Gravitational Wave Imprint of New Symmetry Breaking

It is believed that there are extra fundamental gauge symmetries beyond these described by the Standard Model of particle physics. The scale of these new gauge symmetries are usually too high to be reachable by particle colliders. Considering that the phase transition (PT) relating to the spontaneous breaking of new gauge symmetries to the electroweak symmetry might be strongly first order, we propose in this paper taking the stochastic gravitational waves (GW) arising from this phase transition as an indirect way of detecting these new fundamental gauge symmetries. As an illustration, we explore the possibility of detecting the stochastic GW generated from the PT of $\mathbf{ B-L}$ in the space-based interferometer detectors. Out study shows that the GW energy spectrum is reachable by the LISA, BBO, Taiji and DECIGO experiments only for the case where the spontaneous breaking of $\mathbf{B-L}$ is triggered by at least two electroweak singlet scalars.


Introduction
Although predictions of the Standard Model (SM) of particle physics remarkably agree with almost all experimental observations, we never stop exploring new fundamental gauge symmetries beyond these described by the SM, which are usually motivated by the neutrino masses, dark matter, baryon asymmetry of the universe and the gauge couplings unification at a Grand Unified Theory (GUT). Scales relevant to the spontaneous breaking of new symmetries are usually too high to be accessible by colliders in a foreseeable future. How to probe them is an open question.
The observation of Gravitational Wave (GW) signal at the Laser Interferometer Gravitational Wave Observer (LIGO) [1] has opened a new window to explore the universe and various mysteries of particle physics [2][3][4][5][6][7][8][9][10][11][12][13][14]. There are usually two sources of GW [4]: (1) cosmological origin, such as inflation and phase transition (PT); (2) relativistic astrophysical origin ( Binary systems etc.). If phase transitions related to the spontaneous breaking of the new gauge symmetries are strongly first order, bubbles of broken phase may nucleate in the background of symmetric phase when the universe cools down to the bubble nucleation temperature. Bubbles expand, collide, merge and finally fill the whole universe to finish the PT, and stochastic GW signals can be generated via the bubble collisions, sound waves after the bubble collision and turbulent motion of bulk fluid [15]. In this paper we propose taking GW as an indirect way of exploring new gauge symmetries, supposing the PT of new gauge symmetry breaking is strongly first order.
Considering the complexity of the non-Abelian gauge group extended models, we study GWs generated from PTs of Abelian gauge group extended models in this paper. There are many possible U (1) extensions of the SM [16], of which gauged B − L [17][18][19], B, L [20][21][22], B + L [23,24], L i − L j [25] (Here B and L are the baryon number and lepton number, respectively) have received great attentions. Since U (1) B−L only need minimal extensions to the SM for anomalies cancellation, it is believed to be the most natural one according to Occam's Razor 1 . We investigate conditions for the bubble nucleation during the PT of U (1) B−L , then calculate the energy spectrum of GWs generated from this process. Notice that the higher the energy scale of PT is, the larger peak frequency of GW energy spectrum it has [27]. If U (1) is broken at the TeV scale, its GW can be detected at the space-based laser interferometer detectors such as the Laser Interferometer Space Antenna(LISA), Big Bang Observer (BBO), Taiji and Tianqin projects. Alternatively if U (1) is broken at a scale approaching to the GUT, its GW is sensitive to the ground-based Laser interferometer such as aLIGO. Our results show that it is difficult to get large enough GW energy spectrum reachable by the space-based Laser interferometer if the B − L is broken by only one electroweak scalar singlet. Alternatively if B − L is broken by at least two electroweak scalar singlets, its GW energy spectrum is detectable by the LISA detector, ALIA, DECIGO, BBO and Ultimate-DECIGO. For GWs from the spontaneous breaking of non-Abelian symmetries, we refer the reader to Ref. [28] for the case of 3-3-1 model [29,30].
The remaining of the paper is organized as follows: In section 2 we give a brief introduction to the Abelian gauge group extensions to the SM and describe the U (1) B−L model in detail. Section 3 is focused on the GW signals from the PT of U (1) B−L . The last part is concluding remarks.

