Gravitational transverse-momentum distributions

We start with a reminder on the TMD correlators. The fully unintegrated quark-quark correlator for a spin-$\frac{1}{2}$ target Meissner:2009ww is defined in the forward limit as

$$W^{[\Gamma]}(P,k,N,S;\eta)=\frac{1}{2}\int\frac{\mathrm{d}^{4}z}{(2\pi)^{4}}\,e^{ik\cdot z}\,\langle P,S|\overline{\psi}(-\tfrac{z}{2})\Gamma\mathcal{W}(-\tfrac{z}{2},\tfrac{z}{2}|n)\psi(\tfrac{z}{2})|P,S\rangle,$$

(1)

where $\Gamma$ stands for a generic matrix in Dirac space, e.g. $\Gamma=\gamma^{\mu},\gamma^{\mu}\gamma_{5},\cdots$. For a target of mass $M$ and four-momentum $P$, the covariant spin vector $S$ defined via $\overline{u}(P,S)\gamma^{\mu}\gamma_{5}u(P,S)=2MS^{\mu}$ satisfies $P\cdot S=0$ and $S^{2}=-1$. The quark average four-momentum $k$ is defined as the Fourier conjugate variable to the space-time distance $z$ between the two quark operators. Gauge invariance is preserved by the inclusion of a Wilson line $\mathcal{W}$ connecting the points $-\frac{z}{2}$ and $\frac{z}{2}$ via an infinitely long staple-shaped path along the lightlike direction $n$. Since the same Wilson line is unchanged under the rescaling $n\mapsto\alpha n$ with $\alpha>0$, the correlator depends in fact on the rescaling-invariant four-vector

$$N=\frac{M^{2}n}{P\cdot n}.$$

(2)

The parameter $\eta=\text{sign}(n^{0})$ indicates whether the Wilson line is future-pointing ($\eta=+1$) or past-pointing ($\eta=-1$).

For convenience, we choose the coordinate system and the rescaling factor $\alpha$ such that

	$\displaystyle P^{\mu}$	$\displaystyle=\left[P^{+},\frac{M^{2}}{2P^{+}},\bm{0}_{\perp}\right],$		(3)
	$\displaystyle k^{\mu}$	$\displaystyle=\left[xP^{+},k^{-},\bm{k}_{\perp}\right],$
	$\displaystyle n^{\mu}$	$\displaystyle=\left[0,\eta,\bm{0}_{\perp}\right],$

where $v^{\mu}=[v^{+},v^{-},\bm{v}_{\perp}]$ with the light-front components defined as $v^{\pm}=(v^{0}\pm v^{3})/\sqrt{2}$. The quark TMD correlator (see e.g. Bacchetta:2006tn ) is then obtained by integration over the quark light-front energy

	$\displaystyle\Phi^{[\Gamma]}(P,x,\bm{k}_{\perp},N,S;\eta)$	$\displaystyle=\int\mathrm{d}k^{-}\,W^{[\Gamma]}(P,k,N,S;\eta)$		(4)
		$\displaystyle=\frac{1}{2}\int\frac{\mathrm{d}z^{-}\,\mathrm{d}^{2}z_{\perp}}{(2\pi)^{3}}\,e^{ik\cdot z}\,\langle P,S|\overline{\psi}(-\tfrac{z}{2})\Gamma\mathcal{W}(-\tfrac{z}{2},\tfrac{z}{2}|n)\psi(\tfrac{z}{2})|P,S\rangle\Big{|}_{z^{+}=0}.$

In QCD, the local gauge-invariant EMT operator for quarks is given by

$$T^{\mu\nu}_{q}(r)=\overline{\psi}(r)\gamma^{\mu}\tfrac{i}{2}\overset{\leftrightarrow}{D}\!\!\!\!\!\phantom{D}^{\nu}\psi(r)$$

