Abstract
Leptonic CP violating phase in the light neutrino sector and leptogenesis via present matter-antimatter asymmetry of the Universe entails each other. Probing CP violation in light neutrino oscillation is one of the challenging tasks today. The reactor mixing anglemeasured in reactor experiments, LBL, and DUNE with high precision in neutrino experiments indicates towards the vast dimensions of scope to detect . The correlation between leptonic Dirac CPV phase , reactor mixing angle , lightest neutrino mass , and matter-antimatter asymmetry of the Universe within the framework of symmetry breaking assuming the type I seesaw dominance is extensively studied here. Here, a SO(10) GUT model with flavor symmetry is considered. In this work, the idea is to link baryogenesis through leptogenesis and the hint of CP violation in the neutrino oscillation data to a breaking of the mu-tau symmetry. Small tiny breaking of the symmetry allows a large Dirac CP violating phase in neutrino oscillation which in turn is characterized by awareness of measured value of and to provide a hint towards a better understanding of the experimentally observed near-maximal value ofmixing angle. Precise breaking of the symmetry is achieved by adding a 120-plet Higgs to the-dimensional representation of Higgs. The estimated three-dimensional density parameter space of the lightest neutrino mass , , and reactor mixing angle is constrained here for the requirement of producing the observed value of baryon asymmetry of the Universe through the mechanism of leptogenesis. Carrying out numerical analysis, the allowed parameter space of , , and is found out which can produce the observed baryon to photon density ratio of the Universe.
1. Introduction
In 1950, Bruno Pontecorvo for the first time emphasized the idea of neutrino oscillations which resembled oscillations. In neutrino oscillations, a neutrino originated with a definite flavor, () oscillates to a distinct contrasting lepton flavor. Neutrino oscillation reveals that each of the three states of neutrino in flavor basis is a superposition of three mass eigen states () [1]. Neutrinos are massive, and they mix with each other. The massive neutrinos are formed in their gauge eigen states which are linked to their mass eigen states . Gauge eigen states participate in gauge interactions as where , is the neutrino of distinct mass . is parameterised as where , and [2] are the solar, atmospheric, and reactor angles according to the global fits, respectively. The Majorana phases and dwell in , where
is known as the Pontecorvo-Maki-Nakagawa-Sakata matrix [3]. Since a of a given flavor is a mixed state of at least three with distinct masses, this three-generation mixing could result into the flavor mixing mass matrix or PMNS matrix possessing an irreducible imaginary component. This irreducible imaginary component is responsible for CP asymmetry. CP violation interchanges every particle into its antiparticle. in PMNS matrix can induce CP violation. CP asymmetry can be observed in neutrino oscillations. phase measures the amount of asymmetries between lepton oscillations and antilepton oscillations. Neutrinos are massive, and they mix with each other. This may be a source of CP violation if . The amount of violation phase in this case is estimated by the Jarkslog invariant [4]. when . In leptogenesis, lepton-antilepton asymmetry is explained if there are complex imaginary irreducible terms in the Yukawa couplings of lepton mass matrices. The lepton number generation of the early Universe can be estimated by the complex CPV phase term in the fermion mass matrices. The paper gives the impression that the neutrino Dirac CP phase is automatically connected to the baryon asymmetry of the universe; in some special cases, e.g., in flavored leptogenesis, such a connection exist. Also, softly broken mu-tau symmetry provides also another way (in addition to flavored leptogenesis) to connect the two important observables (Dirac CP phase and baryon asymmetry). The ongoing T2K experiment has reported that CP violating phase,, which excludes the value [5] at the 2 confidence interval for either of the mass orderings, normal ordering, or inverted ordering. The value of Dirac CPV phase, is preferred in [6]. The neutrino mass matrix is invariant under exchange symmetry, in a basis where the charged leptons are mass eigen states. Under the exchange symmetry, the 2-3 mixing is maximal, i.e, and the 1-3 mixing is zero, i.e, . The deviation of from the maximal angle , the explanation of reactor angle , and the existence of CP violating phase necessitate the spontaneous breaking of the exchange symmetry in the neutrino sector. The measurement of the neutrino mixing angle in concurrence with a measurement of the departure from maximality of the atmospheric mixing angle can be a very strong way to probe any possible symmetry present in the neutrino mass matrix. Different types of plausible mechanism of breaking of symmetry and the possible types of resultant gauge symmetry for generation of nonzero are introduced in [7].
symmetry is an important idea in neutrino physics in view of the near-maximal atmospheric mixing angle. The original papers which introduced the concept of symmetry are cited in [8–10].
Here, in this work, an explicit form of the Dirac neutrino mass matrix in broken [11] symmetry framework in type I seesaw mechanism is used in our calculation for generating baryon asymmetry of the Universe via leptogenesis. This scenario is characterized by small divergence of from the maximal angle , which is consistent with a liberal size of and a large phase in the neutrino sector. The renormalisable Dirac neutrino Yukawa couplings of the Dirac mass matrices are determined from the fermion Yukawa couplings to the 10, , and 120 dimensional fields of Higgs multiplet in the SO(10) group. Higgs field under the 10 and representations is symmetrical under the generalized symmetry, while the 120-dimensional representation changes sign. This spontaneously breaks the invariant symmetry, which in turn allows a generalized phase in the PMNS matrix.
