The light-front gauge-invariant energy-momentum tensor

We provide for the first time a complete parametrization for the matrix elements of the generic asymmetric, non-local and gauge-invariant canonical energy-momentum tensor, generalizing therefore former works on the symmetric, local and gauge-invariant kinetic energy-momentum tensor also known as the Belinfante-Rosenfeld energy-momentum tensor. We discuss in detail the various constraints imposed by non-locality, linear and angular momentum conservation. We also derive the relations with two-parton generalized and transverse-momentum dependent distributions, clarifying what can be learned from the latter. In particular, we show explicitly that two-parton transverse-momentum dependent distributions cannot provide any model-independent information about the parton orbital angular momentum. On the way, we recover the Burkardt sum rule and obtain similar new sum rules for higher-twist distributions.


I. INTRODUCTION
Following the canonical procedure based on Noether's theorem, one ends up with a canonical energy-momentum tensor which is usually neither symmetric nor gauge invariant. Because of these pathologies, one often abandons the canonical energy-momentum tensor in favor of the Belinfante-Rosenfeld improved energy-momentum tensor [1][2][3] which is both symmetric and gauge invariant. The Belinfante-Rosenfeld tensor differs from the canonical tensor by a so-called superpotential term which modifies the definition the momentum density but leaves both the total linear and angular momenta unchanged. It has the peculiar feature that it denies the mere existence of spin density. Indeed, from the conservation of total angular momentum ∂ µ J µνρ = 0 where J µνρ = r ν T µρ − r ρ T µν + S µνρ with T µν the conserved total energy-momentum tensor and S µνρ the spin density tensor, one deduces that the antisymmetric part of the energy-momentum tensor is intimately related to the quark spin density T νρ − T ρν = −∂ µ S µνρ . So, in the Belinfante-Rosenfeld approach, what is usually refered to as "spin" is simply described as a flow of momentum. There is therefore no clear distinction between spin and orbital angular momentum (OAM) in this approach, just like there is no clear distinction between energy flow T i0 and momentum density T 0i .
On the other hand, spin is an intrinsic property of a particle defined as one of the two Casimir invariants of the Poincaré group (the other Casimir invariant being the mass). Contrary to OAM, one cannot change the spin of a particle by changing the Lorentz frame. Spin and OAM are distinguishable, and so dealing with a symmetric energy-momentum tensor is not very natural in Particle Physics. Where does this symmetry requirement come from? It is mainly motivated by General Relativity where gravity couples to a symmetric energy-momentum tensor. It is however important to notice that General Relativity is a classical theory while spin is fundamentally a quantum concept. Moreover, the symmetry of the energy-momentum tensor in General Relativity follows from the postulated absence of space-time torsion. More general theories relax the no-torsion assumption and do not require the energymomentum tensor to be symmetric. The gravitational effects of the antisymmetric part of the energy-momentum tensor are however extremely small and are expected to show up only under extreme conditions, see e.g. [4][5][6] and references therein. Finally, we note that the classical argument in favor of a symmetric energy-momentum tensor based on dimensional analysis and presented e.g. in section 5.7 of [7], is valid only for the orbital form of angular momentum.
The early papers about the proton spin decomposition [8][9][10] start with the Belinfante-Rosenfeld tensor, but then add appropriate superpotential terms to decompose the quark angular momentum into spin and orbital contributions. According to textbooks [11,12], no such decomposition is possible for the gauge field angular momentum. Though, photon spin and OAM are routinely measured in Quantum ElectroDynamics, see e.g. [13] and references therein. In Quantum ChromoDynamics (QCD), a quantity called ∆G which can be interpreted in the light-front gauge A 0 +A 3 = 0 as the gluon spin [8] has been measured in polarized deep inelastic and proton-proton scatterings, see [14] for a recent section III, we provide for the first time the parametrization of the generic non-local light-front gauge-invariant energymomentum tensor and discuss various constraints in section IV. In section V, we derive the relations between the scalar functions appearing in this parametrization and derive the relations with the two-parton generalized and transversemomentum dependent distributions, obtaining on the way new sum rules. Finally, we gather our conclusions in section VI. Some details about the parametrization are given in Appendix A

