NMSSM Higgs Benchmarks Near 125 GeV
The recent LHC indications of a SM-like Higgs boson near 125 GeV are consistent not only with the Standard Model (SM) but also with Supersymmetry (SUSY). However naturalness arguments disfavour the Minimal Supersymmetric Standard Model (MSSM). We consider the Next-to-Minimal Supersymmetric Standard Model (NMSSM) with a SM-like Higgs boson near 125 GeV involving relatively light stops and gluinos below 1 TeV in order to satisfy naturalness requirements. We are careful to ensure that the chosen values of couplings do not become non-perturbative below the grand unification (GUT) scale, although we also examine how these limits may be extended by the addition of extra matter to the NMSSM at the two-loop level. We then propose four sets of benchmark points corresponding to the SM-like Higgs boson being the lightest or the second lightest Higgs state in the NMSSM or the NMSSM-with-extra-matter. With the aid of these benchmark points we discuss how the NMSSM Higgs boson near 125 GeV may be distinguished from the SM Higgs boson in future LHC searches.
The ATLAS and CMS Collaborations have recently presented the first
indication for a Higgs boson with a mass in the region
GeV [1, 2]. An excess of events is
observed by the ATLAS experiment for a Higgs boson mass hypothesis
close to 126 GeV with a maximum local statistical significance of 3.6
above the expected SM background and by the CMS experiment at 124 GeV with
2.6 maximum local significance. If the ATLAS and CMS signals
are combined the statistical significance increases, but is still
less than the 5 required to claim a discovery. Interestingly,
the ATLAS signal in the decay channel by itself has a
local significance of 2.8 whereas a SM-like Higgs boson would
only have a significance of half this value, leading to speculation
that the observed Higgs boson is arising from beyond SM physics.
In general, these results have generated much excitement in the
community, and already there are a number of papers discussing the
implications of such a Higgs boson
[3, 4, 5, 6, 7].
In the Minimal Supersymmetric Standard Model (MSSM) the lightest Higgs boson is lighter than about 130-135 GeV, depending on top squark parameters (see e.g.  and references therein). A 125 GeV SM-like Higgs boson is consistent with the MSSM in the decoupling limit. In the limit of decoupling the light Higgs mass is given by
where is dominated by loops of heavy top quarks and
top squarks and is the ratio of the vacuum expectation
values (VEVs) of the two Higgs doublets introduced in the MSSM Higgs
sector. At large , we require GeV
which means that a very substantial loop contribution, nearly as large
as the tree-level mass, is needed to raise the Higgs boson mass to 125
The rather complicated parameter dependence has been studied in
 where it was shown that, with “maximal stop
mixing”, the lightest stop mass must be
GeV (with the second stop mass considerably larger) in the MSSM in order to achieve
a 125 GeV Higgs boson. However one of the motivations for SUSY is to
solve the hierarchy or fine-tuning problem of the SM
. It is well known that such large stop masses
typically require a tuning at least of order 1% in the MSSM,
depending on the parameter choice and the definition of fine-tuning
In the light of such fine-tuning considerations, it has been known for some time, even after the LEP limit on the Higgs boson mass of 114 GeV, that the fine-tuning of the MSSM could be ameliorated in the Next-to-Minimal Supersymmetric Standard Model (NMSSM) . With a 125 GeV Higgs boson, this conclusion is greatly strengthened and the NMSSM appears to be a much more natural alternative. In the NMSSM, the spectrum of the MSSM is extended by one singlet superfield [12, 13, 14] (for reviews see [15, 16]). In the NMSSM the supersymmetric Higgs mass parameter is promoted to a gauge-singlet superfield, , with a coupling to the Higgs doublets, , that is perturbative up to unified scales. In the pure NMSSM values of do not spoil the validity of perturbation theory up to the GUT scale only providing , however the presence of additional extra matter  allows smaller values of to be achieved. The maximum mass of the lightest Higgs boson is
where here we use GeV. For , the tree-level
contributions to are maximized for moderate values of
rather than by large values of as in the MSSM. For example, taking
and , these tree-level
contributions raise the Higgs boson mass to about 112 GeV,
and is required. This is to be compared
to the MSSM requirement . The difference
between these two values (numerically about 30 GeV) is significant since
depends logarithmically on the stop
masses as well as receiving an important contribution from stop
mixing. This means for example, that, unlike the MSSM, in the case of
the NMSSM maximal stop mixing is not required to get the Higgs heavy
The NMSSM has in fact several attractive features as
compared to the widely studied MSSM. Firstly, the NMSSM naturally solves in an
elegant way the so–called problem  of the MSSM, namely that
the phenomenologically required value for the supersymmetric Higgsino mass
in the vicinity of the electroweak or SUSY breaking scale is not explained.
This is automatically achieved in the NMSSM, since an effective -parameter is
dynamically generated when the singlet Higgs field acquires a vacuum expectation
value of the order of the SUSY breaking scale, leading to a
fundamental Lagrangian that contains no dimensionful parameters apart
from the soft SUSY breaking terms. Secondly, as compared to the MSSM,
the NMSSM can induce a richer phenomenology in the Higgs and
neutralino sectors, both in collider and dark
matter (DM) experiments. For example, heavier Higgs states can decay into
lighter ones with sizable rates [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30].
