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

This paper presents a feasible static and cylinder-like symmetric thin shell wormhole model within the exponential form of <italic>f</italic>( <italic>R, T</italic>) gravity with normalized Ricci scalar relative to the cosmological constant and this gravity model is coherent with local gravity constraints and cosmologically viable. In this study, the cut-and-paste approach joins two identical spacetimes along a hypersurface &#931;. This model assumes isotropic fluid representing the matter sector, with quintessence polytropic energy sustaining matter at the shell. To ensure consistent spacetime matchings, the investigation establishes certain constraints for stable wormhole formations and examines the impact of mass, equation of state parameters, and selected gravitational theory. Stability is ascertained by evaluating the positive value of the potential&#8217;s second derivative at the throat radius.

Azerbaijan

Pakistan

M.Z. Bhatti

A. Adeel

M. Yousaf

Z. Yousaf

A study of stability and quintessence polytropic matter with thin shell wormholes in exponential gravity

By means of the zero-curvature equation and two sets of Lenard recursion sequences, we construct a nonisospectral generalized 3 × 3 Ablowitz–Kaup–Newell–Segur (AKNS) integrable hierarchy. The general inhomogeneous Dirac–Manakov equations with temporally and spatially modulated coefficients are derived. Under some special reductions, the two-component nonlinear Schrödinger (NLS) or Manakov equations with helicoidal spin-orbit (SO) coupling, Rabi coupling, and mixed helicoidal SO and Rabi couplings are obtained. In particular, we propose several novel inhomogeneous two-component NLS equations which have variable helicoidal SO and Rabi couplings as well as external potentials. The <italic>N</italic>-fold Darboux-dressing transformation related to the nonisospectral generalized 3&#8239;&#215;&#8239;3 AKNS matrix eigenvalue problem is established, and further the <italic>N</italic>-soliton solution represented in a compact determinant form is given. The classification of solitons for a two-component NLS equations with temporally modulated mixed helicoidal SO and Rabi couplings is discussed in detail. Specifically, three types of solitons including beating, bell-shaped and multi-peak solitons as well as their parabolic, periodic, exponential and kinked deformations, and some unconventional nonlinear superpositions are presented.

Jiao Wei

Xin Wang

Yanpei Zhen

Chendi Zhu

Tengjin Zhao

The general integrable inhomogeneous Dirac–Manakov equations: Darboux-dressing transformation and <italic>N</italic>-soliton solutions

The exact localization result for the expectation value of the <math altimg='si1.svg'><mfrac><mn>1</mn><mn>2</mn></mfrac></math> BPS circular Wilson loop in <math altimg='si2.svg'><mrow><mi mathvariant='script'>N</mi><mo linebreak='goodbreak'>=</mo><mn>4</mn></mrow></math> SYM theory is given in the planar limit by the famous Bessel function expression: <math altimg='si3.svg'><mrow><mrow><mo>&#9001;</mo><mi>W</mi><mo>&#9002;</mo></mrow><mo linebreak='goodbreak'>=</mo><mfrac><mrow><mn>2</mn><mi>N</mi></mrow><msqrt><mi>&#955;</mi></msqrt></mfrac><msub><mi>I</mi><mn>1</mn></msub><mrow><mo>(</mo><msqrt><mi>&#955;</mi></msqrt><mo>)</mo></mrow></mrow></math>. Expanded in large <italic>&#955;</italic> and expressed in terms of the AdS <inf>5</inf>&#8239;&#215;&#8239; <italic>S</italic> 5 string tension <math altimg='si4.svg'><mrow><mi>T</mi><mo linebreak='goodbreak'>=</mo><mfrac><msqrt><mi>&#955;</mi></msqrt><mrow><mn>2</mn><mi>&#960;</mi></mrow></mfrac></mrow></math> this gives <math altimg='si5.svg'><mrow><mrow><mo>&#9001;</mo><mi>W</mi><mo>&#9002;</mo></mrow><mo linebreak='goodbreak'>=</mo><mfrac><msqrt><mi>T</mi></msqrt><mrow><mn>2</mn><mi>&#960;</mi><msub><mi>g</mi><mi>s</mi></msub></mrow></mfrac><msup><mi>e</mi><mrow><mn>2</mn><mi>&#960;</mi><mi>T</mi></mrow></msup><mrow><mo>(</mo><mn>1</mn><mo linebreak='badbreak'>&#8722;</mo><mfrac><mn>3</mn><mrow><mn>16</mn><mi>&#960;</mi></mrow></mfrac><msup><mi>T</mi><mrow><mo>&#8722;</mo><mn>1</mn></mrow></msup><mo linebreak='badbreak'>+</mo><mo>&#8230;</mo><mo>)</mo></mrow></mrow></math>.The exponential is matched by the value of the action of the string with the AdS <inf>2</inf> world volume while the prefactor comes from the 1-loop GS string correction. Here we address the question of how the subleading <math altimg='si6.svg'><msup><mi>T</mi><mrow><mo>&#8722;</mo><mn>1</mn></mrow></msup></math> term could be reproduced by the 2-loop correction in the corresponding partition function of the AdS <inf>5</inf>&#8239;&#215;&#8239; <italic>S</italic> 5 GS string expanded near the AdS <inf>2</inf> minimal surface. We find that the string correction contains a non-zero UV logarithmic divergence implying that comparison with the SYM result requires a particular subtraction prescription. We discuss implications of this conclusion for checking the AdS/CFT duality at strong coupling.

