Higgs phenomenology in the Peccei-Quinn invariant NMSSM

We study the Higgs phenomenology in the Peccei-Quinn invariant NMSSM (PQ-NMSSM) where the low energy mass parameters of the singlet superfield are induced by a spontaneous breakdown of the Peccei-Quinn symmetry. In the generic NMSSM, scalar mixing among CP-even Higgs bosons is constrained by the observed properties of the SM-like Higgs boson, as well as by the LEP bound on the chargino mass and the perturbativity bound on the singlet Yukawa coupling. In the minimal PQ-NMSSM, scalar mixing is further constrained due to the presence of a light singlino-like neutralino. It is noticed that the $2\sigma$ excess of the LEP $Zb\bar b$ events at $m_{b\bar b}\simeq$ 98 GeV can be explained by a singlet-like 98 GeV Higgs boson in the minimal PQ-NMSSM with low $\tan\beta$, stops around or below 1 TeV, and light doublet-higgsinos around the weak scale.


I. INTRODUCTION
There are many reasons to anticipate new physics beyond the standard model (SM), including the naturalness problems such as the hierarchy problem and the strong CP problem, and a variety of cosmological observations such as the existence of dark matter, the matter-antimatter asymmetry, and the evidences for inflation in the early Universe.
Among the known scenarios of new physics, a particularly compelling possibility is a supersymmetric extension of the SM [1] incorporating also the axion solution to the strong CP problem through a spontaneously broken Peccei-Quinn (PQ) symmetry [2,3]. While solving the two major naturalness problems of the SM, such an extension of the SM provides an attractive candidate for dark matter, either the lightest supersymmetric particle or the axion, or both. It also offers an interesting possibility that the PQ scale is generated by an interplay between supersymmetry (SUSY) breaking effect and Planckscale suppressed effect, yielding an intermediate PQ scale v PQ ∼ √ m soft M P l in a natural manner [4,5], where m soft is a soft SUSY breaking mass presumed to be of the order of the weak scale. In such a scenario, the PQ phase transition takes place in the early Universe at a temperature T ∼ m soft . This results in a late thermal inflation over the period m soft < T < v PQ , with which dangerous cosmological relics such as the moduli and gravitinos are all diluted away [5,6].
The scalar boson with a mass m h ≃ 125 GeV, which was recently discovered in the LHC experiments, has been found to behave like the SM Higgs boson [7,8]. On the other hand, a SM-like Higgs boson at 125 GeV in the minimal supersymmetric standard model (MSSM) requires that stops have either a heavy mass in multi-TeV range or maximal LR-mixing, which would cause a fine-tuning worse than 1 % in the electroweak symmetry breaking. This fine-tuning can be ameliorated in the next-to-minimal supersymmetric standard model (NMSSM) involving a singlet superfield S with the superpotential coupling λSH u H d . In the NMSSM, the SM-like Higgs boson h gains an additional tree-level mass from the F -term scalar potential λ 2 |H u H d | 2 , or from scalar mixing if the singlet scalar s is lighter than h. This makes it possible to have m h ≃ 125 GeV even when stops are relatively light and stop mixings are small, and therefore reduces the amount of fine-tuning required for the electroweak symmetry breaking [9,10]. Furthermore the Higgs and neutralino sector of the NMSSM have a richer structure than the MSSM. If all the Higgs bosons in the NMSSM have masses in sub-TeV range, e.g. below 500 GeV, there can be sizable mixings among the three CP-even Higgs bosons, leading to interesting phenomenological consequences as discussed in [10][11][12][13].
