# Ambiguity in Beta Functions (2-loop)

Beyond one-loop, the beta function of a QFT is scheme dependent. I would like to understand better this ambiguity.

The easiest thing to say is that you haven't calculated something physical, so of course it does not need to be scheme independent. However, the anomalous dimension of operators is I think an observable quantity since we may measure critical exponents in the lab, and the anomalous dimension results from the same sort of calculation.

Moreover, I can relate the beta functions to the trace anomaly. Schematically, $\langle \partial_\mu j_{dilation}^\mu \rangle=\langle T^\mu_\mu \rangle \approx \beta$ (See Peskin 19.5 for the case of QED). If i couple some field to the trace of $T$ I think I should be able to turn this anomaly into a cross-section for some process which would be measurable (think of ABJ anomaly and $\pi^0 \to \gamma \gamma$ for instance).

So the questions is:

1) Is it known how the terms in the beta function may differ between regularization schemes? If I try to calculate the couplings at the fixed point using different schemes, will I get the same answer (I am aware the location of the fixed point is not physical, but if I use the same field variables I could imagine this being scheme independent)? How may I see that although the beta function and location of the fixed point are ambiguous, the anomalous dimensions are not?

2) How would this ambiguity cancel out if I have a theory where I can turn the trace anomaly into a prediction of a scattering amplitude? Or can this simply not be done?

Any clarification or suggestion for references is appreciated.

• You cannot couple any normal fields to $T = T_{\mu \mu}.$ By construction (or definition) $T$ couples to the metric, i.e. under a variation $g \rightarrow g + \delta g,$ $\delta S = \int T_{\mu \nu} \delta g^{\mu \nu}.$ So the trace of $T$ measures how the theory responds to a rescaling. This is closely related to the dilaton scattering amplitudes in the Luty-Polchisnki-Rattazzi paper and the two papers by Komargodski et al. and Luty et al. about scale invariance that appeared last week. – Vibert Sep 19 '13 at 23:24

## 1 Answer

The beta function beyond 1-loop is scheme dependent, but the physical quantities you can extract from it are scheme independent (at least if you can compute the beta function at all order). For instance, even though the fixed point position is scheme dependent, the critical exponent are not. On the other hand, if you stop at a given order in the loop expansion, it is possible that physical quantities depends on the scheme.

In the case of the functional RG (such as Wilson-Polchinski or Wetterich's version), the physical quantities should be regulator independent, but as you make approximations, a spurious dependence on the regulator appears (for instance, when you change a parameter B of the regulator, the critical exponent eta depends on B). A way to cope with this, it to use the PMS (principle of minimum sensitivity), which tells you that when the real value of eta is given by the extremum of eta(B).

One last point : when one does a loop expansion (say a 4-epsilon expansion), one needs to resum the series, which involve some (non-physical) parameters. The final result should be independent of the resummation techniques, but because one knows only a finite number of terms, one gets again a spurious dependence on the resummation parameters.

• Thanks! Do you know any paper/textbook treatment where this is spelled out explicitly? I've never seen it addressed. – Dan Sep 23 '13 at 0:32
• Halas, no, at least not a reference which talks about all this. For instance, the epsilon expansion is always described as "controlled", but you will rarely hear anything about the dependence on the resummation parameters. In another context (fixed dimension, but perturbation in the interaction strength), you can have a superficial look at arXiv:1009.1492. In the context of the FRG, a technical discussion of the PMS and the dependence on the regulator is given in arXiv:hep-th/0211055. (Disclaimer : no self-citation) – Adam Sep 23 '13 at 14:52