Higgs-Portal Dark Matter in the Nonlinear MSSM

PHYSICAL REVIEW D 103, 015027 (2021)

Higgs-portal dark matter in the nonlinear MSSM

† Masato Arai1,* and Nobuchika Okada 2, 1Faculty of Science, Yamagata University, Yamagata 990-8560, Japan 2Department of Physics and Astronomy, University of Alabama, Tuscaloosa, Aalabama 35487, USA

(Received 17 August 2020; accepted 8 January 2021; published 25 January 2021)

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 X2 ¼ 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 (Φ). We show that the interaction of the scalar component in Φ with the MSSM Higgs doublets is induced after eliminating F component of the Goldstino superfield, and the lightest real scalar in Φ 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. In addition, if SUSY breaking scale lies in the multi-TeV range, the SM-like Higgs boson mass of 125 GeV can be achieved by the tree-level Higgs potential through the low-scale SUSY breaking effect.

DOI: 10.1103/PhysRevD.103.015027

I. INTRODUCTION In phenomenologically viable models, SUSY is sponta- Although the current experimental data show no plau- neously broken in the hidden sector, and the SUSY sible evidence of new physics beyond the Standard Model breaking effects are mediated to the MSSM sector by a (SM), the minimal supersymmetric (SUSY) extension of certain mechanism for generating soft SUSY breaking the SM (MSSM) is still a primary candidate for new terms in the MSSM. Associated with spontaneous SUSY physics. As has been well-known and intensively studied, breaking, a massless fermion called Goldstino emerges due the MSSM not only provides us with a solution to the gauge to the Nambu-Goldstone theorem, and it is absorbed into hierarchy problem but also offers a variety of interesting the spin-1=2 component of the spin-3=2 massive gravitino phenomenologies, such as the origin of the electroweak in supergravity. The gravitino mass is characterized by the symmetry breaking from SUSY breaking, the SM-like SUSY breaking order parameter f and the reduced Planck 19 Higgs boson mass prediction with soft SUSY breaking mass of MP ¼ 2.43 × 10 GeV as m3=2 ≃ f=MP.Itis parameters, the lightest superpartner (LSP) as a natural possible that SUSY breaking occurs at a very low energy candidate for the dark matter (DM) in our Universe, and the (see, for example, Ref. [1]). If this is the case, the gravitino grand unified theory paradigm with the successful uni- becomes the LSP and is involved in phenomenology at low fication of the three SM gauge couplings at the scale of energies. For example, if the SUSY breaking scale lies in 16 Oð10 GeVÞ. Many ongoing and planned experiments the multi-TeV range, the LSP gravitino is extremely light will continue searching for the MSSM, or in more general, with its mass of OðmeVÞ. Assuming the decoupling of supersymmetric theories beyond the SM. 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 nilpo- *[email protected] † 2 [email protected] tent condition X ¼ 0 [2–4]. With this formalism, the phenomenology of the MSSM with the Goldstino super- Published by the American Physical Society under the terms of field has been studied in detail [5–7] (see also Ref. [8] for the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the phenomenology in a more general setup). This frame- the author(s) and the published article’s title, journal citation, work is the so-called nonlinear MSSM (NL-MSSM). and DOI. Funded by SCOAP3. Interestingly, it has been shown that if the SUSY breaking

