Symmetry, Integrability and Geometry: Methods and Applications Non-Perturbative Asymptotic Improvement of Perturbation Theory and Mellin–Barnes Representation

Using a method mixing Mellin-Barnes representation and Borel resummation we show how to obtain hyperasymptotic expansions from the (divergent) formal power series which follow from the perturbative evaluation of arbitrary"$N$-point"functions for the simple case of zero-dimensional $\phi^4$ field theory. This hyperasymptotic improvement appears from an iterative procedure, based on inverse factorial expansions, and gives birth to interwoven non-perturbative partial sums whose coefficients are related to the perturbative ones by an interesting resurgence phenomenon. It is a non-perturbative improvement in the sense that, for some optimal truncations of the partial sums, the remainder at a given hyperasymptotic level is exponentially suppressed compared to the remainder at the preceding hyperasymptotic level. The Mellin-Barnes representation allows our results to be automatically valid for a wide range of the phase of the complex coupling constant, including Stokes lines. A numerical analysis is performed to emphasize the improved accuracy that this method allows to reach compared to the usual perturbative approach, and the importance of hyperasymptotic optimal truncation schemes.


Introduction
The divergent behavior of a (divergent) asymptotic expansion does not at all detract from its computational utility. This statement is corroborated by the fact that, in what concerns its first few partial sums, a divergent asymptotic expansion of a given quantity "converges" in general much faster to the exact result than a convergent series representation of the same quantity does. In the case of the Standard Model quantum field theories, we may therefore even say that, regarding the extreme difficulty to go beyond the first few perturbative orders when computing observables in QCD or in the electroweak theory, it is an advantage, for phenomenology, to deal with a formalism that leads to presumably 1 asymptotic power series, diverging for all values of the coupling constants, rather than convergent ones.
However, one has of course to keep in mind that when dealing with divergent asymptotic perturbative power expansions, there always remains a finite limit of precision beyond which the usual asymptotic theory cannot go, even when the objects that one wants to compute are well-defined 2 . All ways to break open this precision limit are welcome. In the beginning of the 1990's, new asymptotic objects, which have in general a larger region of validity (a larger domain of definition in the complex expansion parameter) and a greater accuracy than conventional asymptotic expansions, appeared in the mathematical litterature [2]. With them, a new asymptotic theory emerged: exponential asymptotics (or hyperasymptotics 3 ). These asymptotic objects (hyperasymptotic expansions) are very interesting since they correspond, for some optimal truncation schemes to be defined later, to what we could call in physics a non-perturbative asymptotic improvement of a perturbative (asymptotic) power series. Our aim in this paper is to show how hyperasymptotic expansions appear naturally in the simplest example one may have in mind in "particle physics", namely zero-dimensional φ 4 field theory. In particular, we show that in the course of the study of the N -point functions of this theory one may obtain hyperasymptotic expansions directly from the formal (divergent) expansions which follow from their perturbative evaluation and without further information (i.e. without using the fact that by their integral representation, for instance, we have a rigorous definition for these objects).
The paper is organized as follows. In the introductive section 2.1 where basic facts are recalled, the perturbative approach is detailed and, to fix ideas, numerical results are given for a particular value of the coupling constant λ. The main part of the paper is section 2.2. In section 2.2.1 we present the formal approach which allows to rewrite the perturbative results in terms of hyperasymptotic expansions (at first level in the hyperasymptotic process). The calculations are based on so-called inverse factorial expansions of the ratios of Euler Gamma functions which constitute the coefficients of the perturbative terms forming the tail of the perturbative series. This makes appear the Mellin-Barnes representation into the game. After term by term Borel resummation of the reexpanded tail (this strategy is inspired from [5]), a new expansion of the tail emerges which, added to the perturbative partial sum, form the hyperasymptotic expressions at first hyperasymptotic level. At this level one may already notice a resurgence phenomenon that links the perturbative coefficients with those of the tail's new expansion, and which will also be observed at each higher hyperasymptotic level. Next, section 2.2.2 explains in detail how the hyperasymptotic theory of Mellin-Barnes integrals [4] leads to the proof that the hyperasymptotic results obtained in section 2.2.1 are correct, and this for a wide range of the phase of the complex coupling constant (taking into account, this time, the integral representation of N -point functions that was avoided until here), in particular on Stokes lines where the perturbative expansions are not Borel summable. In section 2.2.3, one is concerned with optimal truncation schemes of the hyperasymptotic expansions at zeroth and first hyperasymptotic levels. Their link with the non-perturbative interpretation of our results are underlined. Then follows section 2.2.4 where higher order hyperasymptotic levels are obtained. One ends the main body of the paper in section 2.2.5 by performing a numerical analysis which allows to compare the hyperasymptotic expansions for different optimal truncation schemes with the perturbative results. Two appendices give the proofs of some results quoted in the text.

