2004, Torino
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Transcript 2004, Torino
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
2004, Torino
Aram Kotzinian
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Neutrino-electron scattering
Tree level Feynman diagrams:
e
e e e e
e
e
W
e
H eff
e
Z0
e
e
e
Effective Hamiltonian:
GF
e (1 5 )e e (1 5 ) e e (1 5 ) e e ( gV g A 5 )e
2
GF
e (1 5 ) e e (1 gV (1 g A ) 5 )e
2
(through a Fierz transformation)
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Only charged current:
s p( ) p(e) 2me E ( ) (in LAB)
2
t q 2 p( ) p( )
2
W
e
e
e e
y
p(e) p( ) p( )
p(e) p( )
E( ) E ( )
E ( )
(in LAB)
Inelasticity variable (0<y<1)
d CC ( e )
2
2
2
mW
2GF me
E ( ) (in LAB)
2
2
dy
q mW
2
G
E 2
43
F s
cm
0.4 10
Total cross-section: CC ( e )
10 MeV
GF s
(cross-section proportional to energy!)
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( )
()
e ( gV g A 5 )e g Le (1 5 )e g Re (1 5 )e
()
1
1
( gV g A ) sin 2 W
2
2
1
g R ( gV g A ) sin 2 W
2
gL
Z0
e
e
d NC ( e )
dy
d NC ( e )
dy
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( )
e e
Only neutral current:
2
GF s
GF s
2
2
mZ
2
q 2 mZ
2
mZ
2
q 2 mZ
2
1
2
4
2
sin W sin W (1 y )
2
2
1
2
2
4
sin W (1 y ) sin W
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( )
Only neutral current (total cross-section):
( )
e e
2
2
E 2
G
s
1
1 4
2
43
F
cm
NC ( e )
sin W sin W 0.1510
2
3
10 MeV
2
E 2
GF s 1 1
2
4
43
cm
NC ( e )
sin
sin
0
.
14
10
W
W
3 2
10 MeV
2
Can obtain value of sin2W
from neutrino electron
scattering (CHARM II):
sin 2 W 0.2324 0.0058 0.0059
Ee2 2me (1 y)
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Back to
e e e e
(charged and neutral currents)
1
1
1
(1 gV 1 g A ) sin 2 W 1 sin 2 W
2
2
2
1
g R (1 gV (1 g A )) sin 2 W
2
gL
Then:
2
2
d ( e e ) GF s 1
2
2
4
sin W sin W 1 y
dy
2
2
E 2
GF s 1
1 4
2
43
cm
( e e )
sin W sin W 0.9 10
2
3
10 MeV
2
This cross-section is a consequence of the interference of the
charged and neutral current diagrams.
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Neutrino pair production:
e e e e
Contribution from both W and Z graphs.
e
e
W
e
e
e
e
e
Z
e
2
GF s 1
1
2
(e e e e )
2 sin W
12 2
4
2
Then:
Only neutral current contribution to:
e e
2
2
G
s
1
1
2
F
(e e )
2 sin W
12 2
4
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Neutrino-electron scattering
Summary neutrino electron scattering processes:
Process
Total cross-section
e e
e e e e
e e e e
e e
e e
e e e e
e e
2
GF s
2
GF s
4 4
2
2
sin
1
sin W
W
4
3
2
2
GF s 1
2
2
sin
1
4 sin 4 W
W
4 3
2
2
GF s
4
2 sin 2 W 1 sin 4 W
4
3
2
2
GF s 1
2
4
2
sin
1
4
sin
W
W
4 3
2
GF s 1
2 sin 2 W 4 sin 4 W
12 2
2
GF s 1
2 sin 2 W 4 sin 4 W
12 2
2
s 2me E( ) (in the LAB frame)
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Neutrino-nucleon quasi-elastic scattering
Quasi-elastic neutrino-nucleon scattering reactions (small q2):
()
()
n p
p n p p
W
n
()
W
p
Z0
n
p
()
p
p
M , p H eff , n
GF cos c
(1 5 ) p FV (q 2 ) FA (q 2 ) 5 n
2
FV (q2 ) vector form factor
cosC 0.975(Cabbiboangle)
FA (q 2 ) axial vector form factor
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Neutrino-nucleon quasi-elastic scattering
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):
FV (0)
FV (q )
1 q
2
FA (q 2 )
1 q
2
2
FV (0) 1
/ 0.71
FA (0)
2
FA (0) g A 1.2573 0.028
/ 1.065
2
For low energy neutrinos (E<<mN):
2
2
GF cos C E
( e n) ( e p)
FV (0) 2 3FA (0) 2
2
42
9.7510
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E
cm2
10 MeV
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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.
nucleon X
nucleon 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 = Q2/2M and y = /E can be used.
