**1. Introduction**

The theory of General Relativity (GR) is the best tested theory of gravity developed up to date. The classical tests, within the solar system, where the deviations from Newtonian gravity are small, helped to promote GR to one of the fundamental theories of nature. For strong gravitational fields, i.e., when the deviations from Newtonian gravity become important, observations and tests have now become available. An excellent laboratory to consider strong gravitational field effects are black holes (BHs) and neutron stars (NSs). It is a common belief that BHs are "simpler" to describe than NSs partly due to the lack of knowledge of the equation of state of the matter making up the NS. BHs, in fact, are still believed to follow the so-called no-hair conjecture [1], which states that all stationary asymptotically flat BHs are fully characterized by global charges associated with a Gauss law. There are numerous counter-examples for the conjecture when considering nonlinear matter fields. However, the situation for scalar fields is very different. A number of no-scalar-hair theorems were put forward (see [2] for a recent review) and scalar fields, apparently, should be trivial around a stationary and asymptotically flat BH spacetime.

One process where a scalar field can become non-trivial in a BH spacetime is through spontaneous scalarization. It was first discussed in a scalar-tensor theory of gravity around a NS [3], in which the scalar field couples to the trace of the energy-momentum tensor and can obtain non-vanishing values even if its asymptotic value is zero. The corresponding phenomenon has gained considerable attention in black hole physics [4].

The view on no-hair theorems for minimally coupled scalar fields has changed since the discovery of hairy Kerr BHs in a model where a massive complex scalar field is minimally coupled to gravity [5,6]. To circumvent the no-scalar-hair theorems, it is necessary to assume a harmonic dependence on the time and azimuth coordinates. Then, the so-called synchronization condition *ω*/*m* = Ω *H* must be

imposed, where *ω* is the scalar field frequency, *m* an integer, and Ω *H* the horizon angular velocity. The configuration of the test scalar field outside the event horizon was dubbed "scalar cloud" and when taking backreaction into account leads to the existence of hairy Kerr BH solutions. As the horizon radius approaches zero, the solution is reduced to a spinning *boson star*.

Boson stars (BSs) are regular, stationary, and localized solutions to the Einstein–Klein–Gordon system of equations, formed by a complex scalar field with a continuous U(1) symmetry, which gives rise to a globally conserved Noether charge. They are the self-gravitating counterparts of *Q*-balls [7].

Their size can range from the atomic scale up to the size of supermassive BHs, depending on the choice of the scalar potential (see [8] for a review), and can be used as models for dark matter particles [9] and BH mimickers [10], for example. The absence of an event horizon, however, can lead to significant changes in the propagation of light rays when compared to the spacetime of a BH [11]. Charged BSs were studied in [12] for a self-interacting scalar potential whose motivation comes from supersymmetric extensions of the standard model and, originally, uncharged *Q*-balls had been discussed [13–15]. A combination of the attractive effects of gravity and the repulsive effects of electromagnetism can render the charged BS stable. Most work that deals with boson stars is concerned with infinitely extended boson stars, e.g., also when considering these compact objects as black hole mimickers. Moreover, the boson stars that are actually compact do need a very specific potential that is not differentiable at vanishing scalar field value; see, e.g., [16]. In this latter case, the exterior of the boson star would simply be given by the Schwarzschild solution with a scalar field identically zero. Thus, in this paper, we consider only the boson stars with scalar field falling off exponentially at infinity.

Further studies concerning scalar clouds were considered subsequently for Reissner–Nordström (RN) spacetime [17], where, for a non-trivial configuration of the gauged scalar field, it was shown that it is necessary to add self-interactions in the scalar potential and the resonance condition to be satisfied, *ω* = *q<sup>V</sup>*(*rh*), where *q* is the scalar coupling constant and *<sup>V</sup>*(*rh*) the electric potential on the horizon.

Gauged scalar clouds in the Schwarzschild BH were considered in [17,18]. In this case, the background is fixed by the Schwarzschild metric and the differential equations for the electric potential and the scalar field are coupled. It was shown that the scalar clouds exist for some range in the gauge coupling, and it was also found numerically in [18] that two different solutions exist for the same values of the gauge coupling constant and the electric potential at infinity. When backreaction is taken into account, the solutions exist up to a maximal value of the gravitational constant and lead to two distinct situations: (i) an extremal BH with a diverging derivative of the scalar field at the horizon and (ii) a RN–de Sitter solution with a screened electric charge.

In this paper, we extend the results of [18] to include globally regular space-times with the same matter field content. The corresponding solutions are charged *Q*-balls (in a Minkowski space-time) and boson stars. In the following, we will demonstrate that the phenomenon described above doesn't depend on the details of the scalar self-interaction or on the fact that the space-time possesses a priori a horizon. Our paper is organized as follows: in Section 2, we discuss the model and equations of motion, while Section 3 contains our results on black hole and globally regular space-times, respectively. We end with a discussion in Section 4.
