Domain walls and CP violation with left right supersymmetry:implications for leptogenesis and electron EDM

Low scale leptogenesis scenarios are difficult to verify due to our inability to relate the parameters involved in the early universe processes with the low energy or collider observables. Here we show that one can in principle relate the parameters giving rise to the transient $CP$ violating phase involved in leptogenesis with those that can be deduced from the observation of electric dipole moment (EDM) of the electron. We work out the details of this in the context of the left right symmetric supersymmetric model (LRSUSY) which provides a strong connection between such parameters. In particular, we show that baryon asymmetry requirements imply the scale $M_{B-L}$ of $U(1)_{B-L}$ symmetry breaking to be larger than $10^{4.5}~\mathrm{GeV}$. Moreover the scale $M_R$ of $SU(2)_R$ symmetry breaking is tightly constrained to lie in a narrow band significantly below $M_{B-L}^2 / M_{EW}$. These are the most stringent constraints on the parameter space of LRSUSY model being considered.

without R-parity violation, low scale supersymmetry with exact R-parity and existence of a stable lightest supersymmetric particle.
Breaking of the Z 2 discrete symmetry of left-right symmetric models in the early universe ensures the occurrence of domain walls, but also begs for the Z 2 symmetry to be not exact to avoid conflict with the observed homogeneous universe. However an approximate Z 2 is adequate to generate the domain walls which are a robust topological prediction independent of details of parameters, and can play the same role as the phase transition bubble walls for the purpose of leptogenesis.
In the context of non-supersymmetric LRSM, fate of the CP violating phase was studied in [13] for the purpose of explaining leptogenesis. It was shown that a spatially varying CP violating phase occurs inside the domain walls separating the left handed and right handed domains. They showed that in order to explain the observed baryon asymmetry of the universe, the Yukawa coupling of the right handed neutrino to the SU(2)-triplet Higgs must be larger than 10 −2 . They also obtained some heuristic constraints on the mass of the left handed neutrinos or alternatively, the temperature scale of LR symmetry breaking. Since the LRSUSY model has additional Higgs bosons as well as additional CP violating phases, it may be able to generate the necessary conditions for successful baryogenesis in the early universe.
The first step in this direction was taken in [14] by showing the possibility of having a spatially varying CP violating phase in the domain walls in the context of LRSUSY. However they did not provide any quantitative estimates and left open the question of whether LRSUSY is actually capable of generating the baryon asymmetry of the universe.
The core idea of such proposals is that the spatially varying phase implies a spatially varying complex mass for left handed neutrinos inside the wall. Studying the diffusion equation for lepton number density with spatially varying complex mass results in the preferential transmission of left handed neutrinos across a slowly moving thick domain wall. The moving wall encroaches upon the energetically disfavoured right handed domain. We solve the diffusion equation numerically for various wall speeds and thicknesses, obtaining excesses of almost massless left handed neutrinos inside the left handed domain. After the wall disappears, electroweak sphaelerons convert a part of the neutrino excess to baryon excess. We calculate the amount of baryon excess that survives the washout processes. Requiring the surviving baryon excess to be not much above the experimental limit of around 6 × 10 −10 for the baryon asymmetry to entropy ratio [15] allows us to constrain the (M R , M B−L ) parameter space of LRSUSY.
A smoking gun signature of CP violation in a theory is the presence of a non-zero EDM of the electron and neutron. The Standard Model predicts a non-zero electron EDM at the three loop level but the effect is estimated to be very small, around 1.9 × 10 −39 e cm [16]. This is way below the current experimental upper bound of 1.1 × 10 −29 e cm obtained by the ACME II experiment [17]. Left right symmetric theories contain many additional sources of CP violation as compared to the Standard Model, and so predict larger electron and neutron EDMs. Thus the experimental bound on EDMs serve to constrain the parameters of these theories. Electron and neutron EDMs in the non-supersymmetric LRSM have been studied in several earlier works e.g. [18,19], obtaining lower bounds on the scale M B−L of SU(2) R × U (1) B−L symmetry breaking and on the mass M H of the heavier scalar Higgs in the two Higgs doublets arising from breaking of left right symmetry in the bidoublet Higgs of LRSM. Electron and neutron EDMs have been studied in the SLRSUSY model by [20,21], obtaining bounds on the masses of certain superpartners and some other parameters of SLRSUSY.
