1. Hampton University / Jefferson Lab
Quark-Hadron Duality
Cynthia Keppel
Hampton University / Jefferson Lab
CIPANP 2003

2. Quark-Hadron Duality complementarity between quark and hadron descriptions of observables
At high energies: interactions between quarks and gluons become weak
(“asymptotic freedom”)
efficient description of phenomena afforded in terms of quarks
At low energies: effects of confinement make strongly-coupled QCD highly non-perturbative
collective degrees of freedom (mesons and baryons) more efficient
Duality between quark and hadron descriptions
reflects relationship between confinement and asymptotic freedom
intimately related to nature and transition from non-perturbative to perturbative QCD

3. Example: e+e- hadrons / m+m- lim. s(e+e- X) = NC S eq2 E. 
Example: e+e hadrons / m+m- lim s(e+e X) = NC S eq2 E  s(e+e m+m-) q

4. Duality in Inclusive electron scattering
12/8/2018
Duality in Inclusive electron scattering
Single Photon Exchange
Elastic
Resonance
Deep Inelastic
In terms of g coupling:
ds = G [s T(x,Q2) + e s L(x,Q2)}
dWdE'
Where
G : flux of transversely polarized virtual photons
e : relative longitudinal polarization
ds ∝ G [2xF1(x,Q2) + e FL(x,Q2)]
Alternatively:
dW dE'
FL = F2 – 2xF1 + 2MpF2
R = s L / s T = FL / 2xF1
F2 prop s L + s T

5. Duality in the F2 Structure Function
First observed ~1970 by Bloom and Gilman at SLAC
Bjorken Limit: Q2, n  
Empirically, DIS region is where logarithmic scaling is observed: Q2 > 5 GeV2, W2 > 4 GeV2
Duality: Averaged over W, logarithmic scaling observed to work also for Q2 > 0.5 GeV2, W2 < 4 GeV2, resonance regime

6. 12/8/2018
What about the other structure functions FL, F1, g1,….? World's L/T Separated Resonance Data (before 2002):
R = sL/sT
Not able to study the Q2 dependence of individual resonance regions!
No resonant behaviour can be observed!
(All data for Q2 < 9 (GeV/c)2)
JLab E94-110: a global survey of longitudinal strength in the resonance region…...

7. 12/8/2018
JLab Hall C E94-110: Global Survey of Longitudinal Strength in Nucleon Resonance Region
Covers 0.4 < Q2 < 5.0 (GeV/c)2, Mp < W2 < 4.0 GeV2
Clear resonant behaviour can be observed!
Now able to study the Q2 dependence of individual resonance regions!
R = sL/sT
<
(All data for Q2 < 9 (GeV/c)2)
Now able to extract F2, F1, FL and study duality!...

8. Rosenbluth Separations
12/8/2018
Rosenbluth Separations
180 L/T separations total (most with 4-5 e points)
Spread of points about the linear fits is fairly Gaussian with s ~ 1.6 %- consistent with the estimated pt-pt experimental uncertainty
a systematic “tour de force”

9. Duality now observed in all unpolarized structure functions

10. …and in Nuclei (F2)
x = 2x[1 + (1 + 4M2x2/Q2)1/2]
GRV curve p d Fe

11. Scaling (F2) in Nuclei

12. Duality and the EMC Effect
Medium modifications to the pdfs are the same in the resonance region
Rather surprising (deltas in nuclei, etc.)
J. Arrington, et al., in preparation

13. …and in Spin Structure Functions
HERMES
JLab Hall B
A1p
A1p g1

14. Qualitatively, duality is observed to hold in all unpolarized structure functions, in nuclei, and in tested spin structure functions down to surprisingly low Q2
Apparently a non-trivial property of nucleon structure
If we had used only scintillators, scaling would be thought to hold down to low Q2!

