Spontaneous charge breaking in the NMSSM: dangerous or not?
Eur. Phys. J. C
Spontaneous charge breaking in the NMSSM: dangerous or not?
Manuel E. Krauss 2
Toby Opferkuch 2
Florian Staub 0 1
0 Institute for Nuclear Physics (IKP), Karlsruhe Institute of Technology , Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen , Germany
1 Institute for Theoretical Physics (ITP), Karlsruhe Institute of Technology , Engesserstraße 7, 76128 Karlsruhe , Germany
2 Bethe Center for Theoretical Physics and Physikalisches Institut der Universität Bonn , Nußallee 12, 53115 Bonn , Germany
We investigate the impact of charge-breaking minima on the vacuum stability of the NMSSM. We concentrate on the case of vanishing A-terms in the sfermion sector, i.e. the only potentially dangerous sources of charge breaking are vacuum expectation values of the charged Higgs fields. We find that, in contrast to Two-Higgs-Doublet Models like the MSSM, at both tree and loop level there exist global charge-breaking minima. Consequently, many regions of parameter space are rendered metastable, which otherwise would have been considered stable if these chargebreaking minima were neglected. However, the inclusion of these new scalar field directions has little impact on otherwise metastable vacuum configurations.
At first glance, the discovery of a standard model (SM)-like
Higgs boson with a mass of approximately 125 GeV [1,2]
appears to be a huge success of supersymmetry (SUSY)
and in particular of the minimal supersymmetric standard
model (MSSM). In contrast to other ideas to extend the SM,
SUSY predicts that the Higgs boson should not be
significantly heavier than the Z -boson if new physics is around the
TeV scale; see e.g. Ref.  and the references therein. Other
avenues such as technicolour prefer the natural mass range
for the Higgs to lie at scales well above the measured mass.
On the other hand, closer investigation shows that the
situation is more complicated in the MSSM as the Higgs mass
requires large radiative corrections to be compatible with
experimental data. The main source of these corrections are
the superpartners of the top, the stops. In order to maximise
their contributions to the Higgs mass, one needs to consider
scenarios in which they are maximally mixed [4–7]. This can
be dangerous because it can lead to the presence of
chargeand colour-breaking vacua whereby the stops receive
vacuum expectation values (VEVs) [8–11]. Since the tunnelling
rate to these vacua is typically large, this results in tension
between an acceptable Higgs mass and a sufficiently
longlived electroweak (EW) breaking vacuum. Consequently,
SUSY models which can enhance the Higgs mass at tree
level are especially appealing. The simplest such extension
is to add a scalar singlet, resulting in the next-to-minimal
supersymmetric standard model (NMSSM), yields F -term
contributions, which raise the tree-level Higgs mass [12,13].
This significantly reduces the need for large loop corrections.
As a result, large stop mixing is no longer necessary.
Therefore, the vacuum stability problems of the MSSM are cured
as well as reducing the EW fine-tuning [14–21]. However, the
extended Higgs sector in the NMSSM introduces new
couplings which can potentially destabilise the EW vacuum. The
vacuum stability in the NMSSM has been studied in the past
at tree level [22–26], and also with one-loop corrections .
Potentially dangerous parameter ranges have been identified
in this work. However, all these studies made the assumption
that charge is conserved at the global minimum of the scalar
potential, i.e. the charged Higgs boson VEVs were neglected.
This was motivated to some extent as it has been shown that
the global minimum of two-Higgs-doublet models, if they
have a minimum with correct electroweak symmetry
breaking, is always charge conserving at tree level [28–30].
However, for non-vanishing singlet–doublet interactions this is no
longer the case  and one must in principle always take
these VEVs into account. The aim of this letter is to discuss
the impact of charged Higgs VEVs on the vacuum
stability in the NMSSM. We start in Sect. 2 with a discussion of
the scalar potential, before we show the numerical results in
Sect. 3. We conclude in Sect. 4.
2 Spontaneous charge breaking in the NMSSM
We consider in the following the NMSSM with a Z3 to
forbid all dimensionful parameters in the superpotential. The
WNMSSM = λHˆd Hˆu Sˆ + 3 κ Sˆ3 + WY ,
with the standard Yukawa interactions WY as in the MSSM.
The additional soft-terms in comparison to the MSSM are
where we have used the common parametrisation for the
trilinear soft terms Tλ = Aλλ, Tκ = Aκ κ . Note that we assume
in the following that all A-terms in the sfermion sector vanish
or are sufficiently small such that the colour- and/or
chargebreaking minima in the respective field directions cannot be
deep enough to destabilise the scalar potential. After
electroweak symmetry breaking, the scalar singlet S obtains a
VEV vS which generates an effective Higgsino mass term
μeff = λ√v2S .
