MHV amplitudes at strong coupling and linearized TBA equations

The maximally helicity violating (MHV) amplitudes of ${\cal N} =4$ super Yang-Mills theory at strong coupling are obtained by solving auxiliary thermodynamic Bethe ansatz (TBA) integral equations. We consider a limit where the TBA equations are linearized for large chemical potentials and masses therein. By solving the linearized equations, we derive analytic expansions of the 6-point MHV amplitudes in terms of the ratio of the chemical potential $A$ and the mass $M$. The expansions are valid up to corrections exponentially small in $A$ or inversely proportional to powers of $A$. The analytic expansions describe the amplitudes for small conformal cross-ratios of the particle momenta in a standard basis, and interpolate the amplitudes with equal cross-ratios and those in soft/collinear limits. The leading power corrections are also obtained analytically. We compare the 6-point rescaled remainder functions at strong coupling and at 2 loops for the above kinematics. They are rather different, in contrast to other kinematic regions discussed in the literature where they are found to be similar to each other.


Introduction
Developments in the study of the gauge-string duality and perturbative gauge theory have merged into a deep understanding of the four-dimensional maximally supersymmetric Yang-Mills theory (N = 4 SYM). It is now possible to address its dynamics even at finite coupling based on the underlying integrability in the planar limit [1]: one can find the spectrum of single-trace operators [2][3][4], and a formulation has been given [5] to obtain the scattering amplitudes, or equivalently [6][7][8][9][10], the expectation values of null-polygonal Wilson loops.
Such integrability-based approaches have also been extended to the correlation functions of single-trace operators [11,12].
Focusing on the maximally helicity violating (MHV) amplitudes, the form of the npoint amplitudes is almost fixed by the Bern-Dixon-Smirnov (BDS) expression [13] due to the anomalous conformal Ward identity [9,10]. The evaluation of the full amplitudes thus reduces to finding its remainder which exists for n ≥ 6 beyond one loop [14][15][16]. A notable fact to this end is that the analytic structure of the amplitudes is well controlled by the transcendentality and the associated symbol of loop integrals. This enables us to "bootstrap" the amplitudes. The complete and concise expression of the 6-point amplitudes at 2 loops has been obtained in this way [17], in agreement with the direct computation [18].
The bootstrap method has been extended up to 5 loops for n = 6 [19][20][21], and up to 2 loops for n = 7 [22]. For the restricted kinematics where the momenta of particles are two dimensional, analytic results are also given in [23,24].
At strong coupling, the MHV amplitudes are obtained [6] by evaluating the area of the minimal surfaces in the five-dimensional anti-de Sitter space (AdS 5 ) whose boundary ends on the corresponding null-polygonal Wilson loops at the boundary of AdS 5 . Although only a few exact solutions are known for such null-polygonal minimal surfaces [6,[25][26][27][28], one can evaluate their area by solving auxiliary integral equations [29][30][31][32]. These integral equations take the form of the thermodynamic Bethe ansatz (TBA) equations, which appear in the analysis of finite-size effects in integrable models.
The equations for n = 6 are indeed identified [30] with the TBA equations of the Z 4integrable model [33]. When the momenta of particles are restricted to three-dimensional spacetime, the minimal surfaces are embedded in the AdS 4 subspace. In this case, the integral equations for the n-point amplitudes are identified [32] with the TBA equations of the su(n − 4) 4 /u(1) n−5 homogeneous sine-Gordon (HSG) model [34]. For the two-dimensional kinematics for which the corresponding minimal surfaces are embedded in AdS 3 , they are identified [32] with the TBA equations of the su(n/2 − 2) 2 /u(1) n/2−3 HSG model. The inte-grable models in the general AdS 5 case have not been identified yet for n ≥ 7, in particular due to the unusual nature of the integral equations.
The integral equations for the amplitudes can be solved numerically by iteration. In addition, the equations can be simplified by taking limits of the parameters therein. For example, when the "mass" parameters are large, the integrable model reduces to a free massive theory in the infrared regime. The solution to the TBA equations is then expressed by iterative multiple integrals [35]. On the SYM side, the cross-ratios formed by particle momenta may become large or small in this limit. This includes the collinear limit for which the OPE/flux-tube expansions of the amplitudes are formulated at finite coupling [5,36].
On the other hand, in the strict limit of vanishing masses (the ultraviolet regime), the TBA equations are solved analytically [35]. From the point of view of SYM, it is the limit where the corresponding Wilson loops form regular polygons. Around this regular-polygonal limit, the amplitudes are expanded analytically by small masses for n = 6 in the AdS 5 case [37,38], and for general n in the AdS 4 [39] and the AdS 3 [40,41] case.
In this paper, we consider another limit of the TBA equations for the amplitudes, where the "chemical potentials" as well as the masses are large and hence the equations are linearized. The corrections in the linearization are exponentially small in the chemical potentials, or suppressed by their powers as understood by the Sommerfeld type argument [42].
In the case of the 6-point amplitudes, which we discuss concretely below, the linearized equations are solved following [43] as expansions in the ratio of the mass M and the chemical potential A to any order. The power corrections are analyzed by extending the analysis in [43], and we show that the leading power corrections are analytically evaluated. The results are checked against the numerical solutions, to be found in agreement. These analyses of the corrections assure that the linearization gives a controlled approximation of the original TBA equations for large A and M.
