In QFT under the $\overline{\text{MS}}$ subtraction scheme the renormalized mass $m_R$ becomes a $\mu$-dependent quantity which "runs" via a differential equation $$\frac{1}{m_R^2}\frac{d m_R^2}{d\log\mu}=\gamma_m(g,m_R^2,..),$$ what is the physical significance of this scale-dependence? With the renormalized coupling $g(\mu)$, one could interpret this scale dependence as giving a general indiciation of the strength of the interaction at a momentum scale $\mu$, however I don't see how I could apply this logic to $m_R^2$ since the physical mass of the spectrum is fixed at the pole mass and cannot depend on any momentum scale. How then should I interpret $m_R^2$?
Put another way, say we have a theory were the pole mass is non-zero but the renoramlized mass flows to zero in some limit of $\mu$, does this actually mean that the spectrum is massless in that limit? If so how can that be true since the pole mass remains non-zero?
