Simona MalaceUniversity of South Carolina

Overview

Standard pQCD fits and their limitations (example => CTEQ6)

Another kind of QCD fits: extension of fits at larger x in the nonperturbative region (example => Alekhin)

Can we go even further? Quark-hadron duality: => experimental observation & working hypothesis for PDFs extension at large x => recent results from Jlab on quark-hadron duality in the F2

p,d structure function => Quark-hadron duality in F2

n

Plans for future

Complete picture(or closer to …)

Naive picture

Operator Product Expansion in pQCD:

leading-twist higher-twist

The quark and gluon structure of the Proton in QCD

1 ( ) 2( ) 2 2 22 2 2

2,4,..0

( ( ))( ) ( , ) , 2, 4,6,..

nn n sA Q

M Q dx x F x Q nQ

q

xqxexF )(22

q

Qxqxqxe

QxF22

22

,

,

perturbative ln(Q2) corrections

nonperturbative corrections

1 (2) 2(2) 2 2 2 42 2

0

( )( ) ( ( , ) ( , )) ...q

q

A QM Q e dx x q x Q q x Q

Q

How does it compare to data?

Very good, where only the leading twist is expected to contribute

Most cases, parton distribution functions (PDFs) are extracted from data from “safe kinematic regions”, only (no nonperturbative effects) What is the price to pay? A: Unconstrained PDFs outside the “safe kinematic regions”

Let’s see why….Let’s see why….

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PDFs Extraction in pQCD

Two basic ideas of QCD:

Factorization: separate the long-distance from short-distance dependence

Gffj x

pjsijpi QQx

Pd

QxdQ

dQ

,,

1222

22 ),())(,(),(

),())(,(, 222

,,

1

0

22 QQ

xCdQxF pis

i

Gffi

p

perturbative

nonperturbative input (PDF)

Evolution: knowledge of implies knowledge of ……………at all Q2 > (<) Q2

0, where a perturbative expansion is still appropriate => DGLAP equations

),( 20Qxpi

),( 2Qxpi

splitting functions

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PDFs Extraction in pQCD: Recap Three basic quantities needed for pQCD calculation of F2 : ))(,( 2

2 Qx

C si

))(,( 2Qx

P sij

Computed perturbatively as power series in s

Examples of parameterizations for nonperturbative input: Only requirements: flexible enough to accommodate small/large x behavior + obey the sum rules

Q2 evolution of PDF calculated via DGLAP equationsx dependence of PDF assumed and constrained by data

),( 2Qxpi

CTEQ6:

MSTW:

Alekhin:

54321 )1()1(),( 02

0AAxAAA xeexxAQxx

)1()1(),( 2120 xxxAxQxx

)1()1(2

),( 20 xxx

NQxx ba

To constrain the x dependence is evolved to all Q2 where data exist in the “safe kinematic regions”

),( 20Qx

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Standard pQCD fits: PDFs from CTEQ6

CTEQ6: pQCD fit to hard scattering and DIS data with Q2 > 4 GeV2 and W2 > 12.25 GeV2; the x dependence of PDFs parameterized at Q2 = 1.3 GeV2; evolution up to NLO

JHEP 0207:012, 2002

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CTEQ6: Comparison to Data

Good fit to data in the “safe kinematic regions” but beyond …

/ndf = 1.1 /ndf = 1.52

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CTEQ6: Large Uncertainties at Large x

Large x region important for (see Alberto’s talk): - study the mechanism of spin-flavor symmetry breaking in valence …..quark distributions - determining high-energy cross sections at collider energies - quantification of quark-hadron duality, etc.

but what’s involved in extending PDFs validity to larger x?

Large uncertainties where there are no constraints from data

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Complete picture

1 (2) 2(2) 2 2 2 42 2

0

( )( ) ( ( , ) ( , )) ...q

q

A QM Q e dx x q x Q q x Q

Q

Corrections beyond leading twist

PDFs at Large x and low Q2

))(~

)(2

1()( 2)2(

42)2(

,422)2(

4 QAQAMQA TMC

1) Higher-Twists: kinematic and dynamical

Kinematic HT – associated to twist-2 operator => no additional information on the quark dynamics

Dynamical HT – contains information about the valence quarks dynamics (confinament)

2) Large-x resummation

3) Nuclear Corrections – for the neutron

Messy but needs to be done to achieve exhaustive knowledge of the dynamic of the nucleon!

