Self-gravitating Higgs field of scalar charge

We study the self-gravitating Higgs field of a scalar charge. We show that in the zeroth and first approximation in the smallness of the scalar charge, the gravitational field of the scalar charge is described by the Schwarzschild–de Sitter metric with a cosmological constant determined by the vacuum potential of the Higgs field. An equation for the perturbation of the vacuum potential is obtained and studied. Particular exact solutions of the field equation are given. It is shown that in the case of a naked singularity, solutions of the field equation have the character of microscopic oscillations with a Compton wavelength. Asymptotic limit cases of the behavior of solutions are studied and their comparative analysis is carried out in relation to the Fisher solution. The averaging of microscopic oscillations of the scalar field is carried out and it it shown that at $\Lambda>0$ they make a negative contribution to the macroscopic energy of the scalar field, reducing the observed value of the black hole mass. A computer simulation of a scalar field demonstrates various types of the behavior of solutions.