$$ \frac{\mathbf{H}(p=0)}{Q(g)} \approx \frac{1}{g^2} \int dt' \int dt'' J(t', t'', t'') \\ = \frac{H_1^2 H_2}{g^2} = \frac{8\pi^2}{g^2}, \qquad (3.107) $$

confirming the leading-order result given in Eq. (3.101). One can conveniently express the correction as the sum of seven terms, which contribute up to relative order g,

$$ \frac{\mathbf{H}(p=0)}{Q(g)} = \sum_{i=1}^{7} \mathcal{U}_i. \qquad (3.108) $$

We have the first correction “for free,” because it is just the multiplicative correction $\mathcal{F}_Z$, multiplying the leading-order instanton result,

$$ \mathcal{U}_1 = \frac{8\pi^2}{g^2} \mathcal{F}_Z = \frac{8\pi^2}{g^2} + \frac{2\pi^2}{g} \left( \frac{5}{6} + \frac{9}{4}N + \frac{7}{8}N^2 \right). \qquad (3.109) $$

Just as we did in our analysis of the four-point function, we now replace, in the Green function in coordinate space, two classical fields by fluctuations and obtain two terms,

$$ [\mathcal{U}_2]_1 = -\frac{1}{g} \int dt' \int dt'' \Delta_L(t', t'') \int dt'''[\xi_{cl}(t'')]^2 \\ -\frac{4}{g} \int dt' \xi_{cl}(t') \int dt'' \Delta_L(t', t'') \int dt''' \xi_{cl}(t'') \\ -\frac{1}{g} \int dt' \Delta_L(t', t') \int dt'' \xi_{cl}(t'') \int dt''' \xi_{cl}(t'')], \qquad (3.110) $$

$$ [\mathcal{U}_3]_1 = -\frac{N-1}{g} \int dt' \int dt'' \Delta_T(t', t'') \int dt'''[\xi_{cl}(t'')]^2 \\ -\frac{N-1}{g} \int dt' \Delta_T(t', t') \\ \times \int dt'' \xi_{cl}(t'') \int dt''' \xi_{cl}(t''). \qquad (3.111) $$

From the perturbative compensating term in Eq. (3.65), we have

$$ [\mathcal{U}_2]_2 = \frac{1}{g} \int dt' \int dt'' \Delta_0(t', t'') \int dt'''[\xi_{cl}(t'')]^2 \\ + \frac{4}{g} \int dt' \xi_{cl}(t') \int dt'' \Delta_0(t', t'') \int dt''' \xi_{cl}(t'') \\ - \frac{1}{g} \int dt' \Delta_0(t', t') \int dt'' \xi_{cl}(t'') \int dt''' \xi_{cl}(t''), \qquad (3.112) $$

$$ [\mathcal{U}_3]_2 = \frac{N-1}{g} \int dt' \int dt'' \Delta_0(t', t'') \int dt'''[\xi_{cl}(t'')]^2 \\ + \frac{N-1}{g} \int dt' \Delta_0(t', t') \\ \times \int dt'' \xi_{cl}(t'') \int dt''' \xi_{cl}(t''). \qquad (3.113) $$

With the help of the integrals listed in Appendix A, we can express $\mathcal{U}_2$ and $\mathcal{U}_3$ as follows:

$$ \mathcal{U}_2 = -\frac{1}{g} H_2 I_1 - \frac{4}{g} H_1 (J_1 - J_1^{(0)}) - \frac{1}{g} H_1^2 N_1 \\ = \frac{48 + 31\pi^2}{3g}, \qquad (3.114) $$

$$ \mathcal{U}_3 = -\frac{N-1}{g} H_2 I_2 - \frac{N-1}{g} H_1^2 N_2 \\ = \frac{2(N-1)}{g} [\pi^2 - 2 + 7\zeta(3)]. \qquad (3.115) $$

We now consider the term

$$ \sqrt{-g} \left[ \int dt \xi_{cl}(t) \cdot \underline{\chi}(t) \underline{\chi}^2(t) \right] \qquad (3.116) $$

in the Jacobian Eq. (3.25). After the application of the Wick theorem and suitable variable changes, one arrives at a contribution to $\mathbf{H}(p=0)/Q(g)$, consisting of the sum of two terms, $\mathcal{U}_4$ and $\mathcal{U}_5$,

$$ \begin{aligned} \mathcal{U}_4 &= -\frac{6}{g} \int dt \xi_{cl}(t) \Delta_L(t, t) \\ & \times \int dt' \Delta_L(t, t') \int dt'' [\xi_{cl}(t'')] \int dt''' [\xi_{cl}(t'')]^2 \\ & -\frac{6}{g} \int dt \xi_{cl}(t) \Delta_L(t, t) \int dt' \xi_{cl}(t') \Delta_L(t, t') \left[ \int dt'' \xi_{cl}(t'') \right]^2 \\ &= -\frac{6}{g} H_1 H_2 J_3 - \frac{6}{g} H_1^2 M_3 = -\frac{19\pi^2}{3g}, \end{aligned} \qquad (3.117) $$

$$ \begin{aligned} \mathcal{U}_5 &= -\frac{2(N-1)}{g} \int dt \xi_{cl}(t) \Delta_T(t, t) \int dt' \Delta_L(t, t') \\ & \times \int dt'' [\xi_{cl}(t'')] \int dt''' [\xi_{cl}(t'')]^2 \\ & -\frac{2(N-1)}{g} \int dt \xi_{cl}(t) \Delta_T(t, t) \int dt' \xi_{cl}(t') \Delta_L(t, t') \\ & \times \left[ \int dt'' \xi_{cl}(t'') \right]^2 \\ &= -\frac{2(N-1)}{g} H_1 H_2 J_4 - \frac{2(N-1)}{g} H_1^2 M_4 = 0. \end{aligned} \qquad (3.118) $$

From the term
$$ \sqrt{-g} \left[ -\frac{3}{4} \int dt \underline{\chi}(t) \cdot \xi_{\text{cl}}(t) \right] \quad (3.119) $$

in the Jacobian given in Eq. (3.25), one has

$$ \begin{aligned} U_6 &= \frac{3}{2g} \int dt \xi_{\text{cl}}(t) \int dt' \Delta_{\text{L}}(t, t') \int dt'' \xi_{\text{cl}}(t'') \int dt'''[\xi_{\text{cl}}(t'')]^2 \\ & \quad + \frac{3}{2g} \int dt \xi_{\text{cl}}(t) \int dt' \xi_{\text{cl}}(t') \Delta_{\text{L}}(t, t') \left[ \int dt'' \xi_{\text{cl}}(t'') \right]^2 \\ &= \frac{3}{2g} H_1 H_2 J_2 + \frac{3}{2g} H_1^2 M_2 = \frac{9\pi^2}{g}. \end{aligned} \quad (3.120) $$

The last correction is from the term

$$ \sqrt{-g} \frac{N-1}{4} \int dt \underline{\chi}(t) \cdot \xi_{\text{cl}}(t) \quad (3.121) $$

in Eq. (3.25) and leads to the correction

$$ \begin{aligned} U_7 &= -\frac{N-1}{2g} \int dt \xi_{\text{cl}}(t) \int dt' \Delta_{\text{L}}(t, t') \\ & \quad \times \int dt'' \xi_{\text{cl}}(t'') \int dt'''[\xi_{\text{cl}}(t'')]^2 \\ & \quad -\frac{N-1}{2g} \int dt \xi_{\text{cl}}(t) \int dt' \xi_{\text{cl}}(t') \Delta_{\text{L}}(t, t') \left[ \int dt'' \xi_{\text{cl}}(t'') \right]^2 \\ &= -\frac{N-1}{2g} H_1 H_2 J_1 - \frac{N-1}{2g} H_1^2 M_1 = \frac{\pi^2(N-1)}{g}. \end{aligned} \quad (3.122) $$

The final result [see Eq. (3.79)] for order-g corrections to the imaginary part of the wigglet insertion into the two-point function reads as follows:

$$ \frac{\mathbf{H}(p=0)}{Q(g)} = \sum_{i=1}^{7} U_i = \frac{8\pi^2}{g^2} + \frac{1}{g} \left[ 20 + \frac{35\pi^2}{3} - 14\zeta(3) + N \left( \frac{15\pi^2}{2} - 4 + 14\zeta(3) \right) + \frac{7\pi^2}{4} N^2 \right]. \quad (3.123) $$

It has the same structure as the result for the four-point function, given in Eq. (3.96).

# IV. CONCLUSIONS

## A. Large-order behavior: A summary

In this article, we have concentrated on the one-dimensional field theory, with an internal $O(N)$ symmetry group, in the normalization [see Eq. (2.1)]

$$ S[\underline{q}(t)] = \int dt \left[ \frac{1}{2} \left( \frac{\partial \underline{q}(t)}{\partial t} \right)^2 + \frac{1}{2} \underline{q}^2(t) + \frac{g}{4} \underline{q}^4(t) \right], \quad (4.1) $$

where $\underline{q}(t) = \{q_1(t), \dots, q_N(t)\} = \sum_{\alpha=1}^{N} q_{\alpha}(t) \underline{\epsilon}_{\alpha}$. The start time $t_0$ of the instanton and the coordinate $\tau_i$ with

$i=1, \dots, N-1$ are the collective coordinates of the problem. Furthermore, we have analyzed the functional determinant of the transformation of the path integral, into integrals over the collective coordinates and path integrals over paths orthogonal to the zero modes, in Sec. II C, with the result for the Jacobian given in Eq. (2.21).

