The contribution of scalars to ${\cal N}=4$ SYM amplitudes II: Young tableaux, asymptotic factorisation and strong coupling

We disentangle the contribution of scalars to the OPE series of null polygonal Wilson loops/MHV gluon scattering amplitudes in multicolour ${\cal N}=4$ SYM. In specific, we develop a systematic computation of the $SO(6)$ matrix part of the Wilson loop by means of Young tableaux (with several examples too). Then, we use a peculiar factorisation property (when a group of rapidities becomes large) to deduce an explicit polar form. Furthermore, we emphasise the advantages of expanding the logarithm of the Wilson loop in terms of 'connected functions' as we apply this procedure to find an explicit strong coupling expansion (definitively proving that the leading order can prevail on the classical $AdS_5$ string contribution).


D Connected functions 49 1 Introduction and summary
In the last years there has been much interest in SU(N c ) N = 4 Super Yang-Mills (SYM) theory, especially in the so-called planar limit N c → ∞, g Y M → 0, and fixed 't Hooft coupling final outcome is (in both cases) a Thermodynamic Bethe Ansatz (TBA) system. We have already reproduced it by re-summing the contributions of gluons and fermions/antifermions bound states to the OPE series at LO [30,31]. This kind of computation is of very different nature (w.r.t. string calculations), and besides it is a genuine re-summation (among very rare cases) of a FF series which, very surprisingly and interestingly, generates a fully different integrability scenario, namely a TBA set-up (this happens for the first time, to our knowledge). However, despite being worth further investigation, this is only part of the story concerning the strong coupling regime. After considering also scalars (which from the string side correspond to fluctuations on the five sphere S 5 ), the situation is even more intriguing. In fact, for very large coupling λ the scalars decouple in a O(6) Non Linear Sigma Model (NLSM) with an exponentially small (dynamically generated) mass m gap ∼ e − √ λ 4 [47,48,49,50,51], so that the theory is almost conformal. Therefore, the scalar contribution to the WLs can be guessed to be given in the limit by the conformal correlation functions of some pentagonal twist field. This makes concrete calculations possible and a surprising contribution proportional to √ λ has been found for the logarithm of the WL [42]. This result needed a corroboration by Monte Carlo simulations on the few-particles terms of the series [42], which was bolstered by [52] (see also [53] where equivalence of sigma models to twisted parafermions is proposed as a calculation tool). Nevertheless, this behaviour was asking for a definitive and stringent proof directly from the OPE series because many subtleties are to be considered. Moreover, this contribution can be dominant on the AdS 5 string action, which gives 2π , decaying with the hexagon area A 6 . In fact, the latter is exponentially small ) at large τ (collinear limit) and thus the only multiplicative contribution to the WL is coming from the five sphere S 5 , and it is actually exponentially large W O(6) ∼ e J 4 √ λ at least for small m gap τ ∼ e − √ λ 4 τ 1 (see (4.33) which is valid for z ∼ e − √ λ 4 τ ≪ 1). Similar considerations hold if we include string one loop corrections (whose details are still unknown but, relying on [30,31], may have an analogue in the one-loop for in N = 2 partition function [54,55]): in the collinear limit the scalar contribution (given for any τ and σ by (4.38)) is actually the dominating one. Of course, being a purely quantum effect (moreover in another sector of the theory), the leading term from scalars is missed by the classical minimisation of [19,44,45].
Interestingly, we have found that a simple analytic derivation of this behaviour may be possible [32], although in that letter some issues could not receive the attention they deserve: here this lack will be solved along with new results. In fact, the idea anticipated in [32] is that of passing from the OPE series for the expectation value of the WL, W , to the series for ln W : this change corresponds, as for the generic term of the series, to passing from the (non-connected) multiparticle 2n function of W to the 2n 'connected' multiparticle function of ln W . Upon integrating each term on one of the 2n rapidities and expanding for large λ inside the multi-integral (this is a very delicate point as will be shown below in Subsection 4.2 in the part between formula (4. 30) and (4.33)), the √ λ factorises in front of the LO (of each term of the series of ln W , cf. below formula (4.33)). Despite the simplicity of this idea, a very essential point for its effectiveness is 1 So that we can conjecture that it dominates or contributes still at finite m gap τ as long as e −mgapτ is not too small.
where each term W (2n) denotes the contribution of 2n scalars with rapidities {u i }.
