Excited Hexagon Wilson Loops for Strongly Coupled N=4 SYM

This work is devoted to the six-gluon scattering amplitude in strongly coupled N=4 supersymmetric Yang-Mills theory. At weak coupling, an appropriate high energy limit of the so-called remainder function, i.e. of the deviation from the BDS formula, may be understood in terms of the lowest eigenvalue of the BFKL hamiltonian. According to Alday et al., amplitudes in the strongly coupled theory can be constructed through an auxiliary 1-dimensional quantum system. We argue that certain excitations of this quantum system determine the Regge limit of the remainder function at strong coupling and we compute its precise value.

On the gauge theory side, the discussion was stimulated by the intriguing BDS formula of Bern, Dixon and Smirnov [14]. It encapsulates the known infrared and collinear behavior of n-particle maximally helicity violating (MHV) amplitudes in the planar approximation. The authors of [14] conjectured the BDS formula to determine the amplitudes at each loop order L ≥ 2 up to some additive finite function R (n) of the kinematic variables. R (n) has been dubbed the 'remainder function'. Both string and gauge theory arguments suggest that the remainder function R (n) vanishes for n = 4, 5. In the weakly coupled theory, perturbative computations have uncovered an important new symmetry of scattering amplitudes [15]. Assuming this so-called dual conformal symmetry, the remainder functions therefore should be connected with the remainder function R (6) . The BFKL color-octet Hamiltonian coincides with the Hamiltonian of an integrable open spin chain [25]. Similar results hold for scattering amplitudes involving n > 6 gluons. In a processes 2 → 2n there exist Regge cut contributions with up to n gluons and the corresponding spin chains consist of n spins. Hence, the weak coupled theory provides direct evidence for integrability in the high energy behavior of planar scattering amplitudes.
Having reviewed all these results from gauge theory it is natural to ask what string theory has to say about the high energy limit of the remainder function R (6) . In order to understand how the issue can be addressed, we need to briefly sketch the development that was initiated by the work [26] of Alday and Maldacena. The main insight of this paper was to identify the leading strong coupling contribution to an n-gluon amplitude with the area A n of some 2-dimensional surface S n inside AdS 5 . According to the prescription of [26], S n ends on a piecewise light-like polygon in the boundary of AdS 5 . The light-like segments of this polygon are given by momenta p j of the external gluons. For n = 4 it is possible to find the surface explicitly and the resulting amplitude matches the prediction from the BDS formula. Constructing S n for n > 5, however, turned out to be a rather difficult problem, at least for finite n and generic choice of the external momenta. The issue was resolved through a series of papers [27][28][29] in which the area of S n is related to the free energy of some auxiliary quantum integrable system. More precisely, it was argued that A n may be computed from a family of functions Y s,j with s = 1, 2, 3 and j = 1, . . . , n−5. The latter can be determined by solving a set of coupled non-linear integral equations. Very similar mathematical structures are actually familiar from the study of ground states in 1-dimensional quantum integrable systems. Moreover, the functional A n resembles expressions for the free energy of such systems.
We employ this auxiliary quantum integrable system to study the Regge limit of the remainder function R (6) at strong coupling. The non-linear integral equations for the Yfunctions we have mentioned in the previous paragraph provide an explicit set of equations which enable us to take the high energy limit and to investigate the analytic continuation between the physical and the mixed region. As the main result, we determine the value of the exponent E 2 at strong coupling, In passing to the mixed regime, we need to analytically continue the parameters of the auxiliary quantum integrable system. This continuation leads to a new system of equations with additional terms. The structure of the new integral equations and the corresponding modifications of the relevant energy functional are again familiar from the theory of quantum integrable systems. In that context they are used to describe excitations of the system. Our formula for the value of E 2 at strong coupling assumes that we can perform the analytic continuation into the mixed regime within the parameter-space of the quantum integrable system and that the corresponding amplitude is given by the exponential of the energy functional, as in the Euclidean regime.
This paper is organized as follows. In section 2 we define the kinematic regions and the path of analytic continuation for the kinematic variables. Section 3 starts with a brief summary of relevant results from the work of Alday et. al. As a first exercise we then verify that the Regge limit of the remainder function R (6) vanishes in the physical regime (up to an irrelevant constant). Sections 4 and 5 contain the main part of our analysis. In these two sections we perform the analytic continuation, discuss the origin of the additional terms and calculate their contribution to the amplitude. For pedagogical reasons, we first illustrate the general arguments with a simpler example that is taken from the work [30,31] of Dorey et al. The generalization to the system of Alday et al. is straightforward. Section 5 is devoted to the analysis of the resulting equations. With strong support from numerics we argue that the transition to the mixed regime is associated with the simplest kind of excitations in the auxiliary integrable system. Once this is established, the precise value of E 2 can be calculated analytically. A few technical details are contained in two appendices.

