Fermion Masses and Mixings, Leptogenesis and Baryon Number Violation in Pati-Salam Model

In this work we study a predictive model based on a partially unified theory possessing the gauge symmetry of the Pati-Salam group, $SU(2)_L\times SU(2)_R\times SU(4)_C$ supplemented by a global Peccei-Quinn symmetry, $U(1)_{PQ}$. A comprehensive analysis of the Higgs potential is carried out in a minimal set-up. The assumed Peccei-Quinn symmetry along with solving the strong CP problem, can provide axion dark matter candidate. This minimal set-up with limited number of Yukawa parameters can successfully incorporate the hierarchies in the charged fermion masses and mixings. The automatic existence of the heavy Majorana neutrinos generate the extremely small light neutrino masses through the seesaw mechanism, which is also responsible for producing the observed cosmological matter-antimatter asymmetry of the universe. We find interesting correlation between the low scale neutrino observables and the baryon asymmetry in this model. Baryon number violating nucleon decay processes mediated by the scalar diquarks and leptoquarks in this framework are found to be, $n,p\to \ell+m, \ell^c + m$ ($m=$ meson, $\ell=$ lepton, $\ell^c=$ antilepton) and $n,p\to \ell+\ell^c+\ell^c$. For some choice of the parameters of theory, these processes can be within the observable range. Another baryon number violating process, the neutron-antineutron oscillation can also be in the observable range.


Introduction
Despite being a very successful theory, the Standard Model (SM) of particle physics has many shortcomings. Such as, the SM does not provide any insights for understanding the hierarchical pattern of the masses and mixings of the charged fermions. Also the origin of the neutrino oscillations is unexplained in the SM. The observed quantization of electric charge in the SM is also not obvious. To explain these shortcoming of the SM, extensive search for finding new physics beyond the SM has been carried out. One of the most attractive extensions of the SM proposed in Refs. [1][2][3][4] are based on partial unification with non-Abelian gauge group G 224 = SU (2) L × SU (2) R × SU (4) C . This Pati-Salam (PS) group is the most minimal quarklepton symmetric model based on the SU (4) C group with the lepton number as the fourth color [3]. The minimal gauge group respecting symmetry between the left and right representations and that contains SU (4)-color and ensures electric charge quantization is the PS gauge group. Due to quark-lepton unification, one can hope to understand the flavor puzzle in the PS model. The fermion multiplets of this theory automatically contain the right-handed neutrino which is SM singlet, this is why seesaw mechanism [5] is a natural candidate in the PS model to explain the tiny masses of the SM light neutrinos. Furthermore, our universe does not show symmetry between matter and antimatter. The origin of this matter-antimatter asymmetry may have link with the origin of neutrino mass. In seesaw scenario, the Majorana mass term violates the lepton number conservation, so employing the seesaw mechanism in this PS framework, the observed baryon asymmetry of the universe can be incorporated by the Baryogenesis via Leptogenesis mechanism.
In such a framework, the lepton asymmetry that is generated dynamically, later converted into the baryon asymmetry by the (B + L)-violating sphaleron interactions that exist in the SM. In the SM, conservation of baryon number and lepton number are accidental, however, violation of these quantum numbers are natural in the PS model and baryon number violation induces interesting processes like nucleon decay and neutron-antineutron (n − n) oscillation.
In this paper, we construct a minimal realistic model based on the PS gauge group augmented by a global U (1) P Q Peccei-Quinn (PQ) symmetry in the non-supersymetric framework. Assuming an economical Higgs sector, we construct the complete Higgs potential and analyse it. This minimal set of Higgs fields is required not only to realize successful symmetry breaking of the PS group down to the SM and further down to SU C (3) × U em (1), but also to reproduce realistic fermion masses and mixings. We discuss the possibility of baryon number violating processes such as nucleon decay and n − n oscillation in this set-up. Nucleon decay processes in this framework are found to be, nucleon → lepton + meson, nucleon → antilepton + meson and nucleon → lepton + antilepton + antilepton. Relative branching fractions of different modes of nucleon decay processes arising in this theory are computed on the dimensional ground.
We also analyze the predictions of this model for quark and lepton masses and mixings. Our numerical study shows full consistency with the experimental data. In addition to unifying quarks and leptons, seesaw mechanism arises in G 224 framework naturally due to the automatic presence of the right-handed neutrinos. To solve the matter-antimatter asymmetry of the universe, we implement the novel idea of Baryogenesis via Leptogenesis. Utilizing the type-I seesaw scenario, the Baryogenesis via Leptogenesis mechanism can link between the matter-antimatter asymmetry and the CP violation in the neutrino sector. In search of successful baryon asymmetry, we scan over the relevant parameter space and, present the predictions of our model of the neutrino observables. In this work, on top of the PS gauge symmetry, we impose a global U (1) P Q PQ symmetry, that solves the strong CP problem. If the PQ symmetry is broken at the high scale ∼ 10 11−12 GeV, then the pseudo-scalar Goldstone boson associated with this breaking can explain the observed dark matter relic density of the universe, so the dark matter candidate in this model is the axion. The presence of this global U (1) symmetry puts some additional restrictions on the Higgs potential and and also in the Yukawa Lagrandian, hence reduce the number of parameters in the theory significantly. We discuss the implications of both the high scale and low scale PS breaking scenarios. With the economic choice of Higgs multiplets, we do a general study in SU (2) L × SU (2) R × SU (4) C × U (1) P Q set-up; a special case with the imposed discrete parity symmetry that demands g L = g R at the PS symmetric phase, is also considered and additional restrictions due to the consequence of this discrete symmetry are mentioned explicitly through out the text. We also explore another interesting possibility, where with the absence of the discrete parity symmetry, g L = g R unification can still be realized at the PQ breaking scale, by extending the minimal Higgs sector if the PS breaking scale is low.
The rest of the paper is organized as follows. In Sec. 2 we give the detail of the model. In Sec. 3 we discuss the mass generation of the charged fermions as well as the neutrinos, then we briefly review the leptogenesis mechanism in Sec. 4. Detail numerical analysis of the charged fermion masses and mixings and also leptogenesis is performed in Sec. 5. Comprehensive analysis of the Higgs potential and calculating the detail mass spectrum of the Higgs is carried out in Sec. 6. In Sec. 7 we find the baryon number violating processes within the model and construct the effective higher dimensional operators responsible for such processes and finally conclude in Sec. 8. We work on a left-right symmetric partial unification theory based on the PS gauge group, SU (2) L × SU (2) R × SU (4) C . SU (4) C is an extension of the QCD gauge group, SU (3) C with lepton as the fourth color and SU (2) R is right-handed gauge group similar to the SM SU (2) L weak interac-tions. Starting from this gauge group, to break it down to the SM group, several different breaking chains are possible, but in this paper we assume the one step spontaneous symmetry breaking (SSB) of the PS group to that of the SM group, In our model, we assume the existence of the following Higgs multiplets (under the PS group): Φ = (2, 2, 1), Σ = (2, 2, 15), ∆ R = (1, 3, 10).
Instead of G 224 , if left-right parity symmetry is also preserved (in this case we denote the group as G 224P ), the existence of the Higgs field ∆ L = (3, 1, 10) is needed due to the presence of the parity symmetry. This choice of the Higgs multiplets is the minimal set. This one step breaking of PS group to the SM can be achieved by the VEV of the (1,3,10) multiplet, v R = ∆ R [6].
If the group is G 224P then, in general the breaking of the parity scale may not coincide with the breaking of the PS symmetry. However, breaking the G 224P group by the VEV of (1, 3, 10) automatically breaks the parity symmetry. The multiplet ∆ R , breaking SU (4) C , B − L and leftright symmetry spontaneously also provides masses to the heavy right-handed neutrinos. In an alternative approach the parity symmetry can be broken before breaking the PS group by a parity odd singlet Higgs and then the PS symmetry can be broken by the usual (1, 3, 10) VEV. The SM group can be broken by the scalar field Φ that contains the SM doublet. The VEV of Φ field, is responsible for generating Dirac mass terms for the SM fermions. But if only Φ is responsible for generating charged fermion masses, one gets the unacceptable relations, m e = m d , m µ = m s and m τ = m b . These lead to m e /m µ = m d /m s , which are certainly not in agreement with experimental measured values. These bad relations are the consequence of the multiplet Φ being color singlet (in the SU (4) C space the fourth entry is also 1) and cannot differentiate fermions with different colors. To cure these bad relations, the existence of the Higgs multiplet Σ is assumed which is not color blind, and by acquiring VEV of the form can correct these bad mass relations [2,3,6], m e = m Φ e − 3m Σ e , m d = m Φ d + m Σ d and so on. Even though the field Φ treats quarks and lepton on the same footing, Σ field being color non-singlet, distinguishes them and brings additional (−3) Clebsch factor for the leptons.

