PeV scale Left-Right symmetry and baryon asymmetry of the Universe

We study the cosmology of two versions of supersymmetric Left-Right symmetric model. The scale of the $B-L$ symmetry breaking in these models is naturally low, $10^4 - 10^6$ GeV. Spontaneous breakdown of parity is accompanied by a first order phase transition. We simulate the domain walls of the phase transition and show that they provide requisite conditions, specifically, $CP$ violating phase needed for leptogenesis. Additionally soft resonant leptogenesis is conditionally viable in the two models considered. Some of the parameters in the soft supersymmetry breaking terms are shown to be constrained from these considerations. It is argued that the models may be testable in upcoming collider and cosmology experiments.


I. INTRODUCTION
Left-Right symmetric model [1,2,3,4,5] is a simple extension of the Standard Model (SM) [6,7,8]. As for the fermion sector the presence of right handed neutrino states in the theory allows the possibility of explaining the smallness of the observed neutrino masses [11,12,13,14] from the see-saw mechanism [15,16,17,18]. While the scale of Majorana masses is no longer as high as in the conventional see-saw expectations, the PeV scale still permits [19] explaining the smallness of the light neutrino mass scale for at least certain textures of fermion mass parameters. It is therefore worth exploring the possibility that the scale of Left-Right symmetry be the PeV scale, potentially testable in colliders.
Whether we follow the GUT proposal or the PeV scale possibility, the large hierarchy between the mass scales M EW ∼ 250GeV of electroweak symmetry and M GU T ∼ 10 15 GeV is difficult to understand within the Higgs paradigm. While the Higgs sector of the Standard Model is poorly understood, it is nevertheless very successful. We therefore speculate that the breaking of both the SU(2) L and SU(2) R being at a comparable scale will have a similar explanation, possibly a comprehensive one including both. There remains the need to understand the hierarchy with respect to a larger mass scale either the GUT scale or the Planck scale. In this paper we assume supersymmetry (SUSY) to be the mechanism to stabilize the hierarchy beyond the electroweak scale [20,21], in other words we assume TeV scale SUSY 1 . We study what has been called the minimal supersymmetric Left-Right symmetric model (MSLRM) [25] with the gauge group SU(3) c ⊗ SU(2) L ⊗ SU(2) R ⊗ U(1) B−L augmented 1 See for instance [22,23,24] and references therein. by parity P exchanging L and R sectors. Lee et al. [26] have studied a similar model with the gauge group SU(4) c ⊗ SU(2) L ⊗ SU(2) R and connected it to cosmological phenomena, specifically inflation. Our discussion differs in being specifically PeV scale.
In the MSLRM class of Left-Right symmetric models, spontaneous gauge symmetry breaking required to recover SM phenomenology also leads to observed parity breaking.
However, for cosmological reasons it is not sufficient to ensure local breakdown of parity.
We have earlier proposed [27] that the occurrence of the SM like sector globally is connected to the SUSY breaking effects from the hidden sector. Another approach to implementing the global uniformity of parity breaking is to have terms induced by gauge symmetry breaking which signal explicit parity breaking [28,29]. This model has been dubbed MSLR/ P. In earlier papers we have explored the overall cosmological setting for these models and traced issues such as removal of unwanted relics and a successful completion of the first order phase transition. Here we show that sufficient conditions exist in the model to provide for the leptogenesis required to explain the baryon asymmetry of the Universe.
A possible implementation of this idea follows the thermal leptogenesis [30] route. This however has been shown to generically require the scale of majorana neutrino mass, equivalently, in our model the scale of B − L breaking to be 10 11 -10 13 GeV [31,32], with a more optimistic constraint M B−L > 10 9 GeV [33,34]. This situation is not improved [35,36,37,38] by assistance from cosmic string induced violation [39,40,41] of lepton number [42]. On the other hand, it has been shown [19,43] that the only real requirement imposed by Leptogenesis is that the presence of heavy neutrinos should not erase lepton asymmetry generated by a given mechanism, possibly non-thermal. This places the modest bound M 1 > 10 4 GeV, on the mass of the lightest of the heavy majorana neutrinos. A scenario which exploits this window and relies on supersymmetry is the "soft leptogenesis", [44,45,46,47] relying on the decay of scalar superpartners of neutrino and a high degree of degeneracy [48] in the mass eigenvalues due to soft SUSY breaking terms.
Another possibility for leptogenesis arises from the fact that generically the Left-Right breaking phase transition is intrinsically a first order phase transition. Due to the presence of lepton number violating processes, the problem of leptogenesis then becomes analogous to that explored for the electroweak phase transition [49], provided a source for CP asymmetry can be found. It has been shown [50] that the domain walls arising during the phase transition generically give spatially varying complex masses to neutrinos. Here we explore the parameter space required in the two variants of Left-Right symmetric model to ensure the required leptogenesis.
The paper is organized as follows. In the next sections II and III we review the models being considered. In sec. IV we discuss the cosmological evolution characteristic of each of the models, along with the constraints that can be obtained on the soft parameters of the models by the demand that the phase transition is completed successfully. In V we identify the soft parameters in the model that can be constrained by the demand for soft leptogenesis. In sec.s VI and VII we detail the mechanism of leptogenesis by the domain wall (DW) structure of the phase transition and then obtain numerical solutions which support the possibility of this mechanism to operate in the two models. Conclusions are summarized in sec. VIII.
where we have suppressed the generation index. The minimal set of Higgs superfields required is, Under discrete parity symmetry the fields are prescribed to transform as, However, this minimal model is unable to break parity spontaneously [51,52]. A parity odd singlet solves this problem [53], but this also breaks electromagnetic charge invariance [51].
Breaking R parity and introducing non-renormalizable terms solves this problem. A more appealing way out is to introduce a pair of scalar triplets (Ω, Ω c ), which are even under parity viz., Ω ↔ Ω * c [27,29,54]. The quantum numbers for the two fields are, The superpotential for this model was given in [54]. It is almost the same as the superpotential given later in this paper, in sec. III, eq. (11) from which it can be obtained with The stages of breaking required to implement parity breaking and avoid electromagnetic charge breaking vacua, are as follows: first the Ω's get a vev at a scale M R , which breaks From the F and D flatness conditions we are led to the following solution for the vev's [27,29,54] Due to the possibility of alternative set of Higgs vacua, in the early universe, parity breakdown does not select unique ground state and formation of domain walls (DW) is inevitable.
As this contradicts present observable cosmology the model must have an inbuilt asymmetry to remove the domain walls. Since the superpotential doesn't allow such asymmetry in the present model, we depend on the soft terms to do the job.
The mechanism which induces the soft terms can arise due to gravitational effects in the gravity mediated supersymmetry breaking. In gauge mediated supersymmetry breaking (GMSB), the soft terms can arise due to the messenger sector, the hidden sector or both. In the next section III however, we look for an alternative possibility for the breaking parity, which arises naturally out of the Higgs sector.

