Soft leptogenesis in the NMSSM with a singlet right-handed neutrino superfield

In this work, we explore soft leptogenesis in the NMSSM framework extended by a right-handed neutrino superfield. We calculate the CP asymmetry, ε, and find it to be non-zero at tree-level without using thermal effects for the final state particles. This is in contrast to soft leptogenesis in the MSSM extended by a right-handed neutrino superfield where thermal effects are essential. The difference arises due to the presence of a 3-body decay of the sneutrino in the NMSSM that violates lepton number at tree-level. Apart from this, we also find that ε ≠ 0 if the additional singlet scalar has a complex vacuum expectation value while all the other NMSSM parameters including the soft SUSY breaking ones relevant for CP asymmetry remain real. We estimate the order of magnitudes of these parameters to produce sufficient baryon asymmetry of the Universe.


Introduction
It is well known that the observable Universe has an asymmetry between baryons and antibaryons [1], often called the problem of baryogenesis. Over the years, many mechanisms have been proposed to create this baryon asymmetry. More recently, leptogenesis [2,3] has become a highly favoured model for baryogenesis, specially because this mechanism is naturally linked to neutrino masses. Adding right-handed (RH) singlet heavy neutrinos to the standard model (SM) generates neutrino masses by the seesaw mechanism [4][5][6][7][8][9]. These RH neutrinos can also decay to produce a scalar and SM leptons and if the decay violates CP for example due to interference between tree-level and loop-level decays owing to complex couplings, a lepton asymmetry is generated. Since, in the SM, the B − L symmetry is exact while the B + L symmetry is broken by the electroweak (EW) sphaleron processes [10], these sphaleron processes can convert the generated lepton asymmetry to baryon asymmetry.
Soft leptogenesis [11][12][13] pertains to generating lepton asymmetry at the tree-level itself due to mixing between the particle and anti-particle states of the RH singlet sneutrino,Ñ , because of the presence of soft SUSY breaking terms. 1 In the most minimal soft leptogenesis setup using minimal supersymmetric standard model (MSSM) [25,26] extended by one RH neutrino superfield,N , a CP asymmetry in the RH sneutrino sector is created only when JHEP04(2020)065 thermal masses for the final products are considered. The asymmetry is present because the thermal phase space factors are different for bosons and fermions. Another work featuring soft leptogenesis looks at CP violation not just due to mixing between particle and antiparticle initial states but in decays and in the interference of mixing and decay [27]. In ref. [28], it is shown that considering most generic soft trilinear couplings and one loop self energy contributions for sneutrino decay it is possible to generate CP violation even without finite temperature effects within the same setup.
However, the MSSM suffers from the so-called µ-problem [29] -there is no explanation to why the SUSY scale preserving µ-term (a direct SUSY mass term for the Higgs fields) should be of the same order as the soft SUSY breaking terms. The most straightforward solution to the µ-problem comes by promoting the µ-parameter into a field whose vacuum expectation value (vev) is determined, like the other scalar field vevs, from the minimization of the scalar potential along the new field direction [30]. Naturally, it is expected to fall in the range of the other vevs, i.e., of order O(M SUSY ). The next-to-minimal supersymmetric standard model (NMSSM) (for review see [30,31]) is the most simple and elegant model to solve this problem, where a singlet superfieldŜ is introduced to the MSSM superfields which gets non-zero vev. The NMSSM can be extended by a set of RH neutrino superfields to generate masses for the SM light neutrinos by the type-I seesaw (see [32][33][34] for the MSSM extended by RH neutrino superfield). This has been explored earlier in ref. [35]. This extension also keeps the R-parity conserved if the sneutrinos do not get vevs [35].
In this work, by using the NMSSM extended by the RH neutrino superfield, we present a soft leptogenesis scenario that creates a lepton asymmetry at the tree-level decay of the RH sneutrino without using thermal mass factors. The CP violation is achieved by the mixing between the particle and anti-particle states. This is due to the presence of the soft terms and the trilinear coupling between the additional singlet superfield which takes a vev and the RH neutrino superfield. A similar non-SUSY setup with such a trilinear term can be found in [36]. We also show that it is possible to obtain non-zero CP asymmetry even when all the soft parameters are real. Since the soft terms are responsible for creating the CP asymmetry instead of needing flavour effects as in usual leptogenesis, using only one generation of the RH neutrino superfield is enough. Even so, the setup can be easily extended to get the experimentally observed SM neutrino mass hierarchies and their mixing angles pattern [37,38].
The paper is organised in the following manner. We setup the model and segregate the parts required for soft leptogenesis in the next section (section 2). In the one following that, i.e., section 3, we calculate the CP asymmetry produced by decays of the various particle present in the model that contribute to non-zero CP asymmetry parameter ε at the treelevel. We talk about the decays ofÑ as well as the scalar S in the model. In section 4, we discuss the most crucial and important constraints and give a simple expression for ε. In section 5, we give and discuss the results of our calculation. We find that for successfully generating the observed baryon asymmetry of the Universe, we need ε ≈ O(10 −6 ). We also discuss what this could mean for various parameters of the model including the soft ones. We finally conclude in section 6.

