Baryogenesis from inverted. hierarchical mass models with

Chapter 5 Baryogenesis from inverted hierarchical mass models with tribimaximal mixings 5.1 Introduction There are several interesting ansatz for neutrino mixings. Two most familiar ansatz are bimaximal and tribimaximal mixings. The bimaximal mixing predicts maximum values of solar and atmospheric mixings i.e., On = 623 = 45 and 0]3 = 0, whereas the tribimaximal mixing represents tan2923 = 1, tan2on = 0.5 and sin&xz = 0. Corresponds to bimaximal and tribimaximal mixing patterns of light neutrinos, the heavy masses can be constructed via inverse seesaw relation. It is well known that heavy masses have serious implication towards baryon asymmetry. In the present work, we investigate the possible baryogenesis 95

t scenario for two specific structures of inverted hierarchical mass models: bimaximal mixings(bm) and tribiinaxirnal mixings(tbm), both with opposite CP-parity in the first two mass eigenvalues to,- = ( toi,to2, to3). 5. 2 M i x i n g M a t r i x f o r N e u t r i n o s The unitary matrix U ( = Umns) eq.(l.l) carries the information of mixing angles. The mixing angles are expressed in terms of the matrix elements in eq.(1.33) parameterised by three rotations. For explicit structure of Umns we refer to eq.(3.23). 5. 2. 1 B i m a x i m a l M i x i n g s The Bimaximal mixing(bm) ansatz of neutrinos predicts maximum values for both solar and atmospheric mixing angles. For 0 u = #23 = 45 and 0i3 = 0 the corresponding mixing matrix is expressed as c; to I N > 1 h * O <N _ I m 3 i g to o» 6 O- 3 5. 2. 2 T r i b i m a x i m a l M i x m g s It is a specific leptonic mixing ansatz proposed by Harrison, Perkins and Scott [95], which gathers momentum in the light of SNO results [96]. The 9 6

Um n s matrix, which connects the flavor and mass eigenstates takes the following form in HPS scheme: Uh p s = V V F X V3 0 N 1 1 V 3 1 v'a 1 s/g 1 V3 L v/2 / (5.2) The matrix (5.2) has some significant features: (а ) The intermediate mass neutrino v2 is trimaximally mixed between all three lepton flavors i.e., \Ue2\2 = [U^ ] 2 = t/t2[2 = - (б) The heavy neutrino 1/3 is bi-maximally mixed between vtl and vr i.e., M = PrZ? = The various mixing angles corresponds to the above Uh p s are tan'2823 1 and tan2812 = 0.5. Whereas SNO results shows slight deviation from tribimaxima! mixings with tan2812 = 0.45ion8» there is no reported deviation from atmospheric mixings. The latest data shows deviation of tan2023 from 1. The matching of H PS mixings with experiments can not be treated as a mere coincidence. There must be some deeper underlying symmetry, which can be the backbone of tribimaxima! mixings. To understand H P S scheme, we first consider, how the mixing matrix is controlled by flavor symmetry [97]. The leptonic mixing matrix Umns (eq.(3.23)) is expressed in terms of two unitary matrices: Um n s = U}LUV1 where, UiL and Uv are connected to the diagonalisation of charged lepton and neutrino mass matrix in the following way Diag(me, mt) = U}LMiUtB, Diag(m,i, m2, m3) = (5.3) 97

For Majorana neutrinos the structure of Mv is constrained to be symmetric and the charged lepton mass matrix M; can be of any 3 x 3 matrix. If this is the case the mixing matrix can be of any unitary form, and the mass are unrestricted by Standard Model symmetry. The presence of generation symmetry, restricts the structure of mass matrices which in turn fixes the mixing matrix. The associated Lagrangian has to be invariant under the following transformations: v -» X v, Ir > X ril, hi >X rir where Xz(i = v, ll, lr} are three 3 x 3 unitary matrices. This type of transformation results in the specific structure of mass matrices: Mv = XlM Xl, Mt = XlM,XR. (5.4) In UPS scheme, the charged lepton mass matrix was taken to be circulant form: * a b e ^ Mi c a b ) (5.5) ^ b c a j where a, 6, c axe related to the three masses of charged leptons. The above Mi can be generated by using cyclic permutation C3 symmetry [98, 99]. Again the neutrino mass matrix M was expressed in terms of x,y,z which are associated with the three independent neutrino masses: (5.6) (x 0 Mv = 0 2 0 l y 0 * / 98

