RG and Perturbation Theory
In this chapter we consider how to implement RG ideas in the context of perturbation theory, and in a way which will be easier to generalize to other theories including gauge theories which we treat in the following chapter.
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RG and Perturbation Theory
In this chapter we consider how to implement RG ideas in the context of perturbation theory, and in a way which will be easier to generalize to other theories including gauge theories which we treat in the following chapter. We have seen that it is the remarkable focussing properties of the RG flows and the consequent universality that allows one to formulate theories in terms of simple actions. The action only needs to include a kinetic term and the relevant interactions; in particular, all the irrelevant couplings can be taken to vanish. The argument runs as follows: in order to take a continuum limit, we need to let the “bare couplings” gi = g˜i (μ ) depend on the cut off μ in such a way that the phenomena at a physically relevant scale μ < μ remains fixed. This is guaranteed if we use the RG equation (1.2) and take μ → ∞ with the couplings gi = gi (μ ) following the RG flow into the UV along a renormalized trajectory. However, this would mean fine tuning all the irrelevant couplings. As we described in Chap. 1, there is no need to do this. It is sufficient to keep only the relevant couplings g2 (μ ) and g4 (μ ) in addition to the kinetic term and choose all the irrelevant couplings to vanish. The resulting behaviour of all the couplings, that is g˜ 2 (μ ) = g2 (μ ), g˜ 4 (μ ) = g4 (μ ), g˜ 2n (μ ) = 0, n > 2,
(3.1)
is not an RG flow but rather is a projection of the RG flow onto the finite-dimensional subspace (g2 , g4 ) of the relevant couplings. At low energies the flows from the subspace (3.1) are drawn to the renormalized trajectory and the true continuum theory is recovered. In particular, irrelevant couplings are generated but they are not free parameters in the low energy theory, on the contrary, they are fixed by the relevant couplings. The RG flow is illustrated below for the scalar field theory in dimension d = 3 with the magnitude of the irrelevant couplings exaggerated.
T. J. Hollowood, Renormalization Group and Fixed Points, SpringerBriefs in Physics, DOI: 10.1007/978-3-642-36312-2_3, © The Author(s) 2013
25
26
3 RG and Perturbation Theory g2
renormalized trajectory
theories with relevant coupling only
g6,8,...
g4
In many situations in QFT, perturbation theory is a powerful tool for uncovering the RG flows of the relevant couplings. In particular, even the one-loop approximation yields a wealth of insight. However, in the context of the RG, the use of perturbation poses an important question. Perturbation Theory in Which Coupling? According to the RG, the couplings flow with the scale of interest μ and this poses the question as to which coupling should be used as the perturbative parameter? For example, in d = 4, g4 runs according to (2.26). So the coupling at the scale μ is an infinite perturbative series of the coupling at μ: g4 (μ ) =
1+
g4 (μ) 3 g (μ) log μμ 16π 2 4
= g4 (μ)−
3 μ g4 (μ)2 log +· · · . (3.2) 16π 2 μ
If we could completely sum all of the perturbative expansion in g4 (μ), then the resulting physical observables would, b
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