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Lecture 3 – neutrino oscillations and hadrons

● Review of leptons● Neutrino oscillations● Introduction to hadrons

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The lepton family

Lepton

(antilepton)Charge

(e)Mass (GeV)

e- (e+) -1 (+1) 0.0005

e,(e) 0 0(+) -1 (+1) 0.105

0 0(+) -1 (+1) 1.8

0 0

Spin 1/2

6 leptons + antileptons. Divided into 3 families/flavours/multipletes: electron,muon tau

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- -

Lepton interactions

1; 1; 0

1; 1; 0

Electromagnetic

In:

Out: e

e

L L L

L L L

1; 0; 0

1; 0; 0

In:

Out: e

e

L L L

L L L

e- e-

e-

W-

+

Charged leptons interact via the em and weak forces.Neutrinos only interact via the weak force.

Lepton number is always conserved at a vertex and in the whole process.

As for all forces: Charge conservation and energy-momentum conservation for incoming and outgoing particles.

Charge is conserved at a vertex though energy can appear to beviolated when dealing with ”internal lines” (lecture 1).

2; 0; 0

2; 0; 0

Weak: neutral current

In:

Out: e

e

L L L

L L L

2; 0; 0

2; 0; 0

Weak; Neutral current

In:

Out: e

e

L L L

L L L

1; 1; 0

1; 1; 0

Weak: charged current

In:

Out: e

e

L L L

L L L

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Neutrino masses3 3Use decays eg

Mass and momenta of all particles except neutrino known and

used to infer unobserved neutrino mass.

eH He e

Lepton Mass (MeV)

e < 0.0003

<0.2

<18

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Neutrino oscillations

1 2 3 1 2 3 1 2 3

1 2

, ,

,

Hypothesis: are eigenstates of flavour (lepton number) but not "mass eigenstates".

+ ; ; (3.1)

are mass eigenstates with definite energy:

e

e

i

a b c d e f g h k

2 2

0

, , ....

(3.2)

(1.1) ; Implicit notation: ; are co-efficients which must be measured.

Remarks:

(1) It is nonsense to ask "what is the mass of, eg ?"

Any measurement wou

iiE ti

i i i

e

t e

E p m a b c

1 2 3

1 2 3

.

, ,

ld return a value of , or .

(2) The weak force produces a flavour eigenstate, eg

That particle then propagate through space as a mixed state of

, each with different energies. If

m m m

m

1 2 3 , and have

different masses, this allows interference effects and the

conversion of neutrino species, eg (next slides).

can then be measured in a dector via a weak interaction.

Lepton nu

m m

mber is violated. The oscillation length determines

whether or not this would be observed in a laboratory experiment.

Strategy:

(1) Neutrino oscillation theoretical framework

(2) How neutrinos are produced and measured.

(3) Interpretation and final results.

Whole process violates: LL

Conserves L

Conserves L

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Neutrino oscillation formalism

sin cos cos sin

sin cos cos sin

,

Simplicity - consider two neutrino eigenstates, and eg and

and (3.3) Obs: .

and (3.4)

- mass eigens

e

i j

i j i j

i j

0 0

.

0 1 ; 0 0 0 sin 0 cos

sin cos

tates with definite energy: ; (3.2)

is a mixing angle - must be measured.

Particle starts off as a

;

ji

i

iE tiE ti i j j

i j

iE ti j

t e t e

t e t e

22 2

2 22

cos sin sin cos

sin 2sin cos 1 1

4

sin 2 sin 22 2cos 4sin

4 4 2

=

Probabili

j

ji

j i j ij ji i

iE t

iE tiE ti j

i E E t i E E tiE t iE tiE t iE t

j i

j i

t t t e e

t e e e e e e

E E tE E t

2

sin 2 sin2

ty of : (3.5)

Neutrino oscillations!!

j iE E tP

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22 2 2 2

2

2 2

2

2 2 2 2

2

1

12 2

2 2

sin 2 sin

(highly relativistic particle)

For a flavour neutrino consisting of two neutrino mass states and i j

j i j ij i

j

mE p m p

p

m mE p p

p p

m m m mE E

p E

m mP

22

2 22 2

0 2 20

4

4sin 2 sin sin 2 sin

4

Alternatively if neutrino travels a distance (nu)

= (3.6) Oscillation length

i

j iij

j i

tE

L ct t

m m L EP L L

E L m m

2 2 2 0

0

(3.7)

For oscillations - a mass difference and

a non-zero mixing angle are needed. j im m m

Real(i)

probability prob. prob.

