Hadron Tomography. Matthias Burkardt. New Mexico State University Las Cruces, NM, 88003, U.S.A. Hadron Tomography p.

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1 Hadron Tomography Matthias Burkardt New Mexico State University Las Cruces, NM, 88003, U.S.A. Hadron Tomography p.1/27

2 Outline GPDs: probabilistic interpretation as Fourier transforms of impact parameter dependent PDFs H(x, 0, 2 ) q(x,b ) H(x, 0, 2 ) q(x,b ) E(x, 0, 2 ) deformation of unpol. PDFs in pol. target physics: orbital motion of the quarks intuitive explanation for SSAs exclusive SSAs Sivers effect 2 H T + E T deformation of pol. PDFs in unpol. target correlation between quark angular momentum and quark transversity Boer-Mulders function h 1 (x,k ) Summary Hadron Tomography p.2/27

3 Generalized Parton Distributions (GPDs) GPDs: decomposition of form factors at a given value of t, w.r.t. the average momentum fraction x = 1 2 (x i + x f ) of the active quark dxh q (x, ξ, t) = F q 1 (t) dxe q (x, ξ, t) = F q 2 (t) dx H q (x, ξ, t) = G q A (t) dxẽ q (x, ξ, t) = G q P (t), x i and x f are the momentum fractions of the quark before and after the momentum transfer 2ξ = x f x i GPDs can be probed in deeply virtual Compton scattering (DVCS) Hadron Tomography p.3/27

4 Generalized Parton Distributions (GPDs) dx formal definition (unpol. quarks): 2π eix p + x ( p q x 2 ) ( ) x γ + q p 2 = H(x, ξ, 2 )ū(p )γ + u(p) +E(x, ξ, 2 )ū(p ) iσ+ν ν 2M u(p) in the limit of vanishing t and ξ, the nucleon non-helicity-flip GPDs must reduce to the ordinary PDFs: H q (x, 0, 0) = q(x) Hq (x, 0, 0) = q(x). Hadron Tomography p.4/27

5 Form Factors vs. GPDs operator forward matrix elem. off-forward matrix elem. position space qγ + q Q F(t) ρ( r) dx e ixp+ x 4π ( q x 2 ) ( γ + x q 2 ) q(x) H(x, ξ, t)? Hadron Tomography p.5/27

6 Form Factors vs. GPDs operator forward matrix elem. off-forward matrix elem. position space qγ + q Q F(t) ρ( r) dx e ixp+ x 4π ( q x 2 ) ( γ + x q 2 ) q(x) H(x, 0, t) q(x,b ) q(x,b ) = impact parameter dependent PDF Hadron Tomography p.6/27

7 Impact parameter dependent PDFs define localized state [D.Soper,PRD15, 1141 (1977)] p +,R = 0, λ N d 2 p p +,p, λ Note: boosts in IMF form Galilean subgroup this state has R 1 P dx d 2 x + x T ++ (x) = i x ir i, = 0 (cf.: working in CM frame in nonrel. physics) define impact parameter dependent PDF dx q(x,b ) p +,R = 0 q( x 4π 2,b )γ + q( x 2,b ) p +,R = 0 e ixp + x q(x,b ) q(x,b ) = d 2 (2π) e i b H(x, 0, 2 2 ), = d 2 (2π) e i b H(x, 0, 2 2 ), Hadron Tomography p.7/27

8 u(x,b ) u X (x,b ) d(x,b ) d X (x,b ) x = 0.1 x = 0.1 x = 0.1 x = 0.1 x = 0.3 x = 0.3 x = 0.3 x = 0.3 x = 0.5 x = 0.5 x = 0.5 x = 0.5 Hadron Tomography p.8/27

9 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): dx dx 4π eip+ x x P+, q(0) γ + q(x ) P, = H(x,0, 2 ) 4π eip+ x x P+, q(0) γ + q(x ) P, Consider nucleon polarized in x direction (in IMF) X p +,R = 0, + p +,R = 0,. unpolarized quark distribution for this state: = x i y 2M E(x,0, 2 ). q(x,b ) = H(x,b ) 1 2M d 2 (2π) 2 E(x, 0, 2 )e ib Physics: j + = j 0 + j 3, and left-right asymmetry from j 3! [X.Ji, PRL 91, (2003)] Hadron Tomography p.9/27

10 physical origin for distortion a) b) Consider nucleon moving in ẑ-direction. quarks orbiting around the axis of motion (long. pol. nucleon), the orbital motion does not affect the longitudinal momentum distribution. quarks orbiting around direction ( pol.nucleon) orbital motion adds/subtracts to long. momentum for y > 0 and y < 0 respectively PDFs rapidly fall with x boost/de-boost on ±ŷ side results in enhancement/suppression of q(x,b ). details described by E(x, 0, 2 ). c) Hadron Tomography p.10/27

