Twofield cosmological phase transitions and gravitational waves in the singlet Majoron model
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Abstract
In the singlet Majoron model, we study cosmological phase transitions (PTs) and their resulting gravitational waves (GWs), in the twofield phase space, without freezing any of the field directions. We first calculate the effective potential, at one loop and at finite temperature, of the Standard Model Higgs doublet together with one extra Higgs singlet. We make use of the public available Python package ‘CosmoTransitions’ to simulate the twodimensional (2D) cosmological PTs and evaluate the gravitational waves generated by firstorder PTs. With the full 2D simulation, we are able not only to confirm the PTs’ properties previously discussed in the literature, but also we find new patterns, such as strong firstorder PTs tunneling from a vacuum located on one axis to another vacuum located on the second axis. The twofield phase space analysis presents a richer panel of cosmological PT patterns compared to analysis with a singlefield approximation. The PTGW amplitudes turn out to be out of the reach for the spaceborne gravitational wave interferometers such as LISA, DECIGO, BBO, TAIJI and TianQin when constraints from colliders physics are taken into account.
1 Introduction
Strong firstorder phase transitions (PTs) can be involved in a rich phenomenology for cosmology. In particular, they are conducive to successful baryogenesis [1]—the mechanism which explains why there is more matter than antimatter in our universe. On the other hand, the first discovery of gravitational waves (GWs), by the Advanced Laser Interferometer Gravitational Wave Observatory (aLIGO) [2], has pushed us to a new era of GW astronomy. We are now in the good position of studying GW generation in various physical processes, and searching them on multiband frequencies. Today the stochastic GW background generated during firstorder PTs [3, 4, 5, 6] has become more and more popular due to their rich phenomenology. Unfortunately, the electroweak phase transition (EWPT) in standard model (SM) turns out to be a crossover [7, 8, 9], which is too weak to get baryogenesis or PTGW. In order to realize strong firstorder PTs and probe new physics, various extensions of the SM have been studied, such as models with extra scalar singlet(s) [10, 11, 12, 13, 14, 15], with dimensional six effective operators [16, 17, 18, 19, 20, 21], 2HDMs [22, 23, 24, 25, 26], NMSSM [27, 28, 29]; see also [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41].
The singlet Majoron model is a simple extension of the SM [42] (see also the famous GR model [43]), which was originally proposed to solve the neutrino mass problem. In this model, an additional complex Higgs singlet is introduced, which breaks the global \(U(1)_{BL}\) symmetry with its VEV and generates mass for the righthanded (RH) neutrino. Recently, there has arisen strong interest in the possibility of firstorder phase transitions [44, 45, 46, 47], GWs in radio astronomy [48] and dark matter physics [49]. In Refs. [44, 45] the authors claim that a flat direction exists on the surface of the effective Majoron potential, allowing one to reduce the twofield problem to a oneeffectivefield problem. Their analysis shows that strong firstorder PTs may be realized. Another study [46], further confirmed PTs can typically proceed in two steps, i.e., a very weakly firstorder transition from the \(U(1)_{BL}\) breaking followed by the EWPT. In this earlier work, the Yukawa couplings between the singlet Higgs field and the righthanded neutrinos was not considered. Later on, a very comprehensive work was completed [47] by a full numerical simulation to handle the twofield problem. Their work shows the existence of two step PTs—and once again the PT due to the \(U(1)_{BL}\) symmetry breaking is above the EWPT. They also find large values of the two Higgs interactive coupling while the Yukawa coupling between the Higgs singlet and the righthanded neutrinos are required in order to obtain strong firstorder PTs.
In this paper, we conduct a full numerical simulation to analyze the twofield cosmological PTs and evaluate the resulting GWs, by using the public available Python package CosmoTransition [50]. Thanks to the full 2D simulation, we are able to not only confirm the PT patterns which have been found in [46, 47], but also find more new patterns, see Sect. 3 for more details. For each PT pattern we have found, we discuss the possibility to detect PTGWs with spaceborne gravitational wave interferometers such as LISA [51], DECIGO [52, 53, 54, 55], BBO [56], TAIJI (ALIA descoped) [57] and TianQin [58, 59].
