March 7, 2005 Benasque

Neutrinos Theory Neutrinos Theory

Carlos Pena Garay

IAS, Princeton

~

PlanPlan

I. Neutrinos as Dirac Particles

II. Neutrinos from earth and heavens

III. Neutrinos as Majorana Particles

PlanPlan

I. Neutrinos as Dirac Particles

II. Making Neutrinos

III. Neutrinos as Majorana Particles

Nuclear Nuclear -decay-decay

Neutrinos are left-handedNeutrinos are left-handed

All neutrinos left handed => massless

Neutrinos must be masslessNeutrinos must be massless

If neutrinos are massive

neutrino right-handed => Contradiction!

Helicity = Chirality (massless)

Standard Model Standard Model

Exact Lepton number symmetry : terms never arise in perturbation theoryRenormalizability : All gauge currents satisfy anomaly cancellationWhat about global symmetries : L is broken non perturbatively through the anomaly, but B-L is conserved to all orders : terms never arise in the Standard Model

annihilate , create ,

create , annihilate ,c

c

Tc C

cm

cm

CPT : L => R

Anti-neutrinos are right-handedAnti-neutrinos are right-handed

-

3 neutrino families (flavors) 3 neutrino families (flavors)

Standard Model Standard Model

...

h.c. 2

-

cos2i 0

ii

iiW

ii

Wlg

Zg

L

Global symmetriesGlobal symmetries

0,...,;1,;1, eeeee LeLeL

LLLL e

0,...,;1,;1, eeLLL

The results of charged-current experiments (until recently) were consistent with the existence of three different neutrino species with the separate conservation of Le, L and L.

Massive Neutrinos : Dirac Massive Neutrinos : Dirac

(p,h) (p,-h)CPT

(p,h)(p,-h)CPT

Boost, if m ≠ 0

Boost, if m ≠ 0

Standard Model + right handed Standard Model + right handed

... h.c. m

h.c. cos2

-

cos2i

i

0

ii

iijiW

iiW

ii

WUlg

Zg

L

)1(2

1

)(2

vm

5,

LR

RLLR

P

vvvvy

Standard Model + right handed Standard Model + right handed

... h.c. m

h.c. cos2

-

cos2i

i

0

ii

iijiW

iiW

ii

WUlg

Zg

L

)1(2

1

)(2

vm

5,

i

LR

RLLRii

P

vvvvy

Standard Model + right handedStandard Model + right handed

Masses : Several distinctive features I am going to concentrate today is neutrino oscillations

Parameters : Dirac Parameters : Dirac

100

0

0

0

010

0

0

0

001

1212

1212

1313

1313

2323

2323

321

321

321

cs

sc

ciδes

iδesc

cs

sc

UUU

UUU

UUU

U

τττ

μμμ

eee

i

im

20

ij 002 mΔ 02 MΔ0m

Exercise

Neutrino Oscillations Neutrino Oscillations

tiEjkkji

j

ijeUUllA *

li

W X

lk

j

Uij Ujk

Neutrino Oscillations in Vacuum Neutrino Oscillations in Vacuum

tiEjkkji

j

ijeUUllA *

Due to different masses, different phase velocities

2

2

2

2

i 211v

i

i

i

i

i

i

E

m

p

m

p

E

2

2

i 2v

E

mi

2osc

4l

m

E

Osc. Length : distance to come back to the initial state

Production, propagation and detection as a unique process, where neutrinos are virtual particles propagating between production and detection : Neutrinos are described by propagators S (xP-xD)

Consider finite production and detection regions and finite energy and momentum resolution

pxx PD

1

||If neutrinos can be considered real (on shell) and production, propagation, and detection can be treated separately

Match initial and final conditionsWave packet formalism

)(v

|| )(

2

2v

||)(

2

22

2

PDg

PDPD

g

PDPD

ttxx

EttE

m

p

mEm

dm

dpE

dE

dpp

xxpttE

Correct calculation of the phaseCorrect calculation of the phase

Phases calculated in the same space-time point

Second summand is small by either one and/or the other term => standard result Oscillation effects disappear if neutrino masses are all equal

t

E

mi

jjii

tiEii

ji

i

eUUUU

eUUvP

2**

2*

2

||

100

0

0

0

010

0

0

0

001

1212

1212

1313

1313

2323

2323

321

321

321

cs

sc

ciδes

iδesc

cs

sc

UUU

UUU

UUU

U

τττ

μμμ

eee

i

Exercise : Write Pee, Pe

Exercise : Minimal number of Pto know all the others

One example : One example :

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2 3 4 5 6 7 8

"pee_kland.txt"

E

LmP f

4sin2sin

2222

Some real cases :Some real cases :

Neutrino Oscillations in matter Neutrino Oscillations in matter

After a plane wave pass through a slab, the phase is shifted : p (x+(n-1)R)

Net effect :

p

NRfe

edrNRfee

ipx

ipx

Rx

ipxRnxip

)0( 211

)0( 2 ))1((

2

)0( 21

p

fNn

Only the difference of potential is relevant

Net effect :

eFF NGeevv

GH 2)1( )1(

255int

i

0

vei

e

Nee

Nee

eF NG2

2lmatt

cosVesin 2

sin 2

m2

m2

m2

i =

ddt

e

e

Two Neutrino Oscillations in matterTwo Neutrino Oscillations in matter

sin22m =sin22

( cos2GFNeE/m2)2 + sin 22

Difference of the eigenvalues

H2 - H1 = m2

2E( cos2GFneE/m2)2 + sin 22

l / l0

sin2 2m

sin2 2 = 0.08 sin2 2 = 0.825

~ n E

Resonance width: nR = 2nR tan2

Resonance layer: n = nR + nR

ResonanceResonance

sin2 2m = 1

Flavor mixing is maximalLevel split is minimal

In resonance:

l = l0 cos 2

Vacuumoscillation length

Refractionlength~~

l/ l0

l/ l0

2m

2m

1m

e

1m

e

sin2 2 = 0.825 sin2 2 = 0.08

Large mixing

Small mixing

Level crossingLevel crossing

(t) = cosa 1m + sina2m e-i(t)

= (Re e

+, Im e+, e

+e - 1/2) = (Re e+, Im e

+, e+e - 1/2)

B = (sin 2m, 0, cos2m) 2 lm

elements of density matrix

lm = 2H oscillation length

= ( B x ) d dt

Evolution equation

Coincides with equation for the electron spin precession in the magnetic field

= 2t/ lm - phase of oscillations

P = e+e = Z + 1/2 = cos2Z/2P = e+e = Z + 1/2 = cos2Z/2

Graphical interpretationGraphical interpretation

A real case: solar neutrinosA real case: solar neutrinos

The Neutrino Matrix :SM + The Neutrino Matrix :SM + massmassALL oscillation neutrino data BUT LSND

)2.8( 3.9eV10

3.725

2m

Δ

)2.2( 6.3eV10

M6.1

23

2

Δ

3 ranges

81.056.074.044.053.020.0

82.058.073.042.052.019.0

20.000.061.047.088.079.0

||||||

||||||

||||||

||

321

321

321

τττ

μμμ

eee

i

UUU

UUU

UUU

U

(0.39) 60.0tan0.28 122

(1.0) 1.2tan0.51 232

(0.005) 041.0sin 132