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
1
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)
2004, Torino
Aram Kotzinian
2

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!)
2004, Torino
Aram Kotzinian
3
( )
()

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
2004, Torino
( )
   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 

 2

Aram Kotzinian
4
( )

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   0.1510 
  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 sin2W
from neutrino electron
scattering (CHARM II):
sin 2 W  0.2324 0.0058 0.0059
Ee2  2me (1  y)
2004, Torino
Aram Kotzinian
5

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.
2004, Torino
Aram Kotzinian
6
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 

2004, Torino
Aram Kotzinian
7
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)
2004, Torino
Aram Kotzinian
8
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
cosC  0.975(Cabbiboangle)
FA (q 2 )  axial vector form factor

2004, Torino
 
Aram Kotzinian

9
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
 9.7510
2004, Torino
 E 

 cm2
 10 MeV 
Aram Kotzinian
10
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.
2004, Torino
Aram Kotzinian
11
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.
2004, Torino
Aram Kotzinian
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Scaling behaviour
Compilation of the
data on structure
functions in deep
inelastic neutrino
scattering (1983)
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Aram Kotzinian
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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
2004, Torino
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

Aram Kotzinian
1  y 2
14
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)
F2p ( x)  2xd ( x)  u ( x)  s( x)  c ( x)
xF3p ( x)  2xd ( x)  u ( x)  s( x)  c ( x)
F2p ( x)  2x u( x)  c( x)  d ( x)  s ( x)
xF3p ( x)  2x u( x)  c( x)  d ( x)  s ( x)




Neutron (isospin symmetry):





F2n ( x)  2x u( x)  d ( x)  s( x)  c ( x)
xF3n ( x)  2x u( x)  d ( x)  s( x)  c ( x)
2004, Torino
Aram Kotzinian
15
Scattering off isoscalar target (equal number neutrons and protons):
q ud sc
q u d s c
F2N ( x)  xq( x)  q ( x)
xF3N ( x)  xq( x)  q ( x)  2s( x)  c( x)
xF3N ( x)  xq( x)  q ( x)  2s( 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 1038 cm 2 / GeV
 CC (  N ) / E  F
Q

Q

 
3
2
GF M  1
  0.34 1038 cm 2 / GeV
 CC (  N ) / E 
Q

Q

  3
2
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Aram Kotzinian
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2004, Torino
Aram Kotzinian
17
Rise of mean q2 with energy
Mean q2 was found to be linear function in neutrino (antineutrino) energy.
2004, Torino
Aram Kotzinian
18
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
QQ
QQ
QQ
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
F2N ( x)dx  Q  Q  0.49
Quarks and antiquarks carry 49% of proton momentum, valence
quarks only 33% and sea quarks only 16%.
2004, Torino
Aram Kotzinian
19
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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Aram Kotzinian
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To obtain these expressions we have used
2004, Torino
Aram Kotzinian
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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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Aram Kotzinian
22
We now consider the scaling limit
Substituting in terms of the scaling variables
we find the result
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Aram Kotzinian
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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.
2004, Torino
Aram Kotzinian
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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
2004, Torino
Aram Kotzinian
25
Gross-Llewellyn-Smith (2 names) sum rule
In terms of the parton distributions in the proton we have
Thus we have
and hence
2004, Torino
Aram Kotzinian
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