Revealing Generalized Parton Distribution

zup.physik.uni.muenchen.de

Revealing Generalized Parton Distribution

Revealing

Generalized Parton Distributions

Dieter Müller

Ruhr-Universität Bochum

K. Kumerički, DM, K. Passek-Kumerički (KMP-K), hep-ph/0703179

GPD fits at NLO and NNLO of H1/ZEUS data

KMP-K, 0805.0152 [hep-ph]

constructive critics on ad hoc GPD model approach [lot of good news]

first applications of dispersion integral approach

KMP-K, 0807.0159 [hep-ph]; KM 0904.0458 [hep-ph]

flexible GPD model for small x and fits of H1/ZEUS data

dispersion integral fits of HERMES and JLAB data

1


• Where one can reveal GPDs

• GPDs: : Can one ``measure’’ them

• Small x and global DVCS fits

• Summary/Conclusions

2


GPDs embed non-perturbative

physics

GPDs appear in various hard exclusive processes,

e.g., hard electroproduction of photons (DVCS)

[DM et. al (90/94)

Radyushkin (96)

Ji (96)]

p


γ (q)

DVCS

γ

p'

x + ξ

Q 2 > 1GeV 2

GPD

t = ∆ 2

x − ξ

− fix

F(ξ, Q 2 , t) = R 1

−1 dx C(x, ξ, α s(μ), Q/μ)F (x, ξ, t, μ) + O( 1 Q 2 )

CFF

Compton form factor

observable

hard scattering part

perturbation theory

(our conventions/microscope)

GPD

universal

(conventional)

higher twist

depends on

approximation


GPD related hard exclusive processes

scanned area of the surface as

a functions of lepton energy

• Deeply virtual Compton scattering (clean probe)

ep → e 0 p 0 γ

ep → e 0 p 0 μ + μ −

γp → p 0 e + e −

e

e'

γ ∗ (*)

γ

• Hard exclusive meson production (flavor filter)

ep → e 0 p 0 π

ep → e 0 p 0 ρ

ep → e 0 nπ +

ep → e 0 nρ +

• etc.

e

p

e'

γ ∗

M

p'

p'

+

μ


μ

η

x

+ −

ep → e' p' μ μ

twist-two observables:

cross sections

transverse target spin

asymmetries

4


Photon leptoproduction

e ± N → e ± Nγ

measured by H1, ZEUS, HERMES, CLAS, HALL A collaborations

planed at COMPASS, JLAB@12GeV, perhaps at EIC,

Ã


dx Bj dyd|∆ 2 |dφdϕ = α3 x Bj y

1 + 4M 2 x 2 Bj

16 π 2 Q 2 Q 2

! −1/2 ¯¯¯¯

2

T

e

¯¯¯¯ 3

,

x Bj =

Q 2

2P 1 · q 1



1 + ξ ,

y = P 1 · q 1

P 1 · k ,

∆ 2 = t (fixed, small),

Q 2 = −q 2 1 (> 1GeV2 ),

5


interference of DVCS and Bethe-Heitler

processes


γ (q)

γ

p

T μν

p'

H, E, H e · · ·

J μ

J μ

12 Compton form factors elastic form factors

(helicity amplitudes)

(

)

|T BH | 2 e 6 (1 + ² 2 ) −2

2X

=

x 2 Bj y2 ∆ 2 c BH

0 + c BH

n cos (nφ) ,

P 1 (φ)P 2 (φ)

|T DVCS | 2 = e6

y 2 Q 2 (

c DVCS

0 +

2X

n=1

n=1

£

c

DVCS

n cos(nφ) + s DVCS

n sin(nφ) ¤) ,

F 1 , F 2

exactly known

(LO, QED)

harmonics

1:1

helicity ampl.