Multi-Regge kinematics of six-gluon amplitudes
We consider the scattering of six gluons with momenta p j , j = 1, . . . , 6 satisfying the onshell condition p 2 j = 0. From these momenta we can form Lorentz invariant combinations of the form Throughout this note we shall use the metric η =diag(−, +, +, +). Furthermore, we extend the range of the index j on p j to the integers with the periodicity condition p j ≡ p j+6 . Due to momentum conservation the quantities x 2 ij are symmetric in their indices, i.e. the obey x 2 ij = x 2 ji . Six-point amplitudes of a 4-dimensional gauge theory depend on eight kinematical invariants. Hence, there exist many relations between the x 2 ij we introduced above.
In the literature, six-point amplitudes are usually parametrized in terms of the following nine invariants, as shown schematically in figure 1, Variables denoted by s or t are referred to as energies and momentum transfers, resp. In the Regge limit, energies are large (and can be positive or negative), whereas momentum transfer variables are negative and finite. As an example, a positive energy s implies the negative distance x 2 26 . Since we have only eight independent variables, one of the above can be eliminated with the help of the Gram determinant, namely det([p i · p j ] i,j=1,...,5 ) = 0. Since each of the products [p i · p j ] may be expressed through the kinematical invariants defined in eqs. (2.2), we can use the Gram determinant relation to eliminate one of these nine invariants.
In the physical regime, all the 's-like' kinematical invariants are positive while the 't-like' variables are negative, i.e. s, s i , s 456 , s 345 > 0 and t i , s 234 < 0 . (2.3) Finite quantities in a theory with dual conformal invariance can only depend on crossratios. For a six-point amplitude, there are three independent conformal cross-ratios which are defined as (2.4) For later use we have expressed them both in terms of the general kinematical invariants x ij and in terms of the nine special invariants that were defined in eq. (2.2).
In the next section we shall look at the multi-Regge limit for six-point functions which, by definition, explores the regime (2.5) The conformal cross-ratios are functions of the kinematical invariants and one can study there behavior in the Regge regime. It turns out that vanish in the same order (see Appendix A). Hence, the leading asymptotics of any conformal invariant of the momenta p i may be parametrized by the two quantities In this simple multi-Regge limit, the six gluon scattering amplitude actually agrees with the answer that is predicted by the BDS formula. This is because the limiting values (u 1 , u 2 , u 3 ) Regge = (0, 0, 1), corresponds to a special case of the collinear limit, namely u 1 = 0, u 3 = 1 − u 2 where u 2 → 0. In the collinear limit, the six-point amplitude reduces a 5-point amplitude which is known to be BDS exact. Hence, the Regge limit (2.5) of the BDS formula for six gluons needs no corrections and therefore is of no further interest to us, at least for the moment.
As we have outlined in the introduction, however, there exists another Regge limit in which the BDS formula requires corrections, i.e. the remainder function does not vanish. Reaching this limiting regime requires to pass to a new kinematic region which is defined as s, s 2 > 0 and s 1 , s 3 , s 345 s 456 < 0 , t 1 , t 2 , s 234 < 0 . (2.8) In the corresponding multi-Regge limit all energy variables are much larger than the momentum transfer variables, x 1 x 2 x 3 x 4 x 5 x 1 x 2 x 3 x 4 x 5 In order to reach this regime, one either starts from the region where all invariants are negative and then continues in s and s 2 or, alternatively, one first goes to the physical region and then continues all the following energy variables analytically along the following simple path At the final point ϕ = π all three cross-ratios come back to their original values. Along the way, the cross-ratio u 3 moves along a full circle. Our analysis below starts with some conformal amplitude A(u i ) in the physical regime. Its precise form will be described in the next section. In a first step we want to continue this amplitude analytically along the path (2.11). Since A(u i ) is not single valued in the complex u−space, the resulting amplitude A C (u i ) differs from the amplitude we started with. Then we perform the Regge limit. This leaves us with a function a(ũ 1 ,ũ 2 ) that describes the leading Regge asymptotics of A C . It is this quantity we are actually after.
The amplitude we shall subject to this analysis is taken from the recent work of Alday et al. By construction, it is initially defined in the Euclidean region in which all the kinematical invariants (2.2) are negative. In order to go to the physical region (2.3) one needs to continue s, s i , s 345 and s 456 analytically while keeping t i and s 234 fixed. This continuation may be performed such that all three conformal cross-ratios u i remain fixed.
Hence, conformal amplitudes do not require analytical continuation in order to pass from the Euclidean to the physical region.

Six gluon amplitudes at strong coupling
In the first part of this subsection, we shall briefly recall the prescription of Alday et al. for six gluon amplitudes at strong 't Hooft coupling. The second subsection contains an analysis of the Regge limit in the physical regime. We shall demonstrate that the Regge limit of the six gluon amplitude is BDS exact, in agreement with general arguments.

Review of results from Alday et al.
According to the work of Alday et al. [28], the strong coupling limit of a six gluon amplitude in N = 4 supersymmetric Yang-Mills theory is obtained through exponentiation of the quantity at least up to a constant that is independent of the kinematics. The various pieces on the right hand side are all known explicitly. In particular the divergent contribution is given by where ǫ is the cut-off. The BDS-like amplitude is also relatively simple. In the case of six gluon scattering, it is given by For us, it is most relevant to have an expression for the difference between the usual BDS and the BDS-like amplitudes. This difference is given by The conformal cross-ratios u i that appear in this expression were defined in the previous section.
The two remaining contributions to the total amplitude cannot be written directly in terms of the three cross-ratios u i . Instead, they involve a new set of parameters m, C, φ that are related to the u i through some functional relations to be spelled out below. Given m, C, φ Alday et al. instruct us to solve the following system of non-linear integral equations for three functions These formulas can be used as long as | Im(θ − iφ)| < π/2. For values of θ outside this strip, one can employ certain recurrence relation to determine Y i ; details are given in the Appendix B. The equations (3.5) and (3.6) involve two integral kernels K 1 and K 2 that are given by Suppose for the moment that we can solve these equations somehow. The resulting functions Y i (θ) clearly depend on the parameters m, C and φ in some complicated way. It is now possible to express m, C, φ through the cross ratios u i by solving the following three equations The last equation contains the function Y 2 outside the strip |Im(θ − iφ)| < π/2. With the recurrence relations described in the Appendix B we rewrite this equation in the following form Once m = m(u i ), C = C(u i ) and φ = φ(u i ) are known, we can construct the two missing terms of the total amplitude. The so-called free energy is given by The final contribution to the amplitude depends on the cross ratios only through the parameter m = m(u i ), This concludes our description of the input from the work of Alday et al.

