Resurgence in sine-Gordon quantum mechanics: Exact agreement between multi-instantons and uniform WKB

# Resurgence in sine-Gordon quantum mechanics: Exact agreement between multi-instantons and uniform WKB

## Abstract

We compute multi-instanton amplitudes in the sine-Gordon quantum mechanics (periodic cosine potential) by integrating out quasi-moduli parameters corresponding to separations of instantons and anti-instantons. We propose an extension of Bogomolnyi–Zinn-Justin prescription for multi-instanton configurations and an appropriate subtraction scheme. We obtain the multi-instanton contributions to the energy eigenvalue of the lowest band at the zeroth order of the coupling constant. For the configurations with only instantons (anti-instantons), we obtain unambiguous results. For those with both instantons and anti-instantons, we obtain results with imaginary parts, which depend on the path of analytic continuation. We show that the imaginary parts of the multi-instanton amplitudes precisely cancel the imaginary parts of the Borel resummation of the perturbation series, and verify that our results completely agree with those based on the uniform-WKB calculations, thus confirming the resurgence : divergent perturbation series combined with the nonperturbative multi-instanton contributions conspire to give unambiguous results. We also study the neutral bion contributions in the model on with a small circumference, taking account of the relative phase moduli between the fractional instanton and anti-instanton. We find that the sign of the interaction potential depends on the relative phase moduli, and that both the real and imaginary parts resulting from quasi-moduli integral of the neutral bion get quantitative corrections compared to the sine-Gordon quantum mechanics.

## I Introduction

In the recent study on quantum field theories and quantum mechanics, topologically neutral soliton molecules, which are locally composed of (fractional) instantons and anti-instantons, have been attracting a great deal of attention in relation to the IR-renormalonUnsal:2007vu (); Unsal:2007jx (); Shifman:2008ja (); Poppitz:2009uq (); Anber:2011de (); Poppitz:2012sw (); Argyres:2012vv (); Argyres:2012ka (); Dunne:2012ae (); Dunne:2012zk (); Dabrowski:2013kba (); Dunne:2013ada (); Cherman:2013yfa (); Basar:2013eka (); Dunne:2014bca (); Cherman:2014ofa (); Behtash:2015kna (); Bolognesi:2013tya (); Misumi:2014jua (); Misumi:2014raa (); Misumi:2014bsa (); Nitta:2015tua (); Nitta:2014vpa (); Shermer:2014wxa (); Dunne:2015ywa (). Imaginary ambiguities arising in amplitudes of such topologically neutral configurations can cancel out those arising in non-Borel-summable perturbative series (IR-renormalon) in quantum theories under certain conditions on the spacetime manifold Argyres:2012vv (); Argyres:2012ka (); Dunne:2012ae (); Dunne:2012zk (); Dabrowski:2013kba (); Dunne:2013ada (); Cherman:2013yfa (); Basar:2013eka (); Dunne:2014bca (); Cherman:2014ofa (); Bolognesi:2013tya (); Misumi:2014jua (); Misumi:2014raa (); Misumi:2014bsa (); ’tHooft:1977am (); Fateev:1994ai (); Fateev:1994dp (). In field theories on compactified spacetime with a small compact dimension, these objects are termed as “bions” Argyres:2012vv (); Argyres:2012ka (); Dunne:2012ae (); Dunne:2012zk (). It is expected that full semi-classical expansion including perturbative and non-perturbative sectors as bions, which is called “resurgent” expansion Ec1 (); Marino:2007te (); Marino:2008ya (); Marino:2008vx (); Pasquetti:2009jg (); Drukker:2010nc (); Aniceto:2011nu (); Marino:2012zq (); Hatsuda:2013gj (); Schiappa:2013opa (); Hatsuda:2013oxa (); Aniceto:2013fka (); Santamaria:2013rua (); Kallen:2013qla (); Honda:2014ica (); Grassi:2014cla (); Sauzin (); Kallen:2014lsa (); Couso-Santamaria:2014iia (); Honda:2014bza (); Aniceto:2014hoa (); Couso-Santamaria:2015wga (); Honda:2015ewa (); Hatsuda:2015owa (); Aniceto:2015rua (); Dorigoni:2015dha (), leads to unambiguous and self-consistent definition of field theories in the same manner as the conjecture in quantum mechanics Bogomolny:1980ur (); ZinnJustin:1981dx (); ZinnJustin:1982td (); ZinnJustin:1983nr (); ZinnJustin:2004ib (); ZinnJustin:2004cg (); Jentschura:2010zza ().