(5)

with $\overset{\leftrightarrow}{D}\!\!\!\!\!\phantom{D}^{\nu}=\overset{\rightarrow}{\partial}\!\!\!\!\phantom{\partial}^{\nu}-\overset{\leftarrow}{\partial}\!\!\!\!\phantom{\partial}^{\nu}-2igA^{\nu}(r)$. In order to define the EMT for a quark with average four-momentum $k$, we need to consider a bilocal generalization of this expression. Unfortunately, the covariant derivative does not commute with the Wilson line, making the bilocal generalization of Eq. (5) ambiguous Lorce:2012ce . The problem can be traced back to the fact that $[D_{\mu},D_{\nu}]\neq 0$ whereas $[k_{\mu},k_{\nu}]=0$, which implies that $k$ cannot be identified with the quark kinetic four-momentum. However, if we work in the gauge where the Wilson line reduces to the identity (namely the light-front gauge with appropriate advanced of retarded boundary conditions depending on the value of $\eta$ Belitsky:2002sm ), the four-vector $k^{\mu}$ can be represented by the partial derivatives $i\partial^{\mu}$, and hence be interpreted as the quark canonical four-momentum. Therefore, instead of looking for the bilocal generalization of the kinetic EMT operator (5), we should rather be looking for the bilocal generalization of the light-front gauge-invariant canonical (gic) EMT operator Lorce:2012rr ; Leader:2013jra ; Lorce:2015lna

$$T^{\mu\nu}_{q,\text{gic}}(r)=\overline{\psi}(r)\gamma^{\mu}\tfrac{i}{2}\overset{\leftrightarrow}{D}\!\!\!\!\!\phantom{D}^{\nu}_{\text{pure}}\psi(r),$$

(6)

where $D^{\mu}_{\text{pure}}=\partial^{\mu}-igA^{\mu}_{\text{pure}}$ is known as the pure-gauge covariant derivative Chen:2008ag ; Wakamatsu:2010cb , corresponding in the present context to the covariant derivative reducing in the light-front gauge $A^{+}=0$ (with appropriate boundary conditions) to $\partial^{\mu}$ Hatta:2011zs ; Hatta:2011ku ; Lorce:2012ce . Note that by definition $A^{+}_{\text{pure}}(r)=A^{+}(r)$, meaning that $T^{\mu+}_{q}(r)=T^{\mu+}_{q,\text{gic}}(r)$. Therefore, as far as the longitudinal light-front momentum is concerned, there is no difference between the kinetic and the gauge-invariant canonical definitions.

Following the spirit of Refs. Ji:2003ak ; Belitsky:2003nz ; Lorce:2011kd ; Lorce:2011ni , it is natural to define the bilocal gauge-invariant canonical (gic) EMT operator for quarks as Lorce:2012ce

$$T^{\mu\nu}_{q,\text{gic}}(r,k)=k^{\nu}\int\frac{\mathrm{d}^{4}z}{(2\pi)^{4}}\,e^{ik\cdot z}\,\overline{\psi}(r-\tfrac{z}{2})\gamma^{\mu}\mathcal{W}(r-\tfrac{z}{2},r+\tfrac{z}{2}|n)\psi(r+\tfrac{z}{2}).$$

(7)

Integrating by parts, we can write

	$\displaystyle T^{\mu\nu}_{q,\text{gic}}(r,k)$	$\displaystyle=\int\frac{\mathrm{d}^{4}z}{(2\pi)^{4}}\,e^{ik\cdot z}\,i\partial^{\nu}_{z}\!\left[\overline{\psi}(r-\tfrac{z}{2})\gamma^{\mu}\mathcal{W}(r-\tfrac{z}{2},r+\tfrac{z}{2}|n)\psi(r+\tfrac{z}{2})\right]$		(8)
		$\displaystyle=\int\frac{\mathrm{d}^{4}z}{(2\pi)^{4}}\,e^{ik\cdot z}\left[\overline{\psi}(r-\tfrac{z}{2})\gamma^{\mu}\mathcal{W}(r-\tfrac{z}{2},r+\tfrac{z}{2}|n)\tfrac{i}{2}\overset{\rightarrow}{D}\!\!\!\!\!\phantom{D}^{\nu}_{\text{pure}}(r+\tfrac{z}{2})\psi(r+\tfrac{z}{2})\right.$
		$\displaystyle\qquad\qquad\qquad\quad\left.-\overline{\psi}(r-\tfrac{z}{2})\tfrac{i}{2}\overset{\leftarrow}{D}\!\!\!\!\!\phantom{D}^{\nu}_{\text{pure}}(r-\tfrac{z}{2})\gamma^{\mu}\mathcal{W}(r-\tfrac{z}{2},r+\tfrac{z}{2}|n)\psi(r+\tfrac{z}{2})\right],$

