l6 neutrino

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12004, Torino Aram Kotzinian

Neutrino Scattering

Neutrino interactions

Neutrino-electron scattering

Neutrino-nucleon quasi-elastic scattering

Neutrino-nucleon deep inelastic scattering

Variables

Charged current

Quark content of nucleons

Sum rules

Neutral current




NeutrinoNeutrino--electron scatteringelectron scattering

Tree level Feynman diagrams:

0 Z

eR eR

e

e

W

eR

eR

e

e

p ee ee R R

Eff ective Hamiltonian:

? A? A _ ae g g eG

AV ee F ))1(1()1(2

55 K K R K K R Q Q!

(through a Fierz transf ormation)

? A? A ? A? A _ ae g g eeeG

H AV eeee F

eff )()1()1()1(

2

5555 K K R K K R R K K K K R Q Q

Q Q!




Only charged current:

W

QR

eR e

Q

p QR R Q ee

)()(2)()( 2 L ABin E me p p s e Q Q R R !!

22)()( QR Q p pqt !!

)(

)(

)()(

)()(

)()()( LABin

pe p

p pe p y

Q

Q

Q

Q

R

QR

R

QR

)()(2)( 2

22

22

LABin

q

s

dy

ed e

W

W CC

Q Q R

T T

R W }

Inelasticity variable (0<y<1)

243

2

10104.0)( cm

MeV

E sGe F

¹¹ º

¸©©ª

¨v!!

T R QTotal cross-section:

(cross-section proportional to energy!)




Only neutral current:

p ee Q Q R R )()(

¼¼½

»

¬¬-

«¹

º

¸©ª

¨

24

2

2

22

22

)1(sinsin2

1)( y

q

s

dy

ed W W

NC UU

T

R W Q

0 Z

QR

)(

e

QR

)(

e

¼¼½

»

¬¬-

« ¹ º ¸©

ª¨

W W

Z

Z F NC y

q

s

dy

ed UU

T R W Q 42

22

22

22

sin)1(sin2

1)(

ee g ee g e g g e R L AV )1()1()( 555 K K K K K K Q Q Q

W AV Lg g g U2sin

2

1)(

2

1!!

W AV g g g U2sin)(2

1!!




Only neutral current (total cross-section):

2434

2

2

2

101015.0sin

31sin

21)( cm

M eV

E

se W W F

NC ¹¹ º

¸©©ª

¨!

¼¼½

»

¬¬-

«

¹ º ¸©ª̈ ! R Q UUT R W

2434

2

2

2

101014.0sinsin

2

1

3

1)( cm

M eV

E se W W

F

NC ¹¹ º

¸©©ª

¨!

¼¼½

»

¬¬-

«¹

º

¸©ª

¨ ! R Q UU

T R W

Can obtain value of sin2 UW

f rom neutrino electron

scattering (CHARM II):

0059.00058.02324.0sin 2 ss!W U

p ee Q Q R R )()(

)1(22 ym E ee !5




Back to (charged and neutral currents)

Then:

p ee ee R R

W W AV L g g g UU

22

sin2

1

1sin2

1

)11(2

1

!!!W AV g g g U2sin))1(1(

2

1!!

¼¼½

»

¬¬-

«¹

º

¸©ª

¨!

24

2

2

2

1sinsin2

1)( y

sG

dy

ed

W W

F e UUT

R

This cross-section is a consequence of the inter f erence of the

charged and neutral current diagrams.

2434

2

2

2

10109.0sin

3

1sin

2

1)( cm

M eV

E se W W

F

e ¹¹ º

¸©©ª

¨!

¼¼½

»

¬¬-

«¹

º

¸©ª

¨ ! R UUT

R W




Neutrino pair production:

Then:

eeee R R p

¼¼½

»

¬¬-

«¹

º

¸©ª

¨!p

4

1sin2

2

1

12)(

2

2

2

W F

ee

see U

T R R W

Contribution f rom both W and Z graphs.

W

e

e

eR

eR Z

e

e

eR

eR

Only neutral current contribution to: Q Q R R p

ee

¼¼½

»

¬¬-

«¹

º ¸

©ª¨ !p

4

1sin2

2

1

12)(

2

2

2

W F sG

ee UT

R R Q Q




NeutrinoNeutrino--electron scatteringelectron scattering Summary neutrino electron scattering processes:

p ee Q Q R R

¼½

»¬-

« W W

F sGUU

T 422

2

sin3

41sin2

4

Process Total cross-section

p ee Q Q R R

p ee ee R R

ee R QR Q p

p ee ee R R

eeee R R p

Q Q R R p ee

¼½

»¬-

« W W

F sGUU

T 422

2

sin41sin23

1

4

¼

½

»¬

-

« W W F s

UU

T

422

2

sin

3

41sin2

4

¼½

»¬-

« W W

F sGUU

T 422

2

sin41sin23

1

4

T

sG F

2

¼½»¬

-« W W

F sG UUT 42

2

sin4sin221

12

¼½

»¬-

« W W

F sGUU

T 42

2

sin4sin22

1

12

)()(2 frame LABthein E m s e QR !




Neutrino-nucleon quasi-elastic scattering

Quasi-elastic neutrino-nucleon scattering reactions (small q2

):

W

QR

pn

Q

pn p QR

Q

!! n H p M eff ,,

QR Q

p p p

Q Q R R )()(

n p p QR Q

W

QR

n

Q

0 Z

QR

)(

p p

QR

)(

factor formv ctor q F V !)( 2

factor formvector axial q F A )( 2

)(975.0cos ang l eC abbiboC !U

? A ? AnqqG

V 5225 )()()1(2

K K R K K QU

Q Q Q




Ne tri - cle asi-elastic scatteri

F r l w e er y e tri s ( E

R <<m

N ):

.73.1)( s!! A A F

? A22

22

)0(3)0(cos

)()( AV C F

ee F F E G

pn !!T

UR W R W

R

2

2

42

101075.9 cm

M e ¹¹ º

¸©©ª

¨v} R

Form f actors introduced since proton, neutron not elementary.