The resurgence in theoretical physics was at first investigated in the matrix model and topological string theory Marino:2007te (); Marino:2008ya (); Marino:2008vx (); Pasquetti:2009jg (); Marino:2012zq (); Hatsuda:2013oxa (). Then, the study on the topic has been extended to ABJM theory Drukker:2010nc (); Hatsuda:2013gj (); Honda:2014ica (); Kallen:2014lsa (), string and supersymmetric gauge theories Aniceto:2011nu (); Grassi:2014cla (); Couso-Santamaria:2014iia (); Aniceto:2014hoa (), and general quantum systems Schiappa:2013opa (); Aniceto:2013fka (); Santamaria:2013rua (); Kallen:2013qla (); Sauzin (); Honda:2014bza (); Couso-Santamaria:2015wga (); Honda:2015ewa (); Hatsuda:2015owa (); Aniceto:2015rua (); Dorigoni:2015dha (). Bions and resurgence in non-SUSY field theories, especially in the low-dimensional models, have been extensively investigated for the model Dunne:2012ae (); Dunne:2012zk (); Dabrowski:2013kba (); Bolognesi:2013tya (); Misumi:2014jua (); Shermer:2014wxa (), the Grassmann sigma model Misumi:2014bsa (); Dunne:2015ywa (), the principal chiral model Cherman:2013yfa (); Cherman:2014ofa (); Nitta:2015tua (), and the model Nitta:2014vpa (); Dunne:2015ywa (). According to these studies, the leading-order renormalon ambiguity arising in non-Borel-summable perturbative series, which corresponds to the singularity closest to the origin on the Borel plane, is compensated by the amplitude of neutral bions. On the other hand, it is expected but not verified that the ambiguities corresponding to singularities further from the origin (, ,…) are cancelled by amplitudes of bion molecules with more than four instanton constituents.

In the case of quantum mechanics, not only the sector of zero instanton charge but also those of nonzero instanton charge contribute to physical observables such as the energy levels. The authors in Refs. ZinnJustin:1981dx (); ZinnJustin:1982td (); ZinnJustin:1983nr (); ZinnJustin:2004ib (); ZinnJustin:2004cg (); Jentschura:2010zza () investigated quantum mechanics with several types of potential including the sine-Gordon type. They showed that the leading instanton contributions are consistent with the perturbative calculation, and conjectured the explicit equation connecting perturbative and instanton contributions, which they call the generalized quantization condition. Recently the authors in Refs. Dunne:2013ada (); Dunne:2014bca () adopted the uniform-WKB method based on the boundary condition, which is equivalent to the quantization condition in Refs. ZinnJustin:1981dx (); ZinnJustin:1982td (); ZinnJustin:1983nr (); ZinnJustin:2004ib (); ZinnJustin:2004cg (); Jentschura:2010zza (), and pointed out the general relation between perturbative and non-perturbative contributions. Explicit calculations of multi-instanton amplitudes at each configuration level are expected to clarify the structure of resurgence and to verify the conjectured relation between perturbative and non-perturbative contributions ZinnJustin:1981dx (); ZinnJustin:1982td (); ZinnJustin:1983nr (); ZinnJustin:2004ib (); ZinnJustin:2004cg (); Jentschura:2010zza (); Escobar-Ruiz:2015nsa (); Dunne:2013ada (); Dunne:2014bca ().

