Narrow Results

The general form of the Bethe–Salpeter wave functions for bound states comprising one scalar constituent and one fermion, or two scalars is presented. Using the reduced Salpeter equation obtained, we can work out the effective nonrelativistic potentials. And one new version of reduced Bethe–Salpeter equation is proposed by extending Gross approximation.

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.

We reproduce masses of the self-conjugate and non-self-conjugate mesons in the context of the spinless Salpeter equation taking into account the relativistic kinematics and the quark spins. The hyperfine splittings for the 2S charmonium and 1S bottomonium are also calculated. Further, the pseudoscalar and vector decay constants of the B_c meson and the unperturbed radial wave function at the origin are also calculated. We have obtained a local equation with a complete relativistic corrections to a class of three attractive static interaction potentials of the general form V(r) = -Ar^-β + κr^β + V₀, with β = 1, 1/2, 3/4 decomposed into scalar and vector parts in the form V_V(r) = -Ar^-β + (1-∊)κr^β and V_S(r) = ∊κr^β + V₀; where 0≤∊≤1. We have used the shifted large-N-expansion technique (SLNET) to solve the reduced equation for the scalar (∊ = 1), equal mixture of scalar-vector (∊ = 1/2), and vector (∊ = 0) confinement interaction kernels. The energy eigenvalues are carried out up to the third order approximation.

We study pseudoscalar and scalar mesons using a practical and symmetry preserving truncation of QCD's Dyson-Schwinger equations. We investigate and compare properties of ground and radially excited meson states. In addition to exact results for radial meson excitations we also present results for meson masses and decay constants from the chiral limit up to the charm-quark mass, e.g., the mass of the χ_c0(2P) meson.

We consider an imaginary time functional integral formulation of a two-flavor, 3+1 lattice QCD model with Wilson's action and in the strong coupling regime (with a small hopping parameter, κ > 0, and a much smaller plaquette coupling, , so that the quarks and glueballs are heavy). The model has local SU(3)_c gauge and global SU(2)_f flavor symmetries, and incorporates the corresponding part of the eightfold way particles: baryons (mesons) of asymptotic mass ≈-3 ln κ(≈-2 ln κ). We search for pentaquark states as meson–baryon bound states in the energy–momentum spectrum of the model, using a lattice Bethe–Salpeter equation. This equation is solved within a ladder approximation, given by the lowest nonvanishing order in κ and β of the Bethe–Salpeter kernel. It includes order κ² contributions with a exchange potential together with a contribution that is a local-in-space, energy-dependent potential. The attractive or repulsive nature of the exchange interaction depends on the spin of the meson–baryon states. The Bethe–Salpeter equation presents integrable singularities, forcing the couplings to be above a threshold value for the meson and the baryon to bind in a pentaquark. We analyzed all the total isospin sectors, I = 1/2, 3/2, 5/2, for the system. For all I, the net attraction resulting from the two sources of interaction is not strong enough for the meson and the baryon to bind. Thus, within our approximation, these pentaquark states are not present up to near the free meson–baryon energy threshold of ≈-5 ln κ. This result is to be contrasted with the spinless case for which our method detects meson–baryon bound states, as well as for Yukawa effective baryon and meson field models. A physical interpretation of our results emerges from an approximate correspondence between meson–baryon bound states and negative energy states of a one-particle lattice Schrödinger Hamiltonian.

We determine two-baryon bound states in a 3+1 lattice QCD model with improved Wilson action and two flavors. We work in the strong coupling regime: small hopping parameter κ > 0 and much smaller plaquette coupling β > 0. In this regime, it is known that the low-lying energy–momentum spectrum is comprised of baryons and mesons with asymptotic masses -3 ln κ and -2 ln κ, respectively. We show that the dominant baryon–baryon interaction is an order κ² space-range-one -exchange potential. We also show that this interaction has an important and novel isospin–spin interchange symmetry relating the various possible bound states, and then governing the two-baryon spectral structure. Letting S(I) denote the total spin (total isospin) of the two-baryon bound states, S, I = 0, 1, 2, 3, we find bound states with asymptotic binding energy κ²/4, for I+S = 1, 3, and 4 (here, with I = S = 2); κ²/12, for I+S = 0, 2, 4 and 3 (here, with I = 1, 2). In particular, we show that the two-baryon spectrum contains deuteron (I = 0), diproton (I = 1) and dineutron (I = 1)-like bound states. Using the isospin–spin symmetry, we can circumvent the lack of spin symmetry of the lattice action and show they all have the same asymptotic binding energy, namely κ²/4. Our analysis uses convenient two and four-baryon correlations, their spectral representations and a lattice Bethe–Salpeter equation, which is solved in a ladder approximation. For the isospin, spin part of the interaction, we obtain a permanent representation which describes the interaction of the individual spins and isospins of the quarks of one baryon with those of the other baryon.

