# running couplings in the on-shell scheme

I am trying to understand exactly in which way running coupling constants arise in the "on-shell" renormalization scheme and which variations of this scheme are useful if I want a clean, physical interpretation of the renormalized couplings and the renormalized mass.

For sake of definiteness, lets take $\phi^4$-theory and look at the quartic vertex. I am defining my renormalized coupling by the requirement that the amputated 4-point Green's function satisfies $$G_4(s_0,t_0,u_0) = -i\lambda$$ at $s_0 = 4m^2$, $t_0 = u_0 = 0$, where $\lambda$ is the quartic coupling.

Now, when I compute the running of the coupling $\lambda$, I find that it doesn't run at all. This makes sense to me; after all, the scale at which I define it is absolutely fixed.

This means, if I want to see a running of my coupling constant, I should define it within an "on-shell-like" scheme at some arbitrary scale $\mu_R$, such as $$G_4(s_0,t_0,u_0) = -i\lambda$$ with $s_0 = t_0 = u_0 = \frac{4\mu_R^2}{3}$, correct?

This makes sense to me, physically: $\lambda$ is just the "effective" coupling strength I would observe in the process at the scale $\mu_R$, and is supposed to change as I repeat the experiment at a different $\mu_R$.

Now, I am wondering how to sensibly implement the analog of the second condition (in a clear, experimentally accessible way) for a cubic vertex or the propagator.

For the $\phi^3$-interaction, which I can interpret as a decay process $\phi\rightarrow\phi\phi$: if $p_1$ is the momentum of the incoming $\phi$, and $p_2$, $p_3$ the momenta of the outgoing particles, does it make sense to demand that $$G_3\big\rvert_{p_1^2 = \mu_R^2,p_2 = p_3} = -i\alpha$$ if we take $\alpha$ to be the cubic coupling? (I imagine that with this definition, the running of $\alpha$ would give me information about how the coupling behaves as I sweep over the offshellness of the incoming $\phi$, which is possibly an intermediate state in some bigger diagram)

I am having trouble picturing an equally accessible definition of the renormalized mass entering into the propagator. If we write the full propagator as $$\frac{i}{p^2-m^2-\Sigma(p^2)}$$ a common prescription seems to be (at least according to Peskin & Schröder) to require a pole of unit residue at $p^2 = -\mu_R^2$. It is not obvious to me why using a spacelike momentum is a sensible way to do it.

Can somebody enlighten me or provide an alternative, in line with the other two examples?