Leptonic $CP$ Violation and Leptogenesis in Minimal Supersymmetric SU(4)$_c \times$SU(2)$_L \times$SU(2)$_R$

We consider a supersymmetric SU(4)$_c \times$SU(2)$_L \times$SU(2)$_R$ model with a minimal number of Higgs multiplets and Dirac and Majorana $CP$-violating phases in the neutrino flavor mixing matrix. The model incorporates the charged fermion masses and quark mixings, and uses type I seesaw to explain the solar and atmospheric neutrino oscillations. With the neutrino oscillation data of two mass squared differences and three flavor mixing angles, we employ thermal leptogenesis and the observed baryon asymmetry to find the allowed regions for the Dirac and Majorana phases. For a normal neutrino mass hierarchy, we find that the observed baryon asymmetry can be reproduced by a Dirac phase of around $\delta_{CP}=3 \pi/2$, which is strongly indicated by the recent T2K and NO$\nu$A data. For the case of inverted neutrino mass hierarchy, the predicted baryon asymmetry is not compatible with the observed value.

The solar and atmospheric neutrino oscillation have established non-zero neutrino masses and mixings between different neutrino flavors. Two mass squared differences and two mixing angles relevant for the solar and atmospheric neutrinos are measured with good accuracy. The neutrino oscillation parameters to be determined in future experiments include the socalled reactor angle θ 13 , Dirac CP-phase δ CP in the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix, and the ordering of the neutrino mass eigenvalues. The T2K collaboration [1] has recently reported a non-zero value for θ 13 at the 2.5σ confidence level. Following the T2K results, the MINOS collaboration [2] has reported that their data is consistent with the T2K results, while disfavoring θ 13 = 0 at the 89% confidence level. In the not too distant future, the accuracy of measurements for masses, mixing angles and CP-phase in the lepton sector should be comparable to that in the quark sector. A precise information of quark and lepton mass matrices could provide important clues regarding the origin of fermion masses, flavor mixings and CP-violations which, most likely, comes from new physics beyond the standard model (SM).
In order to explain the observed neutrino masses and flavor mixings, we need to extend the SM. The type I seesaw mechanism [3] is one of the more promising ways not only to incorporate the neutrino masses and flavor mixings but to also explain the tiny of neutrino masses naturally. A class of supersymmetric (SUSY) grand unified theories (GUT) has attracted much interest in this regard. In addition to providing a resolution of the gauge hierarchy problem, the paradigm of SUSY grand unification is also supported by the successful unification of the three SM gauge couplings at the GUT scale, M GU T ≃ 2 × 10 16 GeV. Among several possibilities, SO (10) unification is one of the more compelling ones, with the quark and lepton multiplets of each generation unified in a 16 dimensional representation along with a right-handed neutrino. The seesaw mechanism is also automatically implemented, being associated with the breaking of SO(10) symmetry to the SM gauge group at M GU T , which is fairly close to the desired seesaw scale.
The so-called minimal SUSY SO(10) model [4] with the minimal set of Higgs multiplets (10+126) relevant for fermion mass matrices is a natural extension of non-supersymmetric SO(10) models considered a long time ago [5]. Because of the unification of quarks and leptons in the 16 representation and the minimal set of Higgs multiplets, the fermion Yukawa matrices are highly constrained with the quark and lepton mass matrices related to each other. Note that the Higgs 10-plet has been used to implement t-b-τ Yukawa unification in SO(10) [6].
There have been several efforts within the SO(10) framework to simultaneously reproduce the observed quark-lepton mass matrix data as well as the neutrino oscillation data [7,8]. It is quite interesting that after the data fitting, essentially no free parameter is left and all fermion Yukawa matrices, in particular the neutrino Dirac Yukawa matrix, are unambiguously determined. The neutrino Dirac Yukawa matrix allows us to provide concrete predictions for proton lifetime [9] and the rate of lepton flavor violations [10].
