Observations of Chiral Odd GPDs and their Implications...Presentation for QCD-Evolution 2013! Jefferson Lab! These ideas were developed in Trento ECT*, INT, Jlab, DIS2011, Frascati

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Observations of Chiral Odd GPDs and their Implications

Gary R. Goldstein !Tufts University!

Simonetta Liuti,!

Osvaldo Gonzalez-Hernandez!

University of Virginia!

!

Presentation for QCD-Evolution 2013!Jefferson Lab!

These ideas were developed in Trento ECT*, INT, Jlab, DIS2011, Frascati INF, Transversity 2011, PANIC, POETIC, QCD-Evol2013 & in consultation with many of you !

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Thomas Jefferson

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Outline !   Hadron Spin Structure from GPD & TMD à GTMD & Fracture functions

perspectives !!   Our “Flexible” parameterization for Chiral Even GPDs!

! Regge ✖ diquark spectator model: R✖Dq!! Why Regge? à product form!!   Satisfying constraints: pdfs, EM Form Factors!!   Results for DVCS (transverse γ* à transverse γ) !!   EM Form factors!

! Extend to Chiral Odd GPDs via diquark spin relations! Transversity!

!   Model relations between Chiral even & odd helicity amps!!   π0 production sizable γ*

Transverse!

! Helicity & Transversity Amplitudes, unintegrated GPDs & TMDs !!   Spin amps <-> Spin bilinears !

!   Extend R✖Dq to TMDs ! !See S. Liuti’s talk!!   Trans even & odd!

! Wigner Distributions à GTMD à TMDs & GPDs !!   What is measurable?!

QCD-Evol2013 GR.Goldstein

Distributions – “landscape” WignerDist. X(x,ξ,kT

2,bT2,kT⋅bT)

GTMD X(x,ξ,kT

2,ΔT2,kT⋅ΔT)

TMD h1(x,kT)

GPD H(x, ξ, t)

fracture functions Gp,h(x,z,pT, Q2 ) PDF

q(x) FF F(ΔT

2)

Spin densities q(x,bT)

Charge densities F(bT

2)

FT ΔT / bT

FT ΔT / bT

FT ΔT / bT

ξ=0, Integrate bT

4 4

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); x=ζ/(2-ζ) see Ahmad, GG, Liuti, PRD79, 054014, (2009) for first chiral odd GPD parameterization Gonzalez, GG, Liuti PRD84, 034007 (2011)

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


Jqx= 1

2 dx H (x, 0, 0)+ E(x, 0, 0)[ ]x!

5

6

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

T-reversal at ξ =0


7

Invert to obtain model for GPDs

7

S=0 diquark Spectator model A++,-+= - A++,+-

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

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

double flip


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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

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

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

λ’

Λ’


!H !"++

*(k ',P ')"

++(k,P) #"

#+

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

!E!"++

*(k ',P ')"

+# (k,P) #"+#

*(k ',P ')"

++(k,P)

λ

9

S=0 Chiral even <-> odd

Invert to get GPDs – same helicity amp sets


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S=1 Chiral even <-> odd

Invert to get GPDs


Chiral odd integrated amplitudes

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-


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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 - Phys.Rev. D84, 034007 (2011))!

§  2. getting 8 spin dependent GPDs!•  Chiral Odd GPDs π0 production is testing ground (Ahmad, GG, Liuti, PRD79,054014 (2009),

Gonzalez, GG, Liuti, arXiv:1201.6088 [hep-ph] J. Phys. G: Nucl. Part. Phys. 39 115001 (2012)!

•  => Chiral even AND Chiral odd GPDs normalizations!! !H, E, . . ß helicity amp relations à HT, ET, . . !

•  Small x & Regge behavior in pdf’s & DVCS!


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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 (MXq, Λq, αq)

+ initial Qo2

✔  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 of GPDs) Evolution

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

Regge factor R ~ X-α (t)

Quark-Diquark

EM Form Factors

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0.2

0.4

0.6

0.8

1

1.2

u

d

Diquark

Regge

Total u

Total dt2

F1

d

u

u

d

-t (GeV2)

κq-1 t

2 F

2

0

0.05

0.1

0.15

0.2

0.25

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

O.Gonzalez, GG, S.Liuti arXiv:hep/1206.1876; data: G.D. Cates, et al. PRL106,252003 (2011).

