RG evolution of neutrino parameters

Transcription

1 RG evolution of neutrino parameters ( In TeV scale seesaw models ) Sasmita Mishra Physical Research Laboratory, Ahmedabad, India Based on arxiv: November 12, 2013 Institute of Physics, Bhubaneswar. p.1/27

2 Outline of the talk Introduction Neutrino masses and mixing Seesaw mechanism Renormalization group effet (RG) RG and Tribimaximal mixing matrix Analytical and numerical results Conclusion. p.2/27

3 Evidence for Neutrino Mass Experimental observations using solar, atmospheric, reactor and accelerator neutrinos have confirmed that neutrinos oscillate between flavours ( ) This is possible if neutrinos have mass and mixing Neutrino mass cannot be accommodated naturally in Standard Model Need to go for a theory Beyond Standard Model Only those theories which can successfully explain all the neutrino oscillation parameters are allowed. p.3/27

4 The Neutrino Mass Matrix The neutrino mass matrix at low energy: m ν = U PMNS Diag(m 1,m 2,m 3 )U PMNS U PMNS = R 23 (θ 23 )R 13 (θ 13,δ)R 12 (θ 12 )P(σ,ρ) Leptonic mixing matrix relating the flavour and mass eigenstates U PMNS = c 12 c 13 s 12 c 13 s 13 e iδ s 12 c 23 c 12 s 23 s 13 e iδ c 12 c 23 s 12 s 23 s 13 e iδ s 23 c 13 P s 12 s 23 c 12 c 23 s 13 e iδ c 12 s 23 s 12 c 23 s 13 e iδ c 23 c 13. p.4/27

5 Parameters of the Neutrino Mass Matrix 9 unknown parameters for three neutrino flavours 3 masses, m 1, m 2 and m 3 3 mixing angles 3 phases, 1 Dirac, 2 Majorana Oscillation experiments sensitive to 2 mass squared differences ( m 2 21, m 2 31 m 2 32) 3 mixing angles (θ 12, θ 13, θ 23 ) 1 Dirac phase. p.5/27

6 Neutrino Masses: Ordering Normal Hierarchy Inverted Hierarchy Solar data tells m 2 21 > 0 But sign of m 2 31 not known m 3 m 2 m 1 Normal Ordering : m 2 31 >0 m 2 31 <0 m 3 >> m 2 >> m 1 m 2 Inverted Ordering : m 1 m 3 m 2 3 << m2 2 m2 1 m2 atm Quasi-Degenerate m 3 m 2 m 1 >> m 2 atm. p.6/27

7 Neutrino Masses For normal hierarchy: m 1 0, m ev and m ev fermion masses d s b (large angle MSW) ν 1 ν 2 ν 3 u c e µ τ t TeV GeV MeV kev ev mev µev Neutrino masses much smaller than quark and charged lepton masses Hierarchy of neutrino masses not strong : m 3 /m 2 6 Completely different from quark sector Inverted hierarchy and quasi-degeneracy has no analogue in quark sector. p.7/27

8 Models for Neutrino Mass: Key Issues Why two large and one small mixing angle unlike in quark sector where all mixing angles are small U PMNS V CKM Current data sin 2 θ 23 = 0.5, sin 2 θ 12 = 0.33, sin 2 θ 13 = 0.02( 0) Tri-Bimaximal Form. p.8/27

9 Seesaw Mechanism Relates the smallness of neutrino mass to some new physics at high scale M = m T D M 1 R m D; m D vy ν m ν 0.05 ev for M R = GeV, m D 100 GeV, Y ν 1. p.9/27

10 Origin of Seesaw Heavy field present at high scale Λ Tree level exchange of this heavy particle = effective dimension 5 operator at low scale L = κ 5 l L l L φφ, κ 5 = κ/λ φ a φ d v v Weinberg, 79 κ 5 EW symmetry breaking κ 5 l g Lc l f Lb ν g L ν f L Majorana Mass : κ 5 v 2 In the seesaw approximation κ = Y ν T Λ 1 Y ν κ depends on Heavy particles and their couplings. p.10/27

11 Three types of Seesaw L eff = κ 5 l l φ φ Three ways to form a gauge singlet with the SM doublets l andφ Type-I : l and φ form a singlet [ 2 2 = 3 1] Mediated by singlet Fermions Type-II: l and l (and φ and φ) form a triplet [ 3 3 = 5 3 1] Mediated by SU(2) triplet Higgs Type-III: l and φ form a triplet [ 3 3 = 5 3 1] Mediated by SU(2) triplet fermions Mediated by tree level exchange at a high scale heavy particle mass Predictive theory at high scale ( Type I, II SO(10); Type III SU(5)) How heavy? TeV scale seesaw Low energy signatures? Probing seesaw at LHC?. p.11/27