Renormalization group equations and the v R scale
According to phenomenological consideration, the required hierarchical pattern of the VEVs is: (2.6) As previously mentioned, in the model without the parity symmetry ∆ L field need not to be present. Even when this field is present, we assume that this field does not get any explicit VEV.
However, this field does get small induced VEV due to the presence of specific types of quartic terms in the Higgs potential that are linear in ∆ L . After the EW symmetry breaking such acquired VEV is of the form, ∆ L ∼ λ v 2 ew /v R (where λ is the relevant quartic coupling). The fields Φ and Σ containing the weak doublets acquire VEVs around the electro-weak scale.
If parity is assumed to be a good symmetry, v R can be fixed by the renormalization group equations (RGEs) running of the gauge coupling constants by using low energy data. This additional discrete symmetry on top of the PS symmetry demands g L = g R . The one-loop RGEs for the gauge couplings are given by [7] For the SM group, G 321 b i = (−7, −19/6, 41/10) [8]. Applying proper matching condition for the coupling constants, and using the low energy data, α s (M Z ) = 0.1184, α −1 (M Z ) = 127.944 and s 2 θ W = 0.23116 taken from Ref. [9] (only the central values are quoted here), we find M X = 10 13.71 GeV. From now on, for models with parity symmetry broken by the ∆ R VEV, we set v R = 10 14 GeV for the rest of the analysis. Specially when we will discuss the high scale leptogenesis, we stick to this value of v R . On the other hand, If left-right parity symmetry is absent, then the scale v R is not fixed by the RGEs running. The differences in results for the cases with G 224 and G 224P are mentioned explicitly through out the text when needed.

Left-right gauge coupling unification at the Peccei-Quinn scale
In this subsection, we explore an alternative realization of g L = g R unification without the presence of the left-right parity symmetry. As explained above, breaking the parity symmetry that demands g L = g R along with the breaking of the PS symmetry by the (1, 3, 10) multiplet restricts the PS breaking scale to be high ∼ 10 14 GeV. If parity symmetry is absent, this scale is not determined by the RGEs running from the low energy experimental data and the PS breaking can happen at much higher or even at much lower scale. The experimental limits on the branching ratio for K 0 L → µ ± e ∓ processes, mediated by the new gauge bosons X a (a is the Lorentz index) with (B − L) charge of (4/3), implies that the v R scale that breaks the SU (4) C must be greater than about 1000 TeV [10,11]. Here we explore the possibility of low scale PS scale breaking where g L = g R unification can still be realized at the PQ scale ∼ 10 11−13 GeV. However, this requires extension of the minimal Higgs sector. For example, by including an extra (1, 3, 10) multiplet and a real (1, 3, 15) multiplet on top of the minimal Higgs content that are a complex (2, 2, 1), a complex (2, 2, 15) and a (1, 3, 10) multiplet, left-right unification can happen at the PQ scale as shown in Fig. 1  μ GeV that are a complex (2, 2, 1), a complex (2, 2, 15) and a (1, 3, 10) multiplet, g L = g R unification at the PQ scale ∼ 10 11−13 GeV can be realized.

