1 Introduction

One of the main challenges of hadronic physics is to fully understand the structure and characteristics of nucleons through the measurement of their structure functions. Of particular interest is the origin of the nucleon spin, a key question at the core of the experimental program of the future Electron-Ion Collider (EIC) at the Brookhaven National Laboratory [1, 2]. This requires a proper decomposition of the nucleon total angular momentum (AM) into orbital motion and intrinsic spin of its constituents. The most common decompositions of AM are the Jaffe-Manohar (JM) [3] and Ji [4] decompositions, which are based on two different notions of orbital angular momentum (OAM). Variations and extensions of these have been proposed and discussed at length in the literature. For further details, we refer the interested reader to the reviews by Leader and Lorcé [5], and Wakamatsu [6].

For the purpose of the present work, it suffices to recall that the Ji decomposition is based on the kinetic momentum defined by the covariant derivative \(D_\mu =\partial _\mu -ig A_\mu \), whereas the JM decomposition is based on the canonical momentum \(\partial _\mu \) defined in the light-front (LF) gauge \(A^+=(A^0+A^3)/\sqrt{2}=0\). More specifically, we are interested in the difference between JM and Ji quark OAM, known as potential AM [7]

$$\begin{aligned} L^z_\text {pot}\equiv L^{q,z}_\text {JM}-L^{q,z}_\text {Ji}. \end{aligned}$$
(1)

From a physical point of view, this potential AM has been interpreted as the accumulated change in OAM experienced by the struck quark due to the color Lorentz forces as it leaves the target in high-energy scattering processes [8]. Similarly, the potential transverseFootnote 1 momentum (TM) is defined as the difference between JM and Ji notions of the quark transverse momentum inside the nucleon [7]

$$\begin{aligned} \varvec{k}_{\perp ,\text {pot}}\equiv \varvec{k}^q_{\perp ,\text {JM}}-\varvec{k}^q_{\perp ,\text {Ji}}. \end{aligned}$$
(2)

This potential TM can be related to the Sivers shift (as justified with more detail in Sect. 3), which is the non-vanishing average parton transverse momentum in a transversely polarized target resulting from the Sivers mechanism [9, 10].

The JM and Ji OAM have both been computed in QED at one-loop order and found to be identical [11]. However, there is no reason for the potential AM to vanish in general. Indeed, some recent lattice QCD calculations [12, 13] found a sizeable value for the potential AM. Moreover, a numerical study of the renormalization scale dependence of \(L^z_\text {pot}\) [14] and a detailed discussion of the Landau problem [15] also hint towards a non-vanishing potential AM.

Both the potential TM and potential AM can in principle be experimentally observed. A comparison of their magnitudes is motivated by Burkardt’s proposal of a lensing mechanism due to soft gluon rescattering in deep-inelastic and other high-energy scattering [16, 17]. Originally, such mechanism describes the asymmetry in transverse momentum space (i.e. Sivers effect) as originating from an asymmetry of the parton distribution in the impact-parameter space (due to the orbital motion of partons) convoluted with a lensing function that accounts for the effects of attractive initial/final-state interactions (ISI/FSI). Even if the notion of a lensing function is intuitive and is supported by some model calculations [17,18,19,20], its factorization depends on strong assumptions that cannot be strictly satisfied in QCD [21]. Nevertheless, a non-zero Sivers function necessarily requires a non-vanishing orbital motion of the partons. The existence of a non-vanishing potential TM hints towards a non-vanishing potential AM, since both quantities find their common origin in the ISI/FSI. It is therefore interesting to compare them quantitatively within the same approach.

In the present paper, we compute both the potential TM and the potential AM in the framework of a simple scalar diquark model (SDM) of the nucleon at order \({\mathcal {O}}(\lambda ^2 e_q e_s)\) in perturbation theory. The model is based on the assumption that the nucleon splits into a quark and a scalar diquark structure, which are regarded as elementary fields of the theory [22]. More details on the model are presented in Appendix A.

Despite being a simple model, the SDM provides analytic results that have been broadly explored in the literature. Additionally, it has the feature of maintaining explicit Lorentz covariance. For these reasons, the SDM provides an interesting framework for studying the relative magnitudes between the potential AM and the potential TM. A more realistic description of the nucleon would naturally require to add both a vector diquark and an internal structure to the diquarks. These contributions go beyond the purpose of the present work and are left for future investigations.

The manuscript is organized as follows. We start with a reminder of the gauge-invariant definition of canonical momentum. In Sect. 3 we define and compute the potential TM of an unpolarized quark in a transversely polarized nucleon. Thereafter we report the potential AM in Sect. 4 for a quark in a longitudinally polarized nucleon. Finally, we summarize our results in Sect. 5. Some details about the two-loop calculations are collected in the Appendices.

