Naturalness and Higgs Decays in the MSSM with a Singlet
Abstract:
The simplest extension of the supersymmetric standard model  the addition of one singlet superfield  can have a profound impact on the Higgs and its decays. We perform a general operator analysis of this scenario, focusing on the phenomenologically distinct scenarios that can arise, and not restricting the scope to the narrow framework of the NMSSM. We reexamine decays to four b quarks and four ’s, finding that they are still generally viable, but at the edge of LEP limits. We find a broad set of Higgs decay modes, some new, including those with four gluon final states, as well as more general six and eight parton final states. We find the phenomenology of these scenarios is dramatically impacted by operators typically ignored, specifically those arising from Dterms in the hidden sector, and those arising from weakscale colored fields. In addition to sensitivity of , there are potential tunings of other aspects of the spectrum. In spite of this, these models can be very natural, with light stops and a Higgs as light as 82 GeV. These scenarios motivate further analyses of LEP data as well as studies of the detection capabilities of future colliders to the new decay channels presented.
1 Introduction
Since 1934 when Fermi wrote down his theory of the weak interactions, we have singled out the weak scale as an interesting scale for new physics. Within the last two decades, we have finally reached this energy scale at LEP and the Tevatron, and with the LHC we should be able to probe this scale thoroughly.
The most pressing question at the weak scale is the origin of electroweak symmetry breaking (EWSB). Within the GlashowSalamWeinberg theory of weak interactions, it is broken by a fundamental scalar doublet of , the Higgs field. In this framework, all precision quantities are calculable and agree with present experimental limits.^{1}^{1}1We are of course neglecting dark matter here, neutrino masses, the LSND anomaly and the NuTeV anomaly, all of which are beyond the scope of this paper.
The major theoretical shortcoming of the standard model (SM) is the question of the stability of the Higgs mass. Scalars in general acquire quadratically divergent mass, suggesting that new physics should cut off this divergence and appears near the weak scale. Numerous solutions to this theoretical problem have been proposed, most notably supersymmetry (SUSY).
2 Naturalness in the MSSM
In this paper we shall focus on supersymmetric solutions of the hierarchy problem. In these theories the introduction of superpartners cancels the quadratic divergences of the standard model with loops of opposite statistics particles, leaving only logarithmically divergent contributions to the Higgs mass, proportional to the SUSY breaking soft mass parameters.
Despite its excellent success in controlling divergences from very high scales, in recent years SUSY has become far more constrained, with superpartners pushed to higher scales, resulting in tunings which are typically in the minimal supersymmetric extension of the standard model (MSSM).^{2}^{2}2For an excellent discussion of the details of fine tunings in SUSY models, see [1]. The reason for this is quite simple: the success of the LEP program has pushed the lower limit on a SM Higgs boson above 114 GeV, a limit which applies in a large region of the MSSM parameter space. In the MSSM, the Higgs has a mass which has a tree level upper limit of . To achieve a mass of 115 GeV, one must invoke large radiative corrections due to (s)top loops. At one loop, the correction to the Higgs mass is[2]
(1) 
requiring a stop mass of roughly 500 GeV. Unfortunately, such a large stop mass also feeds into the soft mass squared of the Higgs,
(2) 
where is the scale where the stop masses are generated. Even for low scale mass generation such as gauge mediation, where , this results in a radiative correction to the uptype Higgs soft mass squared of . This must then be cancelled against a positive mass squared, for instance via a term, in order to achieve the appropriate vev, . This cancellation must be tuned at the level of in order to achieve this. If the stops are heavier, or the mediation scale is higher, the tuning becomes worse.
The root of this problem is the small quartic of supersymmetric models, which is fixed by the electroweak gauge couplings. There are many proposals to enlarge the quartics, for instance by adding terms in the superpotential [3, 4, 5, 6, 7, 8, 9, 10], with nondecoupling Dterm quartics [11, 12, 13], or through strong dynamics at an intermediate scale [14, 15, 16].
New Higgs Decays and Naturalness
An alternative approach is to evade the LEP limits indirectly. For instance, if the Higgs field decays in a nonstandard way, LEP may not have been as sensitive to the decay. This is not as trivial as it may appear, in particular because limits on invisible Higgs decays are nearly as strong as those of bquarks. If these nonstandard decays are present in nature, there must be new states lighter than the Higgs boson, which themselves decay into ordinary SM fields. The simplest possibility is the presence of an additional singlet superfield to which the Higgs can decay, and which then decays into SM particles, such as bjets. In this vein, it has been recently argued that within the NMSSM, where a singlet field acquires a vev to supply the term of the Higgs sector, certain nonstandard decays are possible, leading to more natural theories [17].
However, there is still a strong need for additional analyses, for many reasons.

