Neutrino Masses and Mixing:

Neutrino Masses and Mixing: New Symmetry of Nature? A. Yu. Smirnov International Centre for Theoretical Physics, Trieste, Italy Institute for Nuclear Research, RAS, Moscow, Russia Summarizing achievements and highlights Pattern of lepton mixings: New theoretical puzzle? Towards the underlying physics New symmetry of Nature? Quarks - Lepton symmetry Flavor symmetry or family symmetry? Mass spectrum: Degenerate or hierarchical?

Flavors, Masses, Mixing Flavor states: Mixing: = Flavor states Mass eigenstate ne nm nt Eigenstates of the CC weak interactions m3 n3 ? ns Sterile neutrinos no weak interactions mass m2 n2 Mass eigenstates: m1 n1 n1 n2 n3 m1 m2 m3 Neutrino mass and flavor spectrum A Yu Smirnov

Mixing and Flavor states n2 = sinq ne + cosq nm n1 = cosq ne - sinq nm vacuum mixing angle coherent mixture of mass eigenstates ne = cosq n1 + sinq n2 n1 wave packets ne n2 Dm 2 2E Df = ---- l Interference of the parts of wave packets with the same flavor depends on the phase difference Df between n1 and n2 Dm2 = m22 - m12

Reactor experiments CHOOZ, PaloVerde KamLAND Solar neutrinos Accelerator experiments Masses, mixing Dmij2, qi j Oscillations Conversion in matter Supernova neutrinos, SN 1987A Cosmology Neutrinoless double beta decay Atmospheric neutrinos Direct kinematic measurements of neutrino masses A Yu Smirnov

Vacuum Oscillations Flavors of mass eigenstates do not change Determined by q Admixtures of mass eigenstates do not change: no n1 <-> n2 transitions n2 ne n1 Df = Dvphase t Df = 0 Dm2 2E Dvphase = Dm2 = m22 - m12 Due to difference of masses n1 and n2 have different phase velocities: Oscillation length: ln = 2p/Dvphase= 4pE/Dm2 oscillations: Amplitude (depth) of oscillations: effects of the phase difference increase which changes the interference pattern A = sin22q A. Yu. Smirnov

Adiabatic conversion n1m <-->n2m P = sin2q n0 >> nR Non-oscillatory transition n2m n1m n2 n1 interference suppressed Resonance Mixing suppressed n0 > nR Adiabatic conversion + oscillations n2m n1m n2 n1 n0 < nR Small matter corrections n2m n1m n2 n1 ne A. Yu. Smirnov

Atmospheric neutrinos nm - nt vacuum oscillations ne p p m nm nm - ne oscillations in matter N e nm n core cosmic rays At low energies: r = Fm /Fe = 2 atmosphere n mantle Parametric effects in nm - ne oscillations for core crossing trajectories detector

Oscillation parameters SuperKamiokande 1.9x10-3 < m2 < 3.0x10-3 eV2 0.90 < sin22at 90% C.L. Preliminary m2=2.4x10-3,sin22=1.00 2min=37.8/40 d.o.f (sin22=1.02, 2min=37.7/40 d.o.f) L/E analysis Zenith angle analysis 90% allowed regions

Solar Neutrinos 4p + 2e- 4He + 2ne + 26.73 MeV Adiabatic conversion in matter of the Sun electron neutrinos are produced F = 6 1010 cm-2 c-1 r : (150 0) g/cc total flux at the Earth Oscillations in vacuum n Oscillations in matter of the Earth J.N. Bahcall

After SNO salt results P. de Holanda, A.S. solar data solar data + KamLAND sin2q13 = 0.0 KL Dm2 = 6.3 10-5 eV2 Dm2 = 7.1 10-5eV2 tan2q = 0.40 tan2q = 0.39

KamLAND Kamioka Large Anti-Neutrino Detector 1 kton of LS Reactor long baseline experiment 150 - 210 km Liquid scintillation detector ne + p---> e+ + n Epr > 2.6 MeV Total rate energy spectrum of events LMA precise determination of the oscillation parameters 10% accuracy Detection of the Geo-neutrinos Epr > 1.3 MeV

U_e3 Has important implications for phenomenology and theory: supernova neutrinos: can lead to O(1) effect; atmospheric neutrinos (resonance enhancement, parametric effects); allows to establish mass hierarchy; LBL experiments; CP-violations requires Ue3 = 0; key test of models of large lepton mixing Ue3 = sin q13 e-id = <ne | n3 > CHOOZ atmospheric CHOOZ+atmospheric tan q13 sin22q13

