Contents

July 13, 2019

UMD-PP-012-021

Natural Islands for a 125 GeV Higgs in the scale-invariant NMSSM

Kaustubh Agashe, Yanou Cui, Roberto Franceschini

Maryland Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742, U.S.A.

Abstract

We study whether a 125 GeV standard model-like Higgs boson can be accommodated within the scale-invariant NMSSM in a way that is natural in all respects, i.e., not only is the stop mass and hence its loop contribution to Higgs mass of natural size, but we do not allow significant tuning of NMSSM parameters as well. We pursue as much as possible an analytic approach which gives clear insights on various ways to accommodate such a Higgs mass, while conducting complementary numerical analyses. We consider both scenarios with singlet-like state being heavier and lighter than SM-like Higgs. With -terms being small, we find for the NMSSM to be perturbative up to GUT scale, it is not possible to get 125 GeV Higgs mass, which is true even if we tune parameters of NMSSM. If we allow some of the couplings to become non-perturbative below the GUT scale, then the non-tuned option implies that the singlet self-coupling, , is larger than the singlet-Higgs coupling, , which itself is order 1. This leads to a Landau pole for these couplings close to the weak scale, in particular below TeV. In both the perturbative and non-perturbative NMSSM, allowing large gives “more room” to accommodate a 125 GeV Higgs, but a tuning of these -terms may be needed. In our analysis we also conduct a careful study of the constraints on the parameter space from requiring global stability of the desired vacuum fitting a 125 GeV Higgs, which is complementary to existing literature. In particular, as the singlet-higgs coupling increases, vacuum stability becomes more serious of an issue.

1 Introduction

Very recently, the ATLAS and CMS collaborations at the large hadron collider (LHC) have discovered a resonance with a mass of around 125 GeV which is consistent with a standard model (SM) Higgs boson [1]. Here, we assume it is a SM-like Higgs.

In the SM, the quartic coupling for the Higgs and hence its mass is a free parameter so that the above observation can be easily “accommodated”. However, the SM is plagued by the Planck-weak hierarchy problem for which weak-scale supersymmetry (SUSY) is perhaps the most studied solution. In turn, in the minimal supersymmetric SM (MSSM), the Higgs tree-level quartic coupling is given by the electroweak (EW) gauge coupling so that there is actually a prediction for the Higgs mass. The flip side is that the Higgs mass then has a tree-level upper bound of , which is well below the above observation of 125 GeV. It is well-known that the contribution of the superpartner of top quark (stop) at the loop-level can raise the Higgs mass in the MSSM, but reaching 125 GeV this way requires stops to be as heavy as TeV or tuned, large  [2, 3, 4, 5]. The heaviness of the stop results in reintroduction of fine-tuning of the EW scale. This issue is embodied in the the relation

(1)

that is one of the minimization constraints of the MSSM potential. Here the soft mass is largely determined by the stop mass via RGE, resulting in [6]. Given the above stop mass, such large value of needs to be canceled by other contributions at the right-hand side of eq. (1), in particular, either against the supersymmetric mass for Higgs doublets (-term) or the other Higgs soft mass () [7, 8, 9]. At any rate the need for a cancellation of the large signals that a new “little hierarchy” needs to be explained in the MSSM. This little hierarchy problem in MSSM can be quantified with a fine-tuning measure [10]:

(2)

with , i.e., less than tuning, conventionally taken as typical for a natural theory.

The MSSM also has another drawback, partly connected with the issue of having a cancellation in eq. (1) involving . The issue is about why is the -term close to the weak scale as required phenomenologically: there is lower bound on it of GeV from chargino mass [11], whereas much larger values would require tuning in eq. (1). An attractive solution to this -problem is the scale-invariant NMSSM (for a review see [12]). Here, an explicit -term is forbidden in the superpotential, and instead is generated by the vacuum expectation value (VEV) of an added SM gauge singlet, , which is coupled to Higgs doublets, and , via the superpotential term

(3)

In turn, this VEV for the singlet is driven by soft SUSY-breaking mass terms for singlet. As there are no explicit mass scales in the superpotential, this model can be referred to as the scale-invariant NMSSM. This form of superpotential can be ensured by a -symmetry, which can be extended to the soft SUSY breaking terms as well. Thus it is often also referred to as the invariant NMSSM 111The original NMSSM typically suffers from the difficulty of simultaneously solving domain wall problem and tadpole problem which destabilizes electroweak scale [13]. A simple way to resolve this issue is to assume a low-scale inflation such that the domain walls are significantly diluted [12]. Alternatively, a suitable R-symmetry can be imposed to constrain the form of explicit -breaking terms which may reconcile the tension of solving domain wall and tadpole problems [14, 15]..

