Jaffe-Manohar decomposition is the so-called canonical OAM, which is not gauge-invariant. On the other hand, what appears in the Ji decomposition is the so-called dynamical (or mechanical) OAM, which is manifestly gauge-invariant [@BookSakurai95]. As is well-known, the gauge principle in physics dictates that observables must be gauge-invariant. Because of this reason, the observability of the canonical OAM has been questioned for a long time. On the other hand, Ji showed that the gauge-invariant dynamical quark OAM can be extracted from the combined analysis of unpolarized generalized parton distributions and the longitudinally polarized parton distributions [@Ji97PRL],[@Ji97].

Some years ago, however, Chen et al. proposed a new gauge-invariant decomposition of nucleon spin [@Chen08],[@Chen09]. The basic idea is to decompose the gluon field $\bm{A}$ into two parts, i.e. the physical part $\bm{A}_{phys}$ and the pure-gauge part $\bm{A}_{pure}$, which is a generalization of the decomposition of the photon field $\bm{A}$ in QED into the transverse component $\bm{A}_\perp$ and the longitudinal component $\bm{A}_\parallel$. In addition to general conditions of decomposition, by imposing one plausible theoretical constraint, Chen et al. proposed a new decomposition of the nucleon spin. A prominent feature of their decomposition is that each term is separately gauge-invariant, while allowing the decomposition of 