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In the Wick-Cutkosky model, where two scalar massive constituents interact by means of the exchange of a scalar massless particle, the Bethe-Salpeter equation has solutions of two types, called "normal" and "abnormal". In the non-relativistic limit, the normal solutions correspond to the usual Coulomb spectrum, whereas the abnormal ones do not have non-relativistic counterparts -- they are absent in the Schr\"odinger equation framework. We have studied, in the formalism of the light-front dynamics, the Fock-space content of the abnormal solutions. It turns out that, in contrast to the normal ones, the abnormal states are dominated by the massless exchange particles (by 90 \% or more), what provides a natural explanation of their decoupling from the two-body Schr\"odinger equation. Assuming that one of the massive constituents is charged, we have calculated the electromagnetic elastic form factors of the normal and abnormal states, as well as the transition form factors. The results on form factors confirm the many-body nature of the abnormal states, as found from the Fock-space analysis. The abnormal solutions have thus properties similar to those of hybrid states, made here essentially of two massive constituents and several or many massless exchange particles. They could also be interpreted as the Abelian scalar analogs of the QCD hybrid states. The question of the validity of the ladder approximation of the model is also examined., Comment: 22 pages, 20 figures, 2 tables. Accepted in Eur. Phys. J. C

The Bethe-Salpeter equation for two massive scalar particles interacting by scalar massless exchange has solutions of two types, which differ from each other by their behavior in the non-relativistic limit: the normal solutions which turn into the Coulomb ones and the "abnormal" solutions. The latter ones have no non-relativistic counterparts and disappear in the non-relativistic limit. We studied the composition of all these states. It turns out that the normal states, even for large binding energy, are dominated by two massive particles. Whereas, the contribution of the two-body sector into the abnormal states, even for small binding energy, is of the order of 1% only; they are dominated by an indefinite number of the massless particles. The elastic electromagnetic form factors for both normal and abnormal states, as well as the transition ones between them, are calculated., Comment: 5 pages, 3 figures, Contribution in proceedings of the conference: Light Cone 2019, 16-20 September 2019, Ecole Polytechnique, Palaiseau, France

Bethe-Salpeter equation, for massless exchange and large fine structure constant $\alpha>\pi/4$, in addition to the Balmer series, provides another (abnormal) series of energy levels which are not given by the Schr\"odinger equation. So strong field can be created by a point-like charge $Z>107$. The nuclei with this charge, though available, they are far from to be point-like that weakens the field. Therefore, the abnormal states of this origin hardly exist. We analyze the more realistic case of exchange by a massive particle when the large value of coupling constant is typical for the strong interaction. It turns out that this interaction still generates a series of abnormal relativistic states. The properties of these solutions are studied. Their existence in nature seems possible., Comment: 13 pages, 10 figures, to be published in Proceedings of the International Conference: Nuclear Theory in the Supercomputing Era-2018 (NTSE-2018), Daejeon, South Korea, October 29 - November 2, 2018; eds. A. M. Shirokov and A. I. Mazur. Pacific National University, Khabarovsk, Russia, 2019

The bound state Bethe-Salpeter amplitude was expressed by Nakanishi in terms of a smooth weight function g. By using the generalized Stieltjes transform, we derive an integral equation for the Nakanishi function g for a bound state case. It has the standard form g= Vg, where V is a two-dimensional integral operator. The prescription for obtaining the kernel V starting with the kernel K of the Bethe-Salpeter equation is given., Comment: 7 pages

The bound state Bethe-Salpeter amplitude was expressed by Nakanishi using a two-dimensional integral representation, in terms of a smooth weight function $g$, which carries the detailed dynamical information. A similar, but one-dimensional, integral representation can be obtained for the Light-Front wave function in terms of the same weight function $g$. By using the generalized Stieltjes transform, we first obtain $g$ in terms of the Light-Front wave function in the complex plane of its arguments. Next, a new integral equation for the Nakanishi weight function $g$ is derived for a bound state case. It has the standard form $g= N g$, where $N$ is a two-dimensional integral operator. We give the prescription for obtaining the kernel $ N$ starting with the kernel $K$ of the Bethe-Salpeter equation. The derivation is valid for any kernel given by an irreducible Feynman amplitude., Comment: 12 pages, 1 figure, to appear in Phys. Lett. B

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., Comment: 11 pages, 12 figures