Using the three minimisation conditions of the potential,
the Higgs sector in the NMSSM is specified at tree level
by six parameters: λ, κ, Aλ, Aκ , μeff , tan β, with the ratio
vu of the doublet VEVs.
tan β = vd
However, we have so far neglected the possibility that
charged Higgs bosons can acquire VEVs. In order to include
this possibility, one needs to check for the global minimum
of the scalar potential resulting from the following VEVs:
S = √ .
One can reduce this five-dimensional problem via an SU (2)
gauge transformation to eliminate one of the charged Higgs
VEVs. This turns out to be more robust for the numerical
evaluation, but for the current discussion we keep the more
intuitive form with all five VEVs.
The scalar potential of the Higgs sector in the NMSSM
with these five VEVs consists of F -, D- and soft-terms
λvS2 λ vd + vm + v p + vu
2 2 2 2
+ λ2(vm v p − vd vu )2 + κ v
2 S4 ,
+ g22 vd4+vm4 +2vd2 vm2 +v2p−vu2 +8vd vm v pvu
2 2 2 2 2
− 2vm2 v p − vu + v p + vu
In what follows we shall always use the equations which
determine the stationary points with respect to vu , vd and
vS (while simultaneously setting vm = v p = 0) to
eliminate the soft SUSY-breaking masses m2Hu , m2Hd and ms2 from
the potential. In doing so we insist upon the existence of an
appropriate electroweak vacuum through the introduction of
the input parameters μeff , tan β and the electroweak VEV
v. These input parameters only fix the soft SUSY-breaking
masses and retain the same values irrespective of the
specific minimum under consideration. To emphasise, if one
considers a generic minimum of the potential, these input
parameters only enter in the scalar potential as a substitute
for the soft SUSY-breaking masses while the free directions
in field space, vu,d, p,m,S , are varied to determine other
minima of the theory. Consequently, all minima which we find
in addition to the desired EW vacuum configuration occur
Before we continue, we can check if parameter points
exist for which the global minimum of the potential is charge
breaking. In order to do so, we compute
V = VFull − VFull vm=vp=0.
where tβ = tan β, we get in the limit tβ → 1, vm → 01
V = 312 v2p g22 2vd2−2v2+v2p+2vu2 −16μe2ff +8λ2vS2
Thus, one can see that in particular for large μeff it is possible
to get very deep charge-breaking (CB) minima below those
which are charge conserving (CC).
1 Again, the choice vm → 0 can always be made using a SU (2) gauge
We now seek to gain some additional insight into the
behaviour of the potential and, in particular, regions where
the CB minima are potentially dangerous. The most
promising directions in field space to discover deep minima are
those in which either the F - or D-terms vanish. Since we
are in general interested in points with sizeable λ couplings
in order to get a large enhancement for the Higgs mass, the
most stabilising effect of the potential can be expected to
come from the F -terms. It is actually not possible to find any
F -flat directions which are charge conserving. However, in
the charge-breaking case the F -terms vanish for
vm = vu , v p = vd , vS = 0.
In this direction in VEV space the value of the potential is
V = 8 (vd2 + vu2)(4m2Hd + 4m2Hu + g22(vd2 + vu2)),
V = 8
8 Aλμeff + g22 vd + vu
Defining x CB = vd2 + vu2 + vm2 + v2p, we find that new
minima develop in the direction vu = vm , vd = v p at the point
xmCBin = ±
−4 Aλλμeff + 4μe2ff (λ − κ) + λ3v2
at which the value of the potential is
V xmCBin = −
3 2 2
−4 Aλλμeff + 4μe2ff (λ − κ) + λ v
From these expressions one sees that the following
conditions characterise the potentially dangerous regions in which
CB minima might develop: (i ) large |λ| and |μeff |, (ii) either
opposite signs for λ and κ or |κ/λ| < 1 as well as (iii)
opposite signs for Aλ and μeff . Equation (13) has to be combined
with the condition that all Higgs masses are non-tachyonic
at the electroweak vacuum. The condition to have a positive
charged Higgs mass is
which for large μe2ff , prefers λ and κ of same signs and also
either equal signs for Aλ and μeff or small Aλ compared
to μeff . From the positivity condition on the pseudo-scalar
Fig. 1 Value of the scalar potential in the direction of vanishing
Fterms for three different values of μeff . Here, we have chosen κ = 21 λ,
Aλ = 100 as well as λ = −1 (full lines) and λ = −2 (dashed lines)
masses one can further see that opposite signs for Aκ and
μeff are preferable. Therefore, combined with Eq. (13), we
see that CB minima are likely to occur if:
It is important to note that in these regions, the mostly
singletlike scalar is heavy therefore, the SM-like Higgs is always
the lightest CP-even scalar state.
