Notes on the orbital angular momentum of quarks in the nucleon

We discuss the orbital angular momentum of partons inside a longitudinally polarized proton in the recently proposed framework of spin decomposition. The quark orbital angular momentum defined by Ji can be decomposed into the `canonical' and the `potential' angular momentum parts, both of which are represented as the matrix element of a manifestly gauge invariant operator.


Introduction
Recent polarized beam experiments and global QCD analyses suggest that the contribution of the gluon helicity ∆G to the spin of the proton is rather small [1]. This observation, together with the inexorable fact that the quark helicity contribution ∆Σ is also small (less than 30%), lead one to suspect that the key to understand the proton spin puzzle is the orbital angular momentum (OAM) of quarks and gluons. However, progress in this direction has been hindered by a number of difficulties in measuring, and even defining the OAM. So far, the only well-recognized, gauge invariant definition of the quark OAM is the one by Ji [2] which can be measured, indirectly, as the difference between a certain moment of the generalized parton distribution and ∆Σ. Although generally accepted, this approach may be criticized on the basis that the corresponding operator is not the 'canonical' one that satisfies the fundamental commutation relation of the angular momentum operator in quantum mechanics. Efforts to improve upon this point have led Chen et al. to propose a completely new decomposition scheme of the QCD angular momentum tensor [3,4] which has triggered a flurry of activity lately [5,6,7,8,9,10,11,12,13,14]. However, the issue still remains very controversial, and the overarching impact of this new formalism as well as its practical usefulness in phenomenology are yet to be clarified.
In this work, we investigate the quark OAM along the line of our previous work [10] which we view as the proper rendition of the formalism [3,4] in the context of high energy QCD. We shall show that one can represent the canonical OAM as the matrix element of a manifestly gauge invariant operator which turns out to be equivalent to that obtained in the Wigner distribution approach [16]. This paves the way to measure the canonical OAM experimentally or numerically on a lattice, and thus helps to mitigate the criticism that the matrix elements defined in [3] do not have known physical measurements [17]. Actually, in the gluon helicity sector Ref. [10] has already shown how one can reconcile the gluon helicity defined in [3] with ∆G which is measurable. We now extend this finding to the OAM sector.

Decomposition of the QCD angular momentum operator
The main idea of [3,4] is that one can achieve a complete, gauge invariant decomposition of the QCD angular momentum operator by identifying the 'physical' and 'pure gauge' components of the gauge field which transform differently under gauge transformations The QCD angular momentum tensor M µνλ can then be written as a sum of the helicity and the orbital angular momentum of quarks and gluons. In the 'covariant' form [5,6] useful for high energy experiments, the original proposal by Chen et al. [3,4] reads where D ν pure ≡ ∂ ν + igA ν pure , and a, b = 1, 2, · · · , 8 are the color indices. 1 Using the transformation rule (3), it is easy to check that each of the above components is gauge invariant. Such a 1 Our convention is ǫ 0123 = +1 , γ 5 = −iγ 0 γ 1 γ 2 γ 3 . We shall use the light-cone coordinates x ± = 1 √ 2 (x 0 ± x 3 ) and denote the transverse coordinates with latin indices x T = {x i }, (i, j, · · · = 1, 2). The two-dimensional antisymmetric tensor is defined as ǫ ij = −ǫ +−ij , ǫ 12 = ǫ 12 = −ǫ 21 = 1.
complete decomposition goes beyond Ji's framework in which the gluonic part cannot be separated into the helicity and orbital parts. The price to pay, however, is that the decomposition is not local in the sense that A µ phys is in general nonlocally related to the total A µ . Moreover, it is not entirely covariant, either, because A µ phys actually depends on the frame as we shall soon see. An alternative decomposition of the orbital part was suggested by Wakamatsu [5] M ′ µνλ The second term of (9) is gauge invariant on its own. Using the equation of motion D α F αµ a = gψγ µ t a ψ, one sees that it accounts for the difference between D ν pure in (5) and D ν in (8). There is no consensus as to which definition, (5) or (8), is more appropriate for the quark orbital angular momentum. To some extent, it is a matter of choice. It is (5), but not (8), that is compatible with the (equal-time) canonical commutation relation of the angular momentum operator L × L = i L, The pure gauge condition (2) is crucial for this. (5) may thus be called the canonical angular momentum. 2 On the other hand, (8) is the same as Ji's definition [2] and is accessible from the analysis of the generalized parton distribution (GPD), whereas it has not been known how to measure (5). We will derive an explicit expression of the canonical angular momentum (5) in terms of a manifestly gauge invariant operator whose matrix element is, in principle, related to experimental processes or observables in lattice QCD simulations. For this purpose, one must specify what A µ phys is. There are several proposals for A µ phys in the literature [4,7,10,13]. These definitions are not equivalent as suggested by the work of Ref. [8] which showed that they give different values of the gluon helicity (the proton matrix element of (6)). Here we employ the one proposed in [10] A µ where we use the notation in the fundamental representation. The superscript '−' denotes that the path ordering is in the where It represents the residual gauge symmetry of the light-cone gauge A + = 0, and is fixed by specifying the boundary condition of the gauge field at x − → ±∞ mentioned above. It has been shown in [10] that (11) and (13) are a viable decomposition of the total gauge field A µ . We wish to stress that this particular choice is singled out among others by the criterion of measurability: The corresponding gluon helicity coincides with the usual gluon helicity ∆G that has been measured in experiments.
Note that the definition (11) already selects a particular frame-the infinite momentum frame where the partonic interpretation of hadrons is clearest. As emphasized in [15], the decomposition of spin into the helicity and the orbital parts cannot be made entirely covariant, but depends on the frame of reference.