The paper is organized as follows: in Sect. 2 we write down the singlet Majoron model and set up our notations; in Sect. 3 we calculate the effective potential of the two classic fields within the finite temperature field theory, and present patterns of cosmological PTs produced by our simulation that we classify; in Sect. 4 we evaluate the GWs generated from firstorder PTs and put them in the light of future spaceborne GW detectors; finally, in Sect. 5 we summarize our main results and conclude.
2 The singlet Majoron model
3 Patterns of cosmological phase transitions in the singlet Majoron model

Step I: start from the minima of \(v_{BL}\), \(\lambda _s\), and \(\lambda _h\).

Step II: evaluate the mass of the (non)SM Higgs from Eq. (22) or Eq. (23), \(\lambda _{sh}\) from Eq. (24), \(\sin \theta \) and \(\tan \beta \) from Appendix A.

Step III: compare \(( \tan \beta , m_1^2, m^2_2, \sin \theta )\) with the theoretical and experimental constraints. If they can pass, then go to Step IV, otherwise iterate \(v_{BL}\), \(\lambda _s\) and \(\lambda _h\) and go to Step II.

Step IV: input the parameters \((v_{BL}, \lambda _{s}, \lambda _{h}, \lambda _{sh}, g)\) (g starts form its initial value) into the model and calculate the PTs, iterate g.

Step V: iterate \(v_{BL}\), \(\lambda _s\) and \(\lambda _h\), initialize g, go to Step II, or end when the all the parameter points have been exhausted.
In Fig. 1 a higherorder PT occurs around \(T\simeq 182~\mathrm{GeV}\), along the s field direction, followed by a strong firstorder PT at the temperature \(T\simeq 118.89~\mathrm{GeV}\). We see that as temperature falls, the original vacuum located at the origin point splits into two symmetric new vacua on the Higgs axis; this is a higherorder PT. When the temperature drops even further, two new vacua on the h axis are formed. These coexist with the old vacua until the tunneling occurs at \(T\simeq 118.89~\mathrm{GeV}\). This firstorder PT is very strong; here it tunnels from one vacuum located on the first axis directly to the second one located on another axis. Due to the mixing effect between Higgs and the s field, hereafter we shall refer to these as ’hybrid PTs’. In our simulation, we do find some parameters which can lead to very strong hybrid PTs and detectable GWs for the mentioned interferometers. Unfortunately, they also lead to unacceptable branch ratio for the \(H\rightarrow hh\) decay channel. For the parameter in Fig. 1, it predicts \(m_1 = 7.9~\mathrm{GeV}\), and \(\mathrm{BR}_{H\rightarrow hh}\simeq 0.99\), which definitely has been ruled out by collider experiments. We present it here in order to completely show the 2D PT properties of the singlet Majoron model.
Figure 2 shows a firstorder PT along the sdirection (notice the zoomedin plots) followed by a higherorder PT along the mixing direction. The occurrence of the sdirection PT can also be attributed to the mixing coupling \(\lambda _{sh}\), from Fig. 2 one can see \(s/T \sim 100/357 < 1\), which means the PT is weak. Indeed, it proceeds very fast, leads to very high frequency GWs which go beyond the detectable range of spatial GW interferometers. As mentioned before, the final vacuum is located at \((v_\mathrm{BL}, ~v_\mathrm{ew})\) obviously when the temperature gets close to 0. This parameter gives \(m_2 \simeq 328\, \mathrm{GeV}\) and thus belongs to the high mass region (see the details in Table 3).
It is even more interesting to find multiple firstorder PTs from some parameters. In Fig. 3 we show an example with a hybrid firstorder PT happening after a sdirection firstorder PT. Near \(T\simeq 294.06~\mathrm{GeV}\), there exist three vacua separated by barriers, one located at the zero point; the other two are the nonzero ones with opposite signs according to the symmetry. As temperature drops, the field tunnels from the origin to the nonzero VEV. Notice here we also employ zoomedin plots to better show the two first PTs. As above the PTGWs have too high frequency and too weak amplitude to be detectable. The other firstorder PT happens to be a hybrid one at \(T\simeq 132.23~\mathrm{GeV}\); its GWs are also too weak to be detectable. This point also belongs to the low mass region (see the details in Table 2).