It is well known that a PQ-symmetry spontaneously broken at v PQ ∼ √ m soft M P l can explain why the doublet-higgsino mass µ in the MSSM is comparable to m soft [4,5,14,15]. Similarly, if the singlet superfield S is PQ-charged, the low energy mass parameters of S in the effective superpotential of the PQ-invariant NMSSM are induced by a spontaneous breakdown of the PQ symmetry, and so can have a value comparable to m soft , while the singlet cubic coupling is always negligible [16,17]. In this paper, we wish to examine the Higgs phenomenology in such a PQ-invariant NMSSM while focusing on the phenomenological consequences of scalar mixing. The Higgs boson masses and mixing angles in the neutral CP-even Higgs sector crucially depend on the coupling λ and the doublet-higgsino mass µ. As a result, scalar mixing is constrained not only by the observed mass and signal strengths of the SM-like Higgs boson, but also by the perturbativity bound on λ and the LEP bound on the chargino mass. We will examine first the constraints on scalar mixing in the context of the general NMSSM, and then consider a specific minimal PQ-invariant NMSSM which is further constrained by the presence of a light singlino-like neutralino.
If the singlet-like Higgs boson s has a mass near the weak scale, it can have a large mixing with the SM-like Higgs boson h. We identify the parameter region of the sizable singlet-doublet mixing that is compatible with all the LHC and LEP data available at present, as well as with the perturbativity bound on λ and a stop mass between 600 GeV and a few TeV. We explore also the possibility that the 2σ excess of the LEP Zbb events at m bb ≃ 98 GeV is explained by e + e − → Zs → Zbb within the framework of the minimal PQ-NMSSM. 1 We then find that it requires low tan β smaller than about 2, a light doublethiggsino mass around the weak scale, and stop masses around or below 1 TeV. For the case with m s > m h , it is found that s decays dominantly into a neutralino pair in most of the viable parameter region, which would make its detection at collider experiments difficult. We examine also the signal rates of the SM-like Higgs boson in the bb (ττ ) and di-photon channels over the phenomenologically viable parameter region which gives the signal rate of the W W/ZZ channel close to the SM value.
This paper is organized as follows. In section II, we discuss how the constraints on λ and µ, and the observed properties of the SM-like Higgs boson translate into the constraints on the scalar mixing angles. In section III, we discuss some generic features of the PQinvariant NMSSM, and present a specific model which is considered to be a minimal The neutralino sector of the minimal PQ-NMSSM is also discussed with a focus on the additional constraints arising due to a light singlino-like neutralino in the model. In section IV, we apply the results of the section II to the Higgs phenomenology in the minimal PQ-NMSSM. We present first the results that hold in the general NMSSM, and then impose additional constraints specific to the minimal PQ-NMSSM. Section V is the conclusions.

II. CONSTRAINTS ON HIGGS MIXING IN THE NMSSM
In this section, we briefly discuss phenomenological consequences of Higgs mixing in the general NMSSM and the resultant constraints on the model. Let us begin with the Higgs sector superpotential of the general NMSSM, which is given by in an appropriate basis of the singlet superfield S. The first term is responsible for the higgsino mass parameter µ and Higgs bilinear coupling Bµ: where A λ is the soft SUSY breaking parameter for the superpotential term SH u H d . There is one combinationĥ of CP-even neutral Higgs bosons which corresponds to the fluctuation of Re(H 0 u ) and Re(H 0 d ) in the vacuum value direction, and therefore behaves like the SM Higgs boson in the limit when the other Higgs bosons are decoupled. In the NMSSM, it generally mixes with the other CP-even neutral Higgs bosons, and the SM-like Higgs boson in the mass-eigenstate is given by with c θ i = cos θ i and s θ i = sin θ i for the mixing angles θ i defined in appendix A, whereĤ is the fluctuation of Re(H 0 u ) and Re(H 0 d ) orthogonal toĥ, andŝ is the CP-even fluctuation of the singlet scalar.
Around the weak scale, the SM-like Higgs boson interacts with the SM particles through the terms 2 [19], where f denote the SM fermions, and v ≃ 174 GeV is the Higgs vacuum expectation value. At tree level, the Higgs couplings to the massive SM particles are determined by the mixing angles as 2 This should be understood as an 1PI effective Lagrangian including quantum corrections for the SM-like Higgs boson near the mass-shell.
On the other hand, the Higgs couplings to massless gluons and photons are radiatively induced. The dominant contribution comes from the W -boson and top quark loops: where we have used the well-known production and decay properties of the SM Higgs boson under the assumption that the Higgs decay rate into non-SM particles is negligible.