Gauge boson mass spectrum
In the PS model, there are in total 21 gauge bosons, The decomposition of these fields under the SM are: {X a (uγ a ν + dγ a e) + X * a (u c γ a ν c + d c γ a e c )}. Since one can assign specific baryon and lepton numbers to these gauge bosons, in contrast to SO(10) model, proton decay does not take place via these gauge bosons. Unification scale in minimal SO(10) model needs to be really high > 5 × 10 15 GeV to save the theory from too rapid proton decay.
The spontaneous symmetry breaking G 224 → G 213 , that does not break the SU (2) L group, the W L a gauge bosons remain massless in this stage. Due to this breaking, among the 18 (15 of SU (4) C and 3 of SU (2) R ) massless gauge bosons, 9 of them become massive after eating up the 9 Goldstone bosons (will be identified at the later part of the text), from the field ∆ R and the other 9 of them (8 of SU (3) C and 1 of U (1) Y ) remain massless. Here we compute the mass spectrum of the gauge bosons. Following Ref. [12] the covariant derivative can be written as where a represents the Lorentz index. When the PS symmetry gets broken by the VEV of the ∆ R field, using this covariant derivative, the gauge boson mass spectrum can be computed  R , V (15) } the mass squared matrix is given by where we have defined g c = 3/2 g C . One can easily calculate the two eigenvalues of this matrix, one of the eigenvalues is zero and the corresponding eigenstate is given by (2.23) This is the massless gauge boson of U (1) Y group. Its orthogonal eigenstate acquires mass given In addition, for the unbroken SU (3) C group, the massless gauge bosons, the gluons are identified with V (i) (i = 1 − 8) fields.

Peccei-Quinn symmetry
On top of the PS gauge symmetry we assume the existence of global Peccei-Quinn (PQ) symmetry, U (1) P Q [13][14][15][16] (for a relation between leptonic CP violation with strong CP phase in the context of left-right symmetric models see Ref. [17]). The PQ symmetry naturally solves the strong CP problem and simultaneously provides the axion solution to the dark matter problem [18,19]. So the complete symmetry of our theories are either G 224 × U P Q (1) or G 224P × U P Q (1). The SM singlet present in ∆ R that break the PS symmetry and the singlet S, each can break one U (1) symmetry. If the VEV of the singlet, S = v S > v R , then this VEV breaks the U (1) P Q . On the contrary, if v R > v S , it leaves U (1) B−L+P Q group unbroken since it carries a PQ charge, which is then further broken via U (1) B−L+P Q → U (1) B−L by the VEV of the S field. Due to the presence of the U (1) P Q symmetry, the complex scalar fields carry PQ charge which consequently puts additional restrictions on the Higgs potential and also in the Yukawa Lagrangian, this reduces the number of parameters in the Higgs potential as well as in the Yukawa sector significantly. For example, if PQ symmetry is not imposed, each of these Φ and Σ fields can have two independent Yukawa coupling matrices. However, the presence of the PQ symmetry restricts one of such Yukawa coupling terms, hence instead of four, only two Yukawa coupling matrices determine the charged fermion spectrum, makes the theory predictive.

Fermion masses and mixings
In this section we discuss the fermion masses and mixings in the PS model. The model under consideration is very predictive in explaining the data in the fermion sector. The Yukawa part of the Lagrangian is given by: where, Y 1 , Y 15 and Y R,L When parity is imposed (see Eq. (6.64)) the matrices Y 1 and Y 15 become Hermitian and Y R,L become identical, i.e, For the analysis of the fermion masses and mixings we restrict ourselves to the case when parity summery is realized since this significantly reduces the number of parameters in the fermion sector due to constraints mentioned in Eq. (3.25), so our model is highly predictive.
The VEV of the (1,3,10) multiplet ∆ R , breaks the G 224 group down to the SM group and generates the right-handed Majorana neutrino masses given by v R Y 10 . The Higgs fields Φ and Σ each contains two doublets of the SM, acquire non-zero VEVs and are responsible for generating charged fermion masses. From the Lagrangian one can write down the fermion mass matrices as: type VEVs of the multiplet Φ(2, 2, 1) (Σ(2, 2, 15)). In general these VEVs are complex and there is one common phase for k u and k d and different phases for each of v u and v d . Only two relative phases will be physical and we bring these phases (θ 1,2 ) with v u and v d . The analysis done in Sec.
6.2 shows that the VEV ratios are complex and can not be made real. One can absorb the VEVs into the coupling matrices and redefine them, leaving two relevant VEV ratios (r 1,2 ). Following these arguments, we can rewrite the mass matrices as, Let us now discuss the neutrino sector. The right-handed Majorana mass matrix is complex symmetric matrix and the corresponding Yukawa coupling matrix Y 10 is arbitrary since it decouples from the charged fermion sector which is unlike the case of SO(10) models. In unified theories due to the presence of right-handed neutrinos seesaw mechanism is a very good candidate to explain the extremely small observed light neutrino masses. One should note that due to the presence of terms linear in ∆ L in the Higgs potential (Eq. (6.55)), this field will acquire a small induced VEV, v L as mentioned above, that would be responsible for generating left-handed Majorana neutrino mass, M L = v L Y 10 , which is the type-II contribution. In this paper, we assume the dominance of type-I seesaw scenario, then the light neutrino mass matrix is given by the type-I seesaw [5] formula, Inverting the type-I seesaw formula one can express M R as, There is no new parameter in the M D matrix and is completely fixed by the charged fermion sector. The light neutrino mass matrix, M ν can be diagonalized as with the eigenvalues being real and in the basis where the charged lepton mass matrix is diagonal, where α and β are Majorana phases and U PMNS is the CKM type mixing matrix with only one Dirac type phase δ in it.
We assume normal hierarchy 2 in the light neutrino sector, which leads upto a good approximation, m 2 ∼ ∆m 2 sol and m 3 ∼ ∆m 2 atm for neutrino masses 3 . The quantities (∆m 2 sol , ∆m 2 atm , θ PMNS ij ) in the neutrino sector have already been measured experimentally with good accuracy.
The quantities m 1 , α, β and δ are yet to be determined experimentally. So in Eq. (3.33), using the experimentally measured quantities in the neutrino sector, the right-handed Majorana mass matrix can be determined as a function of these four unknown quantities. In Sec. 5.2, we will explain the algorithm we follow while searching for the allowed parameter space to reproduce successful leptogenesis in this model and also present our results.

