Commit 1c7f4658 authored by Ben O'Leary's avatar Ben O'Leary
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\section{Discussion and conclusion}
\label{sec:discussion}
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\section{Parameter point selection and stability evaluation}
\label{sec:method}
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We categorize the stability or metastability of a parameter point by a
multi-stage process. First, a consistent set of Lagrangian parameters at a
fixed renormalization scale is generated by
\spheno\ \cite{Porod:2003um,Porod:2011nf}, such that the MSSM physics at the
DSB vacuum is consistent with the SM inputs ($m_{Z}, G_{F}$, \etc), and these
parameters are stored in a file in the \slha format which is passed to \vcs.
\vcs is a publicly-available code \cite{Camargo-Molina:2013qva} that then
prepares the minimization conditions for the tree-level potential as input for
the publicly-available binary \homps\ \cite{lee2008hom4ps} that finds all
possible solutions to the particular minimization conditions of the parameter
point. These are then used by \vcs as starting points for gradient-based
minimization by \minuit\ \cite{James:1975dr} through
\pyminuit\ \cite{pyminuit} to minimize the full one-loop potential with
thermal corrections at a given temperature. If a minimum deeper than the DSB
vacuum is found, the probability of tunneling out of the false DSB vacuum is
then calculated through \cosmotransitions\ \cite{Wainwright:2011kj}.
If a parameter point is found to have a deeper CCB minimum, we label it as
metastable, otherwise we label it stable\footnote{It may be that a parameter
point is actually metastable if other scalar fields such as the partners of
bottom quarks were allowed non-zero \vevs. However, we restrict ourselves to
a region of parameter space where such concerns are negligible as the
relevant trilinear interaction is small, but note also that this restriction
cannot mistakenly label a stable parameter point as metastable.}. We then
divide the metastable points into short-lived points which would tunnel out
of the false DSB vacuum in 3 giga-years or less (corresponding to a
survival probability of lasting 14 Gy of one per-cent or less), and the rest
as long-lived. Finally, we divide the long-lived points into thermally
excluded, by having a probability of the DSB vacuum surviving thermal
fluctuations of one per-cent or less, or allowed, by having a survival
probabiltity of greater than one per-cent.
\subsection{Thermal corrections}
\label{sec:thermal_corrections}
\BOL{\vcs 1.1: thermal}
\subsubsection{Comparison in methodology to previous works}
\label{sec:comparison_to_previous}
\subsection{Parameter scan}
\label{sec:parameter_scan}
\BOL{Parameters of scan}
While spontaneous symmetry breaking in the SM is triggered by a negative
mass-squared term in the Lagrangian for the Higgs
field\footnote{The possibility that it is due to a massless Coleman-Weingberg
model has been ruled out by measurements of the top mass, for
example~\cite{Sher:1988mj}.}, it is neither a necessary nor sufficient
condition for any scalar field in a multi-scalar theory to develop a non-zero
\vev. In particular, a positive mass-squared for the stop fields does not
preclude a parameter point from having a CCB minimum, especially if the
effective trilinear interaction $X_{t}$ between the stops and the Higgs is
large.
We refer the reader to section $2.1$ of \Ref~\cite{Blinov:2013fta} or
section B of \Ref~\cite{Chowdhury:2013dka} for the explicit form of the
relevant part of the tree-level scalar potential. One may also inspect the
file \verbatim{SARAH-SPhenoMSSM_RealHiggsAndStopVevs.vin} provided in the
\verbatim{MSSM} sub-directory of the \vcs download, where the tree-level
potential and also the mass-squared matrices required for assembling the
loop corrections, as described in \cite{Camargo-Molina:2013qva}, are shown.
\section{Constraining relevant regions of the Natural MSSM parameter space}
\label{sec:constraining}
\label{sec:results}
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