In Fig. 1, we show the behaviour of the potential in the
direction x = vd2 + vu2 + vm2 + v2p for different values of
μeff . We see in these examples that the minima are in the
multi-TeV range and move quickly to larger values with
increasing μeff . Thus, it needs to be checked how efficient the
tunnelling to these minima is. In addition, one also needs to
compare the tunnelling to these minima with the tunnelling
to potential CC minima which do not coincide with the
electroweak breaking vacuum. One important VEV direction in
this context is the one with
vu = vm = v p = vS = 0, vd ≡ x CC = 0,
in which the potential is given by
In this direction, new minima appear at
xmCCin = ±
−4 Aλλμeff + 4(λ − κ)μe2ff + λ3v2
Fig. 2 Comparison of the potential in the charge-conserving
(dashdotted) and charge-breaking direction (full) defined by Eqs. (11) and
(17), respectively. The same parameter choices as in Fig. 1 were made
and we show here the case λ = −1
at which the depth of the potential is
V xmCCin = −
deeper than the non-EW but CC as expected. However, the
latter appears at slightly smaller x values. Consequently, it
is not a priori clear to which minima the electroweak state
would tunnel to more effectively—to the deeper one or the
nearer one—as the field space is highly non-trivial. In these
cases, one needs to calculate the tunnelling rate to the
different minima in order to be able to judge if the inclusion of
charged Higgs VEVs yields additional constraints.
In general, the decay rate per unit volume for a false
vacuum is given in [32,33] by /vol. = Ae(−B/h¯) (1 + O(h¯)),
where A is a factor which depends on the eigenvalues of
a functional determinant and B is the bounce action. A is
usually taken to be of order the renormalisation scale and is
less important for the tunnelling rate which is dominated by
the exponent B. In a multi-dimensional space it only makes
sense to calculate B numerically as any approximations,
analytic or otherwise, are simply not accurate enough due to the
huge sensitivity of on B. Of course, there are also other
directions in VEV space where CB minima might establish.
However, an analytical discussion of all these cases does not
give further insights. We therefore turn directly to the
3 Numerical results
The second derivatives of the scalar potential in both the CB
and the CC but non-EW cases are given by
∂ x 2
which, given the above conditions on the parameters, always
turns out to be positive, ensuring that the configurations we
consider here indeed correspond to minima of the potential.
Thus, we find that the conditions to develop additional
charge-conserving and charge-breaking minima in addition
to the one with correct EWSB are very similar and both kind
of minima can appear simultaneously for given input values.
Comparing Eqs. (14)–(15) with Eqs. (19)–(20), we find that
the CB minima are deeper than the CC ones by a factor (g12 +
g22)/g22. At the same time, the CB minimum is further away
in field space by a factor g12 + g22/g2. A one-dimensional
comparison between the behaviour of the potential in this
direction and in the direction defined via Eq. (11) is shown
in Fig. 2.
As a result, we observe in typical regions of parameter
space that CB and CC minima occur at the same time and,
both are usually deeper than the correct electroweak vacuum.
Furthermore, the behaviour indicated in Eqs. (14)–(15) and
(19)–(20) can be seen from Fig. 2 where the CB minimum is
As we have seen so far, one can find new vacua in the NMSSM
when including the possibility of spontaneous charge
breaking. However, it needs to be clarified how important the
study of these minima is. Therefore, we are going to make
a numerical analysis not only of the tree-level potential but
also of the one-loop effective potential with and without the
consideration of charge-breaking VEVs. For doing that, we
use Vevacious , for which we have generated model
files with SARAH [35–40]. We also used SARAH to
generate a SPheno module [41,42] for the NMSSM. With this
module we calculate the SUSY and Higgs masses including
NMSSM-specific two-loop corrections [43–45] which are
important in particular for large |λ| [46,47]. Consequently,
the accuracy in the Higgs mass prediction is similar to the
MSSM and we use 3 GeV for the theoretical uncertainty
in the following. The spectrum file generated by SPheno
is passed to HiggsBounds [48,49] and is also used as
input for Vevacious. Vevacious finds all solutions to
the tree-level tadpole equations by using a homotopy
continuation implemented in the code HOM4PS2 . These
extrema are used as the starting points to find the minima
of the one-loop effective potential using minuit . If
it finds deeper minima than the EW one, Vevacious calls
CosmoTransitions  to get the tunnelling rate.
