This study investigates polarized images of an equatorial synchrotronemitting ring surrounding hairy black holes within the EinsteinMaxwellscalar theory. Our analysis demonstrates qualitative similarities between the polarization patterns of hairy black holes and Schwarzschild black holes. However due to the nonminimal coupling between the scalar and electromagnetic fields an increase in black hole charge and coupling constant can substantially amplify polarization intensity and induce deviations in the electric vector position angle. These effects may offer observational signatures to distinguish hairy black holes from Schwarzschild black holes.
['General Relativity and Quantum Cosmology']
Polarized Image of a Synchrotronemitting Ring in EinsteinMaxwellscalar Theory
{'Polarized Image of a Synchrotron-emitting Ring in Einstein-Maxwell-scalar Theory': 'Yiqian Chen a,b , ∗ Lang Cheng a , † Peng Wang a , ‡ and Haitang Yang a §  \na Center for Theoretical Physics, College of Physics, Sichuan University, Chengdu, 610064, China and School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, Hangzhou, 310024, China  \nb  \nThis study investigates polarized images of an equatorial synchrotron-emitting ring surrounding hairy black holes within the Einstein-Maxwell-scalar theory. Our analysis demonstrates qualitative similarities between the polarization patterns of hairy black holes and Schwarzschild black holes. However, due to the non-minimal coupling between the scalar and electromagnetic fields, an increase in black hole charge and coupling constant can substantially amplify polarization intensity and induce deviations in the electric vector position angle. These effects may offer observational signatures to distinguish hairy black holes from Schwarzschild black holes.  \n§', 'I. INTRODUCTION': 'The groundbreaking images of the supermassive black holes at the centers of M87 and the Milky Way galaxies, obtained by the Event Horizon Telescope (EHT) Collaboration [1-14], have ushered in a new era for testing general relativity in the strong-field regime. These observations, featuring a central brightness depression surrounded by an asymmetric bright ring, are attributed to synchrotron emission originating from relativistic plasma near the black hole. To identify black hole parameters from the data, a comprehensive library of ray-traced General Relativistic Magnetohydrodynamic (GRMHD) images of simulated accretion flows has been constructed. While some models were eliminated by the EHT results, a significant number of GRMHD models were found to be consistent with the total intensity data [5].  \nIn addition to total intensity data, the 2017 EHT observations also captured polarimetric information, which is influenced by the emitting plasma, magnetic field and spacetime curvature. This polarimetric data provides a unique opportunity to gain deeper insights into the properties of black holes and their surrounding environments. Over the past few decades, simulated polarized images have been used to explore various phenomena, including accretion states [15, 16], the jet [17], Faraday rotation [18, 19] and more [20-27]. However, the majority of these studies have  \nrelied on GRMHD, which, while powerful, is computationally intensive and can limit comprehensive parameter surveys. Furthermore, the GRMHD approach primarily focuses on accretion flows, potentially constraining our ability to investigate relativistic effects around black holes.  \nRecently, a simplified model of a synchrotron-emitting fluid ring orbiting a Schwarzschild black hole was proposed to better understand polarized images of black holes [28]. This model, with appropriately chosen parameters, effectively reproduces key features of the polarimetric morphology observed in EHT images of M87 ∗ , such as the Electric Vector Position Angle (EVPA) pattern and the relative polarized intensity. Additionally, it can predict the polarization evolution caused by orbital hotspots during flares, similar to those observed from Sgr A ∗ with GRAVITY [29]. This model was later extended to Kerr black holes [30], leading to further investigations of polarized images using this and similar models within various theoretical frameworks [31-39].  \nMeanwhile, a class of Einstein-Maxwell-scalar (EMS) models, featuring a non-minimal coupling between the scalar and Maxwell fields, has been introduced to investigate the formation of hairy (scalarized) black holes [40]. Through fully nonlinear numerical simulations, these EMS models have revealed the intriguing phenomenon of spontaneous scalarization in Reissner-Nordstrom (RN) black holes, driven by a tachyonic instability. This process has spurred extensive research into the properties and implications of scalarized RN black holes [41-66]. In particular, recent studies have explored the optical appearance of accretion disks surrounding scalarized RN black holes, demonstrating multiple bright rings of varying radii and a noticeable increase in the total intensity flux in black hole images [67, 68]. However, polarized images of scalarized RN black holes remain unexplored. Investigating such images could deepen our understanding of how interactions between scalar and electromagnetic fields influence light propagation, potentially offering a promising avenue for the search of hairy black holes.  \nThis paper investigates polarized images of a synchrotron-emitting ring around scalarized RN black holes, as well as those of RN black holes. The subsequent sections of this paper are structured as follows: Sec. II briefly reviews scalarized RN black holes, discusses geometric optics and introduces the equatorial emission model in the EMS model. Sec. III presents the numerical results of polarized images for both equatorial and vertical magnetic fields. Finally, Sec. IV summarizes our findings. We adopt the convention G = c = 4 πϵ 0 = 1 throughout the paper.', 'II. SET UP': 'This section begins by providing a concise review of the scalarized RN black hole solution within the 4-dimensional EMS model. It then examines light propagation within this framework using geometric optics approximations. Subsequently, we introduce an equatorial emission ring model to analyze polarized images of the black holes.', 'A. Black Hole Solution': 'The EMS model, as detailed in [40], incorporates a gravity theory with a scalar field ϕ and the electromagnetic field A µ through the action,  \nS = 1 16 π ∫ d 4 x √ -g [ R2 ∂ µ ϕ∂ µ ϕ -f ( ϕ ) F µν F µν ] , (1)  \nwhere R is the Ricci scalar and F µν = ∂ µ A ν -∂ ν A µ is the electromagnetic field strength tensor. In the EMS model, the scalar field ϕ is non-minimally coupled to the electromagnetic field A µ via the coupling function f ( ϕ ). For scalar-free black holes, such as RN black holes, to exist, the coupling function must satisfy the condition df ( ϕ ) /dϕ | ϕ =0 = 0 [40, 46]. This study specifically examines the exponential coupling function f ( ϕ ) = e αϕ 2 with α > 0. In the RN black hole background, the equation of motion governing the scalar perturbation δϕ is given by  \n( □ -µ 2 eff ) δϕ = 0 , (2)  \nwhere µ 2 eff = -αQ 2 /r 4 . Notably, a tachyonic instability arises when the effective mass squared µ 2 eff becomes negative. This instability can induce spontaneous scalarization in the scalar-free solution, resulting in the formation of hairy (scalarized) black hole solutions [40].  \nTo obtain scalarized RN black hole solutions, we adopt the following ansatz for the metric, electromagnetic field and scalar field,  \nds 2 = -N ( r ) e -2 δ ( r ) dt 2 + 1 N ( r ) dr 2 + r 2 ( dθ 2 +sin 2 θdφ 2 ) , A µ dx µ = Φ( r ) dt and ϕ = ϕ ( r ) . (3)  \nAppropriate boundary conditions are imposed at the event horizon r h and at spatial infinity as follows,  \nN ( r h ) = 0, δ ( r h ) = δ 0 , ϕ ( r h ) = ϕ 0 , Φ ( r h ) = 0, N ( ∞ ) = 1, δ ( ∞ ) = 0, ϕ ( ∞ ) = 0, Φ ( ∞ ) = Ψ. (4)  \nHere, δ 0 and ϕ 0 characterize the black hole solutions, while Ψ represents the electrostatic potential. By specifying δ 0 and ϕ 0 , we obtain scalarized RN black hole solutions with a non-trivial scalar field ϕ using the shooting method implemented in the NDSolve function of Wolfram Mathematica . The black hole mass M and charge Q are determined from the asymptotic behavior of the metric functions at infinity,  \nN ( r ) = 1 -2 M r + · · · , Φ( r ) = Ψ -Q r + · · · . (5)  \nFor simplicity, all physical quantities are expressed in units of the black hole mass by setting M = 1 throughout the paper.  \nIntriguingly, scalarized RN black holes have been shown to possess two photon spheres outside the event horizon within specific black hole parameter regimes [67]. This distinctive feature gives rise to unique phenomena, including black hole images with intricate structures [68-72] and echo signals [73, 74]. Furthermore, studies on superradiant instabilities and the nonlinear stability of these double photon sphere black holes have been conducted [75, 76]. For a comprehensive exploration of black holes with multiple photon spheres, we direct readers to [77].', 'B. Propagation of Light': 'The non-minimal coupling between the scalar and electromagnetic fields can influence light propagation in scalarized RN black holes. Following [78], we employ the geometric optics approximation to derive the equations governing light propagation. In the Lorentz gauge, ∇ µ A µ = 0, the equation of motion for the electromagnetic field becomes  \n∇ µ f ( ϕ ) ( ∇ µ A ν -∇ ν A µ ) + f ( ϕ ) ( □ A ν -R µ ν A µ ) = 0 , (6)  \nwhere R µ ν is the Ricci tensor. The geometric optics approximation is valid when the electromagnetic wavelength is much smaller than any other relevant scale, including the characteristic scales of the background metric and scalar field. This permits the use of the ansatz  \nA µ ( x ) = Re [ ¯ A µ ( x ) e iθ ( x ) /ϵ ] , (7)  \nwhere ¯ A µ is the slowly evolving complex amplitude vector, and θ is the rapidly oscillating phase. Here, ϵ is a small parameter introduced to track various orders of terms, with the geometric-optics limit corresponding to ϵ ≪ 1. The 4-wavevector k µ is then defined as  \nk µ ≡ 1 ϵ ∂ µ θ, (8)  \nrepresenting the local direction of wave propagation.  \nSubstituting the ansatz (7) into Eq. (6), the term of order ϵ -2 yields the dispersion relation k µ k µ = 0. This dispersion relation, together with the property ∇ µ k ν = ∇ ν k µ , leads to  \nk µ ∇ µ k ν = 0 , (9)  \nindicating that the propagation of light rays is governed by null geodesics. Furthermore, the term of order ϵ -1 gives  \n∇ µ f ( ϕ ) ( k µ ¯ A ν -¯ A µ k ν ) + f ( ϕ ) ( 2 k µ ∇ µ ¯ A ν + ¯ A ν ∇ µ k µ ) = 0 . (10)  \nTo simplify the calculations further, the amplitude vector ¯ A µ can be expressed in terms of a normalized spacelike polarization vector, ξ µ , and an amplitude, ¯ A , as ¯ A µ = ¯ Aξ µ . Here, ξ µ and ¯ A satisfy ξ µ ξ ∗ µ = 1 and ¯ A 2 = ¯ A µ ¯ A ∗ µ , respectively. Additionally, expanding the Lorentz gauge with respect to ϵ , the obtained leading term implies that the polarization vector is orthogonal to the wavevector, k µ ξ µ = 0.  \nContracting Eq. (10) with ¯ A ν ∗ and summing with its complex conjugate, we arrive at  \n∇ µ [ f ( ϕ ) ¯ A 2 k µ ] = 0 , (11)  \nwhich is a conservation equation for the amplitude of ¯ A µ along null geodesics. In the EinsteinMaxwell theory, where f ( ϕ ) = 1, Eq. (11) reduces to the conservation of photon number [79]. However, due to the photon-scalar interaction, the conservation of photon number is no longer preserved in the EMS model. Instead, a new conserved quantity, f ( ϕ ) ¯ A 2 k 0 , emerges. Moreover, using Eqs. (10) and (11), we obtain the equation for the polarization vector ξ µ ,  \n2 f ( ϕ ) k µ ∇ µ ξ ν -k ν ξ µ ∇ µ f ( ϕ ) = 0 , (12)  \nwhich governs the evolution of ξ µ along null geodesics. In the Einstein-Maxwell theory, the second term of Eq. (12) vanishes, leading to the parallel transport of the polarization vector ξ µ along null geodesic. Conversely, the presence of the non-minimal coupling between the scalar and electromagnetic fields prevents the parallel transport of the polarization vector ξ µ in the EMS model. Consequently, a numerical method is necessary to compute the evolution of the polarization vector ξ µ along null geodesics.', 'C. Ring Model': "To investigate polarized images of synchrotron radiation, a simplified model of an axisymmetric, emitting ring surrounding a Schwarzschild black hole has been proposed [28]. Remarkably, this  \ntoy ring model has been demonstrated to effectively capture the polarimetric image morphology for a specific subset of GRMHD simulations of M87 ∗ . In particular, a narrow ring of radius r e is situated on the black hole's equatorial plane, and radiating fluid elements within the ring emit linearly polarized synchrotron radiation in the presence of a local magnetic field. In the following, we briefly review the ring model, incorporating modifications arising from the photonscalar interaction in the EMS model. For a comprehensive treatment of the ring model, please refer to [28].  \nAt a point P on the fluid ring, a set of orthonormal tetrad e µ ( a ) can be constructed to describe the local P -frame,  \ne ( t ) = e δ √ N ∂ t , e ( r ) = √ N∂ r , e ( φ ) = 1 r sin θ ∂ φ , e ( θ ) = -1 r ∂ θ , (13)  \nwhere the negative sign ensures that ( e ( r ) , e ( φ ) , e ( θ ) ) forms a right-handed system. The fluid at the point P moves in the ( r )-( φ ) plane of the P -frame with a velocity ⃗ β , given by  \n⃗ β = β ( cos χe ( r ) +sin χe ( φ ) ) , (14)  \nwhere χ is the angle between the velocity ⃗ β and the basis vector e ( r ) . We assume that β and χ are constant on the ring. Furthermore, the fluid's local rest frame at the point P , denoted as the F -frame, can be obtained via a boost transformation from the P -frame, e µ ˆ a = Λ ( b ) ˆ a e µ ( b ) . Here, Λ is the transformation matrix with γ = 1 / √ 1 -β 2 ,  \nΛ =         γ -βγ cos χ -βγ sin χ 0 -βγ cos χ ( γ -1) cos 2 χ +1 ( γ -1) sin χ cos χ 0 -βγ sin χ ( γ -1) sin χ cos χ ( γ -1) sin 2 χ +1 0 0 0 0 1         . (15)  \nIn the F -frame, the magnetic field around the fluid has only spatial components ⃗ B = ( B ˆ r , B ˆ φ , B ˆ θ ) . We assume that the components of the local magnetic field in the fluid's rest frame are constant at any point on the ring. Given the 3-wavevector ⃗ k = ( k ˆ r , k ˆ φ , k ˆ θ ) in the F -frame, the normalized polarization vector ( ξ ˆ 0 , ⃗ ξ ) of the synchrotron radiation emitted by the rotating fluid can be expressed as  \nξ ˆ 0 = 0 , ⃗ ξ = ⃗ k × ⃗ B ∣ ∣ ∣ ⃗ k × ⃗ B ∣ ∣ ∣ . (16)  \nUsing this equation, we can determine the polarization vector ξ µ e for an emitted light ray of 4wavevector k µ e . Specifically, after acquiring the components of k µ e in the P -frame, the boost transformation (15) yields its 3-wavevector ⃗ k e in the F -frame. Subsequently, the normalized polarization  \n3-vector ⃗ ξ e is determined by Eq. (16). Finally, the inverse of the boost transformation (15) and the tetrad basis (13) provide the components of ξ µ e in the spacetime coordinates.  \nIn the case of an optically thin medium, the emitted intensity in the F -frame depends on the angle ζ between the fluid's 3-wavevector ⃗ k e and the magnetic field ⃗ B , the frequency ν of the emitted photons and the geodesic path length l p in the emitting region. Specifically, the emitted intensity is given by  \nI ν, e ∝ ν -α ν l p ∣ ∣ ∣ ⃗ B ∣ ∣ ∣ 1+ α ν sin 1+ α ν ζ. (17)  \nIn models of M87 ∗ , a dependence of sin 2 ζ is often observed at 230 GHz. Consequently, we set α ν = 1 for the remainder of this paper. In the Einstein-Maxwell theory, the conservation of photon number can be used to calculate the intensity I ν along geodesics. Conversely, in the EMS model, the aforementioned quantity f ( ϕ ) ¯ A 2 k 0 is conserved, resulting in a new invariant along the geodesics, f ( ϕ ) I ν /ν 3 .  \nIf an observer at r = r o receives a light ray with the 4-wavevector k µ o , we can use the backward ray-tracing method to determine the 4-wavevector k µ e of the light ray when it is emitted from the ring. Specifically, the light ray is obtained by numerically integrating the geodesic equations,  \ndx µ dλ = k µ , dk µ dλ = -Γ µ ρσ k ρ k σ , (18)  \nwhere λ represents the affine parameter, and Γ µ ρσ is the Christoffel symbol. Once k µ e is given, the emitted polarization vector ξ µ e can be determined as previously discussed. Eq. (12) is then used to evolve ξ µ forward along the light ray, yielding the observed polarization vector ξ µ o . Additionally, the observed intensity I ν, o is derived from the invariant f ( ϕ ) I ν /ν 3 ,  \nI ν, o = f [ ϕ ( r e )] f [ ϕ ( r o )] δ 3 I ν, e ∝ f [ ϕ ( r e )] f [ ϕ ( r o )] δ 3 k ˆ t e l p ∣ ∣ ∣ ⃗ B ∣ ∣ ∣ 2 sin 2 ζ, (19)  \nwhere δ = ν o /k ˆ t e is the Doppler factor, and ν o is the observed frequency.  \nFurthermore, we assume that the observer is located in a frame with the observer basis ( e ( t ) , e ( r ) , e ( θ ) , e ( ϕ ) ) . In this observer's frame, the direction of the measured polarization vector is indicated by the EVPA, defined as,  \nEVPA = arctan ( -ξ ( φ ) o ξ ( θ ) o ) , (20)  \nwhere ξ ( a ) o represents the components of the polarization vector in the observer's frame. Additionally, the observed frequency ν o corresponds to the ( t )-component of the observed wavevector, k ( t ) o ,  \nwhich can be conveniently set to 1 without compromising generality. The observation angles Θ and Φ of received light rays are defined as [80],  \nsin Θ = k ( θ ) o k ( t ) o , tan Φ = k ( ϕ ) o k ( r ) o . (21)  \nConsequently, the Cartesian and polar coordinates in the image plane are expressed as  \nx = -r o Φ , y = r o Θ , (22)  \nand  \nρ = √ x 2 + y 2 , φ ρ = arccos x ρ , (23)  \nrespectively. In the image plane, the EVPA is measured counterclockwise relative to the positive y -axis.", 'III. RESULTS': "In this section, we investigate polarized images of the aforementioned emitting ring around scalarized RN black holes. Our numerical simulations model the ring as a torus with a minor radius of H = 0 . 