Significance of broken $ \mu-\tau $ Symmetry in correlating $ \delta_{CP} $, $ \theta_{13} $, Lightest neutrino Mass and neutrinoless double beta decay $ 0\nu\beta\beta $

Leptonic CP Violating Phase $ \delta_{CP} $ 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 angle $ \theta_{13} $ measured in reactor experiments, LBL, DUNE with high precision in neutrino experiments indicates towards the vast dimension of scope to detect $ \delta_{CP} $. The correlation between leptonic Dirac CPV phase $ \delta_{CP} $, reactor mixing angle $ \theta_{13} $, lightest neutrino mass $ m_{1} $ and matter antimatter asymmetry of the Universe within the framework of $ \mu-\tau $ symmetry breaking assuming the type I seesaw dominance is extensively studied here. Small tiny breaking of the $ \mu-\tau $ symmetry allows a large Dirac CP violating phase in neutrino oscillation which in turn is characterised by awareness of measured value of $ \theta_{13} $ and to provide a hint towards a better understanding of the experimentally observed near maximal value of $ \nu_{\mu} -\nu_{\tau} $ mixing angle $ \theta_{23}\simeq \frac{\pi}{4}$. Precise breaking of the $ \mu-\tau $ symmetry is achieved by adding a 120 plet Higgs to the 10 $+$ $\bar{126}$ dimensional representation of Higgs. The estimated three dimensional density parameter space of lightest neutrino mass $ m_{1} $, $ \delta_{CP} $, reactor mixing angle $ \theta_{13} $, 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 $ m_{1} $, $ \delta_{CP} $, $ \theta_{13} $, is found out which can produce the observed baryon to photon density ratio of the Universe.

massive neutrinos are formed in their gauge eigen states (ν α ) which is linked to their mass eigen states ν i . Gauge eigen states participate in gauge interactions as where, α = e, µ, τ , ν i is the neutrino of distinct mass m i .U is parameterised as where, θ 12 = 33 0 , θ 23 = 38 0 − 53 0 , θ 13 = 8 0 [2] are the solar, atmospheric and reactor angles according to the global fits respectively. The Majorana phases α, β dwells in P, where U * P is known as the Pontecorvo-Maki-Nakagawa-Sakata U P MN S matrix [3]. Since a ν of a given flavor α is a mixed state of atleast three ν 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. δ CP in PMNS matrix can induce CP violation. CP asymmetry can be observed in neutrino oscillations. δ CP phase measures the amount of asymmetries between lepton oscillations and antilepton oscillations. neutrinos being massive and they mix with each other. This may be a source of CP violation if Sinδ CP = 0. The amount of δ CP violation phase in this case is estimated by the Jarkslog invariant [4].
Cosθ 13 Sin2θ 12 Sin2θ 23 Sin2θ 13 Sinδ CP (4) when Sinδ CP = 0 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 ongoing T2K experiment have reported that CP violating phase, δ CP excludes the value δ CP = 0, π [5] at the 2σ confidence interval for either of the mass orderings, Normal ordering and Inverted ordering. The value of Dirac CPV phase, δ CP = 276.5 0 is preferred in [6].
The neutrino mass matrix is invariant under µ − τ exchange symmetry. In a basis where the charged leptons are mass eigen states. Under the µ − τ exchange symmetry, the 2-3 mixing is maximal, i.e, θ 23 = π 4 and the 1-3 mixing is zero, i.e, θ 13 = 0. The deviation of δθ 23 from the maximal angle θ 23 , the explanation of reactor angle θ 13 and the existence of CP violating phase neccessitates of the spontaneous breaking of the µ−τ exchange symmetry in the neutrino sector.