The LRSUSY model [12] contains essentially only one major unknown, the M R scale and a trilinear Higgs coupling parameter α that is used to ensure that only two out of four SM type Higgs doublets arising after SU(2) R × U(1) B−L symmetry breaking remain lighter, with a relative phase between their vacuum expectation values. In this paper we study the CP violating phase of the bidoublet fields in LRSUSY by setting up the domain wall solutions.
The parameters relevant for the domain wall solutions are understood to involve the high temperature corrections needed in the early Universe, at the temperature T B−L ∼ M B−L . We separately investigate the contribution of the lighter two mass eigenstates of the bidoublets to the electron EDM at zero temperature, which differ only by the T B−L corrections. We compute the contribution of this low energy phase to the EDM at one loop and two loop levels as a function of α. The two loop computation follows along the lines of the seminal work of Barr and Zee [22] on the electron EDM in multi Higgs doublet models. A similar calculation of electron EDM arising as a residual effect of domain wall collapse in two Higgs doublet models was done in [23]. It turns out that successful leptogenesis in LRSUSY requires α 0.1. Combining this with the requirement that the EDM obtained be less than the experimental limit of 1.1 × 10 −29 e cm, we get an allowed region in the (M B−L , M R )parameter space of LRSUSY. It turns out that the limit on the parameter space arising from baryon asymmetry is more stringent than the limit from electron EDM. Further requiring that the observed baryon asymmetry be explained to within an order of magnitude by LRSUSY puts stringent constraints on the (M B−L , M R )-parameter space of LRSUSY. In particular M B−L < 10 4.5 GeV is ruled out. These are the most stringent constraints on the parameter space of LRSUSY by far.
The charge zero condition forces the vevs of the Higgs fields to be where the quantities above are in general complex numbers.
The D-terms are given by (where m = 1, 2, 3 refer to the three generators of SU(2)) [10] After substituting the vevs into the D-term expressions we get Taking |d| = |d| and |d c | = |d c | ensures the vanishing of D B−L -term always. We can use the B − L gauge invariance to ensure that d,d have the same complex phase. Subsequently using SU(2) L invariance, we can ensure that d =d and real positive.
Extending Aulakh et al, we see that the resulting F-terms are: After substituting the vevs, the expressions for the F-terms become We now investigate what vevs ensure flatness conditions for all the F-terms and all the D-terms. For generic values of µ 11 , µ 12 and µ 22 , the entries of F Φ 1 and F Φ 2 are linearly independent. Hence flatness of F Φ 1 and F Φ 2 implies that the vevs of k 1 , k 1 , k 2 , k 2 are all zero. This automatically makes D L,3 and D R,3 flat.
Proceeding ahead, we now see that the F-flatness conditions split into two subsets viz. the left handed conditions and right handed conditions. This allows to conclude, as noted first by Aulakh et al., that the complete solution set for F-flatness and D-flatness is obtained by tak- and (k 1 , k 1 , k 2 , k 2 ) = (0, 0, 0, 0) (note a < 0). Thus, the complete SUSY preserving solution set has, beside the trivial all zero solution, three non-trivial solutions. Of these, the solu- The expressions for the vevs must be related to the two physical scales in LRSUSY. We need to set in the LH regions and likewise in the RH regions. A solution to the proliferation of mass scales was sought in [12] by invoking an R symmetry of the superpotential which forbids the terms Ω 2 and Ω 2 c . The R charge values can be set to where the L, Q etc are matter superfields which are not relevant to this paper. The terms Ω 2 and Ω 2 c can then be introduced only as soft terms, with the coefficients m Ω = m Ωc determined by SUSY breaking scale ∼ = M EW . This leads to an elegant simplification giving rise to the see-saw relation From the cosmological viewpoint, the early universe enters an epoch with two types of domains. In the left handed (LH) domains, the right handed vevs take non-zero values and in the right handed (RH) domains, the left handed vevs take non-zero values. The corresponding SUSY preserving vevs are The formation of the two types of domains also leads to topological domain walls separating them. This is because together with the breaking of the gauge symmetry, a discrete left-right symmetry is also broken. The presence of these walls or energy barriers conflicts with current cosmology. Several earlier works have discussed how such walls may be made to disappear fast enough so as to be consistent with present day observations [24]. The older study [14] demonstrated the existence of domain walls containing a CP violating phase in LRSUSY with implications to leptogenesis, but only as a proof-of-concept study. In this paper, we extend their ideas greatly and come up with quantitative estimates relating the parameter ranges that can give rise to the required spatially varying CP violating phase within the domain wall with the non-zero phase required for electron EDM in zero temperature translation invariant theory. The next section provides the details.
