Phenomenology of NMSSM in TeV scale mirage mediation

We study the next-to-minimal supersymmetric standard model (NMSSM) with the TeV scale mirage mediation, which is known as a solution for the little hierarchy problem in supersymmetry. Our previous study showed that 125 GeV Higgs boson is realized with O\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O} $$\end{document}(10)% fine-tuning for 1.5 TeV gluino (1 TeV stop) mass. The μ term could be as large as 500 GeV without sacrificing the fine-tuning thanks to a cancellation mechanism. The singlet-doublet mixing is suppressed by tan β. In this paper, we further extend this analysis. We argue that approximate scale symmetries play a role behind the suppression of the singlet-doublet mixing. They reduce the mixing matrix to a simple form that is useful to understand the results of the numerical analysis. We perform a comprehensive analysis of the fine-tuning including the singlet sector by introducing a simple formula for the fine-tuning measure. This shows that the singlet mass of the least fine-tuning is favored by the LEP anomaly for moderate tan β. We also discuss prospects for the precision measurements of the Higgs couplings at LHC and ILC and direct/indirect dark matter searches in the model.


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The TeV scale mirage mediation was applied to the NMSSM in ref. [103]. It was found that the fine-tuning is improved in the parameter region to realize m h ≈ 125 GeV. The effective µ can be considerably larger than the EW scale without sacrificing the fine-tuning due to a cancellation mechanism, although it is assumed to be around the EW scale in conventional natural SUSY models [114][115][116][117][118][119][120][121][122]. 2 The mixing between the singlet and doublet Higgs bosons is suppressed. In addition, the Higgs sector as well as the neutralino sector has a rich structure compared with one in the MSSM by adding the singlet. That leads to interesting aspects as shown in ref. [103]. 3 In this paper, we further extend the analysis in [103] including new phenomenological studies. We show that approximate scale symmetries play an important roll to suppress the singlet-doublet mixing in the NMSSM. We discuss the suppression holds as far as κ ≈ 0 and m 2 S,Hu m 2 H d . We introduce a simple formula for the fine-tuning measures and study the fine-tuning of the EW symmetry breaking in detail. We show that the singlet mass of the least fine-tuning is favored by the LEP anomaly [99,103,[125][126][127][128][129][130][131][132] when tan β is moderately large. We also study other phenomenological aspects such as the precision measurement of the Higgs couplings and the dark matter search.
This paper is organized as follows. In section 2, we give a brief review on the mirage mediation, and the TeV scale mirage mediation. We introduce the Z 3 symmetric NMSSM in section 3 and apply the TeV scale mirage mediation in section 4. In section 5, we study phenomenological aspects such as the mass spectrum, fine-tuning in the EW symmetry breaking, Higgs couplings and dark matter. Section 6 is devoted to conclusion and discussion. In appendix A, we show explicitly initial conditions of soft parameters, which are induced through the mirage mediation in the NMSSM.

TeV-scale mirage mediation
Here we give a brief review on the mirage mediation [38,39]. The mirage mediation is the mixture of the modulus mediation and the anomaly mediation with a certain ratio, which would be determined by the modulus stabilization mechanism and SUSY breaking mechanism. In the mirage mediation, the gaugino masses are written by M a (M GUT ) = M 0 + m 3/2 8π 2 b a g 2 a , (2.1) where g a and b a are the gauge couplings and their β function coefficients, and m 3/2 denotes the gravitino mass. We assume that the initial conditions of our SUSY breaking parameters are input at the GUT scale, M GUT = 2 × 10 16 GeV. The first term, M 0 , in the right hand side denotes the gaugino mass due to the pure modulus mediation, while the second term corresponds to the anomaly mediation contribution. In addition, we can write the soft scalar mass m i of matter fields φ i and the so-called A-terms of φ i φ j φ k corresponding to the Yukawa couplings y ijk as Here, γ i denotes the anomalous dimensions of φ i and C a 2 (φ i ) denotes the quadratic Casimir corresponding to the representation of the matter field φ i . In the right hand side, a ijk M 0 and c i M 2 0 denote the A-term and soft scalar masses squared due to the pure modulus mediation. These coefficients, a ijk and c i , are determined by modulus-dependence of the Kähler metric of φ i , φ j and φ k as well as Yukawa couplings. Indeed, by using the treelevel Kähler metric, the coefficient c i is explicitly calculated as a fractional number such as 0, 1, 1/2, 1/3, etc. [64-67, 72, 134]. We would have O(1/8π 2 ) of corrections on c i due to the one-loop corrections on the Kähler metric [135]. Such a correction would be important when c i = 0, but that is model-dependent. Here, we consider the case with Such a relation is often satisfied for O(1) of Yukawa couplings in explicit string-derived supergravity models [64][65][66][67]. 4 It is convenient to use the following parameter [40], α ≡ m 3/2 M 0 ln(M pl /m 3/2 ) , (2.5) to represent the ratio of the anomaly mediation to the modulus mediation. Here M pl is the reduced Planck scale.
The mirage mediation has a very important energy scale, that is, the mirage scale defined by, (2.6) The above spectrum of the gaugino masses at M GUT leads to [40], at the mirage scale. That is, the anomaly mediation contributions and the radiative corrections cancel each other, and the pure modulus mediation appears at the mirage scale. Furthermore, the A-terms and the scalar mass squared also satisfy if the corresponding Yukawa couplings are small enough or if the following conditions are satisfied, a ijk = c i + c j + c k = 1, (2.9) for non-vanishing Yukawa couplings, y ijk [40]. When α = 2, the mirage scale M mir is around 1 TeV. Then, the above spectrum (2.7) and (2.8) is obtained around the TeV scale. That is the TeV scale mirage mediation scenario. In particular, there would appear a large gap between M 0 and the scalar mass m i with c i ≈ 0. We will apply the TeV scale mirage scenario to the NMSSM in section 4. In the TeV scale mirage scenario, the stop mass squared becomes negative at high energy [138], while it is positive at low energy below 10 6 GeV. Thus, the vacuum which breaks the EW symmetry at the EW scale might be a local minimum, but instead there would be a color and/or charge breaking vacuum with field values larger than 10 6 GeV. Here, we assume the thermal history of the Universe such that field values remain around the origin until the EW scale temperature. In addition, we need to confirm that the tunneling rate is small enough, i.e. less than the Hubble expansion rate. In refs. [139][140][141], it has been shown that such a rate is small enough, as long as the squark/slepton masses squared are vanishing or positive around 10 4 GeV. This condition is satisfied in our TeV scale mirage mediation scenario.