In this paper, we focus on a quantum mechanical system with the sine-Gordon potential, and we calculate the multi-instanton amplitude by explicitly integrating quasi moduli parameters corresponding to separations of instanton-constituents in a semi-classical limit, in comparison with the uniform WKB calculations Dunne:2013ada (); Dunne:2014bca (); ZinnJustin:2004ib (); ZinnJustin:2004cg (). We adopt an extension of Bogomolnyi–Zinn-Justin prescription Bogomolny:1980ur (); ZinnJustin:1981dx () for multi-instanton configurations with an appropriate subtraction scheme for divergent parts. We calculate contributions to the energy eigenvalue of the lowest band from each multi-instanton configuration in a semi-classical limit (). For the configurations with only instantons (anti-instantons) such as , and , we have unambiguous results without imaginary parts. Here, we have denoted an instanton (anti-instanton) as (). For configurations containing both instantons and anti-instantons such as , and , the results contain ambiguous imaginary parts, which depend on the path of analytic continuation. These imaginary parts correspond to the large-order behavior of perturbation series around the saddle point without the pair. For instance, we show explicitly that the imaginary part of the multi-instanton amplitude cancels the imaginary part of the Borel resummation of the large-order perturbation series around the nontrivial background with a single instanton . By investigating the uniform-WKB calculations in detail, we verify that all of our results agree completely with those based on the uniform-WKB calculations up to a four-instanton order.

While the sine-Gordon quantum mechanics is worth to study on its own, another strong motivation lies in its close relationship to small circumference limit of the two-dimensional model on (circumference ) with the -symmetric twisted boundary condition Dunne:2012ae (); Dunne:2012zk (). However, we observe that some of the field configurations of the model are not faithfully represented by means of the sine-Gordon quantum mechanics. One can derive the sine-Gordon quantum mechanics from the two-dimensional model by applying the Scherk-Schwarz dimensional reduction, which requires a particular dependence of the phases of fields on the coordinate of compactified dimension (). It is important to realize that only parts of field configuration of model can be consistent with this dependence. For instance, the BPS solution of two fractional instantons is not consistent with the Scherk-Schwarz reduction, and hence its small circumference limit cannot be described by the sine-Gordon quantum mechanics. On the other hand, two adjacent instantons in the sine-Gordon quantum mechanics are mutually non-BPS, although each individual instanton may be understood as a limit of BPS fractional instanton (with a different dependence). Even in the instanton and anti-instanton configurations, model has a significant difference compared to the sine-Gordon quantum mechanics: The phase moduli of the fractional instantons in the model are neglected in the sine-Gordon quantum mechanics. For the configuration of a neutral bion composed of a fractional instanton and an anti-fractional instanton, we find that the interaction between them strongly depends on the relative phase of constituents. We calculate the neutral bion contribution in the model, based on the interaction potential with the quasi moduli parameter corresponding to the relative phase between the fractional instanton and anti-instanton. We find that this calculation gives a correction factor compared to the neutral bion amplitude obtained in the sine-Gordon quantum mechanics Dunne:2012ae (); Dunne:2012zk ().

This paper is organized as follows. In Sec. II, we review instantons and their interactions in the quantum mechanics with sine-Gordon potential and the Borel summation. In Sec. III we calculate amplitudes of multi-instanton configurations in sine-Gordon quantum mechanics by integrating out the moduli parameters. In Sec. IV we discuss the results from the uniform WKB calculations, and show that they completely agree with the instanton moduli calculations. In Sec. V we discuss the neutral bion contributions in the compactified model based on the interaction potential including the relative phase parameter. Section VI is devoted to a summary and discussion. In Appendix A we give some details of four-instanton calculations.

## Ii Quantum mechanics with the sine-Gordon potential

In this article, we focus on the sine-Gordon quantum mechanics described by the Schrödinger equation

 Hψ(x)=−12d2dx2ψ(x)+18g2sin2(2gx)ψ(x)=Eψ(x), (1)

where we follow the notation in Refs. ZinnJustin:2004ib (); Dunne:2013ada () except is replaced by here 1. The Euclidian Lagrangian for the sine-Gordon quantum mechanics is given by 2

 L=12(dxdt)2+V(x),V(x)=18g2sin2(2gx). (2)

In the limit, it reduces to the Schrödinger equation of the harmonic oscillator.

