Hadron Tomography
Matthias Burkardt
burkardt@nmsu.edu

New Mexico State University
Las Cruces, NM, 88003, U.S.A.

Hadron Tomography – p.1/30

Outline
GPDs: probabilistic interpretation as Fourier transforms of impact
parameter dependent PDFs
H(x, 0, −∆2⊥ ) −→ q(x, b⊥ )H̃(x, 0, −∆2⊥ ) −→ ∆q(x, b⊥ )E(x, 0, −∆2⊥ )֒→ ⊥ deformation of unpol. PDFs in ⊥ pol. target
Sivers effect
2H̃T + ET −→ ⊥ deformation of ⊥ pol. PDFs in unpol. target
correlation between quark angular momentum and quarktransversity
Boer-Mulders function h⊥1 (x, k⊥ )Summary

Hadron Tomography – p.2/30

Generalized Parton Distributions (GPDs)
GPDs: decomposition of form factors at a given value of t, w.r.t. theaverage momentum fraction x = 21 (xi + xf ) of the active quark
ZZ

dxHq (x, ξ, t) = F1q (t)dxEq (x, ξ, t) = F2q (t)ZZ

dxH̃q (x, ξ, t) = GqA (t)dxẼq (x, ξ, t) = GqP (t),xi and xf are the momentum fractions of the quark before andafter the momentum transfer
2ξ = xf − xi
GPDs can be probed in deeply virtual Compton scattering (DVCS)

Hadron Tomography – p.3/30

Generalized Parton Distributions (GPDs)
formal definition (unpol. quarks):Z −  −dx− ix− p̄+ x ′xxep q̄ −γ+qp
= H(x, ξ, ∆2 )ū(p′ )γ + u(p)2π22iσ +ν ∆ν
u(p)+E(x, ξ, ∆ )ū(p )2M2′in the limit of vanishing t and ξ, the nucleon non-helicity-flip GPDs
must reduce to the ordinary PDFs:
Hq (x, 0, 0) = q(x)H̃q (x, 0, 0) = ∆q(x).

GPDs are form factor for only those quarks in the nucleon carrying
a certain fixed momentum fraction x
֒→ t dependence of GPDs for fixed x, provides information on theposition space distribution of quarks carrying a certain momentum
fraction x
Hadron Tomography – p.4/30

Form Factors vs. GPDs

operator

forwardmatrix elem.

off-forward
matrix elem.

position spaceq̄γ + q

QF (t)

ρ(~r)

q(x)

H(x, ξ, t)?

R dx− eixp+ x− −x− + x−q̄ 2 γ q 24π

Hadron Tomography – p.5/30

Form Factors vs. GPDs

operator

forwardmatrix elem.

off-forward
matrix elem.

position spaceq̄γ + q

QF (t)

ρ(~r)

q(x)

H(x, 0, t)q(x, b⊥ )R dx− eixp+ x− −x− + x−q̄ 2 γ q 24π

q(x, b⊥ ) = impact parameter dependent PDF

Hadron Tomography – p.6/30

Impact parameter dependent PDFs

define state that is localized in ⊥ position:[D.Soper,PRD15, 1141 (1977)]
p+ , R⊥ = 0⊥ , λ ≡ NZd2 p⊥ p+ , p⊥ , λNote: ⊥ boosts in IMF form Galilean subgroup ⇒ this state has
R − 2P1++R⊥ ≡ P + dx d x⊥ x⊥ T (x) = i xi ri,⊥ = 0⊥
(cf.: working in CM frame in nonrel. physics)
define impact parameter dependent PDF
q(x, b⊥ ) ≡Z−+ −dx− +x−+ x
p , R⊥ = 0⊥ q̄(− , b⊥ )γ q( , b⊥ ) p+, R⊥ = 0⊥ eixp x4π
22

Hadron Tomography – p.7/30

GPDs ←→q(x, b⊥ )֒→ relate GPDs to distribution of partons in ⊥ planeR d2 ∆⊥ i∆ ·bq(x, b⊥ ) = (2π)2 e ⊥ ⊥ H(x, 0, −∆2⊥ ),R d2 ∆⊥ i∆ ·b∆q(x, b⊥ ) = (2π)2 e ⊥ ⊥ H̃(x, 0, −∆2⊥ ),no rel. corrections to this result! (Galilean subgroup of ⊥ boosts)

q(x, b⊥ ) has probabilistic interpretation, e.g.