We also find that, for some parameter space, it would allow weak firstorder PTs above the U(1) symmetry breaking, but the perturbative analysis has become jeopardized if the Higgs VEV in the broken phase is too small. To address this issue, one probably needs to do a lattice analysis (namely following [9]) that would find a crossover in this parameter space. However, as we have seen from the above analyses, the patterns of cosmological PTs in a singlet Majoron model are already very fruitful, which is a distinct feature for 2D problems compared to singlefield ones. We thus believe that the single Majoron model is a useful laboratory to study the properties of 2D cosmological PTs and GWs. In the followup study we plan to develop our numerical method further as regards the lattice analysis on the issue of nonperturbative regime in order to explore more details on cosmological PTs and GWs in higher dimensions.
4 Phase transition gravitational waves in the singlet Majoron model
Carrying out our scan strategy, for parameters with the same \((v_{BL},~\lambda _s)\), we pick out the ones which can produce the largest GW amplitudes and display their signals in Fig. 4 with \(v_{BL} < 246~\mathrm{GeV}\) and Fig. 5 with \(v_{BL} \ge 246~\mathrm{GeV}\), meanwhile confronting them with the spaceborne interferometers such as LISA, DECIGO, BBO, TAIJI and TianQin. Note that there are too many curves in some subplots; in order to save the plots space we will take the unique string notation such as ‘\(\lambda _s\lambda _h\lambda _{sh}g\)’. For the benchmark parameters taken in Figs. 4 and 5, we also make three tables (Tables 1, 2 for \(v_{BL} < 246~\mathrm{GeV}\) and Table 3 for \(v_{BL} \ge 246~\mathrm{GeV}\)) in Appendix B to show more details, especially as regards which mass region (HM, IM and LM) they belong to. We can use the three tables to analyze how they could survive from the theoretical and experimental constraints such as in Ref. [73].
When \(v_{BL} < 100~\mathrm{GeV}\), the most of the benchmark parameters lead to (non)SM Higgs with mass \(m_1 < m_{H, \mathrm{SM}}/2\), and they thus are severely constrained by the collider experiments on the decay channel \(H\rightarrow hh\). Since the upper bound about the decay branch ratio \(\mathrm{BR}_{H\rightarrow hh} \lesssim 0.2\) [84, 85], the coupling \(\lambda _h\) must be quite close to the SM value \(m^2_{H, \mathrm{SM}}/(2\times v^2_{ew})\), and \(\sin \theta \) nearly equals 1. In order to survive from the \(H\rightarrow hh\) constraints, in Table 1 we take three benchmark scales, \(v_{BL}=25~\mathrm{GeV}\), \(50~\mathrm{GeV}\), and \(75~\mathrm{GeV}\), set \(\lambda _h=m^2_{H, \mathrm{SM}}/(2\times v^2_{ew})\), and meanwhile we let the mixing coupling \(\lambda _{sh}\) vanish. From Eq. (23) we can get an upper bound \(\lambda _s \le (1/4) \lambda _{h,\mathrm{SM}}(v_{ew}/v_{BL})^2\), which equals 3.1, 0.78 and 0.35 for \(v_{BL}=25~\mathrm{GeV}\), \(50~\mathrm{GeV}\) and \(75~\mathrm{GeV}\). In the second quadrant of Fig. 4, we show the largest GWs generated for the benchmark scales. All the display curves are due to the hybrid firstorder PTs. Unluckily these GWs are hardly within reach of the planned interferometers.