To see the effect of scalar mixing, it is convenient to factor the signal rate into W W/ZZ as where R V V h | 0 is the signal rate for δC g = 0, i.e. in the limit that all the colored superparticles are heavy. It is important to note that R V V h | 0 depends only on θ 1 , θ 2 , and tan β. In addition, because the effect of colored superparticles is to modify the Higgs production rate in the gluon fusion process, the ratio R ii h /R V V h for each channel is insensitive to the correction δC g . For other channels, we find where R ii h = 1 in the limit of vanishing mixing angles and decoupled superparticles. In the NMSSM, the Higgs quartic coupling receives an additional tree-level contribution proportional to λ 2 , and consequentlyĥ obtains a mass according to where m 0 corresponds to the SM-like Higgs boson mass at large tan β in the decoupling limit of MSSM, including the well-known radiative correction from top and stop loops [20]: for the stop mass mt and the stop mixing parameter X t = (A t − µ cot β)/mt. It is straightforward to see that the mass of the SM-like Higgs boson in the NMSSM reads where the last two terms are due to scalar mixing. Note that the mixing with singlet scalar increases m 2 h if the singlet-like Higgs boson s is lighter than the SM-like Higgs boson h [16,21,22].
In the presence of scalar mixing, the singlet-like Higgs boson s also interacts with the SM particles via the doublet components. Those interactions are obtained from (4) by replacing C i with the effective couplings at tree-level, and the coupling to gluons and photons are radiatively generated depending on the singlet mass m s .
Let us examine how the SM-like Higgs boson in the NMSSM can be arranged to be consistent with the LHC data. The most important constraints come from the mass and signal rates for the various Higgs decay channels observed at the LHC. In particular, the signal rate for the W W/ZZ channel should be close to the SM value, which does not necessarily imply that the h-s mixing angle θ 2 should be small. Keeping in mind that the Higgs coupling to gluons can receive a non-negligible correction from relatively light stops, we impose the condition to account for the observed Higgs signal rate in W W/ZZ. This is the case when the mixing angles obey the relation [12], Here we have used that R V V h | 0 is determined only by θ 1 , θ 2 , and tan β. For such Higgs mixing, the signal rates for the fermionic (bb or ττ ) and di-photon channel are estimated to be with R V V h ≈ 1. This shows that the signal rates for the bb and ττ channel are reduced below the SM prediction as a result of scalar mixing at tree level. The di-photon rate is less affected by scalar mixing. However, in the presence of sizable θ 2 and light charged-higgsinos, it can significantly deviate from the SM value due to the chargino-loop contribution to δC γ , which is given by [19] Note that the charged-higgsino loop can either enhance or reduce the di-photon rate, depending on the sign of θ 2 .
In the NMSSM, for a given value of tan β, the off-diagonal components of the mass matrix of (ĥ,Ĥ,ŝ) are determined by three parameters {λ, µ, Λ} (see appendix A), where is independent from the effective Higgs bilinear coupling Bµ. These parameters can be expressed in terms of the mixing angles θ i and the mass eigenvalues m h , m H and m s . In particular, λ and µ are given by On the other hand, the coupling λ is constrained to be less than about 0.7 at the weak scale, if one wishes to maintain the model to be perturbative up to the GUT scale [23], while the LEP bound on the chargino mass requires |µ| to be larger than about 100 GeV [24]. These constraints on λ and µ can be translated into those on the mixing angles and mass eigenvalues through the above relations.
which amounts to assuming that stops are not significantly heavier than 1 TeV. Note that δC g receives the dominant contribution from stop loops [27,28], which can be sizable for stop masses of our interest. This correction to the Higgs coupling to gluons modifies the Higgs production rate in the gluon fusion, and enhances the Higgs signal rates for the stop mixing parameter X t < √ 2. For instance, taking X t = 0, one finds that the Higgs signal rate in each channel increases universally by about 8 % for the stop mass around 600 GeV, and by less than about 3 % for the stop mass heavier than remains almost the same. The stop contribution to the Higgs-photon coupling C γ is below 1 % even for the stop mass around 600 GeV.

