J. Roche Ohio University and Jefferson Laboratory

Results shown are from the ”DVCS HALL A collaboration” (E00-110 and E03-106)

First steps toward nucleon tomography

usingthe Generalized Parton Distribution

functions

Carlos MUNOZ CEA/SaclayAlexandre CAMSONNE IN2P3/ClermontMalek MAZOUZIN2P3/Grenoble

Pierre BERTIN IN2P3/Clermont Charles HYDE ODU/ClermontRon RANSOMME RutgersFranck SABATIE CEA/SaclayEric VOUTIER IN2P3/Grenoble

Form factorIn 1956, R. Hofstadter confirms that the proton is not a point like objectBut has a spatial extension that can be described in terms of form factors.

Q2=-q2

),( 0 kkkr

)','(' 0 kkkr

),( PMPr

)','(' 0 PPPr

F1(Q2),F2(Q2)

Lowest order

'),( kkqq −=r

ν

In 1972, with the discovery of the weak currents ( Z0, W), the axial () form factor GA(Q2) and the pseudo-scalar form Gp(Q2) are introduced.

00 'kk −=ν

Spin flip

In 1969-1973, the experiments at Stanford (R.Taylor et al.) showed that hadrons are made up of point like parts. These partons wereidentified to quarks and gluons predicted by theorists (M. Gellmann)in 1964. In these experiments an electron scatters inclusively off the nucleonWith a large energy transferred to the target. These experimentsAre know as “Deep Inelastic Scattering”

)','(' 0 kkkr

),( 0 kkkr

00 'kk −=ν

Extensively measured at SLAC, CERN, DESY,JLAB…..Beam: e,νpolarized and non polarizedTarget: P,D,… polarized and non-polarized

⎥⎦

⎤⎢⎣

⎡

−−+−

−= 22

2

12

24

222

)()1(

)(4F

Ms

MxFyFy

Q

Ms

dydx

d απσ

νM

Q

Pq

Qx

2.2

22

==

kPk

Pqy

ν==

.

.q=(k-k’)

k0-k’0=ν Q2=-q2),( 0 kkkr

)','(' 0 kkkr

),( PMPr

DEEP INELASTIC SCATTERING

Note: There are two more structure functions corresponding to the weak exchange (Z0,W)

Structure functions

X

Proton

Particle Data Book

Photon flux before the target - photon flux after the target = all the photons that interacted

€

σX (r k ')

all X

∑ ≡ 4π ℑm[A(kp,kp)]

Optical theorem

X

€

all X

∑p

k

q

The photon couples to a charge

€

≡p p’

k k’

q q

p=p’ k=k’

€

Q2 = −(k − k')2 → ∞

ν = k0 − k0' → ∞

x =Q2

2Mνfixed

In the Bjorken’s limit(no transverse extension)

The scattering happens off a parton (quark or antiquark) carrying a fractionx of the proton momentum

•The structure functions F1,2(x,Q2) ought to be independent of Q2 : there is “scaling” This is wrong but one knows their evolutions through DGLAP

•The structure functions can be de-convoluted in flavor:

Parton distributions

€

q f (x), q f (x)

P P

K K

q q

P P

K K

q q

u

d

g/10

u,d

x

xf(

x)

Polarized structure functions

X

e

P

€

d2 r σ (→,↓)

dx dy−

d2 s σ (←,↑)

dx dy⇒ g1

n,p (x,y),g2n,p (x, y)

⇒ Δu(x),Δd(x),Δu (x),Δd (x),....

= Δq f

Polarized electron

Polarized target: transverse or longitudinal

Neutron, proton….

Proton

Deuteron

Neutron (from 3He)

PDG

If not at the Bjorken’s limit

€

f = u,u ,d,d ....TWISTQTWIST(1/Q4)+….

P P

K K

q q

+ + …=

Sum rules

Nucleon observables

Parton distributions Observables of the partons (charge, spin ….)+

€

dx∫ Q f [q f (x) − q f (x)]f

∑ =1.602 10−19 Coulomb

€

1

2Δq f

f

∑ (x)dx0

1

∫ =0.3± 0.04

2≠

1

2

Charge

Spin

Consider the quarks only:

Add the gluons:

€

1

2Δq f (x) + Δg(x)

⎡ ⎣ ⎢

⎤ ⎦ ⎥

f

∑ dx0

1

∫ ≠1

2

€

[1

2Δq f

f

∑ (x) + Δg0

1

∫ (x)]dx + Lq + Lg =1

2

The quarks orbit !!