NMSSM
Here, we briefly review on the NMSSM, in particular its Higgs sector. We extend the MSSM by adding a singlet chiral multiplet S and imposing a Z 3 symmetry. Then, the superpotential of the Higgs sector is written as Here and hereafter, for S, H u and H d we use the convention that the superfield and its lowest component are denoted by the same letter. The full superpotential also includes the Yukawa coupling terms between the matter fields and the Higgs fields, which are the same as those in the MSSM. The following soft SUSY breaking terms in the Higgs sector are induced, Then, the scalar potential of the neutral Higgs fields is given as where g 1 and g 2 denote the gauge couplings of U(1) Y and SU (2). Similarly, there appear the soft SUSY breaking terms including squarks and sleptons as well as gaugino masses. These are the same as those in the MSSM.

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The minimum of the Higgs potential is obtained by analyzing the stationary conditions of the Higgs potential, Here, we denote VEVs as Using the above stationary conditions, we obtain the Z boson mass m 2 Z = 1 2 g 2 v 2 as where µ = λs. For tan β 1, this equation becomes eq. (1.1). That is, this relation is the same as the one in the MSSM. Thus, the natural values of |m Hu | and |µ| would be of O(100) GeV. Furthermore, the natural value of |m H d |/ tan β would be of O(100) GeV or smaller. Alternatively, |µ| and |m H d |/ tan β could be larger than O(100) GeV when µ 2 and m 2 H d / tan 2 β are canceled each other in the above relation at a certain level. Even in such a case, |m Hu | would be naturally of O(100) GeV. On the other hand, other sfermion masses as well as gaugino masses must be heavy as the recent LHC results suggested. To realize such a spectrum, we apply the TeV scale mirage mediation in the next section, where we take c Hu = 0 to realize a suppressed value of |m Hu | compared with M 0 .

TeV scale mirage mediation in NMSSM
Here, we apply the TeV scale mirage mediation scenario to the NMSSM and study its phenomenological aspects.

Model
Soft SUSY breaking terms are obtained through the generic formulas (2.1) and (2.2) with taking α = 2. For concreteness, we give explicit results of all the soft SUSY breaking terms for the NMSSM in appendix A. We concentrate on the Higgs sector as well as gauginos and stops.
A concrete model in the mirage mediation is fixed by choosing c i . We consider the following values of c i [103],