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Reggeization via spectator or diquark mass formulation

Where does the Regge behavior come from?

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

Diquark spectral function

ρ

MX2

∝ (MX2)α

∝δ(MX2-MX

2)

“Regge”

+ Q2 Evolution


QCD-Evol2013 GR.Goldstein 16

Reggeization

q, Λγ

k’= p –Δ. λ’

q'=q+Δ, Λγ’

k, λ

P’= P-Δ, Λ’ P, Λ

MXà kX,λ’’

Landshoff, Polkinghorn, Short ‘71 Brodsky, Close, Gunion ’71 Regge behavior required for Compton AHLT ’07, ’09 Gorshteyn & Szczepaniak (PRD, 2010) Brodsky, Llanes-Estrada ‘07 Brodsky, Llanes, Szczepaniak arXiv:0812.0395 Gonzalez, GG, Liuti, arXiv:1201.6088 [hep-ph] J. Phys. G: Nucl. Part. Phys. 39 115001 (2012)!

QCD-Evol2013 GR.Goldstein 17

Spectral function leads to small x

X≠0 ~ Factorization with non Regge behavior

X≈0 Regge behavior

QCD Bilbao 10/23/12 18

~Initially introduced by Radyushkin, Burkardt, … to account for coordinate space behavior

Now see as effectively taking into account Regge cuts O. Gonzalez Hernandez, GG, S. Liuti arXiv 1206.1876

P.D.B.Collins and Kearney

Parametric Form

5/8/13 19

R✖Dq

the minimal number of parameters necessary to fit X and t ?” Results of Recursive Fit

5/8/13 20

21

00.20.40.60.8

11.21.4 Q2= Qo

2

Q2 = 2 GeV2Q2= Qo

2 (BCR)

Q2 = 2 GeV2 (BCR)Q2 = 2 GeV2 (Alekhin)x

f 1u

x

x f 1d

00.10.20.30.40.50.60.70.8

10 -3 10 -2 10 -1 1


22 QCD-Evol2013 GR.Goldstein

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Compton Form Factors Real & Imaginary Parts


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Having fit other data we predict Hermes data


25 25

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 fails to describe transverse γ *

N N

πo

+1/2 -1/2

+1,0 b1 & h1

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


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f1 f2 f3 f4 f5 f6

6 helicity amps for π0

The question is: how do we normalize the GPDs? Only Physical constraints on the various chiral-odd GPDs are

HT(x,0,0) = q!

"(x) # q!$(x) = h

1(x)

Forward limit

!ET(x,!,t)dx! = 0

HT(x,!,t)dx! = !

T(t)

ET(x,!,t)dx! = 2 !H

T+ E

T( )dx! = !

T(t)

No direct interpretation of ET

Form Factors

5/8/13 27

5/6/13 QCD-Evol2013 Goldstein 28

-0.10

0.10.20.30.40.50.60.70.8

Q2= 2 GeV2 This paperBCRTorino

x h 1u

x

x h 1d

-0.15-0.125

-0.1-0.075

-0.05-0.025

00.025

10 -1 1

Extraction of transversity after using DVCS data via chiral evenßàodd

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GPDs at fixed Q2 & ς vs. X & t

5/8/13 30

-4-2024

-6-4-202

51015

2468

-80-60-40-20

0

-80-60-40-20

0

02468

0 0.5 1 1.5

Im HTQ2= 1.1 GeV2 xBj=0.13

Re HT

Im 2 H~

T+ET

Re 2 H~

T+ET

Im H~

T Re H~

T

Im E~

T

-t (GeV2)

Re E~

T

-t (GeV2)

05

1015

0 0.5 1 1.5

Chiral odd GPD form factors

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6 helicity amps for π0 after Compton Form Factors

Observables

• Cross sections!• Asymmetries!

