The full analytic trans-series in integrable field theories

We analyze a family of generalized energy densities in integrable quantum field theories in the presence of an external field coupled to a conserved charge. By using the Wiener-Hopf technique to solve the linear thermodynamic Bethe ansatz equations we derive the full analytic trans-series for these observables in terms of a perturbatively defined basis. We show how to calculate these basis elements to high orders analytically and reveal their complete resurgence structure. We demonstrate that the physical value of the generalized energy densities is obtained by the median resummation of their ambiguity-free trans-series.


Introduction
Systems soluble by the Bethe ansatz are relevant in condensed matter systems, in statistical as well as in particle physics [1,2].They provide toy models, where non-perturbative, strongly interacting phenomena such as dynamical mass generation, superconductivity, etc. can be analyzed in simplified circumstances.Additionally, some of them also have experimental realizations.
In most of the applications we are interested in the thermodynamic limit of the Bethe ansatz equations, which can be formulated as linear integral equations.Unfortunately, these integral equations cannot be solved exactly and we have to rely on their perturbative expansions.These perturbative series are asymptotic and their factorial growth signals non-perturbative terms.For a complete description one has to build a multiple series, i.e. a trans-series both in the perturbative coupling and in the exponentially suppressed non-perturbative corrections.This trans-series is understood as Borel resummed, and the requirement of being free of ambiguities requires an intricate interplay between the various perturbative and non-perturbative terms 1 .The theory which formulates this is called resurgence, which lives its renaissance now, see [4,5,6] for recent reviews.
Recently there has been great progress in the perturbative as well as the leading non-perturbative analysis of the linear TBA equations.On the condensed matter side the groundstate energy density of the Lieb-Liniger, Gaudin-Yang and Hubbard models together with their generalizations were investigated [7,8,9,10,11,12].The nonpertubative terms were in many cases related to the superconductive gap as well as to renormalon diagrams.On the particle physics side free energies of asymptotically free integrable quantum field theories in the presence of an external field coupled to a conserved charge were analyzed [13,14,15,16,17,18,19].These included the O(N ) non-linear sigma model and its supersymmetric extension, the Gross-Neveu model and the principal chiral field for which the large order behavior of the perturbative series were also investigated [20,21,22,23,24].In these quantum theories the non-perturbative terms are related to instantons or renormalons, which were further confirmed by large N calculations [25,26] and in the O(3) model by introducing a θ-term [23].
In constructing the ambiguity free trans-series the first problem is to efficiently calculate the perturbative terms.This was first achieved for the energy density of O(N ) models in [27,28].This method was extended for statistical models and for the circular plate capacitor [7,29] and by combining with the Wiener-Hopf technique to integrable quantum field theories [30].The first few exponentially suppressed corrections can be extracted from the asymptotics of the perturbative coefficients [7,30,8,9,31,32,21,22,24]. A systematic treatment based on the Wiener-Hopf approach was presented in [20], which resulted in the precise structure of the trans-series and explicit calculations of the first few non-perturbative corrections.This was further extended to higher orders and improved by introducing the running coupling for the O(N ) models in [24].The aim of our present paper is to solve completely these models by determining the full trans-series, i.e. all the non-perturbative terms together with their perturbative expansions.We are doing this by expressing these higher perturbative expansions in terms of the original perturbative series of generalized observables, which we also determine from the known perturbative series of the energy density.
The paper is organized as follows.In section 2 we introduce the integral equation, the generalized observables and differential equations which relate them to each other.In section 3 we demonstrate how the Wiener-Hopf technique can be used to calculate these generalized observables.This provides a structural result, which we make explicit by introducing a perturbatively calculable basis in section 4 and constructing the full transseries.In section 5 we present a method for determining the basis and investigate how the various parts of the trans-series are connected.We also relate its median resummation to the TBA result.We provide explicit examples in section 6.Finally, we conclude in section 7.