The exponential of the energy E(u i ) and momentum p(u i ) represents the free propagation in the Wick's rotated 2D space [19], while G (2n) correspond to the probability (modulus square) transitions from the vacuum to a (2n scalars) state of the flux-tube [22]. The cross ratios τ, σ determine the conformal geometry of the polygon (of the WL) 6 and thus are related to the dual momenta of the scattering gluons (in the amplitude).
Focusing on the functions G (2n) , they are conveniently factorised into a λ-dependent dynamical part Π (2n) dyn and a λ-independent factor Π (2n) mat reflecting the matrix structure of scalars under the internal SO(6) symmetry [42]: being P ss the pentagonal amplitude and µ s the measure for scalars. 5 They form an antisymmetric representation 6 and produce a singlet 1 and the other two (irreducible) representations in the decomposition of the product of two of them 6 ⊗ 6 = 1 + 20 + 15, so that only the product of an even number of them produces again a singlet. 6 There is another cross ratio necessary to fix the hexagon, φ, but it does not appear in the scalar contribution.
On the other hand, the factor accounting for the matrix structure under SO (6) does not depend on the coupling constant λ and involves integrations over n auxiliary roots of type a, 2n of type b, n of type c: , where the functions f (x) and g(x) are defined as f (x) = x 2 + 1 4 , g(x) = x 2 (x 2 + 1) . (2.5)

A Young tableaux approach
Now we provide a way to compute explicitly the matrix factor (2.4) by residues, eventually based on Young tableaux. No need to say that the matrix factor does not depend on the coupling constant λ, so that the results apply at any coupling.
The variables a, c in (2.4) do not couple to each other, are symmetric and can be integrated over to give us the same contribution , (2.6) where the result of the integrations over the a k (and identically for the c k ) is the symmetric function . (2.7) The integrations in the auxiliary variables a 1 , . . . , a n can be evaluated by residues . (2.8) Each term in the multiple sum depends on a partition of labels α k (with α k ∈ {1, . . . , 2n}), which we indicate as S α = {α 1 , . . . , α n } (making use of the shorthand notation α = α 1 , . . . , α n ). It is convenient to introduce also the complementary setS α = {1, . . . , 2n} − {α 1 , . . . , α n }. Equipped with these notations, we rewrite (2.8) as where we introduced Thanks to the symmetry under permutation of rapidities the function defined above is a polynomial, since a single pole for b i = b j would spoil this symmetry and double poles do not appear. Properties of the polynomials δ 2n are discussed in Appendix A. What is relevant to us now is that the matrix factor is expressed as Π (2n) mat (u 1 , . . . , u 2n ) = 4n 2 (2n)!(n!) 2 (2.11) which shows many similarities with the Nekrasov instanton partition function in N = 2 theories [57]. More precisely, (2.11) can be compared to Z (2n) U (2n) , the 2n-instanton contribution to the partition function of a U(2n) theory, where the physical rapidities u i play the role of the vevs a i of the scalar fields and the instanton positions φ i are represented by the isotopic roots b i . For these integrals, is well-known an evaluation by residues which results in a sum over Young tableaux configurations [58]. This observation allows us to put forward the proposal to compute Π (2n) mat by residues and classify the contributions in Young tableaux, along with further diagrams obtained upon performing permutations on the column index. We must highlight, though, some differences with respect to the Nekrasov partition function. In our case the polar part is somehow simpler, since we do not have two deformation parameters ǫ 1 , ǫ 2 as in Z (2n) U (2n) , but only one 7 , which is fixed to ±i. The polynomial δ 2n , on the contrary, is absent in the Nekrasov function and brings some important effects on the computation. However, the key features of the multiple integrals, which allow us to employ this method, are shared by Π (2n) mat , Z (2n) U (2n) and are basically three: • The poles of the type b i = u j + i 2 , which relate the residues positions to the physical rapidities; • The double zeroes b 2 ij that cancel the contributions in which two or more residues are evaluated at the same point: as an example, if we take the first residue in b 1 = u k + i 2 , the poles in b j =1 = u k + i 2 disappear and we do not have to consider them when we integrate over the other variables b j ; • The polar part 1 b 2 ij +1 , whose effect is to arrange the residues in strings in the complex plane, displaced by +i: considering the example before, the first residue in b 1 = u k + i 2 generates poles in b j =1 = u k + 3i 2 . Eventually, a particular residue configuration is represented by their 2n coordinates, which are all different and arranged in strings in the complex plane starting from u i + i 2 and displaced by +i. The examples n = 1, 2 discussed later will clarify the procedure.