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up to one-loop corrections for H d , H u , S, and left and right-handed (s)top fields, respectively. This is the same assignment as the pattern II in ref. [72] for the MSSM except for c S . Then, the soft parameters due to only modulus mediation contribution are given by up to one-loop corrections. The above assignment of c i (4.1) satisfies the condition, (2.9) for the top Yukawa coupling and the coupling λ, but not for the coupling κ. However, we do not consider a large value of κ to avoid the blow-up of κ and λ as will be shown later. Thus, we obtain the following values, Similarly, at the TeV scale we can obtain Hu and m 2 S are suppressed compared with M 2 0 , and their explicit values depend on the one-loop corrections on the Kähler metric. Thus, we use A κ as a free parameter, which must be small compared with M 0 . In addition, we determine the values of m 2 Hu , m 2 S and µ (= λs) at the EW scale from the stationary conditions, (3.5), where we use the experimental value m Z = 1 .
Then, m 2 Z is almost determined by m 2 Hu alone and insensitive to µ ≈ M 0 / tan β. This means that tan β ≈ 3 is enough to obtain the fine-tuning of TeV. Therefore µ can be as heavy as O(400) GeV without sacrificing the fine-tuning. This cancellation originates in the structure of the doublet mass matrix, (4.10) Calculating the determinant of the mass matrix, it can be easily checked that the modulus mediated contribution M 0 cancels, det(M 2 H ) ≈ µ 4 while the trace of the mass matrix is M 2 0 + 2µ 2 . This means that the heavy mode has mass of O(M 0 ) and the mass of the light mode is suppressed as µ 2 /M 0 ≈ µ/ tan β. Therefore a flat direction appears along H u /H d ≈ M 0 /µ ≈ tan β. This mechanism was previously observed in [72] in the context of the MSSM where A λ is replaced with the B-term. In the mirage mediation, B-term is a remnant of the fine-tuned cancellation between the terms of O(m 3/2 ). Thus the relation m H d ≈ B is subject to uncontrolled corrections. However, in the NMSSM, the relation is well controlled up to the leading contribution of the modulus mediation.
This cancellation is also important for raising the SM-like Higgs boson mass radiatively. In the NMSSM, singlino-higgsino loop with λ = O(1) could contribute to the radiative correction of the Higgs boson mass in addition to the top quark loop [104,142]. If the singlet scalar is as light as the singlino or higgsino, this correction is canceled by the scalar loops because of SUSY. However, the structure of the mass matrix (4.10) tells the coupling |S| 2 h 2 (∼ µ 2 h 2 ) vanishes due to the SUSY breaking effect. This means the correction JHEP02(2016)089 could be sizable even if the singlet scalar is light as in our case. The renormalization group equation in the effective theory roughly gives where m h (m H ) is the mass of the light (heavy) doublet. In our case, m H ≈ A λ ≈ M 0 . This correction amounts ∼ 5 GeV for m h = m Z and λ = 0.7. Potentially, the mixing between the light doublet and the singlet may invalidate the above discussion on the hierarchy of the Higgs boson masses. In models with m 2 S ≈ 0 and κ 1, this mixing is automatically suppressed by λv/A λ / tan β if we neglect the gauge interaction. (This suppression is noticed in [100] without a reason.) In this limit, the Higgs action has the following approximate scaling symmetry: where φ is a free scaling parameter. This symmetry is explicitly broken by the Kähler potential, K S = S † S, (4.13) the D-term potential and all the kinetic terms. After the Higgs bosons develop the vacuum expectation values, the symmetry is spontaneously broken and the Nambu-Goldstone boson appears, which corresponds to the light doublet. 6 Thus it is inherently mass eigenstate without the singlet component. The symmetry breaking Kähler potential generates the F-term potential: which does not affect the singlet sector at all. This potential gives additional terms in the doublet mass matrix which we neglected before: The contributions to the diagonal elements are eliminated by the minimization conditions (3.5a), (3.5b). The CP even part of the mass matrix is given by and we used the minimization condition (3.5c), e.g. µ = 1 2 A λ sin 2β which is not affected by the gauge coupling. The last term which corresponds to the off-diagonal contribution in (4.15) mixes the light and heavy doublets and slightly changes the diagonalization angle of the CP even mass matrix, β ≡ π/2 − α from the direction of the Nambu-Goldstone boson, β as where δβ ≡ β − β. This small rotation generates a non-zero entry in the singlet-light doublet mixing terms (1-3, 3-1 elements) in the CP even Higgs mass matrix. The 2-3 (3-2) element is −λA λ v cos 2β. Thus the rotation yields ≈ 2λ 3 v 3 /A λ sin 2β cos 2 2β in 1-3 (3-1) element. The mass of the light doublet and that of the singlet are order of λv. Then the mixing angle between the light doublet and the singlet is order of λv/A λ / tan β. Similar discussion applies for the D-term potential. The diagonalization matrix S of M 2 S is given by at tree level. Note that the mass of the light doublet originates in the symmetry breaking terms. Non-vanishing κ and m 2 S are expected to add corrections of order κ and m 2 S /λ 2 v 2 to S. Radiative corrections to the Higgs potential are another source of the singlet-doublet mixing because the kinetic terms and the quark-lepton Yukawa couplings also break the scaling symmetry. Actually, this is not the end of story. The Higgs action has another approximate scale symmetry if κ = 0 and m 2 Hu = 0, in which the transformations of H u

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and S are interchanged in eq. (4.12). The VEVs of S and H d spontaneously break this symmetry and the Nambu-Goldstone boson appears, which consists of a combination of S and H d corresponding to the singlet-like Higgs boson. In the vanishing κ, m 2 S,Hu , H d limit, the real parts of S and H u are the Nambu-Goldstone bosons (mass eigenstates). After we turn on the symmetry breaking terms, the 1-3 and 3-2 elements of the mixing matrix S must break the both symmetries because S or H u is still the Nambu-Goldstone boson if one of the symmetries is preserved. In addition, the mixing between S and H u must vanish in the Lagrangian, eq. (3.3) if H d decouples [101]. Therefore the leading contributions to and m 2 Hu m 2 S /M 4 0 at tree level. Our argument on the suppression of the singlet-doublet mixing generally holds for models with small κ and m 2 S, Hu m 2 H d . The CP-odd part of the mass matrix is diagonalized as , Im(S)) and tan γ = λv/A λ . Two Nambu-Goldstone bosons appear for the broken electroweak and Pecci-Quinn symmetries. The former is absorbed by Z boson and the latter acquires mass with non-vanishing κ. In the following section, we show numerically the spectrum of our model.

Spectrum and phenomenological aspects
Here, we study numerically the spectrum and phenomenological aspects of our model. Our numerical calculation is performed with NMSSMTools package [143][144][145]. We choose input parameters as the squark and slepton masses, their tri-linear couplings, and parameters in the Higgs sector (λ, κ, A λ , A κ , m H d and tan β). We first minimize the one-loop effective potential of our model at the SUSY scale and calculate the effective µ-term. Then use it as an input of the NMSSMTools instead of m H d . NMSSMTools allows various levels of sophistication in the calculation of the Higgs boson mass. To keep the consistency with the input, we choose the default option (BLOCK MODSEL 8 0) which corresponds to the one-loop calculation at SUSY scale. The small parameters m 2 S , m 2 Hu are calculated so that we can obtain the correct EW symmetry breaking. In the following, we only adopt the parameters which predict |m 2 S |, |m 2 Hu | M 2 0 /8π 2 within the range of the model ambiguity. We choose the small parameter A κ as an input because the number of minimization conditions is not enough to predict it.