Potential angular momentum
We now focus on the orbital angular momentum of quarks inside a longitudinally polarized proton. It is given by the forward matrix element of the µνλ = +ij component of (5) or (8) where P 2 = −S 2 = M 2 (the proton mass squared) and (2π) 3 δ 3 (0) = dx − d 2 x T is the momentum space delta function. The longitudinal polarization means S µ = (S + , S − , S T ) ≈ (S + , 0, 0 T ). We first observe that, since A µ pure = 0 in the light-cone gauge A + = 0 [10], L Chen is actually identical to the Jaffe-Manohar (JM) definition [18] of the quark orbital angular momentum The subscript LC means that the matrix element is evaluated in the light-cone gauge. In other words, L Chen is the generalization of L JM to arbitrary gauges (see, also, [5]). We thus unify the notations L Chen = L JM ≡ L can by introducing the canonical orbital angular momentum L can , and write where the so-called potential angular momentum [19,5] is, with our choice of A µ phys , Now consider the non-forward matrix element whereP µ = (P µ + P ′µ )/2 and the momentum transfer will be denoted as ∆ µ ≡ P ′µ − P µ . The explicit factor x i can be traded for the derivative with respect to ∆ i such that Parity and time-reversal (P T ) symmetry tells that the following parametrization of the matrix element is possible whereS = (S + S ′ )/2 and ξ ≡ −∆ + /2P + is the skewness parameter. [The dependence of h(ξ) on the renormalization scale is implicit. We also suppress the dependence on ∆ 2 ≈ −∆ 2 T since it is of higher order. Similar comments apply to other distributions defined below.] This leads to (20) thus defines the potential angular momentum as the matrix element of a manifestly gauge invariant operator.
The quark-gluon mixed operator that appears in the matrix element (20) is familiar in the context of the twist-three mechanism of the single spin asymmetry (SSA). Let us pursue this analogy and consider the following non-forward matrix element By symmetry considerations, it follows that Ψ(x 1 , x 2 , ξ) = Ψ(x 2 , x 1 , −ξ) and Φ(x 1 , x 2 , ξ) = −Φ(x 2 , x 1 , −ξ). In the forward limit, and in the transversely polarized case S µ = δ µ i S i , only the Ψ-term survives. The function Ψ(x 1 , x 2 , 0) plays the cental role in the so-called soft gluonic pole mechanism of the SSA [20,21]. In the longitudinally polarized caseS µ ≈ δ µ +S + , the Ψterm vanishes for the relevant component µν = +j. By performing Fourier transformations, we find where we switched to the notation X = x 1 +x 2 2 , x = x 1 − x 2 . The kernel is in the case K(y − ) = 1 2 ǫ(y − ) and in the cases K(y − ) = ±θ(±y − ). Comparing with (20), we obtain an alternative expression for the potential angular momentum Note that, since Φ(X, 0, 0) = 0, different choices for K lead to the same result, as they should.