Observables and their properties
We investigate linear integral equations of the form for |θ| ≤ B, where r n (θ) = cosh(nθ) and the kernel is a symmetric function, which, in most of the applications, is related to the logarithmic derivative of the scattering matrix [13,14,15,16,17,18,19,7].We are interested in the observables as functions of B, but we do not indicate this dependence explicitly.This observable is symmetric in n and m, which are not necessarily integers but non-negative.Its Bderivative (which we denote by a dot) can be written in terms of the boundary values of χ n -s as [24] dO By generalising the manipulation of the integral equation in [12,11] one can show that these boundary values satisfy the differential equation where f (B) is an n-independent function, which can be calculated, for instance, from the n = 1 case.These equations connect all observables to one of them, say to O 1,1 , which is the groundstate energy of the integrable model in a magnetic field coupled to a conserved charge.The observable O n,m for n, m integers can be interpreted as the expectation value of the conserved spin m charge, in the presence of the magnetic field, when the Hamiltonian is given by the conserved spin n charge and we have periodicity in the direction of the corresponding generalized volume.Similar formulae appear in the generalized hydrodynamics for expectation values of conserved charges and currents, although in slightly different settings [33,34] Let us note that although we analyse the case of the r n (θ) = cosh(nθ) source terms, they can be used to express the solutions and observables for the r n (θ) = sinh(nθ) source terms as well.They together form a complete basis, from which by differentiation or analytical continuation and Fourier transform cases with more general sources can be reached [35].

Wiener-Hopf integral equation
The standard way to solve the integral equation is the Wiener-Hopf technique [14,15,20,24].As a first step we extend the source as r n (θ) = Θ(−θ + B) e nθ 2 + Θ(θ + B) e −nθ 2 , (Θ being the Heavyside theta function), as well as the integrations, (but not χ n (θ)), for the whole line by paying the price of introducing an unknown function R(θ), which, however, vanishes for θ < B. In solving the equation in Fourier space the key point is the factorization into factors analytic in the lower and upper half planes.This can be done by taking logarithms and projecting into the appropiate analytic pieces [24].Implementing the separation of the equation into lower and upper half analytical pieces we arrive at where σ(ω) = G+(−ω) G+(ω) and the unknown function X n (ω) is related to the Fourier transform of R(θ) as Except for the explicitly introduced pole at ω = in, X n (ω) is analytic in the upper half plane.We also assume that n > 0. The n = 0 case requires special care [20,24] and we can recover it by solving the differential equations (2.3,2.4).
In the typical applications σ(ω) has a cut and poles at ω = iκ l , l = 1, 2, . . ., on the positive imaginary line.Additionally, we also have the explicit pole at ω = in coming from X n .It is advantageous to disentangle the poles from the cut by moving the cut a bit away from the imaginary line in either direction [20,24].We then deform the integration contour from the real line surrounding the cut and separately the poles whose residues we collect.We can do it in two different ways, by integrating a bit left or right of the poles.The residues will also depend on this choice, but the final result must be the same.This is the manifestation of a Stokes phenomena and the two different choicesare related to the two lateral resummations.This is by no means obvious and we have only a confirmation a posteriori.Indeed, by switching between the contours the residue terms providing the Stokes constants change signs, which is the same which comes out from the lateral Borel resummations of the trans-series solution of the problem.For definiteness, we integrate a bit left of the imaginary line ( in κ a bit above the real positive line): where q n,κ l = X n (iκ l ) and S l is the residue of iσ(iκ is the discontinuity of σ.We included in the sum the contribution of the pole of X n (iκ) at κ 0 = n with residue S 0 = −iσ(in+0) = −iσ + n with the convention that q n,n = 1.Here we assume that all poles κ l are distinct, including κ 0 , i.e. n ̸ = κ l .In the more general case, σ(iκ) can have higher order poles, which could even coincide with the pole at ω = in.In this case S l and q n,κ l are related to the expansion of the functions around the singularity, however, we do not consider these complicated cases in this short letter, see our upcoming paper for further details [35].
Typically, we can introduce a running coupling v with an arbitrary constant L, such that the integral equation for the rescaled variable where and The model-dependent parameter γ has to be chosen such that A(y) = e −vy(γ log v+L) δσ(vy) has a power-series expansion in v without any log v terms: A(y) = ∞ j=0 v j α j (y).By appropriately choosing L the linear y-dependence in log A(y) can be canceled.Here q n,κ l -s (except q n,n = 1) are also unknowns, which have to be calculated by evaluating the integral equation (3.7) at the positions xv = κ l .If Q n (x) including q n,κ l are determined, then the observable O n,m can be written (similarly to [24]) as with where Here we assumed that n ̸ = κ l and n ̸ = m, otherwise we have to calculate the residue of a second order pole.The n = m case can be recovered by taking the n → m limit.For the boundary value of the field, similarly to [24], we obtain with for n̸ = 0.In the case of n = 0 we need to calculate f from χ 1 in (2.4) and solve the equation (2.4) for χ 0 and (2.3) for O 0,0 .

