Puzzles of Neutrinos and Anti-Matter: Hidden Symmetries and Symmetry Breaking. Hong-Jian He

Transcription

1 Puzzles of Neutrinos and Anti-Matter: Hidden Symmetries and Symmetry Breaking Tsinghua University Collaborators: Shao-Feng Ge & Fu-Rong Yin Based on arxiv: (and works in preparation)

2 Outline 1 Motivations Experimental Data Approximating µ τ & Dirac CP Symmetries 2 General Construction The Unique Model Solving Low Energy Parameters Correlation between θ 13 & θ 23 Low Energy Observables J & M ee 3 Connection with Leptogenesis Baryon Asymmetry η B CP Asymmetry Parameter ǫ 1 Prediction of Leptogenesis for Baryon Asymmetry η B Constrains of Leptogenesis on Low Energy Observables J & M ee 4 New Hidden Symmetry Dictating θ 12 Conjecture of Possible Hidden Symmetry Representation of the Hidden Symmetry 5 Summary

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4 Experimental Data Current Neutrino Oscillation Data ν-parameters Lower Limit Best Value Upper Limit (2σ) (2σ) m 2 21 (10 5 ev 2 ) m 2 31 (10 3 ev 2 ) sin 2 θ 12 (θ 12 ) (31.8 ) (34.0 ) (36.4 ) sin 2 θ 23 (θ 23 ) (37.2 ) (43.0 ) (50.9 ) sin 2 θ 13 (θ 13 ) 0 (0 ) (7.3 ) (10.9 )

5 Approximating Symmetry Evidence of µ τ Symmetry at Low Energy Two small deviations (2σ level): 7.8 < θ < 5.9, 0 < θ 13 < 10.9 with Best Fitted Value: θ = 2.0, θ 13 = 7.3. µ τ Symmetric Limit as Good 0th Order Approximation: θ 23 = 45, θ 13 = 0. corresponding to mass matrix with Zero Dirac CP Phase: A B B M ν (0) = C D C

6 Assignment Model Assignment at Zeroth Order M (0) ν Minimal Seesaw & µ τ and CP Symmetries: T µτ (3) = 1 ( ) 1 T µτ (2) 1 = 1 1 T (3) µτ m D T (2) µτ m D & T (2) µτ M R T (2) µτ M R : a a ( ) m D = b c m22 m, M R = 23 m c b 23 m 33 with all elements being REAL. Seesaw Mass Matrix for light neutrinos (M ± m 22 ± m 23 ): m D M 1 R mt D = 2a 2 M + a(b+c) M + [ ] 1 (b+c) 2 2 M + + (b c)2 M a(b+c) M + [ 1 (b+c) 2 2 M + [ (b+c) M + ] (b c)2 M ] + (b c)2 M

7 Assignment Common µ τ & CP Soft Breaking Approximate µ τ symmetry at Zeroth-Order vanishing θ 13 & Dirac CP Phase δ D ; So, µ τ breaking should be Small and Simultaneously generates δ D µ τ & Dirac CP broken by a Common Origin. Natural and Simple, so Tempting, to expect a Common Origin for all CP Phases; Conjecture: µ τ & CP Symmetries are Softly broken from a Common Origin which is Uniquely determined as: ( ) ( 1 R M R = m 22 R 1 ζe iω R m ) 23 m 22 Note: µ τ & CP Recover with ζ 0.

8 Assignment The Consequence: Main Predictions δ a ( θ ) & δ x ( θ 13 ) Common Origin & Linear δ a δ x ; Once θ 23 well measured Predict θ 13! Dirac CP Phase δ D & Majorana CP Phases Common Origin Correlated; Once Dirac CP Phase δ D is measured J & M ee ; Vice Versa, Constrains from Leptogenesis. Normal Hierarchy with m 1 = 0. Fully reconstructed mass spectrum M ee ; Vice Versa.