Exponential asymptotics in zero-dimensional Euclidean φ 4 theory
The 4-dimensional Euclidean φ 4 action is given by Going to the 0-dimensional theory, which consists to reduce space-time to just one point, makes that the x-dependency disappears (φ(x) becomes a simple real variable φ, the derivative term and the overall integral disappear), so that one gets The corresponding 4-dimensional generating functional is therefore also reduced, in the 0-dimensional case, to a usual integral of the form where j is the external source and N a normalization factor. In the following, objects under study are the "N -point" functions, defined as where is the 0-dimensional version of the generating functional of vacuum to vacuum transitions. Notice that Eqs. (2.4) (and (2.5)) are defined for Re λ > 0 and we want to study the small λ case. We will see that, thanks to the Mellin-Barnes representation, the results that will be obtained in this paper are in fact valid for a much wider range of complex values of λ than just the right half complex λ-plane. In particular, our results are valid on Stokes rays where the perturbative expansion is not Borel summable (in our case | arg λ| = π).
Since the theoretical results of interest that we found may already be observed in the study of Z(0) with N = 1 √ 2π , we think that it is more pedagogical to present detailed derivations of our analysis on the simple example of Z(0) rather than on arbitrary N -point functions. This avoids a dependance on the N parameter in the calculations, which would complicate the expressions in the text without really giving new results compared to Z(0) (apart from the form of the resurgence phenomenon, see section 2.2.4). However, the final expressions of the calculations will also be given in the more general N -point functions case and some subtle changes that have to be done in the computation, for our theoretical strategy to be valid in this more general case, will also be explained when necessary.

Perturbative approach
In this introductive section, we recall basic facts about the link between the perturbative expansion of 0-dimensional Euclidean φ 4 theory and the counting of Feynman diagrams in the corresponding 4-dimensional theory. We do this on the example of Z(0) and, to fix ideas before the non-perturbative asymptotic analysis, we also give perturbative numerical predictions for a particular value of the coupling constant.

Perturbative expansion
So our first object of study is Z(0) with 4 m = 1 and N = 1 √ 2π , In this very simple case, one may get a closed expression for (2.7) in terms of special functions or even keep the integral representation and perform the whole (non-perturbative) asymptotic analysis in a fully rigorous way. This will be done as a check, but what we precisely want to show in this paper is a method for performing the analysis in the converse way. Indeed, the question we want to answer is: how can we get a non-perturbative improved expansion from a divergent perturbative expansion when only the latter is available?
Let us compute the perturbative expansion of (2.7) for small λ with Re λ > 0. Replacing e − λ 4 φ 4 by its series representation and performing term by term integration, one finds For further purpose, we define u k so that Z(0) ∼ λ→0 +∞ k=0 u k and the perturbative partial sum S Pert n . = n k=0 u k . The first few terms of (2.8) are given by The expansion in the r.h.s of (2.8) is divergent for any value of λ, as can be seen from the fact that (2.10) and it is easy to prove that it is an asymptotic expansion of Z(0) as we will see later, but for the moment we only suppose that it is so 5 (we do as in the case of Standard Model gauge theories where one assumes that perturbative expansions are asymptotic to objects whose definition is still lacking today). The perturbative result for an arbitrary N -point function (with N = 2p) is similar since (2.11)