11 Transversely Deformed Distributions and E(x, 0, 2 ) q(x,b ) in polarized nucleon is deformed compared to longitudinally polarized nucleons! mean deformation of flavor q ( flavor dipole moment) d q y dx d 2 b q X (x,b ) = 1 2M with κ p u/d F u/d 2 (0) = O(1 2) d q y = O(0.2fm) dxe q (x, 0, 0) = κp q 2M simple model: for simplicity, make ansatz where E q H q with κ p u = 2κ p + κ n = E u (x, 0, 2 ) = κp u 2 H u(x, 0, 2 ) E d (x, 0, 2 ) = κ p d H d(x, 0, 2 ) κ p d = 2κ n + κ p = Model too simple but illustrates that anticipated deformation is very significant since κ u and κ d known to be large! Hadron Tomography p.11/27

12 u(x,b ) u X (x,b ) d(x,b ) d X (x,b ) x = 0.1 x = 0.1 x = 0.1 x = 0.1 x = 0.3 x = 0.3 x = 0.3 x = 0.3 x = 0.5 x = 0.5 x = 0.5 x = 0.5 Hadron Tomography p.12/27

13 Exclusive SSAs (A.Belitsky & D.Müller; see also S.J.Brodsky & A.Mukherjee) incoming γ outgoing γ,π,ρ,.. For simplicity, only momentum transfer A d 2 b q(b, x)t q e i b where T q is the parton-level Compton -amplitude left-right asymmetry of quark distribution translates directly into left-right asymmetry of the scattering amplitude exclusive SSA probe GPDs which describe this deformation ( Huygen s principle! ) Hadron Tomography p.13/27

14 SSAs in SIDIS (γ + p π + + X) e q e π + q(x,k ) D π+ q (z,p ) use factorization (high energies) to express momentum distribution of outgoing π + as convolution of momentum distribution of quarks in nucleon unintegrated parton density f q/p (x,k ) momentum distribution of π + in jet created by leading quark q fragmentation function D π+ q (z,p ) average momentum of pions obtained as sum of average k of quarks in nucleon (Sivers effect) average p of pions in quark-jet (Collins effect) Hadron Tomography p.14/27

15 GPD SSA (Sivers) Sivers: distribution of unpol. quarks in pol. proton f q/p (x,k ) = f q 1 (x,k2 ) f q 1T (x,k2 ) (ˆP k ) S M without FSI, k = 0, i.e. f q 1T (x,k2 ) = 0 with FSI, k 0 (Brodsky, Hwang, Schmidt) FSI formally included by appropriate choice of Wilson line gauge links in gauge invariant def. of f q/p (x,k ) What should we expect for Sivers effect in QCD? Hadron Tomography p.15/27

16 GPD SSA (Sivers) example: γp πx (Breit frame) p γ p N d π + u, d distributions in polarized proton have left-right asymmetry in position space (T-even!); sign determined by κ u & κ d attractive FSI deflects active quark towards the center of momentum FSI translates position space distortion (before the quark is knocked out) in +ŷ-direction into momentum asymmetry that favors ŷ direction correlation between sign of κ p q and sign of SSA: f q 1T κp q f q 1T κp q confirmed by HERMES results (also consistent with COMPASS f1t u + f q 1T 0) u Hadron Tomography p.16/27

17 Chirally Odd GPDs ( dx x 2π eixp+ p q x 2 ) σ +j γ 5 q ( x 2 ) p = H T ūσ +j γ 5 u + H T ū ε+jαβ α P β M u 2 +E T ū ε+jαβ α γ β 2M u + Ẽ T ū ε+jαβ P α γ β M u See also M.Diehl+P.Hägler, hep-ph/ Fourier trafo of Ēq T 2 H q T + Eq T for ξ = 0 describes distribution of transversity for unpolarized target in plane q i (x,b ) = εij 2M b j d 2 (2π) 2 eib Ē q T (x, 0, 2 ) origin: correlation between quark spin (i.e. transversity) and angular momentum Hadron Tomography p.17/27

18 Transversity Distribution in Unpolarized Target Hadron Tomography p.18/27

19 Chirally Odd GPDs J i = 1 2 εijk d 3 x [ T 0j x k T 0k x j] J x q diagonal in transversity, projected with 1 2 (1 ± γx γ 5 ), i.e. one can decompose J x q = J x q,+ˆx + J x q, ˆx where J x q,±ˆx is the contribution (to Jx q ) from quarks with positive (negative) transversity derive relation quantifying the correlation between quark spin and angular momentum [M.B., PRD72, (2006); PLB639, 462 (2006)] J y q,+ŷ = 1 4 dx [ H q T (x, 0, 0) + Ēq T (x, 0, 0)] x (note: this relation is not a decomposition of J q into transversity and orbital) Hadron Tomography p.19/27