I =

±e 6

x Bj y 3 ∆ 2 P 1 (φ)P 2 (φ)

(

c I 0 +

3X £

c

I

n cos(nφ) + s I n sin(nφ)¤) .

n=1

harmonics

1:1

helicity ampl. 6


1/Q

relations among harmonics and GPDs are based on expansion:

(all harmonics are expressed by twist-2 and -3 GPDs)

½ ¾ I c1

∝ ∆ s 1 Q tw-2(GPDs) + O(1/Q3 ), c I 0 ∝ ∆2

Q 2 tw-2(GPDs) + O(1/Q4 ),

[Diehl et. al (97)

Belitsky, DM, Kirchner (01)]

½

c2

s 2

¾ I

∝ ∆2

Q 2 tw-3(GPDs) + O(1/Q4 ),

½

c3

s 3

¾ I

∝ ∆α s

Q (tw-2)T + O(1/Q 3 ),

c CS

0 ∝ (tw-2) 2 ,

½

c1

s 1

¾ CS

∝ ∆ Q (tw-2)(tw-3),

½

c2

s 2

¾ CS

∝ α s (tw-2)(tw-2) GT

setting up the perturbative framework:

twist-two

two coefficient functions at next-to

to-leading

order

[Belitsky, DM (97);

Mankiewicz et. al (97);

Ji,Osborne (98)]

evolution kernels at next-to

to-leading

order

[Belitsky, DM, Freund (01)]

next-to

to-next-to-leading

order in a specific conformal subtraction scheme

[KMP-K &

Schaefer 06]

twist-three

three including quark-gluon-quark correlation at LO

partial twist-three

three sector at next-to

to-leading

order

`target mass corrections’ (not well understood)

[Anikin,Teryaev, Pire (00);

Belitsky DM (00); Kivel et. al]

[Kivel, Mankiewicz (03)]

[Belitsky DM (01)]

7


GPDs: : a partonic duality interpretation

quark GPD (anti-quark x → -x):

F = θ(−η ≤ x ≤ 1) ω ¡ x, η, ∆ 2¢ + θ(η ≤ x ≤ 1) ω ¡ x, −η, ∆ 2¢

ω ¡ x, η, ∆ 2¢ = 1 η

Z x+η

1+η

0

dy x p f(y, (x − y)/η, ∆ 2 )

dual interpretation on partonic level:

η+x

2

η−x

2

support extension

is unique [DM et al. 92]

x+η

2

x−η

2

p

p

central region - η < x < η

mesonic exchange in t-channel

ambiguous (D-term)

[DM, A. Schäfer (05)

KMP-K (07)]

p

p

outer region η < x

partonic exchange in s-channel

8


GPD modeling & Evolution

outer region governs the evolution at the cross-over trajectory

μ 2

d H(x, x, t, μ 2 ) = R 1

dμ 2 x

dy

x V (1, x/y, α s(μ))H(y, x, μ 2 )

GPD at h = x is `measurable’ (LO)

central region follows

(polynomiality of moments)

outer region governs evolution

x

= PV

Z 1

h

PV

0

dx

net contribution of

outer + central region is

governed by a sum rule:

Z 1

0

dx

2x

2x

η 2 − x 2 H− (x, η, t)

η 2 − x 2 H− (x, x, t) + C(t)

9


Can one `measure’ GPDs

• CFF given as GPD convolution:

H(ξ, t, Q 2 ) LO =

Z 1

−1

dx

µ

1

ξ − x − i² − 1

ξ + x − i²

LO

= iπH − (x = ξ, η = ξ, t, Q 2 ) + PV


H(x, η = ξ, t, Q 2 )

Z 1

0

2x

dx

ξ 2 − x 2 H− (x, η = ξ, t, Q 2 )

• H(x,x,t, 2 ) viewed as ”spectral function” (s-channel cut):

H − (x, x, t, Q 2 ) ≡ H(x, x, t, Q 2 ) − H(−x, x, t, Q 2 ) LO = 1 π =mF(ξ = x, t, Q2 )

• CFFs satisfy `dispersion relations’

(not the physical ones, threshold ξ 0

set to 0)


exclusive

processes

@ large t

hard exclusive

processes

form

factors

lattice

simulations

GPDs

LC-wave

functions

QCD-models

Regge-phenom

phenom.