2 Gauge-invariant definition of canonical momentum

The canonical momentum refers to the partial derivative \(\partial _\mu \). Since this is not a gauge-covariant operator, Jaffe and Manohar considered it only in a particular gauge, namely the Light-Front (LF) gauge \(A^+=0\) well-suited for the description of high-energy scatterings. Since the JM decomposition is defined in a fixed gauge, doubts were expressed concerning the measurability of its individual terms, in particular, the JM OAM of quarks and gluons.

A decade ago, a paper by Chen et al. [23] shook the nucleon spin community by showing that the canonical decomposition of angular momentum could be written in a gauge-invariant way. To achieve this goal, one has to separate the gauge potential into the so-called “pure-gauge” and “physical” (or better dynamical) parts [7, 23, 24]

$$\begin{aligned} A_\mu = A^\text {pure}_\mu +A^\text {phys}_\mu . \end{aligned}$$
(3)

By definition, \(A^\text {pure}_\mu \) has a vanishing field strength

$$\begin{aligned} F^\text {pure}_{\mu \nu }\equiv \partial _\mu A^\text {pure}_{\nu }-\partial _\nu A^\text {pure}_{\mu }-ig\,[A^\text {pure}_{\mu },A^\text {pure}_{\nu }]= 0, \end{aligned}$$
(4)

and behaves as a connection

$$\begin{aligned} A^\text {pure}_\mu (x)\mapsto {{\widetilde{A}}}^\text {pure}_\mu (x)=U(x)\left[ A^\text {pure}_\mu (x)+\frac{i}{g}\,\partial _\mu \right] U^{-1}(x)\nonumber \\ \end{aligned}$$
(5)

under gauge transformations. It can then be used to define a new covariant derivative

$$\begin{aligned} D^\text {pure}_\mu =\partial _\mu -ig A^\text {pure}_\mu . \end{aligned}$$
(6)

It is nothing but the gauge-invariant version of the canonical momentum operator, since it satisfies the standard canonical commutation relations

$$\begin{aligned}{}[D^\text {pure}_\mu ,D^\text {pure}_\nu ]=-igF^\text {pure}_{\mu \nu }=0, \end{aligned}$$
(7)

and it generates the gauge-covariant translations of the fields (in the fundamental representation). The dynamical degrees of freedom of the gauge potential are encoded in \(A^\text {phys}_\mu \) which behaves covariantly

$$\begin{aligned} A^\text {phys}_\mu (x)\mapsto {{\widetilde{A}}}^\text {phys}_\mu (x)=U(x)A^\text {phys}_\mu (x)U^{-1}(x) \end{aligned}$$
(8)

under gauge transformations. These are responsible for the non-vanishing of the field strength

$$\begin{aligned} F_{\mu \nu }\equiv {\mathcal {D}}^\text {pure}_\mu A^\text {phys}_{\nu }-{\mathcal {D}}^\text {pure}_\nu A^\text {phys}_{\mu }-ig\,[A^\text {phys}_{\mu },A^\text {phys}_{\nu }], \end{aligned}$$
(9)

where \({\mathcal {D}}^\text {pure}_\mu A^\text {phys}_\nu =\partial _\mu A^\text {phys}_\nu -ig\,[A^\text {pure}_\mu ,A^\text {phys}_\nu ]\) is the pure-gauge covariant derivative in the adjoint representation. Note that since \(A^\text {pure}_\mu \) is a pure-gauge field, there exists a gauge where \(A^\text {pure}_\mu =0\) and hence \(A_\mu =A^\text {phys}_\mu \). We will refer to this gauge as the natural gauge.

The separation of the gauge potential (3) is however not unique [25,26,27] and requires an additional condition to define unambiguously \(A^\text {phys}_\mu \). Such condition is akin to a standard gauge condition, except that it is imposed only on \(A^\text {phys}_\mu \) and not on the full \(A_\mu \). This amounts to defining \(A^\text {phys}_\mu \) as a certain non-local functional of \(A_\mu \). Owing to the gauge principle, no gauge is more fundamental than the others. Accordingly, no separation of the gauge potential (3) is more fundamental than the others.Footnote 2 However, when dealing with a particular problem some separation into pure-gauge and physical parts may appear more convenient to use. This is the case of high-energy scatterings where a particular spatial direction (identified with the z-direction) is determined by the kinematics. In this context, the physical part is defined by the condition \(A^+_\text {phys}=0\) and assumes the following non-local formFootnote 3 [28,29,30,31]

$$\begin{aligned} A^\mu _{\text {phys},\pm }(x)= & {} \int ^{x^-}_{\pm \infty ^-}\mathrm {d}y^-\,{\mathcal {W}}_\text {LF}(x^-,y^-;\tilde{\varvec{x}})\nonumber \\&\times F^{+\mu }(y^-,\tilde{\varvec{x}})\,{\mathcal {W}}_\text {LF}(y^-,x^-;\tilde{\varvec{x}}), \end{aligned}$$
(10)

where the LF Wilson line is defined by the following path-ordered exponential

$$\begin{aligned} {\mathcal {W}}_\text {LF}(x^-,y^-;\tilde{\varvec{x}})\equiv {\mathcal {P}}\left[ \exp \left( ig\int _{y^-}^{x^-}\mathrm {d}z^- A^+(z^-,\tilde{\varvec{x}})\right) \right] .\nonumber \\ \end{aligned}$$
(11)