The energy eigenvalues of periodic potentials split into bands of states. Within each band, they are labeled by the Bloch angle defined by

 ψ(x+π2g)=eiθψ(x). (3)

In this article, we are interested in the lowest band, although excited bands can be treated similarly. The energy eigenvalue of the lowest band can be expressed in terms of the path-integral

 E=limβ→∞−1βTre−βH=limβ→∞−1β∫x(t=−β/2)=x(t=β/2)Dx(t)e−S+iQθ. (4)

For weak coupling, the path-integral has contributions around the perturbative vacuum, as well as contributions from nonperturbative saddle points

 E=Epert(g2)+△E. (5)

Perturbation series in powers of coupling constant in quantum field theories or in quantum mechanics are extremely useful, but are usually factorially divergent

 Epert(g2)=∞∑K=0aK(g2)K,aK∼K!. (6)

It is useful to define the Borel transform

 Bpert(t)=∞∑K=0aKK!tK. (7)

The Borel resummation of the divergent series is defined as an integral of the Borel transform along the positive real axis in the complex Borel plane

 Epert(g2)=∫∞0dte−tBpert(g2t). (8)

If the factorially divergent series is alternating, the Borel transform has no singularities along the positive real axis, and the Borel resummation becomes well-defined (Borel-summable). For the potential with degenerate minima such as the sine-Gordon quantum mechanics, however, the perturbation series is non-alternating factorially divergent. In that case, the Borel transform is convergent with the finite radius of convergence, but the Borel resummation is ill-defined because of singularities in the complex Borel plane. Since the series become alternating and the Borel resummation is unambiguous for , we can analytically continue it from to the physical region to obtain a real analytic function . If there is no complex singularities, we obtain a branch cut along the positive real axis of complex plane. The imaginary part at is related to the large-order behavior () of perturbation series in Eq.(6) through the dispersion relation ZinnJustin:1989mi ()

 aK≈−1π∫∞0dg2[ImEpert(g2)](g2)K+1. (9)

This large-order behavior corresponds to the singularities of the Borel transform in the complex Borel plane . Of course this ambiguous (path-dependent) imaginary part is unacceptable, and should disappear, since the energy eigenvalue should be real, and ambiguity due to the choice of path is unphysical. In fact, it has been found that the leading term of the imaginary ambiguities is cancelled by the contributions from non-perturbative saddle points associated with neutral objects composed of instantons Dunne:2013ada (); Dunne:2014bca (). This phenomenon is called the resurgence of the perturbation series.

Let us now consider non-perturbative saddle points. By rescaling the variable

 2gx=y, (10)

the Euclidean Lagrangian in Eq.(2) can be rewritten as

 L=18g2(dydt)2+V,V=18g2sin2(y). (11)

and the instanton number as a topological charge may be defined by

 Q=1π∫∞−∞dtdydt. (12)

Single instanton solution () is given by3

 yI(t)=2arctanet−t0+nπ,n∈Z, (13)

whereas single anti-instanton solution () is given by

 y¯I(t)=2arctane−(t−t0)+(n−1)π,n∈Z, (14)

with the Euclidean action

 SI=12g2. (15)

The moduli parameter is a zero mode (moduli) associated to the breakdown of translation, representing the location of the (anti-)instanton. For even , the solutions (13) and (14) satisfy the following BPS equation4 saturating the BPS bound for

 dydt=siny,S=∫∞−∞dtdydtsiny. (16)

For odd , they satisfy the anti-BPS equation saturating the anti-BPS bound for

 dydt=−siny,S=−∫∞−∞dtdydtsiny. (17)

By integrating over the translational zero mode , one finds the contribution of single instanton to the energy as

 △E(1,0)=−[I]=−(e−SI√πg2)eiθ. (18)

