We revisit the fundamentals of two different methods for calculating classical observables: the eikonal method, which is a scattering amplitude-based method, and the worldline quantum field theory (WQFT) method. The latter has been considered an extension of the worldline effective field theory. We show that the eikonal and WQFT methods are equivalent and that calculations can be translated freely between them. We also show how the scattering amplitude can serve as the generating function of all the WQFT diagrams. This offers another computational avenue for the WQFT diagrams, which can be useful at higher loop orders. Concretely, we focus on 2-into-2 scattering processes mediated by massless force carriers. On one hand, taking the classical limit of the QFT scattering amplitude leads to the eikonal method. On the other hand, since in the classical limit the scattering particles are almost on-shell throughout the scattering process, the worldline, a first quantized formalism, is the most efficient framework to study the scattering amplitude. This is an alternate but equivalent formalism to the quantum field theoretical (QFT) framework. By taking the classical limit of the scattering amplitude computed in the worldline, we can derive the WQFT rules of Mogull, Plefka and Steinhoff [1]. In WQFT, the Feynman diagrams are reorganized into a new set of diagrams that facilitate the ℏ expansion. Unlike the QFT eikonal method, which identifies the eikonal phase recursively, the worldline-based computation allows to target and systematically extract the classical contributions directly through a specific set of WQFT diagrams. In worldline formalism the perturbative expansion of the scattering amplitude is naturally organized in diagrams which factorize (reducible) and diagrams which are new to that order (irreducible), in a one-to-one map with the structure of the amplitude in the eikonal method. This opened up the possibility to investigate and prove the conjectured exponentiation of the eikonal phase in Du et al.[2].

United States

Siddarth Ajith

China

Yuchen Du

Ravisankar Rajagopal

Diana Vaman

Worldline formalism, eikonal expansion and the classical limit of scattering amplitudes

In this paper we propose two sets of nonlinear integral equations (NLIE) for describing the thermodynamics in the sine-Gordon model, when higher Lorentz spin conserved charges are also coupled to the Gibbs ensemble. We call them NLIE I and II. The derivation of the equations, is based on <math altimg='si7.svg'><mrow><mi>T</mi><mo linebreak='goodbreak'>&#8722;</mo><mi>Q</mi></mrow></math> relations given by the equivalent thermodynamic Bethe ansatz (TBA) formulation of the problem in the repulsive regime. Though the equations are derived in the repulsive regime at discrete values of the coupling constant, a straightforward analytical continuation ensures their validity within the whole repulsive regime of the theory. For the NLIE I formulation, appropriate analytical continuation makes the penetration into the attractive regime also possible. However, the magnitude of this penetration is restricted by the spin of the largest spin conserved charge contained in the Gibbs ensemble. Within their range of validity, these NLIE formulations provide efficient theoretical frameworks for computing expectation values of conserved charge densities, their associated currents, and vertex operators and their descendants, with respect to the generalized Gibbs ensemble.

Hungary

Árpád Hegedűs

NLIE formulations for the generalized Gibbs ensemble in the sine-Gordon model

The interaction problem between solitons and mean flows including the dispersive shock wave, rarefaction wave, kink and contact dispersive shock wave, is discussed in the framework of the Gardner equation, a generic non-convex mathematical model containing quadratic-cubic nonlinear terms, describing the dynamics of internal waves for an arbitrary vertical stratification of the density field. Based on the augmented solitonic modulation system constructed in a Riemann invariant form, two main interaction regimes are identified: soliton tunneling and soliton trapping. Furthermore, kinks and flat-top solitons caused by the non-convexity, play dual roles as both mean flows and localised waves, imparting polarity reversal to solitons and mean flows. Specially, flat-top solitons formed by the quadratic-cubic nonlinear competition, can be considered as a combination of kinks and antikinks, and induce polarity reversal to solitons and mean flows twice, which is an evidence of elastic collisions between flat-top solitons and bell solitons. In addition, the shape of flat-top solitons is changed after the interaction with mean flows. A full classification of the interaction problem is given for both signs of the cubic nonlinear term in the Gardner equation, and numerical simulations agree with modulation theory predictions.

Jia-Xue Niu

Rui Guo

Jian-Wen Zhang

Soliton–mean flow interaction problem for a non-convex system with quadratic-cubic nonlinearity in the stratified fluid

We study the plausibility of a holographic dark energy (HDE) model using the form of horizon entropy proposed by Viaggiu in 2014. This form of entropy is a generalization of the usual Bekenstein-Hawking entropy, having an extra term arising due to the dynamical nature of horizons in an expanding universe. We examine this new HDE model in the context of a flat Friedmann-Lemaître-Robertson-Walker universe filled with two cosmic fluids — dark matter in the form of dust and holographic dark energy generated by Viaggiu entropy. We consider the Hubble horizon and the future event horizon as characteristic length scales and study the evolution of the Universe within these frameworks. Our analysis reveals some intriguing findings that include a possible cosmic doomsday scenario in the future, and certain observations are in striking contrast to other HDE models studied in the literature.

