NEUTRINO COMPLEMENTARITY,SOME RECENT RESULTS. E. Torrente-Lujan. Dept. Fisica, Murcia & Milan University.

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1 NEUTRINO COMPLEMENTARITY,SOME RECENT RESULTS E. Torrente-Lujan. Dept. Fisica, Murcia & Milan University.

2 NEUTRINO COMPLEMENTARITY,SOME RECENT RESULTS E. Torrente-Lujan. Dept. Fisica, Murcia & Milan University. Complementarity (I) Complementarity and CP violation:the complex bimaximal model. (II) Quark lepton Complementariy and predictions for PMNS matrix. Based on works: 1. Complementarity and neutrino oscillations, Chauhan, Pulido (Lisboa), Picariello (Milan), E.T-L, hepph/ CP Complementarity and Complex Bimaximality. E. T-L, hepph/66nnn

3 GENERAL: COMPLEMENTARITY. LEPTONS:θ sun θ ± 2.4, θ 23 45, θ 3 < 9. QUARKS: θ C (M Z ) θ ±.15, θ , θ.2, and δ ckm 65. CONCLUSION: The mechanisms that determine lepton (and quark?) masses are different for charged and neutral leptons. any hope of a link?. Complementarity,Quark-lepton complementarity (QLC): Minakata 24 (C1) θ sun + θ C = θ PMNS 12 + θ CKM 12 = 45.1 ± 2.4 π 4 ; θpmns 23 ± θ CKM o. and (C2) θ PMNS θ CKM, or θ PMNS θ CKM 23 Accidental?, GUT?, Q-L symmetry?,flavor dynamics?, nothing related to θ C?. Maybe (C1), (C2) of different origin?.. U PMNS = U CKM U bimax

4 TYPES OF COMPLEMENTARITY MORE IN GENERAL: Effective description of the (small) deviation from maximal mixing parametrize U PMNS with λ, and U(λ) I. DIFFERENT CASES: U PMNS = U (λ) U bimax QUARK-LEPTON COMPLEMENTARITY (QLC): λ sin θ C ; U(λ) U CKM smallness of U e3 small θ C. GUT theories + flavour symmetries CP COMPLEMENTARITY: λ J, a CP ; U(λ) U(J), smallness of U e3 small a CP. USE COMPLEX BIMAXIMAL MATRICES Torrente-Lujan 1997 LR models with SCPV MIXED POSSIBILITY: U PMNS = U CKM(θ C )U CP(J)U bimax U CP explains smallness θ (C2), U CKM explains (C1). Why we want U bimax instead of, for example, U trimax? theoretical prejudices.

5 GENERAL: MECHANISM TOOLBOX& COMPLEMENTARITY Natural mechanisms to explain (unusual) neutrino flavour properties: 1) SEE-SAW very small masses. 2) Froggatt-Nielsen mechanism: abelian flavor symmetry breaking hierarchy of masses small mixing angles. θ ij m i m j in reasonable agreement with CKM matrix and θ C. fine tuning is needed if want to produce large mixing angles and use see-saw mechanism. 3) Non-abelian flavour SB: continuous or discrete. a robust mechanism, it admits only vanishing or maximal mixing angles (at least in simple SO(3) F, SU(2) F.) Barbieri-Hall-Kane 99,Raidal 4 Realistic models might need all: Abelian: generates light charged fermions, Non-Abelian: generates maximal mixing See-Saw: generates very small neutrino masses U PMNS = U CKM(θ C ) U bimax NEUT=Froggat. X NONABELIAN F

6 GUTS+Flavour QLC GUTS: additional relations. SU(5): Y d Y T l U d V l SO(1): Y u Y D U u U D U PMNS = U l U D V M = U l U d U CKM V M (SU(5)) = V T d U du CKM V M (SO(1)) U PMNS = AU CKM B. Diverse possibilities: 1)Flavor-Gauge SYMs: To include further assumptions, for example Y d symmetric so that U d = V d, or U d U l to obtain, in any case U PMNS = U CKM V M V M contains two large angles (BIMAX?), they could require non-abelian flavour symmetries. 2) No assumptions as before. U PMNS = K U CKM V M V M could contain only one large angle: a more easy alternative (Standard F-N mechanism).