Baryogenesis via Leptogenesis
In unified theories the Baryogenesis via Leptogenesis [20] is a natural candidate to explained the observed matter-antimatter asymmetry [21]. This simple mechanism can be implemented in theories where light neutrino mass is generated via seesaw mechanism. For studies on leptogenesis in the framework of G 224 /SO(10) see for example Refs. [22][23][24][25][26][27][28]. In this mechanism, the baryon asymmetry of the universe is generated by the lepton asymmetry which is initially produced dynamically and later converted into the baryon asymmetry via the (B + L)-violating sphaleron process [29] that exist in the SM. Computing the baryon-asymmetric parameter involves solving the coupled Boltzmann equations. The asymmetry is generated when the decay rates of the heavy neutrinos < H (H being the Hubble expansion rate), so leptogenesis is expected to occur at a temperature of order of the mass of the lightest right-handed heavy neutrino, M 1 . For hierarchical spectrum of the right-handed neutrinos, i.e, M 1 M 2 < M 3 , the lightest heavy neutrino is responsible for generating the baryon asymmetry and known as N 1 -dominated leptogenesis (for reviews on leptogenesis see for example Refs. [30,31]). In this work we concentrate on N 1dominated leptogenesis. In the literature it has been pointed out that flavor can play significant role in the mechanism of leptogenesis. Flavored leptogenesis has been studied with great details in the literature, see for example Refs. [32][33][34][35][36][37] for earlier works.
The minimum required reheat temperature of the universe depends on the details of the flavor structure of the lepton asymmetry. Without taking into account the flavor effects, the lower bound to produce successful baryon asymmetry is M 1 > 10 9 GeV [38]. Including the flavor effects relaxes this lower bound a little bit (for details see Ref. [33]). Approximate analytical solutions of the Boltzman equations have been derived that are in good agreement with the exact solutions (see for example Ref. [34]). While scanning over the parameter space in search for successful leptogenesis we apply these analytical solutions to compute the baryon asymmetry. The analytical formula depends on the interaction rate of the charged lepton Yukawa couplings [37]. We are interested in the two different regions, first, when only the tau Yukawa coupling is in equilibrium which corresponds to the region 10 9 GeV M 1 10 12 GeV. In this first case, the flavor effects play vital role. The second region where no charged lepton Yukawa couplings are in equilibrium that corresponds to the case M 1 10 12 GeV. In this second case all flavors are indistinguishable and is no different than the one flavor scenario.
Here we briefly summarize the required approximate analytical solutions for our analysis that are derived in the literature as mentioned above. In the regime where flavors are indistinguishable, the CP asymmetry generated by the N 1 decay is where, Beside the CP parameter 1 , the final asymmetry depends on the wash-out parameter, with m * ∼ 10 −3 eV and In the strong wash-out regime, i.e, for K >> 1, the lepton asymmetry is given by the following approximate formula with g * being the effective number of spin-degrees of freedom in thermal equilibrium, which is ∼ 108 in the SM with a single generation of right-handed neutrinos. With these the baryon asymmetry is given by the weak wash-out regime, the approximate analytical formula is, On the contrary, the regime where the flavor effects are important, the CP asymmetry in the α-th flavor is given by And the wash-out parameter is that parametrizes the decay rate of N 1 to the α-th flavor. In the strong wash-out regime for all flavor, i.e, K αα >> 1, the total asymmetry generated is given by, Y L = α Y αα , where the approximate analytical formula for each flavor, Y αα is And in the weak wash-out regime the formula is, The Baryon asymmetric parameter has been measure experimentally which is η B = (5.7 ± 0.6) × 10 −10 4 [39,40]. Since this scenario of generating baryon asymmetry requires the righthanded neutrino mass scale to be high, for this analysis we fix the PS breaking scale to be v R = 10 14 GeV as discussed before in the text.
5 Fit to fermion masses and mixings and parameter space for successful Leptogenesis

Numerical analysis of the charged fermion sector
In this sub-section we show our fit results of the fermion masses and mixings in the charged fermion sector. For optimization purpose we do a χ 2 -analysis. The pull and χ 2 -function are defined as:  is assumed in order to take into account the theoretical uncertainties, for example threshold effects at the PS scale. The inputs are shown in the Table 2 where the fit results are presented.
As noted before, for this case, we have 14 parameters, 11 magnitudes and 3 phases. We perform the χ 2 function minimization and the best minimum corresponds to total χ 2 = 1.2 is obtained for 13 observables which is a good fit 5 . The result corresponding to the best fit is shown in Table 2. The values of the parameters corresponding to the best fit are:  Table 2: χ 2 fit of the observables in the charged fermion sector. This best fit correspond to χ 2 = 1.2 for 13 observables. For charged leptons, a relative uncertainty of 0.1% is assumed to take into account the uncertainties, for example threshold corrections at the PS scale.