Applying the solution of the linearized TBA equations, we derive analytic expansions of the 6-point MHV amplitudes to any order. As the parameters are varied, the cross ratios of the particle momenta in a standard basis are kept small and change from the equal value in the UV regime to those for the soft/collinear limits in the IR regime. The amplitudes are well described by the expansion over the corresponding kinematic region from the UV regime to the IR regime. Our results thus provide another concrete example where focusing on the strong coupling enables us to explicitly evaluate the MHV amplitudes. Since the collinear limit can be realized as the IR end point in the present case, it is of interest to consider implications of our expansion to the OPE/flux-tube expansion at finite coupling.
Along the above trajectory of the cross-ratios, we also compare the 6-point remainder functions at strong coupling and at 2 loops which are rescaled/normalized by their UV and IR values. They turn out to be rather different in contrast to the cases where similarities are found between the strong-coupling results and the perturbative results [19,20,[38][39][40][41]44].
This implies that the kinematic region described by our expansion provides a probe to study structural differences of the strong-coupling and the perturbative results. This paper is organized as follows: In section 2, we review the MHV amplitudes at strong coupling and the associated TBA system. In section 3, we solve the linearized TBA equations for the 6-point amplitudes, and analyze the corrections in the linearization inversely proportial to powers of A. In section 4, we check the results in section 3 against numerical solutions. In section 5, we derive analytic expansions of the 6-point MHV amplitudes, and compare the rescaled remainder functions at strong coupling and at 2 loops. We conclude with a summary and discussion in section 6. Two appendices are also attached. In appendix A, we evaluate one of the pseudo energies in a different way from the main text by direct integration. This shows explicitly how the fractional powers of the spectral parameter appear from the summation over its integral powers, in accordance with the periodicity which is required from the algebraic equations (Y-system) associated with the TBA equations. In appendix B, we estimate the derivative of a pseudo energy.
2 Scattering amplitudes at strong coupling Let us consider the MHV amplitude M of N = 4 SYM in the planar limit. This is equivalent/dual to the expectation value of the null-polygonal Wilson loops whose edges correspond to the momenta of scattering particles [6][7][8][9][10]. After the tree amplitude is factored out, the remaining scalar part at strong coupling is thus evaluated by the area A of the minimal surfaces in AdS 5 ending on the null-polygonal Wilson loops at the boundary of AdS 5 [6]. Schematically, where λ ≫ 1 is the 't Hooft coupling. Since N = 4 SYM is conformal, natural kinematical variables to express amplitudes are the cross-ratios formed by momenta of particles.
The Y-functions, Y a,s , are defined through the Stokes data of the auxiliary linear problem associated with the string equations of motion. We have also denoted by * the convolution , with the kernels, Numerically, these equations are solved by iteration where the initial values of log Y a,s are approximated by the "driving terms", i.e. the terms not involving convolutions in (2.1).
Though M s are assumed to be real and positive in (2.1), they are complex in general as M s = |M s |e iϕs . For small ϕ s , the equations (2.1) keep the same form but with M s → |M s |, where K a,a ′ s,s ′ is the kernel K j for the the convolution involving Y a,s and Y a ′ ,s ′ . When |ϕ s −ϕ s+1 | exceeds π/4, extra terms appear as the integrals pick up the poles of the integrands.
This formulation covers signatures of the four-dimensional spacetime other than the usual (3, 1) of R 3,1 . By the reality condition of the minimal surfaces, C s are required to be purely imaginary for the (3,1) and (1,3) signatures, whereas C s are real for the (2,2) signature. The number of |M s |, ϕ s , C s , i.e. 3(n − 5), matches the number of the independent cross-ratios formed by the momenta of the scattering particles.
The Y-system can also be used to obtain the Y-functions with large imaginary shift of the argument, for which the TBA equations (2.1) are modified due to the pole contributions.
For a review of Y-systems, see for example [49].
The integral equations of the type (2.1) appear in the analysis of finite-size effects in integrable models, and are called the thermodynamic Bethe ansatz equations [35]. As mentioned in the introduction, the equations (2.1) are identified [30] with the TBA equations of the Z 4 -integrable model [33] for n = 6. When Y 1,s = Y 3,s and hence C s = 0, the minimal surfaces are embedded in AdS 4 , which describe the scattering of the special kinematics with three-dimensional momenta. In this case, (2.1) are identified [32] with the TBA equations of the su(n − 4) 4 /u(1) n−5 homogeneous sine-Gordon model [34]. By imposing further constraints, the minimal surfaces are embedded in AdS 3 , which describe the scattering for the two-dimensional kinematics. In this case, n is even because of the momentum conservation, and (2.1) become the TBA equations of the su(n/2 − 2) 2 /u(1) n/2−3 HSG model [32]. In the integrable models, the logarithms of the Y-functions Y a,s are pseudo energies, M s are masses of particles (measured in the unit of the inverse system size 1/L), and C s are chemical potentials. The corresponding integrable models in the AdS 5 case have not been identified yet for n ≥ 7. From the TBA point of view, what is unusual in (2.1) in this case is that the kernel K 3 (θ) does not decay for large |θ|.

