JIGSAW 07. Neutrino Mixings and Leptonic CP Violation from CKM Matrix and Majorana Phases. Sanjib Kumar Agarwalla

JIGSAW 07 Neutrino Mixings and Leptonic CP Violation from CKM Matrix and Majorana Phases Sanjib Kumar Agarwalla Harish-Chandra Research Institute, Allahabad, India work done in collaboration with M. K. Parida, R. N. Mohapatra and G. Rajasekaran hep-ph/0301234 & hep-ph/0611225 S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.1/35

Plan Motivation The Model RGE Analysis Results Conclusions S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.2/35

Motivation GUT with quark-lepton unification key ingredients to understand physics at low energies This scheme in the light of SUSY GUTs explains the experimentally measured value of the electro-weak mixing angle The same model also lead to t b τ Yukawa unification which agree with observation for large values of tan β(= v u /v d ) =========================================== An ask What are the other manifestations of quark lepton unification at low energies???? =========================================== S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.3/35

Proposal Recently M. K. P., R. N. M. & G.R. proposed the concept of High Scale Mixing Unification (HUM) hypothesis hep-ph/0301234 & hep-ph/0611225 S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.4/35

HUM : INPUTS At high GUT-seesaw scale (M R = 10 15 to 10 16 GeV), the quark (CKM) and lepton (PMNS) mixing matrices are equal another signature of quark-lepton unification S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.5/35

HUM : INPUTS At high GUT-seesaw scale (M R = 10 15 to 10 16 GeV), the quark (CKM) and lepton (PMNS) mixing matrices are equal another signature of quark-lepton unification 3 mixing angles of leptons = 3 mixing angles of quarks The Dirac phase of leptons is same as that for quarks ============================================= Neutrinos are Majorana fermions with quasi-degenerate masses and with same CP (all Majorana mass eigen values have same sign, eg. positive) S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.5/35

HUM : INPUTS At high GUT-seesaw scale (M R = 10 15 to 10 16 GeV), the quark (CKM) and lepton (PMNS) mixing matrices are equal another signature of quark-lepton unification 3 mixing angles of leptons = 3 mixing angles of quarks The Dirac phase of leptons is same as that for quarks ============================================= Neutrinos are Majorana fermions with quasi-degenerate masses and with same CP (all Majorana mass eigen values have same sign, eg. positive) At high scale, we provide 3 ν mass eigen values S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.5/35

HUM : INPUTS At high GUT-seesaw scale (M R = 10 15 to 10 16 GeV), the quark (CKM) and lepton (PMNS) mixing matrices are equal another signature of quark-lepton unification 3 mixing angles of leptons = 3 mixing angles of quarks The Dirac phase of leptons is same as that for quarks ============================================= Neutrinos are Majorana fermions with quasi-degenerate masses and with same CP (all Majorana mass eigen values have same sign, eg. positive) At high scale, we provide 3 ν mass eigen values Two Maj. phases are additional input parameters at high scale S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.5/35

HUM : INPUTS The common mass of quasi-degenerate neutrinos at high scale required for HUM 0.15 ev m i 0.65 ev S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.6/35

HUM : INPUTS The common mass of quasi-degenerate neutrinos at high scale required for HUM 0.15 ev m i 0.65 ev Overlaps with the values claimed by Heidelberg-Moscow ββ 0ν experiment Satisfies the WMAP bound S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.6/35

HUM : INPUTS The common mass of quasi-degenerate neutrinos at high scale required for HUM 0.15 ev m i 0.65 ev Overlaps with the values claimed by Heidelberg-Moscow ββ 0ν experiment Satisfies the WMAP bound Range is accessible to the KATRIN experiment S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.6/35

HUM : INPUTS The common mass of quasi-degenerate neutrinos at high scale required for HUM 0.15 ev m i 0.65 ev Overlaps with the values claimed by Heidelberg-Moscow ββ 0ν experiment Satisfies the WMAP bound Range is accessible to the KATRIN experiment It works only for reasonably large values of tan β (40-60) S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.6/35

HUM : OUTPUTS Starting with the CKM mixing matrix for neutrinos at the GUT-seesaw scale, renormalization group evolution (RGE) to the weak scale leads to S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.7/35

