Resurgence of the dressing phase for ${\rm AdS}_5\times {\rm S}^5$

We discuss the resummation of the strong coupling asymptotic expansion of the dressing phase of the ${\rm AdS}_5\times {\rm S}^5$ superstring. The dressing phase proposed by Beisert, Eden and Staudacher can be recovered from a modified Borel-Ecalle resummation of this asymptotic expansion only by completing it with new, non-perturbative and exponentially suppressed terms that can be organized into different sectors labelled by an instanton-like number. We compute the contribution to the dressing phase coming from the sum over all the instanton sectors and show that it satisfies the homogeneous crossing symmetry equation. We comment on the semiclassical origin of the non-perturbative terms from the world-sheet theory point of view even though their precise explanation remains still quite mysterious.


Introduction
In recent years a lot of progress has been achieved in the spectral problem of the gaugestring correspondence by using ideas and methods from the theory of integrable models [1,2]. For strings on AdS 5 × S 5 the corresponding light-cone sigma model is quantum integrable which allows one to obtain its spectrum by means of Thermodynamic Bethe Ansatz (TBA) [3]- [7] or the modern incarnation of the latter known as Quantum Spectral Curve [8].
We recall that the construction of the TBA is essentially based on the asymptotic S-matrix for scattering of string world-sheet excitations in the uniform light-cone gauge. This S-matrix is determined by symmetries of the AdS 5 × S 5 light-cone sigma model up to an overall scalar factor called the dressing factor [9]. Thus, determination of the latter quantity and investigation of its properties constitutes an important part of the spectral problem to which many studies has been devoted in the recent past. In this paper we will undertake an effort to complete the existing considerations and clarify some issues related to a perturbative expansion of the dressing factor (phase) at strong coupling.
Before we pass to the discussion of our approach, we briefly recall what is known about the dressing factor σ = e iθ , where θ is the dressing phase. The functional form of σ as a perturbative power series in the inverse string tension g with coefficients written in terms of local conserved charges was conjectured in [9] by discretising equations that encode finitegap solutions of the classical string sigma model. Since g is related to the 't Hooft coupling λ as g = √ λ/2π, from the point of view of gauge theory this power series represents a strong coupling expansion of σ. Further, the asymptotic S-matrix appears to be compatible with crossing symmetry which implies a non-trivial functional equation for the dressing factor -the crossing equation [10]. The found leading (AFS) [9] and sub-leading (HL) [11] terms in the strong coupling expansion of σ were shown to satisfy the crossing equation [12] and an all-order asymptotic solution of the latter was obtained in [13]. The weak coupling expansion for σ was conjectured in [14] (BES) as a sort of analytic continuation of the corresponding strong coupling expansion. In opposite to the latter, the weak coupling expansion of θ(x 1 , x 2 ) has a finite radius of convergence and defines a function which admits an integral representation (DHM) well defined in a certain kinematical region of particle rapidities x 1 , x 2 and for finite values of g [15]. Analytic continuation of the dressing phase to other kinematical regions compatible with crossing symmetry has been constructed in [16], which in fact provides verification of the crossing equation for finite g. Finally, under some assumptions on the analytic structure the minimal solution of the crossing equation has been found and cast precisely in the DHM form [17,18]. Let us also note that the dressing phase admits a representation in terms of a single integral (rather than double integral representation of DHM) which proved to be useful for numerical construction of solutions of the TBA equations [19].
It was soon realised [20] that a non-perturbative resummation prescription must be implemented if we want to extract the weak coupling expansion of σ from the strong coupling data. After a particular non-perturbative prescription to resum the leading order dressing phase contribution at strong coupling, the authors of [20] were able to expand it in a suitable weak coupling regime. In this way they found a connection between the strong and weak coupling coefficients of the dressing phase, reminiscent of the analytic continuation conjectured in [14]. Similarly in [21] the authors expanded the dressing phase of [14] reproducing precisely the asymptotic strong coupling coefficients. To obtain the strong coupling regime, these authors did not use the contour integral type of argument utilized in [14], but rather implemented a suitable ad hoc regularization procedure to expand the integrand of the Beisert-Eden-Staudacher dressing phase and obtained back the formal asymptotic expansion studied in [13]. Although the results of both [20] and [21] suggest that the dressing phase proposed in [14] (that we will call BES in what follows) has the correct properties to interpolate between the weak and strong coupling regime, to our mind no rigorous and complete treatment of the resummation procedure of the full strong coupling asymptotic expansion of σ exists so far.
In this paper we would like to present the discussion of the strong coupling expansion of the dressing phase, and its resummation, in the modern context of resurgence [22]. We show how to resum the strong coupling expansion by using a modified version of the wellknown Borel transform method. Our main result is that, in order to reproduce the dressing phase of [14], we have to modify the perturbative strong coupling expansion studied in [13] to what is called a transseries expansion by adding new, non-perturbative terms of the form e −4πg n with n 1 integer. These exponentially suppressed terms can be associated with ambiguities related to the resummation procedure of the purely perturbative expansion. Having modified the purely perturbative coefficients we need to check once again that the new strong coupling dressing phase satisfies the crossing symmetry equation and indeed we show that these new non-perturbative contributions to σ solve the homogenous crossing symmetry equation.
According to our findings the leading non-perturbative correction to the dressing phase comes with an exponentially suppressed factor e −4πg multiplied by an infinite perturbative expansion starting from three-loops, i.e. with the factor g −2 . From the purely perturbative point of view, the three-loop coefficient is also distinguished because only starting from three-loops the odd coefficients produce contributions to the dressing phase which satisfy the homogeneous crossing equation, while this is not the case for the one-loop perturbative coefficients or the even ones. For the non-perturbative contributions it might be that there is a protection mechanism based on vanishing of the zero mode contributions, forcing perturbation theory on top of these new non-perturbative saddles to start from three-loops, in the same spirit to what has been observed for the case of the instanton corrections for the anomalous dimension of the Konishi operator [23,24].
The origin of these new, non-perturbative effects in the dressing phase is quite mysterious. This story is analogous to the non-perturbative effects [25,26] emergent in the strong coupling expansion, g → ∞, of the cusp anomalous dimension of N = 4. Similarly to the dressing phase, the cusp anomaly has a transseries expansion at strong coupling [27,28] and, perhaps surprisingly, these exponentially suppressed terms have a semiclassical origin that can be understood from the string theory side. In the dual, weakly coupled description the calculation of the cusp anomaly translates into the computation of the spectrum of folded spinning strings on AdS 5 × S 5 , the so-called GKP-strings [29]. At low energies we can describe the world-sheet theory in terms of an effective sigma model, containing an O (6) factor [30], with a non-trivial strongly coupled IR dynamics. In a suitable regime [31], this 2-d quantum field theory contains non-perturbative objects, i.e. finite action solutions to the classical equations of motion, that, in the semiclassical approximation, give rise to exponentially suppressed contributions to the energy levels hence explaining the presence of non-perturbative terms in the cusp anomaly expansion at strong coupling, on the gauge theory side. How precisely these non-perturbative objects translate into the full string theory remains however to be understood.
In the case of the dressing phase the weakly coupled dual side can be most conveniently studied via a different stringy solution: the BMN string [32]. The S-matrix computed from the sigma model perturbation theory has been shown (see e.g. [33,34]) to reproduce the well known first few orders of the dressing phase expansion. For this reason and from the presence of non-perturbative terms in the dressing phase transseries expansion, we predict the existence of new non-perturbative objects in the world-sheet sigma model theory 1 (or possibly a suitable complexification thereof) that hopefully one can construct more easily in one of the Pohlmeyer reduced versions of the world-sheet theory [35]. Note also that the leading non-perturbative effect presents in the cusp anomalous dimension [25,30] takes the form e −πg , or e − √ λ/2 in terms of the 't Hooft coupling, while the leading correction we find in the dressing phase is of the form e −4πg , or e −2 √ λ . This stresses once more that these new non-perturbative corrections we find in the dressing phase should have a different semi-classical origin compared to the cusp anomaly ones.
From a mathematical point of view we perfectly understand why these non-perturbative terms must be incorporated in order to represent a very particular analytic function, i.e. this BES dressing phase, in terms of a transseries expansion, but from a physical point of view it is a very important question to understand the semi-classical origin of these exponentially suppressed contributions in terms of non-perturbative strings configurations.
Finally, we mention the universality of the methods developed in the present paper. Similar results about non-perturbative sectors of the dressing phase might be expected also for the case of q-deformed theories [36]- [42] and for lower dimensional examples of AdS/CFT, like for instance for AdS 3 /CFT 2 , see e.g. [43]- [45]. Furthermore similar type of methods can be applied also to different observables within the context of AdS/CFT correspondence, for example it was realized in [46] that the hydrodynamic gradient series for the strongly coupled N = 4 super Yang-Mills plasma is only an asymptotic expansion leading to the works [47]- [49] dealing with resurgence and resummation issues in the fluid context of AdS 5 /CFT 4 .
The paper is organized as follows. In Section 2 we review some known facts about the dressing phase and its strong coupling expansion while in Section 3 we introduce a modified version of the Borel transform to resum the perturbative coefficients. We prove in Section 4 that the Borel-Ecalle resummation of our proposed transseries expansion matches perfectly the BES dressing phase. The exact form of the non-perturbative terms is related to the ambiguity in the resummation of the perturbative expansion, which is computed explicitly in Section 5 and then expanded at strong coupling in Section 6. In Section 7 we use a standard dispersion-like argument to show how the perturbative coefficients of the non-perturbative sectors can be reconstructed from the large order behaviour of the purely perturbative ones and finally, in Section 8, we use precisely these coefficients to obtain new, non-perturbative contributions to the dressing phase, solutions to the homogeneous crossing symmetry equation. Because of the involved algebraic manipulations, many of the more technical results obtained in this paper are relegated to the appendices.

