1FK7003

Lecture 3 – neutrino oscillations and hadrons

● Review of leptons● Neutrino oscillations● Introduction to hadrons

2FK7003

The lepton family

Lepton (antilepton)

Charge (e)

Mass (GeV)

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

3FK7003

- -

Lepton interactions

1; 1; 0

1; 1; 0

ElectromagneticIn:

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 currentIn:

Out: e

e

L L L

L L L

2; 0; 0

2; 0; 0

Weak; Neutral currentIn:

Out: e

e

L L L

L L L

1; 1; 0

1; 1; 0

Weak: charged currentIn:

Out: e

e

L L L

L L L

4FK7003

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

5FK7003

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 theconversion 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

6FK7003

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 14

sin 2 sin 22 2cos 4sin4 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 ij 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

7FK7003

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 pp

m mE p pp p

m m m mE E

p E

m mP

22

2 22 2

0 2 20

4

4sin 2 sin sin 2 sin4

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 LE 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

8FK7003

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

e utr i

no in

tera

c tio

n le

n gth

i In

wa t

er/k

m

9FK7003

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 Fl

ux/e

nerg

y (c

m-2s-1

MeV

-1)

Neutrino energy (eV)

Relevant for this lecture

10FK7003

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 inphotomultipliers. Energy and dir

e p e X p X p X

ection measured.Expt: 50,000 tons of water, 1200 photomultipliers

11FK7003

Super-K: oscillations2

1.3

flux of muon-neutrinosExpect flux of electron-neutrinos

Observed Additional observationsFlux 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(

(Dat

a/Pr

edic

tion

for o

scill

atio

ns)

oscillations Earth

12FK7003

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

13FK7003

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 Fl

ux/e

nerg

y (c

m-2s-1

MeV

-1)

Neutrino energy (eV)

Relevant for this lecture

14FK7003

Neutrinos in the sun – not for the exam

● Fusion process leading to

● ke measured flux on earth can be predicted

Solar neutrino energies

15FK7003

Solar neutrino problem

37 37

Ray Davis et al. 1968Vat of 600 tons chlorine (cleaning fluid) inHomestead 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

16FK7003

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 neutrinosfrom 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 neutrinoflavour (question) and consistent

e

, with oscillations, e e e

17FK7003

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

18FK7003

Generalise for 3 neutrinos

1

2

3

12 13 12 13 13

"

cos cos sin cos sin

MNS" matrix (3.13)

e

ie

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

19FK7003

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.

20FK7003

Question

.

Using Feynman diagrams explain why the reactions

and are suppressed

with respect to e e

e e e e

e e

21FK7003

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

22FK7003

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

ELm

eVL m E GeVc

EL cm

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

E Lm c

E M L T m M

M L TE M L T M L T c L Tm M

EM L T M L T L T Lc m

2219 19

10 10 2 2 22 22 28 8

34 10

2 41938 228

1.6 10 1.6 101 1.6 10 ( ) ( ) 1.6 10 ( )3 10 3 10

4 1.05 10 1.6 104

1.6 103 103 10

.)

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

EcELm c

m

2 2 2 2

787 0.7871.27

m= km= kmE E Em m m

23FK7003

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.