After the renormalization procedure, fields will gain an anomalous dimension, $\gamma$, which means that their scaling dimension will be different from what we would guess from the dimensional analysis. My question is whether this means that the action will no longer be dimensionless and if so what are the consequences?
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$\begingroup$ Wait, what? Seems pretty sick. Is it in the sense of this paper? arxiv.org/abs/1304.4131 $\endgroup$– user257090Commented May 23, 2020 at 21:42
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$\begingroup$ Do you mean the bare action or the effective action? $\endgroup$– Prof. LegolasovCommented May 23, 2020 at 23:11
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1 Answer
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Yes, the action remains dimensionless. The "failure" of dimensional analysis is due to the fact that dimensional quantities that could be neglected under certain conditions (e.g., at a Gaussian fixed point in the renormalization group flow) cannot be neglected at other non-trivial fixed points in the renormalization group flow. The anomalous dimension comes from the contribution of these neglected quantities to the scaling.
For instance, if we have a quantity that has dimensions of length to some power, say $[Q] = L^d$ for example, then by dimensional analysis this quantity must be some function of the dimensional parameters of the model. For example, in the Ising model we might take the independent the length scales to be the correlation length $\xi$ and the lattice spacing $a$. Thus, it must be that $Q = \xi^d f(a/\xi)$ for some function $f(\cdot)$ that cannot be determined by dimensional analysis. The usual argument is that near a critical point (or in the continuum limit), the lattice spacing is negligible compared to the correlation length of the system, $a/\xi \ll 1$, so we can set $Q = \xi^d f(0)$.
The key assumption here, however, is that we can take the limit $\lim_{x \rightarrow 0} f(x) = f(0)$. It turns out that this limit exists only under certain conditions (such as in dimensions $d > 4$ for the Ising universality class). In general, this limit does not exist, and we actually observe the asymptotic behavior $$\lim_{x \rightarrow 0} f(x) \sim x^{-\gamma} g(x)$$ for some power $\gamma > 0$ and regular function $g(x)$ that is finite at $x = 0$. Thus, $Q = \xi^d f(a/\xi) \sim \xi^{d+\gamma}$ and $\gamma$ appears as an anomalous dimension. However, note that the $\sim$ hides the fact that there is still a factor of $a^\gamma$ in the exact expression---it has just been dropped because it is a constant independent of the temperature (or other variable that tunes the correlation length). Thus, $Q$ still has dimensions of $L^d$, even at a critical point in which the scaling as a function of the correlation length is anomalous.
For more details, see Goldenfeld's book, Lectures on Phase Transitions and the Renormalization Group, in particular Chapter 7.