In addition, a new possibility appears for achieving the correct
cosmological relic density  through the
so-called “singlino”, i.e. the fifth neutralino of the model, which can have
weaker-than-usual couplings to SM particles.
The NMSSM Higgs boson near 125 GeV may also have significantly different properties than
the SM Higgs boson [5, 6], as we
shall discuss in some detail here. Thirdly, as already discussed
at length above, the NMSSM requires less fine-tuning than the MSSM .
Since the NMSSM (as in the case for the MSSM) has a rich parameter space,
it is convenient to consider the standard approach of so-called “benchmark
points” in the SUSY parameter space. These consist of a few “discrete” parameter
configurations of a given SUSY model, which in our case are supposed to lead to
typical phenomenological features. Using discrete points avoids
scanning the entire parameter space, focusing instead on
representative choices that reflect the new interesting
features of the model, such as new signals, peculiar mass spectra, and so on. A
reduced number of points can then be subject to full experimental investigation,
without loss of substantial theoretical information.
While several such benchmark scenarios have been devised for the
MSSM [32, 33] and thoroughly studied in both the
collider and the DM contexts, and even for the NMSSM ,
these need to be revised in the light of the 125 GeV Higgs signal.
The motivation for finding such NMSSM benchmarks is to enable the
characteristics of the NMSSM Higgs boson near 125 GeV to be identified, so
that it may eventually be resolved from that of the SM Higgs boson.
In this paper we present a set of NMSSM benchmark points with a
SM-like Higgs boson mass near 125 GeV, focussing on the cases where
both stop masses are as light as possible in order to reduce the
fine-tuning, in accordance with the above discussion. The tools to
calculate the Higgs and SUSY particle spectra in the NMSSM, in
particular NMSSMTools, have been
available for some time [21, 35, 36],
although these will be supplemented by other codes as discussed later.
The goal of the paper is to firstly find NMSSM benchmark points
which contain a 125 GeV SM-like Higgs boson
and in addition involve relatively light stops. Since the stops
receive radiative corrections to their masses from gluinos, we shall
also require relatively light gluinos as well as having as small an
effective parameter ()
as possible . Having relatively light stops (and
sbottoms), care has to be taken not to be in conflict with direct searches at the Tevatron
and recent searches at the LHC. In addition, light stops appearing in
the triangle loop diagrams can significantly affect Higgs production
via gluon fusion as well as Higgs decay into two photons.
Once a parameter set leading to a 125 GeV NMSSM Higgs boson has been
found it has to be checked if the production cross-sections and
branching ratios are such that they lead to a total Higgs production
cross-section times branching ratios which is compatible with the LHC
searches. The obtained branching ratios can be used to ultimately
distinguish the NMSSM Higgs boson from the SM Higgs state. However, we do
not attempt to simulate the LHC search strategies, our goal being the
much more modest one of providing benchmarks with different
characteristic types of NMSSM Higgs bosons, which can eventually be studied
in more detail. The types of benchmark points considered here all
involve relatively large values of and small values
of , in order to allow for the least fine-tuning
possible involving light stops as discussed above. However we are
careful to respect the perturbativity bounds on up to the
GUT scale, either for the pure NMSSM, or the NMSSM supplemented by
three copies of extra states, where such
bounds are calculated using two-loop renormalisation group equations
(RGEs). We find that the NMSSM Higgs boson near 125 GeV can come in
many guises. It may be very SM-like, practically indistinguishable
from the SM Higgs boson. Or it may have different Higgs production
cross-sections and widely varying decay branching ratios which enable it to be
resolved from the SM Higgs boson. The key distinguishing feature of
the NMSSM, compared to the MSSM, is the presence of the singlet
which may mix into the 125 GeV Higgs mass state to a greater or lesser extent.
As the singlet component of the 125 GeV Higgs boson is increased,
either the or components (or both) must be reduced. If the
component is reduced then this reduces the decay rate into bottom
quarks and can allow other rare decays like to have
larger branching ratios .
In addition there is the effect of SUSY particle and charged Higgs
boson loops in the Higgs coupling to photons which can increase or
decrease the rate of decays into .
In order to put the present work in context, it is worth to emphasise
that this is the first paper to propose “natural” (i.e. involving
light stops) NMSSM benchmark points with a SM-like Higgs boson near
125 GeV. For example, the previous NMSSM benchmark paper
 was concerned with the constrained NMSSM which
does not allow light stops consistent with current LHC SUSY
limits. Also, while this paper was being prepared there have appeared
a number of other related papers, none of which however are focussed
on the task of providing benchmark points. For example, the results in
 were mainly concerned with fine-tuning issues in
SUSY models with a coupling for a SM-like Higgs
boson near 126 GeV and the actual NMSSM was strictly not considered
since an explicit term was included whereas the trilinear
singlet coupling was neglected.
By contrast another recent paper  did consider
the actual NMSSM although no benchmarks were proposed.
As this paper was being finalised, further dedicated NMSSM papers with
a Higgs boson near 125 GeV have started to appear, in particular
 in which the two photon enhancement was
emphasised but without benchmark points, and 
where (versions of) the constrained NMSSM were considered. It should
be clear that the present paper is both complementary and
contemporary to all these papers.