4) …

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Stepping out of the “safe kinematic region” => inclusion of nonperturbative effects (TMC, HT)

(and nuclear effects for nuclear targets)

Example: PDFs from ALEKHIN

22,

22

)(

Q

xHFF TMCLT

Phys. Rev. D 68, 014002 (2003); JETP Lett. 82, 628 (2005) Extension of PDF fits to larger x: kinematic cuts

(W2,Q2,x,) are relaxed to provide more constraints from data

ALEKHIN

CTEQ6

The x dependence of PDFs parameterized at Q2 = 9 GeV2; evolution up to NNLO

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Uncertainties: Alekhin vs CTEQ6 Result: smaller uncertainties at large x

Relative experimental uncertainties of PDFs at a Q2 of 9 GeV2:

full = Alekhin; dotted = CTEQ6

Redu

ctio

n by

~

10 o

f d

unce

rtai

nty

at

larg

e x

Redu

ctio

n by

~ 4

of u

unc

erta

inty

at la

rge

x

Phys. Rev. D 68, 014002 (2003)

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Dynamical Higher Twist Interplay of Higher-Order QCD corrections and dynamical Higher Twists

Decrease of magnitude of HT with increase of pQCD order but HT don’t vanish in NNLO

HT contribution to F2: at most ~10% of Leading Twist (maximal at x~0.6 and Q2 = 5 GeV2)

order S

LO 0.1301 +/- 0.0026

NLO 0.1171 +/- 0.0015

NNLO 0.1143 +/- 0.0014

From extrapolation: HT not expected to vanish in NNNLO either

Phys. Rev. D 68, 014002 (2003)

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How about extending PDFs to even large x?

2nd resonance region at Q2 = 2 GeV2

2nd resonance region at Q2 = 5 GeV2

Q2 = 2 GeV2

Q2 = 5 GeV2

Extending to larger x at finite Q2 => encounter the resonance region Resonances are basically “made” of higher twists

The contribution of higher-twist terms in the resonance region would be expected to be large…

Or is it?

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Bloom-Gilman Duality The resonance region data: - oscillate around the scaling curve - are on average equivalent to the scaling curve. - “slide” along the deep inelastic curve with increasing Q2

222

1 20

222

,2 QmM mm

dWdQWQ

M

Quantitatively: comparing the lhs to the rhs, relative difference 10% for Q2=1 GeV2 to <2% for Q2=2 GeV2.

Phys. Rev. Lett. 25, 1140 (1970)

“… resonances are not a separate entity but are an intrinsic part of the scaling behavior of W2 …”

Yes, but not on average

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Duality in QCD

W2

W2

W2

Q2 = 1 GeV2

Q2 = 3 GeV2

Q2 = 5 GeV2

data

pQCD

De Rujula, Georgi, Politzer:

“The most intriguing aspects of SLAC data on inclusive electroproduction are precocious scaling and local duality ”

Phys. Lett. B 64, 428 (1976)

Duality = higher-twists are either small or cancel on average (on average, the interactions between the valence quarks are suppressed)

Operator product expansion:

1

2''2

''1

'1

5

4

4

22'

2'

4

3

2

22

23

22

2'

),(12),(6),(),(

QFddk

x

Q

mQFd

k

x

Q

mQF

k

xxQW SSS

Mellin transform of twist-2

pQCD calculation of W2

1

0 1

220

22 )()()(),(k

nkk

nn QBQMnQAdQF

twist-2

On average, the resonance region data mimic the twist-2 pQCD calculation

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Quark-Hadron Duality in F2: Recent Experiments at JLab

Jefferson Lab

Electron-beam acceleratorAs of now, beam energies up to 6 GeVAs of now, three experimental halls: A, B, C

Two spectrometers: HMS & SOS

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1996 JLab-96 (I. Niculescu): duality dedicated experiment; measures H(e,e’) & D(e,e’) cross sections

1998 E94-110 (Y. Liang): performs Rosenbluth separation (measures R = L/T); measures H(e,e’) cross sections

2003 E00-116 (S. Malace): duality dedicated experiment, push to larger x and Q2;

measures H(e,e’) & D(e,e’) cross sections

Inclusive Resonance Region Measurements in Hall C

Among other, three experiments: JLab-96, E94-110, E00-116

Kinematics covered: x between ~0.3 and 0.9, Q2 up to 7 GeV2, in the resonance region (mainly)

JLab-96E94-110E00-116

CTEQ6ALEKHIN

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1111

)('' AdEdNN

BGNdEd

d

temeasured

Procedure for F2 extraction Differential one-photon exchange (Born) cross section

background BG

acceptancedetector A

detectionforefficiencytotal

F2 extraction requires the knowledge of cross section and R

2222

21

11

41

1),(

Q

K

R

R

Edd

dQxF

E94-110: measured RJLab-96: used R from E94-110E00-116: used R from R1998 (R < 0.2 @ E00-116 kinematics)

