Gravitational waves from the phase transition in the B-LSSM

Based on the gauge symmetry group $SU(3)_C\otimes{SU(2)_L}\otimes{U(1)_Y}\otimes{U(1)_{B-L}}$, the minimal supersymmetric extension of the SM with local B-L gauge symmetry(B-LSSM) has been introduced. In this model, we study the Higgs masses with the one-loop zero temperature effective potential corrections. Besides, the finite temperature effective potentials connected with two $U(1)_{B-L}$ Higgs singlets are deduced specifically. Then we can obtain the gravitational wave spectrums generated from the strong first-order phase transition. In the B-LSSM, the gravitational wave signals can be as strong as $h^2\Omega_{GW}\sim10^{-11}$, which may be detectable in the future experiments.


I. INTRODUCTION
Based on the gauge group SU(3) C ⊗ SU(2) L ⊗ U(1) Y , the standard model(SM) has been successfully established. Nevertheless, the SM possesses some limitation and additional U(1) B−L gauge interaction will be a promising extension to the SM. Therefore, the minimal supersymmetric extension of the SM(MSSM) [1][2][3][4] with local B-L gauge symmetry (B-LSSM) is introduced, where B represents baryon number and L stands for lepton number [5][6][7][8][9]. In this model, the invariance under U(1) B−L gauge group imposes the R-parity conservation to avoid proton decay [10]. Besides, the right-handed neutrino superfields have been imported in the B-LSSM to obtain the tiny neutrino masses through type-I seesaw mechanism. Thus, B-LSSM provides an elegant solution to the existence of the light left-handed neutrino.
Then the possibility of baryogenesis via leptogenesis will be well explained. Furthermore, due to the introduction of the additional singlet Higgs states and right-handed (s)neutrinos, B-LSSM alleviates the hierarchy problem through additional parameter space released from the LEP, Tevatron and LHC constraints [11,12]. Other than this, the model can also provide much more dark matter candidates [13][14][15][16] than the MSSM.
The gravitational wave(GW) signals have been detected at the Laser Interferometer Gravitational Wave Observer (LIGO) [17,18], which urges physicists to explore the various universe mysteries. The sources of GW signals can be sensitive to arise from the first-order phase transition(PT) in the early universe [19][20][21][22][23]. The electroweak symmetric broken phase is produced in the form of bubbles. When the universe cools down to the nucleation temperature, bubbles of broken phase may nucleate in the background of symmetric phase.
Bubbles expand, collide, merge and finally fill the whole universe to finish the first-order PT. Then, the stochastic GW signals can be generated through the bubble collisions, sound waves after the bubble collision and magnetohydrodynamic turbulent of the surrounding plasma [22,24,25].
Unfortunately, the electroweak PT in the SM turns out to be too weak to obtain GW signals [26][27][28]. In order to realize strong first-order PT and finally obtain the GW spectrums, physicists have studied various extensions of the SM, such as the model with a dimensionsix operator [29][30][31], NMSSM [32][33][34] and other studies [35][36][37]. In this paper, we hope to investigate the GW spectrums produced from the first-order PT of the B-LSSM. We find that the strength of GW signals will be up to h 2 Ω GW ∼ 10 −11 , which can be detected by the future experiments, such as the LISA with N2A5M5L6 design configurations, the Big Bang Observer(BBO), DECi-hertz Interferometer Observatory(DECIGO) and Ultimate-DECIGO [38,39].
In this paper, after discussing the Higgs mass, we mainly study the GW spectrums generated from the strong first-order PT of the B-LSSM. We first introduce characteristics of the B-LSSM, then we discuss the Higgs masses corrected by the one-loop zero temperature effective potential in section II. In section III, we compute the concrete (φ η , φη)-dependent finite temperature effective potential. Meanwhile, we derive the GW generation by cosmological first-order PT in section IV. The GW spectrums are comprised by the bubble collision, sound wave and turbulence contributions. The numerical results of the GW spectrums that depend on the model parameters will be further illustrated in section V. Last but not least, we summarize the conclusion in section VI. The anomaly free of B-LSSM will be demonstrated in Appendix A.