HUM : OUTPUTS Starting with the CKM mixing matrix for neutrinos at the GUT-seesaw scale, renormalization group evolution (RGE) to the weak scale leads to Large solar and atmospheric mixings and small reactor angle in agreement with data small angles become larger radiative magnification Current work (hep-ph/0611225) suggests that HUM hypothesis works quite successfully in the presence of CP violating phases S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.7/35

HUM : OUTPUTS Starting with the CKM mixing matrix for neutrinos at the GUT-seesaw scale, renormalization group evolution (RGE) to the weak scale leads to Large solar and atmospheric mixings and small reactor angle in agreement with data small angles become larger radiative magnification Current work (hep-ph/0611225) suggests that HUM hypothesis works quite successfully in the presence of CP violating phases Provides an alternative way to understand the difficult problem of the diverse mixing patterns between quarks and leptons without relying on new mass textures for neutrinos or new family symmetries S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.7/35

HUM : The Basis Underlying gauge symmetry The basis of HUM hypothesis is quark-lepton unification theory of Pati and Salam through SU(4) C where fourth color is lepton S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.8/35

HUM : The Basis Underlying gauge symmetry The basis of HUM hypothesis is quark-lepton unification theory of Pati and Salam through SU(4) C where fourth color is lepton For first generation, ν e is the partner of three up quarks (red, blue, green) and electron is the partner of three down quarks (red, blue, green) The left-right symmetric Pati-Salam model SU(2) L SU(2) R SU(4) C times a global family symmetry S 4 which leads to quasi-degenerate light neutrinos through the type-ii seesaw mechanism S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.8/35

HUM : The Basis Underlying gauge symmetry The basis of HUM hypothesis is quark-lepton unification theory of Pati and Salam through SU(4) C where fourth color is lepton For first generation, ν e is the partner of three up quarks (red, blue, green) and electron is the partner of three down quarks (red, blue, green) The left-right symmetric Pati-Salam model SU(2) L SU(2) R SU(4) C times a global family symmetry S 4 which leads to quasi-degenerate light neutrinos through the type-ii seesaw mechanism The RH Majorana neutrino mass matrix is proportional to a unit matrix due to an S 4 symmetry S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.8/35

HUM : The Basis The underlying symmetry breaks at high scale and then MSSM is the theory from M R down to 1 T ev S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.9/35

HUM : The Basis The underlying symmetry breaks at high scale and then MSSM is the theory from M R down to 1 T ev Seesaw mechanism allows the Majorana phases at the high scale. Two Majorana phases are additional input parameters at high scale S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.9/35

HUM : The Basis The underlying symmetry breaks at high scale and then MSSM is the theory from M R down to 1 T ev Seesaw mechanism allows the Majorana phases at the high scale. Two Majorana phases are additional input parameters at high scale Susy scale is 1 T ev and below that we have non-susy Standard Model S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.9/35

HUM : Sketch Schematic diagram of HUM S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.10/35

Input parameters Low energy data for the CKM matrix given by PDG : V ub = 0.0037, V cb = 0.0413 V us = 0.2243, δ = 60 ± 14 J CKM CP = 2.89 10 5 S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.11/35

Input parameters Low energy data for the CKM matrix given by PDG : V ub = 0.0037, V cb = 0.0413 V us = 0.2243, δ = 60 ± 14 J CKM CP = 2.89 10 5 Due to the dominance of the top quark Yukawa coupling the one-loop renormalization corrections give, Vub 0 V ub = V 0 [ cb V cb exp y2 top ] 16π ln M 2 R M Z 0.83, while all other elements are almost unaffected S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.11/35

Input parameters Low energy data for the CKM matrix given by PDG : V ub = 0.0037, V cb = 0.0413 V us = 0.2243, δ = 60 ± 14 J CKM CP = 2.89 10 5 Due to the dominance of the top quark Yukawa coupling the one-loop renormalization corrections give, Vub 0 V ub = V 0 [ cb V cb exp y2 top ] 16π ln M 2 R M Z 0.83, while all other elements are almost unaffected Input values for the CKM matrix at the GUT-seesaw scale are, sin θ13 0 = 0.0031, sin θ0 23 = 0.034, sin θ0 12 = 0.224, δ0 = 60 JCP 0 = 2 10 5 S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.11/35

Defn. & Boundary Conditions At µ = M R : U P MNS (M R ) = V CKM (M R ) V D (M R ) S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.12/35