The layout of the remainder of the paper is as follows. In Section 2 we discuss the MSSM and fine-tuning, showing that it leads to constraints on the mass of the heavier stop quark. In Section 3 we briefly review the NMSSM, and provide perturbativity limits of the coupling depending on the coupling as well as . We show how the limit on may be increased if there is extra matter in the SUSY desert. Section 4 is concerned with constraints from Higgs searches, SUSY particle searches and Dark Matter. Section 5 contains the four sets of NMSSM Higgs benchmarks near 125 GeV that we are proposing, including the Higgs production cross-sections and branching ratios which ultimately will enable it to be distinguished from the SM Higgs boson. Section 6 summarises and concludes the paper. Appendix A contains the two-loop RGEs from which the perturbativity bounds on were obtained.
2 The MSSM
The superpotential of the MSSM is given, in terms of (hatted) superfields, by
in which only the third generation fermions have been included
(with possible neutrino Yukawa couplings have been set to zero), and stand for superfields associated with the and
The soft SUSY breaking terms consist of the scalar mass terms for the Higgs and sfermion scalar fields which, in terms of the fields corresponding to the complex scalar components of the superfields, are given by
and the trilinear interactions between the sfermion and Higgs fields,
The tree-level MSSM Higgs potential is given by
where , and are the low energy (GUT normalised)
and gauge couplings, ,
In the MSSM, at the 1-loop level, stops contribute to the Higgs boson mass and three more parameters become important, the stop soft masses, and , and the stop mixing parameter
The dominant one-loop contribution to the Higgs boson mass depends on the geometric mean of the stop masses, , and is given by,
The Higgs mass is sensitive to the degree of stop mixing through the second term in the brackets, and is maximized for , which was referred to as “maximal mixing” above.
The fine-tuning in the MSSM can be simply understood by examining the leading one–loop correction to the Higgs potential,
where the two stop masses are,
The fine-tuning originates from the fact that , where GeV is the combined Higgs VEV. By considering the minimization conditions for the Higgs potential, one finds
In Eq. (2.12) is the mixing angle in the stop sector given by
Here we set the renormalisation scale . From Eq. (2.11) one can see that in order to avoid tuning,
This shows that both stop masses must be light to avoid tuning.
For example, defining the absence
of any tuning requires . This in turn requires
the heavier stop mass to be below about 500 GeV as illustrated in
Fig.1. This constraint on the heavier stop mass has not been
emphasised in the literature, where often the focus of attention is
on the lightest stop mass.
It has been noted that large or maximal stop mixing is associated
with large fine-tuning. This also follows from Fig.1 and
Eq. (2.12). Indeed, Fig.1 demonstrates that the
contribution of one–loop corrections to Eq. (2.11) increases
when the mixing angle in the stop sector becomes larger.
In fact when is close to the last term in
Eq. (2.12) gives the dominant contribution to
enhancing the overall contribution of loop corrections
in the minimization condition (2.11) which determines the
mass of the –boson.
Eq. (2.11) also indicates that in order to avoid tuning one has to ensure that the parameter has a reasonably small value. To avoid tuning entirely one should expect to be less than . However, so small values of the parameter are ruled out by chargino searches at LEP. Therefore in our analysis we allow the effective parameter to be as large as that does not result in enormous fine-tuning.
3 The NMSSM
In this paper, we only consider the NMSSM with a scale invariant
superpotential. Alternative models known as the minimal
non-minimal supersymmetric SM (MNSSM), new minimally-extended supersymmetric SM
or nearly-minimal supersymmetric SM (nMSSM) or with additional gauge
symmetries have been considered elsewhere ,
as has the case of explicit CP violation .
The NMSSM superpotential is given, in terms of (hatted) superfields, by
in which only the third generation fermions have been included. The first two terms substitute the term in the MSSM superpotential, while the three last terms are the usual generalization of the Yukawa interactions. The soft SUSY breaking terms consist of the scalar mass terms for the Higgs and sfermion scalar fields which, in terms of the fields corresponding to the complex scalar components of the superfields, are given by,
The trilinear interactions between the sfermion and Higgs fields are,
In the unconstrained NMSSM considered here, with non–universal soft terms at the GUT scale, the three SUSY breaking masses squared for , and appearing in can be expressed in terms of their VEVs through the three minimization conditions of the scalar potential. Thus, in contrast to the MSSM (where one has only two free parameters at the tree level, generally chosen to be the ratio of Higgs vacuum expectation values, , and the mass of the pseudoscalar Higgs boson), the Higgs sector of the NMSSM is described by the six parameters
We follow the sign conventions such that the parameters
and are positive, while the parameters ,
, and can have both
In addition to these six parameters of the Higgs sector, one needs to specify the soft SUSY breaking mass terms in Eq. (3.16) for the scalars, the trilinear couplings in Eq. (3.17) as well as the gaugino soft SUSY breaking mass parameters to describe the model completely,
Clearly, in the limit with finite , the NMSSM
turns into the MSSM with a decoupled singlet sector. Whereas the phenomenology
of the NMSSM for could still differ somewhat from the MSSM in
the case where the lightest SUSY particle is the singlino (and hence
with the possibility of a long lived next-to-lightest SUSY particle
), we will not consider this situation here.