Italy

M. Beccaria

United Kingdom

S.A. Kurlyand

A.A. Tseytlin

Strong coupling expansion of <math altimg='si1.svg'><mfrac><mn>1</mn><mn>2</mn></mfrac></math> BPS Wilson loop in SYM theory and 2-loop Green-Schwarz string in AdS <inf>5</inf>&#8239;&#215;&#8239; <italic>S</italic> 5

Recently, the BESIII experiment performed a measurement of the energy-dependent Born cross section and the effective form factor for the <math altimg='si2.svg'><mrow><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>&#8722;</mo></msup><mo>&#8594;</mo><msup><mrow><mstyle mathvariant='normal'><mi>&#937;</mi></mstyle></mrow><mo>&#8722;</mo></msup><msup><mover accent='true'><mstyle mathvariant='normal'><mi>&#937;</mi></mstyle><mo>&#175;</mo></mover><mo>+</mo></msup></mrow></math> reaction at the center-of-mass energies ranging from 3.4 to 4.7&#8239;GeV. The energy dependence of the dressed cross section is fitted with the hypothesis of a charmonium(-like) resonance [i.e., <italic>&#968;</italic>(3770), <italic>&#968;</italic>(4040), <italic>&#968;</italic>(4160), <italic>Y</italic>(4230), <italic>Y</italic>(4360), <italic>&#968;</italic>(4415), or <italic>Y</italic>(4660)] combined with a power-law function. The fit is applied to the recent BESIII data, combined with the earlier measurements from CLEO-c data, but no significance is found. The products of the branching fractions and the two-electronic partial widths for the assumed charmonium(-like) states decaying into the <math altimg='si4.svg'><mrow><msup><mrow><mstyle mathvariant='normal'><mi>&#937;</mi></mstyle></mrow><mo>&#8722;</mo></msup><msup><mover accent='true'><mstyle mathvariant='normal'><mi>&#937;</mi></mstyle><mo>&#175;</mo></mover><mo>+</mo></msup></mrow></math> final state are also provided. In addition, by taking the world average value of the two-electronic partial width, the upper limits at the 90% confidence level on the branching fractions of <italic>&#968;</italic>(3770), <italic>&#968;</italic>(4040), <italic>&#968;</italic>(4160), and <italic>&#968;</italic>(4415) decays into <math altimg='si4.svg'><mrow><msup><mrow><mstyle mathvariant='normal'><mi>&#937;</mi></mstyle></mrow><mo>&#8722;</mo></msup><msup><mover accent='true'><mstyle mathvariant='normal'><mi>&#937;</mi></mstyle><mo>&#175;</mo></mover><mo>+</mo></msup></mrow></math> are determined for the first time. These are found to be at least an order of magnitude larger than expected from predictions using a scaling based on the observed electronic widths.