Trans-series ansatz and its solution
We solve the equations (3.7,3.8) for Q n (x) in terms of a trans-series ansatz where the set of nonzero Q n,j -s is model-dependent and encodes via ν ∼ e −1/v the non-perturbative corrections.One has to investigate the set of poles {κ l }, which (up to some isolated cases such as κ 0 ) can be described as a union of finitely many sets of the form {a i l + b i } with l = 1, 2, . . . .We should introduce d l such that all non-peturbative corrections are accounted for.We give concrete examples in section 7.
The generic solution to (4.1) can be calculated iteratively in l.We start with the l = 0 perturbative part, i.e. we have to solve perturbatively the following problem: This can be done by expanding A(y) and the source term in power series in v and iteratively solving at any order based on lower order solutions [20,24].We will see, however, that the explicit solution is not needed.Observe also that originally we needed P α for α > 0, but the equation and the perturbative solution make perfect sense also for α < 0. Using these solutions the unknown Q n (x) can be written as where, from the definition of q n,κs , we obtain a closed system of linear equations of the form with the exception of q n,n = 1.Here we introduced the symmetric building block (for α ̸ = −β) as which contains the moment The symmetric moments ⟨P α ⟩ β are understood perturbatively in v and are well-defined for any signs of α and β.We note that the recursive structure for q n,κ l is the consequence of the integral equation, where the model-specific feature lies in the set κ l (the nonperturbative nature) as well as in A n,m (the perturbative nature).In the following we solve this linear system of equations.Since q n,n = 1 is not an unknown, we regard its contribution as an inhomogeneous source term where s n,−κs = A n,−κs + σ + n ν n A −n,−κs .This linear matrix equation (I − A)q n = s n , with A s,l = iS l ν κ l A −κs,−κ l can be solved by inversion q n = (I − A) −1 s n , which can be represented by the Neumann series q n = (I + A + A 2 + . ..)s n and expanded in ν.Alternatively, we can plug back recursively every lower order solution in to the ν-expansion, leading to q n,κs = paths sA n,path,−κs S path , where a path means a sequence starting from n and ending at −κ s : (n, l 1 , l 2 , . . ., l N , −κ s ).
The contribution of such a path is where the inner indices take only the values l k = 1, 2, . . . .At each non-perturbative order in ν we have only a finite number of terms contributing.With this solution the unknown function Q n m v = q n,m can also be written as The observables W n,m can be obtained in terms of q n,κ l as Clearly, the basic building block A n,m is nothing but the perturbative part of our generic observable W n,m .The boundary value of the field can be expressed as where a α = lim β→∞ βA α,β .By this we provided a complete recursive solution of the problem, i.e. we expressed the observables in terms of the perturbatively defined A n,m -s.In the following we explain how the perturbative expansion of the building blocks can be calculated.

