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% ==============================================================================
Bethe Center for Theoretical Physics \& Physikalisches Institut der
Universit\"at Bonn, \\
53115 Bonn, Germany }
Physik Department T70,
Technische Universit\"at M\"unchen\\
85748 Garching, Germany}
Institut f\"ur Theoretische Physik und Astronomie,
Universit\"at W\"urzburg\\
Am Hubland,
97074 W\"urzburg, Germany}
\title{Constraining the Natural MSSM through tunneling to color-breaking vacua
at zero and non-zero temperature}
\author[a]{J.\ E.\ Camargo-Molina}
\author[b]{F.\ Staub}
\author[a]{B.\ O'Leary}
\author[a]{W.\ Porod}
\author[c]{F.\ Staub}
\keywords{supersymmetry, vacuum stability}
%\pacs{??, ??, ??}
\BOL{We show that important constraints can be placed on the parameter space of
the minimal supersymmetric standard model, in particular the region known as
the Natural MSSM, where the masses of the scalar partners of the top quarks
are within an order of magnitude or so of the electroweak scale. These
constraints arise from the interaction between these scalar tops and the Higgs
fields, which allows the possibility of parameter points having deep charge-
and color-breaking true vacua. In addition to requiring that our
electro-weak-symmetry-breaking, yet QCD- and electromagnetism-preserving
vacuum has a sufficiently long lifetime at zero temperature, demanding
stability against thermal tunneling as well further restricts the allowed
parameter space. We calculate these constraints using code which we have also
made publicly available: \vcs, which now in addition to finding deeper minima
and calculating tunneling times at zero temperature now also calculates a
survival probability against thermal tunneling.}
% ==============================================================================
The authors would like to thank \BOL{some friends, B.O'L. is sure}. This work
has \BOL{probably} been supported by the DFG, research training group GRK 1147
and project No.\ \BOL{??-????/?-?}. FS is \BOL{(still?)} supported by the BMBF
PT DESY Verbundprojekt 05H2013-THEORIE ``Vergleich von LHC-Daten mit
supersymmetrischen Modellen''.
\section{Discussion and conclusion}
\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.}
\BOL{Blah blah LHC blah blah SM Higgs.}
\BOL{Blah blah BSM blah blah SUSY blah blah Natural MSSM.}
\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.}
\BOL{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
critical bubbles of true vacuum forming through quantum fluctuations in our
past light-cone at zero temperature, or by such bubbles forming through
thermal fluctuations during the period when the Universe was at sufficiently
high temperature that such fluctuations could have been sufficiently probable.
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}.}
The standard model of particle physics (SM)
\cite{Glashow:1961tr, Weinberg:1967tq, Salam:1968rm} has proven itself to be an
extremely good description of particle physics all the way up to the
tera-electron-Volt scale. The interpretation of the bosonic resonance at
$125 \gev$ recently discovered at the Large Hadron Collider (LHC)
\cite{Aad:2012tfa,Chatrchyan:2012ufa} as the Higgs boson of the SM completes
the picture, and allows us to probe the mechanism of the spontaneous breaking
of gauge symmetries.
A particular issue, however, of the Higgs boson is that its mass, determining
the electroweak scale, is many orders of magnitude smaller than the ``natural''
value of the order of the Planck scale. This, along with the fact that quantum
corrections to the mass of the Higgs boson are typically of the scale of the
heaviest particles which interact with it, leads to model builders attempting
to resolve this ``hierarchy problem''.
One popular way of ameliorating the hierarchy problem is to promote the SM to a
supersymmetric theory, such as the minimal supersymmetric standard model (MSSM)
(see \cite{Martin:1997ns} for a review). The MSSM contains all the particles
and interactions of the SM as a subset, and the interactions of the
supersymmetric partners are related to the SM interactions.