2470-0010=2021=103(1)=015027(9) 015027-1 Published by the American Physical Society MASATO ARAI and NOBUCHIKA OKADA PHYS. REV. D 103, 015027 (2021) scale lies in the multi-TeV range, the SM-like Higgs boson X2 ¼ 0; ð2Þ receives a sizable contribution to its mass at the tree-level after eliminating F component of the Goldstino superfield, which leads us to the expression of the superfield with the and as a result, the Higgs boson mass of around 125 GeV components, can be achieved by the tree-level Higgs potential. This is in ψ ψ pffiffiffi sharp contrast with the usual MSSM in which the 125 GeV X X X ¼ þ 2θψX þ θθFX: ð3Þ SM-like Higgs boson mass is reproduced by quantum 2FX corrections through scalar top quarks with the mass larger than multi-TeV. In the viewpoint of the collider physics, the The scalar component in the Goldstino superfield is to be NL-MSSM has an advantage that the scalar top quarks can integrated out in the low energy effective theory, and under be sufficiently light to be explored in thepffiffiffi near future. the nilpotent condition, it is replaced by the bilinear term of The SUSY breaking order parameter f ≲ Oð100Þ TeV the Goldstino fields. In fact, substituting Eq. (3) into Eq. (1) gives the extremely light gravitino with mass m3=2 ≲ 10 eV and eliminating the auxiliary field FX, we recover the in the NL-MSSM. Although such a light gravitino is Volkov-Akulov Lagrangian [13]. harmless in the phenomenological point of view (see, In the superfield formalism, the spurion technique is a for example, Ref. [9]), its relic density is far below the simple way to introduce the soft SUSY breaking terms observed dark matter (DM) density. Even if the observed to the MSSM Lagrangian. We introduce a dimensionless ¼ θ2 relic density is achieved by some nonstandard thermal and SM-singlet spurion field of the form, Y msoft, history of the Universe, the very light gravitino is likely to where msoft is a generic notation for the soft terms (denoted be a hot DM and prevents the formation of the observed m1;2;3, mΨ, mλa in the following) and attach it to any structure of the Universe. Therefore, for the completion of SUSY operators in the MSSM. The recipe to obtain the the NL-MSSM, we should consider an extension of the NL-MSSM is to replace the spurion by the Goldstino model which can supplement the model with a suitable DM superfield as [4] candidate. In this paper, we propose a minimal extension of m the NL-MSSM by introducing a Z2-parity odd SM gauge Y → soft X: ð4Þ singlet chiral superfield Φ and show that the lightest scalar f component in Φ plays the role of the Higgs-portal DM [10,11]1 through its coupling with the MSSM Higgs We apply this rule and write the NL-MSSM Lagrangian as doublets induced by the Goldstino superfield. With a follows [5]: suitable choice of the model parameters, we can realize L ¼ L þ L þ L þ L þ L þ L ð Þ a phenomenologically viable Higgs-portal DM scenario. 0 X H m AB g: 5 If SUSY breaking scale lies in the multi-TeV range, the In the right-hand side, the first term L0 denotes the SM-like Higgs boson mass of 125 GeV can be achieved by 2 the tree-level Higgs potential through the low-scale SUSY supersymmetric part of the MSSM Lagrangian given by breaking effect. X Z 4 † V L0 ¼ d θΨ e Ψ Ψ;H ;H II. NL-MSSM AND THE HIGGS BOSON MASS u Z d 2 c c We first present the basic formalism of the NL-MSSM þ d θ½μHHdHu þ λuHuQU þ λdQD Hd and show how the 125 GeV SM-like Higgs boson mass can be achieved in the framework. We begin with the Goldstino þ λ c þ effective Lagrangian of the form [4], eLE Hd H:c: Z Z Z X3 1 4 † 2 2 α L ¼ θ þ θ þ ð Þ þ d θTr½W W αþH:c:; ð6Þ X d X X d fX H:c: ; 1 4 2κ a a a¼1 ga where X is a Goldstino chiral superfield, and f is the SUSY where Ψ ¼ Q, Uc, Dc, L, Ec, the index a ¼ 1, 2, 3 denotes breaking order parameter in the hidden sector. Although the SM gauge groups SUð3Þ, SUð2Þ, and Uð1Þ, ga is the the stability of the hidden sector scalar potential needs an corresponding gauge couplings, and κ ¼ 1 for Uð1Þ and extension of the above minimal Kähler potential, this 1=2 for SUð3Þ and SUð2Þ. The vector superfield V in Lagrangian is enough to understand the essence of the the Kähler potential for the chiral superfields implies, 1 formalism. The Goldstino chiral superfield is subject to the for example, V ¼ 2V3 þ 2V2 þ 3 V1 for Q etc., where nilpotent condition [2–4], 2For a concise review of the MSSM and the standard notation, 1For a recent review, see Ref. [12] and references therein. see, for example, Ref. [14].

