Correlation functions of the anharmonic oscillator: Numerical verification of two-loop corrections to the large-order behavior

Recently, the large-order behavior of correlation functions of the $O(N)$-anharmonic oscillator has been analyzed by us [L. T. Giorgini et al., Phys. Rev. D 101, 125001 (2020)]. Two-loop corrections about the instanton configurations were obtained for the partition function, the two-point and four-point functions, and the derivative of the two-point function at zero momentum transfer. Here, we attempt to verify the obtained analytic results against numerical calculations of higher-order coefficients for the $O(1)$, $O(2)$, and $O(3)$ oscillators, and we demonstrate the drastic improvement of the agreement of the large-order asymptotic estimates and perturbation theory upon the inclusion of the two-loop corrections to the large-order behavior.

Figure 1

Coefficients of the perturbative expansion of the ground state of the energy for $N=1$, 2, 3 are displayed as a function of the inverse of the order of perturbation (blue dots). These coefficients have been divided by their leading asymptotic estimate, given in Eq. (89), and have been compared with their subleading order estimate. Dashed black lines have been obtained while including the $b-1$ term in the denominator of the $1/(K+b-1)$ term in Eq. (91), while solid black lines exclude this term and follow only the $1/K$ term given in Eq. (90). The first 150 perturbative coefficients have been obtained for $N=1$, 2, 3. (a) Ground-state energy for $N=1$. (b) Ground-state energy for $N=2$. (c) Ground-state energy for $N=3$.

Figure 2

Same as Fig. 1, but data are plotted for the correlation functions in $N=1$. The first 63 perturbative coefficients have been obtained for each correlation function. (a) Two-point correlation function ($N=1$). (b) Derivative of two-point correlator ($N=1$). (c) Four-point correlation function ($N=1$). (d) Correlator with wigglet insertion $N=1$).

Figure 3

Same as Fig. 2, but for $N=2$. We numerically compute the first 16, 18, 13, 14 perturbative coefficients, respectively, for panels (a), (b), (c), and (d).

Figure 4

Same as Fig. 2, but for correlation functions in a theory with an internal symmetry group $O(N=3)$. Correspondingly, we numerically compute the first 16, 18, 15, 14 perturbative coefficients, respectively, for panels (a), (b), (c), and (d).

Figure 5

Comparison between the extrapolated (red curves) and predicted (black curves) asymptotic behavior of the perturbative coefficients of the four-point correlation functions for $N=1$, 2, 3. (a) Four-point correlation function $N=1$. (b) Four-point correlation function $N=2$. (c) Four-point correlation function $N=3$.