MASSIVE NEUTRINOS: MODEL BUILDING M. Lola, HEP2006 funded by - PDF document

MASSIVE NEUTRINOS: MODEL BUILDING M. Lola, HEP2006 funded by MEXT-CT-2004-014297 � Neutrino data: By now convincing for m ν � = 0 and physics beyond SM � What do we know? lower limit best value upper limit sun (10 − 5 eV 2 ) ∆ m 2 5.4 6.9 9.5 atmo (10 − 3 eV 2 ) ∆ m 2 1.4 2.6 3.57 sin 2 θ 12 0.23 0.30 0.39 sin 2 θ 12 0.31 0.52 0.72 sin 2 θ 13 0 0.006 0.1 � Open questions: - Is the atmospheric mixing maximal or close to maximal? - How large is the solar mixing? - What about θ 13 ? - What is the pattern of masses? ( we know only ∆ m ) - Phases? How large is leptonic CP-violation?

� Origin of neutrino mass in SM extensions - Family symmetries (abelian? non-abelian?) - GUTs (which one?) / SUSY? Use (i) multiplet structure (ii) known fermion mass and mixing parameters to predict those we know less � Additional aspects to consider: - Stability under quantum corrections (large effects possible) - Leptonic CP-violation/Baryogenesis through leptogenesis? - Rare charged-lepton decays, µ - e conversion on nuclei -Collider Signatures/LHC � Combined analysis of the above for max. information

3 × 3 mixing � | ν a > = U ai | ν i >, a = e, µ, τ ; i = 1 , 2 , 3 i U = diag ( e iδ e , e iδ µ , e iδ τ ) .V.diag ( e − iφ 1 / 2 , e − iφ 2 / 2 , 1)   s 13 e − iδ c 12 c 13 s 12 c 13   V = − c 23 s 12 − s 23 s 13 c 12 e iδ c 23 c 12 − s 23 s 13 s 12 e iδ s 23 c 13       − s 23 c 12 − c 23 s 13 s 12 e iδ c 23 c 13 s 23 s 12 − c 23 s 13 c 12 e iδ U ∗ ii U ij U ji U ∗   jj 13 c 23 s 13 + c 12 c 23 s 13 c 12 c 2 δ = − arg  , i, j = 1 , 2 , 3 and i � = j   s 12 s 23  i.e. CP-violation ∝ Jarlskog invariant J CP = 1 22 ) | = 1 2 | Im ( U ∗ 11 U 12 U 21 U ∗ 2 | Im ( U ∗ 11 U 13 U 31 U ∗ 33 ) | = 1 33 ) | = 1 2 | Im ( U ∗ 22 U 23 U 32 U ∗ 2 | c 12 c 2 13 c 23 sin δs 12 s 13 s 23 | Different models predict different CP violation i.e. zero violation, for models with texture zeros in (1,3) entries

ν R and See-Saw mechanism How can we generate naturally light neutrinos? Combine m D ν and M ν R to write a mass matrix   m D  0 ν M ν =  m D ν M ν R If M ν R ≫ m D ν , a very heavy eigenvalue M N ≈ M ν R and a very light ( m D ν ) 2 � � � � m eff ≈ � � M ν R � � Φ H H m k VR V R i VL j VL ν ) 33 ≈ (200 GeV) ( λ N ≈ λ t ) and M N 3 ≈ O (10 13 GeV), For ( m D m eff ≈ 1 eV

Mass hierarchies and flavour symmetries � Start with L-R symmetric model, assume flavour symmetry (different generations of fermions have different charges) Invariance under symmetry, determines magnitude of masses Q i ¯ ¯ ¯ U i D i L i E i H 2 H 1 U (1) a i a i a i b i b i − 2 a 3 wa 3 � LR + SU (2) ⇒ same charge for Q i , ¯ U i , ¯ D i � Up-mass matrix: Top coupling Q 3 ¯ U 3 H 2 0 charge ⇒ allowed All other couplings forbidden   0 0 0   M up = 0 0 0       0 0 1 Suppose singlets θ with non-0 flavor-charges � (singlets expected in realistic models) Then: invariant terms Q i ¯ U j H 2 ( < θ > /M ) n n depending on flavour charges � Hierarchical mass structures generated for ALL fermions

