1. A viable RS Model for Quarks and Leptons with T´ Flavor Symmetry Felix Yu University of California, Irvine Pheno 2010 M-C. Chen, K. T. Mahanthappa, FY – Phys. Rev. D 81, 036004 (2010) [arXiv: 0907.3963 [hep-ph]]

2. Motivation Fermion mass hierarchy unexplained Gauge hierarchy problem motivates new physics at about TeV Randall-Sundrum (RS1) model with bulk fermions provides a good framework –Can get fermion mass hierarchy with O(1) coefficients –Need to suppress FCNCs

3. Randall-Sundrum RS1 Model – warped geometry –5 th dimension compactified via S 1 /  2 –Higgs field confined to TeV brane (y =  R), other fields propagate in bulk –From compactification and boundary conditions, can find Fourier modes for bulk fields –SM masses and mixings arise from zero modes Integrate out y to find overlap between SM fields and Higgs Gherghetta, Pomarol (2000), Huber, Shafi (2000), Grossman, Neubert (1999) Randall, Sundrum (1999)

4. Flavor Changing Neutral Currents in RS Change from gauge interaction basis to mass basis Generically get FCNCs if bulk masses are not equal Solutions: (1) alignment, (2) degeneracy †

5. The Finite Group T´ Double covering of A4 –A4 is the discrete invariant rotations of a tetrahedron Has two generators: S=(1234)  (4321), T=(1234)  (2314) –S 2 =R, T 3 =1, (ST) 3 =1, R 2 =1 R=1: 1, 1´, 1´´, 3 (vector) [use for leptons] R=-1: 2, 2´, 2´´ (spinorial) [use for quarks] Frampton, Kephart (1995)

6. Assignment of T´ Representations Motivated by neutrino mixing data: assign L ~ 3 (LH lepton doublets), N ~ 3 (RH neutrinos) under T to obtain the tri-bimaximal mixing pattern –Introduce e ~ 1,  ~ 1 ,  ~ 1 for charged lepton masses –Tree-level lepton FCNCs are eliminated via degeneracy (left- handed lepton doublets share a common bulk mass term) and alignment (right-handed lepton singlets can freely rotate) Motivated by quark masses, use 2  1 assignment –Tree-level quark FCNCs involving first and second generations are eliminated via degeneracy (up- and down- type first two generations share a common bulk mass term) Require additional flavon fields to break T symmetry on the IR brane

7. Leptons in T´ Purely Dirac neutrino masses Seesaw type 1 neutrino masses

8. Quarks in T´: 2  1 Framework Down-type Yukawa Lagrangian is exactly analogous

9. Parameter Counting Input parameters (Naïve counting) –Charged lepton: 8 (= 4 bulk + 3 Yukawa + 1 flavon) –Neutrino: [seesaw] 6 [7] (= 2 bulk + 2 [3] Yukawa + 2 flavon) –Quark: 24 = (6 bulk + 8 Yukawa + 10 flavon) Actual number of independent input combinations –16 = Lepton matrix (3) + Neutrino matrix (2) + Quark matrices (6 + 5) Contrast with anarchic case –36 [30] for leptons, 36 for quarks Fit parameters –Lepton and quark masses (3 + 6 = 9) –CKM matrix (+ CP violating phase) (3 + 1 = 4) –Neutrino mixing angles (3) 16 Inputs, 16 Outputs

10. Results –Leptons Set all leptonic Yukawas to 1. Renormalization effects negligible. Gives m e =511 keV, m  =105.7 MeV, m  =1.777 GeV For normal hierarchy For inverted hierarchy Normal, Dc:  m sol 2 = 7.6370  10 -5 eV 2,  m atm 2 = 2.4031  10 -3 eV 2 Inverted, SS:  m sol 2 = 7.6560  10 -5 eV 2,  m atm 2 = –2.4009  10 -3 eV 2 Experimental:  m sol 2 = 7.65  10 -5 eV 2,  m atm 2 = 2.40  10 -3 eV 2 Fusaoka, Koide (1998), Schwetz, Tortola, Valle (2008) Normal, SS:  m sol 2 = 7.6520  10 -5 eV 2,  m atm 2 = 2.4001  10 -3 eV 2 For normal hierarchy

11. Results – Quarks Bulk mass parameters Flavons and Yukawas Prediction (3 TeV)Fit bounds mumu 1.49 MeV0.75-1.5 MeV mdmd 2.92 MeV2-4 MeV mcmc 0.541 GeV0.56 ± 0.04 GeV msms 36.6 MeV47 ± 12 MeV mtmt 134.8 GeV136.2 ± 3.1 GeV mbmb 2.41 GeV2.4 ± 0.04 GeV Csaki, Falkowski, Weiler (2008) Other Yukawas set to 1

12. Results – CKM and Jarlskog Fusaoka, Koide (1998), Charles, et al. (CKMfitter Group) (2009) Corrections to quark mixings from running are small. Perform fit at m Z

13. Leading FCNC Estimate Leading contribution is from dim-6 operators arising from fermion zero-modes mixing with KK modes Scaled to Z-coupling, leading contribution is Using M KK ~ 3 TeV, kR ~ 11, v = 246 GeV: –coefficient is 2.965  10 -6 for u-c transition –coefficient is 4.156  10 -6 for d-s transition

14. Conclusions RS1 + T´ provides a framework for realistic fermion masses and mixings –Motivated by neutrino mixings and quark masses, we choose T´ representations This choice eliminates tree-level lepton FCNCs and first- second generation quark FCNCs –Can fit for all SM fermion masses, CKM matrix, and Jarlskog invariant with 16 input parameter combinations –Allows a low first KK mass scale, testable at colliders

16. Group Algebra of T´ 2 S=A 1, T=  A 2, 2´ S=A 1, T=  2 A 2, 2´´ S=A 1, T=A 2 1 S=1, T=1, 1´ S=1, T= , 1´´ S=1, T=  2 Feruglio, Hagedorn, Lin, Merlo (2007) 3

17. Neutrino Constraints Neutrino measurements (at 2  ) (at 1  ) Well-fit by Tri-Bimaximal Mixing (TBM) Harrison, Perkins, Scott (1999) TBM can be easily obtained from A4 or T´ group symmetries Schwetz, Tortola, Valle (2008) Ma, Rajasekeran (2001)

18. Leptons in T´ T´ contraction: Diagonal charged lepton mass matrix because of T´ assignments and flavon VEVs

19. Quarks in T´: The 2  1 Framework 2  1

20. Quarks in T´: The 2  1 Framework 2  1

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