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þ Va (a ¼ 1, 2, 3) denote the vector superfields of the H L H ¼ ; ð14Þ corresponding SM gauge groups. X is the hidden sector u p1ffiffi ð þ þ Þ 2 vu Ru iIu Lagrangian already introduced in Eq. (1). LH is the Higgs sector Lagrangian involving the Goldstino sueprfield, p1ffiffi ðv þ R þ iI Þ ¼ 2 d d d ð Þ Z Hd − ; 15 m2 H L ¼ − 1 4θð † Þ † V H 2 d X X Hde Hd f Z where vu ¼ v sin β, vd ¼ v cos β with v ¼ 246 GeV, H m2 − 2 4θð † Þ † V ð Þ are charged Higgs fields, and R , I , R , I are real scalar 2 d X X Hue Hu: 7 u u d d f fields. Substituting them into the Higgs potential, we derive the stationary conditions, The matter field Lagrangian involving the Goldstino super- ∂ field is given by V ¼ v 4μ2 β − 2 2 2β β ∂ 4 H sin MZ cos sin Z Ru 0 X 2 m 2 2 2 L ¼ − Ψ 4θð † ÞΨ† VΨ ð Þ 2Am cos β A m sin β m 2 d X X e : 8 − 3 þ 2 ¼ 0; ð16Þ Ψ f B B2 ∂ The bilinear and trilinear SUSY breaking couplings are V ¼ v 4μ2 β þ 2 ð β þ 3βÞ L H cos MZ cos cos given by AB, ∂R 4 d 0 Z 2Am2 sin β A2m2 cos β m2 − 3 þ 1 ¼ 0; ð17Þ L ¼ 3 d2θXH H þ H:c: B B2 AB f d u Z A A where j means that all the fields are taken to be zero, and þ d2θX λ u UcH Q þ λ d DcH Q 0 u f u d f d 1 2 ¼ 2 2 Ae c MZ gZv ; þ λe E HdL þ H:c: ð9Þ 4 f 2 2 2 A ¼ 4f þ m3v sin 2β; B ¼ −2 2 þ 2 2 2β þ 2 2 2β ð Þ The last term Lg denotes the gauge sector Lagrangian f m1v cos m2v sin : 18 given by ∂V The other stationary conditions such as j0 are automati- ∂Iu X3 Z 1 2mλ cally satisfied. The mass matrix of the CP-even Higgs L ¼ a 2θ ½ α þ ð Þ g 4 2κ d XTr WaWaα H:c: 10 bosons is given by ¼1 ga f a 0 1 2 2 ∂ V ∂ V ∂ 2 ∂R ∂R We focus on the Higgs potential in the NL-MSSM, B Rd 0 d u 0 C MCP− ¼ @ A; ð19Þ which is read off from L0 þ L þ L þ L , even 2 2 X H AB ∂ V ∂ V ∂ ∂ ∂ 2 Ru Rd 0 Ru 0 V ¼ V þ V ; ð11Þ SUSY soft while the mass matrices for the CP-odd Higgs bosons and the charged Higgs bosons are where 0 1 ∂2V ∂2V 2 2 ∂ ∂ g B ∂I Id Iu C ¼ μ2 ðj j2 þj j2Þþ Z ðj j2 − j j2Þ2 d 0 0 VSUSY Hu Hd Hu Hd M − ¼ @ A; H 8 CP odd 2 2 ∂ V ∂ V ∂ ∂ ∂ 2 2 Iu Id 0 Iu 0 g2 † 2 0 1 þ jHuH j ; ð12Þ 2 2 2 d ∂ V ∂ V B ∂H−∂H− ∂H−∂Hþ C M ¼ 0 0 ð Þ charged @ A: 20 2 2 ∂2 ∂2 m3 V V jf þ H H j ∂H−∂Hþ ∂Hþ∂Hþ ¼ f u d ð Þ 0 0 Vsoft 2 2 ; 13 m1 2 m2 2 1 − 2 jH j − 2 jH j f d f u By using the above formulas, we numerically calculate the Higgs boson mass spectra.p First,ffiffiffi we choose appropriate 2 ≡ 2 þ 2 β with gZ g1 g2. We express the up-type Higgs and values for m1, m2, tan , and f as the input parameters down-type Higgs doublets as and solve the stationary conditions of Eqs. (16) and (17) to

015027-3 MASATO ARAI and NOBUCHIKA OKADA PHYS. REV. D 103, 015027 (2021) m2 2 m2 ≃ M2 cos 2β þ 2 2 v2sin2β: ð21Þ h Z f

