Chiral properties of the nucleon interpolating current and $\theta$-dependent observables

We revisit the chiral properties of nucleon interpolating currents, and show that of the two leading order currents $j_1$ and $j_2$, only two linear combinations $j_1\pm j_2$ transform covariantly under the anomalous $U(1)_A$ symmetry. As a result, calculations of quantities which vanish by symmetry in the chiral limit may produce unphysical results if carried out with different linear combinations of the currents. This includes observables such as electric dipole moments, induced by the QCD parameter $\theta$, and the $\theta$-dependence of the nucleon mass. For completeness, we also exhibit the leading order results for nucleon electric dipole moments ($d_{n,p}$) induced by $\theta$, and the nucleon magnetic moments ($\mu_{n,p}$), when calculated using QCD sum rules for both the covariant choices of the nucleon interpolating current. The results in each channel, conveniently expressed as the ratios, $d_{n,p}/\mu_{n,p}$, are numerically consistent, and reflect the required physical dependence on $\theta$.