q(x, b⊥ ) ≥ |∆q(x, b⊥ )| ≥ 0 for x > 0q(x, b⊥ ) ≤ |∆q(x, b⊥ )| ≤ 0 for x < 0Note that x already measures longitudinal momentum of quarks
֒→ no simultaneous measurement of long. position of quarks(Heisenberg) −→ no FT w.r.t. ξHadron Tomography – p.8/30GPDs ←→q(x, b⊥ )≡b⊥ distribution measured w.r.t. RCM⊥Pi xi ri,⊥

֒→ width of the b⊥ distribution should go to zero as x → 1, since
the active quark becomes the ⊥ center of momentum in that limit!
֒→ H(x, 0, −∆2⊥ ) must become ∆2⊥ -indep. as x → 1. Confirmedby recent lattice studies (QCDSF, LHPC)
Anticipated shape of q(x, b⊥ ):large x: quarks from localized valence ‘core’,small x: contributions from larger ‘ meson cloud’֒→ expect a gradual increase of the t-dependence (⊥ size) ofH(x, 0, t) as x decreases

Hadron Tomography – p.9/30

q(x, b⊥ ) in a simple model

0.2
0.4
0.60.81864x2010.5

0-0.5-1b?(fm)Hadron Tomography – p.10/30

Transversely Deformed Distributions and E(x, 0, −∆2⊥ )M.B., Int.J.Mod.Phys.A18, 173 (2003)
So far: only unpolarized (or long. pol.) nucleon! In general (ξ = 0):
R dx− ip+ x− xhP+∆,↑ |q̄(0) γ + q(x−)| P,↑i = H(x,0,−∆2⊥ )4π e
R dx− ip+ x− x∆x−i∆y+−2ehP+∆,↑|q̄(0)
γq(x)|P,↓i=−E(x,0,−∆⊥ ).
4π2MConsider nucleon polarized in x direction (in IMF)
|Xi ≡ |p+ , R⊥ = 0⊥ , ↑i + |p+ , R⊥ = 0⊥ , ↓i.

֒→ unpolarized quark distribution for this state:1 ∂q(x,b⊥ ) = H(x,b⊥ ) −2M ∂byZd 2 ∆⊥2−ib⊥ ·∆⊥E(x,0,−∆)e⊥(2π)2Physics: j + = j 0 + j 3 , and left-right asymmetry from j 3 !
[X.Ji, PRL 91, 062001 (2003)]
Hadron Tomography – p.11/30

~qIntuitive connection with L
Electromagnetic interaction couples to vector current. Due tokinematics of the DIS-reaction (and the choice of coordinates —ẑ-axis in direction of the momentum transfer) the virtual photons“see” (in the Bj-limit) only the j + = j 0 + j z component of the quark
current
If up-quarks have positive orbital angular momentum in thex̂-direction, then j z is positive on the +ŷ side, and negative on the−ŷ sideŷp~γ

jz > 0ẑ
jz < 0Hadron Tomography – p.12/30

~qIntuitive connection with L
Electromagnetic interaction couples to vector current. Due tokinematics of the DIS-reaction (and the choice of coordinates —ẑ-axis in direction of the momentum transfer) the virtual photons“see” (in the Bj-limit) only the j + = j 0 + j z component of the quark
current
If up-quarks have positive orbital angular momentum in thex̂-direction, then j z is positive on the +ŷ side, and negative on the−ŷ side
֒→ j + is deformed not because there are more quarks on one sidethan on the other but because the DIS-photons (coupling only toj + ) “see” the quarks on the +ŷ side better than on the −ŷ side.
⊥ deformation described by Eq (x, 0, t)
֒→ not surprising to find that Eq (x, 0, t) enters the Ji relation
Jqi = S iZ

dx [Hq (x, 0, 0) + Eq (x, 0, 0)] x.
Hadron Tomography – p.13/30

Transversely Deformed Distributions and E(x, 0, −∆2⊥ )q(x, b⊥ ) in ⊥ polarized nucleon is deformed compared tolongitudinally polarized nucleons !
mean ⊥ deformation of flavor q (⊥ flavor dipole moment)dqy ≡ZZZpκ
1qdxEq (x, 0, 0) =dx d2 b⊥ qX (x, b⊥ )by =2M2Mu/dwith κpu/d ≡ F2(0) = O(1 − 2)dqy = O(0.2f m)⇒simple model: for simplicity, make ansatz where Eq ∝ Hq
Eu (x, 0, −∆2⊥ ) =Ed (x, 0, −∆2⊥ ) =with κpu = 2κp + κn = 1.673