However, the minimal SO(10) model suffers from a serious problem. The observed neutrino oscillation data suggest the right-handed neutrino mass scale to be around 10 13 − 10 14 GeV, which is a few orders of magnitude below the GUT scale. With fixed Yukawa couplings of right-handed neutrinos in the minimal SO(10) model, this intermediate scale is provided by the vacuum expectation value (VEV) of the 126 Higgs multiplet. This indicates the existence of many exotic states with intermediate mass scale, which significantly alter the running of the MSSM gauge couplings. This has been discussed in [11], where it is shown that the gauge couplings are not unified any more, and even the SU(2) gauge coupling blows up below the M GU T . To solve this problem, we may extend the minimal model or may consider a different direction in constructing GUT models [12].
Assuming appropriate VEVs for the Higgs multiplets, we can parameterize the fermion mass matrices as the follows: Here In the 4-2-2 model, M R is independent of the other mass matrices, while it is proportional to M 15,2,2 in the minimal SO(10) model.
Assuming left-right symmetry 1 , the MSSM gauge couplings unify as usual at M GU T , which we identify with the breaking scale of 4-2-2 down to the MSSM gauge group. Therefore, the procedure for fitting the charged fermion mass matrices is the same as in the minimal SO (10) model. On the other hand, M R being independent of the other mass matrices provides us with the freedom to fit the neutrino oscillation data.
Let us count here the number of free parameters used to fit the charged fermion mass matrices. Because of left-right symmetry, M 1,2,2 and M 15,2,2 are 3 × 3 complex symmetric matrices. Without loss of generality, we take a basis where M 1,2,2 is real and diagonal, so that the number of free parameters in M 1,2,2 and M 15,2,2 is 3 + 12 = 15. The two complex parameters c 1 and c 15 introduce an additional 4 degrees of freedom, and therefore in total we have 19 free parameters. The degrees of freedom of charged fermion mass matrices are decomposed into 3 + 6 = 9 for the lepton and quark mass eigenvalues, and another 9 for a unitary matrix for the quark mixings which consists of 4 parameters in the CKM matrix and 5 diagonal CP-phases. Since the 5 CP-phases are not observable in the SM, we drop these degrees of freedom. Thus, we have 14 free parameters to fit 13 observables [14]. In the minimal SO(10) model, this single free parameter is adjusted to fit the neutrino oscillation data (see [7] for details).
Through the type I seesaw mechanism [3], the light neutrino mass matrix is given by where Y D is the neutrino Dirac Yukawa matrix and v u is the VEV of the up-type Higgs doublet.
For simplicity, we assume here that all Majorana phases are zero, so that the light neutrino mass matrix is transformed by the PMNS matrix U P M N S to the diagonal mass eigenvalue matrix D ν .
Using (Eq. 3), we can express the right-handed neutrino mass matrix as Recall that in this 4-2-2 model, Y D is fixed by fitting the Dirac fermion masses and mixings in the same manner as the minimal SO(10) model. Therefore, once all neutrino oscillation parameters have been measured, we can obtain the complete information of the right-handed neutrino mass matrix M R from Eq. (4). Since θ 13 and the Dirac CP-phase δ CP in the PMNS matrix have not yet been determined, we obtain M R as a function of θ 13 and δ CP from the solar and atmospheric neutrino oscillation data. 1 The left-right symmetry requires us to add Y ij R F i F j H 10,3,1 to Eq. (1) with a Higgs multiplet H 10,3,1 : (10, 3, 1). This term corresponds to type II seesaw [13] once H 10,3,1 develops a non-zero VEV. Since a more complicated Higgs sector seems necessary to induce such a VEV, we do not consider type II seesaw in this paper.
Models with the seesaw mechanism can also account for generating the observed baryon asymmetry in the universe [15], via thermal leptogenesis [16], where Y B is the ratio of the baryon (minus anti-baryon) density (n B ) to the entropy density (s). The out-of-equilibrium decays of heavy Majorana neutrinos in the presence of non-zero CP-violating phase generates a lepton asymmetry Y L in the universe, which is partially converted to the baryon asymmetry through (B+L)-violating sphaleron transitions [17,18]. The conversion rate is given by [19] Here we set N f = 3 and N H = 2 for the numbers of fermion families N f and Higgs doublets N H as in the minimal SUSY SM (MSSM).
