Probing new physics with multi-vacua quantum tunnelings beyond standard model through gravitational waves

We report on a novel phenomenon of particle cosmology, which features specific cosmological phase transitions via quantum tunnelings through multiple vacua. This is inspired by the axion(-like) scalar potential, and enables to probe the associated new physics models, constraining these from the stochastic gravitational waves background. Multiple vacua may induce the nucleation of multiply co-existing bubbles over the phase transition epoch, hence enhancing the overall process of bubbles' nucleation. Our detailed analysis of semi-analytical and numerical solutions to the bounce equations of the path integral, enabled us to determine the existence of three most probable escape paths towards a local maximum of the Euclidean action. These generate non-negligible contributions to the overall decay rate to the true vacuum. This new mechanism of cosmological phase transitions clearly represents a possibly sizable new source of gravitational waves, with its energy spectrum being featured with particular patterns, which could be probed by the future gravitational wave interferometers.

We report on a novel phenomenon of particle cosmology, which features specific cosmological phase transitions via quantum tunnelings through multiple vacua. This is inspired by the axion(-like) scalar potential, and enables to probe the associated new physics models, constraining these from the stochastic gravitational waves background. Multiple vacua may induce the nucleation of multiply co-existing bubbles over the phase transition epoch, hence enhancing the overall process of bubbles' nucleation. Our detailed analysis of semi-analytical and numerical solutions to the bounce equations of the path integral, enabled us to determine the existence of three most probable escape paths towards a local maximum of the Euclidean action. These generate non-negligible contributions to the overall decay rate to the true vacuum. This new mechanism of cosmological phase transitions clearly represents a possibly sizable new source of gravitational waves, with its energy spectrum being featured with particular patterns, which could be probed by the future gravitational wave interferometers. Introduction. -Cosmological phase transitions (PTs) offer an inspiring possibility to probe physics beyond the Standard Model (SM). If first-order PTs took place at early cosmological times, a gravitational waves (GWs) spectrum can be induced, with crucial observational consequences for current and future GW experiments [1][2][3][4]. Such a scenario is traditionally considered in the hot cosmological plasma characterized by a scalar-field effective potential accounting for both the loop and thermal corrections. In such a thermal system, the quantum tunneling is either ignored or only considered as happening between two vacua at a typical time scale of a given PT. [5]. However, in order to realize strong enough first-order PTs, several extended models of particle physics were considered that involve more than two vacua in the effective potential at a given temperature. These are models with extra scalar singlets [6][7][8][9][10][11] and doublets [12][13][14][15][16], dimensional six effective operators [17][18][19][20][21], as well as supersymmetric models [22][23][24], hidden dark sectors [25][26][27][28][29][30][31][32][33][34][35][36], and other SM extensions. [37][38][39][40].
Nonetheless, new physics beyond the SM may arise for example by invoking the axion, originally postulated in Ref. [41][42][43] to address the strong CP problem in quantum chromodynamics (QCD), or axion-like particles. The axion was recently revitalized in the cosmological relaxation model, to dynamically address the electroweak (EW) hierarchy problem [44][45][46], which also naturally yields multiple vacua for a single scalar field. Inspired by this innovative phenomenology, we propose in this Letter to study quantum tunneling transitions, within the case of non-degenerate multiple vacua in a simple model with a single axion-inspired scalar field. Besides, we illustrate the possibility of probing new physics scenarios of this type by analysing the signals of the primordial GWs spectra generated by such transitions.
Instanton methods, initially developed in [47][48][49] to investigate quantum tunnelings in a gravitational environment, are nowadays widely exploited in the community. Even the functional Schrödinger equation, supplied with the WKB approximation, was established to gain further insights into fields' potentials endowed with multi-vacua [50][51][52]. The path integral over the quantum field configurations is addressed in terms of the most probable escape paths (MPEP) dominating it in the classically forbidden region. In this Letter, we perform a complete analysis of quantum tunnelings for the non-degenerate multi-vacua case, which preserves the major characteristics of the cosmological relaxation, and can yield well-behaved approximate solutions to the so-called bounce equations of the path integral. Around the reheating epoch 1 , multiple types of bubbles would run away in a cosmic medium, and generate GWs that are expected to be examined in various observational windows, including GW astronomy and cosmic microwave background (CMB) signals [60].
