aa r X i v : . [ h e p - ph ] N ov Transverse (Spin) Structure of Hadrons
Matthias Burkardt ∗ New Mexico State UniversityE-mail: [email protected]
Parton distributions in impact parameter space, which are obtained by Fourier transforming GPDs,exhibit a significant deviation from axial symmetry when the target and/or quark are transverselypolarized. Connections between this deformation and transverse single-spin asymmetries as wellas with quark-gluon correlations are discussed. The sign of transverse deformation of impact pa-rameter dependent parton distributions in a transversely polarized target can be related to the signof the contribution from that quark flavor to the nucleon anomalous magnetic moment. Therefore,the signs of the Sivers function for u and d quarks, as well as the signs of quark-gluon correla-tions embodied in the polarized structure function g can be understood in terms of the protonand neutron anomalous magnetic moments. Light Cone 2010 - LC2010June 14-18, 2010Valencia, Spain ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ ransverse (Spin) Structure of Hadrons
Matthias Burkardt b x b y b x b y u X ( x, b ⊥ ) d X ( x, b ⊥ ) Figure 1:
Distribution of the j + density for u and d quarks in the ⊥ plane ( x B j = .
3) for a proton polarizedin the x direction in the model from Ref. [2]. For other values of x the distortion looks similar. The signs ofthe distortion are determined by the signs of the contribution from each quark flavor to the proton anomalousmagnetic moment.
1. Impact Parameter Dependent Parton Distributions
Generalized Parton Distributions (GPDs) can be obtained from the same light-cone wave func-tion overlap integrals that yield form factors, except that the momentum fraction x of the activequark is not integrated over, i.e. GPDs can be understood as an x decomposition of form fac-tors. The 2-dimensional Fourier transform of the GPD H q ( x , , t ) yields the distribution q ( x , b ⊥ ) ofunpolarized quarks and target, in impact parameter space [1] q ( x , b ⊥ ) = Z d D ⊥ ( p ) H q ( x , , − D ⊥ ) e − i b ⊥ · D ⊥ , (1.1)with D ⊥ = p ′⊥ − p ⊥ . For a transversely polarized target (e.g. polarized in the + ˆ x -direction) theimpact parameter dependent PDF q + ˆ x ( x , b ⊥ ) is no longer axially symmetric and the transversedeformation is described by the gradient of the Fourier transform of the GPD E q ( x , , t ) [2] q + ˆ x ( x , b ⊥ ) = q ( x , b ⊥ ) − M ¶¶ b y Z d D ⊥ ( p ) E q ( x , , − D ⊥ ) e − i b ⊥ · D ⊥ (1.2) E q ( x , , t ) and hence the details of this deformation are not very well known, but its x -integral, thePauli form factor F , is. Eq. (1.2) allows to relate the average transverse deformation d qy ≡ Z dx Z d b ⊥ q ( x , b ⊥ ) b y = M Z dxE q ( x , , ) = k pq M (1.3)to the contribution from the corresponding quark flavor to the anomalous magnetic moment k pu = k p + k n = ∗ . − . = .
673 and k pd = k n + k p = ∗ ( − . ) + . = − . M ≈ . f m this implies a very significant deformation | d yq | = O ( . f m ) for both u and d quarks and in opposite directions.For example, u quarks in a proton contribute with a positive anomalous magnetic moment and d quarks (after factoring out the negative d quark charge) with a negative value. Eq. (1.2) thusimplies that for a nucleon target polarized in the + ˆ x direction, the leading twist distribution of u quarks is shifted in the + ˆ y direction while that of d quarks is shifted in the − ˆ y direction (Fig. 1).This has important implications for the sign of transverse single-spin asymmetries (SSAs).2 ransverse (Spin) Structure of Hadrons Matthias Burkardt ~p γ du π + Figure 2:
The transverse distortion of the parton cloud for a proton that is polarized into the plane, incombination with attractive FSI, gives rise to a Sivers effect for u ( d ) quarks with a ⊥ momentum that is onthe average up (down).
2. Transverse Single-Spin Asymmetries
In a target that is polarized transversely ( e.g. vertically), the quarks in the target can exhibit a(left/right) asymmetry of the distribution f q / p ↑ ( x B , k T ) in their transverse momentum k T [3, 4] f q / p ↑ ( x B , k T ) = f q ( x B , k T ) − f ⊥ q T ( x B , k T ) ( ˆP × k T ) · S M , (2.1)where S is the spin of the target nucleon and ˆP is a unit vector opposite to the direction of the virtualphoton momentum. The fact that such a term may be present in (2.1) is known as the Sivers effectand the function f ⊥ q T ( x B , k T ) is known as the Sivers function. The latter vanishes in a naive partonpicture since ( ˆP × k T ) · S is odd under naive time reversal (a property known as naive-T-odd), whereone merely reverses the direction of all momenta and spins without interchanging the initial andfinal states. The significant distortion of parton distributions in impact parameter space (Fig. 1)provides a natural mechanism for a Sivers effect. In semi-inclusive DIS, when the virtual photonstrikes a u quark in a ⊥ polarized proton, the u quark distribution is enhanced on the left side ofthe target (for a proton with spin pointing up when viewed from the virtual photon perspective).Although in general the final state interaction (FSI) is very complicated, we expect it to be onaverage attractive thus translating a position space distortion to the left into a momentum spaceasymmetry to the right and vice versa (Fig. 2) [5]. Since this picture is very intuitive, a few wordsof caution are in order. First of all, such a reasoning is strictly valid only in mean field models forthe FSI as well as in simple spectator models [6]. Furthermore, even in such mean field or spectatormodels there is in general no one-to-one correspondence between quark distributions in impactparameter space and unintegrated parton densities (e.g. Sivers function) (for a recent overview, seeRef. [7]). While both are connected by an overarching Wigner distribution [8], they are not Fouriertransforms of each other. Nevertheless, since the primordial momentum distribution of the quarks(without FSI) must be symmetric, we find a qualitative connection between the primordial positionspace asymmetry and the momentum space asymmetry due to the FSI. Another issue concernsthe x -dependence of the Sivers function. The x -dependence of the position space asymmetry isdescribed by the GPD E ( x , , − D ⊥ ) . Therefore, within the above mechanism, the x dependence ofthe Sivers function should be related to that of E ( x , , − D ⊥ ) . However, the x dependence of E isnot known yet and we only know the Pauli form factor F = R d xE . Nevertheless, if one makes theadditional assumption that E does not fluctuate as a function of x then the contribution from eachquark flavor q to the anomalous magnetic moment k determines the sign of E q ( x , , ) and hence ofthe Sivers function. With these assumptions, as well as the very plausible assumption that the FSI3 ransverse (Spin) Structure of Hadrons Matthias Burkardt is on average attractive, one finds that f ⊥ u T <
0, while f ⊥ d T >
0. Both signs have been confirmed bya flavor analysis based on pions produced in a SIDIS experiment by the H
ERMES collaboration [9]and are consistent with a vanishing isoscalar Sivers function observed by C
OMPASS [10].
3. Transverse Force on Quarks in DIS
The chirally-even spin-dependent twist-3 parton distribution g ( x ) = g T ( x ) − g ( x ) is definedas Z d l p e i l x h PS | ¯ y ( ) g m g y ( l n ) | Q | PS i = (cid:2) g ( x , Q ) p m ( S · n ) + g T ( x , Q ) S m ⊥ + M g ( x , Q ) n m ( S · n ) (cid:3) . Neglecting m q , one finds g ( x ) = g WW ( x ) + ¯ g ( x ) , with g WW ( x ) = − g ( x ) + R x dyy g ( y ) [11], where¯ g ( x ) involves quark-gluon correlations, e.g. [12, 13] Z dxx ¯ g ( x ) = d MP + P + S x d = g (cid:10) P , S (cid:12)(cid:12) ¯ q ( ) G + y ( ) g + q ( ) (cid:12)(cid:12) P , S (cid:11) . (3.2)At low Q , g has the physical interpretation of a spin polarizability, which is why the matrixelements (note that √ G + y = B x − E y ) c E M ~ S = h P , S | q † ~ a × g ~ Eq | P , S i c B M ~ S = h P , S | q † g ~ Bq | P , S i (3.3)are sometimes called spin polarizabilities or color electric and magnetic polarizabilities [14]. In thefollowing we will discuss that at high Q a better interpretation for these matrix elements is that ofan average ‘color Lorentz force’ [15].To see this we express the ˆ y -component of the Lorentz force acting on a particle with charge g that is moving with (nearly) the speed of light ~ v = ( , , − ) along the − ˆ z direction in terms oflight-cone variables, yielding g h ~ E + ~ v × ~ B i y = g ( E y + B x ) = g √ G y + , (3.4)which coincides with the component that appears in the twist-3 correlator above (3.2). Thus Eq.(3.2) represents the (twist 2) quark density correlated with the transverse color-Lorentz force thata quark would experience at that position if it moves with the velocity of light in the − ˆ z direction— which is exactly what the struck quark does after it has absorbed the virtual photon in a DISexperiment in the Bjorken limit. Therefore the correct semi-classical interpretation of Eq. (3.2) isthat of an average transverse force F y ( ) ≡ − √ P + h P , S | ¯ q ( ) G + y ( ) g + q ( ) | P , S i (3.5) = − √ MP + S x d = − M d The average is meant as an ensemble average since the forward matrix element in plane wave states automaticallyprovides an average over the nucleon volume. ransverse (Spin) Structure of Hadrons Matthias Burkardt acting on the active quark in the instant right after it has been struck by the virtual photon.Although the identification of h p | ¯ q g + G + y q | p i as an average color Lorentz force due to the finalstate interactions (3.5) may be intuitively evident from the above discussion, it is also instructiveto provide a more formal justification. For this purpose, we consider the time dependence of thetransverse momentum of the ‘good’ component of the quark fields (the component relevant for DISin the Bjorken limit) q + ≡ g − g + q p + ddt h p y i ≡ ddt h PS | ¯ q g + ( p y − gA y ) q | PS i (3.6) = √ ddt h PS | q † + ( p y − gA y ) q + | PS i = p + h PS | (cid:2) ˙¯ q g + ( p y − gA y ) q + ¯ q g + ( p y − gA y ) ˙ q (cid:3) | PS i− h PS | ¯ q g + g ˙ A y q | PS i . Using the QCD equations of motion ˙ q = (cid:16) igA + g ~ g · ~ D (cid:17) q , (3.7)where − iD m = p m − gA m , yields2 p + ddt h p y i = h PS | ¯ q g + g (cid:0) G y + G yz (cid:1) q | PS i + ‘ h PS | ¯ q g + g − g i D i D j q | PS i ′ = √ h PS | ¯ q g + gG y + q | PS i + ‘ h PS | ¯ q g + g − g i D i D j q | PS i ′ , (3.8)where ‘ h PS | ¯ q g + g − g i D i D j q | PS i ′ stands symbolically for all terms that involve a product of g + g − as well as a g ⊥ and only ⊥ derivatives D i .Now it is important to keep in mind that we are not interested in the average force on the‘original’ quark fields (before the quark is struck by the virtual photon), but after absorbing thevirtual photon and moving with (nearly) the speed of light in the − ˆ z direction. In this limit, thefirst term on the r.h.s. of (3.8) dominates, as it contains the largest number of ‘ + ’ Lorentz indices.Dropping the other terms yields (3.5).The identification of 2 M d with the average transverse force acting on the active quark in aSIDIS experiment is also consistent with the Qiu Sterman result [16] for the average transversemomentum of the ejected quark (also averaged over the momentum fraction x carried by the activequark) h k y ⊥ i = − P + (cid:28) P , S (cid:12)(cid:12)(cid:12)(cid:12) ¯ q ( ) Z ¥ dx − G + y ( x + = , x − ) g + q ( ) (cid:12)(cid:12)(cid:12)(cid:12) P , S (cid:29) (3.9)The average transverse momentum is obtained by integrating the transverse component of the colorLorentz force along the trajectory of the active quark — which is an almost light-like trajectoryalong the − ˆ z direction, with z = − t . The local twist-3 matrix element describing the force attime=0 is the first integration point in the Qiu-Sterman integral (3.9).Lattice calculations of the twist-3 matrix element yield [17] d ( u ) = . ± . d ( d ) = − . ± . ‘Right after’, since the quark-gluon correlator in (3.5) is local! ransverse (Spin) Structure of Hadrons Matthias Burkardt renormalized at a scale of Q = for the smallest lattice spacing in Ref. [17]. These numbersare also consistent with experimental studies [18]. Using (3.5) these (ancient) lattice results thusimply F ( u ) ≈ − / fm F ( d ) ≈ / fm . (3.11)In the chromodynamic lensing picture, one would have expected that F ( u ) and F ( d ) are of about thesame magnitude and with opposite sign. The same holds in the large N C limit. A vanishing Siverseffect for an isoscalar target would be more consistent with equal and opposite average forces.However, since the error bars for d include only statistical errors, the lattice result