The baryon asymmetry produced is evaluated by solving the Boltzmann equations with the information of neutrino Dirac Yukawa coupling matrix and M R . Since Y D is fixed and M R is a function of θ 13 and δ CP in our model, the resultant baryon asymmetry is given as a function of θ 13 and δ CP . Therefore, leptogenesis constrains the parameters θ 13 and δ CP so as to reproduce the observed baryon asymmetry.
As mentioned above, the data fitting procedure for the realistic charged fermion mass matrices is the same as in the minimal SO(10) model, and so in our analysis we employ the numerical values in Y D found in [7]. In the basis where the charged lepton mass matrix is diagonal, the neutrino Dirac Yukawa coupling matrix at the GUT scale is unambiguously determined and explicitly given by for tan β = 45 2 . Using Y D , we determine M R from Eq. (4) as a function of θ 13 and δ CP . Since the absolute mass spectrum of light neutrinos has not yet been determined, we consider two cases for it, the normal hierarchical (NH) case and the inverted hierarchical case (IH). For the NH case, the mass eigenvalue matrix D ν is given by 2 Although the output for the neutrino oscillation parameters obtained in [7] is 3σ away from the current neutrino oscillation data [20], the experimental data for charged fermion mass matrices are nicely fitted. Since Y D is determined only by data-fitting the charged fermion mass matrices, we can safely use this Y D data without contradicting any of the experimental results.  . We have set sin 2 (2θ 13 ) = 0.14, which is the best fit value in T2K results [1] for δ CP = 0 in the inverted hierarchical case.
Let us first show the mass spectrum of the heavy Majorana neutrinos (mass eigenvalues of M R ) as a function of m 0 or δ CP . Figure 1 (left panel) shows the mass spectrum M i (i = 1, 2, 3) of the heavy Majorana neutrinos for the NH case as a function of m 0 with δ CP = 0 and sin 2 (2θ 13 ) = 0.11 (the best T2K fit value for δ CP = 0). Since the VEV of H 10,1,3 (which breaks 4-2-2 down to MSSM) is M GU T , we require m 0 10 −4 eV in order to keep Y ij R within the perturbative regime. The right panel shows the mass spectrum as a function of δ CP for m 0 = 10 −3 eV and sin 2 (2θ 13 ) = 0.11. The corresponding results for the IH case are shown in Figure 2. For the IH case, we find M 1 10 9 GeV for any value of m 0 , sin 2 (2θ 13 ) and δ CP .
As shown in Figures 1 and 2, the heavy neutrino masses are hierarchical for both the NH and IH cases. The lepton asymmetry in the universe in this case is dominantly produced by the lightest heavy neutrino decay, since the asymmetry produced by heavier neutrino decays are almost completely washed-out [21]. Thus, we consider the lepton asymmetry produced by only the lightest heavy neutrino decay. In addition, there is a lower bound on the lightest heavy neutrino mass, M 1 10 9−10 GeV, to produce the desired amount of baryon asymmetry [22]. For the IH case, the lightest heavy neutrino mass is always found to be below this bound, and in our numerical analysis we find that the resultant baryon asymmetry is too small in comparison to the observed baryon asymmetry. Therefore, in the following, we present our results only for the NH case.