New combined limits from LEP [18, 19] exclude most of the generic parameter space of these models. Typically, the decay has been used to allow lighter Higgses. However, new LEP combined analyses have basically raised the minimum Higgs mass for this process to 110 GeV, nearly as strong as the limit on the standard model Higgs. This raises the question of whether other nonstandard decays can occur generically, to which LEP analyses would have been less sensitive. We will discuss these limits further in the next section.

The NMSSM is not a fully general scenario. In the NMSSM, the singlet acquires a vacuum expectation value to generate a term. One typically requires that this be the true vacuum of the theory, and additionally makes assumptions of the form of the theory in the ultraviolet (UV). All of these things can distract from – and are beside the point of – the basic phenomenological questions relating to Higgs decays.

The sensitivity of to UV parameters is not the only measure of naturalness in these theories. Although a significant reason to consider these decays is precisely to ameliorate this tuning, one often ends up with additional tunings in order to evade experimental limits. Thus, while may be relatively natural other elements of the spectrum may be highly tuned, and necessary, to evade experimental limits.

The parameter space of new operators has not been fully explored. In the MSSM, the full set of soft breaking operators is considered. However, in adding a singlet to the theory, one can consider both the effects of Dterms in the supersymmetry breaking sector, and nondegenerate soft masses for the scalar and pseudoscalar components of the singlet, both of which can have significant phenomenological consequences.

The sensitivity of the decays of the pseudoscalar to the presence of new fields has essentially been ignored within the context of supersymmetric theories. Such fields can naturally induce the decay of the pseudoscalar to two glue jets, which have far weaker constraints.
There are two important additional points to be made with regard to naturalness. The first is: what do we mean by tuning in the scenarios we are considering? Since we are interested in the lowenergy phenomenological theory, it is impossible to quantify the sensitivity of to the UV parameters. However, we know that the tuning of arises in general due to the large values of the stop squark masses. Therefore, we shall use the stop masses as a proxy for this tuning. However, there is typically a tuning necessary to achieve the proper spectrum for the lighter scalar states, and this is usually the most severe tuning in the models. Therefore, we shall quote both stop masses, as well as the scalar mass spectrum tuning for every model considered.
The second point is: how natural is it to include a soft supersymmetry breaking operator, without an associated superpotential operator? In particular, how reasonable is it to include a trilinear scalar potential Aterm operator without associated Yukawas? This question has been studied previously in the context of sterile neutrino masses in supersymmetry [20, 21]. In fact, if there is a SUSY breaking field which carries a charge, whether Rcharge or PecceiQuinn charge, it is quite natural for these operators to appear without an associated superpotential term. The converse, however, is not true. The presence of the superpotential term will radiatively generate the soft term at a minimum at the loopsuppressed level. Consequently, we will insist on technical naturalness, that these soft operators are present at least at this small level.
In lieu of these points, we will pursue a phenomenological study of the effects of singlets on the decays and properties of Higgs bosons. We will focus on decay modes that have not been considered previously, including final states with 6 and 8 particles. We will often be studying situations when both the new scalar and pseudoscalar states are lighter than the Higgs, and the Higgs is light enough to have been produced through Higgsstrahlung at LEP, although we will consider other scenarios. By also focusing on the relevant parameters, we hope to clearly elucidate the effects of mixing of the singlet with the Higgs, and the effect on the mass of the Higgs boson.
Since this paper has both basic phenomenological points, as well as points related to model building, we have attempted to lay out this paper so that one who is interested principally in the phenomenology can still learn the relevant points. Sections headed “Model Building” are independent, and the paper can largely be read without those.
The layout of the paper is as follows: in the next section, we will review existing limits on Higgs decays, both for SMlike and nonSMlike Higgs bosons. In section 4, we will discuss the effects of singlets on Higgs physics. First, we show how nonstandard decays into scalars can arise and dominate the Higgs decay width. We will also point out how nonstandard decays allow a larger mixing with the Higgs, this mixing can raise the mass of the Higgs boson considerably without resorting to radiative corrections. In its “Model Building” subsection, we present the relevant operators that induce the Higgs mixing and decays. In section 5, we will sketch out the spectra and nonstandard decay modes which are consistent with existing LEP bounds, and lead to a more natural parameter space of the theory. In its “Model Building” subsections, we discuss model realizations of these different spectra and decay scenarios. In doing so, we can assess more simply the degree of tuning required to satisfy the experimental constraints. In section 6, we summarize the presented benchmark points and their relevant phenomenology for future collider experiments. In section 7, we discuss additional directions worthy of further investigation and conclude. In particular, we suggest some new analyses on LEP data which might be useful when considering these scenarios. Finally, in appendix A some calculational results for scalar trilinears are summarized.
3 Summary of LEP limits on Higgs
Most particle physicists are familiar with the LEP2 95% CL limit on the SM Higgs of , but LEP has also produced a multitude of other limits on Higgs physics.
The limits that are most applicable to this work are the 95% CL limits on the so called parameters (often also referred to as ). We are analyzing a SMlike Higgs, which means that Higgsstrahlung production of the Higgs is close to the SM rate.^{3}^{3}3More generally, we will usually be taking the decoupling limit, where the lightest Higgs state is produced by Higgsstrahlung, but not in associated production with the CPodd . Thus, constraints from Higgsstrahlung are the main concern in these scenarios. Then depending on the assumed process of Higgs decay , the defined limits are on . We now list the different limits that LEP has analyzed, based on Higgsstrahlung and the given Higgs decay process (note: all stated mass limits assume ):