Parameter space for successful Leptogenesis
Using the seesaw formula Eq. (3.32), one can in principle fit all the neutrino observables since the matrix M R which is in general a complex symmetric matrix contains 6 complex parameters.
Instead, we will follow an alternative procedure. The right-handed neutrino mass matrix is given by inverting the seesaw formula Eq. (3.33). After the fitting of the fermion masses and mixings has been done, the Dirac neutrino mass matrix gets fixed. For our fit, this Dirac neutrino mass matrix is Ref. [43]). Baryon asymmetric parameter is computed in a basis where both the charged lepton and the right-handed neutrino mass matrices are real and diagonal. We diagonalize these mass matrices as, In this basis, the Dirac neutrino mass matrix is given by which is fixed from the fit parameters in the charged fermions and U ν R can be computed as a function of the free parameters m 1 , α, β, δ. The inputs in the neutrino sector are taken from [44] and shown in Table 3.
Quantity 1σ range 2σ range  Table 3: Observables in the neutrino sector taken from [44].
While scanning over the parameter space, if 10 9 GeV M 1 10 12 GeV, we compute the baryon asymmetric parameter by taking into account the flavor effects and for the regime M 1 10 12 GeV, calculating η B involving the case where flavors are indistinguishable.We recall the readers that for this high scale leptogenesis study, we have fixed the PS breaking scale to be 10 14 GeV. Since the Majorana mass for the right-handed neutrinos are given by v R Y R , for perturbitivity reason, we put a cut-off of M 3 2 · 10 14 GeV. For both the scenarios, unflavored or flavored, we use the formula for the strong wash-out regime when the wash-out parameter > 1 (K and K αα ) and the formula for weak wash-out regime when it is < 1 (instead of 1 and 1 respectively).
It is to be mentioned that our investigation shows that the parameter space only permits solution in the strong wash-out regime, so all the results presented below are solutions in the strong wash-out regime. We now discuss our results of leptogenesis in our framework. In Fig. 2, η B is plotted against α, β and δ phases respectively for the two different values of m 1 = 1, 2 meV. While keeping m 1 fixed, the other three parameters are varied over the whole range as mentioned before. Similar plots for another two fixed values of m 1 = 0.8 and 4 meV are presented in Fig. 7 in Appendix A. From these plots, it is clear that whether or not flavor effects are involved, depending on that, the allowed region in the parameter space is pretty much different. The general behaviour is as follows, for larger values of m 1 , the parameter space gets more populated for both the flavored and of leptogenesis in PS model including the case of g L = g R will be considered elsewhere [46].
In Appendices C and D, we present additional plots Figs. 9 and 10 to show the correlation between some physical quantities to the baryon asymmetric parameter for these two cases with m 1 = 1 and 2 meV. In Fig. 9, the permitted region for m β and m ββ to have successful leptogenesis is shown, where m β = i |U ν ei | 2 m i is the effective mass parameter for the beta-decay and m ββ = | i U 2 ν ei m i | is the effective mass parameter for neutrinoless double beta decay. The correlations between the Dirac phase δ and the angle θ 13 is presented in Fig. 10. All the plots presented here are the result of 10 8 iterations.    In the neutrino sector, among the four different experimentally unmeasured quantities, particularly the Dirac type phase δ is the most important one, since it has the potential to be measured in the upcoming neutrino experiments. In Fig. 3, we present the allowed range for this CP violating phase to have successful leptogenesis is presented for different values of the lightest neutrino mass m 1 . Benchmark points corresponding to few different cases are presented in Table 4.
6 The Higgs potential and scalar mass spectrum 6

.1 The Higgs potential
In this sub-section we construct the complete scalar potential with G 224 × U (1) P Q symmetry. As mentioned earlier, the field ∆ L which is present if the group is G 224P but need not be present if the gauge group is G 224 instead. But for generality, we construct the scalar potential containing is: with, Lζω β µρλζ ντ χω + λ 9 * ∆ * µνβ Rα ∆ * ρτα Rβ ∆ * λχ β Lα ∆ * ζω α Lβ µρλζ ντ χω ), (6.58) To differentiate the complex couplings from the real ones in the potential we put tilde on the top of the complex ones. All the index contractions are shown explicitly. The parameters with dimension of mass are µ φ , µ Σ , µ ∆ , µ S , ζ, ω. To find the maximum possible number of invariants of each kind one needs to use the group theoretical rules of tensor product decomposition (for details see Ref. [45]). Note that in general there can be more gauge invariant terms in the Higgs potential, but those are absent in out theory due to the presence of the global U (1) P Q symmetry.
Below we discuss the constraints on the cubic and quartic couplings in the potential due to additional left-right parity symmetry.

Scalar potential in the left-right parity symmetric limit
If the parity symmetry is assumed to be a good symmetry then there are further restrictions on the potential Eq. (6.55). Under left-right parity, the fermions and the scalar fields transform as The terms that are achieved by R ↔ L in Eq. (6.55) have exactly the same coupling constants, for example, µ 2 ∆ L = µ 2 ∆ R , λ iL = λ iR (i = 1 − 5) and so on. Also due to the invariance under parity, some of the complex couplings in the potential will become real, they are, α 5,6 , β 3 , η 4,5,6 , χ 3 , ζ, ω ∈ R. (6.65) The only six couplings in the potential that remain complex are λ 9 , γ 9,10 , η 1 , χ 1,2 ∈ C.