Ruoyu Zhang

Xiongfei Wang

Search for charmonium(-like) states decaying into the <math altimg='si4.svg'><mrow><msup><mrow><mstyle mathvariant='normal'><mi>&#937;</mi></mstyle></mrow><mo>&#8722;</mo></msup><msup><mover accent='true'><mstyle mathvariant='normal'><mi>&#937;</mi></mstyle><mo>&#175;</mo></mover><mo>+</mo></msup></mrow></math> final states

We analyze the experimental data of <math altimg='si1.svg'><mrow><mi mathvariant='monospace'>p</mi><mo linebreak='goodbreak'>&#8722;</mo><mi mathvariant='monospace'>p</mi></mrow></math> collisions by the ATLAS and confront it with the AGK model developed by two of the authors, the Kharzeev-Levin (KL) model and the simple exponential behavior for the Koba, Nielsen and Olesen (KNO) scaling function. We show that all three models virtually coincide with all available experimental curves crossed at value <math altimg='si4.svg'><mrow><mi>z</mi><mo linebreak='goodbreak'>=</mo><mi>n</mi><mo linebreak='goodbreak'>/</mo><mo>&#9001;</mo><mi>n</mi><mo>&#9002;</mo></mrow></math> of the KNO scaling parameter in the broad range of the center-of-mass energies. The theoretical results and the experimental data show that the KNO violating terms are highly suppressed in the vicinity of <math altimg='si16.svg'><mrow><mi>z</mi><mo linebreak='goodbreak'>=</mo><mn>2</mn></mrow></math>. The special status of the universal scaling point <math altimg='si16.svg'><mrow><mi>z</mi><mo linebreak='goodbreak'>=</mo><mn>2</mn></mrow></math>, where the KNO scaling is restored for <math altimg='si1.svg'><mrow><mi mathvariant='monospace'>p</mi><mo linebreak='goodbreak'>&#8722;</mo><mi mathvariant='monospace'>p</mi></mrow></math> collisions, suggests that it originates from statistical saturation of the optical theorem. We claim that any unitary model that approximates memoryless distribution in the high energy limit must have the same KNO scaling function <math altimg='si8.svg'><msup><mi>e</mi><mrow><mo>&#8722;</mo><mi>z</mi></mrow></msup></math> in the vicinity of <math altimg='si16.svg'><mrow><mi>z</mi><mo linebreak='goodbreak'>=</mo><mn>2</mn></mrow></math> with KNO violating corrections of the order of one over average multiplicity squared. This reflects the maximal entanglement of the final states.

Israel

Mustapha Ouchen

Alex Prygarin

KNO scaling, memorylessness and maximal entanglement at universal fixed point

This is the first of two papers presenting a geometric framework for Planar QCD ( <italic>N<inf>c</inf></italic>&#8239;&#8594;&#8239;&#8734;). In this part, we establish the kinematic foundation of the theory by constructing the unique stable vacuum of the loop equation. We demonstrate that the Makeenko-Migdal loop equation admits a solution of the form <math altimg='si1.svg'><mrow><mi>W</mi><mrow><mo>[</mo><mi>C</mi><mo>]</mo></mrow><mo linebreak='goodbreak'>=</mo><msub><mi>W</mi><mrow><mi>p</mi><mi>e</mi><mi>r</mi><mi>t</mi></mrow></msub><mrow><mo>[</mo><mi>C</mi><mo>]</mo></mrow><mi>exp</mi><mrow><mo>&#8722;</mo><mi>&#954;</mi><mi>S</mi><mo>[</mo><mi>C</mi><mo>]</mo></mrow></mrow></math>, provided <italic>S</italic>[ <italic>C</italic>] is a specific minimal surface possessing a self-dual area derivative. We prove that such a surface exists and corresponds to the Hodge-dual projection of a minimal surface in <math altimg='si2.svg'><mrow><msup><mi mathvariant='double-struck'>R</mi><mn>3</mn></msup><mo>&#8855;</mo><msup><mi mathvariant='double-struck'>R</mi><mn>4</mn></msup></mrow></math>. Crucially, this confinement mechanism relies on the self-duality of the area derivative&#8212;a property that exists exclusively in four dimensions. This geometric constraint ensures stability only in <math altimg='si3.svg'><mrow><mi>D</mi><mo linebreak='goodbreak'>=</mo><mn>4</mn></mrow></math>, distinguishing the resulting theory from standard string models which require higher critical dimensions. We relate the string tension parameter <italic>&#954;</italic> to the gluon condensate via the Operator Product Expansion. The dynamical quantization of the Fermi string on this rigid surface and the resulting meson spectrum are derived in the companion paper [1].

