# Presentation: 754113

## The Parton-Hadron Transition in Structure Functions and MomentsRolf EntJefferson Lab

- The Parton-Hadron Transition in Moments of Structure Functions
- Quark-Hadron Duality: How Local is the Transition?
- Applications and Theoretical Understanding
- Summary

**The Parton-Hadron Transition in Structure Functions and Moments**Rolf EntJefferson Lab

**Science & Technology Review**
**June 2003**

##  How are the Nucleons Made from Quarks and Gluons?

- How do we understand QCD in the confinement regime?
-      A) What are the spatial distributions of u, d, and s quarks in the hadrons?
-       B) What is the excited state spectrum of the hadrons, and what
-            does it reveal about the underlying degrees of freedom?
-       C) What is the QCD basis for the spin structure of the hadrons?
- 	Q2 evolution of structure functions and their moments
- 	Extended GDH sum rule, Bjorken sum rule
-       D) What can other hadron properties tell us about ‘Strong QCD’?
-  	Inclusive Resonance Electroproduction and Quark-Hadron Duality
- 	Q2 evolution of structure functions and their moments
**              **

## QCD and the Strong Nuclear Force QCD has the most bizarre properties of all the forces in nature

**Confinement:**
**restoring force between quarks at large distances equivalent to 10 tons, *****no matter**** ****how far apart***
**Asymptotic freedom:**
**quarks feel almost no strong force when closer together**

**QCD (+ electro-weak) in principle describes all of nuclear physics - at all distance scales - ****	****	*****but how does it work?***

##    QCD and the Parton-Hadron Transition

**   ****QCD and the Parton-Hadron Transition**

-   Hadrons
-      Q < 
******s**(Q) > 1

-  Constituent
-     Quarks
-       Q > 
******s**(Q) large

- Asymptotically
-  Free Quarks
-       Q >> 
******s**(Q) small

- One parameter, QCD,
- ~  Mass Scale or
-     Inverse Distance Scale
-     where as(Q) = 1
- “Separates” Confinement
-   and Perturbative Regions
- Mass and Radius of the Proton are (almost) completely governed by

- 

- QCD213 MeV

##    QCD and the Parton-Hadron Transition

- ~  Mass Scale or
-     Inverse Distance Scale
-     where as(Q) = 1
- “Separates” Confinement
-   and Perturbative Regions
- Mass and Radius of the Proton are (almost) completely governed by

**   ****QCD and the Parton-Hadron Transition**

- QCD213 MeV

## Inclusive Electron Scattering – Formalism

- Polarized structure functions g1(x,Q2) and g2(x,Q2)

- Q2 :Four-momentum transfer
- x : Bjorken variable (=Q2/2M)
-  : Energy transfer
- M : Nucleon mass
- W : Final state hadronic mass

- L

- T

- U

**Inclusive Electron Scattering – Formalism**

## The Parton-Hadron Transition in Structure Functions and Moments

- The Parton-Hadron Transition in Moments of Structure Functions
- Quark-Hadron Duality: How Local is the Transition?
- Applications and Theoretical Understanding
- Summary

**The Parton-Hadron Transition in Structure Functions and Moments**

##    QCD and Moments

**   ****QCD and Moments**

- Moments of the Structure Function   Mn(Q2) = **  **dx** **xn-2F(x,Q2)

- 0

- 1

- Let F(x,Q2) = xf(x) be a probability density distribution,
- describing the velocity distribution in a dilute gas
- 			M1 = 1
- 			M2 = <x>		average velocity
- 			M3 = <x2>	variance
- 			etc.