A. The B-LSSM
In 1967, Sakharov proposed three necessary conditions for dynamics to produce asymmetry between matter and anti-matter in the universe [40]: (1) baryon number(B) nonconservation; (2) charge conjugation(C) transformation and charge conjugate-parity(CP) joint transformation non-conservation; (3) The system deviation from thermal equilibrium.
Through a lot of research, people realize that the SM can satisfy Sakharov's three conditions. However, the CP violation given by the CKM matrix in the SM is not sufficient to account for the observed baryon-photon number density ratio. In addition, in order to produce strong first-order electroweak phase transition, the mass of Higgs particle in the SM must be less than 45 GeV. In 2012, the mass of physical Higgs particle was found to be around 125 GeV by the European Large Hadron Collider (LHC) [41,42], which directly denied the possibility of realizing the electroweak baryon number production mechanism (baryogenesis) in the SM. Therefore, the current SM can not explain the matter and anti-matter asymmetries in the universe.
In the B-LSSM, the baryon number and lepton number are broken by Sphaleron process respectively, but the difference between the two is conserved. This links the change of the baryon number with the change of the lepton number, and the asymmetry of the baryon number can be converted from the asymmetry of the lepton number through the sphaleron process. In general, the mechanism of lepton number asymmetry (leptogenesis) requires the lepton number violating process, the C and CP destruction of the lepton part and the realization of the non-equilibrium state. These conditions can be achieved in the general models within the mass neutrino. The B-LSSM introduces the right-handed neutrinos. Besides, neutrinos are Majorana type and gain mass through the type-I seesaw mechanism, which breaks the lepton number symmetry. In B-LSSM, the C or CP symmetry is broken.
In addition, the decoupling of heavy right-handed neutrinos provides non-equilibrium conditions. Sphhaleron process only acts directly on left-handed fermions, and can partially convert the lepton number of left-handed leptons to the baryon number, thus explains the baryon number asymmetry. Therefore, the possibility of baryogenesis via leptogenesis can be realized within the B-LSSM.
In the B-LSSM, the local gauge group is defined as SU (3) The specification of the quantum numbers of fields in the B-LSSM will be discussed in the TABLE I. Besides, B-LSSM is assumed the gauged one, and we have proved carefully that this model is anomaly free. The concrete demonstrations will be given out in the Appendix A. B-LSSM introduces U(1) B−L gauge field, right-handed neutrinos ν c i and their superpartners. The masses of right-handed neutrinos will be constructed by Y x,ijν c iην c j . Then we can obtain the tiny neutrinos masses through the type-I seesaw mechanism after the righthanded neutrinos and left-handed neutrinos mixing together. The B-LSSM superpotential is deduced as where W M SSM is the superpotential of MSSM. i, j represent the generation indices, Y x,ij where φ d , φ u , φ η , φη represent the CP-even Higgs components and σ d , σ u , σ η , ση correspond to the CP-odd Higgs components. The VEVs of the U(1) B−L Higgs singletsη and While the VEVs of the Higgs doubletsĤ d andĤ u are v d and v u , which satisfy v = v 2 d + v 2 u . We take tb ′ = vη vη by analogy to the definition tb = vu v d in the MSSM.

B. Higgs mass in the B-LSSM
First, in the base (φ d , φ u , φ η , φη), the tree level mass squared matrix for Higgs boson M 2 h is deduced as: Here, g 2 = g 2 and N 2 = ReBη u 2 . Then, we will consider the radiative corrections ∆Π from the one-loop zero temperature effective potential to the tree level Higgs mass squared matrix.
and φ i,j are the CP-even Higgs components φ d , φ u , φ η , φη. ∆V 1 represents the one-loop zero temperature effective potential whose full form has be discussed in the literature [43]. In principle, the radiative correction is dominated by the contributions of top quark, bottom quark, stop quarks and sbottom quarks.
Here, the masses of top quark and bottom quark are respectively m t = 1 Additionally, the mass squared matrix of stop quark and sbottom quark will be deduced respectively in the basis (Ũ L ,Ũ R ) and (D L ,D R ). where, Similarly, the elements in the sbottom mass squared matrix are Then, the mass eigenvalues of stop and sbottom quarks will be given by Therefore, the radiative corrections ∆Π from the one-loop zero temperature effective potential will be deduced as: Here, f (m 2 , Λ is the new physics scale and we take Λ = 1 TeV in the following numerical discussion. m k represent the corresponding particle masses. The square matrix m 2 h will be diagonalized to the mass eigenstate by the unitary matrix Z h i .

