Bethe-Salpeter Amplitude Research Articles

Bethe-Salpeter equation with cross-ladder kernel in Minkowski and Euclidean spaces

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.

Heavy-to-light form factors: Sum rules on the light cone and beyond

We report the first systematic analysis of the off-light-cone effects in sum rules for heavy-to-light form factors. These effects are investigated in a model based on scalar constituents, which allows a technically rather simple analysis but has the essential features of the analogous QCD calculation. The correlator relevant for the extraction of the heavy-to-light form factor is calculated in two different ways: first, by adopting the full Bethe-Salpeter amplitude of the light meson and, second, by performing the expansion of this amplitude near the light cone ${x}^{2}=0$. We demonstrate that the contributions to the correlator from the light-cone term ${x}^{2}=0$ and the off-light-cone terms ${x}^{2}\ensuremath{\ne}0$ have the same order in the $1/{m}_{Q}$ expansion. The light-cone correlator, corresponding to ${x}^{2}=0$, is shown to systematically overestimate the full correlator, the difference being $\ensuremath{\sim}{\ensuremath{\Lambda}}_{\mathrm{QCD}}/\ensuremath{\delta}$, with $\ensuremath{\delta}$ the continuum subtraction parameter of order 1 GeV. Numerically, this difference is found to be $10\textdiv{}20%$.

Relativistic approaches to investigation of few-nucleon systems

Theoretical approaches used in investigation of relativistic effects in high energy lepton and hadron collisions with few-nucleon systems are analyzed. The Bethe-Salpeter formalism for describing interacting systems of two spinor particles both in the continuum and bound state is described in detail. Special attention is paid to partial expansions, taking into account the analytic properties of Bethe-Salpeter amplitudes and unitary transformations correlating different representations of partial amplitudes. Mathematical methods of numerical solution of equations are considered. Results of calculation of relativistic corrections and effects of interaction in the final state in particular reactions with participation of the deuteron, namely, in elastic pD backward scattering and in reactions of deuteron break-up with production of correlated pairs, are presented.

Bethe-Salpeter equation in Minkowski space with cross-ladder kernel

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.

Majorana neutrinos in rare meson decays

The rare meson decays K+ → π−l+l′+ and D+ → K−l+l′+ (l, l′ = e, μ), which are induced by Majorana neutrino exchange and which are accompanied by lepton-number nonconservation, are considered. The effects of the meson structure are taken into account on the basis of the Gaussian model for the respective Bethe-Salpeter amplitudes. It is shown that existing direct experimental constraints on the decay branching ratios are overly lenient and therefore give no way to set realistic limits on effective Majorana masses. On the basis of the constraints on the lepton-mixing parameters and neutrino masses from precision measurements of electroweak processes, neutrino-oscillation experiments, searches for neutrinoless double-beta decay of nuclei, and cosmological data, indirect constraints on the branching ratios for the decays in question are obtained and found to be much more stringent than the above direct constraints.

Solving Bethe-Salpeter equation in Minkowski space

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.

Quarkonium-hadron interactions in perturbative QCD

The next to leading order quarkonium-hadron cross section is calculated in perturbative QCD. The corresponding leading order result was performed by Peskin more than 20 years ago using the operator product expansion. In this work, the calculation is performed using the Bethe-Salpeter amplitude and the factorization formula. The soft divergence appearing in the intermediate stages of the calculations are shown to vanish after adding all possible crossed terms, while the collinear divergences are eliminated by mass factorization. Applying the result to the Upsilon system, one finds that there are large higher order corrections near the threshold. The relevance of the present result to the charmonium case is also discussed.

Electromagnetic Properties of Diquarks

Diquark correlations play an important role in hadron physics. The properties of diquarks can be obtained from the corresponding bound state equation. Using a model for the effective quark-quark interaction that has proved successful in the light meson sector, we solve the scalar diquark Bethe-Salpeter equations and use the obtained Bethe-Salpeter amplitudes to compute the diquarks' electromagnetic form factors. The scalar ud diquark charge radius is about 8% larger than the pion charge radius, indicating that these diquarks are somewhat larger in size than the corresponding mesons. We also provide analytic fits for the form factor over a moderate range in Q^2, which may be useful, for example, in building quark-diquark models of nucleons.

The Bethe-Salpeter equation and the Low Energy Theorems for $\pi N$ scattering

The Bethe-Salpeter (BS) amplitude for $\pi N$ scattering is evaluated at the off mass shell points corresponding to the Low Energy Theorems (LET) based on PCAC and current algebra. The results suggest a way of maintaining consistency between BS equation and LET.

Chiral approach to the two pion decay of the Roper

The analysis of Manley and collaborators [1] of the decay of the Roper into two pions requires the introduction of an effective scalar-isosclar ϵ or σ meson in order to describe the s-wave ππ interaction. Here we try to understand at a more fundamental level the origin of this effective channel. In our model the two pions are produced in single pion-baryon-baryon vertices and their re-scattering in the L=0, I=0 channel is explicitly evaluated. For that purpose we use unitarized chiral perturbation theory to obtain the Bethe-Salpeter amplitude for the two pion re-scattering in the appropriate channel. Our model produces invariant mass distributions in agreement with the ones obtained in Manley's approach.

Frame dependence of the pair contribution to the pion electromagnetic form factor in a light-front approach

The frame dependence of the pair-term contribution to the electromagnetic form factor of the pion is studied within the Light Front approach. A symmetric ansatz for the pion Bethe-Salpeter amplitude with a pseudo scalar coupling of the constituent to the pion field is used. In this model, the pair term vanishes for the Drell-Yan condition, while it is dominant for momentum transfer along the light-front direction.