Defn. & Boundary Conditions At µ = M R : U P MNS (M R ) = V CKM (M R ) V D (M R ) U = µ < M R : we parameterize the PMNS matrix as c 13 c 12 c 13 s 12 s 13 e iδ c 23 s 12 c 12 s 13 s 23 e iδ c 12 c 23 s 12 s 13 s 23 e iδ c 13 s 23 s 12 s 23 c 12 s 13 c 23 e iδ c 12 s 23 c 23 s 13 s 12 e iδ c 13 c 23 diag. ( e iα 1, e iα 2, 1 ) At where c ij = cos θ ij and s ij = sin θ ij (i,j=1, 2,3) S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.12/35

Defn. & Boundary Conditions At µ = M R : U P MNS (M R ) = V CKM (M R ) V D (M R ) U = µ < M R : we parameterize the PMNS matrix as c 13 c 12 c 13 s 12 s 13 e iδ c 23 s 12 c 12 s 13 s 23 e iδ c 12 c 23 s 12 s 13 s 23 e iδ c 13 s 23 s 12 s 23 c 12 s 13 c 23 e iδ c 12 s 23 c 23 s 13 s 12 e iδ c 13 c 23 diag. ( e iα 1, e iα 2, 1 ) At where c ij = cos θ ij and s ij = sin θ ij (i,j=1, 2,3) All the mixing angles and phases are now scale dependent S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.12/35

Defn. & Boundary Conditions At µ = M R : U P MNS (M R ) = V CKM (M R ) V D (M R ) U = µ < M R : we parameterize the PMNS matrix as c 13 c 12 c 13 s 12 s 13 e iδ c 23 s 12 c 12 s 13 s 23 e iδ c 12 c 23 s 12 s 13 s 23 e iδ c 13 s 23 s 12 s 23 c 12 s 13 c 23 e iδ c 12 s 23 c 23 s 13 s 12 e iδ c 13 c 23 diag. ( e iα 1, e iα 2, 1 ) At where c ij = cos θ ij and s ij = sin θ ij (i,j=1, 2,3) All the mixing angles and phases are now scale dependent The Majorana phases are chosen to be two times of that in usual parameterizations for the sake of convenience in computation S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.12/35

Defn. & Boundary Conditions The values of the mixing angles at low energies will be obtained from RG evolution following the top-down approach under the boundary conditions: sin θ12 0 sin θ 12 (M R ) P MNS = sin θ 12 (M R ) CKM, sin θ23 0 sin θ 23 (M R ) P MNS = sin θ 23 (M R ) CKM, sin θ13 0 sin θ 13 (M R ) P MNS = sin θ 13 (M R ) CKM, δ 0 δ(m R ) P MNS = δ(m R ) CKM, S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.13/35

Defn. & Boundary Conditions The values of the mixing angles at low energies will be obtained from RG evolution following the top-down approach under the boundary conditions: sin θ12 0 sin θ 12 (M R ) P MNS = sin θ 12 (M R ) CKM, sin θ23 0 sin θ 23 (M R ) P MNS = sin θ 23 (M R ) CKM, sin θ13 0 sin θ 13 (M R ) P MNS = sin θ 13 (M R ) CKM, δ 0 δ(m R ) P MNS = δ(m R ) CKM, Two unknown Majorana phases at µ = M R will be treated as unknown parameters, α 0 1 and α0 2 S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.13/35

Low Energy Predictions We follow bottom up approach to obtain information on the Yukawa and gauge couplings at the GUT-seesaw scale to serve as inputs in the top down approach S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.14/35

Low Energy Predictions We follow bottom up approach to obtain information on the Yukawa and gauge couplings at the GUT-seesaw scale to serve as inputs in the top down approach The extrapolated values of the gauge and the Yukawa couplings for tan β = 55 at M R = 10 15 GeV are : g 0 1 = 0.6683, g0 2 = 0.6964, g0 3 = 0.7247, h 0 top = 0.8186, h0 b = 0.6437, and h0 τ = 0.7105 Data from neutrino oscillation experiments (within 4σ limit) S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.14/35

