We present a systematic asymptotic small-[Formula: see text] expansion of the Liouville DOZZ three-point structure constant in the light-operator regime [Formula: see text] in the limit [Formula: see text]. In this limit the DOZZ structure constant admits the asymptotic representation [Formula: see text] where [Formula: see text] is an explicitly determined prefactor. Using Thorn’s asymptotic expansion of the [Formula: see text]-function, we derive explicit closed-form expressions for the first coefficients [Formula: see text] and show that they form a symmetric polynomial structure in the variables [Formula: see text]. The resulting asymptotic expansion provides explicit finite-[Formula: see text] corrections to the leading light-limit Liouville three-point data and furnishes a practical framework for applications in celestial holography. In particular, it provides Liouville data relevant for a perturbative program aimed at studying finite-[Formula: see text] corrections to celestial Yang–Mills amplitudes. Finally, we formulate such a perturbative Liouville framework and discuss several directions for its future development.
This paper examines charged anisotropic compact star structures under modified Rastall [Formula: see text] gravity, where [Formula: see text] is the non-metricity scalar. The Karmarkar condition establishes a particular relationship between the metric potentials, making it easier to develop precise solutions to the relevant field equations. Two different methods are used to analyze the behavior of the [Formula: see text] function. To determine the form of [Formula: see text], a hybrid linear equation of state is used in the first method. In the second, [Formula: see text] is kept in the same hybrid form, but [Formula: see text] is given a logarithmic expression. Various gravitational models with hybrid, power-law and exponential forms are examined by changing the parameters [Formula: see text] and [Formula: see text]. The physical plausibility of the obtained configurations is examined using the observed mass and radius measurements of PSR J1614-2230 and PSR J0740+6620. The observational uncertainties are propagated to the derived compactness quantities, while the model parameters are explored within physically viable ranges. A quantitative comparison of the charged and uncharged limits shows that the representative electric charge produces small but non-zero modifications in the surface properties. The resulting configurations satisfy the required physical acceptability and stability conditions.
We investigate strong gravitational lensing by a quantum loop black hole with a cloud of strings (CoS), surrounded by homogeneous plasma. The deflection angle is analytically obtained by evaluating the strong-deflection-limit coefficients [Formula: see text] and [Formula: see text], incorporating the combined effects of the cloud-of-strings parameter and plasma parameter.We numerically compute lensing observables, including angular image positions, angular image separations, the flux ratio between the outermost and remaining images, and the time delay between the first and second relativistic images, for the supermassive black holes M87* and Sgr A*. Our results show that the photon-sphere radius and the strong-deflection coefficient [Formula: see text] increase with the CoS and plasma parameters, whereas [Formula: see text] exhibits weakly non-monotonic, parameter-dependent behavior, thereby significantly modifying the lensing observables. In particular, the cloud-of-strings parameter produces more pronounced deviations than the quantum correction and plasma parameters. Furthermore, by comparing the theoretical shadow angular diameter with observational data from the Event Horizon Telescope (EHT), we obtain approximate constraints on the quantum correction parameter [Formula: see text] in the presence of homogeneous plasma and a cloud of strings. The allowed range of [Formula: see text] decreases as the plasma parameter increases, indicating a strong interplay between the plasma parameter and quantum-gravitational corrections.
In this article we present a framework for the energy spectrum of the Universe by regarding the energy spectrum of the Universe just like an atom, i.e. [Formula: see text], with discrete energy levels, from the Planck scale to the lightest neutrino mass. To establish this spectrum analytically in a heuristic way and starting from two known constants, [Formula: see text] and [Formula: see text], we establish a Hierarchy Equation, [Formula: see text], to bridge them through the Lambert W function to the dimensionless energy geometric structure constant [Formula: see text], which is the geometric fundamental unit for all energy scales. Concerning the detailed energy spectrum within the electroweak vacuum, [Formula: see text], we propose a Mass Generation Equation (MGE), [Formula: see text], and the masses of all the fundamental particles can be obtained analytically. We have proposed also some analytical formulae for the cosmology, CKM and PMNS related quantities, such as the mixing, oscillation angles, and CP phases. By investigating the energy geometrical structure in [Formula: see text] space, we find that there exist inherent fractal and holographic structures within the whole spectrum based on the UV/IR duality connections, such as [Formula: see text], with the lightest neutrino mass [Formula: see text] and the string energy scale [Formula: see text]. In this article, we establish many energy scale formulae, such as [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] (cosmology constant), [Formula: see text], and particle mass predictions, such as [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], which could be measured by the corresponding experiments. These formulae are offered as conjectures to stimulate further theoretical and experimental scrutiny.
We will get the mass spectrum of all the fermions in the context of the supersymmetric [Formula: see text] Model, when all the usual neutral scalars fields obtain vacuum expectation values. In this model, as happen in MSSM, we can define a [Formula: see text]-parity conservation, where our superpotential respect the invariance of the quantum number, [Formula: see text], where [Formula: see text] is the baryon number while [Formula: see text] is the total lepton number. We will present numerical predictions for all the fermions of this model and we will show that all masses and mixing angles are in agreement with current experimental data.
