Phase transition and vacuum stability in the classically conformal B-L model

Within classically conformal models, the spontaneous breaking of scale invariance is usually associated to a strong first order phase transition that results in a gravitational wave background within the reach of future space-based interferometers. In this paper we study the case of the classically conformal gauged B-L model, analysing the impact of this minimal extension of the Standard Model on the dynamics of the electroweak symmetry breaking and within cosmology. Particular attention is paid to the problem of vacuum stability and to the role of the QCD phase transition, which we prove responsible for concluding the symmetry breaking transition in part of the considered parameter space. Finally, we calculate the gravitational wave signal emitted in the process, finding that a large part of the parameter space of the model can be probed by LISA.


I. INTRODUCTION
The recent detection of the first gravitational wave signal by the LIGO collaboration [1] has opened a new observational window into the Universe. An important point of these investigations concerns the dynamics of phase transitions that occurred during the cosmological evolution, which may play a central role in an array of topics spanning from the problem of the baryon asymmetry of the Universe [2][3][4][5][6][7][8][9][10][11] to the quest for an ultraviolet completion of the Standard Model (SM) [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. To provide a concrete example, gravitational wave astronomy has the potential to pinpoint the dynamics of the phase transition behind the generation of the electroweak scale, setting a new important benchmark for particle physics models. In fact, whereas the SM supports a second order electroweak phase transition, many of its extensions instead predict a first order phenomenon. In this case the electroweak phase transition proceeds through the nucleation and consequent expansion of bubbles that contain the true symmetry-breaking vacuum. Collisions between the bubbles and the motion in the plasma after bubble collisions then produce gravitational wave signals that can be detected in the present stochastic background by next-generation experiments such as the satellite-based interferometer LISA [32].
In this work we continue these analyses by considering the classically conformal B-L model introduced originally in Refs. [61][62][63] and further studied in Refs. [64][65][66][67][68] 1 . Differently from previous studies [19], we focus on the impact of the SM QCD phase transition, improving the analysis in Ref. [41] and demonstrating that in part of the considered parameter space it is responsible for concluding the dynamics of the electroweak symmetry breaking. In line with the general results of Ref. [74], we also find that thermal inflation [75,76] is a feature of the model. In fact, the contribution of the potential energy difference between true and false vacuum states is large enough to dominate the Hubble parameter at temperatures below the critical one, and the inflationary regime may last until the onset of the QCD phase transition with non-trivial consequences on additional phenomenology [77].
The structure of the paper is as follows: after introducing the model in Sec. II, we briefly discuss in Sec. III its phenomenology at collider experiments and within cosmology. The effective potential, including the contributions of thermal corrections and QCD phase transition, is presented in Sec. IV, whereas the relative analyses of perturbativity and vacuum stability are detailed in Sec. V. The electroweak phase transition is studied in Sec. VI, and the resulting gravitational signature of the model is computed in Sec. VII. Finally, in Sec. VIII, we gather our conclusions.

II. THE MODEL
The model we consider is based on the symmetry group SU(3) c × SU(2) L × U(1) Y × U(1) B−L , with quarks and leptons having a B -L charge of +1/3 and −1, respectively. The particle content of the SM is extended to include right handed neutrinos (RHN) ν Ri , required by the cancellation of the U(1) B−L anomaly, and a complex scalar φ = (ϕ + iG)/ √ 2 that only carries a +2 U(1) B−L charge. Notice that the SM Higgs doublet The scalar sector of the model is characterised by the following tree-level potential, where we include the so-called 'portal coupling' λ p between the Higgs doublet and the new scalar field. Through the dimensional transmutation mechanism, radiative corrections induce non-trivial solutions for the minimization equation of the scalar potential and, consequently, both h and ϕ develop non-vanishing vacuum expectation values (VEVs): According to the emerging phase transition pattern, the radiative symmetry breaking occurs along the ϕ direction of the field space and is transmitted to the Higgs sector via the (positive) portal coupling. We anticipate that our work focuses on the case where w v, as collider bounds disfavour the complementary setup. In this limit, matching the observed Higgs boson mass m h = 126 GeV and the electroweak VEV v = 246 GeV forces the Higgs boson quartic coupling approximately to its SM value, λ h m 2 h /(2v 2 ). The portal coupling is instead related to the VEV of ϕ by λ p m 2 h /w 2 . The interactions of the RHNs are given by and after the symmetry breaking result in Dirac neutrino masses, as well as Majorana masses for the RHNs. This implements the seesaw mechanism [78][79][80][81] in the model, which ascribes the smallness of the measured (active) neutrino masses to a suppression factor given by the ratio between the neutrino Dirac mass scale and the RHN Majorana mass scale. In order not to clash with the bounds from Big Bang Nucleosynthesis we will consider RHNs with masses above 200 MeV. The final ingredient of the model is a kinetic mixing term for the U(1) Y and U(1) B−L gauge fields, which is generally produced by quantum corrections even if set to zero at a scale. After diagonalising the kinetic term, the U(1) Y × U(1) B−L part of the gauge covariant derivative is given by whereg parametrizes the extent of the kinetic mixing and q j , g j and B µ j are the charges, the gauge coupling and the gauge fields, respectively. Given the charge assignment of the extra scalar field φ, the spontaneous symmetry breaking of the U(1) B−L symmetry will induce a mass for the corresponding gauge boson given after the diagonalization by m Z = 2g B−L w.