Alexander Migdal

Geometric QCD I: the Hodge-dual surface and quark confinement

Recent advances in modeling compact astrophysical objects have focused on incorporating dark energy as a relativistic component within their internal structures, offering novel insights into extreme matter configurations. Theoretical frameworks now explore anisotropic pressure profiles and modified equations of state, enabling predictions about stability and mass-radius relationships under dark energys repulsive influence. Such studies aim to resolve discrepancies in observed compact object populations while probing interactions between high-density matter and cosmological vacuum energy. This study presents a new analytical framework for a two-fluid system consisting of ordinary baryonic matter and dark energy in modified <italic>f</italic>( <italic>Q</italic>) gravity, with <math altimg='si8.svg'><mrow><mi>f</mi><mo>(</mo><mi>Q</mi><mo>)</mo><mo>=</mo><mi>&#958;</mi><mo linebreak='goodbreak'>+</mo><mi>&#967;</mi><mi>Q</mi></mrow></math>, where <italic>&#958;</italic> and <italic>&#967;</italic> are coupling constants. By utilizing the Durgapal-Lake metric potentials (Durgapal et al., Gen. Relativ. Gravit. 17:671, 1985, Lake K, Phys. Rev.D 67:10 2003) as seed solutions and assuming a linear equation of state for dark energy, the modified gravitational field equations are solved to derive exact interior solutions. The model parameters, including the dark energy coupling constant and constants from the metric ansatz, were determined by smoothly matching the interior geometry with the exterior spacetime at the stellar boundary. Observational data from three compact stars <math altimg='si9.svg'><mrow><mo>(</mo><mi>H</mi><mi>e</mi><mi>r</mi><mi>X</mi><mo linebreak='goodbreak'>&#8722;</mo><mn>1</mn><mo>,</mo></mrow></math> <math altimg='si10.svg'><mrow><mn>4</mn><mi>U</mi><mn>1538</mn><mo linebreak='goodbreak'>&#8722;</mo><mn>52</mn><mo>,</mo></mrow></math> <math altimg='si11.svg'><mrow><mi>S</mi><mi>A</mi><mi>X</mi><mi>J</mi><mn>1808.4</mn><mo linebreak='goodbreak'>&#8722;</mo><mn>3658</mn><mo>)</mo></mrow></math> were used to test the model&#8217;s viability and regularity. The study included a thorough physical analysis, examining energy density gradients, anisotropy, stability, equilibrium conditions using the modified TOV equation. The results align with observational data, indicating that the model offers a physically plausible description of compact stars while maintaining stability and equilibrium under the tested conditions.

M.R. Shahzad

Wajiha Habib

Saudi Arabia

Muneerah Alomar

Mona Bin-Asfour

Possible existence and physical features of mathematical models of dark energy ultra-compact objects in modified symmetric teleparallel gravity