Median resummation and alien derivatives
Let us summarize what we have achieved so far.The observables O n,m and χ n (B) can be written in terms of W n,m and w n as (3.10) and (3.12), which satisfy two differential equations (5.1) and have the solutions (4.11) and (4.12) in terms of q n,κs , which is given by (4.8).
The perturbative parts A n,m and a n satisfy the W → A, w → a differential equations (5.1,5.2).Since Volin's method [27,28,30] determines W 1,1 at the perturbative level it provides A 1,1 .We can then extract the perturbative part of w 1 , namely a 1 from (5.1), and by plugging back to eq. ( 5.2) we can extract a n for any n, not necessarily positive integer.These perturbative series then can be used to calculate the expansion of A n,m to the desired order from the perturbative part of (5.1).By using these building blocks the all order solution for q n,κ l , W n,m and w n can be built up.
The results for q n,m , W n,m and w n are given in terms of trans-series, which is understood as laterally Borel resummed.This prescription does not follow from our derivation, although very plausible from the contour shift, see also [20] for comments about this point.Thus we assume that the lateral Borel resummation of the trans-series solution gives the TBA result.Since the TBA result is free of ambiguities, we can calculate the various alien derivatives of A n,m from the ambiguity cancellations 2 .These quantities differ only by the various source terms, thus we expect that the (I − A) −1 operation guarantees the ambiguity cancellation.
We analyze the behaviour of q n,m using resurgence theory and alien derivatives following [5,6,32].Assuming m > 0 and n > κ 1 , the leading singularity of the Borel transform of A n,−m on the positive real line is at κ 1 .The corresponding ambiguity, which is encoded in the alien derivative 3 ∆ κ1 A n,−m has to be canceled by the leading non-perturbative correction of order ν κ1 , i.e. by iS l A n,−κ l A −κ l ,−m leading to ∆ κ1 A n,−m = 2iS 1 A n,−κ1 A −κ1,−m .By moving iteratively further and subtracting the already known alien derivatives one can show that In a similar way one can also show that ∆ n A n,−m = 2iS 0 A −n,−m thus naturally extending the result for l = 0. We can then construct a multi-parameter trans-series for our basic quantity as where σ path = σ l1 ν κ l 1 . . .σ l N ν κ l N .One can even formally replace the σ + n ν n A −n,−m term with iS 0 ν κ0 σ 0 A n,−m and introduce the trans-series parameter σ 0 .As ∆ 2 n q n,m = 0, the 2 In this short note we assume that all Stokes constants, S l are real.This is true for most of the models we analyse here.For complex Stokes constants one has to take the appropriate real part of the expressions, which complicates the discussion and we postpone the detailed analysis for [35]. 3The alien derivative is understood in the running coupling v as: ∆n = ν n ∆n, where [ ∆n, ∂ B ] = 0. Thus it has an extra v −γ factor compared to the standard definition.
trans-series expression is linear in σ 0 .By using (4.8) and (5.3) one can show that the action of the pointed alien derivative ∆κ l = ν κ l ∆ κ l on the trans-series is equivalent to 2iS l times differentiation wrt.σ l : ∆κ l qn,m ({σ}) = 2iS l ∂ σ l qn,m ({σ}) . (5.5) The Stokes automorphism which relates the two lateral Borel resummations S ± ( is the exponentiation of all the alien derivatives, which then acts on the trans-series parameters as Sq n,m ({σ}) = e l=0 ∆κ l qn,m ({σ}) = e l=0 2iS l ∂σ l qn,m ({σ}) = qn,m ({σ l → σ l + 2iS l }) . (5.6) The ambiguity free median resummation is then nothing but the TBA result, if we turn off every σ.Note also that the inverse Stokes automorphism, which relates the two lateral Borel resummations the opposite way is equivalent to σ k → σ k − iS k , which corresponds to the alternative integrations in the contour deformations as we anticipated before.Similarly one can show that W n,m = S − (S 1/2 A m,n ).In particular, for the energy density we obtain W 1,1 = S − (S ), which is nothing but the median resummation of the perturbative series.

