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2004, Torino Aram Kotzinian 1 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

12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

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Page 1: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 1

Neutrino Scattering

Neutrino interactionsNeutrino-electron scattering

Neutrino-nucleon quasi-elastic scattering

Neutrino-nucleon deep inelastic scatteringVariables

Charged current

Quark content of nucleons

Sum rules

Neutral current

Page 2: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 2

Neutrino-electron scatteringNeutrino-electron scattering Tree level Feynman diagrams:

0Z

e e

ee

W

e

ee

e

ee ee

Effective Hamiltonian:

eggeG

AVeeF ))1(1()1(2

55

(through a Fierz transformation)

eggeeeG

H AVeeeeF

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

5555

Page 3: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 3

Only charged current:

W

ee

ee

)()(2)()( 2 LABinEmepps e

22 )()( ppqt

)(

)(

)()(

)()(

)()()(LABin

E

EE

pep

ppepy

)()(2)( 2

22

22

LABinEmG

mq

msG

dy

edeF

W

WFCC

Inelasticity variable (0<y<1)

2432

10104.0)( cm

MeV

EsGe F

CC

Total cross-section:

(cross-section proportional to energy!)

Page 4: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 4

Only neutral current:

ee )()(

24

22

22

22

)1(sinsin2

1)(y

mq

msG

dy

edWW

Z

ZFNC

0Z

)(

e

)(

e

WW

Z

ZFNC ymq

msG

dy

ed

42

22

22

22

sin)1(sin2

1)(

eegeegegge RLAV )1()1()( 555

WAVL ggg 2sin2

1)(

2

1

WAVR ggg 2sin)(2

1

Page 5: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 5

Only neutral current (total cross-section):

24342

22

101015.0sin

3

1sin

2

1)( cm

MeV

EsGe WW

FNC

24342

22

101014.0sinsin

2

1

3

1)( cm

MeV

EsGe WW

FNC

Can obtain value of sin2W from neutrino electron scattering (CHARM II):

0059.00058.02324.0sin 2 W

ee )()(

)1(22 ymE ee

Page 6: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 6

Back to (charged and neutral currents)

Then:

ee ee

WWAVL ggg 22 sin2

11sin

2

1)11(

2

1

WAVR ggg 2sin))1(1(2

1

24

22

2

1sinsin2

1)(y

sG

dy

edWW

Fe

This cross-section is a consequence of the interference of the charged and neutral current diagrams.

24342

22

10109.0sin

3

1sin

2

1)( cm

MeV

EsGe WW

Fe

Page 7: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 7

Neutrino pair production:

Then:

eeee

4

1sin2

2

1

12)(

22

2

WF

ee

sGee

Contribution from both W and Z graphs.

W

e

e

e

eZ

ee

e

e

Only neutral current contribution to: ee

4

1sin2

2

1

12)(

22

2

WF sG

ee

Page 8: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 8

Neutrino-electron scattering Neutrino-electron scattering Summary neutrino electron scattering processes:

ee

WW

F sG

4222

sin3

41sin2

4

Process Total cross-section

ee

ee ee ee

ee ee

eeee

ee

WW

F sG

4222

sin41sin23

1

4

WW

F sG

4222

sin3

41sin2

4

WW

F sG

4222

sin41sin23

1

4

sGF

2

WW

F sG

422

sin4sin22

1

12

WW

F sG

422

sin4sin22

1

12

)()(2 frameLABtheinEms e

Page 9: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 9

Neutrino-nucleon quasi-elastic scatteringQuasi-elastic neutrino-nucleon scattering reactions (small q2):

W

pn

pn

nHpM eff ,,

pp

)()(

np

W

p n

0Z

)(

p p

)(

factorformvectorqFV )( 2

factorformvectoraxialqFA )( 2

)(975.0cos angleCabbiboC

nqFqFpG

AVcF

522

5 )()()1(2

cos

Page 10: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 10

Neutrino-nucleon quasi-elastic scattering

For low energy neutrinos (E<<mN):

028.02573.1)0( AA gF

2222

)0(3)0(cos

)()( AVCF

ee FFEG

pn

2

2

42

101075.9 cm

MeV

E

Form factors 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 (PCAC):

