Gravitational Waves from Phase Transition in Minimal SUSY U ( 1 ) B − L Model

Many extensions of the Standard Model include a new U(1) gauge group that is broken spontaneously at a scale much above TeV. If a U(1)-breaking phase transition occurs at nucleation temperature of O(100)-O(1000) TeV, it can generate stochastic gravitational waves in O(10)-O(100) Hz range if βn/Hn = 1000, which can be detected by ground-based detectors. Meanwhile, supersymmetry (SUSY) may play a crucial role in the dynamics of such high-scale U(1) gauge symmetry breaking, because SUSY breaking scale is expected to be at TeV to solve the hierarchy problem. In this paper, we study the phase transition of U(1) gauge symmetry breaking in a SUSY model in the SUSY limit. We consider a particular example, the minimal SUSY U(1)B−L model. We derive the finite temperature effective potential of the model in the SUSY limit, study a U(1)B−L-breaking phase transition, and estimate gravitational waves generated from it.


Introduction
Many extensions of the Standard Model (SM) include a new U(1) gauge group that is broken spontaneously, important examples being the minimal U(1) B−L model [1,2,3], the left-right symmetric model [4,5] and Pati-Salam model [6]. Usually, there is no theoretical reason to expect that the breaking scale of such U(1) gauge group is at TeV scale. If the breaking scale is beyond the reach of new gauge boson searches at colliders, observation of stochastic gravitational waves generated from a U(1)-breaking phase transition is the key to testing such models [7]. This is because the nucleation temperature of the phase transition is encoded by the peak position of gravitational wave spectrum, and Advanced LIGO [8], Advanced Virgo [9] and KAGRA [10]  that occurs separately from electroweak symmetry breaking includes [11,12,13,14,15].
If the breaking of a U(1) gauge group occurs at a scale much above TeV, supersymmetry (SUSY) may play a crucial role in its dynamics, since we expect SUSY breaking scale to be at TeV to stabilize the electroweak scale with respect to Planck scale. In this paper, therefore, we study the phase transition of a U(1) gauge symmetry breaking in a SUSY model and gravitational waves generated from it. We work in the SUSY limit, namely, we assume that the nucleation temperature is above the SUSY breaking scale so that soft SUSY breaking terms are negligible in the study of phase transition. For concreteness, we focus on the minimal SUSY U(1) B−L model 1 , which is by itself highly motivated because it can explain the origin of the seesaw scale, and if B − L is broken by even charges, R-parity is derived and accounts for the stability of dark matter. To simplify our analysis on U(1) B−L -breaking phase transition, we assume R-symmetry of the model. R-symmetry is well-motivated by itself because one can forbid µH u H d term by R-symmetry thereby solving the µ-problem.
Although we concentrate on the minimal SUSY U(1) B−L model, our study is applicable to a wide class of U(1)-gauge-extended SUSY models that contain superfields with U(1) charge +a and −a and a gauge singlet S to achieve the U(1) breaking. Remarkably, this U(1) need not be visible, i.e. the SM fields need not be charged under it, for the study of gravitational waves.
We comment in passing that SUSY models are more predictive than non-SUSY models about high-scale U(1)-breaking phase transitions. This is because in non-SUSY models where scalar field φ breaks an extra U(1), no symmetry forbids the Higgs portal term (H denotes the SM Higgs field), Suppose φ develops a large vacuum expectation value (VEV) much above the electroweak scale. To achieve the correct electroweak symmetry breaking, one has two options; one fine-tunes the portal coupling λ φH so that the emergent mass term λ φH | φ | 2 H † H is negligible compared to genuine Higgs mass term m 2 H H † H; or one assumes that the genuine Higgs mass term nearly cancels the emergent mass term. In the latter case, the study of the U(1)-breaking phase transition involves the SM Higgs field and depends on unknown genuine Higgs mass term m 2 H , in addition to the U(1)-breaking scale. In SUSY models, such Higgs portal coupling is forbidden at the renormalizable level and hence is justifiably neglected. This paper is organized as follows. In Section 2, we explain the minimal SUSY U(1) B−L model, and derive the finite temperature effective potential for U(1) B−L -breaking VEVs. In Section 3, we numerically compute the O(3)-symmetric Euclidean action for a high-temperature U(1) B−L -breaking phase transition, calculate quantities that determine gravitational wave spectrum, and estimate stochastic gravitational waves generated from a U(1) B−L -breaking phase transition. Section 4 summarizes the paper.