The neutrino mass matrix was generated by using 52 x S2 symmetry. The mass matrices Mi and M axe diagonalised by [97] UlL = Ui Ir _1_ V3 1 1 1 1 it) 10* ^ 1 u* a) j and U = i r i i 71 U 71 V 71 0 1 0 72 J (5.7) where ui = e x p (^ ). Combination of the above two matrices gives j Uhps = K u- 5.3 Inverted hierarchical M odel and Chang- ing scenario The inverted hierarchical patterns of light left-handed neutrinos (mi ~ m2 > m3), are brat understood in terms of two mass models [35, 37]: Inverted hierarchical type-2a(invt2a) and Inverted hierarchical type- 2B (InvT2B) based on the relative CP-parity between mi and m2- For InvT2A the mass pattern is denoted by (m i,m2, m3) whereas for InvT2B the mass pattern is ( m i,m2,m 3). The inverted hierarchical model of neutrinos with odd C P parity ( mi, m2,m 3) generally predicts nearly bimaximal mixings in the diagonal basis of the charged lepton mass matrix. This is specifically true when (ll)-element is nearly zero. A familiar mass matrix of InvT2B is generally given by 99

( 4 1 1 \ mll = 1 4 l 1 4 4, (5.8) where 1,2,3 < 1. The diagonalisation o f mass matrix (5.8) gives following mass eigenvalues, mi,2= ^[(4 + k + 4) ± ]> m3= m0(4-4); x^ = S+{Sf + 6l + S l)-25t82 ~~ 26x63 + 2 8283 and the three mixing angles are calculated as tan2 = 1, sin 0i3 = O,tan2012 = Such a simplest form (5.8) is found to be realised within seesaw framework [22] mu, - mlrmr]imfjr, using diagonal form o f Dirac neutrino mass matrix rnlr diag(xm, A, l)v and a suitable non-diagonal texture of heavy Majoraaa mass matrix Mrr. For each pair of (m,n), the corresponding texture of Mrr are different, and the light neutrino mass matrices, are left unaffected. For example, we use the following form o f ttill from ref. [35] f 0 1 m i x = 1 (A3 A4)/2 - ( A 3 + A4)/2 mo- (5.9) \ 1 (A3 + A4)/2 (A3 A4)/2 j For input values A = 0.3 and m0 = 0.035eV we have the mass eigenvalue: m fls = diay (0.00028,0.04903, -0.04997)eV 100

The corresponding oscillation parameters are calculated as Am i = 9.30 x lo~~5ev 2, Amfg = 2.50 x 10~3eV2.tan2 012 = 0.98, sin2 2023 1.0, sin 013 = 0. The predicted solar angle is nearly maximal. Several attempts have been made to tone down solar mixing angle but the effect is not so satisfactory to the experimentally acceptable level. It has been pointed out [100] that there is also a possibility to realise the tribimaximal mixings [95, 97] within the inverted hierarchical mass matrix in eq.(5.8), provided the numerical values of 1,2,3 are comparatively larger but smaller than 1. As a specific example we follow the ref. [100] and use the following values: 1 = 0.56855627, 2 = 0.71572329, 3 = 0.12477548,too = 0.035eU in eq.(5.8).the corresponding mass eigenvalues are TOi = 0.05378eF, m2 = Q.05299eF and TO3 = 0.02936eF. The predictions on neutrino oscillation parameters are: A to x = 8.34 x 10-5eF 2, Arnig = 1.95 x l O 'W 2, tan2 012 = 0.45, tan2 023 = 1.0, sin 0i3 = 0. The solar angle is slightly smaller than tribimaximal mixing and agrees with recent data. The above prediction does not require any fine tuning of the solar angle from charged lepton sector or from renormalization effects. The inverted hierarchical neutrino mass matrix in eq(5.8) with bimaxima! mixings as well as tribimaximal mixings is independent of pair of (m, n) appeared in m m within the framework of seesaw formula, and this can however be fixed in the estimation of baryon asymmetry via lepton asymmetry which depends on the texture of MRR. For our interest we take up three different cases [80] of (to, n) depending on ttilr and Mrr as allowed by 50(10) GUT. Casel: mj,r = charged lepton mass matrix with (m,n ) = (6, 2) i.e., m LR = diag{ A6, A2, l)w; 101

