Beyond the Standard Model

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1 4 Orsay, 9-12 January '07 Beyond the Standard Model Neutrino Masses & Mixings Guido Altarelli Univ. Roma Tre CERN

2 In the last decade data on ν oscillations have added some (badly needed) fresh experimental input in particle physics ν masses are not all vanishing but they are very small ν mixing angles follow a different pattern from quark mixings For ν masses and mixings we do not have so far a "Standard Model": many possibilities are still open. In fact, this is also the case for quarks and charged leptons; we do not have a theory of flavour that explains the observed spectrum, mixings and CP violation. ν's are interesting because they can provide new clues on this important problem

3 ν Oscillations Imply Different ν Masses flavour mass ν e : same weak isospin doublet as e - ν e ν e ν µ ν τ = U ν 1 ν 2 ν 3 ν e = cosθ ν 1 + sinθ ν 2 ν µ = -sinθ ν 1 + cosθ ν 2 U: mixing matrix e.g 2 flav. ν 1,2 : different mass, different x-dep: ν a (x)=e ip a x ν a p a2 =E 2 -m a 2 e - W - U = U P-MNS Pontecorvo Maki, Nakagawa, Sakata Stationary source: Stodolsky P(ν e <-> ν µ ) = < ν µ (L) ν e > 2 =sin 2 (2θ). sin 2 (Δm 2 L/4E) At a distance L, ν µ from µ - decay can produce e - via charged weak interact's

Solid evidence for solar and atmosph. ν oscillations (+LSND unclear) Δm 2 values fixed: Δm 2 atm ~ 2.

4 Solid evidence for solar and atmosph. ν oscillations (+LSND unclear) Δm 2 values fixed: Δm 2 atm ~ ev 2, Δm 2 sol ~ ev 2 (Δm 2 LSND ~ 1 ev2 ) Also confirmed in laboratory exp.ts KamLAND, K2K mixing angles: θ 12 (solar) large θ 23 (atm) large,~maximal θ 13 (CHOOZ) small

5 ν oscillations measure Δm 2. What is m 2? Δm 2 atm ~ ev 2 ; Δm 2 sun ~ ev 2 Direct limits 0νββ Cosmology m "νe" < 2.2 ev m "νµ" < 170 KeV m "ντ" < 18.2 MeV Σ i m i < ev (dep. on data&priors) Any ν mass < ev End-point tritium β decay (Mainz, Troitsk) m ee < ? ev (nucl. matrix elmnts) Evidence of signal? Klapdor-Kleingrothaus Ω ν h 2 ~ Σ i m i /94eV (h 2 ~1/2) WMAP, 2dFGRS, Ly-α

6 0νββ experiments dd -> uuee Pavan phase space matrix elmnt large uncrtnts m ee = <m ν >= Σ U ej 2 m j e iαj Future: a factor ~ 10 improvement in next decade

7 ν oscillations measure Δm 2. What is m 2? Δm 2 atm ~ ev 2 ; Δm 2 sun ~ ev 2 Direct limits 0νββ Cosmology m "νe" < 2.2 ev m "νµ" < 170 KeV m "ντ" < 18.2 MeV Σ i m i < ev (dep. on data&priors) Any ν mass < eV End-point tritium β decay (Mainz, Troitsk) m ee < ? ev (nucl. matrix elmnts) Evidence of signal? Klapdor-Kleingrothaus Ω ν h 2 ~ Σ i m i /94eV (h 2 ~1/2) WMAP, SDSS, 2dFGRS, Ly-α

8 By itself CMB (eg WMAP) is only mildly sensitive to Σ i m i Only in combination with Large Scale Structure (2dFGRS, SDSS) the limit becomes stronger. And even stronger by adding the Lyman alpha forest data (but some tension among the data). 95%cl Seljac et al 06 Note: for degenerate ν's the mass of each would be Σm νi /3. eg 0.68/3 ~ 0.23 ev to be compared with (Δm 2 atm) 1/2 ~ 0.05 ev

9 Log 10 m/ev d e c s µ u t b τ Neutrino masses are really special! m t /(Δm 2 atm) 1/2 ~10 12 Massless ν s? no ν R L conserved Upper limit on mν (Δ m 2 sol )1/2 WMAP (Δm 2 atm )1/2 KamLAND Small ν masses? ν R very heavy L not conserved

