,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%  
with $m_{1,2}^2=m_{H_{1,2}}^2+\mu^2$ and $t=\ln Q^2$. 
Then up to two loop corrections,
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\frac{dV}{dt}=\frac{d\Omega_\Lambda}{dt}+\frac{dV_0}{dt}+\frac{dV_1}{dt}=0
\;.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%
The field dependent first and second derivatives of the potential are
%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
V_{h_1} &=& m_1^2 h_1 + B_\mu h_2 + \frac{3}{16\pi^2}
\frac{h_t \mu}{\sqrt{2}} \Biggl [ \gi(m_\stl^2)-\gi(m_\str^2) \Biggr ] \\
V_{h_2} &=& m_2^2 h_2 + B_\mu h_1 + \frac{3}{16\pi^2} \Biggl [h_t^2
h_2 \Biggl (\gi(m_\stl^2)+\gi(m_\str^2)-2\gi(m_t^2)\Biggr )\nonumber
\\ 
&+&\frac{h_t
  A_t}{\sqrt{2}}\Biggl ( \gi(m_\stl^2)-\gi(m_\str^2) \Biggr ) \Biggr ] \\
V_{h_1h_1} &=& m_1^2  + \frac{3}{16\pi^2} \frac{h_t^2 \mu^2}{2}
\Biggl [\ln\frac{m_\stl^2}{Q^2}+ \ln\frac{m_\str^2}{Q^2} \Biggr ] \\
V_{h_2h_2} &=& m_2^2  + \frac{3}{16\pi^2} \Biggl \{ \Biggl [
h_t^2 h_2+\frac{h_t A_t}{\sqrt{2}} \Biggr ]^2 \ln\frac{m_\stl^2}{Q^2}
+\Biggl [
h_t^2 h_2-\frac{h_t A_t}{\sqrt{2}} \Biggr ]^2 \ln\frac{m_\str^2}{Q^2}
\nonumber \\
&+&h_t^2 \Biggl (\gi(m_\stl^2)+\gi(m_\str^2) -6 E(m_t^2)\Biggr ) \Biggr
\} \\
V_{h_1h_2} &=& B_\mu  + \frac{3}{16\pi^2} \Biggl [\frac{(h_t^2\mu)(h_t
  h_2)}{\sqrt{2}}\Biggl ( \ln\frac{m_\stl^2}{Q^2}- \ln\frac{m_\str^2}{Q^2}
\Biggr )+\frac{h_t^2\mu A_t}{2} 
\Biggl ( \ln\frac{m_\stl^2}{Q^2}+ \ln\frac{m_\str^2}{Q^2} \Biggr ) \Biggr
]\;,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\gi(m^2)=m^2 \biggl (\ln\frac{m^2}{Q^2}-1 \biggr )$ and 
$E(m^2)=m^2\biggl (\ln\frac{m^2}{Q^2}-\frac{1}{3} \biggr )$.

\section{Two loop corrections}


\section{Numerical Analysis}

\section{Conclusions}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}
%
\bibitem{Espinosa:2000df}
J.~R.~Espinosa and R.~Zhang,
%``Complete two-loop dominant corrections to the mass of the lightest  CP-even Higgs boson in the minimal supersymmetric standard model,''
Nucl.\ Phys.\ {\bf B586} (2000) 3
[hep-ph/0003246].
%%CITATION = HEP-PH 0003246;%%
\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%






\end{document}