22

2

71.0/1

)0()(

q

FqF V

V

1)0( VF

22

2

065.1/1

)0()(

q

FqF A

A

Page 11: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 11

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 = Q2/2M and y = /E can be used.

nucleon X

• Parton model is used to make predictions for deep inelastic neutrino-nucleon scattering. • Neutrino beams from pion and kaon decays, dominated by muon neutrinos are used to study this process.

nucleon X

Page 12: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 12

Weak structure functions

General form for the neutrino-nucleon deep inelastic scattering cross-section, neglecting lepton masses and corrections of the order of M/E:

d,

dxdy

GF2 ME

1 y F2

N y 2xF1N y

y2

2

xF3

N

The functions F1 , F2 and F3 are the functions of Q2 and . In the scaling limit they are the functions of x only.

Page 13: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 13

Scaling behaviour

Compilation of the data on structure functions in deep inelastic neutrino scattering (1983)

Page 14: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 14

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):

Xppp )()(

1

0

1

02)()()( dxxuxudxxuV

Scattering off quarks:

dxxu )(

)()()( xuxuxu SV )()()( xdxdxd SV )()( xuxuS )()( xdxdS

1

0

1

01)()()( dxxdxddxxdV

EmG

dy

qd

dy

qd qFCCCC22)()(

22

12)()(

yEmG

dy

qd

dy

qd qFCCCC

cos12

11

E

Eywith

Page 15: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 15

Scattering off proton:

22

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

yxcxuxsxdxMEG

dxdy

pdFCC

)()()()(2)(2 xcxsxuxdxxF p

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

2

xsxdyxcxuxMEG

dxdy

pdFCC

Structure functions:

Callan-Gross relationship:

)()()()(2)(3 xcxsxuxdxxxF p

)()(2 21 xFxxF

)()()()(2)(2 xsxdxcxuxxF p

)()()()(2)(3 xsxdxcxuxxxF p

Neutron (isospin symmetry):

)()()()(2)(2 xcxsxdxuxxF n

)()()()(2)(3 xcxsxdxuxxxF n

Page 16: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 16

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

22

)1()()()(

yxqxqxMEG

dxdy

NdFCC

csduq csduq

)()()(2 xqxqxxF N

)()(2)()()(3 xcxsxqxqxxxF N

)()(2)()()(3 xcxsxqxqxxxF N

)()1)(()( 2

2

xqyxqxMEG

dxdy

NdFCC

Total cross-section:

GeVcmQQMG

EN FCC /1067.0

31

/)( 238

2

GeVcmQQMG

EN FCC /1034.0

31

/)( 238

2

Page 17: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 17

Page 18: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 18

Rise of mean q2 with energy

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

Page 19: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 19

Quark content of nucleons from CC cross-sectionsDefine:

Experimental values from y distribution of cross-sections yields:

If

.,)(1

0etcdxxxuU

03.015.0 QQ

Q03.000.0

QQ

S01.016.0

QQ

SQ

)(495.0)(

)(measured

N

Nr

CC

CC

19.03

13

r

r

Q

Q

33.0 QQQV08.0 QQQ SS

49.0)(1

0 2 QQdxxF N

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

Page 20: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 20

Some details

Note that for right-handed incident anti-neutrinos the term changes sign. Note also that the term is orthogonal to the asymmetric hadronicterm that is proportional to since q = l – l’ and gives zero when dotted into

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

Page 21: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 21

To obtain these expressions we have used

Page 22: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 22

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.

Page 23: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 23

We now consider the scaling limit

Substituting in terms of the scaling variables

we find the result

Page 24: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 24

For scattering on structureless fermions/antifermions (e.g., point particle quarks) we have

Thus measures the difference between quarks and antiquarks.

Page 25: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 25

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

Page 26: 12004, TorinoAram Kotzinian Neutrino Scattering Neutrino interactions Neutrino-electron scattering Neutrino-nucleon quasi-elastic scattering Neutrino-nucleon

2004, Torino Aram Kotzinian 26

Gross-Llewellyn-Smith (2 names) sum rule

In terms of the parton distributions in the proton we have

Thus we have

and hence