$QQ$-system and non-linear integral equations for scattering amplitudes at strong coupling

We provide the two fundamental sets of functional relations which describe the strong coupling limit of scattering amplitudes in $\mathcal{N} = 4$ SYM dual to Wilson loops in $AdS_3$: the basic $QQ$-system and the derived $TQ$-system. We use the $TQ$ relations and the knowledge of the main properties of the $Q$-function (eigenvalue of some $Q$-operator) to write the Bethe Ansatz equations, {\it viz.} a set of ('complex') non-linear-integral equations, whose solutions give exact values to the strong coupling amplitudes/Wilson loops. Moreover, they have some advantages with respect to the ('real') non-linear-integral equations of Thermodynamic Bethe Ansatz and still reproduce, both analytically and numerically, the findings coming from the latter. In any case, these new functional and integral equations give a larger perspective on the topic.

1 Some background and plan of the paper Maximally supersymmetric N = 4 gauge theory in 4D has been revealing many integrable features, mainly in the planar limit, which also involve its strong coupling dual string theory [1]: for instance, integrable spin-chain hamiltonians and S-matrices appear in the computation of anomalous dimensions of composite operators (cf. e.g. [2] and [3] for the related Thermodynamic Bethe Ansatz to be compared with the following). Furthermore, a gluon scattering amplitude equals its dual, namely the null polygonal Wilson loop (WL) and the latter admits an expression as quantum string partition function [4]. In particular, at strong coupling, the amplitude is dominated by the classical contribution, i.e. the minimal area of the surface in AdS 5 ending on the WL in the (boundary) 4D Minkowski. Now, another integrability issue enters the stage and it will be our main concern in this paper. Albeit our investigation is applicable to the full AdS 5 space, we shall confine our analysis to the easier AdS 3 subspace and its classical string solutions. In fact, the problem of finding global coordinates parametrising AdS 3 is equivalent to solving the two spectral linear problems DΨ(z,z; θ) = 0,DΨ(z,z; θ) = 0, with z,z world-sheet coordinates, θ (complex) spectral parameter and D,D linear differential operators depending on a 'potential' η(z,z) (explicitly given by formulae (2.2, 2.3) below). The latter must satisfy the (Lax) compatibility condition (denoting classical integrability), which in this case is a modified Sinh-Gordon equation -(2.5) below -depending uniquely on an entire function p and its complex conjugate,p. We choose p to be a polynomial of degree 2N (integer) so that we describe light-like WLs with 4N + 4 cusps (4N − 2 real cross ratios out of these points: the number of real parameters in p(z) if the highest coefficient is 1 and the sum of its zeroes is null) or a scattering amplitude among 4N + 4 gluons. Besides the aforementioned classical integrability, the two coupled linear problems show remarkable quantum integrability properties: in a nutshell Wronskians of solutions are Q and T functions (eigenvalues of some Baxter Q-operator and transfer matrix respectively) of a generalised Baxter setup with moduli (e.g. the complex coefficients of the polynomial p(z)). This is clearly an extension of the so-called ODE/IM correspondence [5,6], which still maintains the Q-functions and their functional relations, the QQ-systems, as fundamental building blocks.
Actually, some T s and Y s are already present in previous papers [7], while here and in the forthcoming paper [8] we make the construction more general and extended by founding it on the basic Q functions and their QQ relations. Next, we derive the fundamental generalised Baxter T Q relations and easily from these the Bethe Ansatz equations which are not so tractable because of their infinite number. Therefore, we convert them into a system of 2N non-linear integral equations (NLIEs) depending on the 2N -ple of the coefficients of the polynomial p(z). Crucially, for the whole construction of our theory we make use of two distinct discrete symmetries,Ω andΠ, of the linear associated problems DΨ = 0,DΨ = 0, which, yet, gain their effectiveness by a non-trivial action on the vacuum Ψ (not invariant): in other words these are broken symmetries giving birth to the characteristic quantities of integrability (Qs, T s and Y s, which are generically the generators of the (local and non-local) conserved charges). Eventually, we propose an expression for the free energy in terms of solutions of the NLIEs. We claim and verify both analytically and numerically in many cases that our expression coincides with the free energy of [7] written in terms of solutions of Thermodynamic Bethe Ansatz (TBA) equations.
