Higgs-Portal Dark Matter in Nonlinear MSSM

Supersymmetric (SUSY) extension of the Standard Model (SM) is a primary candidate for new physics beyond the SM. If SUSY breaking scale is very low, for example, the multi-TeV range, and the SUSY breaking sector, except for the goldstino (gravitino), is decoupled from the low energy spectrum, the hidden sector effect in the minimal SUSY SM (MSSM) is well described by employing the goldstino chiral superfield ($X$) with the nilpotent condition of $X^2=0$. Although this so-called"nonlinear MSSM"(NL-MSSM) provides a variety of interesting phenomenologies, there is a cosmological problem that the lightest superpartner gravitino is too light to be the major component of the dark matter (DM) in our universe. To solve this problem, we propose a minimal extension of the NL-MSSM by introducing a parity-odd SM singlet chiral superfield ($\Phi$). We show that the interaction of the scalar component in $\Phi$ with the MSSM Higgs doublets is induced after eliminating F-component of the goldstino superfield and the lightest real scalar in $\Phi$ plays the role of the Higgs-portal DM. With a suitable choice of the model parameters, a successful Higgs-portal DM scenario can be realized while achieving the SM-like Higgs boson mass of 125 GeV from the tree-level Higgs potential through the multi-TeV SUSY breaking effect.


Introduction
Although the current experimental data show no plausible evidence of new physics beyond the Standard Model (SM), the minimal supersymmetric (SUSY) extension of the SM (MSSM) is still a primary candidate for new physics. As has been well-known and intensively studied, the MSSM not only provides us with a solution to the gauge hierarchy problem but also offers a variety of interesting phenomenologies, such as the origin of the electroweak symmetry breaking from SUSY breaking, the SM-like Higgs boson mass prediction with soft SUSY breaking parameters, the lightest superpartner (LSP) as a natural candidate for the dark matter (DM) in our universe, and the grand unified theory paradigm with the successful unification of the three SM gauge couplings at a scale of O(10 16 GeV). Many ongoing and planned experiments will continue searching for the MSSM, or in more general, supersymmetric theories beyond the SM.
In phenomenologically viable models, SUSY is spontaneously broken in the hidden sector and the SUSY breaking effects are mediated to the MSSM sector by a certain mechanism for generating soft SUSY breaking terms in the MSSM. Associated with spontaneous SUSY breaking, a massless fermion called goldstino emerges due to the Nambu-Goldstone theorem, and it is absorbed into the spin-1/2 component of the spin-3/2 massive gravitino in supergravity. The gravitino mass is characterized by the SUSY breaking order parameter f and the reduced Planck mass of M P = 2.43 × 10 19 GeV as m 3/2 f /M P . It is possible that SUSY breaking occurs at a very low energy (see, for example, Ref. [1]). If this is the case, gravitino becomes the LSP and is involved in phenomenology at low energies. For example, if the SUSY breaking scale lies in the multi-TeV range, the LSP gravitino is extremely light with its mass of O(meV). Assuming the decoupling of the hidden sector fields except for the light gravitino (or, equivalently, goldstino) the low energy effective theory involving the very light gravitino can be described by employing a goldstino chiral superfield X with the nilpotent condition X 2 = 0 [2,3,4]. With this formalism, the phenomenology of the MSSM with the goldstino superfield has been studied in detail [5,6,7] (see also Ref. [8] for the phenomenology in a more general setup). This framework is the so-called nonlinear MSSM (NL-MSSM). In particular, it has been shown that if the SUSY breaking scale lies in the muti-TeV range, the SM-like Higgs boson receives a sizable contribution to its mass at tree-level after eliminating F -component of the goldstino superfield and as a result, the Higgs mass of around 125 GeV can be achieved even without the scalar top-quark quantum corrections.
Although the extremely light gravitino in the NL-MSSM is harmless in the phenomenological point of view (see, for example, Ref. [9]), its relic density is far below the observed dark matter (DM) density. Thus, for the completion of the NL-MSSM, we may consider an extension of the model which can supplement the model with a suitable DM candidate. In this paper, we propose a minimal extension of the NL-MSSM by introducing a Z 2 -parity odd SM gauge singlet chiral superfield Φ and show that the lightest scalar component in Φ plays the role of the Higgs-portal DM [10,11] 1 through its coupling with the MSSM Higgs doublets induced by the goldstone superfield. With a suitable choice of the model parameters, we can realize a phenomenologically viable Higgs-portal DM sceanrio while achieving the 125 GeV mass for the SM-like Higgs boson from the Higgs potential at the tree level.

