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Phase Transitions and Gravitational Wave Tests of Pseudo-Goldstone Dark Matter in the Softly Broken Uð1þ Scalar Singlet Model

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Phase Transitions and Gravitational Wave Tests of Pseudo-Goldstone Dark Matter in the Softly Broken Uð1þ Scalar Singlet Model PHYSICAL REVIEW D 99, 115010 (2019) Phase transitions and gravitational wave tests of pseudo-Goldstone dark matter in the softly broken Uð1Þ scalar singlet model † Kristjan Kannike* and Martti Raidal National Institute of Chemical Physics and Biophysics, Rävala 10, Tallinn 10143, Estonia (Received 26 February 2019; published 10 June 2019) We study phase transitions in a softly broken Uð1Þ complex singlet scalar model in which the dark matter is the pseudoscalar part of a singlet whose direct detection coupling to matter is strongly suppressed. Our aim is to find ways to test this model with the stochastic gravitational wave background from the scalar phase transition. We find that the phase transition which induces vacuum expectation values for both the Higgs boson and the singlet—necessary to provide a realistic dark matter candidate—is always of the second order. If the stochastic gravitational wave background characteristic to a first order phase transition will be discovered by interferometers, the soft breaking of Uð1Þ cannot be the explanation to the suppressed dark matter-baryon coupling, providing a conclusive negative test for this class of singlet models. DOI: 10.1103/PhysRevD.99.115010 I. INTRODUCTION Is there any other way to test the softly broken Uð1Þ Scalar singlet is one of the most generic candidates for singlet DM model experimentally and to distinguish the the dark matter (DM) of the Universe [1,2], whose proper- particular model from more general versions of singlet scalar ties have been exhaustively studied [3–6] (see [7,8] for a DM? A new probe of physics beyond the Standard Model recent review and references). However, the recent results (SM) became experimentally available due to the discovery from direct detection experiments [9–11] have pushed the of gravitational waves (GWs) by LIGO experiment [18,19]. singlet scalar DM mass above a TeV-scale (except in a It is well known that first-order phase transitions generate a – narrow region around the Higgs resonance). Thus, the stochastic GW background [20 22], which can potentially singlet scalar models with the simplest scalar potential, in be probed in future space-based GW interferometers [23,24]. While the Higgs phase transition in the SM is of which the DM is stabilized by a Z2 symmetry, appear to be strongly constrained, less natural and less attractive. second order [25,26] and, thus, does not generate the GW This conclusion need not hold for specific realizations of signal, in models with extended scalar sector the first-order the singlet scalar DM idea. A neat observation was made phase transition in the early Universe can become exper- in [12] that for the case of a less general scalar potential imentally testable by the GW experiments. (For a recent obtained by imposing an Uð1Þ symmetry that is softly review on phase transitions and GWs, see [27].) broken, first studied in [13], the direct detection cross GWs from the extension of the SM with a scalar singlet section is strongly suppressed at tree level by the destruc- have been extensively studied. In general a two-step phase tive interference between two contributing amplitudes. This transition will take place in those models that can be of the – – result persists even if loop-level corrections to the direct first order [28 35] and be testable with GWs [36 46]. The detection cross section are considered [14,15], making aim of this work is to study the properties of the phase the softly broken scalar singlet model really interesting. transition in the scalar singlet model with a softly broken ð1Þ This has motivated follow-up studies demonstrating that it U symmetry in order to find out whether the GW signal is possible for pseudo-Goldstone DM to show up at the can distinguish between different versions of the singlet LHC [16] or in indirect detection [17]. DM models. We reach a definitive conclusion: in this class of models with a suppressed direct detection cross section, the phase transition is necessarily of the second order and no testable GW background will be generated. Therefore, if *[email protected] the stochastic GW background characteristic to