JHEP04(2020)065 2 Model
In the NMSSM, an extra singlet superfieldŜ is added to the MSSM Higgs sector [30]. Assuming explicitly Z 3 symmetry, the superpotential for the NMSSM with a singlet RH neutrino superfieldN in terms of the new singlet superfieldŜ and the MSSM doublet superfieldsĤ u andĤ d will be as follows [30]: whereL i andQ i are the SU(2) doublet superfields of leptons and quarks;Ê i andD i (Û i ) denote singlet down (up)-type quark superfields, respectively, and Y 's, λ's and κ are dimensionless couplings with generation indices (i, j = 1, 2, 3). After the singlet S obtains a vacuum expectation value (vev) S , an effective µ-term is generated: µ eff = λ S , which solves the so-called µ-problem [29]. The soft SUSY-breaking Lagrangian is given by whereL i andÑ are the scalar components ofL i andN superfields, respectively. CP is spontaneously violated when the scalars H u , H d , S attain vevs with relative physical phases. The vev of the singlet S is complex: Since leptogenesis occurs above the electroweak (EW) phase transition, we do not give vevs to the two Higgs doublets. In this case, spontaneous CP violation can occur only when sin δ = 0.

Terms relevant for soft leptogenesis
The terms from the superpotential required for leptogenesis via sneutrino decay are: Here we consider λ N , κ to be all real and positive. We also remove the i, j indices from the leptons and the u index from the Higgs superfield for brevity. The scalar potential is obtained using:

JHEP04(2020)065
The fermionic part of the Lagrangian is given by: The soft SUSY-breaking Lagrangian terms that play a role in leptogenesis are: (2.10) The superpotential and the soft breaking terms combine to give the following interactions forÑ and σ which could in principle contribute to soft leptogenesis due to mixing between the particle and anti-particle states through the soft terms: where σ = S − S .

CP asymmetry
Because of the soft terms as well as the vev of S, there is a mixing between particle and anti-particle states of the sneutrino and the singlet scalar σ which is the dynamic part of S. The squared mass matrices for the two of them are given by: If A λ is real, the mass square eigenvalues of the sneutrino are: which have the following eigenstates: Similarly, one can write the mass square eigenvalues and eigenstates for the σ − σ * system. Because of mixing between the particle and anti-particle states of sneutrino and singlet scalar, these systems are similar to K 0 −K 0 and B 0 −B 0 systems [39]. The evolution of these systems in the non-relativistic limit are driven by the Hamiltonian H defined as follows: where M is the mass matrix and Γ is the decay rate matrix of the corresponding system.

JHEP04(2020)065
Finally, the decay rates of the time evolved particle and anti-particle states of thẽ N −Ñ * and the σ − σ * system are calculated to get the final total CP asymmetry. As can be seen from eq. (2.11), bothÑ and σ can decay to produce lepton asymmetry depending on the nature of their couplings. Therefore, we separately consider the two limiting cases where only one of them can decay at a time by fixing their masses. The first case is when M 1 m σ . In this case,Ñ decays to produce the CP asymmetry at the tree-level while decays of σ into a pair of RH (s)neutrinos are kinematically suppressed. The relevant Feynman diagrams for this situation are shown in figures 1 and 2. Figure 1 shows the point interaction, 2 and 3 body decays ofÑ while figure 2 shows the other possible 3-body decays ofÑ that are mediated by an off-shellÑ . The first three point interactions of figure 1 are present in MSSM soft-leptogenesis as well and the observed matter-anti-matter asymmetry requires the decay products to have thermal corrections. The last diagram of figure 1 and those in figure 2 are presnt only in NMSSM. We will see in section 3.1 that the 3-body decay diagram of figure 1 is responsible for creating enough CP asymmetry without any thermal mass corrections. The contribution of diagrams in figure 2 is suppressed by the higher order soft terms. The second possibility to create the CP asymmetry is through decays of σ which occurs when m σ M 1 . The diagrams are shown in figure 3. However, as we will see in section 3.2, these decays can not produce any CP asymmetry at the tree-level via the same mechanism followed byÑ decays. This tells us that we can ignore the decays of σ without any loss of generality for the study of soft leptogenesis in the present setup. We consider these cases one-by-one.

