Abstract
Discrete flavour symmetries have been proven successful in explaining the leptonic flavour structure. To account for the observed mixing pattern, the flavour symmetry has to be broken to different subgroups in the charged and neutral lepton sector. However, cross-couplings via non-trivial contractions in the scalar potential force the group to break to the same subgroup. We present a solution to this problem by extending the flavour group in such a way that it preserves the flavour structure, but leads to an ’accidental’ symmetry in the flavon potential.
We have searched for symmetry groups up to order 1000, which forbid all dangerous cross-couplings and extend one of the interesting groups A 4, T 7, S 4, T′ or Δ(27). We have found a number of candidate groups and present a model based on one of the smallest extensions of A 4, namely \( {{Q}_8} \rtimes {{A}_4} \). We show that the most general nonsupersymmetric potential allows for the correct vacuum alignment. We investigate the effects of higher dimensional operators on the vacuum configuration and mixing angles, and give a see-saw-like UV completion. Finally, we discuss the supersymmetrization of the model. Additionally, we release the Mathematica package Discrete providing various useful tools for model building such as easily calculating invariants of discrete groups and flavon potentials.
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References
T. Schwetz, M. Tortola and J. Valle, Where we are on θ 13 : addendum to ’Global neutrino data and recent reactor fluxes: status of three-flavour oscillation parameters’, New J. Phys. 13 (2011) 109401 [ar**v:1108.1376] [INSPIRE].
T. Schwetz, M. Tortola and J. Valle, Global neutrino data and recent reactor fluxes: status of three-flavour oscillation parameters, New J. Phys. 13 (2011) 063004 [ar**v:1103.0734] [INSPIRE].
G. Fogli, E. Lisi, A. Marrone, A. Palazzo and A. Rotunno, Evidence of θ 13 > 0 from global neutrino data analysis, Phys. Rev. D 84 (2011) 053007 [ar**v:1106.6028] [INSPIRE].
M. Gonzalez-Garcia, M. Maltoni and J. Salvado, Updated global fit to three neutrino mixing: status of the hints of θ 13 > 0, JHEP 04 (2010) 056 [ar**v:1001.4524] [INSPIRE].
T2K collaboration, K. Abe et al., Indication of electron neutrino appearance from an accelerator-produced off-axis muon neutrino beam, Phys. Rev. Lett. 107 (2011) 041801 [ar**v:1106.2822] [INSPIRE].
CHOOZ collaboration, M. Apollonio et al., Search for neutrino oscillations on a long baseline at the CHOOZ nuclear power station, Eur. Phys. J. C 27 (2003) 331 [hep-ex/0301017] [INSPIRE].
P. Harrison, D. Perkins and W. Scott, A redetermination of the neutrino mass squared difference in tri-maximal mixing with terrestrial matter effects, Phys. Lett. B 458 (1999) 79 [hep-ph/9904297] [INSPIRE].
P. Harrison, D. Perkins and W. Scott, Tri-bimaximal mixing and the neutrino oscillation data, Phys. Lett. B 530 (2002) 167 [hep-ph/0202074] [INSPIRE].
P. Harrison and W. Scott, Symmetries and generalizations of tri-bimaximal neutrino mixing, Phys. Lett. B 535 (2002) 163 [hep-ph/0203209] [INSPIRE].
MINOS collaboration, P. Adamson et al., Improved search for muon-neutrino to electron-neutrino oscillations in MINOS, Phys. Rev. Lett. 107 (2011) 181802 [ar**v:1108.0015] [INSPIRE].
C.D. Froggatt H.B. Nielsen, Hierarchy of quark masses, Cabibbo angles and CP violation, Nucl. Phys. B 147 (1979) 277 [INSPIRE].
K. Babu and X.-G. He, Model of geometric neutrino mixing, hep-ph/0507217 [INSPIRE].
E. Ma, A 4 symmetry and neutrinos with very different masses, Phys. Rev. D 70 (2004) 031901 [hep-ph/0404199] [INSPIRE].
K. Babu, E. Ma and J. Valle, Underlying A 4 symmetry for the neutrino mass matrix and the quark mixing matrix, Phys. Lett. B 552 (2003) 207 [hep-ph/0206292] [INSPIRE].
E. Ma and G. Rajasekaran, Softly broken A 4 symmetry for nearly degenerate neutrino masses, Phys. Rev. D 64 (2001) 113012 [hep-ph/0106291] [INSPIRE].
X.-G. He, Y.-Y. Keum and R.R. Volkas, A 4 flavor symmetry breaking scheme for understanding quark and neutrino mixing angles, JHEP 04 (2006) 039 [hep-ph/0601001] [INSPIRE].
