A Flexible Parameterization of GPDs & Their Role in DVCS & - - PowerPoint PPT Presentation

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A Flexible Parameterization of GPDs & Their Role in DVCS & Neutral Meson Leptoproduction

Gary R. Goldstein Tufts University

Simonetta Liuti,

• S. Ahmad, Osvaldo Gonzalez-Hernandez

University of Virginia

Presentation for PANIC MIT July 2011

These ideas were developed in Trento ECT*, INT, Jlab, DIS2011, Frascati INF, . . .

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Outline

New “Flexible” parameterization for Chiral Even GPDs

Regge ✖ diquark spectator model

Satisfies all constraints

Results for DVCS (transverse γ*  transverse γ)

cross sections & asymmetries

Extend to Chiral Odd GPDs via diquark spin relations

Some simple relations between Chiral even & odd helicity amps

π0, η, η’ production data involve sizable γ*

Transverse

(factorization shown at leading twist for γ*

Longitudinal [Collins, Franfurt, Strikman])

γ*

T requires

requires chiral odd GPDs

Q2 dependence for π0 depends on γ*+(ρ, b1)π0

π0 cross sections & asymmetries

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t DVCS & DVMP γ*(Q2)+P→(γ or meson)+P’ 
 partonic picture P'+=(1-ζ)P+ P’T=-Δ k+=XP+ k'+=(X-ζ)P+ q q+Δ k'T=kT-Δ kT P+

} {

ζ→0 Regge

Quark-spectator quark+diquark

Factorized “handbag” picture

}

X>ζ DGLAP ΔT -> bT transverse spatial X<ζ ERBL x=(X-ζ/2)/(1-ζ/2); ξ=ζ/(2-ζ) see Ahmad, GG, Liuti, PRD79, 054014, (2009) for first chiral odd GPD

parameterization focused on pseudoscalar production

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GPD

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Momentum space nucleon matrix elements of quark field correlators

see, e.g. M. Diehl, Eur. Phys. J. C 19, 485 (2001).

Chiral even GPDs

• > Ji sum rule

Chiral odd GPDs

• > transversity

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Jq

x = 1 2

dx H(x,0,0)+ E(x,0,0)

[ ]x

5 5

How to determine GPDs? Flexible Parameterization  Recursive Fit

CHIRAL EVENS: O. Gonzalez Hernandez, G. G., S. Liuti, arXiV:1012.3776 PRD accepted

Constraints from Form Factors

Dirac

Pauli, etc. including Axial & Pseudoscalar

How can these be independent of ξ? Constraints from Polynomiality

dx x nH(x,,t) = An ,k

k =0, 2,.. n

• 1

+1

• (t) k + 1 (1)

n

2 Cn (t) n +1 dxH(x,,t)

1

• = F

1(t)

dxE(x,,t)

1

• = F2(t)

dx x n E(x,,t)

1 1

• =

Bn ,k(t) k

k=0,2,... n

• 1- (-1)

n

2 Cn (t)n +1

Result of Lorentz invariance & causality. Not necessarily built into models

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Constraints from pdf’s:H(x,0,0)=f1(x),H~(x,0,0)=g1(x), HT(x,0,0)=h1(x)

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Spectator inspired model of GPDs

• 2 directions –
• 1. getting good parameterization of H, E & ~H, ~E

satisfying many constraints

(see O. Gonzalez-Hernandez, GG, S. Liuti arXiv: 1012.3776)

• 2. getting 8 spin dependent GPDs
• Chiral Odd GPDs π0 production is testing ground (New results – preliminary)
• Simplest Spectator -- scalar diquark –

helicity amps for Pq ::::: q’P’ Spin simplicity - Pq+diq and q’+diqP’ are spin disconnected => Chiral even related to Chiral odd GPDs Chiral even related to Chiral odd GPDs H, E, . .  helicity amp relations  HT, ET, . .

• Axial diquark: more complex linear relations

& distinction between u & d flavors

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Spectator inspired model (cont’d)

• u + scalar (ud+du) -> u struck quark
• axial (uu, ud-du, dd) -> d and u struck

: 3 – relations among helicity amps => at ξ & t not 0

• E=-ξET + ET~ , ξE~ = - 2HT~ - ET + ξET~ (multiplied by √(t0-t))

HT +(t0-t)HT~/4M2= (H+H~)/2–ξ[(1+ξ/2)E+ξE~]/(1-ξ2) and HT~ in terms of chiral evens

Could apply to u-quark GPDs – part that has scalar if same Regge factors

• Regge-like behavior 1/xα(t) could differentiate among GPDs depending on

t-channel quantum numbers

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k+=XP+ k’+=(X-ζ)P+ P’+=(1- ζ)P+ PX

+=(1-X)P+

Vertex Structures

P+ PX

+=(1-X)P+

λ Λ

S=0 or 1

E

++

* (k',P')+ (k,P) +

+

* (k ', P')+ +(k,P)