As a bonus of the NMSSM, we get an extra tree-level quartic coupling for the Higgs doublets from the same singlet-Higgs coupling which solves the -problem. This effect in raising the Higgs mass relative to its MSSM prediction is well known and has been investigated at length in the literature (see for instance [16, 17, 18]). Here we re-visit this issue in the light of the discovery of a SM-like Higgs boson at 125 GeV.

Variations of the NMSSM may allow dimensionful terms in the superpotential. In this work, we focus on the version of the NMSSM with no mass scales in superpotential so as to keep its other merit of solving the -problem. We assume that the stop contribution to Higgs mass is small, as follows from natural . This means that we aim at using tree-level contributions to get close to 125 GeV, about 110 GeV or more, and use the stop contribution to gain just the “last” 15 GeV or less in mass 222The estimate of 110 GeV lower limit will be detailed in the following and roughly corresponds to small mixing and the contribution of a stop with mass of GeV which is a general upper bound from naturalness according to eq. (2) . .


Our goal is to see if it involves any tuning of parameters of the NMSSM to reach the required tree-level value for Higgs mass. Such a suspicion is rather motivated. For example, it is well known that in a large region of the NMSSM parameter space, the lightest CP even state, , is identified as the SM-like Higgs. The mass of has an upper bound at tree-level

(4)

If we prefer to preserve the conventional unification of the gauge couplings, which is a major success of MSSM, is constrained to remain perturbative up to the GUT scale. This means that the value of at the weak scale which enters the above bound cannot be much larger than [19]. In turn, even for such a maximal value of , the tree-level upper bound for the Higgs mass is very close to 110 GeV. Given that Higgs mass of GeV or more at tree-level is our goal, it is clear that we must actually saturate the upper bound in this case. However, the upper bound is not a generic value of the Higgs mass in the NMSSM, i.e., we expect that the model parameters must be arranged in a specific manner to fit the data in this case. Here we shall investigate in detail what are the conditions that define the locus of maximal Higgs mass in this region of the parameter space of the NMSSM.

A key quantity to identify the regions that lead to the largest Higgs boson mass (when it is the lightest CP-even state) is the mixing of the doublet-like states with the singlet-like states. Indeed it has been known that the extra quartic coupling of the NMSSM for the SM-like Higgs is not the “end of the story” since mixing with the singlet modifies the mass of the SM-like state [16] and in general pushes it away from the value of the right-hand side of eq. (4). When the singlet-like scalar is heavier, as discussed so far, its mixing with the doublet-like state results in a “pull-down” of the mass of the latter. The crucial point is that, even though the singlet-like scalar is heavier than the SM-like Higgs, it turns out that the shift of the mass of the SM Higgs-like state due to mixing with singlet state can still be significant, since this effect does not quite decouple even if singlet VEV is large 333Although we might actually require in order to have a natural -term, . . As mentioned already, if the singlet-Higgs coupling is to remain perturbative up to GUT scale, then it turns out that the maximal tree-level SM-like Higgs mass – before negative mixing effect – can only be barely around or lower than our target of GeV. Thus one has to work hard to minimize the negative mixing effect on the SM-like Higgs mass in order to obtain 125 GeV mass at loop-level.

There is actually a different possibility, namely, that the singlet-like scalar is actually lighter than the SM-like Higgs. In this case, their mixing provides a “push-up” effect, which raises the SM-like Higgs mass. Thus, the tree-level mass, before the mixing is taken into account, can be below GeV, lessening the tension with perturbativity mentioned above for the other case. However, we face a different potential worry: the state lighter than 125 GeV necessarily has a component of Higgs doublet so that it couples to the boson and a bound from LEP2 applies [20]. Thus, in this case, one has to work hard to make sure that the singlet-like scalar is not too light or too strongly coupled to the boson.