For \(100~\mathrm{GeV}\le v_{BL} < 246~\mathrm{GeV}\), all the three cases (LM, IM and HM) can be realized. In Table 2 we show the parameters which have been taken in Fig. 4 for benchmark scales \(v_{BL} = 100\,\mathrm{GeV}, ~150\,\mathrm{GeV}, ~200\,\mathrm{GeV}\). Taking advantage of Fig. 7 in Ref. [73], one can see that a large value (close to unity) for \(\sin \theta \) is needed for the LM cases, while a small value (smaller than \(\sim 0.4\)) is needed for the HM cases. We have compared the parameters in Table 2 to make sure that they are not ruled out by the experiments. Notice again that there exists an upper bound for \(\lambda _s\), i.e., \(\lesssim 0.8\) for \(100\,\mathrm{GeV}\), \(\lesssim 0.5\) for \(150\,\mathrm{GeV}\), and \(\lesssim 0.45\) for \(200\,\mathrm{GeV}\). This is due to the constraint contour on \(\tan \beta \) and \(m_2\) in the HM region; see Fig. 10(b) in Ref. [73]. Compared with the smaller \(v_{BL}\) in Table 1, there tend to be more and more GWs generated by multiple firstorder PTs.
In Fig. 5 we show the GWs generated by parameters with \(v_{BL} \ge 246~\mathrm{GeV}\); see Table 3 for more details as regards the chosen parameters. All except one belong to the HM region, the upper bound for \(\lambda _s\) becomes smaller and drops slower. As we have mentioned, for these high \(v_{BL}\) parameters, the PTs along the sdirection play a more and more important role in generating GWs, with even higher peak frequencies.
5 Conclusion
In this paper, we study the properties of cosmological PTs, especially the resulting GWs, in the singlet Majoron model, using a numerical simulation to treat the twofield problem without freezing any of the field directions. Compared with a singlefield treatment, the patterns of PTs turn out to be much more diverse. We have not only verified the pattern with an EWPT happening after the global U(1) symmetry breaking, but also we find new patterns, such as strong hybrid PTs happening before the U(1) symmetry breaking. Our simulation suggests that the single Majoron model is an ideal benchmark in understanding the phenomenology of twofield cosmological PTs.
The PTGWs are likely not within the reach of detectability of spaceborne interferometers such as LISA, DECIGO, BBO, TAIJI and TianQin—either their amplitudes are too low or their frequencies are too high for those nextgeneration instruments. As a matter of fact, without considering the collider experimental exclusion bounds, we are able to find strong hybrid PTs which can generate detectable GWs. Unfortunately, in such cases, the mixing coupling necessarily has to be large, and the nonSM Higgs need acquire a mass smaller than the half mass of the heavier Higgs. Experimentally this parameter space is ruled out by constraints on the \(H\rightarrow hh\) decay branch ratio. However, we emphasize that the aforementioned conclusion is model dependent and the numerical method developed in the present study shall be widely applied to another model of cosmological PTs, which involves more than one degrees of freedom.
Finally, we would like to highlight the implications of the developed numerical method, which could be extended from several perspectives in future study. Note that we have only considered the bubbles which belong to the types of nonrunaway or runaway in plasma (see Ref. [70]), but we abandon those bubbles with runaway in vacuum (see Ref. [86]; see also Ref. [87] for a most recent study), since they spoil our criterion in Eq. (21). By doing so we expect that the parameter points would be further constrained. Additionally, the same analysis may be applied to some cosmological PTs that occur at extremely low frequency band and hence might be falsifiable in the forthcoming experiments of primordial gravitational waves [88, 89, 90, 91].
Notes
Acknowledgements
We are grateful to Andrea Addazi, Jim Cline, Ryusuke Jinno, Antonino Marciano and Pierre Zhang for valuable comments. We also thank two reviewers for insightful suggestions on the manuscript. YPW would like to thank Carroll L. Wainwright for useful discussions of the package of CosmoTransitions. This work is supported in part by the NSFC (nos. 11722327, 11653002, 11421303, J1310021), by CAST Young Elite Scientists Sponsorship Program (2016QNRC001), by the National Youth Thousand Talents Program of China, by the Fundamental Research Funds for the Central Universities, and by the Postdoctoral Science Foundation of China (2017M621999). All numerical simulations are operated on the computer clusters Linda & Judy in the particle cosmology group at USTC.
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