DESY 16-114

A new view on vacuum stability in the MSSM

Wolfgang Gregor Hollik 111E-mail<EMAIL_ADDRESS>
Deutsches Elektronen-Synchrotron (DESY)

Notkestraße 85, D-22607 Hamburg, Germany

A consistent theoretical description of physics at high energies requires an
assessment of vacuum stability in either the Standard Model or any extension
of it. Especially supersymmetric extensions allow for several vacua and the
choice of the desired electroweak one gives strong constraints on the
parameter space. As the general parameter space in the Minimal Supersymmetric
Standard Model is huge, any severe constraint on it unrelated to direct
phenomenological observations enhances the predictability of the model. We
perform an updated analysis of possible charge and color breaking minima
without relying on fixed directions in field space that minimize certain terms
in the potential (known as “$D$-flat” directions). Concerning the cosmological
stability of false vacua, we argue that there are always directions in
configuration space which lead to very short-lived vacua and therefore such
exclusions are strict. In addition to existing strong constraints on the
parameter space, we find even stronger constraints extending the field space
compared to previous analyses and combine those constraints with predictions
for the light CP-even Higgs mass in the Minimal Supersymmetric Standard Model.
Low masses for supersymmetric partners are excluded from vacuum stability in
combination with the $125\,\mathrm{GeV}$ Higgs and the allowed parameter space
opens at a few $\mathrm{TeV}$.

Keywords: Mostly Weak Interactions: Beyond Standard Model, Supersymmetric
Standard Model
PACS: 11.15.Ex, 11.30.Pb, 12.60.Jv

##  1 Introduction

The Standard Model (SM) of particle physics is completed with the final
discovery of the Higgs boson (the SM scalar) [1, 2] which shows the expected
properties in the experiment [3] and only leaves small room for deviations
from the SM predictions. However, this discovery finalized a set of problems
within the SM from which one is the hierarchy problem of the Higgs mass [4, 5,
6, 7] another one the discussion about the cosmological stability of the
electroweak ground state [8, 9, 10, 11, 12, 13, 14, 15, 16]. Surprisingly, the
most popular extension of the SM to solve the hierarchy problem simultaneously
cures the stability problem, which is the Minimal Supersymmetric Standard
Model (MSSM). Besides the well-known solution of the hierarchy problem by the
existence of bosonic degrees of freedom that cancel loop contributions,
similar contributions render the effective potential stable—besides the
property of the MSSM having an intrinsically stable Higgs potential at the
tree-level. This solution to all problems, however, comes along with a bunch
of new problems from which a prominent one in connection to the stability of
the vacuum state is the possible destabilization of the Higgs potential by
additional scalar degrees of freedom. Finally, the true vacuum of the theory
is related to the absolute ground state of the scalar potential which is not
exclusively dedicated to vacuum expectation values (_vev_ s) of Higgs scalars
anymore but can be due to _vev_ s of the additional scalars that break
electric and/or color charge and/or additionally baryon and lepton number.
While spontaneous breaking of lepton number may be a desired solution to the
origin of neutrino masses [17, 18, 19, 20], the spontaneous breakdown of good
gauge symmetries in the SM should be avoided in a way that
$\mathrm{SU}(3)_{c}\times\mathrm{U}(1)_{\text{em}}$ stays intact.

It was already noticed in the early 1980s [21] that supersymmetric models tend
to have charge breaking minima and in the following rather strong constraints
on the soft breaking terms have been derived [22, 23, 24, 25, 26, 27].
Subsequently, many attempts have been performed to improve such kind of bounds
using several optimization criteria [28], higher loop effects [29, 30, 31,
32], relaxing constraints allowing for metastable states [33, 34],
constraining flavor violation [35] and applying metastability constraints on
the flavor violating bounds [36]. A sophisticated collection of codes checking
for non-standard tree-level minima, improving with the one-loop effective
potential and calculating tunneling rates in presence of finite temperatures
by the help of CosmoTransitions [37] is given by the Vevacious collaboration
[38]. Recently, the old charge and color breaking (CCB) constraints have been
analyzed and tested in the light of the Higgs discovery at the LHC [39, 40]
with an updated tunneling analysis [41]. An investigation of the one-loop
Higgs potential in the MSSM [42] reveals an interrelation of one-loop
stability constraints from the Higgs sector only and tree-level CCB
constraints including colored directions [43]. Considerations of vacuum
stability are a widely used ingredient in studies of MSSM-like scenarios [44,
45, 46, 47, 48].

A general paradigm is that charge and color breaking minima in the MSSM most
probably appear in such directions in field space where the $D$-terms vanish.
$D$-terms are the quadrilinear contributions to the full scalar potential
proportional to squared gauge couplings and therefore always positive and
always seen as to win over any negative contribution. A first more complete
and rather exhaustive analysis taking basically all directions in field space
into account was given about twenty years ago by [27], where a full list of
many special cases had been discussed.

Still, a complete analysis of the problem that somehow resides in a
satisfactory solution is not possible. We provide a possible way to handle the
existence of non-standard vacua in the MSSM scalar potential that follows the
spirit of [27] and goes beyond. The minimization procedure reduces then
effectively to the optimization of the necessary condition for the existence
on non-standard vacua. This optimization, however, is neither unique nor
unambiguously to be determined. Moreover, once the vacuum tunneling
probability is addressed, a new concern for the “optimized” field direction
may arise: to give the strongest bound from the vacuum metastability,
configurations are rather preferred that lead to the _minimal_ tunneling time.
Whether or not this requirement can be exploited in automated computer tools
may be left to the programming skills of the developers. For the pedestrian,
it appears sufficient to have a clear analytical cut although those rules are
indeed not sufficient but necessary. This analytical cut, however, should only
distinguish between a global CCB minimum and a strictly stable “desired”
electroweak vacuum.

Why is a reassessment of this problem needed? Besides the complete analysis of
[27] not so much has been done on the analytical level as it is quite hard and
any access lacks generality. Since this great catalog of dangerous directions
and associated bounds on the parameters has been worked out, the greatest
further achievement is the discovery of the Higgs boson [2, 1] that appears to
be very SM-like and has (for MSSM purposes) a rather high mass of
$m_{h^{0}}=125\,\mathrm{GeV}$ as follows from the combination of ATLAS and CMS
data at $7$ and $8\,\mathrm{TeV}$ [49]. This value requires sizeable radiative
corrections, that are known to be large in the MSSM [50, 51]. However, the
available parameter space gets very much constraint imposing the correct Higgs
mass, even if one allows for a generous theoretical error of about
$3\,\mathrm{GeV}$ in the determination of this mass [52]. Especially, to
achieve this shift a large stop mixing is needed which conversely requires
large trilinear soft SUSY breaking couplings [53, 54], assisted maybe by a
large Higgsino mass parameter. These large trilinear scalar terms, however,
unambiguously lead to CCB minima and render the desired vacuum unstable. It is
therefore necessary and important to put severe constraints on those terms in
order to assure theoretical consistency. As long as there persists to be no
discovery of any sparticles at the LHC, inferring larger lower bounds on the
sparticle masses will also lead to possibly more stable configurations as
larger SUSY masses themselves lead to larger shifts in the Higgs mass [52]
without the need for large left-right squark mixing. Anyhow, compressed
scenarios that might be hidden in the collider searches are likely to be in
trouble with the stability bounds; especially if they tuned [55] in such a way
to reproduce weird signatures [56, 57].

We proceed in this paper as follows: after introducing the four-field scalar
potential, which is basically the necessary object to deal with in connection
to the influence on the Higgs mass, we derive a generic exclusion bound in
Section 2. The anatomy of the CCB states described by this bound is discussed
in Section 3. Finally, we conclude.

##  2 The four-field scalar potential

The MSSM in fact is a multi-scalar theory and its scalar potential is a
complicated object potentially leading to undesired configurations. The
configuration space depends on the vacuum expectation values of each field
that are the field values at the minima of the potential. The potential in
general has multiple minima where only the global one is considered to be the
true ground state of the theory. If in any case the current electroweak vacuum
we are believing to be sitting in is not the true one, this configuration will
only be stable for a certain amount of time and due to quantum tunneling the
global minimum will be reached. Moreover, we have to take care that the
potential is not unbounded from below (constraints known as UFB, i. e.
_unbounded from below_ bounds in the literature). Taking quantum corrections
(and at least the one-loop effective potential) into account, those will
always be rescued and the quantum potential will be bounded from below [27,
42, 58], whereas a new deep minimum will appear at very large field values.
Contrary to large field-valued minima that usually come along with low
tunneling rates into the true ground state, the minima discussed in this paper
are close-by roughly with _vev_ s around the SUSY scale (few $\mathrm{TeV}$).

