We study the symmetries and conserved quantities in $$f(R)$$ gravity for the static, spherically symmetric Reissner–Nordström spacetime using two complementary frameworks: Noether symmetries and Mei symmetries. Starting from a canonical Lagrangian for radial metric functions and the curvature scalar $$R$$ , we derive the associated Hamiltonian and show that the Legendre map is regular whenever both the first derivative of $\textit{f}$($\textit{R}$) with respect to $\textit{R}$ and the second derivative with respect to $\textit{R}$ is non-zero. Within Noether’s approach (variational and Lie-derivative forms), we obtain general, canonical, and internal symmetry classes and identify explicit generators; for the quadratic model $$f(R)=R^{2}$$ these include radial translations and scaling symmetries. We then formulate Mei symmetry conditions as invariance of the Euler–Lagrange equations under the first prolongation, which yields an overdetermined partial differential equation (PDE) system for the generator components. Solving this system for $$f(R)=R^{2}$$ , we find eight independent Mei generators and construct the corresponding conserved currents, some with no direct Noether analogue. The analysis demonstrates that Mei symmetries extend the standard Noether framework for higher-order Lagrangians and provide additional conserved quantities relevant to black-hole dynamics in modified gravity. We conclude with a comparison of the two symmetry schemes and outline applications to broader $$f(R)$$ models and to rotating spacetimes.

Egypt

Tahia Dabash

United States

Moataz Emam

Germany

Lukas Schöppner

Symmetry and conserved quantities in $$ f(R) $$ -gravity: Mei vs. Noether approaches

We consider the slowly rotating black hole with non-flat, non-(A)dS asymptotics and the rotating black holes with AdS asymptotics in the Einstein-Maxwell-dilaton (EMD) theory, and compare the shadows for such black holes with the Kerr-Newman-AdS black holes based on the numerical backward ray-tracing method. We confirm that in the small rotation limits all kinds of non-extreme black holes have similar shadow shapes but for extremal black holes, the shadow shapes in EMD theory are different with the Kerr-Newman-AdS black holes for certain spin parameters. Due to slowly rotating approximation we analytically calculate the photon geodesics using the perturbation method. We find that the similarity of the shadow shapes between EMD black holes and Kerr-Newman-AdS black holes can only happen for small enough spin parameters restricted by the black hole mass.

China

Yong-Zhuang Li

Xiao-Mei Kuang

Effects of (un)usual asymptotics on shadows and lensing of rotating black holes: Numerical and perturbative treatments

In this article, the shadow and the quasinormal modes (QNMs) of an exact asymptotically flat hairy electrically charged black hole solution with a dilaton potential are investigated. Using the constraint equation among the integration constant $$\eta $$ of the gravitational field, the mass $\textit{M}$, the electric charge $\textit{Q}$ and the coupling constant $$\nu $$ between the $\textit{U}$(1) field and the dilaton field, we find that the shadow radii, the Lyapunov exponent $$\lambda $$ and the coordinate angular velocity $$\Omega _{c}$$ only significantly affected by $$\nu $$ if the $\textit{Q}$ is close to the extremal value. Especially, near the $$\nu =1$$ critical point the phenomenon exhibits singular enhancement. Furthermore, the QNMs are numerically computed by using the Hatsuda method and verified with the higher-order WKB approximations with the Padé summation. We find that the QNMs are close to that of the low energy limit of the string theory when $$\nu $$ is large enough. In the eikonal limit, the real and imaginary parts are proved to be approximated by $$\Omega _{c}$$ and $$\lambda $$ , respectively.

Sheng-Han Xiong

Poland

Jerzy Matyjasek

The shadow and quasinormal modes of the asymptotically flat hairy black holes with a dilaton potential

We revisit the large-small black hole phase transitions in Reissner–Nordström (RN) AdS black holes coupled with perfect fluid dark matter. Besides the free energy, we examine the features of Lyapunov exponents for both massive and massless particles, which can characterize the chaos and instability of a particle’s motion orbiting the black hole. The presence of dark matter increases the critical temperature, making it more challenging for the phase transition to occur in comparison to an RN AdS black hole. Our analysis reveals that during the transition in phase structure, the Lyapunov exponents manifest multivalued behavior, with exclusive branches associated with distinct phases of black holes. Additionally, we observe that the discontinuous changes in the Lyapunov exponent act as an order parameter, characterized by a critical exponent of 1/2 near the critical point. These phenomena are similar to those found in RN AdS black holes and further suggest that the Lyapunov exponent can provide a valuable framework for probing the phase structure of black holes.