9 Solving Low Energy Parameters Expanding Mass Matrix M ν Expanding the mass matrix M ν in terms of r & ζ up to Linear Order: M ν = M ν (0) + M ν (1) + O(r 2,rζ,ζ 2 ) M (0) ν = M (1) ν = with: (b c) (2 X)M 10 1 with an Overall Phase (No Physical Consequence) r (2 X) 2 a 2 (2 X)[(1 X)b + c]a (2 X)[b + (1 X)c]a (2 X) 2 [(1 X)b + c] 2 (1 X)(b + c) 2 + X 2 bc M 10 [b + (1 X)c] 2 eµ δm eτ (1) δm µµ (1) δm µτ (1) δm ττ (1) δm (1) ee δm (1) Reconstruction where r 1 R & X ζ r eiω and M 10 is the Zeroth-Order of the Lightest Eigenvalue of M R : M 10 = rm 22

10 Solving Low Energy Parameters Reconstruction of Light Neutrino Mass Matrix Note: Majorana Neutrino s mass matrix is Symmetric: m ee m eµ m eτ 1 M ν V D ν V = m µµ m µτ with D ν m m 2 m ττ where: V U UU, U diag(e iα 1,e iα 2,e iα 3 ), U diag(e iφ 1,e iφ 2,e iφ 3 ); c s c x s s c x s x e iδ D U s s c a c s s a s x e iδ D c s c a + s s s a s x e iδ D s a c x s s s a + c s c a s x e iδ D c s s a s s c a s x e iδ d c a c x (θ x θ 13,θ s θ 12,θ a θ 23 ) Note: of the Six Rephasing Phases, only Five are Independent. m 3

11 Solving Low Energy Parameters Reconstructed Mass Matrix Elements m ee = e i2α [ 1 c 2 s cx 2 m 1 + ss 2 cx 2 m 2 + sxe 2 ] 2iδD m 3, m µµ = e i2α [ 2 (ss c a c s s a s x e iδ D ) 2 m 1 + (c s c a + s s s a s x e iδ D ) 2 m 2 + sa 2 c2 x m ] 3, m ττ = e i2α [ 3 (ss s a + c s c a s x e iδ D ) 2 m 1 + (c s s a s s c a s x e iδ D ) 2 m 2 + ca 2 c2 x m ] 3, m eµ = e i(α 1 +α 2 )[ c s c x (s s c a c s s a s x e iδ D ) m 1 s s c x (c s c a + s s s a s x e iδ D ) m 2 +s a s x c x e iδd m 3 ], m eτ = e i(α 1 +α 3 )[ c s c x (s s s a +c s c a s x e iδ D ) m 1 s s c x (c s s a s s c a s x e iδ D ) m 2 c a s x c x e iδd m 3 ], m µτ = e i(α 2 +α 3 )[ (s s c a c s s a s x e iδ D )(s s s a + c s c a s x e iδ D ) m 1 with m i m i e 2iφ i. + (c s c a + s s s a s x e iδ D )(c s s a s s c a s x e iδ D ) m 2 s a c a c 2 x m 3],

12 Solving Low Energy Parameters Perturbation Parameters in Reconstructed M ν From: M (0) ν = (b c)2 1 1 (2 X)M 10 1 we can get Two Vanishing Mass Eigenvalues: m 1 = m 2 = 0 Normal Hierarchy Nonzero m 2 : Besides: y m 2 m 3 O(r,ζ) δ a,δ x,z δm 3 m 3,δα i (α i α j + φ 3 )

13 Solving Low Energy Parameters Expanding Reconstructed Mass Matrix with: M (0) ν = 1 2 m 30e 2iα 20 M ν M (0) ν + M (1) ν , M ν (1) 1 δm ee (1) δm eµ (1) δm eτ (1) δm µµ (1) δm µτ (1) δm ττ (1) Note: Overall CP Phase (No Physical Consequences!!!) For Linear Order: δm (1) ee = m 30 s 2 s e 2i(α 10 φ 23 ) y δm µµ (1) 1 = 2 m 30e 2iα 20 [cs 2 ] e 2iφ 23y + z + 2δ a 2iδα 2 δm (1) 1 ττ = 2 m 30e 2iα 20 [c 2 ] s e 2iφ 23y + z 2δ a 2iδα 3 δm (1) eµ = δm (1) eτ = δm (1) µτ = 1 m 30 e i(α 10 +α 20 ) [ c ss se 2iφ 23y + e iδ ] D δ x 2 1 m 30 e i(α 10 +α 20 ) [ c ss se 2iφ 23y e iδ ] D δ x m 30e 2iα 20 [ c 2 ] s e 2iφ 23y z + i(δα 2 + δα 3 ) Seesaw

14 Predictions Solutions: Lower Limit on θ 13 Zeroth-Order: m 10 = m 20 = 0, m 30 = 2(b c)2 (2 X)M 10, e2iα30 = r r Overall CP Phase (No Physical Consequence!) Linear Order: yss ζ δ x = 2 [ζ 2 4rζ cosδ D + 4r 2 ] 1/4 Correlations: δ a = yc s cosδ D ζ 2 [ζ 2 4rζ cosδ D + 4r 2 ] 1/4 2 X 2 X δ x = tanθ s δ a δ x tan s δ a cosδ D Solar Mixing Angle θ s Dictated by Dirac Mass Matrix m D : 2a tanθ s = b + c Will be elaborated later.