In fact we shall be interested exclusively in large values of
(i.e. ) in order to increase the tree-level Higgs mass as
in Eq. (1.2). For the same reason we shall also focus on
moderate values of () that result in the relatively
large values of the top quark Yukawa coupling at low energies.
The growth of the Yukawa couplings , and at the
electroweak (EW) scale
entails the increase of their values at the Grand Unification scale resulting
in the appearance of the Landau pole. Large values of ,
and spoil the applicability of perturbation theory at high
energies so that the RGEs cannot be used for an adequate description
of the evolution of gauge and Yukawa couplings at high scales .
The requirement of validity of perturbation theory up to the Grand Unification
scale restricts the interval of variations of Yukawa couplings at the
EW scale. In particular, the assumption that perturbative physics
continues up to the scale sets an upper limit on the low energy value of
for each fixed set of and (or ).
With decreasing (increasing) the maximal possible value of
, which is consistent with perturbative gauge coupling unification,
increases (decreases) for each particular value of .
In Table 1 we display two-loop upper bounds on
for different values of and in the NMSSM.
As one can see the allowed range for the Yukawa couplings varies when
changes. Indeed, for the value of
should be smaller than to ensure the validity of perturbation theory up
to the scale . At large the allowed range for the Yukawa
couplings enlarges. The upper bound on grows with increasing
because the top–quark Yukawa coupling decreases. At large
(i.e. ) the upper bound on
approaches the saturation limit where .
The renormalisation group (RG) flow of the Yukawa couplings depends rather strongly on the values of the gauge couplings at the intermediate scales. To demonstrate this we examine the RG flow of gauge and Yukawa couplings within the SUSY model that contains three extra –plets that survive to low energies and can form three –plets in the SUSY-GUT model based on the gauge group. In this SUSY model the strong gauge coupling has a zero one–loop beta function whereas at two–loop level the coupling has a mild growth as the renormalisation scale increases. Since extra states form complete multiplets the high-energy scale where the unification of the gauge couplings takes place remains almost the same as in the MSSM. At the same time extra multiplets of matter change the running of the gauge couplings so that their values at the intermediate scale rise substantially. In fact the two–loop beta functions of the SM gauge couplings are quite close to their saturation limits when these couplings blow up at the GUT scale. Further enlargement of the particle content can lead to the appearance of the Landau pole during the evolution of the gauge couplings from to . Because the beta functions are so close to the saturation limits the RG flow of the gauge couplings (i.e. their values at the intermediate scale) also depends on the masses of the extra exotic states. Since occurs in the right–hand side of the RGEs for the Yukawa couplings with negative sign the growth of the gauge couplings prevents the appearance of the Landau pole in the evolution of these couplings. It means that for each value of (or ) and the upper limit on increases as compared with the NMSSM. The two-loop upper bounds on for different values of and in the NMSSM supplemented by three –plets, or equivalently three –plets are displayed in Table 2 for the case that the masses of all extra exotic states are set to be equal to and in Table 3 in case they are set to be equal to . The two-loop RGEs used to obtain these results are given in Appendix A. Because the RG flow of the gauge couplings depends on the masses of extra exotic states the upper bounds on presented in Tables 2 and 3 are slightly different. The restrictions on obtained in this section are useful for the phenomenological analysis which we are going to consider next.
4 Constraints from Higgs Boson Searches, SUSY Particle Searches and Dark Matter
Our scenarios are subject to constraints on the Higgs boson masses
from the direct searches at LEP, Tevatron and the LHC. Also the SUSY
particle masses have to be compatible with the limits given by the
experiments. Finally, the currently measured value of the relic
density shall be reproduced. Further constraints arise from the low-energy observables.
4.1 Higgs boson searches
We start by discussing the constraints which arise from the LHC search
for the Higgs boson. At the LHC, the most relevant Higgs
boson production channels for neutral (N)MSSM Higgs bosons are given
by gluon fusion, gauge boson fusion, Higgs-strahlung and associated
production with a heavy quark pair. The two main mechanisms for
charged Higgs boson production are top quark decay and associated
production with a heavy quark pair. For reviews, see
[8, 41, 42]. As in our scenarios the charged Higgs
boson mass is larger than 450 GeV and hence well beyond the sensitivity
of Tevatron and current LHC searches, we will discuss in the following
only neutral Higgs boson production.
Gluon fusion In the SM and in SUSY extensions, such as the (N)MSSM, for low values of , the most important production channel is given by gluon fusion . In the NMSSM we have
Since this is the dominant Higgs production mechanism for a 125 GeV Higgs boson at the LHC, we find it convenient to define for later use the ratio of the gluon fusion production cross-section for the Higgs boson in the NMSSM to the gluon fusion production cross-section for a SM Higgs boson of same mass as ,
Gluon fusion is mediated by heavy quark loops in the SM and
additionally by heavy squark loops in the (N)MSSM.