Area, remainder function and cross-ratios
After a proper regularization, the area is expressed for n / ∈ 4Z as up to a constant. Here, A div is a divergent term, and A BDS−like is the term which satisfies the anomalous conformal Ward identity [9,10], as the Bern-Dixon-Smirnov (BDS) expression [13]. The third term A periods comes from period integrals associated with the underlying auxiliary linear problem or the Hitchin system, and is expressed by the mass parameters M s = |M s |e iϕs . The explicit forms of A div , A BDS−like and A periods are found in [29][30][31]. The last term A free is obtained from the solution to the TBA equations (2.1). It coincides with the free energy of the corresponding integrable model (when it exits). Explicitly, with θ s := θ + iϕ s . For n ∈ 4Z, the expression of the area may be obtained by taking an appropriate limit from n / ∈ 4Z. Given the formula (2.6), the non-trivial part of the area/strong-coupling amplitudes reduces to A free .
Since the structure of the MHV amplitudes is almost captured by the BDS expression, it may be sufficient to consider its remainder [14][15][16], where A BDS is the (finite part of the) BDS expression, and ∆A BDS : This remainder functions is conformally invariant and is a function of the cross-ratios of the particle momenta p µ j , The momenta form null polygons corresponding to the dual Wilson loops due to the momentum conservation. The subscript of the cusp points x j is hence understood modulo n.
These cross-ratios are expressed by the Y-functions at special values of the argument, e.g.
where we have defined The product of these cross-ratios yields generic ones.
In the OPE approach [5,36], these are used to compute the finite coupling amplitudes, which are not restricted to the MHV case. The amplitudes there are expanded by the contributions from the flux-tube excitations around the multi-collinear limit τ s → ∞. Re-summing over such series recovers the TBA equations at strong coupling [51,52]. The explicit relation of (τ s , σ s , φ s ) and the cross-ratios χ ijkl in (2.8) is found by using (2.8), (2.9) and the Y-system (2.4).

Limits of the TBA system
The analysis of the TBA equations (2.1) can be simplified by taking the limits of the parameters therein.
As for the mass parameters |M s |, there are two simple limits where Y a,s and A free are analytically evaluated.
Large mass/IR limit : One is the limit where all |M s | ≫ 1. This is the low temperature/IR limit where the integrable models reduce to free massive theories. The Y-functions are given by the driving terms, and the free energy trivially vanishes. From the point of view of SYM, it is the limit where cross-ratios can be large or small. Around this limit, the Y-functions and the free energy are expanded by iterative multiple integrals [35]. By analytic continuation, this limit is also connected to the Regge limit [53]. Further adjusting the phases ϕ s yields the above-mentioned multi-collinear limit.
Small mass/UV limit : The other is the limit where all |M s | ≪ 1. This is the high temperature/UV limit where the integrable models reduce to conformal field theories (CFTs).
From the point of view of SYM, it is the limit where the dual null polygonal Wilson loops become Z n -symmetric (regular-polygonal). By the standard method on the TBA system [35], or from the Y-system (2.4) with the θ-dependence dropped, the Y-functions and the free energy are evaluated explicitly in the strict limit of vanishing masses. We supply a concrete example for n = 6 shortly in subsection 2.4. Around this limit, the free energy is expanded by conformal perturbation [35].
For the hexagonal minimal surfaces in AdS 5 , the Y-functions and hence the amplitudes are expanded around the small mass limit based on the quantum Wronskian relation [38] following an earlier work [37]. For the minimal surfaces in AdS 3 [40,41] and AdS 4 [39], the small-mass expansions of the Y-functions and the amplitudes are derived based on the relation [54][55][56] between the g-and Y-functions using auxiliary boundary conformal perturbation. The expansions by the conformal perturbation are in terms of the couplings of the relevant operators to the CFT in the UV limit. They are expressed by the mass (and other) parameters of the TBA system via the mass-coupling relation. For the hexagonal minimal surfaces in AdS 5 , the corresponding Z 4 integrable model has a single mass parameter, and the relevant exact mass-coupling relation is given in [57] based on the work [43]. When the models have multi-scales, it is in general difficult to obtain such relations. In [58,59], for the analysis of multi-scale integrable models, the exact mass-coupling relation is obtained in the n = 10 case for the AdS 3 minimal surfaces. (For AdS 3 , the TBA system has multi-scales for n ≥ 10.) Phases : As for the phases ϕ s , they appear as the imaginary shifts of Y a,s (θ) as in (2.3). Their change induces the "wall-crossing" phenomenon, or the change of the form of the TBA equations (2.1). They are restricted to a finite range because of the periodicity of the Y-functions (2.5).
Large chemical potentials : As for the last parameters C s , it follows from the TBA equations that where we have used a rough estimate for large C s and |M s | such that log Y a,s are approximated in (2.1) by the driving terms. From (2.10), one finds that C s are directly related to φ s (Lorentz boost coordinates) in the cross-ratios. In the following, we consider the case of n = 6 and demonstrate that the large potentials C s indeed provide another useful limit in analyzing the TBA system for the amplitudes. Precisely, we combine two limits, large mass and chemical potential, which has not yet been considered so far.

