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Fermion Masses and Unification Steve King University of Southampton

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Lecture III Family Symmetry and Unification I 1.Doublet-triplet splitting 2.Introduction to family symmetry 3.Froggatt-Nielsen mechanism 4.Gauged U(1) family symmetry and unification 5.SO(3) or A 4 family symmetry and unification

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Two possible types of solutions: a Give large GUT scale masses to b Allow TeV scale masses to but suppress interactions Doublet-Triplet splitting Yukawa suppression is required (discussion session?) a ‘Solves’ Proton Decay and Unification problems b ‘Solves’ Proton Decay problem but leaves Unification problem Doublet-triplet splitting or light triplets?

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Nontrivial to give huge masses to but not e.g. most simple mass term would be in Minimal superpotential contains: Need to fine tune = m to within 1 part in to achieve » TeV light Higgs GUTEW scale Doublet-Triplet Splitting Problem

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Pair up H with a G representation (e.g. 50 of SU(5) ) that contains (colour) triplets but not (weak) doublets Suppose superpotential contains: Under : Nothing for Higgs h u, h d to couple to Problems: Large rank representations Then in direction gives mass couplings to problem for Higgs mass… 50 contains (3,1) but not (1,2) Missing Partner Mechanism

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We would like to account for the hierarchies embodied in the textures Introduction to Family Symmetry SUSY GUTs can describe but not explain such hierarchies To understand such hierarchies we shall introduce a family symmetry that distinguishes the three families It must be spontaneously broken since we do not observe massless gauge bosons which mediate family transitions The Higgs which break family symmetry are called flavons The flavon VEVs introduce an expansion parameter = /M where M is a high energy mass scale. Idea is to use to explain the textures.

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In SM the largest family symmetry possible is the symmetry of the kinetic terms In SO(10), = 16, so the family largest symmetry is U(3) Candidate continuous symmetries are U(1), SU(2), SU(3) or SO(3) … N.B. If family symmetries are gauged and broken at high energies then no direct low energy signatures What is a suitable family symmetry?

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Nothing Candidate Family Symmetries (incomplete)

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Simplest example is 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 ( 3 )=0, Q ( 2 )=1, Q ( 1 )=3, Q(H)=0, Q( )=-1,Q( )=1 Then at tree level the only allowed Yukawa coupling is H 3 3 ! The other Yukawa couplings are generated from higher order operators which respect U(1) family symmetry due to flavon insertions: When the flavon gets its VEV it generates small effective Yukawa couplings in terms of the expansion parameter U(1) Family Symmetry

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

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

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Gauged U(1) Family Symmetry Problem: anomaly cancellation of SU(3) C 2 U(1), SU(2) L 2 U(1) and U(1) Y 2 U(1) anomalies implies that U(1) is linear combination of Y and B-L (only anomaly free U(1)’s available) but these symmetries are family independent Solution: use Green-Schwartz anomaly cancellation mechanism by which anomalies cancel if they appear in the ratio: Ibanez, Ross; Kane, SFK, Peddie, Velasco-Sevilla

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Suppose we restrict the sums of charges to satisfy Then A 1, A 2, A 3 anomalies are cancelled a’ la GS for any values of x,y,z,u,v But we still need to satisfy the A 1 ’=0 anomaly cancellation condition.

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The simplest example is for u=0 and v=0 which is automatic in SU(5) GUT since10=(Q,U c,E c ) and 5 * =(L,D c ) q i =u i =e i and d i =l i so only two independent e i, l i. In this case it turns out that A 1 ’=0 so all anomalies are cancelled. Assuming for a large top Yukawa we then have: SO(10) further implies q i =u i =e i =d i =l i

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F=(Q,L) and F c =(U c,D c, E c,N c ) In this case it turns out that A 1 ’=0. PS implies x+u=y and x=x+2u=y+v. So all anomalies are cancelled with u=v=0, x=y. Also h=(h u, h d ) The only anomaly cancellation constraint on the charges is x=y which implies Note that Y f is invariant under the transformations This means that in practice it is trivial to satisfy for any choices of charges

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A Problem with U(1) Models is that it is impossible to obtain For example consider Pati-Salam where there are effectively no constraints on the charges from anomaly cancellation There is no choice of l i and e i that can give the desired texture e.g. previous example l 1 =e 1 =3, l 2 =e 2 =1, l 3 =e 3 =h f =0 gave: Shortcomings of U(1) Family Symmetry The desired texture can be achieved with non-Abelian family symmetry. There is also an independent motivation for non-Abelian symmetry from neutrino physics…

