Revisiting the Wess-Zumino-Witten Term in Nuclear and Quark Matter under Magnetic Fields and Rotation

We study anomalous Wess-Zumino-Witten terms associated with the chiral anomaly in dense QCD matter under magnetic fields and rotation. By introducing electromagnetic, baryon number, and isospin background gauge fields, we write down the topological couplings of neutral mesons for the $N_f=2$ and $N_f=3$ cases. The resulting terms contain characteristic contributions proportional to $\vec{B}\cdot \vec{\mathrm{\nabla }}\phi$ and $\vec{\mathrm{\Omega }}\cdot \vec{\mathrm{\nabla }}\phi$, where $\phi$ denotes ${\pi }^0$, $\eta$, or ${\eta }^{\mathrm{\prime }}$. These results are relevant to chiral soliton lattices in dense rotating matter.