We shortly review different methods to obtain the scattering solutions of the Bethe-Salpeter equation in Minkowski space. We emphasize the possibility to obtain the zero energy observables in terms of the Euclidean scattering amplitude., Comment: 7 pages, 2 figures, contribution to the conference Light-Cone 2015, Frascati, Italy, 21-25 September, 2015. Submitted in Few-Body Systems

We review a method to directly solve the Bethe-Salpeter equation in Minkowski space, both for bound and scattering states. It is based on a proper treatment of the many singularities which appear in the kernel and propagators. The off-mass shell scattering amplitude for spinless particles interacting by a one boson exchange was computed for the first time. Using our Minkowski space solutions for the initial (bound) and final (scattering) states, we calculate elastic and transition (bound to scattering state) electromagnetic form factors. The conservation of the transition electromagnetic current J.q=0, verified numerically, confirms the validity of our solutions., Comment: 4 pages, 4 figures, contribution to 21st International Conference on Few-Body Problems in Physics, Chicago, USA, May 18-21, 2015. To be published by EPJ Web of Conferences

We show that the Bethe-Salpeter equation for the scattering amplitude in the limit of zero incident energy can be transformed into a purely Euclidean form, as it is the case for the bound states. The decoupling between Euclidean and Minkowski amplitudes is only possible for zero energy scattering observables and allows determining the scattering length from the Euclidean Bethe-Salpeter amplitude. Such a possibility strongly simplifies the numerical solution of the Bethe-Salpeter equation and suggests an alternative way to compute the scattering length in Lattice Euclidean calculations without using the Luscher formalism. The derivations contained in this work were performed for scalar particles and one-boson exchange kernel. They can be generalized to the fermion case and more involved interactions., Comment: 11 pages, 3 figures, to be published in Phys. Lett. B

The transition form factor for electrodisintegration of a two-body bound system is calculated in the Bethe-Salpeter framework. For the initial (bound) and the final (scattering) states, we use our solutions of the Bethe-Salpeter equation in Minkowski space which were first obtained recently. The gauge invariance, which manifests itself in the conservation of the transition electromagnetic current Jq = 0, is studied numerically. It results from a cancellation between the plane wave and the final state interaction contributions. This cancellation takes place only if the initial bound state BS amplitude, the final scattering state and the operator of electromagnetic current are strictly consistent with each other, that is if they are found in the same dynamical framework. A reliable result for the transition form factor can be obtained in this case only., Comment: 23 pages, 9 figures, to be published in Phys. Rev. D

Using our solutions of the Bethe-Salpeter equation with OBE kernel in Minkowski space both for the bound and scattering states, we calculate the transition form factors for electrodisintegration of the bound system which determine the electromagnetic current J of this process. Special emphasis is put on verifying the gauge invariance which should manifest itself in the current conservation. We find that for any value of the momentum transfer q the contributions of the plane wave and the final state interaction to the quantity J.q cancel each other thus providing J.q=0. However, this cancellation is obtained only if the initial Bethe-Salpeter amplitude (bound state), the final one (scattering state) and the current operator are strictly consistent with each other. A reliable result for the transition form factor can be found only in this case., Comment: 7 pages, 5 figures. Contribution to the proceedings of the Workshop: Light-Cone 2014, Raleigh, North Carolina, USA, May 26-30, 2014. To be published in Few-Body Systems

We present a method to directly solving the Bethe-Salpeter equation in Minkowski space, both for bound and scattering states. It is based on a proper treatment of the singularities which appear in the kernel, propagators and Bethe-Salpeter amplitude itself. The off-mass shell scattering amplitude for spinless particles interacting by a one boson exchange is computed for the first time., Comment: 24 pages, 12 figures, to be published in Phys. Rev. D

When a two-body system is bound by a zero-range interaction, the corresponding three-body system -- considered in a non-relativistic framework -- collapses, that is its binding energy is unbounded from below. In a paper by J.V. Lindesay and H.P. Noyes it was shown that the relativistic effects result in an effective repulsion in such a way that three-body binding energy remains also finite, thus preventing the three-body system from collapse. Later, this property was confirmed in other works based on different versions of relativistic approaches. However, the three-body system exists only for a limited range of two-body binding energy values. For stronger two-body interaction, the relativistic three-body system still collapses. A similar phenomenon was found in a two-body systems themselves: a two-fermion system with one-boson exchange interaction in a state with zero angular momentum J=0 exists if the coupling constant does not exceed some critical value but it also collapses for larger coupling constant. For a J=1 state, it collapses for any coupling constant value. These properties are called "critical stability". This contribution aims to be a brief review of this field pioneered by H.P. Noyes., Comment: 20 pages, 7 figures, 1 table