For the most general case of 2n scalars, this procedure leads to the formula Formula (2.12) is our sum over Young tableaux. Some explanations are needed. The symbol (l 1 , . . . , l 2n ) represents the contribution of a particular residue pattern: we have l i residues with real coordinate u i arranged in a string in the upper half plane, as described before. The constraint 2n i=1 l i = 2n follows from the fact that we have 2n integrations, while l i < 3 is due to a particular feature of the polynomials δ 2n , see Appendix A. The lower index s in (l 1 , . . . , l 2n ) s means sum over permutation (of inequivalent rapidities) of a single residue configuration: (l 1 , . . . , l 2n ) s ≡ (l 1 , . . . , l 2n ) + permutations of l 1 , . . . , l 2n . (2.13) Finally, the symbol Y (with |Y | = i l i ) is a shorthand for (l 1 , . . . , l 2n ) and will be often used in the following to shorten various expressions. The building block of (2.12) is the contribution of the diagram (l 1 , . . . , l 2n ). Its evaluation gives 8 (2.14) We split (2.14), except for a factor, in two parts, one coming from δ 2n (Y ) and the rest, called [l 1 , . . . , l 2n ]. The notation δ 2n (Y ) means that the arguments of δ 2n are the coordinates of the residues described by the pattern Y = (l 1 , . . . , l 2n ). Formula (2.14) can be specialized for our actual diagrams, with l i ≤ 2: the most generic contribution contains k columns with l i = 2, 2(n − k) with l i = 1 and k with l i = 0, thus the total number of different Young tableaux is n + 1.
To obtain a more compact expression of (2.14) we start with the simplest case which, using 16) and the Pfaffian representation [59] of δ 2n 9 can be recast in the compact form We introduced the subscript 2n in (1, 1, . . . , 1, 1) 2n to highlight the fact that there are 2n variables. At this stage this subscript is redundant, but it will be necessary in the following. The Det above is the determinant of a matrix whose element i, j is u ij /(u 2 ij + 1). On the other hand, the formula (A.10) gives our polynomial computed in a configuration of the type 10 (2, . . . , 2, 0, . . . , 0), which, combined with (always from (2.14)) , (2.19) yields the residues contribution for the other special case . (2.20) The most general configuration, up to a permutation of the rapidities, contains k columns of height two, k of height zero and 2n − 2k with height one. Relation (A.11) in Appendix A gives the polynomial δ 2n corresponding to (2, . . . , 2, 0, . . . , 0 2k , 1, . . . , 1) 2n in terms of the two special cases described before. The intermediate subscript 2k means that the first 2k columns are of the type (2, . . . , 2, 0, . . . , 0) 2k , while 9 We are very grateful to Ivan Kostov and Didina Serban for the various discussions on the subject and for pointing out this specific representation of δ 2n . 10 We permuted the rapidities in order to get all the columns with two boxes on the left, to get (2, . . . , 2, 0, . . . , 0) in place of (2, 0, . . . , 2, 0). the remaining 2n − 2k contains 1. To get an explicit formula of the general contribution we also need the property, again from (2.14) Combining all the pieces, we write down the most general contribution as follows , (2.22) where with (1 2k+1 , 1, . . . , 1, 1) 2n it is intended the contribution of the type (2.18) restricted to the variables u 2k+1 , . . . , u 2n . Analogously, the determinant in (2.22) concerns a matrix whose elements are u ij /(u 2 ij + 1), with 2k + 1 ≤ i, j ≤ 2n. Recalling (2.12), the matrix part is a sum over the Young tableaux configurations, which are in turn given by all the permutations of inequivalent rapidities of (2.22). We perform the sum over (Y ) s considering all the cases k = 0, . . . , n in (2.22); to symmetrize the Young tableaux, we sum over the (2n)! permutations P and divide by the overcounting factor (k!) 2 (2n − 2k)!, to get the final expression for the matrix part Formula (2.23) is the main result of this section: it represents the matrix factor as a finite sum of rational functions. However, the polar structure of Π (2n) mat remains somehow hidden in that representation, since many poles appearing in the sum cancel once we consider all the terms. The exact polar structure of Π (2n) mat will be analysed in next section using another feature, the asymptotic factorisation.
In order to elucidate the method of the Young tableaux, below we outline the computations for the simplest cases.