Low Energy Predictions We follow bottom up approach to obtain information on the Yukawa and gauge couplings at the GUT-seesaw scale to serve as inputs in the top down approach The extrapolated values of the gauge and the Yukawa couplings for tan β = 55 at M R = 10 15 GeV are : g 0 1 = 0.6683, g0 2 = 0.6964, g0 3 = 0.7247, h 0 top = 0.8186, h0 b = 0.6437, and h0 τ = 0.7105 Data from neutrino oscillation experiments (within 4σ limit) m 2 21 = (6.8 9.3) 10 5 ev 2, m 2 31 = (1.8 3.5) 10 3 ev 2, sin 2 θ 12 = 0.22 0.44, sin 2 θ 23 = 0.31 0.71, sin 2 θ 13 0.058 S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.14/35

RGE for masses RGEs are used directly for the mass eigenvalues and mixing angles including phases in the mass basis (S. Antusch etal. and J. A. Casas etal.) For the sake of RG solutions assume the three neutrino mass eigenvalues to be real and positive dm i dt = 2F τ (P i + G i )m i m i F u, (i = 1, 2, 3) S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.15/35

RGE for Mixings ds 23 dt = F τ c 23 sin 2θ 23 2(m 2 3 m2 2 ) [ c 2 12 (m 2 3 + m2 2 + 2m 3m 2 cos 2α 2 ) +s 2 12(m 2 3 + m 2 1 + 2m 3 m 1 cos 2α 1 )/(1 + R) ], proportional to s 23 c 2 23 c2 12 S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.16/35

RGE for Mixings ds 23 dt = F τ c 23 sin 2θ 23 2(m 2 3 m2 2 ) [ c 2 12 (m 2 3 + m2 2 + 2m 3m 2 cos 2α 2 ) +s 2 12(m 2 3 + m 2 1 + 2m 3 m 1 cos 2α 1 )/(1 + R) ], proportional to s 23 c 2 23 c2 12 ds 13 dt = F τ c 13 sin 2θ 12 sin 2θ 23 m 3 2(m 2 3 m2 1 ) [m 1 cos(2α 1 δ) (1 + R)m 2 cos(2α 2 δ) Rm 3 cos δ], S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.16/35

RGE for Mixings ds 23 dt = F τ c 23 sin 2θ 23 2(m 2 3 m2 2 ) [ c 2 12 (m 2 3 + m2 2 + 2m 3m 2 cos 2α 2 ) +s 2 12(m 2 3 + m 2 1 + 2m 3 m 1 cos 2α 1 )/(1 + R) ], proportional to s 23 c 2 23 c2 12 ds 12 dt ds 13 dt = F τ c 13 sin 2θ 12 sin 2θ 23 m 3 2(m 2 3 m2 1 ) [m 1 cos(2α 1 δ) (1 + R)m 2 cos(2α 2 δ) Rm 3 cos δ], = F τ c 12 sin 2θ 12 s 2 [ 23 m 2 1 + m 2 2 + 2m 1 m 2 cos(2α 2 2α 1 ) ] / [2(m 2 2 m 2 1 )] proportional to s 12 s 2 23 S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.16/35

RGE for Mixings where R = (m 2 2 m2 1 )/(m2 3 m2 2 ), P 1 = s 2 12 s2 23, P 2 = c 2 12 s2 23, P 3 = c 2 13 c2 23, G 3 = 0, but G 1 = 1 2 s 13 sin 2θ 12 sin 2θ 23 cos δ + s 2 13 c2 12 c2 23, G 2 = 1 2 s 13 sin 2θ 12 sin 2θ 23 cos δ + s 2 13 s2 12 c2 23, S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.17/35

RGE for Mixings In the case of MSSM with µ M S, F τ = h 2 τ / ( 16π 2 cos 2 β ), ( ) ( ) 1 6 F u = 16π 2 5 g2 1 + 6g2 2 6 h2 t sin 2 β, but, for µ M S, F τ = 3h 2 τ / ( 32π 2), F u = ( 3g2 2 2λ 6h 2 ) ( t / 16π 2 ). Jarlskog invariant : J CP = 1 8 sin 2θ 12 sin 2θ 23 sin 2θ 13 cos θ 13 sin δ S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.18/35

RGE for Dirac Phase dδ dt = F τ m 3 sin 2θ 12 sin 2θ 23 2θ 13 (m 2 3 m2 1 ) [m 1 sin(2α 1 δ) (1 + R)m 2 sin(2α 2 δ) + Rm 3 sin δ] [ m1 m 2 s 2 23 2F sin(2α 1 2α 2 ) τ (m 2 2 m2 1 ( ) +m 3 s 2 m1 cos 2θ 23 sin 2α 1 12 (m 2 3 m2 1 ) + m 2c 2 23 sin(2δ 2α ) 2) (m 2 3 m2 2 ( ) +m 3 c 2 m1 c 2 23 sin(2δ 2α 1) 12 (m 2 3 m2 1 ) + m )] 2 cos 2θ 23 sin 2α 2 (m 2 3 m2 2 ) S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.19/35