Feynman diagrams counting
It is clear that the coefficients in the expansion (2.8) are related to the counting of Feynman diagrams in the 4-dimensional theory (taking into account their symmetry factors), since at first order in the λ-expansion we have, for 4-dimensional Z(0), the contribution and, at second order, where, as an example of how we get the coefficients in the l.h.s of (2.13), the 4! in the numerator of the coefficient in front of the last diagram is the number of different Wick contractions of the fields that lead to this topology (see Figure 1). We see that by summing the coefficients of each different topology (and this defines our counting of Feynman diagrams), one obtains from (2.12), (2.13) and higher orders the  result written in (2.8), see also (2.9), apart from the first term that does not exist in the 4-dimensional theory, since there are no tree vacuum diagrams. Zero-dimensional field theories therefore have a practical interest for 4-dimensional particle phenomenology by the fact that they allow a partial but important check of Feynman diagrams coefficients appearing in perturbative calculations.

Numerical analysis
Let us now perform a bit of numerical analysis to see the efficiency and predictive power of the perturbation theory developed in the preceding subsections. In the following, we choose λ = 1 3 . Imagine that we do not know the value of (2.7) for λ = 1 3 so that the only information that we have for computing Z(0) is the divergent perturbative expansion in (2.8) (remember that we made the hypothesis that it is an asymptotic expansion of Z(0)). Let us see what is the best perturbative prediction that can be obtained from (2.8). In Table 1, we computed the first twenty truncated sums S Pert n−1 (n ∈ {1, ..., 20}) of (2.8) for λ = 1 3 (and the values of the general term of the perturbative series for the same values of n).
It is readily seen that the partial sums rapidly converge to a value around 0.965 from which they finally begin to diverge (see also Figure 2).
In fact, since (2.8) is a divergent alternating asymptotic series, simple general interpretative considerations for its sum lead to the fact that, if we define the remainder R n so that  Table 1: Numerical values of the perturbative general term and partial sums of (2.8), for λ = 1 3 , with an 8 decimal places precision. We may conclude from these inequalities that the best truncation of the series (obtained by minimizing the remainder R n ) is theoretically obtained by truncating before or after the smallest term in magnitude. Therefore, there are two "best predictions" from the perturbative expansion (2.8) with λ = 1 3 , which only differ by their central value (see Table 1): and It is clear that since (2.8) is an alternate series, the exact value has to be between them, from what we conclude that so that the perturbative expansion (2.8) leads to an already very good prediction (from the precision level viewpoint), of the order of 0.15%. In fact, from (2.7), one gets, with an 8 decimal places precision, The central value in (2.19) is therefore very close to the exact value, and corresponds actually to the standard Stieltjes approximative resummation formula for alternating divergent series, which reads [5] where u η is the term of least magnitude.
The point is that one can approach the "right" value (2.20) much closer than what perturbation theory does, by a refined asymptotic analysis that we present in the next sections where we obtain non-perturbative asymptotic improvements of the perturbative expansions (2.8) and (2.11). At the first stage of this analysis, the improvement takes the form of so-called exponentially improved asymptotic expansions [2] but, the process being iterative, it will be possible to get more and more non-perturbative refinements, in terms of hyperasymptotic expansions of higher level.

Exponential asymptotics: non-perturbative asymptotic improvement
To go beyond perturbation theory, one could perform a Borel resummation of (2.8) and, in this simple example, it works: one may reconstruct Z(0) from its perturbative expansion by a Borel resummation, for Re λ > 0.
On the one hand, as we will see, the method we present here also allows to reconstruct Z(0) from its perturbative expansion but, on the other hand, it gives another look at Z(0): as a non-perturbative asymptotic method that reveals interesting effects that would be hidden in a usual Borel resummation, like a resurgence phenomenon.