India

Somnath Saha

Subhajit Saha

Nilanjana Mahata

The peculiar case of the Viaggiu holographic dark energy

In this work, we investigate the influence of quantum-gravity induced corrections on the entanglement entropy associated with two-flavor neutrino oscillations in vacuum. Using the von Neumann entropy as a measure of quantum correlations, we analyze how Planck scale suppressed modifications implemented through quantum-gravity motivated changes to the neutrino dispersion relation affect the evolution of entanglement during successive oscillation cycles. By performing a statistical analysis over a range of oscillation lengths and neutrino energies, we identify characteristic deviations in the entropy profile arising from quantum-gravity effects. Our results suggest that entanglement entropy provides a sensitive probe of small quantum-gravity induced departures from standard neutrino oscillation dynamics, highlighting its potential role in testing Planck-scale physics within neutrino phenomenology.

Bipin Singh Koranga

Baktiar Wasir Farooq

Pranav Kumar

Modified entanglement patterns in two-flavor neutrinos from quantum-gravity interactions

Rimi Debnath

Sanjeeda Sultana

Khandro K. Chokyi

Surajit Chattopadhyay

Corrigendum to “Phenomenological insights into holographically reconstructed fractional action cosmology: Cosmic dynamics, thermodynamics, and statefinder diagnostics” [Nucl. Phys. B 1018 (2025) 117074]

We investigate black hole configurations within the theoretical framework of Cotton gravity coupled to nonlinear electrodynamics, with the aim of exploring how higher-derivative curvature terms and nonlinear electromagnetic interactions influence accretion behavior and optical characteristics. Restricting attention to static and spherically symmetric spacetimes, we examine scalar-field perturbations, test-particle dynamics, analytical bounds on greybody factors, and the optical appearance of the black hole under a static spherical accretion scenario. Our analysis demonstrates that the Cotton gravity contribution induces a genuine long-range deformation of the spacetime geometry, resulting in stronger wave confinement, an increased shadow radius, and outwardly displaced photon-ring structures when compared with the predictions of General Relativity. In contrast, nonlinear electrodynamics mainly affects the near-horizon region, giving rise to strong-field corrections that cannot be reproduced within standard Maxwell theory. We further show that the combined presence of Cotton gravity and nonlinear electrodynamics significantly modifies photon-sphere properties, orbital stability, and the location of the innermost stable circular orbit, thereby impacting accretion dynamics. Overall, these findings extend the Einstein–Maxwell paradigm and indicate that black hole imaging, accretion-related observables, and gravitational-wave measurements offer promising probes of Cotton gravity and nonlinear electrodynamics in the strong-gravity regime.

Saudi Arabia

Ali Ahmad

Pakistan

Azerbaijan

Shahid Chaudhary

Uzbekistan

Farruh Atamurotov

Hamdan Alshehri

Mohammad Shahid Kamal

Perturbation dynamics and optical structure of black hole in cotton gravity coupled to nonlinear electrodynamics

Corrigendum to “Phenomenological insights into holographically reconstructed fractional action cosmology: Cosmic dynamics, thermodynamics, and statefinder diagnostics” [Nuclear Physics B Volume: 1018 (September 2025) 117074]

Assuming the gluon propagator to be finite at <math altimg='si1.svg'><mrow><msubsup><mi>g</mi><mi>s</mi><mn>2</mn></msubsup><mo linebreak='goodbreak'>/</mo><mn>4</mn><mi>&#960;</mi><mo>&#8801;</mo><mi>&#945;</mi><mo>&#8594;</mo><msub><mi>&#945;</mi><mo>*</mo></msub></mrow></math>, where <math altimg='si2.svg'><mrow><msub><mi>&#945;</mi><mo>*</mo></msub><mo linebreak='goodbreak'>=</mo><mi>&#945;</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow></mrow></math>, we derive its low-momentum asymptotics. In some cases the results modify the known ones. In our work, we also establish the field-theoretical nature of a universal phenomenological parameter introduced in works [1] and determining the behaviour of Green&#8217;s functions in the infrared region. The importance of studying the renormalization group parameters in the vicinity of the infrared stable point is emphasized.

Russia

Vladimir A. Petrov

IR fixed point and low-momentum gluon propagator

We investigate the gravitational lensing effects on neutrino oscillations within the framework of Modified Gravity (MOG). By modeling neutrinos as both plane waves and Gaussian wave packets, we analyze how the presence of a massive lensing object between the neutrino source and detector modifies the flavor transition probabilities. In the plane-wave approximation, we derive analytical expressions for lensing-induced interference patterns, showing explicit dependence on the MOG parameter <italic>&#945;</italic>, the impact parameters, and the neutrino mass hierarchy. Extending the analysis to a wave-packet framework, we account for decoherence effects and quantify the damping of oscillation amplitudes due to finite energy and spatial spreads of the wave packets. We compute the decoherence length and damping factor for a realistic Sun-Earth system and demonstrate how both the absolute neutrino mass scale and the MOG parameter influence coherence loss. Our results indicate that gravitational lensing in alternative gravity theories can leave observable imprints on neutrino oscillation patterns, offering a novel probe of both neutrino properties and modifications of General Relativity.