Modelling neutrino cross
sections
Jan T. Sobczyk
Institute of Theoretical Physics, University of Wrocław
(in collaboration with A. Ankowski, K. Graczyk, C. Juszczak, J. Nowak)
Plan of the talk:
1.
Overview, „decomposition” of the total cross section.
2.
Quasi-elastic scattering (form-factors, axial mass).
3.
Single pion production (Rein-Sehgal model, Δ excitation models)
4.
More inelastic channels (DIS formalism, structure functions,
higher twists, target mass corrections, low Q2 limit,
quark-hadron duality)
5.
Nuclear effects (Fermi gas, spectral function,
nuclear effects in DIS)
6.
Outlook.
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
2/44
Total neutrino – nucleon cross sections
Plots from Wrocław MC generator
We distinguish:
• quasi-elastic
• single pion production („RES region”,
e.g. W<=2 GeV)
• more inelastic („DIS region”)
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
Focus od few GeV
neutrino energy region:
all 3 dynamics are relevant.
Focus on MC implementations
3/44
Kinematics
Threshold for the pion production
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
Boundary of RES – DIS regions
4/44
Quasi-elastic reaction

 n  l


p  l

p
 n
F2 (Q 2 )
FP (Q 2 )
2
    F1 (Q )  i  q
    5 FA (Q )   5 q
2M
M

2
CVC – use electromagnetic data
PCAC
2
2
2
M
F
(
Q
)
2
A
FP (Q ) 
2
m  Q 2
We need the axial form-factor; the standard dipole form
FA (Q ) 
2
gA

Q 
1 

 M 2
A 

2
2
gA =1.26 from neutron decay;
MA a free parameter (the only one)
The value of axial mass is obtained from experimental data.
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
5/44
Quasi-elastic reaction
Q 2 (MW ) 2
where:
s  (k  p) 2 ,
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
u  (k ' p) 2
6/44
Quasi-elastic reaction
GD 
(from J.J. Kelly, Phys. Rev. C 70 (2004) 068202)
1
,
2
Q
1
2
MV
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
M V  0.71GeV 2
2
7/44
Quasi-elastic reaction
Dipole electromagnetic form-factors:
1     proton  neutron  2.79  (1.91)  4.7
One can find better fits to the existing data,
BBBA2005
(from R. Bradford talk at NuInt05)
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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Quasi-elastic reaction
Must be taken into account in the discussion of small
Q2 deficit of events in K2K experiment.
Two ways of determining axial mass: the shape of differential
cross section and total cross section.
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
9/44
Quasi-elastic reaction
(from Naumov)
The limiting value depends on
the axial mass
Under asumption of
dipole vector form-factors:
(A. Ankowski)
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
10/44
Single pion production
3 CC channels for neutrino reactions:
  p  l  p  
  n  l  p  0
  n  l  n  
Characteristic feature is that the dominant contribution comes from:
  p  l      l   p   
  n  l   
 l  p  0
  n  l   
 l  n  
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
11/44
Single pion production
BNL data: Kitagaki et al. PRD54 (1986) 2554:
  p  l  p  
  n  l  p  0
  n  l  n  
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
12/44
Single pion production
The overall cross sections for SPP reactions (with the W ≤ 2 GeV cut):
  p  l  p  
  n  l  p  0
  n  l  n  
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
13/44
Single pion production
Theoretical models:
• resonance excitation
• non-resonant background
Resonance excitation:
• in most practical applications: Rein-Sehgal model (18 resonances
up to 2 GeV, interference terms, a fit to data of the non-resonant part)
• recent detail study: Lalakulich, Paschos; leptonic mass terms are kept
• Sato-Lee model (non-resonant background is included)
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
14/44
Single pion production
Predictions of the Rein-Sehgal model
(with nonresonant background):
ANL beam
FNAL beam ~ 30 GeV (antineutrinos)
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
15/44
Single pion production
V
V
 C3V  