Here, mass term µ Φ ΦΦ is absorbed by a redefinition of S and m, κ. Y D is the neutrino Dirac Yukawa coupling, and Y M is the coupling that generates Majorana mass for the right-handed neutrinos after U(1) B−L breaking. By a phase redefinition, we take λ, v 2 , Y M i to be real positive without loss of generality. From now on, we assume |m| 2 ≪ v 2 and |κ| ≪ 1. This limit is obtained when the model has R-symmetry, under which superfield S has R = +2 and Φ, Φ have R = 0, and the matter superfields have R = +1 and the Higgs superfields H u , H d have R = 0. Assuming R-symmetry is advantageous for explaining the smallness of µ in µH u H d . In the rest of the paper, we neglect |m| 2 , κ and work with the R-symmetric superpotential, As the mechanism for SUSY breaking (at zero temperature) is beyond the scope of this paper, we do not discuss soft SUSY breaking gaugino mass. The tree-level scalar potential involving Φ, Φ, S reads 2

Finite Temperature Effective Potential
To compute the one-loop effective potential at zero and finite temperature, we need the fielddependent mass eigenvalues for bosonic and fermionic components. When SUSY is preserved, bosonic and fermionic components have the same set of mass eigenvalues. However, since SUSY is already broken at finite temperature, we must also consider SUSY-breaking configurations of VEVs, e.g., the case with Φ Φ = v 2 2 giving F -term SUSY breaking, and the case with Φ = Φ giving D-term SUSY breaking. So, we study the mass eigenvalues of bosonic and fermionic components separately.
We use Landau gauge for U(1) B−L gauge theory. Before deriving the field-dependent mass eigenvalues, we assume that the VEVs at any temperature satisfy Φ Φ = (real positive), 2 By abuse of notation, we denote the scalar component by the same character as the superfield.
Then, we take advantage of the U(1) B−L symmetry to set both Φ , Φ to be real positive, and rewrite these VEVs as The rest of the section is devoted to the study on the potential for h,h.
The (h,h)-dependent mass eigenvalues for bosonic components are given as follows: We decompose the scalar components of Φ, Φ as Φ = 1 The (h,h)-dependent mass eigenvalues of fermionic components are given as follows. Let ψ Φ , ψΦ, ψ S , ψ N c i denote the fermionic part of Φ, Φ, S, N c i , respectively, and let X denote U(1) B−L gaugino. The (h,h)-dependent Majorana mass matrix for fermionic components is given by The mass eigenvalues are obtained by diagonalizing M † F M F , and are given by . It is easy to verify that when h =h = v so that SUSY is preserved, non-zero mass eigenvalues of bosonic components obtained from Eqs. (11)-(17) coincide with those of fermionic components (with the correct counting of degrees of freedom).
Finally, the finite temperature effective potential [17] for h,h is obtained as Here, Eq. (19) represents the tree-level potential. Eq. (20) is the one-loop effective potential at zero-temperature, with µ being the renormalization scale in DR scheme. Eq. (21) is the temperature-dependent part of the potential, with Bj denote the (h,h)-dependent mass eigenvalues for bosonic components, obtained by diagonalizing Eqs. (11),(12) and from Eqs. (13)- (17), with no duplication for real scalars, 2 duplications for complex scalars and 3 duplications for X µ gauge boson. M 2 F j denote the (h,h)-dependent mass eigenvalues for fermionic components, which are At temperature near or above the critical temperature, daisy diagrams cause breakdown of perturbation theory. This problem is remedied by replacing the tree-level masses of bosonic components M 2 Bj in Eqs. (11)-(17) with loop corrected ones. We follow Ref. [18] and only include T 2 -proportional part of the one-loop correction 3 , and make the following replacements for the mass of the scalar components of Φ, Φ, S, N c i and MSSM fields: 4 Here, the factor 3 2 on the second and third terms on the right hand side reflects the fact that in SUSY theories, a bosonic loop correction is always accompanied by a fermionic loop correction with the same coupling constant, and that T 2 -part of the fermionic one-loop correction to a boson mass is half the bosonic one-loop correction, and hence their sum is 3 2 times the bosonic one. The bosonic part (i.e. part without factor 3 2 ) of the second term comes from one-loop corrections via D-term and F -term quartic couplings, and that of the third term comes from one-loop corrections via gauge couplings. For the longitudinal component of the U(1) B−L gauge boson, we replace its mass, (M 2 X ) L , as in the loop is not included [19,20]. The correct recipe is to solve a self-consistency equation derived from the finite temperature effective potential. Ref. [19] has confirmed the appropriateness of the partial dressing procedure [21]. Unfortunately, this procedure has not yet been extended to a multi-field case, which is our case. 4 M 2 Q1,2 , M 2 U1,2 represent the corrected masses for 1st and 2nd generation quark doublets and up-type singlets, and M 2 Q1,2 , M 2 U1,2 represent those for 3rd generation. Their difference is a large thermal mass via the top quark Yukawa coupling y t .
while the mass of the transverse component is unchanged.
