Positronium and heavy quarkonia as testing case for discretized light-cone quantization

Abstract

A nonperturbative method for solving quantum field theories in three space and one time dimensions is applied to the bound-state problem of positronium and heavy quarkonia. The model includes only one dynamical photon, i.e., the irradiation channels are closed. An integral equation of the Bethe-Salpeter type is derived, being the light-cone analogue of the Tamm-Dancoff equation, and solved numerically. The model accounts for the Bohr-Sommerfeld and Dirac physics such as hyperfine splitting including the correct retardation. Special emphasis is put on the role of the Coulomb singularity in momentum space. Numerical results for the mass spectrum and the wave functions are presented, and compared to analytical results. Agreement is found for the physical value of the coupling constant. For very large coupling constants a -0.3 discrepancies are noted and discussed.

Non-perturbative Calculation of the Positronium Mass Spectrum in Basis Light-Front Quantization

“…It is also interesting to compare our results to the Discretized Light-Cone Quantization (DLCQ) 6 results of Ref. [15], also using α = 0.3. While their method of calculation is different, their interaction incorporates all the same physics and assumptions as ours.…”

“…If this term is simply dropped, the results become convergent. This procedure has some justification, as this divergent piece of the effective interaction will be cancelled when higher Fock sectors are included [15]. Below, we consider the continuum limit using only this modified interaction, where convergence can be expected.…”

On confinement in a light-cone Hamiltonian for QCD

“…In the first place one should address to develop a well-defined effective interaction, such one as proposed for example by Tamm [24] and independently by Dancoff [25], in their paradigmatic treatment of the Yukawa model. But if one does so and adapts the Tamm-Dancoff approach to the light-cone [26], one finds no trace of a possible confinement in these 'mesons'. This is rather disturbing since the same approach applied to positronium yields the Bohr aspects of the spectrum including the correct fine and hyperfine splittings.…”

On zero modes and gauge fixing in light-cone quantized gauge theories

“…)¤ I <·]i)cv (9) Vi(A`)0 -@$(A+)» + 3+Vr -(Ai)!. = (J`).»» (8) Vi(A+)» = (J+)¤» As in the scalar theory, one integrates these over m' and obtains (7) (26-6+ -V?L)AQ + 6,6+.4+ + 6i6-A"-6iVJ_ · Ay : Jl (6) (6-6+-Vi)A"-6 §_A++6+Vy·AL : J`, (5) (6-6+ -Vj)A+ -03,4* + 0-VL -Ay Z J+, tion. It is therefore convenient to write the equations (4) explicitly for u : (+, -, i) We intend to perform on the Maxwell theory the procedure discussed in the introduc compactification leads to a discretization in momentum space.…”

“…As a matter of fact, Franke et al [15] have and van de Sande [12] is particularly lucid on this point. Other theories such as Yukawa symmetry breaking in gb4 theory in 1+1 dimensions [11,7]. The work of Bender, Pinsky independent degrees of freedom.…”