*Article* **Asymptotically Flat, Spherical, Self-Interacting Scalar, Dirac and Proca Stars**

#### **Carlos A. R. Herdeiro and Eugen Radu \***

Departamento de Matemática da Universidade de Aveiro and Centre for Research and Development in Mathematics and Applications (CIDMA), Campus de Santiago, 3810-183 Aveiro, Portugal; herdeiro@ua.pt

**\*** Correspondence: eugen.radu@ua.pt

Received: 14 November 2020; Accepted: 1 December 2020; Published: 8 December 2020

**Abstract:** We present a comparative analysis of the self-gravitating solitons that arise in the Einstein–Klein–Gordon, Einstein–Dirac, and Einstein–Proca models, for the particular case of static, spherically symmetric spacetimes. Differently from the previous study by Herdeiro, Pombo and Radu in 2017, the matter fields possess suitable self-interacting terms in the Lagrangians, which allow for the existence of *Q*-ball-type solutions for these models in the flat spacetime limit. In spite of this important difference, our analysis shows that the high degree of universality that was observed by Herdeiro, Pombo and Radu remains, and various spin-independent common patterns are observed.

**Keywords:** solitons; boson stars; Dirac stars

#### **1. Introduction and Motivation**

#### *1.1. General Remarks*

The (modern) idea of solitons, as extended particle-like configurations, can be traced back (at least) to Lord Kelvin, around one and half centuries ago, who proposed that atoms are made of vortex knots [1]. However, the first explicit example of solitons in a relativistic field theory was found by Skyrme [2,3], (almost) one hundred years later. The latter, dubbed *Skyrmions*, exist in a model with four (real) scalars that are subject to a constraint. They were proposed as field theory realizations of baryons; nonetheless, Skyrmions capture the basic properties of a generic soliton.

In the context of this work, solitons are defined as spatially localized, singularity-free solutions of a field theory model, which possess a finite mass and angular momentum (the fields may possess a non-trivial time-dependence, as for the solitons in this work. However, this dependence is absent at the level of the energy-momentum tensor. Additionally, in our definition, we shall ignore the issue of stability and do not impose the solutions possessing a topological charge). As already suspected in the literature before Skyrme's seminal work (in a historical setting, a concrete realization of the idea of (relativistic) solitons was put forward in 1939 by Rosen [4], who looked for localized configurations which could be used as representations of extended particles), many non-linear field theories possess such solutions. The study of solitons has attracted a lot of attention in the last 50 years; they have important applications in various contexts, ranging from, e.g., the models of condensed matter physics to high energy physics and cosmology. Moreover, solitons are also significant for a nonperturbative quantum description of states. Famous examples of solitons include the sphalerons in the Standard Model [5] and the magnetic monopoles [6,7] in various extensions thereof.

A review of soliton physics in the non-gravitating case can be found in the textbooks [8,9]; also see the reviews [10,11]. In this work, we shall focus on a special class of solitons in four spacetime dimensions that are time-dependent (at the level of the matter field), but non-dissipative (with a time independent spacetime geometry), and carrying a Noether charge *Q* that is associated with a global *U*(1) symmetry. We shall restrict ourselves to models with a single matter field (except for the fermionic case) and possessing a standard kinetic term. These solitons do not possess topological properties, but the field theory Lagrangian must have suitable self-interactions.

Three different models are considered here: (ungauged) field theories describing particles of spin *s* = 0, 1/2, 1, with the goal of providing a comparative analysis of their spherically symmetric solitonic solutions. As such, this work is an extension of that in Ref. [12], which dealt with the same matter fields, but without self-interactions. The latter introduce a novel feature: self-interactions allow for the existence of solitonic solutions, even on a fixed flat spacetime background, i.e., ignoring gravity. As we shall see, self-interactions lead to a more intricate landscape of (gravitating) solutions, with some new qualitative features when compared to the picture that is revealed in [12].

Historically, in the non-gravitating case, solitons in a complex scalar field model have been known for more than 50 years, being dubbed *Q* − *balls* by Coleman [13] (also see the previous work [14]). These are central for our discussion, sharing most of the basic features with the higher spin generalizations. Within a Dirac field model with a positive quartic self-interaction term, Ivanenko [15], Weyl [16], Heisenberg [17], and Finkelstein et al. [18,19] have made early attempts to construct particle-like solutions. However, the first rigorous numerical study of such configurations was been done by Soler in 1976 [20]. The case of a (Abelian) vector field has been ignored until recently; the study of *Q*-ball-like solutions was initiated by Loginov [21].

In all three cases, a large literature has grown based on these early studies, which includes generalizations in various directions. A particularly interesting case concerns the study of backreaction of these solutions on the spacetime geometry. This is a legitimate question, since the existence of solitons relies on the nonlinearity of the field theory and General Relativity (GR) is intrinsically highly nonlinear. Therefore, gravitational interactions could significantly alter the flat spacetime solitons.

The results in the literature prove that all flat space solitons survive when including gravity effects; however, a more complicated picture emerges when compared to the non-self-interacting case. The study of gravitating generalization of the flat spacetime *Q*-balls has started with Friedberg et al. [22]. The higher spin fields are less studied; GR generalization of the self-interacting spinors were only considered recently by Dzhunushaliev and Folomeev [23], while the gravitating *Q*-Proca stars have been studied by Brihaye et al. [24] and Minamitsuji [25].

The main purpose of this work is to review various existing results on this type of solitons, putting them together under a consistent set of notations and conventions. A comparative study is presented, which starts with the generic framework and scaling symmetries (Section 2). There and afterwards, the mathematical description of each of the three models is made in parallel in order to emphasise their similarities. The basic properties of solutions in a flat spacetime background are discussed in Section 3, where we adapt Deser's argumen<sup>t</sup> [26] to find virial identities that yield some insight on the existence of such configurations. Gravity is included in Section 4; a curved spacetime generalization of the Deser-type virial identities is also discussed. Extending the models' framework to include GR effects allows for the possibility to add a black hole (BH) horizon at the center of soliton, a possibility that is indeed realized for other well-known solutions, such as Skyrmions, magnetic monopoles, and sphalerons (see e.g., the review work [27]). However, as we shall discuss in Section 4 (and Appendix B), the situation is different for the considered (spherically symmetric) solutions and there are no BH generalizations, regardless of the presence of self-interactions. We also discuss the physical interpretation of the fermionic solutions. Section 5 presents concluding remarks and some open questions.