Example consistent with charged fermion hierarchies     ǫ 8 ǫ 3 ǫ 4 ǫ 8 ¯ ǫ 3 ¯ ǫ 4 ¯     M u ∝  , M d ∝ ǫ 3 ǫ 2 ǫ ǫ 3 ¯ ǫ 2 ¯ ¯ ǫ            ǫ 4 ǫ ǫ 4 ¯ 1 ¯ ǫ 1   ǫ 5 ǫ 3 ǫ 5 / 2 ¯ ¯ ¯    , V tot = V † ǫ 3 ǫ 1 / 2 M ℓ ∝ ν V ℓ ¯ ǫ ¯ ¯      ǫ 5 / 2 ¯ ǫ 1 / 2 ¯ 1 • If mixing mostly dominated by V ℓ : - naturally large neutrino mass hierarchies - simplest constructions with too small solar mixing (fixed by charged lepton hierarchies) • If R-H neutrino sector relevant for mixing see-saw 0-determinant cancellations, large solar mixing • Models with more than one U(1)’s and more than one singlets -less predictive, but often simulate well non-abelian structures (very instructive to study them) .

Minimal Models with Abelian Flavour Symmetries • Large splitting between fermion masses Naturally leads to large neutrino hierarchies • Unknown phases/order unity coefficients ⇓ Difficult to obtain naturally degenerate neutrinos • In many models lepton hierarchies consistent with mostly SAMSW but LAMSW possible, ie by see-saw conditions Models with non-Abelian flavour symmetries • Degenerate ν and ℓ ± textures assuming ie that the lepton fields are SO(3) triplets • Subsequently break SO(3) so as: large charged lepton splitting/ small neutrino splitting • Favour almost-degenerate neutrino textures • Textures with (almost)-bimaximal mixing predicted LAMSW / VO oscillations for solar neutrinos

SU (5) (i) Assume the family symmetry is combined with SU (5) (ii) Use the GUT structure ONLY to constrain U (1) charges Under this group we have the following relations: Q ( q,u c ,e c ) i = Q 10 i Q ( l,d c ) i = Q 5 i Q ( ν R ) i = Q ν R i • M ℓ ± = M T • M up symmetric down • L lepton mixing ≈ R down-quark one Can we obtain acceptable patterns of masses/mixings? i.e.     ǫ 6 ¯ ǫ 5 ¯ ǫ 4 ¯ ǫ 3 ¯ ǫ 3 ǫ 3 ¯ ¯ M u  , M down     ǫ 5 ¯ ǫ 4 ¯ ǫ 3 ¯ ǫ 2 ¯ = ǫ 2 = ǫ 2  ¯   ¯  m t m b        ǫ 3 ¯ ǫ 2 1 ¯ ǫ ¯ 1 1   ǫ 4 ¯ ǫ 3 ¯ ¯ ǫ M ℓ   ǫ 3 ¯ ǫ 2 1 = ¯   m τ     ǫ 3 ¯ ǫ 2 1 ¯ Solar Mixing: more structure needs to be added! Ellis, Gomez, ML: computerised scanning of viable constructions, in progress

FIT 2 θ 12 θ 12 FIT 3 θ 13 θ 13 1.2 1.2 Neutrino Yukawa mixing (Rad) Neutrino Yukawa mixing (Rad) θ 23 θ 23 10 x JCP 100 x JCP 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 ν (Rad) ν (Rad) φ φ θ 12 θ 12 FIT 5 FIT 4 θ 13 θ 13 1.2 1.2 Neutrino Yukawa mixing (Rad) Neutrino Yukawa mixing (Rad) θ 23 θ 23 100 x JCP 10 x JCP 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 1 2 3 0 0.5 1 1.5 2 2.5 3 ν (Rad) ν (Rad) φ φ Figure 1: Neutrino mixing angles and values J CP vs. the phase φ ν of the Dirac neutrino matrix Y ν . The phase φ X 23 of the Y e is kept constant (and the results are almost independent of it). The texures corresponds to fits 2,3,4,5 of KKPV with the values of coef. a ij provided in their tables 11 and 13.

