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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.
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