To the SUSY scalar potential we add the following soft mass terms for the bidoublets where µ 2 i > 0 and β 3 , β 4 , β 5 are explicit CP phases. Substituting the vevs we get We shall take the fine tuning condition µ 2 12 ≈ µ 11 µ 22 + α 2 M 2 R of Aulakh et el. [12] which ensures that out of the four neutral Higgs scalars that arise from the bidoublets after the breaking of SU (2)

III. SPATIALLY VARYING HIGGS VEVS
The SUSY scalar potential and the soft mass terms for the bidoublets receive temperature corrections determined by the scale M B−L . The temperature dependence of the squared mass term of a Higgs scalar has been evaluated at the one loop level in earlier works [25]. For this indicative study we shall take the temperature correction to each mass matrix element to be [26][27][28], The full temperature dependent mass matrix can be found in Appendix B. Thus, going from temperature M B−L to zero temperature entails the lowering of the mass matrix elements by . For the leptogenesis calculations, we work at the high temperature T = M B−L with the mass matrix arising from the SUSY scalar potential and the soft masses described above.
These choices for the mass parameters ensure that the mass matrix of the bidoublets has a negative eigenvalue inside the wall whereas all its eigenvalues are positive outside. This in turn means that it is energetically favourable for the bidoublet fields to take non-zero vevs inside the wall while continuing to take zero vevs outside, even though the soft terms have negative squared masses. For the zero temperature electron EDM calculation that we do later, we can work in a four Higgs doublet model [26,29] with temperature corrections dropped from the mass matrix of Appendix B.
In this section we shall see that, for a certain choice of soft SUSY breaking mass terms for the bidoublets, it becomes energetically favourable for the bidoublet fields to take non-zero vevs within the wall while continuing to take (almost) zero vevs outside it. Moreover Letḟ denote the derivative of vev of field f with respect to x. Let r 1 , i 1 be the real and imaginary parts of vev of k 1 , r 2 , i 2 , . . . , r 2 , i 2 the real and imaginary parts of the vevs of the respective bidoublet fields. We make the simplifying assumptions that the non-bidoublet Higgs fields are real everywhere. The finite temperature energy per unit area, which is the sum of gradient energies and potential energies of all the fields, plus field dependent temperature corrections can now be taken to be, Here a superscript T on H and V soft is a reminder of the temperature dependence. We determine the domain wall solutions such that SUSY is preserved asymptotically by the vevs upto relatively small temperature correction. It is only in the narrow region of the wall where the omega fields become small that the temperature dependent terms and soft terms become more significant. In the equations below, all the vev are meant to be temperature dependent, though for simplicity of notation we drop the superscript T .
in the left domain and Since Ω, Ω c have the heaviest vevs outside the wall, we fix a natural ansatz for them that smoothly goes from the LH solution to the RH solution while passing through the wall on the way. The wall is assumed to extend from −L to L in units of inverse temperature. The ansatz takes the form of kink functions: The ansatz has the property that ω(x), ω c (x) take the correct limiting values outside the wall in both domains, but are non-zero within the wall. In the example plots later on, we shall be taking L ∼ ( √ λ) −1 ∼ 5 for concreteness.   These considerations show that a spatially varying CP violating phase can indeed be produced by the bidoublet Higgs vevs inside the domain wall in the early universe. This phase persists as a constant non-zero quantity outside the wall. In the next two sections, we investigate the implications of this phenomenon for the electric dipole moment of the electron and the baryon asymmetry of the universe.

