Models of Neutrino Masses & Mixings

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

2 Neutrino oscillation parameters Maltoni et al

3 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 priors) Any ν mass < ? ev End-point tritium β decay (Mainz, Troitsk) Future: Katrin (sub-ev) m ee < ? ev (nucl. matrix elmnts) Evidence of signal? Klapdor-Kleingrothaus Ω ν h 2 ~ Σ i m i /94eV (h 2 ~1/2) Why ν's so much lighter than quarks and leptons? WMAP, 2dFGRS...

4 Lahav 95%cl By itself CMB (WMAP, ACBAR) do not fix M ν Only in combination with galaxy power spectrum (2dFGRS, SDSS) become sensitive.

5 After KamLAND, SNO and WMAP not too much hierarchy is needed for ν masses: r~δm 2 sol/δm 2 Δχ atm~1/35 2 Precisely at 3σ: < 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!

6 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. sinθ 13 ~ 1/2 sinθ 12 not excluded! 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 maximal or θ 13 very small or also: all mixing from the charged lepton sector... U = U e+ U ν

7 The current experimental situation is still unclear LSND: true or false? what is the absolute scale of ν masses? Different classes of models are still possible: If LSND true 3-1 sterile ν(s)?? LSND CPT violat n?? ν sterile If LSND false 3 light ν's are OK m 2 ~1-2eV 2 Degenerate (m 2 >>Δm 2 ) m 2 < o(1)ev 2 Inverse hierarchy sol atm We assume this case here m 2 ~10-3 ev 2 Normal hierarchy sol atm m 2 ~10-3 ev 2

3-ν Models ν e ν e ν µ ν τ flavour = U ν 1 ν 2 ν 3 mass In basis where e -, µ -, τ - are diagonal: U = 1 0 0 0 c 23 s 23 0 - s 23 c 23 s = solar: large ~ c 13 c 12... c 13 s 12... s 13 e -iδ c 13 s 23.

8 3-ν Models ν e ν e ν µ ν τ flavour = U 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 c 13 s 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)

9 m ν ~ U* L T m ν L Note: e iφ 1m e iφ 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

10 3 atm sol 1,2 sol atm 1,2 3 cosmo limit Only moderate degeneracy allowed cosmo limit

0νββ 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 LA:~0.3-1 Degenerate: ~ m c 122 +e iα s 122 m ee ~ m (0.3-1) < 0.

11 0νββ 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 LA:~0.3-1 Degenerate: ~ m c 122 +e iα s 122 m ee ~ m (0.3-1) < ev IH: ~(Δm 2 atm) 1/2 c 122 +e iα s 122 m ee ~ (1.6-5) 10-2 ev m ee Feruglio, Strumia, Vissani 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)

12 Degenerate ν's m 2 >> Δm 2 Apriori compatible with hot dark matter (m~1-2 ev) was considered by many Limits on m ee from 0νββ then imply large mixing also for solar oscillations: (Vissani; Georgi,Glashow) 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 If m 1 ~ m 2 ~ m 2 ~1-2 ev LA solution: sin 2 θ~0.3 m 1 = -m 2 and c 2 12~s 2 12 cos 2 θ sin 2 θ~0.4 a moderate suppression factor! Trusting WMAP&2dF: m < 0.23 ev, only a moderate degeneracy is allowed: for LA, m/(δm 2 atm) 1/2 < 5, m/(δm 2 sol) 1/2 < 30. Less constraints from 0νββ (both m 1 =±m 2 allowed) Recall: leptogenesis prefers m < 0.1 ev

13 Anarchy (or accidental hierarchy): No structure in the leptonic sector Hall, Murayama, Weiner See-Saw: m ν ~m 2 /M produces hierarchy from random m,m r~δm 2 sol/ Δm 2 atm~1/40 r peaks at ~0.1 could fit LA But: all mixing angles should be large sin 2 2θ marginal for LA predicts θ 13 near bound

14 Semianarchy: no structure in 23 Consider a matrix like m ν ~ λ 1 1 λ2 λ λ λ 1 1 Note: θ 13 λ θ 23 1 with coeff.s of o(1) and det23~o(1) [λ~1 corresponds to anarchy] After 23 and 13 rotations m ν ~ λ2 λ 0 λ η Normally two masses are of o(1) and θ 12 λ But if, accidentally, η λ, then the solar angle is also large. The advantage over anarchy is that θ 13 is small, but the hierarchy m 2 3>>m 2 2 is accidental Ramond et al, Buchmuller et al