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Cross sections for π 0

Unpolarized cross section components

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(t0-t)/2M2 = t

HT, ET, !ET, ET

5/6/13 QCD-Evol2013 GR.Goldstein

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data

h1=HTh1T

perp=H~

TBoer Mulders=2H

~T+ET

x

1st m

omen

t TM

D v

s. G

PDs

-1.5

-1

-0.5

0

0.5

1

10-3

10-2

10-1

1


-0.250

0.250.5

0.75

-0.050

0.05

0

0.5

1

0

0.5

1

-3-2-10

-0.5

0

0.5

-0.5

0

0.5

0.2 0.4 0.6 0.8

HTu

=0, t=0, Q2= 2 GeV2 HTd

2 H~

Tu+ET

u 2 H~

Td+ET

d

H~

Tu

H~

Td

E~

Tu

X

E~

Td

X

-0.5

0

0.5

0.2 0.4 0.6 0.8

37

How well do the parameters fixed with DVCS data reproduce πo

electroproduction data?

Hall B data, Kubarovsky& Stoler, PoS ICHEP 2010 & PRL 109, 112001 (2012)

5/6/13 QCD-Evol2013 GR.Goldstein

Comparing to other models

• The t-> 0 feature for us is that f2 dominates & it is driven by HT. But f1 & f4 also contribute as ~√(t0-t), however weaker. !

•  f1 & f4 are not equal in magnitude, especially vs. ζ or ξ. !

•  In ALL ~ |f1|2 + |f2|2 - |f3|2 - |f4|2 !

38

39

-400

-200

0

200

400

600

-400

-200

0

200

400

600

-80-60-40-20

020406080

0 0.5 1 1.5

FUU,T + FUU,LFUU

cos

FUUcos 2

F UU

(nb/

GeV

2 )

f10++

f10+-

f10-+

f10--

xBj= 0.28 Q2= 2.2 GeV2

f10++

f10+-

f10-+

f10--

-t (GeV2)

F UU

(nb/

GeV

2 )

f10 * ++ f10

--

f10 * +- f10

-+

-t (GeV2)

-400-300-200-100

0100

0 0.5 1 1.5

40

-400

-200

0

200

400

600

-400

-200

0

200

400

600

-80-60-40-20

020406080

0 0.5 1 1.5

FUU,T + FUU,LFUU

cos

FUUcos 2

F UU

(nb/

GeV

2 )

f10++

f10+-

f10-+

f10--

xBj= 0.19 Q2= 1.6 GeV2

f10++

f10+-

f10-+

f10--

-t (GeV2)

F UU

(nb/

GeV

2 )

f10 * ++ f10

--

f10 * +- f10

-+

-t (GeV2)

-400-300-200-100

0100

0 0.5 1 1.5

CLAS π0 arXiv:1206.6355v1 [hep-ex] PRL

41 QCD-Evol2013 GR.Goldstein

Vary tensor charge as a parameter to see sensitivity of data

5/8/13 42

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Asymmetries: Longitudinal polarizations

ALL & ALU

44

-t (GeV2)

0 /

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Q2=1.75 GeV2

xBj=0.22

5/8/13 45

η/π

o

Preliminary

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Summarizing chiral odd for π0

47

Spectator inspired model of GPDs •  2 directions – !

§  1. getting good parameterization of H, E & ~H, ~E ! satisfying many constraints ! !Simonetta Liuti’s talk!

(see O. Gonzalez-Hernandez, GG, S. Liuti - Phys.Rev. D84, 034007 (2011))!

§  2. getting 8 spin dependent GPDs!•  Chiral Odd GPDs π0 production is testing ground (Ahmad, GG, Liuti, PRD79,054014 (2009),

Gonzalez, GG, Liuti, arXiv:1201.6088 [hep-ph] J. Phys. G: Nucl. Part. Phys. 39 115001 (2012)!

•  => Chiral even related to Chiral odd GPDs normalizations!! !H, E, . . ß helicity amp relations à HT, ET, . . !

•  Small x & Regge behavior in pdf’s & DVCS!•  Bridge through GPD in helicity or transversity to TMDs? GTMDs!

! ! ! !see S.Liuti talk!