The dressing phase
Here we collect some known facts about the dressing phase which we need to our further discussion. The S-matrix is determined up to an overall scalar function -the dressing factor σ(x ± 1 , x ± 2 ), which satisfies a non-trivial functional equation -the crossing equation. It turns out to be convenient to write the dressing factor in the exponential form σ(x 1 , x 2 ) = e iθ(x 1 ,x 2 ) . Here the dressing phase where x ± are subject to the relation Here g is related to the 't Hooft coupling λ as g = √ λ/2π. The phase θ 12 can be written as where the function χ obtained from (2.1-2.2) is The coefficients c r,s (g) admit an asymptotic large g expansion where the numerical coefficients are given by , (2.6) and for n 2 by . (2.7) Note that for n = 0, 1 this expression is formally 0/0 , but nevertheless (2.6) can easily be recovered from (2.7). At any given order in the asymptotic 1/g expansion the double series defining χ is convergent for |x 1,2 | > 1.
The series (2.5) is divergent and of Gevrey-1 type 2 since the coefficients (2.7) grow as c (n) r,s ∼ n!, for this reason we can thus perform a Borel resummation of series (2.5).
The crossing equation satisfied by the dressing phase has the form . (2.9) Here we have chosen to uniformize x ± in terms of a single variable x via [12] x (2.10)