II. THE GAUGE-INVARIANT LINEAR AND ANGULAR MOMENTUM TENSORS
In order to deal most conveniently with the various gauge-invariant decompositions proposed in the literature, we consider the following five gauge-invariant energy-momentum tensors where ǫ 0123 = +1 and i Similarly, we consider the following seven gauge-invariant generalized angular momentum tensors where x [µ y ν] = x µ y ν − x ν y µ . The standard expressions for the Belinfante-Rosenfeld, Ji, Wakamatsu and Chen et al. decompositions 1 are then obtained by combining these contributions according to Tables II and III, and using the following identities based on the QCD equations of motion where c, c ′ are color indices in the fundamental representation and D µ = ∂ µ − ig[A µ , ] is the adjoint covariant derivative. In particular, because of the first identity in Eq. (3), we can write T µν 4 (r) = − 1 2 T [µν] 1 (r) and therefore discard the tensor T µν 4 (r) in the following discussions. Note that the tensors T (r) and L µνρ 5 (r) + S µνρ 2 (r) have the form of a superpotential ∂ α f [αµ]··· (r) [8]. Assuming as usual that surface terms vanish, this means that we have where n is a timelike or lightlike four-vector and d 3 r = ǫ αβγδ n α dr β ∧ dr γ ∧ dr δ is the volume element. This ensures that the quark and gluon linear and angular momenta are the same in the three kinetic decompositions where J µνρ (r) = S µνρ (r) + L µνρ (r).
The Wakamatsu and Chen et al. decompositions require the introduction of a pure-gauge field where W(r) (called U pure (r) in [19]) is some phase factor transforming as W(r) → U (r)W(r) under gauge transformations. The "physical" gluon field is then defined as In the gauge where W(r) = 1, the Chen et al. decomposition takes the same mathematical form as the Jaffe-Manohar decomposition, and can therefore be considered as a gauge-invariant extension of the latter [16,19,46,47]. The phase factor W(r) is non-locally related to the field strength and is in principle not unique [16,19]. The original Wakamatsu [41] and Chen et al. [15] decompositions correspond to a particular choice of the phase factor which makes the physical field transverse in a given Lorentz frame. Leaving the phase factor unspecified allows us to consider at once two whole classes of decompositions differing simply by the precise form of the non-local phase factor. In order to stress this point, we will follow from now on the terminology of Ref. [16] and refer to the Wakamatsu and Chen et al. decompositions as the gauge-invariant kinetic (gik) and canonical (gic) decompositions, respectively. For a given phase factor, the difference between the gauge-invariant kinetic and canonical decompositions lies in the separation of total linear and orbital angular momentum into quark and gluon contributions. This difference corresponds to which are called potential linear and angular momentum tensors [41,42], respectively.