III. PHENOMENOLOGICAL CONSEQUENCES
The interactions contained in Eq. (2) link the present framework to the problem of the origin of the baryon asymmetry detected in our Universe. In fact, as RHNs acquire a Majorana mass through the symmetry breaking of U(1) B−L , it is possible to implement the leptogenesis mechanism for baryogenesis.
In standard scenarios of thermal leptogenesis [82,83], RHNs with hierarchical Majorana masses M i 10 9 GeV are thermally produced in the plasma after inflation. As the Universe expands, an original lepton, or B -L, asymmetry is generated via the CP-violating out of equilibrium decays of the RHNs, and consequenlty partially converted into a baryon asymmetry by the SM sphaleron processes. Remarkably, possible pre-existing B -L asymmetries can be efficiently washed out owing to the interplay between flavour effects and the RHN mass hierarchy [84][85][86]. In terms of the present analysis, implementing a standard leptogenesis scenario would force the Z mass scale well above the reach of contemporary collider experiments, being this parameter sourced by the same VEV w behind the RHNs mass scale. On general grounds, we also expect sizeable coupling in Eq. (2) that could drive the running of the scalar sector parameters.
As an alternative, it is possible to consider a scenario where RHNs with masses comparable to, or below, the electroweak scale, can produce the required baryon asymmetry via CP-violating flavour oscillations [87]. More in detail, the complex non-diagonal Majorana Yukawa matrices in Eq. (2) induce CP-violating ν Ri ↔ ν Rj transitions, which conserve the overall lepton number but violate the lepton number of individual flavours. In this way, provided that at least one species of RHNs remains out of equilibrium while the SM sphalerons are active, the lepton asymmetry transmitted to the SM by the remaining RHN species will be reprocessed by the same sphaleron processes. Within the context of the conformal B -L model, Ref. [68] adopted this mechanism to explain the observed baryon asymmetry of the Universe, and this is also the case that we implicitly assume in the present work. Given the allowed RHN mass range, we can estimate the typical size of the neutrino Dirac Yukawa couplings via the seesaw formula, where M ν and m ν are the mass scale of RH and active neutrinos, respectively. As a consequence, we can safely neglect the role of these parameters in determining the radiative corrections to the scalar sector. The same holds for the remaining couplings in Eq. (2), provided that RHN do not exceed the electroweak scale: For the above estimate we adopted a conservative bound, m Z 4 TeV, suggested by the current Z searches for g B−L 10 −1 [66,67,73].

IV. EFFECTIVE POTENTIAL
Although the full scalar potential selects a surface in the two-dimensional field space, and both the fields develop a non-zero VEV at the minimum generated by radiative corrections, for w v the properties of the phase transition can be inferred by analysing the dependence on ϕ only [88].
The one-loop finite temperature effective potential along the ϕ direction is given by where the one-loop RG-improved T = 0 contribution is The RG scale is chosen such that t = log(ϕ/µ 0 ), where µ 0 is a reference scale. By using the above expression, we find that the mass of the extra scalar approximately scales as The finite temperature part is where the sum runs over the B -L gauge boson, RHNs, the scalar boson ϕ and the Goldstone boson G, and k j indicates the intrinsic number of degrees of freedom (k Z = 3, k i = 1, k ϕ = 1 = k G ,). The field dependent masses are given by and the Debye masses by Here, the thermal integral J T is defined as with the negative sign for bosons and the positive for fermions.
Depending on the values of these parameters, it is possible that the U (1) B−L breaking phase transition does not occur above the QCD phase transition temperature, T QCD = O(0.1 GeV). In this case, the QCD phase transition induces an additional linear term for the Higgs field, j y j ψ jψj h/ √ 2, which consequently acquires a nonzero VEV, v QCD ≡ h = O(0.1 GeV) [41]. In turn, the portal coupling then induces a negative mass term for ϕ, so that at T < T QCD the effective potential along the ϕ direction becomes where V T >TQCD eff is given by Eq. (6). In our analysis we take v QCD = T QCD = 0.1 GeV.

