Variational Perturbation Theory for Summing Divergent Non-Borel-Summable Tunneling Amplitudes

We present a method for evaluating divergent non-Borel-summable series by an analytic continuation of variational perturbation theory. We demonstrate the power of the method by an application to the exactly known partition function of the anharmonic oscillator in zero spacetime dimensions. In one spacetime dimension we derive the imaginary part of the ground state energy of the anharmonic oscillator for {\em all negative values of the coupling constant $g$, including the nonanalytic tunneling regime at small-$g$. As a highlight of the theory we retrieve from the divergent perturbation expansion the action of the critical bubble and the contribution of the higher loop fluctuations around the bubble.

We present a method for evaluating divergent non-Borel-summable series by an analytic continuation of variational perturbation theory. We demonstrate the power of the method by an application to the exactly known partition function of the anharmonic oscillator in zero space-time dimensions. In one space-time dimension we derive the imaginary part of the ground state energy of the anharmonic oscillator for all negative values of the coupling constant g, including the non-analytic tunneling regime at small −g. As a highlight of the theory we retrieve the divergent perturbation expansion from the action of the critical bubble and the contribution of the higher loop fluctuations around the bubble.

I. INTRODUCTION
None of the presently known resummation schemes [1,2] is able to deal with non-Borel-summable series. Such series arise in the theoretical description of many important physical phenomena, in particular tunneling processes. In the path integral, these are dominated by non-perturbative contributions coming from nontrivial classical solutions called critical bubbles [3,4] or bounces [5], and fluctuations around these.
Any Borel-summable series becomes non-Borel-summable if the expansion parameter, usually some coupling constant g, is continued to negative values. Here we show that non-Borel-summable series can be evaluated with any desired accuracy by an analytic continuation of variational perturbation theory [2,4] in the complex g-plane. This implies that variational perturbation theory can give us information on non-perturbative properties of the theory.
Variational perturbation theory has a long history [6,7,8,9]. It is based on the introduction of a dummy variational parameter Ω on which the full perturbation expansion does not depend, while the truncated expansion does. An optimal Ω is selected by the principle of minimal sensitivity [10], requiring the quantity of interest to be stationary as function of the variational parameter. The optimal Ω is usually taken from a zero of the derivative with respect to Ω. If the first derivative has no zero, a zero of the second derivative is chosen. For Borel-summable series, these zeros are always real, in contrast to statements in the literature [11,12,13,14] which have proposed the use of complex zeros. Complex zeros produce in general wrong results for Borel-summable series, as was recently shown in Ref. [15].
The purpose of this paper is to show that there does exist a wide range of applications of complex zeros if one wants to resum non-Borel-summable series, which have so far remained intractable. These arise typically in tunneling problems, and we shall see that variational perturbation theory provides us with an efficient method for evaluating these series, rendering their real and imaginary parts with any desirable accuracy, if only enough perturbation coefficients are available. An important problem which had to be solved is the specification of the proper choice the optimal zero from the many possible candidates existing in higher orders. A non-Borel-summable series is associated with a function which has an essential singularity at the origin in the complex g-plane, which is the starting point of a left-hand cut. Near the tip of the cut, the imaginary part of the function approaches zero rapidly like exp(−α/|g|) for g → 0−. If the variational approximation is plotted against g with an enlargement factor exp(α/|g|), oscillations become visible near g = 0. The choice of the optimal complex zeros of the derivative with respect to the variational parameter is fixed by the requirement of obtaining, in each order, the least oscillating imaginary part when approaching the tip of the cut. We may call this selection rule the principle of minimal sensitivity and oscillations.
In Section II, we shall explain and test the new principle on the exactly known partition function Z(g) of the anharmonic oscillator in zero space time dimensions. In Section III, we apply the method to the critical-bubble regime of small −g of the anharmonic oscillator and find the action of the critical bubble and the corrections caused by the fluctuations around it. In Section IV we present yet another method of calculating the properties of the critical-bubble regime. This method is restricted to quantum mechanical systems. Its results for the anharmonic oscillator give more evidence for the correctness of the general method of Sections II and III.