The scalar mass spectrum
In this sub-section, we compute the Higgs mass spectrum after the PS symmetry is broken.

Mass spectrum of ∆ R scalar fields
The Yukawa Lagrangian of the theory is given in Eq. (3.24), where the first two terms are the Dirac type Yukawa couplings. The third term generates the right-handed neutrino Majorana masses when the PS symmetry is broken by the VEV (1, 3, 10) . Expanding this term of the Yukawa coupling one gets (here ∆ represents ∆ R ): Only the neutral component of ∆ R gets VEV, v R = ∆ νν . With this identification and by minimizing the potential Eq. (6.55), one can compute the mass spectrum of ∆ R . The PS breaking minimization conditions is Choosing the non-trivial solution with v R = 0, this equation is used to eliminate µ 2 ∆ from the potential. Imposing this extremum condition back to the potential one gets the following mass spectrum for ∆ R : There is a mass relation which is given by There exist seven physical Higgs states ∆ νν , ∆ ee , ∆ uu , ∆ dd , ∆ ud , ∆ de , ∆ ue and three Nambu- Goldstone bosons. These Goldstone bosons correspond to ∆ eν , ∆ uν and ∆ G (real field) fields.
We note that these sextets can have rich phenomenology if their masses are relatively low, for example, these sextets can be responsible for generating baryon asymmetry after the sphaleron decoupling, see Ref. [47][48][49][50]. By considering the sextet masses at the TeV scale, flavor physics constraints are also computed in Ref. [51].
If both the PS and PQ symmetry breaking are taken into account together, where the PQ symmetry is broken by the complex singlet VEV, S = v S the minimization conditions are Assuming the general symmetry breaking solutions v S = 0 and v R = 0, these equations can be used to solve for µ 2 ∆ and µ 2 S . Using these minimization conditions like before one can easily derive the mass spectrum for the ∆ R and S fields. The mass spectrum essentially remain unchanged except ∆ νν mixes with the real part of the singlet field. The two by two mass squared matrix of this mixing in the basis {∆ νν , Re[S]} is given by The imaginary part of S remains massless after the PQ symmetry breaking. After EW symmetry breaking, this field will eventually mix with the components from the four doublets coming from Φ and Σ and receive a mass of the order of v ew /v S . Since v ew v S , this field will remain essentially massless and can be identified as the axion field, which is the dark matter candidate in our model.
the doublet mass squared matrix, D in the flavor basis can be found from the Higgs potential as It is straightforward to compute this doublet mass square matrix, where we have defined Recall that if parity symmetry is imposed, ζ and ω will be real but χ 1,2 entering in this matrix will remain complex, so in general D will have two independent phases entering in this matrix.
The Hermitian matrix, D can be diagonalized as D = U ΛU † , where U is an unitary matrix (Λ is the diagonal matrix containing real eigenvalues) that relates the flavor basis, h (i) α and mass basis, h The doublet mass matrix written here is before the EW phase transition, so the SM Higgs doublet will correspond to the zero eigenvalue solution, which can be found by imposing the fine tuning condition det(D) = 0. One can write the SM Higgs doublet that is a linear combination of the four doublets as, When the SM doublet acquires VEV, H = v EW , the EW phase transition takes place and one gets, By finding the matrix elements U ij it can be shown that the combinations αγ * and βδ * will remain complex and so all the VEVs in Eq. (6.93) cannot be taken to be real. This is why the VEV ratios of the doublets that appear in the fermion mass matrices are in general complex. This conclusion is also applicable for the case with parity symmetry imposed, since χ 1,2 that are complex couplings will introduce two independent phases in D.

(6.95)
Note that if the parity symmetry is imposed, all the matrix elements in this mass squared matrix will become real.
The mass spectrum of ∆ L field The identification of the multiplets of the (3, 1, 10 * ) field under the SM group is (here ∆ represents ∆ L ): The mass spectrum of these fields are given as follows: (3,1,10), nucleon is absolutely stable. The reason of the stability is due to the existence of a hidden discrete symmetry [52] in the model The Lagrangian is invariant under this discrete symmetry even after SSB. But the scalar sector Eq.
(2.3) that we are considering in this work, which also contains (2,2,15) multiplet, in principle can lead to baryon(B) and lepton(L) violating processes by nucleon decay [6,53]. This happens due to the presence of some specific quartic terms in the scalar potential Eq. (6.55). In our model, the part of the potential V Σ∆ in Eq. (6.61) contains terms that can cause the nucleon to decay.
For PS model with this minimal set of scalars, another feynman digram that contributes to the nucleon decay can be constructed by replacing the color octet Σ 8 by a color triplet Σ 3 and the sextet ∆ 6 by color triplet ∆ 3 as shown in Fig. 4 (right diagram). This kind of diagrams will lead to nucleon decay, 3q → c . The processes shown in Fig. 4 are generated by the dimension nine (d = 9) operators. Shortly we will show that in our set-up, d = 9 operators only give rise to the decay processes of the type nucleon→ lepton+ meson(s) but not nucleon → lepton + lepton + antilepton processes since these three lepton decays always involve ν R in the final state and hence are extremely suppressed.
However, three lepton decay processes of nucleon can take place in our model via the d = 10 operators [54][55][56]. The feynman diagrams corresponding to nucleon decay processes mediated by d = 10 operators are shown in Fig. 5. These decay modes are: nucleon→ antilepton + meson and nucleon→ lepton + antilepton+ antilepton. Below we present the effective Lagrangians corresponding to d = 9 and d = 10 and discuss the different nucleon decay modes and compute the branching fractions in certain approximations. For operator analysis regarding baryon and lepton number violation see Ref. [57][58][59][60]. Figure 5: Feynman diagrams for nucleon decay with the SM doublet VEV insertions. The left diagram induces nucleon decay processes like, nucleon → antilepton + mesons and the right digram, nucleon → lepton + antilepton + antilepton processes.