V. VACUUM STABILITY AND PERTURBATIVITY
The high-energy behaviour of a model can be inferred by studying the renormalization flow of its parameters. The requirement of desireable properties, such as stability and perturbativity, then generally result in further constraints on the low-energy parameter space of the framework under examination.
Within the SM, for example, renormalization group methods indicate the allowed top quark and Higgs boson mass windows through the requirements of i) limited interaction couplings (perturbativity), and ii) the absence of scalar background configurations with energies below the EW one (vacuum stability) [89,90], which crucially depends on the value of the top quark mass. In regard of this, according to present measurements of this parameter, the SM vacuum is only metastable [91][92][93][94][95].
Many extensions have been proposed in the attempt to overcome this puzzling feature of the SM. For instance the simplest SM×U (1) B−L framework with explicit symmetry breaking has been investigated up to next-to-leading precision in Refs. [72,[96][97][98][99][100][101][102]. However, all these analyses confirm that the extra Yukawa terms introduced in this simple scenario generally worsen the overall high-energy behavior of the model, in spite of the stabilizing effect of scalar mixing and gauge couplings. An exception to these conclusion is provided by classical conformal models [65][66][67], where the requirement of a radiative U (1) B−L breaking bounds the magnitude of the parameters in the Majorana neutrino mass matrix.
In fact, as shown in Ref. [61], the presence of a radiatively generated minimum breaking the B-L symmetry can be inferred independently from the SM Higgs background. Along the ϕ direction, the minimization of the effective potential reduces to a single scale problem, Notice that in writing Eq. (13) we assumed negligible values of the portal coupling λ p , as required for v w. As a consequence, scalar mixing cannot be invoked here to stabilize the SM vacuum. The analytical minimization of the effective potential, resulting in the boundary condition β λ φ + 4λ φ = 0 for t = log(w/µ 0 ), ensures that the h−independent part of the effective potential reaches its minimum at ϕ = w and increases elsewhere. Therefore, considering also the smallness of λ p , in the conformal B -L model instabilities can only be generated along the Higgs direction.
The sign of the second derivative of the effective potential in the ϕ direction, as computed from the non-trivial solution of Eq. (13), receives a positive contribution from the gauge sector and a negative one from the Majorana Yukawa couplings. Hence, for w to be in correspondence of a minimum of the potential (or, equivalently, to prevent a tachyonic scalar mass), the RHN Majorana mass scale must satisfy M ν < m Z /2 1/4 [66] (enforced in the present case by Eq. (5)), in line with similar bounds concerning the evolution of λ p and the following instability [72,100]. Within the classical conformal case, the high-energy behavior of the model is therefore shaped by the extended gauge sector, which enters the RGE of the scalar sector through the gauge coupling g B−L and the mixing parameterg. The kinetic mixing, in particular, affects the evolution of λ H already at the one-loop level, allowing to solve the issue of the SM instability when the mixing is sizeable. This is explicitly shown in Fig. 1, where λ H is plotted as a function of the renormalization scale for three different values ofg. The indicated values of the couplings have been set at the w scale.
The results of our analysis concerning the perturbativity and stability of the model are presented in Fig. 2   −0.5. We therefore adopt this value for the mixing parameter in the following analysis, anticipating that the phase transition dynamics do not significantly depend on this choice: g does not directly affect the one-loop effective potential, entering only the running of g B−L . To conclude the section, we remark that the sudden change in the stability of the potential for values ofg −0.4 is due to the RG flow of λ h shown in Fig. 1.