As we proceed with our above analysis, we distinguish two cases: the “perturbative” NMSSM that we mentioned above, which requires all the couplings do not hit a Landau pole till GUT scale, and the non-perturbative NMSSM, where such singularity of the couplings can appear below the GUT scale. The motivation is clear: the former case can manifestly maintain the successful gauge coupling unification of the MSSM while in the latter case unification can be achieved only in specific UV completions [21, 22, 23]. Another useful “axis” of our exploration of the parameter space is the size of soft trilinear SUSY breaking terms for Higgs doublets and singlet, and . We consider both cases where they are small and cases where they are large. This division clearly is relevant for some SUSY breaking mediation schemes that predict the -terms to be small, for example, minimal gauge mediation. Finally, as far as possible, we try to come up with analytical insights into these various issues.

The issue of the Higgs mass in the NMSSM has been studied recently, also in connection with the experimental finding of a 125 GeV SM-like Higgs boson. However our study is significantly different in a number of aspects with respect to the current literature. Some of them (see, for example, [3] and [24]) focus on more general singlet extensions of the MSSM, including the addition of explicit mass terms for the singlet and Higgs doublets. Other studies such as [25, 26, 27, 28, 29, 30, 31] are mostly numerical and they lack a clear division into the above cases that we highlight.

Our results can be summarized as follows, focusing on the case with negligible -terms. For the perturbative case, we demonstrate that even if we allow tuning of parameters of NMSSM it is not possible to reach 125 GeV mass: for the sub-case here of pull-down, this limitation is due to a combination of the facts that the singlet-Higgs coupling is small to begin with and that the reduction in Higgs mass due to the mixing cannot be made small. On the other hand, for push-up, the failure to reach 125 GeV Higgs mass is due to the light singlet-like state typically being ruled out by LEP2. In the non-perturbative NMSSM, the singlet-Higgs coupling and singlet self-coupling can be larger. Thus in this case the maximum tree level value itself can be well above 110 GeV. Typically, one then uses the singlet-Higgs mixing to get down from the above bound to 110 GeV at tree-level. In this way we completely avoid the tuning associated with requiring the pull-down mixing effect to be near vanishing that plagued the perturbative case above. Looking closer at the non-perturbative case it turns out that, if one wishes to avoid any kind of tuning of NMSSM parameters, then we end up in a specific region, namely, singlet-Higgs coupling, , is of order one and singlet self-coupling, , is larger.

One of the constraints in addition to that we take into account, both analytically and numerically, is the stability of the vacuum. Although this is a complicated issue to deal with, it is essential for the phenomenological viability of the model: the parameter choice that accommodates the right Higgs mass should correspond to a cosmologically stable vacuum. This issue, however, is either overlooked or only investigated in a partial way in most existing literature trying to explain within NMSSM.

Here is the outline of the rest of the paper. We begin with reviewing the model and giving relevant formulae for our analysis. This is followed by a general discussion of how to get to Higgs mass of 125 GeV, including relevant issues such as perturbativity and vacuum stability. After this setting of the stage, we discuss in detail the viability of the model in the push-up scenario first and then the pull-down case, before concluding.

2 The Model

The scale-invariant NMSSM is a very well-known extension of the MSSM which was originally proposed to overcome the problem and can also serve to lift the tree-level Higgs mass. Nonetheless, in this section, for the sake of completeness and to fix our notation, we give some details of the model relevant for our discussion. For a complete overview on the NMSSM we refer to [12].

The superpotential of the model is

(5)

where is a singlet chiral superfield and and are the MSSM Higgs doublets (unhatted capital letters will indicate their complex scalar components). The soft-SUSY breaking Lagrangian is

(6)

Hence the complete scalar potential in this sector is:

(7)

where the last line is the MSSM -term, and denote the and gauge couplings, and are the -terms.

We expand the neutral scalar fields around the vacuum expectation values (VEVs) as follows (see, for example, [12] ) :

(8)
(9)
(10)

where GeV, and is the neutral would-be Nambu-Goldstone boson.

The extremization conditions of the scalar potential are

(11)

where and . We can use the above three equations in order to replace the parameters , and by the three VEVs.

The scalar Higgs states can be divided into a CP-even and a CP-odd sector. In fact, with the tree-level potential we consider there is neither explicit nor spontaneous CP violation in the Higgs sector [32]. The CP-even mass squared matrix in the above basis of , , is

(12)

where , and .

The basis used in eq. (12) has the advantage that it shows clearly what state couples linearly to the SM gauge bosons, such that it can be produced at LEP2 in association with a boson. In fact is rotated in the same way as , so that it is exactly the linear combination of and which is responsible for the masses of the and gauge bosons. Thus only has tree-level couplings to , while and do not. This implies that only the component of in each CP-even mass eigenstate is relevant for the LEP2 limit.

