Annals of physics

Graph theory and Feynman integrals

Heavy quarkonium in a holographic basis

Electromagnetic form factors have long been used to probe the underlying charge and magnetization densities of hadrons and nuclei. Traditional three-dimensional Fourier transform methods are not rigorously applicable for systems with constituents that move relativistically. The use of the transverse charge density is a new, rigorously defined way to analyze electromagnetic form factors of hadrons. This review is concerned with the following issues: what a transverse charge density is, how one is extracted from elastic scattering data, the existing results, its relationship with other observable quantities, and future prospects.

https://www.annualreviews.org/doi/pdf/10.1146/annurev.nucl.012809.104508

Transverse Charge Densities

Measurements of the deuteron form factors over a wide range of momentum transfer can provide important clues to the role of subnucleon degrees of freedom in nuclear dynamics. For a meaningful calculation of the form factors it is essential that the current density operators and the deuteron wave function transform under Lorentz transformations in a mutually consistent manner. Standard nucleon-nucleon interactions can be used to construct unitary representations of the Poincare group on the two-nucleon Hilbert space. Deuteron wave functions represent eigenstates of the four-momentum operator. Existing parameterizations of measured single-nucleon form factors are used to construct a conserved covariant electromagnetic current operator. The light-front symmetry of the representation allows a clean separation of the effects of one- and two-body currents for arbitrary momentum transfers. Comparison with data indicates that for Q/sup 2/ < GeV/sup 2/ the elastic cross sections are not dominated by two-body currents.

Light-front dynamics of elastic electron-deuteron scattering

For a relativistic system of two scalar particles, we find the Bethe-Salpeter amplitude in Minkowski space and use it to compute the electromagnetic form factor. The comparison with Euclidean space calculation shows that the Wick rotation in the form factor integral induces errors which increase with the momentum transfer Q2. At JLab domain (Q
 2 = 10 GeV^2/c2), they are about 30%. Static approximation results in an additional and more significant error. On the contrary, the form factor calculated in light-front dynamics is almost indistinguishable from the Minkowski space one.

Electromagnetic form factor via Bethe-Salpeter amplitude in Minkowski space

In the light-front dynamics, the wave function equations and their numerical solutions, for two fermion bound systems, are presented. Analytical expressions for the ladder one-boson exchange interaction kernels corresponding to scalar, pseudoscalar, pseudovector, and vector exchanges are given. Different couplings are analyzed separately and each of them is found to exhibit special features. The results are compared with the nonrelativistic solutions.

Two-fermion relativistic bound states in light-front dynamics

The bound state solutions of two fermions interacting by a scalar exchange are obtained in the framework of explicitly covariant light-front dynamics. The stability with respect to the cutoff of the ${J}^{\ensuremath{\pi}}{=0}^{+}$ and ${J}^{\ensuremath{\pi}}{=1}^{+}$ states is studied. The solutions for ${J}^{\ensuremath{\pi}}{=0}^{+}$ are found to be stable for coupling constants $\ensuremath{\alpha}{=g}^{2}/4\ensuremath{\pi}$ below the critical value ${\ensuremath{\alpha}}_{c}\ensuremath{\approx}3.72$ and unstable above it. The asymptotic behavior of the wave functions is found to follow a ${1/k}^{2+\ensuremath{\beta}}$ law. The coefficient $\ensuremath{\beta}$ and the critical coupling constant ${\ensuremath{\alpha}}_{c}$ are calculated from an eigenvalue equation. The binding energies for the ${J}^{\ensuremath{\pi}}{=1}^{+}$ solutions diverge logarithmically with the cutoff for any value of the coupling constant. For a wide range of cutoff, the states with different angular momentum projections are weakly split.

Relativistic bound states in the Yukawa model

For a relativistic system of two scalar particles, we find the Bethe-Salpeter amplitude in Minkowski space and use it to compute the electromagnetic form factor. The comparison with Euclidean space calculation shows that the Wick rotation in the form factor integral induces errors which increase with the momentum transfer Q^2. At JLab domain (Q^2=10 GeV^2/c^2), they are about 30%. Static approximation results in an additional and more significant error. On the contrary, the form factor calculated in light-front dynamics is almost indistinguishable from the Minkowski space one.

We show that in the system of two fermions interacting by scalar exchange, the solutions for Jπ=0+ bound states are stable without any cutoff regularization, for values of the coupling constant α below a critical value αc. The latter one is calculated from an eigenvalue equation.

J. Carbonell

Papers

Electromagnetic form factor via Bethe-Salpeter amplitude in Minkowski space

Two-fermion relativistic bound states in light-front dynamics

Relativistic bound states in the Yukawa model

Electromagnetic form factor via Bethe-Salpeter amplitude in Minkowski space

Stability of bound states in the light-front Yukawa model