This question sounds ridiculous, but bear with me. I am having a hard time reconciling the following two facts:

  • Classical global symmetries can become anomalous upon quantization, and the anomalous non-conservation of the associated current is a one-loop effect (since the effect is based on the path integral measure failing to be invariant, and the transformations in the path integral measure contain no powers of $\hbar$, which is a loop-counting parameter).
  • The non-vanishing of the beta function in a classically scale-invariant theory constitutes anomalous dilitation symmetry violation.

These two facts seem to imply that if the beta-function of a classically scale-invariant theory vanishes at one-loop, then it vanishes at all orders.

This seems obviously not true, otherwise people wouldn't have spent so long proving that $\mathcal{N}=4$ super-Yang-Mills is conformally invariant to all orders, since the 1-loop calculation is relatively easy to check. So, my question is "Where do the above statements fail to apply?"


The path integral measure is not one-loop exact in general. In the example of the anomaly for the chiral symmetry, the path integral measure for the fermions are one-loop exact because the integral is gaussian on the fermions $\psi$ and $\bar\psi$ (with the gauge field $A_{\mu}$ fixed, as well as any scalar in the Yukawa coupling). The anomaly comes from the fact that there is a conflict in trying to preserve both the gauge symmetry and the chiral symmetry in the process of regularization and renormalization of the measure. If you regulate the measure in order to preserve the gauge symmetry, then instead of a smooth cut-off like $f(\displaystyle{\not}\partial)$ you are going to use $f(\displaystyle{\not}\partial+i\displaystyle{\not}A)$ regulator, that breaks the chiral symmetry. After the shift of $k$ you get terms involving $\displaystyle{\not}A$ multiplying $\gamma_5$ what makes the trace in the space of fields being nonzero. For a general regulator, not necessarily a gauge invariant one, the finite counterterms that you are going to add in order to recover the gauge symmetry at the limit will also break the chiral symmetry. You can show that there is no available set of finite counterterms that preserve both the gauge symmetry and the chiral symmetry.

The beta function measures the anomaly of the conformal symmetry. It is the coefficients in an operator expansion for the trace of the energy-momentum tensor. In free theories there is no anomaly (locally), so it means that it is possible to define the path integral measure that is invariant under the conformal symmetry. Once we introduce interactions plus renormalization conditions that guarantee that the theory is not trivial at some particular scale, this renormalzation conditions plus interactions will require to modify the path integral measure in such way that it is no longer invariant under conformal transformations, up to some few exceptions called Conformal Field Theories (e.g. N=4 d=4 Super Yang Mills).

The main point here is that the path integral measure is not entirely fixed. You need to regularize and renormalize it. What fix a unique path integral measure is the renormalization conditions, that presents as requirements for preserving some symmetry, or that some three point-function at some energy scale is finite (the physical coupling), or fixing the location of some poles in the two point function (physical mass), etc. Your loop couting only works for the case of gaussian integrand, where there is no need to introduce more loop dependence on the measure to account for the renormalization conditions. In the case of an interacting theory usually there is a need to introduce higher loop dependence on the definition of the measure to accommodate the renormalization conditions.

This higher loop dependence are nothing more than the higher loop counter-terms that you add in the action to make sense of the theory. Note also that the counter-terms also contribute to the one-loop dependence of the measure. There is nothing prohibiting a theory to have no one-loop counter-terms but having higher loops counter-terms, i.e. being one-loop finite but diverges at higher loops.


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