Examples
In this section we provide explicit examples for our generic solution in various models.
There is a large class of integrable particle-models, where a magnetic field can be coupled to one of the global charges and the energy density can be investigated by the thermodynamic limit of the Bethe ansatz equations.In these cases the kernel is related to the logarithmic derivative of the scattering matrix and the Wiener-Hopf method leads to a generic structure.We focus here on the bosonic models having where γ, b are model-dependent constants, while H(κ) is a model-dependent product of gamma-functions.We indicated the signs of the residues depending on the two possible ways how we can shift the cut away from the imaginary line.The poles on the imaginary line are determined by the careful analysis of H(κ), which we go through model by model.Clearly, the running coupling can be always introduced with γ and b, together with another linear term coming from the H-s, leading to the kernel where z k is proportional to ζ k in a model-dependent way and q is parametrizing the various running couplings, which we fix to the convenient value q = γ E + 2 ln 2.
We start with the observable A 1,1 , which can be calculated by modifying Volin's method to keep track of the ζ-s coming from the kernel.The result is We regard this as an input to our analysis and show how all the perturbative parts can be determined from this.We can calculate this series analytically up to 50 orders while numerically up to few hundred orders for generic N and up to 2000 terms for N = 4 with very high precision [32,31,22].For demonstration, we merely included here the first few terms, and keep doing the same from now on.By using the differential equation 2A 1,1 + Ȧ1,1 = a 2 1 , i.e. the perturbative part of (5.1), one can obtain Then using (5.2) for n = 1 we obtain By solving (5.2) for other n-s we can get Actually a n can be obtained directly from a 1 by the v → v n , γ → γn, z 2k+1 → n 2k+1 z 2k+1 replacements.The exceptional χ 0 is Finally, by solving (5.1) we obtain the basic building blocks In order to get the generic solutions in terms of these A-s as (4.8,4.11,4.12)we need the locations κ l and Stokes constants S l , which should be found model by model.

O(N ) models
The O(N ) non-linear sigma models in a magnetic field coupled to one of the O(N ) charges [14,15] can be analyzed by the thermodynamic limit of the Bethe Ansatz equation, which takes the form of the integral equation ( 1) with the kernel related to the S-matrix [36] S and the kernel is described by 11) in terms of the running coupling 3.6 defined with the specific choice L = b − 4∆ ln 2 for the arbitrary constant.The zeros of σ(iκ) are located N -independently at the positions κ = 2l − 1, while its poles are at κ = l(N − 2), where l ∈ N.This implies that κ l = lκ 1 with κ 1 = N − 2 for N even and κ 1 = 2N − 4 for N odd [20,24].

O(4) model
Let us start with the O(4) model, which is the simplest.In this case the running coupling is v = 1 2B − 2 ln 2 and ∆ = 1 2 .The poles and the zeros do not interact and κ l = 2l; l = 1, 2, . . ., with residues Observe also that σ(in) = 0 for n odd, i.e. S 0 = 0, so in these cases κ 0 = n is not singular and we do not have the l = 0 term in the sums.With these building blocks the trans-series for the observable w 1 takes the form It is free of ambiguities due to the relation Finally W 1,1 can be obtained from eq. (5.1) as l1,l2,... e −4(l1+l2+... )B iS l1 iS l2 . . .A 1,−2l1 A −2l1,−2l2 . . .A −l k ,1 , (6.15) where M is an integration constant, which comes from the zero mode of 2 + ∂ B .By taking the n → 1 and m → 1 limit in (4.11) it can be calculated explicitly to be M = −2i.The first few terms take the form .16) By explicitly investigating the analytic structure of A 1,1 on the Borel plane (displayed on Figure 6.1) we confirmed the perturbative expansion of all these terms up to high orders.We also verified numerically that the median resummation reproduced the TBA result.
These results have a direct extension for N > 4. The only difference is that κ 1 is N -dependent, otherwise the poles form the lattice κ l = lκ 1 and the generic solutions in terms of the A-s in eqs.(4.8,4.11,4.12)applies.The integration constant for W