6-particle amplitudes
When the number of the particles n = 6, there are only three non-trivial Y-functions Thus, L 3s in (2.1) is vanishing, and the ratio Y 3,1 /Y 1,1 is a constant. The TBA equations then reduce to where we have set From now on, we assume |ϕ| < π/4, unless otherwise stated, so that the TBA equations take the form as in (2.1) without the pole contributions. The Y-system (2.4) for n = 6 . (2.13) (2.14) The components of the area or the remainder function in (2.7) are and For the 6-point amplitudes, there are three independent cross-ratios. A standard basis for them is with U := U 1 . They are also parametrized by the variables of the type (2.10), with (τ, σ, φ) := (τ 1 , σ 1 , φ 1 ). From the periodicity (2.14), one has Y −1 2 = Y 5 2 , and u 2 is readily related to τ . Furthermore, u 1,3 are related to (τ, σ, φ) through which follows from the Y-system. We thus find [5], 1 UV and IR limits : For reference for the discussion below, we summarize the behavior of the pseudo energies, the free energy and the remainder function in the UV and the IR limit. First, in the UV limit M → 0 with A and ϕ fixed, 1 Comparing with the parametrization in [36], where c = 1 is the central charge of the CFT in the UV limit of the underlying Z 4 integrable model. These imply all u j →û UV := 1/(µ 1/3 + µ −1/3 ) 2 , and As mentioned above, the boundary of the minimal surface becomes regular-polygonal in this limit.
On the other hand, in the IR limit M → ∞ with A and ϕ fixed, Again from the Y-system, these imply that two of u j → 0 and the other u k → 1 for generic ϕ, and This is the soft limit, e.g. where u 1 ∼ 1, u 2,3 ∼ 0 and both τ, σ ≫ 1 for −π/4 < ϕ < 0 [30].
Sending ϕ → −π/4, one can achieve the collinear limit where u 2 ∼ 0, u 1 + u 3 ∼ 1 and τ ≫ 1 with σ fixed. This is the regime of the OPE expansion [5,36]. By changing ϕ, the roles of u j are permuted. For higher order terms in the UV and the IR limit, we refer to [37,38]. Figure 1 illustrates these limits in the space of the three cross-ratios (u 1 , u 2 , u 3 ). The red straight line represents u 1 = u 2 = u 3 corresponding to the UV limit. The triangle represents the IR limit. Its sides correspond to the collinear limits, whereas its vertices (u 1 , u 2 , u 3 ) = (1, 0, 0), (0, 1, 0), (0, 0, 1) to the soft limit. The yellow surface represents which is discussed in section 5. Based on the linearized TBA equations in the next section, the expansions of Y a , A free and R for large A and M are derived there. They interpolate the kinematic points in the UV and the IR limit, e.g. which are marked by the green blobs for generic −π/4 < ϕ < 0. When the phase is adjusted, e.g. as ϕ → −π/4, the point in the IR limit moves onto a side of the triangle away from its vertices.

Linearized TBA equation
In this section, we consider the TBA equations for the 6-point amplitudes (2.11) for large A and M where the equations are linearized. We first introduce the linearlized TBA equations in subsection 3.1 and summarize the solution to the linearized equations following [43] in subsection 3.2. There are several types of the corrections in the linearization. One is exponentially small in A (as in (3.2)), another is expected to be exponentially small in A Figure 1: Cross-ratios and various limits. The red straight line corresponds to the UV limit. The triangle with vertices (u 1 , u 2 , u 3 ) = (1, 0, 0), (0, 1, 0), (0, 0, 1) corresponds to the IR limit. In particular, its sides and vertices correspond to the collinear and the soft limit, respectively.
(as in ∆ 2 in (3.5)), and the other is suppressed by the powers of A (as in ∆ 1 in (3.5)) due to the Sommerfeld type argument [42]. The exponential behavior of the second type is confirmed by numerically checking the scaling of the pseudo energies in the next section.
As for the third type, the power series corrections are analyzed by extending the analysis in [43]. We explicitly solve the linearized equations including the leading term in the power