For convenience, we introduced the notation \(\tilde{\varvec{x}}=(x^+,\varvec{x}_\perp )\) and the LF components \(x^\pm =(x^0\pm x^3)/\sqrt{2}\). The condition \(A^+_\text {phys}=0\) is not sufficient to define unambiguously the physical part, since it does not fix \(x^-\)-independent contributions. This residual freedom is fixed by adding a boundary condition. The advanced (retarded) boundary condition corresponds to setting \(+\infty ^-\) (\(-\infty ^-\)) in Eq. (10). More generally, we can consider any intermediate boundary condition

$$\begin{aligned} A^\mu _{\text {phys},\eta }(x)=\tfrac{1+\eta }{2}\, A^\mu _{\text {phys},+}(x)+\tfrac{1-\eta }{2}\,A^\mu _{\text {phys},-}(x) \end{aligned}$$
(12)

withFootnote 4\(\eta \in [-1,1]\). The particular value \(\eta =0\) is quite popular and corresponds to the antisymmetric boundary condition.

Interestingly, one can see the expression in Eq. (10) as an infinitesimally thin and (semi-)infinitely long Wilson loop. Indeed, the parallel transport from a point x to an infinitesimally close point \(x+\mathrm {d}x\) is defined by the Wilson line

$$\begin{aligned} {\mathcal {W}}_{\mathcal {C}}(x+\mathrm {d}x,x)=\mathbb {1}+igA^{\mathcal {C}}_\mu (x)\mathrm {d}x^\mu , \end{aligned}$$
(13)

where \({\mathcal {C}}\) is some path connecting the two points. As long as the path remains local, i.e. in the infinitesimal neighborhood of x, the connection does not depend on \({\mathcal {C}}\) and we simply write \(A_\mu (x)\). Typically one associates \(A_\mu (x)\) with a straight line between x and \(x+\mathrm {d}x\). In this geometric picture, the field strength describes a closed infinitesimal square \({\mathcal {W}}_\Box (x)=\mathbb {1}+i\tfrac{g}{2}F_{\mu \nu }(x)\mathrm {d}x^\mu \wedge \,\mathrm {d}x^\nu \). When the path between x and \(x+\mathrm {d}x\) is non-local, the connection retains its path dependence. We can see \(A^{\mathcal {C}}_\mu \) as an explicit realization of \(A^\text {pure}_\mu \). By definition \(A^\text {phys}_\mu =A_\mu -A^\text {pure}_\mu \) corresponds therefore to a closed non-local path. In the context of high-energy scatterings, the natural paths to consider are non-local in the LF direction \(x^-\). We can then understand Eq. (10) as the breaking up of an infinitesimally thin Wilson loop extending from \(x^-\) to \(\pm \infty ^-\) into an infinite sum of parallel-transported infinitesimal squares. Changing boundary conditions corresponds to adding a gauge-invariant contribution proportional to

$$\begin{aligned}&A^\mu _{\text {phys},+}(x)-A^\mu _{\text {phys},-}(x)\nonumber \\&\quad =-\int ^{+\infty ^-}_{-\infty ^-}\mathrm {d}y^-\,{\mathcal {W}}_\text {LF}(x^-,y^-;\tilde{\varvec{x}})\,F^{+\mu }(y^-,\tilde{\varvec{x}})\,{\mathcal {W}}_\text {LF}(y^-,x^-;\tilde{\varvec{x}}), \end{aligned}$$
(14)

i.e. an infinitesimally thin Wilson loop extending from \(-\infty ^-\) to \(+\infty ^-\). In general \(A^\mu _{\text {pure},\eta }(x)\) is associated with an s-shaped Wilson line (see Fig. 1) which reduces to the standard staple when \(\eta =\pm 1\).