V. ELECTRON EDM CONSTRAINTS ON LRSUSY
The zero temperature mass matrix of the neutral components of the bidoublet Higgs fields is given in Appendix B. As there are two neutral complex components in each bidoublet, we get in total eight real neutral fields and so the mass matrix is 8 × 8. It turns out that the neutral mass eigenstates induce complex phases for the bidoublet Higgs fields relative to each other. This is true both within the wall as well as outside it. This feature gives rise to an electric dipole moment (EDM) for the electron at one loop and two loop levels. The maximum effect on the electron EDM is exerted by the lightest mass eigenstate.
The one loop contribution to electron EDM d e is given by [28] (d e /e)| one loop ∼ αm e 4πM 2 h sin δ, where M h is the mass of the lightest eigenstate of the bidoublet Higgs, α is the fine structure constant evaluated at the scale M h and δ is the complex relative phase between the neutral scalars induced by the lightest mass eigenstate. Surprisingly however, for large values of M B−L and M R two loop effects arising from the neutral scalars dominate the one loop effect.
This was first realised by Barr and Zee [22] and then refined by several other authors. We use the formulas of Chang, Keung and Yuan [31] in order to compute the two loop contribution.
where the functions f are certain logarithmically growing functions defined in [22,31].
We calculate the electron EDM numerically as a function of the LRSUSY model pa-  [12]. The lower bound on M B−L ensures that it is above any reasonable supersymmetry breaking scale M S and so one can comfortably break the U (1) B−L gauge symmetry to reduce to the MSSM. In other words, as argued in more detail in [12], the low energy effective theory of LRSUSY turns out to be the MSSM with strictly unbroken R-parity, and so the lightest supersymmetric particle is stable. The upper bound on M B−L follows from the consideration that for M B−L ≥ 10 10 GeV we have to take M R ≥ 10 12 GeV which is rather high for parity breaking. The experimentally allowed region, where the electron EDM, is less than 1.1 × 10 −29 e cm [17], is plotted as the green hatched region in the (M B−L , M R )-plane in Figure 6.
The non-supersymmetric LR model was studied in the context of conventional electroweak baryogenesis mechanisms in [32,33], and in a domain wall mediated baryogenesis via leptogenesis mechanism in [13]. The possibility of extending the latter mechanism to LRSUSY was indicated in [14], but no concrete numerical calculations were performed there. In this paper, we address this deficiency.
Adequate amount of CP violation as well as strong loss of equilibrium conditions have been major challenges for low energy baryogenesis. The presence of a moving domain wall, a topological defect, towards the energetically disfavoured right handed domain immediately guarantees a strong loss of equilibrium. The main LRSUSY model per se does not explain why this happens, but we assume that this occurs because of tiny effects like soft SUSY terms [14] or Planck suppressed non-renormalisable terms [24] breaking exact left-right symmetry. A similar calculation has recently been done [34] showing how Planck suppressed non-renormalisable terms can remove a pseudo domain wall in supersymmetric SO(10) GUT without conflicting with standard cosmology. Given that the domain wall in LRSUSY has somehow disappeared early enough so as not to conflict with present day cosmology, we can exploit it to obtain leptogenesis and consequent baryogenesis consistent with experimental bounds on baryon asymmetry. This is done as follows.
The lepton-Higgs Yukawa part of the superpotential of LRSUSY is [12] where j = 1, 2 and h, f are 3 × 3 real symmetric matrices. The Majorana mass terms above corresponding to the Yukawa coupling matrix f are a source of lepton number violation.
However, in LRSUSY they do not favour conventional thermal leptogenesis because at the usual scale of thermal leptogenesis, the B − L gauged symmetry is unbroken [14]. That is why we have to resort to domain wall mediated leptogenesis in LRSUSY. The lepton number violating decay of the heavy Majorana RH neutrino will instead give rise to a lepton asymmetry washout which will dilute any lepton asymmetry mediated by the domain wall.
Nevertheless, as we will see below, it is possible to obtain baryon asymmetry consistent with experimental data for a certain region of the LRSUSY parameter space.