VI. PHASE TRANSITION
At very high temperatures, thermal corrections dominate the potential and localize the fields at the origin, preventing the formation of new minima that would result in the spontaneous breaking of the symmetries of the model. This configuration is maintained until the temperature decreased enough to allow for the appearance of a second minimum in the potential, corresponding to a non-vanishing value of ϕ. We can therefore define the critical temperature T c as the temperature for which the new, symmetry-breaking, minimum is degenerate with the stationary point at the origin. We find numerically that As the temperature further decreases, the symmetrybreaking minimum becomes a global minimum of the potential, but thermal corrections still result in a potential barrier that prevents the fields from leaving the origin. At temperatures T T c , the potential energy difference between the global minimum and the origin is then sizeable, whereas the height of the potential barrier progressively decreases. Quantum tunneling effects can then drive the fields to the global minimum of the potential, starting a first-order phase transition that proceeds via nucleation and consequent expansion of bubbles inside of which the symmetry is broken.
The bubble nucleation rate per unit of time and volume can be estimated as [103] Γ(T ) T 4 S 3 2πT where is the action for an O(3)-symmetric bubble. The largest contribution into the above quantity arises from the classical path which minimizes S 3 , corresponding to the solution of with boundary conditions dϕ/dr = 0 at r = 0, and ϕ → 0 at r → ∞. The evolution of S 3 /T as a function of temperature is shown for a benchmark case in Fig. 3. As temperature decreases, S 3 /T also decreases and eventually results in a sizeable bubble nucleation rate. However, below T = T QCD , the QCD phase transition changes the behaviour of S 3 /T inducing a negative mass term which cancels the thermal potential barrier, in a way that S 3 /T eventually vanishes.
We can then define the bubble nucleation temperature T n as the temperature at which the probability of producing at least one bubble per horizon volume in a unit of Hubble time approaches unity [104], 4 1 .
Notice that the Hubble rate H includes the contribution due to the energy difference between the symmetryconserving and symmetry-breaking vacua, ∆V (T = 0). This vacuum energy dominates over the radiation contribution at as long as the phase transition is ongoing, and results in an epoch of thermal inflation. Following Ref. [74], we proceed by computing the volume fraction converted to the symmetry-broken phase at temperature T , The percolation temperature T p is defined as I(T p ) = 0.34, and here always satisfies the condition [74] 3 + T dI/dT < 0 (21) enforcing that the physical volume of the patches still in the symmetric phase of the theory decrease. The lines in the bottom panel of Fig. 4 show the behaviour of the different temperatures defined above as a  function of the B-L gauge coupling, assuming m Z = 10 TeV andg = −0.5 and neglecting the Majorana Yukawa couplings. We see that percolation always takes place right after nucleation. The relative difference between the two corresponding temperatures is at its maximum immediately before the QCD phase transition starts to affect the dynamics. The temperature T co , represented by the lowest dashed line, signals the vanishing of the thermal potential barrier. For small values of g B−L this is due to the negative mass term induced by the QCD phase transition, whereas for larger g B−L this is caused by the running of λ φ .
The middle panel of Fig. 4 shows instead that the number of e-folds of thermal inflation, N ≡ log(T v /T p ), significantly increases for lower values of g B−L . Consequently, only for g B−L 0.42 the phase transition concludes during the radiation dominated epoch. We find that N behaves as N log(m Z /GeV) + C for T p < T QCD , where C is a constant that is mildly dependent on g B−L and C(g B−L = 0.1) 1.0. CMB studies then bound m Z from above through the requirement that N 60, as required by the observed perturbation spectra. The bound however is not very efficient: for g B−L = 0.1 we have m Z 10 25 GeV. After the phase transition concludes, the vacuum energy decays into radiation, consequently producing a plasma thermalised at a temperature T v < T reh < T c 3 shown by the red line in the bottom panel of Fig. 4. We find that T reh scales as m Z , and if T p T v , the reheating temperature is T reh T v 0.09m Z . Given the parameter values indicated by collider experiments, the resulting temperature is high enough to restore the electroweak symmetry. This will be then broken in the same way as in the pure SM, as the temperature decreases below T 140 GeV with the expansion of the Universe.
Our results concerning the phase transition dynamics are also presented in the first panel of Fig. 6, as a function of m Z and g B−L . The color code indicates the percolation temperature, the dashed lines show the number of e-folds of thermal inflation, and the dot-dashed lines represent the reheating temperature. The thick black line highlights the contour T p = T QCD , below which the transition happens only after the QCD one.

