Steve King, DCPIHEP, Colima

Transcription

1 !!! 1/6/11 Lecture I. The Flavour Problem in the Standard Model with Neutrino Mass Lecture II. Family Symmetry and SUSY Lecture III. SUSY GUTs of Flavour with Discrete Family Symmetry Steve King, DCPIHEP, Colima 1

2 ! Lecture II. Family Symmetry and SUSY

3 ! GUTs and Family Symmetry See-saw naturally suggests a high scale GUT Flavour problem suggests a family symmetry t e d u s c b Family symmetry GUT symmetry

4 Yukawa matrices

5 Quark mixing matrix V CKM Defined as 5 phases removed Lepton mixing matrix U PMNS Light neutrino Majorana mass matrix Defined as 3 phases removed

6 Thus the origin of the electron mass in the SM are the Yukawa couplings: Yukawa coupling y e must be small since <H 0 >=v=175 GeV Unsatisfactory Introduce right-handed neutrino ν er with zero Majorana mass then Yukawa coupling generates a Dirac neutrino mass Even more unsatisfactory

7 Hierarchical Symmetric Textures Consider the following ansatz for the upper 2x2 block of a hierarchical Y d Gatto et al successful prediction This motivates having a symmetric down quark Yukawa matrix with a 1-1 texture zero and a hierarchical form λ ¼ 0.2 is the Wolfenstein Parameter successful predictions

8 Up quarks are more hierarchical than down quarks G.Ross et al This suggests different expansion parameters for up and down Charged leptons are well described by similar matrix to the downs but with a numerical factor of about 3 in the 2-2 entry (Georgi-Jarlskog) at M U N.B. Electron mass is governed by an expansion parameter ε d» 0.15 which is not unnaturally small providing we can generate these textures from a theory at m b

9 See lecture by Ferruccio Feruglio Textures from U(1) Family Symmetry Consider a U(1) family symmetry spontaneously broken by a flavon vev For D-flatness we use a pair of flavons with opposite U(1) charges Example: U(1) charges as Q (ψ 3 )=0, Q (ψ 2 )=1, Q (ψ 1 )=3, Q(H)=0, Q(φ )=-1,Q(φ)=1 Then at tree level the only allowed Yukawa coupling is H ψ 3 ψ 3! The other Yukawa couplings are generated from higher order operators which respect U(1) family symmetry due to flavon φ insertions: When the flavon gets its VEV it generates small effective Yukawa couplings in terms of the expansion parameter Not quite of desired form non-abelian

10 Froggatt-Nielsen Mechanism What is the origin of the higher order operators? Froggat and Nielsen took their inspiration from the see-saw mechanism Where χ are heavy fermion messengers c.f. heavy RH neutrinos

11 There may be Higgs messengers or fermion messengers Fermion messengers may be SU(2) L doublets or singlets

12 SFK, Ross, Valesco-Sevilla,Varzielas Textures from SU(3) Family Symmetry In SU(3) with ψ i =3 and H=1 all tree-level Yukawa couplings Hψ i ψ j are forbidden. In SU(3) with flavons the lowest order Yukawa operators allowed are: For example suppose we consider a flavon with VEV then this generates a (3,3) Yukawa coupling Note that we label the flavon with a subscript 3 which denotes the direction of its VEV in the i=3 direction.

13 Next suppose we consider a flavon with VEV then this generates (2,3) block Yukawa couplings To complete the model we use a flavon with VEV then this generates Yukawa couplings in the first row and column Taking we generate desired structures

14 Froggatt-Nielsen diagrams Right-handed fermion messengers dominate with M u» 3 M d ε d = <φ 23 >/M d» 0.15 and ε u = <φ 23 >/M u» 0.05 Unsatisfactory features: 1. Suggests y b >y t! 2. First row/column is only quadratic in the messenger mass. To improve model we introduce sextet and singlet flavons

15 Flavon sextets for the third family SFK,Luhn 09 Idea is to use flavon sextets χ = 6 and Higgs messengers to generate the third family Yukawa couplings Third family Yukawas

