Gravitational waves from phase transition in split NMSSM

We discuss gravitational wave signal from the strongly first order electroweak phase transition in the split NMSSM. We find that for sets of parameters predicting successful electroweak baryogenesis the gravitational wave signal can be within the reach of future experiments LISA, BBO and Ultimate DECIGO.

The phenomenological searches for physics beyond the Standard Model (SM) with gravitational wave (GW) interferometers can significantly improve our understanding of matter properties and the early Universe evolution. Recent LIGO and VIRGO detection of the GW signal from binary black hole mergers [1,2,3] opens up opportunities in the field for the future space-based GW interferometers eLISA [4], DECIGO [7,8] and BBO [9]. Among numerous GW signals originated from different sources (for a recent review see, e.g., [10]) one can single out gravitational waves from the first order electroweak phase transition (EWPT). It is known that this transition is a crossover in the SM [11,12]. However, new scalar degrees of freedom introduced in various extensions of the SM at the electroweak scale can turn this transition to the first order phase transition. This possibility is quite interesting because on the one hand it allows for generation of the observed baryon asymmetry of the Universe (BAU). On the other hand, it requires a change of physics around TeV scale which could be potentially probed by the ongoing experiments at the Large Hadron Collider (LHC). GW signal from the EWPT would be an independent signature of such new physics models. In particular, it is important for testing the blind spots of the LHC searches. The GW signal from the EWPT was recently studied within the Next-to-Minimal Supersymmetric Standard Model (NMSSM) [13], in the model with singlet extension of the Higgs sector [15,16], in 6D operator extension of the SM [14], and in the scenarios with a hidden QCD-like sector of the SM [17,18].
In this paper we estimate the GW signal from EWPT within the framework of split NMSSM [19,20]. This model provides with the strongly first order EWPT due to specific singlet extension of the Higgs sector. Percolation of the new phase bubbles eventually produces stochastic background of GWs, which is not flat, but exhibits a characteristic peak. We find that for the benchmark points in parameter space of this model with successful electroweak baryogenesis the GW signals are expected to be right in the sensitivity region of the proposed space-based interferometers eLISA, BBO, DECIGO. Their observation would provide with independent evidence for the new physics playing important role in the early Universe.
We start with recalling basic properties of the split NMSSM [19,20]. This model is an extension of split MSSM [21,22] with an additional singlet superfield. The latter contains CP-even S and CP-odd P scalars as well as singlet fermionñ. Particle spectrum in this model is splitted: all the scalars except for the lightest Higgs boson h and the singlet scalars are heavy with their masses around a splitting scale M S . To be consistent with the measured Higgs boson properties this scale should be about 10-20 TeV [20]. Other superpartners (neutralinos, charginos, etc) are supposed to be around (sub)TeV scale. The low energy Lagrangian can be obtained by integrating out the heavy particles and is presented in [19,20].
It was shown in Ref. [20] that in the framework of split NMSSM the observed baryon asymmetry of the Universe (BAU) can be generated during the strongly first order EWPT. The strengthening of the PT with respect to the SM is a result of modification of the scalar potential, which is more complicated in the model due to presence of new scalar fields. The source of CP-violation in this model is associated with complex chargino mass matrix and in particular with the effective µ-parameter [19]. As the split NMSSM provides strongly first order PT it can be potentially probed with GW echo from colliding bubbles of the new phase. In what follows we consider the benchmark points in the parameter space of the model marked as Setup 1 and Setup 2 in Ref. [20]. Apart from predicting the observed baryon asymmetry of the Universe, they satisfy current experimental constraints from LHC data and the EDM bounds and suggest the lightest neutralino as a viable DM candidate.
The dynamics of cosmological first-order PT via tunneling of the scalar fields from false to true vacuum has been thoroughly studied in literature, for review see e.g. Refs. [23,24,25]. The corresponding probability of bubble nucleation per unit Hubble space-time volume at temperature T is given by where S 3 is the three dimensional Euclidean action or, equivalently, free energy of the scalar bubble configuration with V eff T (h, S, P ) being a one-loop effective potential at the finite temperature [19]. The bubbles of new phase nucleate when P ∼ 1, and eq. (1) yields S 3 /T ∼ 4 log(M * P l /T ) ∼ 150 at the typical electroweak temperature T 100 GeV. The accurate calculation [26] reveals the following nucleation condition for the free energy and nucleation temperature T c : Note that here we are interested in one-step PT for split NMSSM. However, there is another option: the first order PT can go in two steps in models with singlet extension of the Higgs sector if there exists a metastable vacuum of the effective potential. Moreover, such multi-step PT is also associated with a GW signal which can be measured by eLISA and DECIGO, see e.g. Ref. [27]. We do not consider this exotic option in the split NMSSM, which requires special investigation of the electroweak baryogenesis. We use results for the spectrum of GW from the first order EWPT which were obtained from numerical simulations. The GW are originated mainly from three sources (see e.g. [28] for a recent review): bubble collisions [29], sound waves produced by percolation [5], and turbulent motion of plasma originated from the same percolation [6]. Let us introduce key parameters which characterize the GW spectrum; hereafter we adopt the notations of Ref. [28]. The rate of bubble nucleation, i.e. the inverse duration of the PT, is where H c is the Hubble parameter at the nucleation temperature T c . The GW spectrum also depends on the ratio of latent heat released during the PT and radiation energy density ρ γ , Here variation ∆V eff is the potential energy difference between symmetric and broken phase.