III. MSLR/ P
In this section we consider another possibility for parity breaking which takes place within the Higgs sector. The idea was first considered by Chang et al. [28], for the non-susy model To break parity an extra Higgs singlet η which is odd under P parity was introduced .i.e η ↔ −η. As such the potential of the model has a term of the form where the notation is self-evident. Thus, when at a high scale M P , the singlet η gets a vev, the effective masses of the left and right triplet Higgs masses become different, thus explicitly breaking P parity, without affecting SU(2) R . However, in SUSY, a parity odd singlet in the theory would generate the problems of charge breaking vacua as discussed by Kuchimanchi and Mohapatra [51]. To avoid this, but to implement the idea of Chang et al. we propose with a pair triplets (Ω, Ω c ) which are odd under parity. This model was discussed in an earlier paper [29] and was named MSLR/ P . Under parity, The superpotential for this parity symmetry becomes, where color and flavor indices have been suppressed. Further, h Finally, f, h are real symmetric matrices with respect to flavor indices.
The F and D flatness conditions derived from this superpotential are presented in appendix A. However, the effective potential for the scalar fields which is determined from modulus square of the D terms remains the same as for the MSLRM at least for the form of the ansatz of the vev's we have chosen. As such the resulting solution for the vev's remains identical to eq. (6). The difference in the effective potential shows up in the soft terms as will be shown later. Due to soft terms, below the scale M R the effective mass contributions to ∆ and∆ become larger than those of ∆ c and∆ c . The cosmological consequence of this is manifested after the M B−L phase transition when the ∆'s become massive. Unlike MSLRM where the DW are destabilized only after the soft terms become significant, i.e., at the electroweak scale, the DW in this case become unstable immediately after M B−L . Leptogenesis therefore commences immediately below this scale and the scenario becomes qualitatively different from that for the MSLRM.
In the next section we elaborate in detail the areas where the two models MSLRM and MSLR/ P differ from the cosmological point of view.