Similarly, to the extent that ATLAS and CMS collaborations at the LHC have observed a SM-like Higgs at 125 GeV, in particular with SM-like coupling to , this state (denoted by ) should be predominantly , with small admixture of , .

The CP-odd scalar mass squared matrix in the above basis of and is given by

(13)

The Lagrangian is such that we get a light CP-odd scalar in the two approximate symmetry limits: the symmetric limit when the -terms are small; and the Peccei-Quinn symmetric limit when and are small. The light CP-odd scalars then correspond to pseudo-Nambu-Goldstone bosons from spontaneous breaking of these symmetries. If their mass is below about GeV, there can be a constraint coming from -decay into these CP-odd scalars. In general, the CP-odd states can also be produced in association with the CP-even scalars at LEP2.

Finally, there are physical charged Higgs bosons whose mass is given by

(14)

Although it is not directly relevant for our analysis, we refer the reader to [12] for the mass matrix and the properties of the neutralinos and charginos.

Without loss of generality we take while and can have either signs. Meanwhile, in the examples shown in the following sections we take and , as we do not expect the other choices of signs to result in any qualitative change of our findings.

3 Getting SM-like Higgs of mass GeV

As mentioned above, the 125 GeV SM-like Higgs discovered at the LHC is likely to be predominantly , with a small admixture of , . Neglecting this mixing, its mass squared is given by in eq. (12): the piece is the same as in the MSSM and the piece results from the extra quartic of the NMSSM. However, as mentioned in the introduction, the mixing of with and modifies the mass from the above value, lowering (raising) if mixes with heavier (lighter) states. In fact the SM-like Higgs mass can be written as:

(15)

We further discuss below, in turn, each term in the above formula. The first line is the MSSM value, including the stop mixing effect for which we use the approximate formula from eqs. (69-71) of [6]:

(16)

where the “stop-mixing” parameter is given by

(17)

is the top quark mass, being the pole mass, is the QCD coupling and is the geometric average of the two stop mass eigenvalues.

We do not assume ad-hoc large mixing in the stop sector which tends to increase the Higgs mass. We also assume 500 GeV as a rough upper limit of stop mass as it is favored by naturalness of weak scale: in particular, this corresponds to a fine-tuning of if we assume the mediation scale of SUSY breaking of TeV, large and small stop mixing (eq. (6) of [33]444As emphasized in reference [3], for , even TeV stop mass can be natural. However, for such large values of , the tree-level bound is already well above 125 GeV and so a large loop contribution to Higgs mass from a heavy stop is not really needed anyway.. So we take typical values GeV, , GeV and in eq. (16) and find the stop loop contribution to can be up to  GeV. In principle in the NMSSM there are further loop corrections involving the new couplings and [34]. For couplings around the bound imposed by perturbativity up to the GUT scale and stop masses saturating our naturalness upper bound these corrections are in general subdominant to those stemming form the stop sector and we neglect them in the following. This may not be justified when one allows and to become non-perturbative. However in this case the Higgs mass can be larger at tree-level, hence loop corrections are in general less interesting to begin with. Given the loop corrections attainable from the stop sector, our goal here is to achieve a tree-level mass for the SM-like Higgs of GeV or more, up to  GeV. Of course if one would allow a heavier, hence less natural stop our analysis would be overly restrictive.

Moving onto the second line in eq. (15), the first term here is the contribution characteristic of the NMSSM coupling . Notice that for a moderate or large , in the NMSSM, is maximized at in contrast to the case in the MSSM, where is maximized at . Thus, we choose small in our analysis. As it will play an important role in what follows, we have introduced the notation in the second line here to encode the effect from the mixing of with and .

For the discussion here, and for our analytic study in Sections 4 and 5, we shall neglect the mixing and only deal with the mixing part of , reducing the 3-by-3 mixing problem into a simpler two states mixing problem. The justifications of this approximation are as follows. Typically we have to choose parameters so that the and mixings are small effects. The reason is that must be somewhat heavy, because is accompanied by a charged Higgs of similar mass which is constrained by  [35]. As gets heavier its effect on decouples since the mass mixing term does not scale with mass, as shown in eq. (12). Furthermore in the NMSSM small is well motivated and the mixing actually vanishes at . In contrast, the effect of the mixing on SM-like Higgs mass cannot be decoupled. The main reason is that both the mass term and mass scale with the VEV for .