01 and a major radius equal to the ISCO radius, r ISCO , as detailed in [72]. The torus's center lies on the equatorial plane. For simplicity, we approximate the geodesic path length as l p = Hk ˆ t /k ˆ z . The observer is positioned at a radius of r o = 100 with an inclination angle of θ o = 20 · , mimicking the observational setup for M87 ∗ .  \nTo simulate observational images, we vary the observer's viewing angles, Θ and Φ, and numerically integrated at least 2000 × 2000 photon trajectories until they intersected the narrow ring or reached the cutoff radii at r = r + +0 . 001 or 110. In this paper, we focus exclusively on the primary image of the ring, deferring the analysis of higher-order images to future studies. Once the light rays are determined, the polarization vector and intensity are calculated using Eqs. (12) and (19), respectively. To reduce numerical fluctuations arising from the finite grid points, we employed the MeanShiftFilter function of Wolfram Mathematica .", 'A. Equatorial Magnetic Field': "The EHT Collaboration's polarized image of M87 ∗ indicates that, when the black hole's tilt is oriented northward, the polarized flux is concentrated in the right half of the image plane [7].  \nFurthermore, the twist angle between the EVPA and the local radial unit vector remains relatively stable in regions of high polarized flux. Previous studies have shown that an equatorial local magnetic field with non-zero B ˆ r and B ˆ ϕ components can effectively reproduce these key features [28]. Inspired by these findings, we begin our investigation by analyzing simulations of the ring model, where B ˆ r and B ˆ ϕ are non-zero, while B ˆ z is set to zero. Given the probable parallel alignment of the equatorial magnetic field and the fluid motion, we assume that the fluid's velocity is aligned with the magnetic field. Specifically, we define the angle χ as  \nχ = arctan B ˆ φ B ˆ r + π. (24)  \nIt is worth noting that for an equatorial magnetic field, the alternative choice, χ = arctan ( B ˆ φ /B ˆ r ) , results in the same linear polarized emission.  \nFig. 1 presents simulations of the observable polarization for RN and scalarized RN black holes, considering three distinct configurations of purely equatorial magnetic fields. These configurations correspond to fluids flowing inward in a clockwise direction at angles of χ = -120 · , -135 · and -150 · , as illustrated in the top, middle and bottom rows, respectively. The left column displays the polarized images, where polarization vectors are represented by 20 equally spaced ticks along the azimuthal angle φ ρ on the ring image. The length of the polarization vectors signifies the intensity, while their twist around the ring encodes the EVPA. These images are consistent with observations of M87 ∗ , exhibiting the most intense flux concentrated on the right-hand side. Notably, the apparent size of the ring images exhibits a weak dependence on the coupling constant α , resulting in minimal discernible differences in the polarization vectors at a given φ ρ . To highlight these differences, the upper-left panel includes a zoomed-in inset, revealing that a stronger α can yield a larger apparent radius.  \nThe second column from the left presents the polarization intensity as a function of φ ρ for RN and scalarized RN black holes. Although the intensity maintains a largely similar distribution with peak values in the top right quadrant (0 < φ ρ < π/ 2), its magnitude increases with rising coupling constant α or tangential magnetic field component B ˆ ϕ . The third and fourth columns, respectively, illustrate the polarization direction EVPA and the relative direction angle, ∆EVPA ≡ EVPA -EVPA RN , as functions of φ ρ . For scalarized RN black holes, the EVPA deviations from RN black holes are relatively minor. Furthermore, within a given magnetic field configuration, the EVPA deviations increase with growing α , causing the polarization vector to rotate slightly towards the tangential direction. As the radial magnetic field component diminishes, the EVPA at a fixed φ ρ decreases, indicating a growing tangential component of the polarization  \nFig. 2 depicts the polarized images, intensity and EVPA for RN and scalarized RN black holes with Q = 0 . 9, considering the same magnetic field configurations as in Fig. 1. Similar to the Q = 0 . 4 case, the polarized images exhibit flux asymmetry, with the most intense polarized flux concentrated in the upper-right region. However, a larger black hole charge Q introduces several  \n<!-- image -->  \nx  \nFIG. 1. Linear polarization of a synchrotron-emitting ring surrounding RN and scalarized RN black holes for a fixed charge, Q = 0 . 4, in three configurations of equatorial magnetic fields. The configurations correspond to χ = -120 · ( Top Row ), -135 · ( Middle Row ) and -150 · ( Bottom Row ), with the magnetic field oriented opposite to the fluid's velocity. For scalarized RN black holes, the coupling constant α assumes the values of 40, 50, 60, 70, 80, 90 and 100. The first column presents the polarized images, which exhibit flux asymmetry qualitatively consistent with observations of M87 ∗ . The second, third and fourth columns display the polarization intensity, the polarization direction EVPA and the deviation of the EVPA from RN black holes, ∆EVPA, respectively, as functions of the azimuthal angle φ ρ . As α increases, both the intensity and the EVPA of scalarized RN black holes rise.", 'vectors.': 'r  \nFIG. 2. Linear polarization of a synchrotron-emitting ring surrounding RN and scalarized RN black holes for Q = 0 . 9, considering the same equatorial magnetic field configurations as described previously. The pattern of polarized images closely resembles the Q = 0 . 4 case. Compared to the Q = 0 . 4 case, the intensity flux, represented by the length of polarization vectors, increases more substantially with growing α .  \n<!-- image -->  \nx  \nnotable differences. Firstly, the apparent radius of the ring images becomes more sensitive to α , making it easier to distinguish the polarization vectors at a fixed φ ρ for different α values. The ring image with a stronger α exhibits a larger apparent radius. Secondly, due to the smaller ISCO radius, the polarization intensity of scalarized RN black holes is significantly amplified, leading to more pronounced deviations from RN black holes. For instance, the intensity with α = 100 can be up to four times that of the RN black hole. Finally, the discrepancy in the EVPA, ∆EVPA, of scalarized RN black holes also becomes more pronounced. In regions of weak intensity, the EVPA decreases as α increases, rather than increasing.  \nr  \ny  \ny  \nφ  \nθ  \nFIG. 3. Linear polarization of a synchrotron-emitting ring surrounding RN and scalarized RN black holes for Q = 0 . 4 in a vertical magnetic field. The upper and lower rows correspond to purely clockwise rotation ( χ = -90 · ) and purely radial infall ( χ = -180 · ), respectively. The most prominent intensity flux is concentrated at the bottom of the polarized images.  \n<!-- image -->  \nx', 'B. Vertical Magnetic Field': 'As demonstrated in [28], the polarization pattern of the ring model in the Schwarzschild spacetime with a vertical magnetic field is inconsistent with the observations of M87 ∗ , primarily due to aberration effects. To explore whether this pattern persists for scalarized RN black holes, we analyze cases with a purely vertical magnetic field for scalarized RN black holes with Q = 0 . 4 and 0 . 9, as presented in Figs. 3 and 4, respectively. The upper and lower rows in these figures correspond to fluid flows that are clockwise tangential ( χ = -90 · ) and radially inward ( χ = -180 · ), respectively.  \nThe left column of Figs. 3 and 4 presents the corresponding polarized images, with ticks representing polarization vectors. Consistent with the behavior of Schwarzschild black holes, our numerical simulations demonstrate that the regions of strongest intensity flux in these images are concentrated at the bottom, with the intensity diminishing upward. This observation is further supported by the polarization intensity as a function of φ ρ , as depicted in the second column. Similar to the equatorial magnetic field case, the intensity of scalarized RN black holes and their  \nr  \ny  \ny  \nφ  \nθ  \nFIG. 4. Linear polarization of a synchrotron-emitting ring surrounding RN and scalarized RN black holes for Q = 0 . 9 in a vertical magnetic field. The upper and lower rows correspond to purely clockwise rotation ( χ = -90 · ) and purely radial infall ( χ = -180 · ), respectively. Although the polarization pattern closely resembles the Q = 0 . 4 case, the overall intensity and ∆EVPA are significantly greater.  \n<!-- image -->  \n1 ,  \n0.3 ,  \n90  \nx  \ndeviations from RN black holes are more pronounced for larger Q values. However, the overall intensity of the vertical magnetic field is significantly lower than that of the equatorial magnetic field. The third and fourth columns present the EVPA and ∆EVPA as functions of φ ρ for various α values. While the EVPA exhibits minimal sensitivity to α for Q = 0 . 4, its dependence on α becomes more apparent when Q = 0 . 9. Furthermore, as α increases, the EVPA consistently increases in the purely clockwise rotation configuration. In contrast, for the purely radial infall configuration, as α grows, the EVPA increases in regions where the intensity rises with azimuthal angle φ ρ but decreases in regions where the intensity falls with azimuthal angle φ ρ .', 'IV. CONCLUSIONS': 'In this study, we have examined the linear polarization of synchrotron-emitting rings surrounding both RN and scalarized RN black holes, focusing on their observable characteristics in the presence of equatorial or vertical magnetic fields. Our analysis reveals that the polarization patterns of RN and scalarized RN black holes share qualitative similarities with those of Schwarzschild  \nr  \nB  \n0 ,  \nB  \n0 ,  \nB  \nblack holes. Specifically, we observed flux asymmetry in the polarized images, with the most intense flux region typically concentrated in the upper-right and bottom regions for equatorial and vertical magnetic fields, respectively.  \nFurthermore, we investigated the effects of varying black hole charge Q , coupling constant α and magnetic field configurations on the polarization patterns of scalarized RN black holes. For larger Q values, we observed a significant increase in intensity flux and deviations of the EVPA, further differentiating scalarized RN black holes from their RN counterparts. As α increases, the polarized images exhibit notable changes, including larger apparent radii, higher polarization intensity and greater EVPA deviations. Additionally, equatorial magnetic fields generally yield higher intensity and distinct EVPA variations compared to vertical magnetic fields.  \nIn conclusion, this research contributes to our comprehension of polarization signatures of black holes in the presence of scalar fields and magnetic fields. The insights derived from these simulations may serve as valuable benchmarks for interpreting observational data from existing and future astronomical instruments. Future research endeavors could focus on higher-order images, particularly in the context of double photon spheres. Moreover, scalar clouds surrounding Kerr-Newman black holes and scalarized Kerr-Newman black holes have been recently constructed [81, 82]. Consequently, extending our analysis presented in this paper to the rotating case is highly desirable.', 'ACKNOWLEDGMENTS': "We are grateful to Tianshu Wu for useful discussions and valuable comments. This work is supported in part by NSFC (Grant Nos. 12105191, 12275183, 12275184 and 11875196).  \n- [1] Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole. Astrophys. J. Lett. , 875:L1, 2019. arXiv:1906.11238 , doi:10.3847/2041-8213/ab0ec7 . I\n- [2] Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. II. Array and Instrumentation. Astrophys. J. Lett. , 875(1):L2, 2019. arXiv:1906.11239 , doi:10.3847/2041-8213/ab0c96 .\n- [3] Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. III. Data Processing and Calibration. Astrophys. J. Lett. , 875(1):L3, 2019. arXiv:1906.11240 , doi:10.3847/2041-8213/ab0c57 .\n- [4] Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole. Astrophys. J. Lett. , 875(1):L4, 2019. arXiv:1906.11241 , doi:10.3847/ 2041-8213/ab0e85 .  \n- [5] Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring. Astrophys. J. Lett. , 875(1):L5, 2019. arXiv:1906.11242 , doi:10.3847/2041-8213/ ab0f43 . I\n- [6] Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole. Astrophys. J. Lett. , 875(1):L6, 2019. arXiv:1906.11243 , doi:10.3847/ 2041-8213/ab1141 .\n- [7] Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. VII. Polarization of the Ring. Astrophys. J. Lett. , 910(1):L12, 2021. arXiv:2105.01169 , doi:10.3847/2041-8213/abe71d . III A\n- [8] Kazunori Akiyama et al. First M87 Event Horizon Telescope Results. VIII. Magnetic Field Structure near The Event Horizon. Astrophys. J. Lett. , 910(1):L13, 2021. arXiv:2105.01173 , doi:10.3847/ 2041-8213/abe4de .\n- [9] Kazunori Akiyama et al. First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way. Astrophys. J. Lett. , 930(2):L12, 2022. doi:10.3847/2041-8213/ac6674 .\n- [10] Kazunori Akiyama et al. First Sagittarius A* Event Horizon Telescope Results. II. EHT and Multiwavelength Observations, Data Processing, and Calibration. Astrophys. J. Lett. , 930(2):L13, 2022. doi:10.3847/2041-8213/ac6675 .\n- [11] Kazunori Akiyama et al. First Sagittarius A* Event Horizon Telescope Results. III. Imaging of the Galactic Center Supermassive Black Hole. Astrophys. J. Lett. , 930(2):L14, 2022. doi:10.3847/ 2041-8213/ac6429 .\n- [12] Kazunori Akiyama et al. First Sagittarius A* Event Horizon Telescope Results. IV. Variability, Morphology, and Black Hole Mass. Astrophys. J. Lett. , 930(2):L15, 2022. doi:10.3847/2041-8213/ ac6736 .\n- [13] Kazunori Akiyama et al. First Sagittarius A* Event Horizon Telescope Results. V. Testing Astrophysical Models of the Galactic Center Black Hole. Astrophys. J. Lett. , 930(2):L16, 2022. doi:10.3847/ 2041-8213/ac6672 .\n- [14] Kazunori Akiyama et al. First Sagittarius A* Event Horizon Telescope Results. VI. Testing the Black Hole Metric. Astrophys. J. Lett. , 930(2):L17, 2022. doi:10.3847/2041-8213/ac6756 . I\n- [15] Roman V. Shcherbakov, Robert F. Penna, and Jonathan C. McKinney. Sagittarius A* Accretion Flow and Black Hole Parameters from General Relativistic Dynamical and Polarized Radiative Modeling. Astrophys. J. , 755:133, 2012. arXiv:1007.4832 , doi:10.1088/0004-637X/755/2/133 . I\n- [16] Daniel C. M. Palumbo, George N. Wong, and Ben S. Prather. Discriminating Accretion States via Rotational Symmetry in Simulated Polarimetric Images of M87. Astrophys. J. , 894(2):156, 2020. arXiv: 2004.01751 , doi:10.3847/1538-4357/ab86ac . I\n- [17] Monika Moscibrodzka, Heino Falcke, and Hotaka Shiokawa. General relativistic magnetohydrodynamical simulations of the jet in M 87. Astron. Astrophys. , 586:A38, 2016. arXiv:1510.07243 , doi:10.1051/0004-6361/201526630 . I  \n- [18] Monika Moscibrodzka, Jason Dexter, Jordy Davelaar, and Heino Falcke. Faraday rotation in GRMHD simulations of the jet launching zone of M87. Mon. Not. Roy. Astron. Soc. , 468(2):2214-2221, 2017. arXiv:1703.02390 , doi:10.1093/mnras/stx587 . I\n- [19] Alejandra Jim'enez-Rosales and Jason Dexter. The impact of Faraday effects on polarized black hole images of Sagittarius A*. Mon. Not. Roy. Astron. Soc. , 478(2):1875-1883, 2018. arXiv:1805.02652 , doi:10.1093/mnras/sty1210 . I\n- [20] Jeremy D. Schnittman and Julian H. Krolik. X-ray Polarization from Accreting Black Holes: Coronal Emission. Astrophys. J. , 712:908-924, 2010. arXiv:0912.0907 , doi:10.1088/0004-637X/712/2/908 . I\n- [21] Jason Dexter. A public code for general relativistic, polarised radiative transfer around spinning black holes. Mon. Not. Roy. Astron. Soc. , 462(1):115-136, 2016. arXiv:1602.03184 , doi:10.1093/mnras/ stw1526 .\n- [22] Roman Gold, Jonathan C. McKinney, Michael D. Johnson, and Sheperd S. Doeleman. Probing the Magnetic Field Structure in Sgr A ∗ on Black Hole Horizon Scales with Polarized Radiative Transfer Simulations. Astrophys. J. , 837(2):180, 2017. arXiv:1601.05550 , doi:10.3847/1538-4357/aa6193 .\n- [23] Monika Moscibrodzka. General relativistic polarized radiative transfer with inverse Compton scatterings. Mon. Not. Roy. Astron. Soc. , 491(4):4807-4815, 2020. arXiv:1910.02413 , doi:10.1093/mnras/ stz3329 .\n- [24] R. Abuter et al. Detection of the Schwarzschild precession in the orbit of the star S2 near the Galactic centre massive black hole. Astron. Astrophys. , 636:L5, 2020. arXiv:2004.07187 , doi: 10.1051/0004-6361/202037813 .\n- [25] A. Jim'enez-Rosales et al. Dynamically important magnetic fields near the event horizon of Sgr A*. Astron. Astrophys. , 643:A56, 2020. arXiv:2009.01859 , doi:10.1051/0004-6361/202038283 .\n- [26] Monika Moscibrodzka, Agnieszka Janiuk, and Mariefelicia De Laurentis. Unraveling circular polarimetric images of magnetically arrested accretion flows near event horizon of a black hole. Mon. Not. Roy. Astron. Soc. , 508(3):4282-4296, 2021. arXiv:2103.00267 , doi:10.1093/mnras/stab2790 .\n- [27] Zelin Zhang, Songbai Chen, and Jiliang Jing. Image of Bonnor black dihole with a thin accretion disk and its polarization information. 