Expansion for large masses
It is instructive to first study the Regge limit in the physical region and to verify that the remainder function does indeed vanish (up to a constant). Most of the formulas in this subsection have appeared in the literature before, see [32,33]. We repeat them here mostly for pedagogical reasons.
In the region (2.3) one can show that the multi-Regge limit is mapped onto large values of m, while C remains constant and φ goes to zero. To see this we start from eqs. (3.8), (3.9) and observe that our Regge limit (2.6) forces the values Y 2 (±iπ/4), Y 1 (0), Y 3 (0) to approach zero at the same rate, i.e. the right hand sides of eqs. (3.5) and (3.6) must become large and negative. As long as |φ| < π/4, the parameter m is forced to become large and the first terms on the right hand side dominate, i.e.
As we shall demonstrate further below, these approximations are justified since the integrals on the right hand side of eqs. (3.5) and (3.6) remain finite. This will no longer be that case after analytic continuation into the kinematic region (2.8). Taking the difference of the two equations (3.6) we obtain (3.14) As we argued above, the Regge limit requires both Y 1 (0) and Y 3 (0) to approach zero at the same rate so that their ratio stays fixed. Hence, in our Regge limit C is a finite constant. Finally, the ratio of u 1 and u 2 , which should also be nonzero and finite in the Regge limit, is given by Here we have omitted terms of lower order in the Regge limit. Using eq. (3.12) we conclude that m sin φ has to be finite, i.e. that φ tends to zero. For the following analysis of the Regge limit it is convenient to introduce the parametrization ε = e −m cos φ , w = e m sin φ , c = cosh(C) .

(3.15)
We shall expand all equations from the previous subsection in the limit ε → 0 while keeping both w and c fixed. From eqs. (3.8) we find that One easily verifies that the dependence of the conformal cross ratios on ε agrees with their behavior in the Regge limit, i.e. u 1 , u 2 and 1 − u 3 vanish in the same order, when ε is sent to zero. The leading contributions to all conformal invariants in the Regge limit are parametrized by the two functions We can invert these two equations to determine w and c as functions ofũ 1 andũ 2 , Our aim now is to evaluate the dependence of the so-called remainder function R, in the Regge limit as a function ofũ 1 andũ 2 . The function ∆ was spelled out in eq. (3.4).
In our limit we find that it is given by The divergent contribution is actually cancelled exactly when we subtract the period contribution to the remainder function It now remains to compute the contribution from the free energy to leading order in the Regge limit. When we send m to infinity, the functions Y i become small all along the real axis so that we can approximate log(1 + Y i (θ)) ∼ Y i (θ). In this limit, the free energy then takes the form All three integrals can be evaluated easily with the help of the following integral formula such that In the limit of large m we can use the asymptotic form of xK 1 (x) ∼ πx/2 exp(−x). Since the parameter m depends on ε through we conclude that the free energy vanishes when ε is sent to zero. Indeed, once we insert the leading dependence of m ∼ − log ε in ε into the formula (3.23) for the free energy, we obtain where . . . represent lower order terms. Combining this result with our previous equations (3.19), (3.20) and (3.24) for the Regge limit of the amplitude ∆ and the periodic contribution we obtain, As we had anticipated, the leading Regge asymptotics of the remainder function in the physical regime (2.3) is a constant.

Excited states in thermodynamic Bethe Ansatz
In this section we shall perform the analytic continuation of the kinematic variables along the path (2.10), and we show that excited states of the system start to contribute. To explain the basic ideas, we begin the presentation of a toy model [30,31]. The corresponding analysis of the system (3.5) and (3.6) is more involved and is performed in the second subsection.