RGE for Majorana phases dα 1 dt [ (m 1 s 2 12 = 2F τ m 3 cos 2θ sin 2α 1 + (1 + R)m 2 c 2 12 sin 2α 2) 23 (m 2 3 m2 1 ) + m 1m 2 c 2 12 s2 23 sin(2α 1 2α 2 ) (m 2 2 m2 1 ) ] dα 2 dt [ (m 1 s 2 12 = 2F τ m 3 cos 2θ sin 2α 1 + (1 + R)m 2 c 2 12 sin 2α 2) 23 (m 2 3 m2 1 ) + m 1m 2 s 2 12 s2 23 sin(2α 1 2α 2 ) (m 2 2 m2 1 ) ] S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.20/35

Tuning in mass One point of utmost importance in this analysis is the nature of tuning needed in the neutrino mass eigenvalues which are inputs at the GUT seesaw scale. For quasi-degenerate neutrino masses having a common mass m 0 (m 2 m 1 ) = m2 21 2m 0, (m 3 m 2 ) = m2 32 2m 0 Using the experimental data from solar and atmospheric neutrino oscillations, m 2 21 = 8 10 5 ev 2 and m 2 32 = 2.4 10 3 ev 2 yields (m 2 m 1 )(ev ) = 0.004, 0.0002, 0.0001, 0.00004 (m 3 m 2 )(ev ) = 0.12, 0.006, 0.003, 0.0012 m 0 (ev ) = 0.01, 0.2, 0.4, 1.0 S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.21/35

Tuning in mass If m 0 = 0.1 1.0 ev, fitting the experimental data on m 2 requires tuning between m 1 and m 2 at least up to fourth place of decimals while fitting the data on m 2 atm needs tuning at least up to the third place of decimals between m 2 and m 3. During the course of RG evolution in the top-down approach m i m j and the rate of magnification for mixing angles increases Equivalently the solar and atmospheric mass differences decrease from their values at µ = M R to approach their experimental values at µ = M Z and the radiative magnification occurs for all the three mixing angles. S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.22/35

Magnification Damping While sin θ 12 and sin θ 23 attain their respective large values sin θ 13 remains small at low energies in spite of its magnification because its high scale starting value derived from the quark sector is much smaller(= V 0 ub ) Damping in the magnification of the mixing angles θ 12 and θ 23 sets in whenever cos 2α i (i = 1, 2), or cos 2(α 1 α 2 ) deviate from +1 For α 1 α 2 = (2n + 1)π/2 corresponding to the case of opposite CP-Parity of ν 1 and ν 2, magnification of θ 12 is prevented. Similarly if α 1 = α 2 = π/2, the CP-Parity of ν 3 is opposite to that of ν 1 or ν 2 and the mixing angle θ 23 can not be magnified S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.23/35

Results (M Z ) sin θ 13 0.18 0.16 0.14 0.12 0.1 0.08 0 α 2 δ 0 = 0 = 60 0 sinθ CHOOZ < 0.2 ( 3σ ) 1(a) α 1 (M Z ) (degree) 180 160 140 120 100 80 60 40 20 0 α 2 δ 0 = 0 = 60 0 1(b) 0.06 0 20 40 60 80 100 120 140 160 180 0 0 20 40 60 80 100 120 140 160 180 α 1 0 (degree) α 1 0 (degree) Figure 1: 1(a). Prediction of CHOOZ angle as function of input Majorana phase α 0 1 at the GUTseesaw scale while α 0 2 = 0. The vertical dashed lines define the total damping region for sin θ 12 magnification corresponding to α 0 1 = 90 ± 5. 1(b). Output RGE-solution for α 1 at low energies as a function of the input phase at the GUT-seesaw scale. S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.24/35

