neutrino mass and mixing:

A. Yu. Smirnov

International Centre for Theoretical Physics, Trieste, Italy Institute for Nuclear Research, RAS, Moscow, Russia

NO-VE 2006: ``Ultimate Goals’’NO-VE 2006: ``Ultimate Goals’’

Two fundamental issuesTwo fundamental issues

Hierarchy of masses:Hierarchy of masses:

|m2 /m3| ~ 0.2

|sin |0 0.2 0.4 0.6 0.8

1-3

1-2

2-3

1

1-2, 12

2-3, 23

1-3, 13

QuarksQuarks LeptonsLeptons

13o

2.3o 45o

~ 0.5o

34o

<10o

MixingMixing

|m /m| = 0.06

Neutrinos

Chargedleptons

|ms /mb| ~ 0.02 - 0.03Down

quarks|mc

/mt| ~ 0.005Up-quarks

up down charged neutrinosquarks quarks leptons

10-1

10-2

10-3

10-4

10-5

1

mu mt = mc2

mu mt = mc2

Vus Vcb ~ Vub Vus Vcb ~ Vub

at mZ

Regularities?

Koide relationKoide relation

permutation symmetry

Neutrino mass matrix in the flavor basis:

A B B B C DB D C

Discrete symmetries S3, D4

Can bothfeatures be accidental?

Often related to equality of neutrino masses

For charged leptons: D = 0

For charged leptons: D = 0

Are quarks and leptons fundamentally different?

Are quarks and leptons fundamentally different?

Can this symmetry be extended to quark sector?

Can this symmetry be extended to quark sector?

A. Joshipura, hep-ph/0512252

A. Joshipura, hep-ph/0512252

Smallness of VcbSmallness of Vcb Maximal (large)

2-3 leptonic mixing

Maximal (large) 2-3 leptonic mixing

2-3 symmetry2-3 symmetry

X A AA B C A C B

X A AA B C A C B

Universal mass matrices

Universal mass matrices + m+ m

Quarks, charged leptons: B ~ C, X << A << BQuarks, charged leptons: B ~ C, X << A << B

Neutrinos: B >> C, X ~ BNeutrinos: B >> C, X ~ B

- Hierarchical mass spectrum- Small quark mixing

- Hierarchical mass spectrum- Small quark mixing

- Degenerate neutrino mass spectrum; - Large lepton mixing

- Degenerate neutrino mass spectrum; - Large lepton mixing

additional symmetries are needed to explain hierarchies/equalities of parameters

additional symmetries are needed to explain hierarchies/equalities of parameters

2-3 symmetryDoes not contradictmass hierarchy

2-3 symmetryDoes not contradictmass hierarchy

A B B B C DB D C

permutation symmetry

Matrix for the best fit valuesof parameters (in meV)

Matrix for the best fit valuesof parameters (in meV)

3.2 6.0 0.6 24.8 21.4 30.7

Bari groupBari groupsin213 = 0.01sin223 = 0.43

Substantial deviation from symmetric structure

Substantial deviation from symmetric structure

Structure of mass matrix is sensitive to small deviations Of 1-3 mixing from zero and 2-3 mixing from maximal

Structure of mass matrix is sensitive to small deviations Of 1-3 mixing from zero and 2-3 mixing from maximal

Similar gauge structure,correspondence Similar gauge structure,correspondence

Very different massand mixing patternsVery different massand mixing patterns

Particular symmetries in leptonic (neutrino) sector?

Particular symmetries in leptonic (neutrino) sector?

Q-L complementarity?Q-L complementarity?

Additional structure

exists which

produces

the difference.

Additional structure

exists which

produces

the difference.

Is this seesaw? Something beyond seesaw?

Is this seesaw? Something beyond seesaw?

Symmetrycorrespondence

Symmetrycorrespondence

Majorana massesMajorana masses

Q = 0 Qc = 0

Is this enough to explain all salient properties of neutrinos?

mix with singlets of the SM

mix with singlets of the SM

Neutrality

Basis of seesawmechanism

Basis of seesawmechanism

Dynamicaleffects

Dynamicaleffects

l

lL

lR

Standard Model

...

...

H

S

Window to hidden world?

M

S

s

R

A

A

Planck scale physicsPlanck scale physics

s

SS

Screening of the Dirac structureInduced effects of new neutrino states

Correspondence: ur , ub , uj <-> dr , db , dj <-> e

color

Symmetry: Leptons as 4th color

Unification:form multiplet of the extendedgauge group, in particular, 16-plet of SO(10)

Pati-Salam

Can it be accidental?