II. TEST OF RESUMMATION OF NON-BOREL-SUMMABLE EXPANSIONS
The partition function Z(g) of the anharmonic oscillator in zero space-time dimensions is where K ν (z) is the modified Bessel function. For small g, the function Z(g) has a divergent Taylor series expansion, to be called weak-coupling expansion: For g < 0, this is non-Borel-summable. For large |g| there exists a convergent strong-coupling expansion: As is obvious from the integral representation (2.1), Z(g) obeys the second-order differential equation which has two independent solutions. One of them is Z(g), which is finite for g > 0 with Z(0) = a 0 . The weakcoupling coefficients a l in (2.2) can be obtained by inserting into (2.4) the Taylor series and comparing coefficients. The result is the recursion relation

5) {REC}
A similar recursion relation can be derived for the strong-coupling coefficients b l in Eq. (2.3). We observe that the two independent solutions Z(g) of (2.4) behave like Z(g) ∝ g α for g → ∞ with the powers α = −1/4 and −3/4. The function (2.1) has α = −1/4. It is convenient to remove the leading power from Z(g) and define a function ζ(x) such that Z(g) = g −1/4 ζ(g −1/2 ). The Taylor coefficients of ζ(x) are the strong-coupling coefficients b l in Eq. (2.3). The function ζ(x) satisfies the differential equation and initial conditions: The Taylor coefficients b l of ζ(x) satisfy the recursion relation Analytic continuation of Z(g) around g = ∞ to the left-hand cut gives: so that we find an imaginary part where From this we may re-obtain the weak-coupling coefficients a l by means of the dispersion relation , and expanding exp (−x g) into a power series, all integrals can be evaluated to yield: Thus we find for the weak-coupling coefficients a l an expansion in terms of the strong-coupling coefficients coinciding with (2.2). Variational perturbation theory is a well-established method for obtaining convergent strong-coupling expansions from divergent weak-coupling expansions in quantum-mechanical systems such as the anharmonic oscillator [4,16] as well as in quantum field theory [2,17]. We have seen in Eq. (2.8), that the strong-coupling expansion can easily be continued analytically to negative g. This continuation can, however, be used for an evaluation only for sufficiently large |g| where the strong-coupling expansion converges. In the tunneling regime near the tip of the left-hand cut, the expansion diverges. In this paper we shall find that an evaluation of the weak-coupling expansion according to the rules of variational perturbation theory continued into the complex plane gives extremely good results on the entire left-hand cut with a fast convergence even near the tip at g = 0.
The Lth variational approximation to Z(g) is given by (see [2,17])

21) {FP-s}
where q = 2/ω = 4, p = −1 and To apply the principle of minimal sensitivity, the zeros of the derivative of Z (L) var (g, Ω) with respect to Ω are needed. They are given by the zeros of the polynomials in σ: since it can be shown [18] that the derivative depends only on σ: Consider in more detail the lowest non-trivial order with L = 1. From Eq. (2.23) we obtain FIG. 1: Result of the 1st-and 2nd-order calculation for the non-Borel-summable region of g < 0, where the function has a cut with non-vanishing imaginary part: imaginary (left) and real parts (right) of Z var (g) (dashed curve) and Z (2) var (g) (solid curve) are plotted against g and compared with the exact values of the partition function (dotted curve). The root of (2.21) giving the optimal variational parameter Ω has been chosen to reproduce the weak-coupling result near g = 0.
In order to ensure that our method reproduces the weak-coupling result for small g, we have to take the positive sign in front of the square root. In Fig. 1 we have plotted Z var (g) (dashed curve) and Z (2) var (g) (solid curve) and compared these with the exact result (doted curve) in the tunneling regime. The agreement is quite good even at these low orders [19]. Next we study the behavior of Z (L) var (g) to higher orders L. For selected coupling values in the non-Borel-summable region, g = −.01, −.1, −1, −10, we want to see the error as a function of the order. We want to find from this model system the rule for selecting systematically the best zero of P (L) (σ) solving Eq. (2.23), which leads to the optimal value of the variational parameter Ω. For this purpose we plot the variational results of all zeros. This is shown in Fig. 2, where the logarithm of the deviations from the exact value is plotted against the order L. The outcome of different zeros cluster strongly near the best value. Therefore, choosing any zero out of the middle of the cluster is reasonable, in particular, because it does not depend on the knowledge of the exact solution, so that this rule may be taken over to realistic cases.
We wish to emphasize, that for the Borel-summable domain with g > 0, variational perturbation theory has the usual fast convergence in this model. In fact, for g = 10, probing deeply into the strong-coupling domain, we find rapid convergence like ∆(L) ≃ 0.02 exp (−0.73L) for L → ∞, where ∆(L) = log |Z (L) var − Z exact | is the logarithmic error as a function of the order L. This is shown in Fig. 3. Furthermore, the strong-coupling coefficients b l of Eq. var − Zexact| plotted against the order L for different g < 0 in the non-Borel-summable region. All complex optimal Ω's have been used. absolute and relative errors over the order L, and find very good convergence, showing that variational perturbation theory works well for our test-model Z(g).
A better selection of the optimal Ω values comes from the following observation. The imaginary parts of the approximations near the singularity at g = 0 show tiny oscillations. The exact imaginary part is known to decrease extremely fast, like exp (1/4g), for g → 0−, practically without oscillations. We can make the tiny oscillations more visible by taking this exponential factor out of the imaginary part. This is done in Fig. 5. The oscillations differ strongly for different choices of Ω (L) from the central region of the cluster. To each order L we see that one of them is smoothest in the sense that the approximation approaches the singularity most closely before oscillations begin. If this Ω (L) is chosen as the optimal one, we obtain excellent results for the entire non-Borel-summable region g < 0. As an example, we pick the best zero for the L = 16th order. Fig. 5 shows the normalized imaginary part calculated to this order, but based on different zeros from the central cluster. Curve C appears optimal. Therefore we select the underlying zero as our best choice at order L = 16 and calculate with it real and imaginary part for the non-Borel-summable region −2 < g < −.008, to be compared with the exact values. Both are shown in Fig. 6, where we have again renormalized the imaginary part by the exponential factor exp (−1/4g). The agreement with the exact result (solid curve) is excellent as was to be expected because of the fast convergence observed in Fig. 2. It is indeed much better than the strong-coupling expansion to the same order, shown as a dashed curve. This is the essential improvement of our present theory as compared to previously known methods probing into the tunneling regime [19].
This non-Borel-summable regime will now be investigated for the quantum-mechanical anharmonic oscillator.   16) var (g) exp (−1/4g)] to the left and the real part Re[Z (16) var (g)] to the right, based on the best zero C from Fig. 5, are plotted against log |g| as dots. The solid curve represents the exact function. The dashed curve is the 16th order of the strong-coupling expansion Z (L) strong (g) of equation (2.3).