20 Boer-Mulders Function SIDIS: attractive FSI expected to convert position space asymmetry into momentum space asymmetry e.g. quarks at negative with spin in +ŷ get deflected (due to FSI) into +ˆx direction (qualitative) connection between Boer-Mulders function h 1 (x,k ) and the chirally odd GPD Ē T that is similar to (qualitative) connection between Sivers function f 1T (x,k ) and the GPD E. Boer-Mulders: distribution of pol. quarks in unpol. proton f q /p(x,k ) = 1 2 [ f q 1 (x,k2 ) h q 1 (x,k2 ) (ˆP k ) S q M h q 1 (x,k2 ) can be probed in DY (RHIC, J-PARC, GSI) and tagged SIDIS (JLab, erhic), using Collins-fragmentation ] Hadron Tomography p.20/27

21 Boer-Mulders Function Model calculations (bag model, const. quark model, NJL-model) indicate: Ē T > 0 for u and d quarks in nucleon and pion, indicating a universal spin-orbit correlation for valence quarks Ē T > E u, i.e. stronger correlation between L q and quark spin than between L q and the nucleon spin confirmed by lattice calculations (P.Hägler et al.) several interesting predictions: h 1 (x,k ) same sign (negative) as f 1T (x,k ) universal sign for valence h 1 > f 1T h 1 let s measure h 1 to learn more about spin-orbit correlations for quarks! Hadron Tomography p.21/27

22 Summary GPDs FT PDFs in impact parameter space E(x, 0, 2 ) deformation of PDFs for polarized target origin for deformation: orbital motion of the quarks simple mechanism (attractive FSI) to predict sign of f q 1T distribution of polarized quarks in unpol. target described by chirally odd GPD Ēq T = 2 H q T + Ẽq T origin: correlation between orbital motion and spin of the quarks attractive FSI measurement of h 1 (DY,SIDIS) provides information on Ēq T and hence on spin-orbit correlations Hadron Tomography p.22/27

23 Intuitive connection with L q Electromagnetic interaction couples to vector current. Due to kinematics 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 the ˆx-direction, then j z is positive on the +ŷ side, and negative on the ŷ side p γ ẑ ŷ j z > 0 j z < 0 Hadron Tomography p.23/27

24 Intuitive connection with L q Electromagnetic interaction couples to vector current. Due to kinematics 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 the ˆx-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 side than on the other but because the DIS-photons (coupling only to j + ) see the quarks on the +ŷ side better than on the ŷ side. deformation described by E q (x, 0, 2 ) not surprising to find that E q (x, 0, 2 ) enters the Ji relation J i q = S i dx[h q (x, 0, 0) + E q (x, 0, 0)] x. Hadron Tomography p.24/27

25 Single Spin Asymmetry (Sivers) Naive definition of unintegrated parton density f(x,k ) dξ d 2 ξ (2π) 3 e ip ξ P, S q(0)γ + q(ξ) P, S ξ+ =0. Time-reversal invariance f(x,k ) = f(x, k ) Asymmetry d 2 k f(x,k )k = 0 Same conclusion for gauge invariant definition with straight ( Wilson line U [0,ξ] = P exp ig ) 1 0 dsξ µa µ (sξ) Hadron Tomography p.25/27

26 Single Spin Asymmetry (Sivers) Naively (time-reversal invariance) f(x,k ) = f(x, k ) However, including the final state interaction (FSI) results in nonzero asymmetry of the ejected quark! (Brodsky, Hwang, Schmidt) Gauge invariant definition requires quark to be connected by gauge link. Choice of path not arbitrary but must be chosen along path of outgoing quark to incorporate FSI f(x,k ) dξ d 2 ξ (2π) 3 e ip ξ P, S q(0)u [0, ] γ + U [,ξ] q(ξ) P, S ξ+ =0 with U [0, ] = P exp ( ig 0 dη A + (η) ) Hadron Tomography p.26/27

27 Sivers Mechanism in A + = 0 gauge Gauge link along light-cone trivial in light-cone gauge U [0, ] = P exp ( ig 0 ) dη A + (η) = 1 Puzzle: Sivers asymmetry seems to vanish in LC gauge (time-reversal invariance)! X.Ji: fully gauge invariant definition for P(x,k ) requires additional gauge link at x = dy d 2 y f(x,k ) = 16π 3 e ixp+ y +ik y p, s q(y)γ + U [y,y ;,y ]U [,y,,0 ]U [,0 ;0,0 ]q(0) p, s back Hadron Tomography p.27/27