Model (a)
The Higgs potential for the scenario (a) of U (1) B−L can be written as with v Φ the vacuum expectation value (VEV) of Φ. The two parameters µ 2 φ and κ can be replaced by the physical parameters v φ and m φ , where y N is 3 × 3 symmetric Yukawa coupling matrix. The first term generates Majorana masses for right-handed neutrinos as Φ gets non-zero VEV. The tiny but non-zero active neutrino masses arise from the type-I seesaw mechanism [37]. To study properties of the PT, one needs the effective potential at the finite temperature in terms of background field φ, in terms of background field, V CW known as the Coleman-Weinberg potential at the zero temperature, contains one-loop contributions to the effective potential at the zero temperature, V T and V Daisy include the one-loop and the bosonic ring contributions at the finite temperature, n i and s i are the number of degrees of freedom and the spin of the i-th particle, C i equals to 5/6 for gauge bosons and 3/2 for scalars and fermions. Eq. (3) is derived in the Landau gauge. It should be noted that the effective potential is gauge dependent and a gauge invariant treatment of the effective potential is still unknown. We refer the reader to Ref. [38] for a gauge independent approach to the electroweak PT. Thermal masses of scalar singlet φ and gauge boson Z are given by where g B−L is the gauge coupling of U (1) B−L . We list in the Table. 2 the field dependent masses of various particles. One can see from Eq. (3) that the cubic term in the effective potential mainly come from the loop contribution of Z , such that there is strong correlation between the collider constraints on the g B−L , m Z and the strength of the PT.

Model (b)
The correlation of Z with the PT can be loosed in the scenario (b), where an extra scalar singlet, ∆ ≡ (δ + v ∆ + iχ )/ √ 2, is included. For this scenario, the tree-level potential can be written as where Λ is a coupling with energy scale. µ 2 Φ and µ 2 ∆ can be replaced with v φ and v δ via the tadpole conditions The mass matrix for the CP-even scalars follows, which can be diagonalized by a 2 × 2 orthogonal matrix parametrized by a rotation angle θ, where s 1,2 are mass eigenstates with mass eigenvalues m s 1 and m s 2 respectively. Three quartic couplings can now be written in term of physical parameters, For the CP-odd scalars, their mass matrix is given by It can be diagonalized by a rotation matrix with angle θ = arctan[v δ /(2v φ )] and gives the following mass eigenstates where G Z is the Goldstone boson and A is the physical CP-odd scalar with its mass given by m 2 The effective potential of the scenario (b) has the same form as Eq. (3) up to the following replacements: The field dependent masses are tabulated in the second column of Table. 2, while the thermal masses of the various fields are given below, With these inputs, the phase history can be analyzed. A particular advantage of model(b) is that there is a cubic term in Eq. (6) at the tree-level, which can generate a barrier between the broken and symmetric phases without the aid of loop corrections. As a result it is easier to get a first oder PT for this scenario, compared with model(a) where the barrier is provided by Z from loop corrections. We now address collider constraints on the Z mass. A heavy Z with SM Z couplings to fermions was searched at the LHC in the dilepton channel, which is excluded at the 95% CL for M Z < 2.9 TeV from the ATLAS [39] and for M Z < 2.79 TeV from the CMS [40]. The measurement of e + e − → ff above the Z-pole at the LEP-II puts lower bound on M Z /g new , which is about 6 TeV [41]. Further constraint is given by the ATLAS collaboration [42] with 36.1 f b −1 of proton-proton collision data collected at √ s= 13 TeV, which has M Z B−L > 4.2 TeV. We keep these constraints in studying PTs of these models.

Gravitational wave signals
For parameter settings of these two models that can give a first order phase transition, there will be gravitational waves generated, mainly coming from three processes: bubble collisions, sound waves in the plasma and Magnetohydrodynamic turbulence(see Ref. [4,15,43] for recent reviews). The total energy spectrum can be written approximately as the sum of these three contributions: where the Hubble constant is defined following the conventional way H = 100h kms −1 Mpc −1 .
The energy spectrums depend on three important input parameters for each specific par-ticle physics model: the bubble wall velocity(≡ v w ), where ∆ρ is the difference of energy density between the false and true vacua, g * is the number of relativistic degrees of freedom and H n is the Hubble constant evaluated at the nucleation temperature T n , which corresponds approximately to the temperature when S 3 (T )/T = 140 [44]. The parameter α characterizes the strength of the PT while β denotes roughly the inverse time duration of the PT. With these parameters solved numerically, one can obtain the energy spectrum of the gravitational waves for three sources. Firstly for the GW from the bubble collision, it can be calculated using the envelop approximation [45][46][47] either by numerical simulations [48] or by a recent analytical approximation [49]. Both results can be summarized in the following form, Here κ φ is the fraction of latent heat transferred to the scalar field gradient, ∆(v w ) is a numerical factor and S env captures the spectral shape dependence. The two different treatments by Ref. [49] and Ref. [48] lead to slightly different results on the ∆(v w ) and S env . We adopt here the results from the numerical simulation, with f env the peak frequency at present time given by, which is the redshifted frequency of the peak frequency, f * , at the time of the PT, For the spectral shape S env , the analytical treatment in Ref. [49] shows the correct behavior for low frequency S env ∝ f 3 required by causality [50] while the result from the numerical simulations differs from this one in a minor way. According to a more recent paper [51], in which the runaway conclusion [52] of the bubble expansion is ruled-out, the energy deposited in the scalar field is negligible and should be neglected in GW calculations. We therefore neglect the contribution of bubble collision due to the smallness of κ φ .