III. TUNNELING REGIME OF QUANTUM-MECHANICAL ANHARMONIC OSCILLATOR
The divergent weak-coupling perturbation expansion for the ground state energy of the anharmonic oscillator in the potential V (x) = x 2 /2 + g x 4 to order L E (L) 0,weak (g) = L l=0 a l g l ,

1) {WEAK}
where a l = (1/2, 3/4, −21/8, 333/16, −30885/128, . . . ), is non-Borel-summable for g < 0. It may be treated in the same way as Z(g) of the previous model, making use as before of Eqs. (2.20)-(2.23), provided we set p = 1 and ω = 2/3, so that q = 3, accounting for the correct power behavior E 0 (g) ∝ g 1/3 for g → ∞. According to the principle of minimal dependence and oscillations, we pick a best zero for the order L = 64 from the cluster of zeros of P L (σ), and use it to calculate the logarithm of the normalized imaginary part: This quantity is plotted in Fig. 7 against log(−g) close to the tip of the left-hand cut for −.2 < g < −.006. Comparing our result to older values from semi-classical calculations [20] f shown in Fig. 7 as a thin curve, we find very good agreement. This expansion contains the information on the fluctuations around the critical bubble. It is divergent and non-Borel-summable for g < 0. In Appendix A we have rederived it in a novel way which allowed us to extend and improve it considerably. Remarkably, our theory allows us to retrieve the first three terms of this expansion from the perturbation expansion. Since our result provides us with a regular approximation to the essential singularity, the fitting procedure depends somewhat on the interval over which we fit our curve by a power series. A compromise between a sufficiently long interval and the runaway of the divergent critical-bubble expansion is obtained for a lower limit g > −.0229 ± .0003 and an upper limit g = −0.006. Fitting a polynomial to the data, we extract the following first three coefficients: The agreement of these numbers with those in (3.3) demonstrates that our method is capable of probing deeply into the critical-bubble region of the coupling constant.
Further evidence for the quality of our theory comes from a comparison with the analytically continued strongcoupling result plotted to order L = 22 as a fat curve in Fig. 7. This expansion was derived by a procedure of summing non-Borel-summable series developed in Chapter 17 of the textbook [4]. It was based on a two-step process: the derivation of a strong-coupling expansion of the type (2.3) from the divergent weak-coupling expansion, and an analytic continuation of the strong-coupling expansion to negative g. This method was applicable only for large enough coupling strength where the strong-coupling expansion converges, the so-called sliding regime. It could not invade into the tunneling regime at small g governed by critical bubbles, which was treated in [4] by a separate variational procedure. The present work fills the missing gap by extending variational perturbation theory to all g arbitrarily close to zero, without the need for a separate treatment of the tunneling regime.
It is interesting to see, how the correct limit is approached as the order L increases. This is shown in Fig. 8, based on the optimal zero in each order. For large negative g, even the small orders give excellent results. Close to the singularity the scaling factor exp (−1/3g) will always win over the perturbation results. It is surprising, however, how fantastically close to the singularity we can go.

