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In this work, we apply the subtracted kernel method (SKM) to the chiralnucleon-nucleon ($NN$) interaction in the $^3P_0$ channel up tonext-to-next-to-leading-order ($NNLO$). We demonstrate, by explicit numericalcalculations, that the SKM procedure is renormalization group invariant underthe change of the subtraction scale provided the driving-term of the subtractedscattering equation is evolved through a non-relativistic Callan-Symanzik(NRCS) flow-equation.

Renormalization Group Invariance in the Subtractive Renormalization Approach to the NN Interactions