where $A^{\mu}_{\text{pure}}$ is given by

$$A^{\mu}_{\text{pure}}(r)=\mathcal{W}(r,0|n)\tfrac{i}{g}\partial^{\mu}_{r}\mathcal{W}(0,r|n).$$

(9)

Since we obviously have the property

$$D^{\mu}_{\text{pure}}(x)\mathcal{W}(x,y|n)=\mathcal{W}(x,y|n)D^{\mu}_{\text{pure}}(y)$$

(10)

reflecting the commutativity of pure-gauge covariant derivatives, the bilocal operator in Eq. (8) is unambiguous. Moreover, integrating over the quark four-momentum leads to

$$\int\mathrm{d}^{4}k\,T^{\mu\nu}_{q,\text{gic}}(r,k)=T^{\mu\nu}_{q,\text{gic}}(r)$$

(11)

as expected.

We can now define in a natural way the fully unintegrated EMT by considering the forward matrix element²²2The motivation for the factor $\frac{1}{2}$ is the same as for the correlators in Section II.1: the light-front expectation value of an operator $O$ is $\frac{\langle P,S|O|P,S\rangle}{2P^{+}}$ and switching from a distribution in $k^{+}$ to a distribution in $x$ amounts to a multiplication by the Jacobian $P^{+}$. of the operator in Eq. (8)

$$\Theta^{\mu\nu}_{q}(P,k,N,S;\eta)=\frac{1}{2}\,\langle P,S|T^{\mu\nu}_{q,\text{gic}}(0,k)|P,S\rangle,$$

(12)

and the TMD EMT by further integrating over the quark light-front energy

	$\displaystyle\mathcal{T}^{\mu\nu}_{q}(P,x,\bm{k}_{\perp},N,S;\eta)$	$\displaystyle=\int\mathrm{d}k^{-}\,\Theta^{\mu\nu}_{q}(P,k,N,S;\eta)$		(13)
		$\displaystyle=\frac{1}{2}\int\frac{\mathrm{d}z^{-}\,\mathrm{d}^{2}z_{\perp}}{(2\pi)^{3}}\,e^{ik\cdot z}\,i\partial_{z}^{\nu}\langle P,S|\overline{\psi}(-\tfrac{z}{2})\gamma^{\mu}\mathcal{W}(-\tfrac{z}{2},\tfrac{z}{2}|n)\psi(\tfrac{z}{2})|P,S\rangle\Big{|}_{z^{+}=0}.$

This last object can be interpreted as the 3D distribution of the quark EMT in momentum space.

Parity, hermiticity and time-reversal invariance imply that the fully unintegrated EMT satisfies the relations

	$\displaystyle\Theta^{\mu\nu}(k,P,N,S;\eta)$	$\displaystyle=\Theta^{\bar{\mu}\bar{\nu}}(\bar{k},\bar{P},\bar{N},-\bar{S};\eta),$		(14)
	$\displaystyle\Theta^{\mu\nu}(k,P,N,S;\eta)$	$\displaystyle=[\Theta^{\mu\nu}(k,P,N,S;\,\eta)]^{{\dagger}},$
	$\displaystyle\Theta^{\mu\nu}(k,P,N,S;\eta)$	$\displaystyle=[\Theta^{\bar{\mu}\bar{\nu}}(\bar{k},\bar{P},\bar{N},\bar{S};-\eta)]^{*},$

with the notation $v^{\bar{\mu}}=\bar{v}^{\mu}=(v^{0},-\bm{v})$. Since the parametrization should be the same for both quark and gluon contributions to the EMT, we drop the label $q$ in this subsection.