We are especially interested in the cross-relations of current analyses in the
MSSM Higgs sector with the formation of non-standard vacua. The missing
observational evidence for SUSY partners at all paired with a relatively heavy
SM-like Higgs requests extreme parameter configurations. Existing analytical
and semi-analytical bounds on the parameter space from the stability of the
standard electroweak vacuum still are in agreement with what is needed to cope
with the current situation. However, as we will see, most scenarios in the
phenomenological MSSM (pMSSM) where all parameters are defined as input values
at the SUSY scale suffer from charge and color breaking minima already at the
SUSY scale (or slightly above). Moreover, the usual argument that tunneling
rates to the deeper minimum are sufficiently small does not hold as there can
be always a path in field space found where a closer vacuum shows up and fast
tunneling proceeds to rolling down towards the final true vacuum. We shall
explain this further.

Knowing the ground state of the theory means knowing the origin of spontaneous
symmetry breaking means knowing the structure of the _scalar_ potential. Each
non-vanishing _vev_ of fermionic or vector component fields would in addition
break Lorentz symmetry and destroy the structure of space-time. Only the
scalar part can break inner symmetries spontaneously and in a way which keeps
external symmetries intact (not to speak about supersymmetry, but to break it
we rely on soft breaking and stay ignorant about its deeper origin). The
ground state of the theory is given by the state which minimizes the potential
energy density; therefore the relevant object is actually the _effective_
potential, which at tree-level is equivalent to the classical scalar
potential. In principle, quantum (one and higher loop) effects are calculable
[59, 60] and allow for spontaneous breaking radiatively. While the SM
effective potential can be trivially made stable at the tree-level by choosing
the Higgs self-coupling positive, the same coupling runs negative at higher
energies and renders the electroweak vacuum metastable on cosmological scales
[9, 61]. In multi-scalar theories as in the MSSM, the situation is more
involved already at the tree-level; a tree-level analysis of the scalar
potential will result in regions of allowed parameters. Loop corrections are
not expected to make unstable regions more stable around the scale of the
relevant _vev_ , although purely loop-induced minima may be missed.

The MSSM scalar potential is calculated according to some simple rules and
consists of three basic contributions to which we will refer as the soft
breaking, the $F$-term and the $D$-term contribution:

$V=V_{\text{soft}}+V_{F}+V_{D}.$ (1)

The soft breaking part breaks supersymmetry softly and mimics the couplings of
the superpotential plus additional scalar mass terms, where the $F$-terms
basically follow from the superpotential as derivatives with respect to the
scalar components

$V_{F}=\left|\frac{\partial\mathcal{W}}{\partial\phi}\right|^{2},$ (2)

where the sum over all scalar degrees of freedom is implicitly assumed to keep
a plain notation. In our discussion and analysis, we consider only the chiral
supermultiplets of third generation quarks as they couple with comparably
large Yukawa couplings (as superpotential parameters) to the Higgs sector and
also their corresponding trilinear soft SUSY breaking couplings are assumed to
be large. For cleanliness and a first understanding of the “new” phenomena
hidden in an old setup, we leave leptons and their superpartners out of the
game as we are primarily interested in the appearance of color breaking
minima. The inclusion of third generation (s)leptons is, however, trivial and
follows the same procedure. We then define the (reduced) superpotential of
“our” version of the MSSM by

$\mathcal{W}=\mu\;H_{\mathrm{d}}\cdot
H_{\mathrm{u}}+y_{\mathrm{t}}\;H_{\mathrm{u}}\cdot
Q_{\mathrm{L}}\bar{T}_{\mathrm{R}}-y_{\mathrm{b}}\;H_{\mathrm{d}}\cdot
Q_{\mathrm{L}}\bar{B}_{\mathrm{R}},$ (3)

where we denote the left-handed quark doublet as
$Q_{\mathrm{L}}=(T_{\mathrm{L}},B_{\mathrm{L}})$ and the two Higgs doublets as
$H_{\mathrm{d}}=(h_{\mathrm{d}}^{0},-h_{\mathrm{d}}^{-})$ and
$H_{\mathrm{u}}=(h_{\mathrm{u}}^{+},h_{\mathrm{u}}^{0})$, respectively, and
the $\mathrm{SU}(2)_{\mathrm{L}}$-invariant multiplication by the dot product.
The $\mathrm{SU}(2)_{\mathrm{L}}$ singlets are put into the left-chiral
supermultiplets
$\bar{T}_{\mathrm{R}}=\\{\tilde{t}_{\mathrm{R}}^{*},t_{\mathrm{R}}^{c}\\}$ and
$\bar{B}_{\mathrm{R}}=\\{\tilde{b}_{\mathrm{R}}^{*},b_{\mathrm{R}}^{c}\\}$,
respectively.

Additionally, we have to break SUSY softly which is done in the usual way with
scalar mass terms and trilinear couplings:

$\displaystyle V_{\text{soft}}=$
$\displaystyle\;m_{H_{\mathrm{d}}}^{2}|h_{\mathrm{d}}|^{2}+m_{H_{\mathrm{u}}}^{2}|h_{\mathrm{u}}|^{2}-\left(B_{\mu}h_{\mathrm{d}}\cdot
h_{\mathrm{u}}+\text{h.\,c.\ }\right)$ (4)
$\displaystyle+{\tilde{Q}}_{\mathrm{L}}^{*}\tilde{m}_{Q}^{2}\tilde{Q}_{\mathrm{L}}+{\tilde{t}}_{\mathrm{R}}^{*}\tilde{m}_{t}^{2}\tilde{t}_{\mathrm{R}}+{\tilde{b}}_{\mathrm{R}}^{*}\tilde{m}_{b}\tilde{b}_{\mathrm{R}}+\left(A_{\mathrm{t}}h_{\mathrm{u}}{\tilde{t}}_{\mathrm{L}}^{*}\tilde{t}_{\mathrm{R}}+A_{\mathrm{b}}h_{\mathrm{d}}{\tilde{b}}_{\mathrm{L}}^{*}\tilde{b}_{\mathrm{R}}+\text{h.\,c.\
}\right).$

The $D$-term part, finally, gives additional quadrilinear terms for the scalar
potential associated with gauge couplings,

$V_{D}=\frac{g_{1}^{2}}{2}\left(\phi^{\dagger}\frac{\Upsilon_{\phi}}{2}\phi\right)^{2}+\frac{g_{2}^{2}}{2}\left(\phi^{\dagger}\frac{\sigma}{2}\phi\right)^{2}+\frac{g_{3}^{2}}{2}\left(\phi^{\dagger}\frac{T}{2}\phi\right)^{2},$
(5)

with the corresponding hypercharges $\Upsilon_{\phi}$, weak charges $\sigma$
(Pauli matrices for $\mathrm{SU}(2)_{\mathrm{L}}$-doublets $\phi$) and color
charge matrices $T$. Again, summation over all gauge multiplets $\phi$ is
implicitly understood.

The charged Higgs directions play no role in the forthcoming discussion since
on one hand the potential is $\mathrm{SU}(2)$-invariant and may be always
rotated into the desired shape—on the other hand, the soft SUSY breaking terms
also break $\mathrm{SU}(2)$ in the squark sector (as top and bottom squarks
are treated differently and additional left-right mixing is introduced by the
$A$-terms). Any charge breaking Higgs _vev_ will then be related to a color
breaking squark _vev_ anyway and we shall be able to express everything in
neutral Higgs _vev_ s, $h_{\mathrm{d}}^{0}$ and $h_{\mathrm{u}}^{0}$ (for
simplicity, we drop the superscript “0” in the following), as well as stop and
sbottom _vev_ s $\tilde{t}$ and $\tilde{b}$.111The fields are treated as
classical field values, $c$-numbers, and correspond to _vev_ s at the minima
of the effective potential with vanishing external sources—the vacuum
configuration. Finally, we have the combined top/bottom-squark–Higgs scalar
potential