Analytical continuation and excited state TBA
As a warmup example we want to study a rather simple non-linear integral equation that depends on a single complex parameter m. We are interested in the behavior of the energy F 0 (m) as it is continued analytically in the parameter m. Our toy system involves a single function Y (θ) which is determined by We shall think of the kernel function K(θ) as being determined through an S-matrix, If the S-matrix is unitary, i.e. S(θ)S −1 (−θ) = 1, then the function log S(θ) is antisymmetric and so K(θ) is invariant under θ → −θ. Let us furthermore assume that the ground state free energy is given by In order to understand how F 0 (m) depends on the mass, it is quite instructive to study a trivial system with constant S-matrix. In this case, the equation (4.1) determines the solution log Y (θ) = −m cosh θ and we can evaluate the free energy of the ground states exactly, The branch cut of the logarithm is chosen to lie along the negative imaginary axis. For the integration we have used formula (8.526) from [34]. Then we expressed the infinite sum over Bessel functions K 1 through simpler functions with the help of In addition, we used the formula [xK 1 (x)] ′ = −xK 0 (x) for the derivative of xK 1 (x). The free energy F 0 (m) possesses a series of square root singularities at the points x k = i(2k−1)π. Upon analytical continuation along a curve C that encircles the first N of these singularities in the upper half of the complex plane, we obtain We shall reproduce this simple answer down below through an analysis that does not require the S-matrix to be trivial.
When the system possesses a non-trivial S-matrix, the free energy can no longer be evaluated exactly. Nevertheless, we can still analyse how the the free energy F 0 (m) changes under analytic continuation in the mass parameter m. We begin by re-writing our non-linear integral equation (4.1) in the form .
Note that the integrand has poles at those points θ 0 at which the solution Y (θ) assumes the value Y (θ 0 ) = −1. Since the solutions Y (θ) depend on the parameter m, the solutions of Y (θ 0 ) = −1 depend on m, i.e. they move through the complex plane as we vary the parameter m. Whenever one of these solutions approaches the integration contour, we deform it so that solutions never cross. We do that until we get back to the original value of m. At this point we now determine a solution Y C (θ) through an equation of the same form as eq. (4.1) but along some oddly shaped contour that depends on C. We want to move the contour back to the real axis. Along the way we pick up contributions from the solutions of the equations Once the integration contour is back on the real axis, our non-linear integral equation takes the form (4.9) with integers n i depending on the contour. The two equations (4.8) and (4.9) determine the function Y C along with the parameters θ i . Similarly, from the following formula for the free energy . (4.10) we infer that In order to compare with our previous result (4.6) for the trivial system, we set S = 1 Suppose that m is continued along a curve in the upper half of the complex m-plane that encircles the points m = iπ(2k−1) for k = 1, . . . , N . Then there are 2N solutions of the equation Y (θ) = −1 that cross the real line. Since these are located at ±θ k with m cosh θ k = ±i(2k − 1)π, formula (4.11) becomes Here, we expressed sinh θ k through cosh θ k . The sum extends over N pairs of solutions ±θ k . Note that both solutions within each pair contribute the same amount. Therefore, we have simply multiplied the sum over N terms by an overall factor of two. Our result is identical to our previous formula for F 0 in a trivial system. We conclude our analysis of the toy model with a few important comments. The simplest non-trivial case of a continuation occurs when a single pair ±θ 0 of solutions to Y (θ 0 ) = −1 crosses the real axis. In that case, our equations for Y C , F C 0 and θ 0 become The position of the θ 0 can be calculated making use of 14) The first equality is obtained from eq. (4.8) by taking the logarithm. The second equality just repeats the functional equation for Y C . According to the previous equation, the right hand side has to stay finite when we send the mass parameter m to infinity. Since the first term on the right hand side diverges with m, this divergence must be cancelled by one of the other terms. 2 In the limit m → ∞, the integral is a small correction and S(0) does not depend on m at all. Hence, the term log S(θ 0 ) has to diverge with m, i.e. the parameter 2θ 0 has to approach one of the poles θ S of the bare S-matrix. As we go to finite values of the mass parameter, this value of θ 0 will receive corrections, In principle, we can compute such corrections perturbatively in the mass parameter m. It will turn out later that the case of two poles crossing the real line is particularly relevant to the study of the Regge limit.
Before we return to the study of the equations (3.5) and (3.6), let us make one comment on the path of the analytic continuation. In this subsection we assumed the curve C to be closed in the complex m-plane, i.e. in the parameter space of the non-linear integral equation. This is important for the relation with excited states where m plays the role of a physical mass parameter. Our derivations, on the other hand, remain perfectly valid for arbitrary curves C that begin at m and end at a possibly different point m ′ of the m-plane. All we have to do is to replace m by m ′ in our equation (4.9) and all equations after eq. (4.11). This extension shall play an important role in the subsequent analysis.

Excited state TBA for six gluon amplitudes
It is not difficult to extend the analysis of the previous subsection to the system that describes scattering amplitudes of six gluons. In this case, we need to solve the equations (3.5) and (3.6) for three functions Y i , i = 1, 2, 3. The form of these equations is very similar to eq. (4.1), with one notable difference. In the context of six-gluon amplitudes the physical parameters are the cross ratios u i rather than the parameters m, C, φ. Since these two sets of variables are related by some non-linear equations (3.8), an open curve in the space of parameters m, C and φ may be mapped to a closed curve in the cross ratios. Consequently, we shall allow our curve C to end in a point m ′ , C ′ and φ ′ that may differ from starting point m, C and φ. Dealing with the requirement u ′ i = u i is differed to the next section. When we analyzed our toy model is was crucial that the kernel function K could be obtained from an S-matrix through equation (4.2). In complete analogy, we need to find unitary, crossing symmetric S-matrices for the two kernels K 1 and K 2 that were defined in equation (3.7). The solution is given by Up to a common sign, the solution is uniquely fixed by the unitarity condition One may also verify that S i are crossing symmetric, i.e. that they obeȳ whereS 2 (θ) = S 2 (θ) andS 1 (θ) = −S 1 (θ). Once we have rewritten the functional equations in (3.5) and (3.6) in terms of S i , we can perform a partial integration and analyze the behavior of solutions Y i upon analytic continuation of the parameters m, C and φ. The results are and similarly with integers N s and n s,j depending on the path of the analytical continuation. The parameters θ s,j are obtained by solving the equations As before, we can now evaluate the formulas (4.20) and (4.21) at the points θ s,j and insert the conditions to obtain equations for the positions θ s,j . Since the resulting expressions are rather lengthy, we shall refrain from spelling them out. Similarly to the toy model considered in the previous subsection, the position to the nearest poles in the limit of large m are related to poles of the S-matrices S j . Eqs. (4.23) and (4.21), for example, require that for every i at least one of the differences θ 1,i − θ 2±1,j or θ 1,i − θ 2,j must approach a pole of S 1 or S 2 , respectively.
The role of the free energy F 0 is played by the functional A (6) free that was defined in eq. (3.10). During the analytical continuation, this quantity changes according to where variablesθ s,j stand forθ s,j = θ s,j − iφ ′ . The possible θ s,j that contribute terms to the second line are solutions of eq. (4.23) that have passed the line Im θ = −φ upon analytic continuation. For nonzero φ, the number of such solutions and the direction in which they cross the line may differ from the corresponding numbers and directions for the real line Im θ = 0. This is why we had to use new integersÑ s andñ s,j instead of the corresponding parameters N s and n s,j that appeared in the integral equations (4.20) and (4.21). With these comments we conclude our general discussion of analytic continuations in the parameters for six gluon scattering. We shall now return to the Regge limit.