III. PARAMETRIZATION
In practice, since we want to relate the matrix elements of the gauge-invariant energy-momentum tensor to measurable parton distributions, we choose the non-local phase factor W(r) to be a Wilson line W n (r, r 0 ) connecting a fixed reference point r 0 (usually taken at infinity) to the point of interest r. According to the factorization theorems [24], these Wilson lines run essentially in a straight line along the light-front (LF) direction given by a lightlike four-vector n to the intermediate point r n = r ± ∞n, and then in the transverse direction to r 0 . In some sense, these Wilson lines can be viewed as a background gluon field generated by the hard part of the scattering. The Wilson line associated with the first part of the path is responsible for making the LF gauge n · A = 0 special, since this is the gauge where W n (r, r n ) = 1. The transverse Wilson line W n (r n , r 0 ) is associated with the residual gauge freedom and can be set to 1 using appropriate boundary conditions for the gauge field [20,27]. Our gauge-invariant canonical energy-momentum tensor will then be physically equivalent to the Jaffe-Manohar tensor considered in the LF gauge n · A = 0 with appropriate boundary conditions. We will consider in the following the generic LF gauge-invariant energy-momentum tensor of which the Belinfante-Rosenfeld, Ji, gauge-invariant kinetic and canonical energy-momentum tensors represent particular cases. The matrix elements of the generic LF gauge-invariant energy-momentum tensor depends in principle on n. This dependence was overlooked in [45], leading to absurd conclusions. Since any rescaled lightlike four-vector αn specifies the same LF Wilson line, the matrix elements of the generic LF gauge-invariant energy-momentum tensor actually depends, beside the average target momentum P = (p ′ + p)/2 and the momentum transfer ∆ = p ′ − p, also on the following four-vector with M the target mass, and on the parameter η = ±1 indicating whether the LF Wilson lines are future-pointing (η = +1) or past-pointing (η = −1). Note that, contrary to n, the lightlike four-vector N has the same dimension and transformation properties under space-time symmetries as the momentum variables. Since P · ∆ = 0 and M 2 = P · N = P 2 + ∆ 2 /4, the scalar functions parametrizing the matrix elements of the generic LF gauge-invariant energy-momentum tensor are functions of the two scalar variables ξ = −(∆ · N )/2(P · N ) and t = ∆ 2 . Choosing the standard form for the lightlike four-vector n = (1, 0, 0, −1) leads to the usual expression ξ = −∆ + /2P + with a ± = a 0 ± a 3 . Because these scalar functions also depend on the parameter η, they are complex-valued just like the GTMDs [28,48].
Using the techniques from the Appendix A of Ref. [28], we find that the matrix elements of the generic LF gaugeinvariant energy-momentum tensor for a spin-1/2 target can be parametrized as where S and S ′ are the initial and final target polarization four-vectors satisfying p·S = p ′ ·S ′ = 0 and S 2 = S ′2 = −M 2 , and Γ µν a stands for For convenience, we used the notation iσ µb ≡ iσ µα b α . The factors of i have been chosen such that the real part of the scalar functions is η-even and the imaginary part is η-odd X a j (ξ, t; η) = X e,a j (ξ, t) + iη X o,a j (ξ, t) as a consequence of naive time-reversal symmetry. Hermiticity then implies that the real part of B a j with j ≥ 14 is ξ-odd and the imaginary part is ξ-even. For the other functions, the real part is ξ-even and the imaginary part is ξ-odd.
We have found that the parametrization of the matrix elements of the generic LF gauge-invariant energy-momentum tensor for a spin-1/2 target involves 32 complex-valued scalar functions. This number can be obtained from a simple counting. The generic energy-momentum tensor T µν a has 4 × 4 = 16 components. The target state polarizations ±S and ±S ′ bring another factor of 2 × 2 = 4, but parity symmetry reduces the number of independent polarization configurations by a factor 2, leading to a total of 32 independent complex-valued amplitudes p ′ , S ′ |T µν a (0)|p, S . These 32 independent amplitudes correspond to 32 independent Dirac structures, a particular set being given by Eq. (12). Any other Dirac structure like e.g. γ µ , iσ µP or iǫ µνN ∆ γ 5 , can be expressed onshell as a linear combination of these 32 structures, see Appendix A of this paper.

IV. CONSTRAINTS
The parametrization (12) is very general and does not take into account several constraints like linear and angular momentum conservation. We discuss in this section the various constraints and the relation to former works on the local gauge-invariant energy-momentum tensor.
For latter convenience, we introduce the Sudakov decomposition of a generic four-vector a µ = (a · n)n µ + (a ·n)n µ + a µ T .
together with the transverse Kronecker and Levi-Civita symbols wheren is the lightlike four-vector satisfying n ·n = 1 and such that P µ T = 0.

A. Local operators
The energy-momentum tensors T µν 1 (r) and T µν 2 (r) are local. The corresponding matrix elements cannot therefore depend on N or η. All the scalar functions must then vanish except the five real-valued functions A e,a j (0, t) with a = 1, 2. These are related to the standard (local) energy-momentum form factors (FFs) [9,16,44] as follows The first four form factors parametrize the symmetric part of the local gauge-invariant energy-momentum tensor, whereas the last one parametrizes its antisymmetric part. Since T µν 2 (r) is symmetric, we have A e,2 4 (0, t) = A e,2 5 (0, t).

B. Light-front constraints
From our choice of the phase factor (9) it follows that A phys · N = 0 [18,20], leading to Contracting our generic parametrization (12) with N ν , we find the relations for a = 3, 5 which we refer to as the LF constraints.

C. Four-momentum conservation
The total energy-momentum tensor T µν (r) = T µν 1 (r) + T µν 2 (r) and the superpotential terms T and implies the following constraints which are compatible with Eq. (18).