C. Numerical discussion of the Higgs mass in the B-LSSM
The mass of physical Higgs boson reads m h 0 = 125.1 ± 0.14 GeV by the latest LHC experiments [44]. The updated experimental data on the mass of Z ′ boson indicates M ′ Z > 4.05 TeV with 95% confidence level(CL). Refs. [45,46] give us an upper bound on the ratio between the mass of Z ′ boson and its gauge coupling at 99% CL as 2 TeV in our numerical calculation, so the value of parameter g B is restricted in the region of 0 < g B ≤ 0.7. The Yukawa coupling Y b , determined by the parameter tb, is defined and v ≃ 246 GeV, so the parameter tb should be approximatively smaller than 40. Besides, the large tb has been excluded by theB → X s γ experiment. The coupling parameter g Y B will be taken around −0.45 ∼ −0.05, and the reason why constant g Y B is negative has been discussed specifically in Ref. [47]. In addition, LHC searches constrain tb ′ < 1.5 [48].
The physical Higgs mass changing with parameter g B will be shown in FIG.1(a) FIG.1(c). As shown in picture, the physical Higgs mass slightly increases with the enlarging g B , which will be limited in the region 0.3 ∼ 0.7 to obtain suitable Higgs mass.
When the physical Higgs mass satisfies experimental 3σ interval, parameters g Y B , tb and tb ′ will be constrained within the small regions: −0.23 ≤ g Y B ≤ −0.07, 7 ≤ tb ≤ 25 and 1.15 ≤ tb ′ < 1.5. So g Y B , tb and tb ′ are all sensitive parameters. Besides, the physical Higgs mass is mainly determined by the φ u component, which can be up to 99.8% contributions.
Then, The second-light Higgs mass m H 0 will be discussed in FIG.2 with parameters tb ′ and B η . As tb = 22, the m H 0 versus tb ′ will be described in FIG.2(a). We can find that the second-light Higgs mass increases quickly with the increasing tb ′ . So parameter tb ′ affects the second-light Higgs mass smartly. In FIG.2(b), the m H 0 changing with B η will be studied numerical discussions. Then, we consider the tree-level scalar potential including the two singlet superfieldsη andη, which will be written as Here, the contribution of −B η ηη is deduced from the softbreaking terms, and parameter B η possesses mass square dimension..

A. Finite temperature effective potential
The GW spectrum is generated by the first-order PT, which is determined by the finite temperature effective potential. In the B-LSSM, the finite temperature effective potential for both zero and finite temperatures are essential for realizing the first-order PT. We are actually interested in the Reη ≡ 1 √ 2 φ η and Reη ≡ 1 √ 2 φη. Therefore, the (φ η , φη)-dependent finite temperature effective potential can be expressed as (12) where V 0 (φ η , φη) and ∆V 1 (φ η , φη, 0) represent the tree-level and one-loop zero temperature effective potential respectively. ∆V 1 (φ η , φη, T ) are the one-loop finite temperature contributions, and ∆V daisy (φ η , φη, T ) are the daisy corrections. The concrete expressions will be concluded as [49,50] V 0 (φ η , φη) = 1 2  [51][52][53][54]. We will discuss the (φ η , φη)-dependent particle masses square m 2 i and M 2 i specifically as follows. The (φ η , φη)-dependent masses square of the CP-even Higgs and CP-odd Higgs are shown as The (φ η , φη)-dependent masses square of the squarks and sleptons, CP-even and CP-odd sneutrinos can be written as In the B-LSSM, there is a (φ η , φη)-dependent U(1) B−L gauge boson, whose mass square is given by In the basis (B ′ ,η,η), we obtain the neutralino mass matrix, which is a Majorana Diagonalizing the m † χ 0 mχ0, we can obtain the corresponding mass eigenvalues, which are

