cccc sdc

sdu

cossinsincos

013c

baa0

III

II

I

Relations between masses and mixing angles

c

uu m

marctans

dd m

marctan

baa0

c

ui

s

dc m

memm

21.0s

d

mm 07.0

c

u

mm

o13

c

u

mm

090sd mm /

III

II

I

10000

00

00

10000

dd

ddi

uu

uu

cssc

cssc

ecssc

V

du

tan /d d sm m

tan /u u cm m

0 0

0

AA C B

B D

0 0

0

AA C B

B D

tan /d d sm m

tan /u u cm m

0 0

0

AA C B

B D

od 4.00.13

oo

od

Exp 6.27.11:

4.00.13

oou 7.00.5

oo

oou

Exp 1.14.5:

7.00.5

tbtstu

cbcscu

ubusud

VVVVVVVVV

V

c

u

mm

090sd mm /

090

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

254.0

6.0221 108 eVm

235.0

6.0322 104.2 eVm

What type of mass term? How big or small are masses?How big are the mixing angles?Relations among angles and

masses?

Neutrino Masses: Mass terms for charged leptons and neutrinos are not parallel

Neutrino Mixing

321

321

321

VVVVVVVVV

Veee

332211 eeee VVV

332211 VVV

332211 VVV

Neutrino mixing

V=UxP

1000000

i

i

ee

P

10000

00

00

10000

cssc

cssc

ecssc

U

i

ll

ll

sun at anglereactorl

25221 108 eVm

osun

o 3.367.31

oat

o 5238

23322 104.2 eVm

10000

00

00

10000

cssc

cssc

ecssc

U

i

ll

ll

2

1tanmm

0695.0tan

mme

l

Observation:

oo 45__33

12.004.021 42.0/

:

mm

25221 108 eVm

23322 104.2 eVm

:

42.0/ 21 mm

232

221

423

221

422

221

421

2coscos

2coscos

2cossin

mmm

mm

mm

m(1) = 0.0036 … 0.0059 eV

m(2) = 0.0085 … 0.0140 eV

m(3) = 0.044 … 0.058 eV

12

3

e

Neutrino masses less

hierarchical than the masses of the charged leptons

DBBA

AM

00

00

1 2

o

mm

142

o

mm 263

22

o4097.02sin2

321

321

321

VVVVVVVVV

Veee

052.021

3 mmV e

e

Neutrinoless Double Beta Decay

0026.0)054.0()05.0(:.. em

3m 3eV

Conclude:

Fermion masses remain a mystery.Neutrino masses might be Dirac

masses or Majorana masses.

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

agreement with experiment.

0 0

0

AA C B

B D

0 0

0

AA C B

B D

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

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

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

Atmospheric angle about 40 degrees

Neutrinoless double beta decay: difficult