Similar to the mixing we neglect the mixing since it affects the mass of the doublet like state only at higher order and it also vanishes at . We stress that we neglect and mixing only to give analytical arguments, of course, the full mixing is included in our numerical analysis.

As anticipated in the introduction, there are two possibilities for the mass of the SM-like Higgs boson as far as the effect of the mixing effect in eq. (15) is concerned:

  • The “pull-down” case where the singlet-like is heavier than SM-like Higgs, such that the mixing reduces SM-like Higgs mass.
    In this case, (i) and (ii) such that after the pull-down mixing effect, we end up with a SM-like Higgs tree-level mass of GeV. Obviously, eq. (12) then suggests that .

  • The “push-up” case where the singlet is lighter that the SM-like state, so that the mixing increases the SM-like Higgs mass.
    By definition this implies that (i) and (ii) since, after the effect of the mixing is included, we want to get to GeV at tree-level. Therefore, based on eq. (12), we must have .


Elaborating on the above discussion we are in position to outline the following argument. If the model has to be perturbative up to the GUT scale, then in the pull-down scenario, the region of the NMSSM parameter space that can give a SM-like Higgs mass as large as 125 GeV is not at all a generic one. For the push-up case we shall indicate how we get a similar conclusion, though for a different reason.

If the coupling must remain perturbative up to the GUT scale, it cannot exceed a generously estimated upper-bound at about 0.7 555In fact is typically constrained to be below or around 0.6, as in general the need for a non-vanishing reduces the maximal value of compatible with perturbativity up to the GUT scale.. Even plugging in such large value of in eq. (15) one finds that, in the pull-down scenario, effect of the mixing must be minimal to keep the physical Higgs mass above 110 GeV at the tree-level. This fact was already mentioned in the introduction, but now we demonstrate it in Figure 1 and it suggests that the parameters of the model must be arranged in a specific pattern such to have

(18)

This conspiracy of parameters seems rather unnatural, or can be taken as a suggestion for a peculiar relation that must come from a UV construction, and therefore we introduce a measure of “tuning” to quantify to what extent we are requiring cancellations among apparently unrelated parameters of the model.

Figure 1: The green solid line is the maximal mass of the SM-like Higgs boson for from eq. (15) in the “pull-down” scenario. The two horizontal blue dashed lines are the reference values 125 GeV, that is about the observed mass of the SM-like boson, and 110 GeV, that is the minimal value of the tree-level mass to attain 125 GeV after the inclusion of radiative corrections from the stop, taking  GeV. The light (dark) purple shaded regions mark the values of that, according to the one-loop RGE, get larger than , our definition for non-perturbative, below the GUT scale for () at the weak scale. To allow larger values of to be compatible with perturbative unification we have introduced extra matter in 4 copies of of SU(5) at 5 TeV.

In a fashion similar to what is done for the tuning of the EW scale (or boson mass) as given in eq. (2), we introduce the tuning measure for the Higgs mass and expand the expression using eq. (15) and (12):

(19)

where are the various parameters of the NMSSM evaluated at the weak scale, where we compute the Higgs mass. This measure of tuning tries to capture how tightly related must be the parameters of the NMSSM to attain 125 GeV for the mass of the SM-like Higgs state. A large value of corresponds to a fine tuning among the values of two or more parameters.

In the following sections we will discuss both analytically and numerically what relations among the NMSSM parameters are implied by the recent LHC observation of a SM-like Higgs with mass 125 GeV. Furthermore we will quantify using eq. (19) how natural are the identified regions of the parameter space. Altogether we will give the coordinates of the islands in the natural region of the NMSSM parameter space spotted by the LHC Higgs result.

For the push-up scenario the discussion follows a rather different path, but comes to a similar conclusion. The push-up scenario has a singlet-like state at the bottom of the spectrum. This state can be within the reach of LEP2 and to keep it unobservable in the dedicated searches of LEP2 experiments it must be significantly decoupled from the boson, i.e., it must be mostly a pure singlet. In conflict with this is the fact that the lift of the Higgs mass, , goes to zero as the mixing angle vanishes. Consequently in the push-up scenario the need for an almost pure singlet-like state at the bottom of the spectrum and the need to lift the Higgs mass are in tension. This picks out special regions of the parameter space where both the LHC and LEP2 constraints can be accommodated. In a spirit similar to that of the case of pull-down we can quantify how “special” is the choice of the NMSSM parameters to attain such decoupled singlet in association with the needed lifting of the tree-level mass of the SM-like state. Since the issue is also about obtaining a small , we can use the same measure of tuning introduced for the pull-down case as given in eq. (19).