In the rest of the paper, we use the finite temperature effective potential Eqs.
Figs. 1,2,3,4 show that the finite-temperature effective potential is nearly symmetric with respect to h andh at temperature around or below the critical temperature T c . This indicates that even though only Φ, not Φ, couples to the right-handed neutrino through Majorana Yukawa coupling Y M 3 , this asymmetry does not affect the potential.
Since the potential is nearly symmetric with respect to h andh, we can approximate the classical tunneling path from the metastable vacuum (h,h) = (0, 0) to an absolute vacuum (h,h) = (0, 0) by the line h =h, because ∂ h V eff − ∂hV eff ≃ 0 and hence the equation of motion (with −V eff ) for h −h only admits a trivial solution h −h ≃(constant). Under the above approximation, the phase transition is controlled by one-dimensional potential V eff (h, h; µ, T ), which allows a qualitative discussion. The one-dimensional potential reads where M 2 Bj are now obtained by diagonalizing and also from M 2 , while the MSSM particles become irrelevant. One might guess that increasing g B−L and decreasing λ enhances the order of phase transition and hence the amount of latent heat, because the quartic coupling for h is mostly λ, and the field-dependent mass for bosons φ,φ, X µ (which provides h 3 term in high-T expansion) depends on g 2 B−L h 2 times a big factor 4 or 8. However, increasing g B−L also enhances the thermal mass for these bosons (except for the transverse component of X µ ), which diminishes their impact on the finite temperature effective potential. Therefore, we expect that the amount of latent heat (which is related to α θ (T n ) in the next section) is maximized for λ → 0 and for some moderate value of g B−L . On the other hand, Y M i is expected to have a weaker impact on the latent heat because it only appears in the field-dependent mass for N c i and is not accompanied by a big factor. All these expectations will be confirmed by the numerical study in the next section.

O(3)-symmetric Euclidean Action
We calculate the O(3)-symmetric Euclidean action [22,23] for a high-temperature U(1) B−Lbreaking phase transition from the metastable vacuum (h,h) = (0, 0) to an absolute vacuum where (h,h) = (0, 0). Although we have seen in Section 2.3 that the potential is nearly symmetric with respect to h andh, we still consider a multi-field phase transition regarding h andh as being independent. To compute the O(3)-symmetric Euclidean action for a multi-field phase transition, we use CosmoTransitions [24]. From the action computed, we derive the nucleation temperature, T n , the ratio of the trace anomaly divided by 4 over the radiation energy density of the symmetric phase at the nucleation temperature, α θ (T n ), and the speed of the phase transition at the nucleation temperature, β n . They are defined as follows: Let S E (T ) denote the Euclidean action. The tunneling rate per volume at temperature T is is a factor with milder T -dependence than e −S E (T )/T . The nucleation temperature T n satisfies H 4 n = A(T n )e −S E (Tn)/Tn , where H n denotes the Hubble rate at T = T n in the symmetric phase. We estimate A(T n ) as A(T n ) ∼ T 4 n , and further approximate T n by the U(1) B−L -breaking VEV v. Thus, we estimate T n by the relation where M * is the reduced Planck mass and g * = 255 is the effective relativistic degrees of freedom of the SUSY U(1) B−L model (including Φ, Φ, S fields). For example, when v = 100 TeV, the right-hand side of Eq. (40) equals 117, and when v = 1000 TeV, it equals 107. In the following analysis, we fix the right-hand side of Eq. (40) at about 117. α θ (T n ) is given by As with Section 2.3, we take the renormalization scale at µ = v, which does not generate a large logarithm because v is the only mass scale in the model. Given µ = v, a quantity with mass dimension n scales with v n . In particular, T n scales with v, and so we present T n /v in the plots.
In We find that T n /v has little dependence on Y M 3 , and is much affected by λ.
In Fig. 6, we plot g B−L -dependence of the trace anomaly divided by 4 over the radiation energy density α θ (T n ), for λ = 0.01, 0.1 and The dependence on Y M 3 is quite mild compared to those on g B−L and λ.
