G s

w

G s

wu

d

wM

HM

sun

atm

choozl

G

um

s

w

dm

tmbm

em

m

cm

u

d

sm

m

wM

HM2m

1m

3m

sun

atm

choozl

2

3

GeV2462

v

Fermion Masses: ArbitraryFermion Masses: Arbitrary

L

R

lbGeVMeVme30100002246000002105110 ...

GeVMEVm

GeVMeVm

_246...0072.0__1777

_24600043.0_66.105

Quark Masses:Quark Masses:

Observed:Observed:

m(c) : m(t) = m(u):m(c) m(c) : m(t) = m(u):m(c) 1/207 1/2071/207 1/207

m(s):m(b) = m(d):m(s)m(s):m(b) = m(d):m(s) 1/23 1/231/23 1/23

III

II

I

ba

a0

Relations between masses and Relations between masses and

mixing anglesmixing angles

c

uu m

marctan

s

dd m

marctan

c

ui

s

dc m

me

m

m

21.0s

d

m

m07.0

c

u

m

m

o13

c

u

m

m

090sd mm /

2sin / /c d s u cm m m m

0 a

a b

III

II

I

100

0

0

0

0

00

100

0

0

dd

ddi

uu

uu

cs

sc

cs

sc

e

cs

sc

V

tan /d d sm m tan /u u cm m

0 0

0

A

A C B

B D

0 0

0

A

A C B

B D

c

u

m

m

090sd mm /

Thus far neutrinos have been used as tools, Thus far neutrinos have been used as tools, likelike

electrons, to study nuclear matterelectrons, to study nuclear matter(quark distributions etc.)(quark distributions etc.)

New focus now: New focus now: What type of mass? What type of mass? How big or small are masses?How big or small are masses?How big are the mixing angles?How big are the mixing angles?Relations among angles and masses?Relations among angles and masses?

23322 105.2 eVm

25221 108 eVm

Ewigkeit ist lang, speziell gegen dem Ende zu.W.A.

Neutrino masses:Neutrino masses: A: masses about 1 eVA: masses about 1 eV m(1) = 0.94 eV m(1) = 0.94 eV m(2) = 0.95 eV m(2) = 0.95 eV m(3) = 1 eV (nearly m(3) = 1 eV (nearly degenerate)degenerate)

Neutrino masses:Neutrino masses: B: masses much smaller than 1 eV B: masses much smaller than 1 eV m(1) = 0 eV m(1) = 0 eV m(2) = 0.009 eV m(2) = 0.009 eV m(3) = 0.06 eV (strong m(3) = 0.06 eV (strong hierarchy)hierarchy)

Neutrino masses:Neutrino masses: C: neither hierarchy nor degeneracy C: neither hierarchy nor degeneracy m(1) = 0.10 eV m(1) = 0.10 eV m(2) = 0.14 eV m(2) = 0.14 eV m(3) = 0.29 eV m(3) = 0.29 eV

See-saw mechanism:See-saw mechanism:

Mm

mM

0

Mm

Mmm

2

21 /

Ex.: tau-neutrino mass at Ex.: tau-neutrino mass at about 1 eV:about 1 eV:

GeVM

eVMGeV9

2

1016.3

_1/)777,1(

SU(5) theory SU(5) theory with with

supersymmetry supersymmetry okok

Electroweak theory:Electroweak theory:

SU(2) x SU(2) x U(1)SU(2) x SU(2) x U(1)

U(1): (B-L)U(1): (B-L)

Electroweak theory:Electroweak theory:

SU(2) x SU(2) x U(1)SU(2) x SU(2) x U(1)

U(1): (B-L)U(1): (B-L)

O(10)

Neutrino Masses: Neutrino Masses:

Mass terms for Mass terms for charged leptons and charged leptons and neutrinos are not neutrinos are not parallel parallel

Neutrino MixingNeutrino Mixing

321 ......... e

321

321

321

VVV

VVV

VVV

Veee

332211 eeee VVV

332211 VVV

332211 VVV

Neutrino mixingNeutrino mixing

V=UxPV=UxP

100

00

00

i

i

e

e

P

100

0

0

0

0

00

100

0

0

cs

sc

cs

sc

e

cs

sc

U

i

ll

ll

sun at anglereactorl

oat

o

osun

o

2.538.36

2.3930

23322 105.2 eVm 252

21 108 eVm

70.060.038.0

70.060.038.0

05.054.084.0

V

34.6

45

4.6

0.821 0.568 0.08

0.047 0.40 0.03 0.58 0.70

0.047 0.40 0.03 0.58 0.70

osun

oatm

ochooz

i i

i i

V e e

e e

100

0

0

0

0

00

100

0

0

cs

sc

cs

sc

e

cs

sc

U

i

ll

ll

2

1tanm

m 0695.0tan

m

mel

Observation:Observation:

oo 45__33

42.0/

:

21 mm

Neutrino masses fixed, Neutrino masses fixed, sincesincethe mass differences are the mass differences are given by the experiment,given by the experiment,and the angles are fixed and the angles are fixed by the mass matrixby the mass matrix

19.0/ 32

3

2

1

mm

m

m

m

Neutrino masses less Neutrino masses less hierarchical than the hierarchical than the masses of the charged masses of the charged leptonsleptons

Double Beta DecayDouble Beta Decay

DB

BA

A

M

0

0

00

1

2 2 3

2

/ 14

/ 26

40

sin 0.97

o

o

o

m m

m m

1 2

The atm. angle is the sum of the two large The atm. angle is the sum of the two large angles (14 and 26 degrees).angles (14 and 26 degrees).

321

321

321

VVV

VVV

VVV

Veee

052.02

13

m

mV ee

Conclude:Conclude:

Fermion masses remain a mystery.Fermion masses remain a mystery.Neutrino masses might be Dirac masses or Neutrino masses might be Dirac masses or Majorana masses.Majorana masses.

Mixing angles for quarks are fixed by ratios of Mixing angles for quarks are fixed by ratios of quark masses. Very good agreement with quark masses. Very good agreement with experiment.experiment.

This works also well for leptons. One finds, This works also well for leptons. One finds, using the observed mass differences:using the observed mass differences:

m(1)/m(2)=0.42, m(2)/m(3)=0.19.m(1)/m(2)=0.42, m(2)/m(3)=0.19.

m(1): 0.0041 eV m(1): 0.0041 eV m(2): 0.0097 eVm(2): 0.0097 eV m(3): 0.051 eVm(3): 0.051 eV

Atmospheric angle about 40 degreesAtmospheric angle about 40 degreesNeutrinoless double beta decay: difficultNeutrinoless double beta decay: difficultReactor neutrinos: V (3e) ~ 0.052Reactor neutrinos: V (3e) ~ 0.052