We formulate gravitational light deflection in static spherical spacetimes through the amplitude and angle of the null orbit in its phase plane. For asymptoticSchwarzschild scattering, the bending angle equals the excess physical azimuth accumulated over a fixed intrinsic phase interval. The radial first integral reduces the amplitude to the physical branch of a cubic algebraic relation. Lagrange inversion then gives an all-order weak-deflection coefficient formula whose algebraic and trigonometric factors explain the alternating rational and pi-dependent terms. Explicit comparisons recover the known Schwarzschild coefficients through eighth order, the general static spherical coefficients through second order, and the Reissner–Nordström coefficients through third order. These agreements locate the contribution in the phase representation and coefficient factorization rather than in new bending coefficients. We extend the exact phase integral to finite source and receiver radii by including their local propagation angles and recover the leading Schwarzschild finite-distance result. Near the critical photon orbit, we retain the regular part of the complete angular integral and derive the known logarithmic divergence together with its finite constant. The same critical algebraic branch determines the convergence radius and large-order behavior of the Schwarzschild weak series.
Matter density ( ρ ) and radial pressure (p) are often used as the matter Lagrangian density ( ℒ_m ) because both are thermodynamically consistent and produce the same Einstein field equation (EFE) in general relativity (GR). New gravity models with explicit links between matter and geometry instead involve second-order derivatives of ℒ_m relative to the metric tensor. So, picking either p or -ρ for ℒ_m gives different effective EFEs. This confusion appears because one usually treats the four-velocity ( u_μ ) and the metric tensor ( g_μν ) as independent. Here, we revisit the basics and offer a consistent framework by relaxing that assumption, thereby making the modified gravity theory independent of the choice of ℒ_m . Finally, we test this approach on neutron and quark stars (ultraviolet region) and on cosmological situations with matter-dominated or dust ( p=0 ), radiation-dominated ( p=ρ /3 ), and vacuum energy ( p=-ρ ) equations of state (infrared region), showing how it clarifies the ambiguity in picking ℒ_m for gravity models.
The Super Tau-Charm Facility (STCF), with a designed center-of-mass energy range of 2 – 7 GeV , a peak luminosity exceeding 0.5× 10^35 cm^-2s^-1 , a total trigger rate of about 400 kHz, and a total data rate of about 30 GB/s, will impose stringent requirements for fast track finding and data reduction and compression algorithms in the High-Level Trigger. Local track segment finding in the Main Drift Chamber (MDC) underpins subsequent segment combination and full track reconstruction, yet high background rates and limited detection efficiency can significantly increase the risk of false triggers and signal loss in pattern matching algorithms. This paper presents a neural-network-assisted framework for constructing binary template libraries used in matrix-based pattern matching for MDC local track segment finding. The framework formulates template construction as a differentiable multi-objective optimization problem, employing a neural network to jointly learn template parameters under multiple constraints. After training, only binary template pairs are exported and deployed into the existing bitwise pattern matching routine, requiring no neural network inference at runtime and thus preserving the deterministic, fast, and parallelizable nature of the online algorithm. Experimental results based on simulation samples demonstrate that, under limited detection efficiency, the resulting template library maintains relatively high signal retention across different transverse momentum ranges and background levels, and can be flexibly tailored to adjust the coverage range according to practical requirements. The proposed approach combines offline neural-network optimization with a deterministic, fast conventional online algorithm, enabling artificial-intelligence-enhanced online data processing in high-luminosity particle collider experiments.
We investigate early inflation in Einstein Gauss–Bonnet gravity with a scalar field evolving in a plateau modified inverse square potential and non-minimally coupled to the Gauss–Bonnet invariant through an exponentially saturating coupling function. Within the slow-roll approximation, we derive the scalar spectral index, tensor to scalar ratio, and propagation speed of primordial tensor perturbations. Numerical integration of the background equations confirms the existence of stable inflationary trajectories and a graceful exit from the early inflationary phase. In the weak coupling regime, the model reproduces the standard plateau attractor behavior while generating controlled corrections in the tensor sector governed primarily by the ratio λ/b and the effective coupling combination ξ _0 V_0 . The predicted scalar spectral index and tensor to scalar ratio remain compatible with current CMB constraints, while the tensor sector remains free from ghost and gradient instabilities. It is shown that the higher curvature effects can produce a controlled modification of the primordial tensor perturbations without spoiling the successful scalar sector predictions from the plateau inflation.
Abstract We study the $$J/\psi \rightarrow \Lambda \pi \bar{\Sigma }$$ J / ψ → Λ π Σ ¯ reaction by looking at the $$\pi ^+ \Lambda $$ π + Λ mass distribution at low energies, in search of signals for the low lying $$\Sigma ^*$$ Σ ∗ excited states. Apart from a clear signal of the $$\Sigma (1385) (3/2^+)$$ Σ ( 1385 ) ( 3 / 2 + ) state, we find a smaller peak for the predicted $$\Sigma (1430) (1/2^-)$$ Σ ( 1430 ) ( 1 / 2 - ) , which has already been confirmed by the Belle Collaboration. A first analysis, considering only the $$\pi \Lambda $$ π Λ interaction, shows that the low energy part of the spectrum is better reproduced including contributions from the $$\Sigma (1430)$$ Σ ( 1430 ) and the predicted $$\Sigma (1380)(1/2^-)$$ Σ ( 1380 ) ( 1 / 2 - ) state that has been claimed before from analyses of different experiments. However, when we consider the $$\pi \bar{\Sigma }$$ π Σ ¯ interaction the need for the $$\Sigma (1380)$$ Σ ( 1380 ) disappears.