In fact, the NLIEs show many advantages with respect to TBA, at least for the polynomials we studied, which, technically speaking, are in the so-called maximal chamber [7]: for polynomials of degree 2N their number is 2N , whilst the number of TBA equations is 2N (2N −1)/2 (quadratic grow); moreover, the procedure to write NLIEs is undoubtedly simpler than (and should shed light on) the corresponding one for TBA, which passes through (N − 1)(2N − 1) 'wall crossings'. In addition, from the ideological point of view the NLIEs, being equivalent to the infinite Bethe Ansatz equations, lie at the heart of the integrability of the problem, and one can express in terms of their solutions all the relevant quantities, such as the Q functions, the transfer matrices eigenvalues T and the Y functions. We will provide explicit expressions for all that at the end of Section 3.
The paper is organised as follows: in the next Section 2 we sketch quickly the derivation of all the functional relations which will be better analysed in [8]. Then, Section 3 is the core of the paper, as we derive a system of NLIEs and propose an expression for the free energy in terms of solutions of the NLIEs. Our results for the free energy are tested analytically in some simple cases (Section 4) and numerically in many examples (Section 5), finding perfect agreement with analogous computations made by using TBA. Finally in Section 6 we study the 'conformal limit' in which the auxiliary linear problem reduces to a Schrödinger equation: our set of NLIEs reduces -as expected -to the NLIEs introduced in [9, 10].

Functional relations
As sketched in the introduction, we start from the linear problem (the bar means complex conjugation) 2) 3) The polynomial depends on a 2N − 1-ple of complex coefficients c = (c 0 , ..., c 2N −2 ) or moduli. Zero curvature condition [D,D] = 0 constraints the 'potential' η(z,z; c) to be solution of the classical modified Sinh-Gordon equation To completely specify η, we impose the condition η = l ln zz + O(1) as |z| → 0, for |l| < 1/2. Operating in such a way we are extending framework of [12] by introducing the moduli c (besides c 0 ) and the AdS 3 string problem representing the WL [13] which is regular in z = 0, i.e. l = 0. Similar extension is the series of homogeneous Sine-Gordon models [14].
The linear problem has two symmetries. The first one acts aŝ TheΩ symmetry leaves invariant the polynomial p and the function η: p(ze iπ N , c R ) = p(z, c), η(z,z; c) = η(ze iπ N ,ze − iπ N ; c R ). However, it produces a change of sheet and then p(ze iπ N , c R ) = − p(z, c). This symmetry involving rotations of the moduli appeared in the context of a Schrödinger equation with polynomial potential in [15] and, recently, in [16,17]: we will analyse this as limiting 2D conformal case of our massive one in Section 6. Another symmetry of the linear operators D andD is given by the transformationΠ Now, following [12], we can fix two couples of solutions. The first one, Ψ ± (z; θ, c), is completely specified by the expansion around z = ρe iϕ = 0,z = ρe −iϕ = 0 The second couple (a Jost basis) is given by the solutions Ξ(z; θ, c) and Ξ 1 (z; θ, c) with following asymptotic leading expansion around z = +∞ We found convenient to modify the second solution by introducing the phase where the coefficient B −1 ( c) appears in the large |z| expansion of p(z, c): This means that the function Φ(θ, c) = 0 is non zero only if N is integer. Important properties of Φ are . These sets of solutions transforms in a nice way under the symmetriesΩ andΠ: Now, it is natural to connect the two sets of solutions by means of z independent coefficients Q ± (θ, c): Then, applyingΩ to both sides of this equation taking the determinant of both equations and using the explicit results det(Ξ, Ξ 1 ) = −2ie iΦ(θ, c) and det(Ψ + , Ψ − ) = −1/ cos πl, we end up with a functional relation that may be considered the foundation of the integrability of the theory, the QQ-system, (2.20) This form for the QQ-system is very peculiar since it involves rotations of the moduli and the presence of the extra phase Φ as well. Because of the presence of different moduli, relations (2.20) are not closed, similarly to functional relations present at the level of Y -systems in [18]. More in general, we can think of (2.20) as extensions of analogous relations underlying integrability in the case of N = 4 SYM in 4D and N = 6 SCS in 3D and which are often called quantum spectral curves [19].