Results 10 100 α 2 (M Z ) (degree) 5 0 5 0 α 2 δ 0 = 0 = 60 0 2(a) (degree) δ (M Z ) 80 60 40 20 0 20 40 0 α 2 δ 0 = 0 = 60 0 2(b) 60 10 0 20 40 60 80 100 120 140 160 180 80 0 20 40 60 80 100 120 140 160 180 α 1 0 (degree) α 1 0 (degree) Figure 2: 2(a). Output RGE-solution for α 2 at low energies as a function of input α 0 1 at the GUTseesaw scale. 2(b). Output RGE-solution for leptonic Dirac phase at low energies as a function of α 0 1 S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.25/35

Results 0.1 0.01 α 2 0 = 0 δ 0 = 60 0 3(a) 80 60 δ q ( µ ) 3(b) J cp (M Z ) 0.001 1e 04 Phases 40 20 0 20 α ( µ ) 2 α ( µ ) 1 0 ( α 1 =30 0, α 0 2=0 0 ) 1e 05 0 20 40 60 80 100 120 140 160 180 α 1 0 (degree) δ ( µ) 40 2 4 6 8 10 12 14 log 10 ( µ/gev) Figure 3: 3(a).Prediction for the leptonic CP-violation parameter at low energies as a function of input Majorana phase α 0 1 at the GUT-seesaw scale. 3(b). Evolution of the leptonic Dirac phase and Majorana phases from the GUT-seesaw scale down to low energies. The horizontal dotted line shows the constancy of the CKM Dirac phase with unification point at the GUT-seesaw scale. S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.26/35

Results J cp 0.1 0.01 0.001 1e 04 q J cp(µ) 4(a) lepton J cp (µ) α 1 0 ( =120 α 1 0 ( =0 0 ( α 1 =0 0, α 0 0 2=0 ) log 10 ( µ/gev) α 0 0 =0 ) 0, 2 α 0 =30 0 ) 0, 2 1e 05 2 4 6 8 10 12 14 16 J cp 0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 q J cp(µ) 0 ( α 1 =30 0, α 0 0 2=0 ) lepton (µ) 0 0 ( α 1 =60, α 0 0 2=0 ) 0 α 1 α 0 0 = 16 ) J cp ( =60 0, 2 log 10 ( µ/gev) 4(b) 0.04 2 4 6 8 10 12 14 16 Figure 4: 4(a). Evolution of the leptonic CP-violation parameter from the GUT-seesaw scale to low energies for different input values of Majorana phases and for positive values of J CP. Almost horizontal dotted line shows slow evolution of the corresponding baryonic CP-violating parameter in the CKM matrix. 4(b). Same as Fig. 4(a) but only for negative values of the CP-violating parameter J CP obtained for different input values of Majorana phases. S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.27/35

Results 50 δ 0 = 60 0 fig. 5 30 (degree) α 2 0 10 10 30 50 150 100 50 0 50 100 150 α 1 0 (degree) Figure 5: Two branches of the parameter space defined for no threshold corrections through the magnification damping condition Cos 2(α 0 1 α0 2 ) = 0.9 at the GUT-seesaw scale. S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.28/35

Conclusions We have extended the results of the high scale mixing unification (HUM) hypothesis including the effects of the CP phases We have considered the cases with and without the Majorana phases. While the mixing unification hypothesis predicts the Dirac phase to be equal to the CKM phase, it leaves the Majorana phases arbitrary since they have no quark counterpart and will most likely arise from the right handed neutrino sector For both cases, we find consistent quasi-degenerate neutrino mass patterns that lead to desired amount of radiative magnification of the mixings in agreement with data S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.29/35

Conclusions (Contd..) The predictions of the model are as follows: the common mass of the neutrinos must be larger than 0.1 ev; the values of θ 13 and CP phases in the lepton sector are also predicted at low energies In the case without the Majorana phase, the low energy CP violating effect is small with J CP 7.6 10 5, whereas for the case with Majorana phases, J CP can be as large as 0.04 In the case with the Majorana phase, we predict a wider range of θ 13 from 3.5 10 Quasi-degenerate ν masses required to achieve desired radiative magnification, are consistent with the current cosmological bounds including those from WMAP. They are accessible in neutrinoless double beta decay experiments and overlaps the range of KATRIN experiment for beta decay S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.30/35

