Is there a hard gluonic contribution to the first moment of g₁?

We show that the size of the hard gluonic contribution to the first moment of the proton’s spin‐dependent structure function g₁ is entirely a matter of the convention used in defining the quark distributions. If the UV regulator for the spin‐dependent quark distributions respects the gauge invariance of Green’s functions (allows shifts of loop momenta) and respects the analyticity structure of the unregulated distributions, then the hard gluonic contribution to the first moment of g₁ vanishes. This is the case, for example, in dimensional regularization. By relaxing the requirement that the regulator allow shifts of loop momenta, we are able to obtain a nonvanishing hard gluonic contribution to the first moment of g₁. However, the first moments of the resulting quark distributions correspond to matrix elements that are either gauge variant or involve nonlocal operators and, hence, have no analogue in the standard operator‐product expansion.