Higgs mass spectrum
We first discuss the mass spectrum of the Higgs sector. As described in the previous section, our model is in the decoupling region where the heavy doublet cos βH u − sin βH * d orthogonal to the flat direction has a mass much larger than the EW scale. Then this heavy doublet approximately consists of the heaviest CP-even, CP-odd and Charged Higgs bosons and they have degenerate masses ≈ M 0 up to the effect of the EW symmetry breaking. Remaining light degrees of freedom are the first and second lightest CP-even Higgs bosons and the lightest CP-odd Higgs boson.
In figure 1, we show the mass spectrum of the light CP-even Higgs bosons as a function of λ and κ for tan β = 3 and M 0 = 1500 GeV. In addition we present m 2 Hu , m 2 S and µ to check whether the parameters fit in the range favored by the mirage mediation or not. We also plot the coupling squared of the lightest CP-even Higgs boson to the gauge boson normalized with its SM value. This measures the amount of the doublet component in the lightest CP-even Higgs boson. The red curve in the figure indicates the boundary where λ and κ blow up at the Planck scale. Inside this curve the model remains perturbative until the Planck scale. The gray shaded area around the red curve shows the region where our iterative method fails to calculate the minimum of the effective potential. This is due to the tachyonic tree-level Higgs mass which we use as the initial condition of the iteration. The quantum correction may lift it, however, we do not pursue this possibility because the region is already excluded by the LEP Higgs mass limit. The gray shaded area in the small κ shows the region where the lightest CP-even Higgs becomes tachyonic. The yellow region is disfavored due to the false vacuum developed far beyond the EW scale as studied in [146,147].
As we see in the upper left panel, the lightest CP-even Higgs boson mass can reach 125 GeV for large value of λ (≈ 0.7) and small value of κ (≈ 0.1). This region is favored by the mirage mediation as shown in the middle left and middle right panels where the corresponding small mass parameters satisfy m 2 s , m 2 Hu M 2 0 /8π 2 for the observed Higgs boson mass. The lower right panel also indicates h 1 is almost doublet in this region. This is because the mixing between the doublet and the singlet in the mass matrix reduces the smaller eigenvalue while raises the larger eigenvalue [100][101][102][103][104][105][106][107][108][109][110][111][112][113]. Then the mass of the lightest Higgs boson is maximized when the mixing is minimized. The second lightest CP-even Higgs is around 200 GeV in the upper right panel and dominantly consists of the singlet. The higgsino mass parameter µ is around 500 GeV in the lower left panel which is much heavier than the EW scale, however, the cancellation mechanism described in the previous section protects the model from the fine-tuning which we will see numerically later. It is noted that the region is naturally realized in the model where λ has a strong dynamics origin at the Planck scale and an approximate Pecci-Quinn (PQ) symmetry moderately suppresses κ.

Fine-tuning in the electroweak symmetry breaking
Here we numerically estimate the degree of fine-tuning of the EW symmetry breaking in our model. Following the standard lore [148][149][150][151][152][153][154][155][156][157][158][159][160], we define the fine-tuning measure of a observable y against an input parameter x as To evaluate the EW symmetry breaking, the simplest choice of y's are the three Higgs . Instead of them, we choose m 2 Z , tan β and µ, extending the standard choice, y = m 2 Z in the literature [131,132,[161][162][163]. Here we emphasize that in principle all the VEVs must be examined to estimate the degree of fine-tuning once we extend the EW sector of the SM model, including usually disregarded tan β. The maximum of these measures, ∆ = max|∆ y x | could be regarded as the measure of the EW fine-tuning in the model. As x, we take λ, κ and the small parameters, m Hu , m S and A κ at the SUSY

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scale. Note that the large parameters such as m H d , A λ are not free parameters in our model and fixed by the ultraviolet physics.
These measures are easily calculated by the potential minimization conditions eq. (3.5) [156]. If we vary the input parameters x a with ∆x a , the Higgs VEVs v i shift with ∆v i following the potential minimum and they are related as Noticing that the coefficient of the first term is given by the CP even Higgs mass matrix defined in eq. (4.16), this leads to The fine-tuning measures can be constructed from these derivatives as This expression has clear physical meaning. 7 It is obvious that the measures increase if a certain Higgs mass is much lighter than the mass parameters in the potential. In NMSSMTools, the Higgs masses are calculated including the quantum corrections, however, the fine-tuning measures are calculated using the tree-level formula. This overestimates the fine-tuning if any of the tree-level masses vanish although the quantum corrections lift them. We calculate the measure using the above formula with the tree-level potential V and the quantum corrected mass matrix M 2 S . In figure 4, we show the fine-tuning measures, ∆ m 2 Z x for tan β = 3, M 0 = 1500 GeV, A κ = −100 GeV. The red dots satisfy the Higgs mass condition, 124 GeV < m h 1 < 126 GeV. We also plot blue dots representing the region where 124 GeV < m h 2 < 126 GeV. Both of them are favored by the mirage mediation within the 1-loop uncertainty. Here we choose the width, 125 ± 1 GeV as a guide to the eye. Note that currently the theoretical error of the SM-like Higgs mass in the NMSSM could reach 5 GeV [133] and actual bound is somewhat weaker than the bands shown in the figure. 8 The upper left and upper right panels show the case for λ and κ, respectively. Because the effective µ-term is roughly proportional to κ/λ as seen in the figure 2, severe fine-tuning is expected for these parameters. However, ∆ 10 is satisfied for m h 2 ≈ 125 GeV case unless m h 1 is not too light. This is not trivial because the khaki circles which do not necessarily satisfy the conditions required by 7 Essentially the same formula for y = m 2 Z but different presentation is given in the appendix C of [110]. 8 We thank F. Staub for pointing out this issue.  do not work as independent measures. Thus the worst measure is ∆ m Z λ , which is below 10 unless h 1 is extremely light (such a region is actually excluded by phenomenological constraints). Therefore the model requires at most 10% tuning (∆ −1 = 0.1) for tan β = 3 and M 0 = 1500 GeV. In our model the stop mass is √ c t L,R M 0 ≈ 1 TeV. Thus we numerically confirmed the model exhibits surprisingly low level of fine-tuning compared to the conventional SUSY breaking models.

