Theory φ3 in the dimension d = 3 in frames of η-expansion

Abstract

The method of η-expansion calculation in the scalar field model with φ3 interaction in a 3D euclidian space based on conformal bootstrap equations is used in the present paper. As we know, there is an ε-expansion technique that allows us to find the critical exponent in the form of a series in powers of ε, the deviation of the dimension of space from the logarithmic one. However, the logarithmic dimension of the theory φ3 is 6, and the given series in ε have a very small radius of convergence, so that it is not possible to extend it analytically to the dimension d = 3. To solve the problem, we propose using the η-expansion: we construct the series in powers of critical exponent η supposing that it is a small value and obtain some approximate equation for η. If we consider this equation as precise, then it proves that there is no sustainable solution. But using the Pad´e approximant we receive a stable root of equation.