Model Independent Decays: This is the most conservative limit on the Higgs boson. It assumes that the Higgs is produced with a Z boson and looks for electrons and muons that reconstruct to a Z mass, while the Higgs decay process is unconstrained (by theory or the event analysis). The only study of this sort is done by OPAL giving a limit of , see Fig. 11 of [22]. Unfortunately, no other collaboration has released such an analysis.

Standard Model Higgs: LEPwide limits on the SM Higgs are given in Fig. 10 of [23], requiring . This study also includes the strongest limits on rates.

Invisible Decays: In this analysis, the Higgs is assumed to decay into stable (on collider length scales) neutral particles. The implication is that nonstandard Higgs decays have to primarily decay into visible particles. Both L3 and DELPHI have performed such an analysis [24, 25], but the most stringent constraints are from an older preliminary LEPwide analysis that has a limit , see Fig. 4 in [26].

2 Photon Decays (aka Fermiophobic): Fermiophobic typically means a Higgs with standard couplings to gauge bosons, but suppressed couplings to fermions, allowing decays into , as well as photons. If all decay modes are open, there is a limit of , while decays exclusively to two photons have a limit of . See Fig. 2 of [27], which is the LEPwide analysis.

2 parton hadronic states (aka Flavorindependent): In this type of analysis, any two jet decays of the Higgs are allowed. The analyses use the jets that are least sensitive to the candidate Higgs mass and details of the Z decay. Each LEP collaboration has done a study [28, 29, 30, 31]. However, the strongest constraint is the preliminary LEPwide analysis of , see Fig. 2 of [32].

Cascade Decays: These are the constraints that are most relevant for the present study, where cascade decays mean that the Higgs decays into two scalars and those scalars decay into (i.e. ). OPAL [33] and DELPHI [34] looked at b decays (), see their Fig. 12’s and a new LEPwide analysis [18, 19] has constrained both b and decays, with the exclusion plots given on page 8 in the first reference. For the limits are now 110 GeV for a Higgs produced with SM strength. For other intermediate scalar decays, , the best model independent exclusions are from OPAL’s analysis when the mass of the scalar is below threshold, which is given in Fig. 7 of [35] (note: the analysis is restricted for Higgs masses in the range ).
4 Singlets and the MSSM
As we have previously said, the simplest extension of the MSSM is to add a SM singlet superfield , containing a CPeven scalar , and a CPodd pseudoscalar .^{4}^{4}4More precisely, the CPeven properties of are not determined until its interactions and mixings are given. For example, it can pick up a CP transformation if it mixes with a scalar with fixed CP properties (the CPeven or CPodd ) or couples to fermions/gauge bosons in a certain CP fashion. On the other hand, in all cases mixes with and thus is always CPeven. The dominant phenomenological effects of this new field are:

New decays for the Higgs boson. If one or both of the new states are lighter than half the Higgs mass, decays and are possible, followed by or .