Depend on vector and axial weak charges of the proton and neutron.

Two hypotheses:

- Conservation of Vector Current (CVC):

- Partial conservation of Axial Current (PC AC):

22

2

1/1

)(

)( qqV

V

!

1)0( !V F

22

2

65.1/1

)()(

q

F q F A

A

!




Inelastic neutrino-nucleon scattering

Since parity is not conserved in weak interactions, there are

more structure functions for weak processes, like neutrino

scattering, than for electromagnetic processes, like electron

scattering.Again the variables x = Q2 /2 M R and y = R / E can be used.

R Q nucleon p Q

X

Parton model is used to make predictions for deep inelasticneutrino-nucleon scattering.

Neutrino beams from pion and kaon decays, dominated by

muon neutrinos are used to study this process.

Q nucleon p Q

X




Weak structure functions

General form for the neutrino-nucleon deep inelastic scatteringcross-section, neglecting lepton masses and corrections of the

order of M / E :

d W R ,R

d xdy! F

2 M E

T 1 y F 2

R N y2 x F 1

R N m y y2

2

¨ª© ¸

º¹ x F 3

R N «

-¬»

½¼

The functions F 1

, F 2

and F 3

are the functions of Q2 and R . In the scaling

limit they are the functions of x only.




Scaling behaviour

Compilation of the

data on structure

functions in deepinelastic neutrino

scattering (1983)




Neutrino proton CC scattering:

= number of u-quarks in proton between x and x+dx

Some of the quarks are from sea:

For proton (uud):

X p p p dp )()( QR Q

? A´´ ! 00 2)()()(dx xu xudx xu

V

Scattering off quarks:

dx xu )(

)()()( x x xS V

! )()()( x x xS V

!

)()( x xS

! )()( xd xd S

!

? A´´ !!1

0

1

01)()()( dx xd xd dx xd

V

T

R W R W Q Q E mG

dy

qd

dy

qd q F CC CC

22)()(

!!

22

2)()( y

E mG

dy

qd

dy

qd q F CC CC !!

T

R W R W ¡ ¡

Ucos1

2

11 !

d

! E

E ywith




Scattering off proton:

? A ? A_ a22

)1()()()()(2)(

y xc xu x s xd x ME G

dxdy

pd F CC

R W Q

? A)()()()(2)(2 xc x s xu xd x x F p !R

? A ? A_ a)()()()()(2)(

2

2

x s xd y xc xu x E

d xd y

pd F !

T R Q

Structure f unctions:

Callan-Gross relationship:

? A)()()()()(3 xc x s xu xd x x x F p !R

)()(2 21 x F x x F !

? A)()()()(2)(2 x s xd xc xu x x F p !R

? A)()()()(2)( x s x xc x x x x F !¢

Neutron (isospin symmetry):

? A)()()()(2)(2 xc x s xd xu x x F n !R

)()()()(2)(3 xc x s xd xu x x xF n !R




Scattering off isoscalar target (equal number neutrons and protons):

_ a2

2

)()()(

)(

y xq xq x

ME G

x y

N d F CC

! T

R W Q

c sd uq | c sd uq |

? A)()()(2 xq xq x x F

N

!R

? A)()()()()(3 xc x s xq xq x x x F N !R

)()(2)()()(3 xc x xq xq x x xF N !R

_ a)())(()(

2

2

xq y xq x M E G

d xdy

d F CC

!T

R W Q

Total cross-section:

GeV cmQQG

E N F

CC /1067.0

3

1/)( 238

2v¼½

»¬-« R W Q

GeV cQQG

E N F

£ £

/1034.03

1/)( 238

2

v!¼½»

¬-« !

T W Q




R ise of mean q2 with energy

Mean q2 was found to be linear function in neutrino (antineutrino) energy.




Quark content of nucleons from CC cross-sections

Define:

Experimental values from y distribution of cross-sections yields:

If

.,)(1

0etcd uU ´

03.015.0 s! QQ

Q03.000.0 s!

QQ

S 01.016.0 s!

QQ

S Q

)(495.0

)(

)(

mea s r ed N

N

r CC

CC

!| R

R

19.03

13

}

! r

r

Q

Q

33.0}! QQQV 08.0}!! QQQ

S S

49.0)(1

2 }!´ QQd x x F ¤

Quarks and antiquarks carry 49% of proton momentum, valencequarks only 33% and sea quarks only 16%.




Some details

Note that for right-handed incident anti-neutrinos the term changes

sign. Note also that the term is orthogonal to the asymmetric hadronic

term that is proportional to since q = l ± l¶ and gives zerowhen dotted into

where both signs for the last term appear in the literature.




To obtain these expressions we have used




Finally we can put the pieces together to obtain the corresponding cross

sections(in the limit )

We recognize this to be similar to the EM result but with replacements

, an extra factor of 4 and the (new) term.




We now consider the scaling limit

Substituting in terms of the scaling variables

we find the result




For scattering on structureless fermions/antifermions (e.g ., point

particle quarks) we have

Thus measures the difference between quarks and antiquarks.




For elastic neutrino scattering from quark and antiquark we have:

and

Working the details out explicitly in terms of the parton momentum and

mass, we find

Thus for pointlike quarks we have




Gross-Llewellyn-Smith (2 names) sum rule

In terms of the parton distributions in the proton we have

Thus we have

and hence

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