Neutrino Scatteringpersonalpages.to.infn.it/~kotzinia/lectures/pdf/l6-neutrino.pdf · vector form factor F. A (q. 2) = axial − vector form factor. cosθ. C =0.975(Cabbibo angle)

2004,Torino Aram Kotzinian 1

Neutrino Scattering

Neutrino interactionsNeutrino-electron scatteringNeutrino-nucleon quasi-elastic scatteringNeutrino-nucleon deep inelastic scattering

VariablesCharged currentQuark content of nucleonsSum rulesNeutral current


NeutrinoNeutrino--electron scatteringelectron scattering−− +→+ ee ee ννTree level Feynman diagrams:

0Z

eν eν

−e−e

−W

eν

eν−e

−e

Effective Hamiltonian:

[ ][ ]{ }eggeGAVee

F ))1(1()1(2 55 γγνγγν µ

µ +−+−=

[ ][ ] [ ][{ }eggeeeGH AVeeeeF

eff )()1()1()1(2 5555 γγνγγννγγγγν µ

µµ

µ −−+−−= ]

(through a Fierz transformation)


Only charged current:

−W

µν

eν−e

−µ

−− +→+ µνν µ ee

( ) )()(2)()( 2 LABinEmepps e µµ νν =+=

( )22 )()( µν µ ppqt −==

( ))(

)()()(

)()()()()(

LABinE

EEpep

ppepy

µ

µ

µ

µ

νµν

νµν −

=⋅

−⋅=

)()(2)( 2

22

22

LABinEmGmq

msGdy

ed eF

W

WFCCµ

µ νππ

νσ≈

−=

−

Inelasticity variable (0<y<1)

2432

10104.0)( cm

MeVEsGe F

CC

×== −−

πνσ µ

(cross-section proportional to energy!)

Total cross-section:


Only neutral current:−

−−

−

+→+ ee µµ νν)()(

0Z

µν)(−

−e

µν)(−

−e

eegeegegge RLAV )1()1()( 555 γγγγγγ µµµ ++−=−

WAVL ggg θ2sin21)(

21

+−=+=

WAVR ggg θ2sin)(21

=−=

−+

+−

−=

−24

22

22

22

)1(sinsin21)(

ymq

msGdy

edWW

Z

ZFNC θθπ

νσ µ

+−

+−

−=

−

WWZ

ZFNC ymq

msGdy

edθθ

πνσ µ 42

22

22

22

sin)1(sin21)(


Only neutral current (total cross-section):

24342

22

101015.0sin

31sin

21)( cm

MeVEsGe WW

FNC

×=

+

+−= −− ν

µ θθπ

νσ

24342

22

101014.0sinsin

21

31)( cm

MeVEsGe WW

FNC

×=

+

+−= −− ν

µ θθπ

νσ

−−

−−

+→+ ee µµ νν)()(

Can obtain value of sin2θWfrom neutrino electron scattering (CHARM II):

0059.00058.02324.0sin2 ±±=Wθ

)1(22 ymE ee −=Θ


Back to (charged and neutral currents)−− +→+ ee ee νν

WWAVL ggg θθ 22 sin211sin

21)11(

21

+=++−=+++=

1WAVR ggg θ2sin))1(1(

2=+−+=

Then: ( )

−+

+=

−24

22

2

1sinsin21)( ysG

dyed

WWFe θθπ

νσ

24342

22

10109.0sin

31sin

21)( cm

MeVEsGe WW

Fe

×=

+

+=⇒ −− νθθ

πνσ

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


Neutrino pair production: eeee νν +→+ −+

Contribution from both W and Z graphs.

W

+e

−e

eν

eνZ

+e−e

eν

eν

Then:

+

+=→−+

41sin2

21

12)(

22

2

WF

eesGee θ

πννσ

Only neutral current contribution to: µµ νν +→+ −+ ee

+

−=→−+

41sin2

21

12)(

22

2

WF sGee θπ

ννσ µµ


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

−− +→+ ee µµ νν

−− +→+ ee µµ νν

−− +→+ ee ee νν

( )

+− WW

F sG θθπ

4222

sin341sin2

4

( )

+− WW

F sG θθπ

4222

sin41sin231

4

( )

+− WW

F sG θθπ

4222

sin341sin2

4

( )