Suppose, for instance, we have a BPS instanton in Eq.(13) with and wish to place another instanton or anti-instanton to its right, we are forced to take either instanton with in Eq.(13) or anti-instanton with in Eq.(14), both of which are anti-BPS configurations. Therefore two successive (anti-)instantons are inevitably non-BPS. The energy of the non-BPS configuration of two successive instantons should be more than the sum of individual instanton energies. They are found to repel each other with the potential Manton:2004tk () for large separations

 VII(R)=2g2exp[−R]. (19)

The non-BPS configuration of successive instanton and anti-instantons are found to attract each other with the potential for large separations

 VI¯I(R)=−2g2exp[−R]. (20)

For later convenience, we introduce the uniform-WKB ansatz by following Ref. Dunne:2013ada (). With the coordinate variable in Eq.(10), Eq.(1) can be rewritten as

 −g4d2dy2ψ(y)+116sin2(y)ψ(y)=g22Eψ(y). (21)

We define the potential as . By using the parabolic cylinder function satisfying the differential equation

 (22)

we introduce an ansatz for the wave functionalvarez (); langer (); cherry (); millergood (); galindo ()

 ψ(y)=Dν(u(y)/g)√u′(y), (23)

where the parameter is the shift of energy eigenvalue from the ground state energy of the harmonic oscillator ( limit). Then the Schrödinger equation (21) becomes

 U(y)−14u2(u′)2−g2E2+g2(ν+12)(u′)2+g42√u′(u′′(u′)3/2)′=0, (24)

with . In the limit, Eq.(24) just reduces to , whose solution is

 u0(y)2=4∫y0√Udy=2sin2y2→u0(y)=√2siny2, (25)

which gives the zeroth-order argument of the parabolic cylinder function in Eq.(23) and solves the Schrödinger equation of the harmonic oscillator.

## Iii Multi-instanton amplitudes in Sine-Gordon quantum mechanics

### iii.1 General setting

In this section we calculate multi-instanton amplitudes in sine-Gordon quantum mechanics. We need to integrate over the distances between instantons and (anti-)instantons as quasi-moduli. Since the interaction (19) and (20) between instantons and (anti-)instantons vanish at large distances, we need to regulate the integral by introducing a factor into the effective potential

 V[R]=±2g2exp(−R)+ϵR, (26)

where is for the instanton-instanton repulsive interaction and for the instanton–anti-instanton attractive interaction. The regularization parameter can be identified as the number of fictitious fermionsBogomolny:1980ur (); ZinnJustin:1981dx (); ZinnJustin:1982td (); Jentschura:2010zza (); Jentschura:2011zza (); Dunne:2012ae (). After subtracting divergences, we need to take the limit .

Even after eliminating the divergence arising from large separations (), we have another source of divergence for the case of the attractive instanton–anti-instanton interaction: the integrand becomes divergent as , contrary to the repulsive case. Therefore the moduli-integral gets divergent contributions from small regions, and is ill-defined in the semi-classical limit (). This is why we need to introduce the Bogomolnyi–Zinn-Justin (BZJ) prescription Bogomolny:1980ur (); ZinnJustin:1981dx (): We first regard as real positive () to make the integral well-defined in the semi-classical limit, and then we analytically continue back to in the complex plane at the end of the calculation.

The energy eigenvalue of the lowest band has contributions from the amplitude of -instanton and -anti-instanton configuration as

 △E(n,m)=−[II⋅⋅⋅¯I¯I]all, (27)
 [II⋅⋅⋅¯I¯I]all=(e−SI√πg2)n+mei(n−m)θ∫dR1dR2...dRn+m−1e−V[R1]−V[R2]−⋅⋅⋅V[Rn+m−1], (28)

where stands for the sum of configurations which can be composed of instantons and anti-instantons in all possible orderings. As shown in Ref. ZinnJustin:2004ib (), the contribution contains with being the instanton charge since the Bloch angle shows up in a topological term in the Euclidian action in Eq.(4). We perform the quasi-moduli integral taking only interactions between neighboring instantons among the () instantons. We should perform this multi-integral in the semi-classical region , and subtract the divergent parts appropriately at each level of the multi-integral. We will evaluate them explicitly from the next subsection.

