Commit 4201c3bd authored by Ben O'Leary's avatar Ben O'Leary
Browse files

BOL edits

parent 1c7f4658
......@@ -44,10 +44,11 @@
\newcommand{\Tab}[0]{table\xspace}
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\newcommand{\ie}[0]{{i.e}.\xspace}
\newcommand{\Ie}[0]{{I.e}.\xspace}
\newcommand{\etc}[0]{\textit{etc}.\xspace}
\newcommand{\eg}[0]{\textit{e.g}.\xspace}
\newcommand{\Eg}[0]{\textit{E.g}.\xspace}
\newcommand{\ie}[0]{\textit{i.e}.\xspace}
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\newcommand{\Ref}[0]{Ref.\xspace}
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......
......@@ -50,9 +50,6 @@ While the interaction between stops and the Higgs fields allow the mass of the
with metastable DSB vacua should also have a high probability of surviving
tunneling to the true CCB vacua through thermal fluctuations.
\BOL{Something about Rohini and Morrisey:
\cite{Chowdhury:2013dka, Blinov:2013fta}.}
In \Sec~\ref{sec:method} we lay out the algorithm by which we compute
whether a parameter point is excluded by the DSB vacuum having a very low
probability of surviving to the present day either by a high probability of
......@@ -63,69 +60,3 @@ In \Sec~\ref{sec:method} we lay out the algorithm by which we compute
In \Sec~\ref{sec:results}, we show how much of the parameter space is excluded
by such conditions, and compare this to previous work. Finally we discuss the
consequences of the reduced parameter space in \Sec~\ref{sec:discussion}.
\BOL{Keeping the rest for the moment for ease of grabbing appropriate
citations.}
Some of the earliest work investigated the directions of the tree-level
potential where the quartic terms vanish, as soft SUSY-breaking terms could
lead to the potential being unbounded from below in these directions or to CCB
minima deeper than the DSB vacuum developing along these directions
\cite{Nilles:1982dy, AlvarezGaume:1983gj, Derendinger:1983bz, Claudson:1983et,%
Kounnas:1983td, Drees:1985ie, Gunion:1987qv, Komatsu:1988mt, Langacker:1994bc,%
Casas:1995pd, Casas:1996de}. Moreover, studies have been performed on the
tunneling time between different vacua \cite{Kusenko:1996jn}. Certain
conditions relating the trilinear parameters with the mass-squared parameters
have been obtained which ensured that no deeper minimum could develop
{\it along a line where the scalar fields have values in fixed ratios to each
other}. However, it has been known for many years that it is only very special
parameter points where this condition would be sufficient to forbid undesired
minima, and that in general even at tree level it is very difficult to ensure
that there are no undesired minima deeper than the desired minimum
\cite{Abel:1998ie}. Moreover, despite various claims in the literature
\cite{PhysRevD.48.4352, Casas:1995pd}, loop corrections are important: in
\cite{Bordner:1995fh}, a numerical minimization of the one-loop effective
potential including the top-quark Yukawa contributions has been performed
demonstrating the importance of the corrections, and it was noted in
\cite{Ferreira:2000hg} that loop corrections could change the ordering of which
minimum is deepest.
It is possible that CCB minima could be tolerated
\cite{Riotto:1995am, Kusenko:1996xt, Kusenko:1996jn}, if the Universe would
have fallen naturally into the false DSB vacuum as the cosmological temperature
decreased, and if the lifetime of this vacuum for tunnelling into the true CCB
vacuum is much longer than the present age of the Universe. Whether the DSB
vacuum is in fact preferred by cosmology depends, in particular, on the scalar
masses-squared generated during inflation. If these masses-squared are
positive and of the order of the square of the Hubble parameter, the `more
symmetric' DSB vacuum is favored. On the other hand, if these are negative, the
Universe would remain trapped in the true CCB vacuum \cite{Falk:1996zt}. A
detailed discussion on these issues including the case of negative $M^2_0$ at
$M_{GUT}$ and higher dimensional operators can be found in \cite{Ellis:2008mc}.
In the last few years there has been much progress in the field of determining
the global minimum of the potential of a quantum field theory. In particular,
the global minimum of renormalizable tree-level potentials (and other
potentials of a purely polynomial form) can now be found deterministically
with methods such as the Gr{\"{o}}ber basis method
\cite{Maniatis:2006jd, Gray:2008zs} or the homotopy continutation method
\cite{Huang199577, sommesenumerical, li2003numerical}.