Interpretation of the divergent perturbative expansion
Numerically, we saw that the first few perturbative terms already do a very good work for the description of Z(0), but that the divergent character of the perturbative expansion is inavoidable if one includes more and more terms. In order to solve this problem, we have to give a meaning to the tail of the perturbative expansion. With this in mind, we divide the perturbative expansion into two parts, following [5]: The first part is the perturbative contribution that one wants to keep and the second, the tail of the divergent series.
We are now going to perform formal manipulations that will be justified a posteriori, in the next section.
First, it is convenient to use the duplication formula to rewrite the tail as Our main tool is the so-called inverse factorial expansion which may be obtained from Barnes Lemma [6], [4] Γ k and the contour in the Mellin-Barnes integral on the r.h.s. is a straight line with c ∈ ]−1, 0[ (for m = 0, c ∈ − 1 4 , 0 but we always take m > 0). It is important to note that (2.25) is an exact equality 6 only for c + m < k (since in our case m and k are integers, and since min(k) = n and c < 0, this is equivalent to m ≤ n), see Appendix B.
Inserting (2.25) in (2.24), exchanging the two sums and the sum and integral signs, one finds Now we perform Borel resummations using the definition where Λ ℓ (x) is one of the so-called terminant functions [5], defined (when Re ℓ > −1 and | arg x| < π) 7 as which can also be expressed as where we recall that Γ(a, x) is the incomplete gamma function defined, for | arg x| < π, as Notice that the terminant is nothing but the Borel integral of a general UV renormalon pole (see for instance [7]). It may also be seen as a Mellin transform.
At the end, one obtains (for m ≤ n) We therefore conclude that, for m ≤ n and | arg λ| < π, The tail of the divergent perturbative series (2.33) has been rewritten as a partial sum, supplemented by a remainder integral written as a Mellin-Barnes representation. We therefore converted an infinite sum into a finite sum plus a convergent integral and it will be proven in the section 2.2.2 that the formal expression (2.34) is exact for any n and m as long as n ≥ m and constitutes what is called the first level of the hyperasymptotic expansion of Z(0). Better than that, we will also prove in section 2.2.3 that the r.h.s of (2.33) is, at an optimal value of n to be defined later, exponentially suppressed with respect to λ, so that (2.33) gives in fact, for this optimal value of n, the expression of a purely non-perturbative quantity.
Eq. (2.34) is valid in a wider sector (| arg λ| < π) than the usual Borel resummation which is valid for | arg λ| < π 2 . Moreover, we would like the reader to notice the similarity between the perturbative partial sum (first line of (2.34)) and the partial sum in the second line of (2.34). Indeed, choosing n = 5 and m = 5, we explicitly get from (2.34) We see that the coefficients of the second partial sum are the same as the perturbative ones (up to Euler gamma functions that we wrote explicitly to emphasize the symmetry of the formula). We chose n = 5 because, as we saw in (2.17), it is one of the two best orders for truncating the perturbative series for λ = 1 3 . Let us however underline that this interesting phenomenon is independent of the choice of n. Indeed, for example, an equivalent formula to (2.35) is where one sees that although the terms in the second line of (2.36) are not the same as those in the second line of (2.35), their coefficients are still equal to those of the perturbative contributions. This is a so-called resurgence phenomenon [4], which will also manifest itself at higher order in the non-perturbative asymptotic improvement process. It can be understood, at our present level, from the fact that the perturbative series (using the duplication formula and noting that √ 2π = Γ( 1 4 )Γ( 3 4 )) may be rewritten as which has the same coefficient A k than those appearing in the second line of (2.33). An alternative expression for Z(0) (completely equivalent to (2.34)) is obtained by inserting in (2.30) the Mellin-Barnes representation where d = Re t ∈]0, 1[. Performing the y-integral one then finds with Re (l − t) > −1, so that, at the end, Indeed, the inverse factorial that we need is where the contour of the Mellin-Barnes integral in the r.h.s. is a straight line with c a real number so that c + m > − 1 4 + p 2 and c + m < n + p, and from those of Γ(−s)Γ(k + p − s), while going from −i∞ to i∞. Taking into account these facts one finds, at the end, (2.44)

