Models of Neutrino Masses & Mixings

Transcription

1 St. Andrews, August 06 Models of Neutrino Masses & Mixings G. Altarelli Universita' di Roma Tre/CERN Some recent work by our group G.A., F. Feruglio, I. Masina, hep-ph/ , G.A., F. Feruglio, 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/

2 Lecture 2 Survey of basic ideas on model building Part 1: "Normal" models - Degenerate spectrum - Anarchy - Semianarchy - Inverse Hierarchy - Normal Hierarchy - GUT models - SU(5)xU(1) F - SO(10)

3 Still large space for non maximal 23 mixing 2-σ interval 0.32 < sin 2 θ 23 < 0.62 Maximal θ 23 theoretically hard θ 13 not necessarily too small probably accessible to exp. Very small θ 13 theoretically hard "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 very close to maximal and/or θ 13 very small or: a special value for θ 12 or...

4 Degenerate ν's m 2 >> Δm 2 Limits on m ee from 0νββ m ee < ev (Exp) m ee = c 2 13 (m 1 c m 2 s2 12 )+s2 13 m 3 ~ m 1 c m 2 s2 12 are not very demanding: for sin 2 θ~0.3 cos 2 θ sin 2 θ~0.4 and m 1 ~ m 2 ~ m 3 ~1-2 ev (with m 1 = -m 2 ) perfectly fine would be However, WMAP&LSS: m < 0.23 ev, is very constraining Only a moderate degeneracy is still allowed: m/(δm 2 atm) 1/2 < 5, m/(δm 2 sol) 1/2 < 30. If so, constraints from 0νββ are satisfied (both m 1 =±m 2 allowed)

5 It is difficult to marry degenerate models with see-saw m ν ~m DT M -1 m D (needs a sort of conspiracy between M and m D ) So most degenerate models deny all relation to m D and directly work in the L T L Majorana sector Even if a symmetry guarantees degeneracy at the GUT scale it is difficult to protect it from corrections, e.g. from Renormalisation group running

6 For degenerate models there can be large ren. group corrections to mixing angles and masses in the running from M GUT dow to m W In fact the running rate is inv. prop. to mass differences For a 2x2 case: with k = -3/2 (SM), 1 (MSSM) y e = m e /v (SM), m e /vcosβ (MSSM) RG corrections are generally negligible and can only be large for degenerate models especially at large tanβ The observed mixings and splitting do not fit the typical result from pure evolution. See, for example, Chankowski, Pokorski '01

7 In summary: degenerate models are less favoured by now because of: No clear physical motivation: after all quark and charged lepton masses are very non degenerate Upper bounds on m 2 that limit m 2 /Δm 2 atm At present, no significant amount of hot dark matter is indicated by cosmology Only a moderate degeneracy is allowed Can be obtained as a limiting case of hierarchical models. Possible renormalization group instability Disfavoured by see-saw

8 Anarchy (or accidental hierarchy): No structure in the leptonic sector See-Saw: m ν ~m 2 /M produces hierarchy from random m,m r~δm 2 sol/δm 2 atm~1/30 r peaks at ~0.1 Hall, Murayama, Weiner could fit the data But: all mixing angles should be not too large, not too small sin 2 2θ a flat sinθ distrib. --> peaked sin 2 2θ Marginal: predicts θ 13 near bound

9 Semianarchy: no structure in 23 Consider a matrix like m ν ~ ε2 ε ε ε 1 1 ε 1 1 Note: θ 13 ε θ 23 1 with coeff.s of o(1) and det23~o(1) [ε~1 corresponds to anarchy] After 23 and 13 rotations m ν ~ ε2 ε 0 ε η Normally two masses are of o(1) or r ~1 and θ 12 ε But if, accidentally, η ε, then r is small and θ 12 is large. The advantage over anarchy is that θ 13 is small, but θ 12 large and the hierarchy m 2 3>>m 2 2 are still accidental Ramond et al, Buchmuller et al

