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Discrete symmetries

Explore the intricate world of discrete symmetries in modeling flavor mixing, discussing challenges, emerging relations, and recent developments in the field. Dive into theories, data, and evidence shaping our understanding of flavor models.

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Discrete symmetries

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  1. Discrete symmetries and models of flavor mixing A. Yu. Smirnov International Centre for Theoretical Physics, Trieste, Italy DISCRETE 2010

  2. Davide Meloni Luca Merlo De A Toorop Cristoph Luhn C Simoes A Modragon Flavor of flavor models For models see Certain features of the lepton data can be considered as an evidence of discrete symmetries Many various realizations have been proposed • require new extended structures • many assumptions (ad hoc assignments of charges • selection of representations etc..) • additional auxiliary symmetries (which are even • more powerful than the announced symmetry) • no natural (?) extension to quarks • no (direct) relations between masses and mixing No convincing model Introduce symmetry and immediately by any means try to get rid of it… Try further? Be less ambitious? Apply differently?

  3. Plan 1. Data and evidences 2. TBM and ``Symmetry building'' 3. From leptons to quarks and GUT's 4. QLC and other things 5. Perspectives

  4. Data & Evidences

  5. Mass and Mixing ne Origin of exitement nm nt |Ue3|2 |Um3|2 |Ut3|2 tan2q12 =|Ue2|2 / |Ue1|2 n3 bi-maximal sin2q13 = |Ue3|2 tan2q23 = |Um3|2 / |Ut3|2 Dm2atm mass |Ue2|2 deviations ~ 0.01 – 0.05 n2 trimaximal n1 Dm2sun |Ue1|2 nm - nt symmetry which implies q13 = 0 Normal mass hierarchy x 2 / :2 / interchange 1 - 2 Dm2atm = Dm232 = m23 - m22 nf = UPMNSnmass Dm2sun = Dm221 = m22 - m21 UPMNS = U23 Id U13 I-d U12

  6. sinq13 = 0.10 – 0.15 • Correct measure of mixing? • RGE non invariant • New neutrino states ? • 4th generation of fermions? • charge leptons do not show • - such invariance can screw up our constructions Another line of developments

  7. Global fit of oscillation data deviations M. C. Gonzalez-Garcia, M. Maltoni, J. Salvado CL: 90. 95, 99. 99.73 %

  8. 1-3 versus 1-2 mixings Correlate SNO analysis, 2010 KamLAND 1009.4771 TBM

  9. 1-3 vs. 1-2 mixings Combined: solar + KamLand 68, 95 99.7 % CL solar KL Global: M. Mezzetto T. Schwetz ArXiv: 1003.5800 SuperKamiokande arXiv:1010.011

  10. Atmospheric neutrinos SuperKamiokande: the first complete 3n analysis: searches for 1-3 mixing and deviations of 2-3 mixing from maximal R.Wendell et al arXiv:1002.3471 NH In spite of excess of e-like sub-GeV events indication for q23 < p/4 sin2q23 = 0.5 (0.407 – 0.583) 90% sin2q13 = 0.0 ( < 0.04 ) 90%

  11. 12- and 13- mixings Global fit of oscillation data M. C. Gonzalez-Garcia, M. Maltoni, J. Salvado SK09 CHOOZ 90% CL bounds from Different experiments in assumption that true value sin2q13 = 0 QLC Double CHOOZ T2K RENO Daya Bay NOvA Since 2015 dominated by Daya Bay 90, 95, 99, 3s CL contours

  12. 2-3 mixing SK: sin22q23 > 0.93, 90% C.L. T2K maximal mixing QLCn QLCl Gonzalez-Garcia, Maltoni, Salvado 2010 3s 1s SK (3n), 2010 90% 3s Gonzalez-Garcia, Maltoni, A.S. 2s 1s Fogli et al 2008** 0.2 0.3 0.4 0.5 0.6 0.7 sin2q23 * in agreement with maximal, though all complete 3n - analyses show shift * shift of the bfp from maximal is small * still large deviation is allowed: (0.5 - sin2q23)/sin q23 ~ 40% 2s

  13. l Mass hierarchies Solar, KamLAND at mZ up down charged neutrinos quarks quarks leptons 1 m2 m3 Dm212 Dm322 > 10-1 ~ 0.18 10-2 Neutrinos have the weakest mass hierarchy (if any) among fermions mass ratios 10-3 10-4 Related to the large lepton mixing? Koide relation 10-5 sinqC ~ md/ms mu mt = mc2 Gatto–Sartori-Tonin relation