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. Aspffiffiffi shown in Fig. 1,we have obtained mh ¼ 125 GeV for f ¼ 3990 GeV. If the SUSY breaking scale is larger, the hidden sector effect on the SM-like Higgs boson mass is negligible, so that quantum corrections through heavy scalar top quarks play the crucial role to reproduce mh ¼ 125 GeV, as usual in the MSSM. Figure 2 shows the masses of the heavy neutralpffiffiffi Higgs and the charged Higgs bosons as a function of f with the same inputs as in Fig. 1. The solid line depicts to FIG. 1. The SM-likepffiffiffi Higgs boson mass (mh) at the tree-level as a function of f (solid line), along with the standard MSSM the mass of the heavy CP-even Higgs boson (mH) while the prediction at the tree-level (dashed line), and the (green) dashed and dotted lines correspond to the CP-odd Higgs ¼ 125 horizontal line indicting mh GeV. In this plot, we have boson mass (mA) and the charged Higgs boson mass (mH ), 2 2 2 2 2 2 taken m1 ¼ 1000 GeV , m2 ¼ −ð2005Þ GeV , and tan β ¼ 10. respectively.

III. MINIMAL EXTENSION WITH fix the values of μ and m2. We then substitute them into HIGGS-PORTAL DARK MATTER H 3 pffiffiffi the Higgs potential and calculate the Higgs boson mass If the SUSY breaking scale is f ≲ 100 TeV, the eigenvalues from Eqs. (19) and (20). Our results are shown gravitino mass is found to be m3 2 ≲ 10 eV. Although in Figs. 1 and 2. The solid line in Fig. 1 shows the mass of = pffiffiffi such a light gravitino (Goldstino) is harmless in a phe- the SM-like Higgs boson (mh) as a function of f, where nomenological point of view, it is unable to be the dominant 2 ¼ 10002 2 2 ¼ −ð2005Þ2 2 we have fixed m1 pffiffiffi GeV , m2 GeV , component of the DM in our Universe, and therefore, a and tan β ¼ 10.As f decreases, the SM-like Higgs boson suitable DM candidate should be supplemented to the mass increases from the standard MSSM prediction at the NL-MSSM. In order to solve this problem, we propose a ≃ 2β pthreeffiffiffi level mh MZ cos (dashed line) in the limit of minimal extension of the NL-MSSM to incorporate a dark f → ∞. The (green) horizontal line indicates matter candidate, namely, the (scalar) Higgs-portal DM. mh ¼ 125 GeV. We find that the main contribution for The Higgs-portal DM scenario is one of the simplest SM increasing the SM-like Higgs boson mass comes from the extensions to supplement the SM with a dark matter 2 2 2 quartic coupling ðm2jHuj =fÞ in the series of expansion of candidate. For a recent review, see Ref. [12] and references Eq. (13), and the resultant Higgs boson mass is approxi- therein. In the simplest setup, we introduce an SM-singlet mately expressed as real scalar (S) along with a Z2 symmetry. The stability of this scalar is ensured by assigning an odd parity to it, while all the SM fields are Z2 even. At the renormalizable level, the Lagrangian is given by 1 1 1 L ¼ L þ ð∂ Þð∂μ Þ − 2 2 − λ 4 SM 2 μS S 2 MSS 4 SS 1 − λ ð † Þ 2 ð Þ 4 HSS H H S ; 22 L where SM is the Lagrangian of the SM, and H is the SM Higgs doublet field. After the electroweak symmetry breaking, the Lagrangian becomes 1 1 1 L ¼ L þ ð∂ Þð∂μ Þ − 2 2 − λ 4 SM 2 μS S 2 mDMS 4 SS 1 1 − λ vhS2 − λ h2S2; ð23Þ FIG. 2. Same as Fig. 1 but for the CP-even heavy Higgs 4 HSS 8 HSS boson mass (mH) (solid line), the CP-odd Higgs boson mass (mA) (dashed line), and the charged Higgs boson mass (mH ) where h is the physical Higgs boson, and the DM mass (dotted line). mDM is given by

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FIG. 3. Feynman diagrams for dark matter annihilations.