It is subject to important higher-order QCD corrections. For the SM,
they have been calculated at next-to-leading order (NLO) 
full mass dependence of the loop particles and in the heavy top quark
limit, and up to next-to-next-to-leading order (NNLO) in the heavy top
quark limit . The cross-section has been improved by
soft-gluon resummation at next-to-next-to-leading logarithmic (NNLL)
accuracy . Top quark mass effects on the NNLO loop
corrections have been studied in , and the
EW corrections have been provided in . In the MSSM, the
QCD corrections have been calculated up to NLO . The
QCD corrections to squark loops have been first considered in
 and at full NLO SUSY-QCD in the heavy mass limit in
(s)bottom quark contributions at NLO SUSY-QCD have been taken into
account through an
asymptotic expansion in the SUSY particle masses . For
squark masses below GeV, mass effects play a role and can
alter the cross-section by up to 15% compared to the heavy mass limit
as has been shown for the QCD corrections to the squark loops in
[52, 53]. The SUSY QCD corrections including the full mass
dependence of all loop particles have been provided by
. The mass effects turn out to be
sizeable. The NNLO SUSY-QCD corrections from the (s)top
quark sector to the matching coefficient determining the effective
Higgs gluon vertex have been calculated in .
The gluon fusion cross-section has been implemented
in the Fortran code HIGLU  up to NNLO QCD. While at
NLO the full mass dependence of the loop particles is taken into
account the NNLO corrections are obtained in an effective theory
approach. In the MSSM the full squark mass dependence in the NLO QCD
corrections to the squark loops is included . Note, however, that in
the MSSM at NNLO the mismatch in the QCD corrections to the effective
vertex is not taken into account, neither the SUSY QCD corrections to
the effective vertex. The former should be only a minor effect,
though, as the dominant effect of the QCD higher-order corrections stems from
the gluon radiation. Furthermore, the EW corrections to the SM can be
obtained with HIGLU.
In order to check if our scenario is compatible with the recent LHC
results, we need the cross-section of a SM-like Higgs boson of 124 to 126 GeV. The experiments
include in their analyses the NNLO QCD (CMS also the NNLL QCD and NLO
EW) corrections [1, 2] to the
gluon fusion cross-section as provided by the Higgs Cross-Section
Working Group . For SUSY, however, the EW corrections
are not available. In order to be consistent, we therefore
compare in the following the NMSSM cross-section to the SM cross-section
at NNLO QCD. As the QCD corrections are not affected by modifications of
the Higgs couplings to the (s)quarks, the NMSSM cross-section can be obtained with
the program HIGLU by multiplying the MSSM Higgs couplings with the
corresponding modification factor of the NMSSM Higgs couplings to the
(s)quarks with respect to the MSSM case. We have implemented these
coupling modification factors in the most recent HIGLU version 3.11.
-boson fusion Gauge boson fusion  plays an important role for light CP-even Higgs boson production in the SM limit,555The quark stands for a generic quark flavour, which is different for the two quarks in case of -boson fusion. The same notation is applied below in Higgs-strahlung.
Otherwise the (N)MSSM cross-section is suppressed with respect to the SM case by mixing angles entering the Higgs couplings to the gauge bosons. The NLO QCD corrections are of of the total cross-section [58, 59]. The full EW and QCD corrections to the SM are . The NNLO QCD effects on the cross-section amount to % . The SUSY QCD and SUSY EW corrections are small [62, 63]. Once again, as QCD corrections are not affected by the Higgs couplings to the gauge bosons, the QCD corrected NMSSM gauge boson fusion production cross-sections can be derived from the QCD corrected SM cross-section by simply applying the modification factor of the respective NMSSM Higgs coupling to the gauge bosons with respect to the SM coupling,
where denotes the coupling.
The EW corrections, however, cannot be taken over. We have obtained the
SM production cross-section at NLO QCD from the program VV2H
. While the experiments use the SM cross-section at
NNLO QCD (CMS also at NLO EW), the effects of these
additional corrections in the SM limit, where we compare our NMSSM
Higgs cross-section to the SM case, are small.
Higgs-strahlung The CP-even Higgs bosons can also be produced in Higgs-strahlung ,
with the NMSSM cross-section always being suppressed by mixing angles compared to the SM cross-section. The QCD corrections apply both to the SM and (N)MSSM case. While the NLO QCD corrections increase the cross-section by [59, 66] the NNLO QCD corrections are small . The full EW corrections are only known for the SM and decrease the cross-section by . The SUSY-QCD corrections amount to less than a few percent . The NLO QCD SM Higgs-strahlung cross-section has been obtained with the program V2HV . The NMSSM Higgs production cross-sections can be derived from it by applying the Higgs coupling modification factors,
The experiments use the QCD corrected cross-section up to NNLO
(CMS also including the NLO EW corrections). While we neglect the NNLO and EW
corrections, we do not expect this to influence significantly the total cross-section composed of all production channels, in view of
the small size of the total Higgs-strahlung cross-section itself.