IV. GW GENERATION BY COSMOLOGICAL FIRST-ORDER PHASE TRAN-SITION
The breaking of B-L symmetry not only induces first-order PT, but also induces secondorder PT. The first-order PT is from a false vacuum state(electroweak symmetry phase) to a true one(electroweak symmetry broken phase), while the second-order PT is the process from an unstable state to a vacuum state.
The baryon asymmetry of the universe can be generated via the electroweak baryogenesis mechanism, which requires a strong first-order electroweak PT to provide a system deviation from thermal equilibrium. First-order PT will generate bubbles, inside which are electroweak symmetry broken phases, their externals are the electroweak symmetry phases. The universe begins with the electroweak symmetry phase, which is a metastable one. With the expansion of the bubbles, the collision and then fusion, the universe will reside at the electroweak symmetry broken phase. The GW will be generated along with the collision between bubbles, the plasma movement near the bubble wall disturbed by the expanding bubbles, and the magnetofluid turbulence caused by the plasma. The relative height of potential between false vacuum state and true one will determine the strength of GW. The GW can be detectable with the large enough relative height of potential, and the corresponding PT is called as the strong first-order PT.
The second-order PT can also generate bubbles initially. However, the background is unstable and will quickly roll to the vacuum state, so the bubbles and background will be exactly the same soon. The second-order PT will be more synchronous than the first-order one. Therefore, there is almost no bubble collision, plasma turbulence even the GW in the second-order PT. Therefore, we pay attention to the strong first-order PT to study the GW signals within the B-LSSM.

A. Scalar potential parameters related to the GW spectrum
In this section, we briefly discuss the properties of the GWs, which critically depend on two quantities: the ratio of the latent heat energy ρ vac to the radiation energy density ρ rad at the nucleation temperature T n , which is defined as α; β is the speed of the PT at the nucleation temperature T n . The transition from the symmetric phase to the symmetric broken phase takes place via thermal tunneling at the finite temperature. First-order PT pushes the bubbles of the symmetric broken phase to be nucleated, then they expand and eventually fill up the entire universe. The bubble nucleation rate per unit volume at the finite temperature is given by A(T ) is a factor that is roughly proportional to T 4 . S represents the action in the fourdimensional Minkowski space, while S E is the three-dimensional Euclidean action [55,56].
Parameter β can be defined as: where H n denotes the Hubble rate at the nucleation temperature T n . The key parameter that controls the GW signals is β/H n . And the smaller β/H n will lead to the stronger PT and consequently the more sensitive GW signals.
Then, we will consider parameter α, which denotes the strength of PT: where ρ vac represents the latent energy density released in the PT, while ρ rad denotes the radiation energy density.
Here, φ sym (φ bro ) releases the coordinate of the field at the symmetry(symmetric broken) phase. g * corresponds to the relativistic degree of freedom in the thermal plasma at T n . We prefer to a larger value of α, which will produce a much stronger PT, even a stronger GW spectrum.
In FIG.3, we will discuss the β/H n changing with parameters µ η , B η and g B respectively.
The values of β/H n decrease with the enlarging µ η and g B respectively, while increase with the enlarging B η . The strength of PT α versus the parameters m η and mη will be researched in FIG.4 respectively. With the enlarging m η , the values of α decrease slowly. Besides, the values of α are around 0.025 as mη in the region 260 ∼ 310 GeV, and the value of α decreases with the gradually growing mη. We prefer to the smaller β/H n and larger α to obtain the more suitable GW signals. So, we will take µ η = 380 GeV, B η = 8 × 10 5 GeV 2 , g B = 0.6, m η = 1500 GeV and mη = 300 GeV in the following numerical discussion.

B. GW spectrum
In this section, we will discuss the GW spectrums specifically. There are three sources that generate the GW spectrums from the first-order PT: (1) The initial collision of scalar field and relevant shocks in the plasma. The technique of 'envelope approximation' has been widely used to model GW power spectrum from bubble collisions, which can be denoted by h 2 Ω coll . (2) After the bubbles merging, the fluid kinetic energy waves in the plasma go on propagating outward into the broken phase. These waves spread at the sound speed in the plasma without the tractive force of scalar field bubble wall. Sound waves h 2 Ω sw are produced after the bubble collisions but before expansion. (3) The magnetohydrodynamic turbulent h 2 Ω trub of the surrounding plasma will be formed after the bubble collisions. These three contributions linearly combine, then we can obtain the corresponding GW spectrums, which will be expressed as

Bubble collisions
Using the technique of 'envelope approximation', the contribution of the GWs generated from the bubble collisions is [22,57] where a = 2.8 and b = 1.0. κ denotes the efficiency factor of the latent heat deposited into a thin shell with A = 0.715. The concrete expression of κ and ∆ will be discussed as follows [22] κ = 1 1 + Aα Aα + 4 27 here, v b = 0.6 characterizes the bubble wall velocity. The peak frequency f coll will be determined by the characteristic time-scale of the PT. From simulations, the peak frequency at T n is approximately given by fn β = 0.62 , which is then red-shifted to yield the peak frequency today: f coll = 1.67 × 10 −5 fn β β Hn Tn 100 GeV g * 100 1/6 Hz.