Here, we made an effort for correlating or constraining the values of phase, nonzero reactor angle , and the lightest neutrino mass space for both the hierarchies in the context of leptogenesis and current ratio of baryon to photon density of the Universe. Both CPV phase and reactor angle have good vibes between each other. A precise value of plays an imperative role in its CP violation phase measurements. On the basis of this fact, nonzero values of are predicted here in consistency with the phase. Taking into account constraints from the global fit values of oscillation parameters and cosmology, a density plot of the favourable values of the phase, lightest neutrino mass, and is being initiated, which is compatible with the contraints on the sum of the absolute neutrino masses, eV from CMB, Planck 2015 data () [12]. Constraints from the leptonic asymmetry of the Universe are also considered for further restricting the phase space and lightest neutrino mass. The leptonic CP asymmetry is being deliberated via leptogenesis in terms of baryon density to photon density ratio of the Universe accessible as [13]. We also calculate the effective mass spectrum for neutrinoless double beta decay, decay given by for favourable values of the phase and lightest mass explored here in this work. In this paper, we apply the broken symmetry to the Dirac neutrino Yukawa couplings in type I seesaw mechanism in the SO(10) model in predicting favourable values of the phase, lightest neutrino mass, and . We then scan free parameters in these models and search for the allowed region in which the neutrino oscillation data can be fitted. For the allowed parameter sets, we show the predictions of observables like the phase, lightest neutrino mass, and in the neutrino sector. Finally, we show our predictions for the effective mass spectrum for neutrinoless double beta, , decay for favourable values of the phase.
This paper deals with an important aspect of neutrino physics, i.e., its CP violating Dirac phase and its possible connection to the matter-antimatter asymmetry of the universe. In this work, we have used user-defined Dirac Neutrino Yukawa couplings [11] for the Yukawa interactions associated with the broken symmetry model for the generation of nonzero reactor mixing angle and leptonic CP phase in type I seesaw mechanism in the light of leptogenesis; there can be a transformation of the lepton asymmetry into a baryon asymmetry by nonperturbative violating (sphaleron, Sakharov conditions) processes as discussed in [6]. A small explicit breaking of symmetry [11] by hand with the specific numerical value in Equation (30) inherits the property of generating nonzero CP violation in matrices and phase and results in being nonzero. Here, we consider the type I seesaw as the main donor to neutrino mass. We also take into account both inverted and normal ordering of neutrino mass spectrum as well as two different types of the lightest neutrino mass to visualise the results of hierarchical mass spectrum. In the case of normal ordering of masses, the dependance of leptonic CPV phase, , on the lightest mass is predicted in Figures 1–3 (in the light of recent ratio of the baryon to photon density bounds, ). The favoured values of phase is found to lie between for best fit values of corresponding to w/o SK-ATM [14] (in the light of recent ratio of the baryon to photon density bounds ) as a result of contribution of type I seesaw mechanism to neutrino mass matrix. The favoured values of the lightest mass, , in this case come out to be eV. In case of inverted hierarchy, the variation of phase is found to be very intense with the best fit values of corresponding to w/o SK-ATM [14]. Values of the phase favoured are (in the light of recent ratio of the baryon to photon density bounds ) as is evident from Figure 4. The allowed spectrum of the lightest mass is , eV.
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The paper is organized as follows. In Section 2, we introduce our broken symmetry models. In Section 3, we perform parameter scan to fit neutrino oscillation data and provide some information in predicting observables like the phase in the neutrino sector. Also, we show our predictions for the effective mass spectrum for neutrinoless double beta, , decay for favourable values of the phase. Section 4 is our results and conclusions for the Yukawa interactions associated with broken symmetry model discussed here.
2. Broken Symmetry with Type I Seesaw Mechanism
When symmetry is unified with grand unification subsequently, a more general symmetry results that interchanges the second and third generations of fermions. This generalized symmetry is aimed at explaining why the Cabibbo angle is greater than the other two angles, and a soft explicit breaking of this symmetry leads to correct explanation of the quark mixing angles and masses. In this work, we consider a model based on the SO(10) . First, symmetry represents the generalized symmetry. Second, symmetry indicates parity [15] transformation between two components of the 16-dimensional representation of the field acting as (4, 2, 1) and (, 1, 2) under the Pati-Salam group decomposition of SO(10). Sixteen-dimensional representation of fermions gets their masses from coupling to three Higgs multiplets which transform as 10, and 120 representations under SO(10). The SO(10) breaking is fulfilled with 210 plets. In a supersymmetric realm, an additional 126 plets of Higgs conserve the supersymmetry at the GUT breaking scale. These Higgs multiplets have in them altogether six doublets which match with the quantum numbers of the minimal supersymmetric standard model (MSSM) field and six other quantum numbers which complement the quantum field numbers of . Two linear superpositions of these Higgs doublets stay light and play the role of the and fields. This is controlled by the fine-tuning conditions [16, 17]. After this fine-tuning results, the fermion masses can be written as where represents the up and down quarks and the charged leptons, respectively. The mass matrices representing different scalar representations to obtain the flavor texture of interest in the paper appearing in the above Lagrangian can be [18, 19] written as
Here, is the Dirac mass matrix. In the basis where , the Majorana mass matrix for the left(right)-handed neutrinos, gets benefaction only from the vacuum expectation value (vev) of the Higgs field, gauge coupling unification in the minimal model needs that the vacuum expectation value (vev) contributing to is adjacent to the GUT scale. , and are the dimensionless parameters which are determined by the Clebsch-Gordan coefficients, ratios of vevs, and mixing among the Higgs fields [18]. The matrices , and rise from the fermion couplings to the 10, , and 120 Higgs fields, respectively. Normally, are complex (anti) symmetric matrices. Nonetheless, generalized parity symmetry keeps real. Moreover, when all vevs and thus , and are real, then the Dirac masses defined in Equation (10) are Hermitian and and are real. The 10- and -dimensional Higgs field representations are invariant under the generalized symmetry while the 120-dimensional representation changes sign. This allows spontaneous explicit breaking of the symmetry.
