## 1 Introduction  
The polarizability of a nanoparticle is a response function which relates the electric dipole moment produced in it to an externally applied electric field. The polarizability is not an intrinsic property of the nanoparticle, but actually depends on the environment which it is embedded in [1; 2; 3]. As such, a nanoparticle's polarizability will be modified by the presence of an underlying substrate. The study of this problem is of significant interest, since in most experimental setups the nanoparticle (NP) is placed directly onto a dielectric substrate or at a given distance from it. In previous studies in which the radiation scattered by a dielectric NP was measured using dark-field microscopy, it was shown that the presence of the substrate leads to a redshift of the NP's resonance with respect to the situation where the NP is in vacuum [4; 5; 6]. The polarizability of a nanoparticle at a given frequency is a complex quantity, with its real and imaginary parts describing, respectively, the reactive and dissipative responses of a nanoparticle subjected to an electromagnetic field. Therefore, the imaginary part of the polarizability controls the extinction and absorption cross-sections of a nanoparticle subjected to an impinging electromagnetic field [7; 8]. These quantities are essential for the understanding of scattering experiments of electromagnetic radiation involving nanoparticles, either isolated or forming clusters. In particular, the former case has been a topic of much interest in the context of single-molecule or single-particlespectroscopies [9; 10]. The knowledge of the imaginary part of the polarizability is also essential in order to understand the phenomena of blackbody and thermal friction experienced by a neutral nanoparticle in close proximity to an interface between two media [11]. It is therefore of major importance to understand how the imaginary part of the polarizability is renormalized relative to its value in a vacuum when it is near an interface, the most common setup in experiments. It would be of particular relevance (from the device engineering viewpoint), if the dielectric properties of the interface, near which the nanoparticle is located, could be tuned. This would provide a route for controlling the value of the nanoparticle polarizability in real time. Such an approach is not viable when we consider the interface between two conventional dielectrics or between a metal and a dielectric, since the interface has fixed properties by construction. Fortunately, there is a possible and technologically feasible route to overcome this limitation. Adding a graphene sheet between an interface involving two different dielectrics provides an additional degree of freedom to the problem. Indeed, the Fermi energy of a graphene sheet can be controlled in real time using a gate. Tuning the Fermi energy of graphene changes the local dielectric environment around the nanoparticle and therefore the value of the imaginary part of the nanoparticle polarizability. This is the opportunity we will explore in this paper. Incidentally, the problem of nanoparticle polarizability renormalization in the presence of a substrate is also relevant for the characterization of the dielectric properties of a scanning near-field optical microscope (SNOM). Scanning near-field optical microscopy is a technique frequently used to image and characterize surface polaritons in graphene [12] and other two-dimensional materials, such as boron nitride [13]. More recently, exciton-polaritons have also been studied in layered transition metal dichalcogenides using the same method [14]. Indeed, the SNOM tip is frequently modeled as a dipole, as is the nanoparticle [15]. Therefore, understanding how a nanoparticle changes its dielectric properties under illumination allows us to also understand the problem of a SNOM tip illuminated with THz radiation during the excitation of surface polaritons in graphene and other two-dimensional materials.