• Two scalars (n = 1): The simplest case (n = 1) involves just a couple of scalars, u 1 , u 2 . From (2.6) we have , (2.24) which, upon performing the integrations over the variables a and c, turns to This result means that our polynomial is trivial for two particles, i.e. δ 2 = 1. The contour integrals over b 1 , b 2 are easy to perform without any Young tableaux technique (we have just 3 × 2 = 6 residues to evaluate) and we finally get . (2.26) Even though the answer is already known, it is meaningful to solve the n = 1 case within the Young tableaux framework, in order to give a simple illustration of how it works. Afterwards, we will address the first non trivial case, n = 2. We start from the double integral over b 1 and b 2 , We perform the integration on the real axis closing the contour in the upper half plane. Therefore the integral over b 1 gets contributions from the poles b 2 + i, u 1 + i/2 and u 2 + i/2, leading to the following expression are the contributions respectively for b 1 = b 2 + i, u 1 + i/2, u 2 + i/2. As for the integral over b 2 , we see that each term contains two poles. Therefore, in total we have 3 × 2 = 6 residues. We can represent the various contributions by the position of the poles of the isotopic roots (b 1 , b 2 ): it is easy to check that they are (u 1 + i/2, u 1 + 3i/2), (u 1 + 3i/2, u 1 + i/2), (u 1 + i/2, u 2 + i/2), (u 2 + i/2, u 1 + i/2), (u 2 + i/2, u 2 + 3i/2) and (u 2 + 3i/2, u 2 + i/2). The key property that allows us a quick evaluation of the integrals is that the residues are invariant under exchange of isotopic roots, then only three terms are truly different and can be represented by an array of two numbers (l 1 , l 2 ) with l 1 + l 2 = 1, where l i labels the number of roots with real coordinate u i . Therefore, we define the three residues configurations which are the n = 1 version of (l 1 , . . . , l 2n ). Eventually, the total matrix part amounts to , (2.32) which is in agreement with (2.26). As a last step, we note that (2, 0) and (0, 2) are related by the symmetry u 1 ↔ u 2 and thus we define the symmetric (2, 0) s = (2, 0) + (0, 2) which we call Young tableaux, to get our final result Π (2) mat (u 1 , u 2 ) = (1, 1) s + (2, 0) s , (2.33) where (1, 1) s ≡ (1, 1), as it is already symmetric.
• Four scalars (n = 2): When dealing with four scalars (n = 2), the δ-polynomial (2.10) reads Hence, for n = 2, formula (2.6) becomes: mat (u 1 , . . . , u 4 ) = 1 6 A standard evaluation by residues would be very long since there are 7×6×5×4 = 820 contributions (each integration lowers the number of residues by one). The Young tableaux expansion employs two symmetries: under permutations of isotopic rapidities b i (which brings a factor 4! = 24) and under permutations of u i (which is responsible for the subscript s ) to get only 5 different Young tableaux: (1, 1, 1, 1) s , (2, 1, 1, 0) s , (2, 2, 0, 0) s , (3, 1, 0, 0) s and (4, 0, 0, 0) s . Each of them is a sum over the permutations of rapidities of terms (l 1 , . . . , l 4 ), they are respectively 1, 12, 6, 12, 4. As a check, the total number of residues is (1 + 12 + 6 + 12 + 4) ×24 = 840 as stated before, but our method tells us that many of them are either equal (by permutations of b i ) or related by permutations of u i . The latter two diagrams vanish as a result of the property of the δ-polynomial already stated: To sum up, the four scalars matrix part is given by where the building blocks are, according to (2.22) (1, 1, 1, 1) s = (1, 1, 1, 1) = 16 Det (2, 0, 1, 1) = 1 u 12 (u 12 + i) 2 (u 12 + 2i) As an application of the method outlined in this section, we compute the residue of Π (2n) mat in u i = u j + 2i, which can be nicely expressed in terms of the matrix part with 2 scalars less . (2.41) In order to prove (2.41), we start from the sum over Young configurations (2.12) and note that the pole in u 2 = u 1 + 2i appears only in the terms belonging to the type (2, 0, l 3 , · · · , l 2n ), with 2n i=3 l i = 2n − 2. The sum on the RHS reduces then to that of Π (2n−2) mat (u 3 , · · · , u 2n ). To go further we need to work out the expression of (2, 0, l 3 , · · · , l 2n ) and split it in three different contributions where the mixed term M {l i } depends on the specific configuration {l i } and on all the variables u i . We stress that (2.42) is a different and more complicated split than (2.22). The pole for u 2 = u 1 + 2i is contained only in (2, 0) with residue i/2 and, as we will show, when M {l i } is evaluated for u 2 = u 1 + 2i, which we call M * , the dependence on {l i } drops out, thus we get a prefactor multiplying the sum and the matrix part for fewer scalars is recovered By means of (2.20) and (2.22), the mixed contribution can be specified for the configuration (2 3 , · · · , 2 k+1 , 0, · · · , 0 2k , 1, · · · , 1) the other being obtained by a suitable permutation of the variables. By the identification u 2 = u 1 + 2i we find and finally prove the claim. We remark that a residue formula like (2.41) was expected for physical reasons, as Π (2n) mat is a part of the squared form factor of a specific operator, which must satisfy certain axioms. Among them, one concerns the residues of its kinematic poles and relate them to the form factor with two particles less. The kinematic poles are those in u i = u j + 2i, thus (2.41) is nothing but a consequence of the form factor interpretation of the pentagonal transitions.

