MModiﬁed TBM and role of a hidden Z Rome Samanta and Mainak Chakraborty Saha Institute of Nuclear Physics, HBNI, 1/AF Bidhannagar, Kolkata 700064, India [email protected] Centre of Excellence in Theoretical and Mathematical SciencesSOA University, Khandagiri Square, Bhubaneswar 751030, India [email protected]
In a residual Z × Z symmetry approach, we investigate min-imally perturbed Majorana neutrino mass matrices. Constraint relationsamong the low energy neutrino parameters are obtained. Baryogenesis isrealized through ﬂavored leptogenesis mechanism with quasi-degenerateright handed (RH) heavy neutrinos. Keywords:
TBM mixing, Residual symmetry, Leptogenesis.
In the minimally extended Standard Model (SM) with singlet RH neutrino ﬁelds N Ri , O (eV) neutrino masses are generated through Type-I seesaw mechanism.Relevant Lagrangian for the latter can be written as −L ν,Nmass = ¯ ν Lα ( m D ) αi N Ri + 12 ¯ N CiR ( M R ) i δ ij N jR + h . c . (1)with N Ci = C ¯ N Ti . The eﬀective light neutrino Majorana mass matrix M ν whichis then obtained by the standard seesaw formula M ν = − m D M − R m TD , can be putinto a diagonal form as U T M ν U = M dν ≡ diag (m , m , m ) with m i assumedto be real. The eﬀective low energy neutrino Majorana mass term that containsthis M ν comes out as −L νmass = 12 ¯ ν CLα ( M ν ) αβ ν Lβ + h . c .. (2)Now in the basis where the charged lepton mass matrix M (cid:96) is diagonal, U followsthe standard parametrization. Let us now have look at the latest 3 σ ranges for the relevant neutrino parameters obtained from oscillation data. solar: ∆m ≡ m − m : (7 . . × − eV , atmospheric: | ∆m | ≡ | m − m | :(2 . . × − eV , θ : 31 . o –35 . o , θ : 38 . o –53 . o , θ : 7 . o –9 . o .Finally, thanks to the Planck for the observed upper bound on the sum of thelight neutrino masses; Σ i m i < a r X i v : . [ h e p - ph ] J un R. Samanta et al.
In Ref. it is argued that any horizontal symmetry of neutrino Majoranamass matrix M ν is a residual Z × Z ﬂavour symmetry. The symmetry gen-erators G i obey the relation U † G i U = d i with i=2, 3, i.e. there are two inde-pendent G i and hence d i . We can choose these two independent d matrices as d = diag ( − , , −
1) and d = diag ( − , − , U , one can cal-culate G and G corresponding to d and d respectively. In this work we focusparticularly on the TBM mixing and calculate the corresponding G i matrices; G T BM , and G µτ . It can be justiﬁed theoretically as well as phenomenologicallythat G T BM is the only symmetry which is viable one to exist as the unbroken Z generator in the Lagrangian. In the next section, we present an ephemeral discus-sion regarding the implementation of G T BM and G µτ on the neutrino ﬁelds. Forfurther insights related to the application of residual symmetry in the neutrinosector, the readers could have a quick look at Ref.. Z µτ : perturbation to the TBM massmatrices Depending upon the residual symmetries on the neutrino ﬁelds and the phe-nomenological viability of the textures of the mass matrices, we discuss twocases.Case 1. At the leading order G T BM and G µτ transform both the neutrino ﬁelds ν L and N R as ν L → G i ν L and N R → G i N R . Now we choose a perturbation ma-trix M G (cid:15)R which violates µτ interchange in M R but respect G T BM . The leadingorder mass matrices and the perturbation matrix are of forms m D = b − c − a a − aa b c − a c b , M R = y y
00 0 y , M G (cid:15)R = (cid:15) (cid:15) (cid:48) (cid:15) (cid:15) (cid:15) (cid:48) (cid:15) (3)where (cid:15) = (3 (cid:15) + (cid:15) ) and (cid:15) (cid:48) = − (3 (cid:15) + (cid:15) ). Now the eﬀective M ν which isinvariant under G T BM is written as M G TBM ν = − m D M − R ( m D ) T with M R = M R + M G (cid:15)R . Since G T BM invariance of the eﬀective M ν always ﬁxes the ﬁrstcolumn of the mixing matrix to ( (cid:113) , − (cid:113) , (cid:113) ) T up to some phases, a directcomparison of the latter with the U P MNS matrix leads to a constraint relationbetween θ and θ as sin θ = 13 (1 − θ ) . (4)Case 2. In this case, at the leading order, all the neutrino ﬁelds obey G µτ .However G T BM of the over all M ν is ensured only by the transformation ν L → G T BM ν L . Since the RH singlets are free from G T BM , the perturbation matrixwhich is added with M R is now arbitrary. Now the most general Dirac massmatrix m D , the Majorana mass matrix M R and the perturbation matrix are of BM mixing and Z symmetry 3 the forms m D = a ( b − c ) ( c − b ) a b c − a c b , M R = x y
00 0 y , M (cid:15)R = (cid:15)
00 0 (cid:15) . (5)Again the eﬀective M ν is calculated as M G TBM ν = − m D M − R ( m D ) T with M R = M R + M (cid:15)R . Besides reproducing the same relation as obtained in Eq. 4, anotherinteresting point is realized that m D of (5) is of determinant zero due to theresidual G T BM symmetry; thus the M G TBM ν matrix has one zero eigenvalue. Forthe remnant G T BM symmetry, m is of vanishing value. Fig. 1.
Plot in the left side: Variation of δ with θ for diﬀerent values of θ . Plot inthe right side: Variation of δ with θ for diﬀerent values of θ where the green bandrepresents the latest 3 σ range for θ . One can also obtain correlation of δ with the mixing angles. Lepton number, CP violating and out of equilibrium decays of RH neutrinoscreate a lepton asymmetry. A general expression for the CP asymmetry pa-rameter (cid:15) αi for any RH mass spectrum is given by14 πv H ii (cid:104) (cid:88) j (cid:54) = i g ( x ij ) Im H ij ( m D ) † iα ( m D ) αj + (cid:88) j (cid:54) = i r Im H ji ( m D ) † iα ( m D ) αj r + H jj π v (cid:105) . (6)In (6), r = (1 − x ij ), H ≡ m D † m D , x ij = M j /M i and g ( x ij ) is given by g ( x ij ) = √ x ij (1 − x ij )(1 − x ij ) + H jj π v + f ( x ij ) . (7)The term proportional to f ( x ij ) comes from the one loop vertex contributionwhile the remaining are from self energy diagram. Note that in the limit where R. Samanta et al. the RH neutrinos are exactly degenerate, i.e., x ij = 1, the self energy contri-bution vanishes and thus a nonzero value of CP asymmetry parameter ε αi isproduced only through the vertex contribution. In our model RH neutrinos arequasi degenerate and thereby enhances the CP asymmetry parameter signiﬁ-cantly through self energy contributions. Fig. 2.
Variation of Y B with z in the τ -ﬂavored mass regime (left). Upper and lowerbounds on M for the minimal values of the breaking parameters (right). A normalmass ordering for the light neutrinos has been assumed. Another important issue is that the ﬂavor eﬀect to the produced leptonasymmetry. In  we address this issue in detail, theoretically as well as numer-ically. See Fig.2 for a typical variation of Y B with z = M /T . References
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