Lifting degenerate neutrino masses, threshold corrections and maximal mixing

Transcription

1 Lifting degenerate neutrino masses, threshold corrections and maximal mixing Young Scientists Workshop Waldhotel Zollernblick, Freudenstadt (Black Forest) Wolfgang Gregor Hollik Oct 31, 2014 INSTITUT FOR THEORETICAL PARTICLE PHYSICS KARLSRUHE INSTITUTE OF TECHNOLOGY (KIT) KIT University of the State of Baden-Wuerttemberg and National Laboratory of the Helmholtz Association

2 The issue of nearly degenerate neutrinos The neutrino mass spectrum m 1 = m 0, m 2 = m m2 21, m 3 = m m2 31 Oscillations: m 2 21 = ev 2 m 2 31 = ± ev 2 [νfit: Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

3 The issue of nearly degenerate neutrinos The neutrino mass spectrum m 1 = m 0, m 2 = m m2 21, m 3 = m m2 31 Oscillations: m 2 21 = ev 2 m 2 31 = ± ev 2 [νfit: Unknown: Absolute neutrino mass scale Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

4 The issue of nearly degenerate neutrinos The neutrino mass spectrum m 1 = m 0, m 2 = m m2 21, m 3 = m m2 31 Oscillations: m 2 21 = ev 2 m 2 31 = ± ev 2 [νfit: Unknown: Absolute neutrino mass scale Katrin? new limit ( x?): 0.2 ev, discovery (5 σ): 0.35 ev Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

5 Quasi-Degeneration 10 m 1 m 2 m 3 1 neutrino mass [ev] neutrino mass scale m 0 [ev] Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

6 Quasi-Degeneration 10 m 1 m 2 m 3 1 neutrino mass [ev] neutrino mass scale m 0 [ev] Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

7 The history of degenerate neutrinos degenerate patterns follow from S 3 symmetries ( flavour democracy ) [Fitzsch and Xing since 1996] Renormalization Group effects on degenerate neutrinos [Ellis, Lola 1999; Casas, Espinosa, Ibarra, Navarro 1999; Balaji, Dighe, Mohapatra, Parida 2000] Quantum corrections: fixed point solutions Pokorski 2000; Chankowski, Pokorski 2002] Quantum corrections: threshold effects 2001; Chankowski, Ioannisian, Pokorski, Valle 2001] [Chankowski, Krolikowski, [Chun, Pokorski 2000; Chun Issues on degenerate Majorana neutrinos: Majorana phases [Branco, Rebelo, Silva-Marcos 1998; Haba, Matsui, Okamura 2000; Branco, Rebelo, Silva-Marcos, Wegman 2014] Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

8 RGE Corrections [Chankowski, Pokorski 2002] Renormalization Group Equation for ν masses and mixing [ d ( ) T ( ) ] dt C = KC κ Y ey e C + C Y ey e ( ) 16π ln Q 2 M Z t = 1 ) (3Y uy u + 3Y d Y d + Y ey e + 2λ SM: κ = 3 2 and K = 3g Tr ( ) MSSM: κ = +1 and K = 6g2 2 2gY Tr 3Y uy u Solving the RGE C(t) = I K IC(0)I, ( t I K = exp 0 where I = diag(i e, I µ, I τ ) and ) ( t K(t )dt, I ea = exp κ 0 y 2 e A (t )dt ). Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

9 Threshold Corrections Generic treatment m ν AB = m (0) AB + m(0) AC I CB + I AC m (0) CB I: threshold correction I g2 f ( ln(m 2 /Q 2 ) ) (in the SM diagonal). 16π 2 Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

10 Threshold Corrections Generic treatment m ν AB = m (0) AB + m(0) AC I CB + I AC m (0) CB I: threshold correction I g2 f ( ln(m 2 /Q 2 ) ) (in the SM diagonal). 16π 2 tree-level rotation matrix U (0) : U (0)T m (0) U (0) = diagonal Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