Mass spectrum and mixing ne nm nt |Ue3|2 ? n3 n2 Dm2sun n1 mass mass Dm2atm Dm2atm |Ue3|2 n2 Dm2sun n1 n3 Normal mass hierarchy (ordering) Inverted mass hierarchy (ordering) Type of mass spectrum: with Hierarchy, Ordering, Degeneracy absolute mass scale Type of the mass hierarchy: Normal, Inverted Ue3 = ? A Yu Smirnov

Main features m h ~ (0.04 - 0.4) eV m h > Dm232 > 0.04 eV Heaviest mass: Hierarchy of mass squared differences: | Dm122 / Dm232 | = 0.01 - 0.15 No strong hierarchy of masses: + 0.22 - 0.08 |m2/m3| > |Dm122 / Dm232 | = 0.18 Bi-large or large-maximal mixing between neighboring families (1- 2) (2- 3): 1s 2-3 1-2 q12 + qC = q23 ~ 45o 1-3 bi-maximal + corrections? 0 0.2 0.4 0.6 0.8 |sin q|

Toward the underlying physics Neutrino masses and mixing - important message which we can not understand yet In which terms, at which level theory (principles) should be formulated? Observables Mass matrices masses mixing angles are fundamental parameters which show certain symmetry their properties, symmetries at certain energy scale observables are outcome, can be to some extend accidental numbers which do not reflect the underlying symmetry Schemes with bi-maximal mixing, or broken bi-maximal mixing; Tri-bimaximal mixing No regularities - ``anarchy’’ Smallness of neutrino masses and pattern of lepton mixing are related? See-saw

Mixing and Mass Matrix nenm nt ne nm nt Mass matrix for the flavor states nf = ( ne, nm ,nt )T, mf is not diagonal Mixing: nf = Unnmass nmass = ( n1, n2, n3)T (Majorana) Diagonalization: mf = UnT mdiag Un * mdiag = diag ( m1, m2, m3) In general (symmetry) basis the mass matrix of the charged leptons is also non-diagonal Charged leptons lS L = UlL l mass L l = (e, m, t)T Lepton mixing --> the mismatch of rotations of the neutrinos and the charge leptons which diagonalize the corresponding mass matrices Lepton UP-MNS = UlL+ Un mixing matrix

T. Yanagida M. Gell-Mann, P. Ramond, R. Slansky S. L. Glashow R.N. Mohapatra, G. Senjanovic Seesaw Smallness of neutrino masses 1 2 NT Y L H + NTMR N + h.c. mn n L = (n, l)T N = ( nR)c n N 0 mD mDT MR mD mD= Y<H> N If MR >> mD MR mn = - mDTMR-1 mD Zero charges -> can have Majorana mass Right handed components: singlets of the SM symmetry group -> mass is unprotected by symmetry can be large -- at the scale of lepton number violation Neutrality QEM = 0, QC = 0 Smallness of neutrino mass

New Symmetry of Nature? What can testify ? Maximal or close to maximal 2-3 mixing Very small 1 -3 mixing Degenerate or partially degenerate spectrum |0.5 - sin2q23| <<sin2q23 Dm << m q13 << q12 x q23 m1 = m2 = m3 2n data fit: Best fit value: sin2 2q23 = 1.0 m3 = m2 << m1 ( inverted hierarchy) ? sin2 2q23 > 0.91, 90% CL

Degenerate spectrum Degenerate: m0 > (0.08 - 0.10) eV Dm/m0 ~ Dmatm2/2m02 Large or maximal mixing Mass degeneracy Large scale structure analysis including X-ray galaxy clusters Cosmology: m0 = 0.20 +/- 10 eV S.W Allen, R.W. Schmidt, S. L. Bridle Heidelberg-Moscow result mi > mee at least for one mass eigenstate 77.7 kg y 4.2s effect mee = (0.29 - 0. 60 ) eV (3s ) H. Klapdor-Kleingrothaus, et al. mee(b.f.) = 0.44 eV

Absolute scale of mass F. Feruglio, A. Strumia, F. Vissani Sensitivity limit p n H-M e n x e n Neutrinoless double beta decay p mee = Sk Uek2 mk eif(k) Both cosmology and double beta decay have similar sensitivities m1 Kinematic searches, cosmology