``amplitudes’’

3D-picture

spin content

duality

parton

densities

(PDs)

unintegrated

PDs

11


Strategies to analyze DVCS data

GPD model approach:

ad hoc modeling: VGG code [Goeke et. al (01) based on Radyuskin’s DDA]

(first decade) BKM model [Belitsky, Kirchner, DM (01) based on RDDA]

`aligned jet’ model [Freund, McDermott, Strikman (02)]

Kroll/Goloskokov (05) based on RDDA [not utilized for DVCS]

`dual’ model [Polyakov,Shuvaev 02;Guzey,Teckentrup 06;Polyakov 07]

“ -- “ [KMP-K (07) in MBs-representation]

Bernstein polynomials [Liuti et. al (07)]

dynamical models:

not applied [Radyuskin et.al (02); Tiburzi et.al (04); Hwang DM (07)]…

flexible models: any representation by including unconstrained degrees of freedom

(for fits) KMP-K (07/08) for H1/ZEUS in MBs-representation

What is the physical content of `invisible’ (unconstrained) degrees of freedom

Extracting CFFs from data: real and imaginary part

0. analytic formulae [BMK 01]

i. (almost) without modeling [Guidal, Moutarde (08-10)]

ii. dispersion integral fits [KMP-K (08),KM (08/09)]

iii. flexible GPD modeling [KM (08/09)]

12


The fitting problem

experimental data

H1/ZEUS

HERMES, JLAB,…


hypothesis of GPD

(a set of parameters)

asymmetries

cross sections

LO … N(N)LO routines

for the evaluation of gen. CFF

data-filtering

(projection on tw-2)

fitting routine

(optimization

problem)

observables

(in terms of gen. CFF)

• many different observables (formed from cross sections)

• complex theoretical formulae, many modeling possibilities (many parameters)

• GPDs depends on form factors and PDFs, too (known only to a certain extend)

13

• i.e., to pin down GPDs one might fit to FF, structure functions, and exclusive data


Data set for unpolarized proton target

• H1/ZEUS 98 [σ, dσ/dt] +1x6 [BCA(φ)] ≈10 -3 , ≤ 0.8 GeV 2

> ≈ 8 GeV 2

• HERMES(02) 12+3 [BSA, sin(φ)]

• HERMES(08) 12x2 [BCA, cos(0 φ), cos(φ)] 0.05 ≤ ≤ 0.2, ≤ 0.4 GeV 2

12x2 [cos(2 φ), cos(3 φ)] > ≈ 2.5 GeV 2

• HERMES(09) not included new BSA and BCA data

• CLAS(07) 12x12 [BSA(φ)] 0.14 ≤ ≤ 0.35, ≤ 0.3 GeV 2

40x12 [BSA(φ)] (large |t| or bad sta.) > ≈ 1.8 GeV 2

• HALL A(06) 12x24 [Δσ(φ)] =0.36, ≤ 0.33 GeV 2

3x24 [σ(φ)] > ≈ 1.8 GeV 2

How to analyze φ dependence

• fit within assumed functional form [CLAS(07)]

• fit with respect to dominant and higher harmonics [HERMES(08)]

• utilize Fourier analyze (with or without additional weight) [BMK(01)]

equivalent results for CLAS data with small stat. errors

14


DVCS fits for H1 and ZEUS data

DVCS cross section measured at small

x Bj ≈ 2ξ =

2Q2

2W 2 +Q 2

40GeV . W . 150GeV, 2GeV 2 . Q 2 . 80GeV 2 , |t| . 0.8GeV 2

predicted by


dt (W, t, Q2 ) ≈ 4πα2 W 2 ξ 2 ∙

Q 4 W 2 + Q 2 |H| 2 − ∆2

4Mp

2 |E| 2 + ¯ eH

¯ ¡

¯2¸ ξ, t, Q

2 ¢ ¯¯¯ξ= Q 2

2W 2 +Q 2

suppressed contributions

relative O(ξ)

• data could not be described at LO before 2008

• NLO works with ad hoc GPD models [Freund, McDermott (02)]

results strongly depend on employed PDF parameterization

do a simultaneous fit to DIS and DVCS [KMP-K (07)]

use flexible GPD models in a two-step fit [KMP-K (08)]

15


effective functional form at small x:

PDFs:

GPDs:

q sea (ξ, Q) = n(Q)ξ −α(Q) , α ∼ 1, F sea (0) = 1

H = r(η/x = 1, Q)F sea (t)ξ α0 (t,Q) q sea (ξ, Q)

skewness

transverse

distribution


E(ξ, ξ, t, Q)

some evidence from “standard” Regge phenomenology

[e.g. Donnachie 04]

chromo-magnetic “pomeron” might be sizeable

(instantons) [e.g. Diakonov]

pQCD evolution suggests ``pomeron’’ intercept

qualitative understanding of E is needed (not only forJi`s spin sum rule)