1 Higgs-portal DM, g⋆ is the effective number of total degrees m2 ¼ M2 þ λ v2: ð24Þ DM S 4 HSS of freedom for the particles in thermal equilibrium (g⋆ ¼ 106.75 for the SM particles), and K2 is the modified Here, the vacuum expectationpffiffiffi value of the Higgs field is Bessel function of the second kind. The thermal-averaged set to be hHi¼ð0;vÞT= 2 with v ¼ 246 GeV. The DM annihilation cross section is calculated by phenomenology in this Higgs-portal DM scenario is con- Z λ 1 m ∞ trolled by only two free parameters: mDM and HSS. hσ i¼ 2 DM 2ð − 4 2 Þ v gDM 4 ds s mDM The scalar DM S annihilates into the SM particles nEQ 64π x 4m2 pffiffiffiDM through its coupling with the Higgs boson. The annihilation pffiffiffi ð Þ x s processes are shown in Fig. 3, where W Z is the charged × σðsÞ sK1 ; ð27Þ (neutral) weak gauge boson, and f represents quarks and mDM leptons in the SM. We evaluate the DM relic density by σð Þ solving the Boltzmann equation [15], where s is the DM pair annihilation cross section corresponding to the processes in Fig. 3,andK1 is the dY sðm Þ hσv i modified Bessel function of the first kind. By using the ¼ − DM rel ð 2 − 2 Þ ð Þ ð∞Þ ð Þ 2 Y YEQ ; 25 asymptotic value of the yield Y , the DM relic density dx H mDM x Ω 2 DMh is expressed by where the temperature of the Universe is normalized by the ð∞Þ ¼ ð Þ ð Þ 2 mDMs0Y DM mass as x mDM=T, H mDM and s mDM are the Ω h ¼ ; ð Þ DM ρ 2 28 Hubble parameter, and the entropy density of the Universe at c=h ¼ ¼ T mDM, respectively, Y n=s is the DM yield [the ratio −3 of the DM number density (n) to the entropy density (s)], where s0 ¼ 2890 cm is the entropy density of the present ρ 2 ¼ 1 05 10−5 −3 YEQ is the yield of the DM particle in thermal equilibrium, Universe, and c=h . × GeV cm is the critical hσ i density. and vrel is the thermal-averaged DM annihilation cross section times relative velocity (v ). The formulas for the The resultant DM relic density is controlled by two free rel λ quantities in the Boltzmann equation are given as follows: parameters (mDM and HSS), and their relation is determined so as to reproduce the observed DM relic density of rffiffiffiffiffiffiffiffiffiffi 2 2 2 2 Ω h ¼ 0.12 [16]. In addition to the DM relic density, 2π π T DM ð Þ¼ 3 ð Þ¼ the parameter space of m and λ are constrained by the s T 45 g⋆T ;HT 90 g⋆ ; DM HSS MP direct/indirect DM particle search results and the Higgs- 3 gDM mDM portal DM search results by the Large Hadron Collider n ¼ sY ¼ K2ðxÞ; ð26Þ EQ EQ 2π2 x (LHC) experiment. After all the constraints are taken into account, the allowed parameter region is identified. 18 where MP ¼ 2.43 × 10 GeV is the reduced Planck mass, For the result, see, for example, Fig. 19 in Ref. [12]. ¼ 1 gDM is the number of degrees of freedom for the It has been found that the Higgs-portal DM scenario is

015027-5 MASATO ARAI and NOBUCHIKA OKADA PHYS. REV. D 103, 015027 (2021) phenomenologically viable, but the allowed parameter We now read off the scalar potential relevant to the ≃ 2 region is very limited: mDM Mh= with the SM Higgs Higgs-portal DM scenario by eliminating the auxiliary −4 −3 boson mass Mh ¼ 125 GeV and 10 ≲ jλHSSj ≲ 10 . fields, Now we introduce an SM-singlet chiral superfield Φ ¼ þ ð Þ along with a Z2 symmetry and assign odd parity to it V VSUSY Vsoft; 32 while even parity to all the MSSM fields. Hence, the lightest component field in Φ is stable and the DM where candidate. The SUSY Lagrangian L0 in Eq. (6) is then ¼ μ2 ðj j2 þj j2Þþμ2 jΦj2 extended to be VSUSY H Hu Hd Φ Z Z 2 2 þ gZ ðj j2 − j j2Þ2 þ g2 j † j2 ð Þ 4 † 2 2 Hu Hd HuHd ; 33 L0 → L0 þ d θΦ Φ þ d θμΦΦ þ H:c: ; ð29Þ 8 2