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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 ,
GF2 ME
N
2
N
1 y F2 y xF1
dxdy
y 2 N
y xF3
2
The functions F1 , F2 and F3 are the functions of Q2 and . In the scaling
limit they are the functions of x only.
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Scaling behaviour
Compilation of the
data on structure
functions in deep
inelastic neutrino
scattering (1983)
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Neutrino proton CC scattering:
u ( x)dx = number of u-quarks in proton between x and x+dx
( p) p ( p) X
Some of the quarks are from sea:
u( x) uV ( x) uS ( x)
d ( x) dV ( x) d S ( x)
dS ( x) d ( x)
For proton (uud): uS ( x) u ( x)
1
u
d
0
1
0
V
V
( x)dx u( x) u ( x)dx 2
1
0
1
( x)dx d ( x) d ( x) dx 1
0
Scattering off quarks:
d CC ( q)
dy
d CC ( q )
dy
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d CC ( q )
dy
d CC ( q)
dy
2
2GF mq E with y 1 E 1 1 cos
E 2
2
2GF mq E
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1 y 2
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Scattering off proton:
d CC ( p) GF 2 ME
2 x d ( x) s( x) u ( x) c ( x) (1 y) 2
dxdy
d CC ( p)
dxdy
2
GF ME
2 x u( x) c( x) (1 y) 2 d ( x) s ( x)
Structure functions:
Callan-Gross relationship:
2 xF1 ( x) F2 ( x)
F2p ( x) 2xd ( x) u ( x) s( x) c ( x)
xF3p ( x) 2xd ( x) u ( x) s( x) c ( x)
F2p ( x) 2x u( x) c( x) d ( x) s ( x)
xF3p ( x) 2x u( x) c( x) d ( x) s ( x)
Neutron (isospin symmetry):
F2n ( x) 2x u( x) d ( x) s( x) c ( x)
xF3n ( x) 2x u( x) d ( x) s( x) c ( x)
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Scattering off isoscalar target (equal number neutrons and protons):
q ud sc
q u d s c
F2N ( x) xq( x) q ( x)
xF3N ( x) xq( x) q ( x) 2s( x) c( x)
xF3N ( x) xq( x) q ( x) 2s( x) c( x)
d CC ( N )
dxdy
d CC ( N )
dxdy
2
GF ME
GF ME
2
x q( x) q ( x) (1 y) 2
x q( x)(1 y) 2 q ( x)
Total cross-section:
G M
1
0.67 1038 cm 2 / GeV
CC ( N ) / E F
Q
Q
3
2
GF M 1
0.34 1038 cm 2 / GeV
CC ( N ) / E
Q
Q
3
2
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Rise of mean q2 with energy
Mean q2 was found to be linear function in neutrino (antineutrino) energy.
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Quark content of nucleons from CC cross-sections
1
Define:
U xu( x)dx , etc.
0
Experimental values from y distribution of cross-sections yields:
Q
S
Q S
0.15 0.03
0.00 0.03
0.16 0.01
QQ
QQ
QQ
r
If
CC ( N )
0.495 (m easured)
CC (N )
QV Q Q 0.33
1
0
Q 3r 1
0.19
Q 3 r
QS QS Q 0.08
F2N ( x)dx Q Q 0.49
Quarks and antiquarks carry 49% of proton momentum, valence
quarks only 33% and sea quarks only 16%.
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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
hadronic
term 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.
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To obtain these expressions we have used
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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.
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We now consider the scaling limit
Substituting in terms of the scaling variables
we find the result
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For scattering on structureless fermions/antifermions (e.g., point
particle quarks) we have
Thus
measures the difference between quarks and antiquarks.
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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
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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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