16 Flavon singlet for first row/column SFK,Luhn 09 Flavon singlet ξ = 1 leads to first row and column Yukawa couplings involving a cubic messenger mass So finally we arrive at desired quark textures (with y t» y b )

17 The See-Saw Mechanism and Family Symmetry Possible type II contribution (ignored here) Dirac matrix Heavy Majorana matrix Diagonalise to give effective mass Light Majorana matrix A very natural and appealing mechanism! Neutrinos are so light because RH neutrino get heavy Majorana masses (L number violated at HE) Neutrinos are a probe of physics at high energy scales up to M GUT!

18 Sequential Dominance Diagonal RH nu basis columns SFK See-saw Sequential dominance Dominant m 3 Subdominant m 2 Decoupled m 1 Constrained SD Tribimaximal HPS

19 The CSD conditions emerge from a non-abelian family symmetry Need with 2$ 3 symmetry (from maximal atmospheric mixing) 1$ 2 $ 3 symmetry (from tri-maximal solar mixing) Examples of suitable non-abelian Family Symmetries: SFK, Ross; Velasco-Sevilla; Varzelias SFK, Malinsky Discrete subgroups preferred by vacuum alignment

20 Consider the TB neutrino mass matrix in the flavour basis i.e. diagonal charged lepton basis Columns of U TBM

21 Form Dominance Chen,SFK Form Dominance is a mechanism for achieving a form diagonalizable effective neutrino mass matrix starting from the type I see-saw mechanism Form Dominance assumption: in the flavour basis where RH neutrino mass matrix is diagonal the columns of Dirac mass matrix / columns of U TBM with N.B. Only three parameter combinations Form Dominance is generalisation of CSD for any neutrino mass spectrum and applies to e.g. the Altarelli-Feruglio A 4 models

22 Simple toy example: SO(3) Introduce three triplet flavons φ 1, φ 2, φ 3 with VEVs along columns of U TB Consider

23 Vacuum misalignment SFK Consider general perturbations to the vacuum alignment in limit m 1 0 Recall TB deviation parameters s is only sensitive to vacuum misalignments of ϕ 2 proportional to ϕ 1 a,r are only sensitive to vacuum misalignments of ϕ 3 e.g. gives a=s=0 but r 0 tri-bimaximal reactor mixing

24 SUSY Can implement see-saw mechanism in MSSM Superpotential (Yukawa couplings) sneutrinos neutrinos Hats mean superfields which contain both fermion and boson components But SUSY gives no insights into the flavour problem, in fact it makes it worse

25 The SUSY Flavour Problem In SUSY we want to understand not only the origin of Yukawa couplings But also the soft masses See-saw parts

26 SUSY FCNC s result from off-diagonal soft masses in the basis where the charged Yukawas are diagonal (also EDMs result from soft phases) e.g. slepton doublet mass matrix Off-diagonal slepton masses lead to LFV

27 Experimental limits on flavour violation and CP Antusch, SFK, Malinksy 08 Down type squarks Most stringent Up type squarks Charged sleptons

28 In SM we want to understand the origin of Yukawa couplings In SUSY also want to know origin of the soft masses of the squarks and sleptons Both are predicted by family symmetry, leading to approximately universal soft masses: with small off-diagonal elements due to SU(3) breaking flavon corrections flavour changing processes

29 In the SU(3) model the left-handed soft masses are accurately universal While the right-handed soft masses get corrections from universality

30 Buras et al

31 Summary of Lecture 2 Family symmetry with Froggatt-Nielsen mechanism with U(1) family symmetry can provide an explanation of small Yukawa couplings We considered particular symmetric textures which satisfy the Gatto relation, and were led to SU(3) family symmetry Natural implementations of see-saw mechanism with a neutrino mass hierarchy suggests sequential dominance Natural implementation of TB mixing suggests form dominance or CSD which in turn requires a non-abelian family symmetry Non-Abelian Family symmetry such as SU(3) can solve SUSY flavour problem and also SUSY CP problem