Here, in this work, an explicit form of the Dirac neutrino mass matrix in broken µ − τ [7] symmetry framework in type I Seesaw mechanism is used in our calculation for generating baryon asymmetry of the Universe via leptogenesis. This scenario is characterised by small divergence of δθ 23 from the maximal angle θ 23 , which is consistent with a liberal size of θ 13 ∼ 8 0 − 9 0 and a large δ CP phase in the neutrino sector. The renormalisable Dirac neutrino yukawa couplings of the Dirac mass matrices is determined from the fermion yukawa couplings to the 10,1 26 and 120 dimensional fields of Higgs multiplet in the SO(10) group. Higgs field under the 10 and1 26 representations are symmetrical under the generalised µ − τ symmetry, while the 120 dimensional representation changes sign. This spontaneously breaks the µ − τ invariant symmetry, which in turn allows a generalised δ CP phase in the PMNS matrix. Here we made an effort for correlating or constraining the values of δ CP phase, non zero reactor angle θ 13 and 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 δ CP and reactor angle θ 13 have good vibes between each other. Precise value of θ 13 plays an imperative role in its CP violation phase measurements. On the basis of this fact non zero values of θ 13 is predicted here in consistency with δ CP phase. Taking into account constraints from the global fit values of ν oscillation parameters and cosmology a density plot of the favourable values of δ CP phase, lightest neutrino mass, θ 13 is being initiated, which is compatible with the contraints on the sum of the absolute neutrino masses, i m(ν i) < 0.23 eV from CMB, Planck 2015 data (CMB15+ LRG+ lensing + H 0 ) [8]. Constraints fom the leptonic asymmetry of the Universe is also considered for further restricting δ CP V 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 η B accessible as 5.8 × 10 −10 < η B < 6.6 × 10 −10 [9]. We also calculate the effective mass spectrum for neutrinoless double beta decay, 0νββ decay given by for favourable values of δ CP phase and lightest ν mass explored here in this work. In this paper, we apply the broken µ − τ symmetry to the Dirac neutrino Yukawa Couplings in type-I seesaw mechanism in SO(10) model in predicting favourable values of δ CP phase, lightest neutrino mass and θ 13 . 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 δ CP phase, lightest neutrino mass and θ 13 in the neutrino sector. Finally we show our predictions for the the effective mass spectrum for neutrinoless double beta, 0νββ decay for favourable values of δ CP phase.
The paper is organized as follows. In section 2, we introduce our broken µ − τ symmetry models. In section 3, we perform parameter scan to fit neutrino oscillation data and provide some informations in predicting observables like δ CP phase in the neutrino sector. Also we show our predictions for the the effective mass spectrum for neutrinoless double beta, 0νββ decay for favourable values of δ CP phase. Section 4 is our results and conclusions for the Yukawa interactions associated with broken µ − τ symmetry model discussed here.
The Lagrangian of the type I Seesaw model is [10] Here, y Iα is the complex yukawa coupling matrix,L α = (ν L α , L l α ) T is the standard model left handed lepton doublet of flavor α, when α = e, µ, τ andφ is the hypercharge conjugated Higgs Doublet, The Lagrangian describes scenario of generation of ν masses via Higgs mechanism. Electroweak symmetry breaking process allows neutral part of the Higgs field to acquire a VEV, v = 246GeV , In type I Seesaw, the baryon asymmetry of the Universe (BAU) occurs via leptogenesis mechanism via out of equi- where (m D ) αI is the Dirac mass matrix. Eq. 8 shows that in type I seesaw mechanism, SM ν masses are suppressed by the combination of small yukawa couplings and large RH ν masses. neutrino mass matrix on diagonalision, gives two eigen values − light neutrino ∼ m 2 D MR and a heavy neutrino state ∼ M R . This is known as type I seesaw mechanism.
At the end of inflation [21], a certain number density of right-handed neutrinos, n νR , were 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 RH ν mass matrix is diagonal, the type I contribution to ǫ CP l is given by decay of M 1 or the CP violating parameter is given as, where where Γ(ν Ri → l i H u ) means decay rate of heavy Majorana RH ν of mass M 1 to a lepton and Higgs. In electroweak sphaleron process, asymmetries produced by the out of equilibrium decay of M 2 and M 3 gets washed out by lepton number violating interactions after ν R or M 1 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 RH ν mass matrix is diagonal, the type I contribution to ǫ CP l is given by [20], where υ is the Higg's vev. Q is a complex unitary orthogonal matrix where Q is parameterized as [22]:Q = where Y ν is the Dirac neutrino Yukawa couplings. To reproduce the physical, low-energy, parameters, i.e. the light neutrino masses (contained in D K ) and mixing angles and CP phases (contained in U P MN S ), we have taken the most general Dirac neutrino mass matrix in broken µ − τ symmetry framework as [23] In the flavor basis, where the charged-lepton Yukawa matrix, Y e and gauge interactions are flavour-diagonal, In terms of user defined Dirac neutrino mass matrices, [23] U is the PMNS matrix and M R is the RH neutrino Majorana scale. We can always choose to work in a basis of right of both the solar (∆m 2 21 ) and atmospheric (∆m 2 A ) mass squared differences, eq.(12) also reveals that CP asymmetry is linked to Dirac CPV phase. Here we utilse this fact to generate the allowed region of δ CP phase in context of laptogenesis. As has been discussed in [20], the lepton antilepton asymmetry gets connected to both the solar and the atmospheric mass difference square. The transformation of the lepton asymmetry into a baryon asymmetry by non-perturbative B + L 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, FCNC processes, like µ → e + γ processes. In supersymmetric theories like cMSSM, NUHM, NUGM and NUSM where the origin of the ν masses is via see-saw mechanism, it can be shown that the prediction for BR(µ → e, γ), BR(τ → µ, γ) and BR(τ → e, γ) is in general larger than the experimental upper MEG bound [24,25]. Also some studies on decays of b flavoured hadrons in the context of cMSSM/mSUGRA models is being done in [26].
