Spectator Equations Research Articles

Non-perturbative Methods in Relativistic Field Theory

This talk reviews relativistic methods used to compute bound and low energy scattering states in field theory, with emphasis on approaches that John Tjon and I discussed (and argued about) together. I compare the Bethe–Salpeter and Covariant Spectator equations, show some applications, and then report on some of the things we have learned from the beautiful Feynman–Schwinger technique for calculating the exact sum of all ladder and crossed ladder diagrams in field theory.

Comprehensive treatment of electromagnetic interactions and three-body spectator equations

We present a general derivation of the three-body spectator (Gross) equations and of the corresponding electromagnetic currents. As in a previous paper on two-body systems, the wave equations and currents are derived from those for the Bethe-Salpeter equation with the help of an algebraic method using a concise matrix notation. The three-body interactions and currents introduced by the transition to the spectator approach are isolated and the matrix elements of the e.m. current are presented in detail for a system of three indistinguishable particles, namely, for elastic scattering and for two- and three-body breakup. The general expressions are reduced to the one-boson-exchange approximation to make contact with previous work. The method is general in that it does not rely on introduction of the electromagnetic interaction with the help of the minimal replacement. It would therefore work also for other external fields.

Charge-Conjugation Invariance of the Spectator Equations

In response to recent critcism, we show how to define the spectator equations for negative energies so that charge conjugation invariance is preserved. The result, which emerges naturally from the application of spectator principles to systems of particles with negative energies, is to replace all factors of the external energies $W_i$ by $\sqrt{W^2_i}$, insuring that the amplitudes are independent of the sign of the energies $W_i$.

Electromagnetic interactions for the two-body spectator equations

This paper presents a new non-associative algebra which is used to (i) show how the spectator (or Gross) two-body equations and electromagnetic currents can be formally derived from the Bethe-Salpeter equation and currents if both are treated to all orders, (ii) obtain explicit expressions for the Gross two-body electromagnetic currents valid to any order, and (iii) prove that the currents so derived are exactly gauge invariant when truncated consistently to any finite order. In addition to presenting these new results, this work complements and extends previous treatments based largely on the analysis of sums of Feynman diagrams.

Gauging the spectator equations

We show how to derive relativistic, unitary, gauge-invariant, and charge-conserving three-dimensional scattering equations for a system of hadrons interacting with an electromagnetic field. In the method proposed, the spectator equations describing the strong interactions of the hadrons are gauged using our recently introduced gauging of equations method. A key ingredient in our model is the on-mass-shell particle propagator. We discuss how to gauge this on-mass-shell propagator so that both the Ward-Takahashi and Ward identities are satisfied. We then demonstrate our gauging procedure by deriving the gauge-invariant three-dimensional expression for the deuteron photodisintegration amplitude within the spectator approach. {copyright} {ital 1997} {ital The American Physical Society}

Gauging the three-nucleon spectator equation

We derive relativistic three-dimensional integral equations describing the interaction of the three-nucleon system with an external electromagnetic field. Our equations are unitary, gauge invariant, and they conserve charge. This has been achieved by applying the recently introduced gauging of equations method to the three-nucleon spectator equations where spectator nucleons are always on mass shell. As a result, the external photon is attached to all possible places in the strong interaction model, so that current and charge conservation are implemented in the theoretically correct fashion. Explicit expressions are given for the three-nucleon bound state electromagnetic current, as well as the transition currents for the scattering processes \gamma He3 -> NNN, Nd -> \gamma Nd, and \gamma He3 -> Nd. As a result, a unified covariant three-dimensional description of the NNN-\gamma NNN system is achieved.

Covariant equations for the three-body bound state

The covariant spectator (or Gross) equations for the bound state of three identical spin 1/2 particles, in which two of the three interacting particles are always on shell, are developed and reduced to a form suitable for numerical solution. The equations are first written in operator form and compared to the Bethe-Salpeter equation, then expanded into plane wave momentum states, and finally expanded into partial waves using the three-body helicity formalism first introduced by Wick. In order to solve the equations, the two-body scattering amplitudes must be boosted from the overall three-body rest frame to their individual two-body rest frames, and all effects which arise from these boosts, including Wigner rotations and p-spin decomposition of the shell-particle, are treated exactly. In their final form, the equations reduce to a coupled set of Faddeev-like double integral equations with additional channels arising from the negative p-spin states of the off-shell particle.

Heavy mesons in a relativistic model.

Motivated by the present interest in the heavy quark effective theory, the authors use the spectator equations to treat the mesonic bound states of heavy quarks. The kernel they use is based on scalar confining and vector Coulomb potentials. Wave functions are treated to leading order and energies to order 1/m{sub Q} in the heavy-light systems, and order 1/m{sub Q{sup 2}} in heavy-heavy systems. Their results are in reasonable agreement with experimental measurements. They estimate two of the parameters of the heavy quark effective theory, and propose further calculations that may be undertaken in the future.