Results (contd..) Table 1: Solutions of RGEs for neutrino oscillation parameters with radiative magnification including Dirac and Majorana phases. The inputs for high scale mixings and the Dirac phase are from the CKM matrix as defined in the text. M l denotes the mass of slepton used for threshold correction for which we have used the wino mass M w = 150 GeV. The GUT-seesaw scale is M R = 10 15 GeV and tan β = 55. α 0 1 (deg) 40 60 140 45 α 0 2 (deg) 0 0 20 160 m 0 1 (ev) 0.4969 0.4774 0.4975 0.4777 m 0 2 (ev) 0.5 0.48 0.5 0.48 m 0 3 (ev) 0.577 0.554 0.577 0.554 α 1 (deg) 12.3195 17.6606 164.4949 15.4068 α 2 (deg) 5.1266 6.686 8.4804 170.6779 δ(deg) 43.513 54.7767 76.7738 70.0127 S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.31/35

Results (contd..) m 1 (ev) 0.37055 0.356065 0.371006 0.356279 m 2 (ev) 0.37128 0.356461 0.371422 0.35657 m 3 (ev) 0.37467 0.359634 0.3745 0.359535 ( m 2 21 ) RG(eV 2 ) 5.4339 10 4 2.822 10 4 3.085 10 4 2.076 10 4 ( m 2 31 ) RG(eV 2 ) 3.048 10 3 2.554 10 3 2.609 10 3 2.3307 10 3 Mẽ/M µ, τ 1.51 1.39 1.42 1.49 ( m 2 21 ) th(ev 2 ) 4.6339 10 4 2.02 10 4 2.285 10 4 1.1756 10 4 ( m 2 31 ) th(ev 2 ) 0.9712 10 3 0.418 10 3 0.476 10 3 0.272 10 3 S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.32/35

Results (contd..) m 2 21 (ev2 ) 8.0 10 5 8.0 10 5 8.0 10 5 9.0 10 5 m 2 31 (ev2 ) 2.076 10 3 2.1364 10 3 2.1329 10 3 2.0589 10 3 sin θ 12 0.5707 0.5497 0.5639 0.5828 sin θ 23 0.7211 0.7088 0.7145 0.7066 sin θ 13 0.1177 0.1472 0.1512 0.1545 J CP -0.0187-0.027 0.0335-0.0335 m ee (ev) 0.3524 0.322819 0.329399 0.314478 m e (ev) 0.37084 0.356259 0.371215 0.356453 The search for neutrinoless-double beta decay by Heidelberg-Moscow experiment has obtained the upper limit, < m ee >< (0.33 1.35) ev Similarly the current upper bound on the kinematical mass from Tritium beta decay is < m e >< 2.2 ev S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.33/35

Results (contd..) Table 2: Same as Table I. but without the necessity of threshold corrections. The two input Majorana phases satisfy a strong damping condition Cos 2(α 0 1 α0 2 ) = 0.9 which at the electroweak scale becomes moderate with Cos 2(α 1 α 2 ) 0.5 0.6 for different solutions. α 0 1 (deg) 45 60 65 75 α 0 2 (deg) -35-16 -12-2 m 0 1 (ev) 0.4781 0.4251 0.42504 0.43793 m 0 2 (ev) 0.48 0.427 0.427 0.44 m 0 3 (ev) 0.554 0.493 0.493 0.508 α 1 (deg) 20.9358 16.558 17.687 20.4018 α 2 (deg) -9.1444-9.877-9.259-7.6403 δ(deg) -81.3026-74.553-71.27-63.403 Cos 2(α 1 α 2 ) 0.497 0.603 0.589 0.558 m 1 (ev) 0.356619 0.31708 0.32672 0.326665 S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.34/35

Results (contd..) m 2 (ev) 0.356739 0.317208 0.32685 0.32679 m 3 (ev) 0.3593271 0.319905 0.32967 0.329693 m 2 21 (ev2 ) 8.56 10 5 8.12 10 5 8.107 10 5 8.17 10 5 m 2 31 (ev2 ) 1.938 10 3 1.8 10 3 1.939 10 3 1.987 10 3 sin θ 12 0.4924 0.5865 0.572 0.5369 sin θ 23 0.700 0.69023 0.6912 0.69174 sin θ 13 0.165 0.1602 0.1609 0.16026 J CP -0.034-0.03569-0.0348-0.03156 m ee (ev) 0.306543 0.274384 0.283384 0.28645 m e (ev) 0.356721 0.317195 0.326838 0.326778 S. K. Agarwalla JIGSAW 07 TIFR, Mumbai, India 19th Feb, 07 p.35/35