Usually one postulates a conserved parity known as $R$-parity
\cite{Farrar:1978xj} to avoid baryon- and lepton-number violating interactions
which would be otherwise allowed by gauge symmetries and supersymmetry. The
conservation of this parity, which we take here as part of the definition of
the MSSM, implies the stability of the lightest supersymmetric partner (LSP),
which, if uncharged under $SU(3)_{c} \times U(1)_{EM}$, is a candidate particle
to explain the observed dark matter of the Universe.
In principle, the MSSM has {\it less} parameters than the SM, since the quartic
coupling of the Higgs boson is actually a function of the gauge couplings.
However, the mechanism of supersymmetry (SUSY) breaking must introduce more
parameters, and agnosticism of the exact mechanism leads to the common practice
of parametrizing the mechanism by adding {\it soft SUSY-breaking terms} to the
Lagrangian density. The number of parameters specifying the full set of soft
SUSY-breaking terms allowed in the MSSM is rather large, namely 105
\cite{Martin:1997ns}, so often they are taken to be related at a specific
scale. One of the simplest and most popular proposals is the
minimal-supergravity-inspired constrained MSSM (CMSSM), in which all the soft
SUSY-breaking scalar mass-squared terms are taken to be equal to $M_{0}^{2}$
at the scale $M_{GUT}$ where the gauge couplings unify, assuming that somehow
the MSSM is a part of a grand unified theory (GUT). In addition, the soft
SUSY-breaking mass terms for the fermionic partners of the gauge bosons are
also taken to unify at $M_{GUT}$ with a common value $M_{1/2}$. Finally a third
GUT-scale common value is defined: $A_{0}$, a common trilinear scalar
interaction coupling (multiplied by the corresponding Yukawa couplings). In
principle, the $\mu$ parameter coupling the Higgs doublets and the
corresponding soft SUSY-breaking term $B_{\mu}$ could also be defined at the
GUT scale, but for practical reasons they are taken to be engineered by the
requirement of correct electroweak symmetry breaking at a given scale, often
the geometric mean of the masses of the two stops, the scalar partners of the
top quark, referred to as the SUSY scale. The values of $|\mu|$ and $B_{\mu}$
are fixed by requiring that the mass of the $Z$ boson is correct along with
defining the ratio $\tan \beta$ of the {\it vacuum expectation values} (\vevs)
\vd{} and \vu{} of the neutral components of the two Higgs doublets, and the
sign of $\mu$ is given as a final input\footnote{However, it has been noted in
\Ref~\cite{Allanach:2013cda} that defining a Lagrangian partially at different
scales is ambiguous and in the case of the CMSSM, different values of $\mu$
and $B_{\mu}$ at the GUT scale can lead to the same $m_{Z}$ and $\tan \beta$
at the SUSY scale while keeping $M_{0}, M_{1/2}$, and $A_{0}$ the same. This
is indeed interesting, but beyond the scope of this work, where we identify
CCB minima of SUSY-scale Lagrangians which were generated by CMSSM
The constraints of the CMSSM broadly lead to fixed ratios of the gaugino masses
at the SUSY scale, and groupings of the squark masses together and the slepton
masses together. The size of the gaugino masses and the masses of the groupings
of sfermions relative to the gauginos are controlled by $M_{1/2}$ and $M_{0}$
respectively, and the only remaining freedom to change this spectrum lies in
tuning $A_{0}$ and $\tan\beta$ to separate out the third-generation sfermions
from the other generations.
The interplay between this handful of parameters is usually enough to explain
many observations. For instance, requiring that the relic density of dark
matter is within the uncertainties of the observed value might seem to
significantly constrain the allowed parameter space \cite{Olive:1989jg}.
However, if one fixes one or two of the CMSSM parameters, there is typically a
range of the other parameters where the relic density is compatible with
observations \cite{Ellis:1998kh}. One particular region is known as the
``stau co-annihilation region'' \cite{Ellis:1998kh}, where the stau is
marginally heavier than the lightest neutralino, which is the LSP, and freezes
out only slightly earlier, allowing more annihilation before the neutralino
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
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.
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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