7 CP COMPLEMENTARITY: Left Right Models with SCPV LR model with SCPV. EW VEVS Φ:(κe iα κ, κ e iα κ ): VEVS L, R triplets: (v R e iα R, v Le iα L ): SCPV κ, κ : real,positive. Y s: Yukawa matrices, Hermitian, real. Mass matrices: Dirac, Majorana: M u = κy ql + κ e iα κ Y qr ; M ν,d = κy ll + κ e iα κ Y lr M d = κ e iα κ Y ql + κy qr ; M l = κ e iα κ Y ll + κy lr (1) M ν,rr = v R Y M ; M ν,ll = v L e iα LY M κ /κ m t /m b >> 1, κ 2 + κ 2 = v 2, v L v R = β k 2,β 1. CP(quark) related to CP(lepton) through α k, o(κ /κ). α κ (to avoid FCNC). SEE-SAW TYPE II: M ν = Mν II Mν I M I ν M II ν = M t νdm ν,rr M ν,d = M ν,ll = v L e iα L Y M κ << κ M I ν k2 v R Y T ll Y 1 M Y ll = v L β Y T ll Y 1 M Y ll. FULL SEE SAW (II+I), M ν = v L e iα L Y M v L β Y T ll Y 1 M Y ll CP Dirac and Majorana phases δ D, α 1M, α 2M : depend on a single phase α L. GUT embedings,additional relations for Yuks

8 CP COMPLEMENTARITY: MAXIMAL MATRICES. Unitary Mixing matrices depend: on 4 moduli + sgn(j) BASICALLY 2 POSSIBILITIES: SIMPLEST POSSIBILITY: Threefold maximal lepton mixing,no free parameters: all the elements have equal modulus. J=. P ll = P(ν l ν l ) = P = 1/3(SUN). Harrison 1995 NEXT TO SIMPLEST: QUASI MAXIMAL(COMPLEX BIMAXIMAL): maximal in a 2 2 subsector: One free parameter, J connect Ps and CP asymmetry. Torrente-Lujan,1996. U PMNS = 1 k α β γ ǫ exp iλ 1 exp iλ 2 δ exp iλ 3 exp iλ 4 PARTICULAR CASES: BIMAXIMAL ANSATZ:real matrix, ACP=,No free parameters. Barger,Pakvasa BIMAXIMAL WITH CP MAXIMAL: Georgi,Glashow OTHER: Perturbations of the real bimaximal. Rodejohann 25-26,Ferrandis, Pakvasa (22,25). TRIBIMAXIMAL. Mohapatra 1998

9 CP COMPLEMENTARITY:A CONCRETE TOY MODEL. Working examples (α, β real): (I)U MNS (λ) = 1 ik β β α exp( iλ) exp iλ β exp iλ exp( iλ) β ; (II)U MNS (λ) = 1 ik α β β β exp iλ exp( iλ) β exp( iλ) exp(iλ) TO SHOW: U PMNS (λ) = U (λ)u bimax. CASE I: θ 12 = θ 23 = π/4. BUT θ. CASE II: θ 23 = π/4. By unitarity: All related to λ. α = 2 sin(λ), β 2 = 2 cos(2λ), k 2 = 2(1 + cos(2λ)) Det U(λ) = sign cos λ(= 1, if λ < π/2). λ = : BIMAXIMAL, J =, (4s = m 3 + m 2 2m 1,2d = m 2 m 3 ) u = i i 2 1 i i 2 2 ; M = um Du = s s d s s d d d s (2) Aproximate Complex Bimaximal maximal (unitary to 2nd order): U(λ) MNSP = i/ 2 i/ 2 iλ 1/2 iλ/2 1/2 + iλ/2 i/ 2 1/2 + iλ/2 1/2 iλ/2 i/ + O(λ2 ) 2

10 COMPLEMENTARITY: Factorization SHOW COMPLEMENTARITY: U PMNS (λ) = U (λ)u bimax A non trivial property, non shared by approximate bimax perturbations. U(λ) U (λ) = 1. And U() = U bimax. Let us define W(λ) = U bimax U PMNS(λ). W() = I. Stone theorem, apply it to W: dw dλ λ== U bimax du dλ λ= ia; W(λ) = exp(iaλ) Computing,We have where U PMNS (λ) = exp( iaλ)u bimax U PMNS (λ) = exp i λ ( J 12 + J ) 2 1 J 12 = i 1, i/ 2 i/ 2 1 J = i 1 1/2 1/2 i/ 2 1/2 1/2 i/ 2 VERY SIMPLE rotation around the BIMAX(J=): (ij = J 12 J ), symmetry Hint?.