Mellin-Barnes hyperasymptotic theory: first level
As we shall see now, the result (2.34) can be entirely justified from the modern point of view of Mellin-Barnes hyperasymptotic theory [4] if one does not anymore ignore, contrary to what we did since the beginning, the integral representation of Z(0). Indeed, after having computed the asymptotic expansion of Z(0) from its Mellin-Barnes representation, it will be possible to obtain from the expansion of the Mellin-Barnes asymptotic remainder integral, the hyperasymptotic expansion of Z(0) at first hyperasymptotic level. As a by product, it will be easy to show that (2.40) is valid for a wider range of phase than the complex λ-plane with a cut on the negative real axis. In particular it will be proven to be valid for this ray (the negative real axis) on which (2.8) is not Borel summable (and which is a Stokes ray). Let us begin the calculations. Using the Mellin-Barnes representation which is valid in the semi-infinite fundamental strip defined by c = Re s ∈]0, +∞[ (the right half-complex s-plane) and for | arg λ)| < π 2 , one obtains, from (2.7), The φ integral can be computed and we find where c = Re s ∈]0, 1 4 [ so that the φ integral reduced the semi-infinite fundamental strip to a finite one. Moreover our Mellin-Barnes representation increases the λ-domain of validity 8 of Z(0) to | arg λ| < 3π 2 , therefore (2.47) consists in an analytic continuation of (2.7) on the Riemann surface of the logarithm (we recall that (2.7) is defined in the right half complex plane only, i.e. for | arg λ| < π 2 ). Now that we have this simple Mellin-Barnes representation for Z(0), it is straightforward to get its asymptotic expansion when |λ| → 0.
Indeed, for a closed rectangular contour, Cauchy's theorem gives (with n a positive integer and δ ∈]0, 3 4 The λ-domain of convergence of a Mellin-Barnes integral of the type (2.47) is given by | arg λ| < which is equivalent to where d = c + δ ∈]0, 1[. Indeed, the integrals with paths c−n+δ+i∞ c+i∞ and c−i∞ c−n+δ−i∞ are zero due to the exponential decay of the modulus of the integrand when the imaginary part of s = σ + iτ goes to infinity (τ → ±∞) and c − n + δ ≤ σ ≤ c.
This can be proven by using Stirling formula, which says that, for τ → ±∞, One therefore has, for λ = |λ|e iθ , The exponential decay is then obtained for |θ| = | arg λ| < 3π 2 . From now on we define the remainder integral of (2.49) as To show that the exact relation (2.49) (or (2.53)) is an asymptotic expansion when |λ| → 0 (valid, as we will see, for | arg λ| < 3π 2 ), one has to prove that where a > −1. Using the fact that f (z)dz ≤ |f (z)| dz, and performing the change of variable s = d − n + it, we find This proves (2.54) by definition of the O symbol with a = −d, since the integral in the r.h.s of (2.55) is independent of |λ| and converges for |θ| = | arg λ| < 3π 2 (see (2.51)). Therefore, which agrees with (2.8).
Notice that the Mellin-Barnes procedure developed in [8] to compute Feynman diagrams (see also appendices of [9] for details, as well as [10] and [11]) would have given the same result (2.56), but only for λ positive. This is due to the fact that the converse mapping theorem given in [11] has not been established for the complex λ case.
Recall that one wants to prove eq. (2.34). One starts from (2.49). Eq. (2.34) will in fact emerge from an expansion of the remainder integral R n defined in (2.52).
For practical reasons, it will be easier to perform our analysis if we rewrite (2.52) by making the change of variable t = −s and by using the duplication formula (2.23).
This gives where now d = Re t ∈] − 1, 0[. The hyperasymptotic approach developed in [4] for Mellin-Barnes integrals lies on inverse factorial expansions of the integrands. To apply it on (2.57), we use the reflexion formula π sin(πt) The quotient of Gamma functions can be expressed as the inverse factorial expansion (2.25). One therefore has This ends here the first level in the (rather simple) hyperasymptotic process. Now, this last formula is exactly (2.33) if one replaces the incomplete gamma function in the second line of (2.33) by its Mellin-Barnes representation with R n given as in (2.60). Another way to prove it is to simply recover (2.59) from (2.33). This may be done by putting m = 0 in (2.33). This is in fact also equivalent to restart from (2.24) with the m = 0 version of (2.25) which is nothing but Barnes Lemma, see Appendix B. We have therefore proven that the interpretation of the tail of the divergent perturbative expansion given by (2.33) is correct and that (2.34) constitutes the first level of the hyperasymptotic expansion of Z(0). An important remark is that, written in terms of Mellin-Barnes representations, our expansion (2.40) obtained initially for values of λ in the complex plane with a cut on the negative real axis, is now automatically valid in the wider sector | arg λ| < 3π 2 (see footnote 8) where it also gives much more precise results than (2.56).

