Quark-Hadron Duality in Structure Functions

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1 Approaches to QCD, Oberwoelz, Austria September 10, 2008 Quark-Hadron Duality in Structure Functions Wally Melnitchouk

2 Outline Bloom-Gilman duality Duality in QCD OPE & higher twists Resonances & local quark-hadron duality truncated moments in QCD Duality in the neutron extraction of neutron resonance structure from nuclear data comparison with proton Summary

3 Quark-hadron duality Complementarity between quark and hadron descriptions of observables hadrons = quarks Can use either set of complete basis states to describe all physical phenomena

4 Electron scattering e e d 2 σ dωde = 4α2 E 2 cos 2 θ 2 Q 4 ( 2 tan 2 θ 2 F 1 M + F ) 2 ν N γ X ν = E E } x = Q2 Bjorken scaling variable Q 2 = q 2 ν 2 =4EE sin 2 θ 2Mν 2 F 1,F 2 structure functions contain all information about structure of nucleon functions of x, Q 2 in general

5 Bloom-Gilman duality scaling curve ω = 1 x + M 2 Q 2 Bloom, Gilman, Phys. Rev. Lett. 85 (1970) 1185 resonance - scaling duality in proton structure function νw 2 = F 2

6 cf. hadron-hadron scattering s-channel resonances t-channel Regge poles σ π+p σ π p! " j (t) R R(s) j s = =! t Finite energy sum rules Igi (1962), Dolen, Horn, Schmidt (1968)

7 Bloom-Gilman duality 2 Average over (strongly Q dependent) resonances 2 Q independent scaling function Finite energy sum rule for en scattering 2M Q 2 νm 0 dν νw 2 (ν, Q 2 )= ω m 1 dω νw 2 (ω ) measured structure function (function of and ) ν Q 2 hadrons ω = 1 x + M 2 Q 2 scaling function (function of ω only) quarks

8 Bloom-Gilman duality Average over (strongly Q dependent) resonances Q 2 independent scaling function 2 Jefferson Lab (Hall C) Niculescu et al., Phys. Rev. Lett. 85 (2000) 1182 Nachtmann scaling variable 2x ξ = M 2 x 2 /Q 2 see also: Fritzsch, Proc. of the Coral Gables Conf. (1971) Greenberg & Bhaumik, PRD4 (1971) 2048

9 Duality exists also in local regions, around individual resonances F 2 p S local Bloom-Gilman duality

10 Duality in QCD

11 Operator product expansion M n (Q 2 )= Duality in QCD expand moments of structure functions in powers of 1/Q = A (2) n dx x n 2 F 2 (x, Q 2 ) + A(4) n Q 2 + A(6) n Q 4 + matrix elements of operators with specific twist τ τ = dimension spin

12 Duality in QCD (a) (b) (c) τ =2 single quark scattering τ> 2 qq and qg correlations

13 Operator product expansion If moment M n (Q 2 )= 1 0 = A (2) n dx x n 2 F 2 (x, Q 2 ) + A(4) n Q 2 independent of Q2 higher twist terms Duality in QCD expand moments of structure functions in powers of 1/Q 2 A (τ>2) n small + A(6) n Q 4 +

14 Operator product expansion M n (Q 2 )= Duality in QCD expand moments of structure functions in powers of 1/Q = A (2) n dx x n 2 F 2 (x, Q 2 ) + A(4) n Q 2 + A(6) n Q 4 + Duality suppression of higher twists de Rujula, Georgi, Politzer, Ann. Phys. 103 (1975) 315

15 Duality in QCD Much of recent new data is in resonance region, W < 2 GeV common wisdom: pqcd analysis not valid in resonance region in fact: partonic interpretation of moments does include resonance region Resonances are an integral part of deep inelastic structure functions! implicit role of quark-hadron duality

16 Proton moments n relative contribution of resonance region to n-th moment At Q 2 = 1 GeV 2, ~ 70% of lowest moment of F p 2 comes from W < 2 GeV

17 Proton moments total 10% at Q 2 = 1 GeV 2 leading twist higher twist BUT resonances and DIS continuum conspire to produce only ~ 10% higher twist contribution! Ji, Unrau, Phys. Rev. D 52 (1995) 72

18 total higher twist small at Q GeV 2 on average, nonperturbative interactions between quarks and gluons not dominant at these scales suggests strong cancellations between resonances, resulting in dominance of leading twist OPE does not tell us why higher twists are small need more detailed information (e.g. about individual resonances) to understand behavior dynamically

19 Duality & Truncated Moments

20 Truncated moments complete moments can be studied in pqcd via twist expansion Bloom-Gilman duality has a precise meaning (i.e., duality violation = higher twists) for local duality, difficult to make rigorous connection with QCD e.g. need prescription for how to average over resonances truncated moments allow study of restricted regions in x (or W) within pqcd in well-defined, systematic way M n ( x, Q 2 )= dx x n 2 F 2 (x, Q 2 ) x

21 F p 2 resonance spectrum * * JLab Hall C Psaker, WM, et al., Phys. Rev. C 78 (2008) how much of this region is leading twist?