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Lepton mixing is large Harrison, Perkins, Scott e.g. Tri-bimaximal Valle et al Andre de Gouvea

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Each element has three contributions, one from each RH neutrino. If the right-handed neutrino of mass X dominates and A 1 =0 then we have approximately only (2,3) elements with m 1,2 ¿ m 3 and tan 23 ¼ A 2 /A 3 Heavy MajoranaDirac Light Majorana Large Lepton Mixing From the See-Saw

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columns Sequential dominance can account for large neutrino mixing See-saw Sequential dominance Dominant m 3 Subdominant m 2 Decoupled m 1 Diagonal RH nu basis Tri-bimaximal Constrained SD

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Large lepton mixing motivates non-Abelian family symmetry Need Suitable non-Abelian family symmetries must span all three families e.g. 2$ 3 symmetry (from maximal atmospheric mixing) 1$ 2 $ 3 symmetry (from tri-maximal solar mixing) SFK, Ross; Velasco-Sevilla; Varzelias SFK, Malinsky with CSD

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SO(3) family symmetry Suppose that left handed leptons are triplets under SO(3) family symmetry and right handed leptons are singlets Real vacuum alignment (a,b,c,e,f,h real) To break the family symmetry introduce three flavons 3, 23, 123

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But this is not sufficient to account for tri-bimaximal neutrino mixing If each flavon is associated with a particular right-handed neutrino then the following Yukawa matrix results

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For tri-bimaximal neutrino mixing we need This requires a delicate vacuum alignment of flavon vevs – see next lecture

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Extra Slides

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MSSM solves “technical hierarchy problem” (loops) But no reason why » m soft the “ problem”. In the NMSSM =0 but S Hu Hd Hu Hd where » S 3 term required to avoid a massless axion due to global U(1) PQ symmetry S 3 breaks PQ to Z 3 resulting in cosmo domain walls (or tadpoles if broken) The problem One solution is to forbid S 3 and gauge U(1) PQ symmetry so that the dangerous axion is eaten to form a massive Z’ gauge boson U(1)’ model Anomaly cancellation in low energy gauged U(1)’ models implies either extra low energy exotic matter or family-nonuniversal U(1)’ charges For example can have an E 6 model with three complete 27’s at the TeV scale with a U(1)’ broken by singlets which solve the problem This is an example of a model where Higgs triplets are not split from doublets

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E 6 ! SU(5)£U(1) N M GUT TeV U(1) N broken, Z’ and triplets get mass, term generated Incomplete multiplets (required for unification) Right handed neutrino masses M String E 8 £ E 8 ! E 6 Quarks, leptons Triplets and Higgs Singlets and RH s H’,H’-bar MWMW SU(2) L £ U(1) Y broken Right handed neutrinos are neutral under: E 6 SSM= MSSM+3(5+5 * )+Singlets ! SM £ U(1) N

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Family Universal Anomaly Free Charges: Most general E 6 allowed couplings from 27 3 : Allows p and D,D * decay FCNC’s due to extra Higgs SFK, Moretti, Nevzorov term Triplet mass terms

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Rapid proton decay + FCNCs extra symmetry required: Introduce a Z 2 under which third family Higgs and singlet are even all else odd forbids W 1 and W 2 and only allows Yukawa couplings involving third family Higgs and singlet Forbids proton decay and FCNCs, but also forbids D,D* decay so Z 2 must be broken! Yukawa couplings g<10 -8 will suppress p decay sufficiently Yukawa couplings g> will allow D,D* decay with lifetime <0.1 s (nucleosynthesis) This works because D decay amplitude involves single g while p decay involves two g’s

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Unification in the MSSM Blow-up of GUT region M SUSY =250 GeV 2 loop, 3 (M Z )=0.118

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Unification with MSSM+3(5+5 * ) 250 GeV 1.5 TeV Blow-up of GUT region 2 loop, 3 (M Z )=0.118

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SUSY with 3x27’s at TeV scale M GUT TeV U(1) X broken, Z’ and triplets get mass, term generated Right handed neutrino masses M Planck Quarks, leptons Triplets and Higgs Singlet MWMW SU(2) L £ U(1) Y broken E 6 ! SU(4) PS £ SU(2) L £ SU(2) R SU(4) PS £ SU(2) L £ SU(2) R £ U(1) ! SM £ U(1) X x three families £ U(1)

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Planck Scale Unification with 3x27’s Low energy (below M GUT ) three complete families of 27’s of E 6 High energy (above M GUT » GeV) this is embedded into a left-right symmetric Pati-Salam model and additional heavy Higgs are added. M Planck Howl, SFK

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