Color transparency begins with the production of a compact color-singlet configuration in a hard exclusive reaction. In light-front holographic QCD, the leading-twist behavior of the hadronic form factor reflects valence-Fock dominance and encodes the compact hard transverse scale of this configuration. Observable nuclear transparency additionally depends on the reaction-dependent physical-separation distribution, color screening, expansion, and propagation through nuclear matter. We formulate a framework that preserves the exact, external-scale-independent hadronic light-front wave function and distinguishes the elastic current, the exclusive production vertex, and nuclear propagation. The integer-twist form factor is obtained as a finite Regge-pole product, and its positive logarithmic profile slope and exact Beta-distribution ordering are derived analytically. For the published holographic reparametrization, the current-selected physical constituent separation approaches a nonzero endpoint value of approximately 0.457 fm even as the GPD impact profile narrows. Thus, although the form factor supplies the compact hard impact scale, it does not alone determine the reaction-dependent physical-separation distribution required for nuclear attenuation. We specify that mapping through an explicit reaction-dependent production kernel acting on the actual active-constituent/spectator separation. The resulting positive configuration distribution has a physical mean-square radius that scales as the inverse hard scale and determines a saturating color-dipole interaction without modifying the elastic form factor. Nuclear survival is averaged over configurations after exponentiation, the struck nucleon is excluded from the spectator density, the deuteron is treated as a correlated two-body state, and vector-meson decay and pion-pair attenuation are included. All experimental inputs are directly published: 29 pion transparencies, 146 pion/deuteron differential sub-bins, eight resolved proton-shell transparencies, two vector-meson slopes, and the measured vector-meson acceptance and diffraction profiles. Current-derived compact fractions require no fitted event-branch probabilities. With equal reaction-parameter priors, the acceptance-aware pion comparison gives χ ^2=19.05 versus 24.07 for Glauber; withheld-target predictions give 18.91 versus 27.20. Proton electroproduction remains Glauber-like, while channel-specific vector-meson transport improves its two-slope comparison from 6.12 to 2.48. Only published experimental observables and bin boundaries are used; unpublished event populations and bin-by-bin vector-meson transparency values are not inferred or included in the analysis.
A nonlocal two-body scattering problem for optical potentials of the Perey–Buck type is solved within a newly-developed version of the R-matrix approach. The approach is based on harmonic oscillator expansion of the scattering wave functions within the range of the optical potential. It results in structurally identical algebraic representations both for local and nonlocal optical potentials as well as for the kinetic energy and norm operators, which simplifies and optimizes the coding procedure. The method has been tested for proton scattering at 10, 30 and 90 MeV. With a sufficient number of harmonic oscillator basis states the new method reproduces differential cross sections obtained using available reaction codes. The nonlocal calculations were also used as a testing ground for local-equivalent models. It was found that below 30 MeV next-to-leading order models are sufficient to describe elastic scattering, while at 90 MeV this approximation works well only up to 100^∘ . The method developed could be used in reaction calculations within distorted-wave models and in combining scattering calculations with various few- and many-body structure methods that use harmonic oscillator expansions.
The choreographic plane motion of four or more bodies with equal masses has been analytically described with a combination of attractive and repulsive linear forces. In this work, we describe families of choreographic solutions to the equations of motion in a plane with only attractive linear forces along the lines joining the bodies. The conditions that produce collisions between any of the four bodies can be isolated and thus avoided. The expressions of many relevant properties of the system can be obtained in closed form. The angular momentum kernel is evaluated in order to establish the set of choreographic solutions that satisfy the zero angular momentum condition. Depending on the initial parameters, there can be other symmetries and conserved quantities in addition to the energy, momentum and centre of mass. The masses and restoring force coefficients necessary conditions that allow choreographic behaviour are established. The curves admit a larger number of bodies. The analytic choreographic trajectories are similar to those obtained numerically under a Newtonian force. This all attractive forces scheme, albeit linear, moves us closer to the Newtonian gravitational paradigm.
We give a uniform radial formulation of atomic hydrogen and helium in three and two dimensions, solving the 3D S-wave ( l=0 ) and 2D m=0 sectors with a single Laguerre-basis method. A one-parameter transcription, L=0 in 3D and L=-12 in 2D, reduces the dimensional change to one basis parameter, so the same validated code produces both dimensions. For helium, each electron is restricted to the matched radial sector ( l_1=l_2=0 in 3D, m_1=m_2=0 in 2D), which separates the effects of radial measure, residual inverse-square term, and interaction law. A central observation is that dimension-matched electron–electron interactions project to pure outer-radius kernels. In 3D this gives the standard S-wave Coulomb kernel 1/r_12→ 1/r_> ; in intrinsic 2D the corresponding logarithmic interaction gives the less commonly noted kernel -2ln (r_12/r_0)→ -2ln (r_>/r_0) . Each is the Coulomb Green function of its own dimension after angular projection. The planar 2D model, which keeps the 3D 1/r_12 law while restricting the electrons to a plane, is therefore mixed-dimensional and yields an elliptic-integral kernel instead. Analytic log–Coulomb crossover and Rytova–Keldysh potentials are also explored as conceptually intermediate one-electron cases between the pure 3D and planar 2D Coulomb models. The method is validated against exact planar 2D hydrogen energies, published logarithmic spectra, and helium benchmarks, with the restricted radial projection recovering the full correlated energies to within about 1% for the ground configuration and an order of magnitude better for the singly excited states. On this footing, radial m_1=m_2=0 energies for intrinsic Gauss-law 2D helium are reported for the first time. The open-source Python code is publicly available.
We show that one of the models proposed by Ahmed [Few-Body Syst. 66, 28 (2025)] does not support bound states and the eigenfunctions derived by the author are not square integrable. His second example supports bound states but the author failed to truncate the infinite series and consequently he derived functions that are not solutions to the eigenvalue equation. Consequently, the energies reported by Ahmed are in both cases unphysical and of no utility whatsoever.
