Scale-invariant description of the critical region in the method of integral equations for the correlation functions

It is shown that in the framework of the Percus–Lebowitz method of functional expansions it is not possible to obtain an equation closed at the level of the two-particle correlation functions ensuring a realistic description of a large neighborhood of the critical point. A modified variant of the method is used to derive an approximate equation for the twoparticle correlation functions valid both at the critical point and far from it. The $\varepsilon$ expansions of the critical exponents that follow from this equation agree with the wellknown results for the Ising model.