Light states can mix with the Higgs boson. If a light state mixes with the Higgs boson, it can push the mass of the Higgs boson up without large radiative corrections. This is at the cost of having a new light state which can be produced through brehmstrahlung off a Z, for which there are stringent constraints.
We first show that nonstandard decays can easily dominate over SM decays. In the case of cascade decays, for a Higgs produced with SM strength through Higgsstrahlung, the new process must have a width at least^{5}^{5}5This condition is not sufficient if the scalar decays into b quarks, as will be discussed later in section 5.1. . With a term in the Lagrangian
(3) 
the width to two scalars is [36]
(4) 
which one can compare with the dominant width of the Higgs to b quarks (in this mass range)
(5) 
Taking into account higher order effects, a more accurate approximation for the Higgs decay width to SM particles in the mass range of interest is (using HDECAY [37])
(6) 
For a 100 GeV Higgs, in order for scalar decays to sufficiently dominate, one must have
(7) 
The required size of the trilinear indicates that nonstandard cascade decays can easily dominate over standard Higgs decays.
One can also study the effects of mixing with a singlet on the Higgs mass in a model independent fashion. In the presence of mixing, the mass eigenstates and will be related to the interaction eigenstates through a mixing matrix
(8) 
Similarly for the mixing in the CPodd sector, the mass eigenstates and are related to the interaction eigenstates by
(9) 
If the MSSM mass of the Higgs is , then the mass eigenvalue after mixing with the lighter singlet is
(10) 
where is the mass of the mass eigenstate . This increase in mass is only through mixing and not through radiative corrections.
In the subsection below, we discuss some of the operators which can be used in building models with certain phenomenologies. Those readers who are only interested in the descriptions of the phenomenology can proceed to section 5.
Model Building: New Operators with Singlets
The introduction of a singlet superfield allows us to introduce a number of new operators in the theory.
mass for  
mass for  
singlino (Dirac) mass, mixes and  
allows decays, allows Z production  
(with mu term) mixes and , and masses,  
singlino (Dirac) mass, allows , decays  
Real: (with ) mass for decays, 3 coupling  
Imaginary: decays, 3 coupling  
mixes with , with , decays (due to mixing)  
with mixing, allows decays (esp. large )  
Mixes and , allows decays (due to mixing)  
Allows and decays to gluons and photons 
These operators have the effects of introducing into the theory masses, mixing, and decays of the states. The operators are listed in Table 1. We will describe here the effects of these operators when they appear singly. Clearly, the effects of the operators do not necessarily add linearly, particularly when vevs turn on, but it is useful to understand their effects individually. In many scenarios this is sufficient to understand the phenomenology and we will point out the situations where there are interference effects from multiple operators.
, – Scalar and Pseudoscalar Soft Masses
The operators arising from Kähler potential terms, and , determine the spectrum of the theory by adding mass terms to the Lagrangian
(11) 
Most previous analyses include equal soft masses for the scalar and pseudoscalar, although there is often no a priori symmetry reason to do so. All other scalar fields in the MSSM carry a gauged U(1), forcing a degeneracy which is not necessary for singlets. Because our interests are phenomenological, we allow different masses, and thus more interesting spectra. In particular, this allows spectra of the form , leading to events with many final state jets, bquarks and photons, produced through cascades.
The tuning of these theories with singlets, when , is usually encoded in how tuned these additional masses must be in order to achieve a proper spectrum, and not in the sensitivity of or the masses of the stops. When we refer to these operators, a value of or of refers to a contribution of to the scalar or pseudoscalar mass squared.
– Supersoft Operator
is the socalled “supersoft” operator [38] and arises from a SUSY breaking spurion in the superpotential. This operator is particularly interesting, and often neglected. If the hidden sector has a which acquires a Dterm expectation value, it has no effect in the MSSM, in that all soft operators can be generated by Fterm breaking. On the other hand, in the presence of a singlet, Dterm breaking becomes important through the introduction of this new operator[39]. The Lagrangian terms generated are
(12) 
where is the usual hypercharge Dterm, , and . Although it is supersymmetry breaking, it does not feed into the RG flow of any other soft operators, hence the term supersoft. The principal effects are to produce a mass for (but not ), and a coupling which is proportional to the new contribution to the mass. It also generates a Dirac mass for the Bino with equal to . When we refer to a value of of 20 GeV, this implies a contribution to the scalar mass squared of , and the corresponding trilinear term.
