Expansion in Higher Harmonics of Boson Stars using a Generalized Ruffini-Bonazzola Approach, Part 1: Bound States

Abstract

The method pioneered by Ruffini and Bonazzola (RB) to describe boson stars involves an expansion of the boson field which is linear in creation and annihilation operators. In the nonrelativistic limit, the equation of motion of RB is equivalent to the nonlinear Schr\"odinger equation. Further, the RB expansion constitutes an exact solution to a non-interacting field theory, and has been used as a reasonable ansatz for an interacting one. In this work, we show how one can go beyond the RB ansatz towards an exact solution of the interacting operator Klein-Gordon equation, which can be solved iteratively to ever higher precision. Our Generalized Ruffini-Bonazzola approach takes into account contributions from nontrivial harmonic dependence of the wavefunction, using a sum of terms with energy $k\,E_0$, where $k\geq1$ and $E_0$ is the chemical potential of a single bound axion. The method critically depends on an expansion in a parameter $\Delta \equiv \sqrt{1 - E_0{}^2/m^2} < 1$, where $m$ is the mass of the boson. In the case of the axion potential, we calculate corrections which are relevant for axion stars in the transition or dense branches of solutions. We find with high precision the local minimum of the mass, $M_{min}\approx 463\,f^2/m$, at $\Delta\approx0.27$, where $f$ is the axion decay constant. This point marks the crossover from the transition branch to the dense branch of solutions, and a corresponding crossover from structural instability to stability.