Discrete Flavor Symmetries and Origin of CP Violation Mu-Chun Chen, University of California at Irvine Nu@Fermilab, July 23, 2015 Work done in collaboration with Maximilian Fallbacher, K.T. Mahanthappa, Michael Ratz, Andreas Trautner, Nucl. Phys. B883 (2014) 267 K.T. Mahanthappa, Phys. Lett. B681, 444 (2009)
CP Violation in Nature • CP violation: required to explain matter-antimatter asymmetry • So far observed only in flavor sector • SM: CKM matrix for the quark sector • experimentally established δ CKM as major source of CP violation • not su ffi cient for observed cosmological matter-antimatter asymmetry • Search for new source of CP violation: • CP violation in neutrino sector • if found ⇒ phase in PMNS matrix • Discrete family symmetries: • suggested by large neutrino mixing angles • neutrino mixing angles from group theoretical CG coe ffi cients Discrete (family) symmetries ⇔ Physical CP violation Mu-Chun Chen, UC Irvine Discrete Flavor Symmetries and Origin of CP Violation Nu@Fermilab 2015 2
Origin of CP Violation • CP violation ⇔ complex mass matrices CP U R,i ( M u ) ij Q L,j + Q L,j ( M † → Q L,j ( M u ) ij U R,i + U R,i ( M u ) ∗ u ) ji U R,i ij Q L,j − • Conventionally, CPV arises in two ways: • Explicit CP violation: complex Yukawa coupling constants Y • Spontaneous CP violation: complex scalar VEVs <h> Mu-Chun Chen, UC Irvine Discrete Flavor Symmetries and Origin of CP Violation Nu@Fermilab 2015 3
Origin of CP Violation • CP violation ⇔ complex mass matrices CP U R,i ( M u ) ij Q L,j + Q L,j ( M † → Q L,j ( M u ) ij U R,i + U R,i ( M u ) ∗ u ) ji U R,i ij Q L,j − • Conventionally, CPV arises in two ways: • Explicit CP violation: complex Yukawa coupling constants Y • Spontaneous CP violation: complex scalar VEVs <h> Fermion mass and hierarchy problem ➟ Many free parameters in the Yukawa sector Mu-Chun Chen, UC Irvine Discrete Flavor Symmetries and Origin of CP Violation Nu@Fermilab 2015 4
A Novel Origin of CP Violation M.-C.C., K.T. Mahanthappa Phys. Lett. B681, 444 (2009) • Reduce the number of parameters ➪ non-Abelian discrete family symmetry • e.g. A 4 family symmetry ➪ TBM mixing from CG coe ffi cients • Complex CG coe ffi cients in certain discrete groups ⇒ explicit CP violation • real Yukawa couplings, real Higgs VEV • CPV in quark and lepton sectors purely from complex CG coe ffi cients • No additional parameters needed ⇒ extremely predictive model! CG coe ffi cients in non-Abelian discrete symmetries ➪ relative strengths and phases in entries of Yukawa matrices ➪ mixing angles and phases (and mass hierarchy) Mu-Chun Chen, UC Irvine Discrete Flavor Symmetries and Origin of CP Violation Nu@Fermilab 2015 5
A Novel Origin of CP Violation M.-C.C., K.T. Mahanthappa Phys. Lett. B681, 444 (2009) Discrete Basic idea symmetry G C 121 C 112 C 211 C 223 • Scalar potential: if Z 3 symmetric ⇒〈 ∆ 1 〉 = 〈 ∆ 2 〉 = 〈 ∆ 3 〉≡〈 ∆ 〉 real C i j k : complex CG • Complex e ff ective mass matrix: phases determined by group theory coefficients of G ( L 1 L 2 ) ( R 1 R 2 ) C 211 C 112 C 121 C 223 Mu-Chun Chen, UC Irvine Discrete Flavor Symmetries and Origin of CP Violation Nu@Fermilab 2015 6
Physical CP vs. Generalized CP Transformations complex CGs ➪ G and physical CP transformations do not commute L Generalized CP transformation: canonical CP f CP ! U CP Φ ⇤ ( P x ) Φ ( x ) 7� � Generalized CP GCP contains all reps in model Necessary Consistency condition: P � Holthausen, Lindner, Schmidt (2013) L ′ U CP ⇢ ( g ) ⇤ U CP † � � ⇢ u ( g ) 8 g 2 G = outer automorphism u L Mu-Chun Chen, UC Irvine Discrete Flavor Symmetries and Origin of CP Violation Nu@Fermilab 2015 7