Real(i)

Real()

L

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Neutrino Interactions in Matter

Probability of interaction ~ 10-5 / km water at 100 TeV energy

100 billion neutrinos pass through your thumbnail each second but only 1-2 will interact in your body during your lifetime.

1

Beam of neutrinos entering and propagating through matter:

Probability density function for distances between successive collisions:

(3.8)

interaction length mean distance between intera

x

dx e dx

ctions (also called mean free path)

I0x)

interaction

x

0

103

106

109

1 103 106

E (TeV)N

eutr

ino

inte

ract

ion

leng

thi I

n w

ate

r/km

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relics from Big Bang

natural radioactive decay in Earthnuclear reactors

explosions of supernova

remnants of supernova

nuclear reactions in Sun

interactions of cosmic rays in atmosphere

Active Galactic Nuclei

Neutrinos arriving at Earth – not for the exam F

lux/

ener

gy

(cm

-2s-1

MeV

-1)

Neutrino energy (eV)

Relevant for this lecture

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Atmospheric neutrinos and Super Kamiokande

Boat

,

Cosmic rays interact in atmosphere

to produce pions , (this lecture)

(2 muon neutrinos, 1 electron neutrino)

Neutrinos enter Super-K detector.

Underground in Japanese Alps.

ee

, ,

Neutrinos measured in Super-Ks water Cerenkov detector.

Eg (X anything)

electrons and muons distinguished by Cerenkov light - measured in

photomultipliers. Energy and dir

e p e X p X p X

ection measured.

Expt: 50,000 tons of water, 1200 photomultipliers

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Super-K: oscillations

2

1.3

flux of muon-neutrinosExpect

flux of electron-neutrinosObserved

Additional observations

Flux of electron-neutrinos from below=Flux of electron-neutrinos from above

electron neutrinos haven't

R

R

oscillated.

Flux of muon-neutrinos from below< Flux of muon-neutrinos from above

muon neutrinos have oscillated to tau neutrinos:

Super-K

15 km

13000 km

From above

From below

no oscillations

P(

(D

ata/

Pre

dic

tio

n f

or

osc

illat

ion

s)

oscillationsEarth

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Atmospheric neutrino results

2 3 2

sin cos cos sin

45 3 10

Evidence for muon neutrino turning into a tau neutrinos.

Super-K data + constraints from other experiments:

and (3.9)

eV (3.

atm i atm j atm i atm j

oatm atmm

10)

Note: approximate with two flavour and two mass state neutrinos.

E=1 GeV

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relics from Big Bang

natural radioactive decay in Earthnuclear reactors

explosions of supernova

remnants of supernova

nuclear reactions in Sun

interactions of cosmic rays in atmosphere

Active Galactic Nuclei

Neutrinos arriving at Earth – not for the exam F

lux/

ener

gy

(cm

-2s-1

MeV

-1)

Neutrino energy (eV)

Relevant for this lecture

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Neutrinos in the sun – not for the exam

● Fusion process leading to

● ke measured flux on earth can be predicted

Solar neutrino energies

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Solar neutrino problem

37 37

Ray Davis et al. 1968

Vat of 600 tons chlorine (cleaning fluid) in

Homestead gold mine in South Dakota.

Depth reduces background from cosmic rays

Looked for Cl Ar

Collected 33 argon atoms.

(flu

e e

shed out with He and their radioactive decays counted)

Prediction of standard solar model 100.

Solar neutrino problem: 1/3 of expected rate of solar neutrinos

Hypothesis: neutrino oscillations respons

, .

ible: ;

Unable to check since experiment insensitive e e

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SNO

2 ( )

( ) ( )

( ) , ,

Sudbury Neutrino Observatory (SNO)

Ontario, underground water Cerenkov device.