κpu
Hu (x, 0, −∆2⊥ )2
κpd Hd (x, 0, −∆2⊥ )κpd = 2κn + κp = −2.033.

Model too simple but illustrates that anticipated deformation isvery significant since κu and κd known to be large!
Hadron Tomography – p.14/30

u(x, b⊥)uX (x, b⊥)d(x, b⊥)byby

x = 0.1

x = 0.1

x = 0.3

x = 0.3

x = 0.3

bybybxx = 0.5

bx

bx

bybx

byby

x = 0.3byx = 0.5

x = 0.1

bx

bx

bx

bx

bybybybybxbx

x = 0.1

dX (x, b⊥)bxbx

x = 0.5

x = 0.5
Hadron Tomography – p.15/30

GPD ←→ SSA (Sivers)
Sivers: distribution of unpol. quarks in ⊥ pol. proton⊥qfq/p↑ (x, k⊥ ) = f1q (x, k2⊥ ) − f1T(x, k2⊥ )(P̂ × k⊥ ) · SM⊥q
without FSI, hk⊥ i = 0, i.e. f1T(x, k2⊥ ) = 0

with FSI, hk⊥ i =6 0 (Brodsky, Hwang, Schmidt)
FSI formally included by appropriate choice of Wilson line gauge
links in gauge invariant def. of q(x, k⊥ )֒→ Qiu, Sterman; Collins; Ji; Boer et al.;..

Z ∞hk⊥ i ∼ P, S q̄(0)γ +dη − G+⊥ (η)q(0) P, S0R∞− +⊥dη
G (η) is the ⊥ impulse that the active quark acquires
0as it moves through color field of “spectators”

What should we expect for Sivers effect in QCD ?

Hadron Tomography – p.16/30

GPD ←→ SSA (Sivers)
example: γp → πX (Breit frame)p~γp~Nπ+duu, d distributions in ⊥ polarized proton have left-right asymmetry in⊥ position space (T-even!); sign determined by κu & κdattractive FSI deflects active quark towards the center ofmomentum
֒→ FSI translates position space deformation (before the quark is
knocked out) in +ŷ-direction into momentum asymmetry thatfavors −ŷ direction⊥q֒→ correlation between sign of κq and sign of SSA: f1T
∼ −κq
⊥q
f1T
∼ −κq consistent with HERMES & COMPASS results⊥qneed more results for f1T−→DY@J-PARC

Hadron Tomography – p.17/30

Chirally Odd GPDs
 − ER dx− ixp+ x− D ′  x−  +jε+jαβ ∆α Pβx+jp q̄ − 2 σ γ5 q 2u
p = HT ūσ γ5 u + H̃T ū
2π eM2ε+jαβ ∆α γβε+jαβ Pα γβ+ET ū
u + ẼT ū
u2MMFourier trafo of 2H̃Tq + ETq for ξ = 0 describes distribution oftransversity q i (x, b⊥ ) for unpolarized target in ⊥ plane:(M.Diehl+P.Hägler, hep-ph/0504175)
ij∂εiq (x, b⊥ ) =2M ∂bjZid2 ∆⊥ ib⊥ ·∆⊥ h qq
222H̃(x,0,−∆)+E(x,0,−∆e⊥⊥)T
T(2π)2origin: correlation between quark spin (i.e. transversity) andangular momentum

Hadron Tomography – p.18/30

Transversity Distribution in Unpolarized Target

Hadron Tomography – p.19/30

Transversity Decomposition of Jqx
M.B., hep-ph/0505189: ⊥ deformation of q(x, b⊥ ) in ⊥ polarizedtarget described by Eq (x, 0, t)
Rii
֒→ Ji relation: hJq i = S dx x [Hq (x, 0, 0) + Eq (x, 0, 0)]