Fig. 1
figure 1

Path \({\mathcal {C}}\) defining \(A^\mu _{\text {pure},\eta }(0)\equiv A^\mu _{\mathcal {C}}(0)\) via Eq. (13) for an arbitrary value of the parameter \(\eta \in [-1,1]\). An alternative (but equivalent) path would first go to \(-\infty ^-\) and then to \(+\infty ^-\)

Full size image

3 Average transverse momentum of unpolarized quarks inside a transversely polarized nucleon

We are interested in the average TM of quarks inside a nucleon. For the kinetic momentum, it is given by the following expectation valueFootnote 5 at equal LF time \(r^+=0\)

$$\begin{aligned} \langle k^i_\perp \rangle _\text {Ji}&=\frac{\langle P,S|\int \mathrm {d}^4 r\,\delta (r^+)\,{\overline{\psi }}(r)\gamma ^+\tfrac{i}{2}\overset{\leftrightarrow }{D}^i_\perp \psi (r)|P,S\rangle }{\langle P,S|P,S\rangle }\nonumber \\&=\frac{i}{4P^+}\,\langle P,S|{\overline{\psi }}(0)\gamma ^+\overset{\leftrightarrow }{D}^i_\perp \psi (0)|P,S\rangle , \end{aligned}$$
(15)

where \(i=1,2\) is a transverse index, \(\overset{\leftrightarrow }{D}_\mu =\overset{\rightarrow }{D}_\mu -\overset{\leftarrow }{D}_\mu \) is the symmetric covariant derivative, and a translation of the fields to the origin has been used in the second line. The nucleon with mass M has four-momentum \(P^\mu \) and covariant LF polarization \(S^\mu \) satisfying \(P\cdot S=0\) and \(S^2=-1\). Its explicit form is given by

$$\begin{aligned} S^\mu =\left[ s_z \frac{P^+}{M},-s_z \frac{P^-}{M}+\frac{\varvec{P}_\perp \cdot \varvec{S}_\perp }{P^+},\varvec{S}_\perp \right] \end{aligned}$$
(16)

with \(\varvec{S}_\perp =\mathbf {s}_\perp +\frac{\varvec{P}_\perp }{M}\,s_z\) and \(\varvec{s}\) the usual rest-frame spin vector. For convenience, we work in the standard symmetric LF frame defined by \(\varvec{P}_\perp =\varvec{0}_\perp \). For the canonical momentum, we have in a similar way

$$\begin{aligned} \langle k^i_\perp \rangle _{\text {JM},\eta }=\frac{i}{4P^+}\,\langle P,S|{\overline{\psi }}(0)\gamma ^+\overset{\leftrightarrow }{D}^i_{\text {pure},\eta ,\perp }\psi (0)|P,S\rangle . \end{aligned}$$
(17)

We left here the explicit dependence on the choice of the boundary condition. The i-th component of the potential TM is then given by

$$\begin{aligned} \langle k^i_\perp \rangle _{\text {pot},\eta }=\frac{-g}{2P^+}\,\langle P,S|{\overline{\psi }}(0)\gamma ^+A^i_{\text {phys},\eta ,\perp }(0)\psi (0)|P,S\rangle .\nonumber \\ \end{aligned}$$
(18)

The theory being invariant under discrete space-time symmetries, the combined parity and time-reversal transformation (PT) implies thatFootnote 6

$$\begin{aligned} \langle k^i_\perp \rangle _\text {Ji}&=-\langle k^i_\perp \rangle _\text {Ji}, \end{aligned}$$
(19)
$$\begin{aligned} \langle k^i_\perp \rangle _{\text {JM},\eta }&=-\langle k^i_\perp \rangle _{\text {JM},-\eta }. \end{aligned}$$
(20)

It follows that

$$\begin{aligned} \langle k^i_\perp \rangle _\text {Ji}= & {} 0 \Rightarrow \qquad \nonumber \\ \langle k^i_\perp \rangle _{\text {pot},\eta }= & {} \langle k^i_\perp \rangle _{\text {JM},\eta }=-\langle k^i_\perp \rangle _{\text {pot},-\eta }. \end{aligned}$$
(21)

In high-energy scatterings, the LF Wilson lines encode (leading-twist) ISI and FSI [33]. The choice of a particular boundary condition determines therefore the inclusion of these interactions in the definition of the canonical momentum. Except for \(\eta =0\), ISI and FSI are included in an asymmetric way and allow for a nonvanishing average quark TM. Since time-reversal symmetry plays an important role, it is useful to split \(A^\mu _{\text {phys},\eta }\) into PT-even and oddFootnote 7 contributions [29]

$$\begin{aligned} A^\mu _{\text {phys},\eta }(x)=A^\mu _{\text {phys},e}(x)+\eta A^\mu _{\text {phys},o}(x) \end{aligned}$$
(22)

