Critical coupling in (1+1)-dimensional light-front φ theory

The spontaneous symmetry breaking in (1+1)-dimensional φ theory is studied with discretized light-front quantization, that is, by solving the zero-mode constraint equation. The symmetric ordering is assumed for the operator-valued constraint equation. The commutation relation between the zero mode and each oscillator mode is calculated with h̄ expansion. A critical coupling evaluated from the first some terms in the expansion is 28.8μ/h̄ ≤ λcr ≤ 31.1μ/h̄ consistent with the equal-time one 22μ/h̄ ≤ λcr ≤ 55.5μ/h̄. The same analysis is also made under another operator ordering. PACS numbers:11.10.Lm, 11.30.Qc E-mail: [email protected] E-mail: [email protected] 1 The light-front formulation of field theory is formally equivalent to the equal-time formulation within the framework of perturbation theory [1]. Nonperturbative calculations in the light-front formulation are successful in lower-dimensional field theories and yield the correct particle spectra [2, 3, 4]. The light-front formulation is then expected to be a promising method for solving the nonperturbative dynamics such as QCD. Nevertheless, the application to spontaneous symmetry breaking (SSB) is obstructed by the unsolved problem of the vacuum structure on the light cone [5, 6, 7]. It is then an important issue to confirm that for SSB the light-front formulation yields the same result as the equal-time formulation. We study spontaneous breaking of the Z2 symmetry in (1+1)-dimensional φ 4 theory as an instance. It is well known in the equal-time formulation that the model undergoes a phase transition at strong coupling [8]. In light-front field theory, the true vacuum always equals to the Fock vacuum [5], so SSB is believed to occur through the zero mode; a non-zero vacuum expectation value of the zero mode signals SSB. To ascertain this idea, several authors [9, 10, 11, 12, 13] have computed a vacuum expectation value of the zero mode and a critical coupling with either a truncation of Fock space or perturbation. Their results are qualitatively satisfactory. However, it is not clear whether the prescriptions are reliable for nonperturbative phenomena such as SSB, and it is not easy to improve the accuracy of the calculations. Our purpose in this brief report is to propose a different computation rule and obtain more accurate results. This rule yields a systematic way of obtaining a critical coupling λcr and a critical exponent β. We define the light-front coordinates x = (x ± x)/ √ 2. The Lagrangian density of (1 + 1)-dimensional φ theory is described as L = ∂+φ∂−φ− μ 2 φ − λ 4! φ. (1) We put the quantum system in a box of length d and impose periodic boundary conditions.