$\displaystyle V_{\tilde{q},h}=$
$\displaystyle{\tilde{t}}_{\mathrm{L}}^{*}\left(\tilde{m}_{Q}^{2}+|y_{\mathrm{t}}h_{\mathrm{u}}|^{2}\right)\tilde{t}_{\mathrm{L}}+{\tilde{t}}_{\mathrm{R}}^{*}\left(\tilde{m}_{t}^{2}+|y_{\mathrm{t}}h_{\mathrm{u}}|^{2}\right)\tilde{t}_{\mathrm{R}}$
(6)
$\displaystyle+{\tilde{b}}_{\mathrm{L}}^{*}\left(\tilde{m}_{Q}^{2}+|y_{\mathrm{b}}h_{\mathrm{d}}|^{2}\right)\tilde{b}_{\mathrm{L}}+{\tilde{b}}_{\mathrm{R}}^{*}\left(\tilde{m}_{b}^{2}+|y_{\mathrm{b}}h_{\mathrm{d}}|^{2}\right)\tilde{b}_{\mathrm{R}}$
$\displaystyle-\left[{\tilde{t}_{\mathrm{L}}}^{*}\left(\mu^{*}y_{\mathrm{t}}\;h^{*}_{\mathrm{d}}-A_{\mathrm{t}}h_{\mathrm{u}}\right)\tilde{t}_{\mathrm{R}}+\text{h.\,c.\
}\right]-\left[{\tilde{b}_{\mathrm{L}}}^{*}\left(\mu^{*}y_{\mathrm{b}}\;h^{*}_{\mathrm{u}}-A_{\mathrm{b}}h_{\mathrm{d}}\right)\tilde{b}_{\mathrm{R}}+\text{h.\,c.\
}\right]$
$\displaystyle+|y_{\mathrm{t}}|^{2}|\tilde{t}_{\mathrm{L}}|^{2}|\tilde{t}_{\mathrm{R}}|^{2}+|y_{\mathrm{b}}|^{2}|\tilde{b}_{\mathrm{L}}|^{2}|\tilde{b}_{\mathrm{R}}|^{2}$
$\displaystyle+\frac{g_{1}^{2}}{8}\left(|h_{\mathrm{u}}|^{2}-|h_{\mathrm{d}}|^{2}+\frac{1}{3}|\tilde{b}_{\mathrm{L}}|^{2}+\frac{2}{3}|\tilde{b}_{\mathrm{R}}|^{2}+\frac{1}{3}|\tilde{t}_{\mathrm{L}}|^{2}-\frac{4}{3}|\tilde{t}_{\mathrm{R}}|^{2}\right)^{2}$
$\displaystyle+\frac{g_{2}^{2}}{8}\left(|h_{\mathrm{u}}|^{2}-|h_{\mathrm{d}}|^{2}+|\tilde{b}_{\mathrm{L}}|^{2}-|\tilde{t}_{\mathrm{L}}|^{2}\right)^{2}$
$\displaystyle+\frac{g_{3}^{2}}{8}\left(|\tilde{t}_{\mathrm{L}}|^{2}-|\tilde{t}_{\mathrm{R}}|^{2}+|\tilde{b}_{\mathrm{L}}|^{2}-|\tilde{b}_{\mathrm{R}}|^{2}\right)^{2}$
$\displaystyle+(m_{H_{\mathrm{u}}}^{2}+|\mu|^{2})|h_{\mathrm{u}}|^{2}+(m_{H_{\mathrm{d}}}^{2}+|\mu|^{2})|h_{\mathrm{d}}|^{2}-2\operatorname{Re}(B_{\mu}\;h_{\mathrm{d}}h_{\mathrm{u}}).$

Some remarks are necessary on the structure of the scalar potential given
above and how to treat the field values and their possible phases. In the
previous honorable and groundbreaking works introducing charge and color
breaking solutions for the first time [21, 22] it is correctly stated that for
potentials considered in these cases, the trilinear couplings as well as the
corresponding field _vev_ s can always be chosen real and positive. This
obvious observation, however, might be used to overconstrain the field space
and therefore underconstrain the constraints on the involved parameters.
Indeed, the potential of Eq. (6) has some freedom in the field redefinitions;
especially, it is rephasing invariant apart from the trilinear terms and the
Higgs bilinear $\sim B_{\mu}h_{\mathrm{d}}h_{\mathrm{u}}$. The last term is
real by construction, all the others (besides the trilinears) are absolute
squares of field values. Still, we do not have the freedom to rephase the
fields in such a way, that the trilinear terms behave in a well defined way.
In particular, the choice of all fields real and positive is not possible! We
can, for sure, find a convention for the scalar quarks but not anymore for the
Higgs fields. We therefore allow both $h_{\mathrm{u}}$ and $h_{\mathrm{d}}$ to
vary in the positive and negative regime and only constrain
$|\tilde{t}|=\alpha|\phi|$ as well as $|\tilde{b}|=\beta|\phi|$ with a certain
scalar field value $\phi$ (where we choose $h_{\mathrm{u}}=\phi$). Moreover,
we set $h_{\mathrm{d}}=\eta\phi$ with $\eta$ any real number and $\alpha$,
$\beta$ real and positive. In case, we are considering real parameters only
(not the complex MSSM), the potential is symmetric in
$\mathrm{L}\leftrightarrow\mathrm{R}$ exchange of left- and right-handed field
labels. Setting all squark fields
$\tilde{q}_{\mathrm{L}}=\tilde{q}_{\mathrm{R}}$ (with $q=t,b$) simplifies also
the $D$-terms in the sense, that the $g_{3}^{2}$ contribution vanishes and the
$g_{2}^{2}$ and $g_{1}^{2}$ are the same in terms of the fields. The
commitment to real parameters (and fields!) nevertheless is also a severe
constraint, that may be, however, compassed by imposing global
$\mathrm{CP}$-invariance of the theory (i. e. $\mathrm{CP}$-invariance of SUSY
breaking if one refers to the $A$-terms). It is therefore a good assumption to
consider real fields only and just constrain the colored scalars to be
positive (as the colored potential is invariant under
$\tilde{q}\to-\tilde{q}$).222Complex fields in the effective potential mean
spontaneous CP violation.

Applying the considerations from above, we now have

$\displaystyle V_{\phi}=$
$\displaystyle\alpha^{2}(\tilde{m}_{Q}^{2}+\tilde{m}_{t}^{2}+2y_{\mathrm{t}}^{2}\phi^{2})\phi^{2}+\beta^{2}(\tilde{m}_{Q}^{2}+\tilde{m}_{b}^{2}+2y_{\mathrm{b}}^{2}\eta^{2}\phi^{2})\phi^{2}$
(7)
$\displaystyle+\left(m_{H_{\mathrm{u}}}^{2}+\eta^{2}m_{H_{\mathrm{d}}}^{2}+(1+\eta^{2})|\mu|^{2}-2B_{\mu}\eta\right)\phi^{2}$
$\displaystyle-2\alpha^{2}(\mu y_{\mathrm{t}}\eta-
A_{\mathrm{t}})\phi^{3}-2\beta^{2}(\mu y_{\mathrm{b}}-\eta
A_{\mathrm{b}})\phi^{3}$
$\displaystyle(\alpha^{4}y_{\mathrm{t}}^{2}+\beta^{4}y_{\mathrm{b}}^{2})\phi^{4}+\frac{g_{1}^{2}+g_{2}^{2}}{8}\left(1-\eta^{2}+\beta^{2}-\alpha^{2}\right)^{2}\phi^{4},$

where we applied in Eq. (6)

$\displaystyle h_{\mathrm{u}}=\phi,$
$\displaystyle|\tilde{t}_{\mathrm{L}}|=|\tilde{t}_{\mathrm{R}}|=|\tilde{t}|=\alpha|\phi|,$
(8) $\displaystyle h_{\mathrm{d}}=\eta\phi,$
$\displaystyle|\tilde{b}_{\mathrm{L}}|=|\tilde{b}_{\mathrm{R}}|=|\tilde{b}|=\beta|\phi|.$

Rewriting finally the potential, we get

$\displaystyle V_{\phi}=$
$\displaystyle\left(m_{H_{\mathrm{u}}}^{2}+\eta^{2}m_{H_{\mathrm{d}}}^{2}+(1+\eta^{2})\mu^{2}-2B_{\mu}\eta+(\alpha^{2}+\beta^{2})\tilde{m}_{Q}^{2}+\alpha^{2}\tilde{m}_{t}^{2}+\beta^{2}\tilde{m}_{b}^{2}\right)\phi^{2}$
(9) $\displaystyle-2\left(\alpha^{2}(\mu y_{\mathrm{t}}\eta-
A_{\mathrm{t}})+\beta^{2}(\mu y_{\mathrm{b}}-\eta
A_{\mathrm{b}})\right)\phi^{3}+(\alpha^{2}y_{\mathrm{t}}^{2}+\beta^{4}y_{\mathrm{b}}^{2})\phi^{4}$
$\displaystyle+\left(\frac{g_{1}^{2}+g_{2}^{2}}{8}(1-\eta^{2}+\beta^{2}-\alpha^{2})^{2}+2\alpha^{2}y_{\mathrm{t}}^{2}+2\beta^{2}y_{\mathrm{b}}^{2}\right)\phi^{4}$
$\displaystyle\equiv$ $\displaystyle
M^{2}\phi^{2}-\mathcal{A}\phi^{3}+\lambda\phi^{4},$

with

$\displaystyle M^{2}$
$\displaystyle=\;m_{H_{\mathrm{u}}}^{2}+\eta^{2}m_{H_{\mathrm{d}}}^{2}-2B_{\mu}\eta+(1+\eta^{2})\mu^{2}+(\alpha^{2}+\beta^{2})\tilde{m}_{Q}^{2}+\alpha^{2}\tilde{m}_{t}^{2}+\beta^{2}\tilde{m}_{b}^{2}\,,$
(10a) $\displaystyle\mathcal{A}$ $\displaystyle=\;2\alpha^{2}\eta\mu
y_{\mathrm{t}}-2\alpha^{2}A_{\mathrm{t}}+2\beta^{2}\mu
y_{\mathrm{b}}-2\eta\beta^{2}A_{\mathrm{b}}\,,$ (10b) $\displaystyle\lambda$
$\displaystyle=\;\frac{g_{1}^{2}+g_{2}^{2}}{8}(1-\eta^{2}+\beta^{2}-\alpha^{2})^{2}+(2+\alpha^{2})\alpha^{2}y_{\mathrm{t}}^{2}+(2\eta^{2}+\beta^{2})\beta^{2}y_{\mathrm{b}}^{2}\,.$
(10c)