12 Type-I seesaw L = L SM +(Y ν )N R φ l L N R c (M R ) ij N R + h.c The Majorana mass matrix at seesaw scale M ν = 0 m D ; m D = Y ν v m T D M R The light neutrino mass matrix after the seesaw diagonalization assuming M R m D is given by M ν = m T D M 1 R m D The Majorana neutrino mass matrix M ν is symmetric and can be diagonalized as V T ν MV ν = diag(m 1,m 2,m 3 ) V ν the neutrino mixing matrix L CC = g 2 l L γ µ ν L W µ +h.c = l L V l V νν L W µ U PMNS = V l V ν In a basis where the charged lepton mass matrix is diagonal U PMNS = V ν Connects flavour states to mass states. p.12/27

13 Renormalization group effects The neutrino mass matrix at the high scale Λ originates from a dimension-5 operator M Λ αβ = κ αβ (l α.h)(l β.h) Λ. κ αβ runs with energy scale. p.13/27

14 RG equations 16π 2 dκ dt = Cκ(Y l Y l Y νy ν )+C(Y l Y l Y νy ν ) T κ+kκ t = lnµ The first two terms contain the charged lepton Yukawa coupling and thus distinguish between generations Responsible for the running of mixing parameters K is flavour blind One has to consider the running of all the quantities simultaneously Can be solved numerically MSSM C = 1 K = (6g g 2 Y +2S) S = Tr(3Y uy u ) SM C = -3/2 K = (3g2 2 +2λ+2S) S = Tr(Y l Y l +Y νy ν +3Y uy u +3Y d Y d) Babu,Leung,Pantaleone, 1993; Drees, Kersten, Lindner,Ratz, 2001,2002. p.14/27

15 Threshold effect M 1,M 2,M 3 : heavy particles coupled to the theory (SM) At scale µ >> M 2 : particles of masses, >> M 2 coupled. At scale µ << M 2 : particles of masses, >> M 2 decoupled. At µ = M 2, parameters are continuous. Matching conditions are needed. EFT 1 κ EFT 2 (2) κ, Y (2) ν, M (2) EFT 3 (3) κ, Y (3) ν, M (3) Y ν, M Full theory m ν = v2 (2) κ 4 v2 (2) Y T (2) ν M 1(2) Y ν 2 m ν = v2 (3) κ 4 v2 (3) Y T (3) ν M 1(3) Y ν 2 m ν = v2 2 Y T ν M 1 Y ν integrate out N 2 R integrate out NR 1 integrate out NR 3 µ M 1 M 2 M 3 M GUT. p.15/27

16 Analytic Method The RG equation: 16π 2 dκ dt = Cκ(Y l Y l Y νy ν )+C(Y l Y l Y νy ν ) T κ+kκ κ(λ) = I K I T κκ(λ)i κ = M λ ν = I K I T κ M Λ ν I κ, Where we define the integrals : I K exp I κ exp [ t(λ) t(λ) K(t)dt ] [ t(λ) t(λ) (Y l Y l Y νy ν )(t)dt ], with, Y l Y l = Diag (y 2 e,y 2 µ,y 2 τ). t(q) (16π 2 ) 1 ln(q/q 0 ) Assuming ye,µ 2 << yτ 2, I κ Diag(1,1,e τ ) = Diag(1,1,1 τ )+O( 2 τ), τ = t(λ) t(λ) y τ(t) Y νy ν withy ν = without Y ν Ellis and Lola, PLB, 1999; Chankowski & Pokorski MPA, p.16/27

17 Numerical Procedure Taking all the experimental available results; angles and masses, construct U PMNS. Phases varied randomly. Construct κ = 4 v 2 (m ν ); m ν = U PMNS Diag(m 1,m 2,m 3 )U PMNS Run κ from the SM scale to 1 TeV. Matrix equations = diagonalize at each step. Extract the parameters; angles, masses and phases At scale 1TeV, new degrees of freedom appear; Fermion singlets The RG equation changes and matching condition is imposed Presence of fermion singlets = new Yukawa couplings Again run all the parameters, diagonalize at each step and extract the parameters. Our case; upto GeV. p.17/27

18 Tri-bimaximal Mixing Present data gives two large and one small mixing Close to Tri-bimaximal form U TBM = /2 diag(eiα,e iα 2/2,e iα 3/2 ), Harrison, Perkins and Scott sin 2 θ 23 = 0.5, sin 2 θ 12 = 0.33, sin 2 θ 13 = 0.03 Key Question : Is there some underlying Symmetry? Hint of non-zero θ 13 = symmetry is approximate, Renormalization Group effects. p.18/27

