Reduction of the Bethe–Salpeter wave function: Fermion–scalar case and scalar–scalar case

In this paper, the general forms of the nonrelativistic Bethe–Salpeter wave functions for fermion–scalar bound state and scalar–scalar bound state are presented. Using the obtained normalization conditions and the corresponding Schrödinger equations for these bound states, the nonrelativistic Bethe–Salpeter wave functions can be calculated and can be used to compute the amplitude for the process involving these bound states.     

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.     

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.     

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.     

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.     

Solutions of the Bethe–Salpeter Equation in Minkowski Space and Applications to Electromagnetic Form Factors

We present a new method for solving the two-body Bethe–Salpeter equation in Minkowski space. It is based on the Nakanishi integral representation of the Bethe–Salpeter amplitude and on subsequent projection of the equation on the light-front plane. The method is valid for any kernel given by the irreducible Feynman graphs and for systems of spinless particles or fermions. The Bethe–Salpeter amplitudes in Minkowski space are obtained. The electromagnetic form factors are computed and compared to the Euclidean results.     

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.     

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.     

Studying the D ∗ K molecular structure of D^{+}_{s1}(2460) in the Bethe–Salpeter approach

To explain both the possible superluminal neutrino propagation and all the known phenomenological constraints/observations on Lorentz violation, the Background-Dependent Lorentz Violation (BDLV) has been proposed. We study the BDLV in a model-independent way. Assuming that the Lorentz violation on the Earth is much larger than those on the interstellar scale, we automatically escape all the astrophysical constraints on Lorentz violation. For the BDLV from the effective field theory, we present a simple model and discuss the possible solutions to the theoretical challenges of the superluminal neutrino propagation such as the Bremsstrahlung effects for muon neutrinos and the pion decays. Also, we address the Lorentz violation constraints from the LEP and KamLAMD experiments. For the BDLV from the Type IIB string theory with D3-branes and D7-branes, we point out that the D3-branes are flavour blind, and all the SM particles are the conventional particles as in the traditional SM when they do not interact with the D3-branes. Thus, we not only can naturally avoid all the known phenomenological constraints on Lorentz violation, but also can naturally explain all the theoretical challenges. Interestingly, the energy-dependent photon velocities may be tested at the experiments.     

Inverting the Nakanishi Integral Relation for a Bound State Euclidean Bethe–Salpeter Amplitude

The extraction of the weight function g of the Nakanishi integral representation of the Bethe–Salpeter amplitude is investigated. We studied the numerical inversion of the discretized Nakanishi kernel in the case of an Euclidean bound state. The discretized kernel is regularized by adding the identity operator times a small regularisation parameter \({\varepsilon}\) to avoid numerically unstabilities. We have found that the weight function g as well as the associated light-front valence wave function are unstable against variation of \({\varepsilon}\). These results suggest that the extraction of the Nakanishi weight function from an Euclidean amplitude, is an ill-defined problem. Without further assumptions on the solution or/and without developing more elaborate methods, the Nakanishi weight function, as well as the corresponding light-front valence wave function, cannot be safely determined.     

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.     

Decay of the Bethe–Salpeter Kernel and Bound States for Lattice Classical Ferromagnetic Spin Systems at High Temperature

We consider lattice classical ferromagnetic spin systems at high temperature (1) with nearest neighbor interactions and even single-spin distributions (ssd). Associated with each system is an imaginary time lattice quantum field theory. It is known that there is a particle of mass m–ln in the energy-momentum spectrum. If s ⁴–3s ²²<0, where s ^k is the kth moment of the ssd, and is sufficiently small, we show that in the two-particle subspace there is no mass spectrum up to 2m. For >0 we show that the only mass spectrum in (m, 2m) is a bound state of mass m _b=2m+ln(1–)+O(), where =(+2s ²²)^–1. A bound on the decay of the kernel of a Bethe–Salpeter equation is obtained and used to prove these results.     

Separation of variables in the wave equation. Sets of the type (1.1) and Schrödinger equation

All complete sets of symmetry operators of the wave equation in Minkowski space containing one isotropic first-order operator of the form /x⁰+/x³ and two second-order operators are obtained. Privileged coordinates corresponding to these sets are given. The variables are separated in the wave equation and in the free Schrödinger equation in three-dimensional spacetime.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 44–48, February, 1991.     

Wigner function of coherent state of N components

In this paper, we study the Wigner function of coherent state of N components, especially two components and three components. This function consists of two terms： the Gaussian term and the interference term with the negativity. The first term comprises N Gaussian surfaces evenly centred on a circle of radius ｜β｜ = ｜α｜ with a separate angle of 2π/N, and the second term is composed of 1/2N（N - 1） Gaussian-cosine surfaces evenly centred in a circular region of radius ｜β｜ 〈 ｜α｜. Here, a is the eigenvalue of the annihilation operator α, and β is a variable in some complex space in which the Wigner function is defined. We have proved that the essential condition to eliminate the negativity of the Wigner function is that the mean photon count of the coherent state is equal to that of the Glouber coherent state.     

A trial wave function to study of the structure of the molecular dd μ ion

This paper provides a new effort to study of the ddμ structure. The present work is numerically performed using a new trial wave function to the ddμ system in configuration of coupled channels. The present results of energies are more accurate than those of our previous work. The obtained results of formation rates are close to results published by Yu.V. Petrov et al. and giving strong indications that the trial wave function is good enough in determining the resonance states of the mentioned ionic molecule.