SO (10) • All L- and R-handed fermions in the 16 of SO (10) • Both MSSM Higgs fields fit in a single 10 of SO (10) ⇓ For all fermions, L-R symmetric textures , similar structure (different expansion parameters due to Higgs mixing) Flipped SU (5) i , e c singlet of SU (5) Q ( q,d c ,ν c ) i = Q 10 i , Q ( l,u c ) i = Q 5 • m D ν = M T • Symmetric M down up SU (3) c ⊗ SU (3) L ⊗ SU (3) R Particles placed in (3 , 3 , 1), (¯ 3 , 1 , ¯ 3) and (1 , 3 , ¯ 3) as:     ℓ c L e − u     � � L c ℓ u ¯ d ¯  d  ¯ D  ν      L     e + ν c N D L L • Symmetric lepton mass matrices (as in L-R symm. models) • Asymmetric up and down Different predictions and correlations between observables

Baryogenesis through leptogenesis • Neutrinos have masses and mix with each other • Like quarks, CP violation in neutrino sector L + 1 L = ℓ L Φ h ν N c L M N c 2 N c L + h . c . • Lepton-number-violation (i.e. in decays of heavy, RH Majorana neutrinos) N c L → Φ + ℓ N c L → Φ + ℓ REMEMBER: L/B-violating interactions in thermal equilibr. at high T Changes in lepton number ⇒ Changes in baryon number THUS: Generate ∆ L � = 0 , which then transforms to ∆ B � = 0

Out-of-equilibrium condition: Decay rates smaller than Hubble parameter H at T ≈ M N 1 Three-level width of N 1 : Γ = ( λ † λ ) 11 8 π M N 1 T 2 Compare with: H ≈ 1 . 7 g 1 / 2 ∗ M p ( g MSSM ≈ 228 . 75, g SM = 106 . 75) ∗ ∗ ⇒ ( λ † λ ) 11 M p < M N 1 14 πg 1 / 2 ∗ More accurate by looking at Boltzmann equations CP -violating asymmetry, ǫ (interference between tree-level and 1-loop amplitudes) � m 2 � 1 N j � ( λ † λ ) 2 � � ǫ j = Im f 1 j m 2 (8 πλ † λ ) 11 N 1 j � � 1 + y �� f ( y ) = √ y 1 − (1 + y ) ln y M N 1 Plus self-energy corrections ˜ δ ∝ ( M N 2 − M N 1 ) What can leptogenesis tell us about fermion mass patterns?

Effects of radiative corrections on neutrino masses and mixing For i, j , generation indices � � 1 d 1 t + 1 dtm ij − c i g 2 i + 3 λ 2 2( λ 2 i + λ 2 eff = j ) m ij 8 π 2 eff m 33 eff + m 22 16 π 2 d dt sin 2 2 θ 23 =2 sin 2 2 θ 23 (1 − 2 sin 2 θ 23 ) λ 2 eff τ m 33 eff − m 22 eff sin 2 2 θ 23 affected by quantum corrections if: (ii) m 33 eff − m 22 (i) λ τ large (large tan β ) eff small Semi-analytic and numerical studies ⇒ - The mixing can even be amplified/destroyed � 1 � t m ij � �� t + 1 eff − c i g 2 i + 3 λ 2 2( λ 2 i + λ 2 = exp j ) m ij 8 π 2 t 0 eff, 0 � � ≡ I g · I t · I i · I j 1. The relative structure in m eff is only modified by the lep- tonic Yukawa couplings 2. On the contrary, the gauge and top couplings give only an overall scaling factor