Motivated by the observation of the doubly charmed tetraquark $ T_{cc}(3875)^+ $ , we present a systematic study of doubly heavy tetraquarks ( $ T_{QQ'\bar{q}\bar{q}'} $ ) using heavy antiquark-diquark symmetry (HADS) within a constituent quark model. By calibrating model parameters to known hadron spectra and incorporating the effective mass formula, we predict the masses for 38 ground-state tetraquarks with $ cc $ , $ bb $ , and $ bc $ heavy quark pairs, including the non-strange, single-strange, and double-strange configurations with quantum numbers $ J^P = 0^+, 1^+ $ and $ 2^+ $ . Notably, we identify several stable states below the relevant meson-meson thresholds, particularly in the $ bb\bar{q}\bar{q}' $ sector. The explicit connection between the doubly heavy tetraquark and the heavy baryon spectra through HADS reduces model dependence and reveals the fundamental systematics in the heavy-quark hadron landscape.

China

Liu-Yu Zhang

Tian-Wei Wu

Yong-Liang Ma

Bridging doubly heavy tetraquark mass spectrum with heavy baryons utilizing heavy antiquark-diquark symmetry * The work of Tian-Wei Wu is supported by the National Natural Science Foundation of China (12405108). Yong-Liang Ma is supported in part by the National Science Foundation of China (NSFC) (12347103) and Gusu Talent Innovation Program (ZXL2024363)

The photon sphere, a hypersurface of null circular geodesics, plays a fundamental role in characterizing black hole spacetimes, influencing phenomena such as black hole shadows, gravitational lensing, and quasinormal modes. In this work, we derive both upper and lower bounds on the photon sphere radius for static, spherically symmetric, asymptotically flat black holes within $\textit{n}$-dimensional Einstein gravity $$(n\ge 4)$$ , assuming an anisotropic matter field satisfying the weak energy condition and a non-positive trace of the energy–momentum tensor. For the upper bound, we obtain $$r_\gamma \le [(n-1)M]^{\frac{1}{n-3}},$$ where $\textit{M}$ is the ADM mass. In the four-dimensional case $$(n=4)$$ , this reduces to $$r_\gamma \le 3M,$$ in agreement with previous results. For the lower bound, under the additional assumption that $$|r^{n-1}p_r(r)|$$ is monotonically decreasing, we prove $$r_\gamma \ge (\frac{n-1}{2})^{1/(n-3)}r_H,$$ where $$r_H$$ is the radius of the outer event horizon; for $$n=4$$ this gives $$r_\gamma \ge \frac{3}{2}r_H,$$ also consistent with previous four-dimensional result. These results provide dimension-dependent geometric constraints that generalize well-known four-dimensional bounds to a specific class of higher-dimensional black holes (described by a Tangherlini-type metric) and deepen our understanding of spacetime structure in higher-dimensional gravitational theories.

Yong Song

Jiaqi Fu

Yiting Cen

Bounds on the photon sphere radius for spherically symmetric black holes in $${\varvec{n}}$$ -dimensional Einstein gravity

The spin-curvature coupling in the Mathisson–Papapetrou–Dixon (MPD) formalism induces non-geodesic motion, shifting the orbital parameters of spinning test particles in black hole spacetimes. We investigate whether these quantitative shifts alter the qualitative, global structure of the orbit manifold. Using a topological approach, we study timelike circular orbits (TCOs) for spinning particles in static, spherically symmetric spacetimes. By constructing an auxiliary vector field, we compute the topological winding number $\textit{W}$ in horizon-bounded regions of asymptotically flat, anti-de Sitter (AdS), and de Sitter (dS) backgrounds. We find that $\textit{W}$ is robust against both the magnitude and direction of the particle’s spin: between two horizons, $$W = -1,$$ guaranteeing at least one unstable TCO; outside the outermost horizon in asymptotically flat and AdS spacetimes, $$W = 0,$$ enforcing that TCOs must appear in stable–unstable pairs or be absent. This spin independence reveals that the fundamental orbital structure is a property of spacetime geometry itself, not of the particle’s spin. We validate this with quantitative examples in Schwarzschild, Schwarzschild–AdS, and Schwarzschild–dS spacetimes, showing explicit spin-induced TCO shifts while confirming the invariant topology. This result provides a topological foundation for interpreting gravitational waveforms from extreme mass-ratio inspirals involving spinning secondaries.

Robust topological invariants of timelike circular orbits for spinning test particles in black hole spacetimes

The study of circular orbits offers profound insights into the structure of spacetime around black holes. While the topological properties of these orbits are well-established for neutral particles, the influence of electric charge–particularly for massless particles–remains a subject of exploration. In this work, we employ a topological current $$\phi $$ -mapping approach to systematically investigate the circular orbits of charged test particles in static, spherically symmetric black hole spacetimes with flat, anti-de Sitter (AdS), and de Sitter (dS) asymptotics. We demonstrate that the particle’s charge significantly alters the topological classification of both timelike and null circular orbits. A key finding is that for multi-horizon black holes, if a circular orbit with fixed angular momentum and charge exists between two neighboring horizons, there will always be at least one unstable null and one unstable timelike circular orbit. Outside the outermost horizon, the asymptotic behavior of spacetime and the specific charge ratio crucially determine the topological charge $\textit{W}$, dictating the existence and stability of orbits. Our results, validated through Reissner–Nordström (RN), RN-AdS, and RN-dS examples, extend the topological orbit classification framework and provide a foundation for potential applications in environments where effective charge dynamics may be relevant, such as magnetized plasmas around black holes.