10 How to guarantee a massless neutrino? 1) ν R does not exist and No Dirac mass ν L ν R + ν R ν L 2) Lepton Number is conserved No Majorana mass ν c ν >ν Τ R Cν R or ντ L Cν L C=iγ 0 γ 2

11 ν masses: Dirac mass: ν L ν R + ν R ν L (needs ν R ) ν's have no electric charge. Their only charge is lepton number L. IF L is not conserved (not a good quantum number) ν and ν are not really different TCP, "Lorentz" ν, h= -1/2> ν, h= +1/2> Majorana mass: (we omit the charge conj. matrix C) ν T R ν R or ν T L ν L Violates L, B-L by ΔL = 2

12 Weak isospin I ν L => I = 1/2, I 3 = 1/2 ν R => I = 0, I 3 = 0 Dirac Mass: ν L ν R + ν R ν L ΔI =1/2 Can be obtained from Higgs doublets: ν L ν R H Majorana Mass: ν T L ν L ΔI =1 Non ren., dim. 5 operator: ν T L ν L HH ν T R ν R ΔI =0 Directly compatible with SU(2)xU(1)!

13 See-Saw Mechanism Minkowski; Glashow; Yanagida; Gell-Mann, Ramond, Slansky; Mohapatra, Senjanovic.. Mν T R ν R allowed by SU(2)xU(1) Large Majorana mass M (as large as the cut-off) m D ν L ν R ν L ν R Eigenvalues ν light = ν L 0 m D m D M Dirac mass m D from Higgs doublet(s) ν R -m D 2 M, ν heavy = M sign conventional for fermions M >> m D

14 In general ν mass terms are: Dirac m D =hv v=<0 H 0> Majorana More general see-saw mechanism: ν L ν R m light ~ m heavy ~ M R ν L λv 2 /M L m D 2 M R m D ν R m D M R and/or λv 2 M L m eff = ν T Lm light ν L

15 A very natural and appealing explanation: ν's are nearly massless because they are Majorana particles and get masses through L non conserving interactions suppressed by a large scale M ~ M GUT Note: m ν ~ m 2 M m: m t ~ v ~ 200 GeV M: scale of L non cons. m ν (Δm 2 atm) 1/2 ~ 0.05 ev m ~ v ~ 200 GeV M ~ GeV Neutrino masses are a probe of physics at M GUT!

16 The current experimental situation is still unclear LSND: true or false? -> MiniBooNE soon will tell what is the absolute scale of ν masses? no detection of 0νββ (proof that ν s are Majorana) Different classes of models are still possible: If LSND true 3-1 or 3-n sterile ν(s)?? LSND CPT violat n?? ν sterile If LSND false 3 light ν's are OK m 2 ~1-2 ev 2 We assume this case here Degenerate (m 2 >>Δm 2 ) m 2 < o(1)ev 2 Inverse hierarchy sol atm m 2 ~10-3 ev 2 Normal hierarchy sol atm m 2 ~10-3 ev 2

17 3-ν Models ν e ν e ν µ ν τ = U + flavour ν 1 ν 2 ν 3 mass In basis where e -, µ -, τ - are diagonal: U = c 23 s s 23 c 23 s = solar: large ~ c 13 c 12 c 13 s 12 s 13 e -iδ c 13 s c 13 c 23 c 13 0 s 13 e -iδ s 13 e iδ 0 c 13 e - W - U = U P-MNS Pontecorvo Maki, Nakagawa, Sakata c 12 s s 12 c CHOOZ: s 13 <~0.2 atm.: ~ max δ: CP violation ~ (some signs are conventional) In general: U = U + eu ν

18 m ν ~ U* L T m ν L Note: e iα 1m e iα 2m 2 0 U m 3 For s 13 ~ 0: m ν 0νββ m ν is symmetric phases included in m i In general 9 parameters: 3 masses, 3 angles, 3 phases m 1 c 2 +m 2 s 2 (m 1 -m 2 )cs/ V2 (m 1 -m 2 )cs/ V2... (m 1 s 2 +m 2 c 2 +m 3 )/2 (m 1 s 2 +m 2 c 2 -m 3 )/ (m 1 s 2 +m 2 c 2 +m 3 )/2 Relation between masses and frequencies: P(ν e <->ν µ )= P(ν e <->ν τ )=1/2 sin 2 2θ 12. sin 2 Δ sun P(ν µ <->ν τ )=sin 2 Δ atm - 1/4 sin 2 2θ 12. sin 2 Δ sun In our def.: Δ sun >0, Δ atm > or < 0