Based on the results given in Sec. III and the dispersion relations studied in Sec. I B, we are now in the position to write down the large-order behavior of various perturbative expansions, of the partition function and various correlation functions [in the sense of Eqs. (1.8) and (1.9)]. We have from Eq. (2.55), for the imaginary part of the ground-state energy of the $O(N)$ oscillator,

$$ \begin{aligned} \frac{\mathrm{Im} E_0(g)}{Q(g)} &= \lim_{\beta \to \infty} \left( -\frac{1}{Q(g)\beta} \frac{\mathrm{Im} \mathcal{Z}_1(\beta)}{\mathcal{Z}_0(\beta)} \right) \\ &= -\left[ 1 + g \left( \frac{7}{32} N^2 + \frac{9}{16} N + \frac{5}{24} \right) \right]. \end{aligned} \quad (4.2) $$

Here, according to Eq. (3.32), one has

$$ Q(g) = \frac{1}{\Gamma(N/2)} \left( -\frac{8}{g} \right)^{N/2} \exp \left( \frac{4}{3g} \right). \quad (4.3) $$

Based on the formalism outlined in Sec. I B, this can be converted to the asymptotics of the perturbative coefficients of the ground-state energy in large orders [see Eq. (2.55)]. One can identify $E_0(g) \equiv \mathbf{G}^{(0)}(g)$ with the ground-state energy and study the perturbative expansion $\mathbf{G}^{(0)}(g) = \sum_{\mathcal{K}} \mathbf{G}_{\mathcal{K}}^{(0)} g^{\mathcal{K}}$. Within the conventions outlined in Eq. (1.9), one has $n=0, D=1$ and

$$ c^{(0)}(N, 1) = - \frac{8^{N/2}}{\Gamma(N/2)}, \quad (4.4a) $$

$$ d^{(0)}(N, 1) = \frac{7}{32} N^2 + \frac{9}{16} N + \frac{5}{24}. \quad (4.4b) $$

The two-point function has the angular structure [see Eqs. (3.22) and (3.27)]

$$ \mathrm{Im} \mathbf{G}_{\alpha\beta}^{(2)}(t_1 - t_2) = \mathcal{G}_{\alpha\beta}(t_1, t_2) = \frac{\delta_{\alpha\beta}}{N} \mathcal{G}(t_1 - t_2). \quad (4.5) $$

We recall that $\mathcal{G}_{\alpha\beta}(t_1, t_2)$, as defined in Eq. (3.15), is the imaginary part of the two-point function $\mathbf{G}_{\alpha\beta}(t_1 - t_2)$, in the sense in Sec. I B. The result for $\mathcal{G}(p=0) = \int dt' \mathcal{G}(t')$ of the “scalar” two-point function at $p=0$ has been recorded in Eq. (3.53),
$$ \sqrt{-g} \left[ -\frac{3}{4} \int dt \underline{\chi}(t) \cdot \dot{\xi}_{\text{cl}}(t) \right] \quad (3.119) $$

in the Jacobian given in Eq. (3.25), one has

$$ \begin{aligned} U_6 &= \frac{3}{2g} \int dt \xi_{\text{cl}}(t) \int dt' \Delta_{\text{L}}(t, t') \int dt'' \xi_{\text{cl}}(t'') \int dt'''[\xi_{\text{cl}}(t'')]^2 \\ & \quad + \frac{3}{2g} \int dt \xi_{\text{cl}}(t) \int dt' \xi_{\text{cl}}(t') \Delta_{\text{L}}(t, t') \left[ \int dt'' \xi_{\text{cl}}(t'') \right]^2 \\ &= \frac{3}{2g} H_1 H_2 J_2 + \frac{3}{2g} H_1^2 M_2 = \frac{9\pi^2}{g}. \end{aligned} \quad (3.120) $$

The last correction is from the term

$$ \sqrt{-g} \frac{N-1}{4} \int dt \underline{\chi}(t) \cdot \dot{\xi}_{\text{cl}}(t) \quad (3.121) $$

in Eq. (3.25) and leads to the correction

$$ \begin{aligned} U_7 &= -\frac{N-1}{2g} \int dt \xi_{\text{cl}}(t) \int dt' \Delta_{\text{L}}(t, t') \\ & \quad \times \int dt'' \xi_{\text{cl}}(t'') \int dt'''[\xi_{\text{cl}}(t'')]^2 \\ & \quad -\frac{N-1}{2g} \int dt \xi_{\text{cl}}(t) \int dt' \xi_{\text{cl}}(t') \Delta_{\text{L}}(t, t') \left[ \int dt'' \xi_{\text{cl}}(t'') \right]^2 \\ &= -\frac{N-1}{2g} H_1 H_2 J_1 - \frac{N-1}{2g} H_1^2 M_1 = \frac{\pi^2(N-1)}{g}. \end{aligned} \quad (3.122) $$

The final result [see Eq. (3.79)] for order-g corrections to the imaginary part of the wigglet insertion into the two-point function reads as follows:

$$ \frac{\mathbf{H}(p=0)}{Q(g)} = \sum_{i=1}^{7} U_i = \frac{8\pi^2}{g^2} + \frac{1}{g} \left[ 20 + \frac{35\pi^2}{3} - 14\zeta(3) + N \left( \frac{15\pi^2}{2} - 4 + 14\zeta(3) \right) + \frac{7\pi^2}{4} N^2 \right]. \quad (3.123) $$

It has the same structure as the result for the four-point function, given in Eq. (3.96).

# IV. CONCLUSIONS

## A. Large-order behavior: A summary

In this article, we have concentrated on the one-dimensional field theory, with an internal $O(N)$ symmetry group, in the normalization [see Eq. (2.1)]

$$ S[\underline{q}(t)] = \int dt \left[ \frac{1}{2} \left( \frac{\partial \underline{q}(t)}{\partial t} \right)^2 + \frac{1}{2} \underline{q}^2(t) + \frac{g}{4} \underline{q}^4(t) \right], \quad (4.1) $$

where $\underline{q}(t) = \{q_1(t), \dots, q_N(t)\} = \sum_{\alpha=1}^{N} q_{\alpha}(t) \underline{\epsilon}_{\alpha}$. The start time $t_0$ of the instanton and the coordinate $\tau_i$ with

$i=1, \dots, N-1$ are the collective coordinates of the problem. Furthermore, we have analyzed the functional determinant of the transformation of the path integral, into integrals over the collective coordinates and path integrals over paths orthogonal to the zero modes, in Sec. II C, with the result for the Jacobian given in Eq. (2.21).

Based on the results given in Sec. III and the dispersion relations studied in Sec. I B, we are now in the position to write down the large-order behavior of various perturbative expansions, of the partition function and various correlation functions [in the sense of Eqs. (1.8) and (1.9)]. We have from Eq. (2.55), for the imaginary part of the ground-state energy of the $O(N)$ oscillator,

$$ \begin{aligned} \frac{\mathrm{Im} E_0(g)}{Q(g)} &= \lim_{\beta \to \infty} \left( -\frac{1}{Q(g)\beta} \frac{\mathrm{Im} \mathcal{Z}_1(\beta)}{\mathcal{Z}_0(\beta)} \right) \\ &= -\left[ 1 + g \left( \frac{7}{32} N^2 + \frac{9}{16} N + \frac{5}{24} \right) \right]. \end{aligned} \quad (4.2) $$

Here, according to Eq. (3.32), one has

$$ Q(g) = \frac{1}{\Gamma(N/2)} \left( -\frac{8}{g} \right)^{N/2} \exp \left( \frac{4}{3g} \right). \quad (4.3) $$

Based on the formalism outlined in Sec. I B, this can be converted to the asymptotics of the perturbative coefficients of the ground-state energy in large orders [see Eq. (2.55)]. One can identify $E_0(g) \equiv \mathbf{G}^{(0)}(g)$ with the ground-state energy and study the perturbative expansion $\mathbf{G}^{(0)}(g) = \sum_K \mathbf{G}_K^{(0)} g^K$. Within the conventions outlined in Eq. (1.9), one has $n=0, D=1$ and

$$ c^{(0)}(N, 1) = - \frac{8^{N/2}}{\Gamma(N/2)}, \quad (4.4a) $$

$$ d^{(0)}(N, 1) = \frac{7}{32} N^2 + \frac{9}{16} N + \frac{5}{24}. \quad (4.4b) $$

The two-point function has the angular structure [see Eqs. (3.22) and (3.27)]

$$ \mathrm{Im} \mathbf{G}_{\alpha\beta}^{(2)}(t_1 - t_2) = \mathcal{G}_{\alpha\beta}(t_1, t_2) = \frac{\delta_{\alpha\beta}}{N} \mathcal{G}(t_1 - t_2). \quad (4.5) $$

We recall that $\mathcal{G}_{\alpha\beta}(t_1, t_2)$, as defined in Eq. (3.15), is the imaginary part of the two-point function $\mathbf{G}_{\alpha\beta}(t_1 - t_2)$, in the sense in Sec. I B. The result for $\mathcal{G}(p=0) = \int dt' \mathcal{G}(t')$ of the “scalar” two-point function at $p=0$ has been recorded in Eq. (3.53),
$$ \frac{\mathcal{G}(p=0)}{\mathcal{Q}(g)} = -\frac{2\pi^2}{g} \left[ 1 + g \left\{ \frac{5}{24} + \frac{5}{2\pi^2} - \frac{7\zeta(3)}{4\pi^2} + \left( \frac{9}{16} - \frac{1}{2\pi^2} + \frac{7\zeta(3)}{4\pi^2} \right) N + \frac{7}{32} N^2 \right\} \right]. \quad (4.6) $$

For the imaginary part $\mathcal{G}(p=0)$, this translates into the following coefficients in the large-order asymptotics, according to Eq. (1.9):

$$ c^{(2)}(N, 1) = 2\pi^2 \frac{8^{N/2}}{\Gamma(N/2)}, \quad (4.7a) $$

$$ d^{(2)}(N, 1) = \frac{5}{24} + \frac{5}{2\pi^2} - \frac{7\zeta(3)}{4\pi^2} + \left( \frac{9}{16} - \frac{1}{2\pi^2} + \frac{7\zeta(3)}{4\pi^2} \right) N + \frac{7}{32} N^2. \quad (4.7b) $$