### iii.2 2 instantons

The amplitude of two instantons shown in Fig. 1 is obtained as

 [II]e−2iθξ−2=∫∞0dRexp(−2g2e−R−ϵR) =(g22)ϵ∫2/g20dse−ssϵ−1 |g2|≪1⟶(g22)ϵΓ(ϵ) =−(γ+log2g2)+O(1ϵ)+O(ϵ), (29)

where is the Euler constant and is an instanton factor defined by

 ξ≡e−SI/√πg2=e−1/(2g2)/√πg2. (30)

Here we have neglected terms of order or higher. To simplify the formula, we divide the amplitude by and . Precisely speaking, the interaction energy between instantons at small separation may not be precisely represented by the potential in Eq.(26). However, our result is unchanged as long as is satisfied. We need to subtract the divergent term while the term disappears in the limit. The contribution from this amplitude to the energy eigenvalue of the lowest band is then given by

 △E(2,0)=e2iθξ2(γ+log2g2), (31)

where the superscript stands for two-instanton and zero–anti-instanton amplitude. We note that the contribution from the two anti-instanton amplitude is obtained by replacing by .

### iii.3 1 instanton + 1 anti-instanton

The amplitude of one instanton and one anti-instanton amplitude is composed of two configurations and , as shown in Fig. 2. In these cases, the interaction between the two constituents is attractive, and the quasi moduli integral is ill-defined. Therefore we introduce the Bogomolnyi–Zinn-Justin (BZJ) prescription Bogomolny:1980ur (); ZinnJustin:1981dx (): we first evaluate the integral by taking , and then we analytically continue the result from back to in the complex plane. This procedure provides the imaginary ambiguity depending on the path of the analytic continuation as .

The amplitude of one-instanton and one anti-instanton configuration corresponding to the left of Fig. 2 is obtained as

 [I¯I]ξ−2 =∫∞0dRexp(−2−g2e−R−ϵR)|g2|≪1⟶(−g22)ϵΓ(ϵ) −g2=e∓iπg2⟶−(γ+log2e∓iπg2)+O(1ϵ)+O(ϵ) =−(γ+log2g2)∓iπ+O(1ϵ)+O(ϵ), (32)

where we perform the integral in the first line by considering , and in the second line analytically continue back to in the complex plane Bogomolny:1980ur (); ZinnJustin:1981dx (). The third line shows a two-fold ambiguous expression of depending on the path of analytic continuation as . As with the two-instanton case, we have subtracted the divergent part while the term disappears in the limit.

Another amplitude of one-instanton and one anti-instanton configurations corresponding to the right of Fig. 2 turns out to give identical contribution as that in Eq.(32). The total contribution is given by the sum of them with dropping and terms,

 ([I¯I]+[¯II])ξ−2=−2(γ+log2g2)∓2iπ. (33)

Its contribution to the energy eigenvalue of the lowest band is then given by

 △E(1,1)=ξ2[2(γ+log2g2)±2iπ]. (34)

If the resurgence idea is valid, this imaginary ambiguity should cancel the imaginary ambiguity of the non-Borel summable divergent series of the perturbative contribution.

 Im[△E(1,1)]+Im[Epert]=0. (35)

If we insert Eqs.(34) and (35) into the dispersion relation in Eq.(9), we should be able to reproduce the large-order behavior of the perturbation series

 ak =−1π∫∞0dg2Im[Epert(g2)](g2)k+1=1π∫∞0dg2Im[△E(1,1)](g2)k+1 =−1π∫∞0d(g2)2e−1/g2(g2)k+2=−2πk!(k≥2), (36)

in accordance with the leading large-order behavior of the perturbation series ZinnJustin:1981dx (). Thus we find that the imaginary ambiguity of the instanton–anti-instanton amplitude correctly cancels the imaginary ambiguity of the Borel resummation of the (non-Borel summable) perturbation series.