15 Predictions Correlation between θ 13 & θ 23 7 δ x = tanθ s cos δ D δ a Θ Double Chooz RENO Daya Bay Θ Constrained Correlation

16 Low Energy Observables Jarlskog Invariant J J = 1 4 sin2 2θ s sinδ D δ x + O ( ) δx 2,δ xδ a,δa 2 Θ a J Constrained Jarlskog

17 Low Energy Observables 0ν2β Decay Observable m ee M ee m 3 s 4 sy 2 + 2s 2 s cos2(δ D φ 23 )yδ 2 x + (δ 4 x 2s 2 sy 2 δ 2 x) Θ b M ee mev Constrained M ee

18 Origin of Matter Origin of Matter: Baryon Asymmetry via Leptogenesis The Universe contains 4% Matter: η B = n B n B = (6.21 ± 0.16) n γ where n γ is Photon Number Density & n B is Baryon Number Density. Leptogenesis Mechanism generates η B from Lepton Asymmetry Y L via Sphaleron Interactions which violate B + L but preserve B L: η B = ξ f Nf B L = ξ f Nf L = 3ξ 4f κ fǫ f where ξ (8N F + 4N H )/(22N F + 13N H ) = 28/79 for SM, and f = N rec γ /N γ is the Dilution Factor. Efficiency Factor: ( ) 1.16 κ 1 m 1 f ev ev m 1 with m 1 ( m D m D) 11 /M 1 ( m D m D V R ).

19 CP Asymmetry Parameter ǫ 1 CP Asymmetry Parameter ǫ 1 CP Asymmetry Parameter ǫ 1 : ǫ 1 Γ[N 1 lh] Γ[N 1 lh ] Γ[N 1 lh] + Γ[N 1 lh ] = 1 4πv 2 F ( M2 M 1 { ) ) Im [( m D m D ( ) m D m D Complex m D differs Γ[N 1 lh] from Γ[N 1 lh ]. In Minimally Extended SM (Heavy Majorana Neutrinos): [ ( ) 1 + x F(x) x 1 (1 + x 2 2 )ln x ] 1 x 2 = 3 ( ) 1 2x + O x 3 The expansion applies for x M 2 /M ] 2 } In Current Model: ǫ 1 = m 3 (4y ) 2 ζ 3M 2 4rζ cosδ D + 4r 2 1 4πv (ζ 2 4rζ cosδ D + 4r 2 (4r cosδ D ζ)sinδ D ζ 2 ) where m 3 is obtained by RG-running m 3 from M Z to Leptogenesis Scale. RGE

20 Prediction of Leptogenesis for Baryon Asymmetry Prediction of Leptogenesis - η B /M 1 ( ) 2 η B = 3ξ 3 4y ζ M 1 4f κ m 3 M 2 4rζ cos δ D + 4r 2 1 f 4πv ( ζ 2 4rζ cos δ D + 4r 2) (4r cos δ D ζ) sin δ D ζ a Η B M 1 GeV D Constrains on δ D

21 Prediction of Leptogenesis for Baryon Asymmetry Prediction of Leptogenesis - Lower Limit of M 1 M 1 = 4f 4πv 2 128(4r 2 4rζ cos δ D + ζ 2 ) η B [ ] 3ξ κ f m y 4r 2 4rζ cos δ D + ζ 2 (4r cos δ D ζ) sin δ D ζ GeV a M 1 GeV J

22 Constrains of Leptogenesis on Low Energy Observables Upper Limit on Leptogenesis Scale M 1 M 1 = (b c) GeV m M 1 GeV b b c GeV

23 Constrains of Leptogenesis on Low Energy Observables Lower Bound on θ 13 M 1 = 4f 4πv 2 128(4r 2 4rζ cos δ D + ζ 2 ) η B [ ] 3ξ κ f m y 4r 2 4rζ cos δ D + ζ 2 (4r cos δ D ζ) sin δ D ζ GeV 7 Θ Double Chooz RENO Daya Bay Θ Free Correlation

24 Constrains of Leptogenesis on Low Energy Observables Constrained Jarlskog Invariant J 8 6 Θ a J Free Jarlskog

25 Constrains of Leptogenesis on Low Energy Observables Constrained 0ν2β Decay Observable M ee 8 6 Θ b M ee mev Free M ee

26 Conjecture θ s Determined by m D As we have seen: 2a tanθ s = b + c which holds before and after soft breaking! Fully Determined by m D : a a m D = b c c b Not Affected by µ τ and CP symmetry breaking in M R! Protected or Accidental?