This operator directly affects the physics of electroweak symmetry breaking and the properties of the Higgs. However, it is simple to perform the same minimization of the Higgs potential in the presence of this term. Let us consider the usual MSSM neutral Higgs potential with this contribution.
where encodes other contributions to the mass, such as from .
When electroweak symmetry is broken, there is a linear term for , and so we shift by an amount
(14) 
The Higgs potential (neglecting which has already been shifted to its appropriate minimum) now reads
So we see the presence of the vev amounts to a redefinition of the values of and . Thus, we can perform the usual minimization and diagonalization of the MSSM Higgs fields. The new trilinear couplings induce a mixing between the and the other Higgses, which is encoded in the mass matrix (in the basis)
(16) 
Here, as we dial up , which is always positive, we increase the mixing. The values of and are the usual ones from the MSSM, in particular with bounded at tree level to be below .
Since generally, and , we can just focus on the mixing submatrix between and . The mass eigenstates are related to the interaction eigenstates as given in (8). We are principally interested in the trilinear terms, especially the term, which allows decays. There are two contributions to this term, one arising from the new supersoft operator, once we have mixed the CPeven states, and one from the usual Dterm trilinears, once we have mixed. The coefficients of these terms are given in the appendix.
is a commonly studied superpotential operator, as it induces an additional Higgs quartic at small which can raise the tree level Higgs mass [9, 8, 10]. After electroweak symmetry breaking, it generates masses for and of size . With these contributions the potential is given by
Note the explicit term for the Higgses (and the resulting trilinears), this is to be compared to the NMSSM where the vev of gives the term. However, absent any additional mass terms for , i.e. , we can simply shift , removing the entire term. In the presence of of additional soft masses for there is an svev of,
(18) 
The effect of this is to replace by an effective , which we denote , where
(19) 
Through the new quartic and trilinear terms there are mixings between and the MSSM Higgses, giving a mass matrix
(20) 
Notice, however, that in the absence of additional soft masses for () the mixing vanishes, and increases as we deviate from zero. The presence of  mixing allows decays. It is possible for decays to occur with equal amplitude, for instance at small mixing. However, it is also possible to arrange for to give only one dominate decay of . Finally, remember that the presence of naturally requires the presence of , at least at a loopsuppressed level.
Another possible operator to add is a soft trilinear for in the potential, . Alone this has no effect on the Higgs, although it does give equal but opposite contributions to and soft masses with an vev and also induces decays. However, if there is another source of mixing, can generate and decays as well as the original . The effective potential with this contribution is,
Since this operator does little of interest by itself it must be considered in tandem with some other operator that produces mixing and an svev. One combination of note is and . We can analyze the general case under the simplifying assumption, as above, that the mixing involves mainly and and not , so the mass and interaction bases are related as in Eq. (8).
The relevant trilinears are given in the appendix. As in the case of , at small mixing and have equal amplitudes. The size of the svev , generated by the mixing operator will be corrected by but if , as will often be the case, its effects can be ignored.
The potential with included is,
After EWSB this generates a vev for s,
(23) 
This gives a correction to the bterm for the MSSM Higgses, but still allows the usual diagonalization, leading to a mixing matrix of the form,
(24) 
Unlike the previously discussed operators, mixes the CP odd component of with . The mixing matrix, in the basis ( ), is
(25) 
The mass eigenstates are related to the interaction eigenstates as given in (29). This mixing of with means that can decay to 2 bquarks or 2 ’s if its mass is below about 10 GeV. With no additional terms in the Lagrangian, e.g. , this is the only decay path of . The relevant trilinears are given in the appendix.
– CP Mixing Mass
The operator is a CP mixing mass which mixes the wouldbe pseudoscalar with the wouldbe scalar . This operator can be important in allowing decays of the when and are zero and can be particularly important when decays would dominate the Higgs decay, but are kinematically forbidden. In this case, mixing due to can make the dominant decay mode.