3.1Ñ decays
Irrespective of the mass of σ relative to the mass ofÑ , the sneutrino can decay into leptonic (sleptonic) and Higgs (Higgsino) final particles. The CP asymmetry generated from such a scenario was calculated in [11,12]. However, if M 1 m σ , the 3-body decay channel shown in the last Feynman diagram of figure 1 opens up and it leads to interesting consequences for soft leptogenesis as we show below. For the snuetrino system, upto leading order in the off-diagonal terms, 2 the mass matrix can be calculated from the squared mass matrix in eq. (3.1): The decay rate matrix can be written from the eq. (2.11). It contains both diagonal as well as off-diagonal terms becauseÑ can decay into particle as well as anti-particle final states.
The parameter α and β in ΓÑ matrix are associated with the 3-body decay contributions ofÑ given by, The logarithm in β can be expanded for the soft term A N at least an order of magnitude smaller than M 1 (this can be assumed without any loss of generality as later to draw our conclusions and compare with MSSM soft-leptogenesis in [11][12][13], we will be restricting to leading order in soft-terms) to give: Under the same approximation used for β and with M 1 m σ , α becomes: It is important to note that as long as the soft term A λ is sufficiently small compared to M 1 (at least by an order of magnitude) or the phase π/2 < δ < 3π/2, α = 0.

JHEP04(2020)065
There are 4 terms in α of eq. (3.10) which represents the contribution ofÑ → σL * H * decay, where the first two terms come from the 3-body decay arising from the point interaction in figure 1 while the third term comes from the second diagram of figure 2. The last term in α comes from the interference of these two diagrams. The β in eq. (3.11) appears in the decay matrix ΓÑ due to the presence of first and the third diagrams of figure 2. It is pertinent to note here that β is at least quadratic in the soft terms A λ and A N . One could in principle have higher order diagrams relating to 4 or more decay products. However, all such diagrams will be suppressed heavily by higher powers of A λ /M 1 .
The solutions for the time evolution ofÑ andÑ * come from the Schrodinger like equation where ψ = {Ñ ,Ñ * } T . The solutions are obtained as, whereÑ 0 ,Ñ * 0 are the field values at t = 0 and If ∆ΓÑ ∆M as well, the argument of the trigonometric functions becomes ∆M t/2 such that we can write:Ñ where (3.25)

JHEP04(2020)065
The eqs. (3.22) and (3.23) are substituted back in eq. (2.11) for calculating the CP asymmetry factor ε which is defined as the ratio of the difference between the decay rates of N andÑ * into final state particles with lepton number +1 and −1 to the sum of all the decay rates, i.e., , (3.26) where f,f are the final states with lepton number +1 and −1, respectively. This then gives us the following CP asymmetry parameter: The decay rates at the tree-level itself for final states with lepton numbers ±1 are different because the factor p q = 1 as it is not a hermitian quantity. This requires non-zero off-diagonal terms to be present in the mass matrix as well as the decay rate matrix of the system with atleast one of them being complex.
The sum and difference of the ratios |q/p| 2 and |p/q| 2 can be written as: Keeping terms upto the leading order in Γ 1 , we find that the sum of the ratios is 2 while the difference is twice the values of y/(x 2 − y 2 ) 1/2 with, such that the final CP asymmetry can be written as:

JHEP04(2020)065
where Γ is defined as Γ = Γ 1 From eqs. (3.32) and (3.33) it is evident that there is a non-zero CP asymmetry even with real A N , provided δ is sufficiently large.
We also note that while the usual soft leptogenesis done in the MSSM [11,12], to the leading order in soft terms (M 2 M 2 1 , A 2 N M 2 1 ), necessarily requires thermal phase space factors for the final state bosons (c B ) and fermions (c F ) to have ε ∝ ∆ BF = c B −c F c B +c F , we get an asymmetry even without thermal mass corrections to the decay products at the leading order in soft terms. This happens because of the presence of σ in the model which facilitates a 3-body decay ofÑ which is not cancelled by the other terms. If we did not have this, at leading order in soft terms, 1 − It is clear from eq. (3.34) that the only contribution coming from diagrams of figure 2 are coming from the cross term of the second diagram with the last diagram of figure 1 through α. Even in α, the dominant contribution comes through the 3-body decay by the vertex interaction of figure 1. Therefore it is possible to successfully generate non-zero lepton asymmetry fromÑ −Ñ * system at the tree-level without using thermal phase space factors for bosonic and fermionic final states. We discuss this more and give some numerical estimates in section 5 for relevant parameters. For the moment, let's consider the decays of σ.