with

$$\begin{aligned} A^\mu _{\text {phys},e}(x)&=\frac{1}{2}\left[ A^\mu _{\text {phys},+}(x)+A^\mu _{\text {phys},-}(x)\right] \nonumber \\&=-\frac{1}{2}\int _{-\infty ^-}^{+\infty ^-}\mathrm {d}y^-\,\varepsilon (y^-)\,{\mathcal {W}}_\text {LF}(x^-,y^-;\tilde{\varvec{x}})\,\nonumber \\&\quad \times F^{+\mu }(y^-,\tilde{\varvec{x}})\,{\mathcal {W}}_\text {LF}(y^-,x^-;\tilde{\varvec{x}}), \end{aligned}$$
(23)
$$\begin{aligned} A^\mu _{\text {phys},o}(x)&=\frac{1}{2}\left[ A^\mu _{\text {phys},+}(x)-A^\mu _{\text {phys},-}(x)\right] \nonumber \\&=-\frac{1}{2}\int _{-\infty ^-}^{+\infty ^-}\mathrm {d}y^-\,{\mathcal {W}}_\text {LF}(x^-,y^-;\tilde{\varvec{x}})\,\nonumber \\&\quad \times F^{+\mu }(y^-,\tilde{\varvec{x}})\,{\mathcal {W}}_\text {LF}(y^-,x^-;\tilde{\varvec{x}}), \end{aligned}$$
(24)

where \(\varepsilon (y^-)\) is the sign function.

In the following, we will compute directly the potential TM of quarks in the perturbative scalar diquark model (SDM) at order \({\mathcal {O}}(\lambda ^2e_qe_s)\) and then compare it with the result one obtains for the JM quark TM. In this model, only abelian gauge interactions are considered, as any non-abelian effects would appear at higher order in perturbation theory. Expressions for the TM are therefore the same upon the substitution \(g\mapsto -e_q\).

3.1 Potential transverse momentum of quarks

In the SDM at order \({\mathcal {O}}(\lambda ^2e_qe_s)\), the three diagrams shown in Fig. 2 contribute a priori to the quark potential TM.

Fig. 2
figure 2

Diagrams contributing to the quark potential TM and AM at \({\mathcal {O}}(\lambda ^2 e_q e_s)\) in perturbation theory. The dotted line represents the scalar diquark, the solid thin (thick) line represents the quark (nucleon), and the wavy line represents the photon. The crossed blob represents the insertion of an external operator

Full size image

The first two diagrams represent the self-force since the Lorentz force felt by the quark finds its source in the quark itself. The last diagram represents the force exerted by the spectator diquark system. An explicit calculation outlined in Appendix B.1 shows that

$$\begin{aligned} {\mathcal {M}}^i_{A+B}=0, \end{aligned}$$
(25)

which can be understood by the fact that this sum is even under PT, whereas the potential TM is PT-odd. Also we can intuitively expect that there are no Münchhausen quarks, i.e. quarks that can self-accelerate.

All the quark potential TM comes therefore from the third diagram where the photon is attached to the spectator system. It contains both PT-even and odd contributions. A direct calculation confirms that the PT-even contribution vanishes. Using dimensional regularization with \(D=4-2\epsilon \), we find for the PT-odd contribution

$$\begin{aligned} \langle k^i_\perp \rangle _{\text {pot},\eta }=-\eta \,\epsilon ^{ij}_\perp s^j_\perp \,\frac{\pi }{6}\,(3m_q+M)\, \frac{{\mathcal {N}}}{(4\pi \epsilon )^2}+{\mathcal {O}}(1/\epsilon )\nonumber \\ \end{aligned}$$
(26)

with \({\mathcal {N}}=\lambda ^2e_qe_s/(4\pi )^2\) and \(m_q\) the quark mass, see Appendix B.2 for details.

3.2 Canonical transverse momentum of quarks

The JM quark TM can be expressed directly in terms of transverse-momentum dependent distributions (TMDs) [34, 35]. The leading-twist vector TMD correlator is defined as

$$\begin{aligned}&\varPhi ^{[\gamma ^+]}_\eta (P,x,\varvec{k}_\perp ,S) \nonumber \\&\quad = \frac{1}{2}\int \frac{\mathrm {d}z^-\,\mathrm {d}^2 z_\perp }{(2\pi )^3}\,e^{ik\cdot z} \langle P,S | {\bar{\psi }}(-\tfrac{z}{2})\gamma ^+\nonumber \\&\qquad \times {\mathcal {W}}_\eta (-\tfrac{z}{2},\tfrac{z}{2})\psi (\tfrac{z}{2}) | P,S \rangle \Big |_{z^+=0}, \end{aligned}$$
(27)

where \(x=k^+/P^+\) is the fraction of longitudinal LF momentum carried by the quark. The LF Wilson line satisfies the differential equation

$$\begin{aligned} D^{\text {pure},\eta }_\mu {\mathcal {W}}_\eta (z,z_0)=0 \end{aligned}$$
(28)

and ensures that the non-local correlator is gauge invariant. In practice, the boundary condition is determined by the scattering process at hand. In semi-inclusive deep-inelastic scattering (SIDIS) one has \(\eta =+1\), whereas in the Drell–Yan process one has \(\eta =-1\). The Wilson lines \({\mathcal {W}}_\pm (-\tfrac{z}{2},\tfrac{z}{2})\) with finite transverse width look like staples and result from the infinite product of infinitesimally thin staples all with the same orientation. When \(\eta \ne \pm 1\), the Wilson line consists of infinitely many zigzags between \(+\infty ^-\) and \(-\infty ^-\) and looks like a 2D “mille-feuille”, see Fig. 3. No physical process so far has been identified for this case.