Of course, we have $n=2$ in the sense of Eq. (1.9) for the two-point function at zero momentum transfer as well as for its derivative, which gives rise to the following imaginary part, according to Eq. (3.62),

$$ \frac{\partial^2}{\partial p^2} \mathcal{G}|_{p=0} = \frac{\pi^4}{g} \left[ 1 + g \left\{ \frac{5}{24} + \frac{4}{\pi^4} - \frac{21\zeta(3)}{\pi^4} - \frac{93\zeta(5)}{2\pi^4} + \left( -\frac{3}{16} - \frac{6}{\pi^4} + \frac{93\zeta(5)}{2\pi^4} \right) N + \frac{7N^2}{32} \right\} \right]. \quad (4.8) $$

The leading and subleading large-order asymptotics for the perturbative coefficients of $\frac{\partial^2}{\partial p^2} \mathcal{G}|_{p=0}$ are given as follows [see Eq. (1.9)]:

$$ c^{(\partial p)}(N, 1) = -\pi^4 \frac{8^{N/2}}{\Gamma(N/2)}, \quad (4.9a) $$

$$ d^{(\partial p)}(N, 1) = \frac{5}{24} + \frac{4}{\pi^4} - \frac{21\zeta(3)}{\pi^4} - \frac{93\zeta(5)}{2\pi^4} + \left( -\frac{3}{16} - \frac{6}{\pi^4} + \frac{93\zeta(5)}{2\pi^4} \right) N + \frac{7}{32} N^2. \quad (4.9b) $$

For the imaginary part of the four-point function, according to Eq. (3.70), we have

$$ \mathcal{G}_{\alpha\beta\gamma\delta}(t_1, t_2, t_3, t_4) = \frac{\delta_{\alpha\beta}\delta_{\gamma\delta} + \delta_{\alpha\gamma}\delta_{\beta\delta} + \delta_{\alpha\delta}\delta_{\gamma\beta}}{N(N+2)} \times \mathbf{G}(t_1 - t_4, t_2 - t_4, t_3 - t_4). \quad (4.10) $$

One defines according to Eq. (3.76),

$$ \mathbf{G}(p_1 = 0, p_2 = 0, p_3 = 0) = \mathbf{G}(p_{i=1,2,3} = 0) = \int dt' \int dt'' \int dt''' \mathbf{G}(t', t'', t'''). \quad (4.11) $$

The result, to relative order $g$, is given by [according to Eq. (3.96)],

$$ \frac{\mathbf{G}(p_{i=1,2,3} = 0)}{\mathcal{Q}(g)} = \frac{4\pi^4}{g^2} \left[ 1 + g \left\{ \frac{5}{24} + \frac{13}{\pi^2} - \frac{7\zeta(3)}{2\pi^2} + \left( \frac{9}{16} - \frac{1}{\pi^2} + \frac{7\zeta(3)}{2\pi^2} \right) N + \frac{7N^2}{32} \right\} \right]. \quad (4.12) $$

The perturbative expansion of the imaginary part $\mathbf{G}(p_{i=1,2,3} = 0)$ of the four-point function therefore has the following asymptotics of the perturbative coefficients [see Eq. (1.9) with $n=4$]:

$$ c^{(4)}(N, 1) = 4\pi^4 \frac{8^{N/2}}{\Gamma(N/2)}, \quad (4.13a) $$

$$ d^{(4)}(N, 1) = \frac{5}{24} + \frac{13}{\pi^2} - \frac{7\zeta(3)}{2\pi^2} + \left( \frac{9}{16} - \frac{1}{\pi^2} + \frac{7\zeta(3)}{2\pi^2} \right) N + \frac{7}{32} N^2. \quad (4.13b) $$

Finally, for the imaginary part of the two-point function with a wigglet insertion, computed at zero momentum, we have the following expression, according to Eq. (3.103),

$$ \mathcal{G}_{\alpha\beta}^{(1,2)}(t_1, t_2) = -\frac{\delta_{\alpha\beta}}{2N} \mathbf{H}(t_1 - t_2). \quad (4.14) $$

The quantity of interest is [see Eq. (3.106)]

$$ \mathbf{H}(p=0) = \int dt' \mathbf{H}(t'), \quad (4.15) $$

for which we obtain the result [see Eq. (3.123)]

$$ \frac{\mathbf{H}(p=0)}{\mathcal{Q}(g)} = \frac{8\pi^2}{g^2} \left[ 1 + g \left\{ \frac{35}{24} + \frac{5}{2\pi^2} - \frac{7\zeta(3)}{4\pi^2} + \left( \frac{15}{16} - \frac{1}{2\pi^2} + \frac{7\zeta(3)}{4\pi^2} \right) N + \frac{7}{32} N^2 \right\} \right]. \quad (4.16) $$

The large-order asymptotics of the perturbative coefficients for $\mathbf{H}(t_1 - t_2)$ are as follows:

$$ c^{(1,2)}(N, 1) = 8\pi^2 \frac{8^{N/2}}{\Gamma(N/2)}, \quad (4.17a) $$

$$ d^{(1,2)}(N, 1) = \frac{35}{24} + \frac{5}{2\pi^2} - \frac{7\zeta(3)}{4\pi^2} + \left( \frac{15}{16} - \frac{1}{2\pi^2} + \frac{7\zeta(3)}{4\pi^2} \right) N + \frac{7}{32} N^2. \quad (4.17b) $$
$$ \frac{\mathcal{G}(p=0)}{\mathcal{Q}(g)} = -\frac{2\pi^2}{g} \left[ 1 + g \left\{ \frac{5}{24} + \frac{5}{2\pi^2} - \frac{7\zeta(3)}{4\pi^2} + \left( \frac{9}{16} - \frac{1}{2\pi^2} + \frac{7\zeta(3)}{4\pi^2} \right) N + \frac{7}{32} N^2 \right\} \right]. \quad (4.6) $$

For the imaginary part $\mathcal{G}(p=0)$, this translates into the following coefficients in the large-order asymptotics, according to Eq. (1.9):

$$ c^{(2)}(N, 1) = 2\pi^2 \frac{8^{N/2}}{\Gamma(N/2)}, \quad (4.7a) $$

$$ d^{(2)}(N, 1) = \frac{5}{24} + \frac{5}{2\pi^2} - \frac{7\zeta(3)}{4\pi^2} + \left( \frac{9}{16} - \frac{1}{2\pi^2} + \frac{7\zeta(3)}{4\pi^2} \right) N + \frac{7}{32} N^2. \quad (4.7b) $$

Of course, we have $n=2$ in the sense of Eq. (1.9) for the two-point function at zero momentum transfer as well as for its derivative, which gives rise to the following imaginary part, according to Eq. (3.62),

$$ \frac{\partial^2}{\partial p^2} \mathcal{G}|_{p=0} = \frac{\pi^4}{g} \left[ 1 + g \left\{ \frac{5}{24} + \frac{4}{\pi^4} - \frac{21\zeta(3)}{\pi^4} - \frac{93\zeta(5)}{2\pi^4} + \left( -\frac{3}{16} - \frac{6}{\pi^4} + \frac{93\zeta(5)}{2\pi^4} \right) N + \frac{7N^2}{32} \right\} \right]. \quad (4.8) $$

The leading and subleading large-order asymptotics for the perturbative coefficients of $\frac{\partial^2}{\partial p^2} \mathcal{G}|_{p=0}$ are given as follows [see Eq. (1.9)]:

$$ c^{(\partial p)}(N, 1) = -\pi^4 \frac{8^{N/2}}{\Gamma(N/2)}, \quad (4.9a) $$

$$ d^{(\partial p)}(N, 1) = \frac{5}{24} + \frac{4}{\pi^4} - \frac{21\zeta(3)}{\pi^4} - \frac{93\zeta(5)}{2\pi^4} + \left( -\frac{3}{16} - \frac{6}{\pi^4} + \frac{93\zeta(5)}{2\pi^4} \right) N + \frac{7}{32} N^2. \quad (4.9b) $$

For the imaginary part of the four-point function, according to Eq. (3.70), we have

$$ \mathcal{G}_{\alpha\beta\gamma\delta}(t_1, t_2, t_3, t_4) = \frac{\delta_{\alpha\beta}\delta_{\gamma\delta} + \delta_{\alpha\gamma}\delta_{\beta\delta} + \delta_{\alpha\delta}\delta_{\gamma\beta}}{N(N+2)} \times \mathbf{G}(t_1 - t_4, t_2 - t_4, t_3 - t_4). \quad (4.10) $$

One defines according to Eq. (3.76),

$$ \mathbf{G}(p_1 = 0, p_2 = 0, p_3 = 0) = \mathbf{G}(p_{i=1,2,3} = 0) \\ = \int dt' \int dt'' \int dt''' \mathbf{G}(t', t'', t'''). \quad (4.11) $$

The result, to relative order $g$, is given by [according to Eq. (3.96)],

$$ \frac{\mathbf{G}(p_{i=1,2,3} = 0)}{\mathcal{Q}(g)} = \frac{4\pi^4}{g^2} \left[ 1 + g \left\{ \frac{5}{24} + \frac{13}{\pi^2} - \frac{7\zeta(3)}{2\pi^2} + \left( \frac{9}{16} - \frac{1}{\pi^2} + \frac{7\zeta(3)}{2\pi^2} \right) N + \frac{7N^2}{32} \right\} \right]. \quad (4.12) $$

