Neutrino oscillation physics...2 U D3 Q 3 Neutrino mixing •The mixing matrix is appearing in the...

Neutrino oscillation physics

Alberto Gago PUCP

CTEQ-FERMILAB School 2012 Lima, Perú - PUCP

Outline

• Introduction

• Neutrino oscillation in vacuum

• Neutrino oscillation in matter

• Review of neutrino oscillation data

• Conclusions

Historical introduction

Observed Spectrum – Continuos three body decay

(1930) W. Pauli propose a new particle for saving the incompatibility between the observed electron energy spectrum and the expected.

XeZAZA )1,(),(

Expected Spectrum-Monoenergetic two body decay

eZAZA )1,(),(

1,, ZAMZAMQEe

XeEZAMZAMQE 1,,


(1956)- F. Reines y C.Cowan detected for the first time a neutrino through the reaction

This search was called as poltergeist project.

First reactor neutrino experiment

(1962)-Lederman-Schwartz-Steinberger, discovered in Brookhaven the muon antineutrino through the reaction:

they did not observe:

Confirming that

e

First neutrino experiment that used


(2000) In Fermilab the DONUT collaboration observed for the first time events of . They detected four events.

X

SD

Neutrinos

8

2

2

cm

cm10

10

10

)(

)(41

33

e

ee

e

Z

0082.09840.2 N

Active neutrinos

Neutrinos -SM

Therefore in the Standard Model (SM) we have:

Neutrinos are fermions and neutrals

LLL

e

e

Left-handed doublet

Only neutrinos of two kinds have been seen in nature: left-handed –neutrino right-handed- antineutrinos

Neutrino –SM

In the SM the neutrinos are consider massless (ad-hoc assumption) ….BUT we know that they have non-zero masses because they can change flavour or oscillate …

Neutrino oscillations -history

).( to analogy

in of idea the time first the

for suggested Pontecorvo Bruno

0000 KKKK

)1957(

(1962) Maki, Nakagawa and Sakata proposed the mixing

Neutrino oscillation in vacuum

.

.

.

.

2

1

e

Flavour conversion

k

k

kU *

,,e seigenstate flavour 3,2,1k seigenstate mass

ikki

The flavour (weak) eigenstates are coherent superposition of the mass eigenstates

Non-diagonal

332211 UUU

Neutrino mixing

• The mixing matrix is appearing in the charged current interaction of the SM :

,,

*

,,

2

2

e k

kLLkLkLk

CC

SM

e k

LLLL

CC

SM

WlUWlUg

L

WlWlg

L

analog to the quark mixing case

LkL

LkL

l

l

LkL

LkL

l

l

CKM

U

L

D

LL

l

LPMNS UVVVVU

PMNS=Pontecorvo-Maki-Nakagawa-Sakata

D= down quarks U= up quarks

Neutrino oscillation in vacuum the scheme

Detector

Propagation

k

Source

Flavour states Flavour states

v v

Mass eigenstates

*

kU

kU

Coherent superposition of the mass eigenstates The flavour conversion happens

at long distances!


• Starting point :

• Evolving the mass eigenstates in time and position (Plane wave approximation):

seigenstateflavourseigenstatemass α,βi,k

amplitudeyprobabilitxtA

iii

i

i

e

i

iiii

UxpitEiU

xpitEiUxt

),(

*

,,

*

exp

exp),(

0,0,,

xtUe

kk

i

iiU

*

2

*22

exp),(),( iii

i

i UxpitEiUxtxtAP


Analyzing:

onscontributi mass neglecting energy neutrino the is

Lt assumption

E

LE

m

pE

LpELpELptE i

ii

iiiiii

2)(

222

222

jiij mmm

L

E

miUUUUP

ij

ji

jjii2

exp

2

,

**

We are using L instead of x


LE

miUUUUUUELP

ij

ji

jjiii

i

i2

expRe2),(

2

**22

energy neutrino

distance detector - source

:

:

E

L

constant term oscillating term

2

24

22 ij

oscosc

ij

m

ELL

E

m

Oscillation wavelength

1,,,,

ee

PP

Valid if there is no sterile neutrinos


LE

miL

E

mUUUUUUUUELP

UUUUUU

LE

miUUUUUUELP

ijij

ji

jjii

ji

jjii

ji

jjiii

i

i

ij

ji

jjiii

i

i

2sin

2cosRe2Re2),(

Re2

2expRe2),(

22

****

**22

2

**22

us ing

LE

mUUUU

LE

mUUUUELP

ij

ji

jjii

ij

ji

jjii

2sinIm2

4sinRe4,

2

**

2

2**


• For the antineutrino case:

*UUUi

ii

LE

mUUUU

LE

mUUUUELP

ij

ji

jjii

ij

ji

jjii

2sinIm2

4sinRe4,

2

**

2

2**

This is the only difference…we will return to this later

Two neutrino oscillation

cossin

sincos)(U

21

21

cossin

sincos

e

LE

m

LE

m

LE

miL

E

m

LE

mi

LE

miUUUU

ULE

miUP

ee

ii

i

i

ee

4sin2sin

2cos12sin

2

1

2sin

2cos1sincos

2expcossinsincos

2exp

2exp

2

2122

2

212

22

21

2

2122

22

21

2

2

212

*

21

*

1

222

*

,


I

In the two neutrino framework we have:

yprobabi l i t trans ition

yprobabi l i t survival

phase noscillatioamplitude noscillatio

LE

mELP

LE

mELP

e

ee

4sin2sin,

4sin2sin1,

2

2122

2

2122

ELPELP

ELPELP

,,

,,


• Introducing units to L and E

)(

)(27.1)(

)(27.1

4

2

21

2

21

2

21 kmGeV

eVorm

MeV

eV 22

LE

mL

E

mL

E

m

kmeV

GeVm

eV

MeV22 )(

)(47.2

)(

)(47.2

42

21

2

21

2

21 m

E

m

E

m

ELosc

Oscillation regimes

oscL

LL

E

mL

E

m

44

2

21

2

21

• Oscillation starting

• Ideal case

• Fast oscillations

114

2

21

oscL

LL

E

m

2

1~)1(

4

2

21

oscL

LL

E

m

114

2

21

oscL

LL

E

m

04

00

02

2

21

1212

2

21

E

Lm

mmmmm or

sen2

No oscillation

Oscillation regimes

Short distance

Ideal distance

Long distance

2sin2

1 2

'2

'22

212

2

2'

2

2'

27.1

2

dEe

dEeE

Lm

P

E

E

e EE

EE

s in

s in2

Fixed E

Sensitivity plot

c

a

b

Not sensitivity

sensitivity

L

LL

L

L

PL

Em

PL

Em

E

LmP

E

LmP

P

27.1log2log2

1)(log

27.12

127.12

27.12

22

1

22

21

2

21

2

2

21

2

21

sin

sinsinc)

sinsinb)

sina)

2

22

2

LP lower probability limit for having a positive signal in a detector

ysensitivit maximum b @

2~27.1 2

E

Lm

Exclusion plot

allowed

excluded

LPupper limit of the oscillation probability PPL

⟨L/E⟩ = 18km/GeV

Positive signal

⟨L/E⟩ = 1km/GeV

0.05 < P < 0.15 0.001< P < 0.005

The mixing matrix

1

)1(2 22

2

*

UUUU

NNNNN

NU

i

iiUU

NN

condition the satisfies and unitary is matrix ng mixi the

parameters 2 has matrix complex general A

from

rows different between ityorthogonalrow eachof length unitarity

phases complex

angles rotation has matrix mixing unitary

phases complexangles rotation

2

)1(2

)1(

2

)1(

2

)1(2

NN

NN

NNNNN

phases complex )(

angles rotation )(

:scheme 3 In

62

133

32

133

-

The mixing matrix

L

L

L

i

i

i

TYPECKM

i

i

LLL

LkLk

e

e

e

eUe

e

lU

e

100

00

00

100

00

00

2

1

321

* : within tionrepresenta 33 matrix unitary a written

rephas ing by absorbed be can and

nosantineutri neutrinos neutrinos Dirac are IF *

invariant are lagrangian theof terms Other

and

: rephas ing by absorbed be can and , phases The *

phase) complex 1 & angles rotation

21

)(

,

'

3(

s

elel

UU

L

i

LL

i

elL

e

CKM

l

The mixing matrix

• If neutrinos are equal than antineutrinos

condition Majorana

noantineutrineutrino

TC C

2

12

phases physical

2

2

angles mixing

Neutrinos Majorana Neutrinos Dirac

:scheme 3 the In

3)1()2)(1(

3)1(

3)1(

NNNN

NNNN

Charge conjugation operator

fixed are phases neutrino the since absorbed be not can and 21

CiCiii vevevevevv 2if instance, For zero be to has

3-mass scheme

2

1

2

2

m

m

2

3m2

1

2

2

m

m

23m

Normal Hierarchy Inverted Hierarchy

22 eVeV 52

21

32

31 1010 mm

2m

M

D

D

i

i

U

i

i

M

e

e

cs

sc

ces

esc

cs

scU

100

00

00

100

0

0

0

010

0

0

0

001

2

1

1212

1212

1313

1313

2323

2323

The mixing matrix in 3 scheme

03

withii

i

DMDM eUDUU

Dirac CP phase

Majorana phases

…Tomorrow we will see the current measurements of these angles done by the experiments