Other representations
The matrix part, starting from its expression (2.11) as an integral over the isotopic roots b i , enjoys other alternative representations. They are originated form the properties of the δ 2n polynomials.

The matrix factor as an integral of a determinant
As previously anticipated, the δ-polynomials enjoy a nice expression in terms of the Pfaffian of a skew-symmetric matrix [59], which implies a determinant representation for its square It then follows, from (2.11), a determinant representation for the (integrand of) matrix part We can put the functions f (u i − b j ) inside and define the new 2n × 2n matrix A . (2.48) We can go further by using the Vandermonde formula combined with the well-known Cauchy identity or, in terms of the matrix D Π (2n) .

The matrix factor as a scalar product
Another nice representation of the matrix factor is as a scalar product between two symmetric multiparticle wave functions. These are defined by (2.55) 11 For the square of the delta polynomials we have also a representation in terms of determinants: In terms of ψ (2n) the matrix factor (2.11) can be recast as denoting with [ψ (2n) ] * the complex conjugate of ψ (2n) 12 . The recursive behaviour of the δ polynomials (A.7) also affects the functions ψ (2n) : indeed, they enjoy the following recursive relation where Θ(j) stands for the Heaviside step function; the index k can be arbitrarily chosen, then held fixed, with no summation involved. Upon introducing the set whose elements are couples of natural number (from 1 to 2n), of whom at least one is equal to k or l, the relation above finds a more compact expression (k arbitrarily chosen): For clarity, the very first ψ (2n) are listed, making use of the short-hand notation ω ij ≡ u i − b j + i/2 : (in reference to (2.57), we set the fixed index k = 2 to write ψ (4) , as an example).

Asymptotic factorisation
The main aim of this article is to arrive at definite expressions for the contributions of scalars to the expectation values of hexagonal Wilson loop in the strong coupling limit, which is expected to be exponential in √ λ. In the framework of the OPE discussed before, this entails going through integrations involving the functions G (2n) . The different terms of the series W (2n) turn out to be of different orders in √ λ, in particular W (2n) ∼ ( √ λ) n . Given the exponential behaviour of W , what we have in mind is to study its logarithm, whose terms of the series expansion contain the connected counterparts g (2n) of the G (2n) and their order is expected to be √ λ: we will describe the procedure in details in Section 4. This part is devoted to prove some fundamental properties of the function G (2n) , which in turn will apply on the connected parts enabling us to effectively employ the series of the logarithm.
In order to understand the properties of the expansions of W and ln W , it is necessary to analyse the behaviour of these functions when some of their arguments, the rapidities u i , go to infinity. More in detail, we are going to show a factorisation property for any G (2n) when we shift an even number m of its rapidities by large amounts Λ i , while holding fixed the remaining 2n − m.
To be specific, we will prove that factorisation arises when considering where each of the Λ i is parametrised as Λ i = c i R + O(R 0 ), with c i constants and R → ∞. The procedure is an extension of the one discussed in the letter [32], where we proved the factorisation G (2n) → G (2k) G (2n−2k) when the 2n particles are split in two groups composed respectively by 2k and 2n − 2k particles, separated by the large parameter Λ (in this particular case all the c i above are equal). More in general the discussion we are going to expose now are extensions to asymptotically free theories of analogous results [56] found in Form Factor computations. While in [56] the corrections to factorisation are exponentially suppressed, in our case they enjoy a simple power-like decay as a consequence of asymptotic freedom. This very important fact is the core of this section since, as it will be extensively clarified throughout the paper, the integral of the connected functions g (2n) over the 2n − 1 variables on which they depend must be finite. In order to assure that, we need to unravel the properties of G (2n) when some rapidities go to infinity separately.