R. Ali

Probing thermodynamic phase transitions via Lyapunov exponent in AdS black hole with perfect fluid dark matter

A future $ e^+e^- $ collider could run at the $\textit{Z}$-pole to perform important electroweak (EW) precision measurements, but such a run may not be viable for a future muon collider. However, this can be compensated for by measuring other EW processes utilizing the high energy and large luminosity of the muon collider. In this paper, we consider the measurements of the vector boson fusion processes of $ WW/WZ/W\gamma $ to a pair of fermions (along with a $ \nu_{\mu}\bar{\nu}_{\mu} $ or $ \nu_{\mu}\mu^+/\bar{\nu}_{\mu}\mu^- $ pair) at a high-energy muon collider and study their potential for probing EW observables. We consider two run scenarios for the muon collider with center-of-mass energies of 10 and 30 TeV, respectively, and focus on the processes involving $ f=b,c,\tau $ and the dimension-6 operators that directly modify the corresponding fermions coupling to the $ Z/W $ bosons. The invariant mass distribution of the $ f\bar{f} $ pair helps to separate the events from the $ Z/W $ and high-energy resonances, whereas the polar angle of the outing fermion provides additional information. Through a chi-squared analysis on the binned distributions and combining the information from the $\textit{WW}$ and $ WZ/W\gamma $ fusion processes, all relevant Wilson coefficients can be constrained simultaneously. The precision surpasses the current EW measurement constraints and is even competitive with future $ e^+e^- $ colliders. Our analysis can be included in a more complete framework that is required to fully determine the potential of muon colliders in EW precision measurements.

Hao-Qiao Li

Hai-Ning Yan

Jiayin Gu

Xiao-Ze Tan

Probing $\textit{Z}$/$\textit{W}$ pole physics at high-energy muon colliders via vector-boson-fusion processes * Supported by the National Natural Science Foundation of China (NSFC) (12035008, 12375091, 12347171), and the Innovation Program for Quantum Science and Technology (2024ZD0300101). Xiao-Ze Tan also acknowledges the support from Helmholtz – OCPC (Office of China Postdoctoral Council) Postdoctoral Fellowship Program

This paper investigates the observational appearances of an inner extremal regular black hole (IERBH) illuminated by various types of accretion models. The study reveals that when the BH is illuminated by specific accretion flows, the effects of quantum gravity become more pronounced, significantly impacting key observational features such as the shadow radius, photon ring, and total observed intensity. Specifically, the introduction of a more realistic radially infalling spherical accretion flow further accentuates these differences. This dynamic flow results in a darker central region in the BH image due to the Doppler effect, which modulates the observed intensity based on the relative motion of the infalling matter. The shadow radius and total observed intensity are notably affected by the quantum correction parameters, providing additional signatures that distinguish regular BHs from their classical counterparts.

Dan Zhang

Guoyang Fu

Xi-Jing Wang

Qiyuan Pan

Jian-Pin Wu

Observational appearances of an inner extremal regular black hole illuminated by various accretion flows

We study $ \cos2\phi $ azimuthal asymmetry in doubly longitudinally polarized proton-proton Drell-Yan collisions within the transverse momentum dependent factorization framework. The asymmetry arises from the convolution of the longitudinal transversity distribution $ h_{1L}^{\perp} $ for both protons. Using the Bacchetta-Delcarro-Pisano-Radici-Signori parameterization for the nonperturbative Sudakov form factor and the Wandzura-Wilczek approximation for collinear $ h_{1L}^{\perp} $ , we predict the double spin asymmetry $ A_{LL}^{\cos2\phi} $ at RHIC and NICA kinematics. Our results demonstrate sensitivity to sea quark distributions, with the asymmetry reaching up to 25% for maximal sea quark contributions. These predictions highlight the potential of polarized Drell-Yan measurements to probe sea quark dynamics and advance our understanding of nucleon structures.