We calculate the exclusive semileptonic and nonleptonic decays of B_c meson to heavy-light orbitally excited states in the framework of the improved Bethe–Salpeter method. The hadronic matrix elements are calculated and expressed as overlap integrals of the relativistic Salpeter wave functions of corresponding mesons, decay rates of the semileptonic and nonleptonic B_c decays to , and are obtained. Some channels provide us sizable ratios, for examples, and , with such ratios, the missing scalar B_s0 may be detected though its cascade strong decays in the forthcoming LHC experiment.

Describing the lightest pseudoscalar mesons as bound states of quark and antiquark within the framework of an instantaneous Bethe–Salpeter formalism constructed such as to retain (in contrast to Salpeter’s equation) as much information on the relativistic effects provided by the full quark propagator as conceivable allows for a surprisingly simple implementation of their near masslessness mandatory for their interpretability as pseudo-Goldstone bosons related to the spontaneous breaking of the chiral symmetries of quantum chromodynamics.

For a two-particle bound-state equation closer to its Bethe–Salpeter origins than Salpeter’s equation, with effective interaction kernel deliberately forged such as to ensure, in the limit of zero mass of the bound-state constituents, the vanishing of the arising bound-state mass, we scrutinize the emerging features of the lightest pseudoscalar mesons for their agreement with the behavior predicted by a generalization of the Gell-Mann–Oakes–Renner relation.

We perform a Bethe–Salpeter equation (BSE) evaluation of composite scalar boson masses in order to verify how these masses can be smaller than the composition scale. The calculation is developed with a constituent self-energy dependent on its mass anomalous dimension ($\gamma$), and we obtain a relation showing how the scalar mass decreases as $\gamma$ is increased. We also discuss how fermionic corrections to the BSE kernel shall decrease the scalar mass, whose effect can be as important as the one of a large $\gamma$. An estimate of the top quark loop effect that must appear in the BSE calculation gives a lower bound on the composite scalar mass.

In a many-body perturbation treatment of electronic excitations, one has to solve the so-called Bethe–Salpeter equation (BSE) for the kernel (vertex) describing the interaction between electrons. In general, the BSE exhibits a non-local and frequency-dependent screened interaction and consequently it is extremely difficult to solve. We have developed a scheme that solves the BSE equation iteratively, including dynamically screening, by considering the change in the electron self-energy upon applying a time-dependent field. The BSE is obtained from the self-energy in the GW approximation (GWA) using the Schwinger functional technique. We apply our formalism to a simple model system and discuss briefly changes in the charge response, when dynamical screening is taken into account.

Based on the concepts of a superpropagator, multiple Debye temperatures, and equivalence of the binding energy of a Cooper pair and the BCS energy gap, the set of generalized BCS equations obtained recently via a temperature-generalized Bethe–Salpeter equation is employed for a unified study of the following composite superconductors: MgB₂, Nb₃Sn, and YBCO. In addition, we study the Nb–Al system in which Cooper pairs as resonances have recently been reported to have been observed. Our approach seems to suggest that a simple extension of the BCS theory that accommodates the concept of Cooper pairs bound via a more than one phonon exchange mechanism may be an interesting candidate for dealing with high-temperature superconductors.

Dyson–Schwinger equations furnish a Poincaré covariant framework within which to study hadrons. A particular feature is the existence of a nonperturbative, symmetry preserving truncation that enables the proof of exact results. The gap equation reveals that dynamical chiral symmetry breaking is tied to the long-range behavior of the strong interaction, which is thereby constrained by observables, and the pion is precisely understood, and seen to exist simultaneously as a Goldstone mode and a bound state of strongly dressed quarks. The systematic error associated with the simplest truncation has been quantified, and it underpins a one-parameter model efficacious in describing an extensive body of mesonic phenomena. Incipient applications to baryons have brought successes and encountered challenges familiar from early studies of mesons, and promise a covariant field theory upon which to base an understanding of contemporary large momentum transfer data.