U_e3 - expectations Naively excluded which implies dominant structures or/and degeneracy in the mass matrix sinq13 ~ sinq12 x sinq23 ~ 0.3 - 0.5 solar sector A bit seriously Atmospheric sector sinq13 Mass scales: Dmsun2 Dmatm2 sinq13 ~ Dmsun2/Dmatm2 ~ 0.15 - 0.20 With comparable contribution from the charge leptons If there is no cancellation: sin2q13 ~ 0.01 - 0.05

Relations? Deviation of 2-3 mixing Degeneracy from maximal of mass spectrum Symmetry which leads to maximal 2-3 mixing e.g. Permutations, Z2 can give sin q13 = 0 ? Violation of the symmetry 1-3 mixing D23 = 0 sin q13 = 0

Two extreme cases Quasi-degenerate spectrum S Barshay, M. Fukugita, T. Yanagida ... New symmetry SO(3), A4, Z2 , ... Hierarchical Quark -lepton symmetry? Spectrum No particular symmetry Deviation from maximal mixing Quark-lepton analogy/symmetry

Symmetry case ne nm nt In the flavor basis: 1 0 0 mn = m0 0 0 1 + dm 0 1 0 dm << m0 Features: Degenerate spectrum: m1 = m2 = - m3 Maximal or near maximal 2-3 mixing Opposite CP parities of n2 andn3 n2 andn3 form pseudo-Dirac pair Neutrinoless double beta decay: mee~ m0 Most of the oscillation parameters are not imprinted in to the dominant structure: all Dm2 the as well as 1-2 and 1-3 mixings are determined by dm. dm is due to the radiative corrections?

Symmetry E. Ma, G. Rajasekaran K.S. Babu, J. Valle A4 Symmetry group of even permutations of 4 elements Symmetry of tetrahedron: (4 faces, 4 vertices) Plato’s fire Irreducible representations: 3, 1, 1’, 1’’ (i = 1, 2, 3) H1,2 ~ 1 (ni , ei ) ~ 3 (ui , di ) ~ 3 Transformations under A4 u3c, d3c, e3c ~ 1’’ u1c, d1c, e1c ~ 1 u2c, d2c, e2c ~ 1’ Ui, U1c, Di, D1c, Ei, E1c, Nic , ci ~ 3 New heavy quarks leptons and Higgs are introduced Explicit asymmetry of charged fermions and neutrino sector: no nic and Ni Leads to different mixing of quarks and leptons

Model: quarks and leptons Problem to include leptons and quarks in unique model Symmetry between leptons and quarks is explicitly broken Extra symmetries required (e.g. Z3) Neutral and charge lepton sectors are different: ei e1c | | Eic ----- Ei ni | Nic H1 H2 < ci > i = 1, 2, 3 Majorana mass ME MN A4 is broken Seesaw Mixing - from the charged leptons in the symmetry basis Mixing of quarks: UL+UL = I 1 1 1 UL = 1 w w2 1 w2 w Mixing of leptons: ULTUL = M 0 Oscillation parameters are determined by dm - an additional theory independent on A4 required w = exp (-2ip/3) Masses and mixing angles are unrelated

Symmetry approach Charged leptons ? Neutrinos No symmetry Quarks Symmetry Realization requires introduction of - New leptons and quarks; - Extended non-trivial Higgs sector to break symmetry; - Additional symmetries to suppress unwanted interactions of new particles Often - difficult to embed into Grand Unified schemes ? or ? Profound implications Misleading approach ? ?

Do we really see a symmetry? 1). The only serious indication is nearly maximal 2-3 mixing: sin2 2q23 > 0.91, 90 % CL 2). sin2 2q23 is a ``bad’’ parameter from theoretical point of view sin2q23 is better, but for this parameter: sin2q23 = (0.35 - 0.65) 90 % CL Relative deviation from maximal mixing (1/2) can be large Dsin2q23 /sin2q23 ~ 0.7 3). The atmospheric neutrino results may provide some hint that the mixing is not maximal Excess of the e-like events (?) Three neutrino analysis of data is needed

Best fitsin22=1.0, m2=2.0x10-3 eV2 Null oscillation  ~13000km ~500km ~15km Zenith angle distributions Up stop Sub-GeV Multi-R -like Sub-GeV e-like Sub-GeV -like Multi-GeV -like + PC Multi-GeV Multi-R -like Up thru Multi-GeV e-like SuperKamiokande

LMA oscillations of atmospheric neutrinos Excess of the e-like events in sub-GeV Fe Fe0 - 1 = P2 ( r c232 - 1) ``screening factor’’ P2 = P(Dm212 , q12) is the 2n transition probability In the sub-GeV sample r = Fm0 / Fe0 ~ 2 The excess is zero for maximal 23- mixing Searches of the excess can be used to restrict deviation of the 2-3 mixing from maximal Zenith angle dependences of the e-like events 0.Peres, A.S.