d = 9 proton decay
To write down terms responsible for d = 9 proton decay, we expand the scalar potential having couplings withγ 9R ,γ 10R ,η 1 in the potential Eq. (6.61) in terms of the SM multiplets, )}] + h.c. , (7.102) )] + h.c. , )] + h.c. , respectively. From these, the effective Lagrangian describing these d = 9 six-fermion vertex that corresponds to nucleon decay can be written down,   + h.c , (7.106) here k, l, m, n, p, q are the generation indices. The terms involving color octets mediate nutron decay via the channels n → π + e − R , K + e − R , π + µ − R , K + µ − R , π 0 ν R , K 0 ν R and proton decay via p → π + ν R , K + ν R . And the terms where the color triplets replacing the color octets, the decay modes There exist also terms that lead to three quark decay as mentioned above.
Note that for all the three lepton decay modes of the nucleon as well as, some of the two body decay modes with the lepton being the neutrino, these decays can not be observed due to the additional suppressions of large right-handed neutrino mass. For the three lepton decay channels, always one of the leptons is a right-handed neutrino and for the two body decay channels with neutrino as the lepton, it is always the right-handed neutrino. This is not true in general within the PS framework. But in our model due to the additional U (1) P Q symmetry, Σ has coupling with ψ L ψ R and Σ * has coupling with ψ R ψ L , see Eq. (3.24). Also PQ charge conservation does not allow quartic terms of the form Σ 2 ∆ * R 2 , rather allowed term is of the form Σ 2 ∆ R 2 . The combined effect of these two facts restricts nucleon decay modes containing only left-handed neutrinos in the final state in our model. However, neutron decay into a lepton and a meson (n → e − R π + , e − R K + , µ − R π + , µ − R K + ) can be within the observable range with specific choice of the parameter space. There will be similar modes of proton decay (p → e − R π + π + , e − R K + π + , µ − R π + π + , µ − R K + π + ) with an additional pion in the final state and hence will be suppressed compared to neutron decay.
On the dimensional ground the decay rate of these n → lepton + meson processes is given by Here m p is the mass of the proton and the mass of the Higgs bosons involved are taken to be of the same order and is denoted by M . While computing this, the amplitude of such processes gets multiplied by the factor Λ 5 QCD , here a factor of Λ 3 QCD enters due to the hadronization of 3 quarks into a nucleon and a factor of Λ 2 QCD comes into play due to the hadronization of qq c to a meson (for numerical computations, we take Λ QCD = 170 MeV). Assuming the Higgs bosons masses equal to the PS breaking scale, i.e, M = v R , the decay rate (τ = Γ −1 ) of such processes to be within the observables range (τ ∼ 10 34 yrs) requires the PS breaking scale to be as low as v R ∼ 3.5 × 10 5 GeV.
For high scale breaking of PS group, the nucleon decay is completely unobservable. On the other hand, low scale PS symmetry breaking can lead to observable nucleon decay modes. As mentioned above, the PS breaking scale can be as low as 10 3 TeV. From the naive computation of decay rate performed above, it is clear that, for low scale PS breaking, if the scalar masses that mediate the nucleon decay are somewhat smaller than the breaking scale, which can be made by some choice of the parameters of the theory, these processes can be within the observable range.
On the other hand, decay rate of the p → lepton + mesons processes is given by The additional factor of Λ 2 QCD is due to the presence of an extra pion in the final state. By a similar computation one finds that v R ∼ 9.5 × 10 4 GeV is required for such processes to be within the observable range. Again this required v R is computed naively, but in general there is no reason for the masses of the scalar fields that mediate nucleon decay to be degenerate with v R . Even though additional suppression factor is present due to an extra pion in the final state, with some choice of the model parameters can make these proton decay modes observable.

d = 10 proton decay
To write down down the d = 10 proton decay operators, we also expand the relevant terms in the potential that have couplingsγ 9L andγ 10L , )}] + h.c., ) ∆ * γ L κ (3, 3, ) ∆ * γ L α=1 (3, 3, ) ∆ * γ L α=1 (3, 3,  (2,2,15) multiplet. Note, since these terms are generated due to the electroweak symmetry breaking, SU (2) indices are shown explicitly. The effective Lagrangian describing the d = 10 six-fermion vertex that correspond to nucleon decay can be written down, with, The terms involving color octets mediate neutron decay via the channels n → ν L c π 0 , e + L π − , µ + L π − , ν L c K 0 , K − e + L , K − µ + L and proton decay via p → ν L c π + , e + L π 0 , ν L c K + , e + L K 0 , µ + L π 0 , µ + L K 0 . And the terms where the color triplets replacing color octets, the decay modes are, n → ν L ν L c ν L c , Six fermion vertex d = 9 nucleon decay operators mediate processes like n → lepton + meson and p → lepton + mesons, whereas, n → antilepton + meson and p → antilepton + meson processes arise through d = 10 six fermion vertex operators. d = 10 operators also induce processes with three lepton final state, which is not the case with d = 9. The decay width for processes like n, p → antilepton + meson is For n, p → antilepton + meson to be within the observable range (τ ∼ 10 34 yrs [61]), the requirement on the PS scale is v R ∼ 10 5 GeV. The three lepton final state also requires v R ∼ 10 5 GeV (here τ ∼ 10 33 yrs [62]). Again, as mentioned above, by some choice of the quartic couplings involved in these decay rate of these processes can simultaneously satisfy the lower bound of the PS breaking by making the masses of these scalars somewhat smaller than v R , but still be in the interesting observable range.