Experimental natural variables: momentum and angle of scattered electron

+ energy of incoming electron

x, Q2, W2

crap

crap

crap

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Physics Results from JLab-96 Verifying quark-hadron duality “a la Bloom-Gilman”

NMC fit to DIS data at the same but higher W2, Q2 than RES data

The new precision data display the signature oscillation around the DIS curve (the agreement, on average, better than 10%)

JLab-96 conclusively verifies the observations of Bloom and Gilman

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Physics Results from JLab-96 Verifying quark-hadron duality in a pQCD framework: analysis in fixed W2 bins

Averaged RES data

pQCD(NLO) pQCD(NLO)+TMC

larg

e-x

resu

mm

ati

on:

bri

ngs

pQ

CD

ca

lcula

tion in b

ett

er

agre

em

ent

wit

h

data

TMC significant effect: pQCD calculation in better agreement with data LxR: resummation on ln(1-z) in x space => Q2 scale replaced by Q2(1-z)/z HT: in RES region similar to those for W2> 10, with exception of

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Physics Results from E94-110 More precise data from JLab: the resonances average to pQCD+TMC calculations from CTEQ and MRST The resonance data slide with increasing Q2 to higher x always following the pQCD curves

The ratio of F2 integrals data to pQCD better than 5% at Q2 = 0.5 GeV2 but ~ 18% at Q2 = 3.5 GeV2 ?!? Violation of duality, unconstrained PDFs at large x, something else ?

inegrals over

entire RES region

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Physics Results from E00-116

1st 2st 2nd

2st 3rd

4th

DIS

Verify quark-hadron duality at higher Q2

Region

Wmin Wmax

1st 1.3 1.9

2nd 1.9 2.5

3rd 2.5 3.1

4th 3.1 3.9

DIS 3.9 4.5

dxQxF

dxQxF

IM

m

M

m

x

x

param

x

x

data

),(

),(

22

22

Calculate:

Define:

Data from E00-116, E94-110, JLab-96 and SLAC; parametrizations from CTEQ6, MRST, ALEKHIN

Compare:

Larg

e d

iscrep

an

cies in

the

descrip

tion

of F

2 at la

rge

x

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Physics Results from E00-116 Comparison: data [H(e,e’)] to CTEQ6M (NLO) + TMComparison: data [H(e,e’)] to CTEQ6M (NLO) + TM

I ~ 1 at Q2 ~ 1.5 GeV2 then rises with increasing Q2 and reaches a plateau at ~ 4 GeV2; above this value Q2 dependence saturates

This behavior displayed when integrating globally and locally except for first resonance.

Not a failure of pQCD in describing the Q2 evolution but a paucity in the strength of PDFs at large x I becomes constant at different value for each RES region

Related to growing uncertainty of PDFs strength at large x

Phys. Rev. C 80, 035207 2009

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Physics Results from E00-116 Comparison: data [H(e,e’)] to MRST04 (NNLO) + Comparison: data [H(e,e’)] to MRST04 (NNLO) + TMTM The observed Q2 dependence of I

yields similar conclusions as drawn from the CTEQ6

Not surprising: the extraction procedure (and kinematic cuts) of PDFs similar for MRST04 and CTEQ6

Differences:Differences: MRST04 undershoots the data by an even larger amount and I saturates at a larger value of Q2

than for CTEQ6

Possibly results from the difference in modeling the x dependence of PDFs (?) Phys. Rev. C 80, 035207

2009

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Physics Results from E00-116 Comparison: data [H(e,e’)] to ALEKHIN (NNLO) + HT + Comparison: data [H(e,e’)] to ALEKHIN (NNLO) + HT + TMTM Due to cuts employed for data

selection, Alekhin’s fits far better constrained at large x

For the 4th RES region and DIS, I very close to 1 for entire Q2 range analyzed

Good agreement for 3rd and 2nd RES regions: I deviates from 1 by about 5%

HT in RES region, on average, differ by at most 5% from those extracted by Alekhin 1st resonance in disagreement

with Alekhin’s fit: the validity of the fit questionable at these kinematics Averaged RES data could be used to constrain PDF fits

Phys. Rev. C 80, 035207 2009

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Physics Results from E00-116

Good description at Q2= 3,5 GeV2 (except for largest x regime: 1st RES) Q2= 7 GeV2 : probing the largest x regime (ALEKHIN least constrained) => growing discrepancy Q2= 1 GeV2 : discrepancy as x grows reached limits of applicability