Let be any complex symmetric (anti) matrices in general, which are a measure of the fermion Yukawa couplings to the 10, , and 120 Higgs field, respectively. Here,
Similarly,
Similarly,
The matrices , , and originate from the fermion couplings to the 10, 126, and 120 fields, respectively. are complex (anti) symmetric matrices in general. However, generalized parity makes them real. In addition, if all vevs and hence , , , , , , and are real, then all the Dirac masses in Equations (7), (8), (9), (10), (11), and (12) are Hermitian and and are real. Here, the Higgs field in the 10 and 126 representations are symmetric and invariant under the generalized symmetry while the 120-dimensional representation changes sign. This assumption in turn allows spontaneous breaking of the symmetry. In the symmetric mass matrices in Equations (13), (14), and (15), Higgs field 120 induces antisymmetry of the Yukawa matrix. If that matrices as well as the 10 and 126 couplings are complex, it is well known that if there are no extra symmetries, they will induce a Dirac phase. Explicit breaking of symmetry leads to nonzero . Small explicit tiny breaking of the symmetry allows a large Dirac CP violating phase in neutrino oscillation. Fermion mass spectrum can be explained by Hermitian mass matrices derived from the renormalizable Yukawa couplings of the 16 plets of fermions with the Higgs fields transforming as 10, , and 120 representations of the SO(10) group. The symmetry upon spontaneously broken down through the 120 plets leads to nonzero reactor angle , which in turn induces leptonic dirac CPV phase in the matrix. Tiny explicit breaking of symmetry leads to nonzero . This scenario implies a generalized CP invariance of the fermion mass matrices and vanishing CP violating phases if the Yukawa couplings are symmetric under the symmetry. Small tiny breaking of the symmetry allows a large Dirac CP violating phase in neutrino oscillation. Explicit breaking of the symmetry by hand as evident from Equation (30) provides a nice spectrum of all the fermion masses and mixing and leads to nonzero , which in turn induces the phase and allows a large required Dirac CP violating phase in neutrino oscillation. Detailed fits to the fermion spectrum are presented in several scenarios in [11].
, , and are all real. symmetry is invariant under the exchange of second- and third-generation fermions. When symmetry is added with SO(10) grand unified theory, then a general symmetry results which satisfies where
The implicit neutrino mass matrix can be written as
are inversely proportional to the vev of the RH triplet component in dimensional Higgs field.
The Lagrangian of the type I seesaw model is [20, 21]
Here, is the complex Yukawa coupling matrix; is the standard model left-handed lepton doublet of flavor , when ; and is the hypercharge-conjugated Higgs doublet, .
The Lagrangian describes the scenario of generation of masses via Higgs mechanism. Electroweak symmetry breaking process allows neutral part of the Higgs field to acquire a VEV, , and so that the left-handed and right-handed neutrinos form massive Dirac fermions.
In Equation (6), is a symmetric matrix of right-handed violating Majorana masses, where is real and diagonal. Here, right-handed neutrino masses are larger than the electroweak scale. The masses are then suppressed by right-handed neutrino Yukawa couplings and also by
In type I seesaw, the baryon asymmetry of the Universe (BAU) occurs via leptogenesis mechanism via out of equilibrium decay of heavy RH Majorana neutrinos in the early Universe via electroweak sphaleron processes [22]. The resulting Majorana mass matrix of light SM neutrinos is where is the Dirac mass matrix. Equation (8) shows that in the type I seesaw mechanism, SM masses are suppressed by the combination of small Yukawa couplings and large RH masses. Neutrino mass matrix on diagonalision gives two eigen values—light neutrino ~ and a heavy neutrino state ~ . This is known as type I seesaw mechanism.
In SO(10), heavy right-handed Majorana neutrino couples to the left-handed via Dirac mass matrix . Out of the decay of the lightest of the RH Majorana neutrinos, , i.e., , will contribute to CP asymmetry [6, 23] (for leptogenesis), i.e., and leptogenesis [24–30].
At the end of inflation [31], a certain number density of right-handed neutrinos, , was created, which is linked to the present cosmological scenario. Right-handed neutrinos decayed, with a decay rate that reads, at tree level,
It is convenient to work in a basis of right-handed neutrinos, where the RH mass matrix is diagonal, the type I seesaw mechanism contribution to is given by decay of , or the CP violating parameter is given as where where means decay rate of heavy Majorana RH of mass to a lepton and Higgs. In the electroweak sphaleron process, asymmetries produced by the out of equilibrium decay of and get washed out by lepton number violating interactions after or decay. In lepton number violating interactions, decays, inverse decays, and scatterings must be out of equilibrium when the right-handed neutrinos decay. In the basis where the RH mass matrix is diagonal, the type I seesaw mehanism contribution to is given by [32] where is the Higg’s vev. is a complex unitary orthogonal matrix where is parameterized as [33] , where is the Dirac neutrino Yukawa couplings. To reproduce the physical, low-energy, parameters, i.e., the light neutrino masses and mixing angles and CP phases , we have taken the most general Dirac neutrino mass matrix in broken symmetry framework as [11]
The numerical values of the above Dirac Neutrino Yukawa coupling matrix come from the best fit values of the parameters of , and Equation (10) by performing a fit using the best fit solutions for fermion masses and mixing obtained assuming the type I seesaw dominance in [11]. The Dirac mass matrix is expressed in megaelectron volt units.
