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In Jaume Gomis's lecture 5 on CFT at Perimeter Institute, he says (at 27:40 minute mark) that the beta function, classically, of the $m^2$ parameter in massive $\lambda \phi^4$ theory is

$$\beta(m^2) = -2m^2\,.$$

The intuitive reasoning he gives is that since the dimension of $m^2$ is 2, then the beta function should be -2 times m-squared.

I always thought that the beta functions for parameters of the Lagrangian only start at one-loop and are classically zero. What is the rigorous definition of a beta function (including the classical part)?

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The theory is (even classically) not scale invariant. Just by dimensional analysis, you can note that the scalar field has scaling dimension 1, and the mass (as the name suggests) must also have a scaling dimension of 1. So $m^2$ has a ascaling dimension of 2, which suggests the RG equation which you've written in the question. That essentially says that the $m^2$ parameter decreases exponentially, but with scaling dimension 2 as you go to the UV.

Only when you have couplings which are classically dimensionless, do the RG equations have no tree level contributions. eg: coefficients of dimension 4 operators: like $\lambda$ in $\lambda \phi^4$ or $g$ in $g \phi \bar{\psi} \psi $

FYI: Tree level is equivalent to classical field theory and loop contributions are like quantum corrections. You can see that by restoring powers of $\hbar$ in the path integral.

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  • $\begingroup$ Thanks! but I already understand it qualitatively. I want to know the formal definition: why the minus sign? how to connect it to the 't Hooft parameter $\mu$. And it's relation to the 'anomalous dimension' $\gamma(m^2)$... Thanks! $\endgroup$
    – QuantumDot
    May 17, 2013 at 14:42
  • $\begingroup$ By the `t Hooft parameter $\mu$, do you mean the anomalous scale you get while doing dim reg? That is a quantum phenomenon (since you get it from dim reg on loop integrals). In the classical theory, $m^2$ does not have anomalous scaling. Quantum (loop) corrections to the scaling dimension are what we call anomalous. The minus sign is kinda by definition, because something with a positive mass dimension increases as you flow to the IR and decreases as you flow to the UV. Since increasing $\Lambda$ corresponds to going towards the UV, we need a negative sign. $\endgroup$
    – Siva
    May 17, 2013 at 16:17
  • $\begingroup$ Yes $\mu$ is dim-reg parameter. So, if I understand you correctly? Is it true, in all cases the classical beta function for a parameter, $x$, of mass-dimension $[x]$ is ALWAYS $\beta(x) = -[x]\times x$? Thank you again! So how do I write out the full thing with quantum corrections (involving the anomalous part $\gamma$)? Is it $\beta(x) = -[x]\times x + \gamma(x)$? Is that the exact correct formula? $\endgroup$
    – QuantumDot
    May 18, 2013 at 2:46
  • $\begingroup$ Yes, the loop-corrected RGE will become (small correction to what you wrote) $\beta(x) = - ([x]+\gamma(x)) x$. $\gamma(x)$ is called the anomalous scaling dimension, since it's a correction to $[x]$. Here, the RHS shows only the leading order in $x$; in general there could also be higher order terms. $\endgroup$
    – Siva
    May 18, 2013 at 3:16

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