Fig. 3
figure 3

Light-front Wilson line \(W_\eta (z,0)\) for an arbitrary value of the parameter \(\eta \in [-1,1]\). For advanced (\(\eta =+1\)) or retarded (\(\eta =-1\)) boundary condition, the path reduces to a simple staple

Full size image

Since the pure-gauge covariant derivative satisfies Eq. (28), it follows after some algebra that [29, 36]

$$\begin{aligned} \langle k^i_\perp \rangle _{\text {JM},\eta }=\int \mathrm {d}x\,\mathrm {d}^2k_\perp \,k^i_\perp \,\varPhi ^{[\gamma ^+]}_\eta (P,x,\varvec{k}_\perp ,S). \end{aligned}$$
(29)

Using the generic parametrization of the correlator (27) in terms of TMD parton distribution functionsFootnote 8

$$\begin{aligned} \varPhi ^{[\gamma ^+]}_\eta (P,x,\varvec{k}_\perp ,S)= f_1(x,\varvec{k}^2_\perp )-\eta \,\frac{\epsilon ^{ij}_\perp k^i_\perp S^j_\perp }{M}\,f^{\perp }_{1T}(x,\varvec{k}_\perp ^2),\nonumber \\ \end{aligned}$$
(30)

one arrives at the expression for the so-called Sivers shift [29, 36]

$$\begin{aligned} \langle k^i_\perp \rangle _{\text {JM},\eta }=-\eta \,\epsilon ^{ij}_\perp s^j_\perp \int \mathrm {d}x\,\mathrm {d}^2k_\perp \,\frac{\varvec{k}^2_\perp }{2M}\,f^{\perp }_{1T}(x,\varvec{k}^2_\perp ), \end{aligned}$$
(31)

where under the integral one has replaced \(k^i_\perp k^l_\perp \) by \(\delta ^{il}_\perp \,\varvec{k}_\perp ^2/2\). The function \(f^{\perp }_{1T}\) was originally introduced by Sivers [38] to explain large left-right asymmetries observed in pion-nucleus collisions [39]. The Sivers asymmetry has now been experimentally observed in both SIDIS [40,41,42] and Drell–Yan [43] processes. See also [44] for a review on the gluonic counterpart.

The quark Sivers function within the SDM was computed in [33] using the advanced boundary condition

$$\begin{aligned} f^{\perp q}_{1T}(x,\varvec{k}^2_\perp )= & {} \frac{\lambda ^2 e_q e_s (1-x)}{4(2\pi )^4}\, \frac{M(m_q + xM)}{\varvec{k}^2_\perp [\varvec{k}^{2}_{\perp }+\varLambda ^2 (x)]}\nonumber \\&\times \ln \! \left( \frac{\varvec{k}^{2}_{\perp }+\varLambda ^2 (x)}{\varLambda ^2 (x)} \right) , \end{aligned}$$
(32)

where \(\varLambda ^2(x)=xm^2_s+(1-x)m^2_q-x(1-x)M^2\) with \(m_s\) the scalar diquark mass. Inserting this expression into Eq. (31) we obtain using dimensional regularization with \(d\equiv D-2=2-2\epsilon \) for the transverse-momentum integral

$$\begin{aligned} \langle k^i_\perp \rangle _{\text {JM}, \eta } \underset{\text {naive}}{=} -\eta \,\epsilon ^{ij}_\perp s^j_\perp \,\frac{\pi }{3}\,(3m_q+M)\, \frac{{\mathcal {N}}}{(4\pi \epsilon )^2}+{\mathcal {O}}(1/\epsilon ),\nonumber \\ \end{aligned}$$
(33)

i.e. twice \(\langle k^i_\perp \rangle _{\text {pot},\eta }\), in flagrant conflict with the expectation from PT symmetry (21)! The solution to this conundrum is the realization that the expression in Eq. (32) is not the appropriate one in dimensional regularization. In the derivation of the Sivers function, a transverse loop integral

$$\begin{aligned} I_{d}=\int \frac{\mathrm {d}^{d} l_\perp }{ (2\pi )^{d}} \,\frac{\varvec{l}_\perp \cdot \varvec{k}_\perp }{\varvec{l}^{2}_\perp [(\varvec{k}^{2}_{\perp }+\varvec{l}^{2}_{\perp })+\varLambda ^2 (x)]} \end{aligned}$$
(34)

has been performed in \(d=2\) dimensions, leading to the logarithmic factor [36]

$$\begin{aligned} I_2= - \frac{1}{4\pi }\, \ln \!\left( \frac{\varvec{k}^{2}_{\perp }+\varLambda ^2 (x)}{\varLambda ^2 (x)} \right) . \end{aligned}$$
(35)