For convenience, we define the transverse part of a four-vector by $v^{\mu}_{T}=g^{\mu\nu}_{T}v_{\nu}$ using the projector onto the subspace orthogonal to $P$ and $N$

$\displaystyle g_{T}^{\mu\nu}=g^{\mu\nu}-\frac{P^{\mu}N^{\nu}+P^{\nu}N^{\mu}}{M^{2}}+\frac{N^{\mu}N^{\nu}}{M^{2}}.$

(15)

The covariant spin vector can then be expressed as

$\displaystyle S^{\mu}=\frac{\lambda}{M}(P^{\mu}-N^{\mu})+S_{T}^{\mu},$

(16)

where the longitudinal light-front polarization is denoted by the parameter $\lambda$ and the transverse light-front polarization by the four-vector $S^{\mu}_{T}=[0,0,\bm{S}_{\perp}]$. We define also the transverse Levi-Civita pseudotensor

$\displaystyle\epsilon_{T}^{\mu\nu}=\frac{\epsilon^{\mu\nu\alpha\beta}N_{\alpha}P_{\beta}}{M^{2}}$

(17)

with the convention $\epsilon_{0123}=1$ such that $\epsilon_{T}^{12}=1$, and we introduce the compact notation $\epsilon_{T}^{\mu v_{T}}\equiv\epsilon_{T}^{\mu\nu}v_{T\nu}$.

A complete parametrization of the TMD EMT $\mathcal{T}^{\mu\nu}(P,x,\bm{k}_{\perp},N,S;\eta)$ for spin-$0$ and spin-$\frac{1}{2}$ targets can be obtained by writing down all the independent rank-2 tensors built out of $g_{T}^{\mu\nu}$, $\epsilon_{T}^{\mu\nu}$, $P^{\mu}$, $N^{\mu}$, and $k^{\mu}_{T}$, which are at most linear in the polarization and which satisfy the constraints in Eq. (14). We find³³3Note that other possible tensor structures have been discarded thanks to the Schouten identity $$g^{\alpha\beta}\epsilon^{\mu\nu\rho\sigma}+g^{\alpha\mu}\epsilon^{\nu\rho\sigma\beta}+g^{\alpha\nu}\epsilon^{\rho\sigma\beta\mu}+g^{\alpha\rho}\epsilon^{\sigma\beta\mu\nu}+g^{\alpha\sigma}\epsilon^{\beta\mu\nu\rho}=0.$$

	$\displaystyle\mathcal{T}^{\mu\nu}=\frac{1}{P^{+}}\Big{\{}$	$\displaystyle P^{\mu}P^{\nu}a_{1}+N^{\mu}N^{\nu}a_{2}+k_{T}^{\mu}k_{T}^{\nu}a_{3}+P^{\mu}N^{\nu}a_{4}+N^{\mu}P^{\nu}a_{5}$		(18)
		$\displaystyle+P^{\mu}k_{T}^{\nu}a_{6}+k_{T}^{\mu}P^{\nu}a_{7}+N^{\mu}k_{T}^{\nu}a_{8}+k_{T}^{\mu}N^{\nu}a_{9}+M^{2}g^{\mu\nu}_{T}a_{0}$
		$\displaystyle-\frac{\epsilon_{T}^{k_{T}S_{T}}}{M}\left[P^{\mu}P^{\nu}a_{1T}^{\perp}+N^{\mu}N^{\nu}a_{2T}^{\perp}+k_{T}^{\mu}k_{T}^{\nu}a_{3T}^{\perp}+P^{\mu}N^{\nu}a_{4T}^{\perp}+N^{\mu}P^{\nu}a_{5T}^{\perp}\right.$
		$\displaystyle\qquad\qquad\quad\left.+P^{\mu}k_{T}^{\nu}a_{6T}^{\perp}+k_{T}^{\mu}P^{\nu}a_{7T}^{\perp}+N^{\mu}k_{T}^{\nu}a_{8T}^{\perp}+k_{T}^{\mu}N^{\nu}a_{9T}^{\perp}+M^{2}g^{\mu\nu}_{T}a_{0T}^{\perp}\right]$
		$\displaystyle-M\left[P^{\mu}\epsilon_{T}^{\nu S_{T}}a_{1T}+P^{\nu}\epsilon_{T}^{\mu S_{T}}a_{2T}+N^{\mu}\epsilon_{T}^{\nu S_{T}}a_{3T}+N^{\nu}\epsilon_{T}^{\mu S_{T}}a_{4T}+k_{T}^{\mu}\epsilon_{T}^{\nu S_{T}}a_{5T}+k_{T}^{\nu}\epsilon_{T}^{\mu S_{T}}a_{6T}\right]$
		$\displaystyle-\lambda\left[P^{\mu}\epsilon_{T}^{\nu k_{T}}a_{1L}+P^{\nu}\epsilon_{T}^{\mu k_{T}}a_{2L}+N^{\mu}\epsilon_{T}^{\nu k_{T}}a_{3L}+N^{\nu}\epsilon_{T}^{\mu k_{T}}a_{4L}+k_{T}^{\mu}\epsilon_{T}^{\nu k_{T}}a_{5L}+k_{T}^{\nu}\epsilon_{T}^{\mu k_{T}}a_{6L}\right]\Big{\}},$

