Duality in Unpolarized Structure Functions: Validation and

Application

First Workshop on Quark Hadron Duality and the Transition to

Perturbative QCDFrascati June 6-8, 2005 (Italy)

Rolf Ent

First observed ~1970 by Bloom and Gilman at SLAC by comparing resonance production data with deep inelastic scattering data

• Integrated F2 strength in Nucleon Resonance region equals strength under scaling curve. Integrated strength (over all ’) is called Bloom-Gilman integral

Shortcomings:

• Only a single scaling curve and no Q2 evolution (Theory inadequate in pre-QCD era)

• No L/T separation F2 data depend on assumption of R = L/T

• Only moderate statistics

Duality in the F2 Structure Function

’ = 1+W2/Q2

Q2 = 0.5 Q2 = 0.9

Q2 = 1.7 Q2 = 2.4

F 2

F 2

First observed ~1970 by Bloom and Gilman at SLAC

Now can truly obtain F2 structure function data, and compare with DIS fits or QCD calculations/fits (CTEQ/MRST/GRV)

Use Bjorken x instead of Bloom-Gilman’s ’

Bjorken Limit: Q2, 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

(note: x = Q2/(W2-M2+Q2) JLab results: Works

quantitatively to better than 10% at surprisingly low Q2

Duality in the F2 Structure Function

Bloom-Gilman Duality in F2

Today

The scaling curve determines the Q2 behavior of the resonances! … or … the resonances “fix” the

scaling curve!

Low Q2 Too! Even Down to Photoproduction

Bodek and Yang duality-based model works at all Q2 values tested – DIS to photoproduction!

Quantification: Resonance Region F2

w.r.t. Alekhin NNLO Scaling Curve (Q2 ~ 1.5 GeV2)

Quantification in a Fixed W2 Framework(S. Liuti et al., PRL 2002)

F2exp/F2

pQCD+TMC = 1 + CHT(x)/Q2 + H(x,Q2)

Previous Extractions (DIS Only)

Duality not only a leading-twist phenomenon

If one integrates over all resonant and non-resonant states, quark-hadron duality should be shown by any model. This is simply unitarity.However, quark-hadron duality works also, for Q2 > 0.5 (1.0) GeV2, to better than 10 (5) % for the F2 structure function in both the N- and N-S11 region!(Obviously, duality does not hold on top of a peak! -- One needs an appropriate energy range)

One resonance + non-resonant background

Few resonances + non-resonant background

Why does local quark-hadron duality work so well, at such low

energies?~ quark-hadron transition

Confinement is local ….

Quark-Hadron Duality – Theoretical EffortsN. Isgur et al : Nc ∞ qq infinitely narrow resonances qqq only resonances

Distinction between Resonance and Scaling regions is spuriousBloom-Gilman Duality must be invoked even in the Bjorken Scaling region

Bjorken Duality

One heavy quark, Relativistic HO

Scaling occurs rapidly!

Q2 = 5

Q2 = 1

F. Close et al : SU(6) Quark ModelHow many resonances does one needto average over to obtain a completeset of states to mimic a parton model?56 and 70 states o.k. for closureSimilar arguments for e.g. DVCS and semi-inclusive reactions

u

Quark-Hadron Duality - Applications

• CTEQ currently planning to use duality for large x parton distribution modeling

• Neutrino community using duality to predict low energy (~1 GeV) regime Implications for exact neutrino mass Plans to extend JLab data required and to test duality with

neutrino beams• Duality provides extended access to large x regime• Allows for direct comparison to QCD Moments

Lattice QCD Calculations now available for u-d (valence only) moments at Q2 = 4 (GeV/c)2

Higher Twist not directly comparable with Lattice QCD If Duality holds, comparison with Lattice QCD more robust

G0

SOS

HMS

E94-110/E00-116/E00-108 Experiments performed at JLab-Hall C

“The successful application of duality to extract known quantities suggests that it should also be possible to use it to extract quantities that are otherwise kinematically inaccessible.” (CERN Courier, December 2004)

E94-110 : Precise Measurement of SeparatedStructure Functions in Nucleon Resonance Region

The resonance region is, on average, well described by NNLO QCD fits. This implies that Higher-Twist (FSI) contributions cancel, and are on average small. The result is a smooth transition from Quark Model Excitations to a Parton Model description, or a smooth quark-hadron transition. This explains the success of the parton model at relatively low W2 (=4 GeV2) and Q2 (=1 GeV2).

Duality in FDuality in FTT and F and FLL Structure Functions Structure Functions

Duality works well for both FT and FL above Q2 ~ 1.5 (GeV/c)2

QuantificationRatio Resonance Region / Scaling

CurvesLarge x Structure Functions

SLAC DIS dataQ2 = 6 GeV2

xUndershoot at large x

E00-116: Experiment to Push to larger Q2 (larger x)Emphasis: 1) With Duality Verified at Lower Q2, Now Can Use Data to Extend Range in x; 2) Need Large x data to Constrain Higher Moments; 3) Quantify Q2 vs. Q2(1-x) Evolution

Preliminary data again do overshoot QCD calculations

But show great numerical agreement compared to (internal) scaling curve determines Q2 behavior of the resonances

E00-116: Experiment to Push to larger Q2 (larger x)

Same behavior at largest Q2 of E00-116 analyzed to date

Additional data (4 more resonance scans) at higher Q2 for both proton and deuteron reach x ~ 0.9. Data final in 1-2 months.

