Two-dimensional light-front $φ^4$ theory in a symmetric polynomial basis

TL;DR

The paper tackles the nonperturbative spectrum of $φ^4_{1+1}$ by diagonalizing the light-front Hamiltonian in a Fock-space basis, employing a fully symmetric polynomial basis (GenSymPolys) to represent sector wave functions. It derives and solves the coupled integral equations for the wave functions with a dimensionless coupling $g=\frac{\lambda}{4\pi\mu^2}$ and a mass-renormalization scheme that connects light-front and equal-time parameters via $\mu_{\rm LF}^2=\mu_{\rm ET}^2-\frac{\lambda}{4\pi}\Delta$, where $\Delta$ is computed from the eigenstates. The study reports rapid convergence with basis size, estimates a positive-mass critical coupling $g_c=2.1\pm0.05$ (and $\bar{g}_c=1.1\pm0.03$), and finds a renormalization shift $\Delta(g_c)\approx -0.47\pm0.12$, demonstrating how mass renormalization reconciles light-front and equal-time results and highlighting the increasing importance of higher Fock sectors near criticality. The results corroborate the method’s efficiency, compare favorably with LFCC approaches, and point to extensions to higher dimensions and to incorporating sector-dependent renormalization for improved near-critical accuracy.

Abstract

We study the lowest-mass eigenstates of $φ^4_{1+1}$ theory with both odd and even numbers of constituents. The calculation is carried out as a diagonalization of the light-front Hamiltonian in a Fock-space representation. In each Fock sector a fully symmetric polynomial basis is used to represent the Fock wave function. Convergence is investigated with respect to the number of basis polynomials in each sector and with respect to the number of sectors. The dependence of the spectrum on the coupling strength is used to estimate the critical coupling for the positive-mass-squared case. An apparent discrepancy with equal-time calculations of the critical coupling is resolved by an appropriate mass renormalization.

Figures (8)

• Figure 1: Lowest mass eigenvalue for odd numbers of constituents. Results are shown for different Fock-space truncations to three, five, and seven constituents. Also plotted are results for the leading light-front coupled-cluster (LFCC) approximation and for a sector-dependent modification of the three-body truncation. The errors are estimated from extrapolations in polynomial basis size.
• Figure 2: Same as Fig. \ref{['fig:extrap-odd']} but for even numbers of constituents, with Fock-space truncations at four, six and eight.
• Figure 3: Relative Fock-sector probabilities for the lowest mass eigenstate with odd numbers of constituents.
• Figure 4: The lowest masses for the odd and even cases, as used to estimate the critical coupling, including a plot of the threshold for two-particle states at four times the mass-squared of the odd case.
• Figure 5: The renormalization shift $\Delta$, defined in Eq. (\ref{['eq:Delta']}) of the text, as a function of the square of the dimensionless coupling $g$. The points displayed are obtained as extrapolations in the polynomial basis size. The lines are linear and quadratic fits to shifts below $g=1$, extrapolated to the region of the critical coupling.