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I am struggling against the running of coupling constants in QFT. I will consider $\lambda \phi^4_4$-Theory in order to express my point of view and to ask for further explanations.

Since most of the expositions on textbooks and on the web are very synthetical and often refer to things that have been explained before, I tried to sum up the whole picture in order to fix the notation and to clarify all possible obscure points. Please feel free to correct whatever is wrong in the following lines.


Recap: Dimensional Regularization in the Minimal Subtraction Scheme

In a renormalized theory, we can start off with the bare Lagrangian \begin{equation} \mathcal{L} = \frac{1}{2} \partial_\mu \phi_0 \partial^\mu \phi_0 - \frac{1}{2} m_0^2\phi_0^2 - \frac{\lambda_0}{4!} \phi_0^4, \end{equation} and then, in the dimensional regularization framework, get it in the "dressed terms + counterterms form" in dimension $D$, that is \begin{equation} \begin{split} \mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2\phi^2 &- \mu^{4-D}\frac{\lambda}{4!} \phi^4 + \\ & + \frac{1}{2} \delta_Z \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} \delta m^2\phi^2 - \frac{\delta_\lambda}{4!} \phi^4, \end{split} \end{equation} where $\mu$ is an arbitrary mass parameter, with (being $Z = 1 + \delta_Z$) \begin{equation} \phi_0 = Z^{1/2} \phi, \quad\quad m_0^2 = Z^{-1} \left( m^2 + \delta m^2 \right), \quad\quad \lambda_0 = Z^{-2} \left( \mu^{4-D} \lambda + \delta \lambda \right). \end{equation}

Finally, we can get finite proper vertices $\tilde{\Gamma}^{(2)}$ and $\tilde{\Gamma}^{(4)}$ by setting in the Minimal Subtraction Scheme \begin{equation} \begin{split} \delta_Z & = -\frac{\lambda^2}{256 \pi^4} \frac{1}{24\epsilon} + O\left(\lambda^3,\epsilon\right), \\ \delta m^2 & = \frac{m^2}{2} \left[ \frac{\lambda}{16 \pi^2} \frac{1}{24 \epsilon} + \frac{\lambda^2}{256 \pi^4} \left( \frac{1}{\epsilon^2} - \frac{1}{2\epsilon} \right) \right] + O\left(\lambda^3,\epsilon\right), \\ \delta_\lambda & = \mu^{2\epsilon} \frac{3 \lambda^2}{32 \pi^2} \frac{1}{\epsilon} + O\left(\lambda^3,\epsilon\right), \end{split} \end{equation} with $2\epsilon=4-D$. In this way, renormalized proper vertices in dimension $D=4$ (i.e. with $\epsilon=0$) have the form \begin{equation} \begin{split} \tilde{\Gamma}_R^{(2)}\left(k^2\right) & = k^2 - m^2 - M_R^2 \left(k^2;\lambda, m^2;\mu\right), \\ \tilde{\Gamma}_R^{(4)}\left(k_i\right) & = - \lambda - \Sigma_R \left(k_i;\lambda, m^2;\mu\right), \end{split} \end{equation} where $M_R^2$ and $\Sigma_R$ represent the (now) finite 2- and 4-legs loop expansions.


Problem: Running Couplings and Observable Quantities

After this long introduction, I can express my problem. What I have understood is that, because the physical measurements coincide with the "observable" terms \begin{equation} \begin{split} m^2_{\mathrm{eff}} & = m^2 + M_R^2 \left(k^2;\lambda, m^2;\mu\right), \\ \lambda_{\mathrm{eff}} & = \lambda + \Sigma_R \left(k_i;\lambda, m^2;\mu\right), \end{split} \end{equation} since they must be independent of the arbitrary scale $\mu$, we must have $m^2=m^2\left(\mu\right)$ and $\lambda=\lambda \left(\mu\right)$ in such a way as to have the right-handsides independent of $\mu$.
If we expand the bare parameters $Z$, $m^2_0$ and $\lambda_0$ in Laurent series, then we can compute the $\beta$-functions and determine the functions $m^2=m^2\left(\mu\right)$ and $\lambda=\lambda \left(\mu\right)$ (I did it and I got the correct behaviours).

The picture I have described so far is quite clear to me, but maybe I am wrong at some point. If instead what I wrote is correct, then I do not understand if and why the "measurable" parameters $m^2_{\mathrm{eff}}$ and $\lambda_{\mathrm{eff}}$ do or do not run, i.e. if they modify their values as the energy scale of the experiment is changed (noticing the dependence on $k^2$ and $k_i$).

Moreover, I cannot figure out how the parameter $\mu$ can be associated to the energy scale of an experiment. In other words, if we want to compute a scattering amplitude, which value of $\mu$ do we have to choose to "select" the proper mass $m^2(\mu)$ and the proper coupling $\lambda(\mu)$ and why? I cannot see any connection from the above reasoning.

Thank you in advance for your help.

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