More complicated connection between quarks and leptons? Complementarity?

Provide with all the ingredients necessary for seesaw mechanism

Large mass scaleLarge mass scale

Lepton number violationLepton number violation

RH neutrino componentsRH neutrino components

Give relations between masses of leptons and quarks

genericallygenerically

mb = m

b - unification

In general: ``sum rules’’

But - no explanation of the flavor structure

But - no explanation of the flavor structure

large 2-3 leptonic mixing

is realized in terms of the mass matrices (matrices of the Yukawa couplings) and not in terms of observables – mass ratios and mixing angles.

YU = YD = YD = YL = Y0 YU = YD = YD = YL = Y0

Universal structure for mass matrices of all quarks and leptonsin the lowest approximation:

Yf = Y0 + Yf Yf = Y0 + Yf ( Y0)ij >> (Yf)ij

( Y0)ij >> (Yf)ij f = u, d, L, D, M

approximate

Mass matricesM = Y V

Mass matricesM = Y V

diagonalization

Eigenvalues = masses

Eigenstates = mixing

Small perturbations:

I. Dorsner, A.S.NPB 698 386 (2004)

I. Dorsner, A.S.NPB 698 386 (2004)

Small perturbations allow to explain large difference in mass hierarchies and mixings of quarks and leptons

Unstable with respect to small perturbations

Yfij = Y0

ij (1 + fij)

Yf

ij = Y0ij (1 + f

ij)

Universalsingular

f = u, d, e,

Perturbations ~ 0.2 – 0.3

4 3 2

Y0 = 3 2 2 ~ 0.2 - 0.3 ~ 0.2 - 0.3

Form of perturbationsis crucial

Form of perturbationsis crucial

Important example:Important example:

Nearly singular matrix of RH neutrinos leads to - enhancement of lepton mixing - flip of the sign of mixing angle, so that the angles from the charged leptons and neutrinos sum up

Seesaw: m ~ 1/MSeesaw: m ~ 1/M

In some (universality) basis in the first approximation all the mass matrices but Ml (for the charged leptons) are diagonalized by the same matrix V:

In some (universality) basis in the first approximation all the mass matrices but Ml (for the charged leptons) are diagonalized by the same matrix V: V+ Mf V = Df

For the charged leptons, the mass For the charged leptons, the mass VT Ml V* = Dl

is diagonalized by V*is diagonalized by V*

Ml = MdTV for u, d,

V* for l

V for u, d, V* for lDiagonalization:Diagonalization:

SU(5) type relationSU(5) type relation

Quark mixing:Quark mixing:VCKM = V+ V = IVCKM = V+ V = I

Lepton mixing:Lepton mixing:VPMNS = VT V VPMNS = VT V

In the first approximationIn the first approximation

A Joshipura, A.S.hep-ph/0512024

A Joshipura, A.S.hep-ph/0512024

Another version is when neutrinos have distinguished rotation:

Another version is when neutrinos have distinguished rotation:

V for u, d, l V* for

V for u, d, l V* for

VCKM = V’+ V VCKM = V’+ V VPMNS = VT V’ VPMNS = VT V’

In general, up and down fermions can be diagonalized by different matrices V’ and V respectively

In general, up and down fermions can be diagonalized by different matrices V’ and V respectively

VPMNS = VTV VCKM+ = V0

PMNS VCKM

+ VPMNS = VTV VCKM+ = V0

PMNS VCKM

+

VPMNS VCKM = VTV VPMNS VCKM = VTV Quark and lepton rotations are complementary to VVT

Quark and lepton rotations are complementary to VVT

V0PMNS = VTV V0

PMNS = VTV - symmetric, characterized by 2 angles;- close to the observed mixing for /2 ~ ~ 20 – 25o - 1-3 mixing near the upper bound

- symmetric, characterized by 2 angles;- close to the observed mixing for /2 ~ ~ 20 – 25o - 1-3 mixing near the upper bound

- gives very good description of data - predicts sin 13 > 0.08

- gives very good description of data - predicts sin 13 > 0.08 VPMNS

(with CKM corr.)

VPMNS (with CKM corr.)