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In figure 6, we summarize the same fine-tuning measures for the tan β = 10 case. The other input parameters are same as in figure 4. Here, we impose 124 GeV < m h 2 < 126 GeV. There is no parameter region satisfying m h 1 ≈ 125 GeV in this case.  Also we see fine-tuning measure ∆ µ m 2 S 10 if m h 1 60 GeV. It is remarkable that even the 5 TeV case is acceptable in the standard of the conventional models build around 1 TeV. Again the LEP bound for h 1 is weak where the fine-tuning measures are minimized (m h 1 ∼ 100 GeV).

Higgs couplings and collider signature
In this section, we discuss the couplings of the Higgs bosons to the SM particles. They play crucial roles in production and decay of the Higgs bosons at the LHC and future colliders like ILC, FCC-ee(TLEP)/CEPC. The precision measurements of the signal strength for various production and decay channels will constrain the couplings of the SM-like Higgs boson at O(1)% level or below [164][165][166][167][168][169]. The pattern of deviations from the SM predictions will bring hints for underlying models beyond the SM. Also there are chances to detect extra light Higgs bosons if their couplings to the SM particles are strong enough.
The interaction of the CP-even Higgs bosons h i (i = 1 − 3) with the SM fields can be described by the following effective Lagrangian [170]: Table 1. Coupling constants of the CP-even Higgs bosons in the κ = 0, m 2 S = 0 limit.
where we assume the custodial symmetry relating the W and Z couplings. The sum of f runs all the SM fermions. G a µν and A µν are the field strengths of the gluon and photon, respectively. The coupling constants in the first line are normalized so that C SM V = C SM f = 1 for the SM Higgs boson h SM at the tree level. In the NMSSM, they are given by where S ij is the diagonalization matrix for the mass matrix of the CP-even Higgs bosons. C i V is often quoted as ξ i in the literature and satisfies 3 i=1 ξ 2 i = 1. We have already depicted ξ 2 1 in the lower right panels in figure 1 and 2. These couplings are summarized in table 1 in the κ = 0, m 2 S = 0 limit. The singlet-like Higgs h S seems to be the lightest. However, once we include corrections from non-vanishing κ and m 2 S , it could be heavier than h for small tan β. The light doublet h behaves like the SM Higgs boson within the approximation. Note that C h V and C h u are stable against the small mixing between h and H induced by the finite κ and m 2 S , however, C h d,e could be affected considerably by the tan β enhanced C H d,e [107,112,171]. The coupling constants C i g and C i γ are loop induced. Generally, if we have a scalar S, a fermion f and a (color-less) vector boson ρ µ fields with the following couplings with the Higgs bosons [170], their contributions to C g i and C γ i are given by

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where T (r s ) (T (r f )) denotes the Dynkin index of the scalar (fermion) field in the representation r s (r f ) as defined by tr(T a (r)T b (r)) = T (r)δ ab with the SU(3) generator T a (r). N (r s ) (N (r f )) is the number of freedom in the representation r s (r f ). Q s,f,ρ are charges of the scalar, fermion and vector fields, respectively. The mass functions A s,f,v (τ ) are defined as Since A s,f,v (τ ) decouple quickly for τ 1, the standard model contribution is dominated by those of the top quark and the W boson and given by [170], For τ 1, we obtain A s,f,v 1. Then, we have, Together with the relation, we arrive at where M 2 s,f,ρ is the mass matrix for the corresponding field. Noticing that the coefficients of the derivatives are given by the steps of the beta function coefficients after integrating out JHEP02(2016)089 the corresponding fields, this can be cast into the well known low energy theorem [172][173][174][175][176]: if all the integrated fields are much heavier than the renormalization scale, µ. Here, h i dependence of α s and α enters through the mass thresholds. The contribution from the light fields should be included separately.
In SUSY models, additional contribution comes from the charged Higgs boson, sfermion and chargino loops. The charged Higgs contribution to δC h,S γ is calculated as The sfermion and chargino mass matrices are given by where v u,d = H u,d . The determinants of these mass matrices are calculated as Thus the squark-slepton contributions to C i g and C i γ are suppressed by v 2 /m 2 q,˜ at leading order and almost negligible if the squarks and sleptons are above the TeV scale. In addition, they are further suppressed if the following conditions are satisfied by the fields coupling via non-vanishing Yukawa interactions, It is interesting to observe that this is exactly the prediction of the TeV scale mirage mediation, with µ ≈ M 0 / tan β. 9 If any of the squarks or sleptons are light (e.g. ct R = 0), this provides an exception of the anti-correlation between the fine-tuning of the EW symmetry breaking and the new physics contribution to the Higgs couplings [177].
The chargino contribution can be comparable to the SM contribution if the chargino mass is light enough [107,170,178,179]. In the κ = 0, m 2 S = 0 limit, it is calculated as . (5.21) For i = H, we multiplied a suppression factor, A s (τ ) −12/ tan 2 β. If µ ∼ v, λ ∼ 1 and once the finite κ and m 2 S induce a mixing of O(10)% between the singlet and the heavy doublet, then O(10)% deviation is expected in C h γ . The effective Lagrangian describing the CP-odd Higgs bosons is given by tan β sin γ tan β cos γ C i u cot β sin γ cot β cos γ Table 2. Coupling constants of the CP-odd Higgs bosons in the limit κ = 0, where tan γ = λv/A λ .
The renormalizable couplings between the CP-odd Higgs boson and the vector bosons are forbidden by the CP symmetry. We omit the loop induced couplings with W /Z bosons because they do not give sizable effects in the production and decay of the Higgs bosons. The coupling constants with the fermions are given by where P is the diagonalization matrix for the mass matrix of the CP-odd Higgs bosons. These coupling constants in the κ = 0 limit are summarized in the table 2. Note that those of a 1 are highly suppressed by sin γ λv/A λ because they require the mixing with the heavy doublet. The coupling constants with gluon and photon are calculated as [180][181][182], where the mass function is given by In the limit τ i f 1, they are approximated as The charged Higgs contribution is forbidden by the CP symmetry and the sfermion contribution is highly suppressed as in the CP-even case. The chargino contribution to C γ is approximated as . For i = 2, we multiplied a suppression factor, A s (τ )/2 inside the bracket. If µ ∼ v, λ ∼ 1, C γ 1 could be comparable with C SM γ . Next we present the results of our numerical calculation on the coupling constants including the effect of finite κ and m 2 S , the radiative corrections and various phenomenological constraints. Here we use the scale factors of the Higgs couplings relative to the SM instead of the coupling constants themselves [183,184], (5.28) In figure 8, we plot the κ's for tan β = 3, M 0 = 1500 GeV and A κ = −100 GeV. The horizontal axis is chosen as the mass of the singlet-like Higgs boson which is heavier than the SM-like Higgs boson for small tan β. We impose the condition 124 GeV < m h 1 < 126 GeV and various phenomenological constraints built in the NMSSM Tools package. The parameter region with 124 GeV < m h 2 < 126 GeV is excluded by the LEP Higgs boson search [99]. The vertical lines show the expected precision of the coupling constants for LHC (300 f b −1 ) [164], ILC (250 GeV) and ILC (250 GeV+500 GeV) [165]. (The position of the center is arbitrary.) We also quote the numbers in table 3. We plot the precision of κ 2 Z for κ 2 V and take conservative values for the LHC expectation. The pseudo Nambu-Goldstone nature of the SM-like Higgs boson well suppresses the mixing with the singlet and the deviation from the SM is at most 20 %. This suppression is also important for not reducing the lighter SM-like Higgs boson mass due to the mixing. The approximate sum rule (κ 1 X ) 2 + (κ 2 X ) 2 = 1 means that the deviation mainly comes from the mixing with the singlet and the SUSY contribution is negligible except for b (τ ) for which the contamination of the heavy doublet has tan β enhanced coupling. The singletlike Higgs boson is hidden from the collider search and an interesting target for the precision measurement at the LHC and ILC. In this parameter region, the prediction does not change  Table 3. Expected precisions on the Higgs coupling for LHC [164] and ILC [165]. Two numbers for LHC represent different assumptions on the systematic error. The seven parameter HXSWG benchmark parametrization [183,184] is assumed.
significantly even if we adopt a conservative bound, 120 GeV < m h 1 < 130 GeV, taking into account the theoretical error [133].
In figure 9, we plot the same scale factors (normalized by the SM Higgs couplings) for tan β = 10. Increasing tan β, the singlet-like Higgs boson is now lighter than the SM-like Higgs boson and we impose 124 GeV < m h 2 < 126 GeV. We again choose the horizontal axis as the singlet-like Higgs mass. The tree-level contribution to the SMlike Higgs mass is now not effective with larger tan β and small mixing with the singlet is favorable to achieve 125 GeV. However, a strong constraint exists for the light singlet from the LEP Higgs boson search if it couples with the SM particles through the mixing with the SM-like Higgs boson [99]. The khaki shaded region is mostly excluded by this constraint. The region 90 GeV m h 1 100 GeV survives due to a small excess in the LEP measurement [99,103,[125][126][127][128][129][130][131][132]. Around m h 1 120 GeV, the two bosons almost degenerate and the mixing is maximized, but the LEP bound is weak in this region. For large λ, the new radiative corrections to the SM-like Higgs mass, eq. (4.11) could relax the mixing with the singlet (around m h 1 40 GeV). In this parameter region, the lightest neutralino becomes lighter than the half of the Z boson mass and the decay channel, Z → 2χ 0 1 opens. Then the region is excluded by the measurement of the Z boson width [185]. The mixing with the singlet leads to large deviations in the coupling constants of the SMlike Higgs boson observable at the LHC. On the other hand, searching the light singlet boson missed by LEP is important physics at ILC. The sum rule holds again except for b(τ ) and g, for which the mixing with the heavy doublet is not negligible due to the tan β enhancement. The small excess in (κ 2 γ ) 2 around m h 1 30 GeV is contribution from the chargino interfering with that of the SM. The surviving region expands once we raise the SUSY breaking scale M 0 and increase the top radiative correction to the SM-like Higgs mass. Also note that the prediction is sensitive to the theoretical error in the Higgs mass calculation. If we take a conservative bound, e.g. 120 GeV < m h 2 < 130 GeV, much lighter h 1 (m h 1 20 GeV) is allowed because the singlet-doublet mixing is suppressed. For such m h 1 , the h 1 couplings to the SM particles and the deviations of h 2 couplings from those of the SM Higgs boson are highly suppressed except for b and τ . In figure 10, we show the scale factors of the lightest CP-odd Higgs boson for the same parameter set with the CP-even bosons. The horizontal axis is chosen as the mass of a 1 , which is rather sensitive to the choice of the PQ symmetry breaking parameter, A κ . (κ t 1 ) 2 and (κ g 1 ) 2 are doubly suppressed by tan −2 β and sin 2 γ and negligible. The direct production of a 1 at the LHC is almost hopeless and the gamma-gamma option of ILC may help if (κ γ 1 ) 2 is large enough.