Each of the effective parameters in the potential Eq. (9) depends implicitly
on the scaling parameters, so $M^{2}=M^{2}(\eta,\alpha,\beta)$,
$\mathcal{A}=\mathcal{A}(\eta,\alpha,\beta)$ and
$\lambda=\lambda(\eta,\alpha,\beta)$. The minimization of the one field
potential is done trivially and also the requirement for the global minimum at
$\langle\phi\rangle=0$ is known to be

$M^{2}>\frac{\mathcal{A}^{2}}{4\lambda}.$

Knowing about the dependence on the actual field direction, this bound can be
improved as

$4\min_{\\{\eta,\alpha,\beta\\}}\lambda(\eta,\alpha,\beta)M^{2}(\eta,\alpha,\beta)>\max_{\\{\eta,\alpha,\beta\\}}\left(\mathcal{A}(\eta,\alpha,\beta)\right)^{2},$
(11)

or rather

$\min_{\\{\eta,\alpha,\beta\\}}\left[4\lambda(\eta,\alpha,\beta)M^{2}(\eta,\alpha,\beta)-\left(\mathcal{A}(\eta,\alpha,\beta)\right)^{2}\right]>0.$

Note, that we easily recover the famous “traditional” CCB bound by Frère et
al. [21] setting $\eta=\beta=0$ and $\alpha=1$ which corresponds to the ray
$|\tilde{t}_{\mathrm{L}}|=|\tilde{t}_{\mathrm{R}}|=|h_{\mathrm{u}}|$ in field
space:
$M^{2}(0,1,0)=m_{H_{\mathrm{u}}}^{2}+\mu^{2}+{\tilde{m}}_{Q}^{2}+{\tilde{m}}_{t}^{2}$,
$\mathcal{A}(0,1,0)=-2A_{\mathrm{t}}$ and
$\lambda(0,1,0)=3y_{\mathrm{t}}^{2}$, such that333As important side remark, we
have to admit that we defined our soft breaking $A$-terms differently from the
common SUSY literature, where the Yukawa couplings are been factored out (to
recover the original result, one has to replace $A_{\mathrm{t}}\to
y_{\mathrm{t}}A_{\mathrm{t}}$).

$|A_{\mathrm{t}}|^{2}<3y_{\mathrm{t}}^{2}\left(m_{H_{\mathrm{u}}}^{2}+\mu^{2}+{\tilde{m}}_{L}^{2}+{\tilde{m}}_{t}^{2}\right).$
(12)

Similar expressions can be easily achieved for different field directions. The
specific choices have been made to make all gauge coupling contributions in
Eq. (10c) vanish—though the quartics from the Yukawa couplings, which are
numerically much larger, remain. There exists no real solution for $\eta$ with
non-vanishing $\alpha$ and/or $\beta$ to have $\lambda=0$. So there will be
(large) quartics anyway, which on the other hand means that we do not
necessarily need to restrict to the $D$-flat condition which explicitly forces
all $g_{i}^{2}$-terms in the scalar potential to be absent.

Similarly, by employing other alignments, we also recover the recently
proposed [43] $\mu y_{\mathrm{b}}$ bound and the corresponding bound from the
$h_{\mathrm{u}}$-$\tilde{b}$ $D$-flat direction with either $\eta=0$ and
$\beta=1$,

$\frac{\left(\mu
y_{\mathrm{b}}\right)^{2}}{y_{\mathrm{b}}^{2}+(g_{1}^{2}+g_{2})^{2}/2}<m_{H_{\mathrm{u}}}^{2}+\mu^{2}+{\tilde{m}}_{Q}^{2}+{\tilde{m}}_{b}^{2},$
(13)

or $h_{\mathrm{d}}=\pm\sqrt{1+\alpha^{2}}|h_{\mathrm{u}}|$, corresponding to
$|h_{\mathrm{d}}|^{2}=|h_{\mathrm{u}}|^{2}+|\tilde{b}|^{2}$, and leading to

$\frac{\alpha^{2}\mu^{2}}{2+3\alpha^{2}}<(1+\alpha^{2})m_{H_{\mathrm{d}}}^{2}+m_{H_{\mathrm{u}}}^{2}+(2+\alpha^{2})\mu^{2}\pm
2B_{\mu}^{2}\sqrt{1+\alpha^{2}}+\alpha^{2}\left({\tilde{m}}_{Q}^{2}+{\tilde{m}}_{b}^{2}\right),$
(14)

with $\alpha>0$; a reasonable fit to the numerically derived exclusion limits
can be found for $\alpha\approx 0.8$.

In the past, many attempts have been exercised to significantly improve the
stability bound on the trilinear $A$-term according to Uneq. (12). Possible
replacements range from

$|A_{\mathrm{t}}|^{2}<3y_{\mathrm{t}}^{2}\left(m_{H_{\mathrm{u}}}^{2}+{\tilde{m}}_{Q}^{2}+{\tilde{m}_{t}}^{2}\right),$
(15)

which was given (actually for $\mathrm{t}\leftrightarrow\mathrm{u}$ on the
first generation $A$-term) by [27] and improved considering the cosmological
stability of the potential through tunneling effects by [34] to444Uneq. (16)
is sometimes referred to “empirical” bound in contrast to the “traditional”
one of Uneq. (12).

$A_{\mathrm{t}}^{2}/y_{\mathrm{t}}^{2}+3\mu^{2}<7.5\left({\tilde{m}}_{Q}^{2}+{\tilde{m}_{t}}^{2}\right),$
(16)

and recently updated by [41] in the light of the Higgs discovery as

$A_{\mathrm{t}}^{2}/y_{\mathrm{t}}^{2}<3.4\left({\tilde{m}}_{Q}^{2}+{\tilde{m}}_{t}^{2}\right)+60\left(m_{h_{2}}^{2}+\mu^{2}\right),$
(17)

which is more in agreement (numerically) with Uneq. (15) but shall only be
applied to smaller values of $\mu$ and larger pseudoscalar masses $m_{A}$,
whereas moderate $\tan\beta$. How exactly this “small”, “large” and “moderate”
is defined may be left to the _gusto_ of the user. All in all, the bounds
(12)–(17) leave an undecided feeling behind and remain open the question for a
robust, roughly unique and unambiguous constraint (which we also fail to
provide).

We insist on the smaller sign ($<$) in Uneq. (12) and later on because the
smaller or equal ($\leq$) includes a degenerate vacuum with
$\langle\phi\rangle\neq 0$ which also leads to undesired phenomenology (where
we do not want to speculate about multiple degenerate vacua as done for the SM
Higgs case [62]). To be on the safe side, the $<$ is always preferred. The
optimized class of conditions given in Uneq. (11) lead in general to a more
involved interplay of different field directions that cannot be displayed in
such a nice expression like Uneq. (12).

The meaning of such bounds stayed controversial in the literature and history.
One significant improvement has been achieved by the discussion about the
stability on cosmological grounds, the question whether or not the desired
vacuum has had the possibility to decay to the true vacuum within the life-
time of the universe. However, any (semi-)analytical constraint suffers from a
distinct choice of the field configurations as any such choice influences the
tunneling rate, as well.

The main task is now to find the “optimized” directions, meaning certain
combinations of $\eta$, $\alpha$ and $\beta$ that give rise to the most severe
bounds à la Uneq. (11) leading to the deepest CCB minimum (and therefore the
true vacuum of the theory). Numerical minimization (and maximization) can be
efficiently done with many available tools. However, as we will see, the
optimized direction is not necessarily the most dangerous direction as the
former one is in certain cases related to very large field _vev_ s accompanied
with a rather high barrier between the trivial (local) minimum at
$\langle\phi\rangle=0$ and the true vacuum. Those configurations are related
to very large tunneling times for the vacuum-to-vacuum transition and thus
considered to be less dangerous. There are nevertheless slightly tilted or
shifted directions in field space where the non-standard minimum lies closer
and also the barrier is more complanate and therewith easier to be reached by
quantum tunneling. Once the barrier is overcome, the true vacuum can be
approached directly.

Before we continue with the actual analysis of the (reduced) MSSM incarnated
in the full scalar potential of Eq. (6), we make a brief but necessary
digression and discuss the issue of vacuum tunneling.

