Fermion Masses and Unification

Fermion Masses and Unification Steve King University of Southampton

Lecture 4 SU(3), GUTs and SUSY Flavour 1.SU(3) Family Symmetry 2.SU(3) £ SO(10) Model 3.Quark-lepton connections 4. SUSY Flavour Problem 5. SU(5) GUTs and Soft Masses 6. Family Symmetry and Soft Masses 7. Family Symmetry and SUSY CP 8. Do we need a family symmetry?

The family symmetry is spontanously broken by antitriplet flavons Unlike the U(1) case, the flavon VEVs can have non-trivial vacuum alignments. We shall need flavons with vacuum alignments: <3>/ (0,0,1) and <23>/ (0,1,1) in family space (up to phases) so that we generate the desired Yukawa textures from Froggatt-Nielsen: 1. Gauged SU(3) family symmetry Now suppose that the fermions are triplets of SU(3) i = 3 i.e. each SM multiplet transforms as a triplet under a gauged SU(3) with the Higgs being singlets H» 1 This “explains” why there are three families c.f. three quark colours in SU(3)c

In SU(3) with flavons the lowest order Yukawa operators allowed are: Frogatt-Nielsen in SU(3) family symmetry In SU(3) with i=3 and H=1 all tree-level Yukawa couplings Hij are forbidden. 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.

Next suppose we consider a flavon with VEV then this generates (2,3) block Yukawa couplings Writing and these flavons generate Yukawa couplings If we have 3¼ 1 and we write 23 =  then this resembles the desired texture To complete the texture there are good motivations from neutrino physics for introducing another flavon <123>/ (1,1,1)

Varzielas,SFK,Ross 2. SU(3) £ SO(10) Model Majorana Operators Yukawa Operators

Inserting flavon VEVs gives Yukawa couplings After vacuum alignment the flavon VEVs are Writing Yukawa matrices become:

Assume messenger mass scales Mf satisfy Then write Yukawa matrices become, ignoring phases: Where

3. Quark-Lepton Connections . . . . . From above we see that

In a given model we can predict and . Note the sum rule Charged Lepton Corrections and  sum rule SFK,Antusch; Masina,…. Assume II: all 13 angles are very small Assume I: charged lepton mixing angles are small

The Neutrino Sum Rule Predicted by theory e.g.1. bi-maximal predicts 45o e.g.2. tri-bimaximal predicts 35.26o Measured by experiment – how well can this combination be determined?

Tri-bimaximal sum rule Bands show 3  error for a neutrino factory determination of 13cos  Current 3 . . A Prediction Antusch, Huber, SFK, Schwetz

4. 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

The Super CKM Basis Squark superfields Quark mass eigenvalues Quark mass eigenstates

Super CKM basis of the squarks (Rule: do unto squarks as we do unto quarks)

Squark mass matrices in the SCKM basis Flavour changing is contained in off-diagonal elements of Define  parameters as ratios of off-diagonal elements to diagonal elements in the SCKM basis ij = m2ij/m2diag

Down squark mass matrix in SCKM basis

Flavour changing observables in the down sector Ciuchini,Masiero, Paradisi,Silvestrini, Vempati,Vives

Lepton Flavour Violation (LFV) results from off-diagonal soft masses in the basis where the charged Yukawas are diagonal (leptonic analogue of SCKM basis) e.g. slepton doublet mass matrix in charged lepton mass eigenstate basis Off-diagonal slepton masses in this basis lead to LFV

Ciuchini,Masiero, Paradisi,Silvestrini,Vempati,Vives Lepton Flavour Violation

Typical upper bounds on  Quarks Leptons Clearly off-diagonal elements 12 must be very small

Ciuchini,Masiero, Paradisi,Silvestrini,Vempati,Vives LHC connection: need to measure squark and slepton masses to relate quark and lepton flavour violation 5. SU(5) GUTs and Soft Masses

Ciuchini,Masiero, Paradisi,Silvestrini, Vempati,Vives Hadronic constraints Leptonic constraints Quark-lepton connection: LFV processes can constrain Quark Flavour Violation via GUTs Hadronic constraints Leptonic constraints

6. Family Symmetry and Soft Masses An old observation: SU(3) family symmetry predicts universal soft mass matrices in the symmetry limit However Yukawa matrices and trilinear soft masses vanish in the SU(3) symmetry limit So we must consider the real world where SU(3) is broken by flavons

Soft scalar mass operators in SU(3) Using flavon VEVs previously

Recall Yukawa matrices, ignoring phases: Where Under the same assumptions we predict:

In the SCKM basis we find: Yielding small  parameters

Abel, Khalil,Lebedev The SUSY CP Problem • Neutron EDM dn<4.3x10-27e cm • Electron EDM de<6.3x10-26e cm In the universal case Why are SUSY phases so small?

7. Family Symmetry and SUSY CP • Postulate CP conservation (e.g. real) with CP is spontaneously broken by flavon vevs • This is natural since in the SU(3) limit the Yukawas and trilinears are zero in any case • So to study CP violation we must consider SU(3) breaking effects in the trilinear soft masses as we did for the scalar soft masses Ross,Vives

Using flavon VEVs previously Soft trilinear operators in SU(3) N.B parameters cif and if are real

Compare the trilinears to the Yukawas They only differ in the O(1) real dimensionless coefficients

Since we are interested in the (1,1) element we focus on the upper 2x2 blocks The essential point is that  ,  ,  ,  are real parameters and phases only appear in the (2,2,) element (due to SU(3) flavons) Thus the imaginary part of Ad11 in the SCKM basis will be doubly Cabibbo suppressed

To go to SCKM we first diagonalise Yd Then perform the same transformation on Ad Extra suppression factor of c.f. universal case

One family of “messengers” dominates Three families of quarks and leptons then in a particular basis Suppose Accidental sym Not bad! But… Need broken Pati-Salam… 8. Do we need a family symmetry? Ferretti, SFK, Romanino; Barr Conclusion: partial success, but little predictive power esp. in neutrino sector

Fermion Masses and Unification