Transcript Document

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
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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
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Kinematics
Threshold for the pion production
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
Boundary of RES – DIS regions
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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
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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
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Quasi-elastic reaction
GD 
(from J.J. Kelly, Phys. Rev. C 70 (2004) 068202)
1
,
2
Q
1
2
MV
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M V  0.71GeV 2
2
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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)
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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.
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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
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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  
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Single pion production
BNL data: Kitagaki et al. PRD54 (1986) 2554:
  p  l  p  
  n  l  p  0
  n  l  n  
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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  
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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)
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Single pion production
Predictions of the Rein-Sehgal model
(with nonresonant background):
ANL beam
FNAL beam ~ 30 GeV (antineutrinos)
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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



PCAC:



A
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)
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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)
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Single pion production
Results from Lalakulich, Paschos model:
Non-zero leptonic mass
A comparison with
Rein-Sehgal model is missing.
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Single pion production
Very little is known about NC pion production:
(NC π0 production is an important background!)
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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 

W4
Em2
 2  2
2  W3
2
2
2
  EQ  Q  m 
m Q m

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
F j  W j ,
F2
F4 
 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.
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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.
Fj ( x, Q 2 )  Fj ( x, Q 2 ) 
LT
H j ( x, Q 2 )
Q2
 (
1
)
4
Q
Expressed by PDFs; Q2 dependence via Altarelli-Parisi equation.
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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
( x, Q 2 ) 
1
x2
 
2
3
F2
LT
3
2
6
x
M
dz LT
( , Q 2 )  2 4  2 F2 ( z , Q 2 )
Q  z
1
x2
2 x 3 M 2 dz LT
TMC
LT
2
2
xF3 ( x, Q )  2 2 F3 ( , Q )  2 3  2 zF3 ( z , Q 2 )
 
Q  z
4x2M 2
  1
,
2
Q

2x
1 
Jan T. Sobczyk, Epiphany Conference, Kraków, Jan. 6, 2006
is Nachtmann variable
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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
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More inelastic channels
Low Q2 behavior.
In electron scattering gauge invariance implies:
F2 ( x, Q 2 )  Q 2
as
Q 2  0,
 Q2 
FL ( x, Q )  1  2  F2 ( x, Q 2 )  2 xF1 ( x, Q 2 )  Q 4
  


2
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
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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!
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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)
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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)
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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
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Nuclear effects
Fermi gas model – quasi-elastic reaction
E = 1GeV
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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)
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Nuclear effects
Spectral function:

P( E, p) 


 (M A  ER  M  E)  R( pR ) | 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
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Nuclear effects
Spectral function „cuts” the
quasi-elastic peak.
In the second figure
FSI is reduced to
the Pauli blocking
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Nuclear effects - FSI
Distorted wave IA: outgoing nucleon is
a solution of equation:




i   (M  US )  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!
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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
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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
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Conclusions:
A lot of theoretical activity.
Precise data for few GeV neutrino interactions is missing
Future experiment: MINERνA
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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
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MINERνA
Now…
.. after MINERνA measurements
(from V.Naumov, XX Max Born Symposium)
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MINERνA
Now…
(from V. Naumov)
BNL data
.. after MINERνA measurements
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In 6 years a talk on
modelling neutrino cross-sections
will be very different!
THE END
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