- In QCD the moments are also dependent on Q2 due to Q2 evolution of
- probability density distributions  <x>, <x2> vary with Q2

## Moments of the Structure Function    Mn(Q2) =   dx xn-2F(x,Q2)
Operator Product Expansion
	 Mn(Q2) =  (nM02/ Q2)k-1 Bnk(Q2) 
	  		             higher               logarithmic
			              twist                dependence

- Moments of the Structure Function    Mn(Q2) = **  **dx** **xn-2F(x,Q2)
- Operator Product Expansion
- 	 Mn(Q2) = ****** **(nM02/ Q2)k-1 Bnk(Q2)
- 	  		             higher               logarithmic
- 			              twist                dependence

**   ****QCD and the Operator-Product Expansion**

- 0

- 1

- k=1

- 

- At High Q2 : ln(Q2) dependence of
-                       moments one of first
-                       “proofs” of QCD
- 		 (QCD)
- At Low Q2 : Unique regime for JLab
-                       to determine (1/Q2)m
-                          Higher Twist effects
- Lowest moments are calculable in Lattice QCD (LQCD): least computer intensive!

##     Moments of F2p @ Low Q2

**    ****Moments**** of ****F****2****p**** @ Low Q****2**

- n = 2

- n = 4

- n = 6

- n = 8

- 50% of momentum
-  carried by quarks
-  (Momentum Sum Rule)

##     Moments of F2p @ Low Q2

- Proton Charge
- (Coulomb Sum Rule)

-    Elastic
- contribution

- n = 2

- n = 4

- n = 6

- n = 8

- 50% of momentum
-  carried by quarks
-  (Momentum Sum Rule)

- (A combination of Hall C + CLAS data presented by VB to constrain
- the Constituent Quark radius, here only Hall C data are shown)

##     Moments of F2p @ Low Q2

- Proton Charge
- (Coulomb Sum Rule)

-    Elastic
- contribution

- n = 2

- n = 4

- n = 6

- n = 8

- 50% of momentum
-  carried by quarks
-  (Momentum Sum Rule)

- total

- n = 2

- elastic

- -region

- W2 > 4 GeV2 (“DIS”)

- S11-region

- @ Q2 = 2 (GeV/c)2 30% of M2
- comes from the resonance region

## n = 2 Moments of F2, F1 and FL:  Mn(Q2) =   dx xn-2F(x,Q2)

**n = 2 Moments of F****2****, F****1**** and F****L****:****  ****M****n****(Q****2****) =  ** **dx** **x****n-2****F(x,Q****2****)**

_**Elastic**__** **__**Contributions**_

- Flat Q2 dependence   small higher twist!  - not true for contributions from the elastic peak (bound quarks)

**Elastic**** ****contribution**
** ****excluded**

- DIS:  SLAC fit to F2 and R
- Resonances: E94-110 (Hall C) fit

**F****1****EL**** = G****M****2 ********(*****x*****-1)**

**F****2****EL**** = (G****E****2**** +**** ********G****M****2**** )********(*****x*****-1)**

**F****L****EL**** = G****E****2 ********(*****x*****-1)**

**1 +**** ******

****** ****= Q****2****/4M****p****2**

- Preliminary

**F****2**

**F****1**

**F****L**

- 0

- 1

## n = 4 Moments of F2, F1 and FL

**n = 4 Moments of F****2****, F****1**** and F****L**

**Neglecting elastic contribution,**
**n = 4 moments have only a **
**small Q****2**** dependence as well.**

_**Momentum sum rule**_

- This is only at leading twist and for zero proton mass
- ⇒ 	Must remove non-zero proton	mass effects from data to 	extract moment of xG(x,Q2)
- ⇒	Work in progress

- Preliminary

**M****L****(n)**** =  ********s****(Q****2****){ 4****M****2****(n)**** +  2c****∫****dx *****x*****G(*****x*****,Q****2****)} **

**3(n+1)**

**(n+1)(n+2)**

***Gluon distributions!***

**Elastic contribution**
** ****excluded**

##     Moments of g1p (=1p)

-   30% of Spin
- carried by quarks
- (Ellis-Jaffe Sum Rule)

- ~  of proton
- (GDH Sum Rule)

- Elastic not included in
- Moment as shown 
- With Elastic included
-   no zero crossing, and
-   Q2 dependence far
-   smoother

- CLAS EG1 Data

##     Moments of g1p (=1p)

-   30% of Spin
- carried by quarks
- (Ellis-Jaffe Sum Rule)

- ~  of proton
- (GDH Sum Rule)

- Zero crossing mainly due to cancellation of  (negative)
- and S11 Resonances

- Elastic not included in
- Moment as shown 
- With Elastic included
-   no zero crossing, and
-   Q2 dependence far
-   smoother