Using togetherΩ andΠ on equation (2.18) we obtain a particular form of quasi-periodicity involving the rotation of the moduli c Quasi-periodicity (2.21) allows to bring the QQ-system (2.20) in a form which does not involve rotations of the moduli and the phase Φ: Another useful property of Q ± is its behaviour under complex conjugation which derives from the relations In addition, usual real-analyticityQ holds.
In order to write NLIEs we need to introduce the transfer matrix T , which we define as a quadratic construct of Q's: Using the QQ system, we get the T Q-relation or T Q-system, Importantly, it follows from their construction that both T (θ, c) and Q ± (θ; c) are analytic functions of θ. This information will be used in next Section to derive in quite a standard way a system of Bethe equations and NLIEs.
We stress that that the construction of QQ and T Q-systems was possible thanks to the use of theΩ symmetry. This symmetry was not used in [7], but we underline its importance to unveil the basics of integrability of the problem. More importantly,Ω symmetry is essential when differential operators have two irregular singularities, like in Liouville ODE [20].
The last ingredient for the derivation of NLIEs is the leading behaviour of Q ± when Reθ → ±∞. This is obtained by studying the linear problem (2.1) at large θ. Details will be reported in the publication [8]. We give here only the final result: n=0 c n x n by a change of sign and a half rotation of coefficients c: The function q N is another polynomial, defined by the condition which assures convergence of the integration in (2.29).
Remembering the form of the p times rotated 2N − 1-ple we expect 2N NLIEs, since 2N is the number of discrete rotations (including the identity, i.e. the number of independent sets of c R p ) of the group acting on the moduli space. In order to write the NLIEs, we recall the T Q-system (2.27) which indicates, as usual, that the zeroes of Q are indeed the Bethe roots, θ n ( c; k). This suggests us the definition of their counting function [21] so that the Bethe Ansatz equations take the form and equivalent ones obtained by use of the periodicity of the counting function Z(θ; k, c) = Z θ + iπ + iπ N ; k, c R −1 . Therefore it is very relevant to recast the counting function in a more explicit form as (3.6) In order to understand the position of the Bethe roots we refer to the behaviour for large θ of Z(θ; k, c R m ). We claim that, when θ → ±∞, where we introduced the adimensional quantity r( c R m ) = |r( c R m )|e iφm and used the propertyw 0 ( c) = w 0 (¯ c R −1 ). Relation (3.7) comes from (3.4, 2.28). From these definitions it follows that the quantity r( c) is obtained by an integration on a contour Γ defined by ] − i∞, −i0] [0, +∞[: In addition, from the asymptotic behaviour (3.7) we deduce that, for very large θ n , Im θ n ( c R m ; k) = −Imθ n ( c R m ; −k) = −φ m . We suppose that for finite θ n the position of the Bethe roots does not displace much from such lines. Then, a general sum over the Bethe roots θ n ( c R m ; k), −θ n ( c R m ; −k) can be converted into integrals by using Cauchy theorem [21], which fact for Z itself boils down, after standard manipulations, to with > 0 and L ± (x; k, c) = 1 2i ln 1 + e ±iZ(x±i ;k, c) . The positive number is chosen in such a way that integration contour contains all the Bethe roots, i.e. |Im θ n ( c R m ; k) + φ m | < , ∀n, m.
The kernels K m (x) equal 11) and in Fourier transform read Using the periodicity K m (x) = K m+2N (x) of the kernels and bringing the integrations on the real axis, we can also write Passing to Fourier transforms, we define the matrixM i,j (p) = δ i,j − e pφ iK j−i (p)e −pφ j , whose determinant is explicitly computed: The inversion ofM i,j (p) allows to write the final set of NLIEs. The driving term for the n-th equation equals plus zero modes of the matrixM i,j , i.e. solutions of the homogeneous equationM i,j c j = 0. In this respect, we remark the following property which holds for all m = 0, ..., 2N − 1, except when m = N − 1 for δ(p − i) and for m = N + 1 for δ(p + i) (therefore, this two 'problems' appear only for integer N ). This means that zero modes are linear combinations of vectors with the form c (±,m) n = e ∓iφn e iπnm N δ(p ± i), with m = 0, ..., 2N − 1 labeling the independent solutions -with the two exceptions mentioned before. We remark that the quantities r 0 ( c R n )e θ−iφn ,r 0 ( c R n )e −θ+iφn can be constructed out of the zero modes, i.e. we have the relation which we have numerically verified, see Table 3 and Table 6 in Section 5. Adding then the last two terms of (3.14) coming from the extra phase Φ, we completely reconstruct the asymptotic behaviour (3.7) previously announced, i.e.