Linearization
We consider the limit, The TBA equations (2.11) imply the following representation of ε(θ) andε(θ), where r i (θ) are positive numbers. Hence some terms in (2.11) are exponentially small in L for θ ∈ R: By neglecting these terms, we approximate the first equation in (2.11) by the form containing only ε, The other functionε is then evaluated by using ε as We assume that there are unique Fermi points θ = ±B satisfying The explicit expression of B in terms of A and M will be determined a posteriori. The convolution term is then divided into three pieces, Similarly we haveε The eq. (3.6) is identical to the p = 1 2 case of (3.3) in [43] bỹ We can thus follow [43] to treat it analytically.
In the limit (3.1), we expect from the linearized equations (3.6), (3.7) that Under the assumption of this scaling, ∆ 2 is exponentially small in L. The scaling will be confirmed numerically in the next section. Below we are interested in the corrections to the linearized equations due to ∆ 1 (θ). The Sommerfeld type argument results in, as the simplest approximation [42], This turns out to yield an O(L −2 ) correction to ǫ(θ). One can systematically analyze the higher power terms in L −2 in principle by incorporating higher order terms from ∆ 1 . We, however, restrict ourselves to the leading correction below.

Analysis of the linearized TBA
We recall the results in [43] which are necessary in the following discussion. Let us introduce Then the linearized TBA equation (3.6) is equivalent to, The function D ǫ (θ) in the left hand side is defined by Here X(θ) is an unknown function of θ.
The (inverse) Fourier transformation for an arbitrary smooth function f (θ) is defined by The kernelK admits the factorization in the Fourier space, where K + ( K − ) is analytic in the lower (upper) half plane and K ± (w) → 1 as |w| → ∞.
The explicit form of K + (w) reads, Based on the Wiener-Hopf type factorization, it is shown in [43] that eq. (3.10) is casted The integral equation (3.11) concludes that v(w) is an "almost" pole free function in the upper half plane and takes the form, This determines the coefficients v n in a self-consistent way. In order to determine them, we utilize the condition ε(B) = 0. This implies v(w) ∼ O(1/w 2 ) for |w| ≫ 1, which is written Here the integration contour C + encircles the positive imaginary axis of w ′ but it excludes By using the condition (3.12), one rewrites the integral equation (3.11) in terms of u(w) as Here we understand that the pole at w ′ = −w in the integrand is outside of C + .
Then one derives from eq. (3.13), Although the solution to eq. (3.14) is given by an infinite power series in q, it converges quickly for a large a/m.
The quantity of our interest is the pseudo energy, which is given by and its (inverse) Fourier transformation ǫ B (θ). In the inverse Fourier integral over θ with the region −B < θ < B, we close the integration contour by adding the large semicircle in the upper half plane for the first term and in the lower half plane for the second. One immediately sees that the pole of v(w) at w = i is canceled by the zero of K + (w). The first term is thus evaluated by the contributions from the poles at w = i0 + , w = 4 3 ni (n > 0, n = 0 (mod3)). Similarly the second term is evaluated by those at w = i0 − , w = − 4 3 ni (n > 0, n = 0 (mod3)). After simple manipulations, we obtain for |θ| < B, where we have used K + (0) = 2 3 . The coefficients c n are expressed by quantities already introduced in the above as .
The other TBA function,ǫ = L −1ε , is also evaluated within the present linear approxima-tion, which is given bỹ . Now we deform the integration contour by adding a semi-circle in the upper half plane. One easily verifies that K 2 (w) K + (w) is analytic at w = (2n + 1)i. It is also easy to see that the pole contribution at w = i (of v(w)) cancels √ 2m cosh θ. Thus, by calculating other pole contributions at w = i0 + (of v(w)) and at w = 4/3ni (n = 0 mod 3) (of K + (w)), we havẽ Finally we need to represent B in terms of a and m. This is achieved by taking account of the pole contributions in the right hand side of (3.12). The result is This equation fixes q (and thus B) in terms of y. It also implies B = O(L 0 ).
In appendix A, we evaluateǫ by directly plugging the expansion of ǫ (3.17) into (3.7).
From this, one can see explicitly how the fractional powers of e θ appear from the summation over its integral powers, in accordance with the periodicity required from the Y-system.