B = R 1

0 dx xE(x, η, t, Q) (J q = A q + B q )

16


good DVCS fits at LO, NLO, , and NNLO with flexible GPD ansatz

17


quark skewness ratio from DVCS fits @ LO

R = =mA DVCS

=mA DIS

LO

=

H(ξ,ξ)

H(2ξ,0) ≈ 2α r

conformal ratio

W = 82GeV

r = H(ξ,ξ)

H(ξ,0)

ξ ∼ 10 −5 · · · 10 −2

conformal ratio

• @LO the conformal ratio is ruled out for sea quark GPD

• a generically zero-skewness effect over a large Q 2 lever arm

• scaling violation consistent with pQCD prediction

• this zero-skewness effect is non-trivial to realize in conformal space

(SO(3) sibling poles are required)

18


• CFF H posses ``pomeron behavior’’ ξ-α(Q) - α’(Q)t

α increases with growing Q 2

α’ decreases growing Q 2

• t-dependence: exponential shrinkage is disfavored (α’ ≈ 0)

dipole shrinkage is visible (α’ ≈ 0.15 at Q 2 =4 GeV 2 )

• (normalized) profile functions

ρ ∝ R d 2 ~ ∆⊥ e i~ b·~∆ ⊥

H(x, 0, t = −~ ∆

2


)

sea quarks

gluons

essentially differ

for b > 1 fm

19


Beam charge asymmetry

T Interference

BCA = dσ e − dσ + e −

=

dσ e

+ + dσ e


∝ F 1 (t)


Dispersion relation fits to unpolarized DVCS

• model of GPD H(x,x,t) within DD motivated ansatz at Q 2 =2 GeV 2

fixed: PDF normalization eff. Reage pole large t-counting rules

H(x, x, t) = n r 2α

1 + x

µ 2x

−α(t) µ b 1 − x 1

1 + x 1 + x

³

1 − 1−x

1+x

´p .

t

M 2

free:

r-ratio at small x large x-behavior p-pole mass

sea quarks (taken from LO fits)

n = 0.68, r = 1, α(t) = 1.13 + 0.15t/GeV 2 , m 2 = 0.5GeV 2 , p = 2

valence quarks

flexible parameterization of subtraction constant

+ pion-pole contribution

n = 1.0, α(t) = 0.43 + 0.85t/GeV 2 , p = 1

D(t) =

36 + 4 data points quality of global fit is good χ 2 /d.o.f. ≈ 1

−C

(1−t/M 2 c )2

21


Global GPD fit example: HERMES & JLAB

BCA HERMES

BSA CLAS/JLAB

HALL A/JLAB

22


A qualitative interpretation of first global fits

eH(x, x, t) H(x, x, t)

is two x bigger as in valence region (sounds wrong)

‣ ansatz is to improve (or to reinterpret)

‣ longitudinal polarized proton data will help to pin down

‣ real part of is crucial to reveal H(x,x,t)

Q 2 =2 GeV 2

t=-0.2 GeV 2 t=0

t=0


Where is

the zero

t=-0.2 GeV 2

predictions from fits to

H1êZEUS

HERMES

CLAS

H1êZEUS

HERMES

CLAS

Hall A

• real part of has a zero: Can it be revealed by COMPASS

large negative value of e @x=1 arises from substraction constant

(so-called D-term, [Goeke, Polyakov, Vanderhaeghen (01); lattice (≥ 07) ] )

23


DVCS observables at COMPASS (unpolarized(

target)

dσ ↓+ (φ), dσ ↑− (φ), dσ ↓+ (φ) ± dσ ↓+ (±φ), dσ ↓+ (φ) ± dσ ↑− (±φ),

0.20

predictions from fits to

• revealing real part of

dσ ↓+ (φ) − dσ ↑− (φ)

A BCS = dσ↓+ (φ)−dσ ↑− (φ)

dσ ↓+ (φ)+dσ ↑− (φ)

• revealing imaginary part of

dσ ↓+ (φ) − dσ ↓+ (−φ)

A BSA = dσ↓+ (φ)−dσ ↓+ (−φ)

dσ ↓+ (φ)+dσ ↓+ (−φ)