VII. GRAVITATIONAL WAVE SIGNAL
We now discuss the gravitational wave signal emitted at the phase transition, focusing on strongly supercooled dynamics, T p T c , that occur during the vacuum energy dominance. We assume here that the bubble walls do not reach terminal velocity before they collide, as the energy density of the plasma is strongly depleted during the thermal inflation period. Bubble collisions then source the GW spectrum, which in the source frame is given in by [105], as a function of the wave-number k = 2πν. Herek = 3.2/R * corresponds to the peak frequency ν = ν env of the spectrum, H * = H(T p ) is the Hubble rate, is the vacuum energy density parameter, and is the average bubble separation [74] at the percolation temperature. Our computations indicate that R * scales roughly as m −2 Z , so the product H * R * is basically independent of m Z . The dependence of H * R * on the B-L gauge coupling is shown in the top panel of Fig. 4.
In order to predict the corresponding signal detectable at gravitational wave observatories, we let the gravitational waves emitted at the phase transition propagate until today. This amounts to a scaling of amplitude and frequency given by [106] where g * = g(T reh ) is the effective number of relativistic degrees of freedom at the reheating temperature. The gravitational wave spectra generated for three benchmark points are shown in Fig. 5. The signal is the strongest for g B−L = 0.26, as for smaller values of this parameter the phase transition takes place only after the QCD one. In fact, as shown in Fig. 3, the slope of S 3 /T changes after the QCD phase transition has induced a negative mass term for ϕ, which effectively speeds up the process. This is manifest in the top panel of Fig. 4, showing the average bubble separation at T p as a function of g B−L .
Our final results are summarised in Fig. 6. The left panel shows the percolation temperature, the number of e-folds of thermal inflation and the reheating temperature, as discussed in the previous section. In all panels the thick solid black line indicates where T p = T QCD . Below this line, the phase transition happens after QCD has already induced a negative mass term for the ϕ field.
The middle and right panels of Fig. 6 characterise the GW emission consequent to the phase transition. The middle panel shows the peak frequency ν env and the amplitude of the spectrum at the corresponding frequency, Ω GW (ν = ν env ). We see that the strongest GW signal is obtained when the transition takes place immediately before the QCD one. The shape of the peak frequency contours follows from the g B−L dependence of R * , shown in the top panel of Fig. 4, and the peak frequency increases as a function of m Z as the spectrum is redshifted. The blue dashed line highlights the frequency ν = 3 mHz to which LISA is sensitive the most. The mild dependence of H * R * on m Z is also evident from the behaviour of the Ω GW (ν = ν enc ) contours (dot-dashed lines).
Finally, our prospect for the detection at LISA of the GW spectrum emitted in the considered model is shown in the right panel of Fig. 6. The color code indicates here the amplitude of the GW signal relative to the best sensitivity of the experiment. In the A5M5 setup, (see Fig. 5), LISA would be able to probe the whole region between the blue dashed contours.
We remark that our prediction of the GW spectrum is not to be trusted in the region above the black dashed line, where the phase transition concludes in the radiation dominated era. In fact, in this case Eq. (22) is not applicable as GWs originate from sound waves and turbulence in the plasma rather than bubble collisions. Notice however that entering such a region requires a substantial g B−L coupling, which according to the bottom panel of Fig. 2 sets the perturbativity scale of the model below the Planck scale.

VIII. CONCLUSIONS
In this work we furthered the study of the conformal B-L extension of the Standard Model. After introducing the framework and briefly reviewing its general phenomenology, we focused on the phase transition dynamics that the scenario supports and on the high-energy properties of the theory.
With the RG-improved potential for the scalar sector of the theory at hand, we have identified a region in the parameter space of the model that ensures the stability of the potential and the perturbativity of its parameters up to scales well beyond the Planck one. In particular, we have found that the electroweak vacuum instability is here rescued by the effect of the gauge mixing, once the mixing parameter is set tog −0.5.
Assuming this value in the following analysis (which is rather insensitive to this parameter), we then studied the symmetry breaking patter supported by the model, originated by the extra scalar field responsible for the radiative breaking of the B-L symmetry. We find that thermal corrections prevent the transition to the emerging symmetry breaking minimum of the effective potential in a large part of the considered parameter space. As a consequence, we see the rise of an epoch of thermal inflation sourced by the potential energy difference between the false and true vacua of the theory.
At the latest, the inflationary regime concludes soon after the onset of the QCD phase transition, as the additional term induced in the scalar potential weakens the thermal contribution and a first order electroweak phase transition takes place. The potential energy density of the false vacuum is then transferred to radiation in a sec- ond reheating process, which generally is strong enough to restore once again the electroweak symmetry. After the second reheating process concludes, the leptogenesis mechanisms implemented in the framework give possibly rise to the observed baryon asymmetry of the Universe. The electroweak symmetry is instead finally broken after the temperature has dropped below the electroweak scale, in the same way as in the original Standard Model.
To conclude our investigation, we have computed the gravitational signal emitted in the present framework during the first order phase transition that follows the thermal inflation epoch. We find that the amplitude of the spectrum is sizeable enough to fall within the reach of next-generation interferometers. In particular, LISA will probe most of the parameter space considered in the present analysis.