Putting everything together, the final equations are (0 ≤ n ≤ 2N − 1): We now give the explicit form for the kernels: the periodicity G (N ) (3.19) Equations (3.18) with kernels (3.19) are valid in the strip defined by the conditions |Imθ − φ n + φ m ± | < min{π, π/N }, ∀n, m. They can be solved analytically in few cases, which we report in next Section, or alternatively by numerical iteration: we will discuss many examples in Section 5. We observe the simple summation whereĜ LZ (p)| α=N is the Fourier transform of the kernel of equation (3.32) of [12] (and of (2.29) of [22]) to which our equations (3.18) reduce in the case in which Z(θ; k, c R n ) do not depend on n.
Once we know Z we can very reasonably conjecture that the free energy be given by the functional 3 with f n (θ) = |r( c R n )| sinh θ, so that when l = 0 it does coincide with the TBA contribution [7,23] to the area (scattering amplitude) A = A f ree +A periods of 4N + 4 gluons. The functionsỸ AM SV s (θ), whose number N max depends on the parameters c and varies from 2N − 1 (minimal chamber) to (2N − 1)N (maximal chamber), satisfy TBA equations. For parameters in minimal chamber TBA equations have the form (26) of [7]: , s = 1, ..., 2N − 1 .
In next Sections, we test the conjectured coincidence in examples where analytic or numerical computations can be done. For what concerns the precise connexion with TBA equations of [7,23], this will be pursued in more details in a future publication. For the moment we anticipate the relation between the functions Y AM SV [7] and the counting function: Before ending this part, we want to stress that NLIEs (3.18) are valid if |c n |, n = 1, ..., 2N − 2 and, consequently the phases φ n , are not too large: to be precise, remembering what we wrote just after equation (3.19), the condition |φ n − φ m | < min{π, π/N }, ∀n, m has to be satisfied. In this case, we have 2N NLIEs in the form (3.18). For general c n the form of the NLIEs changes and we expect the occurrence of a phenomenon analogous to the 'wall crossing' of TBA equations [7,24,23]: in the context of integrability this phenomenon already appeared and was called 'second determination' of the NLIEs [21].
As we discusses in the introduction, the main importance of the NLIEs in the context of integrability is that one can express in terms of the counting functions all the relevant functions of the problem. The connection to the Y -functions descends from (3.24). On the other hand, the transfer matrix is a functional of the counting functions. For instance, in the simplest case, in which there is no dependence on the moduli, for the ground state eigenvalue we find, following procedures discussed in [25,22], where, when 0 < Imθ < π, and, when −π/N < Imθ < 0, with (3.28) Finally, the ground state eigenvalue of the Q function in the simplest case with no moduli has been written in formulae (3.38-3.40) of [12] and therefore we refer to that paper for explicit expressions. Formulae for the eigenvalues of the T and Q functions in terms of Z in the general case with arbitrary moduli are only technical complications and therefore for the moment we do not write them.

Regular polygons
The regular polygon case corresponds to polynomials This means that c 2N = 1, whilst all the other moduli c n = 0, n = 0, 1, ..., 2N − 1. This is the 'massless' limit w 0 ( c) = 0: in this limit, the evaluation of the free energy is done by using the 'dilogarithmic trick' applied to NLIEs (see [21]). We have that where the effective central charge c ef f is easily computed (following Section 7.1 of the first of [21]) as Then, from (4.3) and (4.2) we get the free energy which agrees with (4.7) of [23], since the number n there is expressed by our N by n = 2N + 2 . The octagon Another case in which explicit analytic computation is possible is the octagon: N = 1, p(z) = z 2 + c 0 . This case corresponds in the ODE limit to the harmonic oscillator and the solutions of the two NLIEs are 4 Z(θ; k, c) = Z(θ; k, c R ) = π|c 0 | sinh θ − 2πk , c = (c 0 , 0, 1) . (4.6) In order to evaluate the free energy it is useful to rewrite (3.21) as: We stick to the case l = 0, that is k = 1/4. Then, we choose the arbitrary constant = π/2: with these positions Expanding the logarithm in series, we find the result as a series of Bessel functions K 1 : which agrees with computations made [26] using minimal area.

