Dyson-Schwinger equations in zero dimensions and polynomial approximations

Abstract

The Dyson-Schwinger (DS) equations for a quantum field theory in $D$-dimensional space-time are an infinite sequence of coupled integro-differential equations that are satisfied exactly by the Green's functions of the field theory. This sequence of equations is underdetermined because if the infinite sequence of DS equations is truncated to a finite sequence, there are always more Green's functions than equations. An approach to this problem is to close the finite system by setting the highest Green's function(s) to zero. One can examine the accuracy of this procedure in $D=0$ because in this special case the DS equations are just a sequence of coupled polynomial equations whose roots are the Green's functions. For the closed system one can calculate the roots and compare them with the exact values of the Green's functions. This procedure raises a general mathematical question: When do the roots of a sequence of polynomial approximants to a function converge to the exact roots of that function? Some roots of the polynomial approximants may (i) converge to the exact roots of the function, or (ii) approach the exact roots at first and then veer away, or (iii) converge to limiting values that are unequal to the exact roots. In this study five field-theory models in $D=0$ are examined, Hermitian $φ^4$ and $φ^6$ theories and non-Hermitian $iφ^3$, $-φ^4$, and $-i φ^5$ theories. In all cases the sequences of roots converge to limits that differ by a few percent from the exact answers. Sophisticated asymptotic techniques are devised that increase the accuracy to one part in $10^7$. Part of this work appears in abbreviated form in Phys.~Rev.~Lett.~{\bf 130}, 101602 (2023).

Figures (19)

• Figure 1: Plot of the parabolic cylinder function $f(x)={\rm D}_{3.5}(x)$ for $-4<x<4$; $f(x)$ is not a harmonic-oscillator eigenfunction because it blows up when $x$ is large and negative. Note that $f(x)$ has four real zeros.
• Figure 2: [color online] Roots of the 9th-degree Taylor-polynomial approximation to ${\rm D}_{3.5}(x)$ plotted in the complex-$x$ plane. Four real roots (red squares) are located at $x=-2.09226...$, $-1.19286...$, $0.39183...$, $1.94724...$, which are moderately close to the exact roots of $f(x)$ given in (\ref{['e2.2']}). The remaining roots (black dots) are spurious zeros that gradually move outward to complex $\infty$ as the degree of the Taylor polynomial increases.
• Figure 3: [color online] Roots of the 17th-degree Taylor polynomial approximation to ${\rm D}_{3.5}(x)$ plotted in the complex-$x$ plane. The real roots (red squares) lie at $x=-2.84103...$, $-1.19090...$, $0.39183...$, $2.04519...$ and are fairly close to their exact values in (\ref{['e2.2']}). Spurious roots (black dots) lie along parenthesis-shaped curves along with an isolated spurious root on the positive-real axis.
• Figure 4: [color online] Roots of the 25th-degree Taylor polynomial approximation to ${\rm D}_{3.5}(x)$ plotted in the complex-$x$ plane. Real roots (red squares) at $x=-3.04510...$, $-1.19090...$, $0.39183...$, $2.04545...$ are closer to their exact values in (\ref{['e2.2']}). All but one of the spurious roots (black dots) lie on parenthesis-shaped curves that expand outward slowly as the degree of the Taylor polynomial increases. The isolated spurious root on the positive-real axis also moves outward.
• Figure 5: [color online] Roots of the 33rd-degree Taylor polynomial approximation to ${\rm D}_{3.5}(x)$ plotted in the complex-$x$ plane. Real roots (red squares) at $x=-3.04735...$, $-1.19090...$, $0.39183...$, $2.04542...$ are now are quite close to their exact values in (\ref{['e2.2']}). Spurious roots (black dots) lie on parenthesis-shaped curves and the isolated spurious root on the positive-real axis continue to move slowly outward.