Flavour and Higgs in Pati-Salam Models

Transcription

1 Flavour and Higgs in Pati-Salam Models Florian Hartmann Universität Siegen Theoretische Physik I Siegen,

2 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 Outline 1 Why Pati-Salam and Flavour SU(3)? Problems of the SM Fermion embedding in Pati Salam 2 Higgs potential Higgs potential of PS Mass spectra 3 Flavour physics Idea of flavour symmetry SU(3) flavour symmetry Possible flavon representations Triplet flavon model Decuplet flavons 4 Conclusions

3 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 Problems of the SM Flavour structure unclear why there are three families Flavour structure is experimental input CKM and PMNS matrix different form not clear CKM PMNS Hypercharge normalisation of the Hypercharge not fixed by theory electric charge depends on Hypercharge Q = T 3 + Y normalisation fixed by charge of particles (experiment) Neutrino mass Neutrinos are massive right handed neutrino needed

4 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 PS SU(3) F as possible solution Embedding of the hypercharge G PS = SU(4) SU(2) L SU(2) R U(1) Y natural subgroup of left-right symmetric theories U(1) Y SU(2) R U(1) B L SU(4) SU(2) R SU(4) leads to unification of leptons and quarks Flavour symmetry consider Yukawa coupling as field (flavon) families correspond to representations flavons break Flavour symmetry Flavour structure of the SM

5 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 Fermion embedding in Pati Salam ( u u ) u d d d L (3, 2) 1 6 ( ) ν e L (1, 2) 1 2 SM: (SU(3) C, SU(2) L ) Y ( u c u c u c) ( R d c d c d c) R (3, 1) 2 3 (3, 1) 1 3 e c R (1, 1) 1 ( u u u ) ν d d d e L (4, 2, 1) PS: (SU(4), SU(2) L, SU(2) R ( u c u c u c ) ν d c d c d c e c R (4, 1, 2) (4, 2, 1) = (3, 2) 1 (1, 2) (4, 1, 2) = (3, 1) 1 (3, 1) 2 (1, 1) (1, 1) 1 natural ν R candidate (type I See-Saw)

6 Higgs potential of PS Used Higgs fields Σ 1 = (15, 1, 1) or Σ = (15, 1, 3) to break SU(4) H u = (4, 1, 2) and H d = (4, 1, 2) to break U(1) B L U(1) R MSSM-Higgs: h = (1, 2, 2) F = (6, 1, 1): unifies together with h to 10 of SO(10) Superpotential with Σ 1 W = 1 2 m h (hh) m H (H u H d ) 1 2 m F FF 1 2 m Σ 1 Σ 1 Σ 1 +c 1 Σ 1 Σ 1 Σ 1 + c 3 H u Σ 1 H d + c 4 H d H d F + c 5 H u H u F Superpotential with Σ W = 1 2 m h (hh) m H (H u H d ) 1 2 m F FF 1 2 m Σ ΣΣ +c 1 ΣΣΣ + c 3 H u ΣH d + c 4 H d H d F + c 5 H u H u F Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20

7 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 Mass spectra of case Σ 1 Figure: Spectrum for v d v Figure: Spectrum for v d v

8 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 Mass spectra of case Σ Figure: Spectrum for v d v Figure: Spectrum for v d v

9 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 Idea of flavons Yukawa coupling Yukawa term with flavons: L Yuk = y ij h Ψ L i Ψ cr j L Yuk = c φ ij h Ψ L i Ψ cr j Aim of flavons reproduce family structure of the SM use only few new flavon fields and a simple symmetry group G E.g.: G = SO(3), G = SU(3) Bonus: additional relations between u, d, e and ν from GUTs Explanation of fermion masses and mixings of the SM

10 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 SU(3) Flavour symmetry SU(3) Ψ L and Ψ R are flavour triplets all fermions can be embedded in one representation Example: SO(10) SU(3) F : Ψ = (16, 3) Flavoured Higgs consider three generations of Higgs fields Higgs is also a flavour triplet Unification of fermions and Higgs possible Example: E 6 SU(3) F Ψ = (27, 3) Attention: new trivial invariant ε ijk Ψ L i Ψ R j Additional discrete symmetry needed h k

11 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 Possible flavon representations Representation possible invariants result singlet ε ijk Ψ L i Ψ R j H k no flavour structure triplet φ l ε ijk Ψ L i H j φ k Ψ R l extension of Ψ L i φ i Ψ R j φ j H k φk existing models sextet Ψ L l φ lm Ψ R i ε ijk φ mj H k bad Yukawa structure octet ε ijk Ψ L i Ψ R j H l φ l k bad Yukawa structure decuplet Ψ L i Ψ R j H k φ ijk easy top Yukawa coupling