V. THE NUMERICAL RESULTS OF GW SPECTRUM IN THE B-LSSM
In our numerical calculation, we consider the constraints of parameter space related to the future experiment within B-LSSM. In section II, we study the Higgs masses with the one-loop zero temperature effective potential corrections. Parameters g B , g Y B , tb and tb ′ have been limited within small regions, which will affect the numerical results of GW spectrum. As well as, we have studied Higgs decay modes, B meson rare decayB → X s γ and the muon anomalous magnetic dipole moment in the B-LSSM [62]. The corresponding parameter constraints will be considered in our numerical discussion. In the following numerical discussion, we use the numerical package CosmoTransitions [63] for analyzing the corresponding PT. After calculating, we obtain the bounce solutions and discover the nucleation temperature T n . The value of S E /T n is around 140 below certain nucleation temperature T n .
Then we can obtain the important results of parameters α and β, which finally determine the GW spectrums.
In this part, we study the dependence of the GW spectrums on the parameters tb, g Y B , presents in the Yukawa couplings Y t and Y b , which affects the the finite temperature effective potential through the temperature corrections of the Higgs boson and third squark masses, so that influence the GWs. We can discover that the GW spectrums can be detected by the LISA, BBO and Ultimate-DECIGO as 8 < tb < 23 and −0.17 < g Y B < −0.08. In our following discussion, we make tb = 10 and g Y B = −0.1.
Parameters m η and mη both exist in the tree-level zero temperature effective potential.
When tb = 10, g Y B = −0.1, g B = 0.6, µ η = 380 GeV and B η = 8 × 10 5 GeV 2 , the GW spectrums versus frequency f with parameters m η and mη will be shown in FIG.7. The smaller m η but larger mη, the larger GW spectrum with relatively smaller peak frequency we can obtain. Other than this, the value of parameter m η is around 1500 GeV, while the value of parameter mη is around 300 GeV. Our numerical simulations reveal that the parameters m η and mη possess tiny parameter spaces to acquire suitable GW spectrums. As µ η is in the region of 360 ∼ 400 GeV, a larger value of µ η tends to produce a larger GW spectrum with a relatively smaller peak frequency. It is worth noting that GW spectrum can be detected by the LISA, BBO and Ultimate-DECIGO with µ η ≃ 400 GeV.
Then the GW spectrum versus parameter B η will be studied. With the increasing value of B η , the GW signal gradually strengthens. As B η ≃ 8 × 10 5 GeV 2 , GW spectrum will be detected by DECIGO, BBO and Ultimate-DECIGO; While B η ≃ 7.8 × 10 5 GeV 2 , GW signal will be detected by LISA, BBO and Ultimate-DECIGO.
Through the above discussion in FIG.7 and FIG.8, we find that parameters m η , mη, µ η and B η fluctuate within a small and fine-tuned region. We expect to find the relationship between these parameters to explain these relatively special parameter values. So, we random scan the parameter space as 7 < tb < 25, −0.23 < g Y B < −0.07, 0.3 < g B < 0.7, 200 GeV < µ η < 500 GeV, 7 × 10 5 GeV 2 < B η < 10 6 GeV 2 , 1400 GeV < m η < 2000 GeV and 100 GeV < mη < 400 GeV. When the numerical results satisfy the strong first-order PT and can obtain the suitable GW signals, we plot the ratio of random parameter B η to 9. We conclude that these parameters satisfy the approximate relation: (4) the gravitational anomaly with one U(1) gauge boson U(1) Y or U(1) B−L is propor- The anomalies that do not relate with U(1) B−L are proved free and they are very similar as the SM condition.
The anomalies including U(1) B−L are also proved free, which are more complicated than the calculation of SM.