In the flavor basis, where the charged-lepton Yukawa matrix and gauge interactions are flavor-diagonal,
In terms of user-defined Dirac neutrino mass matrices, [11]
is the PMNS matrix and is the RH neutrino Majorana scale. We can always choose to work in a basis of right neutrinos where is diagonal where . Equation (12) expresses in terms of both the solar () and atmospheric () mass squared differences. Equation (12) also reveals that CP asymmetry is linked to the Dirac CPV phase. Here, we utilise this fact to generate the allowed region of the phase in the context of leptogenesis. As has been discussed in [32], the lepton-antilepton asymmetry gets connected to both the solar and the atmospheric mass squared differences. The transformation of the lepton asymmetry into a baryon asymmetry by nonperturbative violating (sphaleron) processes is discussed in [6].
Neutrino masses and mixings are connected with the atmospheric and solar neutrino fluxes; this is suitable to explain flavor changing neutral current processes and FCNC processes, like processes. In supersymmetric theories like cMSSM, NUHM, NUGM, and NUSM where the origin of the masses is via the seesaw mechanism, it can be shown that the prediction for , , and is in general larger than the experimental upper MEG bound [34, 35]. Also, some studies on decays of-flavored hadrons in the context of cMSSM/mSUGRA models is being done in [36].
A small explicit breaking of symmetry is put by hand, by inheriting the property in Equation (13).
This introduces CP violation in PMNS matrices and results in being nonzero. Although an explicit breaking of the symmetry is used in [11], the magnitude of the breaking needed in order to get a large CP violating phase is very minute. This tiny amount of breaking which is used here for generating nonzero CP asymmetry producing a measurable CP violating phase via Dirac neutrino Yukawa couplings used from [11] is fixed to a well specific numerical value, which in turn allows one to replicate mixing angles precisely.
The mass matrices defined in Equations (7), (8), (9), (10), (11), and (12) in the model are symmetric under CP invariance if Yukawa couplings are taken to be symmetric. Small explicit breaking of this symmetry defined in Equation (30) is enough to produce the required CP violating phase.
The Higgs field in the 10 and 126 representations are symmetric and invariant under the generalized symmetry while the 120-dimensional representation changes sign. This assumption in turn allows spontaneous breaking of the symmetry. 120 Higgs vev only contributes to off-diagonal elements. Also, one can break the exact symmetry explicitly through the use of Equation (30) which also involves both diagonal and off-diagonal elements. are any complex symmetric (anti) matrices in general, which are a measure of the fermion Yukawa couplings to the 10, , and 120 Higgs field, respectively. So the 22 entries and 33 entries in symmetric matrix can be broken down by explicitly breaking the symmetry by hand. A required amount of the CP violating phase is generated by explicitly breaking the symmetry. This assumption by using Equation (30) leads to and . All these remarks enable one to use Dirac neutrino Yukawa coupling mass matrices reproduced by the explicit use of broken symmetry embedded in Equation (30).
The exchange symmetry in the neutrino mass matrix restricts the 2-3 and 1-3 neutrino mixing angles as and . We find that the symmetry breaking prefers a large CP violation to realize the observed value of and also keeping nearly maximal. We also propose a concrete model to break the exchange symmetry spontaneously, and its breaking is mediated by a Yukawa coupling to the Higgs field transforming in 120 of SO(10). As a result of the explicit breaking in the neutrino mass matrix, a large Dirac CP phase is preferable. As a consequence, nonzero is generated opening up the possibility of having Dirac CP violation in the lepton sector. The latter may be responsible for generation of the observed baryon asymmetry of the Universe (BAU).
The deviations of from maximal and the explanation of nonzero are major predictions, goals, and motivation for breaking the symmetry.
The input parameters defined here are , , , , and ; Equations (7), (8), (9), (10), (11), and (12); the real elements of the matrices , , and ; Equations (13), (14), and (15); and the overall scales . There can be an overall rotation on . This equals to a choice of initial basis for the 16 plets of fermions. We can then set . This is done with a specific choice . Here, symbolises rotation in the plane by an angle and
This rotation equals to reformulation of elements of and which still retain the same form as in Equations (13), (14), and (15). With the option , there are 15 input parameters in the case of type I seesaw mechanism. These input parameters when well organized set up 12 fermion masses and six mixing angles. The exact symmetric and are not able to provoke CP violation. CP violation is introduced by adding a small breaking difference between the 22 and 33 elements in as seen in Equation (30).
We concentrated here in building comprehensive fits to fermion masses and mixings rather than taking into account the whole parameter space of the theory provided by the Yukawa couplings and basic parameters in the superpotential Lagrangian. Parameters in fermion mass matrices are a measure of the strengths of the light Higgs components in different SO(10) Higgs representations.
Many models make use of complex vev to obtain breaking. In our approach, the breaking is by hand; i.e., we introduce small explicit breaking of symmetry in . The models which use complex vev have 20 free parameters compared to 15 used here.
The explicit breaking of the symmetry is methodologically natural in the supersymmetric circumstances. However, one can achieve such breaking by introducing an additional 10 plets of the Higgs field which changes sign under the symmetry. Integrated benefaction of these two 10 plets would then provide an explicitly noninvariant .
Also, if we abide by the best fit values of leptonic CP phase discussed in the literature [14, 37], then our scenario of explicit breaking of the symmetry by hand, evident from Equation (30), leads to for inverted ordering of masses corresponding to eV which exactly matches with the best fit values of with w/o SK-ATM [14].