ABCSHj L

ABSAHj L

0.15

0.10

0.05

0.00

-0.05

0.20

0.15

0.10

0.05

0 50 100 150

0.00

0 50 100 150

24

j

H1êZEUS

HERMES

CLAS

H1êZEUS

HERMES

CLAS

Hall A

x B

=0.05

t=-0.2 GeV 2

Q 2 =2 GeV 2


Towards realistic GPD (TMD) models

L = ¯ψ (i/∂ − m) ψ − 1 2 φ ¡ ∂ 2 + λ 2¢ φ + g ¯ψψφ

struck spin-1/2 quark

Diagrammatic approach:

via covariant time ordered

perturbation theory

LC- Hamiltonian approach

collective scalar

diquark spectator

/k+/p 1 +m

/k+/p 2 +m

(k+p 1 ) 2 −m 2

k μ → (k + , k − , k ⊥ ), k ± = k 0 ± k 3 , k ⊥ = (k 1 , k 2 ).

integrate out minus component to find LCWF

p 1

coupling knows

about spin

δ(xP + − P + − 2k + )

(k+p 2 ) 2 −m

many 2

studies of

spectator quark

1

models

k 2 −λ 2 p 2

[HWANG, DM (07)]

parton number

conserved LCWF

(outer region)

parton number

violating LCWF

(central region)

25


skewness effect :

SHxL= H Hx,x,t minL

H Hx,0,0L -1

(as expected: [Hwang, DM (07)]

3

2

1

0

HERMES

+CLAS

HERMES

+CLAS

+Hall A

@Hwang,DMD

-1

0.0 0.2 0.4 0.6 0.8 1.0

X =2xêH1+xL

enhanced || @ large X=x Bj :

»x Hx,tminL»

[Guidal, Morrow; Hwang, DM (07)] )

3.0

2.5

2.0

1.5

1.0

0.5

qualitative agreement

with ρ 0 production [CLAS]

0.0

0.0 0.2 0.4 0.6 0.8 1.0

X =2xêH1+xL

‣ femto-photography

b pseudo


=

photography [Pire, Ralston (01)]

viewed as transverse `pseudo’ width [DM]

amplitude interpretation [Diehl (02)]

(distance of struck quark to spectator system )

recall of transverse width

b ⊥ =

q

4 d dt

q

4 d dt

ln H(x, x, t)¯¯¯t=0

[Burkardt (02)]

ln H(x, 0, t)¯¯¯t=0

b ¦

pseudo @fmD

1.0

0.8

0.6

0.4

0.2

valence & small X:

b ┴ ≈ b ┴

pseudo

fits give

the same

b ┴

pseudo

! model dependence

0.0

0.0 0.2 0.4 0.6 0.8 1.0

X =2xêH1+xL

large X:

b ┴ < b ┴

pseudo

26


Summary

hard exclusive leptoproduction

• possesses a rich structure, allowing to access various CFFs/GPDs

• is elaborated at NLO

• DVCS is widely considered as a theoretical clean process

• addresses our QCD understanding

GPDs are intricate and (thus) a promising tool

‣ COMPASS DVCS physics is complementary to, e.g., JLAB 12@GeV

‣ unpolarized target: revealing the real and imaginary part of H

‣ needed to reveal t- and skewness dependence of H GPD

‣ transversally polarized target: allows to access E (expected to be sizeable)

tools/technology for next generation of global fits are required:

to quantify the partonic picture and to get a better QCD understanding

27


Back up slides are coming

28


Photon leptoproduction

e ± N → e ± Nγ

measured by H1, ZEUS, HERMES, CLAS, HALL A collaborations

planed at COMPASS, JLAB@12GeV, perhaps at EIC,

Ã


dx Bj dyd|∆ 2 |dφdϕ = α3 x Bj y

1 + 4M 2 x 2 Bj

16 π 2 Q 2 Q 2

! −1/2 ¯¯¯¯

2

T

e

¯¯¯¯ 3

,

x Bj =

Q 2

2P 1 · q 1



1 + ξ ,

y = P 1 · q 1

P 1 · k ,

∆ 2 = t (fixed, small),

Q 2 = −q 2 1 (> 1GeV2 ),

29


interference of DVCS and Bethe-Heitler

processes


γ (q)

γ

p

T μν

p'

H, E, H e · · ·

J μ

J μ

12 Compton form factors elastic form factors

(helicity amplitudes)

(

)

|T BH | 2 e 6 (1 + ² 2 ) −2

2X

=

x 2 Bj y2 ∆ 2 c BH

0 + c BH

n cos (nφ) ,

P 1 (φ)P 2 (φ)

|T DVCS | 2 = e6

y 2 Q 2 (

c DVCS

0 +

2X

n=1

n=1

£

c

DVCS

n cos(nφ) + s DVCS

n sin(nφ) ¤) ,

F 1 , F 2

exactly known

(LO, QED)

harmonics

1:1

helicity ampl.