12 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 Triplet flavon model Aim extend model of King and Ross 1 to triplet Higgses introduce no additional fields or symmetries Reproduce Yukawa matrix of the form ( 0 O(ɛ3 ) ) O(ɛ 3 ) Y = O(ɛ 3 ) O(ɛ 2 ) O(ɛ 2 ) O(ɛ 3 ) O(ɛ 2 ) O(1) m t recover near diagonal Majorana matrix for right handed neutrinos tri-bimaximal Lepton mixing What we not do explain different expansion parameters in up and down sector write down scalar potential of the flavons discuss origin of the flavons 1 (arxiv:hep-ph/ v3)

13 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 Triplet flavon model Field SU(3) F PS U(1) Z 2 Ψ 3 (4, 2, 1) 1 + Ψ c 3 ( 4, 1, 2) 1 + h 3 (1, 2, 2) 1 + φ 3 3 (1, 1, 1) 3 + φ 3 3 (1, 1, 1) 1 + φ 23 3 (1, 1, 1) 2 + φ 23 3 (1, 1, 1) 1 H u 3 ( 4, 1, 2) 2 + H d 3 (4, 1, 2) 2 + φ 3 = ( 0 0 b ) T M φ 3 = ( 0 0 b ) T M φ 23 = ( 0 ɛ ɛ e iθ ) T M φ 23 = ( 0 ɛ ɛ e -iθ ) T M H d = ( 0 0 v d ) T ( Ylead f = 0 by 2 ɛ 2 e -iθ by 2 ɛ 2 0 e -iθ by 2 ɛ 2 b 3 y 1 +e -iθ by 2 ɛ 2 ) W lead = y 1 M 3 Ψ i φ i 23 Ψ c j φ j 23 h k φ k 3 + y 2 M 3 Ψ i φ i 3 Ψ c j φ j 3 h k φ k 3

14 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 Triplet flavon model W sub = y 3 M 4 Ψ i Ψ c j φ 3,k ε ijk h l φ l 23 φ m 23 φ 23,m + y 4 M 4 Ψ i Ψ c j φ 23,k ε ijk h l φ l 23 φ m 23 φ 3,m + y 5 M 4 Ψc i φ 3,j φ 23,k ε ijk h l φ l 23 φ m 23 Ψm + y 6 M 4 Ψ i φ 3,j φ 23,k ε ijk h l φ l 23 φ m 23 Ψc m + y 7 M 4 Ψ i h j φ 23,k ε ijk Ψ c l φ l 23 φ m 23 φ 3,m + y 8 M 4 Ψc i h j φ 23,k ε ijk Ψ l φ l 23 φ m 23 φ 3,m + y 9 M 4 Ψ i Ψ c j h k ε ijk φ l 23 φ 23,l φ m 23 φ 3,m + x 1 M 4 Ψ i Ψ c j φ 3,k ε ijk h l φ l 3 φ m 3 φ 23,m + x 2 M 4 Ψ i Ψ c j φ 23,k ε ijk h l φ l 3 φ m 3 φ 3,m + x 3 M 4 Ψ i φ 23,j φ 3,k ε ijk h l φ l 3 Ψc m φ m 3 + x 4 M 4 Ψc i φ 32,j φ 3,k ε ijk h l φ l 3 Ψmφ m 3 + x 5 M 4 Ψ i h j φ 23,k ε ijk Ψ c l φ l 3 φ 3,mφ m 3 + x 6 M 4 Ψc i h j φ 23,k ε ijk Ψ l φ l 3 φ 3,mφ m 3 + x 7 M 4 Ψ i Ψ c j h k ε ijk φ 23,l φ l 3 φ 3,mφ m 3 x 1 = x 6 x 4 x 7 x 2 = x 4 x 6 x 3 = x 6 + x 5 x 4 0 e -iθ b z 1 ɛ 3 e -2iθ b z 2 ɛ 3 Y f = e -iθ b z 3 ɛ 3 bɛ 2 a 2 bɛ 2 e -2iθ b z 5 ɛ 3 a 2 bɛ 2 b 3 + a 2 bɛ 2