The perturbative expansion of the imaginary part $\mathbf{G}(p_{i=1,2,3} = 0)$ of the four-point function therefore has the following asymptotics of the perturbative coefficients [see Eq. (1.9) with $n=4$]:

$$ c^{(4)}(N, 1) = 4\pi^4 \frac{8^{N/2}}{\Gamma(N/2)}, \quad (4.13a) $$

$$ d^{(4)}(N, 1) = \frac{5}{24} + \frac{13}{\pi^2} - \frac{7\zeta(3)}{2\pi^2} + \left( \frac{9}{16} - \frac{1}{\pi^2} + \frac{7\zeta(3)}{2\pi^2} \right) N + \frac{7}{32} N^2. \quad (4.13b) $$

Finally, for the imaginary part of the two-point function with a wigglet insertion, computed at zero momentum, we have the following expression, according to Eq. (3.103),

$$ \mathcal{G}_{\alpha\beta}^{(1,2)}(t_1, t_2) = -\frac{\delta_{\alpha\beta}}{2N} \mathbf{H}(t_1 - t_2). \quad (4.14) $$

The quantity of interest is [see Eq. (3.106)]

$$ \mathbf{H}(p=0) = \int dt' \mathbf{H}(t'), \quad (4.15) $$

for which we obtain the result [see Eq. (3.123)]

$$ \frac{\mathbf{H}(p=0)}{\mathcal{Q}(g)} = \frac{8\pi^2}{g^2} \left[ 1 + g \left\{ \frac{35}{24} + \frac{5}{2\pi^2} - \frac{7\zeta(3)}{4\pi^2} + \left( \frac{15}{16} - \frac{1}{2\pi^2} + \frac{7\zeta(3)}{4\pi^2} \right) N + \frac{7}{32} N^2 \right\} \right]. \quad (4.16) $$

The large-order asymptotics of the perturbative coefficients for $\mathbf{H}(t_1 - t_2)$ are as follows:

$$ c^{(1,2)}(N, 1) = 8\pi^2 \frac{8^{N/2}}{\Gamma(N/2)}, \quad (4.17a) $$

$$ d^{(1,2)}(N, 1) = \frac{35}{24} + \frac{5}{2\pi^2} - \frac{7\zeta(3)}{4\pi^2} + \left( \frac{15}{16} - \frac{1}{2\pi^2} + \frac{7\zeta(3)}{4\pi^2} \right) N + \frac{7}{32} N^2. \quad (4.17b) $$
We note that the large-order asymptotics have the param-
eter $n = 4$ in the conventions delineated in Eq. (1.9)
because of the additional two fields that have to be inserted
in view of the mass derivative.

B. Interpretation of the results

In this article, we have laid the groundwork for the accurate systematic analysis of the large-order behavior of perturbation theory for the correlation functions in the $N$-vector model. Our paradigm is that once the number of loops in a Feynman diagram becomes very large, the large-order behavior of the $N$-vector model is determined by classical field configurations (instantons), which determine the cut of the correlation functions for negative coupling. They act as a second saddle point of the Euclidean action. The (longitudinal) fluctuation operator around the saddle point has one negative eigenvalue, commensurate with the imaginary square root of the determinant of this operator.

Through the evaluation of corrections to the classical
configurations, we are able to evaluate corrections of
relative order $g$ to the correlation functions, which, via
dispersion relations, immediately lead to the corrections of
relative order $1/K$ to the perturbative coefficients. The
connection is elucidated in great detail in Sec. IB.

In all cases, the leading term of the order-$g$ correction, in
the large-$N$ limit, is given by the multiplicative correction
to the partition function term, i.e., due to the factor $F_Z$
given in Eq. (3.19). One might wonder why the term of
order $N^2$ in the correction to the partition function con-
stitutes the universal leading correction in the large-$N$ limit.
To understand this phenomenon, let us consider the
computational origin of the corrections. The leading term
in a correlation function (for the imaginary part) is given by
a term in which one replaces all field configurations by
instantons. Then, a set of universal corrections is obtained
when one keeps the instanton field configurations inside
the main integrand but considers the corrections due to the
field Jacobian, and due to the effective action around the
instanton, which, together, give rise to the universal
correction factor $F_Z$ given in Eq. (3.19).

Let us now consider the additional corrections obtained
when one replaces, instead, one of the instanton field
configurations by a fluctuation. Then, for the order-$g$
corrections to the imaginary part of the partition function,
one has to combine the fluctuation with the factor $F_J$ from
Eq. (3.25). This combination leads, at most, with regard to
$N$, to the product of a longitudinal fluctuation factor, which
carries no $N$, and one additional factor $N$ due to the
transverse fluctuations encoded in $\chi^2 = \chi_L^2 + \chi_T^2$. Finally,
replacing two instanton configurations by fluctuations, one
obtains at most a single factor of $N$, generated by a term
proportional to $\chi^2$. The universality of the large-$N$ limit of
the correction terms can be justified based on the decou-
pling of expectation values of fields at different spacetime

points in the limit of large $N$, as explained in the text
following Eq. (2.2) of Ref. [26] and Chap. 14 of Ref. [27].
In fact, according to formulas given in Chap. 14 of
Ref. [27], critical exponents reach universal values in the
large-$N$ limit and depend only on the spatial dimension of
the system.

Our expressions for the two-point correlation function,
for its second derivative, for the four-point correlation
function and for the two-point correlation function with a
wigglet insertion, computed at zero momentum, were
obtained by integrating the corresponding correlation
function in coordinate space over the difference of all its
coordinate with respect to one of them. Incidentally, it is
interesting to note that this procedure is completely
equivalent to an alternative procedure where one fixes
one of the time coordinates in a correlation function
to zero, and integrates over all the others. This equivalence
holds due to the time translation invariance of the corre-
lation functions. One of the most important additional
conclusions of the current article is that, for the correlation
functions, the two-loop corrections to the imaginary part,
of relative order $g$, have a much more complex analytic
structure as compared to those of the ground-state
energy.

A remark on our notation is in order. We apologize to the
reader for the many $G$'s in our paper. Generic $n$-point Green
functions are denoted by the sans-serif $G$, according to
Eq. (3.13). We also recall that $G_{a\beta}$, according to Eq. (3.15),
is the imaginary part of the two-point function, while $G_{a\beta\gamma\delta}$,
according to Eq. (3.63), is the imaginary part of the four-
point function. Incidentally, we also have the terms $R_i$
($i = 1, \dots, 7$) for the contributions to the two-point function
at zero momentum transfer, according to Eq. (3.48). Seven
corrections are also incurred for the contributions to the
derivative of the two-point function, summarized in the
terms $S_i$ [see Eq. (3.54)]. For the four-point function one
defines the scalar Green function $G$ according to Eq. (3.76).
Finally, we have the seven terms $T_i$ ($i = 1, \dots, 7$) for the
four-point function at zero momentum transfer, according
to Eq. (3.79), and the seven terms $U_i$ ($i = 1, \dots, 7$) for the
two-point function with a wigglet insertion, according to
Eq. (3.123).

A remark on the character of the factorial divergence
of the perturbation series is in order. According to
Eq. (1.9), all factorially divergent series calculated in this
work are alternating in large order, in view of the factor
$(-1/A)^K$ in Eq. (1.9), where $A = 4/3$. Indeed, it is known
that perturbation series in $\phi^4$ theories are factorially
divergent, Borel summable series [6–8,28–30]. In a more
general context, such series constitute the conceptually
simplest manifestation of so-called resurgent expansions
(transseries), which have recently been shown to lead to
adequate representations of physical quantities of interest in
a number of mathematical investigations and in field
theories [13–15,31–35].
We note that the large-order asymptotics have the param-
eter $n = 4$ in the conventions delineated in Eq. (1.9)
because of the additional two fields that have to be inserted
in view of the mass derivative.

B. Interpretation of the results

In this article, we have laid the groundwork for the
accurate systematic analysis of the large-order behavior of
perturbation theory for the correlation functions in the
N-vector model. Our paradigm is that once the number of
loops in a Feynman diagram becomes very large, the large-
order behavior of the N-vector model is determined by
classical field configurations (instantons), which determine
the cut of the correlation functions for negative coupling.
They act as a second saddle point of the Euclidean action.
The (longitudinal) fluctuation operator around the saddle
point has one negative eigenvalue, commensurate with the
imaginary square root of the determinant of this operator.

Through the evaluation of corrections to the classical
configurations, we are able to evaluate corrections of
relative order $g$ to the correlation functions, which, via
dispersion relations, immediately lead to the corrections of
relative order $1/K$ to the perturbative coefficients. The
connection is elucidated in great detail in Sec. I B.

In all cases, the leading term of the order-g correction, in
the large-N limit, is given by the multiplicative correction
to the partition function term, i.e., due to the factor $F_Z$
given in Eq. (3.19). One might wonder why the term of
order $N^2$ in the correction to the partition function con-
stitutes the universal leading correction in the large-N limit.
To understand this phenomenon, let us consider the
computational origin of the corrections. The leading term
in a correlation function (for the imaginary part) is given by
a term in which one replaces all field configurations by
instantons. Then, a set of universal corrections is obtained
when one keeps the instanton field configurations inside
the main integrand but considers the corrections due to the
field Jacobian, and due to the effective action around the
instanton, which, together, give rise to the universal
correction factor $F_Z$ given in Eq. (3.19).

Let us now consider the additional corrections obtained
when one replaces, instead, one of the instanton field
configurations by a fluctuation. Then, for the order-g
corrections to the imaginary part of the partition function,
one has to combine the fluctuation with the factor $F_J$ from
Eq. (3.25). This combination leads, at most, with regard to
$N$, to the product of a longitudinal fluctuation factor, which
carries no $N$, and one additional factor $N$ due to the
transverse fluctuations encoded in $\chi^2 = \chi_L^2 + \chi_T^2$. Finally,
replacing two instanton configurations by fluctuations, one
obtains at most a single factor of $N$, generated by a term
proportional to $\chi^2$. The universality of the large-N limit of
the correction terms can be justified based on the decou-
pling of expectation values of fields at different spacetime

points in the limit of large $N$, as explained in the text
following Eq. (2.2) of Ref. [26] and Chap. 14 of Ref. [27].
In fact, according to formulas given in Chap. 14 of
Ref. [27], critical exponents reach universal values in the
large-$N$ limit and depend only on the spatial dimension of
the system.