15. Integral Ratio Res / Scaling
Quantification
Integral Ratio Res / Scaling

16. Quantification
The available pdf-based parameterizations significantly undercut the data at large x
SLAC data above W2 = 4 GeV2

17. Duality in QCD Moments of the Structure Function
Mn(Q2) = S dx xn-2F(x,Q2)
If n = 2, this is the Bloom-Gilman duality integral!
Operator Product Expansion
Mn(Q2) =  (nM02/ Q2)k-1 Bnk(Q2)
higher twist logarithmic dependence
(pQCD)
Duality is described in the Operator Product Expansion as higher twist effects being small or cancelling DeRujula, Georgi, Politzer (1977) 1 
k=1

18. 12/8/2018 1
Mn(Q2) = S dx xn-2F(x,Q2) F2
+ elastics……

19. n = 2 Moments of F2, F1 and FL: Mn(Q2) = S dx x2-2F(x,Q2)
12/8/2018 1
n = 2 Moments of F2, F1 and FL: Mn(Q2) = S dx x2-2F(x,Q2)
DIS: SLAC fit to F2 and R
RES: E resonance fit
Elastic Contributions
Preliminary
F1EL = GM2 d(x-1) F2
F2EL = (GE2 + tGM2 )d(x-1)
Elastic contribution
excluded
1 + t F1
t = q2/4Mp2
FLEL = GE2 d(x-1) FL
Flat Q2 dependence  small higher twist! - not true for contributions from the elastic peak (bound quarks)

20. n = 4 Moments of F2, F1 and FL Gluon distributions! Preliminary
12/8/2018
n = 4 Moments of F2, F1 and FL
Preliminary
Neglecting elastics, n = 4
moments have only a small
Q2 dependence as well.
Momentum sum rule
ML(n) = as(Q2){ 4M2(n) + 2c∫dx xG(x,Q2)}
3(n+1)
(n+1)(n+2)
Gluon distributions!
This is only at leading twist and neglecting TM effects.
⇒ Must remove TM effects from data to extract moment of xG…we’re working on it…..

21. For the future….

22. Measuring Neutron Structure Functions: BONUS
12/8/2018
Electron detected in JLab Hall B CLAS spectrometer
Spectator proton detected in RTPC p e- n
Hall B CLAS spectrometer for electron detection
Thin deuterium target (7.5 atm)
Radial Time Projection Chamber (RTPC) for low momentum spectator proton detection
DVCS solenoid to contain Moller background

23. “Very Important Protons”
12/8/2018
“Very Important Protons”
Deuteron ~ free proton + free neutron at small nucleon momenta
Will target Tp ~ 2 – 5 MeV spectator protons
30% of momentum distribution is in
chosen ps range
Tp > 5 MeV spectators will also be detected

24. Duality in Meson Electroproduction Duality and factorization possible for Q2,W2  3 GeV2 (Close and Isgur, Phys. Lett. B509, 81 (2001))
hadronic description quark-gluon description
Requires non-trivial cancellations of decay angular distributions
If duality is not observed, factorization is questionable
ds/dz  iei2qi(x,Q2)Dqim(z,Q2) + qi(x,Q2)Dqim(z,Q2)

25. (Semi-)Exclusive Meson Electroproduction
Large z = Eh/n to emphasize duality and factorization (Berger criterion)
Meson electroproduced along q, i.e. emphasize forward angles
Proposed SHMS in Hall C well suited to detect these mesons (cf. pion form factor)
If Berger criterion and duality  factorization

26. MINERn-A FermiLab proposal en route…..
Can test duality in neutrino scattering!
(Melnitchouk and Close (2003), Beane (2001),….)
Can also help with large x pdfs

27. Summary
Quark-hadron duality is a non-trivial property of nucleon structure
Duality has been shown to hold in all experimental tests thus far
All unpolarized structure functions
Polarized structure functions
Nuclei
More experiments are planned
Neutron
Semi-inclusive
Neutrino scattering
Duality may provide a valuable tool to access high x regime
Duality violations obscure comparison with lattice QCD through the structure function moments

28. “It is fair to say that (short of the full solution of QCD) understanding and controlling the accuracy of quark-hadron duality is one of the most important and challenging problems for QCD practitioners today.” M. Shifman, Handbook of QCD, Volume 3, 1451 (2001)

29. 12/8/2018
RTPC Design