Multi-vacua quantum tunnelings. -According to the relaxation model, the inspired scalar field σ evolves into the classically stable but quantum metastable regions where quantum fluctuations would become dominant in its subsequent evolution. For simplicity, we illustrate the three vacua case, assuming the decay through the fourvacua configuration in a single transition to be exponentially suppressed. Capturing the essence for multi-vacua quantum tunnelings, we consider the scalar potentials as In Eq. (1), four parameters were introduced, a 1 and a 2 roughly denoting the expectation value of the false vacuum and of the true vacuum, respectively, while b 1 and b 2 realizing the energy differences among the three vacua; Λ is an effective energy scale, which, in an axioninspired model, may be identified as Λ QCD 200 MeV. In our analysis, we impose matching conditions, between the two sides of the potential, on the parametric space (a 1 , b 1 , a 2 , b 2 ) of our model, avoiding any discontinuities.
Since this axion-inspired scalar field σ was practically decoupled from any other SM fermions and bosons, while having suppressed scalar self-interactions, the bubble nucleation temperature T n of early Universe would just provide the average kinetic energy available for the scalar fields, without any significant modifications to the potential barrier shape. On the other hand, in our case, it is difficult to find such T n satisfying strong first order PTs condition S 3 (T n )/T n ∼ 140 in hot Universe. Then the dominant PTs contribution will be realized by quantum tunnelings with T n ≈ 0. The small thermal fluctuations estimated by T 2 (σ − 2a 1 ) 2 here reduce the energy difference in Eq.(1), thus leading to the secondary effect against quantum tunnelings. As an example, we choose the parameters so that the effective mass of σ satisfies m σ ∼ < O(10) GeV. This typically corresponds to some moment after inflation, but before reheating with small thermal corrections.
The profile of the potential is sketched in the left top panel of Fig. 1. Ignoring thermal perturbations, quantum tunnelings originate from the homogeneous Universe within a false vacuum, and eventually terminate at a homogenous Universe within a true vacuum.
The semiclassical equation of motion can be both numerically and analytically solved. After inflation, we aplogical PTs, not only the inflaton. For instance, in the early Universe the curvaton may be taken into account [53][54][55][56][57], requiring accordingly a curvaton reheating mechanism [58,59].
proach the lower energy scale Λ, for which suppressed interaction terms of the form σ n , with n > 4, can be ignored. With the functional Schrödinger equation , namely HΨ(σ( x)) = EΨ(σ( x)), and the WKB approximation Ψ(σ) = Ae i S(σ) , with S(σ) = S (0) (σ)+ S (1) (σ)+· · · , one may derive the MPEP semi-classical bounce equation where τ is the parameter of MPEP, ranging from −∞ to +∞, with the critical point, separating the classically forbidden region (τ < 0) and the classically allowed region (τ ≥ 0), being fixed exactly at τ = 0. For cosmological PTs, the former equation can be recognized to describe the bubble nucleation processes, while the latter one drives the bubble evolutions, during which the energy-momentum tensor of the field can evolve and generate GWs.
Following [61], the MPEP solutions to the first bounce equation ought to obey the O(4) invariance and satisfy ∂σ ∂τ τ =0 = 0 as well as σ(τ → −∞) = σ F , with σ F value of the false vacuum, as depicted in Fig. 1. The analytical solutions can be derived through the variational method: with ρ = √ x 2 + τ 2 , µ 1 = 2a 2 1 and µ 2 = 2a 2 2 . R 1 , R 2 are the variational parameters. Then, the Euclidean action can be expressed as Varying the action with respect to R 1 and R 2 , one can get the estimated solutions. Numerical computation can be derived via the Runge-Kutta algorithm for nonlinear ODE based on the package of CosmoTransitions [62]. The process involving three vacua provides us with a novel picture of tunneling transitions. In the previous literature, authors considered only one solution to the bounce equation, which allowed to denote only one MPEP. Our model can now lead to up to three MPEPs at most. If we further require that σ τ =0 = σ M , we can get a trivial solution which is 'Bubble3' in the right small panel in Fig. 1. The existence of this solution is independent on the model parameters a 1 , a 2 , b 1 and b 2 , while the other two are model-dependent.