Associated production with heavy quarks Associated production of (N)MSSM Higgs bosons with top quarks  only plays a role for the light scalar Higgs particle and small values of due to the suppression of the Higgs couplings to top quarks . While associated production with bottom quarks [69, 70] does not play a role in the SM, in the (N)MSSM this cross-section becomes important for large values of and can exceed the gluon fusion cross-section. As our scenarios include small values of we will not further discuss this cross-section here. The values for the SM cross section including NLO QCD corrections , which are of moderate size, can be obtained from the Higgs Cross Section Working Group homepage . From these we derived the NMSSM cross-section values by replacing the SM Yukawa couplings with the NMSSM Yukawa couplings,
The NLO SUSY QCD corrections, which have not been taken into account by the experiments, are of moderate size  .
4.2 Constraints from the LHC Searches
Recent results presented by the ATLAS  and the CMS
 Collaborations seem to indicate a Higgs boson of
mass of 126 and 124 GeV, respectively. Based on the dataset corresponding
to an integrated luminosity of up to 4.9 fb collected at
TeV, an excess of events is observed by the ATLAS
experiment for a Higgs boson mass hypothesis close to GeV
with a maximum local significance of 3.6 above the expected SM
background. The three most sensitive channels in this mass range are
given by , and
. The CMS Collaboration
presented results of SM Higgs boson searches in the mass range 100-600
GeV in 5 decay modes, and
. The data correspond to an integrated luminosity of up to 4.7
fb at TeV. A modest excess of events is observed
for Higgs boson mass hypotheses towards the low end of the
mass range. The maximum local significance amounts to 2.6 for
a Higgs boson mass hypothesis of GeV. For our NMSSM
benchmark scenarios presented below to be consistent with these LHC
results we demand the production cross-section of the SM-like NMSSM
Higgs boson with mass 124 GeV to 126 GeV (depending on the scenario)
to be compatible within 20% with the production cross-section of a SM
Higgs boson of same mass. The 20% are driven by the theoretical
uncertainty on the inclusive Higgs production cross-section given by
the sum of the most relevant production channels at low values of
, i.e. gluon fusion, weak boson fusion,
Higgs-strahlung and Higgs production. The theoretical error
is largest for the gluon fusion cross-section with 10-15% at these Higgs mass
values and TeV , and which contributes
dominantly to the inclusive production. We do not consider any
experimental error since this is beyond our scope.
For simplicity, and since these search channels are common to both experimental analyses, we consider the Higgs decays into , and . In order to get an estimate of how closely the NMSSM Higgs resembles the SM Higgs in LHC searches in these channels we define the ratios of branching ratios into massive gauge boson final states , where ,666The ratio is the same for and final states, respectively, as the NMSSM coupling to and is suppressed by the same factor compared to the SM. and into , respectively, for an NMSSM Higgs boson and the SM Higgs boson of same mass,
We also define analogously for the total widths,
and for the decay into .
Although this final state is not useful for LHC searches, it is interesting to show as in
this mass range and for small values of the decay into
contributes dominantly to the total width. Depending on of how
much component is in the mass eigenstate it is enhanced or
suppressed compared to the SM. This directly influences the total
width and hence the branching ratios of the other final states.
For a crude estimate of the total Higgs cross-section at the LHC, we can combine these channels in quadrature in contrast to the experiments which do a sophisticated statistical combination of the various search channels, which is, however, beyond the scope of our theoretical analysis. Our results should therefore only be regarded as a rough estimate which is indicative enough, however, at the present status of the experimental research, to exclude or not exclude a benchmark scenario. We hence demand for a scenario to be valid that one of the NMSSM Higgs bosons satisfies
The inclusive cross-section is composed of gluon fusion, vector boson fusion, Higgs-strahlung and associated production with ,
with , respectively, subject to the constraint -126 GeV, depending on the scenario under consideration. It is dominated by the gluon fusion cross-section. The factors 16 in Eq. (4.30) arise from the sum of the four possible lepton final states in the decays of the - and -boson pairs, respectively. (We neglect interference effects.) For the gauge boson branching ratios we chose the values given by the Particle Data Group ,
It is useful to define
in order to provide a measure of how closely the NMSSM Higgs resembles the SM Higgs in the most important current LHC search channels. The total cross-section is dominated by the -boson final state due to the large branching ratio. As we will see below, in the NMSSM the branching ratio into can be enhanced for certain parameter configurations compared to the SM. To illustrate this effect, we therefore also calculate separately the ratios of the expected number of events in the NMSSM compared to the SM for the final state and for the () final state, which is the same for or . They are given by
4.3 NMSSM spectrum and NMSSM Higgs boson branching ratios
The SUSY particle and NMSSM Higgs boson masses and branching ratios
are calculated with the program package NMSSMTools
[21, 35, 36]. As for the NMSSM Higgs boson
masses, the leading one-loop contributions due to heavy (s)quark loops
calculated in the effective potential approach , the
one-loop contributions due to chargino, neutralino and scalar loops in leading
logarithmic order in Ref.  and the leading logarithmic
two-loop terms of and , taken over from the MSSM results, have been implemented
in NMSSMTools. The full one-loop contributions have
been computed in the renormalisation scheme
[77, 78] and also in a mixed on-shell (OS) and
scheme as well as in a pure OS scheme
. Furthermore, the corrections have been provided in the approximation of zero
external momentum . The corrections provided by
Ref.  have been implemented in NMSSMTools as well.