  14. TBM and symmetry building

  15. Tri-bimaximal mixing L. Wolfenstein P. F. Harrison D. H. Perkins W. G. Scott 2/3 1/3 0 - 1/6 1/3 - 1/2 - 1/6 1/3 1/2 Utbm = n3 is bi-maximally mixed n2is tri-maximally mixed Utbm = U23(p/4) U12 - maximal 2-3 mixing - zero 1-3 mixing - no CP-violation - sin2q12= 1/3 Uncertainty related to sign of 2-3 mixing: q23 = p/4  - p/4 Symmetry from mixing matrix

  16. The TBM- mass matrix Mixing from diagonalization of mass matrix mTBM = UTBM mdiag UTBMT mdiag = diag (| m1|, |m2|ei2f , |m3|e i2f ) 2 3 a b b … ½(a + b + c) ½(a + b - c) … … ½(a + b + c) mTBM = c = m3 b = (m2 – m1)/3 a = (2m1 + m2)/3 The matrix has S2 permutation symmetry nm<-> nt

  17. TBM-symmetry Key point is that relations are simple and mass matrix has certain symmetry mem = met TBM mass relations mmm = mtt Three angles  three conditions mee + mem = mmm + mmt Sa mea = Sb mmb or ViT mTBM Vi = mTBM Invariance: • 0 0 • 0 0 1 • 0 1 0 -1 2 2 … -1 2 .. … -1 U = S = The mass matrix of the charged leptons is diagonal due to symmetry 1 0 0 … w2 0 … … w w = exp(-2ip/3) T = S, T, U –elements of S4 T+ (me+m e)T = me+m e

  18. Symmery breaking No exact flavor symmetry Mixing appears as a result of different ways of the flavor symmetry breaking in neutrino and charged lepton sectors Symmetry is not broken completely; residual symmetries in the neutrino and charged lepton sectors are different Gf Residual symmetries Gl Gn Mn TBM-type Ml diagonal In turn, this split originates from different flavor assignments of the RH components of Nc and lc and different higgs multiplets

  19. Generic problem m = F(Y, v) Mechanism of mass generation VEV’s Yukawa couplings VEV alignment • different contributions • high order corrections follow from independent sectors Scalar potential Yukawa sector TBM tune by additional symmetries All these components should be correlated ``Natural’’ – consequence of symmetry? one step constructions do not work

  20. at the same time... 0 0 0 0 1 - 1 0 - 1 1 4 - 2 - 2 - 2 1 1 - 2 1 1 • 1 1 • 1 1 1 • 1 1 1 m1 6 m2 3 m3 2 + mTBM= + Three singular matrices If m1 is small 0 1 -1 1 1 1 + B (0, 1, -1) (1, 1, 1) mTBM~ A • - see-saw, D = 5 operatots • universal coupling (which cab be obtained • two product of triplets) • VEV’s of scalar triplet: (1, 1, 1) and (0, 1, -1) • which can be obtained easily in SUSY version This indicates on

  21. Flavons and Flavored higgses Soft via see-saw Flavons Flavored higgses Singlet of gauge symmetry group Many Higgs doublets - tests at LHC Separation of the EW symmetry and flavor symmetry breakings Strongly restricted: - FCNC - anomalous magnetic moment of muon 1 Ln-1 L ec H fn L - above GUT scale?  difficult to test

  22. Small groups with irreducible representations 3 Group Order Representations A4 12 1, 1’ 1’’ 3 S4 24 1 1’ 2 3 3’ T’ 24 1 1’ 1’’ 2 2’ 2’’ 3 T7 21 1 1’ 1’’ 3 3* D(27) 27 11 - 19 3 3’ …

  23. A_4 symmetry E. Ma Symmetry group of even permutations of 4 elements A4 Symmetry of tetrahedron Generators: S, T Presentation of the group: no U = Amt S2 = 1 T3 = 1 (ST)3 = 1 Irreducible representations: 3, 1, 1’, 1’’ Products and invariants 3x3 = 3 + 3 + 1 + 1’ + 1’’ 1’ x 1’’ ~ 1

  24. A_4 symmery breaking A4 ``accidental’’ symmetry due to particular selection of flavon representations and configuration of VEV’s Amt T S Ml diagonal Mn TBM-type In turn, this split originates from different flavor assignments of the RH components of Nc and lc and different higgs multiplets

  25. An A_4 model G. Altarelli D. Meloni Yukawa sectors A4 Z4 Charged lepton Neutrinos 1 i -1 -i 3 L L 1 n k i i hd x’ hu fT i 1’ Nc i 1 i lc -1 at multiplets 1, i, -1 1’’ fS M < fT > = v S (0, 1, 0) x 1 k = 0, … n = 1, … 1 < fS > = v S (1, 1, 1) Particular selection of representations Flavon sector U(1)R x’ x fS fT 0 2 x0 fS0 fT0 Vacuum alignment Driving fields -1 1 1 GUT-scale or higher?