Regge limit of six gluon amplitude
Now we are prepared to combine the results of the previous sections in order to complete our main goal, namely to determine the Regge limit of the six gluon amplitude in the regime (2.8). As explained in section 2, the first step involves an analytic continuation in u-space. The transformations (3.8) describe m, C, φ as a function of the conformal cross ratios u i and hence they determine how the parameters of the integral equations (3.5) and (3.6) change under analytic continuations along C u . Some important features of the curve in the space of parameters m, C, φ are discussed in the first subsection. Numerical results show that very few solutions of the equations (4.22) actually cross the real line while we continue from the physical to the mixed regime. This enables us to determine an explicit formula for the Regge limit of the six gluon amplitude in the regime (2.8).

The path of the analytical continuation
Our first aim is to describe the path of the analytical continuation. As we explained in section 2, we would like to continue analytically along the path C u that was defined in eq. (2.11) with u 1 and u 2 kept fixed. From the first two relations in eq. (3.8) we infer that Note that the left hand side depends on the parameters m, C, φ which enter the nonlinear integral equations for Y i . Since we want to keep the right hand side fixed, eqs. (5.1) provide two constraints on the form of the curve in the space of parameters m, C, φ.
We still need to explore one more relation that involves the cross ratio u 3 . Let us recall that u 3 has been expressed in eq. (3.9) through the values of the functions Y i in the strip |Im(θ − iφ)| < π/2. We can actually turn eq. (3.9) into a formula for C = C(u i ) once we notice that This simple relation follows by subtracting the non-linear integral equations (3.6) for Y 1 and Y 3 from each other. We employ eq. (5.2) to eliminate both Y 3 and Y 1 from the equation for u 3 and arrive at, During our continuation we keep u 1 and u 2 fixed while u 3 is moved along a full circle until it comes back to its original position. Because the left hand side involves the square of c = cosh C, the quantity cosh C has changed its sign once we reach the end-point of the curve C u . Note that eq. (5.3) determines the parameter C explicitly as a function of the cross ratios. The variation of the real and imaginary part of C along the path C u is plotted in fig. 3. All the curves start in the lower half on the left hand side and proceed to the central line at ImC = π/2 and return to the left and into the upper half of the figure. At ϕ = π/2, the real part of C is of order m.
The dependence of m and φ on ϕ is a little more difficult to obtain. It requires inverting the eqs. (5.1) which contain the function Y 2 . The latter must be determined numerically by solving the non-linear integral equation for a large number of values m, φ and C. Given u i we can restrict the values of C to where ϕ runs from ϕ = 0 to ϕ = π. We have performed these numerical studies for special cases in which u 1 = u 2 . The condition amounts to setting φ = 0 at ϕ = 0. It is not difficult to see that φ = 0 will remain true as we move along the contour C u . Hence, it remains to determine m(ϕ) such that Y 2 (±iπ/4), and hence the cross ratios u 1 = u 2 , remain invariant as me change ϕ. An example is shown in fig. 2. The shape of the curve has very little dependence on the parameter C. We observe that the parameter m becomes complex. At ϕ = π, its imaginary part is Im m(π) = −π. The real part also gets shifted by an amount ∆m = Re m(π) − Re m(0) ∼ 1.8. We shall reproduce these shifts through an analytical argument in the next section. Some more comments on the numerical studies can be found in appendix C.

