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
dσ
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
≈
2ξ
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
dσ
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
dσ
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
≈
2ξ
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)
2π
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)
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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)
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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
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