- CLAS EG1 Data

##     Moments of g1p (=1p)

-   30% of Spin
- carried by quarks
- (Ellis-Jaffe Sum Rule)

- ~  of proton
- (GDH Sum Rule)

- Elastic not included in
- Moment as shown 
- With Elastic included
-   no zero crossing, and
-   Q2 dependence far
-   smoother

- = 1/2 – 3/2

- SU(6) unbroken: Mp = M
-  ground state of 3/2

- In this scenario the is
- a  function and all higher
- twist  it plays the same
- role as the elastic in F2

- CLAS EG1 Data

- Zero crossing mainly due to cancellation of  (negative)
- and S11 Resonances

## Moments of g1n and g1p-g1n

- 1n

- Hall A 3He(e,e’) to extract
- g1n and its moment 1n  (E94-010)
- Similar ideas as with proton
- Here, whole region negative

## Moments of g1n and g1p-g1n

**Moments of g****1****n ****and g****1****p****-g****1****n**

- 1n

- Hall A 3He(e,e’) to extract
- g1n and its moment 1n
- Similar ideas as with proton
- Here, whole region negative

- Combine Hall A g1n
- with Hall B g1p data

## Moments of g1n and g1p-g1n

**Moments of g****1****n ****and g****1****p****-g****1****n**

- 1n

- 1p - 1n

- Bjorken Sum Rule
- (Verification of QCD)

- Hall A 3He(e,e’) to extract
- g1n and its moment 1n
- Similar ideas as with proton
- Here, whole region negative

- Combine Hall A g1n
- with Hall B g1p data

- Chiral Perturbation
- Theory (PT)

## Summary I - Moments 

- Resonances are an integral part of the Structure Function
- Moments at Low Q2
- 	(Note: QCD deals with the Moments and does not
- 	care what contributes to the moments!)
- The Structure Function Moments have a smooth behavior
- as a function of Q2, and in fact pick up almost uniquely the
- quark-quark interactions in the SU(6) ground states
- This seems to indicate that the parton-hadron transition
- occurs in a local and small region, with only few resonances
- 	 “Quark-Hadron Duality”
- 	 Let us examine the structure functions
- 	    themselves rather than their moments

## The Parton-Hadron Transition in Structure Functions and Moments

- The Parton-Hadron Transition in Moments of Structure Functions
- Quark-Hadron Duality: How Local is the Transition?
- Applications and Theoretical Understanding
- Summary

**The Parton-Hadron Transition in Structure Functions and Moments**

## Quark-Hadron Dualitycomplementarity between quark and hadron descriptions of observables

- 	Hadronic Cross Sections
- 	averaged over appropriate energy range
-                  hadrons

-     Perturbative
-    Quark-Gluon Theory
- 

**=**

**At ****high enough**** energy:**

- quarks+gluons

***Can use****** ******either****** ******set of complete basis states to describe physical phenomena***
***But why also in limited local energy ranges?***

## 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

***Integrated F******2****** 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**** ****F****2**** ****Structure**** ****Function**

- ’ = 1+W2/Q2

- Q2 = 0.5

- Q2 = 0.9

- Q2 = 1.7

- Q2 = 2.4

- F2

- F2

##  Rosenbluth Separations 

** ****Rosenbluth Separations**** **

**Hall C E94-110: a global survey of longitudinal strength in the resonance region......**

- T

- L +T

- = L T

- /

- (polarization of
-  virtual photon)

## 
Spread of points about the
  linear fits is Gaussian with
   ~ 1.6 % consistent with
  the estimated point-point
  experimental uncertainty
  (1.1-1.5%)
a systematic “tour de force”

**Spread of points about the**
**  ****linear fits is Gaussian with**
**  ******** ****~ 1.6 %**** consistent with**
**  ****the estimated point-point**
**  ****experimental uncertainty**
**  ****(1.1-1.5%)**
**a systematic “tour de force”**