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., Comment: 4 pages, 3 figures. Presented at the 22nd European Conference on Few-Body Problems in Physics, 9-13 September, 2013, Cracow, Poland. Submitted in Few-Body Systems

The off-mass shell scattering amplitude, satisfying the Bethe-Salpeter equation for spinless particles in Minkowski space with the ladder kernel, is computed for the first time., Comment: 9 pages, 4 figures, 2 tables, to be published in Phys. Lett. B

The off-mass shell scattering amplitude, satisfying the Bethe-Salpeter equation for spinless particles in Minkowski space with the ladder kernel, is computed for the first time., Comment: 7 pages, 3 figures. Presented at the XXI International Baldin Seminar on High Energy Physics Problems, September 10-15, 2012, JINR, Dubna, Russia. To be published in the online journal Proceedings of Science - PoS(Baldin ISHEPP XXI)027 (2012)

After a brief discussion of the meaning of the potential in quantum mechanics, we examine the results of the Yukawa model (scalar meson exchange) for the nucleon-nucleon interaction in three different dynamical frameworks: the non-relativistic dynamics of the Schrodinger equation, the relativistic quantum mechanics of the Bethe-Salpeter and Light-Front equations and the lattice solution of the Quantum Field Theory, obtained in the quenched approximation., Comment: 17 pages, 9 figures. Dedicated to Professor Henryk Witala at the occasion of his 60th birthday. Submitted in Few-Body Systems

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., Comment: 4 pages, 2 figures. Presented at the 20th International IUPAP Conference on Few-Body Problems in Physics, 20 - 25 August, 2012, Fukuoka, Japan. Submitted in Few-Body Systems

The masses of the low-lying strange and charm baryons are evaluated using two degenerate flavors of twisted mass sea quarks for pion masses in the range of about 260 MeV to 450 MeV. The strange and charm valence quark masses are tuned to reproduce the mass of the kaon and D-meson at the physical point. The tree-level Symanzik improved gauge action is employed. We use three values of the lattice spacing, corresponding to $\beta=3.9$, $\beta=4.05$ and $\beta=4.2$ with $r_0/a=5.22(2)$, $r_0/a=6.61(3)$ and $r_0/a=8.31(5)$ respectively. %spacings $a=0.0855(5)$ and $a=0.0667(3)$ determined from the pion decay constant. We examine the dependence of the strange and charm baryons on the lattice spacing and strange and charm quark masses. The pion mass dependence is studied and physical results are obtained using heavy baryon chiral perturbation theory to extrapolate to the physical point., Comment: 19 pages, 18 Figures. Typos corrected, reference added, version to appear in Phys. Rev. D. arXiv admin note: text overlap with arXiv:0910.2419

We present results on the lower moments of the nucleon generalized parton distri butions within lattice QCD using two dynamical flavors of degenerate twisted mass fermions. Our simulations are performed on lattices with three different values of the lattice spacings, namely $a=0.089$ fm, $a=0.070$ fm and $a=0.056$ fm, allowing the investigation of cut-off effects. The volume dependence is examined using simulations on two lattices of spatial length $L=2.1$ fm and $L=2.8$ fm. The simulations span pion masses in the range of 260-470 MeV. Our results are renormalized non-perturbatively and the values are given in the $\bar{\rm MS}$ scheme at a scale $ \mu=2$ GeV. They are chirally extrapolated to the physical point in order to compare with experiment. The consequences of these results on the spin carried by the quarks in the nucleon are investigated., Comment: 23 pages, 15 figures

We present results on the nucleon electromagnetic form factors within lattice QCD using two flavors of degenerate twisted mass fermions. Volume effects are examined using simulations at two volumes of spatial length L=2.1 fm and L=2.8 fm. Cut-off effects are investigated using three different values of the lattice spacings, namely a=0.089 fm, a=0.070 and a=0.056 fm. The nucleon magnetic moment, Dirac and Pauli radii are obtained in the continuum limit and chirally extrapolated to the physical pion mass allowing for a comparison with experiment., Comment: ETM Collaboration, 14 pages, 15 Figures