O(L −2 ) corrections
One can extend the analysis by incorporating the O(L −2k ) corrections to ǫ. Here, we consider the O(L −2 ) correction explicitly. The results at the leading order in this subsection reduce to those in the previous subsection.
By substituting eq. (3.9) into the right hand side of (3.4) and by neglecting ∆ 2 (θ), one obtains an equation of the form (3.10). Then Y (θ) is modified by a term, where We argue in appendix B that ǫ ′ (B) is of O(L 0 ). Thus δ 1 represents the correction of O(L −2 ) to the linearized TBA equation. We quantitatively estimate the consequence of this modification.
Note that By using this, one can show that v(w) satisfies eq. (3.11) with The last term is the only modification due to the leading term from ∆ 1 (θ). It however changes the analytic property of v(w). As shown previously, v(w) is almost pole free in the upper half plane of w without δ − (w). This time it has infinitely many poles in both half planes. More explicitly, (3.20) The convolution integral in (3.11) does not produce further terms as α(w)δ − (w) is regular The analogue of (3.12) reads, Here ξ 1 = O(L 0 ), and it is estimated in appendix B. By utilizing this, one derives the equation for u(w), . As mentioned previously, the pole at w ′ = −w in the integrand is outside of C + . From this, we again obtain a set of algebraic equations for w n , where q is defined in (3.15). For a given sequence ω n,0 (n > 0) of O(1), let ω n be a solution to It has a power series solution of the form, The coefficients ω n,ℓ (ℓ ≥ 1) are uniquely determined by ω n,0 . The first few coefficients read explicitly The solution to (3.23) is thus written in the form, In the above we have used γ nℓ = ω n,ℓ | ω n,0 →1/n , κ nℓ = ω n,ℓ | ω n,0 →κ n0 , (3.26) and Note that w (0) n (q) and w (1) n (q) are not decompositions of w n into O(L 0 ) and O(L −2 ) terms. As will be shown below, q is given by a power series in L −2 .
Next we represent q in terms of y defined in (3.19). This is achieved by rewriting (3.21) in terms of w n , Here we introduced which is again of O(L 0 ).
In order to solve this we set respectively. The coefficients w (0) ℓ and w (1) ℓ are given by Note that they are now decompositions of w ℓ into O(L 0 ) and O(L −2 ) terms, We assume expansions and substitute them into (3.30). Then the coefficients q The key quantity ǫ B (w) has formally the same form as (3.16). We take its inverse Fourier transformation. The extra poles of v(w) at w = (2n + 1)i (n ≥ 0) do not play a role as the combination K + (w)v(w) is regular at these points. The integration contour is closed in the upper (lower) half plane for the first (second) term. We then arrive at This is formally identical to the one without ∆ 1 (θ). The coefficients v n , however, contain higher order terms of O(L −2k ) and the coefficients c n also contain them similarly. The O(L 0 ) and O(L −2 ) terms are given explicitly in terms of other coefficients obtained so far, .
(3.33) By using these, we have We can similarly estimate the correction toǫ(θ), The convolution term is again given by As |θ| < B we close the contour in the upper half plane. There are contributions from the poles at w = i0 + , w = i and w = 4 3 ni (n = 0 (mod 3)) as before. In addition, there appear additional contributions at w = (2n + 1)i, originated from the last term in (3.20). After a simple calculation we find that the additional term cancels the last two terms in (3.35), Thus we havẽ

(3.36)
This and (3.34) are consistent with the periodicity property of the Y-system (2.14).

Free energy
The free energy A free is represented in terms of the TBA functions as in (2.15). As before, we drop the terms which are exponentially small by assuming the scaling (3.8), and write For the moment, we concentrate on ∆E. One can show that the following relation, derived in [43], is still valid even in the presence of ∆ 1 (θ), Thanks to the integral equation for ǫ B , we have Here the contour C ′ + encircles the positive imaginary axis, which includes w = i. We use the explicit form of D v (w) and the "boundary condition" (3.21) to represent this as We write ∆E = ∆E 1 + ∆E 2 where ∆E 1 stands for the contribution from the pole at w = i in the integral. Explicitly, Here we used Similarly, The content of the first bracket in the right hand side is evaluated by use of (3.22), The second bracket term is also decomposed into Thus we have As for ∆ F , the lowest order approximation gives We have substituted (3.15) and (3.29) into e B . Thus, we obtain Since we are interested in the expansions for large L, we will focus in section 5 on the expansions at the leading order in 1/L 2 . In this case, one needsw (0) n and q (0) only and does not need to take into account c (1) n , ∆E (1) and δ F . We then have simple expansions of ǫ,ǫ: for |θ| < B, where the coefficients e n for small n are found to be , with b n given in (3.14).
Similarly, the leading order term of the free energy is [43] f where we have introduced f so that the leading terms are of O(L 0 ). The coefficients k n for small n are 2 The O(y 0 ) terms agree with the UV limits in (2.20). Though these expansions are based on the TBA equations of the form (2.11) which are valid for |Im θ| < π/4, they are analytically continued for any Im θ.
Overlap with the small mass expansion : Though we have discussed the case where both A and M are large, the results may be continued to small M, or those of the small M expansions with other parameters fixed [38] may be continued to the regime of our 2 We correct typos in k 3 , k 4 in [43].
expansions. Thus as a non-trivial check, they can be compared. Indeed, taking into account the relation of the notation here and that in [38], we find that the expansions of Y 1,there and A free in [38] precisely reproduce for large A the leading terms of our expansions, a √ πe 1 (0)y 1/2 , − a 2 π k 1 y .