where the real-valued coefficients $a_{i}(x,\bm{k}_{\perp}^{2})$ will be referred to as gravitational TMDs. There are 10 polarization-independent gravitational TMDs (viz. $a_{0-9}$). For a spin-$0$ target, that is all we have. For a spin-$\frac{1}{2}$ target, there are in addition 22 polarization-dependent gravitational TMDs: 6 associated with the longitudinal polarization (viz. $a_{1-6L}$) and 16 associated with the transverse polarization (viz. $a_{1-6T}$ and $a_{0-9T}^{\perp}$). As a result of the discrete symmetries (14), the polarization-independent gravitational TMDs are naive T-even (i.e. independent of $\eta$) whereas the polarization-dependent ones are naive T-odd (i.e. they change sign under $\eta\mapsto-\eta$). Interestingly, the same total numbers of gravitational GPDs for spin-$0$ and spin-$\frac{1}{2}$ targets have been obtained in Ref. Lorce:2015lna . Since $\int\mathrm{d}^{2}k_{\perp}\,\mathcal{T}^{\mu\nu}$ can not depend on $k^{\mu}_{T}$, one may naively think by eliminating all the $k^{\mu}_{T}$-dependent tensors in Eq. (18) that there are only 9 gravitational PDFs. Note however that the combination $k_{T}^{\mu}\epsilon_{T}^{\nu k_{T}}-k_{T}^{\nu}\epsilon_{T}^{\mu k_{T}}=\bm{k}_{\perp}^{2}\epsilon_{T}^{\mu\nu}$ does survive integration over $\bm{k}_{\perp}$, meaning that there are in total 10 gravitational PDFs, in agreement with the results in Section 4.4 of Ref. Lorce:2015lna .

Since the TMD correlator $\Phi^{[\gamma^{+}]}(P,x,\bm{k}_{\perp},N,S;\eta)$ is interpreted as the probability density of finding a quark with three-momentum $[xP^{+},\bm{k}_{\perp}]$, it is natural to interpret

	$\displaystyle\mathcal{T}^{++}_{q}$	$\displaystyle=xP^{+}\Phi^{[\gamma^{+}]}=\left(f_{1}-\frac{\epsilon_{T}^{k_{T}S_{T}}}{M}\,f^{\perp}_{1T}\right)xP^{+},$		(27)
	$\displaystyle\mathcal{T}^{+i}_{q}$	$\displaystyle=k^{i}_{T}\Phi^{[\gamma^{+}]}=\left(f_{1}-\frac{\epsilon_{T}^{k_{T}S_{T}}}{M}\,f^{\perp}_{1T}\right)k^{i}_{T},\qquad\quad i=1,2$

as the quark longitudinal and transverse momentum densities in momentum space. The average quark longitudinal momentum is then obtained by integration over the quark momentum

$$\langle k^{+}\rangle_{q}=\langle x\rangle_{q}P^{+}=\int\mathrm{d}x\,\mathrm{d}^{2}k_{\perp}\,\mathcal{T}^{++}_{q}=P^{+}\int\mathrm{d}x\,\mathrm{d}^{2}k_{\perp}\,xf_{1}.$$

