Commit df23b446 authored by Ben O'Leary's avatar Ben O'Leary
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More intro

parent 85b60248
\section{Introduction}
\label{sec:intro}
\BOL{Something about doing \cite{Chowdhury:2013dka} properly. Don't forget
\cite{Blinov:2013fta} as well!}
\BOL{Intro lifted from CMSSM paper. BOL will edit it down.}
The mechanism of spontaneous symmetry breaking through the vacuum expectation
value for a scalar field is an essential component of the standard model of
particle physics (SM) \cite{Glashow:1961tr, Weinberg:1967tq, Salam:1968rm},
......@@ -16,34 +11,46 @@ The mechanism of spontaneous symmetry breaking through the vacuum expectation
the spontaneous symmetry breaking of the SM, leading one to take the issue of
minimizing the scalar potential seriously.
Though the \vev of the Higgs field is not itself a physical observable, being
gauge- and renormalization-scheme-dependent, it is related to the physical
observable of the Higgs mass, for instance. The low value of the Higgs mass
compared with its ``natural'' value of the Planck mass is the notorious
``hierarchy problem''. A possible solution is to propose that the SM is a
low-energy approximation of a theory which has a symmetry that protects the
mass of the Higgs boson from large loop contributions. One such symmetry is
supersymmetry (SUSY), where every scalar has a fermion partner and
\it{vice-versa}, which leads to exact cancellations of loops.
\BOL{Blah blah SUSY blah blah Natural MSSM.}
Promoting the SM to the minimal supersymmetric standard model (MSSM) (see
\cite{Martin:1997ns} for a review) doubles the particle content and also
requires the addition of a second Higgs $SU(2)_{L}$ doublet and fermionic
partners. While it protects the Higgs boson mass, SUSY also has the serious
flaw that it forces the masses of the scalar partners to be exactly that of
their fermionic partners, and no partners for any of the SM particles have
thus far been found \BOLcite. Hence if SUSY is a symmetry of Nature, it must
be broken spontaneously somehow. However, as we have no clues as to how this
might be exactly, we can parameterize our ignorance by adding soft
SUSY-breaking terms to the Lagrangian.
\BOL{Blah blah other minima blah blah sfermion \vevs blah blah
``desired symmetry-breaking (DSB) vacuum, where only the neutral components
of the Higgs doublets get non-zero \vevs'' blah blah.}
\BOL{Blah blah 14 Gy of $T \simeq 0 \gev$ blah blah thermal fluctuations.}
The minimal supersymmetric extension of the SM (MSSM) has a much more complex
scalar potential by merit of there being many more scalar fields (partners for
each SM fermion as well as a second Higgs $SU(2)_{L}$ doublet) which interact
with the Higgs fields. The large effect of the extra loops on the mass of the
Higgs boson along with the non-observation of supersymmetric partners thus far
has led to the pragmatic region of the MSSM parameter space known as the
Natural MSSM \BOLcite. This is the region where the masses of all the partners
are very large but for those with the largest contributions to the Higgs
mass~\cite{Draper:2011aa, Heinemeyer:2011aa, Brummer:2012ns, Djouadi:2013vqa,%
Arbey:2012bp, Arganda:2012qp}, which should have masses not very far above the
electroweak scale so that there is little finely tuned cancellation between
loop contributions to the Higgs mass, and thus is in some sense natural
\BOLcite. Thus the \it{stops} \tsq (scalar partners of the top quarks) should
have TeV-scale soft supersymmetry-breaking parameters while all others are
assumed to have very large masses. The partners of the bottom quarks and tau
leptons could also be in the TeV-scale, but in this letter we consider only
stops, noting that our algorithm is trivially generalizable and is already
implemented in \vcs, which we use and have made available to the public.
