Transition Electromagnetic Form Factor in the Minkowski Space Bethe–Salpeter Approach

Using the solutions of the Bethe–Salpeter equation in Minkowski space for bound and scattering states found in previous works, we calculate the transition electromagnetic form factor describing the electro-disintegration of a bound system.     

Present Status of the Bethe–Salpeter Approach in Minkowski Space

Recently developed methods allowing to find the solutions of the Bethe–Salpeter equations in Minkowski space, both for the bound and scattering states, are reviewed. For the bound states, one obtains the bound state mass and the corresponding BS amplitude. For the scattering states, the phase shifts (complex above the meson creation threshold) and the half-off-shell BS amplitude are found. Using these solutions, the elastic and transition electromagnetic form factors are calculated.     

Solving the Bethe–Salpeter Equation in Euclidean Space

Different approaches to solve the spinor–spinor Bethe–Salpeter (BS) equation in Euclidean space are considered. It is argued that the complete set of Dirac matrices is the most appropriate basis to define the partial amplitudes and to solve numerically the resulting system of equations with realistic interaction kernels. Other representations can be obtained by performing proper unitary transformations. A generalization of the iteration method for finding the energy spectrum of the BS equation is discussed and examples of concrete calculations are presented. Comparison of relativistic calculations with available experimental data and with corresponding non relativistic results together with an analysis of the role of Lorentz boost effects and relativistic corrections are presented. A novel method related to the use of hyperspherical harmonics is considered for a representation of the vertex functions suitable for numerical calculations.     

Euclidean to Minkowski Bethe–Salpeter amplitude and observables

We propose a method to reconstruct the Bethe–Salpeter amplitude in Minkowski space given the Euclidean Bethe–Salpeter amplitude – or alternatively the light-front wave function – as input. The method is based on the numerical inversion of the Nakanishi integral representation and computing the corresponding weight function. This inversion procedure is, in general, rather unstable, and we propose several ways to considerably reduce the instabilities. In terms of the Nakanishi weight function, one can easily compute the BS amplitude, the LF wave function and the electromagnetic form factor. The latter ones are very stable in spite of residual instabilities in the weight function. This procedure allows both, to continue the Euclidean BS solution in the Minkowski space and to obtain a BS amplitude from a LF wave function.     

Calculation of Kaon Electromagnetic Form Factor in the Framework of Coupled Schwinger—Dyson and Bethe—Salpeter Formulation

The kaon electromagnetic form factor is calculated in the framework of coupled Schwinger-Dyson equation in rainbow approximation and Bethe-Salpeter equation in ladder approximation with the modified flat-bottom potential,which is the combination of the flat-bottom potential with considerations for the infrared and ultraviolet asymptotic behaviours of the effective quark-gluon coupling.All our numerical results give good fit to experimental values and other theoretical results.     

Solving Bethe–Salpeter Equation for Scattering States

We present the Minkowski space solutions of the inhomogeneous Bethe–Salpeter equation for spinless particles with a ladder kernel. The off-mass shell scattering amplitude is first obtained.     

Resolving the Bethe–Salpeter Kernel

A novel method for constructing a kernel for the meson bound-state problem is described.It produces a closed form that is symmetry-consistent(discrete and continuous) with the gap equation defined by any admissible gluon-quark vertex,Γ.Applicable even when the diagrammatic content of Γ is unknown,the scheme can foster new synergies between continuum and lattice approaches to strong interactions.The framework is illustrated by showing that the presence of a dressed-quark anomalous magnetic moment in Γ,an emergent feature of strong interactions,can remedy many defects of widely used meson bound-state kernels,including the mass splittings between vector and axial-vector mesons and the level ordering of pseudoscalar and vector meson radial excitations.     

Separation of different wave components in the Bethe–Salpeter wave function

The scalar products of polarization tensor and unit vectors are presented explicitly in spherical coordinate system expanded in terms of spherical harmonic functions. By applying the obtained formulae, different wave components in the Salpeter wave function can be shown explicitly, and the results are consistent with the results obtained by L–S coupling analysis. The cancelation formula is given, by which the terms with pure L = J + 1 wave components in the Salpeter wave function for the bound state with h_P=(-1)^J\eta_{\rm P}=(-1)^J can be obtained by separating the L = J − 1 wave components from mixing terms. This separation provides the basis for studying higher-order contributions from the coupling of L = J − 1 and J + 1 wave states, and for solving the Salpeter equation exactly without approximation.     

Manifestation of Three-Body Forces in Three-Body Bethe–Salpeter and Light-Front Equations

Bethe–Salpeter and light-front bound state equations for three scalar particles interacting by scalar exchange-bosons are solved in ladder truncation. In contrast to two-body systems, the three-body binding energies obtained in these two approaches differ significantly from each other: the ladder kernel in light-front dynamics underbinds by approximately a factor of two compared to the ladder Bethe–Salpeter equation. By taking into account three-body forces in the light-front approach, generated by two exchange-bosons in flight, we find that most of this difference disappears; for small exchange masses, the obtained binding energies coincide with each other.     

Boundary Solutions of the Quantum Yang–Baxter Equation and Solutions in Three Dimensions

Boundary solutions to the quantum Yang–Baxter (qYB) equation are defined to be those in the boundary of (but not in) the variety of solutions to the modified qYB equation, the latter being analogous to the modified classical Yang–Baxter (cYB) equation. We construct, for a large class of solutions r to the modified cYB equation, explicit boundary quantizations, i.e., boundary solutions to the qYB equation of the form I + tr + t²r₂ +. In the last section we list and give quantizations for all classical r-matrices in sl(3) sl(3).     

Relativity from Light Front and Bethe–Salpeter Equations

We present some results of two independent relativistic approaches to the few-body problem: light-front dynamics and Bethe–Salpeter equation. We show that implementing relativistic invariance leads to new qualitative properties, and that, even driven by the same interaction Lagrangian, both approaches provide different quantitative results, especially in three-body systems. The case of Bethe–Salpeter equation for computing electromagnetic form factors is discussed.     

Tetraquark bound states in a Bethe–Salpeter approach

We determine the mass of tetraquark bound states from a coupled system of covariant Bethe–Salpeter equations. Similar in spirit to the quark–diquark model of the nucleon, we approximate the full four-body equation for the tetraquark by a coupled set of two-body equations with meson and diquark constituents. These are calculated from their quark and gluon substructure using a phenomenologically well-established quark–gluon interaction. For the lightest scalar tetraquark we find a mass of the order of 400 MeV and a wave function dominated by the pion–pion constituents. Both results are in agreement with a meson molecule picture for the _f0(600)$f_0(600)$. Our results furthermore suggest the presence of a potentially narrow all-charm tetraquark in the mass region 5–6 GeV.     

Square Integrability and Uniqueness of the Solutions of the Kadomtsev–Petviashvili-I Equation

We prove that the solution of the Cauchy problem for the Kadomtsev–Petviashvili-I Equation obtained by the inverse spectral method belongs to the Sobolev space H^k(R²) for k 0, under the assumption that the initial datum is a small Schwartz function. This solution is shown to be the unique solution within a class of generalized solutions of the Kadomtsev–Petviashvili-I equation.     

Construction of the Darboux Transformaiton and Solutions to the Modified Nizhnik-Novikov-Veselov Equation

A kind of modified Nizhnik-Novikov-Veselov equation is investigated via the Darboux transformation and some soliton solutions are constructed. The property of excitation is also discussed.