Numerical analysis
In the previous section, the TBA equations for the 6-point amplitudes This also serves as a check of our assumption on the scaling (3.8), which assures that ∆ 2 and a similar term dropped in (3.37) are indeed exponentially small in L. For this purpose, we first consider the pseudo energy at the special value of θ = 0. From the results in the previous section, we have up to (expectedly) exponentially small terms with One has a similar expression forǫ(0). As for the free energy, the L-dependence reads  Table 1: Parameters for the L-dependence of ǫ(0) and f from the results in section 3.
They are again consistent with the values in Table 1. If higher order terms in y are included in evaluating these values, we find a better agreement between Table 1 and 2. Relative corrections for a 2 , d 2 are larger than those for a 1 , d 1 . All in all, we conclude that the scaling    As m/a → 0, they approach the value in the UV limit f UV = −( π 6 + A 2 3π )/L 2 ≈ −0.0346 (blob). As m/a → 1, the numerical data converge to the value in the IR (free field) limit f IR = 0 (dotted line). One can also check that f num − f (k−1) scales as (m/a) 8k/3 , and that it is saturated as L −2 without any power of m/a for small enough m/a. These observations are again consistent with the results in the previous section.

Analytic expansion of 6-point remainder function
Having checked the validity of the expansions and the linearlization against numerical data, we now apply the expansions to the strong-coupling amplitudes. In the following, we focus on the leading expansions in 1/A 2 , though it is straightforward to include the subleading O(1/L 2 ) terms as discussed in section 3. It is thus understood that the equations are valid up to relative corrections, if any, which are exponentially small in A or of O(A −2 ).

Cross-ratios
Let us first discuss how the cross-ratios change according to the changes in the parameters of the TBA equations. From the expansions (3.43) which are valid for any Im θ, one can obtain u j by substituting these into (2.17). Alternatively, one can consider the variables in (2.18) parametrizing the cross-ratios. From (2.12) and (2.18), they are given by For small m/a, the pseudo energies ǫ,ǫ approach their UV values in (2.20), and hence For m/a 1, the pseudo energies approach their IR forms (2.22), and hence variable is a constant, φ = −A = −(4/7) · 10 3 ∼ −571. As ϕ → −π/4, the IR form σ IR is flattened, which corresponds to the collinear limit. In the following, we consider the case where −π/4 < ϕ < 0, unless otherwise stated, as in the above. Other cases are analyzed similarly. In this case, we find that The behavior of (τ, σ, φ) is translated into that of u j by using the relations (2.19). First, it follows from (5.2) that as m/a → 0. To estimate u j for other m/a, we note that σ + τ increases as m/a and approaches M cos ϕ. Thus,   In terms of (τ, σ, φ), these two regimes are smoothly connected. This is also confirmed by the fact that the yellow surface in Figure 1 asymptotes for large A to u 1 u 2 u 3 = 0 which includes the IR end point. Sending ϕ → −π/4, one can adjust u 3 ∼ 1/(1 + e 2σ ) for large enough m/a, so that the end point becomes (1 − u 3 , 0, u 3 ) with non-vanishing u 3 . This corresponds to a collinear limit. Thus, by changing M and ϕ (including generic ϕ), the trajectories of u j for large A form a surface ("cap") which is close to u 1 u 2 u 3 = e −2A but ends on the triangle of the collinear/soft limit.

Expansion of 6-point remainder function
We now focus on the region where m/a < 1 and the expansions of (3.34) and (3.36), or , , for small n. ∆A BDS is symmetric under ϕ → ϕ + 2 · (π/4), which permutes the three crossratios u j . This symmetry strongly constrains the expansion, and only the terms with cos 4n 3 ϕ (n ∈ 3Z) should survive [37]. From the expansions (3.43), such terms are not possible at O(y 1/2 ). The above expansion indeed satisfies this constraint with vanishing β 1 .
Combining the other terms, we obtain the expansion of the 6-point remainder function, where ∆A BDS is given in (  to the truncation of the expansions. By including higher order terms, one obtains a better approximation to the numerical data up to m/a ∼ 1. If we use the expression of ∆A BDS by substituting the expansions (3.43) into (2.16) (without expanding in y n/2 ), higher order terms are partially incorporated and its IR behavior for m/a 1 is improved.