However, in the standard Vevacious package, the calculation
for the tunnelling rate is not done for all minima, but only for
the so called ‘panic’ vacuum. This is the one closest to the
Fig. 3 Stability of the EW vacuum considering the full one-loop
effective potential. Regions shaded in green are stable, indicating that the
desired electroweak breaking minimum is the global minimum. The
yellow and blue regions correspond to metastablity of the desired
electroweak breaking minimum. In particular, the blue region contains only
CB minima that are deeper, while the yellow regions contains both CB
and CC minima. The dashed-grey contours show the equivalent of the
blue CB metastable region assuming only a tree-level potential. Finally,
the region between the black solid contours corresponds to an
acceptable Higgs mass, namely mh ∈ [122, 128] GeV. Here we have chosen
λ = −0.68, tan β = 1.02, Aκ = −700 GeV and Aλ = −300 GeV
EW minimum in field space. We have modified Vevacious
to calculate the tunnelling rate to all minima in order to be
able to compare the different sets of vacua.
We are going to distinguish two cases in the following:
(i) cases in which only CB minima exist which are deeper
than the EW one; (ii) cases in which both deeper CB and CC
minima exist. The results that we show below are particular
points of interest obtained by scans over the parameter space:
− Aλ ∈ [−5, 5] TeV,
− Aκ ∈ [−5, 5] TeV,
−μeff ∈ [−2, 2] TeV.
Also note that in the following numerical examples, we will
minimise the impact of the stop- and sbottom-sector on both
Higgs mass and vacuum stability by assuming negligible
Charge-breaking minima only. Although it is not reflected
in the analytical example discussed in Sect. 2, there also
exist parameter points for which the EW minimum is
only metastable once the possibility of charge breaking
is included. Without the consideration of
charged-HiggsVEVs, the wrong impression of a stable EW minimum would
be obtained. An example is shown in Fig. 3 where the blue
region features a global CB minimum while the next-deepest
minimum is the desired EW one. In the green region, the EW
vacuum is stable whereas in the yellow region, other CC
minima corresponding to Eq. (17) are also deeper than the desired
EW one. In this figure, no parameter point which predicts
the correct Higgs mass features a stable vacuum once the CB
direction is taken into account. As a side remark we note that
one can also see in this example that loop corrections to the
scalar potential can be important when discussing the
vacuum stability: if one would not have included charged Higgs
VEVs, the conclusion whether stable regions in agreement
the Higgs mass measurement exist would have changed from
tree to loop level.
When checking all cases which we found in our scans,
there were no points featuring only CB minima deeper than
the desired EW one which turned out to be short-lived on
cosmological time scales. All points had a lifetime which
was many orders of magnitude longer than the lifetime of
the universe. We therefore conclude that such points are
phenomenologically viable, albeit significantly less appealing
compared with regions where the vacuum is entirely stable.
Charge-breaking and charge-conserving minima. This part
aims to answer the question whether or not CB minima can
further destabilise already metastable regions of parameter
space, reducing the EW vacuum to be dangerously
shortlived on cosmological time scales. As discussed before, this
is not the case in regions where only CB minima are deeper
than the EW minimum, which is why we turn to regions
where also other CC minima are deeper. Indeed we find many
regions of parameter space where the CB vacuum
configuration corresponds to the global minimum, with potential
values up to O(30%) deeper compared to the next deepest
CC minimum, in accordance with the discussion in Sect. 2.
However, as already seen in Fig. 2, other non-EW CC vacua
are nearer to the EW vacuum configuration in field space,
which means that the tunnelling path is reduced compared to
the tunnelling to the global, CB minimum. In practice, it turns
out that this effect is more important than the relative depth of
the minima. Although the global minimum is often CB, we
find that the tunnelling-time to the slightly nearer shallower
CC configuration of Eq. (17) is either shorter or of
comparable size in the regions where the lifetime of the vacuum
is comparable to the lifetime of the universe.2 Furthermore,
we find that in those few cases where the tunnelling to the
CB minimum indeed results in a shorter lifetime, the
differences are typically small. This behaviour is shown Fig. 4.