The novelty of this work lies in the successful explicit breaking of the symmetry within the SO(10) framework in order to obtain a constrained picture of fermion masses. This framework provides a user-defined neutrino Yukawa coupling [11] for the Yukawa interactions associated with the broken symmetry model for the generation of nonzero reactor mixing angle and leptonic CP phase in type I seesaw mechanism; in the light of leptogenesis, there can be a transformation of the lepton asymmetry into a baryon asymmetry by nonperturbative violating (sphaleron, Sakharov conditions) processes as discussed in [6]. A small explicit breaking of the symmetry [11] inherits the property of generating nonzero CP violation in matrices and phase and results in being nonzero. Here, we consider the type I seesaw as the main donor to neutrino mass. We also take into account both inverted and normal ordering of neutrino mass spectrum as well as two different types of the lightest neutrino mass to visualise the results of hierarchical mass spectrum. This scenario is characterized by the predictions that in case of normal ordering of masses, the favoured values of the phase in the light of recent ratio of the baryon to photon density bounds, , are found to lie in the range for best fit values of corresponding to w/o SK-ATM [14] (in the light of recent ratio of the baryon to photon density bounds ) as a result of contribution of the type I seesaw mechanism to neutrino mass matrix. The favoured values of the lightest mass, , in this case come out to be eV. In case of inverted hierarchy, the variation of the phase is found to be very intense with best fit values of corresponding to w/o SK-ATM [14]. Values of phase favoured are (in the light of recent ratio of the baryon to photon density bounds ). The allowed spectrum of the lightest mass is , eV.
We also plot the allowed values of eV for neutrinoless double beta decay and the Jarkslog invariant, , for normal ordering of masses. Prediction of future leptonic CP violation experiments should be able to rule out or take into account some of the results discussed in this work. If we abide by the best fit values of leptonic CP phase discussed in the literature [14, 37], then our scenario, , for inverted ordering of masses corresponding to eV exactly matches with the best fit values of with w/o SK-ATM [14].
The model here is motivated in the sense that it provides a specific form of Dirac neutrino Yukawa coupling matrix in the constraint of explicit breaking of the symmetry as is evident from the relation between the and the elements of the matrix. If we use this specific form of Dirac neutrino Yukawa coupling matrix to calculate baryon asymmetry of the Universe, then in the light of recent ratio of the baryon to photon density bounds as a result of contribution of the type I seesaw mechanism to neutrino mass matrix, we get a range of allowed values of nonzero and large phase.
The motivation of this work is to relate baryogenesis through leptogenesis and the hint of CP violation in the neutrino oscillation data to a breaking of the symmetry. Small explicit tiny breaking of the symmetry allows a large Dirac CP violating phase in neutrino oscillation which in turn is characterized by awareness of measured value of nonzero and to provide a hint towards a better understanding of the experimentally observed near-maximal value of mixing angle . Precise breaking of the symmetry can be achieved by adding a 120-plet Higgs to the -dimensional representation of Higgs.
Here in this work, the estimated three-dimensional density parameter space of the lightest neutrino mass , , and reactor mixing angle is constrained here for the requirement of producing the observed value of baryon asymmetry of the Universe through the mechanism of leptogenesis. Carrying out numerical analysis, the allowed parameter space of , , and is found out which can produce the observed baryon to photon density ratio of the Universe, the details of which are discussed below.
3. Numerical Analysis
In this section, numerical analysis has been carried out. Firstly, the free parameter called the lightest mass, the oscillation parameters like reactor angle , Dirac CPV phase, , Majorana phases and in broken symmetry, and type I seesaw model are scanned to search for dependance of the phase on , (), Jarkslog invariant , and effective mass for decay in case of normal ordering (inverted ordering) in the context of producing correct baryon asyymetry of the Universe [13]. We use the best fit values of oscillation parameters. The two mass square differences and are embedded in neutrino mixing matrix so we are left out with lightest mass as the only free parameter in this model. In the charged lepton basis, we parameterize the PMNS matrix , by diagonalizing the neutrino mass matrix in terms of three mixing angles , one CP violating Dirac CPV phase , and two Majorana phases ( and ) as follows:
The Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix is UP, where U is where , and [14] in case of normal hierarchy (inverted hierarchy) are the solar, atmospheric, and reactor angles, respectively. The Majorana phases reside in P, where
We have taken complex and orthogonal matrix , in terms of user-defined Dirac neutrino Yukawa couplings defined in Equation (13) in order to produce correct baryon asymmetry of the Universe.
For the normally ordered light masses, we have with , , and as is evident from the oscillation data [14], being the lightest of three masses. For the inverted ordered light masses, we have with being the lightest of three masses. Here, we take GeV. For normal ordering, the choices of the lightest neutrino mass is whereas for inverted ordering, the choice of the lightest neutrino mass is . This sustainable allowance of signifies a neutrino mass spectrum where the sum of absolute neutrino masses lies below the cosmological upper bound, [12]. Next, random scan of the mixing matrix parameter space for NH and IH in order to produce correct the baryon asymmetry of the Universe is performed in the following 3 range of with respect to the tabulated map of the Super-Kamiokande analysis of the data within [14]: (i)(ii) for normal ordering(iii) for inverted ordering(iv) for normal ordering for tabulated w/o SK-ATM [14](v) for normal ordering for tabulated w/o SK-ATM [14](vi) for normal ordering for tabulated w/o SK-ATM [14]
While doing parameter scan, we find favoured values of the lightest mass and dirac CPV phase , for producing correct baryon asymmetry of the Universe, .
The lepton flavor effects are significant if the lightest right-handed Majorana neutrino mass is below GeV. Here, GeV. In the type I seesaw mechanism, one can always find the right-handed neutrino mass matrix as where is the Dirac mass matrix. We consider a Dirac neutrino mass matrix defined in Equation (13). Here, when we fix , the remaining free parameter in the neutrino sector within our broken framework is the leptonic CPV phase . When we vary the CPV phase , we compute the favoured regions of . The variations of leptonic CPV phase with , , , and for as shown in the figures discussed here.