  26. Mixing and masses Unrelated or have indirect relations Extend symmetry Additional U(1) or discrete symmetries for mass hierarchies In universal approach: corrected Fritzsch ansatz Discrete symmetries used to get texture zeros (see talk by C. Simoes)

  27. From leptons to quarks Discrete flavor symmetries have not been mentioned in the talk by A Weiler Do quarks need the leptonic discrete symmetries?

  28. Mass & Mixing ne nm nt zero 1-3 mixing? Leptons Quarks n3 t q13 q23 mass mass q12 n2 c n1 u qd = UCKM+ qu nf = UPMNSnmass qu = (u, c, t)

  29. Extending symmetries the same 3,1 different representations representations VCKM = I UPMNS = Utbm 2 + 1 structure  different groups As the lowest order Corrections from high order operators

  30. T' - symmetry Order 24, double covering A4 or binary tetrahedron group Generators: S, T, R 1 odd dimension of -1 even representation (ST)3 = 1 R2 = 1 R = Presentation: T = 1, S2 = R, Irreducible representations: 1, 1’, 1’’ , 2, 2’, 2’’ , 3 complex conjugate Products and invariants 3 x 3 = 1 + 1’ + 1’’ + 3 + 3 as in A4 1’ x 1’’ = 1 2a x 3 = 2 + 2’ + 2’’ 2 x 2 = 1 + 3 (with ``conservation’’ of primes) 1 x 2 = 2 New flavor structure

  31. F. Feruglio, C. Hagedorn Yin Lin L. Merlo T' - symmetry model T’ x Z3 x U(1)FN T’ Z3 Charged lepton Neutrinos 3 1 w w2 2 L L w fT Hu 2’ Hd 1 ec at multiplets x fS 2’’ mc tc w w2 1 1’ w w QD Q3 1’’ Vacuum alignment Hu h fT Hd < fS > = vS (1, 1, 1) bc tc uDc dDc < fT > = vT (1, 0, 0) w2 w2 < h> = v(1, 0) w2 w2 <x> = u <x’’> = < x > = 0

  32. TBM and GUT's Generic problem: In many models, flavor prescription required for explanation of differences of mass and mixing of quarks and leptons prevents from GU Relate this difference to spontaneous breaking of GUT symmetry GUT x G + ... New elements should be added flavor Restrict flavor charge assignments - pair vector-like matter fields - its mixing with usual matter - use seesaw type II for neutrinos - singlet fermions additional higgses

  33. SU(5) x A_4 I. K. Cooper S. K. King, C. Luhn ArXiv1001.2265 A4 x Z2 x Z2 x U(1) x U(1)R Matter: F = 5, T1,2,3 = 10, N = 1, S = 24 Charged leptons / down quarks Neutrinos ++ A4 F F f23 H5 f3 x2 H5 f23 f123 H5 3 H5 +- T1 T3 ++ -+ 1 N S -+ f23 f123 x x2 H45 H5 1’ T2 x4 1’’ f232 f1232 10 plets are F-singlets! Vacuum alignment Z2 x Z2 <f123> ~(1, 1, 1) <f23> ~(0, 1, -1) + - ~ TBM <f1> ~(1, 0, 0) < f3 > ~(0, 0, 1)

  34. ... continued for upper quarks All terms contain H5 1 x A4 T3 3 f23 f3 x2 f3 f123 x2 1 1’ T2 T1 1’’ x f232 x5 x’2 x2

  35. SO(10) x G_f More challenging SO(10) x G + ... New elements should be added flavor - Singlet fermions - 16H - flavons - 126 126 - pair vector-like: 16 16 matter fields - 10’ - Flavons - S4 x Zn C Hagedorn M. Schmidt A.S. - T7 B. Dutta , Y Mimura R. Mohapatra Screening of the Dirac structures Leptons from quarks

  36. SO(10) GUT + ... Hagedorn Schmidt AS Something is missed? RH-neutrino ur , ub , uj , n dr , db , dj , e urc, ubc, ujc, nc drc, dbc, djc, ec 16 S S S S S S S S S S S S S • - Decrease effective scale • Enhance mixing • Produce zero order mixing • - Screen Dirac mass hierarchies • Produce randomness (anarchy) • Seesaw symmetries S S S S S S S S S S S S S S S S Hidden sector

  37. Is TBM accidental? M Abbas, A.S D sin q13 ~ 0.15 D sin2q12 ~ 0.02 D sin2q23 ~ 0.05 Experiment: Allowed deviation from TBM – can lead to strong (maximal) violation of TBM-conditions Strong deviation of mn from TBM form Leading structures are relatively robust sub-leading – can change completely The approximate TBM is accidental or a manifestation of some other symmetries which differ from TBM, or other structures This opens up new new approaches to explain data.