15 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 Majorana potential W Maj = W Majsub = c 1 ( M 5 Ψ c ) 2 ( i Hi d φ 23,j φ j ) 2 c 2 ( 23 + M 5 Ψ c ) 2 ( i Hi d φ 23,j φ j ) c 3 M 5 Ψc i Hi d Ψc j φ j 23 φ 23,k H k d φ 23,l φ l 23 + c 4 M 5 Ψc i Hi d Ψc j φ j 3 φ 23,k H k d φ 23,l φ l 3 + c ( 5 M 5 Ψ c ) 2 ( i φi 23 φ 23,k H k ) 2 c 6 ( d + M 5 Ψ c ) 2 ( i φi 3 φ 3,k H k ) 2 d c 7 M 6 Ψc i φ 3,j φ 23,k ε ijk Ψ c l φ ( m 3,l H d φ ) 2 23,m + c 8 M 6 Ψc i φ 3,j φ 23,k ε ijk Ψ c l Hl d Hm d φ 23,m φ 23,n φ n 3 + c ( 9 M 7 Ψ c i φ 23,j φ 3,k ε ijk ) 2 ( H l ) 2 d φ 23,l c 9 e 2iθ b 2 ɛ 4 0 (c 7 + c 8 )e 2iθ b 2 ɛ 3 M RR = 0 c 1 e 2iθ ɛ 4 (c 1 + 2c 3 )e iθ ɛ 4 (c 7 + c 8 )e 2iθ b 2 ɛ 3 (c 1 + 2c 3 )e iθ ɛ 4 (c 2 + c 4 + c 6 )e -iθ b 2 ɛ 2 Eigenvalues of order: ( ɛ 2 + O ( ɛ 4), ɛ 4, ɛ 4) Eigenvectors: unit vectors up to corrections of order ɛ 2

16 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 Decuplet flavons Field SU(3) F PS U(1) Ψ 3 (4, 2, 1) 1 Ψ c 3 ( 4, 1, 2) 1 h 3 (1, 2, 2) 1 φ 1 10 (1, 1, 1) 3 φ 2 10 (1, 1, 1) 3 φ 3 10 (1, 1, 1) 3 φ 1 10 (1, 1, 1) 5 φ 2 10 (1, 1, 1) 1 φ 3 10 (1, 1, 1) 6 P 1 1 (1, 1, 1) 9 P 2 1 (1, 1, 1) 2 P 3 1 (1, 1, 1) 2 P 4 1 (1, 1, 1) 3 H u 3 ( 4, 1, 2) 2 H d 3 (4, 1, 2) 2 φ M φ ɛ 2 M φ ɛ 2 M φ ɛ 3 M φ ɛ 3 M W lead = Y f lead 3 i=1 y i M ΨΨc hφ i 0 ɛ 3 ɛ 3 ɛ ɛ 2 ɛ 2 ɛ 3 ɛ 2 1

17 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 Majorana mass matrix Neutrino Majorana matrix W Majorana = 1 M 3 Ψc Ψ c ( ) H d H d C1 φ 1 φ 1 + C 2 φ 1 φ 2 + C 3 φ 1 φ 3 + C 4 φ 2 φ 2 c 8 ɛ 4 ν c 3 ɛ 3 ν (c 3 c 7 ) ɛ 3 ν M Maj c 3 ɛ 3 ν c 2 ɛ 2 ν (c 2 c 6 ) ɛ 2 ν (c 3 c 7 ) ɛ 3 ν (c 2 c 6 ) ɛ 2 ν (c 1 + c 4 c 5 ) b Eigenvalues of order: ( ) 1, ɛ 2 ν, ɛ 4 ν Eigenvectors: unit vectors up to corrections of order ɛ 2 near tri-bimaximal lepton mixing

18 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 Scalar potential of the flavons Superpotential W 10 = c M P 1 φ 1 φ 2 φ 3 ; W Ren = 3 i,j P i+1 F-terms F 10 not sufficient for the vacuum alignment D-terms all vevs can be chosen diagonal to each other φ i φ j = 0, i k ( φj φ i µ i ) F Pi+1 φ i φ i µ i = 0 hierarchy of the vevs if µ 1 1, µ 2 ɛ 2 and µ 3 ɛ 6 D-term of only anti-decuplets: D a 10 = g 3 all vevs of same order introduce decuplets with same vev structure k φi k T ij a φcj k

19 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 Conclusions (Part I) Breaking Pati-Salam PS breaking possible with either (15, 1, 1) or (15, 1, 3) no accidental SUSY breaking different mass spectra possible with different ratios v d/v consider running of gauge couplings consider couplings to fermions Flavour SU(3) extension to flavoured Higgses is possible extension for triplet flavons is non trivial but possible decuplet flavons seems to be an interesting theory worth further investigation

20 Florian Hartmann (Uni Siegen) Flavour and Higgs in Pati-Salam Models Siegen / 20 Conclusions (Part II) Further steps explain why ɛ up 3ɛ down reasonable CKM-Mixing include Georgi-Jarlskog mechanism (Σ) consider combination of triplet and decuplet flavons consider phenomenology of the models Combination of both parts mass term of h not invariant under SU(3) F no µ-problem F also flavour triplet (SO(10)) F decouples completely (after SO(10) breaking) no mass term allowed sterile dark matter?