15 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 of predicts: ( a good 1 st approximation) m ν = Um νdiag U T = m 0 c -s c 0 0 -s 0 0 with m νdiag = m m θ 13 = 0 θ 12 = π/4 sin 2 θ 23 = s 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

16 1 st approximation m νdiag = m m m ν = Um νdiag U T = m 0 c -s c 0 0 -s 0 0 Data? This texture prefers θ sol closer to maximal than θ atm i.e θ sol - π/4 small for (Δm 2 sol/δm 2 atm) LA ~ 1/40 In fact: 12-> 0 c c 0 With perturbations: Pseudodirac θ 12 maximal 0 c -s c 0 0 -s > δ η η 1 η η one gets 1- tg 2 θ 12 ~ o(δ + η) ~ (Δm 2 sol/δm 2 atm) LA 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)

17 For the corrections from the charged lepton sector, typically sinθ 13 ~ (1- tan 2 θ 12 )/4cosδ ~ 0.15 GA, Feruglio, Masina 04 Corr. s from s e 12, se 13 to U 12 and U 13 are of first order (2nd order to U 23 )

18 For the corrections to bimixing from the charged lepton sector, typically sinθ 13 ~ (1- tan 2 θ 12 )/4 GA, Feruglio, Masina 04 In general more θ 12 is close to maximal, more is IH likely

19 Lm e R L e -L µ -L τ implies: m e = U + m e V m e m e + = U + m e m e+ U or m e m e + transforms as L L λ,λ from flavons of ±1 charge After diagonalisation of charged leptons θ 23 remains large, while modifications to θ 13 and θ 12 are small. In conclusion IH is viable but prefers θ 12 close to maximal, and given the exp. value of θ 12, needs θ 13 near its upper bound [Both anarchy and IH point to θ 13 near bound]

20 Lindner Measuring θ 13 is crucial for future ν-oscill s experiments (eg CP violation) Present limit

21 Normal Hierarchy atm m 2 ~10-3 ev 2 sol Assume 3 widely split light neutrinos. For u, d and l - Dirac matrices the 3 rd generation eigenvalue is dominant. May be this is also true for m νd : diag m νd ~(0,0,m D3 ). (but not at all necessary!) Assume see-saw is dominant: m ν ~m T DM -1 m D See-saw quadratic in m D : tends to enhance hierarchy Maximally constraining: GUT's relate q, l -, ν masses!

22 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!

23 Examples of mechanisms for Det[23]~0 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 Caution: if 0 -> 0(ε), det23=0 could be spoiled by suitable 1/ε terms in M -1

24 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.

25 Hierarchical ν's and see-saw dominance L T m ν L -> m ν ~m D2 /M allow to relate q, l, ν masses and mixings in GUT models. For dominance of dim-5 operators -> less constraints The correct pattern of masses and mixings, also including ν's, is obtained in simple models based on SU(5)xU(1) flavour λ 2 /M (LH)(LH)-> m ν ~ λ 2 v 2 /M Ramond et al; GA, Feruglio+Masina; Buchmuller et al; King et al; Yanagida et al, Berezhiani et al; Lola et al... SO(10) models could be more predictive, as are non abelian flavour symmetries, eg O(3) Albright, Barr; Babu et al; Buccella et al; Barbieri et al; Raby et al; King, Ross

26 The non trivial pattern of fermion masses and mixing demands a flavour structure (symmetry) (SUSY) SU(5)XU(1) F models offer a minimal description of flavour symmetry A flexible enough framework used to realize and compare models with anarchy or hierarchy (direct or inverse) in ν sector, with see-saw dominance or not. On this basis we found that there is still a significant preference for hierarchy vs anarchy G.A., F. Feruglio, I. Masina, hep-ph/ (v2 Nov 03) Previous related work: Haba,Murayama; Hirsch,King; Vissani; Rosenfeld,Rosner; Antonelli et al.