Modified Borel transform
We start with recalling that the standard Borel transform of a divergent series ∞ n=0 c n z −n−1 (3.1) with coefficients c n growing as n! is defined as The standard Borel image is now convergent to some function ∞ n=0 cn n! ξ n =φ(ξ) and the initial series can be resummed through the "inverse" of the standard Borel transform which is the Laplace transform where ∼ means asymptotic in the standard sense. Typically,φ(ξ) has singularities which lead to ambiguities in the resummation procedure associated with the choice of integration contour in the Laplace transform as we will discuss in full details later on.
Here, to remove an additional Riemann-zeta factor, we consider a modified (similarly to [50] which on a monomial acts as where ζ(n) denotes the Riemann zeta function. This transform can be easily inverted by noticing that the momenta of the measure dµ = 1 4 sinh 2 (ξ/2) dξ (3.6) are precisely As seen before, the "inverse" can be given via According to (2.5) being the generalised hypergeometric function.
Recalling that s + r must be odd and r 2, s r + 1, we can change to s + r = 2p + 1, s − r = 2q + 1, where the integers p, q are q 0, p q + 2, and, with the definition ξ ≡ 4πx, the modified Borel transformφ r,s (ξ) ≡φ p,q (x) (3.11) takes the form In terms of the variables p and q the perturbative coefficients c (n) r,s acquire the form . (3.13) As discussed earlier in this section, we can naively resum the asymptotic power series with coefficients (3.13) via (3.14) To understand the region of analyticity of the function c p,q (g) in the complex coupling constant g-plane, we need first to understand the analytic properties of the modified Borel transform (3.12) in the complex Borel x-plane.
To begin, we note that the first hypergeometric function in (3.12) is a simple polynomial of degree 2q in x. This contribution to the full modified Borel transform is an entire function of x because is coming from the coefficients c (n) p,q with n even which are only finitely many in number: from the explicit expression (3.13), we see that c (2m) p,q = 0 for any m q + 2. The second hypergeometric function in (3.12) , which we denote as Ω(z) := 4 F 3 3 2 − p, 1 2 + p, 1 2 − q, 3 2 + q , 3 2 , 2, 5 2 , z , (3.15) where z = x 2 , has a cut along the real interval (1, +∞). Therefore, the resummation formula (3.14) does not define an analytic function of g, unless we specify a contour of integration that dodges the cut. This introduces an ambiguity in the resummation procedure, related to the particular choice of integration contour, i.e. that is above or below the real line. For the discontinuity of Ω(z) we found in Appendix A the following formula so that combining this with (3.12) we find , |z| > 1 . Note that this discontinuity along the cut (1, +∞) is purely imaginary and also that Discφ p,q (1) = 0, this will shortly be of importance.
One natural way to fix the ambiguity related to the choice of the integration contour is to demand that c p,q (g) must be real for real g. Further analysis reveals that φ p,q (x) has neither pole nor cut on the real line and for generic p and q is a decreasing function as x → ∞, see Figure 1. We thus can define the manifestly real coefficients by whose strong coupling expansion g 1 coincide with the original asymptotic formal power series (2.5). This prescription for the resummation procedure seems somehow ad hoc but in the next section we will show that it corresponds in fact to the median Borel resummation.
To straightforwardly integrate φ p,q is rather difficult because it contains a separate polynomial part. Also the first two terms in (3.18) come apart which suggests that they originate from contour integrals around isolated points, as was explained in [14]. Therefore, to proceed, we show that φ p,q admits another but alternative representation through the functionΦ Namely, both functionsφ p,q andΦ p,q share the same real part a statement which is analytically proven in Appendix B in two different ways 4 . At this point it is gratifying to see that (3.19) is essentially the same formula as equation (3.25) in [14], which has been proposed there to describe a sort of analytic continuation of the coefficients c rs from strong to weak coupling. Note that, contrary toφ p,q , the new function Φ p,q is an even function of x and this property will be crucial in the next section to extend the integration over the whole real line to implement a Cauchy-like argument .

Non-perturbative resummation of the coefficients c r,s (g)
In this section we prove that the manifestly real resummation (3.18) proposed in the previous section does indeed coincide with the coefficients for the BES dressing phase introduced in [14]. Furthermore we show that the proposed real resummation (3.18) can be understood as the Borel-Ecalle resummation of a particular transseries expansion, generalization of the formal power series (2.5) that we started with.

From the Borel sum to the BES dressing phase
According to the discussion in the previous section, the coefficients c p,q can be represented as where the integration was extended to the whole real line since Φ p,q (x) is an even function of x. The rest of the computation follows the same steps as in [14] but now for arbitrary values of r and s and, therefore, we outline it here for completeness. The starting point is to pass from integration ofΦ p,q over the real line to integration ofΦ p,q along the contour depicted on Figure 2. The functionΦ p,q has a cut on the interval (−1, 1) and the integration contour C 1 runs just above this cut. Since the kernel f (z) = Φp,q(z) sinh 2 (2πgz) is symmetric with respect to z → −z, the contribution from two points symmetric around zero amounts to is real analytic. Thus, integration of f (z) above the cut is equivalent to the integration of f (z) over the interval (−1, 1). One has however to take into account that f (z) has a residue at infinity and at z = 0 which lead to additional contributions. In particular, the two isolated terms entering (4.1) can be treated (similarly to [14]) as the following contour integrals: 1) For the integral around the contour C 3 , where |x| → ∞ with < arg x < π − , a non-trivial contribution occurs only due to the leading term inΦ p,q (x) = δ q,0 + . . ., which is present for q = 0 only. One gets 2) To compute the integral around the contour C 2 , we have to expandΦ p,q (x) around zero and we find for the leading behaviour . . i.e. it is purely imaginary for real x. Note that the case q = 0 should be treated with care which results in the absence of the leading δ q,0 when q = 0 in the small x expansion forΦ p,q (x). Hence, the contribution from the contour C 2 is Thus, the resummation formula can be written as the following contour integral . Figure 2. This is evaluated by Cauchy theorem as The derivative is given by (4.7) Thus, for the coefficients c p,q we find Next, we apply the following identity where we identify z = n/g, α = 2p − 1, k = 2 and λ = i. Hence, or, going back to the (r, s)−representation Here one can recognise the well-known formula and use the representation of the Bessel function J ν (t) via the hypergeometric one to get Summing a geometric series up, one finally gets This formula proves that the median Borel resummed formula for c r,s coincides with the coefficients of the BES dressing phase that first appeared in [14].

