1. in the Rein-Sehgal Model
Quark-Hadron Duality
in the Rein-Sehgal Model
須藤　和敬
KEK研究会「レプトン・原子核反応の記述」
22 Dec. 2007
K. Graczyk, C. Juszczak, J. Sobczyk
Nucl. Phys. A781, 227 (2007), hep-ph/
Introduction
What is the Quark-Hadron Duality?
Single Pion Production in nN Scattering
Numerical Results
Summary

2. Hadronic cross section Perturbative quark-gluon theory
Motivation
Complementarity between quark (+gluon) and hadron description of observables
In principle, QCD describes all of hadron physics at all distance scales. – but how does it work?
We want to understand it in language of QCD.
low energy
Hadronic cross section
high energy
Perturbative quark-gluon theory
How connected??

3. Bloom-Gilman Duality
Average over (strongly Q2 dependent) resonance ~ Q2 independent scaling function!!
Finite energy sum rule for eN scattering:
Bloom and Gilman, PRL85,1185(1970)

4. Data from SLAC and J-Lab
Recent Data
Resonance region data oscillate around the scaling curve.
Resonance data are on average equivalent to the scaling curve.
The data “slide” along the DIS curve with increasing Q2.
Data from SLAC and J-Lab
0.06 < Q2 < 3.30 GeV2
Solid curve is DIS fit at Q2=5 GeV2
at the same ξ
F2p
Nachtman variable:

5. Duality in QCD Operator Product Expansion:
Expand moments of structure functions in powers of 1/Q2
If moments are independent of Q2
Duality suppression of higher twist
If higher twists are small (duality “works”)
can use single parton approximation to describe structure function
extract leading twist parton distributions
If duality is violated, and if violations are small
can use to extract higher twist matrix elements
learn about nonperturbative qq or qg corrections
nonperturbative interactions not dominate?
strong cancellations between resonances?
∴ higher twist terms are small.

6. Aim of This Paper Single Pion Production (SPP) in nN scattering
Two different scheme: Resonance & DIS
Rein-Sehgal model: Resonance region
pQCD+PDFs: DIS region
Low Q2
high Q2
discuss that the duality holds in this calculation

7. Rein-Sehgal Structure Function
Single Pion Production in CC and NC reactions:
momentum transfer in LAB frame:
Leptonic Tensor:
Decomposition into 3 Component σL, σR, σS:
cross section of gauge boson in polarization
state scattered off nucleon

8. Cross Section in RS Model:
In language of structure functions
calculated within quark model, given for each SPP channel
See Rein and Sehgal, Ann.Phys.133(1981)79
Hadronic Tensor:

9. Rein-Sehgal Structure Functions:
Energy Range:
σ’s are calculated in the quark model.
In each processes, Δ, S11, P11, D13,,,,,, using Breit-Wigner function and transition matrix elements
Both resonant and non-resonant contributions are shown.

10. DIS Structure Function
DIS structure functions are represented by linear combination of the parton distribution functions (PDFs).
CC nN interaction
NC nN interaction
F1 is obtained from Callan-Gross relation:

11. Single Pion Production and DIS
They evaluate SPP in the QCD framework extrapolated to resonance region.
Introduce 1-pion function: f1π(W)
f1π(W) is obtained from Monte Carlo simulation by LUND algorithm.
It is defined by fragmentation and hadronization routines.
Assume that scaling structure function is expressed as
Artificial, how about the effects beyond the leading order?.
defined as the probabilities that the final state is that of the SPP
strongly model dependent analysis

12. Quark-Hadron Duality The quark-hadron duality holds the relation:
should hold for different values of Q2RES, also at Q2RES < 1 GeV2.
In this paper, Q2DIS is set as Q2DIS = 10 GeV2.
The ratio of these structure functions:

13. RS vs DIS Structure Function
CC nN scattering
xF1 F2
xF3
proton
neutron
Δpeak
GRV94 (LO) PDFs are used: Q2 > 0.23 GeV2, x > 10-5

14. RS vs DIS Structure Function (cont.)
NC nN scattering
xF1 F2
xF3
proton
neutron
Δpeak

15. RS vs DIS Structure Function (cont.)
CC and NC neutrino-isoscalar target scattering
xF1 F2
xF3 CC NC
Δpeak
Duality hold well for xF1 and xF3, while DIS dominates for F2.

16. F2 Ratio: RES to DIS How does 1-pion function affect the duality?
Rci(p)
Rci,1π(p)
Rci(n)
Rci,1π(n)
Rci(I)
Rci,1π(I)
1-pion function restore the duality picture, except Rci,1π(p) in CC.
The quark-hadron duality holds very well in NC.

17. F3 Ratio: RES to DIS Discussion is almost same as for F2. Rci(p)
Rci(n)
Rci,1π(n)
Rci(I)
Rci,1π(I)
It seems that 1-pion function improve the duality.

18. Two-Component Duality
Decomposition of structure functions
Hypothesis of two-component duality
RS resonant non-resonant
DIS valence sea
duality exist?

20. Summary
The quark-hadron duality gives a relation between the structure function of resonance region and that of DIS region.
Powerful tool to constrain the PDFs
Useful to learn about nonperturbative corrections
The quark-hadron duality for single pion production in nN scattering.
Qualitative: compare the RS and DIS structure functions
Quantitative: evaluate the ration of (integrated) structure functions
1-pion function is introduced to describe the SPP.
This function improve the duality. →all of neutron and isoscalar targets
Proton Fi in CC is not the case.