**Hall C E94-110: a global survey of longitudinal strength in the resonance region......**

** ****Rosenbluth Separations**** **

## World's L/T Separated Resonance Data

***(All data for Q******2****** < 9 (GeV/c)******2******)***

**R = ********L****/********T**

- <

**R = ********L****/********T**

**<**

## World's L/T Separated Resonance Data

**Now able**** to study the Q****2**** dependence of individual resonance regions!**
**   **
**Clear resonant behaviour ****can be observed!**
**Use R to extract**
**	****	****	****	****F****2****, F****1****, F****L**

***(All data for Q******2****** < 9 (GeV/c)******2******)***

**R = ********L****/********T**

**<**

**R = ********L****/********T**

## Now can truly obtain F2 structure function data, and compare with DIS fits or QCD calculations/fits (CTEQ/MRST)
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 such low Q2

- Now can truly obtain F2 structure function data, and compare with DIS fits or QCD calculations/fits (CTEQ/MRST)
- Use Bjorken x instead of Bloom-Gilman’s ’
**Bjorken Limit: Q****2****, ******** ******** ******
**Empirically, DIS region is where logarithmic scaling is observed: Q****2 ****> 5 GeV****2****,**
**    ****W****2 ****> 4 GeV****2**
**Duality: Averaged over W, logarithmic scaling**** ****observed to work also for Q****2 ****> 0.5 GeV****2****, W****2 ****< 4 GeV****2****, resonance regime**
**   ****(note: x = Q****2****/(W****2****-M****2****+Q****2****)**
**JLab results: Works quantitatively to better than 10% at such low Q****2**

**Duality**** ****in**** ****the**** ****F****2**** ****Structure**** ****Function**

## Duality in FT and FL Structure Functions

**Duality in F****T**** and F****L**** Structure Functions**

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

## 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)

- 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

- 

## Duality more easily established in Nuclei

**The nucleus does the averaging for you!**

- EMC Effect
- Fe/D

- Resonance
- Region Only

- (Fe/D) IS

- 

- (= x corrected for M  0)

- Nucleons have
- Fermi motion
- in a nucleus

## CLAS: N- transition region turns positive at Q2 = 1.5 (GeV/c)2 
Elastic and N- transition cause most of the higher twist effects

**CLAS: N-******** transition region turns positive at Q****2**** = 1.5 (GeV/c)****2**** **
**Elastic and N-******** transition cause most of the higher twist effects**

**... ****but tougher in Spin Structure Functions**

**g****1****p**

**CLAS EG1**

- 

- Pick up effects of both N and 
- (the  is not negative enough....)

## The Parton-Hadron Transition in Structure Functions and Moments

- The Parton-Hadron Transition in Moments of Structure Functions
- Quark-Hadron Duality: How Local is the Transition?
- Applications and Theoretical Understanding
- Summary

**The Parton-Hadron Transition in Structure Functions and Moments**

## 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 interpretation of neutrino oscillation data
- Proposal to PAC24 to extend JLab data to help with this analysis
- They are also interested in testing 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

## Quark-Hadron Duality – Theoretical Efforts

**Quark-Hadron Duality – Theoretical Efforts**

- _N. Isgur et al : N__c__ ____ ∞_
-         **qq**   infinitely narrow resonances
-         **qqq**             only         resonances

- Distinction between Resonance and
-       Scaling regions is spurious
- Bloom-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 Model_
- How 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
- Similar arguments for e.g. DVCS
-    and semi-inclusive reactions

*u*

## Summary

**Summary**

- Despite the large value of ****s(Q2) at Q2 < QCD2, the transition from a QCD description in terms of hadrons to a QCD description in terms of partons occurs smoothly, in the region Q2 < 2 (GeV/c)2
- Quarks and the associated Gluons (the Partons) are tightly bound in Hadrons due to Confinement
- Still, they rely on camouflage as their best defense:
- 	a limited number of confined states acts as if consisting of	free quarks  Quark-Hadron Duality, which is a non-trivial 	property of QCD, telling us that contrary to naïve expectation 	quark-quark correlations tend to cancel on average
- Integrating over all confined states by forming a moment similarly shows a smooth parton-hadron transition in both spin-averaged and spin-dependent structure functions, to compare with LQCD and PT