11 CP asymmetry: relate λ with CP J and Acp: J = Im[u 11 u 22 u 21 u 12 ] = 1 2 α k 2 β k 2 = 1 sin(4λ) 16cos 4 (λ) λ 4 a CP 2 I(u 11u 22 u 12 u 21 ) u 11 u u 12 u 21 2 = sin 2λ 2λ. J U e3 2 = β k 2, U e2 2 = α k 2, MODEL PREDICTION U e3 2 U e2 2 = 2J Π 4.2 Π 8 Λ Π 8 Π 4 ATM,CHOOZ U e3 2 <.4(95%CL) U e3 2 = α k 2 = 2 sin2 λ 1 + cos(2λ) λ2 Ue3 ^2.1 λ < π/16, sin λ <.2, acp = sin 2λ <.38, J < SinΛ

12 CP COMPLEMENTARITY: EXP EVIDENCE FIT TO DATA?,at least as BIMAX, one free parm better fits ANALYSIS including kamland χ 2 = χ 2 glob,kl ; Take m ev 2. KL global (766 t/yr) R exp =.66 ±.5. RESULTS: Minima χ 2 min better than BIMAX m² ev² m² ev² λ min =.17 m 2 12 = ev 2 Π 16 1 Π 8 Λ 3Π 16 Π SinΛ CP MEASUREMENT J λ/4 =.42 Pee Needed: Compare with solar data complicated, Matter effects, BUT qualitatively one expects good agreement, Pee(Sun).3.5 for L/E. Π 4 Π 8 Λ Π 8 Π 4

13 QUARK-LEPTON COMPLEMENTARITY:, νs and SM θ PMNS = deg. Chauhan,Picariello, Pulido, E. T-L 26, hepph/65123 ASSUME U PMNS = U CKMV M. CORRELATION V M = U CKM U PMNS. V M : Ω = diag(e iω 1, eiω 2, eiω 3 ) V M = U CKM Ω U PMNS V M U 23 (θ)φu (θ)φ U 12 (θ); U PMNS = U 23 Φ U Φ U 12 Φ m. QUESTIONS: BOTTOM-UP:which V M is allowed by data? is this V M ) =?, is this BIMAX, TRIMAX? TOP-DOWN:assuming V M bimax, or V M ) =?, any prediction for the less known θ? Controversy?, Related to accidental complementarity or not. BIMAX 2 large angles non abelian flavour symmetries at work?.

14 BOTTOM-UP:Which V M from DATA?: VM bimaximal? V M, CAN VANISH, without special fine tuning (QLC): V M = U CKM Ω U PMNS sin 2 θ V M = 1 λ2 2 e i(ω 1 ω 2 φ) sin θ PMNS + λ sin θ PMNS 23 cos θ PMNS + O(λ 3 ) 2. sin θ 23 tan θ 1 λ, or, for θ small, θ 23 maximal, θ PMNS λ 2 USING CENTRAL VALUES: 1) CKM WOLFENSTEIN parameters λ =.2237, η =.317, ρ =.225, V cb Aλ 2 =.41, 2) PMNS mixing angles θ PMNS 12 = 34, θ PMNS 23 = 45, θ PMNS = FIXED(3 ). 3) Ω phases: MC flat distributions in the interval [, 2π]. range of values: (V M ) = ) (V M ) 1,3, not bi or tribimaximal. Already obtained. BUT TOO NAIVE. OTHER CENTRAL VALUES? Sin 2 VM Θ , Sin 2 PMNS Θ

15 BOTTOM-UP: A DETAILED STUDY. DETAILED STUDY, USE: V M = U CKM Ω U PMNS 1) Updated CKM, P M N S at 95%CL CKMfitter( Charles:24jd ) 1 V Θ M V 12 Θ M 23 λ = , A = , η = , ρ = with and ρ + iη = 1 A 2 λ 4 (ρ + iη) 1 λ 2 [1 A 2 λ 4 (ρ + iη)] ; sin 2 θ PMNS 12 = , sin2 θ PMNS 23 = , sin2 θ PMNS = Prob Tan 2 Θ 2) Two-sided Gaussian distributions CKM,PMNS parameters Unknown phases: vary in the interval [, 2π] (flat distribution). RESULTS: tan 2 θ V M 23 [.35, 1.4] tan 2 θ V M (= 1. BIMAX), ((=.5 TRIBIMAX). sin 2 θ V M = is preferred. Prob Σ 2Σ Sin 2 V Θ M