3
. From (2.47), after performing the change of variable t = −i(s − c), one gets numerically, with an 8 decimal places precision, Now, from (2.56) which is valid for λ = − 1 3 but not Borel summable, the best prediction is 9 Z(0)| λ=− 1 3 = 1.06098837 ± 0.00140990. (2.63) so that the purely perturbative approach of course completely misses the imaginary part. In fact, imaginary contributions appear from the second sum of (2.40). Indeed, truncating the perturbative series after the fifth term and including contributions of the first two terms of the second line of (2.40) we get We see that even without taking into account the remainder integral (third line of (2.40)), whose contribution would make us fall on the exact result (2.62), we obtain a very good description of Z(0) also for λ < 0, contrary to (2.56) wich is not Borel summable for this value of the coupling constant.

Non-perturbative asymptotic improvement of perturbation theory: optimal truncation schemes
In section 2.2.1, we formally gave a meaning to the tail of the divergent perturbative expansion of Z(0) and of an arbitrary N -point function, in terms of a partial sum added to a Mellin-Barnes remainder integral. This allowed us to obtain eqs. (2.34), (2.40) and (2.44) which are in fact hyperasymptotic expansions at the first hyperasymptotic level. Our interpretation of the tail has been rigorously justified in section 2.2.2 and we saw in (2.64) that numerically the partial sum in the second line of (2.40) gives contributions that perturbation theory (2.56) does not see (in particular an imaginary part, see (2.63)), so that of course there is something beyond perturbation theory in (2.40). In this section we explain the non-perturbative nature of these results. The crucial point is that "by allowing the number of terms in an asymptotic expansion to depend on the asymptotic variable, it is possible to obtain an error term that is exponentially small as the asymptotic variable tends to its limit" [2]. In other words, there is an optimal value of the truncation index of the perturbative series for which the remainder of the expansion is a non-perturbative quantity and in our example this optimal value is obtained by minimizing the remainder with n scaling like 1 |λ| . This will define optimal truncation schemes. Let us see this in more detail.
The perturbative expansion of Z(0) leads us to (2.49) where the integral in the r.h.s. may also be written as (2.59): where d = Re t ∈] − 1, 0[. Now, let us consider the integral where c = Re t ∈] − 1, 0[ and n is a positive integer so that the path of integration in (2.66) lies to the right of the poles of Γ (t + α) Γ (t + β). Then, as n → ∞, we have, for | arg( 3 2λ )| ≤ π and ω = α + β − 1, This is the result of a lemma taken from [4], where cases dealing with more general integrals than (2.66) are also considered. One deduces from it that, since (2.65) fullfills the required condition, Now if, when |λ| is small, we choose n = a 0 |λ| + b 0 (where a 0 > 0 and b 0 is bounded), then Minimizing the remainder then implies that a 0 = 3 2 and one finally has .
(2.70) If one chooses for instance b 0 such that 3 2|λ| + b 0 = [ 3 2|λ| ] (the integer part of 3 2|λ| ), then (2.49) may be written as as |λ| → 0 in arg 3 2λ ≤ π. For this optimal value of the perturbative truncation index the remainder is therefore non-perturbative (exponentially suppressed with respect to the coupling constant |λ|). Eq. (2.71) is called the superasymptotic expansion of Z(0), or optimally truncated hyperasymptotic expansion at zeroth level. Now, the remainder (2.65) has been expanded, see (2.60), and a similar analysis may be done to deal with the remainder R n,m of this expansion. Indeed, (2.60) reads where we recall that f (s) = .