The calculation of the NMSSM Higgs boson decay widths and branching ratios within NMSSMTools is performed by the Fortran code NMHDECAY [21, 35] which uses to some extent parts of the Fortran code HDECAY [80, 81] that calculates SM and MSSM Higgs boson partial widths and branching ratios. The calculation of the SUSY particle branching ratios on the other hand with the Fortran code NMSDECAY  is based on a generalisation of the Fortran code SDECAY [83, 81] to the NMSSM case. NMSSMTools provides the output for the complete NMSSM particle spectrum and mixing angles and for the decays in the SUSY Les Houches format . The latter can be read in by our own Fortran version for NMSSM Higgs boson decays based on an extension of the latest HDECAY version. It reads in the particle spectrum and mixing angles created with NMSSMTools, calculates internally the NMSSM Higgs boson couplings and uses them to calculate the Higgs decay widths and branching ratios. The results for the branching ratios from NMSSMTools and from our own program agree reasonably well and the differences in the total cross-section Eq. (4.30), obtained with the results from the two programs, due to deviations in the branching ratios are in the percent range.
4.4 Constraints from Dark Matter, Low Energy Observables, LEP and Tevatron
Based on an interface between NMHDECAY and micrOMEGAs
 the relic abundance of the NMSSM dark matter
candidate can be evaluated using NMSSMTools.
As an independent check, we also used the stand alone code micrOMEGAs to calculate the relic density. All the relevant cross-sections for the lightest neutralino annihilation and co-annihilation are
computed. The density evolution equation is numerically solved and the relic
density of is calculated. The differences in the
result for the relic density calculated with both tools are negligible.
The results are compared with
the “WMAP” constraint at the level .
When the spectrum and the couplings of the Higgs and SUSY particles are computed with NMSSMTools, constraints from low-energy observables as well as available Tevatron and LEP constraints are checked. The results of the four LEP collaborations, combined by the LEP Higgs Working Group, are included . More specifically, the following experimental constraints are taken into account:
The masses of the neutralinos as well as their couplings to the boson are compared with the LEP constraints from direct searches and from the invisible boson width.
Direct bounds from LEP and Tevatron on the masses of the charged particles (, , , ) and the gluino are taken into account.
Constraints on the Higgs production rates from all channels studied at LEP. These include in particular production, being any of the CP–even Higgs particles, with all possible two body decay modes of (into quarks, leptons, jets, photons or invisible), and all possible decay modes of of the form , being any of the CP–odd Higgs particles, with all possible combinations of decays into quarks, quarks, leptons and jets. Also considered is the associated production mode with, possibly, . (In practice, for our purposes, only combinations of with are phenomenologically relevant.)
Experimental constraints from B physics  such as the branching ratios of the rare decays BR, BR and BR and the mass differences and , are also implemented; compatibility of each point in parameter space with the current experimental bounds is required at the two sigma level.
4.5 Constraints on SUSY particle masses from the LHC
The ATLAS  and CMS  searches in final states with jets and missing transverse energy , with large jet multiplicities and , with heavy flavour jets and within simplified models and mSUGRA/constrained MSSM (CMSSM) models set limits on the masses of gluino and squark masses. Further constraints are obtained from searches in final states with leptons and taus . The precise exclusion limits depend on the investigated final state, the value of the neutralino and/or chargino masses and the considered model. Light gluino (below about 600 GeV) and squark masses (below about 700 GeV) are excluded. The limits cannot be applied, however, to the third generation squarks in a model-independent way. Recent analyses scanning over the physical stop and sbottom masses and translating the limits to the third generation squark sector have shown that the sbottom and stop masses can still be as light as GeV depending on the details of the spectrum . Especially scenarios, where the lightest stop is the next-to-lightest SUSY particle (NLSP) and nearly degenerate with the lightest neutralino assumed to be the lightest SUSY particle (LSP), are challenging for the experiments. Such scenarios can be consistent with Dark Matter constraints due to possible co-annihilation . If the mass difference is small enough, the flavour-changing neutral current decay  is dominating and can compete with the four-body decay into the LSP, a quark and a fermion pair . Limits have been placed by the Tevatron searches  and depending on the neutralino mass still allow for very light stops down to 100 GeV. The authors of Ref.  found that translating the LHC limits to stop searches in the co-annihilation scenario 777For stop searches in scenarios with light gravitinos see . allows for stops lighter than GeV down to 160 GeV. We are not aware, however, of any dedicated LHC analysis which excludes very light stops.
