Fractal dimensions and scaling laws in the interstellar medium: A new field theory approach

We develop a field theoretical approach to the cold interstellar medium (ISM). We show that a nonrelativistic self-gravitating gas in thermal equilibrium with a variable number of atoms or fragments is exactly equivalent to a field theory of a single scalar field φ(x⃗) with an exponential self-interaction. We analyze this field theory perturbatively and nonperturbatively through the renormalization group approach. We show a scaling behavior (critical) for a continuous range of the temperature and of the other physical parameters. We derive in this framework the scaling relation ΔM(R)∼R^d_H for the mass on a region of size R, and Δv∼R^q for the velocity dispersion where q=1/2(d_H-1). For the density-density correlations we find a power-law behavior for large distances ∼|r⃗₁-r⃗₂|^2d_H-6. The fractal dimension d_H turns out to be related with the critical exponent ν of the correlation lenght by d_H=1/ν. The renormalization group approach for a single component scalar field in three dimensions states that the long-distance critical behavior is governed by the (nonperturbative) Ising fixed point. The corresponding values of the scaling exponents are ν=0.631…, d_H=1.585…, and q=0.293…. Mean field theory yields for the scaling exponents ν=1/2, d_H=2, and q=1/2. Both the Ising and the mean field values are compatible with the present ISM observational data: 1.4<~d_H<~2, 0.3<~q<~0.6. As typical in critical phenomena, the scaling behavior and critical exponents of the ISM can be obtained without dealing with the dynamical (time-dependent) behavior.