Consider a domain wall moving slowly with speed v w in the +x direction i.e. encroaching upon the energetically disfavoured RH domain. The wall is assumed to stretch from −L to +L in the x direction and be flat in the yz plane. Slow speed means that v w < 1/ √ 3, the speed of sound in the hot plasma, allowing one to get a solution to the chemical potential of the neutrinos in terms of a fluid approximation [35]. Since the wall is assumed to move due to tiny energy differences between the LH and RH domain, this is a reasonable assumption.
We consider the case of thick walls i.e. 2L > 1/T , the de Broglie wavelength of the neutrinos at temperature T [35]. This is a reasonable assumption ensuring that the mean free path of the neutrinos is smaller than the wall thickness, leading to multiple interactions between neutrinos and the CP violating condensate in the wall and allowing a classical WKB treatment of the neutrinos. A wall thickness of 10/T will be typical in our analysis. We assume that the neutrino diffusion coefficient D < 2L/(3v w ) [35], The diffusion equation for the chemical potential µ of the LH neutrino in the wall rest frame is given by [13] − Dµ where S(x) is the so-called source term defined below, D is the neutrino diffusion coefficient, The source term, which is a CP-violating non-zero force if the neutrino mass m ν (x) and the domain wall CP phase δ(x) are spatially varying, is given by [36] S where p x is the x-component of the LH neutrino's momentum, E is the neutrino energy,Ẽ is the related quantity m ν (x) 2 + p 2 x , and the angular brackets indicate thermal averages. It was shown in [36] that where a = m ν (∞)/T , E 1 is the error function, K 2 is the modified cylindrical Bessel function of the second kind and the LH neutrino mass is evaluated deep inside the RH domain. The source term is zero outside the wall but non-zero inside.
The helicity flipping rate can be calculated by [35] where α w is the weak coupling constant evaluated at temperature T . The diffusion coefficient has the expression [36], where v x is the x-component of the neutrino velocity and a is defined above.
We solve the diffusion equation (Equation 27) numerically the using GSL version 2.6 library under the same settings as before, and take the limiting value µ(−∞) in the LH domain as the steady state chemical potential of the LH neutrino. Then, the steady state neutrino-antineutrino asymmetry in the LH domain becomes [13] To obtain the raw lepton asymmetry to entropy density ratio η raw , we need to divide the above quantity by 2π 2 g * T 3

45
, where g * ≈ 110 is the number of relativistic degrees of freedom.
Doing so gives us Since the heavy neutrinos in this model have mass less than the temperature M B−L , they can easily decay violating lepton number. This process washes out most of the raw lepton asymmetry η raw L . The surviving lepton asymmetry by entropy density ratio becomes [13] η L = η raw · 10 −4·10 −4 mν M Planck v −2 = 3.054 · 10 −11 µ(−∞) T , where m ν is the mass of the heaviest light neutrino and v = 174 GeV is the SM Higgs vev.
We take m ν = 0.05 eV from the Nu-FIT Group [37].
Finally electroweak sphaelerons convert part of the lepton asymmetry to baryon asymmetry starting from the temperature M B−L till the universe cools to the sphaeleron scale of about one TeV. This gives the steady state baryon asymmetry to entropy density ratio [13] η B = 28 51 The yellow strip in Figure 6 shows On the other hand we can reject the R charge proposed in [12], in order to allow the fact that the scale of parity breaking and (B − L)-symmetry breaking are not maximally far apart. The parameter m Ω could then be intrinsic to the superpotential. In this case our results are perfectly consistent with PeV scale supersymmetry [38] also considered in the recent work [39], where the gravitino can be much heavier. This places LRSUSY outside the experimental reach of colliders in the near future. On the other hand, the discovery of a non-zero electron EDM value can be taken to be a hint to a narrow range for the M R scale assuming a world with a renormalisable supersymmetric left-right model. Finally it is interesting that two very different low-energy probes viz. baryon asymmetry and electron EDM provide a strong constraint on the allowed parameter space of LRSUSY.
vevs of the fields ω(x), ω c (x) below take their values from the ansatz in Equation (23).