Secondly, the bulk motion of the fluid in the form of sound wave are produced after the bubble collisions. It also generates GWs and the energy spectrum has been simulated, with [53], Here f sw is the peak frequency at current time redshifted from the one at the phase transition: 2β/( √ 3v w ), then Hz.
Similar to κ φ , the factor κ v is the fraction of latent heat transformed into the bulk motion of the fluid. We use the method summarized in Ref. [54] to calculate κ v as a function of (α, v w ) and note that a fitted approximate formula is given in Ref. [54]. We also note that a more recent numerical simulation by the same collaboration [55] gives a slightly enhanced Ω sw h 2 and a slightly reduced peak frequency f sw . Finally the plasma at the time of phase transition is fully ionized and the resulting MHD turbulence can give another source of GWs. Neglecting a possible helical component [56], the generated GW spectrum can be modeled in a similar way [57,58], with the peak frequency f turb given by, Hz.
We need to know the factor κ turb which is the fraction of latent heat transferred to MHD turbulence. The precise value is still undetermined and a recent numerical simulation shows that κ turb can be parametrized as κ turb ≈ κ v , where the numerical factor varying roughly between 5 ∼ 10% [53]. Here we take tentatively = 0.1. For detection of the GWs, one needs to compare these spectrums with the sensitivity curve of each detector. The LISA detector [59] is currently the most mature experiment and the recently finished LISA pathfinder has confirmed its design goals. We therefore consider the sensitivities of the four LISA configurations N2A5M5L6(C1), N2A1M5L6(C2), N2A2M5L4(C3), N1A1M2L4(C4) presented in Ref. [15,60], which include the instrumental noise of the LISA detector obtained using the detector simulation package LISACode [61] as well as the astrophysical foreground from the compact white dwarf binaries in our Galaxy. We also consider the discovery prospect of several other proposed experiments: the Advanced Laser Interferometer Antenna (ALIA) [62] 2 , the Big Bang Observer(BBO), the DECi-hertz Interferometer Gravitational wave Observatory(DECIGO) 3 and Ultimate-DECIGO [63].
We implement two B − L models in CosmoTransitions [64] which traces the phase history of each model, locates the critical temperature T C and gives the bounce solutions to obtain the bubble nucleation temperature T n . We then use these outputs to calculate the GW energy spectrums and compare them with the listed detector sensitivities.
From an extensive scan over the parameter space of model(a) at the mass scale of O(TeV), we find that a first order PT can occur for a significant proportion of their parameter spaces. However, the resulting GW signals are generally too weak to be discovered where the most optimistic case can marginally be reached by the Ultimate-DECIGO. This is due to the relatively large values of β and small values of α obtained, aside from the enhanced O(TeV) temperature, which reduce the magnitude of GW energy spectrum as well as pushing the peak frequency to higher values. On the other hand, for the parameter space at the electroweak scale, the GWs can generally be reached by most detectors, which is however ruled by collider searches of Z .
Model(b) has a sizable parameter space where the generated GWs from PT falls within the sensitive regions of various detectors, due to the easily realized PT from the tree level barrier with the aid of a negative cubic term in the effective potential in Eq. 6. We show a benchmark point from this parameter space and present the details of the PT and the GW spectrum. This benchmark parameter point is v φ = 4637GeV, v δ = 1902GeV, θ = 0.128, m s 1 = 2400GeV, m s 2 = 1236GeV and Λ = −2143GeV. For this case, the minima in the field space (φ, δ) lie in the direction φ > 0, where the cubic term in Eq. 6 is negative. Due to the reflection symmetry δ → −δ, this occur in a pair. The shape of the effective potential is shown as contours in Fig. 1 where hot regions have larger values of V while cold regions have smaller values. The leftest figure shows the shape at a relatively high temperature where the universe sits at its origin and the two minima in direction φ > 0 are developing. As T drops to the critical temperature T C ≈ 6448GeV, these two minima become degenerate with the one at the origin as shown in the middle figure. As T further drops below the critical temperature, the broken phase begin to nucleate on the background of symmetric phase at T n ≈ 3115GeV, which corresponds to the rightest figure. The details on the evolution of the new phase is shown in Fig. 2 in the plane (φ, δ) where the arrow denotes the direction of time flow and the colors show the value of temperature.