19 Defining: one has: and

20 Neutrino oscillation parameters 2 distinct frequencies 2 large angles, 1 small Maltoni et al 04

21 Fogli et al 05 2σ ranges 95% very precise (KamLAND) very close to 1/3

22 θ 13 bounds Fogli et al 05 λ C 2

23 Measuring θ 13 is crucial for future ν-oscill s experiments (eg CP violation) Triple CHOOZ Double CHOOZ ~Present limit

24 A possible time map for sin22θ13

25 3 atm sol 1,2 sol atm 1,2 3 cosmo limit Only moderate degeneracy allowed cosmo limit

0νββ would prove that L is not conserved and ν s are Majorana Also can tell degenerate, inverted or normal hierarchy m ee =c 13 2 [m 1 c 122 +e iα m 2 s 122 ]+m 3 e iβ s 13 2 Degenerate:~ m c 122 +e

26 0νββ would prove that L is not conserved and ν s are Majorana Also can tell degenerate, inverted or normal hierarchy m ee =c 13 2 [m 1 c 122 +e iα m 2 s 122 ]+m 3 e iβ s 13 2 Degenerate:~ m c 122 +e iα s 122 ~ m (0.3-1) m ee ~ m (0.3-1) < ev m ee Feruglio, Strumia, Vissani IH: ~(Δm 2 atm) 1/2 c 122 +e iα s 122 m ee ~ (1.6-5) 10-2 ev NH: ~(Δm 2 sol) 1/2 s (Δm 2 atm) 1/2 e iβ s 13 2 m ee ~ (few) 10-3 ev lightest m ν (ev) Present exp. limit: m ee < ev (and a hint of signal????? Klapdor Kleingrothaus)

27 Baryogenesis n B /n γ ~10-10, n B >> n Bbar Conditions for baryogenesis: (Sacharov '67) B non conservation (obvious) C, CP non conserv'n (B-B bar odd under C, CP) No thermal equilib'm (n=exp[µ-e/kt]; µ B =µ Bbar, m B =m Bbar by CPT If several phases of BG exist at different scales the asymm. created by one out-of-equilib'm phase could be erased in later equilib'm phases: BG at lowest scale best Possible epochs and mechanisms for BG: At the weak scale in the SM Excluded At the weak scale in the MSSM Disfavoured Near the GUT scale via Leptogenesis Very attractive

28 Possible epochs for baryogenesis BG at the weak scale: T EW ~ TeV Rubakov, Shaposhnikov; Cohen, Kaplan, Nelson; Quiros. In SM: B non cons. by instantons ( t Hooft) (non pert.; negligible at T=0 but large at T=T EW B-L conserved! CP violation by CKM phase. Not enough By general consensus far too small. Out of equilibrium during the EW phase trans. Needs strong 1st order phase trans. (bubbles) Only possible for m H <~40 GeV Now excluded by LEP

29 Is BG at the weak scale possible in MSSM? In principle additional sources of CP violation But so far no signal at beauty factories Constraint on m H modified by presence of extra scalars with strong couplings to Higgs sector (e.g. s-top) Requires: m h < GeV; m s-topl <m t ; tgβ~1.2-5 preferred Espinosa, Quiros, Zwirner; Giudice; Myint; Carena, Quiros, Wagner; Laine; Cline, Kainulainen; Farrar, Losada.. Much disfavoured by LEP

30 Baryogenesis by decay of heavy Majorana ν's BG via Leptogenesis near the GUT scale T ~ 10 12±3 GeV (after inflation) Only survives if Δ(B-L) is not zero (otherwise is washed out at T ew by instantons) Main candidate: decay of lightest ν R (M~10 12 GeV) Buchmuller,Yanagida, Plumacher, Ellis, Lola, Giudice et al, Fujii et al.. L non conserv. in ν R out-of-equilibrium decay: B-L excess survives at T ew and gives the obs. B asymmetry. Quantitative studies confirm that the range of m i from ν oscill's is compatible with BG via (thermal) LG In particular the bound was derived for hierarchy Can be relaxed for degenerate neutrinos So fully compatible with oscill n data!! m i <10-1 ev Buchmuller, Di Bari, Plumacher; Giudice et al; Pilaftsis et al; Hambye et al