22 Truncated moments truncated moments obey DGLAP-like evolution equations, similar to PDFs dm n ( x, Q 2 ) d log Q 2 = α s 2π ( ) P (n) M n ( x, Q 2 ) where modified splitting function is P (n) (z, α s)=z n P NS,S (z, α s ) can follow evolution of specific resonance (region) with in pqcd framework! Q 2 suitable when complete moments not available

23 Data analysis assume data at large enough Q 2 are entirely leading twist evolve fit to data at large Q 2 down to lower Q 2 apply target mass corrections (TMC) and compare with low- Q 2 data TMC W 2 = M 2 + Q2 x (1 x) Psaker, WM, et al., Phys. Rev. C 78 (2008)

24 consider individual resonance regions: W 2 thr <W 2 < 1.9 GeV 2 (1232) 1.9 <W 2 < 2.5 GeV 2 S 11 (1535) 2.5 <W 2 < 3.1 GeV 2 F 15 (1680) as well as total resonance region: W 2 < 4 GeV 2

25 entire resonance region Psaker, WM, et al., Phys. Rev. C 78 (2008)

26 larger S 11 HT small HT Psaker, WM, et al., Phys. Rev. C 78 (2008) higher twists < 10-15% for Q 2 > 1 GeV 2

27 Proton vs. Neutron

28 Is duality in the proton a coincidence? consider symmetric nucleon wave function cat s ears diagram (4-fermion higher twist ~ 1/Q 2 ) ( ) 2 e i e j e 2 i e 2 i i j i i coherent incoherent proton neutron HT 1 HT 0 ( ) 9 ( ) = 0! 0 need to test duality in the neutron!

29 How can the square of a sum become the sum of squares? in hadronic language, duality is realized by summing over at least one complete set of even and odd parity resonances Close, Isgur, Phys. Lett. B509 (2001) 81 in NR Quark Model, even and odd parity states generalize to 56 (L=0) and 70 (L=1) multiplets of spin-flavor SU(6) representation Total p F 1 n F 1 p g 1 n g (3 ) 2 /4 8 2 (3 ) 2 / ( )/ (3 ) 2 /4 4 2 (3 ) 2 / ( )/2 λ (ρ) = (anti) symmetric component of ground state wfn. Close, WM, Phys. Rev. C68 (2003)

30 SU(6) limit λ = ρ relative strengths of N N* transitions: SU(6) : [56, 0 + ] 2 8 [56, 0 + ] 4 10 [70, 1 ] 2 8 [70, 1 ] 4 8 [70, 1 ] 2 10 total F p F1 n g p g1 n summing over all resonances in 56 + and 70 - multiplets R np = F n 1 F p 1 = 2 3 as in quark-parton model (for u=2d)! proton sum saturated by lower-lying resonances expect duality to appear earlier for p than n Close, WM, Phys. Rev. C68 (2003)

31 Extraction of Neutron Structure Function

32 Problem: no free neutron targets! (neutron half-life ~ 12 mins) use deuteron as effective neutron target But: deuteron is a nucleus, and F d 2 F p 2 + F n 2 nuclear effects (nuclear binding, Fermi motion, shadowing) obscure neutron structure information need to correct for nuclear EMC effect

33 nuclear impulse approximation γ d N incoherent scattering from individual nucleons in d (good approx. at x >> 0) F d 2 (x, Q 2 )= x nucleon momentum distribution in d ( smearing function ) N=p+n dy f(y, γ) F N 2 (x/y, Q 2 ) + δ (off) F d 2 off-shell correction (~1%) at finite, smearing function depends also on parameter Q 2 γ = q /q 0 = 1+4M 2 x 2 /Q 2 Kulagin, WM, Phys. Rev. C 77, (2008)

34 N momentum distributions in d broader with increasing γ for most kinematics γ 2

35 Unsmearing - additive method calculated depends on input F d 2 F n 2 extracted n depends on input n... cyclic argument Solution: iteration procedure subtract from d data: 0. δ (off) F2 d F2 d F2 d δ (off) F2 d 1. define difference between smeared and free SFs F d 2 F p 2 = F n 2 f F n 2 F n first guess for F n(0) 2 (0) = F n(0) 2 F n 2 3. after one iteration, gives F n(1) 2 = F n(0) 2 +( F n 2 F n(0) 2 ) 4. repeat until convergence

36 Unsmearing - test of convergence F d 2 constructed from known and inputs F p 2 F n 2 (using MAID resonance parameterization) initial guess F n(0) 2 =0* * even faster convergence if choose F n(0) 2 = F p 2 Kahn, WM (2008) can reconstruct almost arbitrary shape

37 2 Unsmearing - Q dependence important to use correct γ dependence in extraction does not converge to correct shape important also in DIS region (do not have resonance benchmarks ) Kahn, WM (2008)

38 JLab Hall B d & p data quasi-elastic peak Kahn, WM (2008) first extraction of F n 2 in resonance region

39 works also in DIS region Kahn, WM (2008)

40 Structure function integral ratios Data / Leading Twist neutron proton Preliminary largest HT for S 11 smallest HT for Kahn, WM (2008) neutron HT indeed larger than proton!

41 Summary Remarkable confirmation of quark-hadron duality in proton structure functions duality violating higher twists ~ 10% in few-gev range Truncated moments firm foundation for study of local duality in QCD HTs largest in region, smallest in region S 11 Duality in the neutron extraction of neutron structure function from deuteron data neutron HTs larger than proton HTs (as expected from quark models)

42 Future Complete analysis of neutron structure function extraction quantify isospin dependence of HTs Application to spin-dependent structure functions extraction method works also for functions with zeros Cross-check with neutron extracted from 3 He data

43 The End