Physical CP vs. Generalized CP Transformations complex CGs ➪ G and physical CP transformations do not commute L Generalized CP transformation: canonical CP f CP ! U CP Φ ⇤ ( P x ) Φ ( x ) 7� � Generalized CP GCP Necessary Consistency condition: P � Holthausen, Lindner, Schmidt (2013) L ′ U CP ⇢ ( g ) ⇤ U CP † � � ⇢ u ( g ) 8 g 2 G = outer However, GCP may not correspond automorphism u to physical CP transformation ➪ for GCP = physical CP: L more stringent consistency condition Mu-Chun Chen, UC Irvine Discrete Flavor Symmetries and Origin of CP Violation Nu@Fermilab 2015 8
Physical CP vs. Generalized CP Transformations f • generalized CP transformation CP ! U CP Φ ⇤ ( P x ) Φ ( x ) 7� � • Necessary consistency condition P � U CP ⇢ ( g ) ⇤ U CP � � † ⇢ u ( g ) 8 g 2 G Holthausen, Lindner, Schmidt (2013) = • Necessary and su ffi cient consistency condition M.-C.C., M. Fallbacher, K.T. Mahanthappa, M. Ratz, A. Trautner (2014) = U r i ⇢ r i ( g ) ⇤ U † � � physical CP ⇢ r i u ( g ) 8 g 2 G and 8 i r i implies Mu-Chun Chen, UC Irvine Discrete Flavor Symmetries and Origin of CP Violation Nu@Fermilab 2015 9
Physical CP vs. Generalized CP Transformations f • generalized CP transformation CP ! U CP Φ ⇤ ( P x ) Φ ( x ) 7� � • Necessary consistency condition P � U CP ⇢ ( g ) ⇤ U CP � � † ⇢ u ( g ) 8 g 2 G Holthausen, Lindner, Schmidt (2013) = • Necessary and su ffi cient consistency condition M.-C.C., M. Fallbacher, K.T. Mahanthappa, M. Ratz, A. Trautner (2014) = U r i ⇢ r i ( g ) ⇤ U † � � physical CP ⇢ r i u ( g ) 8 g 2 G and 8 i r i implies u has to be a class-inverting, involuntary automorphism of G ➪ non-existence of such automorphism in certain groups ➪ explicit physical CP violation Mu-Chun Chen, UC Irvine Discrete Flavor Symmetries and Origin of CP Violation Nu@Fermilab 2015 10
Three Types of Finite Groups M.-C.C., M. Fallbacher, K.T. Mahanthappa, M. Ratz, A. Trautner (2014) there is a Type II : u defines group G with u for which yes a physical CP automorphisms u no FS ( n ) is 0 transformation u no all FS ( 1 ) u are yes no class- + 1 for a u inverting involutory non-BDA, class- automorphism inverting no BDA automorphism Type I groups G I : Type II A groups G II A : Type II B groups G II B : generic settings based on G I do not allow for a there is a CP basis in there is no basis in which physical CP transformation which all CG’s are real all CG’s are real Mu-Chun Chen, UC Irvine Discrete Flavor Symmetries and Origin of CP Violation Nu@Fermilab 2015 11
A Novel Origin of CP Violation M.-C.C, M. Fallbacher, K.T. Mahanthappa, M. Ratz, A. Trautner, NPB (2014) • For discrete groups that do not have class-inverting, involutory automorphism, CP is generically broken by complex CG coe ffi cients (Type I Group) • Non-existence of such automorphism ⇔ Physical CP violation CP Violation from Group Theory! Discrete (flavor) symmetry G For further insights, see, M. Fallbacher, A. Trautner, NPB (2015) Type II : one can impose a physical CP transformation Type I groups G I : Type II A groups G II A : Type II B groups G II B : generic settings based on there is a CP basis in there is no basis in which G I do not allow for a which all CG’s are real all CG’s are real physical CP transformation Mu-Chun Chen, UC Irvine Discrete Flavor Symmetries and Origin of CP Violation Nu@Fermilab 2015 12
Examples M.-C.C., M. Fallbacher, K.T. Mahanthappa, M. Ratz, A. Trautner (2014) • Type I: all odd order non-Abelian groups group ∆ (27) T 7 5 o 9 o 4 3 SG (20,3) (21,1) (27,3) (27,4) • Type IIA: dihedral and all Abelian groups group T 0 S 3 Q 8 A 4 S 4 A 5 3 o 8 SG (6,1) (8,4) (12,3) (24,1) (24,3) (24,12) (60,5) • Type IIB group Σ (72) (( 3 ) o 4 ) o 3 × 4 SG (72,41) (144,120) Mu-Chun Chen, UC Irvine Discrete Flavor Symmetries and Origin of CP Violation Nu@Fermilab 2015 13
Example for a type I group: ∆ ( 27 ) • decay asymmetry in a toy model • prediction of CP violating phase from group theory Mu-Chun Chen, UC Irvine Discrete Flavor Symmetries and Origin of CP Violation Nu@Fermilab 2015 14
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