Deviation of 2-3 mixing from maximal Dm2 = 0 Dm2 = 0 sin2q23= 0.45 - 0.47 M. C. Gonzalez-Garcia M. Maltoni, A.S.

``No-symmetry'' approach I. Dorsner, A. S 2-3 mixing differs from maximal one No special symmetry for neutrinos or leptons Approximate quark-lepton symmetry (analogy, correspondence) Family structure, weak interfamily connections no large mixing in the original mass matrices All quark and lepton mass matrices have similar structure with ``flavor alignment’’ qij~ mi/mj qatm = 36 - 380 Seesaw mechanism of neutrino mass generation Large lepton mixing is the artifact of seesaw

Quark-Lepton symmetry Quarks-Lepton symmetry is realized in terms of the mass matrices (matrices of the Yukawa couplings). For the Dirac mass matrices of all quarks and leptons: (implies large tan b) YU ~ YD ~ YnD ~ YL ~ Y0 Specifically: Yf = Y0 + DYf ( Y0)ij << (DYf)ij f = U, D, L, nD Y0 is ``unstable’’: det (Y0) ~ 0 (as well as determinants of sub-matrices) small perturbations produce significant change of masses and and mixings a11e4 a12e3 a13e2 Y = a21e3 a22e2 a23e a31e2 a32e 1 e~ 0.2 - 0.3 Assume: |aij|= 1 + O(e) Observables (masses and mixings) appear as small perturbations of the dominant structure given by Y0 .

See-saw Enhances (by factor ~ 2) the mixing which comes from diagonalization of the neutrino mass matrix Changes the relative sign of ``rotations’’ which diagonalize mass matrices of the charge leptons and neutrinos Leads to smallness of neutrino masses For the RH neutrino mass matrix we take for simplicity the same form MR ~ Y0

Flipping the sign of rotation lepton mixing: quark mixing: VP-MNS = U(l)+ U(n) VCKM = U (up)+ U(down) For quarks the up and down rotations cancel each other leading to small mixing whereas for leptons they sum up producing large mixing S. Barshay n L U D qPMNS qCKM It is the see-saw leads to flip of the sign of rotation which diagonalizes the neutrino mass matrix changes the sign of the 23-elements leads to m(22) > m(33) Enhances 22 and 23 elements

Expansion parameter eis determined by inequality of masses of the s-quark and the muon (one would expect mm= ms in the case of exact q-l symmetry ) kf e~ a22 - a23 2 ms/mb = kq e3 mm/mt = kL e3 These mass ratios can be reproduces for aij = 1 + O(e) if e> 0.25

Implications No special symmetry for neutrinos of for lepton sector Y0 can be reproduced with U(1) family symmetry and charges q, q + 1, q + 2 charges, if unut charge is associated with one degree of e ( a la Froggatt-Nielsen) Violation of the symmetry appears at the level e2 -e3 ~ (1 - 3) 10-2 If the flavor symmetry is at GU scale its violation can come e.g. from the string scale

Conclusions One of the main results is an amazing pattern of the lepton mixing which differs strongly from the quark mixing The key question to the underlying physics is if there is a new (different from quark) symmetry behind neutrino mass spectrum and mixing - maximal (near) maximal 2-3 mixing - degenerate spectrum - probably, small 1-3 mixing Nothing special: weakly broken quark-lepton symmetry; family structure, small interfamily mixing. Large lepton mixing -artifact of the seesaw mechanism New symmetry of Nature, which is realized (manifest itself) in the properties of neutrinos or lepton sector. ? Neutrinos can shed some light on the fermion mass problem in general. Hint: flavor alignment, e ~ 0.25 , ``unstable’’ mass matrices, U(1) Priorities

Neutrino Masses and Mixing: