Quark Orbital Angular Momentum in the Wandzura-Wilczek Approximation

We show that quark orbital angular momentum is directly related to off-forward correlation functions which include intrinsic transverse momentum corresponding to a derivative with respect to the transverse coordinates. Its possible contribution to scattering processes is therefore of higher twist and vanishes in the forward limit. The relation of OAM to other twist 2 and 3 distributions known in the literature is derived and formalized by an unintegrated sum rule.


Introduction
It is well known that orbital angular momentum of quarks and gluons can give an important contribution to the total spin of the proton, according to the sumrule where ∆q, L q , ∆g and L g are the first moments of the corresponding quark spin ∆q(x), quark OAM L q (x), gluon spin ∆g(x) and gluon OAM distribution L g (x).Further on one knows since long how to measure polarized distribution functions and much work was invested into reliable estimates of their leading moments.Still, despite enormous theoretical and experimental efforts only ∆q is known to be rather small, [1,2,3], and the largest value given in the literature is of the order of ∆q(Q 2 ≃ 1GeV 2 ) ≃ 0.30 in the MS scheme, while our knowledge of ∆g is extremely limited [4,5].Finally, for a long time it was not known at all how to access the quark and gluon OAM pieces experimentally.With the advent of studies of off-forward scattering processes like DVCS, L q and L g came for the first time within reach, although not directly, but via the sumrule [6] 1 2 dxx (H q,g (x, ξ) + E q,g (x, ξ)) = J q,g , where H(x, ξ) and E(x, ξ) are off-forward distributions in the limit of vanishing momentum transfer squared ∆ 2 → 0. This sumrule allows to extract e.g. the quark OAM contribution, assuming knowledge of the quark spin ∆q, via While an experimental determination of H q,g (x, ξ, ∆ 2 ) and E q,g (x, ξ, ∆ 2 ) with an accuracy which allows to estimate J q and J g with interesting precision is still a long way to go, eq. ( 2) was used recently to calculate J q on the lattice [7], [8].In ref. [9] a sum rule has been presented relating quark OAM to the second moment of the twist 3 off-forward distribution G 3 (x) (now G 2 (x)).Our notation is that GPDs depending only on x denote the GPDs taken in the forward limit, e.g.G i (x) = G i (x, ξ = 0, ∆ 2 = 0).Including the new G 4 (x) distribution [10], this sumrule reads It is useful to notice that the identification eq.( 4) has been made using the sumrule eq.( 2).On the level of distributions, Hoodbhoy et al. [11] showed that the sumrule eq.( 2) is valid for higher moments as well and therefore in the forward limit where it has been used that H(x) is equal to the unpolarized quark distribution q(x).Furthermore Hoodbhoy et al. defined quark spin and OAM distributions (in terms of inverse Melin-transformed higher moments) which are evidently interrelated by Combining eq.( 5) and eq.( 6), it is possible to determine the quark OAM distribution directly from measurable quantities using the sumrule [11] L q (x) = 1 2 x (q(x) + E(x)) − 1 2 ∆q(x).(7) In this study we show that the parton model definition of the quark OAM distribution can be related to a certain off-forward matrix element as soon as intrinsic transverse momenta are taken into account.Using the equation of motion, we derive an unintegrated sum rule in the Wandzura-Wilczek approximation which relates L q (x) to some of the known twist 2 and 3 off-forward distributions.
2 Quark OAM, intrinsic transverse momentum and off-forward correlators

Rewriting Quark OAM
We begin by recalling the definition of the quark OAM distribution in the light cone gauge according to [12] f Lq (x) = dx − e ix P + x − with ψ + = 1/2γ − γ + ψ and where we have replaced the residual gauge covariant derivative by the partial derivative, D = ∂ − igA → ∂.If the boundary conditions can be fixed so that the A ⊥ -fields vanish at infinity, then the residual gauge field A would be exactly equal to zero.However in the light cone gauge, non-vanishing gluon fields at the boundary can give rise to topological effects, as is discussed in [13].How this could effect our calculation lies beyond the scope of this presentation.First we observe that one easily runs into problems due to the explicit factors and the integration of x ⊥ in case that one takes the definition eq.( 8) literally, see below.Our plan is now to reformulate eq.( 8) in order to circumvent this from the beginning by incorporating an additional transverse vector which gives us some handle on the transverse direction.For this purpose we introduce the function f Lq (x, ∆ ⊥ ) defined as which differs from eq.( 8) mainly in that the matrix element is slightly off-forward in the transverse direction, i.e.P ′ = P + ∆ ⊥ /2, P = P − ∆ ⊥ /2.The function f Lq (x, ∆ ⊥ ) is well defined when acting on a test function T (∆ ⊥ ), and we therefore define the quark OAM as We see that the original definition eq.( 8) can be reproduced by choosing T (∆ ⊥ ) = δ 2 (∆ ⊥ ).Shifting the fields and writing the factor x ⊥ as derivative with respect to ∆ ⊥ , we find after partial integration where the antisymmetric ǫ jk = 1 for j = 1, k = 2 and 0 for j = k.Since we get no contribution in case that the derivative acts on the testfunction (e.g. if T (∆ ⊥ ) is symmetric in ∆ ⊥ ), we end up with The choice T (∆ ⊥ ) = δ 2 (∆ ⊥ ) would result in an integral over the square of the delta function, which is not well defined.This is an indication for the difficulties associated with a naive use of (8).
Another way to circumvent these potential problems is to introduce wave packets |φ , e.g.similar to the discussion in [14].We then would start with the forward distribution f (x) and make the integration over the transverse coordinate x ⊥ as in eq.( 8) more explicit by writing The function g(x, x ⊥ ) can now be treated in exactly the same way as the impact parameter dependent distribution in [14], and by replacing the momentum eigenstates in eq.( 13) by wave packets, we find where N is normalization of the wave packet.This has to be compared with the corresponding result using the testfunction (see the structure of eq.( 11) ), From eqs. ( 14) and ( 15) we find that T (∆ ⊥ ) plays a role similar to ). Orbital angular momentum distribution functions similar to eq. ( 8) have been used in refs.[15] and [16] to calculate their respective evolution equations.A study of the evolution of off-forward correlators and the sumrule eq.( 1) for dressed quark states in light-front pQCD can be found in [17].

Quark OAM in terms of proton wave functions
We now reexamine the above considerations using proton wave functions.Starting from the decomposition of the proton state |P in terms of proton wave functions [18] (for the overlap representation of GPDs see also [19]) where Ψ n (x i , k i⊥ , λ i , λ) is the n-particle fock state wave function of the proton.The normalization is given by We choose but work most of the time in the limit ξ = 0. Inserting eq.( 16) in eq.( 9) we end up with where qa⊥ = q a⊥ + (1 We see immediately that the rhs of eq.( 18) vanishes if we take the naive limit ∆ ⊥ → 0, which would reproduce the definition eq.( 8).Using instead the definition eq.( 10), rewriting x ⊥ as a derivative and performing a partial integration gives where we have dropped all terms ∝ ∆ ⊥ .Since the second term in ( 19) vanishes, we end up with which is the quark OAM in terms of proton wave functions.This can be written as which is just equal to eq.( 12).

Off-forward correlators and intrinsic transverse momenta
First we introduce the slightly more general kinematics (In terms of light cone coordinates one can choose Now let us consider the generic off-forward correlator In order to get a gauge invariant correlator, we included a link operator U which runs in particular along the transverse direction connecting the points (−λ/2 n, −x ⊥ /2) and (λ/2 n, x ⊥ /2).For recent discussions of these transverse gauge links and their implications see e.g.[21,22].Neglecting the intrinsic transverse momentum k ⊥ in a given hard scattering amplitude involving eq.( 22) allows for a direct integration of eq.( 22) over k ⊥ , and in this case we end up with an expression which is local in the transverse direction, Such correlations functions have been parametrized in terms of twist 2 and 3 distributions, see e.g.[6,9].Taking into account terms linear in the intrinsic transverse momentum k ν ⊥ and performing the integration leads to a correlator with one derivative [20] dλ 2π where The partial derivative, when acting on the link operator U in eq.( 22) leads to terms with explicit transverse gluon operators A ⊥ .These additional contributions will be neglected in our approximation.In any case, there are no transverse links left in eqs.(23,24) since the relevant operators are local in transverse direction.For the following it is important to observe that the correlator eq.( 24) must be proportional to the components of the only remaining transverse vector, the momentum transfer ∆ ⊥ .We concentrate now on the part which is proportional to the combination ǫ ν ⊥ σ ⊥ ∆ σ ⊥ .Then it is possible to parametrize eq.( 24) by where we have already multiplied with n µ .The last two terms on the rhs of eq.( 25) indicate the presence of contributions proportional to ∆ ν ⊥ and ξ which vanish after taking the derivative ǫ ν ⊥ σ ⊥ ∂ σ ⊥ ∆ (see below) and the forward limit.Of course one can write down other terms for the parametrization in eq.( 25) which only implicitely give rise to a factor ǫ ν ⊥ σ ⊥ ∆ σ ⊥ .Using (generalized) Gordon identities (see discussion below eq.( 36)) these contributions can be, however, reduced to the term on the rhs of eq.( 25), up to the indicated terms which are irrelevant for our calculation.We now rewrite the lhs and obtain The function L q in eq.( 25) can then be extracted by taking the derivative ǫ ν ⊥ σ ⊥ ∂ σ ⊥ ∆ of both sides.For the rhs of (25) we get in the forward limit 2L q (x) = 2L q (x, ξ = 0, ∆ 2 = 0), and the lhs is given by Comparing eq.( 21) and eq.( 27), we see that in the forward limit we have the identification Thus the off-forward correlator (24) is directly related to the quark OAM distribution of the proton.

