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 phi((x) over right arrow) 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 Delta M(R)similar to R(dH) for the mass on a region of size R, and Delta v similar to 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 similar to\(r) over right arrow(1)-(r) over right arrow(2)\(2dH-6). The fractal dimension d(H) turns out to be related with the critical exponent v of the correlation length by d(H)=1/nu. 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 nu=0.631..., d(H)=1.585..., and q = 0.293.... Mean field theory yields for the scaling exponents nu = 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 less than or equal to d(H) less than or equal to 2, 0.3 less than or equal to q less than or equal to 0.6. As typical in critical phenomena, the scaling behavior and critical exponents of the ISM can be obtained without dealing with the dynamical (One-dependent) behavior.