ALEKHINALEKHIN

CTEQ6CTEQ6 Fails to describe x dependence of data

Better data description by ALEKHIN than CTEQ6

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Physics Results from E00-116 Comparison: data [D(e,e’)] to CTEQ6 and ALEKHINComparison: data [D(e,e’)] to CTEQ6 and ALEKHIN

F2d(ALEKHIN,CTEQ6) = F2

p(ALEKHIN,CTEQ6) * d/p (from empirical fit)

The Q2 dependence of I: similar characteristics as in the study of H(e,e’) ALEKHIN offers better description of averaged RES data than CTEQ6

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Is Quark-Hadron Duality Verified in the Proton? Duality is an experimental observation and could be a working hypothesis for extending PDFs at large x => needs to be verified and quantified

It has been observed to work better 5% down to a Q2 as low as 1 GeV2 when compared to pQCD fits from MRST: E94-110

Surprisingly, it has been observed that the violation of duality becomes more pronounced as x and Q2 increase

Our studies indicate that this increasing violation of duality with Q2 is very likely only APPARENT: duality studies involve extrapolations of pQCD fits (unconstrained at large x)

The unconstrained PDFs at large x pose problems for quantifying how well duality holds in this kinematic regime (and that’s not good)

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Extraction of the Neutron Structure Function F2n

F x Q dy f y Fxy

QA N A N

x

M M

N p n

A

22

0 22( , ) ( , ) ,//

,

smearing functions (can be calculated from nuclear wave function)

Impulse Approximation (IA) – virtual photon scatters incoherently from individual nucleons

Beyond IA: nuclear shadowing, MEC, FSI, relativistic effects, off-shell corrections (most not addressed in present analysis)

New method of extracting neutron SF from inclusive SFs of New method of extracting neutron SF from inclusive SFs of nucleinuclei: employs iterative procedure of solving integral convolution equations (Phys. Rev. C 79, 035205 2009)

Can write the nuclear structure functions as convolutions of nucleon structure functions

Present analysis does not attempt to provide a complete Present analysis does not attempt to provide a complete description of nuclear SFs (yet)description of nuclear SFs (yet)

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Extraction of F2n

( )( ) ( , ) / )/

f x dy f y x yx

M MA F F (

NdNNdoffnpd FfFFFFF 2/

22)(

222

~,

~~

In the Weak Bound Approximation (WKA): the deuteron SF is sum of smeared proton and neutron SF and an additive term to account for modifications of SF off-shell

The effective smeared neutron SF:

pdoffQEddn FFFFF 22)()(

222

~~ Need to solve equation:

nn FfF 22

~ MethodMethod- Parameterize the nuclear corrections by an additive term )(

~22 Nn FF

- F2n extracted using an iterative

procedure which gives after first iteration)

~(

1 )0(22

)0(2

)1(2

nnnn FfFFF

assumed

Study sensitivity of extraction to: number of iterations, first guess for neutron SF etc.

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Results from E00-116: Extraction of F2n

The resonances are obvious in the extracted F2

n

After only two iterations F2d

reconstructed from F2p data

and extracted neutron F2n

agrees well with the F2d

data

The extracted F2n yields

similar results after two iterations when different inputs are used [F2

n(0) = F2p

& F2n(0) = F2

p/2]

Both F2p and F2

n average to the QCD fit from Alekhin suggesting the onset of duality How well?

Application of method to data (Application of method to data (Phys. Rev. Lett., xx, to be submittedPhys. Rev. Lett., xx, to be submitted))

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Quark-Hadron Duality in F2n

Compare integrals of neutron “data” to integrals of Alekhin’s newest fit (arXiv:0908.2766, August 2009) Without HT: agreement at the level of 10-15% for Q2 < 3 GeV2 (except for )

covers the highest x regime (the fit least constrained)

The discrepancy increases with increasing Q2 (unconstrained PDFs at larger x?, …) … sounds familiar?

With HT: good agreement; deviation less than 10% in most cases

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Neutron/Proton vs Q2 & x

Good agreement between data and pQCD fits, except for region which is somehow underestimated

The agreement slightly worsens as we go to larger Q2 and x

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Need more data at large x and “low” Q2?

We can help …

To be proposed at the next PAC in January 2010:

CTEQ6ALEKHIN

Measurements at 11 GeV @ JLab

Extend RES region and low W2 DIS region measurements at even higher x and Q2 at JLab

Systematic study of quark-hadron duality; extraction of dynamical HT (interesting in their own right); additional constraints for PDFs at large x; extract the neutron SF at even larger x (and maybe constrain the d quarks distribution better) …