2 2 j þ m3 − BΦ Φ2j f f HuHd 2f μΦ L L ¼ ð Þ where is a mass parameter. Similar to H and m, a new Vsoft 2 2 2 : 34 Φ 1 − m1 j j2 − m2 j j2 − mΦ jΦj2 Lagrangian for involving the Goldstino chiral superfield f2 Hd f2 Hu f2 is given by Although the complete form of the scalar potential includes 2 Z mΦ 4 † † all the sfermions in the MSSM, we have considered the LΦ ¼ − d θðX XÞΦ Φ; ð30Þ f2 potential terms involving only the MSSM Higgs doublets and the SM-singlet scalar Φ. This is because the sfermions where mΦ denotes a soft SUSY breaking mass. Finally, should be heavy to satisfy the current LHC constraints and LAB is extended to be their couplings with the Higgs-portal DM have little effects ≃ 2 Z on the DM physics for mDM Mh= . For the physics of BΦ the Higgs-portal DM scenario, only the bilinear terms with L → L þ − d2θXΦ2 þ H:c: : ð31Þ AB AB 2f respect to Φ are important. To extract them from the scalar Oð1 2Þ potential, we expand Vsoft up to the order of =f and In the following, we assume that BΦ is real and positive. then obtain

m2 m2 m2 V ⊃ ðμ2 þ m2 Þþ 3 H H þ H:c: þ 2 1 jH j2 þ 2 2 jH j2 m2 jΦj2 Φ Φ f2 u d f2 d f2 u Φ 2 2 2 m m m BΦ − 1 þ 3 H H þ 1 jH j2 þ 2 jH j2 Φ2 þ H:c: : ð35Þ f2 u d f2 d f2 u 2

Substituting 1 Φ ¼ pffiffiffi ðϕ þ iηÞð36Þ 2 into Eq. (35), we can find the mass spectrum of the real scalars, ϕ and η, and their couplings with the Higgs bosons. First, we obtain the mass spectrum to be m2 m2 m2 m2 m2 ¼ μ2 þ m2 þ 1 cos2β þ 2 sin2β þ 3 sin β cos β Φ v2 ϕ=η Φ Φ f f f f 2 2 2 2 m1 2 m2 2 m3 v ∓ 1 þ cos β þ sin β þ sin β cos β BΦ f f f f 2 2 ≃ μΦ þ mΦ ∓ BΦ: ð37Þ

j 2 j ≫ 2 In the last expression, we have used m1;2;3 , f v Since all the Higgs bosons except for the SM-like Higgs 2 and mΦ