We should point out that this operator does not, by itself, violate CP. As mentioned earlier, without any other interactions, there is nothing to prevent us from assigning CPeven transformation properties to both and . This remains true with the inclusion of or the supersoft operator in the potential. Only when couples to fermions or gauge bosons as a pseudoscalar, or mixes with the is a CPodd property forced upon it. The scenarios which we will study will not have such a mixing and hence no actual CP violation is introduced into the theory, making such a term safe from edm searches, for example.
One requires either or to induce Higgs decays (through direct couplings or mixings). In the case that is present, is already significant, so typically cannot compete. However, in the case, where has no couplings, such a scenario is viable. We thus consider the mass matrix (in the () basis)
(26) 
Because the mixing will allow the to be produced through strahlung from the Z, LEP limits become very severe. Ultimately, this implies that must be somewhat small, and for our purposes here, we will treat it as a perturbation. Ultimately, it will be necessary to calculate mixings precisely, but for estimates and intuition, it is easiest to study in the perturbative limit. We can perform a rotation in the sector by an angle , which may be large. This leaves the following matrix
(27) 
The mass matrix can then be diagonalized by a series of rotations
(28) 
where diagonalizes the mass matrix, and the are perturbative rotations where and , where are the and entries after the rotation.
The relevant mixing angle which enters into strahlung is
(29) 
– Fermiophobic Decay Operator
If the singlet superfield couples to new vector matter, there are loop induced decays of the scalars into gauge bosons. The presence of the new fields corrects the beta function of the SM gauge groups, with a superpotential coupling
(30) 
the gauge kinetic term becomes
(31) 
where is the gauge field strength, and . This results in a coupling between and two gauge fields, as well as and two gauge fields which explains why we refer to this as a fermiophobic decay operator. Because generally mixes with the Higgs, it is difficult for a loop suppressed decay to compete. However, , which mixes with the often heavy , and often through the loop suppressed operator , can have its dominant decay mode through this operator. Expanding out the expression above, we find a term in the Lagrangian
(32) 
This induces decays into photons or gluons, with the decay width
(33) 
where is the number of “colors” in the final state (i.e. 1 for photons and 8 for gluons).
With the convention above,
(34) 
where labels the gauge group and is it’s beta function where
(35) 
We can compare the decays into gluons and those into b quarks through mixing,
(36) 
which for the particular case of a , becomes
(37) 
For decays to glue to dominate over b’s, one requires small mixing angles in the CPodd sector, which can be achieved simply with loop suppressed , or a heavy . Alternatively, one can have lighter squarks, for example, with the values one requires to get comparable rates.
When induced decays for dominate, we can get an estimate on the branching ratios into gluons and photons in the case of coupling to a complete multiplet. For this, the branching ratios are . If the Higgs dominantly decays into 2, this gives branching ratios of . The channels with photons may be enhanced by including which only induces decays into photons, via Higgsino loops. As a final comment on this operator, we point out that if this is the only allowed decay, as increases, will first decay with a visible displaced vertex and eventually will decay outside of the detector. Both of these possibilities are highly constrained by LEP searches if the Higgs can be produced, ruling out such a Higgs that cascade decays into two such scalars.^{6}^{6}6Note added: It has recently been emphasized that similar Higgs decays with highly displaced vertices could well be visible at Tevatron and LHC, enhancing its detection prospects [40]. Therefore, we only consider values of this operator where decays promptly.
Other Operators
Note that we ignore here the effects of a term for the field, and a superpotential term. Since we are mostly concerned with the scalar sector of the theory, and aren’t concerning ourselves with the cosmological implications at this time, the mass of the singlino is irrelevant for our purposes. The only effect of is then to generate an term in the potential in a cross term with . This has the same phenomenological effect as , so including should not change the basic outcome of our study. Including in the superpotential can induce invisible decays of the Higgs to a singlino, but these are highly constrained and not interesting for our purposes. It can also induce a term in the potential, but this acts simply like in the presence of an vev, and like in the presence of an vev in allowing decays. Hence, we do not consider either operator here.
5 Scenarios
Although we have expanded the MSSM only by a single superfield, there is a remarkably large number of new scenarios of Higgs decays which arise, with large variations in the number of particles in the final state, as well as types of particles in the final state.
We can group the scenarios, in increasing order of complexity, into three categories:

The Higgs is mixed heavy. In the presence of large mixing, the Higgs mass can be increased significantly, in spite of small radiative corrections. In this scenario, the Higgs is at least 114 GeV in the case of SMlike decays, or 106 GeV in the case of cascade decays to 4 b quarks.

The Higgs decay is dominated by a single stage decay. Here or , where is composed of a pair of standard model fields. As above, if , one requires , however, if or , the Higgs can be considerably lighter.

The Higgs decay is dominated by a twostage cascade. That is or . Such two stage cascades generally do not occur if one restricts oneself to the scenario of the NMSSM. Here , resulting in 6 and 8 particle final states, which have not been constrained by LEP analyses other than the model independent ones.
Additionally, we can split the scenarios further into two cases: “largemixing” and “smallmixing”. We define these as follows: the small mixing case arises when the light CPeven singlet has a sufficiently small Higgs component that sstrahlung limits do not constrain it, even with conventional decays. In the largemixing case, the singlet has sufficient Higgs component that nonstandard decays are needed to evade limits.
With large mixing case, there are three real scalars that we are concerned with: , , and . These are defined to be the field which couples to the Z in Higgsstrahlung, the field which mixes with the Higgs, and the field which does not mix with the Higgs, respectively. In many cases can be thought of as the scalar and as the pseudoscalar. The mass eigenstates are , and , which are the fields mostly made of the untilded field of the same character, and usually the heaviest, intermediate, and lightest mass eigenstates, respectively. As a note of caution, we will often refer to the still as the Higgs, CPeven singlet and CPodd singlet and have tried to ensure that the meaning is clear given the context.
First, we will discuss the necessary operators in all scenarios and then we will proceed to study the three basic scenarios, including the existing limits and model building possibilities.
Model Building: Necessary Operators in All Scenarios
The basic requirements on any viable scenario are threefold, and can be satisfied with various combinations of operators. The requirements [operators satisfying the requirements] are:

must decay [, , ]: This is not absolutely true, for instance if is never produced, but almost always is.

must decay into 2 or 2 or [, , (+ mixing), ]: For the mass range we are interested in, these nonstandard decays for must exist.