Our expressions for the two-point correlation function,
for its second derivative, for the four-point correlation
function and for the two-point correlation function with a
wigglet insertion, computed at zero momentum, were
obtained by integrating the corresponding correlation
function in coordinate space over the difference of all its
coordinate with respect to one of them. Incidentally, it is
interesting to note that this procedure is completely
equivalent to an alternative procedure where one fixes
one of the time coordinates in a correlation function
to zero, and integrates over all the others. This equivalence
holds due to the time translation invariance of the corre-
lation functions. One of the most important additional
conclusions of the current article is that, for the correlation
functions, the two-loop corrections to the imaginary part,
of relative order $g$, have a much more complex analytic
structure as compared to those of the ground-state
energy.

A remark on our notation is in order. We apologize to the
reader for the many $G$'s in our paper. Generic $n$-point Green
functions are denoted by the sans-serif $G$, according to
Eq. (3.13). We also recall that $G_{a\beta}$, according to Eq. (3.15),
is the imaginary part of the two-point function, while $G_{a\beta\gamma\delta}$,
according to Eq. (3.63), is the imaginary part of the four-
point function. Incidentally, we also have the terms $R_i$
$(i = 1, \dots, 7)$ for the contributions to the two-point function
at zero momentum transfer, according to Eq. (3.48). Seven
corrections are also incurred for the contributions to the
derivative of the two-point function, summarized in the
terms $S_i$ [see Eq. (3.54)]. For the four-point function one
defines the scalar Green function $G$ according to Eq. (3.76).
Finally, we have the seven terms $T_i$ $(i = 1, \dots, 7)$ for the
four-point function at zero momentum transfer, according
to Eq. (3.79), and the seven terms $U_i$ $(i = 1, \dots, 7)$ for the
two-point function with a wigglet insertion, according to
Eq. (3.123).

A remark on the character of the factorial divergence
of the perturbation series is in order. According to
Eq. (1.9), all factorially divergent series calculated in this
work are alternating in large order, in view of the factor
$(-1/A)^K$ in Eq. (1.9), where $A = 4/3$. Indeed, it is known
that perturbation series in $\phi^4$ theories are factorially
divergent, Borel summable series [6–8,28–30]. In a more
general context, such series constitute the conceptually
simplest manifestation of so-called resurgent expansions
(transseries), which have recently been shown to lead to
adequate representations of physical quantities of interest in
a number of mathematical investigations and in field
theories [13–15,31–35].
In principle, based on the results presented in the current paper, one could go further and calculate the large-order behavior of the renormalization constants of the one-dimensional $O(N)$ field theory (see Sec. B). We recall that the renormalization constants $Z_\phi$ (wave function), $Z_{\phi^2}$ (wigglet insertion), $Z_g$ (coupling constant), and $\delta m^2$ (mass counterterm) are determined by the renormalization conditions imposed on the vertex functions, which can be obtained from the correlation functions (calculated here) via a Legendre transformation. The renormalization constants, in turn, determine the large-order behavior of the beta function $\beta(g)$, the anomalous dimension function $\eta(g)$, and the correlation length function $\eta_2(g)$, which enter the Callan-Symanzik equation. The Callan-Symanzik equation is a RG equation fulfilled by the vertex functions of the theory (a minireview on this aspect is given in Appendix B). However, in one dimension, we refrain from engaging in this endeavor because of the absence of a second-order phase transition due to the low dimensionality of the system under study, which prevents the system from undergoing a phase transition to the low-temperature phase. In higher dimensions, the critical exponents can be studied on the basis of the Callan-Symanzik equation [9–12].

One of the main conclusions of the current paper is that the calculation of corrections to the large-order growth of perturbation theory for correlation functions, beyond the plain calculation of the partition function, is possible for field theories with a nontrivial internal structure [here, the $O(N)$ symmetry group]. The results presented here are a first step toward the evaluation of subleading corrections to the factorial asymptotics of perturbative coefficients in perturbative field theory, for physical quantities of interest beyond the partition function. The evaluation of subleading corrections to the large-order asymptotics of Feynman diagram coefficients in large loop-order provides us with an alternative method to enhance our understanding of the predictive limits of field theory. The ultimate goal of the endeavor is to “interpolate” between (necessarily finite-order) perturbative Feynman diagram calculations and the large-order asymptotics (about “infinite loop” order), the latter being enhanced by the evaluation of the subleading corrections about the instantons. The latter, in turn, lead to corrections of relative order $1/K$ (two-loop order discussed here) to the large-order behavior of perturbation theory. The generalization to relative order $1/K^2$ (four-loop order about the instanton) and the consideration of field theories in higher dimensions are the natural next steps in this program and are currently under investigation.

## ACKNOWLEDGMENTS

This work has been supported by the National Science Foundation (Grant No. PHY-1710856) and by the Swedish Research Council (Grant No. 638-2013-9243). Support from the Simons Foundation (Grant No. 454949) is also gratefully acknowledged.

## APPENDIX A: INTEGRAL TABLE

We first list integrals that involve only the instanton,

$$H_1 = \int dt \xi_{\text{cl}}(t) = \sqrt{2\pi}, \qquad (\text{A1})$$

$$H_2 = \int dt [\xi_{\text{cl}}(t)]^2 = 4, \qquad (\text{A2})$$

$$H_3 = \int dt t^2 \xi_{\text{cl}}(t) = \frac{\pi^3}{2\sqrt{2}}. \qquad (\text{A3})$$

Integrals involving the subtracted longitudinal and transverse propagators read as follows:

$$I_1 = \int dt_1 \int dt_2 [\Delta_L(t_1, t_2) - \Delta_0(t_1, t_2)] = -4, \qquad (\text{A4})$$

$$\begin{aligned} I_2 &= \int dt_1 \int dt_2 [\Delta_T(t_1, t_2) - \Delta_0(t_1, t_2)] \\ &= 1 - \frac{\pi^2}{4} - \frac{7}{2}\zeta(3). \end{aligned} \qquad (\text{A5})$$

In addition, we have integrals involving longitudinal and transverse propagators and instantons,

$$J_1 = \int dt_1 \xi_{\text{cl}}(t_1) \int dt_2 \Delta_L(t_1, t_2) = 0, \qquad (\text{A6})$$

$$J_2 = \int dt_1 \ddot{\xi}_{\text{cl}}(t_1) \int dt_2 \Delta_L(t_1, t_2) = \frac{\pi}{\sqrt{2}}, \qquad (\text{A7})$$

$$J_3 = \int dt_1 \xi_{\text{cl}}(t_1) \Delta_L(t_1, t_1) \int dt_2 \Delta_L(t_1, t_2) = \frac{\pi}{4\sqrt{2}}, \qquad (\text{A8})$$

$$J_4 = \int dt_1 \xi_{\text{cl}}(t_1) \Delta_T(t_1, t_1) \int dt_2 \Delta_L(t_1, t_2) = \frac{\pi}{8\sqrt{2}}. \qquad (\text{A9})$$

Integrals involving the subtracted longitudinal and transverse propagators, and powers of the Euclidean time variable, are needed for the second derivative of the two-point function,

$$\begin{aligned} K_1 &= \int dt_1 \int dt_2 t_1^2 [\Delta_L(t_1, t_2) - \Delta_0(t_1, t_2)] \\ &= -\frac{\pi^2}{3}, \end{aligned} \qquad (\text{A10})$$

$$\begin{aligned} K_2 &= \int dt_1 \int dt_2 t_1^2 [\Delta_T(t_1, t_2) - \Delta_0(t_1, t_2)] \\ &= 2 + \frac{\pi^2}{12} - \frac{\pi^4}{16} - \frac{7}{24} \pi^2 \zeta(3) - \frac{31}{2} \zeta(5). \end{aligned} \qquad (\text{A11})$$

Furthermore, we have the following integrals with mixed contributions:
In principle, based on the results presented in the current paper, one could go further and calculate the large-order behavior of the renormalization constants of the one-dimensional $O(N)$ field theory (see Sec. B). We recall that the renormalization constants $Z_\phi$ (wave function), $Z_{\phi^2}$ (wigglet insertion), $Z_g$ (coupling constant), and $\delta m^2$ (mass counterterm) are determined by the renormalization conditions imposed on the vertex functions, which can be obtained from the correlation functions (calculated here) via a Legendre transformation. The renormalization constants, in turn, determine the large-order behavior of the beta function $\beta(g)$, the anomalous dimension function $\eta(g)$, and the correlation length function $\eta_2(g)$, which enter the Callan-Symanzik equation. The Callan-Symanzik equation is a RG equation fulfilled by the vertex functions of the theory (a minireview on this aspect is given in Appendix B). However, in one dimension, we refrain from engaging in this endeavor because of the absence of a second-order phase transition due to the low dimensionality of the system under study, which prevents the system from undergoing a phase transition to the low-temperature phase. In higher dimensions, the critical exponents can be studied on the basis of the Callan-Symanzik equation [9–12].