But around what ? ?

Orbital momentum

In order to determine Lq, one needs to observe2 partons interacting that is

the correlations between partons.

To describe the waltz, one needs to count the number of dancers, their sexes, their race, their individual spin but none of that tells enough

about the spinning of the whole dance group.

Only a measure of the correlation between all the possible couples gives that information.

DIS only measures the parton distributions anddoesn’t give access to the correlations between the partons therefore

doesn’t give access to the orbital momentum.

The waltz

In 1996, D. Mueller, A. Radyushkin, and X. Ji generalized the partondistributions by breaking the handbag symmetry

),,(~

),,(

),,(~

),,(

5

txEtxEflipspin

txHtxHflipspinno

ff

ff

ξξ

ξξ

4 functions per flavor

Off Forward Parton Distributions

Skewed Parton Distributions

Generalized Parton Distributions

x+ξ x-ξ

t

GPDs

P P’

2)'( PPt −=

GPDs

)()()0,0,(

)()()0,0,(

xqxqxE

xqxqxH

fff

fff

−Δ−=Δ=

−−==

)(),,(~

)(),,(~

)(),,(

)(),,(

1

1

1

1

2

1

1

1

1

1

tGtxEdx

tGtxHdx

tFtxEdx

tFtxHdx

pf

f

Af

f

ff

ff

−=

−=

−=

−=

∑∫

∑∫

∑∫

∑∫

−

−

−

−

ξ

ξ

ξ

ξ

)]0,,()0,,([2

1

2

1 1

1

ξξ xExHxdxLJ ffff

f +=+ΔΣ= ∫−

),,( txGPD ξ€

t = 0

ξ = 0

Limits and sum rules

Ji’s sum rule

There are many GPDs modelsIn all cases, the models predictions are valid if the hand bag dominates Q2, S >>M2 et t << Q2.

The results presented today will be compared to the VGG (M. Vanderaghen, M. Guidal et P.A.M. Guichon ) model.

• x dependency is constrained by the form factors and the structure functions• ξdependency is constrained by mathematical conditions (positivity, polynomiality…)• t dependency is factorized Hq(x)=q(x) F1(t)• inspired by the quark-soliton model

• is a code that you can use to do your plots…

GPDs models

From theory to experiment

Theory

x+ξ x-ξ

t

GPDs

Experiment

But nothing is that simple

Handbag Diagram

Collins, Freund

Physical process

This needs to be tested !!!

Factorisation theorem : If one is close enough from the asymptotic limit then the handbag diagram is the mainContribution to DVCS.(leading twist)

Q2 and ν largewith xB and t fixed

The GPDs enter in the DVCS amplitude as integrals over x

• GPDs appear in the real part through a Principal-value integral over x• GPDs appear in the imaginary part along the line x=+/-ξ

GPDs in the DVCS amplitude

VGG model

€

T DVCS = dxH(x,ξ , t)

x −ξ + iε+L =

−1

+1

∫

= P dxH(x,ξ , t)

x −ξ− i

−1

+1

∫ πH(x = ξ ,ξ , t) + L

x-ξ x+ξ

GPD observables

Experimentaly, DVCS has the same final state than the Bethe-Heitlerprocess. The BH is known if the form factors are known.

With a longitudinaly polarized beam, the GPD observables are:

At JLab energies,|TDVCS|2 is small (?)

( )

( )

42 2

2

4 42 2

2

2

Im2

Re DVCBH BH DVCS

B

BH DVCS DVCSDVCS

B

SdT T T

dx dQ dtd

d dT T T

dx dQ

T

tT

d d

σ

ϕ

σ σ

ϕ

→ ← → ←

≈ + ⋅ +

⎡ ⎤−≈ ⋅ + −⎢ ⎥

⎢ ⎥⎣ ⎦

P.Kroll, P.A.M. Guichon, M. Diehl, B. Pire, …

Parametrization of the cross sections

Terms from the handbag (Twist 2) should dominate σ and ΣTheir dependence in Q2 is predicted by the handbag and can be

tested

Belitsky, Mueller, Kirchner, Nucl. Phys. B629,323(2002)

