Commit 85b60248 authored by Ben O'Leary's avatar Ben O'Leary
Browse files

Wrote some intro

parent eade42d2
......@@ -6,9 +6,38 @@
\BOL{Intro lifted from CMSSM paper. BOL will edit it down.}
\BOL{Blah blah LHC blah blah SM Higgs.}
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},
which has proven itself to be an accurate description of Nature all the way to
the tera-electron-Volt scale. The discovery of the bosonic resonance at
$125 \gev$ at the Large Hadron Collider (LHC)
\cite{Aad:2012tfa,Chatrchyan:2012ufa} is consistent with the Higgs boson of
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 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
......@@ -16,7 +45,7 @@
\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
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
......@@ -25,92 +54,10 @@
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
consequences of the reduced parameter space in \Sec~\ref{sec:discussion}.
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
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