Hui Li

Ting Hu

Liang-Liang Liu

Xiao-Yu Wang

Zhun Lu

Asymmetry ${{\boldsymbol A}_{\boldsymbol{LL}}^{\bf\cos2\boldsymbol\phi}} $ in the polarized proton-proton Drell-Yan process within TMD factorization * Hui Li is supported by Fundamental Research Program of Shanxi Province (202203021222224) and Natural Science Foundation of Shanxi Normal University (JCYJ2023021). Xiao-Yu Wang is supported by the Natural Science Foundation of Henan Province (242300421377, 232300421140)

The Einstein-Maxwell-dilaton model exhibits a first-order phase transition curve that terminates at a holographic critical endpoint, offering intriguing insights into the phase diagram of the dual system living on the boundary. However, the specific instability within the underlying spacetime that triggers the formation of the hairy black hole remains somewhat obscure. This raises the question of whether one of the hairy black hole phases represents a superconducting or scalarized state and how the two distinct phases merge into a indistinguishable one at the critical point. This work investigates the associated black hole phase transition and the underlying instabilities by exploring a specific Einstein-Maxwell-dilaton model. The approach aims to provide transparent insights into the black hole phase transitions in the bulk. By introducing a nonminimal coupling between a massive real scalar field and a Maxwell field in a five-dimensional anti-de Sitter spacetime, we identify two types of scalarization corresponding to tachyonic instabilities in the ultraviolet and infrared regions. These distinct instabilities lead to a first-order phase transition between two phases in the $\textit{μ}$ – $\textit{T}$ phase diagram. Furthermore, this first-order transition terminates at a critical point, beyond which the curve turns back, and the transition becomes a numerically elusive third-order one. Although one does not encounter a critical endpoint, the model still offers a consistent interpretation for the observed $\textit{cross-over}$ in the low baryon density region. We analyze the thermodynamic properties of the scalarized hairy black holes and discuss the implications of our findings.

Brazil

Hong Guo

Wei-Liang Qian

The underlying black hole phase transitions in an Einstein-Maxwell-dilaton model with a holographic critical point

In this paper, we investigate the shadow and optical appearance of the hairy Kerr black hole illuminated by a thin accretion disk. The deformation parameter $$\alpha $$ and hair parameter $$l_o$$ are found to influence the motions of accretion as well as the redshift effect of the photon, such that they significantly affect the shadow and image of the hairy Kerr black hole. In particular, $$l_o$$ suppresses the shadow size of the black hole while $$\alpha $$ enhances it. We then analyze the widths of the direct, lensed ring and photon ring emission regions for the thin accretion disk. It is found that the widths of the direct emission and lensed ring emission change slightly with the parameters, whereas the width of the photon ring emission increases dramatically as $$l_o$$ decreases. This feature provides more favorable conditions for observing the photon rings of black hole. Finally, we study the redshift of the direct and lensed images, showing that the maximal blueshift point is shifted by the change of the parameters. We argue that our study provides a preliminary theoretical prediction that the image of the hairy Kerr black hole, especially the photon ring structure, may be used to constrain the hair parameters with future high-precision astronomical observation.

Yuan Meng

Effects of hair on the image of a rotating black hole illuminated by a thin accretion disk

We study the precession features of a test particle and gyroscope attached to a stationary observer in Horndeski rotating spacetime. First, we study the precession orbit, Lense-Thirring (LT) precession as well as periastron precession frequencies of the test particle, which are helpful to test the strong gravity of the central object and to distinguish Horndeski rotating black holes from naked singularities. We find that both LT frequency and periastron precession frequency of Horndeski rotating black hole and naked singularity behave differently when the particle's orbit approaches the innermost stable circular orbit. Also, the effects of Horndeski parameter on the two precession frequencies are explored. Second, we investigate the spin precession of a test gyroscope in the Horndeski rotating spacetime by analyzing the LT frequency, the geodetic precession frequency, and the general spin precession frequency for a stationary observer in the spacetime. The angle, spin parameter and Horndeski parameter modify all these frequencies, which make the results shift from those in Kerr spacetimes. Furthermore, these precession frequencies for the Horndeski rotating black hole and naked singularity are distinguishable. In particular, as the observer gets close to the central object in an arbitrary direction, the frequencies of Horndeski black hole grow significantly and become divergent, while for naked singularity they are finite. These theoretical results in some sense provide a potential method to “detect” event horizon, and to distinguish Horndeski theory from Einstein's general relativity.

Wen-Qi Zhen

Meng-He Wu

Orbital precession and Lense-Thirring effect of Horndeski rotating spacetimes