### Instability vs. metastability

The process of finding the global minimum of a complicated potential is
hazardous, even more the interpretation of the newly found configuration. Is
the standard (local) vacuum stable against quantum tunneling towards this
preferred true vacuum—or may there even be a path to gently roll down into the
desired state? The estimate of the tunneling rate via the so-called bounce
action itself is a tricky business, however, for a wide class of potentials a
very pictorial approximation can be used where only the position (i.e. _vev_)
of the deeper minimum and the maximum in between and the height of the wall is
needed. For a thick wall separating the false from the true vacuum, a very
convenient approximation formula was provided by [63] which is an exact
solution for a triangular shape of the potential. The difficult part lies in
the calculation of the bounce action $B$ [64], the decay rate per unit volume
is then given by

$\frac{\Gamma}{V}=Ae^{-B},$

where $A$ is an undetermined amplitude factor, usually approximated by the
false _vev_ to the fourth power or the barrier height (as the uncertainty goes
into the exponent, this does not really matter). The bounce itself depends in
this approximation only on the true and the false _vev_ , $\phi_{+}$ and
$\phi_{-}$, respectively, the field value of the maximum in between
$\phi_{M}$, and the values of the effective potential at the false vacuum
$V_{+}=V(\phi_{+})$ as well as the peak of the wall $V_{M}=V(\phi_{M})$. The
difference $\Delta V_{+}=V_{M}-V_{+}$ gives the height of the wall;
furthermore, we define $\Delta\phi_{+}=\phi_{M}-\phi_{+}$ and
$\Delta\phi_{-}=\phi_{-}-\phi_{M}$ and have the bounce action of [63]

$B=\frac{2\pi^{2}}{3}\frac{\left[\left(\Delta\phi_{+}\right)^{2}-\left(\Delta\phi_{-}\right)^{2}\right]^{2}}{\Delta
V_{+}}.$ (18)

Eq. (18) is very convenient to check the stability of a given configuration in
the reduced one-field potential without going into the details of the non-
perturbative calculation. In comparison to the life-time of the universe, one
finds metastable vacua for $B\gtrsim 400$, see [34].

Table 1: Input values and derived quantities for the two parameters points
illustrated in Figs. 1 and 2.

| $M_{\text{SUSY}}$ | $\tan\beta$ | $\mu$ | $A_{\mathrm{t}}=A_{\mathrm{b}}$ | $m_{h^{0}}$ | $B_{\text{global}}$
---|---|---|---|---|---|---
Fig. 1 | $1\,\mathrm{TeV}$ | $40$ | $500\,\mathrm{GeV}$ | $1500\,\mathrm{GeV}$ | $126\,\mathrm{GeV}$ | $354$
Fig. 2 | $1\,\mathrm{TeV}$ | $10$ | $500\,\mathrm{GeV}$ | $500\,\mathrm{GeV}$ | $113\,\mathrm{GeV}$ | $2568$

It is not necessarily the global minimum that determines the tunneling rate to
a non-standard minimum. Numerical procedures may overlook the vacuum on one
hand, but on the other hand the decay time to a local minimum may be much
smaller and the transition to the deeper one does not play a role
anymore.555Quantum mechanics knows about all paths. We want to illustrate at
two sample points with different phenomenology that both show deeper charge
and color breaking minima. The first point accounts for the proper Higgs mass
with $m_{h^{0}}\approx 126\,\mathrm{GeV}$ where the other one would be
discarded because it has $m_{h^{0}}\approx 113\,\mathrm{GeV}$. However, the
nature of the global minimum is different for both points: while the first has
a short-lived electroweak vacuum with $B\lesssim 400$, the other has an
extremely long-lived false vacuum. All relevant parameters are given in Tab.
1. In all our analyses, we keep the pseudoscalar heavy to comply with the
recent exclusions by collider searches for $A,H\to\tau\bar{\tau}$ [65, 66] and
take $m_{A}=800\,\mathrm{GeV}$ (which is very borderline with the respect to
the 2014 analyses up to $\tan\beta=40$ but unconstrained for smaller
$\tan\beta=10$). The pseudoscalar mass has anyway only a mild impact on the
charge and color breaking potential as it enters via the determination of the
soft SUSY breaking Higgs masses $m_{H_{\mathrm{u}}}^{2}$,
$m_{H_{\mathrm{d}}}^{2}$ and $B_{\mu}$ and we can easily set
$m_{A}=M_{\text{SUSY}}$ without changing the results. These three mass
parameters can be related and constrained demanding the Higgs potential being
bounded from below at the tree-level _and_ triggering electroweak symmetry
breaking such that $h_{\mathrm{u}}=h_{\mathrm{d}}=0$ is unstable [27]:

$m_{H_{\mathrm{d}}}^{2}+m_{H_{\mathrm{u}}}^{2}+2|\mu|^{2}\geq
2|B_{\mu}|\geq\sqrt{\left(m_{H_{\mathrm{d}}}^{2}+|\mu|^{2}\right)\left(m_{H_{\mathrm{u}}}^{2}+|\mu|^{2}\right)}.$

As we only check for CCB minima, we do not impose this constraint in addition;
a parameter point excluded by non-vanishing squark _vev_ s is excluded
anyways. For the allowed points in the following numerical evaluation, this
consideration should be applied. Most points do not recover the correct light
CP-even Higgs mass in the MSSM, not even within an error range of about $\pm
3\,\mathrm{GeV}$. If nothing else is quoted, we employed the latest version
(2.11.3) of FeynHiggs [67, 68, 69, 70, 71] to determine its numerical value.
We include a discussion of the influence of a $125\,\mathrm{GeV}$ Higgs in
Sec. 3.

Figure 1: The field configurations for a certain sample point
($\mu=500\,\mathrm{GeV}$, $A_{\mathrm{b}}=A_{\mathrm{t}}=1500\,\mathrm{GeV}$,
$\tan\beta=40$ and $M_{\text{SUSY}}=1\,\mathrm{TeV}$), which yields
$m_{h^{0}}=126\,\mathrm{GeV}$ with $m_{\tilde{g}}=1.5M_{\text{SUSY}}$ and
$m_{A}=800\,\mathrm{GeV}$ but is already excluded by the traditional CCB bound
for $A_{\mathrm{b}}$ (the $A_{\mathrm{t}}$-bound is passed) lead to very
different conclusions on the stability of the desired vacuum on cosmological
scales. While the shape of the potential is qualitatively very much the same
over the excluded field space (roughly $\eta\in[-1.8,0.55]$ and
$\beta\in[0.13,2.2]$ and the non-standard _vev_ varying within maybe a
$1000,\dotsc,2000\,\mathrm{GeV}$ range, the bounce action (shown in contours
on the left panel) indicates cosmologically stable and long-lived (blue:
$B>400$, light blue: $B>1000$) configuration as well as meta-stable and very
short-lived (red: $B<400$, dark red: $B<230$, corresponding to a life-time of
less then a second). The crosses on the left-side plot denote positions of the
three choices in $\eta$ and $\beta$ shown on the right; the yellow one
corresponds to the yellow line, for the others we have
$\text{orange}=\text{blue}$ and $\text{purple}=\text{red}$.

Figure 2: The same as for Fig. 2 but a point which has a long-lived desired
vacuum w.r.t. the true vacuum and a too light Higgs of
$m_{h^{0}}=113\,\mathrm{GeV}$. We have
$A_{\mathrm{t}}=A_{\mathrm{b}}=\mu=500\,\mathrm{GeV}$, $\tan\beta=40$. All
other parameters and color coding as in Fig. 2. Here, the global minimum
(indicated by the yellow cross in the light blue area) would suggest the
desired vacuum to be extremely long-lived. This conclusion may be misleading
as there are other configuration with a much shorter tunneling time.

To check for metastability, one may be tempted to define the field
configuration and the specific ray that shows the deepest non-standard vacuum
as the ideal or optimal one. However, as the new _vev_ appears at say
$\mathcal{O}(10\,\mathrm{TeV})$ and the barrier in between gets sufficiently
high, say $\mathcal{O}(\text{few}\leavevmode\nobreak\ \mathrm{TeV}^{4})$, $B$
is $\gg 400$ in that specific direction as for the point in Fig. 2. However,
there are other directions via which the global minimum can be accessed with a
much smaller tunneling time. For the sample point of Fig. 1 from above, we
show a tomographic view of the scalar potential in the
$\tilde{b}$-$h_{\mathrm{u}}$ plane for increasing
$\eta=h_{\mathrm{d}}/h_{\mathrm{u}}$ in Fig. 3 and the same potential sliced
differently for increasing $\beta=\tilde{b}/h_{\mathrm{u}}$ in the
$h_{\mathrm{d}}$-$h_{\mathrm{u}}$ plane in Fig. 4. This is to illustrate that
there is no unique choice for some fixed values of $\eta$ and $\beta$ that
exclusively show a non-standard vacuum. There are wide regions in field space
and all paths should be treated equal to estimate the tunneling rate. The
“optimal” direction for the determination of the bounds on the potential
parameters (masses, trilinear and quadrilinear couplings) should be rather
given by the shortest tunneling time. As recommendation how to deal with _any_
CCB exclusion, we declare each point that fails the condition

$\mathcal{A}(\eta,\alpha,\beta)^{2}<4\lambda(\eta,\alpha,\beta)M^{2}(\eta,\alpha,\beta)$

for _any_ specific combination of $\eta$, $\alpha$ and $\beta$ as clearly
unstable. An easy (but maybe CPU intensive) way to check this is to scan over
a reasonable range, e. g. $\eta\in[-3,3]$ and $\alpha,\beta\in[0,2]$; with a
binning of $0.1$ this procedure should find CCB configurations (since the
field space regions are quite extended, even coarser binnings should lead to a
trustable result).