σ decays
Unlike theÑ decays where most of the decay products were massless, the final products of σ decay are massive, as shown in figure 3. This creates two possible decay modes of σ according to the condition satisfied.
1. σ decays to N N andÑÑ . This happens when m 2 σ > 4M 2 1 , 2. σ decays only to N N . This happens when 16λ 2 The mass matrix and the Γ matrix of the σ − σ * system are respectively:  A non-relativistic Hamiltonian can be defined following eq. (3.6). Immediately it can be seen that because of the absence of an off-diagonal term in the decay rate matrix of σ, the ratio corresponding to (p/q) 2 ofÑ decay, If we solve for the evolution of the σ − σ * system, the CP asymmetry parameter computed exactly analogously to theÑ −Ñ * system will be zero because of exact cancellation between the ratios |r/s| 2 and |s/r| 2 . Thus ε σ = 0. (3.39)

General constraints
The CP asymmetry in this model depends on a lot of parameters. However, we can constrain some of them by various considerations. In deriving the following general constraints, we take M 1 M ⇒ M 1 2λ N v S , M 1 m σ and A λ κv S for reasons that will become clear later in section 5.
• The condition of out-of-equilibrium decay at T = M 1 is given by comparing the decay rate of theÑ with the Hubble parameter at T = M 1 : where Γ is the diagonal component of eq. (3.8). Substituting it in eq. (4.1) and neglecting the contribution of β we get For M 1 m σ , α ≈ O(10 −2 ) and we may write eq. (4.2) as

JHEP04(2020)065
• The way we derived the CP asymmetry requires well separated states [12], i.e., Γ ∆M as well as ∆Γ ∆M as stated before. This gives us two self-consistent limits: • Neutrino mass upper limits (m ν 0.1 eV [1,40]) put constraints on the Yukawa coupling strength Y N and the mass of the RH (s)neutrino.
• Electric dipole moment (EDM) calculations can constrain the CP violating phases that appear in the vevs of the two Higgs doublets and the scalar singlet S. In ref. [41], they show that in principle δ u (the phase in the vev of H u should we go below the EW scale) and δ could be large as long as the relative phase is kept small. For more details about the EDM constraints on the NMSSM, see [42].

A simpler form for ε
We can write a simpler form for the CP asymmetry by using the approximations made and the general relationships between various parameters given above. The set of parameters governing ε are: This along with κ λ N means both M, m σ 2λ N v S M 1 . Therefore, from eq. (4.6), one can put an upper limit on Y N , i.e., Y N O(10 −4 ). With these choices and approximations the form of ε can be simplified to . (4.8)