G. Altarelli and F. Feruglio, Tri-bimaximal neutrino mixing, A 4 and the modular symmetry, Nucl. Phys. B 741 (2006) 215 [hep-ph/0512103] [INSPIRE].
G. Altarelli and F. Feruglio, Tri-bimaximal neutrino mixing from discrete symmetry in extra dimensions, Nucl. Phys. B 720 (2005) 64 [hep-ph/0504165] [INSPIRE].
C. Luhn, S. Nasri and P. Ramond, Tri-bimaximal neutrino mixing and the family symmetry semidirect product of Z(7) and Z(3), Phys. Lett. B 652 (2007) 27 [ar**v:0706.2341] [INSPIRE].
S. Pakvasa and H. Sugawara, Mass of the t quark in SU(2) × U(1), Phys. Lett. B 82 (1979) 105 [INSPIRE].
Y. Yamanaka and H. Sugawara, Permutation symmetries and the fermion mass matrix, Phys. Rev. D 25 (1982) 1895 [Erratum ibid. D29 (1984) 2135+ [INSPIRE].
T. Brown et al., CP nonconservation and rare processes in S 4 model of permutation symmetry, Phys. Lett. B 141 (1984) 95 [INSPIRE].
T. Brown et al., Neutrino masses, mixing and oscillations in S 4 model of permutation symmetry, Phys. Rev. D 30 (1984) 255 [INSPIRE]
D.-G. Lee and R. Mohapatra, An SO(10) × S 4 scenario for naturally degenerate neutrinos, Phys. Lett. B 329 (1994) 463 [hep-ph/9403201] [INSPIRE].
E. Ma, Neutrino mass matrix from S 4 symmetry, Phys. Lett. B 632 (2006) 352 [hep-ph/0508231] [INSPIRE].
C. Hagedorn, M. Lindner and R. Mohapatra, S 4 flavor symmetry and fermion masses: towards a grand unified theory of flavor, JHEP 06 (2006) 042 [hep-ph/0602244] [INSPIRE].
Y. Cai and H.-B. Yu, A SO(10) GUT model with S 4 flavor symmetry, Phys. Rev. D 74 (2006) 115005 [hep-ph/0608022] [INSPIRE].
F. Caravaglios and S. Morisi, Gauge boson families in grand unified theories of fermion masses: \( E_6^4 \times {{S}_4} \), Int. J. Mod. Phys. A 22 (2007) 2469 [hep-ph/0611078] [INSPIRE].
H. Zhang, Flavor S 4 × Z(2) symmetry and neutrino mixing, Phys. Lett. B 655 (2007) 132 [hep-ph/0612214] [INSPIRE].
Y. Koide, S 4 flavor symmetry embedded into SU(3) and lepton masses and mixing, JHEP 08 (2007) 086 [ar**v:0705.2275] [INSPIRE].
M. Parida, Intermediate left-right gauge symmetry, unification of couplings and fermion masses in SUSY SO(10) × S 4, Phys. Rev. D 78 (2008) 053004 [ar**v:0804.4571] [INSPIRE].
F. Bazzocchi and S. Morisi, S 4 as a natural flavor symmetry for lepton mixing, Phys. Rev. D 80 (2009) 096005 [ar**v:0811.0345] [INSPIRE].
H. Ishimori, Y. Shimizu and M. Tanimoto, S 4 flavor symmetry of quarks and leptons in SU(5) GUT, Prog. Theor. Phys. 121 (2009) 769 [ar**v:0812.5031] [INSPIRE].
F. Bazzocchi, L. Merlo and S. Morisi, Phenomenological consequences of see-saw in S 4 based models, Phys. Rev. D 80 (2009) 053003 [ar**v:0902.2849] [INSPIRE].
G. Altarelli, F. Feruglio and L. Merlo, Revisiting bimaximal neutrino mixing in a model with S 4 discrete symmetry, JHEP 05 (2009) 020 [ar**v:0903.1940] [INSPIRE].
H. Ishimori, Y. Shimizu and M. Tanimoto, S 4 flavor model of quarks and leptons, Prog. Theor. Phys. Suppl. 180 (2010) 61 [ar**v:0904.2450] [INSPIRE].
W. Grimus, L. Lavoura and P. Ludl, Is S 4 the horizontal symmetry of tri-bimaximal lepton mixing?, J. Phys. G 36 (2009) 115007 [ar**v:0906.2689] [INSPIRE].
G.-J. Ding, Fermion masses and flavor mixings in a model with S 4 flavor symmetry, Nucl. Phys. B 827 (2010) 82 [ar**v:0909.2210] [INSPIRE].