Even if this integral is finite in \(d=2\) dimensions for finite \(\varvec{k}^2_\perp \), we are actually interested in the leading divergence and hence in the large-\(\varvec{k}^2_\perp \) region, which interferes with the \(\epsilon \)-expansion and must be treated with care [45,46,47]. In dimensional regularization we need to keep \(d\equiv D-2=2-2\epsilon _l\), leading to

$$\begin{aligned} I_{2-2\epsilon _l}= & {} -\frac{\varvec{k}^2_\perp }{(4\pi )^{1-\epsilon _l}}\,\varGamma (1+\epsilon _l)\nonumber \\&\times \int _0^1\mathrm {d}y\,y^{-\epsilon _l}\left[ (1-y)\varvec{k}^2_\perp +\varLambda ^2(x)\right] ^{-1-\epsilon _l}. \end{aligned}$$
(36)

Proceeding with this expression instead of (35) and integrating over \(\varvec{k}_\perp \) in \(d=2-2\epsilon \) dimensions, we findFootnote 9

$$\begin{aligned} \langle k^i_\perp \rangle _{\text {JM}, \eta }= & {} -\eta \,\epsilon ^{ij}_\perp s^j_\perp \,\frac{\pi }{3}\,(3m_q+M)\nonumber \\&\times \frac{{\mathcal {N}}}{(4\pi )^2\epsilon (\epsilon +\epsilon _l)}+{\mathcal {O}}(1/\epsilon ). \end{aligned}$$
(37)

The naive result (33) corresponds to \(\epsilon _l\rightarrow 0\), while the correct result \(\langle k^i_\perp \rangle _{\text {JM},\eta }=\langle k^i_\perp \rangle _{\text {pot},\eta }\) is obtained when \(\epsilon _l\rightarrow \epsilon \). The same conclusion is reached using Weinberg’s theorem to extract the correct large-\(\varvec{k}^2_\perp \) behavior of \(I_d\). More generally, this is in line with the observation from Refs. [48, 49] that Eq. (31) holds only if the same regularization and renormalization procedures are applied on both sides of the identity. Note however that the explicit check of the Burkardt sum rule of Ref. [50] is not impacted since the same subtlety applies equally to the gluon contribution.

4 Potential angular momentum of quarks

Similarly to the quark po tential TM, we can now consider the quark potential longitudinal AM [7, 30] at equal LF time \(r^+=0\)

$$\begin{aligned}&\langle L_z\rangle _{\text {pot},\eta } =\lim _{\varDelta \rightarrow 0}\nonumber \\&\quad \times \frac{\langle p',S'|\int \mathrm {d}^4 r\,\delta (r^+)\,(-g){\overline{\psi }}(r)\gamma ^+[\varvec{r}_\perp \times \varvec{A}_{\text {phys},\eta ,\perp }(r)]_z\psi (r)|p,S\rangle }{\langle P,S|P,S\rangle }\nonumber \\&\quad =\frac{-g\,\epsilon ^{ij}_\perp }{2P^+}\left[ -i\nabla ^i_{\varDelta _\perp }\langle p',S'|{\overline{\psi }}(0)\gamma ^+A^j_{\text {phys},\eta ,\perp }(0)\psi (0)|p,S\rangle \right] _{\varDelta =0}, \end{aligned}$$
(38)

where \(p=P-\tfrac{\varDelta }{2}\) and \(p'=P+\tfrac{\varDelta }{2}\). The covariant LF polarization vectors S and \(S'\) refer to the same rest-frame spin vector \(\varvec{s}\). In order to deal properly with the explicit factor of \(\varvec{r}_\perp \) in the operator, the matrix element should first be considered with \(p'\ne p\), and the forward limit \(\varDelta \equiv p'-p\rightarrow 0\) has to be taken at the end [51, 52]. This potential AM represents the difference between the JM and Ji definitions of quark OAM, obtained by replacing \(-g A^j_{\text {phys},\eta ,\perp }\) in Eq. (38) with \(\frac{i}{2}\overset{\leftrightarrow }{D}\!\!\!\!\!{D}^j_{\text {pure},\eta ,\perp }\) and \(\frac{i}{2}\overset{\leftrightarrow }{D}\!\!\!\!\!{D}^j_\perp \), respectively. Discrete space-time symmetries indicate that both JM and Ji OAM are PT-even, and so is the potential AM

$$\begin{aligned} \langle L_z\rangle _{\text {pot},\eta }=\langle L_z\rangle _{\text {pot},-\eta }. \end{aligned}$$
(39)

This means that the boundary condition has no effects on the value of the longitudinal OAM [30, 31] and we can drop the reference to the parameter \(\eta \).