Numerical computations
We report some numerical computations of free energy. We test our NLIEs against the TBA equations in [7,16]. Let us first consider the polynomial p(z, c) = z 4 + c 2 z 2 + c 1 z − 1, where c 1 and c 2 are chosen to keep the TBA equations in maximal chamber. In this case, the TBA equations are given by 5 and denotes the convolution. The kernels in the integral equations are .

(5.3)
When c 1 = 0, we get the double well potential, where m 1 = m 3 and m 12 = m 23 . The r in NLIEs can be computed explicitly by using (3.8). In Table 1, we list several examples for r( c) and r( c R 1 ). Other r( c) are obtained by r( c R 2 ) = r( c) and r( c R 3 ) = r( c R ). The free energy can be computed by solving the NLIEs (3.18) and then by substituting the numerical solution into equation (3.21) for the free energy. In Table 2 we show the numerical results of 6 π F and 6 π A free for several values of c 2 . As expected, F matches A free with a very high precision.     We then consider the polynomial with c 1 = 0, for which the phase Φ(θ, c) in equation (2.11) is nontrivial. Evaluating the masses (period integrals), we find m 1 = m 3 and m 12 = m 23 . In Table 3 and Table 4, we list r 0 ( c) and r( c), respectively, as defined in (3.8). In Table 5, we show the 6 π F and 6 π A free for the polynomial p(z, c) = z 4 + c 2 z − z 200 − 1 with several c 2 . Table 3: 3.4841068384904066' 3.5080872072271383' − 0.000024977324456942895'i 3.4841567931393205' Table 4: r( c), r( c R 1 ), r( c R 2 ) and r( c R 3 ) for the polynomial p(z, c) = z 4 +c2z 2 − z 2.5881765299655615' 5.049378533767665' − 0.010172223518009327'i 2.5771050504656823'  We then fix c 2 to be 1, and vary c 1 . In Table 6 and Table 7, we list the r 0 ( c) and r( c) respectively for this case. In Table 8, we show the 6 π F and 6 π A free for the polynomial p(z, c) = z 4 + z 2 + c 1 z − 1. Table 6: r0( c), r0( c R 1 ), r0( c R 2 ) and r0( c R 3 ) for the polynomial p(z, c)  Table 7: r( c), r( c R 1 ), r( c R 2 ) and r( c R 3 ) for the polynomial p(z, c) = z 4 +z 2 +c1z −1.  In a similar way, we have tested our NLIEs (3.15) for the polynomial p(z, c) = z 3 + c 1 z − 1 against the TBA equations in the maximal chamber 6 . In Table 9, we present 6 π F and 6 π A free for several values of c 1 . Table 9: 6 π F and 6 π A free for p(z, c) = z 3 + c1z − 1. The NLIEs are solved by Fourier discretization with cutoff (−12, 12) and 2 12 points. The parameter in eq.  Our numerical results show that F = A free , (5.4) at least in maximal chamber. In other chambers, we need to consider the wall crossing of NLIEs and TBA equations, which will be pursued in a future publication. At the end of section, it is worth to note that our NLIEs are much simpler than the TBA equations, especially when the number of cusps of Wilson loop is large. For the lightlike Wilson loop with 4N + 4 cusps, one needs 2N (2N − 1)/2 TBA equations to compute the free energy in maximal chamber, while the number of NLIEs is only 2N . Moreover, one needs to perform (N − 1)(2N − 1) times wall crossing to derive the TBA equations in the maximal chamber, which is quite complicated. Therefore, it is much convenient to use the NLIEs instead of TBA equation, especially in maximal chamber.