I =

±e 6

x Bj y 3 ∆ 2 P 1 (φ)P 2 (φ)

(

c I 0 +

3X £

c

I

n cos(nφ) + s I n sin(nφ)¤) .

n=1

harmonics

1:1

helicity ampl. 30


1/Q

relations among harmonics and GPDs are based on expansion:

(all harmonics are expressed by twist-2 and -3 GPDs)

½ ¾ I c1

∝ ∆ s 1 Q tw-2(GPDs) + O(1/Q3 ), c I 0 ∝ ∆2

Q 2 tw-2(GPDs) + O(1/Q4 ),

[Diehl et. al (97)

Belitsky, DM, Kirchner (01)]

½

c2

s 2

¾ I

∝ ∆2

Q 2 tw-3(GPDs) + O(1/Q4 ),

½

c3

s 3

¾ I

∝ ∆α s

Q (tw-2)T + O(1/Q 3 ),

c CS

0 ∝ (tw-2) 2 ,

½

c1

s 1

¾ CS

∝ ∆ Q (tw-2)(tw-3),

½

c2

s 2

¾ CS

∝ α s (tw-2)(tw-2) GT

setting up the perturbative framework:

twist-two

two coefficient functions at next-to

to-leading

order

[Belitsky, DM (97);

Mankiewicz et. al (97);

Ji,Osborne (98)]

evolution kernels at next-to

to-leading

order

[Belitsky, DM, Freund (01)]

next-to

to-next-to-leading

order in a specific conformal subtraction scheme

[KMP-K &

Schaefer 06]

twist-three

three including quark-gluon-quark correlation at LO

partial twist-three

three sector at next-to

to-leading

order

`target mass corrections’ (not well understood)

[Anikin,Teryaev, Pire (00);

Belitsky DM (00); Kivel et. al]

[Kivel, Mankiewicz (03)]

[Belitsky DM (01)]

31


Overview: GPD representations

``light-ray spectral functions’’

≡ R ∞ dκ

−∞ 2π eiκ(xP + −P + −2k + )

diagrammatic α-representation k + p 1 k + p 2

DM, Robaschik, Geyer,

Dittes, Hoŕejśi (88 (92) 94)

called double distributions

A. Radyushkin (96)

X

diagrams

p 1

p 2

light cone wave function overlap

(Hamiltonian approach in light-cone quantization)

SL(2,R) (conformal) expansion

(series of local operators)

one version is called Shuvaev transformation,

used in `dual’ (t-channel) GPD parameterization

Diehl, Feldmann,

Jakob, Kroll (98,00)

Diehl, Brodsky,

Hwang (00)

Radyushkin (97);

Belitsky, Geyer, DM, Schäfer (97);

DM, Schäfer (05); ….

Shuvaev (99,02); Noritzsch (00)

Polyakov (02,07)

each representation has its own advantages,

32

however, they are equivalent (clearly spelled out in [Hwang, DM 07])


A partonic duality interpretation

quark GPD (anti-quark x → -x):

F = θ(−η ≤ x ≤ 1) ω ¡ x, η, ∆ 2¢ + θ(η ≤ x ≤ 1) ω ¡ x, −η, ∆ 2¢

ω ¡ x, η, ∆ 2¢ = 1 η

Z x+η

1+η

0

dy x p f(y, (x − y)/η, ∆ 2 )

dual interpretation on partonic level:

η+x

2

η−x

2

support extension

is unique [DM et al. 92]

x+η

2

x−η

2

p

p

central region - η < x < η

mesonic exchange in t-channel

ambiguous (D-term)

[DM, A. Schäfer (05)

