the physical- and pure-gauge components of gluon fields proposed by himself in [@Hatta11]. Starting from the gauge-invariant expression of the Wigner distribution also called the generalized transverse-momentum-dependent distributions (GTMD), which depends not only the longitudinal and transverse momenta but also the momentum transfer of the target nucleon, he showed that the nucleon matrix element of the [*generalized canonical*]{} OAM can be related to a weighted integral of a certain GTMD. It is important to recognize that this quantity does not appear in the standard classification of TMDs by the following reason. To explain it, we first recall the definition of the most fundamental GTMD appearing in the classification given in [@MMS09] : $$\begin{aligned}
 &\,& W^{[\gamma^+]} (x, \xi, \bm{q}^2_T, 
 \bm{q}_T \cdot \bm{\Delta}_T,\bm{\Delta}^2_T ; \eta) \nonumber \\
 &=& \frac{1}{2} \,\int \,\frac{d z^- \,d^2 z_T}{(2 \,\pi)^2} \,
 e^{k \cdot z} \,\langle p^\prime, \lambda^\prime \,| \,
 \bar{\psi} \left(- \,\frac{z}{2} \right) \,\gamma^+ \,
 {\cal W} \left(- \,\frac{z}{2}, \frac{z}{2} \,| \,n \right) \,
 \psi \left( \frac{z}{2} \right) \,| \,p, \lambda 
 \rangle_{z^+ = 0} \nonumber \\
 &=& \frac{1}{2 \,M} \,\bar{u} (p^\prime, \lambda^\prime) \,
 \left[\,F_{1,1} \, + \, 
 \frac{i \,\sigma^{i +} \,q^i_T}{P^+} \,
 F_{1,2} \, + \, 
 \frac{i \,\sigma^{i +} \,\Delta^i_T}{P^+} \,
 F_{1,3} \, + \, 
 \frac{i \,\sigma^{ij} 