Decomposed the amplitudes in terms of angular harmonicsand limited Fourier expansions

Twist 2 Twist 3 azimuth between the virtual and real photon

€

d4σ = d4σ BH2

( )

€

+Γ0

ℜ − cosφ( ) ℜe C I( ) + Γ0,Δ

ℜ ℜe C I + ΔC I( ) + cos2φ Γ2

ℜ ℜe CeffI

( )

1+ cosφ + cos2φ+ ΓηCDVCS

|DVCS|2Interference BH-DVCS

€

d4Σ = 12 d4 r

σ − d4 s σ ( ) = sinφ Γ1

ℑ ℑm C I( ) − sin2φ Γ2

ℑ ℑm CeffI

( ) + sinφ η Γ1ℑ ℑm CDVCS

( )

CI’s are linear combinations of the GPDs integrals and discrete values (x=ξ)

The Jlab/Hall A DVCS experiments

The cross section is 4 times differential, so one needs to measure 4 variables:

Q2, x

The Scattered electronIs detected in the Jlab/Hall A HRS:

High precision determination of the quadri-vector of the *

Δp/p~5 10-5 Δrd

t,

The outgoing real photon is detected in an

high precisionelectromagnetic calorimeter:

This allows an excelentdetermination of the directionof the real photon Δ=2. 10-3 rd

The Jlab/Hall A DVCS experiments

Scattered electron The HRS acceptance

Is well known(standard equipmennt)

1~2 %

Emitted photonThe calorimeter is a

simple box

e p → e (p)

The ‘matching’ of the two acceptances is perfect

The virtual photon pointsTo the center of the calorimeter

*

One of the key strength of this experimental program is the excellentKnowledge of the acceptance which allow the extraction of cross sections.

The Jlab/Hall A experimental apparatus

• 75% polarized 2.5uA electron beam• 15cm LH2 target• Left Hall A HRS with electron package• 11x12 block PbF2 electromagnetic calorimeter• 5x20 block plastic scintillator array

at

2.7%

4.2

E

EGeV

σ=

2.5x y mmσ σ; ;

The Jlab/Hall A experimental apparatus

• 75% polarized 2.5uA electron beam• 15cm LH2 target• Left Hall A HRS with electron package• 11x12 block PbF2 electromagnetic calorimeter• 5x20 block plastic scintillator array

Δt (ns) for 9-blockaround predicted« DVCS » block

E00-110 custom electronics and DAQ scheme

1. Electron trigger starts the game

2. Calorimeter trigger (350ns):

- selects clusters- does a fast energy reconstruction- gives a read-out list of the modules which enter clusters over a certain threshold- gives the signal to read-out and record all the experiment electronics channels

3. Each selected electronics channel is digitized on 128ns by ARS boards

t (ns)

4. Offline, a waveform analysis allows to extractreliable information from pile-up events

H(e,e’)X

H(e,e’p)

H(e,e’)X - H(e,e’’)X'

H(e,e’)N

DVCS : exclusivity

•Good resolution : no need for the proton array•Remaining contamination 1.7%

HRS+calorimeterep -> epep -> ep0 0->ep -> ep0ep -> ep0N…

HRS+calorimeter + proton array

E00-110: difference of cross-sections

2 22.3 GeV

0.36B

Q

x

=

=

Corrected for real+virtual RCCorrected for efficiencyCorrected for acceptanceCorrected for resolution effectsChecked elastic cross-section @ ~1%

Twist-2Twist-3

Extracted Twist-3contribution small !

PRL97, 262002 (2006)

E00-110: Q2 dependence and test of scaling

<-t>=0.26 GeV2, <xB>=0.36

No Q2 dependence: strong indication forscaling behavior and handbag dominance

Twist-2Twist-3

0 electro-production cross section:• Dominance of twist 2 (isolation of leading twist)• Sensitive to different linear combinations of GPDs than DVCS

NEW: E12-06-114: Jlab/Hall A 12 GeV experiment

Q2 variation:• 2:1 range at each xBj

• Accurate measurement of the twist-2 dominance

xBj variation (dependence):• Precision data on variation of

t-dependence with xBj

• Study of transverse correlations

t variation:• 5 bins in 0 < t-tmin < 1 GeV2

• Fourier-conjugate to the spatial distributions of quark as a function of their momentum fraction xBj

E00-110: Total cross-sections

Extracted Twist-3contribution small !

