The remnant CP transformation Felix Gonzalez Canales Departamento de F´ ısica CINVESTAV-IPN XXX Reuni´ on Anual DPyC-SMF Puebla, M´ exico May 24, 2016 F. Gonzalez-Canales (CINVESTAV) Remnant CP 1/18 May 24, 2016 1 / 18
Peng Chen, Gui-Jun Ding, FGC and J. W. F. Valle, Generalized µ − τ reflection symmetry and leptonic CP violation Phys. Lett. B 753 (2016) 644-652 arXiv:1512.01551 Peng Chen, Gui-Jun Ding, FGC and J. W. F. Valle, Classifying CP transformations according to their texture zeros: theory and implications arXiv:1604.03510 F. Gonzalez-Canales (CINVESTAV) Remnant CP 2/18 May 24, 2016 2 / 18
Redefinition of lepton mixing matrix We adopt the charged lepton diagonal basis, m l ≡ diag ( m e , m µ , m τ ) . F. Gonzalez-Canales (CINVESTAV) Remnant CP 3/18 May 24, 2016 3 / 18
Redefinition of lepton mixing matrix We adopt the charged lepton diagonal basis, m l ≡ diag ( m e , m µ , m τ ) . The neutrino mass matrix m ν can be expressed via the mixing matrix U as m ν = U ∗ diag ( m 1 , m 2 , m 3 ) U † under the assumption of Majorana neutrinos. F. Gonzalez-Canales (CINVESTAV) Remnant CP 3/18 May 24, 2016 3 / 18
Redefinition of lepton mixing matrix We adopt the charged lepton diagonal basis, m l ≡ diag ( m e , m µ , m τ ) . The neutrino mass matrix m ν can be expressed via the mixing matrix U as m ν = U ∗ diag ( m 1 , m 2 , m 3 ) U † under the assumption of Majorana neutrinos. The invariance of the neutrino mass matrix under the action of a CP transformation ⇒ X T m ν X = m ∗ ν ⊤ ν L �→ i X γ 0 C ¯ ν , L X should be a symmetric unitary matrix to avoid degenerate neutrino masses. F. Gonzalez-Canales (CINVESTAV) Remnant CP 3/18 May 24, 2016 3 / 18
Redefinition of lepton mixing matrix We adopt the charged lepton diagonal basis, m l ≡ diag ( m e , m µ , m τ ) . The neutrino mass matrix m ν can be expressed via the mixing matrix U as m ν = U ∗ diag ( m 1 , m 2 , m 3 ) U † under the assumption of Majorana neutrinos. The invariance of the neutrino mass matrix under the action of a CP transformation ⇒ X T m ν X = m ∗ ν ⊤ ν L �→ i X γ 0 C ¯ ν , L X should be a symmetric unitary matrix to avoid degenerate neutrino masses. The lepton mixing matrix U = Σ O 3 × 3 Q ν , Σ is the Takagi factorization matrix of X fulfilling X = ΣΣ T , � e − ik 1 π/ 2 , e − ik 2 π/ 2 , e − ik 3 π/ 2 � Q ν = diag , the entries of Q ν are ± 1 and ± i which encode the CP-parity or CP-signs of the neutrino states and it renders the light neutrino mass eigenvalues positive. F. Gonzalez-Canales (CINVESTAV) Remnant CP 3/18 May 24, 2016 3 / 18