### iii.4 3 instantons

For the three-instanton amplitude shown in Fig. 3, we have two quasi modulus corresponding to the separations between adjacent instantons. For multiple moduli integral of each given configuration, we need to specify subtraction scheme explicitly, and propose the following:

1. Enumerate possible ordering of quasi moduli integrations, such as and for the three instanton case.

2. Subtract possible poles like for the first integration, and then perform the next integration successively, and retain the finite piece.

3. Average the results of all possible orderings.

Incorporating the repulsive potentials between adjacent instantons, we obtain the three instanton amplitude as5

 [III]e−3iθξ−3 =∫∞0dR1∫∞0dR2exp[−2g2(e−R1+e−R2)−ϵ(R1+R2)] |g2|≪1⟶(g22)ϵΓ(ϵ)[(g22)ϵΓ(ϵ)−1ϵ] =32(γ+log2g2)2+π212+O(1ϵ)+O(ϵ). (37)

In the three instanton case, we have two possible orderings of the multi integral, each of which gives the identical contribution. Then the three instanton contribution to the energy eigenvalue of the lowest band is given by

 △E(3,0)=−e3iθξ3[32(γ+log2g2)2+π212]. (38)

We note that the contribution from the three anti-instanton amplitude is obtained by replacing by .

### iii.5 2 instantons + 1 anti-instanton

The two-instanton and one–anti-instanton amplitudes consist of three types of configurations, as shown in Fig. 4.

The first one is , where the anti-instanton is sandwiched between two instantons. For this type of configuration, adjacent constituent (anti-)instantons attract each other. Therefore we first take in order to apply the BZJ prescription to the integral, and follow our subtraction prescription

 [I¯II]e−iθξ−3 =∫∞0dR1dR2exp[−2−g2(e−R1+e−R2)−ϵ(R1+R2)] |g2|≪1⟶(−g22)ϵΓ(ϵ)[(−g22)ϵΓ(ϵ)−1ϵ] −g2=g2e∓iπ⟶32(γ+log2g2)2−17π212±3iπ(γ+log2g2)+O(1ϵ)+O(ϵ), (39)

where we again subtract in the first integral, before integrating the second quasi-moduli.

The second type of configuration is , where one pair of constituents is repulsive and the other attractive. In order to apply the BZJ prescription only to the attractive part of the interaction, we temporarily distinguish two coupling constants for the repulsive interaction and for the attractive interaction, and apply the BZJ prescription to , but not to . After analytic continuation, we identify as the original at the end of the calculation. The moduli-integral of is given by

 [II¯I]e−iθξ−3=∫∞0dR1dR2exp[−2g2e−R1−2−~g2e−R2−ϵ(R1+R2)]. (40)

For this configuration , we have two possible orderings of moduli integral. The first ordering is to integrate over and to subtract the pole, before integrating over . We call this ordering as

 F1(g2) |g2|,|~g2|≪1⟶(g22)ϵΓ(ϵ)[(−~g22)ϵΓ(ϵ)−1ϵ] −~g2=~g2e∓iπ⟶32(γ+log2g2)2−5π212±2iπ(γ+log2g2)+O(1ϵ)+O(ϵ). (41)

We call the result of another ordering as , which is given by integrating over first and then over

 F2(g2) |g2|,|~g2|≪1⟶(−~g22)ϵΓ(ϵ)[(g22)ϵΓ(ϵ)−1ϵ] −~g2=~g2e∓iπ⟶32(γ+log2g2)2+π212±iπ(γ+log2g2)+O(1ϵ)+O(ϵ). (42)

In the final expressions for , we implicitly put back to after analytic continuation. We obtain the amplitude of as an average of and with dropping and terms

 [II¯I]e−iθξ−3 =(F1+F2)/2 =32(γ+log2g2)2−π26±32iπ(γ+log2g2). (43)

The third type of configuration is , which gives the identical result as the second type in Eq.(43) : .