All numerical solutions to the bounce equation are shown in the small panel at the middle top region of Fig. 1, having fixed the values of the four parameters. The three tunneling processes depicted can occur simultaneously in the Universe, at an estimated rate per volume Γ ∼ e −S E . In our case, 'Bubble1' and 'Bubble2' solutions are driven by quantum effects, adding two new MPEPs which can not appear in the classical case. Since the length of bubble walls is small enough δl/l 1, the solutions can be further simplified in thin-wall approximation with the tanh function, in analogy with the solution of the quartic potential with degenerate vacua. Additionally, we see that the two radii of 'Bubble1' are nearly the same, rendering negligible the existence of σ M ; while, for 'Bubble2', one can simply sum together the two parts of the contributions, which arise from (σ F , σ M ) and (σ M , σ T ) respectively. Although we can also get more complicated cases like nested and reoccurring [63] bubble profiles by only preserving O(3) symmetry and solving both of the two equations in Eq.(2) simultaneously, these solutions are not from pure quantum effects, inconsistent with our main consideration. Note that the Universe was extremely empty after inflation and before reheating. In this epoch, the plasma effect is negligible, and the bubbles expand and collide with each other to reach thermal equilibrium. Since both 'Bubble1' and 'Bubble2' trigger the tunnelings towards σ T , the whole Universe shall reach the final state represented by the true vacuum, eventually.
Astonishingly, for specific model parameters, 'Bubble1' and 'Bubble2' can vanish at the same time, and then 'Bubble3' is the only solution. We name this novel phe-nomenon as "Two Step Tunneling" (TST). This is the first time that someone realized TST in cosmological PTs within the context of single field models 2 . Looking at the 'Bubble1' and 'Bubble2' solutions in Fig. 1, the radii R 1 and R 2 show a decreasing behaviour when ∆b, which controls the energy difference separation between vacua, decreases. Numerically, we scan the ∆b parameter space from 0.014 GeV 3 /Λ 3 to 0.043 GeV 3 /Λ 3 . Below this range both the solutions vanish. The variation of R 1 and R 2 provides the solutions to Eq. (2). Then S E approaches its static points, of which 'Bubble2' corresponds to the maximum point and 'Bubble1' to the saddle point. This = 0. When the action of each of the two bubbles reaches the same value, the maximum point and the saddle point coincide. Afterwards, for smaller ∆b, there is no static point for any bubble radius, and then, 'Bubble3' is the only MPEP ensuring the quantum decay. The false vacuum σ F would first tunnel to the intermediate vacuum σ M , and then experience the second tunneling to reach the true vacuum σ T .
The occurrence of TST can actually impose very strict constraints on the model parameters. For instance, in the very early Universe the curvaton could decay into radiation, and hence raise the temperature of the Universe thus approaching the reheating regime. In this case, a finite temperature correction ∼ T 2 σ 2 should be taken into account in the effective potential. After its quantum decay into the intermediate vacuum σ M , along with the increase of the temperature, the true vacuum σ T would become degenerate with σ M , and then even vanish. If one still requires TST to occur in the specific example of Fig. 1, the reheating temperature is expected to be below 0.5 GeV, so that σ M and σ T would not be degenerate. As in this Letter we focus on a preliminary analysis of quantum tunnelings via multiple vacua, we leave to forthcoming studies more detailed investigations.
GW signals. -For 'Bubble1' and 'Bubble2' solutions, tunneling rates increase when the energy difference separation ∆b becomes smaller. The magnitude of S E stays of the same order for 'Bubble1' and drops down an order of magnitude for 'Bubble2'. This effect becomes important for an efficient generation of GWs. In general, there exist three major sources for GWs from cosmological PTs [67,68], which respectively are collisions of vacuum bubbles [69,70], sound waves [71][72][73] due to bubbles' expansions inside the plasma, and MHD turbulence [74][75][76] after collisions. Concerning the PTs dynamics of axioninspired model in the vacuum-dominated epoch, which is not significantly affected by the thermal corrections we have discussed, we naturally explore bubbles dynamics in run-away regime, where it is very well known that contributions to GWs spectrum from MHD turbulence and sound waves are negligible compared to collisional contributions [67,68]. Nevertheless, the multi-nucleation phenomena may be recast in a more general contest, well far from the run-away condition, as understood. As for the GWs production, it is sensitive to the temperature at reheating due to those different types of bubble dynamics and cosmological background. We assume that the GWs are produced in a thermal bath at temperature T * which approximately equals the reheating temperature T * ∼ T reh .