C5
C4
V
 

 
  | V | N    
qg  q   2 q  pg  q p  2 q  p' g   q  p'  C6 g   5u
M
M
M








A
A
 C3 A  

C6  
C4
A
 

 
  | A | N    
qg  q   2 q  pg  q p  2 q q  C5 g  u
M
M
M






A
PCAC:
C6
Adler model:
A
C5
2
M
2
m  Q 2
C3  0,
A
C6  0
V
CVC:
C4   C5 4
A
A
1
1

,
2
DA 
Q2 
1 

 M 2
A 

A
C5 (0)
A
2
C5 (Q ) 

DA
1
1
Q 2
MA
2
(from O. Lalakulich, XX Max Born Symposium)
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
16/44
Single pion production
New data on helicity amplitudes and
rather then C5V=0 (magnetic
multipole dominance)
(from L.Tiator et al., EPJA 19 (2004) 55)
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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Single pion production
Results from Lalakulich, Paschos model:
Non-zero leptonic mass
A comparison with
Rein-Sehgal model is missing.
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
18/44
Single pion production
Very little is known about NC pion production:
(NC π0 production is an important background!)
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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More inelastic channels

Structure functions for inclusive cross section:
W   g W1 

p p
2
M
p q  q p
2M
2
W2  i
W5  i
  p q 
2
W3 
2M
p q  q p
2M
2
q q
M
2
W 2 Q 2
2M
W4 
W6
d
GW

  MW

L
W
 2 2
2
2 
dWdQ
4 M E
 MW Q 
2
2
2
2
2
2


Q

m

2
2
LW  Q  m W1  2 E ( E  ) 
W2 
2 

2
W
W
Em
 2  2

4
  EQ  Q  m 2  3  m 2 Q 2  m 2

W5
2
2
2M
M

M






Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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More inelastic channels
Dimensionless structure functions:
F1  MW1,
W4, W5 in terms with m2 only
F4 
F j  W j ,
F2
 F1 ,
2x
j  2,3,4,5,6.
xF5  F2
L
R
,
T
F1 
Instead of Callan-Gross relation:
1  Q2
1
1 R 
 2
 F2

 2x
R is measured experimentally.
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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More inelastic channels
Computation of F2 and F3
In the scaling limit:
PM
 2 xd ( x)  s ( x)  b( x)  u ( x)  c ( x)  t ( x) 
PM
 2d ( x)  s ( x)  b( x)  u ( x)  c ( x)  t ( x) 
p
F2
p
F3
Q2 dependence.
Large Q2, perturbative QCD, twist expansion.
F j ( x, Q 2 )  F j ( x, Q 2 ) 
LT
H j ( x, Q 2 )
Q2
 (
1
)
4
Q
Expressed by PDFs; Q2 dependence via Altarelli-Parisi equation.
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
22/44
More inelastic channels
All above (twist expansion) in the massless limit. Target mass corrections
(TMC) are necessary.
For small x2 M2 /Q2 TMC modify LT
F2
TMC
xF3
( x, Q ) 
TMC
1
x2
2
 
( x, Q 2 ) 
2
3
F2
x2
 
4x2M 2
  1
,
2
Q
2
2
LT
F3
6 x 3 M 2 dz LT
( , Q )  2 4  2 F2 ( z , Q 2 )
Q  z
2
1
LT

2 x 3 M 2 dz LT
2
( , Q )  2 3  2 zF3 ( z , Q 2 )
Q  z
2x
1 
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
is Nachtmann variable
23/44
More inelastic channels
E=1 GeV
(from Wrocław MC)
E=5 GeV
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
E=3 GeV
E=10 GeV
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More inelastic channels
E=10 GeV
E=3 GeV
It is clear that low Q2
(Q 2 ≤ 1GeV2) are important
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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More inelastic channels
Low Q2 behavior.
In electron scattering gauge invariance implies:
F2 ( x, Q 2 )  Q 2
 Q2
FL ( x, Q )  1  2
 

2
as
Q 2  0,

 F2 ( x, Q 2 )  2 xF1 ( x, Q 2 )  Q 4


as
Q2  0
In neutrino scattering from PCAC:




2
2
2
1 
F2 ( x, Q )  AQ  ( f  )  N  2 
 1 Q2 
  
2
PCAC
There is a lot of theoretical activity!
and similarly for FL
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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More inelastic channels
In the few GeV region an important idea from both theoretical
and practical (MC) point of view is quark-hadron duality
MC: how to combine
smoothly RES and DIS regions?
Recent JLab electron data
(from I. Niculescu et al. PRL 85 (2000) 1184, 1187)
In neutrino physics a lot of activity – see the next slide!
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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More inelastic channels
Quark-hadron duality in neutrino scattering:
(from K. Matsui, T. Sato,and T.-S. H. Lee, PRC 72 (2005) 25204)
(from O.Lalakulich)
(from K.Graczyk, C.Juszczak, JS)
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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Nuclear effects
Red line – Relativistic Shell Model
Blue and pink lines – Fermi Gas Model
At E=1 GeV a general picture:
• neutrino interacts with an individual (bound)
nucleons
• „final state interactions” (FSI) follow
Impulse approximation:
In MC codes FSI is usually taken into
account numerically in nuclear cascade.
(from Ch. Maieron, XX Max Born Symposium)
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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Nuclear effects
Fermi gas model – quasi-elastic reaction
Fermi momentum
average binding energy
off shell matrix element
PWIA – „plane wave impulse approximation”:
outgoing nucleon – plane wave Dirac spinor
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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Nuclear effects
Fermi gas model – quasi-elastic reaction
E = 1GeV
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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Nuclear effects
Realistic distribution of momenta
Short range correlations (SRC):
correlated pairs of nucleons
(from A. Ankowski)
(from O. Benhar et al. hep-ph/0516116)
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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Nuclear effects
Spectral function:

P( E , p ) 


 (M A  ER  M  E )  R( p R ) | a( p) | i(M A )
2
R
In the PWIA we obtain:
d
d 3k '
W

 

(GF cos( C )) 2
(2 ) 2 E E '
LW 
 dE d p  (  M  E  E p' ) H
3

  

( p  q; p) P( E, p)
Precise computations for A≤16.
Computations combine mean field part and SRC part.
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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Nuclear effects
Spectral function for oxygen
1p (1/2)
1p (3/2)
(from O. Benhar)
1s
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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Nuclear effects
Spectral function „cuts” the
quasi-elastic peak.
In the second figure
FSI is reduced to
the Pauli blocking
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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Nuclear effects - FSI
Distorted wave IA: outgoing nucleon is
a solution of equation:




i   (M  U S )  E  UV  UC  (r )  0
Scalar and vector complex optical potentials:
rescattering and absorption into other channels.
RSM – relativistic shell model
RFG – reletivistic Fermi gas
RMF – relativistic mean field
Real ROP – no absorption
ROP – full optical potential
(from Ch. Maieron, XX Max Born Symposium)
FSI effects are important!
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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Nuclear effects
Kulagin, Petti approach to deal with DIS nuclear effects
Scales:
x>0.2 → incoherent sum of contributions from bound nucleons (spectral function)
x<0.3 → correction from scattering on pions
x<<0.2 → coherent effects (shadowing, multiple scattering on nucleons)
FMB – Fermi motion, nuclear binding
OS – off-shell corections
PI – nuclear pion excess
NS – nuclear coherent processes
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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Nuclear effects
Kulagin, Petti approach to deal with DIS nuclear effects
The model contains 3 free parameters fitted to the data
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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Conclusions:
A lot of theoretical activity.
Precise data for few GeV neutrino interactions is missing
Future experiment: MINERνA
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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MINERνA
MINERvA is a neutrino detector to study neutrino-nucleus interactions.
It will be placed in the NuMI beam line.
3 energy beams
Commissioning Fall 2008
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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MINERνA
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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MINERνA
Now…
.. after MINERνA measurements
(from V.Naumov, XX Max Born Symposium)
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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MINERνA
Now…
(from V. Naumov)
BNL data
.. after MINERνA measurements
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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In 6 years a talk on
modelling neutrino cross-sections
will be very different!
THE END
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
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