Recently the power of the homotopy continuation method for finding tree-level
minima combined with gradient-based minimization with loop corrections has
been combined in the publicly-available code \vcs\
\cite{Camargo-Molina:2013qva}. We use this tool to investigate regions of the
CMSSM which, despite having local minima with the desired breaking of
$SU(2)_{L} \times U(1)_{Y}$ to $U(1)_{EM}$ while preserving $SU(3)_{c}$, might
have global minima with a different breaking of the gauge symmetries. This
allows us to update existing studies
\cite{Baer:1996jn, Strumia:1996pr, Ferreira:2000hg, Cerdeno:2003yt} by
calculating {\it all} the tree-level extrema and the complete one-loop
effective potential in the neighborhood of these extrema. This also allows us
to check to what extent the existing rules are useful at all. This is
particularly important in view of the fact that the explanation of the observed
Higgs mass requires special parameter combinations.
......@@ -31,20 +31,26 @@ If a parameter point is found to have a deeper CCB minimum, we label it as
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.
probabiltity of greater than one per-cent, as described in more detail in the
following subsection.
\subsection{Thermal corrections}
\label{sec:thermal_corrections}
\BOL{\vcs 1.1: thermal}
\BOL{Lifted from CMSSM paper intro:}
Whether the DSB
vacuum is in fact preferred by cosmology depends, in particular, on the scalar
masses-squared generated during inflation. If these masses-squared are
positive and of the order of the square of the Hubble parameter, the `more
symmetric' DSB vacuum is favored. On the other hand, if these are negative, the
Universe would remain trapped in the true CCB vacuum \cite{Falk:1996zt}. A
detailed discussion on these issues including the case of negative $M^2_0$ at
$M_{GUT}$ and higher dimensional operators can be found in \cite{Ellis:2008mc}.
\subsubsection{Comparison in methodology to previous works}
\label{sec:comparison_to_previous}
\subsection{Parameter scan}
\label{sec:parameter_scan}
\BOL{Parameters of scan}
......@@ -60,12 +66,43 @@ While spontaneous symmetry breaking in the SM is triggered by a negative
effective trilinear interaction $X_{t}$ between the stops and the Higgs is
large.
\subsubsection{Comparison in methodology to previous works}
\label{sec:comparison_to_previous}
\BOL{Something about Rohini and Morrisey:
\cite{Chowdhury:2013dka, Blinov:2013fta}.}
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
file\\
\texttt{SARAH-SPhenoMSSM\_RealHiggsAndStopVevs.vin} provided in the
\texttt{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.
\BOL{Lifted from CMSSM paper intro:}
Some of the earliest work investigated the directions of the tree-level
potential where the quartic terms vanish, as soft SUSY-breaking terms could
lead to the potential being unbounded from below in these directions or to CCB
minima deeper than the DSB vacuum developing along these directions
\cite{Nilles:1982dy, AlvarezGaume:1983gj, Derendinger:1983bz, Claudson:1983et,%
Kounnas:1983td, Drees:1985ie, Gunion:1987qv, Komatsu:1988mt, Langacker:1994bc,%
Casas:1995pd, Casas:1996de}. Moreover, studies have been performed on the
tunneling time between different vacua \cite{Kusenko:1996jn}. Certain
conditions relating the trilinear parameters with the mass-squared parameters
have been obtained which ensured that no deeper minimum could develop
{\it along a line where the scalar fields have values in fixed ratios to each
other}. However, it has been known for many years that it is only very special
parameter points where this condition would be sufficient to forbid undesired
minima, and that in general even at tree level it is very difficult to ensure
that there are no undesired minima deeper than the desired minimum
\cite{Abel:1998ie}. Moreover, despite various claims in the literature
\cite{PhysRevD.48.4352, Casas:1995pd}, loop corrections are important: in
\cite{Bordner:1995fh}, a numerical minimization of the one-loop effective
potential including the top-quark Yukawa contributions has been performed
demonstrating the importance of the corrections, and it was noted in
\cite{Ferreira:2000hg} that loop corrections could change the ordering of which
minimum is deepest.
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