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Figs. 1 and 2, the up-type Higgs doublet is approximately odd parity to Φ while even parity for all the MSSM identified as the SM-like Higgs doublet. By employing this superfields. We have shown that in the decoupling limit approximation Hu ≃ H, we can easily extract the coupling of the sparticles and heavy Higgs bosons, our low energy of ϕ with the SM-like Higgs doublet from Eq. (35) such effective theory is nothing but the Higgs-portal DM that scenario with the lightest Z2-odd real scalar being the DM candidate. With a suitable choice of the model 2 m2 BΦ parameters, we can reproduce the allowed parameter L ≃− m2 − ðH†HÞϕ2: ð38Þ int f2 Φ 2 region of the Higgs-portal DM scenario. Here, we give a comment on a general property of our This is the formula to be compared with Eq. (22) with the model. Since the main point of this paper is to propose identification of S ¼ ϕ. Therefore, in the decoupling limit the minimal extension of the NL-MSSM to incorporate a of the heavy Higgs bosons and all the MSSM sparticles, we suitable DM candidate, we have focused on a special have obtained the Higgs-portal DM scenario as the low parameter region, for which our model at low energies is energy effective theory. In terms of our model parameters, reduced to the simplest Higgs-portal DM scenario (plus a ¼ λ the two parameters mDM mϕ and HSS, which control the extremely light gravitino), namely, the SM with the real Higgs-portal DM physics, are approximately expressed as scalar DM being odd under the Z2 symmetry. In general, we have a wide variety of the parameter choices to realize 2 2 2 m ≃ μϕ þ mΦ − BΦ; a viable dark matter scenario. For example, we may take a DM 2 very small value of BΦ in Eq. (37) so that the mass m BΦ λ ≃ 4 2 2 − ð Þ splitting between ϕ and η is negligibly small. In this case, HSS 2 mΦ 2 : 39 f we identify the complex scalar Φ with the DM particle. pffiffiffi This is a complex scalar extension of the simplest Higgs- 2 ¼ −ð2005Þ2 2 ≥ As in Fig. 1, we may fix m2 GeV and f portal DM scenario with only one real scalar. This 3990 ≤ 125 GeV so as to yield mh GeV at the tree level. extension has been studied in [10,17–22]. Since the Even after this choice, we still have three free parameters, MSSM includes two Higgs doublets, our Higgs-portal μ Φ, mΦ, and BΦ, and we can arrange them to satisfy the DM scenario is basically two Higgs doublet extension of ≃ 2 10−4 ≲ phenomenological constraints, mDM Mh= and the Higgs-portal DM scenario. The two Higgs doublet jλ j ≲ 10−3 3 HSS for the Higgs-portal DM scenario. For extension of the SM supplemented by the Higgs-portal μ2 ≃ 2 ≃ 2 ¼ Oð1 2Þ example, we may set Φ mΦ BΦ= TeV but DM has been studied in [23–38]. In the model, the heavy tune their differences so as to reproduce the allowed values Higgs bosons can play an important role for the DM 2 ≪ 1 2 jλ j ≪ 1 of mDM TeV and HSS . physics, for example, an enhancement of the DM pair annihilations through the heavy Higgs boson resonances. IV. CONCLUSION While the allowed parameter region of the simplest Higgs-portal DM scenario is very limited, a wide param- If SUSY is broken at a low energy, the NL-MSSM with eter space can be phenomenologically viable in the two- the Goldstino chiral superfield is a very useful description Higgs doublet extension. Although we focused on the for taking the hidden sector effect into account to the interaction between the DM particle and the Higgs boson, MSSM. The NL-MSSM may be particularly interesting if the full Lagrangian includes interactions between the the SUSY breaking scale lies in the multi-TeV range. In ¼ 125 DM particles and sfermions, which are also derived by this case, the SM-like Higgs boson mass mh GeV integrating out the F component of the Goldstino super- is achieved by the Higgs potential at the tree level after field. If the DM particle is heavier than sfermions, the eliminating the F component of the Goldstino superfield. DM pair annihilation processes through the interaction However, such a low scale SUSY breaking predicts a between the DM particle and sfermions become impor- milli-eV gravitino LSP, which is too light to be the main tant in evaluating the DM relic density. We leave such component of the DM in our Universe. Thus, a suitable general analysis for future work. DM candidate is missing in the NL-MSSM. To solve this Finally, let us consider a crucial difference of our model problem, we have proposed a minimal extension of the from standard neutralino dark matter scenario in the NL-MSSM by introducing the SM-singlet chiral super- MSSM. Neutralino dark matter is an R-parity odd particle, Φ Z2 field ( ) along with the symmetry. The stability of the while in our scenario the dark matter is an R-parity even lightest component field in Φ is ensured by assigning particle. This fact leads to distinctive phenomenologies. For example, in collider phenomenology, a neutralino dark

3 matter is produced through a cascade decay of heavier For a parameter choice to predict mh < 125 GeV at the 2 2 sparticles due to the R-parity conservation. On the other tree level, Mh ¼ 125 GeV should be reproduced by M ¼ m þ Δ 2 Δ 2 h h hand, Higgs-portal dark matter in our scenario is not mh with mh from quantum corrections through scalar top quarks, as usual in the MSSM. produced by sparticle decays. A pair of Higgs-portal dark

015027-7 MASATO ARAI and NOBUCHIKA OKADA PHYS. REV. D 103, 015027 (2021) matters can be produced from the decays of Higgs bosons. ACKNOWLEDGMENTS Since gravitino is an R-parity odd particle, it is produced N. O. would like to thank the High Energy Theory Group from a cascade decay of sparticles. In our scenario, the in Yamagata University for the hospitality during his visit. gravitino is almost massless, and missing energy distribu- This work is supported in part by the United States tions associated with gravitino production are very different Department of Energy Grant No. DE-SC0012447 (N. O.). from those associated with neutralino production.

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