While the interaction between stops and the Higgs fields allow the mass of the
Higgs boson to reach $125 \gev$ in the MSSM, it also leads to the possibility
of the scalar potential having undesired minima apart from the desired
symmetry-breaking (DSB) vacuum, where only the neutral components
of the Higgs doublets get non-zero \vevs \BOLcite. Even though a parameter
point may be chosen where the scalar potential has a minimum where the stops
do not have non-zero \vevs, there is no guarantee that this is the global
minimum: there may be deeper charge- and color-breaking (CCB) minima to which
the Universe may tunnel \cite{Nilles:1982dy, AlvarezGaume:1983gj,%
Derendinger:1983bz, Claudson:1983et, Kounnas:1983td, Drees:1985ie,%
Gunion:1987qv, Komatsu:1988mt, Langacker:1994bc, Casas:1995pd, Casas:1996de}
\BOLcite. However, even if the DSB vacuum is only metastable, the parameter
point is still acceptable if the expected tunneling time is of the order of
the age of the known Universe
\cite{Riotto:1995am, Kusenko:1996xt, Kusenko:1996jn} \BOLcite. Also, given
the convincing success of the Big Bang theory, acceptable parameter points
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
......@@ -57,38 +64,10 @@ In \Sec~\ref{sec:method} we lay out the algorithm by which we compute
consequences of the reduced parameter space in \Sec~\ref{sec:discussion}.
\BOL{Keeping the rest for the moment for ease of grabbing appropriate
citations.}
At the same time a mass of $125 \gev$ for the lightest MSSM Higgs boson is
difficult to achieve without at least one stop with a multi-TeV mass
\cite{Draper:2011aa, Heinemeyer:2011aa, Brummer:2012ns, Djouadi:2013vqa,%
Arbey:2012bp, Arganda:2012qp}. One way to achieve this is to have a
sufficiently high $M_{0}$ so that the stop mass is large enough, and a
sufficiently high $M_{1/2}$ to bring the mass of the lightest neutralino up to
just below the lightest stau mass (though for sufficiently high masses, the
stau co-annihilation mechanism cannot reduce the relic density to the observed
value, even for a vanishing mass difference between the stau and the lightest
neutralino \cite{Ellis:1998kh, Citron:2012fg}). However, this region of
parameter space has rather bleak prospects for the discovery of sparticles. The
only other way that this can be achieved within the CMSSM is through a large
$A_{0}$, inducing a large splitting between the two mass eigenstates of both
the staus and the stops. This allows the loop corrections to the Higgs mass to
be large through both the existence of a heavy stop and the large splitting
between the stop mass eigenstates, while allowing at least some potentially
LHC-accessible sparticles \cite{Bechtle:2012zk}.
The presence of many additional scalar partners for the SM fermions raises the
question of whether they too could develop \vevs. If, for example, the
potential were such that stops would develop non-zero \vevs, then that would
be disastrous as these \vevs would spontaneously break $SU(3)_{c}$ and
$U(1)_{EM}$! Unfortunately until recently it was quite impractical to search
for other vacua to see whether the desired vacuum is stable, or whether there
are charge- and/or color-breaking (CCB) minima.
Since there are many different possibilities for vacua in the MSSM,
spontaneously breaking any or all of the gauge symmetries, we denote the
vacuum that should describe the Universe in which we live as the
desired-symmetry-breaking (DSB) vacuum.
Some of the earliest work investigated the directions of the tree-level
potential where the quartic terms vanish, as soft SUSY-breaking terms could
......@@ -149,19 +128,3 @@ Recently the power of the homotopy continuation method for finding tree-level
particularly important in view of the fact that the explanation of the observed
Higgs mass requires special parameter combinations.
This paper is organized as follows: in \Sec~\ref{sec:rules} we collect the
existing analytical approximations to determine color- and/or
charge-breaking minima. In \Sec~\ref{sec:method}, we briefly explain the tools
that we used to generate the spectra of CMSSM points and to evaluate the
stability of such points against undesired vacua. Then we consider how robust
our results are against which scalars are allowed non-zero \vevs and against
variations in scale, how dependent they are on loop corrections, and how the
results might depend on the precise values of the Lagrangian parameters as
evaluated by different spectrum generators; we also examine the usefulness of
the tree-level conditions mentioned above. In \Sec~\ref{sec:constraining}, we
investigate part of the stau co-annihilation region to demonstrate that its
parameter points generally are only metastable, and we also demonstrate that it
is difficult to get light stops in the CMSSM without rapid tunneling to CCB
vacua. In addition, we show the stability of regions compatible with the
measured mass of the Higgs boson, and further demonstrate the irrelevance of
the ``thumb rules'' to the parameter space regions of interest.
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