Evaluation of the Regge limit
In the previous section we have investigated the image of curve C u in the space of parameters m, φ and C. For φ = 0, the corresponding values of m(ϕ) and C(ϕ) are depicted in figs. 2 and 3. These numerical data provide us with a good qualitative understanding of the curve along which we would like to continue the parameters of the non-linear integral equations (3.5) and (3.6). As we discussed in section 4, we need to watch the solutions of Y i (θ) = −1 while continuing m, φ and C. Our numerical calculations show that, for large values of m, very few poles get close to the integration contour. In the case of Y 3 , the numerical results are displayed in fig. 4. The plot was produced for a continuation along the paths described by figs. 2 and 3. We see a single pair of poles that crosses the real line. The symmetry under reflection θ → −θ is a direct consequence of our special choice φ = 0.
Analogous results for the solutions of Y 2 = −1 are displayed in fig. 5. In this case, one pair of solutions approaches the real line while all others stay away from it. We did not display the corresponding plot for Y 1 since it looks very similar to that for Y 3 , except for an overall shift by ±iπ.
With these results in mind we can actually spell out a system of equations that should determine the solutions Y i at the end of the analytic continuation, i.e. in the mixed regime, One might have expected to see additional terms from the solutions of Y 2 = −1 on the right hand side. As one may easily see, these take the form Hence, we there are no additional terms on the right hand side of eqs. (5.4) and (5.5). In the above equations and throughout the entire discussion below we place a ′ on all quantities that have been continued to the endpoint of the curve C = C u that we described in the previous subsection. The parameters θ 1 and θ 2 appearing in the arguments of the two S-matrices (4.17) must solve one of the equations depending on value of C ′ . In addition we shall use the convention that, after crossing the contour, θ 1 has positive imaginary part while θ 2 has negative imaginary part. For φ = 0 we have θ 1 = −θ 2 . In the leading approximation for large values of the mass parameter m, the non-linear integral equations simplify, As we explained earlier, the solutions to eq. (5.8) are determined by the poles of the Smatrix, at least to leading order in large m. Since S 2 has poles at θ S with exp iθ S = i, the solutions θ ν ∼ θ S /2 are well approximated by with corrections of order exp(−m). We can insert the formulas for Y ′ i with θ 1 = −θ 2 = iπ/4 into the equations (5.1) and (5.3) that determine the conformal cross ratios u ′ i at the endpoint of the analytic continuation. A short computation gives Here, the parameters w ′ , c ′ and ε ′ are determined by our formulas (3.15) only that we have to replace m, φ, C by m ′ , φ ′ , C ′ . At the end of the analytic continuation we want to return to the original values of the cross rations, i.e. we need to impose u i = u ′ i . This leads to Obviously, there exists a second solution of the conditions u i = u ′ i . This solution, however, violates the condition c = −c ′ that we have argued for in the previous subsection.
It is instructive to re-express our relations (5.12) through the parameters m, C and φ. We have seen already that cosh C ′ = − cosh C. For the other two parameters we obtain where s h = log γ = −iπ + log(3 + 2 √ 2). Let us recall that these relations between the value of m, C and φ in the initial and final point of our curve C were derived under the assumption of single pole crossing. The numeric results we displayed in the previous section were obtained for φ = 0. In this case, the assumption of single pole crossing gives φ ′ = 0 and This coincides with the shift we found in our numerical studies, see fig. 2, and provides indirect evidence for the assumption made in the analytic treatment.
Let us now turn to the main goal, namely to the calculation of the remainder function where the individual terms are not evaluated after analytic continuation along the curve C. Let us begin with the function ∆ ′ that was spelled out in eq. (3.4). Upon continuation along the curve (2.11) we find In passing from the first to the second line we have inserted the asymptotic form (5.11) of the cross ratios u ′ i = u i . The period contribution is even easier to find, Finally, we also have to determine the contribution from the free energy. Specializing our general result (4.24) to the present case we find that Note that the second line contains a contribution from one solution of the equation Y 2 = −1.
As we explained at the end of section 4.2, the contributions to the free energy arise from solutions θ * that cross the line Im θ * = φ. As one can read off from fig. 5, one of the solutions of Y 2 = −1 does cross this line, at least when φ = 0.
In the limit of large m, the integral in eq. (5.19) gives a vanishing contribution as one may see by repeating the analysis we have carried out in great detail in section 3.2. So the only terms that are relevant in the Regge limit come from the second line of the previous equation. This gives In passing to lower line, we have inserted eq. (5.14) and also added the term A free that was evaluated in section 3.2. Since the free energy before analytic continuation vanishes in the limit of ε → 0, the additional term is actually zero. We only put it back in to show that the leading term should be regarded as a difference of energies. Using the final result (3.25) of our computation in section 3.2, we are now prepared to determine the Regge limit of the remainder function in the regime (2.8), Here, we have moved all the constant terms into a redefinition of the 'constant' and defined the constant e 2 by e 2 = √ 2 + 1 2 log(3 + 2 √ 2) . In a final step, we want to express the variables w, c and ε through the conformal cross ratios. Using eqs. (3.17) along with we find that After exponentiation of this result we obtain up to some irrelevant constant prefactor. This formula is the main result of our analysis. It describes the Regge limit of the six-gluon amplitude in the mixed regime and at strong coupling. As we have stressed before, our calculation assumes that the analytical continuation of the amplitude may be performed within the prescription of Alday et al., see our discussion in the introduction and the next section. Before we conclude, we would like to compare our result with the form of the answer at weak coupling. The relevant expression is known in the leading logarithmic approximation [19]. The following formula gives the correction to the BDS expression: All relevant notations are explained in [19]. In particular, the q i are the transverse momenta of the t i invariants, the parameter a is given by a = λ/8π 2 and λ is the 't Hooft coupling λ = g 2 N c . The exponent ω of s 2 is given by In the weak coupling expression (5.27) the new term, ∆ 2→4 , comes as an additive correction.
There is no indication that this correction should be written as an exponential. From the point of view of Regge theory, it seems more natural to sum over the different singularities in the complex angular momentum plane, and it is not clear whether exponentiation would still be consistent with Regge factorization. However, the leading logarithmic approximation alone is not sufficient to exclude the possibility of exponentiation.
The function ∆ 2→4 may be re-expressed in terms of the cross ratios u i that we have built in section 2. In these variables, (5.28) assumes the form where we have used 1 − u 3 ∼ 1/s 2 (appendix A), and cos arg q 2 p 4 The leading power of s 2 belongs to n = 1 and ν = 0 so that the intercept at weak coupling becomes Consequently, up to logarithmic corrections, the leading asymptotic behavior of the remainder function at weak coupling is given by In this leading logarithmic approximation, it is not possible to determine the scale of the energy s 2 . Put differently, the factor 1 − u 3 may be accompanied by some undetermined finite function ofũ 1 andũ 2 . Let us attempt a first comparison of our strong coupling result eq. (5.26) with the weak coupling amplitude in eqs. (5.27),(5.28) and eq. (5.34). In both cases, the energy dependence of the correction is power-behaved (i.e. Regge-behaved). We first face the fact that, at weak coupling, the correction is additive whereas, at strong coupling, the remainder function in the exponent leads to a multiplicative correction to the BDS amplitude. As a possible interpretation, we might assume that in eq. (5.27) the correction, which at weak coupling is of order λ, grows with λ and, at strong coupling, dominates over the 1. We are then led to confront ∆ 2→4 in eqs. (5.30) and (5.34) with eq. (5.26).
A striking observation is the change in the sign of the exponent E 2 : The intercept of the energy s 2 which on the weak coupling side is positive appears to turn negative when the coupling is increased. Also, the strong coupling expression exhibits an energy scale depending upon the product ofũ 1 andũ 2 . Next let us look at the ratioũ 1 /ũ 2 which is raised to power iν on the weak coupling side. Since at large s 2 the leading behavior comes from iν = 0, the dependence onũ 1 /ũ 2 drops out of eq. (5.34). In contrast to this, on the strong coupling side, the ratioũ 1 /ũ 2 comes with an exponent of order √ λ. In addition, the factor cosh C that multiplies the answer at weak coupling, is absent at strong coupling. Finally, in eq. (5.26), we see a phase factor which also depends upon the product ofũ 1 and ũ 2 . It should be kept in mind, however, that the full BDS amplitude at strong coupling, A (6) div +A (6) BDS , when continued into the mixed region, also contains a phase: A straightforward calculation shows that this phase exactly cancels the phase in eq. (5.26). As a result, the strong coupling result has no such phase factor. 3 Let us try a further interpretation and assume that, at strong coupling, the correction factor exp( √ λR ′ /2π) can also be written in the form of eq. (5.30) as an expansion in the twodimensional conformal group (Möbius group), with some strong coupling analogueω(ν, n) of the function ω(ν, n). Then, in eq. (5.26), the exponent ofũ 1 /ũ 2 could be interpreted as coming from a dominant ν-value ν s that is of the order √ λ. At the same time, the exponent of s 2 would come from the functionω(ν, n), evaluated at the value ν s . Finally, the absence of the factor cosh C might result from the fact that, at large ν s ,ω(ν s , 0) dominates over ω(ν s , 1). In pursuing such an interpretation one would perform a saddle point analysis of the ν-integral: At weak coupling, the large parameter is ln s 2 which leads to the dominance of the point ν = 0. In contrast, at strong coupling it is √ λ which plays the role of the large parameter, leading to a value ν 2 = O( √ λ). In this sense, the form of our strong coupling results seems perfectly consistent with the general features of Regge theory.

