Bethe-Salpeter Formalism Research Articles

Tagging the reaction pn → dϕ by backward protons in pd → dϕpsp processes

The reaction pd → dϕpsp is studied within the Bethe–Salpeter formalism. Under special kinematical conditions (slow backward spectator proton psp and fast forward deuteron) relevant for forthcoming experiments, e.g. at COSY, the cross section and a set of polarization observables are shown to factorize in the contribution of the pure subprocess pn → dϕ and a contribution stemming from deuteron quantities and kinematical factors. This provides a theoretical basis for studying threshold-near processes at quasi-free neutrons.

Relating the proton, neutron and deuteron structure functions in the covariant Bethe–Salpeter formalism

The neutron structure function F2n(x) is evaluated within the kinematic range 10−3<x<1 from the deuteron and proton data by employing relativistic theoretical description of F2D(x) and several assumptions on the high-x asymptotics of F2n(x)/F2p(x). It is shown that new measurements of F2D(x) in the range 0.6<x⩽0.8 would substantially improve understanding of the relation between d and u valence quarks in the limit x→1.

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.

Study of relativistic bound states for scalar theories in the Bethe-Salpeter and Dyson-Schwinger formalism

The Bethe-Salpeter equation for Wick-Cutkosky like models is solved in dressed ladder approximation. The bare vertex truncation of the Dyson-Schwinger equations for propagators is combined with the dressed ladder Bethe-Salpeter equation for the scalar S-wave bound state amplitudes. With the help of spectral representation the results are obtained directly in Minkowski space. We give a new analytic formula for the resulting equation simplifying the numerical treatment. The bare ladder approximation of Bethe-Salpeter equation is compared with the one with dressed ladder. The elastic electromagnetic form factors is calculated within the relativistic impulse approximation.

Current in the light-front Bethe-Salpeter formalism. II. Applications

We pursue applications of the light-front reduction of current matrix elements in the Bethe-Salpeter formalism. The normalization of the reduced wave function is derived from the covariant framework and related to non-valence probabilities using familiar Fock space projection operators. Using a simple model, we obtain expressions for generalized parton distributions that are continuous. The non-vanishing of these distributions at the crossover between kinematic regimes (where the plus component of the struck quark's momentum is equal to the plus component of the momentum transfer) is tied to higher Fock components. Moreover continuity holds due to relations between Fock components at vanishing plus momentum. Lastly we apply the light-front reduction to time-like form factors and derive expressions for the generalized distribution amplitudes in this model.

Current in the light-front Bethe-Salpeter formalism. I. Replacement of non-wave-function vertices

We apply the light-front reduction of the Bethe-Salpeter equation to matrix elements of the electromagnetic current between bound states. Using a simple (1+1)-dimensional model to calculate form factors, we focus on two cases. In one case, the interaction is dominated by a term instantaneous in light-cone time. Here effects of higher Fock states are negligible and the form factor can be effectively expressed using non-wave function vertices and crossed interactions. If the interaction is not instantaneous, non-wave function vertices are replaced by contributions from higher Fock states. These higher Fock components arise from the covariant formalism via the energy poles of the Bethe-Salpeter vertex and the electromagnetic vertex. The replacement of non-wave function vertices in time-ordered perturbation theory is a theorem which directly extends to generalized parton distributions, e.g., in (3+1) dimensions.

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.

Infrared behavior of the running coupling constant and bound states in QCD

The perturbative expression of the running strong coupling constant \alpha_{\rm s}(Q^{2}) has an unphysical singularity for Q^{2} = \Lambda^{2}_{\rm QCD}. Various modification have been proposed for the infrared region. The effect of some of such proposals on the quark-antiquark spectrum is tested on a Bethe-Salpeter (second order) formalism which was successfully applied in previous papers to an overall evaluation of the spectrum in the light-light, light-heavy and heavy-heavy sectors (the only serious discrepancy with data being for the light pseudoscalar meson masses). In this paper only the {\rm c} \bar{\rm c}, {\rm b} \bar{\rm b} and {\rm q} \bar{\rm q} (q = u or d) cases are considered and fine structure is neglected. It is found that in the {\rm b} \bar{\rm b} and {\rm c} \bar{\rm c} cases the results are little sensitive to the specific choice. In the light-light case the Dokshitzer et al. prescription is again essentially equivalent to the truncation prescription used in the previous calculation and it is consistent with the same a priori fixing of the quark light masses on the typical current values m_{\rm u} = m_{\rm d} = 10 MeV (only the pion mass resulting completely out of scale of about 500 MeV). With the Shirkov-Solovtsov prescription, on the contrary, a reasonable agreement with the data is obtained only at the price of using a phenomenological momentum dependent effective mass for the quark. The use of such an effective mass should amount to a correction of the free quark propagator. It is remarkable that this has also the effect of bringing the pion mass in the correct range.