First focus e.g. on S=0 pure spectator

H

+ +

* (k ', P')++(k, P) +

+

* (k ',P') +(k, P)

Vertex functions Φ

λ’ Λ’

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• H

++

* (k',P')++(k,P)

+

* (k',P')+(k,P)

• E

++

* (k',P')+(k,P)

+

* (k',P')++(k,P)

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k+=XP+ k’+=(X-ζ)P+ P’+=(1- ζ)P+ PX

+=(1-X)P+

Vertex Structures

P+ PX

+=(1-X)P+

λ Λ

S=0 or 1

E

++

* (k',P')+ (k,P) +

+

* (k ', P')+ +(k,P)

First focus e.g. on S=0 pure spectator

H

+ +

* (k ', P')++(k, P) +

+

* (k ',P') +(k, P)

Vertex function Φ Note that by switching λ- λ & Λ  -Λ (Parity) will have chiral evens go to ± chiral odds giving relations – before k integrations A(Λ’λ’;Λλ) ±A(Λ’,λ’;-Λ,-λ) but then (Λ’-λ’)-(Λ-λ) ≠(Λ’-λ’)+(Λ-λ) unless Λ=λ

λ’ Λ’

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• H

++

* (k',P')++(k,P)

+

* (k',P')+(k,P)

• E

++

* (k',P')+(k,P)

+

* (k',P')++(k,P)

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Helicity amps (q’+N->q+N’) are linear combinations of GPDs

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In diquark spectator models A++;++, etc. are calculated directly. Inverted -> GPDs

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Invert to obtain model for GPDs

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A++,-+= - A++,+- A-+,++ = - A+-,++ A++,++ = A++,--

double flip

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Fitting Procedure e.g. for H and E

✔

Fit at ζ=0, t=0 ⇒ Hq(x,0,0)=q(X)

✔

3 parameters per quark flavor (MX

q, Λq, αq) + initial Qo 2

✔

Fit at ζ=0, t≠0 ⇒

✔

2 parameters per quark flavor (β, p)

✔

Fit at ζ≠0, t≠0 ⇒ DVCS, DVMP,… data (convolutions of GPDs with Wilson coefficient functions) + lattice results (Mellin Moments

• f GPDs)

✔

Note! This is a multivariable analysis ⇒ see e.g. Moutarde, Kumericki and D. Mueller, Guidal and Moutarde

Regge factor Quark-Diquark

R = X [+ '(1X)p t+ ( )t]

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Reggeized diquark mass formulation

Following DIS work by Brodsky, Close, Gunion (1973)

Diquark spectral function ρ MX

2

∝ (MX

2)α

∝δ(MX

2-MX 2)

“Regge”

+ Q2 Evolution

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Flexible Parametrization of Generalized Parton Distributions from Deeply Virtual Compton Scattering Observables Gary R. Goldstein, J. Osvaldo Gonzalez Hernandez , Simonetta Liuti arXiv:1012.3776 PANIC2011 GR.Goldstein

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Compton Form Factors  Real & Imaginary Parts

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Observables

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Note GPD & Bethe-Heitler separate amplitudes

• > interference linear in GPD

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Hall B data ALU(90o)

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Hermes data

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• 0.6
• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1

• 0.6
• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.1 10

• 2

10

• 1

1 1

Hermes kinematics x=0.097 Q2=2.5 GeV2

• t=0.118 GeV2

x=0.097

• t=0.118 GeV2

Q2=2.5 GeV2 Hermes kinematics no DVCS

• t (GeV2)

Q2 (GeV2) xBj 10

• 1

ALU(90o)

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Interference contribution to cos(φ) term in DVCS cross section

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N N

πo

+1,0 +1/2

• 1/2

+1/2 +1/2 +1/2

• 1/2
• 1/2

e.g. f+1+,0-(s,t,Q2)

+1,0

g+1+,0- A++,- - HT

q nos of C-odd 1- - exchange 1+- exchange b1 & h1 What about coupling of π to q→q′ ? Assumed γ5 vertex Then for mquark=0 has to flip helicity for q→π+q′ and q⋅q′ ≠ 0.

Naïve twist 3 ψbar γ5 ψ

Rather than γµγ5 – does not flip twist 2. But q’ γµγ5q will not contribute to transverse γ. Differs from t- channel approach to Regge factorization

N N

πo

+1/2

• 1/2

+1,0

b1 & h1

Exclusive Lepto-production of πo or η, η’ to measure chiral odd GPDs

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Q2 dependent form factors t-channel view

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see Belitsky, Ji, Yuan (2007) & Ng (2007)

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New Chiral Odd GPDs & exclusive π0 electroproduction

xBj=0.41 Q2=3.2 GeV 2

• t (GeV 2)

Cross Section

• 20
• 10

10 20 30 40 50 0.2 0.4 0.6 0.8 1 1.2 1.4

σT +εσL σLT σTT

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New Chiral Odd GPDs & exclusive π0 electroproduction

xBj=0.41 Q2=3.2 GeV 2

• t (GeV 2)