Comparison with 2-loop results
In [40,41,44], the remainder functions corresponding to the minimal surfaces in AdS 3 were compared at strong coupling and at 2 loops [17,18] by changing the mass parameters of the TBA system with other parameters fixed. It was found that they are close to each other (but different) after appropriately rescaled/normalized. Such an observation was also made for the AdS 4 case [39]. In the present 6-point case with the general (AdS 5 ) kinematics, the comparison was extended and the similarity was confirmed up to 4 loops [38] based on the perturbative results [19,20]. The comparisons for other kinematics are found in [19,20]. In this subsection, we compare the remainder functions at strong coupling and at 2 loops for the cross-ratios discussed in the previous subsections.
For this purpose, we introduce the remainder function at strong coupling which is rescaled by its UV and IR values,R  6 where λ is the 't Hooft coupling, the rescaled remainder functionsR ℓ-loop 6 are defined similarly. In particular at 2 loops, 6,UV = R 6 (û UV ,û UV ,û UV ), R 6,IR = 0. The analytic expression of R (2) 6 is given [17,18] by and  To avoid loss of significant digits in the numerical evaluation of R 6 , we have made an approximation, which is valid for (1 − u 1 − u 2 − u 3 ) 2 ≫ u 1 u 2 u 3 , and hence both for (5.4) and (5.5) up to exponentially small terms in A. By using the analytic expansion of R 6 at strong coupling, one can draw a plot ofR strong 6 which well approximates the data points as in Figure 5 (a).
Since the cross-ratios u j are well approximated by the analytic expansions, substituting them into R . This is in contrast to the results mentioned above by varying the mass parameters with others fixed.

Conclusions
Through the gauge-string duality, the MHV amplitudes of N = 4 SYM at strong coupling are obtained by solving auxiliary integral equations of the TBA type. In this paper, we considered the limit where chemical potentials in addition to masses are large, and hence the TBA equations are linearized. Large chemical potentials, together with large masses, thus provide another useful limit. In particular, the linearized TBA equations for the 6point amplitudes are solved analytically according to [43] as expansions to any order in terms of the ratio of the mass and the chemical potential M/A. The relative corrections in the linearization are exponentially small in A or of O(1/A 2 ). The inverse power corrections are analyzed by extending the analysis in [43], and we obtained the explicit forms of the leading corrections. We checked that our results agree with numerical solutions. We also compared the 6-point rescaled remainder functions at strong coupling and at 2 loops along the trajectory of u j mentioned above. We observed that they are rather different, in contrast to the cases where similarities are observed between the strong-coupling results and the perturbative results [19,20,[38][39][40][41]44]. This implies that the kinematic region of the small cross ratios provides a useful probe to study structural differences of the strong-coupling and the perturbative results. For example, since the perturbative results share the property that they are controlled by the transcendentality and the associated symbol [17,[19][20][21], the difference from the strong-coupling case may persist for higher loops.
The actual comparison with higher-loop results in [19][20][21] is thus an interesting future problem.
The extension of our analysis to the general n-point amplitudes would be an important future direction. For n > 6, the TBA system has more than one mass scale, and the generalization of the work [43] is not straightforward. This is closely related to the problem of finding the exact mass-coupling relation, i.e. the relation between the physical mass and the coupling in the Lagrangian, for multi-scale integrable models. As mentioned in section 2, this problem has been solved [58,59] for a simple multi-scale integrable model, i.e. the su(3) 2 /u(1) 2 HSG model, which is relevant for the 10-point amplitudes for two-dimensional kinematics. The issue of the multi-scales has been overcome there by comparing the picture of the conformal perturbation on the UV side and that of the form-factor bootstrap on the IR side.
By adjusting the phase ϕ, the IR end point of the trajectory of u j can be set to the point of the collinear limit, around which the OPE expansion at finite coupling is derived [5,36].
It would be of interest to consider if our expansion at strong coupling provides useful data for the OPE expansion.

A Evaluation ofǫ from direct integrals
In this appendix, we evaluate the expansion ofǫ in (3.18) by plugging that of ǫ (3.17) into (3.7). For this purpose, we note the formula, βz n β + n =: Φ β (z) , for |z| < 1, and its analytic continuation in z, from which we find Taking the derivative of the right hand side of (A.1) with the help of (A.2) and setting z = −e 2(ax+b) , β = (a + c)/2a, we obtain the formula. Furthermore using the inversion formula, Comparing this with (3.18), we observe that the summation over Φ (2+sn)/4 in I n cancels the driving term √ 2m cosh θ, which leaves the first terms (−1) n cosh 4nθ/3 in accordance with the periodicity required from the Y-system. These fractional powers of e θ appeared by summing up integral powers of e θ in Φ β and analytically continuing it by the inversion (A.3).
By repeating the same argument as in section 3, we obtain an integral equation for τ (w), The solution is formally given by where α n = res w= 4 3 ni α(w). Thus we obtain a set of algebraic equations forτ ℓ (ℓ ∈ N), In particular, at θ = B, we deform in the first term the finite part of the integration contour to a semi-circle in the lower half plane, and take the large radius limit of the semi-circle.
Then, we obtain