We present results on the nucleon form factors and moments of generalized parton distributions obtained within the twisted mass formulation of lattice QCD. We include a discussion of lattice artifacts by examining results at different volumes and lattice spacings. We compare our results with those obtained using different discretization schemes and to experiment., Comment: 4 pages, 3 Figures, Invited talk presented at the 35th International Conference of High Energy Physics - ICHEP 2010 July 22-28, 2010, Paris, France

We present results on the nucleon axial form factors within lattice QCD using two flavors of degenerate twisted mass fermions. Volume effects are examined using simulations at two volumes of spatial length $L=2.1$ fm and $L=2.8$ fm. Cut-off effects are investigated using three different values of the lattice spacings, namely $a=0.089$ fm, $a=0.070$ fm and $a=0.056$ fm. The nucleon axial charge is obtained in the continuum limit and chirally extrapolated to the physical pion mass enabling comparison with experiment., Comment: ETM Collaboration, 14 pages, 12 figures

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., Comment: 20 pages, 14 figures, contribution to proceedings of the workshop: "Relativistic Description of Two- and Three-Body Systems in Nuclear Physics", ECT*, October 19-23, 2009. To be published in Few-Body Systems

The method of solving the Bethe-Salpeter equation in Minkowski space, which we developed previously for spinless particles, is extended to a system of two fermions. The method is based on the Nakanishi integral representation of the amplitude and on projecting the equation on the light-front plane. The singularities in the projected two-fermion kernel are regularized without modifying the original Bethe-Salpeter amplitudes. The numerical solutions for the J=0 bound state with the scalar, pseudoscalar and massless vector exchange kernels are found. The stability of the scalar and positronium states without vertex form factor is discussed. Binding energies are in close agreement with the Euclidean results. Corresponding amplitudes in Minkowski space are obtained., Comment: 11 pages, 9 figures, to be published in Eur. Phys. J. A

The method of solving the Bethe-Salpeter equation in Minkowski space, developed previously for spinless particles, is extended to a system of two fermions. The method is based on the Nakanishi integral representation of the amplitude and on projecting the equation on the light-front plane. The singularities in the projected two-fermion kernel are regularized without modifying the original BS amplitudes. The numerical solutions for the J=0 bound state with the scalar, pseudoscalar and massless vector exchange kernels are found. Binding energies are in close agreement with the Euclidean results. Corresponding amplitudes in Minkowski space are obtained., Comment: 8 pages, 5 figures. Contribution to the proceedings of the Workshop: Light-Cone 2010, "Relativistic Hadronic and Particle Physics", June 14-18, 2010, Valencia, Spain. To be published in the online journal "Proceedings of Science" - PoS

We present results on the electromagnetic and axial nucleon form factors using two degenerate flavors of twisted mass fermions on lattices of spatial size 2.1 fm and 2.7 fm and a lattice spacing of about 0.09 fm. We consider pion masses in the range of 260-470 MeV. We chirally extrapolate results on the nucleon axial ch arge, the isovector Dirac and Pauli root mean squared radii and magnetic moment to the physical point and co mpare to experiment., Comment: Report no. added; typos corrected. ETM Collaboration, 7 pages, 10 figures, Talk given at the XXVII International Symposium on Lattice Field Theory - LAT2009, July 26-31 2009, Peking University, Beijing, China

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., Comment: 8 pages, 7 figures, to be published in Eur. Phys. J. A; Reference [15] is added

The electromagnetic form factors calculated through Euclidean Bethe-Salpeter amplitude and through the light-front wave function are compared with the one found using the Bethe-Salpeter amplitude in Minkowski space. The form factor expressed through the Euclidean Bethe-Salpeter amplitude (both within and without static approximation) considerably differs from the Minkowski one, whereas form factor found in the light-front approach is almost indistinguishable from it., Comment: 3 pages, 2 figures. Contribution to the proceedings of the 20th International Conference on Few-Body Problems in Physics (FB20), Pisa, Italy, September 10-14, 2007. To be published in "Few-Body Systems"