Close and Isgur Approach to Duality

Phys. Lett. B509, 81 (2001):

q =h

Relative photo/electroproduction strengths in SU(6)

“The proton – neutron difference is the acid test for quark-hadron duality.”

How many states does it take to approximate closure?

Proton W~1.5 o.k.

Neutron W~1.7 o.k.

Detect 70 MeV/c spectator proton at backward angles in “BONuS” detector

Scatter electrons off deuteron

The BONuS experiment will measure neutron structure

functions…….

Commissioning Run in CLAS now!

Neutron Quark-Hadron Duality – Projected Results

(CLAS/BONUS)

Sample neutron structure function spectra

Plotted on proton structure function model for example only

Neutron resonance structure function essentially unknown

Smooth curve is NMC DIS parameterization

Systematics ~5%

Duality in Meson Electroproduction(predicted by Afanasev, Carlson, and Wahlquist, Phys. Rev. D 62, 074011 (2000))

(e,e’)

(e,e’m)

W2 = M2 + Q2 (1/x – 1)

W’2 = M2 + Q2 (1/x – 1)(1 - z) z = Em/

For Mm small, pm collinear with , and Q2/2 << 1

Hall C Experiment E00-108 Spokespersons:Hamlet Mkrtchyan (Yerevan)Gabriel Niculescu (JMU)Rolf Ent (JLab)H,D(e,e’+/-)X

Duality in Meson Electroproduction

hadronic description quark-gluon description

TransitionForm Factor

DecayAmplitude

Fragmentation Function

Requires non-trivial cancellations of decay angular distributions If duality is not observed, factorization is

questionable

Duality and factorization possible for Q2,W2 3 GeV2 (Close and Isgur, Phys. Lett. B509, 81 (2001))

Factorization

P.J. Mulders, hep-ph/0010199 (EPIC Workshop, MIT, 2000)

At large z-values easier to separate current and target fragmentation region for fast hadrons factorization (Berger Criterion) “works” at lower energies

At W = 2.5 GeV: z > 0.4 At W = 5 GeV: z > 0.2 (Typical JLab) (Typical HERMES)

E00-108 data Experiment ran in August 2003 Close-to-final data analysis Small Q2 variation not corrected yet

No visual “bumps and valleys” in pion ratios off deuterium

x = 0.32

x = 0.32

W’ > 1.4 GeV

W’ < 2.0 GeV

From deuterium data: D-/D+ = (4 – N+/N

-)/(4N+/N

- - 1)

D-/D+ ratio should be independent of x

but should depend on z

Find similar slope versus z as HERMES, but slightly larger values for D-/D+, and definitely not the values expected from Feynman-Field independent fragmentation at large z!

z = 0.55

x = 0.32

W’ = 2 GeV ~ z = 0.35 data predominantly in “resonance region”What happened to the resonances?

From deuterium data: D-/D+ = (4 – N+/N

-)/(4N+/N

- - 1)

The Origins of Quark-Hadron Duality – Semi-Inclusive Hadroproduction

Destructive interference leads to factorization and duality

Predictions: Duality obtained by end of second resonance region Factorization and approximate duality for Q2,W2 < 3 GeV2

F. Close et al : SU(6) Quark ModelHow many resonances does one need to average over to obtain a complete set of states to mimic a parton model? 56 and 70 states o.k. for closure

How Can We Verify Factorization?Neglect sea quarks and assume no pt dependence to parton distribution functions

Fragmentation function dependence drops out

[p(+) + p(-)]/[d(+) + d(-)]

= [4u(x) + d(x)]/[5(u(x) + d(x))]

= p/d independent of z and pt

Dependence on z

Still to quantify (preliminary data only) …

Lund MC

Berger Criterion W’ > 1.4

GeV

Dependence on pt

… but data not inconsistent with factorization

Lund MC

Dependence on x

Lund MC

… trend of expected x-dependence, but slightly low?

Duality in Unpolarized Structure Functions: Validation and

Application

“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)

Duality was well established in the ’70s in the pre- and early-QCD era.

The existence of duality is well known, based on Regge theory, and these ideas have not changed since that time.

Modern studies show that duality works quantitatively better than expected, and works in all observables tested to date.

- Hence, duality can be used as a tool for other studies.

The origin of duality is the subject of modern theoretical research.

• Moments of the Structure Function Mn(Q2) = 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

• Duality is described in the Operator Product Expansion as higher twist effects being small or canceling

DeRujula, Georgi, Politzer (1977)

QCD and the Operator-Product Expansion

0

1

k=1

Example: e+e- hadrons Textbook Example

Only evidence of hadrons produced is narrow states oscillating around step function

R =

Factorization

Both Current and Target Fragmentation Processes Possible. At Leading Order:

Berger Criterion > 2 Rapidity gap for factorization

Separates Current and Target Fragmentation Region in Rapidity