In Finally, we study how the above quantities vary with λ. We concentrate on an interesting case where g B−L = 0.4 and Y M 3 = 1, which has given the largest α θ (T n ) in the above plots when λ = 0.01. The dependence of T n , α θ (T n ), β n /H n on λ for g B−L = 0.4 and Y M 3 = 1 is found in Fig. 8. It is observed that T n and β n /H n increase linearly with λ, while α θ (T n ) decreases much rapidly.

Gravitational Waves
We estimate gravitational waves generated from a U(1) B−L -breaking phase transition in the early Universe. In this subsection, we exclusively study the case with λ = 0.1, which gives α θ (T n ) 0.1 (see the right panel of Fig. 6). This selection is because the study on gravitational wave production in a strong phase transition α θ (T n ) > 0.1 is currently under development (see, e.g., Refs. [25,26]), while that in a weaker phase transition is relatively well established.
The sources of gravitational waves from a finite-temperature phase transition are (i) the energy momentum tensor of scalar field in colliding bubbles, (ii) that of sound waves of a surrounding plasma, and (iii) that of magnetohydrodynamic turbulence of a surrounding plasma. Source (i) is negligible unless the bubble wall runs away [28], and we expect non-run away behavior in a U(1) B−L -breaking phase transition and hence neglect (i). For source (iii), the study on the efficiency of converting latent heat to the energy of magnetohydrodynamic turbulence is at its early stage, and so we omit (iii) making a conservative estimate. Thus, we only consider source (ii).
It is claimed in Ref. [28] that the energy spectrum of gravitational waves generated by sound waves in a hot plasma in a phase transition with α θ (T n ) 0.1 can be expressed as (we rewrite the formula for gravitational wave energy over the critical density we observe today) where L f,n is a typical length scale of fluid motion at the nucleation temperature, L f is the redshifted value of L f,n today, andP gw is a function only of the product kL f . U f is the enthalpyweighted root mean square four-velocity of fluid at the nucleation temperature, and 1 + p/ǫ is the ratio of enthalpy over energy. In this paper, we adopt Eq. (43). We further identify L f,n with the mean bubble separation (8π) 1/3 v w /β n [30] (v w denotes the bubble wall speed), and forP gw , we use a fitting of the simulation results in Ref. [29], which has improved on earlier works [27,28]. For (1 + p/ǫ)U 2 f , we use a fitting formula for the ratio of bulk kinetic energy over vacuum energy κ(α θ , v w ) derived in Ref. [31], and evaluate it as The calculation of the bubble wall speed v w is beyond the scope of the current paper, and we simply assume various values of v w that appear in the simulations of Ref. [29] and evaluate gravitational wave spectrum in each case.
Since we take the renormalization scale at µ = v, v is the only mass scale and a quantity with mass dimension n scales with v n . Accordingly, we present the gravitational wave spectrum in terms of frequency k divided by the tree-level U(1) B−L -breaking VEV v.
Our estimate on gravitational wave spectrum is presented in Fig. 9, for λ = 0. We find that stochastic gravitational waves are out of reach of Advanced LIGO and Virgo [32] for all values of g B−L , provided λ = 0.1. However, if λ is smaller than 0.1 and if the U(1) B−Lbreaking VEV is about 100 TeV or below, stochastic gravitational waves can be detected by these detectors. This is inferred from the fact that as seen in Fig. 8, T n and β n /H n have linear, and thus mild dependence on λ, and so the shape and position of the spectrum does not change significantly with λ. In contrast, α θ (T n ) has violent dependence on λ, and so slight decrease in λ can enhance the energy of the spectrum at the detectable level. To make a reliable prediction in the case with λ < 0.1, we need a further understanding of gravitation wave production in a phase transition with α θ (T n ) > 0.1.

Summary
We have studied the phase transition of a U(1) gauge symmetry breaking in a SUSY model and the production of stochastic gravitational waves associated with it. We have concentrated on a particular model, which is the minimal SUSY U(1) B−L model with R-symmetric superpotential. We have worked in the SUSY limit by assuming that the nucleation temperature is above SUSY breaking scale so that soft SUSY breaking terms are negligible. We have derived the finite temperature effective potential for the U(1) B−L VEVs h,h, and computed the O(3)-symmetric Euclidean action of a high-temperature U(1) B−L -breaking multi-field phase transition. We have estimated stochastic gravitational waves generated from the phase transition, and discussed its detectability.