Non-perturbative ambiguities and median resummation
Let us go back to the initial problem of going from the modified Borel transform to a suitable analytic continuation (3.14) of the original asymptotic formal power series (2.5).
To properly define the inverse transform (3.14), we need to integrate over a contour where the modified Borel transformφ p,q (x) is not singular. As shown above, the singular directions in the complex x Borel plane, also called Stokes directions, for the case under considerations are Arg x = 0 and Arg x = π.
We can thus introduce the directional Borel resummation via which defines an analytic function in the wedge of the complex coupling constant plane given 5 by D θ = {g ∈ C | (e iθ g) > 0}, provided that θ is a regular direction, i.e. θ / ∈ {0, π}. For every θ for which the above integral exists, if we expand for g 1 we obtain precisely the original asymptotic, formal power series expansion (2.5). Furthermore This allows us to analytically continue the function S θ 1 [c p,q ] (g) on a wider wedge of the complex g-plane, i.e. on the union of the Due to the presence of singularities in the Borel plane, if we keep on increasing Arg g, or equivalently θ, we will necessarily encounter branch cut singularities for the analytic continuation of the purely perturbative asymptotic power series (2.5). To understand the reason for that, we pick > 0 and small, and consider the two lateral resummations across the Stokes line θ = 0 given by S ± [c p,q ] (g), a similar story holds for the other Stokes line θ = π. These two analytic functions, although having the same asymptotic expansion (2.5), differ from one another on the intersection of their domains of analyticity. Their difference (related to the so called Stokes automorphism) can be written as an integration over the Hankel contour C shown in Figure 3, originating from infinity below the positive real axis, circling the origin and then going back to infinity above the positive real axis: where we used the fact that the discontinuity (3.17) starts at x = 1 and, in the last step, we expanded the sinh for g 1 to make explicit the exponentially suppressed factor e −4πng , benchmark of non-perturbative physics. 5 The integral (4.16) is well-defined for g ∈ C such that | sinh 2 (2πgx)| > 1 for |x| large enough. This leads to two disjoint domains of analyticity separated by the line (e iθ g) = 0. We decided to restrict our attention to the upper domain but one could have directly worked with the union of the two disjoint domains analyzing the discontinuity across them.
We specify the determination of the analytic continuation of (2.5) as follows where the suffix P reminds us that this is only the resummation of the perturbative power series (2.5). This analytic function has two branch cuts, one for Arg g = 0 and the other for Arg g = π. The two discontinuities are easy to obtain using the formula (4.17) where for the discontinuity along the direction Arg g = π, we used the results, proven in the previous section, that the discontinuity of the modified Borel transform is a function of x 2 over the Borel plane. We will study in detail these discontinuity in the following section. Note that using the function c P p,q (g) to obtain the analytic continuation of the perturbative power series (2.5) for real and positive coupling yields two different and complex results, depending whether we reach g ∈ R + from the upper or lower complex half plane. This "ambiguity" in our resummation procedure suggests that despite (4.18) has the correct asymptotic power series expansion it misses nonetheless crucial non-perturbative contributions and leads to the wrong (i.e. non-physical) analytic continuation.
To obtain an analytic continuation that is real for real coupling we make use of the median resummation [51], i.e. the appropriate, unambiguous, analytic continuation that is real for real coupling.
In the case at hand the median resummation of (2.5) is given by The superscript is to remind us that, upon expansion of the analytic function c T S p,q (g) for g 1, we do not obtain just the power series (2.5) but rather the full transseries representation 6 (see [52]) where the first term denotes the purely perturbative power series (2.5), and the transseries parameter s = −i/2 for 0 < Arg g < π/2 and s = +i/2 for −π/2 < Arg g < 0. The function c T S p,q (g), when expanded at strong coupling, contains infinitely many exponentially suppressed, i.e. non-perturbative, terms of the form e −4πng . Each of these non-perturbative contributions is multiplied by a formal power seriesΦ NP p,q (4πgn) whose standard Borel transform (3.2) can be extracted easily from (4.17) The transseries (4.22) is a formal representation of the analytic function (4.21) that encodes all of its monodromies in the complex g-plane. The Borel-Ecalle resummation of (4.22) gives precisely (4.21), in particular for real and positive coupling g we have that the two lateral resummation of the transseries coincide where L denotes the standard Laplace integral (3.3), inverse of the standard Borel transform. Furthermore, by combining (4.17) and (4.21), the Borel-Ecalle resummation of the transseries gives which is precisely the integral form (3.18) used in the previous section that we proved coinciding with the coefficients (4.15) of the BES dressing phase. So we learn that the correct strong coupling expansion of the BES coefficients (4.15) is not simply given by the asymptotic power series (2.5) but rather from the transseries (4.22) which coincides with (2.5) perturbatively but it contains infinitely many new exponentially suppressed terms. Note that in the present case the median resummation is very simple and ultimately consists in taking the real part of the modified Borel transform of the purely perturbative expansion. Generically, physical observables are represented with multiple parameter transseries and the actual implementation of the median resummation can be very complicated. We refer to [53] for a comprehensive discussion on the cancellation of nonperturbative ambiguities and the construction of the median resummation in one-and two-parameters transseries, relevant for more general physical observables than the one discussed in the present paper.
Another important thing to keep in mind is that the problem under consideration is indeed a linear problem which roughly means that each instanton sector does not "communicate" in an intricate way with all the others. This is a very lucky case which simplifies dramatically the Borel-Ecalle resummation procedure. One of the central points in Ecalle's works [22] is precisely the decodification of the complicated set of relations connecting the different perturbative coefficients in different sectors and the deep intertwining between all sectors: perturbative and non-perturbative. In the present case this could go underappreciated due to the linearity of the problem and perhaps one of the nicest illustrations where the full power of Ecalle's work can be better appreciated is shown in a nonlinear case [54] within the context of large-N dualities where the authors are also able to obtain a very explicit strong-weak coupling interpolation similar to the one described in our paper.
As already shown in the previous section, equation (4.26) coincides with physical answer given by the coefficients of the BES dressing phase (4.15), but in order to obtain (4.26) we had to pass from the formal power series (2.5) to the transseries (4.22). This amounted to introduce infinitely many non-perturbative contributions and ultimately means that the initial purely perturbative formal power series (2.5) is not enough to reconstruct the physical answer.
It is worth emphasizing that, due to its asymptotic nature, the strong coupling transseries representation (4.22) is only a formal object but its Borel-Ecalle resummation defines a perfectly good analytic function in a wedge of the complex g-plane. In particular, this means that the weak coupling expansion coefficients, obtainable from the gauge theory side, must be encoded in some intricate way in the strong coupling transseries coefficients (4.22). We do not know how to read this weak coupling expansion directly from the strong coupling transseries, but as proven above, the median resummation of the strong coupling coefficients yields precisely the coefficients of the BES dressing phase (4.15), which directly allow for a weak coupling expansion that matches precisely the gauge theory results as shown in [14].

