Two-loop RG functions of the massive φ field theory in general dimensions

The dependence of critical exponents of statistical mechanical systems on the physical space dimension is one of the most fundamental features of critical phenomena [1]. Continuous change of the space dimension d has been the basic principle for constructing the famous Wilson-Fisher epsilon expansion [2] with ε = 4−d. However, the Renormalization Group (RG) [3], which provides the fundamental theoretical basis for such calculations, is nonperturbative in its nature. It does not require that the critical exponents or any other universal quantities should be expanded in some space deviation, coupling constant, inverse number of order-parameter components, etc. While investigating the critical phenomena, the RG allows, in principle, to work in any setting of interest provided that we are smart enough to apply the appropriate analytical or numerical tools. The ε-expansion has turned out to be a very useful tool in studying the qualitative features of systems in critical state even in its lowest-order approximations (see, e.g., [4]). Being pushed to higher orders, it allowed to produce very accurate numerical extrapolations for critical exponents of three-dimensional N -vector models [5]. An alternative calculational scheme used with comparative success is the so-called g-expansion within the massive field theory in fixed dimension put forward by Parisi [6]. This approach has been mainly applied directly in three dimensions to systems of different complexity [7–14] (for a recent review see [15]). It was used in calculating the exponents of φ models in two dimensions in [8, 9], albeit with somewhat less accuracy. In [16], the massive theory approach was applied to disordered spin systems in two and three dimensions, while Yu. Holovatch and the present author analyzed the critical behavior of pure and disordered Ising systems in general dimensions 2 < d < 4 [17]. A natural access to non-integer dimensions is also provided by the well-known large-N expansion [18–21]. In its general scope, it gives critical exponents as functions of d. These are valid in the whole range between the lower and upper critical dimensionalities, and can be handled analytically in lower-order approximations. The large-N expansion is capable of yielding information on dimensional dependencies that are hardly accessible by other means, for example, from the epsilon expansion. Unfortunately, it is very hard to obtain such results in higher orders in 1/N , while short series expansions in 1/N usually fail to give very accurate numerical estimates for relatively small values of N , say N = 3. The convergence of truncated 1/N expansions has been analyzed on the basis of the field-theoretical approach at d = 3 in [22].