may not beinconsistent with d ( d ) ∼ − d ( u ) .The average transverse momentum from the Sivers effect is obtained by integrating the trans-verse force to infinity (along a light-like trajectory) h k y i = R ¥ dtF y ( t ) . This motivates us to definean ‘effective range’ R e f f ≡ h k y i F y ( ) . (3.12)Note that R e f f depends on how rapidly the correlations fall off along a light-like direction and itmay thus be larger than the (spacelike) radius of a hadron. Of course, unless the functional form ofthe integrand is known, R e f f cannot really tell us about the range of the FSI, but if the integrand in(3.5) does not oscillate, (3.12) provides a reasonable estimate for the range over which the integrandin (3.5) is significantly nonzero.Fits of the Sivers function to SIDIS data yield about |h k y i| ∼
100 MeV [19]. Together with the(average) value for | d | from the lattice this translates into an effective range R e f f of about 1 fm. Itwould be interesting to compare R e f f for different quark flavors and as a function of Q , but thisrequires more precise values for d as well as the Sivers function.A relation similar to (3.5) can be derived for the x moment of the twist-3 scalar PDF e ( x ) . Forits interaction dependent twist-3 part ¯ e ( x ) one finds for an unpolarized target [20]4 MP + P + e = g h p | ¯ q s + i G + i q | P i , (3.13)where e ≡ R dxx ¯ e ( x ) . The matrix element on the r.h.s. of Eq. (3.13) can be related to the averagetransverse force acting on a transversely polarized quark in an unpolarized target right after beingstruck by the virtual photon. Indeed, for the average transverse momentum in the + ˆ y direction, fora quark polarized in the + ˆ x direction, one finds h k y i = P + Z ¥ dx − g h p | ¯ q ( ) s + y G + y ( x − ) q ( ) | p i . (3.14)A comparison with Eq. (3.13) shows that the average transverse force at t = + ˆ x direction reads F y ( ) = √ p + g h p | ¯ q s + y G + y q | p i = √ MP + S x e = M e . (3.15)The impact parameter distribution for quarks polarized in the + ˆ x direction [21] is shifted inthe + ˆ y direction [22, 23]. Applying the chromodynamic lensing mechanism implies a force in6 ransverse (Spin) Structure of Hadrons Matthias Burkardt the negative ˆ y direction for these quarks and one thus expects e < u and d quarks.Furthermore, since k ⊥ > k , one would expect that in a SIDIS experiment the ⊥ force on a ⊥ polarized quark in an unpolarized target on average to be larger than that on unpolarized quarks ina ⊥ polarized target, and thus | e | > | d | .
4. Summary
The GPD E q ( x , , − D ⊥ ) , which arises in the ‘ x -decomposition’ of the contribution from quarkflavor q to the Pauli form factor F q describes the transverse deformation of the unpolarized quarkdistribution in impact parameter space. That deformation provides a very intuitive mechanism fortransverse SSAs in SIDIS. As a result, the signs of SSAs can be related to the contribution fromquark flavor q to the nucleon anomalous magnetic moment. Quark-gluon correlations appearing inthe x -moment of the twist-3 part of the polarized parton distribution g q ( x ) have a semi-classicalinterpretation as the average (enemble average) transverse force acting on the struck quark in DISfrom a transversely polarized target in the moment after it has absorbed the virtual photon. Sincethe direction of that force can be related to the transverse deformation of PDFs, one can thus alsorelate the sign of these quark-gluon correlations to the contribution from quark flavor q to thenucleon anomalous magnetic moment.Such a correlation between observables that at first appear to have little in common also occursin the chirally odd sector: the impact parameter space distribution of quarks with a given transver-sity in an unpolarized target can be related to the Boer-Mulders function describing the left-rightasymmetry of quarks with a given transversity in SIDIS from an unpolarized target. Furthermore,semi-classically, the quark-gluon correlations appearing in the x -moment of the twist-3 part of thescalar PDF e ( x ) describes the average transverse force acting on a quark with given transversityimmediately after it has absorbed the virtual photon. Acknowledgements:
I would like to thank A.Bacchetta, D. Boer, J.P. Chen, Y.Koike, andZ.-E. Mezziani for useful discussions. This work was supported by the DOE under grant numberDE-FG03-95ER40965.
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