In our case the surface of expanding bubble efficiently interacts with particles in the plasma. This friction limits the bubble wall velocity, which we expect to be subsonic, v w < c s ≡ 1/ √ 3 ≈ 0.57. It makes the contribution to the GW spectrum from the bubble collisions negligible. At the same time, both the sound waves and the turbulence survive in the plasma for quite a long time after the bubble collisions at the PT, that makes their contributions dominant.
The GW signal is usually presented as an integrated contribution of the logarithmic wave interval to the total energy density of the present Universe. The latter is characterized as Ω GW h 2 , where Ω GW is a relative contribution of the GW, while h = 0.68 refers to the present value of the Hubble parameter. Then the main sources of the GW signal -the sound waves Ω sw h 2 (f ) and the magnetohydrodynamic turbulence in plasma Ω m h 2 (f ) -read where Ω sw h 2 (f ) and Ω m h 2 (f ) are, respectively, given by [28] Ω sw h 2 (f ) = 1.23 where v w is the average bubble wall velocity and g * is the effective number of degrees of freedom in the plasma. Corresponding functions of frequency f in Eqs. (7) and (8) read as which peak, respectively, at frequencies where The coefficients k sw and k m entering eqs. (7) and (8) are fractions of the released vacuum energy transformed to the bulk motion of the medium and magnetic turbulence of the plasma, respectively:  Table 1: Benchmark points of the EWPT predicted in the split NMSSM [20].
here the parameters k a and k b are given by k a = 6.9v Numerical values of the parameters characterizing the EWPT for the two chosen benchmark models are shown in Table 1. Namely we present here the critical temperature T c and S 3 /T c which were calculated previously in [20]. Using the Euclidean action (2) and numerical procedure to find the bounce solution which were employed in [20] we numerically calculate the derivative d   Table 1 we obtain the spectra of the GW for two values v w = 0.1 and v w = 0.5. Precise calculation of the bubble wall velocity entering Eqs. (7) and (8) is rather involved (see, e.g. [30,31]). Using the condition suggested in [32] we checked that the expanding bubbles do not run away for the chosen benchmark models but approach a finite value.
Here we consider an optimistic subsonic case for the bubble wall velocity v w < 1/ √ 3 ≈ 0.57. Let us note that the baryon asymmetry of the Universe depends rather weakly [20] on v w .
Our results for the GW spectra as well as numerical values of α and β c are presented in Fig. 2. The sensitivities of eLISA, BBO and DECIGO space-based interferometers to these benchmark points are also shown in Fig. 2. The central part of the GW spectra containing a maximum is dominated by the acoustic wave contribution. At very low frequencies both spectra (9) grow as f 3 but contribution from MHD turbulence dominates. At high frequencies the latter again takes the lead and reveal the expected Kolmogorov turbulent behaviour with (-5/3) power. One can see from Eqs. (7) and (8) that the GW signal scales as ∼ β −1 c α 2 . This means that small values of β c and large α provide a higher GW signal. Therefore the GW signal in Fig. 2 for Setup 1 is enhanced for slightly larger value α = 6.2 × 10 −3 with respect to α = 4.5 × 10 −3 from Setup 2. Comparing the predictions with the expected sensitivities of different experiments we find that the most sensitive eLISA configuration [28] can test all these setups for v w 0.5.
Let us note that current bound on tensor-to-scalar perturbation ration r < ∼ 0.1 implies that contribution of gravitational waves from inflation to Ω GW h 2 should be smaller than about 3 × 10 −16 assuming flat perturbation spectrum. Thus the GW signal from EWPT predicted within split NMSSM can be distinguished from this irreducible background.
Finally, let us make a brief comment on neutralino-chargino sector of the model in context of latest results of searches for superpartners at the LHC experiments. For the two benchmark models in question the soft parameter M 2 is taken to be 1 TeV for concreteness although our results concerning the phase transition and BAU [20] are almost independent of this parameter if it is considerably larger than the critical temperature. This leaves only a tiny wino contribution in the lighter chargino and neutralino species which decreases sensitivity of corresponding LHC searches (see e.g. [34]) with the massive gauge bosons in the final state due to lower coupling of higgsino-like fermions to the gauge bosons. We take it as an additional motivation for the present study.
The work was supported by the RSF grant 14-22-00161.