"Running with momentum $p$" v.s. "running with renormalization scale $mu$"

The renormalized charge/coupling in QFT is usually phrased as renormalization scale $mu$ dependent $alpha(mu)$ in the renormalization group setting. But can we take the more elucidating angle of "momentum $p$ or $p^2$ dependent" $alpha(p^2)$? The renormalization scale $mu$, as it is taught in most QFT text books (often introduced un-intuitively as the scale parameter in dimensional regularization), is baffling to new learners rather than clarifying.

Let’s shed some light on the renormalization scale $mu$ with a simple example of
$$
x(t) = ln(t/t_0) + x_0.
$$
(in the physics context, translated to
$$
alpha(p) = ln(p/mu) + alpha_0
$$
with $alpha$ being the coupling constant, $p$ being momentum , $mu$ being renormalization scale, respectively)

The variable $x$ is the solution to a first-order differential equation ($beta$-function) of
$$
beta (x) = dx(t)/dln(t) = 1,
$$
with the initial condition
$$
x(t)|_{t = t_0} = x_0.
$$

The "running with renormalization scale $mu$" approach is tantamount to regarding $x(t, t_0, x_0)$ as the solution to an alternative differential equation (differentiating against the initial condition point $t_0$, which is $mu$ in physics context)
$$
beta ‘(x) = dx(t_0)/dln(t_0) = -1,
$$
with the initial condition
$$
x(t_0)|_{t_0 = t} = x_0.
$$
Is this wicked and naughty way of looking at the original differential equation really helpful (or just add to the confusion)?

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Let’s take a look at another example of self-energy $Sigma(not{p})$ in the fermion propagator
$$
G = frac{i}{not{p}-m_0 – Sigma(not{p})+iepsilon}
$$
where self-energy $Sigma(not{p})$ can be generally expressed as
$$
Sigma(not{p}) = a(p^2) + b(p^2)not{p}.
$$
To simplify our discussion, let’s assume that (which means there is no wave function renormalization)
$$
b(p^2) = 0.
$$
If we further expand self energy as
$$
Sigma(p^2) = a(p^2) = m_0′ + c_1p^2 + c_2p^4 + …
$$
we will find out that $m_0’$ is divergent, while $c_1$ and $c_2$ are finite. The whole (mathematically shady) mass renormalization business is hinging on the assumption that
$$
m_r = m_0 + m_0′
$$
is finite (or equivalently, $m_0 = m_r – m_0’$, regarding $m_0’$ as mass counter term), so that the fermion propagator
$$
G = frac{i}{not{p}-m_0 – Sigma(p^2)+iepsilon}
$$
$$
= frac{i}{not{p}- (m_r + c_1p^2 + c_2p^4 + …) + iepsilon}
$$
is finite and well defined.

Note that while $m_0$ and $m_0’$ are divergent, finite $m_r$ (it’s not the physical pole mass $m_p$, unless $c_1= c_2 = 0$) can be determined by experiment.

On the other hand, the finite coefficients $c_1$ and $c_2$ can be calculated ($dSigma(p^2)/dp^2$ and $d^2Sigma(p^2)/(dp^2)^2$ are finite, is that cool! It has to do renormalizability/local counter terms of renormalizable QFT), so that we know how self-energy $Sigma(p^2)$ (or more precisely, the finite and well defined $m_0 + Sigma(p^2) = m_r + c_1p^2 + c_2p^4 + …$) runs with momentum/energy $p^2$.

The whole discussion above about running of $Sigma(p^2)$ does NOT depend on the renormalization scale $mu$ at all!

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Update:

"Can you use renormalization schemes without $mu$"? Surely one can, without resorting to any kind of RG (be it Wilsonian/Polchinskian/Wetterichian RG or perturbative QFT RG). Just resume the geometric series (that is how Landau pole was found by Landau!) of Feynman diagrams a la, 1/N (t’Hooft), rainbow/ladder approximation, etc. There are tons of alternative ways of achieving this so called RG enhancement without invoking RG accompanied by the illusive $mu$.