Mu, ~ m D* A D*

Universal mixing and universal matricesUniversal mixing and universal matrices

Md ~ m D*A D Ml ~ m D A D*

D = diag(1, i, 1)D = diag(1, i, 1)A is the universal matrix: A is the universal matrix:

A ~ A ~

i ~ 0.2 – 0.3i ~ 0.2 – 0.3

Can be embedded in to SU(5) and SO(10) with additional assumptions

Can be embedded in to SU(5) and SO(10) with additional assumptions

1

lq

12 lq

12 solC = 46.7o +/- 2.4oA.S.M. RaidalH. Minakata

H. Minakata, A.S.Phys. Rev. D70: 073009 (2004) [hep-ph/0405088]

lq

23 lq

23 atmVcb = 45o +/- 3o

2-3 leptonic mixing is close to maximalbecause 2-3 quark mixing is small

2-3 leptonic mixing is close to maximalbecause 2-3 quark mixing is small

Difficult to expects exact equalities but qualitatively

Difficult to expects exact equalities but qualitatively

1-2 leptonic mixing deviates from maximal substantially because 1-2 quark mixing is relatively large

1-2 leptonic mixing deviates from maximal substantially because 1-2 quark mixing is relatively large

Quark-lepton symmetryQuark-lepton symmetry

Existence of structure which produces bi-maximal mixing

Existence of structure which produces bi-maximal mixing

sin C = 0.22 as ``quantum’’ of flavor physics

sin C = 0.22 as ``quantum’’ of flavor physics

sinC = m /m

sinC = m /msinC ~ sin 13sinC ~ sin 13

Appears in differentplaces of theory

Mixing matrix weakly depends on mass eigenvalues

In the lowest approximation:

Vquarks = I, Vleptons =Vbm

m1 = m2 = 0

``Lepton mixing = bi-maximal mixing – quark mixing’’``Lepton mixing = bi-maximal mixing – quark mixing’’

F. VissaniV. Barger et al

UPMNS = Ubm

Ubm = U23mU12

mUbm = U23

mU12m

Two maximal rotations

½ ½ -½ ½ ½ ½ -½ ½

- maximal 2-3 mixing- zero 1-3 mixing- maximal 1-2 mixing- no CP-violation

Contradictsdata at (5-6) level

In the lowest order?Corrections?

UPMNS = U’ Ubm

U’ = U12()

Generates simultaneously

Deviation of 1-2 mixing from maximal

Non-zero 1-3 mixing

0

Ubm =

As dominant structure?Zero order?

Charged leptons Charged leptons

sin(C) + 0.5sin C( 2 - 1)

NeutrinosNeutrinos q-l symmetry

tan212 = 0.495

H. Minakata, A.S.R. Mohapatra, P. Frampton, C. W.Kim et al.,S. Pakvasa …

Maximal mixing

CKM mixing

mD ~ mu

Maximal mixingml ~ md q-l symmetry

mDT M-1 mD

sin(C

)

CKM mixing

QLC-2QLC-1

sin12 =sin12 =

sin13 =sin13 =

sin C/ 2 ~ Vub~ Vub

29 31 33 35 37 39 12

Fogli et al

Strumia-Vissani

321

SNO (2)

QLC2 tbmQLC1

- analysis does change bft but error bars become smaller 12+ C ~/4

90%99%

Utbm = Utm Um13

UQLC1 = UC Ubm

Give the almost same 12 mixing

Give the almost same 12 mixingcoincidencecoincidence

P. F. HarrisonD. H. PerkinsW. G. Scott

Utbm = U23(/4)U12

Utbm = U23(/4)U12

- maximal 2-3 mixing- zero 1-3 mixing- no CP-violation

Utbm = 2/3 1/3 0- 1/6 1/3 1/2 1/6 - 1/3 1/2

2is tri-maximally mixed 3 is bi-maximally mixed

sin212 = 1/3 in agreement with 0.315

Mixing parameters - some simple numbers 0, 1/3, 1/2

S3 group matrixRelation to group matrices?

In flavor basis…relation to masses?No analogy in theQuark sector? Implies non-abelian symmetry

L. Wolfenstein

0 0.01 0.02 0.03 0.04 0.05 sin213

Fogli et al

Strumia-Vissani

32

1

T2K

90%99%

Double CHOOZ

Cm21

2/m322

Non-zero central value (Fogli, et al): Atmospheric neutrinos, SK spectrum of multi-GeV e-like events

Lower theoretical bounds: Planck scale effects RGE- effects

V.S. Berezinsky F. Vissani

M. Lindner et al

In agreement with 0 value

QLC1QLC1

1). Superheavy MS >> vEW - decouple1). Superheavy MS >> vEW - decouple

2). Heavy: vEW >> mS >> m 2). Heavy: vEW >> mS >> m

3). Light: mS ~ m play role in dynamics of oscillations3). Light: mS ~ m play role in dynamics of oscillations