Exclusive charge exchange reaction pD↦n(pp) within the Bethe-Salpeter formalism

The exclusive charge exchange reaction pD↦n(pp) at intermediate and high energies is studied within the Bethe-Salpeter formalism. The final-state interaction in the detected pp pair at nearly zero excitation energy is described by the 1 S 0 component of the Bethe-Salpeter amplitude. Results of numerical calculations of polarization observables and differential cross-section persuade that, as in the non-relativistic case, this reaction i) can be utilized as a “relativistic deuteron polarimeter” and ii) delivers further information about the elementary nucleon-nucleon charge exchange amplitude.

Deuteron photodisintegration within the Bethe-Salpeter formalism: Effects of negative-energy components

Relativistic covariance requires that an analysis of the reaction γd → pn within a field-theoretical model of the nucleon-nucleon interaction involve negative-energy states. Relativistic effects and the contributions of negative-energy states in the Bethe-Salpeter amplitude to the differential cross section, the asymmetry of linearly polarized photons, and tensor asymmetries of the target deuteron are estimated on the basis of the Bethe-Salpeter formalism as applied to describing the interaction of an electromagnetic field with a two-nucleon system. It is found that the contribution of such states becomes sizable with increasing photon energy, but that it does not change the general qualitative behavior of observables as functions of angles.

Relativistic effects in proton-induced deuteron break-up at intermediate energies with forward emission of a fast proton pair

Recent data of the reaction $pd \to (pp) n$ with a fast forward pp pair with small excitation energy is analyzed within a covariant approach based on the Bethe-Salpeter formalism. Relativistic effects in cross-section are extracted and found to be large. It is demonstrated that the node of the non-relativistic amplitude is shifted and masked by relativistic effects from Lorentz boost and the negative-energy P components in the 1 S 0 Bethe-Salpeter amplitude of the pp pair.

Pair term in the electromagnetic current within the Front-Form dynamics: spin-0 case

The frame and scale dependence of the pair-term contribution to the electromagnetic form factor of a spin-zero composite system of two-fermions is studied within the Light Front. The form factor is evaluated from the plus-component of the current in the Breit frame, using for the first time a nonconstant, symmetric ansatz for the Bethe–Salpeter amplitude. The frame dependence is analyzed by allowing a nonvanishing plus component of the momentum transfer, while the dynamical scale is set by the masses of the constituents and by mass and size of the composite system. A transverse momentum distribution, associated with the Bethe–Salpeter amplitude, is introduced which allows to define strongly and weakly relativistic systems. In particular, for strongly relativistic systems, the pair term vanishes for the Drell–Yan condition, while is dominant for momentum transfer along the light-front direction. For a weakly relativistic system, fitted to the deuteron scale, the pair term is negligible up to momentum transfers of 1 (GeV/ c) 2. A comparison with results obtained within the Front-Form Hamiltonian dynamics with a fixed number of constituents is also presented.

Quarkonium-hadron interactions in QCD

The leading order non-relativistic QCD result for the quarkonium-hadron interaction cross section is derived by using the QCD factorization formula combined with the Bethe-Salpeter amplitude for the heavy quark bound state. Our result shows the equivalence of this approach with the operator product expansion even for the bound state scattering to leading order in QCD. We also apply the formula to the charmonium system and investigate the relativistic correction coming from the phase space of the reaction. We find that although $J/\psi$-gluon cross-section gets a large relativistic correction at higher energies, the correction becomes small in the $J/\psi$-hadron cross section due to the increase of the gluon distribution at smaller $x$ inside the hadron.

Effective Masses of Diquarks

We study meson and diquark bound states using the rainbow-ladder truncation of QCD's Dyson-Schwinger equations. The infrared strength of the rainbow-ladder kernel is described by two parameters. The ultraviolet behavior is fixed by the one-loop renormalization group behavior of QCD, which ensures the correct asymptotic behavior of the Bethe-Salpeter amplitudes and brings important qualitative benefits. The diquark with the lowest mass is the scalar, followed by the axialvector and pseudoscalar diquark. This ordering can be anticipated from the meson sector.

Goldstone boson's valence-quark distribution

Dynamical chiral symmetry breaking (DCSB) is one of the keystones of low-energy hadronic phenomena. Dyson-Schwinger equations provide a model-independent quark-level understanding and correlate that with the behaviour of the pion's Bethe-Salpeter amplitude. This amplitude is a core element in the calculation of pion observables and combined with the dressed-quark Schwinger function required by DCSB it yields a valence-quark distribution function for the pion that behaves as (1 – x ) 2 for x ∼ 1, in accordance with perturbative analyses. This behaviour can be verified at contemporary experimental facilities.

ABSSM—a new field-theoretic method of describing scattering/production processes of composite particles by asymptotic in-/outgoing Bethe–Salpeter states

A new relativistic field-theoretic formalism called "Asymptotic Bethe-Salpeter State Method" (ABSSM) is presented, which allows to calculate both electromagnetic formfactors and hadronic scattering and production amplitudes of elementary and composite systems (being described by Bethe-Salpeter amplitudes) within one theoretical framework. The ABSSM is then applied to the deuteron as a composite system of two nucleons.

Deuteron photodisintegration in the Bethe-Salpeter formalism

A relativistically invariant analysis of the deuteron-photodisintegration reaction is performed. The reaction γd → pn is considered in the plane-wave approximation on the basis of the Bethe-Salpeter equation with a separable kernel. The results obtained in this way are compared with the results based on nonrelativistic models and on the quasipotential approximation. It is shown that the most important relativistic contributions to the relevant angular distributions are associated with the dependence of the Bethe-Salpeter amplitude on the relative energy of nucleons and on the boost transformation of this amplitude.