5 The benchmark points
In this section we present benchmark points for the NMSSM with a SM-like Higgs boson near 125 GeV. The Higgs sector of the NMSSM has a rich parameter space including and the effective parameter. According to the SLHA format , these parameters are understood as running parameters taken at the SUSY scale TeV while is taken at the scale of the boson mass, . In order not to violate tree-level naturalness we set GeV for all the considered points. The Higgs sector is strongly influenced by the stop sector via radiative corrections where we further need to specify the soft SUSY breaking masses and the mixing parameter defined in Eq. (2.7). The main advantage of the NMSSM over the MSSM, regarding a SM-like Higgs boson near 125 GeV, is that the stop masses are allowed to be much lighter, making the NMSSM much more technically natural than the MSSM. Thus all of the benchmarks discussed here will involve relatively light stops, with masses well below 1 TeV. We choose low values of in order to maximise the tree-level contribution to the Higgs boson mass, allowing the stops to be lighter. For very light stops, in order not to be in conflict with the present exclusion limits, the difference between the and masses should be less than GeV to be in the co-annihilation region. By choosing the right-handed stop to be the lightest top squark the sbottoms are still heavy enough to fulfill the LHC limits of about 300 GeV  also in this case. On the other hand, the remaining squarks and sleptons may be heavier without affecting fine-tuning. In order to satisfy in particular the LHC search limits for the squarks of the first two generations, we set all their masses to be close to 1 TeV and, for simplicity, also those of the sleptons. To be precise, for the first and second squark and slepton families and for the stau sector we always take the soft SUSY breaking masses and trilinear couplings to be 1 TeV, and furthermore, the right-handed soft SUSY breaking mass and trilinear coupling of the sbottom sector is set to 1 TeV, i.e. (, , )
This results in physical masses of 1 TeV for the first and second family squarks and sleptons as well as the heavier sbottom. These masses can readily be increased without affecting the properties of the quoted benchmark points appreciably. The gaugino mass parameters have been set such that they fulfill roughly GUT relations. Special attention has been paid, however, not to choose the gluino mass too heavy in order to avoid fine-tuning. Before discussing the benchmark points, a few technical preliminaries are in order. The masses for the Higgs bosons and SUSY particles have been obtained using NMSSMTools. In the presentation of our benchmark scenarios, for the SM-like Higgs boson we furthermore include the ratios of the branching ratios, cf. Eq. (4.27), into , and () as well as the ratio of the total widths Eq. (4.28) for the NMSSM and SM Higgs boson having the same mass. The NMSSM branching ratios and total width have been obtained with NMSSMTools and cross-checked against a private code whereas the SM values have been calculated with HDECAY. The latter are shown separately in Table 4 for a SM Higgs boson with the mass values corresponding to the various masses of the SM-like NMSSM Higgs boson in our benchmark scenarios. The NMSSM values can be obtained by multiplication with the corresponding ratios presented in the benchmark tables, although it is mainly their relative values, compared to the SM, that concern us here. Note that HDECAY includes the double off-shell decays into massive vector bosons, whereas NMSSMTools does not. We therefore turned off the double off-shell decays also in HDECAY. This explains why the SM values given in Table 4 differ from the values given on the website of the Higgs Cross Section Working Group . There are further differences between the two programs in the calculation of the various partial widths. Thus HDECAY includes the full NNNLO corrections to the top loops in the decay into gluons. Also it uses slightly different running bottom and charm quark masses. Taking all these effects into account we estimate the theoretical error on the ratios of branching ratios, which are calculated with these two different programs, to be of the order of 5%. This should be kept in mind when discussing the benchmark scenarios.
As mentioned above, in order to be compatible with the recent LHC results
for the Higgs boson search we demand the total cross-section as
defined in Eq. (4.30) to be within 20 % equal to the
corresponding SM cross-section. We therefore include in our tables for
the benchmark points the ratio , Eq. (4.33), of the
NMSSM and SM total cross-section for TeV and for
completeness also the ratio , Eq. (4.21), of the NMSSM and
SM gluon fusion cross-sections since gluon fusion is the dominant
contribution to inclusive Higgs production for low values of
at the LHC. The latter has been obtained with the Fortran code HIGLU at NNLO QCD and includes the squark loop
contributions. Note, that we did not include electroweak
corrections. Furthermore, we explicitly verified that is
practically the same using NLO or NNLO QCD gluon fusion cross-sections.
The values of the gluon fusion cross-section and of the total cross-section at NLO and NNLO QCD are shown separately for the SM in Table
5. However, the ratios for the cross-sections which we present in the tables, ,
, are for the case of gluon fusion production at
NNLO QCD. With the total cross-section being dominated by the -boson final
state, in order to illustrate interesting effects in the NMSSM branching
ratios compared to the SM ones, for the SM-like Higgs boson
we also give separately the ratios of the
expected number of events in the and () final states
as given by Eq. (4.34).
All our presented scenarios fulfill the cross-section constraint
Eq. (4.29) and are hence compatible with the
LHC searches, keeping in mind though that as theorists we can do only a rough
estimate here. Furthermore, they fulfill the constraints from low-energy parameters
as specified in section 4 and are compatible with
the measurement of the relic density. In principle they could also account for
the deviation of the muon anomalous magnetic momentum from
the SM if we were to lower the smuon mass, but for clarity we have
taken all first and second family squark and slepton masses to be
close to 1 TeV, as discussed above.
We consider four different sets of NMSSM benchmark points as follows:
NMSSM with Lightest Higgs being SM-like near 125 GeV
NMSSM with Second Lightest Higgs being SM-like near 125 GeV
NMSSM-with-extra-matter and Second Higgs being SM-like near 125 GeV
NMSSM-with-extra-matter and Lightest Higgs being SM-like near 125 GeV
This is achieved with and such that does not blow up below the GUT scale in the NMSSM supplemented by three