IV. DYNAMIC APPROACH TO THE CRITICAL-BUBBLE REGIME
Regarding the computational challenges connected with the critical-bubble regime of small g < 0, it is worth to develop an independent method to calculate imaginary parts in the tunneling regime. For a quantum-mechanical system with an interaction potential g V (x), such as a the harmonic oscillator, we may study the effect of an infinitesimal increase in g upon the system. It induces an infinitesimal unitary transformation of the Hilbert space. The new Hilbert space can be made the starting point for the next infinitesimal increase in g. In this way we derive an infinite set of first order ordinary differential equations for the change of the energy levels and matrix elements (for details see Appendix B): This system of equations holds for any one-dimensional Schroedinger problem. Individual differences come from the initial conditions, which are the energy levels E n (0) of the unperturbed system and the matrix elements V nm (0) of the 0,var (g) −1/3g, plotted against small negative values of the coupling constant −0.2 < g < −.006 where the series is non-Borel-summable. The thin curve represents the divergent expansion around a critical bubble of Ref. [20]. The fat curve is the 22nd order approximation of the strong-coupling expansion, analytically continued to negative g in the sliding regime calculated in Chapter 17 of the textbook [4]. 0,var (g)) − 1/3g, plotted against log (−g) for orders L = 4, 8, 16, 32 (curves). It is compared with the corresponding results for L = 64 (points). This is shown for small negative values of the coupling constant −0.2 < g < −.006, i.e. in the non-Borel-summable critical-bubble region. Fast convergence is easily recognized. Lower orders oscillate more heavily. Increasing orders allow closer approach to the singularity at g = 0−. interaction V (x) in the unperturbed basis. For a numerical integration of the system a truncation is necessary. The obvious way is to restrict the Hilbert space to the manifold spanned by the lowest N eigenvectors of the unperturbed system. For cases like the anharmonic oscillator, which are even, with even perturbation and with only an even state to be investigated, we may span the Hilbert space by even basis vectors only. Our initial conditions are thus for n = 0, 1, 2, . . . , N/2: V 2n,2n (0) =3(8n 2 + 4n + 1)/4 (4.5) V 2n,2n±2 (0) =(4n + 3) (2n + 1)(2n + 2)/2 (4.6) V 2n,2n±4 (0) = (2n + 1)(2n + 2)(2n + 3)(2n + 4)/4 (4.7) (4.8) For the anharmonic oscillator with a V (x) = x 4 potential, all sums in equation (4.1) are finite with at most four terms due to the near-diagonal structure of the perturbation. In order to find a solution for some g < 0, we first integrate the system from 0 to |g|, then around a semi-circle g = |g| exp (iϕ) from ϕ = 0 to ϕ = π. The imaginary part of E 0 (g) obtained from a calculation with N = 64 is shown in Fig. refVIx, where it is compared with the variational result for L = 64. The agreement is excellent. It must be noted, however, that the necessary truncation of the system of differential equations introduces an error, which cannot be made arbitrarily small by increasing the truncation limit N . The approximations are asymptotic sharing this property with the original weak-coupling series. Its divergence is, however, reduced considerably, which is the reason why we obtain accurate results for the critical-bubble regime, where the weak-coupling series fails completely to reproduce the imaginary part.

V. APPENDIX A
We determine the ground state energy function E 0 (g) for the anharmonic oscillator on the cut, i.e. for g < 0 in the bubble region, from the weak coupling coefficients a l of equation (3.1). The behavior of the a l for large l can be cast into the form a l /a l−1 = − L j=−1 β j l −j .
(5.1) {B1} The β j can be determined by a high precision fit to the data in the large l region of 250 < l < 300 to be β −1, 0, 1, ... = 3, − Given this, we take the derivative of (6.1) with respect to g and multiply by m, g| from the left to obtain: m, g|V − E ′ n (g)|n, g = k =n u nk m, g|H 0 + g V − E n (g)|k, g .

(6.4) {A4}
Setting now m = n and m = n in turn, we find: E ′ n (g) =V nn (g) (6.5) {A5} V mn (g) =u nm (E m (g) − E n (g)) , (6.6) {A6} where V mn (g) = m, g|V |n, g . Equation (6.5) governs the behavior of the eigenvalues as functions of the coupling constant g. In order to have a complete system of differential equations, we must also determine how the V mn (g) change, when g changes. With the help of equations (6.3) and (6.6), we obtain: Equations (6.5) and (6.8) together describe a complete set of differential equations for the energy eigenvalues E n (g) and the matrix-elements V nm (g). The latter determine via (6.6) the expansion coefficients u mn (g). Initial conditions are given by the eigenvalues E n (0) and the matrix elements V nm (0) of the unperturbed system.