31 Large neutrino mixings can induce observable τ -> µγ and µ -> eγ transitions In fact, in SUSY models large lepton mixings induce large s-lepton mixings via RG effects (boosted by the large Yukawas of the 3rd family) Detailed predictions depend on the model structure and the SUSY parameters. Lopsided models tend to lead to the largest rates. Typical values: Β(µ -> eγ) (now: ~10-11 ) Β(τ -> µγ) < 10 7 (now: ~10-7 ) See, e.g., Lavignac, Masina, Savoy'02 Masiero, Vempati, Vives'03; Babu, Dutta, Mohapatra'03; Babu, Pati, Rastogi'04; Blazek, King '03; Petcov et al '04; Barr '04

32 Model building Some recent work by our group G.A., F. Feruglio, I. Masina, hep-ph/ , G.A., F. Feruglio, hep-ph/ ,hep-ph/ , hep-ph/ G.A, R. Franceschini, hep-ph/ Reviews: G.A., F. Feruglio, New J.Phys.6:106,2004 [hep-ph/ ]; G.A., hep-ph/ ; F. Feruglio, hep-ph/ G.A, hep-ph/

33 General remarks After KamLAND, SNO and WMAP... not too much hierarchy is needed for ν masses: r~δm 2 sol/δm 2 Δχ atm~1/30 2 Precisely at 2σ: < r < or m heaviest < ev m next > ~ ev r For a hierarchical spectrum: Comparable to: Suggests the same hierarchy parameters for q, l, ν e.g. θ 13 not too small!

34 Still large space for non maximal 23 mixing 3-σ interval 0.31< sin 2 θ 23 < 0.72 Maximal θ 23 theoretically hard θ 13 not necessarily too small probably accessible to exp. Very small θ 13 theoretically hard Many viable solutions Normal models: θ 23 large but not maximal, θ 13 not too small (θ 13 of order λ C or λ C2 ) Exceptional models: θ 23 maximal and/or θ 13 very small and/or a special value for θ 12...

35 Natural models of the normal type are not too difficult to build up It is reasonable to attribute hierarchies in masses and mixings to differences in some flavour quantum number(s). A simplest flavour (or horizontal) symmetry is U(1) F For example, some simple models based on see-saw and U(1) F work for all quark and lepton masses and mixings, are natural and compatible with (SUSY) GUT s, e.g SU(5)xU(1) F. Larger flavour symmetry groups have been studied. They are more predictive but less flexible. The problem of the "best" flavour group is still open. The most ambitious models try to combine (SUSY) SO(10) GUT's with a suitable flavour group

36 Hierarchy for masses and mixings via horizontal U(1) F charges. Froggatt, Nielsen '79 Principle: A generic mass term R 1 m 12 L 2 H is forbidden by U(1) if q 1 +q 2 +q H not 0 q 1, q 2, q H : U(1) charges of R 1, L 2, H U(1) broken by vev of "flavon field θ with U(1) charge q θ = -1. If vev θ = w, and w/m=λ we get for a generic interaction: R 1 m 12 L 2 H (θ/m) q1+q2+qh Δ charge m 12 -> m 12 λ q1+q2+qh Hierarchy: More Δ charge -> more suppression (λ small) One can have more flavons (λ, λ',...) with different charges (>0 or <0) etc -> many versions

37 For example: Normal hierarchy A crucial point: in the 2-3 sector we need both large m 3 -m 2 splitting and large mixing. m 3 ~ (Δm 2 atm) 1/2 ~ ev m 2 ~ (Δm 2 sol) 1/2 ~ ev The "theorem" that large Δm 32 implies small mixing (pert. th.: θ ij ~ 1/ E i -E j ) is not true in general: all we need is (sub)det[23]~0 Example: m 23 ~ x2 x x 1 So all we need are natural mechanisms for det[23]=0 Det = 0; Eigenvl's: 0, 1+x 2 Mixing: sin 2 2θ = 4x 2 /(1+x 2 ) 2 For x~1 large splitting and large mixing!