In this paper, the general structure of leptonic decay constants of vector mesons is evaluated in the framework of the Bethe–Salpeter equation under the Covariant Instantaneous Ansatz (CIA), which is a Lorentz-invariant generalization of Instantaneous Approximation (IA). The numerical values of f_V in this CIA framework are on the low side in comparison to recent calculations of these quantities. However, the overall asymptotic behavior of f_V is in conformity with QCD predictions.

We investigate the influence of retardation effects on covariant 3-dimensional wave functions for bound hadrons. Within a quark-(scalar) diquark representation of a baryon, the four-dimensional Bethe–Salpeter equation is solved for a 1-rank separable kernel which simulates Coulombic attraction and confinement. We project the manifestly covariant bound state wave function into three dimensions upon integrating out the non-static energy dependence and compare it with solutions of three-dimensional quasi-potential equations obtained from different kinematical projections on the relative energy variable. We find that for long-range interactions, as characteristic in QCD, retardation effects in bound states are of crucial importance.

Two photon decays of pion are studied in the framework of the Bethe–Salpeter equation under Covariant Instantaneous Ansatz where the structure of hadron-quark vertex function Γ is generalized to include various Dirac covariants (other than γ₅) from their complete set. These covariants are incorporated in accordance with a power counting rule (which was recently employed to calculate leptonic decays constants of pseudoscalar mesons and vector mesons) order by order in powers of the inverse of the meson mass. Pion-photon coupling constant F_π is calculated with the incorporation of leading order covariants.

We employ the framework of Bethe–Salpeter equation under Covariant Instantaneous Ansatz to study the leptonic decays of pseudoscalar mesons. The Dirac structure of hadron-quark vertex function Γ is generalized to include various Dirac covariants besides γ₅ from their complete set. The covariants are incorporated in accordance with a power counting rule, order-by-order in powers of the inverse of the meson mass. The decay constants are calculated with the incorporation of leading order covariants. Most of the results are dramatically improved.

We have employed the framework of Bethe–Salpeter equation under covariant instantaneous ansatz to calculate leptonic decay constants of unequal mass pseudoscalar mesons like π^±, K, D, D_S and B, and radiative decay constants of neutral pseudoscalar mesons like π⁰ and η_c into two photons. In the Dirac structure of hadronic Bethe–Salpeter wave function, the covariants are incorporated from their complete set in accordance with a recently proposed power counting rule. The contribution of both leading order and next-to-leading order Dirac covariants to decay constants are studied. The results are found to improve and hence validating the power counting rule which provides a practical means of incorporating Dirac covariants in the Bethe–Salpeter wave function for a hadron.

In this work we study the process e⁺+e^-→J/Ψ+η_c at energy observed recently at B-factories whose measurements were made by Babar and Belle groups. We calculate the cross-section for this process in the Bethe–Salpeter formalism under Covariant Instantaneous Anstaz. To simplify our calculation, the heavy quark approximation is employed in the quark and gluon propagators. In the exclusive process of e⁺e^- annihilation into two heavy quarkonia, the cross-section calculated in this scenario is compatible with the experimental data of Babar and Belle.

We investigate relativistic bound states for a hypothetical light scalar gluino pair (gluinonium), in the framework of the covariant Bethe-Salpeter equation (BSE). In this paper, we derive, from the covariant BSE for a fermion-anti-fermion system, using charge conjugation, the corresponding bound-state equation for a gluino pair and we then formulate, for a static harmonic kernel, the coupled differential equations for the corresponding static Bethe-Salpeter amplitude. The steps of our approach then include a numerical solution of the Bethe-Salpeter amplitude for a two-body interaction consisting of scalar, pseudo-scalar, and four-vector components and the determination of the energy spectrum for the ground and the radially excited states of massive gluinonium. We found the energy spectrum and radial distributions of fundamental and excited states of gluinonium. The comparison of the values obtained in the extreme relativistic case with the corresponding values predicted by a harmonic oscillator potential model shows that there is good agreement between the two formulations. The predictions of the binding energy of glunionium in the non-relativistic model are however systematically higher.