IV. COSMOLOGY OF BREAKING
In this section we recapitulate the cosmology of these models. In the two models MSLRM and MSLR/ P the stages of breaking are slightly different as shown in Table (I A handle on the explicit symmetry breaking parameters of the two models can be obtained by noting that there should exist sufficient wall tension for the walls to disappear before a desirable temperature scale T D . It has been observed in [56] that the free energy density difference δρ between the vacua, which determines the pressure difference across a domain Onset of wall dominated secondary inflation. Higgs triplet (∆ ′ s) End of inflation and beginning of L-genesis in order for the DW structure to disappear at the scale T D .
We can determine the differences between the relevant soft parameters for a range of permissible values of T D .
In Table II we have taken d ∼ 10 4 GeV, ω ∼ 10 6 GeV and T D in the range 100 MeV − 10 GeV [57]. The above differences between the values in the left and right sectors is a lower bound on the soft parameters and is very small. Larger values would be acceptable to low energy phenomenology. However if we wish to retain the connection to the hidden sector, and have the advantage of secondary inflation we would want the differences to be close to this bound. As pointed out in [56,58] an asymmetry ∼ 10 −12 is sufficient to lift the degeneracy between the two sectors.
These terms remain unimportant at first due to the key assumption leading to MSSM as the effective low energy theory. The SUSY breaking effects become significant only at the electroweak scale. However, below the scale M R , Ω and Ω c acquire vev's given by eq. (5) or (8). Further, below the scale M B−L the ∆ fields acquire vev's and become massive. The combined contribution from the superpotential and the soft terms to the ∆ masses now explicitly encodes the parity breaking, where M 2 ∆ is the common contribution from the superpotential. The difference in free energy across the domain wall is now dominated by the differential contribution to the ∆ masses where we have considered ω c ∼ ω, d ∼d ∼ d c ∼d c . Now using eq (12) for a range of where we have considered |ω| ≃ M R , |d| ≃ M B−L . However, supersymmetry provides new channels for thermal leptogenesis via out of equilibrium decay of scalar superpartners of leptons [44,45,46]. Leptogenesis from scalar sector is free of strong constraints on the Yukawa couplings as happens in thermal leptogenesis from fermion decay [32]. In the mechanism to be discussed, the sneutrino splits into two distinct mass eigenstates due to soft supersymmetry breaking terms. The relevant terms in the superpotential are The relevant soft terms (V ls ) in our model are given by Mixing between the two states of sneutrino generates the CP violation.
Consider the generic model introduced by [45], where the superpotential is given by where, L, H and N are the left-handed lepton doublet, the Higgs and the right handed neutrino respectively. Here we have omitted the generation index for simplicity of notation.
The SUSY soft breaking terms are given by, The mixing between the two eigenstates in the decay of the right-handed sneutrino ( N) produces the required CP violation (ǫ). The two eigenstates N 1 and N 2 of the sneutrino, Due to the near degeneracy of these masses the CP asymmetry can be large. The mechanism has been studied in detail in [59] where it is shown that the constraint on the soft parameter This is the same as the B parameter in our model introduced in eq. (19). In [59] it is shown that this constraint can be corroborated by collider experiments involving Z ′ decays.
The Z ′ sector of the model we are considering is similar and similar collider constraints are applicable.
Further, we see that the B required is O(10 −12 ) relative to the electroweak scale. This smallness of the value is possible in certain scenarios [60] and is expected in models of hidden sector supersymmetry breaking. Here we see a correspondence between the smallness of this parameter and the parameters in the Higgs sector as determined from the cosmological constraint of disappearance of the DW summarized in sec. IV A. This is a strong indication that we may be able to test the validity of MSLRM by ascertaining its hidden sector breaking scheme and correlating the two cosmological requirements determined from smallness of otherwise unrelated parameters arising from the same mechanism.