From the same lemma as before and another one that may be found in [4], one finds that when n and m become infinite (with c = d and n ≥ m) (2.74) Since we found that n = 3 2|λ| + b 0 was the optimal value for the truncation index of the perturbative series, one may keep this value for n and choose m = a 1 |λ| + b 1 (where a 1 > 0 and b 1 are bounded), so that which constitutes the optimally truncated hyperasymptotic expansion of Z(0) at first level, in the optimal truncation scheme that we call OTS1 in the following. In fact, in OTS1, eq. (2.72) constitutes an asymptotic expansion which, being extracted from the exponentially suppressed remainder R 3 2|λ| +b 0 , allows us to conclude that at the first stage of the OTS1 hyperasymptotic approach, we extracted from the perturbative remainder a (leading) non-perturbative asymptotic series.
Let us stress that at first hyperasymptotic level there exists another optimal truncation scheme that leads to even better results. Indeed, instead of keeping the superasymptotic value n (2.79) which is more exponentially suppressed than (2.76). From this, one deduces that (choosing b 0 and b 1 such that 3 which constitutes the first level of the hyperasymptotic expansion of Z(0) in the optimal truncation scheme that we call OTS2 in the following.
Comparing (2.71) and (2.80), one sees that the first level of the hyperasymptotic expansion in OTS2 therefore leads to an exponential improvement of the superasymptotic expansion. This exponential improvement is of the same order than the exponential improvement that was obtained by the superasymptotic expansion of the perturbative series. In this sense eq. (2.80) is non-perturbative.
The non-perturbative results (2.77) and (2.80) have been obtained from truncation indices n = [ a 0 |λ| ] and m = [ a 1 |λ| ] and we saw that the values of the coefficients a 0 and a 1 depend on the chosen optimal truncation scheme.
It is interesting to note that although (2.80) gives a more accurate description of Z(0) than (2.77) since the remainder in OTS2 is more exponentially suppressed than the remainder in OTS1, the number of perturbative terms retained in (2.80) is at least twice the number of those retained in (2.77), which means that taking into account perturbative terms that make the perturbative series begin to diverge from the exact result leads to an improvement 10 !

hyperasymptotics at higher levels and non-perturbative asymptotic refinements
Our starting point to get the hyperasymptotic expansion of Z(0) at second level is the battle-horse formula (2.40) obtained by the formal strategy developed in section 2.2.1 and which has been rigorously demonstrated in section 2.2.2, namely Noticing that where h + m ′ < Re s and h ∈]0, 1[, the final result reads The constraint h + m ′ < Re s implies h + m ′ < c + m in (2.84).
We have now three interwoven partial sums where the third one is expressed in terms of so-called hyperterminants [4] (here defined as double Mellin-Barnes integrals).
For the N -point function case, we have, at second hyperasymptotic level 11 , The resurgence phenomenon is clearly apparent in (2.84) since the coefficient A k appears at each hyperasymptotic order. One may however notice that for the N -point functions case, resurgence manifests itself in a different way: the coefficients B This difference comes from the fact that for Z(0) the function f (s) in the l.h.s. of (2.83) appears also under the integral sign in the r.h.s., whereas for the N -point functions the function g p (s) in the l.h.s. of (2.86) appears as g −p (s) under the integral sign in the r.h.s.
Thanks to these interesting symmetries that are reflected by the resurgence phenomenon, one may deduce that in our cases of study, hyperasymptotic expansions at arbitrary hyperasymptotic levels may be obtained in a straightforward way.
Let us now discuss optimal truncation schemes at second hyperasymptotic level. In OTS1, one finds that the optimal values for the truncation indices are n = [