Conclusions and Outlook
In this paper we have studied the Regge limit of the planar six-point function at strong coupling. In the weak coupling regime this high energy limit, taken in a very special kinematic region ('mixed region') where some energies are positive while others are negative, is known to be particularly sensitive to the analytic properties of scattering amplitudes. Making use of the set of equations for Y -functions, we have been able to perform the analytic continuation of the scattering amplitude at strong coupling into this kinematic region and to calculate new corrections which are given by excitations of the auxiliary quantum integrable system. A comparison of the strong coupling result with the expressions for weak coupling shows interesting features but needs further investigations. At the moment, nothing is known about how the quantity E 2 may be interpolated between weak and strong coupling.
As we pointed out in the introduction, our calculation of the parameter E 2 at strong coupling was based on one important assumption, namely that we can simply continue the prescription of Alday et al. into the mixed regime. In other words, we have assumed that the amplitude in the mixed regime is determined by the exponential of the free energy in the auxiliary quantum integrable system. One may certainly wonder whether this assumption was justified. As we sketched in the introduction, scattering amplitudes at strong coupling do possess a geometric origin. According to the original prescription of Alday and Maldacena, they are determined by the area of a 2-dimensional surface S 6 which stretches to the boundary of AdS 5 . This surface emerges as the unique saddle point that dominates the string theoretic path integral in the limit of strong 't Hooft coupling. The kinematic invariants of the scattering process are encoded in the shape of the 1-dimensional boundary ∂S 6 of S 6 in the boundary of AdS 5 . Our analytic continuation in the space of kinematic invariants amounts to a continuation in the boundary conditions for a (path) integral. In any saddle point approximation, the contributing saddle points can be very sensitive to boundary conditions. In fact, there could even exist points, i.e. values of the kinematic invariants in our case, at which several saddle points contribute to the same order. From this point of view it seems far from obvious that amplitudes in the mixed regime can be obtained by simple analytic continuation and that they contain a single exponential term. This issue certainly deserves further investigation.
There are a few more or less obvious extensions of this work. It seems worthwhile exploring the relation between the analytic structure of scattering amplitudes and excited states of the auxiliary quantum system for more general cases, including other regimes and for a larger number of external gluons. In this context it might be useful to work with an alternative version of the non-linear integrable equations [32] in which only the geometric cross ratios appear as fundamental parameters. 4 Another question addresses the relation between excited states and analytic continuation. For other quantum integrable systems it is well-known that there exist excited states that are not related to analytic continuation in the parameters. The associated integral equations are of the same form, but without any constraints in the numbers N of driving terms. One may wonder whether such excited states could also have some interpretation in the context of scattering amplitudes.
Note that all transverse momenta must be finite to satisfy the on-shell condition. The dependence of kinematic invariants on ω is now easy to compute s ∼ ω 2 , s 456 ∼ ω (1+a) , s 345 ∼ ω (1+b) , Depending on our choice of a and b we can realize various asymptotics of the kinematic invariants. If a = b = 1, for example, one finds s 2 ∼ s. A regime with s 1 ∼ s 2 ∼ s 3 is reached in case we set a = b = 1/3. Setting a = b = 0, on the other hand, allows us to perform a quasi multi-Regge limit in which s 2 ∼ 1.
From the scaling properties of kinematic invariants we can easily deduce the following behavior of the cross-ratios (2.4), Thereby we have shown that u 1 and u 2 vanish at the same order when we take our Regge limit. Their ratio u 1 /u 2 remains constant. It remains to study the properties of u 3 . Using the Gram determinant relation to express s in terms of the other eight kinematic invariants we find After substitution of the gluon momenta we obtain Hence, both u 1 /(1 − u 3 ) and u 2 /(1 − u 3 ) remain finite for ω → ∞. We have used these two ratios to parametrize amplitudes in the Regge limit.