Figure 3: A tomographicly sliced view of the CCB potential in the
$\phi=h_{\mathrm{u}}$ and $\tilde{b}=\beta\phi$ direction for
$\eta={-1.5,-1.0,-0.8,-0.5,0,0.5,0.8,1.0,1.5}$ (reading single plots from left
to right and top to bottom), where $h_{\mathrm{d}}=\eta\phi$. The numbers at
the contour lines represent the scaled potential value
$V(\phi,\eta,\beta)/\mathrm{TeV}^{4}$ to enhance readability. Negative regions
within the $0$-contour indicate the existence of a non-standard true vacuum.

Figure 4: A similar tomographic view as in Fig. 3 in the perpendicular
direction with $\eta=h_{\mathrm{d}}/\phi$ on the vertical axis and scanning
$\beta=\tilde{b}/\phi$ for $\beta={0,0.4,0.6,0.8,1.0,1.2,1.5,2.0,2.6}$ from
left to right and top to bottom with $\phi=h_{\mathrm{u}}$. The existence of
up to two non-standard vacua reflects the actually broken reflection symmetry,
$\eta\to-\eta$ and $\phi\to-\phi$, as can be seen that there no such exact
symmetry.

##  3 Anatomy of Charge and Color Breaking

The main stability condition is given by the unequation

$M^{2}(\eta,\alpha,\beta)>\frac{A(\eta,\alpha,\beta)^{2}}{4\lambda(\eta,\alpha,\beta)},$
(19)

depending on the field misalignments, as well as on all relevant model
parameters. We distinguish between parameters in configuration space that
independently of the model parameters can lead to instable configurations,
such as $\alpha$, $\beta$ and $\eta$—and the model parameters that change the
shape of the scalar potential as whole object (such as the soft masses, the
trilinear couplings $A_{\mathrm{t}}$ and $A_{\mathrm{b}}$ as well as the
$\mu$-parameter). Furthermore, we request the parameters of the one-loop Higgs
potential (i. e. the genuine type-II 2HDM of the MSSM) to allow for
spontaneous electroweak breaking with the correct _vev_ s. The ratio of the
_standard_ $v_{\mathrm{u}}/v_{\mathrm{d}}$ is what we call $\tan\beta$,
obeying $v_{\mathrm{d}}^{2}+v_{\mathrm{u}}^{2}=v^{2}=(246\;\mathrm{GeV})^{2}$.
Note that finally, the “true” $\tan\beta$ will be given by $1/\eta$. We
neither keep $\tan\beta$ fixed in a sense that the true vacuum has to respect
this relation, nor do we infer that from an original $\tan\beta>1$ the ratio
$\langle h_{\mathrm{u}}\rangle/\langle h_{\mathrm{d}}\rangle$ has to have the
same property. What we actually find is that for most configurations the true
vacuum seems to have $\langle h_{\mathrm{d}}\rangle>\langle
h_{\mathrm{u}}\rangle$ and the CCB _vev_ typically shows
$\langle\tilde{q}\rangle\gtrsim 0.7\langle h_{\mathrm{u}}\rangle$.
Unfortunately, for the bounds deduced numerically by attempting to find the
global minimum of parameter points violating (19), no expression of the
scaling parameters $\alpha$, $\beta$ and $\eta$ can be found in terms of the
relevant potential parameters although they crucially depend on the MSSM
parameter point. The ideal solution would be an exclusion of the form (19)
with $\eta$, $\alpha$ and $\beta$ given in terms of the model parameters.
Similar attempts have been achieved by [27] where one trilinear operator at a
time was considered only. For four operators this appears to be impossible.

For the numerical analysis in the following, we consider a very
phenomenological version of the MSSM with all SUSY breaking parameters defined
at the low (SUSY) scale without referring to any high-scale unified scenario.
As the developing CCB minima typically also show up around the same _low_
scale, we ignore any effects from the renormalization group as the
corresponding logarithms are small and only have a mild impact on the shape of
the potential (see e. g. [34] and the reference therein to [72]). For the
qualitative discussion this point is irrelevant anyway. Quantitatively, if
desired, parameters at the relevant scale can be employed as input values for
the analytical bounds.

We determine the soft SUSY breaking Higgs masses $m_{H_{\mathrm{u}}}^{2}$ and
$m_{H_{\mathrm{d}}}^{2}$ requiring electroweak symmetry breaking via the
conditions $\partial V_{1}/\partial
h_{\mathrm{u}}\big{|}_{h_{\mathrm{u}}=v_{\mathrm{u}},h_{\mathrm{d}}=v_{\mathrm{d}}}=0$
and $\partial V_{1}/\partial
h_{\mathrm{d}}\big{|}_{h_{\mathrm{u}}=v_{\mathrm{u}},h_{\mathrm{d}}=v_{\mathrm{d}}}=0$
with the one-loop Higgs potential $V_{1}$ [42]. The bilinear soft breaking
term is related to the pseudoscalar mass at tree-level via
$B_{\mu}=m_{A}^{2}\sin\beta\cos\beta$. Our free parameters are the ratio of
the two Higgs _vev_ s at tree-level,
$\tan\beta=v_{\mathrm{u}}/v_{\mathrm{d}}$, the soft squark masses, which we
for simplicity set to
${\tilde{m}}_{Q}^{2}={\tilde{m}}_{t}^{2}={\tilde{m}}_{b}^{2}=M_{\text{SUSY}}^{2}$,
and the superpotential parameter $\mu$ as well as the soft breaking trilinear
Higgs–squark couplings $A_{\mathrm{t}}$ and $A_{\mathrm{b}}$. The gaugino
masses $M_{1},M_{2}$ and $M_{3}$ enter only indirectly and play a less crucial
role, where the gluino mass $M_{3}$ can be more important for the threshold
effects on the bottom Yukawa coupling and in the two-loop light Higgs mass as
provided by FeynHiggs. If nothing else is stated, we set
$M_{1}=M_{2}=M_{\text{SUSY}}=1\,\mathrm{TeV}$ and $M_{3}=1.5M_{\text{SUSY}}$.

Including bottom Yukawa effects in the analysis of CCB minima has not been
done to great extend in the literature, as $y_{\mathrm{b}}$ usually is
neglected because of its smallness. However, for large $\tan\beta$ and certain
other regions in parameter space this cannot be done anymore. Especially the
$\Delta_{b}$ resummation for the bottom quark mass effectively changes the
bottom Yukawa coupling dramatically for such regions. While $y_{\mathrm{b}}$
gets lowered compared to $m_{\mathrm{b}}/v_{\mathrm{d}}$ for large
$\tan\beta$, it grows severely for negative $\mu$ as can be seen from the
expressions and even runs into a non-perturbative region (what is a well-known
behavior). The reduction at large $\tan\beta$ and small but positive $\mu$
keeps this window open in the following analysis. We include the dominant
contributions to $\Delta_{b}$ from the gluino and the higgsino loop [73, 74,
75, 76],666 $\displaystyle
C_{0}(x,y,z)=\frac{x^{2}y^{2}\log\frac{y^{2}}{x^{2}}+y^{2}z^{2}\log\frac{z^{2}}{y^{2}}+x^{2}z^{2}\log\frac{x^{2}}{z^{2}}}{(x^{2}-y^{2})(x^{2}-z^{2})(y^{2}-z^{2})}$.

$\displaystyle\Delta_{b}^{\text{gluino}}$
$\displaystyle=\frac{2\alpha_{s}}{3\pi}\mu M_{\tilde{G}}\tan\beta
C_{0}(\tilde{m}_{\tilde{b}_{1}},\tilde{m}_{\tilde{b}_{2}},M_{\tilde{G}}),$
(20a) $\displaystyle\Delta_{b}^{\text{higgsino}}$
$\displaystyle=\frac{y_{\mathrm{t}}^{2}}{16\pi^{2}}\mu A_{\mathrm{t}}\tan\beta
C_{0}(\tilde{m}_{\tilde{t}_{1}},\tilde{m}_{\tilde{t}_{2}},\mu),$ (20b)

and get the corrected bottom Yukawa coupling with
$\Delta_{b}=\Delta_{b}^{\text{gluino}}+\Delta_{b}^{\text{higgsino}}$ as

$y_{\mathrm{b}}=\frac{m_{\mathrm{b}}}{v_{\mathrm{d}}(1+\Delta_{b})}.$ (21)

Another remark is inevitable on the relevance of the parameters in the
discussion. Usually, when MSSM effects on the Higgs mass are discussed, the
“stop mixing parameter” $X_{t}=A_{\mathrm{t}}/y_{\mathrm{t}}-\mu\cot\beta$ is
used to measure the strength of the corrections rescaled with the SUSY scale,
$X_{t}/M_{\text{SUSY}}$. In the maximal mixing scenario, this parameter is set
to a large value, $X_{t}=\sqrt{6}M_{\text{SUSY}}$ what appears to be in
trouble with the exclusions presented here. Although it would be desired to
directly impose constraints on $X_{t}$, we are unable to do so because the two
independent Higgs _vev_ s $v_{\mathrm{d}}$ and $v_{\mathrm{u}}$ have to be
treated as dynamical variables and $\tan\beta$ cannot be kept fixed. Doing so
would lead to wrong conclusions. However, we can translate the final
exclusions we found to an effective exclusion on $X_{t}$ where
$\tan\beta=v_{\mathrm{u}}/v_{\mathrm{d}}$ gives the ratio of the two Higgs
_vev_ s in the desired electroweak state. This is especially important in
connection to the importance of $X_{t}$ for the light Higgs mass.