Heavy water

Sensitive to:

( )

Reaction (a) ins

X X e

X X X e

D O D deuteron

a D p n b D e p p

c e e

ensitive to neutrino flavour

(lepton universality, lecture 2)

- reaction rate agrees with predictions - nothing

wrong with prediction of total number of neutrinos

from the sun!

Reaction (b) measured at 1

, .

/3 rate of theoretical prediction

(solar neutrino problem again).

Hypothesis: oscillated to

Check hypothesis with reaction (c) . Rate is sensitive to neutrino

flavour (question) and consistent

e

, with oscillations, e e e

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Results

sin cos cos sin

Electron neutrino oscillates into muon and/or tau neutrino.

Solar neutrino + other experiments:

and (3.11)

arbitary combination of and , treat as

e sol i sol j x sol i sol j

X

2 235 0.0001

one neutrino

(3.12)osol solm eV

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Generalise for 3 neutrinos

1

2

3

12 13 12 13 13

"

cos cos sin cos sin

MNS" matrix (3.13)

e

i

e

U U

e

12 23 12 23 13 12 23 12 23 13 23 13

12 23 12 23 13 12 23 12 23 13 23 13

sin cos cos sin sin cos cos - sin sin sin sin cos

sin sin - cos cos sin - cos sin sin cos sin cos cos

i i

i i

e e

e e

1

2

3

12 23 13

12 23 13

(

, ,

35 45 10

Maki Nakagawa Sakata NMS) matrix

Mixing defined by three angles and one phase factor .

Global fits to many experiments:

o o osol atm

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0.003

0.0001

Mass2 (eV2)

2 2 2 2 2 2 2 2 2 2 2 231 3 1 32 3 2 21 2 1 31 32 21

2 2 2 2 2 232 210.003 0.0001

Three neutrinos mass states three mass splittings (not independent)

(3.14)

eV eV (3.atm sol

m m m m m m m m m m m m

m m m m

0.0005 , 0.1 2

15)

Two possibilities:

(a)normal (similar mass hierarchy to charged lepton:

GeV GeV GeV

(b) inverted.em m m

Flavour composition of mass state.

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Question

.

Using Feynman diagrams explain why the reactions

and are suppressed

with respect to e e

e e e e

e e

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Question

With the Minos experiment a beam of muon-neutrinos of energy 1-5 GeV

(assume a flat, top-hat distribution of energy) is fired 750km from the

Fermilab laboratory in Chicago to the SOUDAN mine in Minnes

2 23 2

2 2 2 23 2

- .

sin 2 0.9 - 0.003 .

ota.

The purpose of the experiment is to measure

Sketch the expected energy spectrum of muon neutrinos at the SOUDAN

mine if and eV

m m

m m

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Question

0 2

22

0 2

0 2

1.27

.

4, .

Show that the oscillation length may be written where

is expressed in km, in and in

nu: : restore units with factors

Working in MKS-SI we expect an

EL

m

eVL m E GeV

c

EL c

m

2

2 2 22

2 21 2 2 2 1 1

22

31 2 2 2 1 1

3 2

4

4

expression: with dimension

;

=

Now decide on the units: start with SI (m,s,J,kg.

a b

EL

m c

E M L T m M

M L TEM L T M L T c L T

m M

EM L T M L T L T L

c m

2219 19

10 10 2 2 22 22 28 8

34 10

2 419

38 228

1.6 10 1.6 101 1.6 10 ( ) ( ) 1.6 10 ( )

3 10 3 10

4 1.05 10 1.6 104

1.6 103 10

3 10

.)

eV eV GeV ; 1 kg J GeV ; J E E m kg m

c c

EcEL

m cm

2 2 2 2

787 0.787

1.27m= km= km

E E E

m m m

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Summary of neutrino oscillations

● Neutrinos produced in the weak force are flavour eigenstates but not mass eigenstates.

● Neutrino oscillations between flavours occur as a consequence of non-zero mixing angles and a non-zero difference between the mass states

● Experiments on atmospheric and solar neutrinos demonstrate neutrino oscillations.