27 Hierarchy for masses and mixings via horizontal U(1) 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. The coupling is allowed: if vev θ = w, and w/m=λ we get: 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

28 With suitable charge assignments all relevant patterns can be obtained Recall: u~ d=e T ~ 510 ν D ~ 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

29 All entries are a given power of λ times a free o(1) coefficient m u ~ v u λ 10 λ 8 λ 5 λ 8 λ 6 λ 3 λ 5 λ 3 1 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) 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 for each model the λ,λ values are optimised < r < U e3 < < tan 2 θ 12 < < tan 2 θ 23 < 2.57

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

31 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

32 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"

33 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

34 What if θ 23 is really maximal? Would be challenging! All existing models invoke peculiar symmetries (non abelian or discrete are crucial) A set of recent models are based on obtaining, in the basis of (nearly) diagonal charged leptons Grimus, Lavoura..., Ma,... Early models: Barbieri et al, Wetterich... This predicts θ 13 =0 and θ 23 max. Imposing a 2-3 perm. symmetry on L T m ν L does not work, because R L then produces a charged lepton mixing that spoils θ 23 max. Rather, discrete broken symmetries are used to make charged leptons and Dirac neutrino masses diagonal, while the perm. symmetry is in the Majorana RR matrix

35 Can ν mixings arise only from the charged lepton sector? ν e ν µ ν τ flavour G.A., Feruglio, Masina 04 = U ν 1 ν 2 ν 3 mass U = U e+ U ν diag of ch leptons m ν =U* m ν diag U + m e =V e m e diag U e + Assume that, in the lagrangian basis where all symmetries are specified, we have: U ν ~ 1. Then: U ~ U e+ ~ (small effects like s 13 can be thought to arise from U ν - 1. Phases dropped for simplicity) Rm e L L diag = U e L R diag = V e R

36 Given m e diag ~m τ diag[0,η,1] (with η=m µ /m τ ) we obtain: m e = V e m e diag U ~ V e m τ For V e ~1 this is a generalisation of lopsided (s large) but with det 12 =0 Independent of V e : m e+ m e ~ U + (m e diag ) 2 U ~m τ 2 all matrix elements of same order (because s is large) democratic (hierarchy of masses non trivial) s 13 =0 (i.e. eigenvector (c,s,0) T ) -> first two columns proportional

37 Note: in minimal SU(5) models m e = m dt. This implies V e = U d Quark mixings are small: V CKM = U u+ U d Two possibilities: Both U u and U d nearly diagonal -> V e ~ 1 U u ~ U d nearly equal and non diagonal This is the way of democratic models: U u ~ U d ~ U e -> V e ~ U e V e ~ 1 V e ~ U e m e = m τ m e = m τ The first two columns are proportional

38 Our general conclusion: From the charged lepton sector: a large s 23 can easily be produced example: lopsided models U e m e but different orders for s 12 and s 13 is not simple

39 Still we have formulated a model where all mixings arise naturally from the charged lepton sector. A set of U(1) charges garantees that m ν is diagonal The spectrum of one family is like in the 27 of E6 27 = = 1 + (5 + 5 ) +( ) E6 SO(10) SU(5) charged leptons A see-saw mechanism involving the two sets of 5 leeds to the required zero determinant condition in m e The model works but requires a complicated setup of charges and flavons. Note that it borrows the see-saw tricks from the neutrino model building

40 To make m ν ~1 a single U(1) is not enough: We need a flavour group F i act on different light ν s In fact as r~ξ 4 ~1/40 then θ 23 ~ξ would be large F 0 fixes quark and lepton hierarchies The model is natural but cumbersome! flavons

41 We obtain a matrix of the form m e :m µ :m τ = λ 4 : λ 2 :1 det We need x 21 x 32 -x 22 x 31 = 0 to guarantee an eigenvector of m e+ m e [c,s,0(λ 4 )] with eigenvalue 0(λ 8 ): s/c = -x 31 /x 32 The hierarchy in the rows is from the U(1) F0 det=0 is arranged by a see-saw with dominance of a single heavy state in M -1 guaranteed by U(1) F1 x U(1) F2 xu(1) F3 Note that θ 13 ~ λ 4 in this model

42 Conclusion We favour: Normal models: θ 23 large but not maximal, θ 13 not too small (θ 13 of order λ C or λ C2 ) - Semi anarchy - Inverse hierarchy In particular - Normal hierarchy with suppressed 23 determinant Exceptional models: θ 23 maximal or θ 13 very small or also: all mixing from the charged lepton sector... are interesting but not very plausible

Models of Neutrino Masses & Mixings

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/0402155, G.A., F. Feruglio, hep-ph/0504165,hep-ph/0512103