INSTANTANEOUS BETHE–SALPETER EQUATION: IMPROVED ANALYTICAL SOLUTION

Studying the Bethe–Salpeter formalism for interactions instantaneous in the rest-frame of the bound states described, we show that, for bound-state constituents of arbitrary masses, the mass of the ground state of a given spin may be calculated almost entirely analytically with high accuracy, without the (numerical) diagonalization of the matrix representation obtained by expansion of the solutions over a suitable set of basis states.

Effects of negative-energy components in two-body deuteron photodisintegration

Several observables in two-body deuteron photodisintegration are investigated in the framework of the Bethe-Salpeter formalism. Apart from keeping the Lorentz covariance throughout, special attention is paid to inclusion of both the positive-energy and negative-energy partial-wave components of the deuteron state. Using the Bethe-Salpeter equation for the deuteron in the ladder approximation with one-boson exchanges as a driving term, the contribution of the negative-energy states is studied for the unpolarized differential cross section as well as for linear photon and tensor target asymmetries. These states are found to have an impact on the observables and, thus, should be taken into account in a complete theoretical treatment of the reaction in the intermediate energy regime.

Light-front quark model analysis of rareB→Kl+l−decays

Using the light-front quark model, we calculate the transition form factors, decay rates, and longitudinal lepton polarization asymmetries for the exclusive rare $B\to K\ell^+\ell^-(\ell=e,\mu,\tau$) decays within the standard model. Evaluating the timelike form factors, we use the analytic continuation method in $q^+=0$ frame to obtain the form factors $F_+$ and $F_T$, which are free from zero-mode. The form factor $F_-$ which is not free from zero-mode in $q^+ = 0$ frame and contaminated by the higher(or nonvalence) Fock states in $q^+ \neq 0$ frame is obtained from an effective treatment for handling the nonvalence contribution based on the Bethe-Salpeter formalism. The covariance(i.e. frame-independence) of our model calculation is discussed. We obtain the branching ratios for ${\rm BR}(B\to K\ell^+\ell^-)$ as $4.96\times10^{-7}|V_{ts}/V_{cb}|^2$ for $\ell=e,\mu$ and $1.27\times 10^{-7}|V_{ts}/V_{cb}|^2$ for $\ell=\tau$.

Bethe—Salpeter approach with the separable interaction for the deuteron

Recent developments of the covariant Bethe—Salpeter (BS) approach with the use of the separable interaction for the deuteron are reviewed. It is shown that the BS formalism allows a covariant description of various electromagnetic reactions like the lepton-deuteron scattering, deuteron electro-disintegration, deep inelastic scattering (DIS) of leptons on light nuclei. The procedure of the construction of the separable nucleon-nucleon ( N N) interaction is discussed. The BS formalism facilitates analysis of the role of the P-waves (negative energy components) in the electromagnetic properties of the deuteron and its comparison with the nonrelativistic results. Furthermore the covariant BS approach makes it possible to analyze DIS of leptons from the deuteron in a model independent way and to extend the formalism to DIS reactions on the light nuclei.

Instantaneous Bethe-Salpeter equation: Analytic approach for nonvanishing masses of the bound-state constituents

The instantaneous Bethe-Salpeter equation, derived from the general Bethe-Salpeter formalism by assuming that the involved interaction kernel is instantaneous, represents the most promising framework for the description of hadrons as bound states of quarks from first quantum-field-theoretic principles, that is, quantum chromodynamics. Here, by extending a previous analysis confined to the case of bound-state constituents with vanishing masses, we demonstrate that the instantaneous Bethe-Salpeter equation for bound-state constituents with (definitely) nonvanishing masses may be converted into an eigenvalue problem for an explicitly---more precisely, algebraically---known matrix, at least, for a rather wide class of interactions between these bound-state constituents. The advantages of the explicit knowledge of this matrix representation are self-evident.

Neutron and proton structure today

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Sciulli Frank 2001Neutron and proton structure todayPhil. Trans. R. Soc. A.359241–256http://doi.org/10.1098/rsta.2000.0724SectionRestricted accessNeutron and proton structure today Frank Sciulli Frank Sciulli Columbia University Physics, Nevis Laboratory, Box 137, 136 South Broadway, Irvington, NY 10533, USA Google Scholar Find this author on PubMed Search for more papers by this author Frank Sciulli Frank Sciulli Columbia University Physics, Nevis Laboratory, Box 137, 136 South Broadway, Irvington, NY 10533, USA Google Scholar Find this author on PubMed Search for more papers by this author Published:15 February 2001https://doi.org/10.1098/rsta.2000.0724AbstractWe present an abbreviated and personal view of the development of our experimental knowledge of nucleon structure. The fundamentals were verified over 20 years, but the past decade has provided many interesting, even exciting, surprises. Perhaps more surprises are to come. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Burov V, Molochkov A, Smirnov G and Toki H (2004) Relating the proton, neutron and deuteron structure functions in the covariant Bethe–Salpeter formalism, Physics Letters B, 10.1016/j.physletb.2004.03.011, 587:3-4, (175-183), Online publication date: 1-May-2004. This Issue15 February 2001Volume 359Issue 1779Discussion Meeting Issue ‘The quark structure of matter’ organized by I. Butterworth, J. Ellis and E. Gabathuler Article InformationDOI:https://doi.org/10.1098/rsta.2000.0724Published by:Royal SocietyPrint ISSN:1364-503XOnline ISSN:1471-2962History: Published online15/02/2001Published in print15/02/2001 License: Citations and impact Keywordsstructurescatteringparticlesnucleon