In the following sections we will organize our analytical and numerical discussion on how to attain the mass of the SM-like Higgs at 125 GeV according to the push-up versus pull-down classification discussed above, paying particular attention to the size of . Before getting into this detailed analysis, we would like to comment on two other issues of the NMSSM, which become especially relevant for such a large mass of the SM-like Higgs. Namely, the stability of the desired vacuum and the presence of Landau poles for the couplings, i.e., perturbativity of the model up to high scales.

3.1 Perturbativity

As we noted when commenting about eq. (15), must be “small” in order to make use of the extra quartic to increase the SM-like Higgs mass beyond its MSSM value. In fact for moderate or large as used in this case, the tree-level upper bound on Higgs mass is itself maximized at . However, such low implies that the top Yukawa coupling is larger than already at the weak scale. In addition to the danger that top Yukawa hits Landau pole below GUT scale, such a choice, in turn, makes blow up faster 666On the other hand negative stop 2-loop contribution is a bit smaller in size due to a cancellation between the now larger top Yukawa coupling and QCD coupling, see eq. (16).. Therefore we shall choose as a “representative” value for our analysis.

Also, in order to ameliorate the potential problem with Landau poles below GUT scales, we allow the possibility for extra matter at low scales. Such matter should appear in complete representations of SU(5) so as to preserve gauge coupling unification, and tends to give larger MSSM gauge couplings in the UV. The gauge couplings, in turn, prevent top Yukawa and singlet-Higgs couplings from blowing up too quickly.

3.2 Constraints from unrealistic minima

A set of important constraints on the values of the NMSSM parameters is that the ones fitting should correspond to a viable realistic vacuum. Basic checks include requiring it to be a minimum with all scalar masses squared being positive. On top of this the desired vacuum there should be either a global minimum or a meta-stable minimum at cosmological time scale. Because of more parameters in NMSSM scalar sector, the check of vacuum stability is more sophisticated than in MSSM, in general with no neat analytic condition. Existing works often overlook the issue of the global stability that, in particular, may become a crucial one when is large [36, 37]. In this work we take this issue seriously and conduct a thorough analysis on how vacuum stability may constrain the parameter space that fits . In this section we first give a general analytic formulation for such an analysis. Then, we show a complete analytic study for a particular yet practically interesting limit where , which, in turn, leads to . Recall that this is the condition which was identified as the key to attaining the desired Higgs mass while preserve perturbativity up to GUT scale: see the discussion after eq. (15).

Guided by this analytic understanding, in the numerical results contained in the following Sections 4 and 5 we will impose vacuum stability constraints on the parameter space that has an otherwise acceptable vacuum, i.e., which gives  GeV and, is compatible with the LEP bound when it is relevant. In the numerical studies the assessment of the global or local nature of the desired minimum is carried on by a global and unconstrained numerical search of possible extra minima of the potential. This must be contrasted with the checks performed by standard tools [38] that evaluate analytical expressions for a (in general incomplete) set of the extremal points of the one-loop effective potential. Although limited by the precision of the numerical scan and by the fact that we use the tree-level potential, our method is in a sense more general as it can capture extra minima in any direction in field space. As we will see later, the requirement of absolute vacuum stability may rule out significant portions of the parameter space that are allowed by all other constraints. Our result gives a warning regarding the viability of some regions of parameters space and motivates a more detailed analysis at loop-level, that we leave for future work.

We start from stationary conditions of the NMSSM potential which determine the VEVs at extremum points, as given in eqs. (11). In general these 3 equations are coupled and the solution of each equation depends on the solution for the others. The equations are cubic in , and therefore we count the following 6 extremum solutions:

  • , ;

  • , ;

  • , ,

where the “r” label denotes the desired quantities at the realistic vacuum with , , and corresponding to natural . On the other hand, the “u” label denotes the nonzero quantities in extra unrealistic vacua. With there is no runaway direction in NMSSM potential. Therefore in order to decide whether the desired vacuum is a global minimum, it is sufficient to check if is deeper than all other 5 extremum points. As already mentioned, in order to keep our analytic study at a manageable level, we will work with the tree-level potential. We are aware that, due to the importance of loop effects, this may not be accurate enough in some situations. In our numerical results presented later we shall consider the comparison of minima as “unresolved” when the value of the potentials at the two points differ by less than 5% 777This threshold of 5% is our rough estimate of size of EW loop effects, which are expected to play a significant role in breaking the tree-level degeneracy of the two vacua. Some numerical investigation of the impact of loop-corrections computed using NMSSMTools [38] confirm that our estimate is in the right ballpark. In fact the smallness of loop correction is expected once one chooses a natural stop mass..

