Renormalizability of leading order covariant chiral nucleon-nucleon interaction

In this work, we study the renormalization group invariance of the recently proposed covariant power counting in the case of nucleon-nucleon scattering [Chin. Phys. C 42 (2018) 014103] at leading order. We show that unlike the Weinberg scheme, renormalizaion group invariance is satisfied in the begin{document}$^3P_{0}$end{document} channel. Another interesting feature is that the begin{document}$^1S_{0}$end{document} and begin{document}$^3P_{1}$end{document} channels are correlated. Fixing the relevant low energy constants by fitting to the begin{document}$^1S_{0}$end{document} phase shifts at begin{document}$T_mathrm{lab.}=10$end{document} and 25 MeV with cutoff values begin{document}$Lambda = 400-650$end{document} MeV, one can describe the begin{document}$^3P_{1}$end{document} phase shifts relatively well. In the limit of begin{document}$Lambdarightarrow infty$end{document}, the begin{document}$^1S_0$end{document} phase shifts become cutoff-independent, whereas the begin{document}$^3P_{1}$end{document} phase shifts do not. This is consistent with the Wigner bound and previous observations that the begin{document}$^{3}P_1$end{document} channel is best treated perturbatively. As for the begin{document}$^1P_{1}$end{document} and begin{document}$^3S_{1}$end{document}-begin{document}$^3D_{1}$end{document} channels, renormalization group invariance is satisfied.