We improve the theoretical predictions for rare <italic>Z</italic>-boson decays, <math altimg='si1.svg'><mrow><mi>Z</mi><mo>&#8594;</mo><mi>V</mi><msup><mi>&#8467;</mi><mo>+</mo></msup><msup><mi>&#8467;</mi><mo>&#8722;</mo></msup></mrow></math> ( <math altimg='si9.svg'><mrow><mi>&#8467;</mi><mo linebreak='goodbreak'>=</mo><mi>e</mi></mrow></math> or <italic>&#956;</italic>), where <italic>V</italic> denotes a heavy vector quarkonium including <italic>J</italic>/&#936;, &#936;(2 <italic>S</italic>), and &#933;( <italic>nS</italic>) with <math altimg='si10.svg'><mrow><mi>n</mi><mo linebreak='goodbreak'>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></math>. These processes are thought to be dominated by the electromagnetic fragmentation transition, i.e., <math altimg='si11.svg'><mrow><mi>Z</mi><mo>&#8594;</mo><msup><mi>&#947;</mi><mo>*</mo></msup><msup><mi>&#8467;</mi><mo>+</mo></msup><msup><mi>&#8467;</mi><mo>&#8722;</mo></msup></mrow></math> followed by <italic>&#947;</italic>*&#8239;&#8594;&#8239; <italic>V</italic>. The present study includes all of the relevant tree-level Feynman diagrams, which contribute to these decays in the standard model. Our analysis shows that, for the charmonium final states, the fragmentation transition almost saturates the whole contribution and the other diagrams can be neglected; while for the bottomonium final states, the inclusion of other diagrams can increase their branching fractions by 4%&#8239;&#8764;&#8239;9%. Further investigation of the differential distributions, especially the angular distributions, indicates that forward-backward asymmetries for final leptons in these processes would be zero in the standard model. Therefore, in future experimental facilities with large number of <italic>Z</italic>-boson events accumulated, studies of these rare <italic>Z</italic> decays may help both to test the standard model and to probe its interesting extensions.

Li Ang

Dao-Neng Gao

Improved analysis of rare <italic>Z</italic>-boson decays into a heavy vector quarkonium plus lepton pair

Phantom divide line transitions are not possible in the context of single scalar field scalar-tensor theories. In this article we study a combined framework of a tachyonic minimally coupled single scalar field theory in the presence of an <math altimg='si1.svg'><msup><mi mathvariant='script'>R</mi><mn>2</mn></msup></math> correction term and with a rescaled Einstein-Hilbert term of the form <math altimg='si6.svg'><mrow><mo>&#8764;</mo><mi>&#955;</mi><mfrac><mi mathvariant='script'>R</mi><mrow><mn>16</mn><mi>&#960;</mi><mi>G</mi></mrow></mfrac></mrow></math>. Such terms can be part of an <math altimg='si7.svg'><mrow><mi>f</mi><mo>(</mo><mi mathvariant='script'>R</mi><mo>)</mo></mrow></math> gravity which in the large curvature regime yields such correction terms effectively. Alternatively, such terms can simply be quantum corrections to the scalar field action. We aim to answer two questions, firstly if this framework can lead to phantom divide line transitions and secondly whether the resulting model can be compatible with the ACT data. The model we studied is an inverse square power-law model, well known from tachyon inflation models. As we show, the field equations can be cast in terms of the scalar field solely, however the resulting theory is distinct from a single scalar field theory, because the phantom divide line is crossed during inflation. Thus initially the tachyonic nature of the scalar field generates a phantom equation of state parameter, and during inflation the phantom divide line is crossed, with the effective equation of state parameter at the end of inflation being <math altimg='si8.svg'><mrow><mi>w</mi><mo linebreak='goodbreak'>=</mo><mo>&#8722;</mo><mn>1</mn><mo linebreak='goodbreak'>/</mo><mn>3</mn></mrow></math> which corresponds to the non-accelerating state of the Universe. The model is proved to be compatible with the ACT data, only when the gravity during inflation is stronger than Einstein-Hilbert gravity, with the effective gravitational constant during inflation being <math altimg='si9.svg'><mfrac><mi>G</mi><mi>&#955;</mi></mfrac></math>. The effective theory is valid only during inflation, thus Big-Bang nucleosynthesis is not affected by the rescaling of the Einstein-Hilbert gravity. The feature of a phantom crossing in <math altimg='si10.svg'><mrow><mi>f</mi><mo>(</mo><mi mathvariant='script'>R</mi><mo>,</mo><mi>&#981;</mi><mo>)</mo></mrow></math> frameworks is new in the literature.

Spain

S.D. Odintsov

Greece

V.K. Oikonomou

<math altimg='si1.svg'><msup><mi mathvariant='script'>R</mi><mn>2</mn></msup></math>-corrected Tachyon Scalar Field Inflation, the ACT Data, and Phantom Transition