11 Threshold Corrections Generic treatment m ν AB = m (0) AB + m(0) AC I CB + I AC m (0) CB I: threshold correction I g2 f ( ln(m 2 /Q 2 ) ) (in the SM diagonal). 16π 2 tree-level rotation matrix U (0) : U (0)T m (0) U (0) = diagonal Mass basis ( ) m ν ab = m(0) a δ ab + m (0) a + m (0) b I ab I ab = AB I ABU (0) Aa U (0) Bb Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

12 Threshold Corrections Generic treatment m ν AB = m (0) AB + m(0) AC I CB + I AC m (0) CB I: threshold correction I g2 f ( ln(m 2 /Q 2 ) ) (in the SM diagonal). 16π 2 tree-level rotation matrix U (0) : U (0)T m (0) U (0) = diagonal Mass basis ( ) m ν ab = m(0) a δ ab + m (0) a + m (0) b I ab I ab = AB I ABU (0) Aa U (0) Bb assumption: degenerate tree-level masses, m (0) 1 = m(0) 2 = m(0) 3 Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

13 Comments on degenerate masses if m (0) = m 0 1: U (0)T m (0) U (0) = m 0 1 for any (real) U (0) Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

14 Comments on degenerate masses if m (0) = m 0 1: U (0)T m (0) U (0) = m 0 1 for any (real) U (0) if e.g. m (0) = diag(1, 1, 1) this is not true Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

15 Comments on degenerate masses if m (0) = m 0 1: U (0)T m (0) U (0) = m 0 1 for any (real) U (0) if e.g. m (0) = diag(1, 1, 1) this is not true in general: Majorana phases! Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

16 Comments on degenerate masses if m (0) = m 0 1: U (0)T m (0) U (0) = m 0 1 for any (real) U (0) if e.g. m (0) = diag(1, 1, 1) this is not true in general: Majorana phases! phase matrix U (0) U (0) P with P = diag(e iα1, e iα2, 1) Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

17 Comments on degenerate masses if m (0) = m 0 1: U (0)T m (0) U (0) = m 0 1 for any (real) U (0) if e.g. m (0) = diag(1, 1, 1) this is not true in general: Majorana phases! phase matrix U (0) U (0) P with P = diag(e iα1, e iα2, 1) m (0) P T U (0)T m (0) U (0) P = m 0 diag(e 2iα1 e 2iα2, 1) Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

18 Comments on degenerate masses if m (0) = m 0 1: U (0)T m (0) U (0) = m 0 1 for any (real) U (0) if e.g. m (0) = diag(1, 1, 1) this is not true in general: Majorana phases! phase matrix U (0) U (0) P with P = diag(e iα1, e iα2, 1) m (0) P T U (0)T m (0) U (0) P = m 0 diag(e 2iα1 e 2iα2, 1) redefine masses Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

19 Comments on degenerate masses if m (0) = m 0 1: U (0)T m (0) U (0) = m 0 1 for any (real) U (0) if e.g. m (0) = diag(1, 1, 1) this is not true in general: Majorana phases! phase matrix U (0) U (0) P with P = diag(e iα1, e iα2, 1) m (0) P T U (0)T m (0) U (0) P = m 0 diag(e 2iα1 e 2iα2, 1) redefine masses m 1 = e 2iα 1 m 0, Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

20 Comments on degenerate masses if m (0) = m 0 1: U (0)T m (0) U (0) = m 0 1 for any (real) U (0) if e.g. m (0) = diag(1, 1, 1) this is not true in general: Majorana phases! phase matrix U (0) U (0) P with P = diag(e iα1, e iα2, 1) m (0) P T U (0)T m (0) U (0) P = m 0 diag(e 2iα1 e 2iα2, 1) redefine masses m 1 = e 2iα 1 m 0, m 2 = e 2iα 2 m 0, Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

21 Comments on degenerate masses if m (0) = m 0 1: U (0)T m (0) U (0) = m 0 1 for any (real) U (0) if e.g. m (0) = diag(1, 1, 1) this is not true in general: Majorana phases! phase matrix U (0) U (0) P with P = diag(e iα1, e iα2, 1) m (0) P T U (0)T m (0) U (0) P = m 0 diag(e 2iα1 e 2iα2, 1) redefine masses m 1 = e 2iα 1 m 0, m 2 = e 2iα 2 m 0, m 3 = m 0. Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