The background colours depict the ratio of the lifetimes when
considering both CC and CB minima (denoted as τ4−VEV)
versus when considering only CC minima (τ3−VEV). Purple
(τ4−VEV/τ3−VEV 1) correspond to regions where the
tunnelling rate of the EW vacuum is unchanged when also
considering the charged-Higgs VEVs. Deviations from the
2 Note that one cannot generalise the statement that tunnelling to the
nearer minimum is more effective: if we were to always consider the
nearest minimum to the EW one, we would often underestimate the
actual tunnelling rates by several orders of magnitude, as is also reflected
in the numerical example shown in Fig. 4.
Fig. 4 Ratio of the lifetimes τ4−VEV and τ3−VEV. Here, τ4−VEV and
τ3−VEV are the lifetimes for the most unstable minima of the respective
systems. The regions above the red (both solid and dashed) and grey
lines correspond to at least 99% survival probabilities of the desired
symmetry breaking (DSB) vacuum. The dashed red and solid grey
contours correspond to the most unstable minima of the three and
four VEV systems, respectively. The solid red contour corresponds to
the stability of the panic vacuum (the minimum closest in field space
to the DSB vacuum) in the 3 VEV system. Once again the region
between the black solid contours corresponds to an acceptable Higgs
mass, mh ∈ [122, 128] GeV. Here we use λ = −0.81, tan β = 1.02,
Aκ = −1400 GeV and Aλ = −580 GeV
ple background colour indicate that including the
chargedHiggs direction leads to a more effective tunnelling than only
considering the neutral Higgs directions. Regions above and
to the left of the red dashed and grey lines correspond to
parameter space where the vacuum is sufficiently long-lived
for the 3- and 4-VEV systems, respectively. Here, we have
used a 99% survival probability to calculate these lines. To
emphasise regions in this plane below the dashed red and
grey lines correspond to model points where the EW
vacuum lifetime is too small such that the probability of the EW
vacuum surviving as long as the age of the universe is below
Note that we see a slight difference between the grey and
red dashed lines in the upper right part of the figure. This is
where the tunnelling to the CB minimum is more efficient
than the tunnelling to the CC one. The area which the two
lines enclose is, however, very small. Therefore the inclusion
of the charged-Higgs direction in the vacuum stability
calculation results in only a tiny strip of parameter space where
the EW vacuum lifetime decreases below the 99% survival
probability threshold. In contrast, other regions of parameter
space (red regions in Fig. 4) show significant changes when
including the charged Higgs direction. However, the charged
Higgs direction does not decrease the EW vacuum lifetime
below the survival probability threshold and is therefore, on
phenomenological grounds, uninteresting. Finally, the red
solid line depicts the instability bound we would arrive at
if we considered only the panic vacuum, i.e. the minimum
nearest to the EW one in field space. It is therefore evident
that a naïve check for the vacuum stability can severely
underestimate the excluded parameter ranges.
In the parameter space scanned, we find that although the
global minimum of an NMSSM parameter point can feature
a global minimum where the charged Higgs develops a VEV,
it is not necessary to check for this extra field direction as the
constraints on the model parameters remain approximately
unchanged if one ensures that the tunnelling rate to all
possible minima are calculated.
We considered the possibility of spontaneous charge
breaking in the NMSSM via VEVs of the charged Higgs
components. We found that in contrast to models without singlets it
is possible that charge is broken at the global minimum of the
potential. We could identify two different kinds of parameter
regions. First, regions in which all vacua deeper than the EW
minimum have broken electric charge. These points would
give the wrong impression of a stable EW vacuum if charged
Higgs VEVs were not included in the study. However, in
all examples we found for these scenarios, the lifetime of
the EW vacuum is sufficiently long on cosmological time
scales. The second possibility is that charge-breaking and
-conserving minima beside the EW one are present at the
same time. Here, the charge-breaking minima could be
significantly deeper than the charge-conserving ones. However,
we found that the parameter regions which are excluded due
to an increased tunnelling rate to these deeper vacuum states
are hardly affected when considering the extra charged Higgs
field direction. Thus, the inclusion of charge-breaking
minima does not drastically change the conclusion of a
‘longlived’ vacuum to a ‘short-lived’ one. All in all, despite the
presence of deep charge-breaking minima in the NMSSM,
their phenomenological impact is rather limited. However,
we want to stress that the usual practice of checking only
the tunnelling rate to the deeper minimum nearest to the EW
vacuum is insufficient for obtaining reliable bounds on the
NMSSM parameter space.
Acknowledgements M.E.K is supported by the DFG Research Unit
2239 “New Physics at the LHC”. T.O is supported by the
SFBTransregio TR33 “The Dark Universe”.
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