A further study on the renormalization group aspect of perturbative corrections

I perform a further study regarding a renormalization-group (RG) issue -- which concerns a wide variety of the so-called perturbative power counting under effective field theories (EFT) -- as pointed out by A. M. Gasparyan and E. Epelbaum [Phys. Rev. C 107, 034001 (2023)]. I show that the issue could originate from a wrong power counting, or from treating those incomplete, truncated amplitudes beyond the degree to which they should be trusted. Meanwhile, under EFT principles, one should always associate the result with an uncertainty that is adequate to its EFT order. One way to accommodate this is to encode its effect in a more general form of contact terms. In this regard, no RG issue is found in the $^3$P$_0$ nucleon-nucleon scattering under the Long and Yang power counting. In contrast, the RG issue under Weinberg's pragmatic proposal remains a problem even with uncertainty taken into account.
Comment: 9 pages, 6 figures