Basis of symmetric polynomials for many-boson light-front wave functions

We provide an algorithm for the construction of orthonormal multivariate polynomials that are symmetric with respect to the interchange of any two coordinates on the unit hypercube and are constrained to the hyperplane where the sum of the coordinates is one. These polynomials form a basis for the expansion of bosonic light-front momentum-space wave functions, as functions of longitudinal momentum, where momentum conservation guarantees that the fractions are on the interval $[0,1]$ and sum to one. This generalizes earlier work on three-boson wave functions to wave functions for arbitrarily many identical bosons. A simple application in two-dimensional ${\varphi }^4$ theory illustrates the use of these polynomials.

Figure 1

Comparison of convergence rates for the fully symmetric polynomial basis (filled circles) and DLCQ (filled squares). The dimensionless eigenmass $M^2/{\mu }^2$ is plotted versus $1/K$, the reciprocal of the basis order and of the DLCQ resolution, for the case where the coupling is $\lambda =4\pi {\mu }^2$. The lower resolution DLCQ results cover a range up to $K=24$; the higher resolution results, a range of 80 to 200. The higher resolution points are used for an extrapolation to $K=\infty$, and the horizontal line is at the value of that limit.