O(3) model
The O(3) model is the most complicated among the O(N ) models.This is due to the fact that σ(i) ̸ = 0, and we have to carry the l = 0 term in the sums for W 1,1 and w 1 .The poles of σ(iκ) are again located at κ l = 2l.The building blocks A n,m can be used here  with ∆ = 1.By focusing on w 1 the main difference compared to the O(4) model is that additionally to the O(4) like sums, we also have others starting at ν: What is interesting is that the two parts are not related by any resurgence relations, i.e. the ∆ 1 alien derivative of the first line is not related to the second line.This can be also seen by noting that S 0 is imaginary and the real leading term cannot be related to the purely imaginary alien derivative of a real series.The even alien derivatives satisfy the relations as before ∆ 2l A n,m = 2iS l A n,−2l A −2l,m .The observable W 1,1 can be obtained by integrating the differential equation (5.1).Again the constant term should be fixed from taking the n → 1 and m → 1 limit.

Principal chiral models
The SU (N ) principal chiral model can be described by ) where ∆ = 1/N .In order to use the generic forms we need the replacements and the running coupling is defined with L = b.The poles of σ(iκ) again form a lattice κ l = lκ 1 with κ 1 = N N −1 .This model is very similar to the O(4) model, which is the SU (2) case here.

Supersymmetric O(N ) models
In this model we have The running coupling is defined with L = b − 4∆ ln 2. The first case is N = 5 for which κ 1 = 6.For N > 5 we have to distinguish between the even and odd cases just as we did for the O(N ) models.We actually have the same set as for the O(N ) models and additionally µ l = N −2 N −4 (2l − 1), although for odd N s some of the residues are zero.This is a very complicated pattern and, additionally, the Stokes constants are complex.We are planning to investigate these cases in detail in our forthcoming publication [35].

Conclusion
In this paper we developed a method to solve completely the integral equations (2.1) in terms of a trans-series.By taking the perturbative energy density A 1,1 as an input we determined a set of observables A n,m which constitute a complete basis in the transseries solution.We used these building blocks to construct the full trans-series for various other observables including the generalized energy densities and the boundary values of the Bethe Ansatz densities.We also revealed the analytical structure of all A n,m -s on the Borel plane by determining their (positive) alien derivatives.The singularities on the positive real lines are interplayed such a way that the TBA result agrees with the median resummation.We checked some of our calculations with the explicit examples of the bosonic integrable models, in particular of the O(N ) non-linear sigma models and this provides further support to our basic assumption of lateral Borel resummation.To our knowledge these are the first explicitly solved asymptotically free quantum field theories.
In the statistical physical applications the systems are not relativistically but Galieaninvariant.This implies that the source terms and the moments has to be changed from cosh nθ to θ j , see [11,12] for details in constructing the analogue of our basis and for describing observables at the perturbative level.Our formulas can be related to those by differentiating wrt.n and putting n to zero.However, here we go beyond the perturbative level, and construct the full non-perturbative trans-series.We think that by making the appropriate differentiation the non-perturbative parts of the non-relativistic moments can also be extracted based on our formulae.
We thus hope that our generic solution of the TBA equation will be fruitful and pave the way of further applications both in statistical and particle physics.

Figure 6 . 1 :
Figure 6.1: Analytic structure of the Borel transform of A 1,1 , B(A 1,1), with its convergence radius indicated.The first singularity is a pole at 2 with residue M .The second singularity is a logarithmic cut starting at 4 multiplied with a function whose perturbative expansion is 2iS 1 A 2 1,−2 .This function also has a logarithmic cut starting at 4 with the function (2iS 1 ) 2 B(A 2 1,−2 A −2,−2 ) multiplying it.This cut in B(A 1,1 ) shows up at 8, which coincides with another logarithmic cut with function 2iS 2 B(A 4 1,−4 ).These two cuts can be disentangled by the different reality properties.