38 Examples of mechanisms for Det[23]~0 based on see-saw: m ν ~m T DM -1 m D 1) A ν R is lightest and coupled to µ and τ M ~ m ν ~ King; Allanach; Barbieri et al... ε a b c d 1/ε M -1 ~ a c b d 2) M generic but m D "lopsided" m ν ~ Albright, Barr; GA, Feruglio,... 0 x 0 1 a b b c 0 0 x 1 1/ε ~ 1/ε m D ~ = c ~ a 2 ac ac c 2 x 2 x x x 1 1/ε 0 0 0

39 An important property of SU(5) Left-handed quarks have small mixings (V CKM ), but right-handed quarks can have large mixings (unknown). In SU(5): LH for d quarks RH for l - leptons 5 m d ~d R d L 10 m e ~e R e L : (d,d,d, ν,e - ) R L m d = m e T cannot be exact, but approx. Most "lopsided" models are based on this fact. In these models large atmospheric mixing arises (at least in part) from the charged lepton sector.

40 The correct pattern of masses and mixings, also including ν's, is obtained in simple models based on SU(5)xU(1) flavour Ramond et al; GA, Feruglio+Masina; Buchmuller et al; King et al; Yanagida et al, Berezhiani et al; Lola et al... Offers a simple description of hierarchies, but it is not very predictive (large number of undetermined o(1) parameters) SO(10) models could be more predictive, as are non abelian flavour symmetries, eg O(3) F, SU(3) F Albright, Barr; Babu et al; Bajic et al; Barbieri et al; Buccella et al; King et al; Mohapatra et al; Raby et al; G. Ross et al

41 Example: Normal Hierarchy 1st fam. 2nd q(10): (5, 3, 0) q(5): (2, 0, 0) q(1): (1,-1, 0) 3rd q(h) = 0, q(h)= 0 q(θ)= -1, q(θ')=+1 In first approx., with <θ>/m~λ~ λ '~0.35 ~o(λ C ) 10 i 10 j 10 λ 10 λ 8 λ 5 i 5 j λ 7 λ 5 λ 5 m u ~ v u λ 8 λ 6 λ 3, md =me T ~v d λ 5 λ 3 λ 3 λ 5 λ 3 1 λ i 1 j m νd ~ v u λ 3 λ λ 2 λ λ' 1 λ λ' 1, G.A., Feruglio, Masina Note: not all charges positive --> det23 suppression 1 i 1 j M RR ~ M λ 2 1 λ 1 λ' 2 λ' λ λ' 1 Note: coeffs. 0(1) omitted, only orders of magnitude predicted "lopsided"

42 5 i 1 j m νd ~ v u λ 3 λ λ 2 λ λ 1 λ λ 1, 1 i 1 j M RR ~ M λ 2 1 λ 1 λ 2 λ λ λ 1 see-saw m ν ~m νdt M RR -1 m νd m ν ~ v u2 /M λ 4 λ 2 λ 2 λ λ det 23 ~λ 2 The 23 subdeterminant is automatically suppressed, θ 13 ~ λ 2, θ 12, θ 23 ~ 1 This model works, in the sense that all small parameters are naturally due to various degrees of suppression. But too many free parameters!!,

43 A very exceptional model Harrison, Perkins, Scott A simple mixing matrix compatible with all present data In the basis of diagonal ch. leptons: m ν =Udiag(m 1,m 2,m 3 )U T Eigenvectors: Note: mixing angles independent of mass eigenvalues

44 Comparison with experiment: At 1σ: Fogli et al 05 sin 2 θ 12 =1/3 : sin 2 θ 23 =1/2 : sin 2 θ 13 = 0 : < 0.02 The HPS mixing is clearly a very good approx. to the data! Also called: Tri-Bimaximal mixing

45 For the HPS mixing matrix all mixing angles are fixed to particularly symmetric values It is interesting to construct models that can naturally produce this highly ordered structure Models based on the A4 discrete symmetry (even permutations of 1234) are very interesting Ma...; GA, Feruglio hep-ph/ , hep-ph/ Alternative models based on SU(3) F or SO(3) F Verzielas, G. Ross King