By taking the sum of all three types of configurations, we end up with

 (44)

Its contribution to the energy eigenvalue of the lowest band is then given by

 △E(2,1)=−92eiθξ3[(γ+log2g2)2−7π218±43iπ(γ+log2g2)]. (45)

We note that the two–anti-instanton and one-instanton contribution is obtained by replacing by .

Resurgence implies that the two–anti-instanton and one-instanton amplitude should correspond to the large-order behavior of the perturbation series around the one-instanton saddle point. Using the cancellation between imaginary ambiguities of Borel resummed perturbation series around one-instanton saddle point and of the two–anti-instanton and one-instanton amplitude, the large-order behavior of the perturbation series around the one instanton saddle point can be estimated from the imaginary part of (45) by means of the dispersion relation as

 ak ≈1π∫∞0dg2Im[△E(2,1)e−iθ/ξ](g2)k+1=6π∫∞0dg2e−1/g2(γ+log(2/g2))(g2)k+2 =6πk!(log2+s(k+1,2)k!)(k≥2), (46)

where is the Stirling number of the first kind, which is the solution of the recurrence relation . The first few numbers are given as for . In Ref. ZinnJustin:1981dx (), the large-order perturbative series around one instanton is numerically calculated as

 apert.k=6πk!(γ+log2k)+O(logkk), (47)

which is consistent with Eq. (46) for large .

### iii.6 4 instantons

For the four-instanton amplitude, the interaction between each pair of instantons is repulsive, as shown in Fig. 5. Since all the possible orderings of multi-moduli integral has the identical contribution, we easily obtain the amplitude using the polygamma function defined as

 [IIII]e−4iθξ−4 =∫∞0dR1dR2dR3exp[−2g2(e−R1+e−R2+e−R3)−ϵ(R1+R2+R3)] |g2|≪1⟶(g22)ϵΓ(ϵ)[(g22)ϵΓ(ϵ){(g22)ϵΓ(ϵ)−1ϵ}+γ+log(2/g2)ϵ] =−83[(γ+log2g2)3+π28(γ+log2g2)−116ψ(2)(1)]+O(1ϵ)+O(ϵ), (48)

where we have first subtracted in the first integral , and subtracted in the next integral , and finally dropped the term in the integral . Its contribution to the eigenvalue of the lowest band is then given by

 △E(4,0)=83e4iθξ4[(γ+log2g2)3+π28(γ+log2g2)−116ψ(2)(1)]. (49)

The four anti-instanton contribution is obtained by replacing by .

### iii.7 3 instantons + 1 anti-instanton

The three-instanton and one–anti-instanton amplitudes consist of four types of configurations , , and , as shown in Fig. 6.

In the first configuration , two pairs of adjacent instantons have repulsive interactions, and the other is attractive. We again apply the BZJ prescription only to the integral for the pair with the attractive interaction, where we denote the coupling as , evaluate the integral at , and analytically continue to , while keeping for the repulsive interactions. The multi-moduli integral is given by

 [III¯I]e2iθξ4=∫∞0dR1dR2dR3exp[−2g2e−R1−2g2e−R2−2−~g2e−R3−ϵ(R1+R2+R3)]. (50)

Here we have three orderings of the multi-integral, distinguished by the ordering of (attractive interaction) relative to . We denote the results , , and as the first, the second, and the third integral, respectively. Referring the calculations given in Appendix A, we just show the result

 G1(g2) =−83(γ+log2g2)3+76π2(γ+log2g2)+16ψ(2)(1)∓4iπ(γ+log2g2)2, G2(g2) =−83(γ+log2g2)3+16π2(γ+log2g2)+16ψ(2)(1)∓iπ[52(γ+log2g2)2+π212], G3(g2) =−83(γ+log2g2)3−13π2(γ+log2g2)+16ψ(2)(1)∓iπ[32(γ+log2g2)2+π212]. (51)

We obtain the amplitude of as the average of , and ,

 [III¯I]e2iθξ4 =(G1(g2)+G2(g2)+G3(g2))/3