content_image/331541.jpg content_image/331542.jpg content_image/331543.jpg content_image/331544.jpg
in the vicinity of T = T$_{Z}$ then obviously the infinitesimal condition: 

<img src='content_image/331541.jpg'>

should be fulfilled. Using Eq. (12) we get that T$_{Z}$ is the root of the equation: 

<img src='content_image/331542.jpg'>

This is the rigorous definition of the limiting Zeno state at n → 0 from which the Zeno line emerges. Note that for vdW equation with B$_{2}$ = b − a $_{T}$, B$_{3}$ = b 2 and n$_{b}$ = 1 /b , Eq. (17) gives the standard result T$_{Z}$ = a/b . Therefore this is the specific property of the vdW equation that the temperatures T$_{Z}$ and T$_{B}$ coincide. Actually the value T$_{Z}$ depends on the dimensionless packing parameter η$_{b}$ = n$_{b}$ σ d in d dimensions, where σ is the characteristic scale of the potential, e.g. the hard core diameter. This parameter is determined for the opposite limiting Zeno state with T → 0. Though Eq. (17) formally coincide with Eq. (6), the approach proposed for the determination of the limiting Zeno-state at n → 0 is different from that used in [9]. Thus the situation is inverse: rather the value n$_{b}$ determines the value T$_{Z}$ and not vice versa. In order to make this consistently with the definition of the Zeno-line we choose n$_{b}$ so that T$_{Z}$ = T (vdW) B , where T ( vdW ) B = a/b is the Boyle temperature in the van der Waals approximation. We use this to calculate the locus of the critical point in Section . Using the standard expressions for the virial coefficients B$_{k}$ for Lenard-Jones potential 

<img src='content_image/331543.jpg'>

the solution of (17) for 3-dimensional system gives n$_{b}$ σ 3 ≈ 0 . 965 and T$_{Z}$ = 4 while T$_{B}$ ≈ 3 . 42. 

## SCALE INVARIANT MEAN-FIELD THEORY 

In this section using the proposed isomorphism we propose the variant of the mean-field approach to calculate the locus of the critical point. It exploits explicitly the scale symmetry which inherent to the power-like interaction potentials. We call it scale-invariant mean-field theory. 

Let us show that the parameter z can be determined if the attractive part of the inter- action potential possesses the scaling symmetry. The starting point is that using (10) it is easy to derive the following relation: 

<img src='content_image/331544.jpg'>