Applying stationary conditions on one obtains two practically interesting variables for each of the above solutions as functions of input parameters and the corresponding :

(20)
(21)

Here in order to clarify things, we use the label “eff” as a general notation for output parameters from stationary conditions. Input parameters are those intrinsic to a model: , , , , , , . Of course one set of needs to be the values at the desired, realistic vacuum, which as mentioned we will label with the label “r”.

Fixing we can retrieve the mass squared model parameters from the quantities at realistic vacuum

Then one can (in principle) solve for other 5 extremum solutions and their corresponding . Finally we can plug all the relevant quantities into the potential and compare the depths at different extrema. To achieve this goal we rewrite the potential as:

(24)

where

(25)

Here a few comments are in order. Based on the first two terms in eq. (24), among solutions with the same , the one with nonzero always has deeper potential. Therefore, among the 5 other extremum solutions we mentioned before, it is actually sufficient to just check if or has a deeper than desired vacuum. Therefore later in this section we focus on these two unrealistic minima.

The solution of is a special case, which is easily obtained from eqs. (21) and (20):

(26)

where we make the conventional ansatz . This choice is motivated by radiative symmetry breaking, however choosing does not change the result.

On the other hand, the solution is hard to solve because this requires fully solving the 3 coupled cubic equations. However, there is a special limit where doublets can decouple from the stationary point equation of , which greatly simplifies the process of obtaining the solution. To see this we recast the stationary point equation of , the last one of eq. (11), in the form

(27)

Apparently, doublet VEVs can be eliminated from eq. (27) when or the expression in the parenthesis vanishes. Remarkably this requirement coincides with which in turn implies the small singlet-doublet mixing eq. (18). As we discussed before this is necessary to attain a 125 GeV mass for the SM-like Higgs while being compatible with perturbative unification.

In our following discussions we will first consider the competition between realistic vacuum and the generic solution given in eq. (3.2), in particular, the dependence on model parameters. Then we conduct a complete analysis for the simple special case mentioned above, i.e., when doublets decouple from stationary equation of .

3.2.1 vs.

At the realistic vacuum, , , . One can then plug the expression of in eq. (3.2) into eq. (24), and simplify as:

Notice that the second line in eq. (3.2.1) vanishes in the limit when the doublets decouple from the stationary point equation for .

At the unrealistic extremum point , we clearly have in eq. (24). Thus, the only relevant quantity here is which can be written in terms of quantities at the realistic minimum using eqs. (3.2,3.2). Finally, we get the following potential at this point:

So, the question of whether the realistic minimum is the global one or not reduces to a comparison of the value of the potential in eq. (3.2.1) and eq. (3.2.1). We first make some general qualitative comments that will be later confirmed by our quantitative studies. With all the other parameters fixed, at a value of much larger than , the unrealistic extremum tends to have deeper than . A general “hint” of this feature, that may also apply to the other unrealistic extrema, comes from the fact that the term(s) in eq. (24) give negative contribution to . At the realistic vacuum has the value 91 GeV, achieving which, according to eq. (21), requires tuning when is large. On the other hand, at the other extrema with non-zero , the size of is typically . Therefore gets a stronger negative push from larger terms. To be more specific to this extremum we can see from eq. (3.2.1) and eq. (3.2.1) that, at large it is the dominant negative contribution of in the potential which tends to make deeper. In fact to be confronted with the contribution in the potential at the realistic minimum . With as a relevant example, we find the coefficient of term in is larger in magnitude (but negative) than in , except for a small intermediate region of .

Furthermore one can study the dependence on with all other parameters fixed. At larger , the unrealistic extremum tends to have deeper than . This can be seen by comparing the leading terms in the two potentials. These are and where the term has an coefficient depending on . Clearly, the leading term gives a significant negative contribution to which is absent in .