Distributions – “landscape” What happens at the unintegrated level

WignerDist. X(x,ξ,kT

2,bT2,kT⋅bT)

GTMD X(x,ξ,kT

2,ΔT2,kT⋅ΔT)

TMD h1(x,kT)

GPD H(x, ξ, t)

fracture functions Gp,h(x,z,pT, Q2 ) PDF

q(x) FF F(ΔT

2)

Spin densities q(x,bT)

Charge densities F(bT

2)

FT ΔT / bT

FT ΔT / bT

FT ΔT / bT

ξ=0, Integrate bT

we are here

49

1

2

3

4

5x=0.01x=0.1x=0.01 (BCR)x=0.1 (BCR)

f 1u (x,

kT2 )

kT2 (GeV2)

f 1d (x,

k T2 )

1

2

3

4

5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Extend R✖Dq to unintegrated kT


Spin amplitudes, GPDs à TMDs

•  F T ‘s are Diagonal in transversities ⇒ probabilistic interpretations !

w/o b-space!•  HT & HT’ Same spin form as TMDs h1T(x,kT

2) combined with h1T

⊥(x,kT2) !

h1T (x,kT2) compare HT(x,0,ΔT

2) ! or unintegrated HT(x,0, ΔT

2,k) !

!

50

h1(x,

!kT2 ) = h1T (x,

!kT2 )+

!kT2

2M 2 h1T! (x,!kT2 )

f1T!(1)(x) = d 2"

!kT

!kT2

2M 2 f1T! (x,!kT2 ) = # g

2MT (x,ST )


51

-1-0.8-0.6-0.4-0.2

-0

-xE, Q2 = Q2o

-xE, Q2 = 2 GeV2

xf1T, Q2 = Q2o (BCR)xf1T, Q2 = 2 GeV2 (BCR)xf1T, Q2 = Q2o

x f 1T

u , -xE

u

x

x f 1T

d , -xE

d

-0.250

0.250.5

0.751

1.251.5

1.752

10 -2 10 -1

Beyond R✖Dq to get trans-odd f1T⊥ need f.s.i./gauge link

Same spin structure as E for this model (0th moment)


52

-1-0.8-0.6-0.4-0.2

-0

-xE, Q2 = Q2o

-xE, Q2 = 2 GeV2

xf1T, Q2 = Q2o (BCR)xf1T, Q2 = 2 GeV2 (BCR)xf1T, Q2 = Q2o

x f 1T

u , -xE

u

x

x f 1T

d , -xE

d

-0.250

0.250.5

0.751

1.251.5

1.752

10 -2 10 -1

Beyond R✖Dq to get trans-odd f1T⊥ need f.s.i./gauge link

Same spin structure as E for this model

All GPDs’ spin structures have corresponding TMDs Is this more general than R✖Dq or simpler quark models? Wigner distributions or Generalized TMDs?


53

-1.8-1.6-1.4-1.2

-1-0.8-0.6-0.4-0.2

0

Q2 = Q2oQ2 = 2 GeV2

Q2 = Q2o (BCR)Q2 = 2 GeV2 (BCR)x

h 1u

x

x h 1d

-0.8

-0.6

-0.4

-0.2

-0

10 -2 10 -1

h1⊥(0)(x)

A++,+! ! A+!,++ " 2 !HT + ET

A++,!+ ! A!+,++ " E!m F"+,"#"$ % h1

& (x,kT2 )

!m F+!,#!!$ % f1T& (x,kT

2 )

Is this more general than R✖Dq or simpler quark models? Wigner distributions or Generalized TMDs? See S. Liuti talk


Summary

• Flexible parameterization for chiral even from form factors, pdfs & DVCS R✖Dq !

• Extended R✖Dq to chiral odd sector!• DVMP – π0 many dσ ‘s & Asymmetries!• Connect to TMDs and beyond!

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Backup Slides

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56

Scalar diquark contribution to H(X,0,0) à G(X,0,0) for MX

=0.6 & 1.7 How to “Reggeize” for small x while keeping large x spectator behavior?

57

Spectral distribution of form ρ(MX

2,k2)≈ ρ(MX2) β(k2)

ρ(MX

2) =(MX2/M0

2) β/(1+MX2/M0

2)β-α+1 β(k2) chosen to give large kT

2 falloff behavior

ρ(MX2)

Polynomiality! Goldstein et al. arXiv:1012.3776

5/8/13 58

Observations of Chiral Odd GPDs and their Implications...Presentation for QCD-Evolution 2013! Jefferson Lab! These ideas were developed in Trento ECT*, INT, Jlab, DIS2011, Frascati

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