Hadron Tomography – p.20/30

Transversity Decomposition of Jqx
similarily ⊥ deformation distribution for quarks with given
transversity described by 2H̃T + ETxx+ Jq,−x̂
֒→ new relation: transversity decomposition of Jqx = Jq,+x̂
xhJq,±x̂i =x ZSdx x [H(x, 0, 0) + E(x, 0, 0)]2Zhi
1±dx x HT (x, 0, 0) + 2H̃T (x, 0, 0) + ET (x, 0, 0) .
4x
is angular momentum carried by quarks with spin
Jq,±x̂

(transversity) in ±x̂-direction; P±x̂ = 21 (1 ± γ x γ5 )
R 3x0z0y
Decomposition is possible since Jq ∼ d r Tq y − Tq z isdiagonal in transversity!
hi
Rx± 14 dx x HT + 2H̃T + ET
unpol. target: Jq,±x̂
hi
Rdx x 2H̃T + ET ) from lattice (hep-ph/0511047), or ...

Hadron Tomography – p.21/30

Boer-Mulders function
Boer-Mulders: distribution of ⊥ pol. quarks in unpol. proton"1 q⊥q
2 (P̂ × k⊥ ) · Sq
2fq↑ /p (x, k⊥ ) =f (x, k⊥ ) − h1 (x, k⊥ )2 1
M#can be probed in Drell-Yan (→ J-PARC)
physical mechanism for BM-function: attractive FSI expected toconvert position space asymmetry for ⊥ polarized quark
distributions into momentum space asymmetry
֒→ e.g. quarks at negative bx with spin in +ŷ get deflected (due toFSI) into +x̂ direction
֒→ (qualitative) connection between Boer-Mulders function h⊥
1 (x, k⊥ )and the chirally odd GPD 2H̃T + ET that is similar to (qualitative)⊥(x, k⊥ ) and the GPD E.
connection between Sivers function f1THadron Tomography – p.22/30

Transversity Distribution in Unpolarized Target
attractive FSI expected to convert position space asymmetry intomomentum space asymmetry
֒→ e.g. quarks at negative bx with spin in +ŷ get deflected (due toFSI) into +x̂ direction
֒→ (qualitative) connection between Boer-Mulders function h⊥1 (x, k⊥ )and the chirally odd GPD 2H̃T + ET that is similar to (qualitative)⊥
connection between Sivers function f1T(x, k⊥ ) and the GPD E.
֒→ qualitative predictions for h⊥
1 (x, k⊥ )sign of h⊥
1 opposite to sign of 2H̃T + ETh⊥1
“ 2H̃ +ETT

⊥
f1T
≈ E ”

use measurement of h⊥
1 to learn about spin-orbit correlation innucleon wave function
use LGT calcs. of 2H̃T + ET to make qualitative prediction for h⊥
1
Hadron Tomography – p.23/30

SummaryGPDs provide decomposition of form factors w.r.t. the momentumof the active quarkZ −  −xdx ixp+ x−x′+ep q̄ −γ qp2π
22−GPDs resemble both PDFs and form factors: defined throughmatrix elements of light-cone correlator, but ∆ ≡ p′ − p 6= 0.
t-dependence of GPDs at ξ = 0 (purely ⊥ momentum transfer) ⇒Fourier transform of impact parameter dependent PDFs q(x, b⊥ )֒→ knowledge of GPDs for ξ = 0 provides novel information about
nonperturbative parton structure of nucleons: distribution ofpartons in ⊥ planeR d2 ∆⊥q(x, b⊥ ) = (2π)2 H(x, 0, −∆2⊥ )eib⊥ ·∆⊥R d2 ∆⊥∆q(x, b⊥ ) = (2π)2 H̃(x, 0, −∆2⊥ )eib⊥ ·∆⊥q(x, b⊥ ) has probabilistic interpretation, e.g. q(x, b⊥ ) > 0 for x > 0
Hadron Tomography – p.24/30