must decay to 2 [ (+ mixing), , (+ mixing), ]: When is light (i.e. ) and in “largemixing”, constraints from SM decays are usually quite stringent. Conceivably, could decay directly primarily to ’s if the mixing is small enough, but in general, the cascade decays for are necessary to evade LEP limits, unless it is heavier than .
5.1 Higgs Decays Through a Single Stage Cascade Into b Quarks
As we have already described, the only means in the MSSM to push up the mass of the Higgs above LEP limits is to introduce very heavy top squarks, which then introduce tunings into the theory. However, as described in section 4, if the Higgs mixes with a lighter field , the mass of the field which is produced strongly through strahlung from a Z can have its mass pushed up, simply by mass mixing, without introducing unnaturally heavy top squarks. However, this mixing allows the lighter field to be produced weakly through strahlung, trading off the heavier for stronger limits on the lighter .
Now that contains a Higgs component, we must consider LEP limits on it. With standard model decays for , the limits are typically . The large mixing necessary to achieve changes in the mass requires that decays dominantly through cascades, in this case, one typically requires if the cascades end in b quarks for lighter and for heavier . These, in general, also imply that the dominant decays of the Higgs are nonstandard.
If these decays are or , then the Higgs can be considerably lighter than 115 GeV, and it is essentially unnecessary to push the Higgs mass up. These scenarios will be discussed in section 5.2.
We show the allowed regions (dark and light shaded) with decays to four b jets in figures 1a and 1b. In these plots, we assume a value for (where kinematics no longer strongly favor decays to ’s) where constraints are strongest. We show the regions which are allowed when accounting for constraints on decays (light and dark shaded regions), assuming a typical . For comparison, we also plot the region where , where is essentially unconstrained (dark shaded regions).
These plots have assumed a 100% branching ratio for , so the only applicable limits are the exclusions given in the LEPwide analysis [18, 19]. However, it is an important question what the LEP constraints are on a scenario with both and rates. In a recent paper on these decays in the NMSSM [41], the authors only required that the rate be consistent with the LEP exclusions, thus they assumed that these analyses are independent of each other. However, upon closer inspection of the details of the analyses, it appears such an approach may be too generous.
In the higher mass () range, the combined LEP analyses are dominated by OPAL [33] and DELPHI [34]. A close reading of these papers suggests that the and decays of the Higgs are generally reconstructed together. For instance, in studying , in the analysis by OPAL, the neural nets trained to capture to the decays of the Higgs are also reasonably efficient in capturing the decays. In the all jet () analysis, OPAL uses the same analysis procedure for both and , forcing the six jet event into a four jet topology via the DURHAM algorithm, hence this analysis efficiently reconstructs both types of decays. As for DELPHI, they clearly state that the analyses are not independent of each other [42], and both decays are reconstructed via the same analysis procedure.
Beyond efficiencies, there are still differences between and events. For example, the distribution of discriminating variables will be broader in decays (e.g. the reconstructed Higgs mass), making it difficult to know how to constrain the scenario when there are significant levels of both and decays. If and signals were indistinguishable, the correct limit would be is the experimental bound of the individual analysis. As an attempt to combine these limits, when there are rates for and decays, in addition to applying the individual limits, we will also require that , where
(38) 
This additional requirement, on the effective , accounts for the redundancy of the analyses, and which should forbid situations where the naive combination of analyses is significantly excluded. Thus, we find it to be a good compromise between the assumptions of complete independence/interdependence ( respectively) of the separate analyses. Note that one can also interpret this as only requiring a 99.5% CL limit, if the two decays give indistinguishable events. More rigorously, a combined analysis should be done to find the proper constraints.
For our purposes here, where we take the branching ratio to be one, the results are relatively simple. Even with relatively light stops (270 GeV), the small suppressions of the Higgs couplings due to mixing are sufficient to allow such a Higgs to have been undetected. However, at this mass, this is achieved by strongly mixing and pushing the physical Higgs above 110 GeV(out of the LEP constraints). At , a region opens up below 110 GeV, but the overall allowed space is still quite narrow. Moving to higher Higgs mass (roughly 325 GeV stops) the model independent parameter space opens up significantly (about twice as large as at 300 GeV). However, even with 300 GeV stops, the tuning of is already expected to be O(15%), which begins to reintroduce fine tuning from another direction.
Note that in all of these cases we are considering to be lighter than . The reason for this is simple: although mixing the Higgs with a heavier singlet can suppress couplings of the , if it is heavier than the Higgs, the effect is to push the mass of down, aggravating naturalness issues. Thus, we should view a light as a natural consequence of this model with . Such a scenario is somewhat distinct from previously discussed scenarios [43, 41] because of the presence of the light which can be easily produced at rates comparable to that of the SM (in general, roughly a factor of 10 smaller).
Let us further note that we have been discussing the model independent tuning. Whether or not one can achieve these mixings in a given model without, e.g., tuning to prevent a tachyonic is a separate, often more stringent constraint.
As a final comment, we note that these limits are based on the best available limits of , which are still preliminary. Final limits may further constrain this scenario.
Model Building: Single Stage Cascades into b Quarks
It is straightforward to construct models in which the Higgs is mixed strongly with a lighter scalar. Because of this mixing, if the Higgs decays , then generally decays to as well. Here we will discuss the model building and tunings associated with large mixings, and situations where . Situations with , where Higgs mass may be below 110 GeV, will be deferred to subsequent discussions.
There are three operators which can induce significant mixing with the Higgs: , , and . principally mixes with rather than because is small, so we focus on the other two.
is unique in that while it induces mixing, it also adds a diagonal mass for , so that a tachyon never appears. As a consequence, with this operator, it is simple to get large mixing without having to tune masses to a high degree. In figures 2 we see that one can easily achieve large Higgs masses and large mixings over broad ranges of the parameter space.
is somewhat more challenging, because we require a sufficiently large term from experiment. From searches for the chargino, we typically require [44]. In figure 3 we consider this scenario for . Here we see that it is very challenging to have sufficiently large while keeping the light state dominantly (i.e., ). One can tune this scenario to achieve this, but generally, it is most natural to have and the decoupled, and (as is required with no singlet component). Naturalness here is not considerably improved from the MSSM, unfortunately.
Since is in general more tuned, we list a benchmark with below (masses in GeV and in the decoupling limit which we define as and large )