In comparison with earlier work on the vacuum stability issue in the MSSM, we
are now able to exclude a wider region of parameter space which gets accessed
when _both_ stop and sbottom _vev_ s and non-standard values for the two Higgs
doublets are considered. Non-standard in this respect means $\langle
h_{\mathrm{u}}\rangle\neq v_{\mathrm{u}}=v\sin\beta$ and $\langle
h_{\mathrm{d}}\rangle\neq v_{\mathrm{d}}=v\cos\beta$. Excluded regions in the
$\mu$-$\tan\beta$ plane have been derived from an analysis of the one-loop
Higgs potential in the MSSM, where loop effects of third generation sfermions
have been included [42]. An additional minimum seems to appear at a larger
field value $h_{\mathrm{u}}$ which is driven by the $\mu$-term and therefore
the requirement is that this non-standard (apparently charge and color
conserving!) _vev_ does not lead to a minimal value of the potential that is
lower than at the electroweak _vev_. Actually, this behavior is an artifact of
neglecting colored directions in the potential already at the tree-level
leading to an imaginary part in the example of Ref. [42] that was not
understood (and therefore just ignored). As this imaginary part is related to
a tachyonic sbottom mass at the new _vev_ , this indicates a CCB global
minimum, where the “one-loop global minimum” is rather a saddle point of the
potential in the Higgs–sbottom field configuration. For the same configuration
(basically $h_{\mathrm{d}}=0$ and $\tilde{b}=h_{\mathrm{u}}$) the shape of the
exclusion is very much the same but a bit tighter and shown in the upper left
plot of Fig. 5. The choice of $h_{\mathrm{d}}=0$ basically follows from the
consideration that if $\langle h_{\mathrm{d}}\rangle=v_{\mathrm{d}}$ kept
fixed, this value can be neglected for large $\tan\beta$ with respect to the
much larger $\langle h_{\mathrm{u}}\rangle>v_{\mathrm{u}}$. However, this
choice (as well as $\tilde{b}\sim h_{\mathrm{u}}$) does not resemble the true
behavior of the potential as can be seen, when both $h_{\mathrm{d}}$ and
$\tilde{b}$ are treated as independent dynamical variables as described above.
If one commits to the genuine $D$-flat direction only (say
$|h_{\mathrm{d}}|^{2}=|h_{\mathrm{u}}|^{2}+|\tilde{b}|^{2}$), similarly wrong
exclusions (conclusions?) can be drawn. The comparison of these two choices
has been elaborated in [43] together with the corresponding analytic bound
Uneqs. (13) and (14). The combined exclusion limit interpolates between the
two and is shown in the upper right plot of Fig. 5. Again, the artificial
constraint $\tilde{t}=0$ leads to weaker exclusions than under a non-vanishing
stop _vev_. This behavior finally is shown in the lower left plot of Fig. 5
and excludes large parts of the $\mu$-$\tan\beta$ plane. So far, we also have
kept the trilinear soft SUSY breaking sbottom parameter $A_{\mathrm{b}}$ to
zero and employed a large but moderate $A_{\mathrm{t}}=-1500\,\mathrm{GeV}$
(the negative sign was chosen to enhance the sbottom-_vev_ bound related to
$y_{\mathrm{b}}$). As a non-vanishing $A_{\mathrm{b}}$ drives the formation of
vacua with $\langle\tilde{b}\rangle\neq 0$ and similarly $\langle
h_{\mathrm{d}}\rangle\gg v_{\mathrm{d}}$, we close the remaining allowed
parameter space to values of $\tan\beta\gtrsim 40$ and $\mu\lesssim
700\,\mathrm{GeV}$ (in a world with
${\tilde{m}}_{Q}={\tilde{m}}_{t}={\tilde{m}}_{b}=M_{\text{SUSY}}=1\,\mathrm{TeV}$
and $m_{A}=800\,\mathrm{GeV}$ as the relevant further input parameters).

Figure 5: Growing exclusion limits if more _vev_ s are allowed; red points are
excluded by vacuum stability, blue points are still allowed. The light blue
points indicate a light Higgs mass within $125\pm 1\,\mathrm{GeV}$ (FeynHiggs
with a gluino mass at $3M_{\text{SUSY}}$). All plots have
$A_{\mathrm{t}}=-1500\,\mathrm{GeV}$ and $M_{\text{SUSY}}=1\,\mathrm{TeV}$ for
comparison with previous works. Top left: similar configuration as in [42, 43]
with no stop and $h_{\mathrm{d}}$ _vev_ s; top right: including also
$h_{\mathrm{d}}\neq 0$, where the exclusion is now interpolating between the
two scenarios of [43] and is a bit stronger (as not necessarily
$|h_{\mathrm{d}}|^{2}=|h_{\mathrm{u}}|^{2}+|\tilde{b}|^{2}$ is fixed). Down
left: including all field directions discussed in this paper, as in the upper
row we kept $A_{\mathrm{b}}=0$; down right: now switching on
$A_{\mathrm{b}}=A_{\mathrm{t}}$, nearly the complete area seems to be
excluded. To compare with the usual (p)MSSM literature, we have to rescale the
$A$-terms $A_{\mathrm{t}}\to A_{\mathrm{t}}/y_{\mathrm{t}}$ and
$A_{\mathrm{b}}\to A_{\mathrm{b}}/y_{\mathrm{b}}$. In this area,
$y_{\mathrm{b}}$ ranges from $\sim 0.12$ to $\sim 0.8$ and gets large in the
upper left corner of the $\mu$-$\tan\beta$ plane including the $\Delta_{b}$
resummation but less large than $m_{\mathrm{b}}/v_{\mathrm{d}}$ (which seems
to rescue this corner once $A_{\mathrm{b}}$ is switched on).

How much is the interplay of stop and sbottom _vev_ s? The exclusion from stop
_vev_ s only has a rather circular shape in the $\mu$-$A_{\mathrm{t}}$ plane,
illustrated in Fig.6. This shape does not change much with $\tan\beta$ as long
as $A_{\mathrm{b}}$ is switched off. In the upper left corner, we show exactly
this for $\tan\beta=40$ and $A_{\mathrm{b}}=0$. For the purposes of Figs. 5
and 6, we relied on our own determination of the Higgs potential parameters
(basically $m_{H_{\mathrm{u}}}^{2}$ and $m_{H_{\mathrm{d}}}^{2}$ to have the
correct $v_{\mathrm{d},\mathrm{u}}$ in presence of the one-loop corrected
potential). Comparison of public codes doing the same (SPheno [77, 78],
softsusy [79] and SuSpect [80] with the convenient Mathematica interface SLAM
[81]) shows very similar shapes, where the border lines get less sharp due to
several effects we do not have under control. For aesthetic reasons, we show
the (slightly wrong but nicer) plots determined with our own algorithm. The
color coding in Fig. 6 shows allowed regions in blue, excluded by stop _vev_ s
in red and sbottom _vev_ appearing in orange. As can be seen by turning on
$A_{\mathrm{b}}$, a larger portion of the previously allowed parameter space
is excluded. The allowed parameter space for $m_{h^{0}}$ within a
$1\,\mathrm{GeV}$ interval around $125\,\mathrm{GeV}$ is shown in light blue.

Figure 6: We show exclusion limits from the formation of non-standard vacua in
the $A_{\mathrm{t}}$-$\mu$ plane. The central blue area appears to be allowed,
where the red and orange regions are excluded by the existence of stop and
sbottom _vev_ s, respectively. Light blue points indicate the region of the
correct light Higgs mass as in Fig. 5. In all plots, we assigned
$M_{\text{SUSY}}=1\,\mathrm{TeV}$. The soft SUSY breaking trilinear coupling
is set to $A_{\mathrm{b}}=0\,\mathrm{GeV}$ (upper left),
$A_{\mathrm{b}}=500\,\mathrm{GeV}$ (upper right),
$A_{\mathrm{b}}=1000\,\mathrm{GeV}$ (lower left) and
$A_{\mathrm{b}}=1500\,\mathrm{GeV}$ (lower right).