Nucleon decay relative branching fractions
By using the formulae as mentioned above one can compare the decay widths of the different modes. A naive estimation of the relative branching fractions reveal Here we have chosen v R = 10 6 GeV. This estimation shows that for the d = 9 case, neutron decay will be dominating over the proton decay due the the presence of extra pion in the final state for the proton decay modes. Again d = 10 processes are suppressed compared to the d = 9 processes due to the extra suppression factor of v 2 ew /v 2 R . We remind the readers that these results are not general, since the Higgs boson mass spectrum is expected to be non-degenerate and appropriate hierarchical pattern can be realized to make these two processes comparable. Also note that the detail of the structure of the Yukawa couplings are ignored for this analysis. Since nucleon decay processes involve more than one quartic coupling, definite predictions about the relative branching fractions of different decay channels can not be firmly predicted.

Comment on d = 7 B-violating operators
In unified theories, another interesting B-violating operators involving Higgs bosons that can mediate nucleon decay correspond to the case of d = 7. In addition to the leptoquark color triplets present in our theory, if also diquark color triplets exist, then d = 7 operators can mediate nucleon decay. For example, quartic terms in the Higgs potential involving a triplet leptoquark, a triplet diquark, a Higgs doublet and the neutral component from ∆ R is responsible for generating nucleon decay processes [63] when the B − L violating VEV of ∆ R is inserted. In our minimal model due to the absence of diquark color triplets, d = 7 operators are not present.

n − n Oscillation
Another phenomenologically interesting process that can take place in PS model is the ∆B = 2 interactions that leads to n−n oscillation. A PS model with the presence of only ∆ R (1,3,10) scalar can have nucleon transition at the tree level that includes six-fermion ∆B = 2 vertex [52,64].
Such transitions are again lead by a specific type of term in the scalar potential and have the form ∆ 4 R . Note that, due to the additional U (1) P Q symmetry, terms of this form are not present in our theory since this field carries non-zero PQ charge. However, there is a quartic term involving both ∆ R and ∆ L in our potential which is, Lζω β µρλζ ντ χω + h.c. (7.127) Interactions generated by Eq. (7.127) and Eq. (3.24) after the spontaneous PS symmetry breaking by ∆ R cause baryon number violating n − n oscillation as shown in Fig. 6. The existing term is of the form ∆ 2 R ∆ 2 L , which indicates that if ∆ L field is not present, n − n transition is forbidden in this set-up due to the added U (1) P Q symmetry. Expanding in terms of the SM multiplets, this term gives, 2λ * 9 v R 3 3 ∆ * R (6, 1, 2/3) ∆ * L (6, 3, −1/3) ∆ * L (6, 3, −1/3) + h.c. . (7.128) From this, the effective Lagrangian describing the six-fermion vertex (d = 9 operators) that corresponds to n − n oscillation can be written down, here k, l, m, n, p, q are the generation indices.
Again on the dimensional ground, the n − n oscillation transition time can be computed as (7.130) The present limit on this transition time is constraint by the matter disintegration, which is τ n−n ≥ 2 × 10 8 sec. [65]. A slightly weaker bound but with less uncertainty is obtained from the free neutron oscillation search, τ n−n ≥ 10 8 sec. [66]. By taking τ n−n = 10 8 sec. one can find the lower bound on the scale v R ∼ 3.2 × 10 5 GeV (like before M = v R = 10 3 TeV is assumed).
So if the scalar fields responsible for this oscillation have masses somewhat smaller than 10 3 TeV, which is certainly possible with some choice of the quartic couplings, n − n transition time can be within the interesting observable range.

Conclusion
In this work, we have presented a minimal renormalizable model based on the Pati-Salam gauge group, SU (2) L × SU (2) R × SU (4) C that unifies quark and leptons by treating leptons as the fourth color. We extend the symmetry of our theory by imposing a global U (1) P Q Peccei-Quinn symmetry, that automatically solves the strong CP problem and provides axion as a dark matter candidate. The minimal Higgs sector consists of (2,2,1), (2,2,15) and (1,3,10) multiplets under the Pati-Salam group, is fixed by the requirement of consistent symmetry breaking pattern and by the phenomenological constraint of reproducing the observed fermion spectrum. This theory is highly predictive and with only 14 parameters in the Yukawa sector a good fit to the charged fermion masses and mixings are obtained. The origin of the baryon asymmetry of the universe is linked to the seesaw mechanism that is also responsible for the observed neutrino oscillations. Detail search of the parameter space for successful generation of matter-antimatter asymmetry is carried out that makes connection between the low scale and high scale parameters of the theory. Comprehensive analysis in the Higgs sector is carried out by constructing the complete Higgs potential with the minimal scalar content and then by computing the mass spectrum of the Higgs fields. We also studied two baryon number violating processes present in our theory that are nucleon decay and  C Relation between baryon asymmetric parameters and m β and m ββ for the case with m 1 = 1 and 2 meV In this appendix we present the permitted region for m β and m ββ to have successful leptogenesis in Fig. 9. m 1 = 1 meV (Flavored) m 1 = 2 meV (Flavored) m 1 = 2 meV (Unflavored) Figure 9: The correspondence between the baryon asymmetry and m β,ββ are plotted, where m β = i |U ν ei | 2 m i is the effective mass parameter for the beta-decay and m ββ = | i U 2 ν ei m i | is the effective mass parameter for neutrinoless double beta decay. Color code is the same as in Fig. 2.

D The correlation between the CP-violating Dirac phase δ and θ 13
The correlations between the phase δ and the angle θ 13 permitted by reproducing correct baryon asymmetry is presented in Fig. 10.