Discontinuity of the Laplace transform
As we have just seen, the ambiguity in the Borel resummation procedure comes from the discontinuity of the Laplace transform. In this section we therefore compute this discontinuity explicitly by analyzing The important property of the discontinuity is that Discφ p,q (1) = 0. Since we have we can write Substituting here the explicit formula for the discontinuity we get where we have introduced a concise notation We proceed integrating by parts and noting that boundary terms always vanish we arrive at the following expression where for conciseness we omitted the parameters of 2 F 1 and introduce the following function In Appendix C we show that Q n (z) has the following representation as a double sum Thus, we can represent Q n (z) as a certain differential operator acting on e −hn √ z : with the whole z dependence just sitting in the exponent. Then further computation reduces to the following integral where a = p + 1 2 , b = q + 3 2 and c = p + q + 1. Thus, we are led to compute the integral For generic values of a, b, c this integral is given in [55]. Keeping for the moment p and q generic (non-integer), the answer is given by the following formula where we have introduced a concise notation t = h 2 /4. This formula can be obtained by using the Mellin transform technique, see e.g. [56]. While well-defined for generic p and q, the above expression becomes nonsensical for p and q being positive integers. In the latter case the answer can still be found from (5.15) by using the continuity principle -first one starts from generic p, q close to integer values by introducing a kind of regularisation and then takes a limit to these values. A regularisation parameter controls the apparent singularities which are supposed to cancel in the final expression.
To proceed, we introduce the following shorthand notation and consider the power series expansion for the second hypergeometric function Denote byp andq positive integers to which p and q are close by. Then this sum can be split into three parts Here to obtain the second line we made a shift of the original summation label k as k → k +p −q − 1, while to get the third line we shifted as k → k +p. Note that the first line of H 2 is finite in the limit p →p, q →q, while the second and the third lines are "linearly" and "quadratically" divergent, respectively, cf. the factors in front of the corresponding sums. Analogously, we consider and split the sum into two parts To obtain the second line we made a shift of the original summation label as k → k +q + 1.
The first line in the expression above is finite in the limit p →p, q →q, while the second one is "linearly" divergent. Now we put everything together and simplify the factors in front of the sums The second line in this expression is finite (it comes from the first line of H 2 ) and we can therefore put there p =p, q =q. This gives the first contribution I 1 to f (h) corresponding to integer values of p, q Obviously, I 1 contains inverse powers of t from t −p up to t −q−2 . The rest of (5.21) is divergent. To proceed, we introduce the following regularisation To take the limit, we need the formulae valid for any integer m. The second contribution to f (h) comes therefore from finite sums Substituting here the formulae (5.23) and taking the limit → 0, we find 26) where after the computation we replacedp → p andq → q. Note that this term contains inverse powers of t from t −q−1 up to t −1 . Finally, the third contribution comes from infinite sums The expression I 3 delivers the most complicated contribution which upon taking the limit and renamingp → p andq → q reads In this way we have found that the original integral is given by the sum of three terms In fact, the whole expression I 3 can be written as

Strong coupling expansion of the discontinuity
Here we show how to obtain an asymptotic expansion at large g starting from the exact answer for the difference ∆S p,q (g). To this end we have to analyse the expansion of I 3 when t → ∞. The simplest way to proceed is to use the formula (5.29), where we keep finite and send t → ∞. The corresponding expansion of 2 F 3 is known to be where 2) and F 3 will be discussed later.
For the case at hand we identify We start with analysis of the contribution of F 1 into I 3 , which we denote as I 3 . We have . (6.5) Further simplification gives Now it is important to realise that the expression in the brackets above has different behaviour in the limit → 0 depending on the value of the summation variable k. If k p then due to the factor sin 2 (π ) tan(π ) the expansion starts from 3 and therefore it does not produce any contribution at order 2 . This means that we can cut the infinite sum at k = p − 1. Then we naturally spilt it into two parts Then we make a replacement in both sums and in the second one we also replace . (6.8) This gives (6.9) In the first sum sin(π ) tan(π ) ∼ π 2 2 in the limit → 0 which allows one to immediately find the corresponding contribution. To proliferate a comparison with the finite contributions delivered by I 1 and I 2 , it is convenient to implement in the first sum the change of the summation variable k → −k + p − 1, while in the second one k → −k + q, correspondingly. This gives we observe that the first sum just becomes −I 1 , while differentiation over in the second one leaves us with the following answer Now taking into account eq.(6.11) as well as the fact that ψ( 3 2 − k + q) = ψ(− 1 2 + k − q), we see that the second sum is nothing else but −I 2 . Thus, we have found, that that is in the strong coupling expansion the contribution of I (1) 3 cancels out against the sum I 1 + I 2 .
Now we analyse the contribution of the terms F 2 , which we denote as I 3 . We have Finally, the contribution F 3 is given by the following formula Determination of the asymptotic coefficients c represents a rather non-trivial task which we undertake in Appendix D. There we show that the coefficients c do not depend on and are given by the following explicit formula . (6.16) Hence we have the following contribution of F 1 which we denote I 3 , . Differentiating over and taking the limit → 0 leaves us with the following expression Note that the growing exponent e 2 √ t does not enter the asymptotic expansion. Recalling that t = h 2 /4 we arrive at the following strong coupling asymptotic expansion of the integral (5.14) With this expression at hand we can now find the asymptotic expansion of ∆S p,q . According to eq.(5.13) we have Performing differentiations we get Due to the gamma function standing in the middle of the denominator in the second line of the above formula, the sum over s can be extended down to zero. Next we introduce a "loop" parameter L = + s + 3 and change the order of summation arranging the sum over L to precede the one over : Here in the last line the lower integration bound L − 3 − p − q 0 of the variable can be extended down to zero without changing the answer because of the gamma function Γ(4 + p + q − L + ). This allows one to combine two sums over L and obtain a formula where we have taken into account that In Appendix D by using the explicit form (6.16) of the coefficients c we bring the expression for discontinuity ∆S p,q (g) found above to the following form where the coefficients c L (p, q) are given by .
In Appendix D we also provide an alternative method to compute the discontinuity ∆S p,q (g) and find the same expression (D.37).