case2: m m = up quark mass m atrix with (m,n) = (8,4 ) i.e., m m = diag{a8, A4, l)u ; cases: m m = down quark mass m atrix with (m,n) = (4,2 ) i.e., m m = diag{a4, A2, i)v. For a particular choice o f mm one can have three possible structure o f M RR depending on the above choice o f (m, n). The choice o f non-diagonal m m in the basis o f diagonal M rr plays a crucial role in the calculation o f baryon asymmetry via lepton asymmetry produced by the decay o f lightest o f heavy M ajorana neutrino M i [64, 101, 102]. 5.4 Numerical calculation and results W e start with light Majorana neutrino mass m atrix mm and translate this matrix to M RR via the inversion o f seesaw formula, M RR = rn[rm m m m Using the right-handed Majorana mass m atrix MRR, we estimate the baryon asymmetry for bimaximal and tribimaximal cases for three different choices o f Dirac neutrino mass m atrix mm- For the expression o f lepton asymmetry we refer to eq(3.6) and baryon asymmetry is calculated from Y sm = 0.0216^6!. (5.10) Again we follow the same procedure for numerical calculation as in subsection- 3.5.2. As an example we consider tribimaxim al pattern o f mm discussed in section 5.3. The light neutrino mass m atrix is given by(in e V ) 102

h i l l ( -0.0198595-0.0349297 ^ -0.0349297-0.0349297-0.0349297 > 0.025-0.00435837-0.00435837 0.025 } Taking Dirac neutrino mass matrix as down quark mass matrix (case-3) i.e., nilr diag{xa, A2, l)n, with A = 0.3 and v = 174GeV we have the corresponding Mrr (in GeV) as M r r = f -1.44 x 1010-2.70 x 10n ^ -3.01 x 1012-2.70 x 10u -3.01 x 1012 N 5.03 x 1012-3.69 x 1013-3.69 x 1013 6.21 x 1014, The mass eigenvalues are = diag(9.7q x 1010, 2.89 x 1012, 6.23 x xlo14). The corresponding diagonalising matrix is ' 0.988403* -0.151777-0.004787 ^ u = 0.151227* 0.986701-0.059593 ^ 0.013768* 0.058177 0.9982il, The neutrino Yukawa coupling becomes ( 0.008006* -0.001229-0.000039 ^ h = 0.013611* 0.088803-0.005363 ^ 0.013768* 0.058177 0.998211 j ( Jn Jn Jn J = (h*mllhf) = J21 J'2'2 J2& 103 KJn J32 Jz3 j