B Hirota equations
There exists another important set of functions T s,m that appear in the context of gluon scattering. Our discussion in this appendix will be kept general, i.e. it applies to an arbitrary number n of gluons. The functions T s,m = T s,m (θ) are parametrized by two integers s, m.
They are subject to the following set of Hirota equations, Using the set of Y −functions one may relate the parameters m m , φ m , C m of the Y −system through cross ratios. In order to do so, one has to invert the following relations .
(B.7) Our equations (3.8) are recovered as a special case when n = 6 with k = 1 and p = −1, 0, 1. We have pointed out before that the Y -functions satisfy the usual non-linear equations as long as their argument θ remains inside the strip | Im(θ)| ≤ π/4. Whenever θ crosses one of the lines Im(θ) = irπ/4, one has to add contributions from poles of the kernel functions K i . To avoid these complications, we can also solve the original equations inside the fundamental strip and extend the solutions to the complex plane recursively, with the help of the Y −system equation, i.e. Y s,m = (1 + Y [1] s,m+1 )(1 + Y [1] 4−s,m−1 ) Y [2] 4−s,m (1 + 1/Y [1] s+1,m )(1 + 1/Y [1] s−1,m ) . (B.8) We have used this recursive construction of Y s = Y s,1 is section 3.1 to obtain the formula (3.9) for the cross ratio u 3 .

C Comments on numerical computations
In this work, numerical calculations were used to find the path of the analytical continuation in the space of parameters m, φ, C and to check which solutions of Y i = −1 cross the integration contour. Most of this was done with the help of Mathematica. In order to determine the Y −functions for fixed m, φ and C we use an algorithm similar to one presented in [29] for the case of AdS 3 . It involves solving the non-linear integral equations (3.5) and (3.6) perturbatively for large enough m. The leading solution is given by expressions that do not contain integrals, namely our eqs. (3.12) and (3.13). To obtain the solution with better precision we insert the k th iteration into the integrals to obtain, i.e. schematically log Y (k+1) i (θ) = −m i cosh(θ − iφ) + C i + (K ij ⋆ log(1 + Y (k) i ))(θ) . (C.1) We perform k such iterations until the Y −functions stabilize. As a result we obtain Y −functions of real θ, or rather a discrete subset thereof. Moreover, using the integral equations we can calculate the Y −functions for complex value of θ by convoluting the kernel functions K i with the Y −functions of real θ. Due to existence of the poles in the kernels, we must limit this evaluation of the Y -functions in the complex θ-plane to the strip | Im θ| < π 4 . For values of θ outside this strip, we recursively use Hirota equations (B.8). In this way, we can construct Y −functions on the whole complex θ−plane.
Next, we evaluate the Y −functions for special values of θ = ±πi/4, 3πi/4 which determine the cross ratios u i defined in eq. (3.8). Actually, for these values of θ the numerical algorithms are not well convergent and it is easier at these points to interpolate the Y −functions. But at the end one can relate values of m, φ and C to values u i . In order to find Y −functions along our analytical continuations contour defined by eq. (2.11) one has to invert the relation between the parameters m, φ and C and the cross ratios u i . The values of C along the path can be found analytically, namely it is given by eq. (5.3). The other two parameters m and φ, on the other hand, require numerical calculations. An example of such path is shown in figs. 2,3. Here, it has been performed by solving a set of nonlinear equations (3.8) using the Newton method.
Furthermore, one can also determine the positions of solutions to the equations Y i = −1 numerically, see figs. 4,5. These solutions are moving while we are continuing along the contour in parameter space. In the process of this continuation, some solutions of Y 3 = −1 and Y 1 = −1 cross the real line. This happens roughly when ϕ ∼ π/2. At this point we must replace the original non-linear equations by those spelled out in eqs. (5.4) and (5.5).
The numerical results show that during analytical continuations only from of the solutions to Y i = −1 cross the integral contour, see figs. 4 and 5. Moreover, to a very good approximation, the imaginary parts of m and φ behave linearly with ϕ while their real parts change quadratically, compare also fig. 2. At the end of the contour path, m and φ go to values which are captured by our eqs. (5.13) and (5.14).