Instantaneous Bethe-Salpeter equation: Utmost analytic approach

The Bethe-Salpeter formalism in the instantaneous approximation for the interaction kernel entering into the Bethe-Salpeter equation represents a reasonable framework for the description of bound states within relativistic quantum field theory. In contrast to its further simplifications (like, for instance, the so-called reduced Salpeter equation), it allows also the consideration of bound states composed of "light" constituents. Every eigenvalue equation with solutions in some linear space may be (approximately) solved by conversion into an equivalent matrix eigenvalue problem. We demonstrate that the matrices arising in these representations of the instantaneous Bethe-Salpeter equation may be found, at least for a wide class of interactions, in an entirely algebraic manner. The advantages of having the involved matrices explicitly, i.e., not "contaminated" by errors induced by numerical computations, at one's disposal are obvious: problems like, for instance, questions of the stability of eigenvalues may be analyzed more rigorously; furthermore, for small matrix sizes the eigenvalues may even be calculated analytically.

Relativistic contributions to the deuteron photodisintegration in the Bethe-Salpeter formalism

In plane wave one-body approximation the reaction of deuteron photodisintegration is considered in the framework of the Bethe-Salpeter formalism for two-nucleon system. Results are obtained for deuteron vertex function, which is the solution of the homogeneous Bethe-Salpeter equation with a multi-rank separable interaction kernel, with a given analytical form. A comparison is presented with predictions of non-relativistic, quasipotential approaches and the equal time approximation. It is shown that important contributions come from the boost in the arguments of the initial state vertex function and the boost on the relative energy in the one-particle propagator due to recoil.

The solar proton burningprocess revisited within a covariant model based on the Bethe-Salpeterformalism

A covariant model based on the Bethe-Salpeter formalism is proposedfor investigating the solar proton burning processpp→De+νe and the near-threshold deuteron disintegration viaelectromagnetic and weak interactions.Results of numerical calculations of the energy dependence of relevantcrosssections and the astrophysical low-energy cross section factorSpp of the proton burning process are presented.Our results confirm previous canonical values, and the energy dependenceofthe Spp factor is rather close to phenomenological extrapolationscommonly adopted in computations of solar nuclear reaction rates.

Space-time structure of a bound nucleon

Distortions of bound nucleon space-time structure, measured in deep inelastic scattering off nuclei, are discussed from view-point of a theoretical approach based on the Bethe-Salpeter formalism. It is shown that modification of the structure function F 2 N results from relative time dependence in Green functions of bound nucleons. The modification plays fundamental role in analysis of deep inelastic scattering experiments and allows one to obtain new information about nucleon structure.

3D–4D INTERLINKAGE OF qqq WAVE FUNCTIONS UNDER 3D SUPPORT FOR PAIRWISE BETHE–SALPETER KERNELS

Using the method of Green functions within the framework of a Bethe–Salpeter formalism characterized by a pairwise qq interaction under the Markov–Yukawa transversality principle (MYTP) which gives Lorentz-covariant 3D support to the BS kernel, the 4D BS amplitude for three identical spinless quarks is reconstructed from the corresponding 3D quantities which satisfy a fully connected 3D BSE. This result is a three-body generalization of a similar 3D–4D interlinkage for the corresponding two-body wave functions found earlier under identical conditions of a Lorentz-covariant 3D support to the corresponding BS kernel, ("CIA" for short), for the [Formula: see text] interaction. (The generalization from spinless to fermion quarks is straightforward). To set "CIA" for 3D kernel support in the context of contemporary approaches to the qqq baryon problem, a model scalar 4D qqq BSE with pairwise contact interactions simulating the NJL–Faddeev equations, is worked out fully and compared with the "CIA" vertex function which reduces exactly to the 4D NJL–Faddeev form in the limit of zero spatial range. This consistency check is part of a detailed accounting of the CIA vertex structure whose physical motivation stems from the role of spectroscopy when considered as an integral part of any QCD-motivated dynamical investigation.