19 TBM: Radiative corrections to mixing angles At low scale : θ ij = θ Λ ij +k eij e +k µij µ +k τ ij τ δ = δ Λ +d e e +d µ µ +d τ τ αi λ = αλ i +a ei e +a µi µ +a τi τ ; i = 1,2 sin 2 θ 12 sin 2 θ12 Λ k e12( e cos 2 θ23 Λ µ sin 2 θ23 Λ ) τ sin 2 θ 23 sin 2 θ23 Λ k µ 23 ( µ τ ) sin 2 θ 13 sin 2 θ13 Λ k µ 23 ( µ τ ) Two cases: Hierarchical Neutrinos; NH and IH Adding two singlets with opposite lepton number to the SM. Quasidegenerate Adding three fermion singlets to the SM Both models give TeV scale seesaw. p.19/27

20 Normal ordering: NH 0.38 NH NH sin 2 θ sin 2 θ sin 2 θ log(µ(gev)) log(µ(gev)) log(µ(gev)) sin 2 θ ( 1 2 sin2θλ 12 e cos 2 θ23 Λ µ sin 2 ) θ23 Λ τ sin 2 θ sin2θλ 12sin2θ23[r Λ + rcos(α2)]( Λ µ τ ) sin 2 θ sin2θλ 23[1+2cos 2 θ12(r+ Λ rcos(α2))]( Λ µ τ ) r = m2 sol m 2 atm p.20/27

21 Inverted ordering: sin 2 θ IH sin 2 θ IH sin 2 θ IH log(µ(gev)) sin 2 θ sin2θλ log(µ(gev)) ( 1+cos(α Λ 1 α Λ 2 ) r log(µ(gev)) ) ( e cos 2 θ Λ 23 µ sin 2 θ Λ 23 τ ) sin 2 θ sin2θλ 12sin2θ23 Λ m 3 [cosα m 2 2 Λ cosα1]( Λ µ τ ) 13 sin 2 θ sin2θλ 23[1+ 2m 3 (cosα m 2 2 Λ cos 2 θ12 Λ +cosα1 Λ sin 2 θ12)]( Λ µ τ ) 13. p.21/27

22 Quasidegenerate neutrinos: sin 2 θ 13 sin 2 θ 12 sin 2 θ 23 QD sin 2 θ ij log(µ(gev)) sin 2 θ sin2θ Λ 12 m 0 2 m 2 21 [1+cos(α Λ 1 α Λ 2)] ( e cos 2 θ Λ 23 µ sin 2 θ Λ 23 τ ) sin 2 θ sin2θλ 12sin2θ Λ 23 sin 2 θ 13 m2 0 m 2 32 m 2 0 [cosα m 2 2 Λ cosα1]( Λ µ τ ) 32 sin2θ Λ 23[1+cosα Λ 2 cos 2 θ Λ 12 +cosα Λ 1 sin 2 θ Λ 12]( µ τ ). p.22/27

23 General remark In general the runnig effect on angles is more in quasidegenerate case than in the hierarchical case. So starting with a zero value for θ 13 it is possible to generate nonzero θ 13 at low scale compatible with experiments. In hierarchical case: in IH θ 12 runs considerably. Other angles do not change even taking threshold effects into account.. p.23/27

24 Ongoing work τ MSSM,tanβ = SM = O( 2 τ) terms negligible Running is opposite for SM and MSSM Running is very small in SM, in MSSM enhancement by tan 2 β. p.24/27

25 Uplifted Spersymmetry Couplings constants in MSSM to be perturbative up to Planck scale requires 2 tanβ 50 MSSM is a viable theory when tanβ 50 The holomorphy of the superpotential demands that the up-type fermions couple to H u and down-type fermions couple only to H d Once supersymmetry is broken the non-holomorphic terms of the form Qu c H d and QDc H u are allowed Yukawa couplings get modified: y l = y lα 8π G The resulting lepton mass is given by : m l = y l v d +y l v u τ = m2 τ (1+tan 2 β) 8π 2 v 2 (1+ αg 8π tanβ)2 lnλ/λ But αg 8π = Can be constrained from neutrino data.. p.25/27

26 Conclusions Global oscillation data give Two large (θ 12 and θ 23 ) and one small (θ 13 ) angle Weak hierarchy of masses (for NH) can also be inverted hierarchy or quasi degenerate Seesaw mechanism can explain smallness of neutrino mass Renormalization group effects including threshold effect are important in view of the onset of precision era in neutrino physics In MSSM enhancement by tan 2 β The non-zero hint of θ 13 cannot be accommodated in TBM + RG effects if the effective theory is SM Runnig effect is more in quasidegenerate case than in the hierarchical case.. p.26/27

27 Future plan Implementing the threshold effect in left right symmetric models. Also in versions of two higgs doublet models and supersymmetric models. Thank you. p.27/27