Jia Li

Kai Diao

Xiaofeng Zhao

Shunping Shi

A unified topological classification of circular orbits for charged particles in black hole spacetimes

The study of circular orbits is fundamental in gravitational physics, yet their definition in dynamical spacetimes remains challenging due to the lack of temporal symmetry. In this work, we establish a unified framework by commencing from the geometrically invariant quasi-local definition of a particle surface. We demonstrate that this definition naturally leads to a set of conditions that can be recast into the language of a coordinate-dependent dynamical potential. This potential serves as a practical computational tool for locating evolving circular orbits within a specific coordinate system. We rigorously prove the equivalence between the quasi-local and dynamical potential approaches in dynamical spherically symmetric spacetimes. The efficacy and self-consistency of the dynamical potential method are explicitly verified through its application to the Oppenheimer–Snyder dust collapse model, where it correctly reproduces the established evolution equations for null and timelike circular orbits. This work bridges the gap between abstract geometric definitions and concrete calculations, providing a robust and adaptable framework for analyzing orbital dynamics in time-dependent gravitational fields.

From quasi-local definitions to a dynamical potential: a unified framework for evolving circular orbits in dynamical spacetimes

In this work, we establish the existence conditions and a universal upper bound on the radius of stable photon spheres (SPS) outside the event horizons of static, spherically symmetric, asymptotically flat black holes surrounded by matter fields. We prove that stable photon spheres exist if the external matter satisfies specific conditions. Furthermore, under the additional assumption of a monotonically decreasing mass-radius ratio $$m(r)/r^3$$ outside the horizon, we derive a strict upper bound on the radius $$r_{\textrm{sps}}$$ of any stable photon sphere: $$r_{\textrm{sps}}<6M$$ , where $\textit{M}$ is the asymptotic mass of the black hole. This bound is independent of specific black hole solutions and broadly applies to hairy black holes and other configurations with external matter fields meeting the stated energy conditions. Our results resolve fundamental questions regarding the existence and spatial constraints on stable photon orbits, with implications for gravitational lensing, accretion disk dynamics (e.g., the Aschenbach effect), and black hole shadow observations.

The existence and upper bound for stable photon spheres in static spherically symmetric black holes

Chuanyu Zhang

Erratum to: Quasi-local studies of the particle surfaces and their stability in general spacetimes

In this work, we investigate a universal upper bound on the radius of the innermost stable circular orbit (ISCOs) for massive particles in static, spherically symmetric, and asymptotically flat black hole spacetimes. By analyzing the spacetime metrics with external matter fields, we derive the characteristic equation for ISCO via the effective potential method. By imposing appropriate energy conditions for the matter fields, we rigorously demonstrate that the ISCO radius is bounded by <math altimg='si1.svg'><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant='normal'>ISCO</mi></mrow></msub><mo>&#8804;</mo><mn>6</mn><mi>M</mi></math>, where <italic>M</italic> is the total ADM mass of the black hole. The Schwarzschild black hole saturates this bound ( <math altimg='si2.svg'><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant='normal'>ISCO</mi></mrow></msub><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>6</mn><mi>M</mi></math>), and the Reissner-Nordstr&#246;m black hole, supergravity black holes, fluid sphere models, which satisfy the imposed energy conditions, also obey <math altimg='si1.svg'><msub><mrow><mi>r</mi></mrow><mrow><mi mathvariant='normal'>ISCO</mi></mrow></msub><mo>&#8804;</mo><mn>6</mn><mi>M</mi></math>. The universality of this upper limit provides a theoretical benchmark for observational astrophysics: deviations from 6 <italic>M</italic> in accretion disk observations or gravitational wave signals could indicate the presence of exotic matter fields. This work highlights the interplay between black hole geometry and external matter fields, paving the way for future studies in compact object dynamics.

Upper bound on the radius of the innermost stable circular orbit of black holes

In this work, inspired by the definition of the photon surface given by Claudel, Virbhadra, and Ellis, we give an alternative quasi-local definition to study the circular orbits of single-pole particles. This definition does not only apply to photons but also to massive point particles. For the case of photons in spherically symmetric spacetime, it will give a photon surface equivalent to the result of Claudel, Virbhadra, and Ellis. Meanwhile, in general static and stationary spacetime, this definition can be regarded as a quasi-local form of the effective potential method. However, unlike the effective potential method which can not define the effective potential in dynamical spacetime, this definition can be applied to dynamical spacetime. Further, we generalize this definition directly to the case of pole–dipole particles. In static spherical symmetry spacetime, we verify the correctness of this generalization by comparing the results obtained by the effective potential method.

Leilei Tang

Jiabao Hu

The particle surface of spinning test particles

In this paper, enlightened by the definition of the photon surface given by Claudel, Virbhadra and Ellis, we give a quasi-local definition of the particle surface. From this definition, one can study the evolution of the circular orbits in general spacetime. Especially, we pointed out that this definition can be used to get the spherical circular orbits in stationary spacetimes which cannot be got by the definition of Claudel, Virbhadra and Ellis. Further, we give a condition to exclude the particle surface in spacetime without gravity. Simultaneously, we give a quasi-local definition of the stability of the particle surface in general spacetime. From this definition, one can get the evolution equation of the innermost stable circular orbit (ISCO) in general spacetime. To verify the correctness of these definitions, we studied the circular orbits in some special cases and the results are all consistent with the previous results.