D. Forward limit and momentum
In the forward limit ∆ → 0, the parametrization of the generic LF gauge-invariant energy-momentum tensor reduces to Since T µν 5 (r) is a total divergence, its matrix elements are proportional to ∆ and therefore vanish in the forward limit, leading to Moreover, since the tensors T µν 1 (r) and T µν 2 (r) are local, the only non-vanishing scalars B a j (0, 0) arise from the potential term T µν 3 (r). This means in particular that naive T-odd effects in the forward limit are necessarily associated with the canonical momentum and disappear when summed over all partons.
Contracting now Eq. (21) with 1 2M 2 N µ gives the average four-momentum in the LF form of dynamics In particular, using Eq. (16) we recover the standard expression for the gauge-invariant kinetic four-momentum in terms of the energy-momentum FFs Interestingly, the last term in Eq. (23) is naive T-odd and can be interpreted as the spin-dependent contribution to the momentum arising from initial and/or final-state interactions. Because of the structure ǫ νS T , this naive T-odd contribution is transverse and requires a transverse target polarization. As we will see in section V B, this is related to the Sivers effect [49]. The combination of scalars A e,a 1 (0, 0) + B e,a 2 (0, 0) contributes only to the energy and is therefore related to the interaction term in the Hamiltonian. In the forward limit, the LF constraints (18) imply that unless B e,3 14 (ξ, t) behaves as 1/ξ near ξ = 0. This suggests that the scalars A e,1 2 (0, 0) and A e,2 2 (0, 0), and hence A q (0) and A G (0), can be interpreted as the interaction-independent contributions of, respectively, quarks and gluons to the four-momentum. This is further supported by the observation that A e,a 2 (0, 0) is the only contribution to the longitudinal momentum p n a , which is purely kinematical in LF quantization. In other words, there is no difference between the longitudinal component of the average kinetic and canonical momenta.
Finally, since the total four-momentum is p ν = P ν , we obtain from Eq. (23) the momentum constraints which are consistent with Eq. (20). In particular, the vanishing of the average total transverse momentum, known as the Burkardt sum rule [50,51], is trivially taken into account in our parametrization because the potential term

E. Angular momentum
Since we have a complete parametrization of the matrix elements of the generic LF gauge-invariant energymomentum tensor, we can easily compute the matrix elements of the corresponding OAM tensor L µνρ a (r) given by Eq. (2). Because of the explicit factors of position r, the matrix elements of the generic LF gauge-invariant OAM tensor need to be handled with care [16,44]. Focusing on the longitudinal component of OAM, we find For a longitudinally polarized target, we have S · N = M 2 and so A e,a 4 (0, 0) can be interpreted as the average fraction of target longitudinal angular momentum carried by the OAM associated with the energy-momentum tensor T µν a (r) in the LF form of dynamics. We confirm in particular that the integrated OAM does not receive any naive T-odd contribution [20,27,47,52]. Using Eq. (16), we also recover the standard expressions for the Belinfante and Ji forms of longitudinal OAM in terms of the energy-momentum FFs [9,10,16,44] Remarkably, thanks to Eq. (4) we can also express the quark and gluon spin contributions in terms of the scalar functions parametrizing the generic LF gauge-invariant energy-momentum tensor. From the integral relations in Eq. (4), we find that the quark and gluon longitudinal spin contributions are given by where we have used Eq. (3) to express L µνρ 4 (r) in terms of T µν 1 (r). The scalars − 1 2 [A e,1 4 (0, 0) − A e,1 5 (0, 0)] = − 1 2 D q (0) and −A e,5 4 (0, 0) can therefore be interpreted as the average fraction of target longitudinal angular momentum carried by the spin of quarks and gluons, respectively. It is easy to check that the following relations are satisfied and that the longitudinal component of the potential OAM is given by Note that the differential relations in Eq. (4), which translate at the level of matrix elements as do not provide additional constraints. Indeed, at O(∆ 0 ), they just reduce to the antisymmetric part of the forward limit (21). At higher orders in ∆, the identification of coefficients between the LHS and the RHS of Eq. (32) are spoiled by the condition P · ∆ = 0 which follows from the onshell relation for the target (P ± ∆ 2 ) 2 = M 2 . Finally, since the total angular momentum is 1/2, we obtain from Eqs. (27) and (29) known as the anomalous gravitomagnetic moment sum rule [53,54].