For global fit values of oscillation parameters, we compute the Jarlskog invariant, , given by PMNS matrix elements . We also compute the Jarkslog invariant for allowed values of the phase, , and lightest mass explored here in this work for both normal ordering and inverted ordering.
We also calculate the favourable space of the effective mass for decay for favourable values of the phase, , and lightest mass given by
The colour coding in the different figures imply the values of baryon asymmetry of the Universe in the allowed range, .
In Figure 1, we have presented the predictions in the broken symmetry model for normal ordering. Panel (a) conveys the predicted favoured values of the plane for the best fit values of with w/o SK-ATM [14] (allowed by updated values of correct baryon asymmetry of the Universe) as a result of contribution of the type I seesaw mechanism to neutrino mass matrix. The allowed spectrum of the lightest mass here lies in the range, 0.09 eV–0.095 eV corresponding to favoured values of reactor angle, , in the interval, in light of the correct baryon to photon density bounds . Panel (b) manifests itself in the predicted favoured value plane, for the best fit value of with [14]. The favoured value of is around in the light of the correct baryon asymmetry of the Universe which can be manifested from the colour coding in the figure.
We have shown the contour plot for predicted favoured values of plane for the best fit values of of w/o SK-ATM [14] (as allowed by updated values of correct baryon asymmetry of the Universe) as a result of contribution of the type I seesaw mechanism to neutrino mass matrix in Figure 5. The colour coding in the contour plot also shows that for to lie in the interval, , the allowed lightest neutrino mass must lie around 0.095 eV.
In Figure 2, we have speculated the predictions in the broken symmetry model for normal ordering. The (a) presents the three-dimensional plot of preferred values of the plane for the best fit values of of w/o SK-ATM [14] (in the light of recent ratio of the baryon to photon density bounds ) as a result of contribution of the type I seesaw mechanism to neutrino mass matrix. The preferred value of the leptonic CPV phase came out to be around 304°-307°. Similarly, the panel (b) communicates the three-dimensional plot of favourable values of the plane, for the best fit value of of [14] (in the light of recent ratio of the baryon to photon density bounds ).
In Figure 3, we have depicted the predictions in the broken symmetry model for normal ordering: panel (a) favours preferred values of for the best fit values of of w/o SK-ATM [14] (in the light of recent ratio of the baryon to photon density bounds ) as a result of contribution of the type I seesaw mechanism to neutrino mass matrix. The preferred value of the leptonic CPV phase came out to be around 304°-307° as is obvious from the figure. Similarly, panel (b) shows the favoured values of the lightest neutrino mass, , for the best fit value of of [14] (in the light of recent ratio of the baryon to photon density bounds ). The favoured values of is around 0.09 eV eV.
Figure 6 reveals the predictions in the broken symmetry model for normal ordering: panel (b) shows the preferred three-dimensional regions of the plane for the best fit values of of w/o SK-ATM [14]. Panel (a) presents allowed two-dimensional space of the (, ) plane for the best fit values of of w/o SK-ATM [14].
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In Figure 7, we present the predictions in the broken symmetry model for normal ordering. Panel (b) favours allowed two-dimensional space of the plane for an absolute range of .
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In Figure 8, we have speculated the predictions in the broken symmetry model for normal ordering. Panel (b) conveys preferred three-dimensional regions of the plane for favoured values of (in the light of recent ratio of the baryon to photon density bounds ) for best fit values of of w/o SK-ATM [14]. Panel (a) presents allowed two-dimensional space of the (, ) plane for favoured values of (in the light of recent ratio of the baryon to photon density bounds ) for the best fit values of of w/o SK-ATM [14] (in the light of recent ratio of the baryon to photon density bounds ).
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In Figure 9, we have shown predictions in the broken symmetry model for normal ordering. Panel (b) presents preferred three-dimensional regions of the (, , ) plane for the best fit values of of w/o SK-ATM [14]. The left panel communicates the allowed two-dimensional space of the (, ) plane for the best fit values of of w/o SK-ATM [14].
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Figure 10 depicts the predictions in the broken symmetry model for normal ordering. The variation of as a function of , for normal ordering for tabulated [14] is speculated.
In Figure 11, we have presented predictions in the broken symmetry model for normal ordering. Panel (a) depicts three-dimensional space of for [eV], decay for favoured values of , , and (in the light of recent ratio of the baryon to photon density bounds, ). Panel (b) demonstrated three-dimensional space of () for [eV], decay for values of , , and in the given three range, corresponding to and w/o SK-ATM [14].
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In Figure 12, we have introduced predictions in the broken symmetry model for normal ordering. The figure depicts the density plot of for [eV], decay for favoured values of , , and (in the light of recent ratio of the baryon to photon density bounds, )
In Figure 13, we have presented predictions in the broken symmetry model for normal ordering. Panel (a) depicts predicted three-dimensional space of for [eV], decay for favoured values of , (in the light of recent ratio of the baryon to photon density bounds, ). Panel (b) depicts predicted three-dimensional space of for [eV], decay for favoured values of , for values of in the given three range, corresponding to w/o SK-ATM [14].
In Figure 14, we have conveyed the three-dimensional space of for [eV], decay for favoured values of , (in the light of recent ratio of the baryon to photon density bounds, ) for the lightest mass eV.
In Figure 15, we have shown the predictions in the broken symmetry model for inverted ordering. Panel (a) predicts favoured values of (in the light of recent ratio of the baryon to photon density bounds, ) for the lightest mass eV, as a result of contribution of the type I seesaw mechanism to neutrino mass matrix. Similarly, panel (b) shows predicted allowed three-dimensional space of the plane for allowed regions of Jarkslog invariant, , values for the best fit value of of [14] as a result of contribution of the type I seesaw mechanism to neutrino mass matrix.