Canonical angular momentum
Next we exploit the relation between the twist-three approach to the SSA and the approach based on the transverse momentum dependent distribution (TMD). 4 In the longitudinally polarized and non-forward case, we define where W T is the Wilson line in the transverse direction at x − = ±∞. In the forward case ∆ = 0, the matrix element (27) reduces to the usual TMD. As is well-known in that context, there is freedom in choosing the path connecting the points (−z − /2, −z T /2) → (z − /2, z T /2).
Using the Wilson line that goes to future infinity and then comes back −z − /2 → +∞ → z − /2, one takes care of the final state interaction. The TMD in this case is relevant to the semi-inclusive DIS (SIDIS). The other case −z − /2 → −∞ → z − /2 includes the initial state interaction relevant to the Drell-Yan process. For the present purpose, one may as well take the average of the two cases. The relation between (27) and (20) is revealed by taking the second moment of f in q T [23] 5 where the kernel K is in one-to-one correspondence with the choice of the Wilson line path in (27). In the forward case, (28) vanishes for the longitudinal polarization by rotational symmetry in the transverse plane. In the non-forward case, however, the following structure 4 The following discussion is similar to the works of Ref. [22,16]. We improve upon these works by fully taking into account the gauge field and the issue of gauge invariance. 5 Cf. Eq. (39) of [23]. Via partial integration, q i T is replaced by the spatial derivative ∂ i T . When acting on the Wilson line, it brings down the factor ∂ i T A + = D + A i + F i+ which reduces to the two terms in (28).
is allowed, so that (28) is not necessarily zero. Note that, because of an extra minus sign from ∆ j under the P T transformation, the functionf does not change signs when changing the directions of the Wilson line, 6 in contrast to the known sign flip of the spin-dependent TMDs in the SIDIS and Drell-Yan reactions [24]. We then take the first moment in x The matrix element (20) indeed appears in the second term of the first equality, but somewhat remarkably, it is absorbed by the covariant derivative in the first term. The final expression features precisely the 'pure gauge' part of the covariant derivative D pure . Differentiating with respect to ∆, we arrive at This is a formula relating the canonical OAM of quarks to the matrix element of a well-defined, manifestly gauge invariant operator. 7 The final expression in (31) agrees with the OAM constructed by Lorcé and Pasquini [16] from the Wigner distribution (neglecting gauge invariance).
We have thus established a gauge invariant link between the Wigner distribution approach and the spin decomposition framework of Chen et al.
Similarly, for the gluon orbital angular momentum one can define where A phys is as in (11), and now the Wilson lines are in the adjoint representation. Under the P T transformation, one gets the same operator back (up to the direction of the Wilson line) provided the skewness parameter ξ vanishes, which we assume here. In deeply virtual Compton scattering (DVCS), ξ = 0 corresponds to the elastic scattering of the photon. Proceeding as before, one finds the double moment in x and q T The matrix element of (7) thus becomes where we parameterized, as ∆ → 0,

Conclusions
We have shown that the physical part of the gauge field A phys proposed in [10] leads to welldefined expressions for the canonical and potential angular momenta in terms of the matrix element of certain gauge invariant operators. If one defines ∆G as the gluon helicity, consistency requires that (31) is the corresponding canonical angular momentum. Now that we can, at least in principle, measure L Ji , L can and the difference L pot = L Ji − L can separately, it seems more legitimate to call L can , or equivalently the OAM from the Wigner distribution [16,25], the quarks' genuine orbital angular momentum since it satisfies the fundamental commutation relation.
Regarding measurability, it should be possible to compute (31) in lattice QCD simulations as in the case of the ordinary (forward) TMD (see, e.g., [26]). Since there is no sign flip in (27) when changing the directions of the Wilson lines, the matrix element may not be very sensitive to the choice of the path. If so, and if one is not interested in the x-dependence, one may first integrate over x in (27) and connect the quark operators (at the same value of z − = 0) by a purely spatial Wilson line in the transverse plane. This avoids the introduction of lightlike Wilson lines on a Euclidean lattice which appears to be a vague issue. For the gluon orbital angular momentum (34) with g defined in (32), one has to deal with lightlike Wilson lines even after the x-integration.
Finally, from the experimental point of view, the matrix elements such as (22) are related to the twist-three GPDs [27,28] which are hard to extract. It would be interesting to see if there are processes in which these functions contribute to the cross section at leading order as in the single spin asymmetry.