Dispersion relation and the non-perturbative sector
Having computed the discontinuities of the modified Borel transform across the two Stokes directions 0 and π, we can obtain the asymptotic expansion of the perturbative coefficients . .
.g c (n) p,q for n 1, via a standard dispersion-like type of argument [57,58]. The way to proceed is to consider the Cauchy integral for the analytic continuation (4.18) of the purely perturbative series c P p,q (g): where the contour is around the complex point g.
Making use of valid for g → ∞, we can read the perturbative coefficients c (n) p,q from the contour integral where we pushed the contour of integration to infinity as depicted in Figure 4, under that assumption that the residue at infinity of c P p,q vanishes. We know the discontinuities across the singular directions 0 and π from (4.19-4.20) so To compute these two integral we make use of the perturbative expansion (6.24), and, by analyzing loop order, L, by loop order we simply need to evaluate So, by putting everything together, we obtain the asymptotic expansion valid for n 1 where the first three coefficients are Note that the n even coefficients completely disappear from this analysis because, as explained before, the c (n) p,q with n even are non-vanishing only for a finite number of terms. The large order behaviour of the perturbative coefficients captures precisely the lower order perturbative coefficients on top of the non-perturbative contributions in the transseries (4.22), i.e. the coefficients for the strong coupling expansion ofΦ NP p,q (g). In Figure 5 we show how well, at large n, the perturbative coefficients c (n) p,q can be approximated by even their leading asymptotic expansion The formula (7.6) allows us to obtain an explicit formula for the polynomials c L (p, q) by comparing the large n asymptotic expansion of the coefficients c (n) p,q with the right hand side of (7.6). Hence, we need to asymptotically expand formula (3.13) for large n. To this end we consider the ratio between c (n) p,q and its leading asymptotic coefficient which for n odd takes the form In what follows it appears advantageous to use the change of variables n = 2(m + 1) where m is half-integer and replace q → q − 1. Then for R (n) (p, q − 1) we get where B(a, b) is the Euler beta integral We observe that in the formula (7.10) contribution of p and q completely factorises and comes in a symmetric fashion. Therefore, our task now is to find an asymptotic expansion of the Euler integral when m → ∞. First we compute the integral by means of the saddle point method. Consider For large m the dominant contribution to this integral comes from the critical point v = 1 2 for which the "action" is log(v(1 − v))| v=1/2 = − log 4. This motivates to perform the following change of integration variable which converts the integral to Now using binomial expansions twice we rewrite the integrand as a double sum Evaluating this integral in the regime m 1 and further performing one summation we arrive at The ratio of two gamma function has the known asymptotic expansion in the limit m → ∞, namely (7.17) where B (1/2) l (−r) are the generalised Bernoulli polynomials also known as Norlund polynomials, see e.g. [59].
Using this result we can obtain the asymptotic expansion of the Euler beta for m 1:  The function R n (p, q − 1) can be expanded for large n, using the variable n = 2(m + 1), via the convolution of the above coefficients This is not quite the expansion we sought for, as shown in equation (7.9) we want to express this ratio as We can easily relate one expansion to the other via L denotes the Stirling number of the first kind. As we will shortly see, these coefficients c L (p, q), polynomials in p and q, will give rise to important non-perturbative contribution to the dressing phase (2.1).

Non-perturbative contributions to the dressing phase
We have shown in Section 4, that the purely perturbative asymptotic power series expansion (2.5), is not enough to reconstruct the coefficients of the BES dressing phase (4.15). We need to replace the perturbative expansion by the transseries representation (4.22) whose Borel-Ecalle resummation (4.26) matches precisely the non-perturbative result (4.15). In this section we see the effects that our transseries expansion produces to the dressing phase.

Effects of the non-perturbative sector to the dressing phase
Our replacement from the perturbative power series (2.5) to the transseries (4.22) is not without consequences. In [13] the authors showed that if we restrict the sum (2.5) to even n, we obtain a strong coupling solution to the crossing symmetry equation (2.8). This particular solution does not have the correct weak coupling limit and for this reason the authors considered the analytic continuation of the series (2.5) by summing over all the integers n. This amounts to adding to the dressing phase a solution to the homogeneous crossing symmetry equation The BES coefficients (4.15) proposed in [14] thus interpolates between the formal power series expansion (2.5) at strong coupling and the correct gauge theory weak coupling limit. The crucial point is that this integral representation for the BES coefficients is not quite equivalent to the formal power series (2.5), but rather it is obtained via the Borel-Ecalle resummation of the transseries (4.22). This means that the non-perturbative terms we added must lead to additional contributions to the dressing phase, solutions to the homogeneous crossing symmetry equation (8.1).
Let us compute this additional non-perturbative contributions to the dressing phase. Since we know from [13] that the formal power series (2.5) solves the full crossing symmetry equation (2.8) we can just focus on the purely non-perturbative terms (6.24) of our transseries ansatz (4.22). The non-perturbative contributions to the function χ(x 1 , x 2 ), given by equation (2.4) and written using p, q variables is where s is once again the transseries parameter discussed in Section 4.2, i.e. s = −i/2 for 0 < Arg g < π/2 and s = +i/2 for −π/2 < Arg g < 0.
We do not know how to perform this double sum (8.2) using the exact integral representation (5.5) for ∆S p,q (g), but we can easily compute it loop order by loop order using the strong coupling expansion (6.24).
Using (6.24) we can rewrite the above equation in the form of the loop expansion with the definition where the coefficients c L (p, q) are given for example by (6.25) (see also the other equivalent forms presented in equation (7.21) and Appendix D). Note that the coupling constant, g, only appears in front of the series (8.3). It is surprising that the L th loop contribution to χ N P coming from all instantons sectors can be fully resummed giving rise to the exponentially suppressed factor Li L−1 e −4πg .
From the explicit coefficients (6.25)-(7.7) it is a straightforward calculation to obtain the first few N P (x 1 , x 2 ) of the g −L+1 term is already on its own a solution to the crossing symmetry equation coming from the resummation of infinitely many instanton sectors.
Note that the first non-perturbative contribution is given by χ N P , which corresponds to a three-loop perturbative correction g −2 , on top of a non-perturbative background. As mentioned in the Introduction, the vanishing of the tree level, one-and two-loops contributions might be explained by a protection mechanism based on vanishing of the zero mode factors, forcing perturbation theory on top of these mysterious non-perturbative saddles to start from three-loops .
We claim that the complete non-perturbative correction (8.2) to the dressing phase, since it is a formal sum (8.3) of homogeneous solutions, gives also rise to a solution to the homogeneous crossing symmetry equation (8.1), very likely not of the simple rational form in x 1 , x 2 as the coefficients χ (L) N P just encountered.

