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Recently, seven produced hadron species have been used to construct multiple hadron sets with given differences in the net electric charge (∆q) and strangeness (∆S) between the two sides. A nonzero directed flow difference △v₁has been proposed as a consequence of the electromagnetic field produced in relativistic heavy ion collisions. Previously, we have shown with quark coalescence that Av1 and the slope difference △v′₁depend linearly on both △qand ∆Swith zero intercept. Here we emphasize that a two-dimensional function or fit is necessary for extracting the △q- and △S-dependences of △v₁. On the other hand, a one-dimensional fit gives a different value for the slope parameter of the ∆q- or ∆S-dependence. Furthermore, a one-dimensional fit is incorrect because its slope parameter depends on the arbitrary scaling factor of a hadron set and is thus ill-defined. We use test data of △v₁to explicitly demonstrate these points. 

Abstract We study the production of$$D^0$$ $D^0$meson inp+pandp-Pb collisions using the improved AMPT model considering both coalescence and independent fragmentation of charm quarks after the Cronin broadening is included. After a detailed discussion of the improvements implemented in the AMPT model for heavy quark production, we show that the modified AMPT model can provide a good description of$$D^0$$ $D^0$meson spectra inp-Pb collisions, the$$Q_{\textrm{pPb}}$$ $Q_{\text{pPb}}$data at different centralities and$$R_{\textrm{pPb}}$$ $R_{\text{pPb}}$data in both mid- and forward (backward) rapidities. We also studied the effects of nuclear shadowing and parton cascade on the rapidity dependence of$$D^{0}$$ $D^0$meson production and$$R_{\textrm{pPb}}$$ $R_{\text{pPb}}$. Our results indicate that using the same strength of the Cronin effect (i.e$$\delta $$ $\delta$value) as that obtained from the mid-rapidity data leads to a considerable overestimation of the$$D^0$$ $D^0$meson spectra and$$R_{\textrm{pPb}}$$ $R_{\text{pPb}}$data at high$$p_{\textrm{T}}$$ $p_{\text{T}}$in the backward rapidity. As a result, the$$\delta $$ $\delta$is determined via a$$\chi ^2$$ ${\chi }^2$fitting of the$$R_{\textrm{pPb}}$$ $R_{\text{pPb}}$data across various rapidities. This work lays the foundation for a better understanding of cold-nuclear-matter (CNM) effects in relativistic heavy-ion collisions.