22 Comments on degenerate masses if m (0) = m 0 1: U (0)T m (0) U (0) = m 0 1 for any (real) U (0) if e.g. m (0) = diag(1, 1, 1) this is not true in general: Majorana phases! phase matrix U (0) U (0) P with P = diag(e iα1, e iα2, 1) m (0) P T U (0)T m (0) U (0) P = m 0 diag(e 2iα1 e 2iα2, 1) redefine masses m 1 = e 2iα 1 m 0, m 2 = e 2iα 2 m 0, m 3 = m 0. taking CP as good symmetry: α 1,2 {0, ± π 2 } Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

23 Comments on degenerate masses if m (0) = m 0 1: U (0)T m (0) U (0) = m 0 1 for any (real) U (0) if e.g. m (0) = diag(1, 1, 1) this is not true in general: Majorana phases! phase matrix U (0) U (0) P with P = diag(e iα1, e iα2, 1) m (0) P T U (0)T m (0) U (0) P = m 0 diag(e 2iα1 e 2iα2, 1) redefine masses m 1 = e 2iα 1 m 0, m 2 = e 2iα 2 m 0, m 3 = m 0. taking CP as good symmetry: α 1,2 {0, ± π 2 } choice: m 1 = m 2 = m 3 : 1 + 2U α1 U β1 I αβ 0 2U α1 U β3 I αβ m ν = m U α2 U β2 I αβ 0 2U α1 U β3 I αβ U α3 U β3 I αβ Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

24 Same CP Trivial tree-level I 11 I 12 I 13 m ν = m m 0 I 12 I 22 I 23 I 13 I 23 I 33 requirements for I ij : 1 θ 23 π/ U 23 = I 12 = I 13 θ 13 ( = 0 3 θ 12 = 1 2 arctan 2 ) 2I 12 2I 22 I 11 get m 1,2 in terms of I ij, m 3 = m 0 Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

25 Same CP Trivial tree-level I 11 I 12 I 13 m ν = m m 0 I 12 I 22 I 23 I 13 I 23 I 33 requirements for I ij : 1 θ 23 π/4 I = U T 23IU 23 = 2 I 12 = I 13 θ 13 ( = 0 3 θ 12 = 1 2 arctan 2 ) 2I 12 2I 22 I 11 get m 1,2 in terms of I ij, m 3 = m 0 I I 12+I I 12+I 13 I12 I I I12 I13 Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

26 Nonzero θ 13 Deviations θ 13 0 I 13 I 12 θ 23 π 2 m 0 m 2 I 33 = I 22 + ε I 13 = I 12 + δ 0 + m2 21 m m I 11 I 12 I 12 + δ = m U(θ 12, θ 13, θ 23 ) T I I 22 I 22 U(θ 12, θ 13, θ 23 ) I 12 + δ I I 22 + ε Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

27 Numerical example Inputs (central values, old numbers ), Outputs θ , θ , θ , m 0 = 0.35 ev m ev 2, m ev 2 m 0.35 ev, I , I , I , δ , ε I = Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

28 Non-diagonal threshold corrections MSSM with righthanded neutrinos W µh 1 H 2 +Y ν ij H 2 L L,i N R,j Y l ij H 1 L L,i E R,j M R ij N R,i N R,j Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

29 Non-diagonal threshold corrections MSSM with righthanded neutrinos W µh 1 H 2 +Y ν ij H 2 L L,i N R,j Y l ij H 1 L L,i E R,j M R ij N R,i N R,j New soft SUSY breaking terms ) Vsoft (m ν = 2 L ij ν L,i ν L,j + + ( ) m 2 R ν R,i ν R,j ij ( A ν ij h 0 2 ν L,i ν R,j + ( B 2) ij ν R,i ν R,j + h.c. ) Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

30 Non-diagonal threshold corrections MSSM with righthanded neutrinos W µh 1 H 2 +Y ν ij H 2 L L,i N R,j Y l ij H 1 L L,i E R,j M R ij N R,i N R,j New soft SUSY breaking terms ) Vsoft (m ν = 2 L ij ν L,i ν L,j + + seesaw type I: ( ) m 2 R ν R,i ν R,j ij ( A ν ij h 0 2 ν L,i ν R,j + ( B 2) ij ν R,i ν R,j + h.c. m (0) ν = v 2 uy T ν M 1 R Y ν + O(v 4 u/m 3 R) adding SUSY 1-loop [Dedes, Haber, Rosiek 2007] ( ) [ m 1 loop ν = (m ν) ij + Re Σ (ν),s ij + m ν i ij 2 Σ(ν),V ij + m ] ν j 2 Σ(ν),V ji ) Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

