Simple argument for critical exponent

[paralleltransport]

Hello.

I wanted to construct a simple and clear explanation for the relation between the beta function and the critical exponents (divergence of correlation length) close to a critical point.

I wanted to check if this reasoning is valid. This is my own rewording of complicated arguments I see in standard QFT texts.

 

[BigJDubs]

The beta function is a measure of how the physical properties of a system change as we vary the coupling constants. At a critical point, the beta function will be zero and the coupling constants remain unchanged. At the critical point, there is a divergence in the correlation length, which is measured by the critical exponents. The critical exponents measure the scaling behavior near the critical point, and this scaling behavior is related to the behavior of the beta function. As the coupling constants approach the critical point, the beta function approaches zero. This implies that the scaling behavior near the critical point is determined by the behavior of the beta function as it approaches zero.