Summary∆⊥
22M E(x, 0, −∆⊥ )describes how the momentum distribution ofunpolarized partons in the ⊥ plane gets transversely deformed
when is nucleon polarized in ⊥ direction.
(attractive) final state interaction in semi-inclusive DIS converts ⊥position space asymmetry into ⊥ momentum space asymmetry
֒→ simple physical explanation for observed Sivers effect inγ ∗ p → πX
2H̃T + ET measures correlation between ⊥ spin and ⊥ angularmomentum (M.B., hep-ph/0505189)
physical explanation for Boer-Mulders effect; relation between h⊥
1and the GPDs 2H̃T + ET
GPDs vs. q(x, b⊥ ): M.B., PRD 62, 71503 (2000), Int. J. Mod.
Phys. A18, 173 (2003); see also D. Soper, PRD 15, 1141 (1977).
Connection to SSA in M.B., PRD 69, 057501 (2004); NPA 735, 185(2004); PRD 66, 114005 (2002).
Hadron Tomography – p.25/30

Q2
Physical Meaning of xBj = 2p·qqpk0k

Go to frame where q⊥ = 0, i.e.
Q2 = −q 2 = −2q + q −Bjorken limit:

q− → ∞,2p · q = 2q − p+ + 2q + p−
q + fixed

֒→q+q+ q−→− +xBj = − − ++−q p +q pp
Hadron Tomography – p.26/30

Q2
Physical Meaning of xBj = 2p·q
q+xBj = − p+LC energy-momentum dispersion relation
m2 + k2⊥
k =2k +−′

֒→ struck quark with k − = k − + q − → ∞ can only be on mass shell if′k+ = k+ + q+ ≈ 0
֒→+k = −q+⇒k+x ≡ + = xBjp

֒→ xBj has physical meaning of light-cone momentum fraction
carried by struck quark before it is hit by photon
back
Hadron Tomography – p.27/30

Impact parameter dependent PDFs
use translational invariance to relate to same matrix element thatappears in def. of GPDs−+ −x−+ xq(x, b⊥ ) ≡ dx p , R⊥ = 0⊥ q̄(− , b⊥ )γ q( , b⊥ ) p+, R⊥ = 0⊥ eixp x22ZZ
Z−+ −x−222 ′− + ′+ x
= |N | d p⊥ d p⊥ dx p , p⊥ q̄(− , b⊥ )γ q( , b⊥ ) p+, p⊥ eixp x22Z−+Hadron Tomography – p.28/30

Impact parameter dependent PDFs
use translational invariance to relate to same matrix element thatappears in def. of GPDs−+ −x−+ xq(x, b⊥ ) ≡ dx p , R⊥ = 0⊥ q̄(− , b⊥ )γ q( , b⊥ ) p+, R⊥ = 0⊥ eixp x
22ZZZ−+ −x−
222 ′+ x− + ′= |N | d p⊥ d p⊥ dx p , p⊥ q̄(− , b⊥ )γ q( , b⊥ ) p+, p⊥ eixp x22ZZ
Z−x−2+ x+ixp+ x−
22 ′− + ′= |N | d p⊥ d p⊥ dx p , p⊥ q̄(− , 0⊥ )γ q( , 0⊥ ) p , p⊥ e
22Z−+ib⊥ ·(p⊥ −p′⊥ )×e

Hadron Tomography – p.29/30

Impact parameter dependent PDFs
use translational invariance to relate to same matrix element thatappears in def. of GPDs−+ −x−+ xq(x, b⊥ ) ≡ dx p , R⊥ = 0⊥ q̄(− , b⊥ )γ q( , b⊥ ) p+, R⊥ = 0⊥ eixp x22ZZ
Z−+ −x−2+ x
22 ′− + ′= |N | d p⊥ d p⊥ dx p , p⊥ q̄(− , b⊥ )γ q( , b⊥ ) p+, p⊥ eixp x22ZZZ−x−2− + ′+ x
22 ′+ixp+ x−= |N | d p⊥ d p⊥ dx p , p⊥ q̄(− , 0⊥ )γ q( , 0⊥ ) p , p⊥ e22Z+−ib⊥ ·(p⊥ −p′⊥ )×e= |N |

֒→2Z2d p⊥

Zd2 p′⊥ Hq(x, b⊥ ) =′2′x, 0, − (p⊥ − p⊥ ) eib⊥ ·(p⊥ −p⊥ )Zd 2 ∆⊥2−ib⊥ ·∆⊥H(x,0,−∆)e⊥(2π)2Hadron Tomography – p.30/30