Dark matter
The existence of the dark matter (DM) is one of the few compelling evidences of physics beyond the SM. In SUSY, the Lightest Supersymmetric Particle (LSP) is stable and good JHEP02(2016)089 candidate of the dark matter, once we assume the R-parity to forbid the dimension 4 baryon/lepton number violating operators to prevent rapid nucleon decay. If LSP is weakly interacting massive particle (WIMP), its annihilation cross section predicts the thermal relic abundance in the ballpark of the observed DM density. This "WIMP miracle" scenario provides a valuable constraint on the SUSY models in light of the recent progress of precision cosmology (for more details, see e.g. [186]). However, in the mirage mediation, the oscillation of the modulus field after the inflation once dominates the energy density of the universe. The reheating temperature of the modulus field is O(100) MeV. It is much larger than the temperature at which the Big Bang Nucleosynthesis (BBN) takes place, however, lower than the decoupling temperature of LSP. Huge entropy produced by the modulus decay dilutes the preexistent thermal LSP abundance to negligible level. The

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modulus decay produces non-thermal LSP. In particular, daughter gravitino later decays at the decay temperature O(10) MeV and overproduces LSP (moduli-induced gravitino problem [188,189]). This non-thermal overproduction of DM does not mean the model is excluded, if we consider a specific non-minimal cosmological scenario or modify the DM sector of the model. For example, 1. Suppressing the modulus oscillation or reducing the gravitino branching ratio (e.g. enhanced symmetry [190], adiabatic oscillation [191,192], alignment [193]).
Explicit model building is beyond the scope of this paper. In this section, we just present the prediction of our model with the minimal DM sector assuming the first or the third scenario. The calculation is performed based on micrOMEGAs built in NMSSMTools [143][144][145].
In our model, LSP is given by mixture of the higgsino and singlino. Generally, the thermal relic of the pure higgsino of O(100) GeV cannot saturate the observed DM density. However, mixing with the singlino reduces the annihilation cross section and enables to explain the observed amount of DM (Well-tempered neutralino) in the first scenario as in the case of bino and higgsino in MSSM [103,200]. (There is no constraint in the third scenario.) The searches of DM are classified into the direct and indirect detections. The former detects the scattering of target nucleus by DM at underground laboratories. The latter observes the annihilation/decay products of DM in celestial/astrophysical circumstances such as the center of the earth/sun or galaxies.
We first discuss the prediction for the direct detection. In the direct detection experiments, the scattering between (slow) neutralino and target nucleus proceeds through the scalar or pseudo-vector interactions due to its Majorana nature. The former gives the spin-independent cross section and the latter gives the spin-dependent cross section [186]. In our model, the scalar interaction mainly comes from the exchange of the CP-even Higgs bosons and the pseudo-vector interaction mainly comes from the exchange of the Z boson. The spin-independent nuclear cross section is enhanced by the square of the mass number because the process is coherent, while the spin-dependent cross section does not have such a factor. Thus the experiments are much sensitive to the spin-independent cross section than the spin-dependent ones at nucleon level. On the other hand, the Z boson interacts with nucleon much stronger than the Higgs bosons. Therefore both cross sections could give meaningful constraints on our model, depending on the amount of higgsino component in LSP.
In the upper left panel of figure 11, we show the spin-independent cross section of LSP for tan β = 3 case. The red region indicates the region where the lightest CP-even boson corresponds to the SM Higgs boson and the blue region indicates the region where the second lightest CP-even boson is the SM-like Higgs boson (although excluded by the other phenomenological constraints). In the pink region, the thermal abundance saturates the observed value of DM density. Around m LSP ≈ 100, 350 GeV, the scattering amplitudes JHEP02(2016)089  [100,201]. The neutralino-Higgs boson coupling involves λ, therefore the red region is on the upper edge of the distribution because almost maximum λ is required to raise the Higgs mass to 125 GeV. With the particular model parameters chosen in the figure, the red region overlaps with the pink region where the thermal abundance saturates the DM, however, is almost excluded by the LUX experiment [202]. If we increases M 0 , the cross section decreases while the pink region remains in the almost same position. In the upper right and lower left panels, we present the spin-dependent cross sections for proton and neutron, respectively. The red region is again on the edge of the distribution because the singlino-higgsino mixing terms in the mass matrix are proportional to λ. The current experimental bounds are 1-2 order above the model prediction [203,204]. Next we discuss the indirect detection. We consider the constraint by the gamma ray from the dwarf spheroidal galaxies observed by Fermi satellite [205]. The annihilation of the neutralinos mainly produces W + W − , Z 0 Z 0 pairs, tt, bb pairs, h i a j and h i Z 0 j pairs if they are kinematically available. Other channels are strongly suppressed due to the Majorana nature of the neutralino and CP conservation [186]. In the lower right panel of figure 11, we show the total annihilation cross section of the neutralino and the experimental bound assuming the decay proceeds through W + W − and bb pairs. In our model, tt decay dominates the process above the threshold. The bound is one order larger than the prediction assuming that the gamma ray spectrum from tt is not much different from those of W W , bb pairs.