One of the main conclusions of the current paper is that the calculation of corrections to the large-order growth of perturbation theory for correlation functions, beyond the plain calculation of the partition function, is possible for field theories with a nontrivial internal structure [here, the $O(N)$ symmetry group]. The results presented here are a first step toward the evaluation of subleading corrections to the factorial asymptotics of perturbative coefficients in perturbative field theory, for physical quantities of interest beyond the partition function. The evaluation of subleading corrections to the large-order asymptotics of Feynman diagram coefficients in large loop-order provides us with an alternative method to enhance our understanding of the predictive limits of field theory. The ultimate goal of the endeavor is to “interpolate” between (necessarily finite-order) perturbative Feynman diagram calculations and the large-order asymptotics (about “infinite loop” order), the latter being enhanced by the evaluation of the subleading corrections about the instantons. The latter, in turn, lead to corrections of relative order $1/K$ (two-loop order discussed here) to the large-order behavior of perturbation theory. The generalization to relative order $1/K^2$ (four-loop order about the instanton) and the consideration of field theories in higher dimensions are the natural next steps in this program and are currently under investigation.

## ACKNOWLEDGMENTS

This work has been supported by the National Science Foundation (Grant No. PHY-1710856) and by the Swedish Research Council (Grant No. 638-2013-9243). Support from the Simons Foundation (Grant No. 454949) is also gratefully acknowledged.

## APPENDIX A: INTEGRAL TABLE

We first list integrals that involve only the instanton,

$$H_1 = \int dt \xi_{\text{cl}}(t) = \sqrt{2\pi}, \qquad (A1)$$

$$H_2 = \int dt [\xi_{\text{cl}}(t)]^2 = 4, \qquad (A2)$$

$$H_3 = \int dt t^2 \xi_{\text{cl}}(t) = \frac{\pi^3}{2\sqrt{2}}. \qquad (A3)$$

Integrals involving the subtracted longitudinal and transverse propagators read as follows:

$$I_1 = \int dt_1 \int dt_2 [\Delta_L(t_1, t_2) - \Delta_0(t_1, t_2)] = -4, \qquad (A4)$$

$$ \begin{aligned} I_2 &= \int dt_1 \int dt_2 [\Delta_T(t_1, t_2) - \Delta_0(t_1, t_2)] \\ &= 1 - \frac{\pi^2}{4} - \frac{7}{2}\zeta(3). \end{aligned} \qquad (A5) $$

In addition, we have integrals involving longitudinal and transverse propagators and instantons,

$$J_1 = \int dt_1 \xi_{\text{cl}}(t_1) \int dt_2 \Delta_L(t_1, t_2) = 0, \qquad (A6)$$

$$J_2 = \int dt_1 \ddot{\xi}_{\text{cl}}(t_1) \int dt_2 \Delta_L(t_1, t_2) = \frac{\pi}{\sqrt{2}}, \qquad (A7)$$

$$J_3 = \int dt_1 \xi_{\text{cl}}(t_1) \Delta_L(t_1, t_1) \int dt_2 \Delta_L(t_1, t_2) = \frac{\pi}{4\sqrt{2}}, \qquad (A8)$$

$$J_4 = \int dt_1 \xi_{\text{cl}}(t_1) \Delta_T(t_1, t_1) \int dt_2 \Delta_L(t_1, t_2) = \frac{\pi}{8\sqrt{2}}. \qquad (A9)$$

Integrals involving the subtracted longitudinal and transverse propagators, and powers of the Euclidean time variable, are needed for the second derivative of the two-point function,

$$ \begin{aligned} K_1 &= \int dt_1 \int dt_2 t_1^2 [\Delta_L(t_1, t_2) - \Delta_0(t_1, t_2)] \\ &= -\frac{\pi^2}{3}, \end{aligned} \qquad (A10) $$

$$ \begin{aligned} K_2 &= \int dt_1 \int dt_2 t_1^2 [\Delta_T(t_1, t_2) - \Delta_0(t_1, t_2)] \\ &= 2 + \frac{\pi^2}{12} - \frac{\pi^4}{16} - \frac{7}{24} \pi^2 \zeta(3) - \frac{31}{2} \zeta(5). \end{aligned} \qquad (A11) $$

Furthermore, we have the following integrals with mixed contributions:
$$ 
\begin{aligned}
\tilde{K}_1 &= \int dt_1 \int dt_2 t_1 t_2 [\Delta_L(t_1, t_2) - \Delta_0(t_1, t_2)] \\
&= -\frac{7\pi^2}{12} - 1 - \frac{21}{2}\zeta(3),
\end{aligned}
\quad (A12) $$

$$ 
\begin{aligned}
\tilde{K}_2 &= \int dt_1 \int dt_2 t_1 t_2 [\Delta_T(t_1, t_2) - \Delta_0(t_1, t_2)] \\
&= -1 + \frac{\pi^2}{12} - \frac{7}{24}\pi^2\zeta(3) + \frac{31}{4}\zeta(5).
\end{aligned}
\quad (A13) $$

Among all integrals considered, the integrals $K_2$ and $\tilde{K}_2$ are by far the most difficult to evaluate.

One may derive a relatively compact representation for $K_2$ via the substitutions $t_2 = 2 \ln(v)$, followed by the $t_2$ integral, then followed by $t_1 = 2 \ln(u)$, and $u = \sqrt{x}$, so one would effectively substitute $t_1 = \ln(x)$. This results in

$$ K_2 = \int_0^\infty dx \left\{ -\frac{\ln^2(x)}{x(x^2+1)} - \frac{\ln^2(x)(x^2-1) \arctan(x)}{x^2(x^2+1)} - \frac{i \ln^2(x)}{1+x^2} [\mathrm{Li}_2(-ix) - \mathrm{Li}_2(ix)] \right\}. \quad (A14) $$

This representation is seen to involve Legendre's $\chi_2$ function [36,37],

$$ \chi_2(z) = \frac{1}{2}(\mathrm{Li}_2(z) - \mathrm{Li}_2(-z)). \quad (A15) $$

Formula (A11) was found by the PSLQ algorithm [20–23]. One can form the combinations

$$ 
\begin{aligned}
\bar{K}_1 &= \int dt_1 \int dt_2 (t_1 - t_2)^2 [\Delta_L(t_1, t_2) - \Delta_0(t_1, t_2)] \\
&= 2(K_1 - \bar{K}_1) = 2 + \frac{\pi^2}{2} + 21\zeta(3),
\end{aligned}
\quad (A16) $$

$$ 
\begin{aligned}
\bar{K}_2 &= \int dt_1 \int dt_2 (t_1 - t_2)^2 [\Delta_T(t_1, t_2) - \Delta_0(t_1, t_2)] \\
&= 2(K_2 - \bar{K}_2) = 6 - \frac{\pi^4}{8} - \frac{93}{2}\zeta(5).
\end{aligned}
\quad (A17) $$

We have additional reference integrals with a second derivative insertion, which are important for the calculation of the derivative $\partial^2 G/\partial p^2$, at $p = 0$,

$$ L_1 = \int dt_1 \xi_{cl}(t_1) \int dt_2 t_2^2 \Delta_L(t_1, t_2) = \frac{\pi^3}{2\sqrt{2}}, \quad (A18) $$

$$ L_2 = \int dt_1 \ddot{\xi}_{cl}(t_1) \int dt_2 t_2^2 \Delta_L(t_1, t_2) = \frac{3\pi^3}{4\sqrt{2}}, \quad (A19) $$

$$ 
\begin{aligned}
L_3 &= \int dt_1 \xi_{cl}(t_1) \Delta_L(t_1, t_2) \int dt_2 t_2^2 \Delta_L(t_1, t_2) \\
&= \frac{5\pi^3}{16\sqrt{2}} - \frac{\pi}{12\sqrt{2}},
\end{aligned}
\quad (A20) $$

$$ 
\begin{aligned}
L_4 &= \int dt_1 \xi_{cl}(t_1) \Delta_T(t_1, t_2) \int dt_2 t_2^2 \Delta_L(t_1, t_2) \\
&= \frac{9\pi^3}{32\sqrt{2}}.
\end{aligned}
\quad (A21) $$

For the wigglet insertion, i.e., the Green function $G^{(1,2)}$ at zero momentum, we also need the integrals

$$ M_1 = \int dt_1 \xi_{cl}(t_1) \int dt_2 \xi_{cl}(t_2) \Delta_L(t_1, t_2) = -1, \quad (A22) $$

$$ M_2 = \int dt_1 \ddot{\xi}_{cl}(t_1) \int dt_2 \xi_{cl}(t_2) \Delta_L(t_1, t_2) = 1, \quad (A23) $$

$$ 
\begin{aligned}
M_3 &= \int dt_1 \xi_{cl}(t_1) \Delta_L(t_1, t_2) \int dt_2 \xi_{cl}(t_2) \Delta_L(t_1, t_2) \\
&= \frac{1}{36},
\end{aligned}
\quad (A24) $$

$$ 
\begin{aligned}
M_4 &= \int dt_1 \xi_{cl}(t_1) \Delta_T(t_1, t_2) \int dt_2 \xi_{cl}(t_2) \Delta_L(t_1, t_2) \\
&= -\frac{1}{4}.
\end{aligned}
\quad (A25) $$

In view of the confluence of arguments of the propagators in the wigglet insertion, additional integrals are required,

$$ N_1 = \int dt_1 [\Delta_L(t_1, t_1) - \Delta_0(t_1, t_1)] = -\frac{7}{6}, \quad (A26) $$

$$ N_2 = \int dt_1 [\Delta_T(t_1, t_1) - \Delta_0(t_1, t_1)] = -\frac{1}{2}. \quad (A27) $$

Finally, we also need the integral

$$ J_1^{(0)} = \int dt_1 \xi_{cl}(t_1) \int dt_2 \Delta_0(t_1, t_2) = \sqrt{2\pi}, \quad (A28) $$

which is obtained from (A6) by the replacement of the longitudinal propagator with its free counterpart.