:Notation ijijijij sc sincos

Majorana phases and neutrino oscillations

2

*

2

*

2

*2

),(

D

i

xpitEi

i

D

i

iD

i

xpitEi

i

iD

i

M

i

xpitEi

i

M

i

UeU

eUeeUUeUxtA

ii

i

ii

i

ii

Only the dirac phase is observable in neutrino oscillation

CP violation

,, 3,2,1

*1

,, 3,2,1

*

2

2

e k

LkLkkLLkCP

CC

lCP

e k

kLLkLkLk

CC

l

WlUWlUg

ULU

WlUWlUg

L

UUCP * needs

CP violation

• What does CP mean in neutrino oscillation ?:

PPCP

RR

P

LL

C

LL

violated)is

andif

CPPP

UU ii

(

*

CP violation

LE

mUUUUPPP

ij

ji

jjii2

sinIm4,

2

**

Jarlskog invariant CPjjii JUUUU **Im

Independent of the mixing matrix parameterization =rephasing invariant

ji

ji

,,)(

,,)(

andin nspermutatio anticycl ic

andin nspermutatio cycl ic

CP violation

sen sencossencossencos 2323121213132CPJ

CP violation phase All the mixing angles have to be

non-zero to have CP violated.

013

CPS

CP

CP

LE

mL

E

mL

E

mJ

LE

mL

E

mL

E

mJP

44416

2224

2

31

2

23

2

12

2

31

2

23

2

12),(

s ins ins in

s ins ins in

In particular we just found out that

CP violation

• It is interesting to note that :

• Checking for CPT

conserved is CPT

PPCPT

LL

T

LL

P

RR

C

RR

,, ODDTCP PPPPPP

ODDTCP PP

Useful approximations

• In the three neutrino framework we have:

• Case 1 : sensitive to the large scale :

• Case 2: sensitive to the small scale

2

2

31

2

212

21

2

31

2

32 10~

m

mmmm with

neglect we 2

21

22

21

2

32 10)1(~4

)1(~4

mLE

mL

E

m

out averaged are related terms

22

32

2

21 10)1(1

~4

)1(~4

LE

mL

E

m

2

21

2

31

2

32 mmm


• Case 1: sensitive to the large scale

LE

mUU

LE

mUUUUUU

LE

mUUUUL

E

mUUUUP

Smmm

eff

CP

4sin4

4sinRe4

4sinRe

4sinRe4

)0(0

2

312

2sin

2

3

2

3

2

312*

22

*

113

*

3

2

312*

223

*

3

2

312*

113

*

3

2

21

2

32

2

31

2

and

Similar structure to the two generation formula


• Case 1: sensitive to the large scale

LE

mcscsP

LE

msP

LE

mcP

LE

mP

u

e

ee

441

42

42

421

2

312

13

2

23

2

13

2

23

2

3113

2

23

2

3123

4

13

2

3113

2

22

22

22

sen-1

sensen

sensen

sensen

Explicit formulas



LE

m

UU

UUUU

LE

mUUUU

UUUULE

mUUP

eff

4sin411

4sin4121

2

1

2

14

4sin41

2

212

2sin

22

1

2

2

2

1

2

222

3

4

3

2

2122

1

2

2

2

3

2

3

2

1

2

3

2

2

2

3

2

2122

1

2

2

2

Similar structure for the two generation formula

averaged out

2

1

4sin

2

32312

L

E

m



24

13

4

13

3

2

2112

4

13

4

13

3

421

eeee

ee

PcsP

LE

mcsP

22 sensen

Approximation

Full Probability

scale2

31m

scale2

21m


• We can get a better approximation for the oscillation formula when we expand this in small parameters up to second order. We are in the case of

sensitivity of the large scale.

• These small parameters are :

23121313

2

13

2

12

2

13

2

122

12

2

2

132

13

22

23

2sin2sin2sin~

4sin

44cos

~

4sin2sin

4sin2sin

cJ

LE

m

E

mL

E

mJ

LE

m

LE

msP

ee

c

223

E

Lm

m

m

2,

2

21

2

31

2

2113

,

leading term –P2 approximation

Now it is appearing and 2

12m

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