where, Ju = (1.31-0.71i)10-15, J12 = J21 = (-0.0 6 + 2.27t)10-14, Ji8 = J3 1 = (0-05 + 2.95i)10-13, J22 = (2.06 + 0.79t)10~13, </23 = J32 = ( 3.86+5.42i)10-13 and J33 = (2.45+0.10i)10~n. For the above structures of m u, h and J we have (hh^)u = 6.56 x 10-5, Im{h"miJ[Jh^)n = 7.09 x 10-16, mi = 2.03 x 10~2eV, m* = 1.08 x 10~3eV, K = 18.73, and K\ 0.008. The lepton asymmetry is found to be = 2.08 x 10-6. Following t eq(5.10) we found YgM = 3.78 x 10" 10 which is consistent with the experimental bound [39]: Y MB = (6.lio; )10-1. We follow the same procedure to estimate lepton and baryon asymmetry for other cases. For quasi-degenerate structure which appears in bimaximal case, we use eq.(3.7) to calculate lepton asymetry. Such degeneracy is found to be lifted in tribimaximal case and the expression in eq.(3.6) is used. For all the cases under consideration the heavy mass eigenvalues, are collected in table-5.1, the effective mass parameters and dilution factors are presented in table-5.2. Finally the estimated lepton and baryon asymmetry are collected in table-5.3. Table 5.1: The three right-handed Majorana neutrino masses in GeV for bimaximal case{bm) and tribimaximal case(tbm). (m,n) BM Mf * TBM \MA (4.2) (6.2) (8,4) 6.2783 x 10u, 6.2838 x 10n,5.38 x 1016 5.6527 x 1010, 5.6532 x 10lo,5.38 x 1016 4.5971 x 108, 4.5974 x 108,5.34 x 1016 9.76 x 1010,2.89 x 1012,6.23 x 1014 8.10 x 108, 2.83 x 1012, 6.23 x 1014 6.56 x 10, 2.30 x 1010,6.21 x 1014 104

T a b le 5.2: Contains the values of effective mass parameter m i in ev, decay parameter K and dilution factor «i for bimaximal and tribimaximal case. BM TBM (m, n) r h i K «i r h i K K i (4,2) 3.16 X 10~3 2.91 0.12 2.03 X 10~2 18.73 8.40 X I Q - 3 (6,2) 2.85 X 10~4 0.26 0.17 1.98 X 10-2 18.37 8.66 X 10~3 (8,4) 2.83 X 10 4 0.26 0.17 1.98 X i o - 2 18.37 8.66 X 10 3 T able 5.3: Calculation of lepton asymmetry ei and baryon asymmetry Yg for bimaximal case(bm) and tribimaximal case(tbm) for three choices o f (m,n). BM TBM (m,n) Cl YB Cl Yb (4,2) 1.64 x 10 2 4.25 x IO'5 2.08 x 10 8 3.78 x IO"10 (6,2) 1.47 x 10~2 5.40 x 10-5 1.55 x 10 9 2.89 x 10-13 (8,4) 1.62 x 10~4 5.94 x 10~7 1.28 x IO"11 2.40 x 10 15 105

5.5 Summary and discussion We start with the bimaximal and tribimaximal mixing pattern of inverted hierarchical light neutrino mass matrix niu,. Heavy right-handed Major an a neutrino mass matrices are constructed via inverse seesaw relation. In case of bimaximal mixing pattern of light neutrinos we observe that the corresponding heavy right-handed neutrino masses manifest quasi-degenerate structure i.e., Mx ~ M2 < M3. This peculiar structure of heavy masses enhance the produced asymmetry known as resonance enhancement by modifying the propagator. This scenario completely changes [103] when we come to tribimaximal mixing(tbm) pattern, with hierarchical pattern of heavy neutrinos i.e., < M2 < M3. This type of hierarchical structure will bypass the resonance enhancement effect and this is clearly seen in the produced asymmetry. For bimaximal case the range of lepton asymmetry is found to be 10~4 < ti < 10~2 whereas in tribimaximal scenario the range is 10~11 < t\ < 10~6. The estimated baryon asymmetry Yg = 3.78 x 10 10 for tribimaximal mixing pattern with down quark mass matrix taken as Dirac neutrino mass matrix is consistent with the experimental value [39]. For hierarchical structure of heavy Majorana neutrino masses, the condition Mi > 4 x 108GeV satifies the famous Davidson-Ibarra bound [87]. In the present calculation we are able to establish the validity of tribimaximal mixing in inverted hierarchical mass model, and also to discriminate the three possible choices of Dirac neutrino mass matrix through the estimation of baryon asymmetry. 106