Generating solutions to the homogenous crossing symmetry equation
From the large order behaviour (7.6) of the perturbative coefficients (2.7) we can construct a generating functional to obtain solutions to the homogeneous crossing symmetry equations (8.1). In [13] the authors noticed that the perturbative coefficient c (n) p,q , with n odd, n > 1, generates a contribution to the dressing phase that solves (8.1). Similarly, for n 1, we can consider the asymptotic expansion (7.6) and, as we have just seen, for each loop order L, the perturbative coefficient c L (p, q) yields once again solutions to the homogeneous crossing symmetry equation.
Thus we can consider, similarly to (2.4), the expression When n = 2z + 1 is an odd integer, this function reproduces the known perturbative contributions to the dressing phase. Viceversa, when z 1, we know from (7.6) that the perturbative coefficients c (2z+1) p,q can be written as an asymptotic expansion in z −1 in terms of the non-perturbative sector's coefficients c L (p, q) (7.21). Thanks to the analysis of the previous section, each one of these terms will produce a solution to (8.1) and equation (8.6) will basically sum up all of these contributions and it will still solve the homogeneous problem since it is a linear problem. Hence equation (8.6) is somehow interpolating between the perturbative and the non-perturbative solutions to (8.1).
Discarding from equation (8.6) an overall factor which is only z-dependent, we consider the generating functional for homogenous solutions to the crossing symmetry equation (8.1) given by The sum over q can be easily performed giving with and (8.10) For I 1 the sum over p is straightforward The first term in S 1 is obviously symmetric in x 1 ↔ x 2 , while for z ∈ N one can easily show using the inversion formula for the hypergeometric function 7 that also the second fraction is symmetric. This means that, for z ∈ N, the contribution of S 1 to the dressing phase (2.3) is actually zero. The second contribution to Ξ comes from This sum is trickier than S 1 because the index of summation p appears in the parameters of the hypergeometric function. We notice that, for z ∈ N, the difference between the two parameters c and b of the hypergeometric is p + 1/2 − z − (p − 1/2 + z) = 2z − 1 ∈ N. This allows us to use the reduction formula, see e.g. [55], The sum over p can be now performed where 3F2 denotes the generalized hypergeometric function regularized. For z ∈ N, our generating functional Ξ(z; x 1 , x 2 ) produces only rational functions of x 1 , x 2 . In particular, using the explicit formulas (8.11)-(8.13), we can easily check that Ξ(1; x 1 , x 2 ) coincides precisely (modulo an overall numerical factor) with the three world sheet loops contribution χ (3) (x 1 , x 2 ) presented in equation (5.6) of [13].
Similarly, from our studies of large order behaviour (7.6), we expect the following behaviour of the generating functional for z 1 where the rational function χ (L) N P (x 1 , x 2 ) are precisely the non-perturbative contributions to the dressing phase (8.5) previously computed.
It would be interesting to obtain an analytic expression for Ξ(z; x 1 , x 2 ) for arbitrary values of z and show that it solves the homogenous crossing symmetry equation.

Acknowledgements
We would like to thank Benjamin Basso, Lorenzo Bianchi, Davide Fioravanti, Marco Rossi and Ricardo Schiappa for useful discussions. We specially thank Sergey Frolov for his careful read of our manuscript. The work of G.A. and S.S. is supported by the German Science Foundation (DFG) under the Collaborative Research Center (SFB) 676 Particles, Strings and the Early Universe and the Research Training Group 1670. D.D. is grateful to Hamburg University and DESY for their hospitality and to the Collaborative Research Center SFB 676 for financial support during the final stages of this project.

A Derivation of the discontinuity of Ω
To obtain the discontinuity of Ω, , cf. (3.15), we first apply the reduction technique [55] which allows one to reduce our particular 4 F 3 to a multiple derivative of 2 F 1 with respect to the variable z by means of the following formula where we have introduced a concise notation First the sum over m can be straightforwardly taken leaving behind And further summation gives where 3F2 stands for the regularised hypergeometric function, which is given in fact by the finite sum .
The function f has a branch cut on the interval [1, ∞) and the corresponding discontinuity is known to be . (A.5) Using the series representation for 2 F 1 we find from this formula Plugging everything together we get .
Because of Γ(1 + − q + r) in the denominator, the sum over can be restricted to runs from q − r to p + q − 2. We therefore make a change of variable = q − r + s, so that s runs from 0 to p + r − 2. Then we get . (A.8) Here in the denominator of the last sum the term Γ(p − 1 − s) cuts the summation range for s at p − 2. Therefore, we can change the order of summation in r and s and write The sum over r yields . (A.10) Next with the help of the well-known transformation formula we can write Next, the following identity holds Hence, we get . (A.14) Further, summing over s results into where to obtain the last expression we again used the transformation formula (A.11). Now, taking into account that the expression for S acquires the form Thus, for the discontinuity we have Summing up we finally get the desired formula (3.16).

B.1 First proof
The main ingredient of the formula (3.12) is its non-polynomial part represented by the hypergeometric function Ω(z), cf. (3.15), where we have introduced a variable z = x 2 . To proceed, we will use representation (A.3), where we analytically continue the function f in the complex plane for the values |Arg (−z)| < π. The corresponding formula is well known and reads Obviously, the function f (z) has a cut on the real axis. Taking into account that p and q are positive integers it is elementary to find the real part for f (z) for z positive. Using the fact that log(−z) = log |z| + iπ, we find that As a next step we compute Substituting this result into the real part of (A.3) and replacing the regularised hypergeometric function via its normal counterpart, we obtain where we need to compute the following sum where the coefficient κ is chosen for convenience to be .

(B.6)
Here for the hypergeometric function we can substitute its definition .
After the change of order of summation eq.(B.5) acquires the form where the restrictions on the summation variable are clear from the arguments of the Γfunctions entering the denominators of the second sum. We further shift the sum variable as = q − r + s, and get the double sum, which we write in the following order The internal sum is given by Finally, to perform the last sum over r we have to carefully distinguish two cases: q = 0 and q = 0. We treat these cases in turn.

1)
Case q = 0. We have , n p + q. (B.9) Now we are ready to compute the real part of Ω Ω = 3 2 Changing the summation indices appropriately, we find the final answer Now we recall that function φ p,q reads as Substituting here (B.11) we find that the polynomial part cancels out completely and we are left with the desired result (3.20).
2) Case q = 0. In this situation we have , n p.

(B.13)
We therefore find which gives Then we specify (B.12) for q = 0 and get which finally boils down to Thus, we have proved that in all the cases φ p,q is equivalent to eq.(3.19).
The method is thus build on the further calculation of the residues of the integrand at each pole. The full expression for the residues includes a huge amount of summands. Therefore, we present only the needed real part which comes from the terms of the form (−z) 1/2+n log(−z), n ∈ N.

The 1 st order poles
In this case the residue is given by 18) and is purely imaginary. Thus, Res 1 = 0.