III. PECCEI-QUINN INVARIANT NMSSM
In this section we discuss the generic low energy limit of the PQ-invariant NMSSM, and present a specific model considered to be a minimal PQ-NMSSM. As we will see, a key feature of the minimal PQ-NMSSM is the presence of a light singlino-like neutralino, with which the model is severely constrained by the Higgs invisible decay and the LEP bound on neutralino productions.

A. Low energy limit of the generic PQ-NMSSM
At energy scales below the PQ-breaking scale v PQ , the PQ-NMSSM can be described by a low energy effective theory with a non-linear U(1) PQ symmetry, under which the NMSSM Higgs superfields and the axion superfield A transform as Throughout this paper, we assume that the PQ-breaking scale is generated by competition between SUSY breaking effect and Planck scale suppressed effect, so that Here the axion superfield A is composed of a pseudo-scalar axion a solving the strong CP problem, its scalar partner saxion ρ, and the fermionic partner axinoã: The PQ-invariant Kähler potential and superpotential below v PQ are generically given by in which Φ i denote the NMSSM chiral superfields. Here ∆K and ∆W stand for the terms induced by a spontaneous breakdown of the PQ symmetry, where the ellipses denote higher dimensional terms, and for model-dependent non-negative integers k j and n j .
Including the effects of soft SUSY-breaking, the vacuum value of the axion superfield can be determined to be 3 where ξ 1,2 = O(1) in general. To examine the particle physics phenomenology at scales below v PQ , it is convenient to replace the axion superfield with its vacuum expectation value. After this replacement, one can make an appropriate field redefinition together with a Kähler transformation to arrive at the following form of the effective Kähler potential and superpotential: where with a PQ scale given by v PQ ∼ √ m soft M P l . A simple generic feature of the PQ-NMSSM is that the singlet cubic coupling κ is always negligible, while the singlet mass parameters µ 1,2 can be either of the order of m soft or negligibly small compared to m soft , depending on the relative charge between S and PQ-breaking fields.

B. A minimal PQ-NMSSM
In this subsection, we present a specific model which is considered to be a minimal PQinvariant NMSSM, and discuss the neutralino sector of the model. At high scales above v PQ , but below the Planck scale M P l , the model includes the PQ-breaking superfields X and Y , as well as the NMSSM Higgs superfields S, H u and H d , with the following PQ-charges: ).
The model can include also exotic gauge-charged matter superfields Ψ I , Ψ c I (I = 1, 2), which are vector-like under the SM gauge group, e.g. 5 +5 of SU(5), and carry a PQcharge which allows renormalizable Yukawa couplings to X or Y , e.g. ).
Then the most general PQ-invariant Kähler potential and superpotential are written as where the ellipses denote higher dimensional operators suppressed by higher powers of 1/M P l . Including soft SUSY breaking terms, the scalar potential of the PQ-breaking fields takes the form Assuming m 2 Y < 0 and m 2 X > 0 around the renormalization point ∼ √ m soft M P l , which can be a consequence of either a D-term induced soft SUSY breaking or the radiative correction due to a large Yukawa coupling λ ′′ , one finds assuming that Now we can replace the PQ-breaking superfields X and Y with their vacuum expectation values while including the soft SUSY-breaking terms explicitly. Making further a field redefinition of (29) and a Kähler transformation of (30), we find that the resulting low energy effective theory takes the form where the quadratic and cubic terms of S in W eff are omitted since their coefficients are negligibly small: Although the effective superpotential of this minimal PQ-NMSSM takes a simple form, 4 the Higgs sector of the model is not distinctive as the Higgs mixing parameter Λ = On the other hand, the neutralino sector of the model is quite distinctive since ∂ 2 S W eff = 0, and therefore the singlino gains a mass only through the mixing with other neutralinos: where the ellipsis denotes the gaugino mass and gaugino-higgsino mixing terms. It can be shown that the lightest neutralino, has a mass lighter than λv cos β in the limit when the mixing with gauginos is ignored [17].
