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Suppose you have $\lambda \phi^3$ theory, and that you renormalize the 2 and 3 point one-particle irreducible graphs, $\Pi_R(p^2)$ and $\Gamma_R(p_1,p_2,p_3)$, by Taylor expanding about $p=\mu_0$ for the 2-pt function, and $p_1=\mu_1$, $p_2=\mu_2$, $p_3=\mu_3$ for the 3-pt function, and using counter-terms to cancel the first few divergent terms of those Taylor series.

Then does the renormalization group equation become multi-dimensional:

$ \frac{\partial}{\partial \mu_i}\left(\text{bare quantity}\right)=0, \quad i=0,1,2,3. $

where the derivative is now a partial derivative due to the different i's?

If this is the case, then in a dimensional regularization scheme where there is only one value of $\mu$ for both the 2 and 3 pt functions, should we view this as really having two different values of $\mu$ ($\mu_1$ and $\mu_2$) and then taking derivative of $\lambda_R$ and $m_R$ along the trajectory $\mu_1=\mu_2=\mu$?

Similarly, would we have to define multiple beta functions, $\beta_i=\beta_i(\lambda_R, m_R,\mu_0,\mu_1,...)=\mu_i\frac{\partial \lambda_R}{\partial \mu_i}$ and $\delta_i=\delta_i(\lambda_R, m_R,\mu_0,\mu_1,...)=\mu_i\frac{\partial m_R}{\partial \mu_i}$?

When using dimensional regularization, the same $\mu$ shows up for both the 2 and 3pt functions. However, if using a Taylor expansion method like above, there doesn't seem to be a way to relate the 2 and 3 pt functions with the same parameter.

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When renormalizing a theory you only introduce a single scale, $\mu$, regardless of the counter terms which require renormalizing. The reason this is a valid approach is because when a process occurs it has one typical energy scale. For a particle decay it will be the mass of the particle decaying, for a $2 \to 2 $ process it will be the center of mass energy. Its true that one of the particles may have a very different energy then the other particles, but the probability of this happening is small and to a good approximation a process occurs at a single energy scale.

As you may imagine, this simplifies the work needed to renormalize a theory.

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