5 2022. arXiv:2205.13696 . I\n- [28] Kazunori Akiyama et al. The Polarized Image of a Synchrotron-emitting Ring of Gas Orbiting a Black Hole. Astrophys. J. , 912(1):35, 2021. arXiv:2105.01804 , doi:10.3847/1538-4357/abf117 . I, II C, II C, III A, III B\n- [29] R. Abuter et al. Detection of the gravitational redshift in the orbit of the star S2 near the Galactic centre massive black hole. Astron. Astrophys. , 615:L15, 2018. arXiv:1807.09409 , doi:10.1051/0004-6361/ 201833718 . I\n- [30] Zachary Gelles, Elizabeth Himwich, Daniel C. M. Palumbo, and Michael D. Johnson. Polarized image of equatorial emission in the Kerr geometry. Phys. Rev. D , 104(4):044060, 2021. arXiv:2105.09440 , doi:10.1103/PhysRevD.104.044060 . I  \n- [31] Xin Qin, Songbai Chen, and Jiliang Jing. Polarized image of an equatorial emitting ring around a 4D Gauss-Bonnet black hole. Eur. Phys. J. C , 82(9):784, 2022. arXiv:2111.10138 , doi:10.1140/epjc/ s10052-022-10753-8 . I\n- [32] Valentin Delijski, Galin Gyulchev, Petya Nedkova, and Stoytcho Yazadjiev. Polarized image of equatorial emission in horizonless spacetimes: Traversable wormholes. Phys. Rev. D , 106(10):104024, 2022. arXiv:2206.09455 , doi:10.1103/PhysRevD.106.104024 .\n- [33] Xin Qin, Songbai Chen, Zelin Zhang, and Jiliang Jing. Polarized Image of a Rotating Black Hole in Scalar-Tensor-Vector-Gravity Theory. Astrophys. J. , 938(1):2, 2022. arXiv:2207.12034 , doi: 10.3847/1538-4357/ac8f49 .\n- [34] Tsai-Chen Lee, Zezhou Hu, Minyong Guo, and Bin Chen. Circular orbits and polarized images of charged particles orbiting a Kerr black hole with a weak magnetic field. Phys. Rev. D , 108(2):024008, 2023. arXiv:2211.04143 , doi:10.1103/PhysRevD.108.024008 .\n- [35] Hu Zhu and Minyong Guo. Polarized image of synchrotron radiations of hotspots in SchwarzschilldMelvin black hole spacetime. 5 2022. arXiv:2205.04777 .\n- [36] Xueyao Liu, Songbai Chen, and Jiliang Jing. Polarization distribution in the image of a synchrotron emitting ring around a regular black hole. Sci. China Phys. Mech. Astron. , 65(12):120411, 2022. arXiv:2205.00391 , doi:10.1007/s11433-022-1946-2 .\n- [37] Zezhou Hu, Yehui Hou, Haopeng Yan, Minyong Guo, and Bin Chen. Polarized images of synchrotron radiations in curved spacetime. Eur. Phys. J. C , 82(12):1166, 2022. arXiv:2203.02908 , doi:10.1140/ epjc/s10052-022-11144-9 .\n- [38] Xin Qin, Songbai Chen, Zelin Zhang, and Jiliang Jing. Polarized image of a rotating black hole surrounded by a cold dark matter halo. Eur. Phys. J. C , 83(2):159, 2023. arXiv:2301.01551 , doi: 10.1140/epjc/s10052-023-11300-9 .\n- [39] Valentin Deliyski, Galin Gyulchev, Petya Nedkova, and Stoytcho Yazadjiev. Polarized image of equatorial emission in horizonless spacetimes: Naked singularities. Phys. Rev. D , 108(10):104049, 2023. arXiv:2303.14756 , doi:10.1103/PhysRevD.108.104049 . I\n- [40] Carlos A.R. Herdeiro, Eugen Radu, Nicolas Sanchis-Gual, and Jos'e A. Font. Spontaneous Scalarization of Charged Black Holes. Phys. Rev. Lett. , 121(10):101102, 2018. arXiv:1806.05190 , doi:10.1103/ PhysRevLett.121.101102 . I, II A, II A, II A\n- [41] Yun Soo Myung and De-Cheng Zou. Quasinormal modes of scalarized black holes in the Einstein-Maxwell-Scalar theory. Phys. Lett. B , 790:400-407, 2019. arXiv:1812.03604 , doi:10.1016/ j.physletb.2019.01.046 . I\n- [42] Yun Soo Myung and De-Cheng Zou. Instability of Reissner-Nordstrom black hole in EinsteinMaxwell-scalar theory. Eur. Phys. J. C , 79(3):273, 2019. arXiv:1808.02609 , doi:10.1140/epjc/ s10052-019-6792-6 .\n- [43] Yves Brihaye, Carlos Herdeiro, and Eugen Radu. Black Hole Spontaneous Scalarisation with a Positive Cosmological Constant. Phys. Lett. B , 802:135269, 2020. arXiv:1910.05286 , doi:10.1016/j.", 'physletb.2020.135269 .': "- [44] Yves Brihaye, Betti Hartmann, Nath'alia Pio Aprile, and Jon Urrestilla. Scalarization of asymptotically anti-de Sitter black holes with applications to holographic phase transitions. Phys. Rev. D , 101(12):124016, 2020. arXiv:1911.01950 , doi:10.1103/PhysRevD.101.124016 .\n- [45] Pedro G.S. Fernandes, Carlos A.R. Herdeiro, Alexandre M. Pombo, Eugen Radu, and Nicolas SanchisGual. Charged black holes with axionic-type couplings: Classes of solutions and dynamical scalarization. Phys. Rev. D , 100(8):084045, 2019. arXiv:1908.00037 , doi:10.1103/PhysRevD.100.084045 .\n- [46] Pedro G. S. Fernandes, Carlos A. R. Herdeiro, Alexandre M. Pombo, Eugen Radu, and Nicolas SanchisGual. Spontaneous Scalarisation of Charged Black Holes: Coupling Dependence and Dynamical Features. Class. Quant. Grav. , 36(13):134002, 2019. [Erratum: Class.Quant.Grav. 37, 049501 (2020)]. arXiv:1902.05079 , doi:10.1088/1361-6382/ab23a1 . II A\n- [47] Yun Soo Myung and De-Cheng Zou. Stability of scalarized charged black holes in the Einstein-Maxwell-Scalar theory. Eur. Phys. J. C , 79(8):641, 2019. arXiv:1904.09864 , doi:10.1140/ epjc/s10052-019-7176-7 .\n- [48] Yan Peng. Scalarization of horizonless reflecting stars: neutral scalar fields non-minimally coupled to Maxwell fields. Phys. Lett. B , 804:135372, 2020. arXiv:1912.11989 , doi:10.1016/j.physletb.2020. 135372 .\n- [49] De-Cheng Zou and Yun Soo Myung. Scalarized charged black holes with scalar mass term. Phys. Rev. D , 100(12):124055, 2019. arXiv:1909.11859 , doi:10.1103/PhysRevD.100.124055 .\n- [50] Dumitru Astefanesei, Carlos Herdeiro, Jo˜ao Oliveira, and Eugen Radu. Higher dimensional black hole scalarization. JHEP , 09:186, 2020. arXiv:2007.04153 , doi:10.1007/JHEP09(2020)186 .\n- [51] Jose Luis Bl'azquez-Salcedo, Carlos A.R. Herdeiro, Sarah Kahlen, Jutta Kunz, Alexandre M. Pombo, and Eugen Radu. Quasinormal modes of hot, cold and bald Einstein-Maxwell-scalar black holes. 8 2020. arXiv:2008.11744 .\n- [52] Jose Luis Bl'azquez-Salcedo, Carlos A.R. Herdeiro, Jutta Kunz, Alexandre M. Pombo, and Eugen Radu. Einstein-Maxwell-scalar black holes: the hot, the cold and the bald. Phys. Lett. B , 806:135493, 2020. arXiv:2002.00963 , doi:10.1016/j.physletb.2020.135493 .\n- [53] Pedro G.S. Fernandes. Einstein-Maxwell-scalar black holes with massive and self-interacting scalar hair. Phys. Dark Univ. , 30:100716, 2020. arXiv:2003.01045 , doi:10.1016/j.dark.2020.100716 .\n- [54] Hong Guo, Xiao-Mei Kuang, Eleftherios Papantonopoulos, and Bin Wang. Topology and spacetime structure influences on black hole scalarization. 12 2020. arXiv:2012.11844 .