To see qualitatively the properties of the lightest neutralino in the minimal PQ-NMSSM, one can take the limit of µ ≫ λv and the gaugino masses M i ≫ v. Then the neutralino mixing coefficients are found to be while the mass eigenvalue is given by The gaugino components in χ 0 1 are generally small because they are further suppressed by v/M i ≪ 1: for i = 1, 2 with M i being the corresponding gaugino mass.
There are important constraints on the minimal PQ-NMSSM associated with the small mass of the lightest neutralino. One is from the LEP bound on the neutralino production via the Z-boson exchange [34]: which applies for m χ 0 2 + m χ 0 1 < 208 GeV and m χ 0 1 > 60 GeV. This puts an upper bound on the Z-boson coupling to χ 0 2 χ 0 1 . In addition, the global fit analysis excludes an invisible decay of the SM-like Higgs boson with a branching ratio greater than 0.38 at 95% confidence level, if one allows its couplings to the SM particles to deviate from the SM values [35,36]: In the NMSSM, the Higgs coupling for this process is given by which can have a sizable value in the limit of low tan β, large λ, and light µ. To avoid a dangerous Higgs invisible decay when the Higgs coupling y hχ 0 1 χ 0 1 is sizable, one needs 2m χ 0 1 > m h so that the process is kinematically forbidden. On the other hand, if 2m χ 0 1 < m h , one needs to adjust the model to suppress y hχ 0 1 χ 0 1 . However, with small y hχ 0 1 χ 0 1 , it is difficult to have a sizable NMSSM contribution to the tree level mass of the SM-like Higgs boson, which is the feature that we like to keep to avoid too severe fine-tuning of the model. Note that, since we are assuming m 0 120 GeV, a sizable NMSSM contribution is required to get m h ≃ 125 GeV. Actually, as we shall see in the next section, this makes it difficult to suppress the branching fraction for the Higgs invisible decay below 0.38 in most of the parameter space of our interest once the mode is kinematically open. We therefore require 2m χ 0 1 > m h to prohibit the decay process h → χ 0 1 χ 0 1 .

IV. HIGGS PHENOMENOLOGY OF THE PQ-NMSSM
The SM-like Higgs boson observed at the LHC can be accommodated in the NMSSM while satisfying the constraints on scalar mixing discussed in sec. II. We will first examine how large the mixing between the SM-like Higgs boson and the singlet-like Higgs boson is allowed in the general NMSSM, and then move on to the minimal PQ-NMSSM where the mixing is further constrained due to a light singlino-like neutralino. As we will see, a SM- in the expansion in powers of θ i . Here we have taken into account that the charged Higgs scalar, whose mass is similar to m H , should be heavier than about 350 GeV to satisfy the b → sγ constraint, barring cancellation with other superparticle contributions [38].
Similarly, one also finds (53) One can see that the perturbativity bound λ 0.7 can be easily satisfied for the scalar mixing angles and tan β satisfying (52) and (53). Note also that, because of the constraint (52), positive θ 2 θ 3 is favored for the mixing angle θ 2 to be sizable.
