observations of chiral odd gpds and their implications...presentation for qcd-evolution 2013!...
TRANSCRIPT
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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
QCD-Evol2013 GR.Goldstein
Jqx= 1
2 dx H (x, 0, 0)+ E(x, 0, 0)[ ]x!
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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
T-reversal at ξ =0
QCD-Evol2013 GR.Goldstein
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Invert to obtain model for GPDs
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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’
QCD-Evol2013 GR.Goldstein
!H !"++
*(k ',P ')"
++(k,P) #"
#+
*(k ',P ')"#+ (k,P)
!E!"++
*(k ',P ')"
+# (k,P) #"+#
*(k ',P ')"
++(k,P)
λ
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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
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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!
QCD-Evol2013 GR.Goldstein
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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
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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)!
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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
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Compton Form Factors Real & Imaginary Parts
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Having fit other data we predict Hermes data
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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 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
QCD-Evol2013 GR.Goldstein
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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
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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
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-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
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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 !
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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
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Vary tensor charge as a parameter to see sensitivity of data
5/8/13 42
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Asymmetries: Longitudinal polarizations
ALL & ALU
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-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
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η/π
o
Preliminary
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Summarizing chiral odd for π0
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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!
QCD-Evol2013 GR.Goldstein
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
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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
QCD-Evol2013 GR.Goldstein
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) !
!
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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 )
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-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)
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-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?
QCD-Evol2013 GR.Goldstein
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-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
QCD-Evol2013 GR.Goldstein
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?
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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