Redefinition of lepton mixing matrix The matrix O 3 × 3 = O 1 O 2 O 3 is a generic three dimensional real orthogonal matrix, and it can be parameterized as 1 0 0 cos θ 2 0 sin θ 2 , O 2 = O 1 = 0 cos θ 1 sin θ 1 0 1 0 0 − sin θ 1 cos θ 1 − sin θ 2 0 cos θ 2 cos θ 3 sin θ 3 0 . O 3 = − sin θ 3 cos θ 3 0 0 0 1 F. Gonzalez-Canales (CINVESTAV) Remnant CP 4/18 May 24, 2016 4 / 18
The neutrino oscillation data Parameter 1 BFP ± 1 σ 2 σ range 3 σ range � 10 − 5 eV 2 � 7 . 60 +0 . 19 ∆ m 2 7 . 26 − 7 . 99 7 . 11 − 8 . 18 21 − 0 . 18 ∆ m 2 � 10 − 3 eV 2 � 2 . 48 +0 . 05 (NH) 2 . 35 − 2 . 59 2 . 30 − 2 . 65 31 − 0 . 07 � 10 − 3 eV 2 � ∆ m 2 2 . 38 +0 . 05 (IH) 2 . 26 − 2 . 48 2 . 20 − 2 . 54 13 − 0 . 06 sin 2 θ 12 / 10 − 1 3 . 23 ± 0 . 16 2 . 92 − 3 . 57 2 . 78 − 3 . 75 sin 2 θ 23 / 10 − 1 (NH) 5 . 67 +0 . 32 4 . 14 − 6 . 23 3 . 93 − 6 . 43 − 1 . 24 sin 2 θ 23 / 10 − 1 (IH) 5 . 73 +0 . 25 4 . 35 − 6 . 21 4 . 03 − 6 . 40 − 0 . 39 sin 2 θ 13 / 10 − 2 (NH) 2 . 26 ± 0 . 12 2 . 02 − 2 . 50 1 . 90 − 2 . 62 sin 2 θ 13 / 10 − 2 (IH) 2 . 29 ± 0 . 12 2 . 05 − 2 . 52 1 . 93 − 2 . 65 1 . 41 +0 . 55 δ/π (NH) 0 . 0 − 2 . 0 0 . 0 − 2 . 0 − 0 . 40 δ/π (IH) 1 . 48 ± 0 . 31 0 . 00 − 0 . 09 & 0 . 86 − 2 . 0 0 . 0 − 2 . 0 The allowed ranges of | ( U PMNS ) ij | are explicitly given at the 3 σ level: NH IH 0 . 780 − 0 . 842 0 . 520 − 0 . 607 0 . 137 − 0 . 162 0 . 779 − 0 . 842 0 . 520 − 0 . 607 0 . 139 − 0 . 163 0 . 207 − 0 . 555 0 . 395 − 0 . 714 0 . 618 − 0 . 794 0 . 207 − 0 . 554 0 . 397 − 0 . 710 0 . 626 − 0 . 792 0 . 226 − 0 . 566 0 . 420 − 0 . 731 0 . 590 − 0 . 772 0 . 229 − 0 . 566 0 . 426 − 0 . 729 0 . 592 − 0 . 765 1D. V. Forero et al. Phys. Rev. D 90 , 093006 (2014) F. Gonzalez-Canales (CINVESTAV) Remnant CP 5/18 May 24, 2016 5 / 18
The neutrino oscillation data NH IH 0 . 780 − 0 . 842 0 . 520 − 0 . 607 0 . 137 − 0 . 162 0 . 779 − 0 . 842 0 . 520 − 0 . 607 0 . 139 − 0 . 163 �� � � �� �� � � 0 . 207 − 0 . 555 0 . 395 − 0 . 714 0 . 618 − 0 . 794 � 0 . 207 − 0 . 554 0 . 397 − 0 . 710 0 . 626 − 0 . 792 � �� � � �� �� � � 0 . 226 − 0 . 566 0 . 420 − 0 . 731 0 . 590 − 0 . 772 � 0 . 229 − 0 . 566 0 . 426 − 0 . 729 0 . 592 − 0 . 765 � ⋆ The | U µi | ≃ | U τi | relation. Approximate µ − τ relation. ⋆ Exact µ − τ relation | U µi | = | U τi | . This equality holds if either of the following two sets of conditions can be satisfied. θ 23 = π 4 , θ 13 = 0; | U µi | = | U τi | ⇔ θ 23 = π 4 , δ CP = ± π 2 . It is clear that θ 13 has already bee ruled out, but θ 23 = π 4 and δ CP = − π 2 are both allowed at the 1 or 2 σ level (and δ CP = π 2 is also allowed at the 3 σ ). F. Gonzalez-Canales (CINVESTAV) Remnant CP 6/18 May 24, 2016 6 / 18
