# Renormalization group equation, the Callan-Symanzik equation, and renormalization group flow

I am learning about the renormalization group and I am getting confused on some terminology.

For the massless $$\phi^4$$ theory the Callan-Symanzik equation is: $$\big[ M \frac{\partial}{\partial M} + \beta(\lambda)\frac{\partial}{\partial \lambda} + n\gamma(\lambda)\big]G^{(n)}(\{x_i\}; M \lambda) = 0$$ where $$M$$ is the renormalization/momentum scale, $$\lambda$$ the coupling, and $$G^{(n)}$$ the $$n$$-point Green's function.

What is the difference between the above and the renormalization group equation? Are they the same?

I am also confused on the renormalization group flow. From what I understand, it is the flow/trajectory of the coupling constant as the energy scale changes. Thus it should map $$M$$ to $$\lambda$$. Is this map obtained from the $$\beta$$ function? If so, how?

Note I have already seen this question: Renormalization Group Flow. It was helpful, but I am still unclear on the above.

• WP, and RG. Commented Apr 5 at 1:49
• @CosmasZachos I have taken a look at the Wikipedia pages but they are not very clear. For example the renormalization group wiki says the $\beta$ function is the renormalization group equation. However the renormalization group equation wiki redirects to both the Callan-Symanzik equation and the $\beta$ function. So I am not clear on what exactly is meant by the renormalization group equation. Commented Apr 5 at 2:23

You must stick to your QFT text, or, if confusing, to Chapter 19 of From Classical to Quantum Fields, by L Baulieu, J Iliopoulos, & R Seneor, (2017, Oxford University Press) .

What is the difference between the above and the renormalization group equation? Are they the same?

Yes and no: formally they are the same, because they were derived in the same context with the same methods. Their symbols, however, mean different things.

The Callan—Symanzik equation is the actual Ward identity of broken scale invariance, involving a physical mass altered among different theories (on Mt Olympus or somewhere), not the renormalization scale.

By contrast, the RG equation describes the response of the Green's functions on variations of an unphysical parameter all the while describing the same physical theory, cf Gell-Mann―Low.

Varying the physical energy in an experiment allows one to be cavalier about which one to apply, in similar ways in applications.

• Thank you for your answer. I think I will have to read further and consult a few other textbooks to fully understand everything. Commented Apr 5 at 17:13