Charged Scalar Mediator

• Charged scalar mediator is a spin-0 bosonic field carrying gauge charge, fundamental for forming non-topological solitons and boson star configurations.
• It balances nonlinear scalar self-interactions with electromagnetic repulsion, establishing critical bounds on field amplitudes for stable solutions.
• Gravitational effects in boson star models further stabilize structures, influencing particle trajectories and offering insights into dark matter and astrophysical phenomena.

A charged scalar mediator is a bosonic field carrying both electric (or more generally gauge) charge and scalar (spin-0) quantum numbers, entering as a fundamental or composite degree of freedom in a wide class of extensions to the Standard Model (SM). Such mediators play a central role in models of non-topological solitonic objects (e.g., charged Q-balls and boson stars), dark sector phenomenology, flavour theory, and scenarios involving new macroscopic quantum states. Their unique feature is the dual function of binding structure through their self-interactions and coupling to gauge sectors (particularly U(1) electromagnetism), which enables distinctive interplay between attractive and repulsive forces at both classical and quantum levels. Below is a comprehensive overview of the charged scalar mediator concept, its theoretical formulation, physical implications, and associated phenomenology, with particular reference to the detailed paper in charged Q-ball and boson star models (Brihaye et al., 2014).

1. Fundamental Lagrangian and Scalar Self-Interactions

The charged scalar field, usually denoted $\Phi$, is taken as a complex field minimally coupled to a $U(1)$ gauge field $A_\mu$ with a Lagrangian density: $$\mathcal{L} = - (D_\mu \Phi)^* (D^\mu \Phi) - U(|\Phi|) - \frac{1}{4} F_{\mu\nu}F^{\mu\nu}$$ where $D_\mu = \partial_\mu + i e A_\mu$ is the gauge-covariant derivative, and $U(|\Phi|)$ is the scalar potential. In the context of gauge-mediated supersymmetry breaking, the potential is given by: $$U(|\Phi|) = m^2 \eta_{\rm susy}^2 \left(1 - \exp\left(-\frac{|\Phi|^2}{\eta_{\rm susy}^2}\right)\right)$$ where $m$ is the scalar mass and $\eta_{\rm susy}$ is the messenger scale. This nonlinear, non-polynomial potential is crucial for the existence and stability of non-topological soliton solutions by controlling the balance between nonlinearity (favoring localization and bound structure) and gradient (dispersive) terms.

The minimal coupling to $U(1)$ ensures the scalar field carries an electric charge, enabling long-range electromagnetic interactions and self-consistent coupling to electromagnetism.

2. Existence Criteria and Maximum Scalar Charge in Charged Q-balls

In the flat space (non-gravitating, $\alpha=0$) regime, the model admits Q-ball solutions—coherent, stationary non-topological solitons with charge-stabilized core. The solvability of the field equations is controlled by a finite range of allowable field central amplitudes $\phi(0)$, determined by the interplay between scalar self-interactions and electromagnetic self-repulsion. Specifically, for fixed gauge coupling $e$, solutions exist only for $\phi(0) \in [\phi(0)_{\min},\, \phi(0)_{\max}]$, where both bounds depend on $e$.

As $e$ increases—or equivalently, as the scalar charge density rises—the allowed interval for $\phi(0)$ shrinks. When $e$ reaches a critical value (numerically found to be $e \approx 0.1262$ in rescaled units of the referenced paper), no Q-ball solutions exist, indicating the point at which electromagnetic repulsion overwhelms the binding effect of the scalar potential. This defines a fundamental upper bound on the allowed charge per field quantum for non-topological solitons stabilized purely through self-interaction and electromagnetic forces.