Numerical analysis
In this section we want to compare numerical results that may be obtained from the hyperasymptotic expansion of Z(0) at various hyperasymptotic levels, in both cases of optimal truncation schemes OTS1 and OTS2, with those obtained from perturbation theory in section 2.1.3, i.e. for λ = 1 3 . Recall that from (2.7) one gets, with an 8 decimal places precision, the "exact" result

Third hyperasymptotic level
At this level (for which we do not give the expression of the corresponding hyperasymptotic expansion of Z(0)), one has no prediction from OTS1, since it would imply m ′′ = [ 3 16|λ| ] = 0 (where m ′′ is the truncation index of the partial sum appearing at third hyperasymptotic level).
In OTS2, however, one has n = [ which gives the exact result with an 8 decimal places precision, although we see that n = 18, which means that the perturbative partial sum contribution to (2.93) is (see Table 1) S Pert 17 = −27.6964871, (2.94) a value extremely far from the exact result (2.87)! It is clear from this analysis that OTS2 is a much better optimal truncation scheme than OTS1 and that the results obtained in OTS2 give much closer value to the exact result than what perturbation theory does.

Conclusions
This paper was aimed to show how an interpretative procedure of divergent perturbative series, involving Mellin-Barnes representation and Borel resummation, leads to a nonperturbative asymptotic improvement of the N -point functions perturbative expansions, for the simple case of zero-dimensional euclidean φ 4 theory, by the appearance of hyperasymptotic expansions. We saw that these hyperasymptotic expansions are composed of interwoven partial sums whose coefficients, in our cases of study, are linked together by a resurgence phenomenon. The non-perturbative interpretation of our results relies crucially on so-called optimal truncation schemes of these partial sums. A numerical analysis has been performed, showing the much more accurate results that one may obtain with the non-perturbative hyperasymptotic expansions, compared to the traditional perturbative approach. We also saw that on the Stokes line λ < 0, one may find, from hyperasymptotic expansions, imaginary contributions that are not obtainable from the truncated perturbative expansions (which are non-Borel summable for these particular values of the coupling constant).
One of the important conclusions concerns the two different optimal truncation schemes that were studied in this paper. Indeed, the best one (i.e. the one that leads to the best predictions) implies a truncation of the perturbative series that depends on the level at which the hyperasymptotic improvement is performed. This leads to the striking result that the corresponding (best) predictions, which are of an amazing accuracy (see eq. (2.93) compared to the exact result (2.87)), involve perturbative contributions that, taken independently of the non-perturbative corrections, would lead to desastrous results (see eq. (2.94)). This is due to the fact that the more one wants to increase the exponential improvement (i.e. the more one wants to include non-perturbative contributions in the analysis), the more one also has to include perturbative contributions far in the divergent tail of the perturbative series.
These results clearly show that hyperasymptotic expansions are tools that could have a lot of applications in high energy physics. In this respect, concerning the study of Stokes phenomenon (and the smoothing of the Borel ambiguity), we would like to mention an interesting paper [12] which appeared a few days before the end of the writting of our manuscript and dealing, among others things, with a hyperasymptotic approach of instantons in topological string theory and c = 1 matrix models. It is however not based on Mellin-Barnes hyperasymptotic theory but rather on the hyperasymptotic approach developped by Berry and Howls [13].
It is clear that one could also find this result by starting the resummation of the tail (2.24) with Barnes lemma instead of the inverse factorial expansion (2.25) (since we see that putting m = 0 in (2.25), which is nothing but Barnes lemma (B.1), is equivalent to have m = 0 in (2.33)).