At one-loop level in the SDM, it has been shown that JM and Ji notions of OAM (computed independently) do coincide [53, 54], implying therefore a vanishing potential AM. This is of course expected since at that order no gauge field is involved. Perhaps more surprisingly, for an electron in QED where the gauge field appears already at the one-loop level, one obtains also a vanishing potential AM at that order [11]. So the presence of a gauge field is not a sufficient condition for inducing a difference between JM and Ji OAM. As already mentioned in the introduction, calculations on the lattice indicate a clear non-vanishing potential AM [12, 13]. This motivates us to investigate the potential AM at the two-loop level within the SDM.

To the best of our knowledge, JM and Ji notions of OAM have never been calculated in the SDM at order \({\mathcal {O}}(\lambda ^2 e_qe_s)\). Since we are in fact interested in their difference, we compute directly the quark potential AM from the diagrams shown in Fig. 2. We have seen that at order \({\mathcal {O}}(\lambda ^2 e_q e_s)\) in the SDM, the potential TM receives some \(1/\epsilon ^2\)-contribution. If the same underlying mechanism is at play for the potential AM, we may expect it to generate also some \(1/\epsilon ^2\)-contribution to the latter.

Contrary to the potential TM, the potential AM is PT-even and can a priori receive a non-vanishing contribution from both mechanisms \({\mathcal {M}}_{A+B}\) and \({\mathcal {M}}_C\). An explicit calculation outlined in Appendix C.1 shows that at best

$$\begin{aligned} \left[ -i\epsilon ^{ij}_\perp \nabla ^i_\perp {\mathcal {M}}^j_{A+B}(\varDelta )\right] _{\varDelta =0}={\mathcal {O}}(1/\epsilon ). \end{aligned}$$
(40)

The absence of \(1/\epsilon ^2\)-divergences can intuitively be understood by the fact that a quark should not be able to exert a self-torque and hence change its own OAM, which would lead to a violation of AM conservation.Footnote 10 All the \(1/\epsilon ^2\)-contributions to the quark potential AM come therefore from the third diagram where the photon is attached to the spectator system. Using dimensional regularization, we actually find that all \(1/\epsilon ^2\)-divergences cancel out, so that we have at best

$$\begin{aligned} \langle L_z\rangle _\text {pot}= {\mathcal {O}}(1/\epsilon ), \end{aligned}$$
(41)

see  Appendix C.2 for more details. This is somewhat surprising since ISI/FSI, represented at this order by the photon exchange with the spectator system, have been shown to exert a Lorentz force \(\propto 1/\epsilon ^2\) on the struck quark. Our results jeopardize therefore the existence of a simple lensing mechanism relating Sivers effect and potential AM. A posteriori, we may argue that a non-vanishing potential AM requires a Lorentz torque defined relative to the center of inertia of the system [16, 55]. In the SDM, the nucleon is described as a two-body system. The positions of the quark, the diquark, and the center of inertia (or \(P^+\)) all lie on the same line, preventing therefore classically any Lorentz torque. This simple argument suggests that \(1/\epsilon \)-divergences may also cancel. Although interesting on its own, an explicit check of this cancellation is non-trivial. It goes beyond the purpose of the present work and is left for future investigations.

5 Summary and outlook

The potential transverse momentum and potential longitudinal angular momentum, describing the difference between the corresponding Jaffe-Manohar and Ji definitions of linear and angular momenta, have been computed for the first time at order \({\mathcal {O}}(\lambda ^2 e_q e_s)\) in the scalar diquark model. At that order, two mechanisms are possible: one where a gauge boson is exchanged between the struck quark and the spectator system, and one where the struck quark exchanges a gauge boson with itself. As expected by conservation of linear and angular momentum, the latter mechanism does not contribute (at least to the \(1/\epsilon ^2\)-divergence). The former mechanism is responsible for the Sivers shift and represents the accumulated exchange of momentum between the struck quark and the spectator system in a high-energy process. Our explicit model calculation confirms that the potential transverse momentum coincides with the average Jaffe-Manohar (or canonical) transverse momentum of the quark, as expected by parity and time-reversal symmetries. For the potential angular momentum, we found that all \(1/\epsilon ^2\)-divergent terms cancel out in the scalar diquark model at two loops, contrasting with the expectation based on the lensing mechanism picture. We proposed that this surprising result could be the reflection of the fact that in a two-body description of a system, there cannot be classically any Lorentz torque exerted by the spectator on the struck constituent despite the presence of a Lorentz force. A confirmation of these observations in QED or the quark-target model would be very interesting, and is left for future works.