\n- [55] Zhan-Feng Mai and Run-Qiu Yang. Stability analysis of a charged black hole with a nonlinear complex scalar field. Phys. Rev. D , 104(4):044008, 2021. arXiv:2101.00026 , doi:10.1103/PhysRevD.104. 044008 .\n- [56] Yun Soo Myung and De-Cheng Zou. Scalarized charged black holes in the Einstein-Maxwell-Scalar theory with two U(1) fields. Phys. Lett. B , 811:135905, 2020. arXiv:2009.05193 , doi:10.1016/j. physletb.2020.135905 .  \n- [57] Yun Soo Myung and De-Cheng Zou. Scalarized black holes in the Einstein-Maxwell-scalar theory with a quasitopological term. Phys. Rev. D , 103(2):024010, 2021. arXiv:2011.09665 , doi:10.1103/ PhysRevD.103.024010 .\n- [58] Yun Soo Myung and De-Cheng Zou. Onset of rotating scalarized black holes in Einstein-Chern-SimonsScalar theory. Phys. Lett. B , 814:136081, 2021. arXiv:2012.02375 , doi:10.1016/j.physletb.2021. 136081 .\n- [59] De-Cheng Zou and Yun Soo Myung. Radial perturbations of the scalarized black holes in EinsteinMaxwell-conformally coupled scalar theory. Phys. Rev. D , 102(6):064011, 2020. arXiv:2005.06677 , doi:10.1103/PhysRevD.102.064011 .\n- [60] Guangzhou Guo, Peng Wang, Houwen Wu, and Haitang Yang. Scalarized Einstein-Maxwell-scalar black holes in anti-de Sitter spacetime. Eur. Phys. J. C , 81(10):864, 2021. arXiv:2102.04015 , doi: 10.1140/epjc/s10052-021-09614-7 .\n- [61] Peng Wang, Houwen Wu, and Haitang Yang. Scalarized Einstein-Born-Infeld black holes. Phys. Rev. D , 103(10):104012, 2021. arXiv:2012.01066 , doi:10.1103/PhysRevD.103.104012 .\n- [62] Cheng-Yong Zhang, Peng Liu, Yunqi Liu, Chao Niu, and Bin Wang. Dynamical charged black hole spontaneous scalarization in anti-de Sitter spacetimes. Phys. Rev. D , 104(8):084089, 2021. arXiv: 2103.13599 , doi:10.1103/PhysRevD.104.084089 .\n- [63] Cheng-Yong Zhang, Qian Chen, Yunqi Liu, Wen-Kun Luo, Yu Tian, and Bin Wang. Critical Phenomena in Dynamical Scalarization of Charged Black Holes. Phys. Rev. Lett. , 128(16):161105, 2022. arXiv: 2112.07455 , doi:10.1103/PhysRevLett.128.161105 .\n- [64] Cheng-Yong Zhang, Qian Chen, Yunqi Liu, Wen-Kun Luo, Yu Tian, and Bin Wang. Dynamical transitions in scalarization and descalarization through black hole accretion. Phys. Rev. D , 106(6):L061501, 2022. arXiv:2204.09260 , doi:10.1103/PhysRevD.106.L061501 .\n- [65] Jia-Yan Jiang, Qian Chen, Yunqi Liu, Yu Tian, Wei Xiong, Cheng-Yong Zhang, and Bin Wang. Type I critical dynamical scalarization and descalarization in Einstein-Maxwell-scalar theory. Sci. China Phys. Mech. Astron. , 67(2):220411, 2024. arXiv:2306.10371 , doi:10.1007/s11433-023-2231-5 .\n- [66] Qian Chen, Zhuan Ning, Yu Tian, Bin Wang, and Cheng-Yong Zhang. Nonlinear dynamics of hot, cold, and bald Einstein-Maxwell-scalar black holes in AdS spacetime. Phys. Rev. D , 108(8):084016, 2023. arXiv:2307.03060 , doi:10.1103/PhysRevD.108.084016 . I\n- [67] Qingyu Gan, Peng Wang, Houwen Wu, and Haitang Yang. Photon spheres and spherical accretion image of a hairy black hole. Phys. Rev. D , 104(2):024003, 2021. arXiv:2104.08703 , doi:10.1103/ PhysRevD.104.024003 . I, II A\n- [68] Qingyu Gan, Peng Wang, Houwen Wu, and Haitang Yang. Photon ring and observational appearance of a hairy black hole. Phys. Rev. D , 104(4):044049, 2021. arXiv:2105.11770 , doi:10.1103/PhysRevD. 104.044049 . I, II A\n- [69] Guangzhou Guo, Xin Jiang, Peng Wang, and Houwen Wu. Gravitational lensing by black holes with multiple photon spheres. Phys. Rev. D , 105(12):124064, 2022. arXiv:2204.13948 , doi:10.1103/", 'PhysRevD.105.124064 .': '- [70] Yiqian Chen, Guangzhou Guo, Peng Wang, Houwen Wu, and Haitang Yang. Appearance of an infalling star in black holes with multiple photon spheres. Sci. China Phys. Mech. Astron. , 65(12):120412, 2022. arXiv:2206.13705 , doi:10.1007/s11433-022-1986-x .\n- [71] Yiqian Chen, Peng Wang, and Haitang Yang. Interferometric Signatures of Black Holes with Multiple Photon Spheres. 12 2023. arXiv:2312.10304 .\n- [72] Yiqian Chen, Peng Wang, and Haitang Yang. Observations of orbiting hot spots around scalarized Reissner-Nordstrom black holes. Eur. Phys. J. C , 84(3):270, 2024. arXiv:2401.10905 , doi:10.1140/ epjc/s10052-024-12635-7 . II A, III\n- [73] Guangzhou Guo, Peng Wang, Houwen Wu, and Haitang Yang. Quasinormal modes of black holes with multiple photon spheres. JHEP , 06:060, 2022. arXiv:2112.14133 , doi:10.1007/JHEP06(2022)060 . II A\n- [74] Guangzhou Guo, Peng Wang, Houwen Wu, and Haitang Yang. Echoes from hairy black holes. JHEP , 06:073, 2022. arXiv:2204.00982 , doi:10.1007/JHEP06(2022)073 . II A\n- [75] Guangzhou Guo, Peng Wang, Houwen Wu, and Haitang Yang. Superradiance instabilities of charged black holes in Einstein-Maxwell-scalar theory. JHEP , 07:070, 2023. arXiv:2301.06483 , doi:10.1007/ JHEP07(2023)070 . II A\n- [76] Guangzhou Guo, Peng Wang, and Yupeng Zhang. Nonlinear Stability of Black Holes with a Stable Light Ring. 3 2024. arXiv:2403.02089 . II A\n- [77] Guangzhou Guo, Yuhang Lu, Peng Wang, Houwen Wu, and Haitang Yang. Black holes with multiple photon spheres. Phys. Rev. D , 107(12):124037, 2023. arXiv:2212.12901 , doi:10.1103/PhysRevD. 107.124037 . II A\n- [78] Dominik J. Schwarz, Jishnu Goswami, and Aritra Basu. Geometric optics in the presence of axionlike particles in curved spacetime. Phys. Rev. D , 103(8):L081306, 2021. arXiv:2003.10205 , doi:10.1103/ PhysRevD.103.L081306 . II B\n- [79] Pierre Fleury. Light propagation in inhomogeneous and anisotropic cosmologies . PhD thesis, Paris U., VI, IAP, 2015. arXiv:1511.03702 . II B\n- [80] Pedro V. P. Cunha, Carlos A. R. Herdeiro, Eugen Radu, and Helgi F. Runarsson. Shadows of Kerr black holes with and without scalar hair. Int. J. Mod. Phys. D , 25(09):1641021, 2016. arXiv:1605.08293 , doi:10.1142/S0218271816410212 . II C\n- [81] Guangzhou Guo, Peng Wang, Houwen Wu, and Haitang Yang. Scalarized Kerr-Newman black holes. JHEP , 10:076, 2023. arXiv:2307.12210 , doi:10.1007/JHEP10(2023)076 . IV\n- [82] Guangzhou Guo, Peng Wang, Tianshu Wu, and Haitang Yang. Stationary Scalar Clouds around KerrNewman Black Holes. 8 2024. arXiv:2408.09243 . IV'}