V. LINK WITH MEASURABLE PARTON DISTRIBUTIONS
Now we are going to see how the scalar functions parametrizing the matrix elements of the generic LF gaugeinvariant energy-momentum tensor are related to GPDs accessed in exclusive scatterings [55] and TMDs accessed in semi-inclusive scatterings [24]. For convenience, we shall focus in the following on the quark sector. The gluon sector proceeds analogously.

A. Generalized Parton Distributions
The quark vector GPD correlator is defined as Remarkably, its second Mellin moment is related to the matrix elements of the quark LF gauge-invariant energymomentum tensor [16,20,55] Note that we do need to specify whether this corresponds to the kinetic or canonical version of the LF gauge-invariant energy-momentum tensor simply because T µN gik,q (r) = T µN gic,q (r) owing to Eq. (17). Up to twist 4, the quark vector GPD correlator (35) is parametrized as [28] From Eq. (36), we then find the relations between the second Mellin moment of vector GPDs and the energymomentum FFs in the quark sector The relations involving twist-3 GPDs are consistent with those found in Ref. [56] where the parametrization is related to the one we used as follows The explicit relations involving twist-4 GPDs are new but somewhat academical as these functions are much harder to access experimentally 2 . Thanks to Eq. (32), the antisymmetric part of the LF gauge-invariant kinetic energy-momentum tensor can be related to the local axial-vector correlator gik,q (0)|p, S .
From the parametrization [55] dx F where G q A (t) = dxH q (x, ξ, t) is the axial-vector FF and G q P (t) = dxẼ q (x, ξ, t) is the induced pseudoscalar FF, it is easy to show using onshell identities that [10,16,44] G q A (t) = −D q (t).
Together with Eq. (38), these relations are consistent with the fact that the GPD and TMD correlators have the same collinear forward limit which implies More interesting are the relations involving the second transverse moment of TMDs. From Eq. (45) and the forward limit ∆ → 0 of the LF constraints (18), we find Anticipating that similar results hold in the gluon sector, we sum over all partons and obtain the following sum rules The first sum rule is known as the Burkardt sum rule [50,51] and simply expresses the fact that the total momentum transverse to the target momentum has to vanish. The other three sum rules are to the best of our knowledge new. They express the fact that the total flow of transverse momentum has also to vanish. They involve higher-twist TMDs and are therefore much harder to test experimentally. Nevertheless, it would be very interesting to test them using phenomenological models, Lattice QCD and perturbative QCD. As a final remark, we would like to stress that the above results show explicitly that TMDs cannot provide any information about the scalar A e,a 4 (0, 0), which means no quantitative model-independent information about the parton OAM, as anticipated e.g. in [65] VI. CONCLUSIONS There has been a prejudice against the canonical form of the quark and gluon energy-momentum tensor, and consequently of the corresponding linear and orbital angular momenta, due to the fact that it cannot be written locally in a gauge-invariant way. A gauge-invariant expression can however be obtained by relaxing the locality requirement in a way that does not harm causality. This indicates that the canonical energy-momentum tensor can be considered as a physical object and measured experimentally. In particular, it can be accessed via particular moments of two-and three-parton correlators which are extracted from numerous physical processes.
In this study, we provided for the first time a complete parametrization for the matrix elements of the generic asymmetric, non-local and gauge-invariant canonical energy-momentum tensor. We found that a generic canonical energy-momentum tensor for a spin-1/2 target consists in 32 independent complex amplitudes. We discussed in detail the various constraints on these amplitudes imposed by non-locality, linear and angular momentum conservation. This generalizes therefore former works on the symmetric, local and gauge-invariant kinetic energy-momentum tensor also known as the Belinfante-Rosenfeld energy-momentum tensor.
We also showed that some of the amplitudes can be expressed in terms of particular moments of two-parton generalized and transverse-momentum dependent distributions, and are therefore clearly measurable. In particular, we proved explicitly that two-parton transverse-momentum dependent distributions cannot provide any quantitative model-independent information about the parton orbital angular momentum. On the way, we recovered the Burkardt sum rule, expressing basically conservation of transverse momentum, and derived three new sum rules invvolving higher-twist distributions. We obtained these results by choosing the non-local phase factors defined by a lightlike four-vector n, in order to make contact with parton physics and factorization theorems.
We believe the present paper will help clarify the differences between canonical and kinetic energy-momentum tensors, and their links with parton distributions. We also expect getting more insights into these matters in a near future coming from explicit results obtained within covariant models, Lattice QCD and perturbative QCD.