To calculate the GWs from this model, we need the input κ v which we calculate following Ref. [54]. For benchmark given in Fig. 1, we find α = 0.09 and κ v depends on one free parameter v w . For different values of v w , the motion of the plasma surrounding the bubble takes different forms and the value of κ v is shown in the left panel of Fig. 3, where representative points are selected marked as A, B, C and D shown as green points in the figure. The velocity profiles of the plasma is shown in the right panel of Fig. 3 as a function of r/t, where r is the radial distance from the bubble center and t starts at T n . For case A, v w is smaller than the sound speed in the plasma(≡ c s = 1/ √ 3, the vertical dashed line in left panel), and the bubble proceeds as deflagrations, with a velocity profile shown by the dotted lines in the right panel. For case B, v w is larger than c s , a rarefaction wave develops behind the bubble wall, yet the fluid has non-zero velocity ahead of the wall, corresponding to the solid lines in the right panel. This falls within the hybrid region of the left panel, denoting supersonic deflagration [65]. For case C, v w is increased to the Jouguet detonation [66]  The resulting GW energy spectrums for these four points from sound waves(blue dashed) and turbulence(brown dotted) are shown in Fig. 4, where their sum corresponds to the red solid line. The color-shaded regions at the top are the experimental sensitivity regions for the four LISA configurations C1-C4(red), ALIA(gray), DECIGO(yellow), BBO(green) and Ultimate-DECIGO(purple). It is observed that for all four cases, the spectrum at around the peak frequency is dominated by sound waves while turbulence becomes more important for large and small frequencies. The total GW spectra all fall within the experimental sensitive regions of the LISA configurations C1, C2, C3 as well as other experiments. For case B, corresponding to the peak of κ v in the left panel of Fig. 3, the least sensitive configuration of LISA C4 can also reach some proportion of the GW spectrum even though the resulting signal-to-noise ratio might be too small.
To assess the discovery prospect of the GWs, we quantify the detectability of the GWs using the signal-to-noise ratio adopted in Ref. [15]: where h 2 Ω exp is the experimental sensitivity shown in Fig. 4 and T is the mission duration of the experiment in years. With this formula, we calculate SNR as a function of v w for each experiment and show the results in Fig. 5. We also show two representative SNR thresholds SNR thr = 10, 50 as suggested by Ref. [15] with horizontal black lines for comparison. From this figure, we can see that all SNR curves have a peak at v w ≈ 0.67. This peak corresponds to the maximum of κ v ≈ 0.44 in the left panel of Fig. 3, represented by case B in previous discussions, which has supersonic deflagration profile of the plasma surrounding the bubble. It is clear from this figure that for a wide range of v w , the SNR for the LISA configuration C1, BBO and UDECIGO is above the two thresholds SNR thr = 10, 50. For DECIGO, there is also a range 0.5 v w < 0.8 above the threshold 50 and this range becomes much wider for the threshold 10. For the LISA configuration C2 with six links, the GW for a wide range 0.4 < v w < 1.0 is above the threshold value 10 and can therefore be detected according to Ref. [15]. For the LISA configurations C3 and C4, both of which have four links, the uncorrelated noise reduction technique used in the six-link cases is not available and therefore the SNR needs to be larger than 50 to be detectable [15]. So in this case, the GW is not reachable by C3 and C4 for any v w . For ALIA, there is a window at v w ≈ 0.7 where the SNR is above 10.

Discussion
The discovery of GW at the LIGO initiates a new era in high energy physics and gravity. In this paper we propose the stochastic GW as an indirect way of probing the spontaneous breaking new gauge symmetry beyond the SM. Working in models with gauged B − L extension of the SM, we studied the strength of PT relating to the spontaneous breaking of the B − L as well as the stochastic GW signals generated during the same PT in the space based interferometer. We find that the power spectrum of GW generated is reachable by the LISA , BBO, ALIA, DECIGO and Ultimate-DECIGO for the case where the spontaneous breaking of B − L is triggered by at least two electroweak scalar singlets. It should be mentioned that there is no way to identify its intrinsic physics if any stochastic GW signal is observed. But it provides a guidance for new physics hunters since stochastic GW signal with peak frequency at near 0.01 Hz is a hint of new scalar interactions or new symmetry at the TeV scale. This work make sense on this point of view. Although we only focused on the U (1) case in this paper, our studies can be easily extended to the non-Abelian case since it contains all ingredients for the GW calculation.