Freedom in the Choice of a Beta Functions in RG

Question

Assume we're given a certain statistical model, say the infinite range Ising model

\begin{equation} H_{N}\{\vec\sigma_{N}\}~=~ - \frac{x_{N}}{2N} \sum_{i,j =1}^{N} \sigma_{i} \sigma_{j} \end{equation}

with $N\rightarrow\infty$, where $N$ is the number of spins, $\sigma_{i} \in \{ \pm 1 \}$ for all $i\in\{ 1..N \}$, $\vec\sigma_{N} = (\sigma_{1}, ... , \sigma_{N})$ and $x_{N}:=\beta J$ with J being the coupling strength between the spins. In addition $\beta := \frac{1}{k_{B}T}$ with $k_{B}$ the Boltzmann constant and T the temperature.

In the limit $mx\ll1$ one can implement a renormalization procedure of the form $Z_{N}(x_{N}) = \exp(-f_{N}(x_{N})) \cdot Z_{N-1}(x_{N-1})$ where intuitively one step in the RG process consists of the reduction of one spin site. In the limit as $N\rightarrow\infty$ one can write down the flow of the coupling constants as a differential equation of the form $\frac{dx}{dl} = (-x + x^{2})$ where the vectorfield $\beta(x):=-x + x^{2}$ is usually called the beta-function.

I was told that the main "physics" (meaning most likely critical exponents) of the RG is contained entirely in the critical points of the beta function, i.e. the x for which $\beta(x)=0$. In particular I was told that one can change the $\beta$-function "arbitrarily" as long as the critical points are the same. So in principle one could use $\beta'$ instead of $\beta$ where $\beta'$ has the same zeros as $\beta$ and obtain the same physics. In particular I could for example to a rescaling, i.e. choose $\beta' := \lambda \cdot \beta$ for some real number $\lambda$.

I don't really understand why this is true? In the case of one coupling constants it's a bit understandable since one only decreases the "speed" so to say as one moves from one critical point to another. But when one has more coupling constants in the RG process (such as a magnetic field) I don't see why this should be the case? Is there a certain general freedom in the choice of a beta-function?

I'm looking forward to your responses:)

Answer 1

There is, because there is not a unique way to define a renormalised coupling constant. In general, if you have a system with one bare coupling $g_0$, you define a renormalised coupling in a sensible way, i.e. such that $g = g_0 + \text{loop/quantum corrections} \sim \mathcal{O}(g_0^2).$ However, there is not a unique way to renormalise a theory. In a different scheme, you could shift the renormalisation conditions in a way that $g' = g_0 + \text{other loop/quantum corrections} \sim \mathcal{O}(g_0^2).$ It is best to think of some examples you know from field theory ($\phi^4$ at different momenta, QED in different schemes etc) to clarify this.

If the running depends on $x$, then for the first coupling $\beta(g) \equiv \partial g/\partial x,$ for the second one $\beta(g') \equiv \partial g'/\partial x = \partial g'/\partial g \, \beta(g).$ So if $g'(g)$ is a smooth function without zeroes (and $\partial g'/\partial g > 0$), the two beta functions have the same zeroes and qualitative behaviour. You can straightforwardly adapt this to multiple couplings, and the idea is called 'covariance of the beta function/RG flow.' For a reference, see Zinn-Justin section 10.11.