(a)
(b)
In Figure 4, we have shown predictions in the broken symmetry model for inverted ordering. The three-dimensional plot for predicted favoured values of plane for the best fit values of of w/o SK-ATM [14] (in the light of recent ratio of the baryon to photon density bounds, ) as a result of contribution of the type I seesaw mechanism to neutrino mass matrix is presented. The favoured values of the phase predited here are and in the light of recent ratio of baryon to photon density bounds, . The value of the lightest mass depicted here from the figure is around 0.005 eV.
In Figure 16, panel (a) presents the density plot of predicted favoured values of the plane for the best fit values of of w/o SK-ATM [14] (allowed by updated values of correct baryon asymmetry of the Universe) as a result of contribution of the type I seesaw mechanism to neutrino mass matrix. The favoured values of the lightest mass depicted here are eV eV for producing correct baryon asymmetry of the Universe. Also, the favoured value reactor angle, , lies in the range 8°–9°. Similarly, in panel (b), we have shown the three-dimensional plot for predicted favoured values of the plane for the lightest mass, (in the light of recent ratio of the baryon to photon density bounds, ) as a result of contribution of the type I seesaw mechanism to neutrino mass matrix.
(a)
(b)
We also plot the allowed values of eV for neutrinoless double beta decay and the Jarkslog invariant, , in Figures 15 and 17–20 for inverted ordering of masses.
(a)
(b)
(a)
(b)
In Figure 17, we have shown the predictions in the broken symmetry model for inverted ordering. Panel (a) shows the density plot of predicted favoured values of the plane for the best fit values of of w/o SK-ATM [14] (allowed by updated values of correct baryon asymmetry of the Universe) as a result of contribution of the type I seesaw mechanism to neutrino mass matrix. Similarly, panel (b) depicts the predicted two-dimensional space of for [eV], decay, for the best fit values of of w/o SK-ATM [14]. The favoured values of the lightest mass depicted here are eV–0.047 eV for producing correct baryon asymmetry of the Universe. Also, the favoured value reactor angle, , lies in the range 8°–9°. In Figure 18, we show the predicted allowed three-dimensional space of the plane for allowed regions of Jarkslog invariant, , values for the best fit value of of [14] as a result of contribution of the type I seesaw mechanism to neutrino mass matrix for inverted ordering. Figure 19 depicts in panel (a) the predicted three-dimensional space of for values of eV and in the given 3 range, corresponding to , and the best fit values of corresponding to w/o SK-ATM [14]. The favoured values of the lightest mass depicted here are eV–0.06 eV corresponding to a value of eV [38]. The favoured values of the leptonic CPV phase, , predicted here are 235°–237° consistent with eV. Panel (b) shows the predicted three-dimensional space of for values of the lightest mass, , , and in the given 3 range, corresponding to and w/o SK-ATM [14] for inverted ordering. The favoured values of the reactor angle predicted here are corresponding to a value of eV [38]. The favoured values of the leptonic CPV phase, , predicted here are 222° consistent with, limits on, , eV [38]. In Figure 20, we have depicted the predicted density plot of for values of eV and in the given 3 range, corresponding to , and the best fit values of corresponding to w/o SK-ATM [14] for inverted ordering of hierarchical masses.
4. Results and Conclusion
Fermion mass spectrum can be explained by Hermitian mass matrices derived from the renormalizable Yukawa couplings of the 16 plets of fermions with the Higgs fields transforming as 10, , and 120 representations of the SO(10) group. The symmetry upon spontaneously broken down through the 120 plets leads to nonzero reactor angle , which in turn induces the leptonic dirac CPV phase, , in the matrix. This scenario implies a generalized CP invariance of the fermion mass matrices and vanishing CP violating phases if the Yukawa couplings are symmetric under the symmetry. Small explicit tiny breaking of the symmetry (evident from Equation (30)) allows a large Dirac CP violating phase in neutrino oscillation. Explicit breaking of the symmetry by hand as seen from Equation (30) provides a nice spectrum of all the fermion masses and mixing and leads to nonzero , which in turn implies phase and allows a large required Dirac CP violating phase in neutrino oscillation. Detailed fits to the fermion spectrum are presented in several scenarios in [11].
The model considered here is motivated in the sense since it provides a constrained fit of fermion masses and explains the largeness of the atmospheric mixing angle, [11]. Neutrino Yukawa coupling matrices considered here exhibit a generalized CP invariance if Yukawa couplings are symmetric. Small explicit breaking of this symmetry by hand as is evident from Equation (30) with the specific numerical value 0.0045 is sufficient to generate the large required CP violating phase discussed here. We have considered the type I seesaw model with very tiny explicit symmetry breaking. This model with very tiny explicit symmetry breaking is motivated in the sense as it is decribed by the predictions and nonzero reactor angle and large required CP violation in neutrino oscillations.