(28)

Similarly, the average quark transverse momentum Boer:2003cm ; Burkardt:2003yg ; Meissner:2007rx ; Amor-Quiroz:2020qmw is given by

$$\langle k^{i}_{\perp}\rangle_{q}=\int\mathrm{d}x\,\mathrm{d}^{2}k_{\perp}\,\mathcal{T}^{+i}_{q}=\epsilon_{T}^{iS_{T}}\int\mathrm{d}x\,\mathrm{d}^{2}k_{\perp}\,\frac{\bm{k}^{2}_{\perp}}{2M}\,f_{1T}^{\perp}.$$

(29)

We illustrate in Fig. 1 the two contributions to the transverse momentum density at some fixed value of $x$ using a simple gaussian model for the transverse momentum dependence $f(\bm{k}^{2}_{\perp})\propto e^{-\bm{k}_{\perp}^{2}/\langle\bm{k}^{2}_{\perp}\rangle}$ with the typical value $\langle\bm{k}^{2}_{\perp}\rangle\approx 0.6$ GeV² for the gaussian width Anselmino:2013lza . Since $\mathcal{T}^{+i}_{q}\propto k^{i}_{T}$, it is natural that the transverse momentum density looks like a hedgehog. The unpolarized contribution driven by $f_{1}$ is necessarily axially symmetric for there is no preferred transverse direction. However, the combination of target transverse polarization and initial/final state interactions breaks axial symmetry. The magnitude of this effect is quantified by the Sivers function $f^{\perp}_{1T}$ Sivers:1989cc .

Because of the Galilean subgroup exhibited by the light-front coordinates, the light-front longitudinal momentum plays the role of inertia in the transverse plane Susskind:1967rg . The transverse flux of longitudinal momentum $\mathcal{T}^{i+}$ can therefore be thought of as the transverse flux of inertia, suggesting the definition of an effective quark transverse velocity via the ratio

$$v^{i}_{\perp}=\frac{\mathcal{T}^{i+}_{q}}{\mathcal{T}^{++}_{q}}.$$

(30)

It is often thought that the EMT is symmetric, and hence that momentum density $\mathcal{T}^{+i}_{q}$ equals flux of inertia $\mathcal{T}^{i+}_{q}$. In that case, the quark transverse velocity is simply given by $\bm{v}_{\perp}=\bm{k}_{\perp}/(xP^{+})$. In a gauge theory, velocity and canonical momentum are however usually not parallel and we should expect in general⁵⁵5Quark spin may also make the EMT asymmetric, but the antisymmetric contribution vanishes when initial and final target momenta are the same Leader:2013jra . $\mathcal{T}^{i+}_{q}\neq\mathcal{T}^{+i}_{q}$. Indeed, we find that

$$\mathcal{T}^{i+}_{q}=xP^{+}\Phi^{[\gamma^{i}_{T}]}=\left(xf^{\perp}-\frac{\epsilon_{T}^{k_{T}S_{T}}}{M}\,xf^{\perp}_{T}\right)k^{i}_{T}-M\epsilon_{T}^{iS_{T}}xf^{+}_{T}-\lambda\epsilon_{T}^{ik_{T}}xf^{\perp}_{L}.$$

(31)

In the case of a symmetric TMD EMT, we should have

	$\displaystyle xf^{\perp}$	$\displaystyle=f_{1},$		(32)
	$\displaystyle xf^{\perp}_{T}$	$\displaystyle=f^{\perp}_{1T},$
	$\displaystyle f^{+}_{T}$	$\displaystyle=f^{\perp}_{L}=0.$

Interestingly, the first relation was found in Ref. Lorce:2014hxa using the free quark equation of motion. In QCD, these relations are not expected to hold in general and their violations are a direct measure of the interaction between quarks and gluons.