Let H, F (G) be any complex symmetric (anti) matrices in general, which are a measure of the fermion yukawa couplings to the 10,1 26 and 120 Higgs field respectively. Here, Similarly Similarly h,f,g are all real. µ − τ symmetry is invariant under the exchange of second and third generation fermions. When µ − τ symmetry is added with SO(10) grand unified theory then a general symmetry results which satisfies where A small explicit breaking of µ − τ symmetry is put by hand, by inheriting the property, This introduces CP violation in PMNS matrices and results in θ 13 being non zero. The amount of breaking used here for generating non zero CP asymmetry producing a measurable CP violating phase via Dirac neutrino Yukawa Couplings used from [23] is Exact µ − τ symmetry imposes Sin 2 θ 23 = 0. One can break the symmetry spontaneously through the vev of the 120 plet and by use of Eq. (22). A required amount of CP violating phase δ CP is generated by explicitly breaking baryon asyymetry of the Universe [9]. We use best fit values of ν oscillation parameters. The two mass square differences ∆m 2 12 , ∆m 2 13 are embedded in neutrino mixing matrix so we are left out with lightest ν mass as only free parameter in this model. In the chargd lepton basis, we parameterize the PMNS matrix U P MN S , by diagonalizing the neutrino mass matrix m ν in terms of three mixing angles θ ij (i, j = 1, 2, 3; i < j), one CP violating Dirac CPV phase δ CP , and two Majorana phases (α 21 and α 31 ) as follows: The Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix is UP, where U is where, θ 12 = 33.82 0 , θ 23 = 48.3 0 (48.6 0 ), θ 13 = 8.61 0 (8.65 0 ) [27] 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 R = U P MN S , in terms of user defined Dirac neutrino Yukawa Couplings defined in Eq. (13) in order to produce correct baryon asymmetry of the Universe.
For the Normally ordered light ν masses, we have With m 1 ∈ [10 −6 eV, 10 −1 eV ], and, m 2 2 − m 2 1 = 7.39 × 10 −5 eV 2 , m 2 3 − m 2 1 = 2.48 × 10 −3 eV 2 as is evident from the ν oscillation data [27], m 1 being the lightest of three ν masses. For the inverted ordered light ν masses, we have with m 3 being the lightest of three ν masses. Here we take M 1 ∼ 10 12 GeV. For normal ordering, the choices of lightest neutrino mass is m 1 = 0.07118eV whereas for inverted ordering, the choice of lightest neutrino mass is m 3 = 0.0657eV . This sustainable allowance of m 1 (m 3 ) = 0.07(0.065)eV signifies a neutrino mass spectrum where the sum of absolute neutrino masses lies below the cosmological upper bound, i m(ν i ) < 0.23eV [28]. Next random scan of the ν mixing matrix parameter space for NH, IH in order to produce correct baryon asymmetry of the Universe 5.8 × 10 −10 < Y B < 6.6 × 10 −10 is performed in the following 3 σranges of δ CP with respect to the tabulated χ 2 map of the SuperKamiokande analysis of the data within ∆χ 2 = 6.2 [27]: While doing parameter scan, we find favoured values of lightest ν mass, dirac CPV phase δ CP , for producing correct baryon asymmetry of the Universe, 5.8 × 10 −10 < η < 6.6 × 10 −10 .