In Figure 1, for the best fit values of with w/o SK-ATM [14] (allowed by updated values of correct baryon asymmetry of the Universe), the allowed spectrum of the lightest mass here lies in the range 0.09 eV eV corresponding to favoured values of reactor angle, , in the interval, 8.1°–8.35° in light of the correct baryon to photon density bounds . Panel (a) manifests itself in the predicted favoured value plane, for the best fit value of with [14]. The favoured value of is around in the light of the correct baryon asymmetry of the Universe which can be manifested from the colour coding in the figure. We have shown for the best fit values of of w/o SK-ATM [14] (as allowed by updated values of correct baryon asymmetry of the Universe) in Figure 5, (for to lie in the interval, ), must lie around 0.095 eV. In Figure 2, for the best fit values of of w/o SK-ATM [14] (in the light of recent ratio of the baryon to photon density bounds ), the preferred value of the leptonic CPV phase came out to be around 304°-307°. Similarly, panel (b) communicates the three-dimensional plot of favourable values of the plane, for the best fit value of of [14] (in the light of recent ratio of the baryon to photon density bounds ). In Figure 3, for the best fit values of of w/o SK-ATM [14] (in the light of recent ratio of the baryon to photon density bounds ), the preferred value of the leptonic CPV phase came out to be around 304°-307° as is obvious from the figure. Similarly, panel (b) shows the favoured values of the lightest neutrino mass, , for the best fit value of of [14] (in the light of recent ratio of the baryon to photon density bounds ). The favoured values of is around 0.09 eV eV. In Figure 8, panel (a) presents the allowed two-dimensional space of the (, ) plane for favoured values of (in the light of recent ratio of the baryon to photon density bounds ) for the best fit values of of w/o SK-ATM [14] (in the light of recent ratio of the baryon to photon density bounds ). In Figure 4, for the best fit values of of w/o SK-ATM [14] (in the light of recent ratio of the baryon to photon density bounds, , the favoured values of phase presented here are and in the light of recent ratio of baryon to photon density bounds, . The value of the lightest mass depicted here from the figure is around 0.005 eV. In Figure 16, for the best fit values of of w/o SK-ATM [14] (allowed by updated values of correct baryon asymmetry of the Universe), the favoured values of the lightest mass depicted here are eV eV for producing correct baryon asymmetry of the Universe. Also, the favoured values reactor angle, , lies in the range 8°–9°. Similarly, in panel (b), we have shown the three-dimensional plot for the predicted favoured values of the plane for the lightest mass, (in the light of recent ratio of the baryon to photon density bounds, ) as a result of contribution of the type I seesaw mechanism to neutrino mass matrix. Figure 19 depicts in panel (a) the predicted three-dimensional space of for values of eV and in the given 3 range, corresponding to , and the best fit values of corresponding to w/o SK-ATM [14]. The favoured values of the lightest mass depicted here are eV eV corresponding to a value of eV. The favoured values of the leptonic CPV phase, , predicted here are 235°–237° consistent with eV. Panel (b) shows the predicted three-dimensional space of for values of the lightest mass, , , and in the given 3 range, corresponding to and w/o SK-ATM [14] for inverted ordering. The favoured values of the reactor angle predicted here are corresponding to a value of eV. The favoured values of the leptonic CPV phase, , predicted here are consistent with, limits on, , eV.
In this work, we learn that by using user-defined Dirac Neutrino Yukawa couplings [11] for the Yukawa interactions associated with the broken symmetry model for the generation of the nonzero reactor mixing angle and leptonic CP phase in the type I seesaw mechanism in the light of leptogenesis, there can be a transformation of the lepton asymmetry into a baryon asymmetry by nonperturbative violating (sphaleron, Sakharov conditions) processes as discussed in [6]. A small explicit breaking of symmetry [11] inherits the property of generating nonzero CP violation in matrices and phase and results in being nonzero. Here, we consider the type I seesaw as the main donor to neutrino mass. We also take into account both inverted and normal ordering of neutrino mass spectrum as well as two different types of the lightest neutrino mass to visualise the results of hierarchical mass spectrum. In the case of normal ordering of masses, the dependance of the phase on the lightest mass is predicted in Figures 1–3 (in the light of recent ratio of the baryon to photon density bounds, ). The favoured values of the phase is found to lie between for the best fit values of corresponding to w/o SK-ATM [14] (in the light of recent ratio of the baryon to photon density bounds ) as a result of contribution of the type I seesaw mechanism to neutrino mass matrix. The favoured values of the lightest mass, , in this case come out to be eV. In the case of inverted hierarchy, the variation of the phase is found to be very intense with the best fit values of corresponding to w/o SK-ATM [14]. Values of the phase favoured are (in the light of recent ratio of the baryon to photon density bounds ) as is evident from Figure 4. The allowed spectrum of the lightest mass is , eV.
We also plot the allowed values of eV for neutrinoless double beta decay and the Jarkslog invariant, , in Figures 6–14 for normal ordering of masses. Prediction of future leptonic CP violation experiments should be able to rule out or take into account some of the results discussed in this work. If we abide by the best fit values of leptonic CP phase discussed in the literature [14, 37], then our scenario, , for inverted ordering of masses corresponding to exactly matches with the best fit values of with w/o SK-ATM [14]. We show the variation of baryon asymmetry with the leptonic phase in Table 1.
Future LBL experiments will hunt for the leptonic CP phase and potentially will measure it with precision. Neutrinoless double beta decay will indicate towards the Majorana CPV phase. The fundamental mysteries in the Universe are about the findings of the nature of the massive neutrinos-Dirac or Majorana. This may be sorted out by the experiments like GERDA, CUORE, KamLAND-Zen, EXO, LEGEND, and nEXO. Determination of the status of leptonic CP asymmetry (T2K, NOA, T2HK, DUNE), determination of the type of neutrino mass ordering (, JUNO, PINGU, ORCA, T2HKK, DUNE), and determination of the order of absolute neutrino mass scale (KATRIN, cosmology) are few of the most challenging tasks today. The ideas presented in this work may definitely will rule in or rule out some of the favoured space in few of the above experiments.
Conflicts of Interest
The author declares that she has no conflicts of interest.
Acknowledgments
I would like to thank my supervisor Prof. Kalpana Bora for useful discussion on this topic.