Decomposing $\epsilon_{T}^{iS_{T}}$ onto components parallel and orthogonal to $k^{i}_{\perp}$, we can rewrite Eq. (31) as

$$\mathcal{T}^{i+}_{q}=\left(xf^{\perp}-\frac{M\epsilon_{T}^{k_{T}S_{T}}}{k^{2}_{T}}\,xf^{-}_{T}\right)k^{i}_{T}-\left(\lambda\,xf^{\perp}_{L}+\frac{M(k_{T}\cdot S_{T})}{k^{2}_{T}}\,xf^{+}_{T}\right)\epsilon_{T}^{ik_{T}}.$$

(33)

The first two terms have the same structure as $\mathcal{T}^{+i}_{q}$ in Eq. (27) and lead to similar hedgehog distributions as in Fig. 1. The last two terms indicate that besides modifying the magnitude of the quark velocity, QCD interactions can also modify its direction relative to $\bm{k}_{\perp}$. The corresponding distributions are illustrated in Fig. 2 using the same simple gaussian model as for $\mathcal{T}^{+i}_{q}$.

The notions of 2D spatial distributions of pressure (or isotropic stress) $\sigma$ and shear forces (or pressure anisotropy) $\Pi$ have been introduced in Ref. Lorce:2018egm . Similarly, we introduce here the distributions of transverse pressure and shear forces in momentum space

$$\mathcal{T}^{ij}=-g^{ij}_{T}\,\sigma+\left(\frac{1}{2}\,g^{ij}_{T}-\frac{k^{i}_{T}k^{j}_{T}}{k^{2}_{T}}\right)\Pi+\frac{k^{i}_{T}\epsilon^{jk_{T}}+k^{j}_{T}\epsilon^{ik_{T}}}{2k^{2}_{T}}\,\Pi^{S}+\epsilon^{ij}_{T}\,\Pi^{A}.$$

(34)

The first two transverse tensors are similar to those found in position space, with transverse momentum $\bm{k}_{\perp}$ replacing impact parameter $\bm{b}_{\perp}$. The last two transverse tensors are new, and are allowed provided that $\Pi_{S}$ and $\Pi_{A}$ are linear in the target polarization and naive T-odd. Using the particular structure of the quark EMT

	$\displaystyle\mathcal{T}^{ij}_{q}=k^{j}_{T}\Phi^{[\gamma^{i}_{T}]}$	$\displaystyle=\tfrac{k^{j}_{T}}{xP^{+}}\,\mathcal{T}^{i+}_{q}$		(35)
		$\displaystyle=\frac{1}{P^{+}}\left[\left(f^{\perp}-\frac{M\epsilon_{T}^{k_{T}S_{T}}}{k^{2}_{T}}\,f^{-}_{T}\right)k^{i}_{T}k^{j}_{T}-\left(\lambda\,f^{\perp}_{L}+\frac{M(k_{T}\cdot S_{T})}{k^{2}_{T}}\,f^{+}_{T}\right)\epsilon_{T}^{ik_{T}}k^{j}_{T}\right],$

we find

	$\displaystyle\sigma_{q}$	$\displaystyle=\tfrac{1}{2}\Pi_{q}=-\frac{1}{2P^{+}}\left[k^{2}_{T}f^{\perp}-M\epsilon^{k_{T}S_{T}}f^{-}_{T}\right],$		(36)
	$\displaystyle\Pi^{A}_{q}$	$\displaystyle=\tfrac{1}{2}\Pi^{S}_{q}=-\frac{1}{2P^{+}}\left[\lambda\,k^{2}_{T}f^{\perp}_{L}+M(k_{T}\cdot S_{T})f^{+}_{T}\right].$

This is to be compared with the free quark case given by $\mathcal{T}^{ij}_{q,\text{free}}=f_{1}\,k^{i}_{T}k^{j}_{T}/(xP^{+})$.