27 Conjecture New Hidden Symmetry Dictating θ 12 θ s is Solely determined by m D ; Soft symmetry breaking comes from M R, m D is not affect; If extra symmetry exists, it shouldn t be affected by soft breaking; It only applies on m D, not M R. Can be realized by: ν L T s ν L, T s m D = m D N N Also respected by light neutrino s mass matrix M ν : T T s M ν T s = M ν which is Independent of M R.

28 Representation Representation of the Hidden Symmetry Neutrino mass matrix Invariant under transformation: Diagonalization Scheme: T T s M νt s = M s V T M ν V = D ν The effect of transformation is just a Diagonal Rephasing: V T T T s M νt s V = d ν D ν d ν = d ν V T M ν Vd ν with d 2 ν = I 3 which constrains d ν = diag(±, ±, ±). General consequence: T s V = Vd ν T s = Vd ν V

29 Representation Representaion of the Hidden Symmetry Two Nontrivial Independent possibilities of d ν : d (1) ν = 1 1, d (2) 1 ν = Mixing matrix with θ s parameterized in terms of k: V(k) = k 2+k k k k 0 2 k 1 2(2+k 2 ) k 2(2+k 2 ) Two Independent symmetry transformations: T s = 1 2 k2 2k 2k 2k k 2 2, 2 + k 2 T µτ = 1 1 2k 2 k 2 1 T µτ is 3D Representation of µ τ symmetry. T (3) µτ

30 Summary Summary Oscillation Data strongly support µ τ symmetry as a Good Approximate Flavor Symmetry. The µ τ symmetry predicts (θ 23,θ 13 ) = (45,0 ) & Vanishing Dirac CP Phase. Conjecture: both µ τ and CP are Softly Broken by a Common Origin in M R. With this conceptually Simple and Attractive construction, θ 13 is Correlated with θ 23 (Lower Bound on δ x / Upper Bound on δ a ). Strong supports for up-coming experiments. Predictions on Baryon Asymmetry through leptogenesis. Constrain by leptogenesis scale: Lower Bound on θ 13. Extra Z 2 dictating solar mixing angle θ 12.

31 Thank You Thank You!

32 Spare Slides - CP Phase Constrained by Leptogenesis Constrained CP Phase δ D by Leptogenesis r = ζ 2 η B = 3ξ M 1 4f κ m 3 M 1 3 (4y ζ 2 4rζ cos δ D + 4r 2) 2 f 4πv ( ζ 2 4rζ cos δ D + 4r 2) (4r cos δ D ζ) sin δ D ζ 2 [ η B > 0 (4r cos δ D ζ)sinδ D > 0 cos δ D ± ] ss 4 y 2 ζ 2 16 δx 4 sin 2 δ D These two inequalities will lead to: ζ 4 δx 2 ss 2 y sin δ D cos 2 δ D 4 s 4 s δ 4 x y 2 ζ 2 Leptogenesis

33 Spare Slides - CP Phase Constrained by Leptogenesis Constrained CP Phase δ D v.s. J 0.02 a 0.01 J D Leptogenesis

34 Spare Slides - CP Phase Constrained by Leptogenesis Constrained CP Phase δ D v.s. M ee 3.5 b Mee mev D Leptogenesis

35 RGE RG Running Effect Low Energy Observables RGE High Energy Observables Only mass eigenvalues are obviously affected: m j (µ) = χ(µ,µ 0 )m j (µ 0 ) which can be expressed as: [ 1 t ] χ(µ,µ 0 ) exp 16π 2 α(t )dt 0 For leptogenesis: m j (M 1 ) = χ(m 1,M Z )m j (M Z ) with α 2g 2 2+6y 2 t +λ CP Asymmetry ǫ 1

36 RGE RG Running Effect ΧΑ Μ m i m i Μ GeV M H 115GeV CP Asymmetry ǫ 1