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Light-front $φ^4_{1+1}$ theory using a many-boson symmetric-polynomial basis

S. S. Chabysheva

TL;DR

This work introduces a high-order light-front method using fully symmetric multivariate polynomials (GenSymPolys) to solve φ^4_{1+1} with many-boson Fock sectors. The per-Fock-sector polynomial expansion yields a generalized eigenvalue problem that is solved stably via a singular-value decomposition of the overlap matrix, enabling independent control of resolution in each sector. The method achieves rapid convergence and agrees with the LFCC approach, producing an estimate of the critical coupling for symmetry breaking at positive mass-squared and clarifying differences with equal-time renormalization. The results demonstrate a scalable, DLCQ-independent framework that can be extended toward higher dimensions and transverse discretization for light-front field theories.

Abstract

We extend earlier work on fully symmetric polynomials for three-boson wave functions to arbitrarily many bosons and apply these to a light-front analysis of the low-mass eigenstates of $φ^4$ theory in 1+1 dimensions. The basis-function approach allows the resolution in each Fock sector to be independently optimized, which can be more efficient than the preset discrete Fock states in DLCQ. We obtain an estimate of the critical coupling for symmetry breaking in the positive mass-squared case.

Light-front $φ^4_{1+1}$ theory using a many-boson symmetric-polynomial basis

TL;DR

This work introduces a high-order light-front method using fully symmetric multivariate polynomials (GenSymPolys) to solve φ^4_{1+1} with many-boson Fock sectors. The per-Fock-sector polynomial expansion yields a generalized eigenvalue problem that is solved stably via a singular-value decomposition of the overlap matrix, enabling independent control of resolution in each sector. The method achieves rapid convergence and agrees with the LFCC approach, producing an estimate of the critical coupling for symmetry breaking at positive mass-squared and clarifying differences with equal-time renormalization. The results demonstrate a scalable, DLCQ-independent framework that can be extended toward higher dimensions and transverse discretization for light-front field theories.

Abstract

We extend earlier work on fully symmetric polynomials for three-boson wave functions to arbitrarily many bosons and apply these to a light-front analysis of the low-mass eigenstates of theory in 1+1 dimensions. The basis-function approach allows the resolution in each Fock sector to be independently optimized, which can be more efficient than the preset discrete Fock states in DLCQ. We obtain an estimate of the critical coupling for symmetry breaking in the positive mass-squared case.

Paper Structure

This paper contains 4 sections, 9 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Solution of the integral equations
  3. Results
  4. Summary

Figures (2)

  • Figure 1: Mass squared vs coupling strength for an (a) odd and (b) even number of constituents. The different Fock-space truncations in (a) are the three-body (triangles), five-body (squares), and seven-body (diamonds) Fock sectors. Results for the LFCC method (circles) are also included. In (b) the different truncations are the four-body (triangles), six-body (squares), and eight-body (diamonds) Fock sectors. Error bars are determined by the fits to extrapolation in the polynomial basis size.
  • Figure 2: Mass squared vs coupling strength, as used to estimate the critical coupling. The odd (triangles) and even (circles) cases are represented by the seven-body and eight-body truncations, respectively. Points at twice the mass of the odd case (squares) are also plotted.