Conformal limit
Consider the NLIEs in the limit with c cf t n , θ cf t finite. Then, we have that The linear problem in the variable z, after the scaling e θ cf t , equals −π/2N times the coefficient of the term x N −1 in V (x), in agreement with [9] and [10].
To complete our comparison between our NLIEs in conformal limit and the NLIEs found in [9] and [10], we report here some numerical results. We consider as an example the polynomial p(z, c) = z 4 + |c 0 | 3/4 c cf t 1 z − |c 0 | with a small negative c 0 . Under the limit (6.1), w 0 ( c)e θ becomes where z = |c 0 | 1/4z and q 2 (z) is the related polynomial which makes the integration convergent. In Table  10 and 11, we list r 0 (c cf t 1 ) and r(c cf t 1 ) Table 10: r0((c cf t   Moreover, the driving terms in NLIEs (3.18) become 1 2 |r((c cf t 1 ) R n )|e θ cf t in the conformal limit. With these driving terms we evaluate (3.18) in the conformal limit numerically.
(6.10) C 1 and C 2 are integration contours just below and just above the real θ cf t axis with a distance . We introduce the related free energy obtained from these NLIEs by c cf t 1 )e θ cf t +i ln 1 + 1 a − (θ cf t + i ) .
(6.11) Then, our free energy in conformal limit F cf t and F lead to the same result when l = 0. In order to understand the precise relationship between our NLIEs and the ones of (6.9), we solve both numerically. We evaluate the two type of NLIEs by Fourier discretization with a cutoff (−12, 12) and 2 10 points. The parameter related with integration contour is fixed to be 1/2 4 . In Fig.1 and Fig.2, we plot the real and imaginary parts of the functions −iZ(θ − iφ n , c R n ) and ln a ± (θ), respectively, with fixed c cf t 1 = − 1 100 and l = 0.
In Fig.3 and Fig.4, we plot the real and imaginary parts of the functions −iZ(θ−iφ n , c R n ) and ln a ± (θ), respectively, with fixed c cf t 1 = − 1 10 and l = 0.

Conclusions
We have constructed the main functional relations, QQ-and T Q-systems, underlying the (integrable) computations of the gluon scattering amplitudes in (planar) N = 4 SYM at strong coupling. Kinematical data of scattered gluons are encapsulated in complex quantities, the moduli, which appear 'dynamically' in the functional relations. In addition, we have provided a system of coupled NLIEs which renders effective the analytical and numerical explicit computation of these amplitudes. We tested our equations in the case of regular polygons (conformal limit) and the octagon ('harmonic oscillator' limit) by means of explicit analytic computations. In addition we provided various numerical tests for polynomials corresponding to polygons with ten and twelve sides.
For the future, the principal aim is that our disentanglement of the integrability structure of the problem could shed light on the string quantisation, which, in the end, should reduce to a suitable deformation of the classical model (2.5, 2.1, 2.2, 2.3) or the derived functional equations or the Bethe Ansatz equations (3.5).
Then, it would be of interest to connect the linear problems with the strong coupling limit of an OPE series [27,28], possibly its restriction to AdS 3 . On the other way around, it is very useful to generalise our method, based on SU (2), to the full AdS 5 (at strong coupling), the SU (4) Hitchin system indeed.
In addition, the formalism discussed in this paper could be applied also to Wilson loops in Euclidean AdS 3 , which problem becomes, after Pohlmeyer reduction, the modified cosh-Gordon equation [29].
An important final remark concerns the solution of classical equation (2.5). In a forthcoming publication [8] we show that it is obtained as a solution of an equation with the form of Gelfand-Levitan-Marchenko [30] for which the inhomogeneous term (scattering data) is provided by T , the eigenvalue of the transfer matrix. This form of the equation is particularly important as it admits the solution to be written in terms of Fredholm determinants (i.e. tau functions) η =η + 1 4 ln PP ,η = ln det(1 + K) det(1 − K) (7.15) and then, for instance, as a series expansion (7.16) η is real due to the propertyT (θ, c) = T (−θ, c), which comes from (2.23, 2.26). And this is in fact the explicit solution of the initial problem, for l = 0 the string problem 7 , and we may think that a similar solution should exist for the more general AdS 5 case in terms of classical tau functions for the SU (4) Hitchin system.