The 2 nd order poles
At the k−th pole, 0 k 2q, we have the following expression for the real part of the residue: Note, that the residue is non-zero only for 0 k q. After changing factorials by Gamma functions, transforming k → q − s, 0 s q, and using Γ(1 − z) = π/(Γ(z) sin(πz)), one gets Summing over all the second order poles (0 s q), we arrive at Putting all the coefficients from (B.17) and (3.12), one gets the polynomial part in φ p,q (x) with a minus sign. This means that the first term in (3.12) is cancelled by this residue term.

The 3 rd order poles
Here 0 k p − q − 2 and the the real part of the residue at the k−th pole is given by Note that 1/∞ 2 behavior of 1/(Γ(−k − q)Γ(1 − k − q)) is not compensated by the rest part. This leads to Res 3 = 0.

The 4 th order poles
In this case the expression for the real part of the residue Res 4 is huge. Nevertheless, again thanks to the Γ(1 − k − p)Γ(2 − k − p) factor in the denominator most of the terms vanish. The rest is given by This can be simplified with the help of the following identities: ψ(1−z) = π cot(πz)+ψ(z), ψ (1 − z) = π 2 / sin 2 (πz) − ψ (z) and Γ(1 − z) = π/(Γ(z) sin(πz)): Summing over all the forth order poles (0 k < ∞), taking into account all the coefficients from (B.17) and (3.12), one comes to the desired expression for Φ p,q which proves (3.20) for q = 0.
Strictly speaking, the direct substitution q = 0 into the results obtained above leads to the wrong answer. In this case one has to perform the same procedure from the very beginning. This happens because (3.15) reduces to 3 F 2 ({ 3 2 − p, 1 2 , p + 1 2 }, {2, 5 2 }, z) and now we have a different system of poles. The final result gives an additional factor 1 which completely corresponds to (3.20).

C Derivation of Q
In Section 5 we introduced the following function Performing straightforward differentiations and then summation over k we arrive at the following formula Further one can show that for p q + 2, q 0, the following identity takes place .
Thus, Q(z) is essentially written as the sum of two 1 F 2 functions that both have the same characteristic feature. Namely, if the upper parameter is ρ + q, where q 0, then among the lower parameters there is ρ. In this situation we can apply the following reduction formula, where 1 F 2 gets replaced by a finite sum of 0 F 1 , the latter being expressed via the modified Bessel function of the first kind I ν ,

D Details for the construction of the asymptotic expansion for ∆S p,q (g)
Here we resolve several technical issues concerning construction of the asymptotic expansion of the discontinuity ∆S p,q (g) and also present an alternative method to derive the same asymptotic expansion.

D.1 Solution of the difference equation for c
The coefficients c which arise in the asymptotic expansion of the function 2 F 3 can be determined recurrently by using Riney's method. The corresponding recurrence formula reads [60] Here the coefficients a 1 , . . . , b 3 are given by (6.4) and For our purposes, however, this recurrence formula is not enough as we need to determine these coefficients in the closed form, i.e. without referring to any recurrence procedure.
We start our analysis with some observations. First, computing e ,k explicitly we note that they do not depend on and, as a consequence, c are also independent. Second, the coefficients c satisfy certain difference equations. To understand this issue, consider the differential equation for the function 2 F 3 : To derive the difference equations for c , it is enough to substitute in this equation the part of the asymptotic expansion for 2 F 3 which contains either damping or growing exponent Both lead to the same difference equations, so it is enough to consider only one of them. Substituting for instance F − 3 , we get an expression which contains -independent term and the term proportional to . Since the original equation is valued for arbitrary these terms must separately vanish. Collecting terms proportional to t /2 in the first, -independent, term we get the following difference equation: (1 + q)(2 + q) + 2p(2 + q) 2 + + 6 2 (2 + p + q) + 2 (5 + 3q + p(7 + 4q)) c + (D.4) From the second term proportional to we find a simpler difference equation, namely, In fact, the second equation (D.5) implies the first. Shifting in eq.(D.5) the variable → + 1, solving for c +2 and plugging this solution into (D.4), we observe that the last equation factorises and it contains the left hand side of eq.(D.5) as a factor. Thus, fulfilment of (D.5) implies the fulfilment of (D.4). Now we explain how to find a closed formula for the coefficients c . It is not difficult to see that these coefficients must arise in the large s-expansion of the following ratio of the gamma functions Indeed, according to the discussion in chapter 2.2.2. by [61], the numbers c can be computed from the following recursion formula where . (D.8) Here b 1 = 1 + p, b 2 = 2 + q, b 3 = 1, and the prime signifies omission of the term with i = j.
One can now directly verify that the coefficients c given by the formula above satisfy the difference equation (D.5) and coincide with those found through the recursion formula.

D.2 Simplifying the expression for ∆S p,q (g)
We start with the expression (6.22) for ∆S p,q (g), isolate the sum over and substitute there the coefficients c in the form of the sum (D.17). We get the following double sum . (D.18) Now we isolate from ∆S p,q (g) the sum over and substitute there c in the form of the sum (D.17). We get the following double sum As a next step we interchange the order of summation and get . (D.20) Taking the sum over we obtain Note that for the allowed values of k and m the function 4 F 3 is always finite, but the gamma function Γ(k − m − q) which divides it is always infinite because k q and m 0. Thus, the term of W containing 4 F 3 does not contribute to the discontinuity ∆S p,q and we can safely omit it. We therefore consider where for further clarity we replace the index k by s and j by k. Further, the hypergeometric function V =3 F2 ({−1 − k − p − q, 3 − L − s + k + m + q, 2 + k + p + q} , {1 + k + p + q, 4 + k + p + q − L} , 1/2) featuring in the last formula can be reduced, namely, To each of these two 2 F 1 's we apply an identity 2 F 1 (a, b, c, z) = (1 − z) −a 2 F 1 (a, c − b, c, z/(z − 1)) and get Therefore, For the discontinuity we therefore find We note as an interesting fact that the hypergeometric function entering in the last expression is expressible via the following Jacobi polynomial Expanding 2 F 1 into the hypergeometric series, one comes to another representation ∆S p,q (g) =4ig (p − q − 1)(p + q) ∞

L=3
Li L−1 (e −4πg ) (4πg) L−1 C L (p, q) , (D. 28) where the coefficients C L (p, q) are where " ∼ " denotes that the hypergeometric function is regularised. In Appendix D.3 we provide an alternative but simpler expression for C L (p, q).