0 mD 0 m = mD

T 0 MD

T

0 MD MS

If MD = A-1mD

m = mD MD-1

MS MD-1mD

m = A2 MSm = A2 MS

Structure of the neutrino mass matrix is determined by MS

-> physics at highest (Planck?) scale immediately

mD similar (equal) to quark mass matrix - cancels

mD << MD << MS

Double (cascade) seesaw

MR = - MDT MS

-1 MD

Additional fermions

A ~ vEW/MGU

NS

R. MohapatraPRL 56, 561, (1986)

MS – Majorana mass matrix of new fermions S

M. LindnerM. SchmidtA.S. JHEP0507, 048 (2005)

A.S.PRD 48, 3264 (1993)

R. Mohapatra. J. Valle

R. Mohapatra. J. Valle

leads to quasi-degenerate spectrum if e.g. MS ~ I, leads to quasi-degenerate spectrum if e.g. MS ~ I,

Structure of the neutrino mass matrix is determined by

origin of ``neutrino’’ symmetry

origin of ``neutrino’’ symmetry

origin of maximal (or bi-maximal) mixing origin of maximal (or bi-maximal) mixing

MS

MS ~ MPl ?

Q-l complementarityQ-l complementarity

Reconciling Q-L symmetry and different mixings of quarks and leptons

Seesaw provides scale and not the flavor structure of neutrino mass matrix

Mixing with sterile states change structure of the mass matrix of active neutrinos

Consider one state S which has - Majorana mass M and - mixing masses with active neutrinos, miS (i = e, , )

Active neutrinos acquire (e.g. via seesaw) the Majorana mass matrix ma

After decoupling of S the active neutrino mass matrix becomes

(m)ij = (ma )ij - miSmjS/M(m)ij = (ma )ij - miSmjS/M

induced mass matrixinduced mass matrixsinS = mS/M

mind = sinS2 M mind = sinS2 M

mind = sinS2 M mind = sinS2 M

sinS2 M > 0.02 – 0.03 eV

Induced matrix can reproduce the following structures of the active neutrino mass

Induced matrix can reproduce the following structures of the active neutrino mass

sinS2 M ~ 0.003 eV Sub-leading structures for normal

hierarchy

Sub-leading structures for normal hierarchy

sinS2 M < 0.001 eV Effect is negligibleEffect is negligible

Dominant structures for normal and inverted hierarchy

Dominant structures for normal and inverted hierarchy

In the case of normal mass hierarchy

mtbm ~ m2/3 + m3/2

1 1 11 1 11 1 1

0 0 00 1 -10 -1 1

m2 = msol

2

Assume the coupling of S with active neutrinos is flavor blind (universal):

miS = mS = m2 /3

Then mind can reproduce the first matrixThen mind can reproduce the first matrix

mtbm = ma +mind ma is the second matrix

Two sterile neutrinos can reproduce whole tbm-matrixTwo sterile neutrinos can reproduce whole tbm-matrix

R. Zukanovic-Funcal, A.S.in preparation

R. Zukanovic-Funcal, A.S.in preparation

Two regions are allowed: Two regions are allowed:

MS ~ 0.1 – 1 eV MS > (0.1 – 1) GeVandand

Q & L:- strong difference of mass and mixing pattern;- possible presence of the special leptonic (neutrino) symmetries; - quark-lepton complementarity

Still approximate quarks and leptons universality can be realized.

Mixing with new neutrino states can play the role of this additional structure:- screening of the Dirac structure - induced matrix with certain symmetries.

This may indicate that q & l are fundamentally differentor some new structure of theory exists (beyond seesaw)

0.2 0.3 0.4 0.5 0.6 0.7 sin223

Fogli et al

32

1

T2K

SK (3)90%

QLC1

SK (3) - no shift from maximal mixing

1). in agreement with maximal2). shift of the bfp from maximal is small3). still large deviation is allowed: (0.5 - sin223)/sin 23 ~ 40% 2

maximal mixing

Gonzalez-Garcia, Maltoni, A.S.

sin2223 > 0.93, 90% C.L.

QLC2

R. Zukanovic-Funcal, A.S.in preparation

R. Zukanovic-Funcal, A.S.in preparation

Smallness of massLarge mixing

Possibility that theirproperties are relatedto very high scale physics

Can propagate in extra dimensions

Manifestations ofnon- QFT features?

Violate fundamental symmetries, Lorentz inv.CPT, Pauli principle?