## APPENDIX B: CALLAN-SYMANZIK EQUATION

In principle, the application of the Callan-Symanzik equation [24,25] to the calculation of critical exponents has been described in a number of monographs [27,30,38,39]. However, to put the calculations reported here into perspective, we should include a few remarks regarding the relation of the quantities calculated here to the RG functions that enter the Callan-Symanzik equation.
$$ 
\begin{aligned}
\tilde{K}_1 &= \int dt_1 \int dt_2 t_1 t_2 [\Delta_L(t_1, t_2) - \Delta_0(t_1, t_2)] \\
&= -\frac{7\pi^2}{12} - 1 - \frac{21}{2}\zeta(3),
\end{aligned}
\quad (A12) $$

$$ 
\begin{aligned}
\tilde{K}_2 &= \int dt_1 \int dt_2 t_1 t_2 [\Delta_T(t_1, t_2) - \Delta_0(t_1, t_2)] \\
&= -1 + \frac{\pi^2}{12} - \frac{7}{24}\pi^2\zeta(3) + \frac{31}{4}\zeta(5).
\end{aligned}
\quad (A13) $$

Among all integrals considered, the integrals $K_2$ and $\tilde{K}_2$ are by far the most difficult to evaluate.

One may derive a relatively compact representation for $K_2$ via the substitutions $t_2 = 2 \ln(v)$, followed by the $t_2$ integral, then followed by $t_1 = 2 \ln(u)$, and $u = \sqrt{x}$, so one would effectively substitute $t_1 = \ln(x)$. This results in

$$ K_2 = \int_0^\infty dx \left\{ -\frac{\ln^2(x)}{x(x^2+1)} - \frac{\ln^2(x)(x^2-1) \arctan(x)}{x^2(x^2+1)} - \frac{i \ln^2(x)}{1+x^2} [\mathrm{Li}_2(-ix) - \mathrm{Li}_2(ix)] \right\}. \quad (A14) $$

This representation is seen to involve Legendre's $\chi_2$ function [36,37],

$$ \chi_2(z) = \frac{1}{2}(\mathrm{Li}_2(z) - \mathrm{Li}_2(-z)). \quad (A15) $$

Formula (A11) was found by the PSLQ algorithm [20–23]. One can form the combinations

$$ 
\begin{aligned}
\tilde{K}_1 &= \int dt_1 \int dt_2 (t_1 - t_2)^2 [\Delta_L(t_1, t_2) - \Delta_0(t_1, t_2)] \\
&= 2(K_1 - \tilde{K}_1) = 2 + \frac{\pi^2}{2} + 21\zeta(3),
\end{aligned}
\quad (A16) $$

$$ 
\begin{aligned}
\tilde{K}_2 &= \int dt_1 \int dt_2 (t_1 - t_2)^2 [\Delta_T(t_1, t_2) - \Delta_0(t_1, t_2)] \\
&= 2(K_2 - \tilde{K}_2) = 6 - \frac{\pi^4}{8} - \frac{93}{2}\zeta(5).
\end{aligned}
\quad (A17) $$

We have additional reference integrals with a second derivative insertion, which are important for the calculation of the derivative $\partial^2 G/\partial p^2$, at $p = 0$,

$$ L_1 = \int dt_1 \xi_{cl}(t_1) \int dt_2 t_2^2 \Delta_L(t_1, t_2) = \frac{\pi^3}{2\sqrt{2}}, \quad (A18) $$

$$ L_2 = \int dt_1 \ddot{\xi}_{cl}(t_1) \int dt_2 t_2^2 \Delta_L(t_1, t_2) = \frac{3\pi^3}{4\sqrt{2}}, \quad (A19) $$

$$ 
\begin{aligned}
L_3 &= \int dt_1 \xi_{cl}(t_1) \Delta_L(t_1, t_2) \int dt_2 t_2^2 \Delta_L(t_1, t_2) \\
&= \frac{5\pi^3}{16\sqrt{2}} - \frac{\pi}{12\sqrt{2}},
\end{aligned}
\quad (A20) $$

$$ 
\begin{aligned}
L_4 &= \int dt_1 \xi_{cl}(t_1) \Delta_T(t_1, t_2) \int dt_2 t_2^2 \Delta_L(t_1, t_2) \\
&= \frac{9\pi^3}{32\sqrt{2}}.
\end{aligned}
\quad (A21) $$

For the wigglet insertion, i.e., the Green function $G^{(1,2)}$ at zero momentum, we also need the integrals

$$ M_1 = \int dt_1 \xi_{cl}(t_1) \int dt_2 \xi_{cl}(t_2) \Delta_L(t_1, t_2) = -1, \quad (A22) $$

$$ M_2 = \int dt_1 \ddot{\xi}_{cl}(t_1) \int dt_2 \xi_{cl}(t_2) \Delta_L(t_1, t_2) = 1, \quad (A23) $$

$$ 
\begin{aligned}
M_3 &= \int dt_1 \xi_{cl}(t_1) \Delta_L(t_1, t_1) \int dt_2 \xi_{cl}(t_2) \Delta_L(t_1, t_2) \\
&= \frac{1}{36},
\end{aligned}
\quad (A24) $$

$$ 
\begin{aligned}
M_4 &= \int dt_1 \xi_{cl}(t_1) \Delta_T(t_1, t_1) \int dt_2 \xi_{cl}(t_2) \Delta_L(t_1, t_2) \\
&= -\frac{1}{4}.
\end{aligned}
\quad (A25) $$

In view of the confluence of arguments of the propagators in the wigglet insertion, additional integrals are required,

$$ N_1 = \int dt_1 [\Delta_L(t_1, t_1) - \Delta_0(t_1, t_1)] = -\frac{7}{6}, \quad (A26) $$

$$ N_2 = \int dt_1 [\Delta_T(t_1, t_1) - \Delta_0(t_1, t_1)] = -\frac{1}{2}. \quad (A27) $$

Finally, we also need the integral

$$ J_1^{(0)} = \int dt_1 \xi_{cl}(t_1) \int dt_2 \Delta_0(t_1, t_2) = \sqrt{2\pi}, \quad (A28) $$

which is obtained from (A6) by the replacement of the longitudinal propagator with its free counterpart.

## APPENDIX B: CALLAN-SYMANZIK EQUATION

In principle, the application of the Callan-Symanzik equation [24,25] to the calculation of critical exponents has been described in a number of monographs [27,30,38,39]. However, to put the calculations reported here into perspective, we should include a few remarks regarding the relation of the quantities calculated here to the RG functions that enter the Callan-Symanzik equation.
The Callan-Symanzik equation is a RG equation fulfilled by one-particle irreducible vertex functions $\Gamma^{(a,n)}$, derived via Legendre transformation from the connected correlation functions (see Chap. 7 of Ref. [30]).

The Callan-Symanzik equation is obtained by differentiating the vertex functions with respect to the renormalized mass parameter while holding the bare parameters constant and reads as follows:

$$ \left[ m_R \frac{\partial}{\partial m_R} + \beta(g_R) \frac{\partial}{\partial g_R} - \frac{n}{2} \eta(g_R) - a \eta_2(g_R) \right] \Gamma_R^{(a,n)}(\{\vec{q}_i\}_{i=1}^a; \{\vec{p}_i\}_{i=1}^n; m_R, g_R) = [2 - \eta(g_R)] \Gamma_R^{(a+1,n)}(\{\vec{q}_i\}_{i=1}^a, \vec{0}; \{\vec{p}_i\}_{i=1}^n; m_R, g_R). \quad (B1) $$

Here, $\Gamma_R^{(a,n)}$ is the n-point vertex function with a wigglet insertions. In the term with $\Gamma_R^{(a+1,n)}$, we have to set the momentum argument corresponding to the wigglet insertion number $a+1$ (the one created by the action of the operator $\partial/\partial m_R^2$ on $\Gamma_R^{(a,n)}$) to zero. As already mentioned, the vertex functions are obtained from the connected correlation functions via a Legendre transformation. For example, the vertex function $\Gamma^{(2)} = \Gamma^{(0,2)}$ is the inverse of the two-point correlation function $W^{(2)}$ [see the discussion following Eq. (7.80) of Ref. [30]]. The four-point function $\Gamma^{(4)} = \Gamma^{(0,4)}$ is obtained from $W^{(4)}$ via “amputation” of the external legs and sign inversion [see the discussion following Eq. (7.80) of Ref. [30], or Eq. (4.24) of Ref. [39]].

Bare (index zero) and renormalized vertex functions are related by [see Eq. (10.20) of Ref. [30]]

$$ \Gamma_0^{(a,n)}(\{\vec{q}_i\}_{i=1}^a; \{\vec{p}_i\}_{i=1}^n; m_0, g_0) = Z_\phi^{-n/2} \left(\frac{Z_\phi}{Z_{\phi^2}}\right)^a \Gamma_R^{(a,n)}(\{\vec{q}_i\}_{i=1}^a; \{\vec{p}_i\}_{i=1}^n; m_R, g_R). \quad (B2) $$

Here, $Z_\phi$ is the wave function renormalization, $Z_{\phi^2}$ is the renormalization of the wigglet insertion, and the bare and renormalized mass parameters are related by

$$ m_0^2 = (m_R^2 + \delta m^2)/Z_\phi, \quad (B3) $$

where $\delta m^2$ is the mass counterterm.