So far, we only analyzed the very generic potential of Eq. (6), rewritten as
single-field potential (9), without any reference to current phenomenology of
the MSSM. It appears that the pure theoretical consideration to have a self-
consistent theory (especially having no deeper minimum than the electroweak
ground state) already excludes wide regions of the available parameter space.
The constraints are even stronger than the well-known strong constraints of
[27]. Reasons are that we do not insist on $\tan\beta>1$ for the new vacuum
and include simultaneously stop and sbottom _vev_ s. Unfortunately, for direct
comparison with the Vevacious collection, the corresponding model file
treating non-vanishing stop and sbottom _vev_ s at the same time without the
need of the _full_ squark potential including the first two generations is
missing (although there $|h_{\mathrm{d}}|>|h_{\mathrm{u}}|$ is allowed and
field values can also acquire negative values with respect to $h_{\mathrm{u}}$
as we do; the usage of the full squark potential appears to be less stable and
requires very long running times for each data point). In its full generality,
however, Vevacious is not constrained to the MSSM and can be used to check the
stability of any desired beyond the SM physics—if the user is willing to
produce the necessary model file. Even more constrained gets the leftover
parameter space when we in addition impose the light Higgs mass
$m_{h^{0}}=125\,\mathrm{GeV}$ which was not available 20 years ago and on its
own narrows down the allowed region. Of course, a detailed analysis of the
MSSM parameter space can only be done in terms of a global fit including
various other constraints (collider data, Higgs properties, dark matter
constraints) and goes much beyond the scope of this work. The analysis of the
CCB anatomy, however, is interesting by itself and even more in connection to
the determination of the Higgs mass.

Figure 7: The allowed and excluded regions for varying SUSY masses
${\tilde{m}}_{Q}={\tilde{m}}_{t}=M_{\text{SUSY}}$ and $\tan\beta=40$ (left) as
well as $\tan\beta=10$ (right) and $\mu=350\,\mathrm{GeV}$ (both). Blue points
do not show any deeper non-standard vacua whereas red and orange do. Orange
points have explicitly non-vanishing sbottom _vev_ s. For the purpose of these
plots, we have employed SPheno to calculate the spectra.

Figure 8: For the inclusion of the Higgs mass prediction in the MSSM, we show
the value of $X_{t}/M_{\text{SUSY}}$ varying with the SUSY scale (at
$A_{\mathrm{b}}=A_{\mathrm{t}}$ as usually done). On the left side, we kept a
low $\mu$-value, $\mu=350\,\mathrm{GeV}$, fixed; whereas on the right side,
$\mu$ scales with the SUSY mass, $\mu=M_{\text{SUSY}}$. In both cases we
employed $\tan\beta=40$. The color coding shows the compatibility with the
Higgs mass measurements: the darker the color the more compatible (the dark
blue and red regions indicate $m_{h^{0}}\in[124,126]\,\mathrm{GeV}$ as
produced by SPheno corresponding to a $1\,\mathrm{GeV}$ uncertainty; the
neighboring region has the $3\,\mathrm{GeV}$ uncertainty). The blueish region
is allowed by vacuum stability considerations whereas the reddish region is
excluded. For fixed $\mu=350\,\mathrm{GeV}$ that means roughly $|X_{t}|\leq
1.1M_{\text{SUSY}}$; in the case with $\mu=M_{\text{SUSY}}$ the allowed area
shrinks with increasing SUSY scale.

### SUSY hides behind the corner

The question that remains is how much do these bounds depend on the SUSY
scale. So far, we have employed $M_{\text{SUSY}}=1\,\mathrm{TeV}$ which needs
large $A_{\mathrm{t}}$ close to the border line in order to get $m_{h^{0}}$
right and what points towards near-criticality also in the MSSM. There is of
course one way out to still consider the MSSM (with the assumptions applied in
this work) as valid and alive. We find that the constraints get weaker with
increasing $M_{\text{SUSY}}$, especially the value of the ratio
$X_{t}/M_{\text{SUSY}}$ for which a model point would be excluded stays rather
constant or even grows for fixed $\mu$ whereas it shrinks with larger $\mu$.
This shrinking is not surprising, as $\mu$ enters $X_{t}$ and can enhance its
value for various configurations. Similarly, the value of the tree-level
$\tan\beta$ enters severely as can be seen from Fig. 7, where we compare the
allowed and excluded regions of $A_{\mathrm{t}}=A_{\mathrm{b}}$ with respect
to $M_{\text{SUSY}}$ for $\tan\beta=40$ and $10$.. In addition, the proper
value for the lightest Higgs mass, $m_{h^{0}}\approx 125\,\mathrm{GeV}$
selects a small band in $X_{t}$-$M_{\text{SUSY}}$. We clearly see from Fig. 8,
where the spectrum has been determined with the help of SPheno, that only for
SUSY masses that are anyway not yet excluded by experiment in the simplified
analyses $M_{\text{SUSY}}\geq 1500\,\mathrm{GeV}$ (for a small $\mu$-term and
rather large $\tan\beta$), we can enter the correct range. Increasing $\mu$
shifts the allowed regime to even larger $M_{\text{SUSY}}$. It is therefore
with hindsight not surprising at all, that there have been no signals of SUSY
found so far in combination to the measured light Higgs mass. Without this
additional crucial ingredient one might get depressed seeing the parameter
space being closed, especially when the trilinear soft SUSY breaking couplings
are taken equally large, $A_{\mathrm{b}}=A_{\mathrm{t}}$, as usually done. One
the other hand, this is exactly what is observed by the non-observation of
light stops so far. Very light squarks (below say $1\,\mathrm{TeV}$) in
connection with large squark mixing are inconsistent with a stable electroweak
vacuum. In that sense, SUSY awaits her discovery in the very near future.

##  4 Conclusions

We have reported on a new view of charge and color breaking minima in the
Minimal Supersymmetric Standard Model and consequently derived novel bounds on
the parameter space from the self-consistency of the theory. In order to avoid
any configurations that lead to an unstable electroweak ground state, large
portions of the available parameter space are excluded. We have argued that
the exclusions cannot be treated as metastability bounds requiring only a
life-time of the false vacuum of about the age of the universe and any CCB
exclusion an MSSM parameters is to be seen strict. We have extended the
exhaustive work of Ref. [27] mostly by relaxing the constraint on
$h_{\mathrm{d}}$ to be strictly smaller than $h_{\mathrm{u}}$ but lacking
simple analytic expressions to cover the numerical exclusions. By analyzing
both Higgs and third generation squark directions simultaneously (four
fields), we cannot fix the signs of the trilinear terms to make them positive.
This in addition opens a new window to exclude larger parameter regions as
$h_{\mathrm{d}}=-h_{\mathrm{u}}$ is allowed and particularly enhances the
effect for certain sign combinations. A generic analytic bound on the four-
field level is rather impossible; the remaining freedom, however, allows not
to be too restrictive and especially allow for short-lived vacua in formerly
metastable parameter regions.

Finally, we have included the determination of the light Higgs mass in the
MSSM and find that a low superpartner spectrum (especially light stops) in
combination with a $125\,\mathrm{GeV}$ Higgs is excluded by the formation of
non-standard vacua around the SUSY scale. A stable electroweak vacuum at the
low scale requests (depending on the specific scenario) SUSY masses to be in
the multi-TeV regime, $M_{\text{SUSY}}\gtrsim 1.5,\dotsc,6\,\mathrm{TeV}$, for
positive $\mu$-values and sizeable $A_{\mathrm{b}}$. Exclusions get weakened
for smaller or vanishing $A_{\mathrm{b}}$. Variations on the bounds with
rising $M_{\text{SUSY}}$ are given in Fig. 9. Further investigation is needed
in very special corners of the parameter space see Fig. 10: _negative_ and
small values of $\mu$ keep the window for a $125\,\mathrm{GeV}$ Higgs and
squark masses below $1\,\mathrm{TeV}$ open (say
$0\geq\mu\geq-1000\,\mathrm{GeV}$, as already indicated in Fig. 6 for
$A_{\mathrm{t}}\approx A_{\mathrm{b}}\approx\pm 1500\,\mathrm{GeV}$).

Figure 9: Variations of Fig. 7, all with $\tan\beta=40$ and
$m_{A}=800\,\mathrm{GeV}$. All but the lower right have
$A_{\mathrm{b}}=A_{\mathrm{t}}$, in the lower right plot we have set
$A_{\mathrm{b}}=0\,\mathrm{GeV}$. For the upper left, we keep a negative
$\mu$-value $\mu=-500\,\mathrm{GeV}$ fixed, where the upper right has a
varying $\mu=M_{\text{SUSY}}$, where the lower left has
$\mu=350\,\mathrm{GeV}$ fixed. Color coding as in Fig. 7.

Figure 10: Variations of Fig. 8 with $\tan\beta=40$ and
$\mu=350\,\mathrm{GeV}$ (left) as well as $\mu=-500\,\mathrm{GeV}$ (right).
The light CP-even Higgs mass has been calculated with the help of FeynHiggs
and $m_{\tilde{g}}=1.5M_{\text{SUSY}}$, $m_{A}=800\,\mathrm{GeV}$. The white
stripes are left blank because the one-loop effective potential of [42]
develops an imaginary part already at the standard _vev_ s _and_ a tachyonic
sbottom mass there.

## Acknowledgments

The author acknowledges support by the DESY fellowship program and discussion
with E. Bagnaschi, S. Di Vita, S. Passehr, and G. Weiglein. We thank F. Staub
for communications on Vevacious.

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