++ WW

F sG θθπ

4222

sin41sin231

4

πsGF

2

++ WW

F sG θθπ

422

sin4sin221

12

+− WW

F sG θθπ

422

sin4sin221

12

Total cross-sectionProcess

−− +→+ ee ee ννee νµν µ +→+ −−

eeee νν +→+ −+

µµ νν +→+ −+ ee

)()(2 frameLABtheinEms e µν=


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

−W

µν

pn

−µ

pn +→+ −µν µ pp +→+−−

µµ νν)()(

np +→+ +µν µ

+W

µν

p n

+µ0Z

µν)(−

p p

µν)(−

== − nHpM eff ,, µνµ

factorformvectorqFV =)( 2

factorformvectoraxialqFA −=)( 2)(975.0cos angleCabbiboC =θ

[ ] ( )[ ]nqFqFpGAV

cF5

225 )()()1(

2cos γγνγγµθ

µµµ +−


Neutrino-nucleon quasi-elastic scattering

028.02573.1)0( ±−== AA gF

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()(

qFqF V

V−

= 1)0( =VF

( )22

2

065.1/1)0()(

qFqF A

A−

=

For low energy neutrinos (Eν<<mN):( ) [ ]22

22

)0(3)0(cos)()( AVCF

ee FFEGpn +==πθνσνσ ν

22

42

101075.9 cm

MeVE

×≈ − ν


Inelastic neutrino-nucleon scattering• 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.

νµ + nucleon → µ+ + Xνµ + nucleon → µ− + X

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 = Q2/2Mν and y = ν /E can be used.


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

dσν,ν

dxdy=

GF2 MEπ

1− y( )F2νN + y 2xF1

νN m y −y2

2

xF3

νN

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


Scaling behaviour

Compilation of the data on structure functions in deep inelastic 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):

Xppp +′→+ − )()( µν µ

[ ]∫∫ =−=1

0

1

02)()()( dxxuxudxxuV

Scattering off quarks:

dxxu )(

)()()( xuxuxu SV += )()()( xdxdxd SV +=)()( xuxuS = )()( xdxdS =

[ ]∫∫ =−=1

0

1

01)()()( dxxdxddxxdV

πνσνσ µµ EmG

dyqd

dyqd qFCCCC

22)()(==

( )22

12)()(

yEmG

dyqd

dyqd qFCCCC −==

πνσνσ µµ

( )θcos1211 −=

′−=

EEywith


Scattering off proton:

[ ] [ ]{ }22

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

yxcxuxsxdxMEGdxdy

pd FCC −+++=π

νσ µ

[ ] [ ]{ })()()1()()(2)( 2

2

xsxdyxcxuxMEGdxdy

pd FCC ++−+=π

νσ µ

Structure functions:Callan-Gross relationship: )()(2 21 xFxxF =

[ ])()()()(2)(2 xcxsxuxdxxF p +++=ν

[ ])()()()(2)(3 xcxsxuxdxxxF p −+−=ν

[ ])()()()(2)(2 xsxdxcxuxxF p +++=ν

[ ])()()()(2)(3 xsxdxcxuxxxF p −−+=ν

Neutron (isospin symmetry):[ ])()()()(2)(2 xcxsxdxuxxF n +++=ν

[ ])()()()(2)( xcxsxdxuxxxF n −+−=ν3


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

csduq +++≡ csduq +++≡

[ ])()()(2 xqxqxxF N +=ν

( )[ ])()(2)()()(3 xcxsxqxqxxxF N −+−=ν

( )[ ])()(2)()()(3 xcxsxqxqxxxF N −−−=ν

{ }22

)1()()()(

yxqxqxMEGdxdy

Nd FCC −+=π

νσ µ

{ })()1)(()( 2

2

xqyxqxMEGdxdy

Nd FCC +−=π

νσ µ

Total cross-section:

GeVcmQQMGEN FCC /1067.0

31/)( 238

2−×=

+=

πνσ µ

GeVcmQQMGEN FCC /1034.0

31/)( 238

2−×=

+=

πνσ µ



Rise of mean q2 with energy

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


Quark content of nucleons from CC cross-sectionsDefine:

Experimental values from y distribution of cross-sections yields:

If

.,)(1

0etcdxxxuU ∫=

03.015.0 ±=+ QQQ

03.000.0 ±=+ QQS

01.016.0 ±=++

QQSQ

)(495.0)()( measured

NNr

CC

CC =≡νσνσ

19.03

13≈

−−

=⇒r

rQQ

33.0≈−= QQQV08.0≈== QQQ SS

49.0)(1

0 2 ≈+=∫ QQdxxF Nν

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


Some details

Note that for right-handed incident anti-neutrinos the e term changes sign. Note also that the e 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.


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

Documents

Neutrino Scatteringpersonalpages.to.infn.it/~kotzinia/lectures/pdf/l6-neutrino.pdf · vector form factor F. A (q. 2) = axial − vector form factor. cosθ. C =0.975(Cabbibo angle)