The left plot of Fig. 1 illustrates the range of the singlet fraction (= c 2 θ 1 s 2 θ 2 ) of the SMlike Higgs boson and tan β for which the LEP excess of Zbb at m bb ≃ 98 GeV is explained by a singlet-like scalar s, while satisfying the perturbativity bound on λ, the LEP bound on chargino mass, and 105 GeV ≤ m 0 ≤ 120 GeV. Here we have imposed the relation (16) to have R V V h ≈ 1, 5 and used |θ 3 | = 0.1, m H = 350 GeV, and the gaugino masses 2M 1 = M 2 = 300 GeV for the purpose of illustration. The LEP bound on the chargino mass puts a lower bound on |µ|, which can be relaxed if the wino mass M 2 is around a few hundred GeV and µM 2 < 0. One can see that the allowed blue-shaded region is determined mainly by the constraints associated with µ and m 0 , whose characteristic features can be understood by the relation (50) and (51). Note that tan β is bounded from above by the constraint on |µ| according to (53), while the constraint on m 0 explains the allowed range of θ 2 2 for a given value of θ 3 and tan β through the relation (52). The allowed region becomes smaller if one increases the wino mass or changes its sign, because 5 One may consider a case where the Higgs signal rate into W W/ZZ deviates from the SM value by an amount δR V V h due to scalar mixing. Then, the relation (16) should be modified as and the mixing effects can be examined by taking the replacement θ 2 2 → θ 2 2 + δR V V h in the relations (49) and (51). As a result, the region consistent with 105 GeV ≤ m 0 ≤ 120 GeV will move horizontally to the left (right) in Fig. 1  then the lower bound on µ from the chargino mass bound is strengthened. We also present in Fig. 1  Now we impose the additional constraints that are particularly relevant for the minimal PQ-NMSSM which predicts a light singlino-like neutralino: As shown in the right plot of Fig 1,   We close this subsection by pointing out that the minimal PQ-NMSSM requires stops around or below 1 TeV, and higgsinos around the weak scale. If m 0 is larger than about 6 The Higgs coupling to photons receives a loop contribution also from the hH +W − interaction, which becomes important when both higgsinos and winos have masses not much above the weak scale. Such an effect has been included in our analysis. 110 GeV, it is difficult to have 2m χ 0 1 > m h , which would be necessary to forbid h → χ 0 1 χ 0 1 . This means that h can be identified as the SM-like Higgs boson observed at the LHC only when stops are not significantly heavier than 1 TeV. In addition, combined with m 0 105 GeV, the requirement 2m χ 0 1 > m h constrains µ to be around the weak scale. As we will see in the next subsection, these features hold also for the case that s is heavier than h.

B. Singlet-like Higgs boson above 125 GeV
Let us move to the case where the singlet-like Higgs boson s is heavier than the SM-like Higgs boson h. One of the main differences from the opposite case with m s < m h is that It is clear that the singlet-doublet mixing angle θ 2 can be sizable if s is not much heavier than h. As in the previous case, the effect of scalar mixing can be understood qualitatively by using the approximated relations (49) and (51) after multiplying the second term with , and the relation (50) after multiplying the first term with m 2 s /(98GeV) 2 . Then it follows that a viable region for the general NMSSM with m s > m h appears at lower tan β compared to the case with m s = 98 GeV. Hence, it becomes relatively easy to satisfy the condition 2m χ 0 1 > m h in the minimal PQ-NMSSM where m χ 0 1 is proportional to sin 2β.
In the left panel of Fig. 3, we show the region of (c 2 θ 1 s 2 θ 2 , tan β) compatible with the constraints on {λ, µ, m 0 } for the general NMSSM with m s = 150 GeV, m H = 350 GeV, and |θ 3 | = 0.1. Here, for simplicity, we have assumed that the gauginos are much heavier On the other hand, if the singlet-like Higgs becomes heavier, only a smaller value of θ 2 will be allowed. For instance, the singlet fraction of h should be less than about 0.1 for m s > 250 GeV and m H = 350 GeV. Fig. 4 illustrates how the allowed region of (c 2 θ 1 s 2 θ 2 , tan β) varies when m H becomes heaver (left panel) or θ 3 becomes larger (right panel). We can see that the allowed region varies in the same way as explained in the previous subsection, so that the phenomenologically viable region of the minimal PQ-NMSSM becomes smaller or vanishes as m H or θ 3 gets larger.
It is worth noting that, if s is heavy enough, it can dominantly decay into a pair of the lightest neutralino. For instance, for the minimal PQ-NMSSM depicted in the right panel of Fig. 3, the region above the thin black line gives m s > 2m χ 0 1 , for which the invisible decay s → χ 0 1 χ 0 1 is open. This region covers all the parameter space satisfying the phenomenological constraints discussed here. We find that the branching fraction of the invisible decay s → χ 0 1 χ 0 1 is about 0.7 − 0.8 over the viable region, which would make it difficult to discover s at collider experiments.