46 A4 is the discrete group of even perm s of 4 objects. (the inv. group of a tetrahedron). It has 4!/2 = 12 elements. An element is abcd which means > abcd C 1 : 1 = 1234 C 2 : T = 2314 ST = 4132 TS = 3241 STS = 1423 C 3 : T 2 = 3124 ST 2 = 4213 T 2 S= 2431 TST = 1342 C 4 : S = 4321 T 2 ST = 3412 TST 2 = 2143 Thus A4 transf.s can be written as: 1, T, S, ST, TS, T 2, TST, STS, ST 2, T 2 S, T 2 ST, TST 2 with: S 2 = T 3 = (ST) 3 = 1 [(TS) 3 = 1 also follows] x, x in same class if C 1, C 2, C 3, C 4 are equivalence classes [x ~ gxg -1 ] g: group element

47 A4 has only 4 irreducible inequivalent represt ns: 1,1,1,3 Table of Multiplication: 1 x1 =1 ; 1 x1 =1 ;1 x1 =1 3x3= In the (S-diag basis) consider 3: (a 1,a 2,a 3 ) For 3 1 =(a 1,a 2,a 3 ), 3 2 =(b 1,b 2,b 3 ) we have in 3 1 x3 2 : A4 is well fit for 3 families! Ch. leptons l ~ 3 e c, µ c, τ c ~ 1, 1, 1 S (a 1,-a 2,-a 3 ) T (a 2,a 3,a 1 ) T e.g. 1" = a 1 b 1 +ωa 2 b 2 +ω 2 a 3 b 3 --> a 2 b 2 +ωa 3 b 3 +ω 2 a 1 b 1 = = ω 2 [a 1 b 1 +ωa 2 b 2 +ω 2 a 3 b 3 ] while, under S, 1" is inv.

48 Three singlet inequivalent represent ns: Recall: S 2 = T 3 = (ST) 3 = 1 1: S=1, T=1 1 : S=1, T= ω 1 : S=1, T= ω 2 The only indep. 3-dim represent n is obtained by: (S-diag basis) An equivalent form: (T-diag basis)

49 What can be the origin of A4? G.A., F. Feruglio,hep-ph/ A4 (or some other discrete group) could arise from extra dimensions (by orbifolding with fixed points) as a remnant of 6-dim spacetime symmetry. x 6 z=x 5 +ix 6 x 5 A torus with identified points: z -> z + 1 z -> z + γ γ=exp(iπ/3) and a parity z -> -z leads to 4 fixed points (equivalent to a tethraedron). There are 4D branes at the fixed points where the SM fields live (additional gauge singlets are in the bulk) A4 interchanges the fixed points

50 Under A4 lepton doublets l ~ 3 e c, µ c, τ c ~ 1, 1, 1 respectively gauge singlet flavons φ, φ, ξ, (ξ ) ~ 3, 3, 1,(1) respectively driving fields (for SUSY version) φ 0, φ 0, ξ 0 ~ 3, 3, 1 Additional symmetries: broken U(1) F symmetry (ch. lepton masses) with e c, µ c, τ c charges (3 or 4,2,0) and a discrete symmetry (dep. on versions) : for example Z: (e c, µ c, τ c )-> -i (e c, µ c, τ c ), l -> il, φ -> φ, (ξ,φ ) -> - (ξ,φ ) The Yukawa interactions in the lepton sector are: Here is without see-saw (with see-saw is also OK: wait!)

51 Structure of the model shorthand: Higgs and cut-off scale Λ omitted, e.g.: ~ ~!!! the big plus of A4 Spectrum free. Diagonalized by U e : =Vl From here it follows that U HPS is the mixing matrix

52 m ν in the basis of diagonal charged leptons is: = which in turn can be written as: with:

53 The crucial issue is to guarantee the strict alignment We have constructed a number of completely natural versions of the model, e.g.: a version in 5 dimensions (economic in flavon fields) a SUSY version in 4-dim (with more fields) We first briefly discuss the 5-dim version

54 GA, Feruglio hep-ph/ The model has 1 compactified extra dim. and 2 branes (crucial issue: guarantee and protect the vev alignment) <φ > = v (1,0,0) <φ> = v(1,1,1) different U(1) charges for e c,µ c,τ c lead to the ch. lept. mass hierarchy 3 <ξ> = u

