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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
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
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!)
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
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
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
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
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
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
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
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
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.
2004, Torino Aram Kotzinian 13
Scaling behaviour
Compilation of the data on structure functions in deep inelastic neutrino scattering (1983)
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
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
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
2004, Torino Aram Kotzinian 17
2004, Torino Aram Kotzinian 18
Rise of mean q2 with energy
Mean q2 was found to be linear function in neutrino (antineutrino) energy.
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
S01.016.0
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%.
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.
2004, Torino Aram Kotzinian 21
To obtain these expressions we have used
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.
2004, Torino Aram Kotzinian 23
We now consider the scaling limit
Substituting in terms of the scaling variables
we find the result
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.
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
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