# In general, do critical points of continuous phase transitions have $\beta =0$?

Consider a phenomenological modelling of a continuous phase transition, where the Lagrangian of the system is given by

$$L=\frac{a}{2}\phi^2+\frac{\lambda}{4}\phi^4-h\phi.$$

Here $$\phi$$ is the order parameter, $$h$$ is the external field, $$a=a_0(T-T_c)$$, and $$a_0>0$$ and $$\lambda>0$$. I have not written the kinetic term here, but one can always put in that. At $$T =T_c$$ we have a second order phase transition.

An oft-quoted statement is that at $$T=T_c$$, we have self similarity at all scales. How do I see this statement? Is it the same statement that the $$\beta$$ function of the above theory vanishes? If so, is there a reference where this calculation for this model is performed in detail?

Assume that I am working in 2 dimensions, where $$\lambda$$ has negative mass dimension. In this case, is it correct to say that a CFT describes the critical point? What happens to the coupling with negative mass dimension here?

• You've probably found the answer already, but if you're working in $2$ dimensions, where $\lambda$ has negative mass dimension, the critical point is not described by that $L$. The actual critical point has to include all the powers of $\phi^2$, and it's that theory that is described by a CFT. Sep 5 at 5:32