D. Meloni, A see-saw S 4 model for fermion masses and mixings, J. Phys. G 37 (2010) 055201 [ar**v:0911.3591] [INSPIRE].
S. Morisi and E. Peinado, An S 4 model for quarks and leptons with maximal atmospheric angle, Phys. Rev. D 81 (2010) 085015 [ar**v:1001.2265] [INSPIRE].
B. Dutta, Y. Mimura and R. Mohapatra, An SO(10) grand unified theory of flavor, JHEP 05 (2010) 034 [ar**v:0911.2242] [INSPIRE].
C. Lam, The unique horizontal symmetry of leptons, Phys. Rev. D 78 (2008) 073015 [ar**v:0809.1185] [INSPIRE].
R. Mohapatra, M. Parida and G. Rajasekaran, High scale mixing unification and large neutrino mixing angles, Phys. Rev. D 69 (2004) 053007 [hep-ph/0301234] [INSPIRE].
G.-J. Ding, Fermion mass hierarchies and flavor mixing from T -prime symmetry, Phys. Rev. D 78 (2008) 036011 [ar**v:0803.2278] [INSPIRE].
P.H. Frampton and S. Matsuzaki, T -prime predictions of PMNS and CKM angles, Phys. Lett. B 679 (2009) 347 [ar**v:0902.1140] [INSPIRE].
P.H. Frampton and T.W. Kephart, Flavor symmetry for quarks and leptons, JHEP 09 (2007) 110 [ar**v:0706.1186] [INSPIRE].
A. Aranda, Neutrino mixing from the double tetrahedral group T -prime, Phys. Rev. D 76 (2007) 111301 [ar**v:0707.3661] [INSPIRE].
P.D. Carr and P.H. Frampton, Group theoretic bases for tribimaximal mixing, hep-ph/0701034 [INSPIRE].
F. Feruglio, C. Hagedorn, Y. Lin and L. Merlo, Tri-bimaximal neutrino mixing and quark masses from a discrete flavour symmetry, Nucl. Phys. B 775 (2007) 120 [Erratum ibid. 836 (2010) 127-128] [hep-ph/0702194] [INSPIRE].
M.-C. Chen and K. Mahanthappa, CKM and tri-bimaximal MNS matrices in a SU(5) × (d) T model, Phys. Lett. B 652 (2007) 34 [ar**v:0705.0714] [INSPIRE].
F. Bazzocchi and I. de Medeiros Varzielas, Tri-bi-maximal mixing in viable family symmetry unified model with extended seesaw, Phys. Rev. D 79 (2009) 093001 [ar**v:0902.3250] [INSPIRE].
C. Luhn, S. Nasri and P. Ramond, The flavor group Δ(3n 2), J. Math. Phys. 48 (2007) 073501 [hep-th/0701188] [INSPIRE].
W. Grimus and L. Lavoura, A model for trimaximal lepton mixing, JHEP 09 (2008) 106 [ar**v:0809.0226] [INSPIRE].
I. de Medeiros Varzielas, S. King and G. Ross, Neutrino tri-bi-maximal mixing from a non-abelian discrete family symmetry, Phys. Lett. B 648 (2007) 201 [hep-ph/0607045] [INSPIRE].
Y. Shimizu, M. Tanimoto and A. Watanabe, Breaking tri-bimaximal mixing and large θ 13, Prog. Theor. Phys. 126 (2011) 81 [ar**v:1105.2929] [INSPIRE].
R.d.A. Toorop, F. Feruglio and C. Hagedorn, Discrete flavour symmetries in light of T2K, Phys. Lett. B 703 (2011) 447 [ar**v:1107.3486] [INSPIRE].
T. Kobayashi, Y. Omura and K. Yoshioka, Flavor symmetry breaking and vacuum alignment on orbifolds, Phys. Rev. D 78 (2008) 115006 [ar**v:0809.3064] [INSPIRE].
K. Babu and S. Gabriel, Semidirect product groups, vacuum alignment and tribimaximal neutrino mixing, Phys. Rev. D 82 (2010) 073014 [ar**v:1006.0203] [INSPIRE].
GAP group, GAP — Groups, Algorithms, and Programming. Version 4.4.12, http://www.gap-system.org.
H.U. Besche, B. Eick and E.O’Brien, SmallGroups — Library of all ’small’ groups. GAP package, version included in GAP 4.4.12 http://www.gap-system.org/Packages/sgl.html.
K.M. Parattu and A. Wingerter, Tribimaximal mixing from small groups, Phys. Rev. D 84 (2011) 013011 [ar**v:1012.2842] [INSPIRE].