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In figure 12, we present the same set of the predictions for tan β = 10 case. The blue and cyan regions indicate the region where the second lightest CP-even boson corresponds to the SM Higgs boson. The blue region is already excluded by the phenomenological constraints explained in the previous section. In the pink region, the thermal LSP abundance saturates the observed DM density. In the surviving cyan region, the annihilation cross section is larger than the value saturating the observed DM density. The spin-independent cross section (upper left panel) shows a dip around m LSP ≈ 100 GeV due to the cancellation between the amplitudes mediated by the two CP even bosons. While the spin-dependent cross sections (upper right and lower left panels) do not show such a behavior because only one diagram dominates the process. They are enhanced with tan β because µ ≈ M 0 / tan β and the higgsino component in LSP increases with tan β. The cyan region is marginally excluded by the spin-independent cross section and the spin-dependent cross section for neutron. Discounting the constraints considering the astrophysical, hadronic and nuclear physics ambiguities in the calculation (see e.g. recent analysis in [187]), it is interesting that the surviving region is exactly located around the dip structure where the fine-tuning of the EW symmetry breaking is also minimized. Once observed, this feature can be checked by the experiment using the target nucleus insensitive to the spin-dependent cross section (enriched even-even nucleus) or combining the results of different target nuclei. In the lower right panel, the resonance structure in the annihilation cross section appears due to the pseudo-scalar mediated bb decay, however, this region is excluded by the phenomenological constraints. Above the W + W − , Z 0 Z 0 threshold, the process is dominated by W + W − and Z 0 Z 0 decays. The surviving region is about a factor 2 below the current bound. Once we raise the SUSY scale, M 0 , the constraint from the LEP Higgs boson search becomes weaker until the radiative correction overshoots 125 GeV and the cyan regions in the figures expand, while the EW fine-tuning deteriorates as shown in figure 7. The conservative Higgs mass bound, 120 GeV < m h 1 < 130 GeV opens a lighter LSP region (m LSP 50 GeV). However, this region is excluded by the upper bound of the spin-independent cross section, if the dominant component of the DM consists of the lightest neutralino.
In this section, we discussed the DM prediction of the model assuming the LSP saturates the observed DM abundance. We stress that the novel structure of the EW symmetry breaking in our model is not directly connected to the DM physics and the model will not be excluded even if the prediction of the minimal model is not confirmed by the experiments.

Conclusion
We have studied the NMSSM with the TeV scale mirage mediation. Our choice of the modular weights (4.1) realizes a little hierarchy m 2 Hu M 2 0 /8π 2 M 2 0 at the EW scale and automatic cancellation between µ 2 and m 2 H d / tan 2 β in (1.1). This leads to a natural EW symmetry breaking with O(10)% tuning for the TeV scale stop mass. The µ term can be as large as √ m Z M 0 without introducing a fine-tuning despite conventional wisdom tells that it lies around the EW scale. The Higgs boson mass is lifted to 125 GeV by the new quartic coupling (small tan β) or small mixing with the singlet (moderate tan β) in addition to the radiative corrections (from the top and singlino sectors). The suppression of the JHEP02(2016)089 singlet-doublet mixing is important for raising the Higgs boson mass in the first case and hiding the singlet from the LEP Higgs boson search in the second case. In the NMSSM, if κ = 0 and m 2 S = 0, the doublet Higgs VEVs spontaneously break an approximate scale symmetry (4.12). The resultant pseudo Nambu-Goldstone boson (mass eigenstate) almost consists of the doublets and corresponds to the SM-like Higgs with a suppressed singlet component. There is another scale symmetry if κ = 0 and m 2 Hu = 0 in which S and H u are interchanged. The corresponding Nambu-Goldstone boson is the singlet-like Higgs boson with suppressed H u component. The mixing with the suppressed components must break the two symmetries and disappear if H d decouples. Thus it is highly suppressed although the symmetries are broken at the EW scale. To evaluate the fine-tuning, we introduced a simple formula for the fine-tuning measure having a clear physical meaning which calls for careful treatment of the radiative corrections. Using it, we performed a comprehensive analysis of the fine-tuning in the EW symmetry breaking which shows the model is tuned at most O(10)% for M 0 = 1.5 TeV. For moderate tan β, the least tuned region is realized with the singlet Higgs boson mass around 100 GeV which overlaps with the anomaly found in the LEP Higgs boson search.
We further investigated the coupling constants of the Higgs bosons in our model. The radiative corrections from the SUSY particles and heavy Higgs bosons are highly suppressed due to the little hierarchy. Thus the deviation from the SM occurs through the suppressed mixing between the SM-like and singlet Higgs bosons. The numerical calculation showed that 20 % deviations are possible in the squares of the scale factors, (κ i X ) 2 and the sum rule (κ 1 X ) 2 + (κ 2 X ) 2 = 1 holds well except for those of b and τ . The mixing with the heavy doublet is also important for b and τ due to tan β enhancement and an O(1) deviation is possible for their couplings to the SM-like Higgs boson. Future precision measurements at the LHC and the ILC will provide good opportunities to explore these deviations and the hidden singlet-like scalar.
We also studied prospects for the direct/indirect dark matter searches assuming the minimal dark matter sector in our model. The spin-independent neutralino-nucleus scattering is mainly mediated by the SM-like and singlet-like Higgs bosons. The two amplitudes interfere destructively in some parameter regions. This suppresses the cross section below the current experimental bound. While the spin-dependent scattering is dominated by the Z boson exchange and does not show such a feature. The surviving regions from various phenomenological constraints are almost excluded for tan β = 3 and on the verge of exclusion for tan β = 10. The suppression of the spin-independent cross section in the surviving region is characteristic signature to identify the model. The bound on the neutralino annihilation cross section from the gamma ray emission by the dwarf spheroidal galaxy is an order above the prediction and will soon start to explore our model.

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Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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