Dyson-Schwinger equations in zero dimensions and polynomial approximations

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 ${\varphi }^4$ and ${\varphi }^6$ theories and non-Hermitian $i{\varphi }^3$, $-{\varphi }^4$, and $-i{\varphi }^5$ 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. 130, 101602 (2023).

Figure 1

Plot of the parabolic cylinder function $f(x)={\mathrm{D}}_{7/2}(x)$ scaled down by a factor of ${\pi }^{1/2}{2}^{7/4}=5.9618\dots$ 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

Roots of the 9th-degree Taylor-polynomial approximation to ${\mathrm{D}}_{7/2}(x)$ plotted in the complex-$x$ plane. Four real roots (red squares) are located at $x=-2.09226\dots$, $-1.19286\dots$, 0.39183…, 1.94724…, which are moderately close to the exact roots of $f(x)$ given in (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

Roots of the 17th-degree Taylor polynomial approximation to ${\mathrm{D}}_{7/2}(x)$ plotted in the complex-$x$ plane. The real roots (red squares) lie at $x=-2.84103\dots$, $-1.19090\dots$, 0.39183..., 2.04519... and are fairly close to their exact values in (2). Spurious roots (black dots) lie along parenthesis-shaped curves along with an isolated spurious root on the positive-real axis.

Figure 4

Roots of the 25th-degree Taylor polynomial approximation to ${\mathrm{D}}_{7/2}(x)$ plotted in the complex-$x$ plane. Real roots (red squares) at $x=-3.04510\dots$, $-1.19090\dots$, 0.39183..., 2.04545... are closer to their exact values in (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

Roots of the 33rd-degree Taylor polynomial approximation to ${\mathrm{D}}_{7/2}(x)$ plotted in the complex-$x$ plane. Real roots (red squares) at $x=-3.04735\dots$, $-1.19090\dots$, 0.39183..., 2.04542... are now are quite close to their exact values in (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.

Figure 6

All eight roots of the five-term polynomial obtained from the truncated asymptotic expansion (4) plotted in the complex-$x$ plane. The two roots on the positive axis are numerically quite accurate. The other six roots are spurious. Dashed lines indicate the edges of the Stokes sector in which the asymptotic series is valid.

Figure 7

All eighteen roots of the ten-term polynomial obtained from the truncated asymptotic expansion (4). The roots are shown as dots in the complex-$x$ plane. Compared with Fig. 6, the ring of spurious roots has moved outward past the smaller positive zero of the parabolic cylinder function ${\mathrm{D}}_{7/2}(x)$ at $x=0.39183$ but the second positive zero is given accurately.

Figure 8

All 28 roots of the fifteen-term polynomial obtained from the truncated asymptotic expansion (4). The ring of spurious roots in the complex-$x$ plane has now expanded past the actual zeros of the parabolic cylinder function. Thus, without summation techniques the asymptotic series in (4) is no longer useful.

Figure 9

The root near $x=2$ of the polynomial obtained from the truncated asymptotic series approximation to the parabolic cylinder function plotted as a function of the degree of the polynomial. The six-term polynomial gives optimal accuracy after which the accuracy rapidly decreases.

Figure 10

Zeros of $P_n(x)$ in (19) plotted as a function of $n$. The zeros are nondegenerate and range from 0 up to just below 0.675978..., the exact value of $G_2$ in (14) (heavy horizontal line) and they become more dense at the upper end of this range. The zeros of successive polynomials interlace.

Figure 11

Results for $G_2$ plotted as a function of $n$ ($1\le n\le 30$) calculated using the asymptotic approximation for $G_{2n}$ in (21). We observe a dramatic improvement over Fig. 10; there is no longer a dense concentration of roots below the exact answer but rather an isolated root that is six orders of magnitude closer to the exact answer.

Figure 12

All solutions $G_1$ to the DS equations from the third to the 200th truncation plotted in the complex plane. The exact value of $G_1$ is $-0.72901113\dots i$. The full set of solutions has threefold symmetry; solutions lie on a three-bladed propeller with each blade having a small dense loop of solutions at the end. The inset shows that the exact solution (red square) lies on the negative-imaginary axis inside this loop.

Figure 13

Closeup of the negative imaginary axis for the solution $n=200$ showing the loop-shaped concentration of solutions for $G_1$ around the exact value of $G_1$, which lies in the interior of the loop as shown in Fig. 12.

Figure 14

Closeup of the negative imaginary axis for the $n=200$ truncation showing solutions for $G_1$, obtained from (i) the unbiased (blue) and (ii) the asymptotic approximations (red) for $G_n$. The blue and red loops are almost the same size, but there is a new red dot that is almost exactly equal to $G_1$. These dots also appear in the two other loops at the ends of the three-bladed propeller.

Figure 15

Absolute values of the solutions for $G_1$ on the imaginary axis, obtained using the asymptotic approximation to $G_n$ for $n$ ranging from 1 up to 200. The heavy red line shows the exact value of $|G_1|$.

Figure 16

All solutions for $G_1$ in the complex plane up to $n=40$. The roots exhibit fourfold symmetry. However, $\mathcal{P}\mathcal{T}$ symmetry requires that $G_1$ be negative imaginary, so only the roots on or near the negative-imaginary axis are physically acceptable. The exact value of $G_1$, given in (36), lies inside the concentration of roots near $-i$.

Figure 17

All solutions for $G_1$ in the complex plane, up to $n=11$. The roots exhibit fivefold symmetry but $\mathcal{P}\mathcal{T}$ symmetry requires that $G_1$ be negative imaginary, so only the roots on or near the negative-imaginary axis are physically acceptable. The exact value of $G_1$, given in (43), lies inside the concentration of roots near $-i$.

Figure 18

Roots of the DS equations for the Hermitian sextic case obtained by means of the unbiased truncation scheme. There are three concentrations of solutions for $G_2$ that differ from the exact values in (48) and (49) (red squares) by a few percent.

Figure 19

Positive values of the roots of the SD equations for the Hermitian sextic case (dots). The exact value of $G_2=0.578616\dots$ is indicated by the red line and the DS results converge to a number that differs from the exact value by about 20%.