16 TOP-DOWN APPROACH: Predictions V PMNS. θ PMNS 1 V M BIMAXIMAL OR TRIBIMAXIMAL, (V M ). PMNS θ PMNS USE U PMNS = (U CKM Ω) 1 V M CONDITIONS: 1)variation U CKM two-sided Gaussians, 2) Ω phases: flat distributions in [, 2π]. 3) FIX V M =BIMAX,TRIBIMAX. FIX tan 2 θ V M 12 = 1.. (No dependence on tan 2 θ V M 12 ) FIX tan 2 θ V M 23 {.5, 1., 1.4}. FIX sin 2 θ V M =. Prob. Prob Tan 2 PMNS Θ Tan 2 PMNS Θ 23

17 RESULTS: sin 2 θ PMNS strongly peaked at (7.3, 8.9, 9.8 ). FOR ALL the physical span of tan 2 θ V M 23 {.5, 1., 1.4}. Prob. 1.5 force θ PMNS to be between 7 and 1. VM BIMAX: sin 2 θ V M =, tan 2 θ23 V M = 1: θ PMN 9 ± 1. ANOTHER LOOK: [ ( ) (U PMNS ) = e iω 1 1 λ2 2 sin θ V M e iφvm λsin θ V M 23 cos θ V M +Aλ 3 ( ρ + i η + 1)cos θ V M 23 cos θ V M + O(λ 4 ) so that, imposing sin 2 θ V M =, A O(1). sin 2 θ PMNS = sin 2 θ V M 23 λ 2 + O(λ 3 ), ], Sin 2 PMNS Θ Sin 2 MNS Θ 2Σ 1Σ Tan 2 V Θ M 23

18 Summary/Conclusions COMPLEMENTARITY: neutrino angles are nearly maximal, but not exactly maximal. DIFFERENT SCENARIOS: CP COMPLEMENTARITY. U PMNS = U (J) V M Complex Bimaximal matrices, one parameter: λ J, a CP, smallness of U e3 small a CP. U e3 J. Estimation of Acp from A) CHOOZ J < B) Kamland: J =.42 any link to LR SCPV models?. QUARK-LEPTON COMPLEMENTARITY (QLC) smallness of U e3 small θ C. a) DATA(NEUTRINOS+CKM) favour (V M ) 1,3. V M can be bimax or tribimax (Non abelian F symms?). b) (V M ) = + CKM data θ PMNS = deg. CKM-PMNS correlation from GUT+Non Abelian Flavour symmetries. A THIRD POSSIBILITY? CP +CKM complementarity?: two small parameters fits. U PMNS = U CKMU CP(J) V M

19 SOME FORMULAS,SEE SAW QUARKS, LEPTONS: M u, M d, M l, M ν,d : Dirac mass matrices for the up, down sectors. They are diagonalized by M u = U u Mu DV u ; M l = U l Ml D V l M d = U d Md D V d ; M (3) νd = U ν,d (M ν,d ) D (V ν,d ) T CKM and lepton (DIRAC) mixing U CKM = U u U d. U PMNS = U l U ν,d. MAJORANA NEUTRINOS: (SEE SAW TYPE I), the light neutrino mass matrix M ν : complex, symmetric. M ν = M D (M R ) 1 M T D = U ν M D ν U T ν (4) FULL PMNS: lepton mixing matrix U PMNS = U l U ν. DEFINE New matrix C: M ν U ν,d CU T ν,d then C (M ν,d ) D (V ν,d ) M 1 R (V ν,d) (M ν,d ) D. The matrix V M diagonalizes this matrix C = V M M D ν V T M, U PMNS = U l U ν,d V M. We can ask: where the large angles reside?, none,one,two, in V M?. can we relate the hierarchy nature of M R to the number of small or large angles in V M? if M R M R I C diagonal V M = I. V M measures the degeneracy of M R. If signficant nonunitarity: until which degree V M is non-unitary?.

20 QLC and renormalization QLC holds at low energies, but Q-L symmetry or unification is most probable and some high energy GUT scale. are renorm effects small enough to keep this relation from high to low scales?. For θ C : in SM or MSSM the effect is small. in MSSM For tanβ = 5, sin θ C (M Z ) =.2225, sin θ C (GUT) = For νs: strong dependence on the mass hierarchy. normal mass hierchy m 1 < m 2 << m 3 : very small effect. inverted mass hierarchy,degenerate or quasi degenerate spectrum: the effect can be large (depending on the y τ yukawa). Specially in MSSM with large tanβ. dθ 12 dt Cy2 τ 32π = 2 (5) = 31 7 (SM) (6) = 31 7 (1 + tan 2 β) (MSSM) (7)