Do RG fixed points depend on the 'rescaling' convention at every RG step?

Question

In statistical mechanics, the RG procedure for a theory with a scalar field $\phi$ and cutoff $\Lambda$ is defined in three steps.

1. Integrate out all the degrees of freedom with momentum over $\Lambda/b$.
2. Rescale all distances back down by $b$.
3. Rescale the field by some other factor $\zeta(b)$.

Typically, the factor $\zeta$ is chosen so that the coefficient of the "kinetic" term $(\partial \phi)^2$ is always $1/2$. However, this seems to me to be an arbitrary choice.

What is particularly worrying is that the RG fixed points change depending on the rescaling procedure used. For example, if we chose to rescale to keep the coefficient of $\phi^4 (\partial^2 \phi^2)$ fixed, then we'd have a rather exotic looking fixed point. While I've been told that the physical predictions must come out exactly the same, I've never seen an explicit demonstration of this. Is the physics really independent of the choice of $\zeta(b)$, and if so, how can one see this?

Answer 1

The way to see that rescaling choice cannot affect the physics is the following:

• The critical exponents of a theory cannot depend on the choice of units you use to measure the fields. For example, consider theory where the field has units of Volts, denoted by $\phi(x)$. The divergence of the correlation function, the critical exponent $\nu$, will not depend on whether you formulate the theory in terms of $mV$ or $V$ or $MV$ units: $$ \langle V(x) V(x-y) \rangle \propto {1 \over |y|^{\nu}}$$. Under rescaling, or changing the units of my fields, $V'(x) = z V(x)$, I get a correlation function: $$ \langle \lambda^2 V(x) V(x-y) \rangle \propto {1 \over |y|^{\nu}} $$ $$ \langle V'(x) V'(x-y) \rangle \propto {1 \over |y|^{\nu}}$$

An important caveat:

• Some rescaling choices can be "sick". In other words, if you try to fix a relevant parameter near a critical point in your rescaling choice, it will be hard to see the physics of criticality because you keep rescaling away the coefficient that tune your system to be critical in the first place.

• To correctly diagnose critical behavior, it seems wise to then pick a rescaling convention that does not fix relevant operators. Another way to say it is if you are on the critical surface, you can flow to the RG fixed point. However, if keep rescaling to stay away from the critical surface, it may be very difficult to "flow" to the fixed point.

• There's a caveat to this. While the scale factor choice is arbitrary, the evolution of the scale factor to keep a particular term fixed is physical. For example, it is common to keep the kinetic coefficient canonically normalized, or in other words, choose $\zeta$ such that:

$$ \zeta(b)^2 K(b) = K(1)$$ where K is the kinetic coefficient.
In that case, the flow of $\zeta$ as a percent change over length scale is physical. In fact: $$ {\partial \ln(\zeta) \over \partial \ln(b)} = {d-2 \over 2} + \gamma_\phi$$

$\gamma_{\phi}$ is the anomalous dimension.

Physically, the anomalous dimension measures how much the kinetic term grows as you "zoom" outward.