Since our model is supersymmetric, we need to consider the lepton asymmetry generated by the decays of both the lightest heavy neutrino and sneutrino. From Figure 1, the lightest heavy neutrino mass is far below M GU T , and so the effective theory for leptogenesis contains the MSSM and three light neutrinos, as well as the lightest heavy neutrino superfield. Although the complete Boltzmann equations for this system is quite involved (see [23] for complete formulas), because of supersymmetry the lepton asymmetry stored in the SM particles is exactly the same as that stored in the sparticles [23]. Since the heavy neutrino mass scale is much higher than the typical sparticle mass scale ∼ TeV, our system is supersymmetric to a very good approximation. Among the many decay and scattering processes involved in the Boltzmann equations, it is known that the (inverse) decay process of the lightest heavy (s)neutrino plays the most important role in determining the resultant baryon asymmetry, while the others are negligible in most of the parameter space [21]. Including only the decay process greatly simplifies the Boltzmann equations, so that for the heavy neutrino they are exactly the same as in the non-supersymmetric case: where Y N 1 is the yield (the ratio of the number density to the entropy density s) of the lightest heavy neutrino, Y eq N 1 is the yield in thermal equilibrium, the temperature of the universe is normalized by the mass of the heavy neutrino z = M 1 /T , H(M 1 ) is the Hubble parameter at T = M 1 , and Y L f is lepton asymmetry stored in the SM particles. The CP-asymmetry parameter, ǫ 1 , is given by [24] where and Y ν is the neutrino Dirac Yukawa coupling matrix in the basis where both the charged lepton matrix and M R are diagonalized. Using Eqs. (4) and (7), we can obtain Y ν as a function of θ 13 and δ CP .
The space-time density of the heavy neutrino decay in thermal equilibrium, γ N 1 is given by where K 1 and K 2 are the modified Bessel functions, and is the decay width of the heavy neutrino. Then, we solve the Boltzmann equations with the boundary conditions Y N 1 (0) = Y eq N 1 (0) and Y L f (0) = 0. The lepton asymmetry generated by the right-handed neutrino decays is converted to the baryon asymmetry via the sphaleron process with the rate of Eq. (6) and hence, we evaluate the resultant baryon number as where the factor 2 takes into account the baryon number stored in sparticles.
For various values of sin 2 (2θ 13 ) and δ CP , we numerically solve the Boltzmann equations. In our analysis, we fix m 0 = 10 −3 eV. For m 0 10 −2 eV, M 1 is almost independent of m 0 , and we find that the results for the generated baryon asymmetry are almost the same. Figure 3 shows the resultant baryon asymmetries as a function of δ CP for three different values of sin 2 (2θ 13 ) = 0.025 (dashed), 0.1 (solid) and 0.3 (dotted), along with the observed value (red horizontal line).
The allowed region for sin 2 (2θ 13 ) and δ CP compatible with the observed baryon asymmetry is depicted in Figure 4. The desired baryon asymmetry is reproduced along the two contours.
In the above analysis, as previously mentioned we have dropped all Majorana phases and kept only the Dirac phase. The general neutrino mixing matrix is given by U = U P M N S × diag(e iφ 1 , e iφ 2 , 1) with two Majorana phases. We have also analyzed this more general scenario and found that the observed baryon asymmetry is realized in this case even for θ 13 = 0. Therefore, our ansatz that φ 1 = φ 2 = 0 is crucial in obtaining the bound sin 2 (2θ 13 ) 0.03.
In summary, we have considered a supersymmetric SU(4) c ×SU(2) L ×SU(2) R model with a minimal number of Higgs multiplets and a single CP violating parameter δ CP in the neutrino flavor mixing matrix. The model has the same structure in the Yukawa couplings for the charged fermions as the supersymmetric minimal SO(10) model, so that the neutrino Dirac Yukawa coupling matrix is unambiguously determined by fitting the experimental data for charged fermion mass matrices. Using the seesaw formula with the neutrino Dirac Yukawa coupling matrix, the right-handed Majorana neutrino mass matrix is given as a function of θ 13 and δ CP . We have employed leptogenesis and the observed baryon asymmetry to identify the relation between θ 13 and δ CP . Only the normal hierarchical case for the light neutrino mass spectrum can reproduce the observed baryon asymmetry, and we find a lower bound of 0.03 on sin 2 (2θ 13 ). The recent results from T2K and MINOS indicate a non-zero θ 13 whose value should be more precisely known within the next few years. If θ 13 is sizable, δ CP and the pattern of mass hierarchies can be determined in future experiments. Our results therefore have interesting implications for future neutrino oscillation experiments.