Some results obtained by a new method for solving the Bethe-Salpeter equation are presented. The method is valid for any kernel given by irreducible Feynman graphs. The Bethe-Salpeter amplitude, both in Minkowski and in Euclidean spaces, and the binding energy for ladder + cross-ladder kernel are found. We calculate also the corresponding electromagnetic form factor., Comment: 4 pages, 3 figures. Contribution to the proceedings of the 18th International IUPAP Conference on Few-Body Problems in Physics (FB18), Santos, Brasil, August 21-26, 2006. To be published in Nucl. Phys. A

A new method for solving the Bethe-Salpeter equation is developed. It allows to find the Bethe-Salpeter amplitudes both in Minkowski and in Euclidean spaces and, as a by product, the light-front wave function. The method is valid for any kernel given by irreducible Feynman graphs. Bethe-Salpeter and Light-Front equations for scalar particles with ladder + cross-ladder kernel are solved., Comment: 7 pages, 5 figures, to appear in the proceedings of the Workshop on Light-Cone QCD and Nonperturbative Hadron Physics, Cairns, Australia, July 7-15, 2005

Bethe-Salpeter (BS) equation in Minkowski space for scalar particles is solved for a kernel given by a sum of ladder and cross-ladder exchanges. The solution of corresponding Light-Front (LF) equation, where we add the time-ordered stretched boxes, is also obtained. Cross-ladder contributions are found to be very large and attractive, whereas the influence of stretched boxes is negligible. Both approaches -- BS and LF -- give very close results., Comment: 11 pages, 7 figures

We develop a new method of solving Bethe-Salpeter (BS) equation in Minkowski space. It is based on projecting the BS equation on the light-front (LF) plane and on the Nakanishi integral representation of the BS amplitude. This method is valid for any kernel given by the irreducible Feynman graphs. For massless ladder exchange, our approach reproduces analytically the Wick-Cutkosky equation. For massive ladder exchange, the numerical results coincide with the ones obtained by Wick rotation., Comment: 10 pages, 4 figures

For zero-range interaction providing a given mass M_2 of the two-body bound state, the mass M_3 of the relativistic three-body bound state is calculated. We have found that the three-body system exists only when M_2 is greater than a critical value M_c (approx. 1.43m for bosons and approx. 1.35m for fermions, m is the constituent mass). For M_2=M_c the mass M_3 turns into zero and for M_2

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 non relativistic solutions., Comment: 40 pages, to be published in Phys. Rev. C, .tar.gz file

The light-front dynamics is an efficient approach to study of field theory and of relativistic composite systems (nuclei at relativistic relative nucleon momenta, hadrons in the quark models). The explicitly covariant version of this approach is briefly reviewed and illustrated by some applications., Comment: 16 pages, 8 figures. Talk at the 21-st International Workshop on Nuclear Theory, Rila Mountains, Bulgaria, June 10-15, 2002

For the zero-range interaction providing a given mass M_2 of the two-body bound state, the mass M_3 of the relativistic three-boson state is calculated. We have found that the three-body system exists only when M_2 is greater than a critical value M_c approximately 1.43 m (m is the constituent mass). For M_2=M_c the mass M_3 turns into zero and for M_2

The bound state solutions of two fermions interacting by a scalar exchange are obtained in the framework of the explicitly covariant light-front dynamics. The stability with respect to cutoff of the J$^{\pi}$=$0^+$ and J$^{\pi}$=$1^+$ states is studied. The solutions for J$^{\pi}$=$0^+$ are found to be stable for coupling constants $\alpha={g^2\over4\pi}$ below the critical value $\alpha_c\approx 3.72$ and unstable above it. The asymptotic behavior of the wave functions is found to follow a ${1\over k^{2+\beta}}$ law. The coefficient $\beta$ and the critical coupling constant $\alpha_c$ are calculated from an eigenvalue equation. The binding energies for the J$^{\pi}$=$1^+$ 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., Comment: 22 pages, 13 figures, .tar.gz file

We show that in the system of two fermions interacting by scalar exchange, the solutions for J$^{\pi}$=$0^+$ bound states are stable without any cutoff regularization, for values of the coupling constant $\alpha$ below a critical value $\alpha_c$. This latter is calculated from an eigenvalue equation., Comment: 4 pages, 3 figures, .tar.gz file

We show that in the system of two fermions interacting by scalar exchange, the solutions for J$^{\pi}$=$0^+$ bound states are stable without any cutoff regularization for coupling constant below some critical value., Comment: 5 pages, 2 figures, submitted to publication