10 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

11 q(5)~(2, 0, 0) with no see-saw --> no structure in 23 Consider a matrix like m ν ~L T L ~ λ4 λ 2 λ 2 λ λ Note: θ 13 λ 2 θ 23 1 with coeff.s of o(1) and det23~o(1) [semianarchy, while λ~1 corresponds to anarchy] After 23 and 13 rotations m ν ~ λ4 λ 2 0 λ 2 η Normally two masses are of o(1) or r ~1 and θ 12 λ 2 But if, accidentally, η λ 2, then r is small and θ 12 is large. The advantage over anarchy is that θ 13 is naturally small, but θ 12 large and the hierarchy m 2 3>>m 2 2 are accidental Ramond et al, Buchmuller et al With see-saw, one can do much better (see later)

12 Inverted Hierarchy Zee, Joshipura et al; Mohapatra et al; Jarlskog et al; Frampton,Glashow; Barbieri et al Xing; Giunti, Tanimoto... sol atm 2 1 An interesting model: 3 m 2 ~10-3 ev 2 An exact U(1) L e -L µ -L τ symmetry for m ν predicts: ( a good 1 st approximation) m ν = Um νdiag U T = m 0 1 x x 0 0 with m νdiag = m' m' θ 13 = 0 θ 12 = π/4 tan 2 θ 23 = x 2 θ sun maximal! θ atm generic Can arise from see-saw or dim-5 L T HH T L 1-2 degeneracy stable under rad. corr.'s

13 1 st approximation m νdiag = m' m' m ν = Um νdiag U T = m 0 1 x x 0 0 Data? This texture prefers θ sol closer to maximal than θ atm In fact: 12-> With HO corrections: one gets Pseudodirac θ 12 maximal 0 1 x x > δ η η 1 η η 1- tg 2 θ 12 ~ o(δ + η) ~ (Δm 2 sol/δm 2 atm) Exp. (3σ): In principle one can use the charged lepton mixing to go away from θ 12 maximal. In practice constraints from θ 13 small (δθ 12 θ 13 ) Frampton et al; GA, Feruglio, Masina 04 θ 23 ~o(1) (modulo o(1) coeff.s)

14 For the corrections from the charged lepton sector, typically sinθ 13 ~ (1- tan 2 θ 12 )/4cosδ ~ 0.15 GA, Feruglio, Masina 04 Feruglio Corr. s from s e 12, se 13 to U 12 and U 13 are of first order (2nd order to U 23 ) Thus approximate L e -Lµ-Lτ favours θ 13 near its bound

15 There is an intriguing empirical relation: θ 12 + θ C = (47.0±1.7) o ~ π/4 Raidal Suggests bimaximal mixing in 1st approximation, corrected by charged lepton diagonalization. Recall that While θ 12 + o(θ C ) ~ π/4 is easy to realize, exactly θ 12 + θ C ~ π/4 is more difficult: no compelling model Minakata, Smirnov

16 A realistic model (eg tan 2 2θ 12 large ) with IH, θ 13 small can be obtained from see-saw, if L e -L µ -L τ is badly broken in M RR Grimus, Lavoura; G.A., Franceschini As ν R are gauge singlets the large soft breaking in M RR does not invade all other sectors when we do rad. corr s By adding a small flavon breaking of U(1) F symmetry with parameter λ~ m µ /m τ the lepton spectrum is made natural and leads to θ 13 ~ m µ /m τ ~ 0.05 or even smaller.

17 U(1) F charges GA, Franceschini; hep-ph/ Charged lepton sector Diagonalisation θ l --> large shift to θ 23, 0(λ 2 ) contrib ns to θ 13, θ 12

18 Neutrino sector Q R =1 Dirac: Majorana: a,b,d W,Z do not break U(1) no soft breaking with soft breaking after see-saw

19 Various stages: exact U(1) r=0 pure L e -L µ -L τ only soft breaking (λ=0) with θ 12 r~1/30 requires large soft breaking requires some fine tuning

20 Summarising: this model with IH In the limit of exact U(1) F θ 12 =π/4 and r, θ 13, as well as m e /m τ and m µ /m τ (for our choice of charges) are all zero. In general a small symmetry breaking will make them different from zero but small. And θ 12 will only be sligthly displaced from π/4 (bad) A large soft explicit mixing in the M RR sector can decouple θ 12, which gets a large shift, from θ 13, m e /m τ and m µ /m τ which remain small. The only remaining imperfection is that a moderate fine tuning is needed for r.