We can also study the dependence on with all other parameters fixed. At larger , the unrealistic extremum tends to be shallower than . This can be seen by comparing coefficients of the leading dependent term of at large . These are and . With as an example, gets a larger negative contribution at larger .

3.2.2 A complete vacuum analysis at doublet-singlet decoupling limit

As mentioned earlier, it is much easier to solve for all extrema in the NMSSM potential in the limit of decoupling doublet from singlet. Strikingly, this is also the interesting limit for obtaining  GeV while being compatible with perturbative unification and LEP limits. Here we shall show that there are parameter regions where the singlet-doublet decoupling holds and the realistic vacuum is a global minimum. Despite the special nature of the limit where our finding applies, this constitutes a proof of existence of a vacuum that is both stable and able to accommodate . The results may be generalized to other cases as well.

Notice that in order for our approach to be self-consistent, other extrema, i.e., including , also needs to satisfy the decoupling condition , just like the realistic minimum:

(30)

This more restrictive condition of vanishing mixing entry also disfavors tachyons at other extrema which makes them more “competitive” compared to realistic vacuum. Meanwhile such additional condition constrains input parameters to satisfy certain relations with and output quantities at realistic vacuum.

In this limit the stationary point equation of eq. (27) reduces to

(31)

and the solutions take a simple form. Evidently satisfies:

(32)

We solve for nonzero using eqs. (20,21,30). First, is fixed by the decoupling relation eq. (30) and . Then based on eq. (21) and eq. (30) we can rewrite and as function of , then plug in eq. (20). A consistency equation for only can be obtained. We solve this equation and pick out 3 real solutions: . The corresponding solutions for can be easily derived.

In order to compute the potentials we need to retrieve the remaining model parameters . We plug into eq. (32) to find the model parameter . Plugging in at realistic vacuum and into eq. (3.2), we retrieve the last model parameter . Now we have obtained all 5 other sets of stationary solutions and can compute potentials using eq. (24) and see when the realistic vacuum has the deepest potential, i.e., it is the global minimum.

The result of our analytic study is shown in Figure 2, where we evaluate the value of the potential at all the stationary points as a function of , i.e., with all the values of the input parameters fixed as functions of as explained above. The red line corresponds to the value of the potential at the realistic realistic minimum. For different choices of in the two panels this figure illustrates for what range of the realistic minimum is the global one.

In the figure we observe the danger of taking large discussed in Section 3.2.1 above. The left panel shows that the value of the potential at the extremal point gets deeper than the realistic minimum when is large. The trend as we increase is as expected from the earlier discussion: the realistic minimum is shallower at smaller , and larger is found to disfavor the unrealistic extremal point .

Figure 2: The value of the potential evaluated at all possible stationary points defined by the requirement of having at the realistic minimum. The parameters of the potential are fixed as functions of as discussed in the text. In red and thick line we show the realistic minimum . The other extremal points of the potential are detonated by dashed line with the following color code: orange for ; blue for ; brown for the origin ; cyan for ; green for . The orange line is close to red line in particular at large as a result of the fact that the potentials of the corresponding extrema only differs by dependent terms in the potential eq. (24) which is a sub-leading contribution.

4 Push-up Scenario

As mentioned, one possibility of alleviating the tension between perturbativity up to GUT scale and is the “push-up” scenario. As discussed earlier, is more decoupled from and , in particular at small as favored in the NMSSM to obtain large enough Higgs mass. Therefore we focus on the 2-by-2 mass matrix in the basis of . In this region of parameter space the SM Higgs-like state (mostly ) is typically the second lightest state of the spectrum 888We find that in the alternative case where the SM Higgs-like state is the heaviest state of the spectrum, the charged Higgs would be too light to be consistent with direct searches and flavor constraints such as from ., heavier than the singlet-like state (mostly ), thus the mixing with the lightest state induces an increase in its mass. Thus, we must have

(33)

Apparently, it seems plausible to avoid the tuning associated with a small in this case. However, as we shall demonstrate in this section, this scenario is strongly constrained by the LEP bound. In the end, the allowed push-up effect is very limited which corresponds to fine-tuning worse than approximately 20% 999See Ref. [39] for a similar analysis..

In detail, we have the following sub-mass matrix for :

(34)

This tree-level mass matrix has two eigenvalues, and , that are the masses of the doublet-like and singlet-like state, respectively. In this case . Also, in the push-up scenario