55 In lowest approximation the action is: a Z-parity has also been imposed Z After integrating out of the F fields one obtains the required effective 4-dim action

56 In the flavour basis: m ν = U diag(a+d,a,-a+d)u T (in units of v u2 /Λ) and U=U HPS In terms of physical param.s (moderate normal hierarchy): ~(0.017 ev) 2 ~(0.017 ev) 2 A moderate fine tuning is needed for r ~(0.053 ev) 2

A version with see-saw is also possible ν R is a triplet of A4: Dirac No change for ch leptons Majorana [Discrete parity Z: ω,ω 2,ω 2,ω 2 for l, ν c, φ T, ξ respectively] m ν D ~ 1 m ν = m ν DT M RR

57 A version with see-saw is also possible ν R is a triplet of A4: Dirac No change for ch leptons Majorana [Discrete parity Z: ω,ω 2,ω 2,ω 2 for l, ν c, φ T, ξ respectively] m ν D ~ 1 m ν = m ν DT M RR -1 m νd ~ M RR -1 The mass matrix appears just as the inverse of what was before, so that the mixing matrix is the same. Eigenvalues are the inverse: one can produce inverse hierarchy with realistic θ 12, θ 23 and very small θ 13

58 The 4-dim SUSY version (written in the T-diag basis) In this basis the ch. leptons are diagonal! One more singlet is needed for vacuum alignment The superpotential (at leading order): and the potential The assumed simmetries are summarised here U(1) F 2q q 1

59 The driving field have zero vev. So the minimization is: Solution: In the paper w at NLO is also studied

60 NLO corrections studied in detail to m l LO is 1/Λ 1st non trivial correction at o(1/λ 3 ) to m ν LO is 1/Λ 2 to vevs LO is 1 δv T, δv S, δv i, δu ~ o(1/λ) All observables get a correction of order 1/Λ From exp (eg θ 12 ) must be less than 5% In particular θ 13 < ~0.05, tg 2 θ 23-1 < ~0.05

61 Extension to quarks If we take all fermion doublets as 3 and all singlets as 1, 1, 1 (as for leptons): Q i ~3, u c,d c ~1, c c,s c ~1, t c,b c ~1 Then u and d quark mass matrices are BOTH diagonalised by U u,u d ~ As a result VCKM is unity: V CKM = U u+ U d ~ 1 So, in first approx. (broken by loops and higher dim operators), ν mixings are HPS and quark mixings ~identity Corrections are far too small to reproduce quark mixings eg λ C (for leptons, corrections cannot exceed o(λ C2 ). But even those are essentially the same for u and d quarks)

62 Note: it not straightforward to embed these models in a GUT: with these assignments A4 does not commute with SU(5) If l ~ 3 then all 5bar ~3, so that d c i ~ 3 if e c, µ c, τ c ~ 1, 1', 1" then all 10 i ~ 1, 1', 1" Realistic quark mass matrices are not easy to obtain from these assigments For example, for u quarks at leading order: m u ~ '.1" + 1".1' ~a u 1 u 1 +b (u 2 u 3 +u 3 u 2 ) or a 0 0 m u ~ 0 0 b 0 b 0 Which implies m c = m t and maximal U 23

63 Main lessons from ν masses and mixings ν s are not all massless but their masses are very small probably masses are small because ν s are Majorana particles then masses are inv. prop. to the large scale M of L n. viol. M~m νr is empirically close to GeV ~ M GUT -> ν masses fit well in the SUSY GUT picture decays of ν R with CP & L violation can produce a B-L asymm. -> baryogenesis via leptogenesis detecting 0νββ would prove ν s are Majorana and L is viol. ν mixing angles are large except for θ 13 that is small ν s are not a significant component of dark matter in Universe there is no contradiction between large ν mixings and small q mixings, even in GUT s

64 Conclusion From experiment: a good first approximation for quarks: V CKM ~ and for neutrinos Models based on A4 indeed lead to this pattern All this is highly non trivial but no real illumination has followed!!

Models of Neutrino Masses & Mixings

Frascati, 27 November '04 Models of Neutrino Masses & Mixings CERN Some recent work by our group G.A., F. Feruglio, I. Masina, hep-ph/0210342 (Addendum: v2 in Nov. 03), hep-ph/0402155. Reviews: G.A., F.