Cross Section

• 20
• 10

10 20 30 40 50 0.2 0.4 0.6 0.8 1 1.2 1.4

σT +εσL σLT σTT data:Q2≈3.2 GeV2, x ≈0.41

preliminary data courtesy

• V. Kubarovsky, HallB

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New Chiral Odd GPDs & exclusive π0 electroproduction

σT +εσL σLT σTT

xBj=0.19 Q2=2.3 GeV 2

• t (GeV 2)

Cross Section

• 40
• 20

20 40 60 80 100 0.2 0.4 0.6 0.8 1 1.2 1.4 PANIC2011 GR.Goldstein

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New Chiral Odd GPDs & exclusive π0 electroproduction

σT +εσL σLT σTT

xBj=0.19 Q2=2.3 GeV 2

• t (GeV 2)

Cross Section

• 40
• 20

20 40 60 80 100 0.2 0.4 0.6 0.8 1 1.2 1.4

data:Q2≈2.3 GeV2, x ≈0.34

preliminary data courtesy

• V. Kubarovsky, HallB

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• 1.00E-04
• 5.00E-05

0.00E+00 5.00E-05 1.00E-04 1.50E-04 2.00E-04 2.50E-04 3.00E-04

• 2
• 1.5
• 1
• 0.5

sigma_L sigma_T sigma_TT simga_LT sigma_LT'

Q2=1.3 GeV2, x =0.13

data:Q2≈1.6 GeV2, x ≈0.19

preliminary data courtesy

• V. Kubarovsky, HallB

large s, small t, but Q2 too small for GPDs

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data:Q2≈2.3 GeV2, x ≈0.32

preliminary data courtesy

• V. Kubarovsky, HallB
• 4.00E-05
• 2.00E-05

0.00E+00 2.00E-05 4.00E-05 6.00E-05 8.00E-05 1.00E-04

• 2
• 1.5
• 1
• 0.5

sigma_L sigma_T sigma_TT sigma_LT

Q2=2.3 GeV2, x =0.27

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• delicate interplay among amps, GPDs & Compton Form

factor phases. very sensitive to physical parameters – tensor charges, “anomalous transversity”

• AUT (sin(Φ-Φs)) AUT’ (sinΦs) α (beam pol’z’n) AUL (“long.”)

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• 0.08
• 0.06
• 0.04
• 0.02

0.02 0.04 0.06 0.08 0.1 0.12 0.5 1 1.5 2

Asymmetries Q^2=3.5, x=0.36 Asymmetries Q^2=3.5, x=0.36

A_UT A_UT' alpha A_UL

• 0.06
• 0.04
• 0.02

0.02 0.04 0.06 0.08 0.5 1 1.5 2

Asymmetries Q^2=2.5, Asymmetries Q^2=2.5, x=0.25 x=0.25

A_UT A_UT' alpha A_UL

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Conclusions



Flexible Model GPDs → phenomenology (DVCS & DVMP)



Spectator models relate Chiral even to Chiral odd GPDs. How far broken? Regge behavior



Exclusive π0 electroproduction observables (depend on axial vector 1+- exchange quantum numbers)



Pseudoscalar form factor without π0 limits E~ coupling for π0



GPD HT yield values of δu & δd also have κT

u & κT d .



dσT/dt, dσTT/dt, AUT, beam asymmetry, beam-target correlations, dσL/dt, dσLT/dt



FUTURE: DVCS & π0 along with η, ρ, ω production will narrow range of basic parameters of GPDs, transversity & hadronic spin.

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backup slides

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Spectator inspired model (cont’d)

• recall that TMDs f1T

1T perp perp = h

= h1

perp perp in scalar diq model with gluon

exchange/gauge link/f.s.i. (Gamberg & Goldstein 2002)

• E=-ξET + ET~ integrated over x at ξ=0 
• ξE = - 2HT - ET + ξET integrated over x at ξ=0,t=0 

No – overall √(t0-t) on both sides => only sensible for non-zero t

• at t->t0 HT = (H+H )/2-ξ[(1+ξ/2)E+ξE ]/(1-ξ2) integrated
• ver x at ξ=0 

for u with“scalar diquark” remnant & 2 . . . . (this roughly Torino analysis) Saturates Soffer bound at low Q2

• Departure from these simple results ->

contribution

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Observables

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more observables

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All GPDs

Ahmad, GRG, Liuti, PRD79, 054014 (2009) 7/24/11 PANIC2011 G.R.Goldstein

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All GPDs

Ahmad, GRG, Liuti, PRD79, 054014 (2009) 7/24/11 PANIC2011 G.R.Goldstein

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Regge-cut model Beam-spin asymmetry α data R. De Masi et al.,Phys.Rev.C77, 042201 (2008). Regge-cut predictions - comparisons involve εL, ε

Ahmad, GRG, Liuti, PRD79, 054014 (2009) blue

GPD predictions (preliminary) red

σLT’~ Im [ f5*(f2+f3) + f6*(f1-f4)]

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