The lepton flavor effects are significant if the lightest right handed Majorana neutrino mass M νR1 is below 10 12 GeV. Here M 1 = 10 12 GeV. In the type I seesaw mechanism one can always find the right handed neutrino mass matrix as; where (m D ) αI is the Dirac mass matrix. We consider a Dirac neutrino mass matrix defined in Eq. For global fit values of ν oscillation parameters, we compute the Jarlskog invariant, δ CP given by PMNS matrix elements U αi . We also compute the Jarkslog invariant for allowed values of δ CP phase, θ 13 and lightest ν mass explored here in this work for both normal ordering and inverted ordering.

IV. RESULTS AND CONCLUSION
In this work, we have used user defined Dirac Neutrino Yukawa couplings for the yukawa interactions associated with the broken µ − τ symmetry model for the generation of non-zero reactor mixing angle θ 13 , leptonic CP phase δ CP in type I seesaw Mechanism in the light of leptogenesis and then the transformation of the lepton asymmetry into a baryon asymmetry by non-perturbative B + L violating (sphaleron, sakharov conditions) processes as discussed (a) Figure 11: Predictions in broken µ − τ symmetry model for Normal ordering: depicts density plot of (m1, δCP ) for mee) [eV], 0νββ decay for favoured values of m1,δCP , θ13 (in the light of recent ratio of the baryon to photon density bounds, 5.8 × 10 −10 < η < 6.6 × 10 −10 ). .  Figure 12: Predictions in broken µ − τ symmetry model for Normal ordering. Left panel depicts predicted three dimensional space of (mee, δCP , m1) for mee [eV], 0νββ decay for favoured values of m1,δCP , θ13 (in the light of recent ratio of the baryon to photon density bounds, 5.8 × 10 −10 < η < 6.6 × 10 −10 ). Left panel depicts predicted three dimensional space of (mee, δCP , m1) for mee [eV], 0νββ decay for favoured values of m1,δCP , mee for values of δCP in the given three σ range, corresponding to ∆χ 2 = 9.5 w/o SK-ATM [27] in [6]. A small explicit breaking of µ − τ symmetry [23] inherits the property of generating non zero CP violation in U P MN S matrices, δ CP phase and results in θ 13 being non zero. 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 lightest neutrino mass m 1 (m 3 = 0.07118eV (0.0657eV )) to visualise the results of hierarchical ν mass spectrum. In case of normal ordering of ν masses, the dependance of δ CP phase on lightest ν mass is predicted in figures 1,3,4 (in the light of recent ratio of the baryon to photon density bounds, 5.8 × 10 −10 < η < 6.6 × 10 −10 ). The favoured values of δ CP phase is found to lie between δ CP ∈ [302 0 , 304 0 ] for best fit values of θ 13 = 8.41 Figure 13: The predicted three dimensional space of (mee, δCP , θ13) for mee [eV], 0νββ decay for favoured values of m1,δCP , θ13 (in the light of recent ratio of the baryon to photon density bounds, 5.8 × 10 −10 < η < 6.6 × 10 −10 ) for lightest ν mass m1 = 0.07118 eV.
(a) (b) Figure 14: Predictions in broken µ − τ symmetry model for Inverted ordering. The left panel: predicted favoured values of δCP (in the light of recent ratio of the baryon to photon density bounds, 5.8 × 10 −10 < η < 6.6 × 10 −10 ) for lightest ν mass m3 = 0.0657 eV, as a result of contribution of type I Seesaw mechanism to neutrino mass matrix. Similarly, The right panel: predicted allowed three dimensional space of (δCP , θ13, JCP ) plane for allowed regions of Jarkslog invariant, JCP values for for best fit value of θ23 = 48.6 0 of ∆χ 2 = 6.2 [27] as a result of contribution of type I Seesaw mechanism to neutrino mass matrix.
(a) Figure 18: Predictions in broken µ − τ symmetry model for Inverted ordering: predicted allowed three dimensional space of (deltaCP , θ23, JCP ) plane for allowed regions of Jarkslog invariant, JCP values for for best fit value of θ13 = 8.49 of ∆χ 2 = 9.5 [27] as a result of contribution of type I Seesaw mechanism to neutrino mass matrix .