  38. QLC and QL- symmetries

  39. QLC relations A.S. M. Raidal H. Minakata ql12 + k qq12 ~ p/4 ql23 + qq23 ~ p/4 k = 2-1/2 or 1 qualitatively: 2-3 leptonic mixing is close to maximal because 2-3 quark mixing is small 1-2 leptonic mixing deviates from maximal substantially because 1-2 quark mixing is relatively large

  40. Complementarity or Cabibbo ``haze '' P. Ramond Deviations from BM due to high order corrections Altarelli et al Complementarity: implies quark-lepton symmetry or GUT, or horizontal symmetry Weak complementarity or Cabibbo haze Corrections from high order flavon interactions which generate simultaneously Cabibbo mixing and deviation from BM, GUT is not necessary mm sinqC = mt or sin qC = 0.22 as ``quantum’’ of flavor physics

  41. Possible implications ``Lepton mixing = bi-maximal mixing – quark mixing’’ Unification or family symmetry Quark-lepton symmetry Existence of structure which produces bi-maximal mixing See-saw? Properties of the RH neutrinos

  42. Bi-maximal mixing Ubm = U23mU12m • - maximal 2-3 mixing • - zero 1-3 mixing • maximal 1-2 mixing • - no CP-violation ½ ½ -½ ½ ½ ½ -½ ½ 0 Two maximal rotations Ubm = F. Vissani V. Barger et al In seesaw: structure of Majorana mass matrix of RH neutrinos In the lowest approximation: Vquarks = I, Vleptons =Vbm m1 = m2 = 0 Corrections generate - mass split - CKM and - deviation from bi-maximal Deviation

  43. BM - symmetry mem = met BM mass relations mmm = mtt mee = mmm + mmt ViT mBM Vi = mBM Invariance: - ½ - ½ - ½ ½ -½ - ½ -½ ½ 0 • 0 0 • 0 0 1 • 0 1 0 U= SBM= The mass matrix of the charged leptons is diagonal due to symmetry with respect to transformations: T = diag (- 1, -i, i ) T, SBM generators of S4

  44. S -symmetry 4 Order 24, permutation of 4 elements Generators: S, T, U or (ST)3 = 1 U2 = 1 A3 = B4 = (BA2)2 = 1 Presentation: T3 = 1, S2 = 1, Irreducible representations: 1 , 1’, 2, 3, 3’ 3 x 3 = 3’ x 3’ = 1 + 2 + 3 + 3’ 3 x 3’ = 1’ + 2 + 3 + 3’ Products and invariants 1’ x 1’ = 1 New flavor structure 2 x 3 = 2 x 3’ = 3 + 3’ 2 x 2 = 1 + 1’ + 2 1’ x 2 = 2

  45. Model for BM-mixing G. Altarelli F. Feruglio, L. Merlo * S4 Z4 Charged lepton Neutrinos 1 i -1 -i 3 * L L 3’ n k m 1 i cl hd q hu fl 2 i Nc i 1 i ec 1 mc 1 at multiplets tc -1, -i, -i 1’ fn M < fl > = vl (0, 1, 0) x 1 < cl > = vl (0, 0, 1) 1 No doublet representations * < fn> = vn(0, 1, -1) Flavon sector U(1)R Froggatt-Nielsen sector cl fl fn x 0 2 ec, mc,t c q i i fn0 x0 cl0 yl0 2, 1, 0 -1 U(1)FN Driving fields -1 1 -1

  46. Perspectives

  47. 1-3 mixing Deviation from Maximal 2-3 Precision measurements Tests Leptogenesis FCNC Can be affected by Discrete symmetries LHC m  e g Models with flavored Higgses

  48. Going beyond More fermions. Singlets Sterile neutrinos 4th generation screening of dirac structures Discrete symmetry – gives similar structures of two Dirac matrices in double see saw mechanism

  49. Conclusions Features which hint some symmetry – accidental Deviations from TBM can be significant, Realizations are too complicated… Alternative approaches to explanation of the observed features Some version of broken discrete symmetries give correct explanation of data. Physics behind neutrino has rich structure and it leads to rich phenomenology. Observed symmetry of lepton is related to symmetry of Hidden sector at very high scales. It is communicated (via mixing) to neutrinos No analogy in quark sector. Another (unique) physics is involved in CKM and PMNS deviations from symmetric structure.

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