NL-MSSM and the Higgs boson mass
We first present the basic formalism of the NL-MSSM and show how the 125 GeV SM-like Higgs boson mass can be achieved in the framework. We begin with the goldstino effective Lagrangian of the form [4]: where X is a goldstino chiral superfield, and f is the SUSY breaking order parameter in the hidden sector. Although the stability of the hidden sector scalar potential needs an extension of the above minimal Kähler potential, this Lagrangian is enough to understand the essence of the formalism. The goldstino chiral superfield is subject to the nilpotent condition [2,3,4], which leads us to express the superfield with the components, The scalar component in the goldstino superfield is to be integrated out in the low energy effective theory, and under the nilpotent condition, it is replaced by the bilinear term of the goldstino fields. In fact, substituting Eq. (3) into Eq. (1) and eliminating the auxiliary field F X , we recover the Volkov-Akulov Lagrangian [13].
In the superfield formalism, the spurion technique is a simple way to introduce the soft SUSY breaking terms to the MSSM Lagrangian. We introduce a dimensionless and SM-singlet spurion field of the form, Y = θ 2 m soft , where m soft is a generic notation for the soft terms (denoted m 1,2,3 , m Ψ , m λa in the following), and attach it to any SUSY operators in the MSSM. The recipe to obtain the NL-MSSM is to replace the spurion by the goldstino superfield as [4] Y → m soft f X .
We apply this rule and write the NL-MSSM Lagrangian as follows [5]: In the right-hand side, the first term L 0 denotes the supersymmetric part of the MSSM Lagrangian given by 2 where Ψ = Q, U c , D c , L, E c , the index a = 1, 2, 3 denotes the the SM gauge groups SU (3), SU (2) and U (1), g a is the corresponding gauge couplings, and κ = 1 for U (1) and 1/2 for SU (3) and SU (2). The vector superfield V in the Kahler potential for the chiral superfields implies, for example, where V a (a = 1, 2, 3) denote the vector superfields of the corresponding SM gauge groups. L X is the hidden sector Lagrangian already introduced in Eq. (1). L H is the Higgs sector Lagrangian involving the goldstino sueprfield: The matter field Lagrangian involving the goldstino superfield is given by The bilinear and trilinear SUSY breaking couplings are given by L AB : The last term L g denotes the gauge sector Lagrangian given by We focus on the Higgs potential in the NL-MSSM, which is read off from L 0 +L X +L H +L AB : where with g 2 Z ≡ g 2 1 + g 2 2 . We express the up-type Higgs and down-type Higgs doublets as where GeV, H ± are charged Higgs fields, and R u , I u , R d , I d are real scalar fields. Substituting them into the Higgs potential, we derive the stationary conditions: where | 0 means that all the fields are taken to be zero, The other stationary conditions such as ∂V ∂Iu 0 are automatically satisfied. The mass matrix of the CP-even Higgs bosons is given by while the mass matrices for the CP-odd Higgs bosons and the charged Higgs bosons are By using the above formulas, we numerically calculate the Higgs boson mass spectra.
Therefore, if the SUSY breaking scale is low enough, the SM-like Higgs boson mass of 125 GeV is achieved by the Higgs potential at the tree-level. As shown in Fig. 1, we have obtained m h = 125 GeV for √ f = 3990 GeV. Fig. 2 shows the masses of the heavy neutral Higgs and the charged Higgs bosons as a function of √ f with the same inputs as in Fig. 1. The solid line depicts to the mass of the heavy CP-even Higgs boson (m H ) while the dashed and dotted lines correspond to the CP-odd Higgs boson mass (m A ) and the charged Higgs boson mass (m H ± ), respectively.

Minimal extension with Higgs-portal dark matter
As we have shown that if the SUSY breaking scale lies in the multi-TeV range, the SM-like Higgs boson mass of 125 GeV can be achieved even at the tree-level. Such a low SUSY breaking scale sets the gravitino mass to be O(meV). Although this extremely light gravitino (goldstino) is harmless in phenomenological point of view, it is unable to be the dominant component of the DM in our universe and a suitable DM candidate should be supplemented. In order to solve this problem, we propose a minimal extension of the NL-MSSM to incorporate a dark matter candidate, namely, the (scalar) Higgs-portal DM.
The Higgs-portal DM scenario is one of the simplest SM extensions to supplement the SM with a dark matter candidate. In the simplest setup, we introduce an SM-singlet real scalar (S) along with a Z 2 symmetry. The stability of this scalar is ensured by assigning an odd-parity to it, while all the SM fields are Z 2 -even. At the renormalizable level, the SM gauge and Z 2 symmetries allow the scalar S to have only one (non-self) interaction term, where H is the SM Higgs doublet field. The scalar DM S communicates with the SM sector only through this Higgs-portal interaction and the DM phenomenology in this Higgs-portal DM scenario is controlled by only two free parameters: λ HSS and the DM mass (m S ). Phenomenological constraints on the two free parameters have been intensively studied, and the allowed parameter region has been identified to be consistent with the cosmological observations, the direct/indirect DM particle search results and the Higgs-portal DM search results by the Large Hadron Collider (LHC) experiment. The Higgs-portal DM scenario is phenomenologically viable, but the allowed parameter region is very limited [12]: m S M H /2 with the SM Higgs boson mass M H = 125 GeV and 10 −4 |λ HSS | 10 −3 . Now we introduce an SM-singlet chiral superfield Φ along with a Z 2 symmetry and assign odd-parity to it while even-parity to all the MSSM fields. Hence, the lightest component field in Φ is stable and the DM candidate. The SUSY Lagrangian L 0 in Eq. (6) is then extended to be where µ Φ is a mass parameter. Similar to L H and L m , a new Lagrangian for Φ involving the goldstino chiral superfield is given by where m Φ denotes a soft SUSY breaking mass. Finally, L AB is extended to be In the following, we assume that B Φ is real and positive. We now read off the scalar potential relevant to the Higgs-portal DM scenario by eliminating the auxiliary fields: where Although the complete form of the scalar potential includes all the sfermions in the MSSM, we have consider the potential terms involving only the MSSM Higgs doublets and the SMsinglet scalar Φ. This is because the sfermions should be heavy to satisfy the current LHC constraints and their couplings with the Higgs-portal DM have little effects on the DM physics. For the physics of the Higgs-portal DM scenario, only the bilinear terms with respect to Φ are important. To extract them from the scalar potential, we expand V soft up to the order of O(1/f 2 ) and then obtain we identify the complex scalar Φ with the DM particle. This is a complex scalar extension of the simplest Higgs-portal DM scenario with only one real scalar. Since the MSSM includes two Higgs doublets, our Higgs-portal DM scenario is basically two Higgs doublet extension of the Higgs-portal DM scenario. In general, the heavy Higgs bosons can play an important role for the DM physics, for example, an enhancement of the DM pair annihilations through the heavy Higgs boson resonances. We leave such a general analysis for future work.