21 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!

22 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

23 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.

24 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

25 SU(5)xU(1) Recall: m u ~ m d =m et ~ 5 10 m νd ~ 5 1; M RR ~ 1 1 1st fam. 2nd 3rd Ψ 10 : (5, 3, 0) Ψ 5 : (2, 0, 0) Ψ 1 : (1,-1, 0) Equal 2,3 ch. for lopsided No structure for leptons No automatic det23 = 0 Automatic det23 = 0 all charges positive not all charges positive With suitable charge assignments all relevant patterns can be obtained

26 All entries are a given power of λ times a free o(1) coefficient m u ~ v u In a statistical approach we generate these coeff.s as random complex numbers ρe iφ with φ =[0,2π] and ρ= [0.5,2] (default) or [0.8,1.2], or [0.95,1.05] or [0,1] (real numbers also considered for comparison) λ 10 λ 8 λ 5 λ 8 λ 6 λ 3 λ 5 λ 3 1 For each model we evaluate the success rate (over many trials) for falling in the exp. allowed window: (boundaries ~3σ limits) Maltoni et al, hep-ph/ r~δm 2 sol /Δm2 atm < r < U e3 < < tan 2 θ 12 < < tan 2 θ 23 < 2.57 for each model the λ,λ values are optimised

27 The optimised values of λ are of the order of λ C or a bit larger (moderate hierarchy)

28 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"

29 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!!,

30 Results with see-saw dominance (updated in Nov. 03): 1 or 2 refer to models with 1 or 2 flavons of opposite ch. With charges of both signs and 1 flavon some entries are zero Scale: Σrates=100 A: Anarchy SA: Semi-anarchy H: Normal Hierarchy IH: Inv. Hierarchy Errors are linear comb. of stat. and syst. errors (varying the extraction procedure: interval of ρ, real or complex) H2 is better than SA, better than A, better than IH

31 With no see-saw (m ν generated directly from L T m ν L~ is better than A [With no-see-saw H coincide with SA] 5 5) IH Note: we always include the effect of diagonalising charged leptons

32 SO(10) in principle has several advantages vs SU(5). More predictive but less flexible. 16 SO(10) = (10 + 5bar +1) SU(5) 1 is ν R : important for see-saw The Majorana term Mν RT ν R is SU(5) but not SO(10) invariant: M could be larger than the scale where SU(5) is broken, while, in SO(10), M should be of order of the scale where B-L is broken [SO(10) contains B-L]

33 Masses in SO(10) models 16x16 = If no non-ren mass terms are allowed a simplest model needs a 10 and a 126: leading to Bajc, Senjanovic, Vissani '02 Goh, Mohapatra, Ng '03 and 126 In the 23 sector, both m d and m e can be obtained (by U(1) F ) as: m d,e ~ Then b-τ unification forces a cancellation 1->λ 2, which in turn makes a large 23 neutrino mixing. Also predicts θ 12 large, r ~ λ 2, θ 13 near the bound Problems: Doublet-Triplet splitting worse, some fine tuning

34 In other SO(10) models one avoids large Higgs represent'ns (120, 126) by relying on non ren. operators like 16 i 16 H 16 j 16' H or 16 i 16 j 10 H 45 H In the F-symmetry limit, the lowest dimension mass terms H is only allowed for the 3rd family. In particular, both lopsided and L-R symmetric models can be obtained in this way Babu, Pati, Wilczek Albright, Barr Ji, Li, Mohapatra I do not know a GUT model which is exempt from some ad hoc ansatz or fine tuning. On the other hand the goal is very ambitious.

35 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

36 Conclusion "Normal" models are not too difficult to build In fact there are quite a number of different examples Some of them require θ 13 near the bound All of them prefer θ 13 >~ λ 2 C Good chances for next generation of experiments!