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/0512015 Introduction What is the Quark-Hadron Duality? Single Pion Production in nN Scattering Numerical Results Summary

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??

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)

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:

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.

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

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

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:

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.

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:

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

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:

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

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

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.

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.

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.

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

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.