W. Grimus and P.O. Ludl, Finite flavour groups of fermions, ar**v:1110.6376 [INSPIRE].
S.F. King and C. Luhn, On the origin of neutrino flavour symmetry, JHEP 10 (2009) 093 [ar**v:0908.1897] [INSPIRE].
B. Brahmachari, S. Choubey and M. Mitra, The A 4 flavor symmetry and neutrino phenomenology, Phys. Rev. D 77 (2008) 073008 [Erratum ibid. D 77 (2008) 119901] [ar**v:0801.3554] [INSPIRE].
J. Barry and W. Rodejohann, Deviations from tribimaximal mixing due to the vacuum expectation value misalignment in A 4 models, Phys. Rev. D 81 (2010) 093002 [Erratum ibid. D 81 (2010) 119901] [ar**v:1003.2385] [INSPIRE].
G. Altarelli, F. Feruglio and Y. Lin, Tri-bimaximal neutrino mixing from orbifolding, Nucl. Phys. B 775 (2007) 31 [hep-ph/0610165] [INSPIRE].
M. Honda and M. Tanimoto, Deviation from tri-bimaximal neutrino mixing in A 4 flavor symmetry, Prog. Theor. Phys. 119 (2008) 583 [ar**v:0801.0181] [INSPIRE].
S. Antusch, J. Kersten, M. Lindner, M. Ratz and M.A. Schmidt, Running neutrino mass parameters in see-saw scenarios, JHEP 03 (2005) 024 [hep-ph/0501272] [INSPIRE].
M. Lattanzi and J. Valle, Decaying warm dark matter and neutrino masses, Phys. Rev. Lett. 99 (2007) 121301 [ar**v:0705.2406] [INSPIRE].
P. Minkowski, μ → eγ at a rate of one out of 1-billion muon decays?, Phys. Lett. B 67 (1977) 421 [INSPIRE].
T. Yanagida, Horizontal symmetry and masses of neutrinos, in the proceedings of the Workshop on the unified theory and the baryon number in the universe, O. Sawada ed., KEK, Tsukuba Japan (1979).
S.L. Glashow, The future of elementary particle physics, in the proceedings of the 1979 Cargèse summer institute on quarks and leptons, M. Levy et al. eds., Plenum Press, New York U.S.A. (1980).
M. Gell-Mann, P. Ramond and R. Slansky, Complex spinors and unified theories, in Supergravity, P. van Nieuwenhuizen and D.Z. Freedman eds., North Holland, Amsterdam The Netherlands (1979).
R.N. Mohapatra and G. Senjanović, Neutrino mass and spontaneous parity violation, Phys. Rev. Lett. 44 (1980) 912 [INSPIRE].
J. Schechter and J.W.F. Valle, Neutrino masses in SU(2) × U(1) theories, Phys. Rev. D 22 (1980) 2227 [INSPIRE].
J. Schechter and J.W.F. Valle, Neutrino decay and spontaneous violation of lepton number, Phys. Rev. D 25 (1982) 774 [INSPIRE].
M. Malinsky, J. Romao and J. Valle, Novel supersymmetric SO(10) seesaw mechanism, Phys. Rev. Lett. 95 (2005) 161801 [hep-ph/0506296] [INSPIRE].
F. Feruglio, C. Hagedorn and L. Merlo, Vacuum alignment in SUSY A 4 models, JHEP 03 (2010) 084 [ar**v:0910.4058] [INSPIRE].
G. Giudice and R. Rattazzi, Theories with gauge mediated supersymmetry breaking, Phys. Rept. 322 (1999) 419 [hep-ph/9801271] [INSPIRE].
S. Antusch, S.F. King, M. Malinsky and G.G. Ross, Solving the SUSY flavour and CP problems with non-abelian family symmetry and supergravity, Phys. Lett. B 670 (2009) 383 [ar**v:0807.5047] [INSPIRE].
V. Dabbaghian, REPSN — For constructing representations of finite groups, GAP package, Version 3.0.2, http://www.gap-system.org/Packages/repsn.html.
A. Merle and R. Zwicky, Explicit and spontaneous breaking of SU(3) into its finite subgroups, ar**v:1110.4891 [INSPIRE].
P.M. van Den Broek and J.F. Cornwell, Clebsch-Gordan coefficients of symmetry groups, Phys. Status Solidi B 90 (1978) 211.
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Holthausen, M., Schmidt, M.A. Natural vacuum alignment from group theory: the minimal case. J. High Energ. Phys. 2012, 126 (2012). https://doi.org/10.1007/JHEP01(2012)126
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DOI: https://doi.org/10.1007/JHEP01(2012)126