3. Gravitational Interplay: Charged Boson Stars and the Balance of Forces

Activating gravity ($\alpha \neq 0$) transforms charged Q-balls into boson stars—self-gravitating, bound states of a charged scalar field. The Einstein-Maxwell-scalar system admits spherically symmetric, globally regular solutions with metric: $$ds^2 = -f(r)dt^2 + l(r)dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2)$$ Here, gravitational attraction provides an additional force counteracting electromagnetic repulsion. Unlike the purely self-interacting Q-ball case, charged boson star solutions can reach larger central field densities before destabilization, and the limiting behavior for high central densities involves both a collapse of the metric functions (the so-called thin-wall limit) and a subtle interplay with the electromagnetic field. The solution exists as long as the total gravitational binding can balance the scalar and electromagnetic pressures. There is no strictly analogous sharp charge cutoff as in Q-balls, but there is a delicate dependence of global properties on the gravitational coupling, gauge coupling, and central field values.

4. Stability, Pair Creation, and Vacuum Robustness

A central question in the formation of charged, self-bound scalar structures is the stability of the vacuum in the presence of strong electromagnetic fields. Charged objects with a sufficiently large number of constituent particles could, in principle, reach electric field strengths capable of spontaneous pair production (Schwinger process). The paper establishes that for charged boson star configurations, the maximal number of bound bosons $N$ remains below the critical value set by $1/\alpha_{\rm fs}$ (the inverse fine-structure constant). For the explicit parameter space explored (maximal $N$ of order $10^2$), the objects remain subcritical, and no vacuum instability (as measured by pair creation threshold) emerges. This ensures that the charged scalar mediator mechanism does not lead to catastrophic decay of the vacuum under the physical conditions considered.

5. Charged Test Particle Dynamics: Effective Potentials and Trajectory Characteristics

Charged test particle motion in the background of a boson star is governed by the Hamilton-Jacobi equation: $$-2\frac{\partial S}{\partial\tau} = g^{\mu\nu}\left(\frac{\partial S}{\partial x^\mu} - q A_\mu\right) \left(\frac{\partial S}{\partial x^\nu} - q A_\nu\right)$$ leading to radial and angular equations of motion after separation of variables. The effective radial "momentum" is: $$\left(\frac{dS_r}{dr}\right)^2 = \frac{l(r)}{f^2(r)}\, (E + q A(r))^2 - \delta\,\frac{l(r)}{f(r)} - \frac{K}{r^2}$$ with associated effective potentials: $$V^\pm_{\text{eff}} = -q A(r) \pm \sqrt{\frac{f^2(r)}{l(r)}\left(\delta\,\frac{l(r)}{f(r)} + \frac{K}{r^2}\right)}$$ These expressions encode the influence of both gravitational and electromagnetic potentials as shaped by the profile of the scalar field. Unlike the Reissner–Nordström charged black hole, where orbits of charged particles generically are not planar, charged boson stars uniquely give rise to planar test particle motion due to the symmetry of the solution and the regular scalar field distribution. The scalar field's backreaction on the metric is crucial, substantially modifying the qualitative form of the effective potentials and thus the nature of particle trajectories.

These results have potential astrophysical applications, e.g., modeling EMRIs, accretion disc formation, and polar jet launching in compact objects where boson star configurations could be realized.

6. Extrapolation to Other Physical Systems and Broader Implications

The framework established by the detailed analysis of charged scalar mediators in non-topological solitons and boson stars has direct implications for multiple domains:

• Formation of compact, horizonless objects: Charged scalar fields with suitable self-interaction can generate macroscopic bound objects different from black holes, with potential observational signatures.
• Dark matter modeling: Scalar mediator models underpin a class of dark matter scenarios where DM self-interaction and stability are established via charge and scalar interactions, fine-tuned to evade vacuum instability and produce novel kinetic signatures [as similarly seen in Double-Dark Portal scenarios, (Curtin et al., 2014)].
• Fundamental limits from gauge couplings: The maximum charge criterion defines an upper bound on the viable parameter space for charged-field-based solitons, constraining model building.

In summary, the charged scalar mediator operates as both a stabilizing agent—through self-interaction and collective charge effects—and as a key determinant of the structure and dynamics of composite objects in models with extended scalar and gauge sectors. Its dual role in governing binding and long-range interaction defines sharp physical limits and imprints specific signatures on particle and field dynamics. The interplay between scalar potential form, gauge coupling, gravitational effects, and vacuum stability constitutes the essential theoretical architecture for any application or phenomenological analysis involving charged scalar mediators (Brihaye et al., 2014).