The µ − τ Flavor Symmetry We claim that there must be a partial or approximate µ − τ flavor symmetry behind the observed pattern of the PMNS matrix. The µ − τ symmetry gives the constraint that Lagrangian is invariant under transformation of µ and τ neutrinos states. ⋆ The µ − τ permutation symmetry The neutrino mass term is unchanged under the transformations; ν e − → ν e , ν µ − → ν τ , ν τ − → ν µ . ⋆ The µ − τ reflection symmetry The neutrino mass term is unchanged under the transformations 2 ; → ν c → ν c → ν c ν e − e , ν µ − τ , ν τ − µ . 2 The superscript c denotes the charged conjugation. F. Gonzalez-Canales (CINVESTAV) Remnant CP 7/18 May 24, 2016 7 / 18
Generalized µ − τ reflection This interesting CP transformation takes the following form: e iα 0 0 e iβ cos Θ ie i ( β + γ ) X = 0 sin Θ , 2 ie i ( β + γ ) e iγ cos Θ 0 sin Θ 2 where the parameters α , β , γ , and Θ are real. F. Gonzalez-Canales (CINVESTAV) Remnant CP 8/18 May 24, 2016 8 / 18
Generalized µ − τ reflection This interesting CP transformation takes the following form: e iα 0 0 e iβ cos Θ ie i ( β + γ ) X = 0 sin Θ , 2 ie i ( β + γ ) e iγ cos Θ 0 sin Θ 2 where the parameters α , β , γ , and Θ are real. The corresponding Takagi factorization matrix is given by e i α 0 0 2 e i β ie i β Σ = 2 cos Θ 2 sin Θ . 0 2 2 ie i γ e i γ 2 sin Θ 2 cos Θ 0 2 2 F. Gonzalez-Canales (CINVESTAV) Remnant CP 8/18 May 24, 2016 8 / 18
Generalized µ − τ reflection This interesting CP transformation takes the following form: e iα 0 0 e iβ cos Θ ie i ( β + γ ) X = 0 sin Θ , 2 ie i ( β + γ ) e iγ cos Θ 0 sin Θ 2 where the parameters α , β , γ , and Θ are real. The corresponding Takagi factorization matrix is given by e i α 0 0 2 e i β ie i β Σ = 2 cos Θ 2 sin Θ . 0 2 2 ie i γ e i γ 2 sin Θ 2 cos Θ 0 2 2 As a result the resulting lepton mixing angles are determined as sin 2 θ 13 = sin 2 θ 2 , sin 2 θ 12 = sin 2 θ 3 , sin 2 θ 23 = 1 2 (1 − cos Θ cos 2 θ 1 ) , F. Gonzalez-Canales (CINVESTAV) Remnant CP 8/18 May 24, 2016 8 / 18
Generalized µ − τ reflection This interesting CP transformation takes the following form: e iα 0 0 e iβ cos Θ ie i ( β + γ ) X = 0 sin Θ , 2 ie i ( β + γ ) e iγ cos Θ 0 sin Θ 2 where the parameters α , β , γ , and Θ are real. The corresponding Takagi factorization matrix is given by e i α 0 0 2 e i β ie i β Σ = 2 cos Θ 2 sin Θ . 0 2 2 ie i γ e i γ 2 sin Θ 2 cos Θ 0 2 2 As a result the resulting lepton mixing angles are determined as sin 2 θ 13 = sin 2 θ 2 , sin 2 θ 12 = sin 2 θ 3 , sin 2 θ 23 = 1 2 (1 − cos Θ cos 2 θ 1 ) , while the CP violation parameters are predicted as 4 sin Θ sin θ 2 sin 2 θ 3 cos 2 θ 2 , J CP = 1 sin δ CP = sin Θ sign[sin θ 2 sin 2 θ 3 ] √ , 1 − cos 2 Θ cos 2 2 θ 1 φ 12 = k 2 − k 1 φ 13 = k 3 − k 1 δ CP = k 3 − k 2 tan δ CP = tan Θ csc 2 θ 1 , π , π , π − φ 23 . 2 2 2 F. Gonzalez-Canales (CINVESTAV) Remnant CP 8/18 May 24, 2016 8 / 18
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