KMP-K (07)]

p

p

outer region η < x

33

partonic exchange in s-channel


(partonic) `quantum’ numbers in GPD representations

spectator model

leading SO(3) PW

t-factorized (DD)


about representation

is not so essential

should be replaced by

How a GPD looks like on its

cross-over trajectory

34


SL(2,R) representations for GPDs

• support is a consequence of Poincaré invariance (polynomiality)

H j (η, t, μ 2 ) =

Z 1

−1

dx c j (x, η)H(x, η, t, μ 2 ) , c j (x, η) = η j C 3/2

j (x/η)

• conformal moments evolve autonomous (to LO and beyond in a special scheme)

μ d

dμ H j(η, t, μ 2 ) = − α s(μ)


γ(0) j H j (η, t, μ 2 )

• inverse relation is given as series of mathematical distributions:

∞X

H(x, η, t) = (−1) j p j (x, η)H j (η, t) , p j (x, η) ∝ θ(|x| ≤ η) η2 − x 2

η

j=0

C3/2

j+3 j

• various ways of resummation were proposed:

• smearing method [Radyushkin (97); Geyer, Belitsky, DM., Niedermeier, Schäfer (97/99)]

• mapping to a kind of forward PDFs [A. Shuvaev (99), J. Noritzsch (00)]

• dual parameterization (a mixture of both) [M. Polyakov, A. Shuvaev (02)]

• based on conformal light-ray operators [Balitsky, Braun (89); Kivel, Mankewicz (99)]

• Mellin-Barnes integral [DM, Schäfer (05); A. Manashov, M. Kirch, A. Schäfer (05)]

(−x/η)

35


GPD ansatz at small x from t-channel view

at short distance a quark/anti-quark state

is produced, labeled by conformal spin j+2

they form an intermediate mesonic state

with total angular momentum J

strength of coupling is fj J, J ≤ j + 1

γ ∗

q

f J j

¯q

γ (∗)

mesons propagate with

1

m 2 (J)−t ∝ 1

J−α(t)

decaying into a nucleon anti-nucleon pair

with given angular momentum J,

described by an impact form factor

¯P 1 P 2

F J j (t) =

f J j

J − α(t)

1

(1 − t

M 2 (J) )p

! GPD E is zero if chiral symmetry holds

(partial waves are Gegenbauer polynomials with index 3/2)

D-term arises from the SO(3) partial wave J=j+1 (j -1)

36


Can the skewness function be constrained from lattice

• relation among measurable and GPD Mellin moments at h=0:

Z 1

0

dξ ξ j =mF(ξ, t, Q 2 ) LO = πf j (t, Q 2 ) £ 1 + δ j (t, Q 2 ) ¤

• deviation factors:

δ j (t, μ 2 ) =

R 1

0 dx xj S(x, t, μ 2 )F (x, η = 0, t, μ 2 )

R 1

0 dx xj F (x, η = 0, t, μ 2 )

are given by a series of operator expectation values with increasing spin j+n+1

∞X

δ j (t, μ 2 f (n)

j+n

) =

(t, μ2 )

, f (n)

f j (t, μ 2 j

(t, μ 2 ) = 1 d n

)

n! dη n f j(η, t, μ 2 ¯

) ¯η=0

n=2

even

• lattice can evaluate j=0,1,2,(3), i.e., n=2:

• wrong expectation from evolution:

the analog small x prediction is ruled out

[Shuvaev et al. (99)]

δ 0 (t, μ 2 = 4 GeV 2 ) ≈ 0.2+

δ j ∼ 2j+1 Γ(5/2 + j)

Γ(3/2)Γ(3 + j) − 1

δ 0 ∼ 0.5 δ 1 ∼ 1.5

thanks to

Ph. Hägler

37


A GPD fit agenda

(a personal view)

decomposition of twist-two

two CFFs (with and without twist-two dominance hypothesis)

• dispersion integral fits (least square method) on the full set of fixed target data

• data filtering is crucial to get rid of the twist-two dominance hypothesis

GPD model fits (based on the least square method)

• fully flexible GPD model in conformal moment space up to NNLO

• (reggeized) spectator quark models up (to NLO)

(given in double distribution representation, positivity constraints are implemented)

• holographic GPD models, such as Radyushkin double distribution model

neural networks (representing and extracting CFFs or GPDs )

• to get rid of theoretical biases

• error propagation/estimates

38

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