Recall that, the observational constraint upon reheating temperature is pretty loose, namely, the Big Bang Nucleosynthesis (BBN) yields a lower bound T reh > 1 MeV [77,78]. In this case, the GW intensity Ω σ and the related peak frequency f σ caused by σ can be approximated as follows, with the two key parameters α and β. The former one, α, is defined at the PT temperature T n by α ≡ (T n )/ρ rad (T n ), and introduces the ratio between the energy difference among two vacua and the thermal energy density of the plasma ρ rad (T n ) ∝ T 4 n . The latter one, β, is the bubbles nucleation rate parameter, captured by β ≡ −dS E /dt| tn 1/Γ (dΓ/dt) tn , which is often rescaled byβ ≡ β/H n = T n (dS E /dT )| T =Tn , so to account for the concurring expansion of the Universe. Finally, κ σ characterizes the fraction of the latent heat for the energy transfer.
PTs driven by σ in our model are due to quantum mechanical effects, implying T n ≈ 0 and thus α → ∞. In this limit, one gets Ω GW ≈ Ω σ , with κ σ ∼ 1 and v w ∼ 1. We present the results in Fig. 2. The total GW signals can be separated into three components, corresponding to the contributions of three types of vacuum bubbles in Fig. 2(a). For relatively large energy difference separations, i.e. ∆b > 0.035 GeV 3 /Λ 3 , the signals from 'Bub-ble2' and 'Bubble3' are severely suppressed, resulting in the fact that the total spectrum is given by the 'Bubble1' contribution. Indeed, a large energy difference separation implies an enhanced S E value in 'Bubble2', and similarly in 'Bubble3', which then leads to a rather low tunneling rate. When the energy difference separation is around 0.020 GeV 3 /Λ 3 , and above 0.014 GeV 3 /Λ 3 , S E1 and S E2 decrease, and their ratio S E1 /S E2 is order of unity. The related tunneling rates will be of the same magnitude, giving rise to similar contributions from 'Bubble1' and (b) GW intensity bands with different value choices of ∆b/a 3 (which, divided by Λ −3 , are 0.022, 0.018, 0.014, 0.01 and 0.006 respectively), b2 (which, divided by GeV 3 /Λ 3 , is smaller than 0.075) and a2 = −a1 = a. The solid lines are for the maximum intensities achieved at fixed ∆b/a 3 while leaving b2 and a free. The top coloured regions correspond to the expected sensitivities of space-borne GW experiments including LISA [67], BBO, DECIGO [79], U-DECIGO [80], TAIJI [81] and TianQin [82].
'Bubble2'. The tiny differences between these three contributions can only influence the fine structure of the GW spectra, which could be examined thanks to data from the future GW interferometers. Since S E significantly depends on ∆b, the amplitude of GW spectra can vary up to O(10 2 ) in the considered example. We emphasize that, the result predicted in our case is fundamentally different from other cosmological PTs, due to the fact that the profile of the GW spectrum is unique. For instance, in the standard case of thermal PTs there are extra contributions from sound waves and MHD turbulence, but all these become secondary in our case, which indicates that our scenario can be probed or falsified by the future high-sensitive GW interferometers.
Conclusions.-In this Letter we put forward a novel mechanism to generate cosmological PTs via quantum tunneling transitions that may arise due to new physics described in terms of axion-inspired scalar field models with multiple vacua. Accounting for a specific parameterization of the field potential, we made first semianalytical and numerical analyses, providing an explicit solution involving three MPEPs, and calculating the quantum decay rates. Our mechanism provides a platform for phenomenological investigations of the rich structure of quantum tunnelings, namely, an innovative realization of the TST phenomenon within the single field scenario in cosmology. This process can lead to a spectrum of induced stochastic GWs, of which the profile is uniquely predicted. Due to the fact that its origin is different from those arisen from sound waves and MHD turbulence, this newly proposed GW source is observationally distinguishable in the future GW astronomy.
We end by discussing several implications of the novel mechanism that could inspire forthcoming studies. From theoretical perspective, our study illustrates that new physics beyond SM could be accessible through cosmological PTs if multiple vacua are allowed. This may be also related to the SM hierarchy problem, through embedding into the relaxation model. Phenomenologically, we report a new paradigm of quantum tunnelings, leading to fruitful phenomena in cosmological PTs. We have neglected tunnelings along more consecutive vacua, but this theoretical possibility deserves further investigations, as a pathway to get better understanding on the new physics related to axion(-like) particles. Furthermore, the physical picture of quantum tunnelings depicted in our mechanism can be also related to the inhomogeneous initial conditions that arise because of thermal perturbations, which may result in resonant tunnelings with higher decay rates. Although it may be challenging to test the heuristic example we are focused on within this Letter, with the resolutions of current GW experiments, our study can either be extended to several theoretical scenarios, or provide detection targets for the next generation of GW instruments.