PRL97, 262002 (2006)

With these data, it is impossible to disentangle DVCS2 from the interference term: they mix in the azimuthal analysis

22

2

4

)Re(2 DVCSDVCSBHBH TTTT

dtddxdQ

d++≈

σ

large

€

Q2 = 0.23GeV 2 x = 0.36

NEW: E07-007: Complete separation of the observables.

22

2

4

)Re(2 DVCSDVCSBHBH TTTT

dtddxdQ

d++≈

σ

Azimuthal analysis and 2 beam energies,the 2 contributions can be separatedRosenbluth type separation

Goal 1: measure the total cross section d4σ using two beam energiesat fixed xBj=0.36 for three Q2=2.3, 1.9 and 1.5 GeV2.

Ebeam ranging from 3.5 to 6 GeV

• will provide a test of the scaling of the total cross section• will allow to separated the interference BH-DVCS from DVCS2

Goal 2: measure the 5 response functions of ep->ep0 at Q2=2.3, 1.9, 1.5 GeV2

–Separate σLT, σTT , σLT’ from σL+ σT by azimuthal variation

and σL from σT by Rosenbluth method

–First test of factorization in this deep meson production channel –If scaling is observed can extract flavor information on GPDs

Some of the topics I did not talk about

• The same collaboration published results on the neutron (E03-106) got some electro-pion production data

• Other collaborations have published dedicated and non-dedicated DVCS data. DESY (HERMES and HERA) JLab/Hall B: large x and Q2 data but asymmetry data only

• Many DVCS experiments have been approved to run in a near future:COMPASS: using as beamJLab/Hall B

• Information about the structure of the nucleon can be obtained by measuringDeep Meson Production (, , etc..)

The Jlab/Hall A DVCS collaboration• has measured DVCS cross sections with high statistical and systematic precision using a HRS and a calorimeter both off the proton and the neutron

•

€

p(e,e'γ) pσσ sr

−σσ sr

+

•Tests of the dominance of the handbag are positive•The VGG model is too small by 30%•The DVCS2 cannot be neglected

In summary

The GPDs provide a new window on the nucleon structure that allow to study the correlation between its constituents BUT

•One needs to test the regime of applicability of the formalism (handbag dominance)

•The interpretability of the cross-section in term of GPDs is still in its infancy

Many groups have and will continue to perform DVCS/GPD experiments. The Jlab/Hall A DVCS collaboration will study

• E07-007: complete separation of the cross section observables (DVCS2)• E12-06-114: extended Q2 and x ranges• data coming up off proton, neutron (deuterium) on electro production of photon and 0 WANT TO JOIN US?

END

n-DVCS is sensitive to Jd

p-DVCS is sensitive to Ju

Complementarity between neutron and proton measurements

E03-106 results

Model dependent extraction of

Ju and Jd

€

Jq = 12 ΔΣq + Lq

= 12 dx x Hq (x,ξ , t = 0) + Eq (x,ξ , t = 0)[ ]

−1

+1

∫

E00-110 kinematics

The calorimeter is centeredon the virtual photon direction

50 days of beam time in the fall 2004, at 2.5A intensity

113294 fbLu dt −⋅ =∫

Remaining contribution: ~1.7%(all non-o electroproduction)

Analysis – Looking for DVCS events

MM2 cut

MM2 cut

However: One needs to do a thorougho subtraction if the only (e,) system is used to select DVCS events !!!

(e,g,p) events

Testing the handbag dominance

Symmetry around thevirtual photon

Broken by : the photon polarization the beam polarization

If the handbag dominates in the case of a real photon

σλ sin)1(2 '

dt

d TL−+

td

d

dtdxdQ

d

Bj

σσ×Γ=

σσσσσ2coscos)1(2

dt

d

dt

d

dt

d

dt

d

dt

d TTLTLT ++++=

virtual photon polarization longitudinal polarization of the beam

There are 12 independant helicity amplitudes but only 5 observables in the case of a polarized beam and an unpolarized target.

e-’

pe-*

hadronic plane

leptonic plane