The wave function renormalization constant $Z_{\phi^2}$ is fixed by the condition

$$ \left. \frac{\partial}{\partial p_1^2} \Gamma_R^{(2)}(\vec{p}_1, \vec{p}_2; m_R^2, g_R) \right|_{\vec{p}_i=\vec{0}} = 1. \quad (B4) $$

The renormalization condition

$$ \Gamma_R^{(2)}(\vec{p}_1, \vec{p}_2; m_R^2, g_R)|_{\vec{p}_i=\vec{0}} = m_R^2 \quad (B5) $$

fixes the mass counterterm $\delta m^2$. The coupling constant renormalization constant $Z_g$ is fixed by the condition

$$ \Gamma_R^{(4)}(\vec{p}_1, \vec{p}_2; \vec{p}_3, \vec{p}_4, m_R^2, g_R)|_{\vec{p}_i=\vec{0}} = m_R^{4-D} g_R, \quad (B6) $$

where $g_R$ is the dimensionless, renormalized coupling, and $D$ is the spatial dimension. Finally, the wigglet insertion renormalization constant $Z_{\phi^2}$ is determined by

$$ \Gamma_R^{(1,2)}(\vec{q}; \vec{p}_1, \vec{p}_2; m_R^2, g_R)|_{\vec{p}_i=\vec{0}} = 1. \quad (B7) $$

The renormalization-group functions are obtained as follows:

$$ \beta(g_R) = -(4-D) \left[ \frac{\partial}{\partial g_R} \ln \left( g_R \frac{Z_g(g_R)}{Z_\phi(g_R)^2} \right) \right]^{-1}, \quad (B8) $$

$$ \eta(g) = \beta(g_R) \frac{\partial}{\partial g_R} \ln(Z_\phi), \quad (B9) $$

$$ \eta_2(g) = \beta(g_R) \frac{\partial}{\partial g_R} \ln\left(\frac{Z_{\phi^2}}{Z_\phi}\right). \quad (B10) $$

From these relations, one calculates the critical value $g^*$ determined by the condition $\beta(g^*) = 0$ which determines the infrared non-Gaussian fixed point of the RG flow, which is approached by the system because of the relevance of the $\phi^4$ interaction in dimensions lower than four. Critical exponents are determined by the values of the RG functions at the critical point, $\eta = \eta(g^*)$ and $\eta_2 = \eta_2(g^*)$, via well-known hyperscaling relations. For example, the critical exponent $\nu$ of the correlation length is determined as $\nu = 1/(2 + \eta_2)$, while the critical exponent $\alpha$ of the heat capacity is $\alpha = 2 - D\nu$, and the critical exponent $\gamma$ of the magnetic susceptibility is $\gamma = \nu(2 - \eta)$.

[1] F. J. Dyson, Divergence of perturbation theory in quantum electrodynamics, Phys. Rev. **85**, 631 (1952).

[2] C. M. Bender and T. T. Wu, Anharmonic oscillator, Phys. Rev. **184**, 1231 (1969).

[3] C. M. Bender and T. T. Wu, Large-Order Behavior of Perturbation Theory, Phys. Rev. Lett. **27**, 461 (1971).

[4] C. M. Bender and T. T. Wu, Anharmonic oscillator. II. A study of perturbation theory in large order, Phys. Rev. D **7**, 1620 (1973).
The Callan-Symanzik equation is a RG equation fulfilled by one-particle irreducible vertex functions $\Gamma^{(a,n)}$, derived via Legendre transformation from the connected correlation functions (see Chap. 7 of Ref. [30]).

The Callan-Symanzik equation is obtained by differentiating the vertex functions with respect to the renormalized mass parameter while holding the bare parameters constant and reads as follows:

$$ \left[ m_R \frac{\partial}{\partial m_R} + \beta(g_R) \frac{\partial}{\partial g_R} - \frac{n}{2} \eta(g_R) - a \eta_2(g_R) \right] \Gamma_R^{(a,n)}(\{\vec{q}_i\}_{i=1}^a; \{\vec{p}_i\}_{i=1}^n; m_R, g_R) = [2 - \eta(g_R)] \Gamma_R^{(a+1,n)}(\{\vec{q}_i\}_{i=1}^a, \vec{0}; \{\vec{p}_i\}_{i=1}^n; m_R, g_R). \quad (B1) $$

Here, $\Gamma_R^{(a,n)}$ is the n-point vertex function with a wigglet insertions. In the term with $\Gamma_R^{(a+1,n)}$, we have to set the momentum argument corresponding to the wigglet insertion number $a+1$ (the one created by the action of the operator $\partial/\partial m_R^2$ on $\Gamma_R^{(a,n)}$) to zero. As already mentioned, the vertex functions are obtained from the connected correlation functions via a Legendre transformation. For example, the vertex function $\Gamma^{(2)} = \Gamma^{(0,2)}$ is the inverse of the two-point correlation function $W^{(2)}$ [see the discussion following Eq. (7.80) of Ref. [30]]. The four-point function $\Gamma^{(4)} = \Gamma^{(0,4)}$ is obtained from $W^{(4)}$ via “amputation” of the external legs and sign inversion [see the discussion following Eq. (7.80) of Ref. [30], or Eq. (4.24) of Ref. [39]].

Bare (index zero) and renormalized vertex functions are related by [see Eq. (10.20) of Ref. [30]]

$$ \Gamma_0^{(a,n)}(\{\vec{q}_i\}_{i=1}^a; \{\vec{p}_i\}_{i=1}^n; m_0, g_0) = Z_\phi^{-n/2} \left(\frac{Z_\phi}{Z_{\phi^2}}\right)^a \Gamma_R^{(a,n)}(\{\vec{q}_i\}_{i=1}^a; \{\vec{p}_i\}_{i=1}^n; m_R, g_R). \quad (B2) $$

Here, $Z_\phi$ is the wave function renormalization, $Z_{\phi^2}$ is the renormalization of the wigglet insertion, and the bare and renormalized mass parameters are related by

$$ m_0^2 = (m_R^2 + \delta m^2)/Z_\phi, \quad (B3) $$

where $\delta m^2$ is the mass counterterm.

The wave function renormalization constant $Z_{\phi^2}$ is fixed by the condition

$$ \left. \frac{\partial}{\partial p_1^2} \Gamma_R^{(2)}(\vec{p}_1, \vec{p}_2; m_R^2, g_R) \right|_{\vec{p}_i=\vec{0}} = 1. \quad (B4) $$

The renormalization condition

$$ \Gamma_R^{(2)}(\vec{p}_1, \vec{p}_2; m_R^2, g_R)|_{\vec{p}_i=\vec{0}} = m_R^2 \quad (B5) $$

fixes the mass counterterm $\delta m^2$. The coupling constant renormalization constant $Z_g$ is fixed by the condition

$$ \Gamma_R^{(4)}(\vec{p}_1, \vec{p}_2; \vec{p}_3, \vec{p}_4, m_R^2, g_R)|_{\vec{p}_i=\vec{0}} = m_R^{4-D} g_R, \quad (B6) $$

where $g_R$ is the dimensionless, renormalized coupling, and $D$ is the spatial dimension. Finally, the wigglet insertion renormalization constant $Z_{\phi^2}$ is determined by

$$ \Gamma_R^{(1,2)}(\vec{q}; \vec{p}_1, \vec{p}_2; m_R^2, g_R)|_{\vec{p}_i=\vec{0}} = 1. \quad (B7) $$

The renormalization-group functions are obtained as follows:

$$ \beta(g_R) = -(4-D) \left[ \frac{\partial}{\partial g_R} \ln \left( g_R \frac{Z_g(g_R)}{Z_\phi(g_R)^2} \right) \right]^{-1}, \quad (B8) $$

$$ \eta(g) = \beta(g_R) \frac{\partial}{\partial g_R} \ln(Z_\phi), \quad (B9) $$

$$ \eta_2(g) = \beta(g_R) \frac{\partial}{\partial g_R} \ln\left(\frac{Z_{\phi^2}}{Z_{\phi}}\right). \quad (B10) $$

From these relations, one calculates the critical value $g^*$ determined by the condition $\beta(g^*) = 0$ which determines the infrared non-Gaussian fixed point of the RG flow, which is approached by the system because of the relevance of the $\phi^4$ interaction in dimensions lower than four. Critical exponents are determined by the values of the RG functions at the critical point, $\eta = \eta(g^*)$ and $\eta_2 = \eta_2(g^*)$, via well-known hyperscaling relations. For example, the critical exponent $\nu$ of the correlation length is determined as $\nu = 1/(2+\eta_2)$, while the critical exponent $\alpha$ of the heat capacity is $\alpha = 2 - D\nu$, and the critical exponent $\gamma$ of the magnetic susceptibility is $\gamma = \nu(2-\eta)$.

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[5] E. Brézin, G. Parisi, and J. Zinn-Justin, Perturbation theory at large orders for a potential with degenerate minima, Phys. Rev. D **16**, 408 (1977).

[6] E. Brézin, J. C. Le Guillou, and J. Zinn-Justin, Perturbation theory at large order. I. The $\phi^{2N}$ interaction, Phys. Rev. D **15**, 1544 (1977).

[7] E. Brézin, J. C. Le Guillou, and J. Zinn-Justin, Perturbation theory at large order. II. Role of the vacuum instability, Phys. Rev. D **15**, 1558 (1977).

[8] E. Brézin and G. Parisi, Critical exponents and large-order behavior of perturbation theory, J. Stat. Phys. **19**, 269 (1978); Reprinted in E. Brézin and G. Parisi, Field Theory, Disorder and Simulations, edited by G. Parisi (World Scientific, Singapore, 1992), pp. 119–142.

[9] J. C. Le Guillou and J. Zinn-Justin, Critical Exponents for the *n*-Vector Model in Three Dimensions from Field Theory, Phys. Rev. Lett. **39**, 95 (1977).

[10] J. C. Le Guillou and J. Zinn-Justin, Critical exponents from field theory, Phys. Rev. B **21**, 3976 (1980).

[11] Large-Order Behaviour of Perturbation Theory, edited by J. C. Le Guillou and J. Zinn-Justin (North-Holland, Amsterdam, 1990).

[12] R. Guida and J. Zinn-Justin, Critical Exponents of the *N*-vector model, J. Phys. A **31**, 8103 (1998).

[13] U. D. Jentschura and J. Zinn-Justin, Instanton effects in quantum mechanics and resurgent expansions, Phys. Lett. B **596**, 138 (2004).

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