VI. LEPTOGENESIS FROM FIRST ORDER PHASE TRANSITION
In addition to the resonant leptogenesis considered in previous section, the models considered here also include natural possibility of non-thermal leptogenesis. The spontaneous breaking of a discrete symmetry automatically makes the Left-Right symmetry breaking phase transition a first order phase transition. The idea is similar to electroweak baryogenesis proposals [49,61] where there are spontaneously formed bubbles which expand to complete the phase transition, a mechanism also considered in the case of Left-Right symmetric model in [62]. it is the ν L whose fate we keep track of. To get leptogenesis, one needs an asymmetry in the reflection and transmission coefficients from the wall between ν L and its CP conjugate (ν c L ). This can happen if a CP-violating condensate exists in the wall. This comes from the Dirac mass terms as discussed in [63,64,65,66,67]. Then there will be a preference for transmission of, say, ν L . The corresponding excess of antineutrinos (ν c L ) reflected in front of the wall will quickly equilibrate with ν L due to helicity-flipping scatterings, whose amplitude is proportional to the large Majorana mass. However the transmitted excess of ν L survives because it is not coupled to its CP conjugate in the region behind the wall, where the majorana mass contribution from ∆ and ∆ c vanishes.
A quantitative analysis of this effect can be made either in the framework of quantum mechanical reflection, valid for domain walls which are narrow compared to the particles' thermal de Broglie wavelengths, or using the classical force method [63,64,65,66,67] which gives the dominant contribution for walls with larger widths. We adopt the latter here. The thickness of the wall depends on the shape of the effective quartic potential and we shall here treat the case of thick walls. Further, we assume that the potential energy difference between the two kinds of vacua is small, for example suppressed by Planck scale effects. In this case the pressure difference across the phase boundary is expected to be small, leading to slowly moving walls. The classical CP-violating force of the condensate on a fermion (in our case a neutrino) with momentum component p x perpendicular to the wall can be shown to be The sign depends on whether the particle is ν L or ν c L , m 2 ν (x) is the position-dependent mass, E the energy and χ is the spatially varying CP-violating phase. One can then derive a diffusion equation for the chemical potential µ L of the ν L as seen in the wall rest frame: Here D ν is the neutrino diffusion coefficient, v w is the velocity of the wall, taken to be moving in the +x direction, Γ hf is the rate of helicity flipping interactions taking place in front of the wall (hence the step function θ(x)), and S is the source term, given by where v is the neutrino velocity and the angular brackets indicate thermal averages. The net lepton number excess can then be calculated from the chemical potential resulting as the solution of eq. (25).
In order to use this formalism it is necessary to establish the presence of a positiondependent phase χ. This is what we turn to in the following discussion of the nature of domain walls in the L-R model.

VII. WALL PROFILES AND CP VIOLATING CONDENSATE
In order for nontrivial effects to be mediated by the walls, the fermion species of interest should get a space-dependent mass from the wall. Furthermore, the CP-violating phase χ should also possess a nonvanishing gradient in the wall interior. We study the minimization of the total energy functional of the scalar sector with this in mind.
The vev's introduced in eq. (5) are in general complex. Some of them can be rendered real by global SU(2) transformations [5,68] according to The vev's of the triplets Ω and Ω c being diagonal are not affected by these transformations.
Their phases if any do not enter fermion or sfermion masses. We choose their phases to be real. This leaves us with 16 degrees of freedom in the Higgs sector. These can be parameterized by allowing three of the vev's in the four ∆ fields and three of the vev's in the two bidoublets Φ to be complex. Here we present a simpler model. As shown in eq.s (32) and (33) The effective potential obtained by substituting these vev's is given in eq. (B1) in the appendix B. In accordance with the discussion accompanying eq.s (6) and (7) Electroweak symmetry is unbroken at the epoch under consideration and hence the asymptotic values for k 1 and k 2 are zero. Since both k 1 and k 2 approach the same values asymptotically, the effective asymptotic value of χ is π/4. The departure from this value at the maxima of the graphs are listed in table III. It was observed that the difference in k 1 and k 2 profiles, the source of spatially varying CP violating phase χ arises from the terms 16 µ 2 k 1 k 2 The parameter α entering the superpotential is the least controlled by the fundamental symmetries and phenomenological considerations, and plays a very significant role. Small values of α make the difference between k 1 and k 2 indistinguishable in the graphs. Since the final baryon symmetry after conversion from the lepton asymmetry is a small number, such parameter ranges are also of relevance. Mid-range values of α are favorable to make the phase of χ = tan −1 (k 2 /k 1 ) more pronounced as can be seen from table III.
We see in table III that the CP phase values in both models are identical, other parameters remaining the same. This can be seen from the effective potential for MSLR/ P worked out in the appendix B. The corresponding expression for the effective potential for MSLRM can be obtained by simply reversing the sign of ω c . However, upon minimizing, the vev for ω c also has opposite signs in the two models and hence the k 1 , k 2 see the same effective potential in the two cases. There are general arguments based on intrinsic reasons suggesting that TeV scale leptogenesis if true cannot be verified in colliders in the near future [32]. We have adopted the approach of [59] wherein cosmology requirements arising from soft resonant leptogenesis are correlated with collider observables. Furthermore, the occurrence of a phase transition accompanied by domain walls may be verifiable in upcoming and planned gravitational wave experiments [69]. An open question for this class of models is a comprehensive analysis of the two different potential sources of leptogenesis, from phase transition DW and from the resonant thermal mechanism. Successful cumulative leptogenesis and subsequent dilution to required baryon asymmetry can further constrain the parameters of the models.

IX. ACKNOWLEDGMENT
This work is supported by a grant from the Department of Science and Technology, India.
The work of AS is supported by Council of Scientific and Industrial Research, India. Tr Ω c∆c ) = 0 The D-flatness conditions for MSLR/ P are given by Since the Leptons L and L c are considered to have zero vev, we omit them from the F and D flat conditions. The above conditions are same for MSLRM with only Ω c replaced by −Ω c .