Results and discussions
To obtain the baryon asymmetry of the Universe, η B , we solve the simultaneous Boltzmann equations for theÑ number density, NÑ , and the B − L number density, N B−L , which are as follows [43][44][45]: where KÑ = Γ H(z=1) is the Hubble parameter at z = 1 with z = M 1 T and N eq is the equilibrium number density ofÑ . They take the following forms: with m P l = 1.22 × 10 19 GeV being the Planck mass and g s is the number of relativistic degrees of freedom in the NMSSM which we take ≈ 225 except for theÑ which is nonrelativistic. In writing the Boltzmann equation for B − L number density, we neglect the ∆L = 2 scattering processes for washout and assume it is dominated mostly by inverse decays. The contribution to washout from the scattering processes is small because we are in the weak washout regime with KÑ 1. The final B − L number density thus created, N f B−L then converts to the baryon asymmetry by the sphaleron processes such that the ratio of the baryon number density to the photon number density, η B , is: where g * g s 225, g 0 * is the effective number of relativistic degrees of freedom at recombination and a sph is the sphaleron conversion factor. Since we will solve the Boltzmann equations numerically, we use the complete form of ε given in eq. (3.33).
In figure 4, we show the typical value of the CP asymmetry that satisfies the observed baryon asymmetry of the Universe. It turns out that we need ε O(10 −6 ) to get the correct observed baryon asymmetry while satisfying neutrino mass bounds. This value of JHEP04(2020)065 ε is similar to the one obtained by other vanilla leptogenesis scenarios in the weak washout regime. The only difference is that usual leptogenesis occurs with decays at the loop-level interfering with tree-level decays due to complex Yukawa couplings that violate CP. In the soft leptogenesis, the Yukawa parameter could very well remain real as the source of CP asymmetry lies elsewhere -in the time-varying mixing betweenÑ andÑ * states due to the complex nature of S and the soft SUSY breaking parameters. For simplicity, we also assume A λ κv S (such that α ≈ λ 2 N 2π ) in finding the correct set of values for other parameters. Using these, we show the relation between A N and κ for different values of Y N and v S (satisfying the SM neutrino mass bounds) in figures 5 and 6 as contour plots in log ε.
In figure 5 we take a complex A N with ImA N = ReA N and vary κ and ReA N for Y N = 10 −4.8 , v S = 10 7 GeV (left panel) and Y N = 10 −4 , v S = 10 8.3 GeV (right panel). As can be seen from the left figure, we need large A N ( M 1 ) to satisfy correct order of ε while ensuring that Γ H(z = 1). However from the right panel, it is clear that sufficient CP asymmetry can be created even at leading order in A N without using thermal phase space factors.
We keep A N real in figure 6 and vary κ and ReA N for similar values of Y N and v S as before. It's clear that non-zero asymmetry can be created even with real A N as long as δ = nπ (n ∈ Z). However compared to figure 5, we find that the bounds on A N and κ in JHEP04(2020)065

Case 2: δ = π
If the phase of the vev of S is large, specially at δ = π, we get a resonance behaviour in ε at A λ κv S . In the limit of δ → π, the CP asymmetry parameter of eq. (4.8) can be written as: Eq. (5.7) also justifies assuming A λ κv S to derive the general constraints on the various parameters in section 4. The behaviour of ε versus A λ is shown in figure 7. For the plot, we take M = m S = 1 TeV. The values of the other relevant parameters are shown in the figure itself. Since δ = π, there is no contribution from the real part of A N in ε. This means that A N necessarily needs to be complex contrary to the case where δ is small. Without resonance, it was found in section 5.1 that A N needs to be several orders larger than A λ for correct amount of ε O(10 −6 ). However, the resonance effect at δ = π mitigates this requirement allowing A N to be of the same order or smaller than A λ .

Conclusion
We have presented a new mechanism for soft leptogenesis in the context of the NMSSM with a singlet RH neutrino superfield. Similar to soft leptogenesis in the MSSM, we also generate CP asymmetry at the tree-level owing to the CP violation occuring due to the difference between the mass and CP eigenstates similar to the K 0 −K 0 or the B 0 −B 0 systems. The difference lies in the fact that MSSM soft leptogenesis requires using thermal JHEP04(2020)065 Figure 7. Variation of ε versus A λ in the δ = π limit. The resonance in ε occurs when A λ = κv S . masses and phase space factors for boson and fermion final states without which there is no asymmetry. In the NMSSM where the singlet scalar S takes a vev, an asymmetry can be generated even without any thermal phase space/mass corrections to the decay products. Further if there is spontaneous CP violation in the system with sin δ = 0, lepton asymmetry can be created without using any other complex parameter. In the numerical analysis for small δ case, we considered the mass scale of the RH sneutrino to be 10 7 -10 8 GeV. We found that to generate sufficient asymmetry, one of the soft trilinear coupling A N needs to be 10 7 GeV and κ O(10 −5 ). This also tells us that A λ O(10 2 ) GeV. However, if δ → π, there occurs a resonance in the system which helps to obtain ∼ O(10 −6 ) even with A N A λ and the value of κ can be comparatively larger ( O(10 −3 )). The mass of the RH sneutrino came out in the range O(10 7 -10 8 ) GeV which lies below the cosmological gravitino overproduction bound of T reheat O(10 9 ) GeV [46][47][48][49]. This mass scale for the sneutrino (which depends on the vev of S, v S ) in the NMSSM also could favour gravitational wave detection at LIGO [50] provided a strong first order phase transition occurs in the scalar sector. It would be interesting to explore the flavor effects in the present scenario that we leave for a future study.