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 $\phi \left(\overrightarrow{x}\right)$ 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 $\Delta M\left(R\right)\mathrm{}\sim R^{d_H}$ for the mass on a region of size $R$, and $\Delta v\sim R^q$ for the velocity dispersion where $q=\frac{1}{2}(d_H-1)$. For the density-density correlations we find a power-law behavior for large distances $\sim {|{\overrightarrow{r}}_1-{\overrightarrow{r}}_2|}^{2d_H-6\mathrm{}}$. The fractal dimension $d_H$ turns out to be related with the critical exponent $\nu$ of the correlation lenght by $d_H=\frac{1}{\nu }$. 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 $\nu =0.631\mathrm{}\dots$, $d_H=1.585\mathrm{}\dots$, and $q=0.293\mathrm{}\dots$. Mean field theory yields for the scaling exponents $\nu =\frac{1}{2}$, $d_H=2\mathrm{}$, and $q=\frac{1}{2}$. 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.