31 Constraints on A ν Random scan M SUSY [500, 5000] GeV M 1 [0.3, 3] M SUSY M 2 [1, 5] M SUSY µ [ 15, 15] TeV tan β [10, 60] Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

32 Constraints on A ν 1e A 13 / GeV e+06 A 12 / GeV Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

33 Constraints on A ν 1e+07 1e+06 A 23 / GeV e+06 A 12 / GeV Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

34 Constraints on A ν 1e+07 1e A ij / GeV A 12 A 13 1 A 23 A 11 A 22 A M SUSY / GeV Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

35 Conclusions if tritium decay measures a neutrino mass: spectrum quasidegenerate large effects from renormalization group running threshold effects become important for degenerate masses update on m 1 = m 2 = m 3 : sin θ 13 ok, but not m 2 ij same CP eigenvalues: threshold corrections have the power to completely determine the mixing and deviations from degenerate masses non-diagonal corrections needed: e.g. supersymmetric requirement for off-diagonal A ν Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

36 Backup Slides

37 Side Note the same follows from a special tree-level mass matrix m ν tree = x y z x y y = y x + z z, y z x + z which can be diagonalized by c 12 s 12 0 U tree = s 12 c with s 12 = sin θ 12, c 12 = cos θ 12, tan 2θ 12 = 2 y z s 2 12 c Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

38 Side Note (cont d) can be inverted m ν tree = m 1 ± i m ± i m m 2 21 m1+m3 m1+m2 m 2 21 m1+m3 m1+m2 ± i 2 m 2 31 m 2 21 m1+m3 m1+m2 m 2 +m ( i m i 3m 1 ) ± i 2 m 2 31 m 2 21 m1+m3 m1+m2 1 2 ( i m i 3m 1 ) m 2 +m 3 2 The philosophy behind threshold corrections exact tree-level: trivial mass matrix m (1) = m (0) + m (0) I, I π small perturbation Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

39 Update on the + + scenario Brief review of [Chankowski, Pokorski 2002] degeneracy leaves freedom of rotation U (0) U (0) R 13 U (0) α1 U (0) β3 I αβ = 0 αβ flavour diagonal corrections: I αβ = I α δ αβ explain deviation from (tri-)bi-maximal mixing: s 13 = sin θ 13 0 s 13 = s 12 c 12 s 23 c 23 I τ I e, where I µ = 0 and I e I τ Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

40 Update on the + + scenario Brief review of [Chankowski, Pokorski 2002] degeneracy leaves freedom of rotation U (0) U (0) R 13 U (0) α1 U (0) β3 I αβ = 0 αβ flavour diagonal corrections: I αβ = I α δ αβ explain deviation from (tri-)bi-maximal mixing: s 13 = sin θ 13 0 s 13 = s 12 c 12 s 23 c 23 I τ I e, where I µ = 0 and I e I τ I µ 0 try to accommodate s and m 2 31 / m s 13 = c 23 s 23 s 12 c 12 I µ I τ I e s 2 23 I µ c 2 23 I τ Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

41 Update on the + + scenario (cont d) m 2 ab = m2 ( [1 + 2U 2 αai α ] 2 [1 + 2U 2 αb I α] 2) m 2 overall scale use relation for s 13 to get correlation between I e and I µ, I τ try to fit ( [1 + 2U m 2 31/ 2 2 m21 = α3 I α ] 2 [1 + 2Uα1 2 I α] 2) ( [1 + 2U 2 α2 I α ] 2 [1 + 2Uα1 2 I α] 2) Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

42 Update on the + + scenario (cont d) I tau x I e x 10 3 Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13

43 Update on the + + scenario (cont d) I tau I mu Wolfgang Gregor Hollik degenerate neutrinos Oct 31, /13