Relation to other twist 2 and 3 off-forward distributions
Since the correlator eq.( 24) involves a transverse derivative, it corresponds to (kinematical) twist 3. Similar to the investigations in [20,23] we apply now the identity coming from the equations of motion, where and where [• • • ] stays for the antisymmetric combination of the indices, to the correlator eq.( 25).We contract again with n µ and choose ν = ν ⊥ .This leads to the following relation Using the third line in eq. ( 31) can be rewritten Using partial integration (when possible) and translations, one can show that the three different derivatives occurring in eq. ( 31) and eq.( 33) can be substituted by Following this we get for the rhs of eq. ( 31) The correlators in eq. ( 35) can be completely parametrized in terms of the known twist 2 and 3 off-forward distributions in the WW-approximation, see [10].This gives where all GPDs are functions of x, ξ and ∆, and where we do not show the terms proportional to ∆ ν ⊥ , because they are not directly related to the L q -term in eq.( 25).Using some Gordon-identities [24] we have with our kinematics and in the limit ξ → 0 Taking this together with eq.( 25) we end up with the following sumrule between the distribution L q (x) from and the distribution functions from eq. (36) in the forward limit, where L q (x) is, according to the upper analysis, identified as (forward) quark OAM in the parton model.

Comparing with the integrated G 2 -sumrule
For reasons of comparison let us now recalculate the integrated sumrule.Following ref. [25] (eq.50), we have due to equations of motions According to the parametrization in [10], this is equal to Taking both equations together we get which leads to the sumrule On the other hand, dxxG 4 (x) = 0, giving the known [9,10] result This sumrule is gauge-invariant, and the possible contributions including explicit gluon operators drop out for the second moment over x.Furthermore, integrating (37) over x and using Ji's sumrule (2) [6] we find L q = dxx (q(x) + E(x) + G 2 (x) − 2G 4 (x)) − dx∆q(x) which is perfectly consistent with the sumrule eq.( 40) and confirms our identification eq.( 28) on the level of integrated distributions.

Conclusions
The quark OAM distributions in eq. ( 7) and eq.(37) as well as the definitions in [15,16] all coincide when contributions are neglected which contain explicit transverse gluon operators A ⊥ .This has been discussed in [11], see eqs.( 20)-( 24) and the paragraph below eq.( 28) therein.Dropping the A ⊥ -terms, we can therefore combine eq. ( 7) and eq.(37) to get the interesting and simple relation which is obviously a generalization of the integrated sumrule eq.( 40).In summary we have shown that in the framework of the WW-approximation the quark orbital angular momentum distribution is directly related to the twist-3 GPDs G 2 (x) and G 4 (x), taken in the forward limit, in form of the sumrule eq.(42).Our results represent only a small step towards solving the notorious problem of a direct measurement of the quark OAM contribution to the nucleon spin.At least it leads to a new and nice interpretation of the above mentioned twist-3 GPDs.It has to be seen if the sumrule eq.( 42) or a similar expression holds outside the WW-approximation.The main obstacle in this regard will be the use of an accurately defined gauge invariant OAM distribution.