the first † [email protected] order phase transition due to scalar singlets will be discovered, the softly broken singlet model cannot be Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. responsible for that. In this case, as a consequence, the Further distribution of this work must maintain attribution to negative results from DM direct detection experiments the author(s) and the published article’s title, journal citation, cannot be explained with the ideas presented in [12]. On the and DOI. Funded by SCOAP3. other hand, note that not discovering a GW signal would 2470-0010=2019=99(11)=115010(5) 115010-1 Published by the American Physical Society KRISTJAN KANNIKE and MARTTI RAIDAL PHYS. REV. D 99, 115010 (2019) not rule out models with first-order phase transitions, 02 2 μ ¼ −mχ: ð8Þ because to generate a large signal, the phase transition S must be strongly first-order. The tree-level direct detection DM amplitude vanishes at This paper is organized as follows. We describe the zero momentum transfer, model in Sec. II. The phase transition in this framework is studied in Sec. III. We conclude in Sec. IV. 2 2 A ð Þ ∝ θ θ m2 − m1 ≃ 0 ð Þ dd t sin cos 2 2 ; 9 t − m2 t − m1 II. THE MODEL which allows one to explain the negative experimental We consider the scalar potential of the SM Higgs boson results from DM direct detection experiments, while still H S together with a complex singlet , keeping the pseudo-Goldstone DM mass in the reach of 1 1 1 collider searches. ¼ μ2 j j2 þ μ2j j2 þ μ02ð 2 þ Ã2Þ V 2 H H 2 S S 4 S S S 1 1 þ λ j j4 þ λ j j2j j2 þ λ j j4 ð Þ III. PHASE TRANSITION 2 H H HS H S 2 S S ; 1 In the high temperature limit, the Uð1Þ-symmetric mass μ02 where the S term is the only one that softly breaks the terms take on temperature-dependent corrections: Uð1Þ symmetry S → eiαS. Without loss of generality, the μ02 μ2 ð Þ¼μ2 ð0Þþ 2 parameter S can be taken to be real and positive. H T H cHT ; We decompose the fields in the electroweak vacuum as μ2ð Þ¼μ2ð0Þþ 2 ð Þ S T S cST ; 10 þ þ χ 0 vs s i where S ¼ pffiffiffi ;H¼ þ : ð2Þ 2 vph ffiffih 2 1 c ¼ ð9g2 þ 3g02 þ 12y2 þ 12λ þ 4λ Þ; Note that both the Higgs boson and the singlet will get a H 48 t H HS vacuum expectation value (VEV) (the Higgs VEV is 1 cS ¼ ðλS þ λHSÞ: ð11Þ vh ¼ 246.22 GeV). The mixing of the CP-even states h 6 and s will yield two CP-even mass eigenstates h1 and h2. μ02 We identify h1 with the SM Higgs boson with mass m1 ¼ The thermal correction to S is zero, because the quartic 125.09 GeV [47]. Notice that the pseudo-Goldstone χ is the couplings do not break the Uð1Þ symmetry. μ02 The cancellation mechanism (9) works only if the CP- DM candidate with a mass determined by S . We express the potential parameters in terms of physical even scalar states mix with each other. The mass matrix quantities in the zero-temperature vacuum, such as the is nondiagonal only if both h and s get VEVs. For that, 2 θ the fields must end up in the ðvh;vs; 0Þ vacuum at zero masses m1;2 of the real scalars, their mixing angle , the 2 temperature. Note that we use e.g., v as a label to indicate pseudoscalar mass mχ, and the VEVs v and v : h h s a nonzero VEV of the Higgs boson, not as a particular 2 2 2 2 solution in terms of the potential parameters. Then the m þ m þðm − m Þ cos 2θ λ ¼ 1 2 1 2 ; ð Þ phase transition pattern consistent with the DM relic H 2 2 3 vh density is 2 þ 2 þð 2 − 2Þ 2θ λ ¼ m1 m2 m2 m1 cos ð Þ ð0; 0; 0Þ → ð0;v ; 0Þ → ðv ;v ; 0Þ: ð12Þ S 2 ; 4 s h s 2vs Both steps are second-order phase transitions. 2 2 ðm1 − m2Þ sin 2θ There is no possibility to engineer a first-order phase λHS ¼ ; ð5Þ 2vsvh transition. The only alternative second step, which could potentially be first-order [48], would be 1 1 μ2 ¼ − ð 2 þ 2Þþ ð 2 − 2Þ H m1 m2 m2 m1 ð0 0 Þ → ð 0Þ ð Þ 2 2vh ; ;vχ vh;vs; : 13 × ðv cos 2θ þ v sin 2θÞ; ð6Þ h s For a first-order phase transition, however, both extrema must be minima at the same time. But if the ðv ;v ; 0Þ 2 1 2 2 2 1 2 2 h s μ ¼ − ðm1 þ m2Þþ2mχ þ ðm1 − m2Þ vacuum is a minimum, the ð0; 0;vχÞ vacuum can only be a S 2 2v s saddle point or maximum, because the mass squared of the × ðvs cos 2θ − vh sin 2θÞ; ð7Þ μ02 0 s particle is S < in this vacuum. When the potential 115010-2 PHASE TRANSITIONS AND GRAVITATIONAL … PHYS. REV. D 99, 115010 (2019) FIG. 1. Phase diagram and thermal evolution of VEVs in the considered model. Left panel: Thermal evolution of the field to T ¼ 0 (dot) is shown by the black line.
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