We start our analysis from the dynamical factor (2.3) which, in the strong coupling (non-scaling) regime, can be given an explicit form through when the displacements Λ i are sent to infinity, as a consequence of the asymptotic behaviour We remark that Π (m) as a consequence of (3.4).
In order to show how to work out the matrix part (2.4) instead, it is convenient to tackle the simplest non trivial case first, i.e. the asymptotic factorisation Π mat : eventually, the procedure can be straightforwardly adapted to the general case Π (2n) To start with, we perform the shift on the rapidities u 1 → u 1 + Λ 1 , u 2 → u 2 + Λ 2 ; then, for large Λ 1 , Λ 2 the integrals (2.4) receive the main contribution from the region in which two roots b, one a and one c are comparable with Λ i . Therefore, we write (2.4) after shifting, for instance, a 1 by Λ a 1 , c 1 by Λ c 1 and b 1 , b 2 by Λ b 1 , Λ b 2 , respectively, where the large shifts of isotopic variables can be equal to Λ 1 or Λ 2 . Actually, we have to sum over all possible choices for shifts Λ α i , α = a, b, c,, which give the same result. We indicate shortly shifts this sum in formula (B.1). Eventually, the resulting expression has to be multiplied by a multiplicity factor taking into account the where, if c 1 = c 2 , Π mat (u 1 + Λ 1 , u 2 + Λ 2 ) = 6/Λ 4 12 + . . ., whilst if c 1 = c 2 this function is finite. On the other hand, for the dynamical parts we have . ., otherwise if c 1 = c 2 this function is finite. Putting the dynamical and matrix parts together (2.2), we get the following asymptotic factorisation when the first two rapidities get large: is a symmetric function of the four u i , property (3.9) is indeed valid when any couple of rapidities is very large.
We now consider a more general case: we shift an even number m = 2k of rapidities by amounts Λ i . Because of the symmetry of the function G we can stick to the case in which the first m rapidities are shifted: In order to get the main contribution to the integrals for large Λ i , we also shift a i → a i + Λ a i and which is an extension to the case in which the Λ i can be different to analogous results [32] holding when all the Λ i are equal. We remark that Π (2k) If some of the Λ i are equal, the function Π (2k) mat (u 1 + Λ 1 , . . . , u 2k + Λ 2k ) goes to zero less rapidly than (3.11).
On the other side, the dynamical part enjoys the behaviour (3.3). Putting dynamical and matrix part together, we get the sought factorisation property: which extends the known factorisation property of [56]. We remark that the function : some of these cases and more general configurations will be examined in Appendix D. Now, we spend a few words to discuss the behaviour of G (2n) when we shift an odd number m of rapidities. This discussion is necessary in order to study the behaviour for large rapidities of the integrands appearing in the matrix part. Obviously, in this case there is no factorisation, since there are no functions G with an odd number of arguments.
When m is odd, we find convenient to define m = 2k − 1. Then, we shift a i → a i + Λ a i and With these positions, formula (B.5) still holds, along with (B.6). Sending Λ i → +∞ inside the integrals, we get Therefore, sticking only to the leading term, when m = 2k − 1 for the matrix part we have the behaviour for different Λ i Π (2n) where for 'finite function' we mean the third line of (B.5) excluding the function R (2n,m) . We have to multiply the matrix part by the dynamical part which behaves -also for odd m -as (3.3). Doing this we get Therefore, we conclude that, if all the Λ i are different and of order R as before, the behaviour of It is worth to point out that there exists a different method to prove the factorisation properties obtained so far, which makes use of the Young tableaux representation (2.23). To give a sketch, let us consider the n = 2 split 4 → 2 + 2, which corresponds to (3.7) with equivalent shifts Λ 1 = Λ 2 ≡ Λ. We thus observe that many diagrams of Π (4) mat split into a product of two diagrams already encountered when computing Π (2) mat , weighted by a factor Λ −8 where the right hand side (RHS) members sum up to Λ −8 Π We remark that other diagrams of Π (4) mat are of subleading order o(Λ −8 ) and thus they do not contribute to the factorisation. The procedure can be extended to the general split 2n → 2(n − k) + 2k, allowing also to have different shifts Λ i .
To summarize, we addressed some 13 different splits of the rapidities and obtained the corresponding asymptotic behaviours of the G (2n) . The main results are equations (3.12), (3.14) and (3.16), they will turn out to be useful in section 4 where we study the connected functions g (2n) .

Polar structure of the matrix factor
We are now coming to an important point of this article. Indeed, making use of the asymptotic factorisation discussed in this section, we can now prove in general that the matrix part can be written as follows where P 2n is a symmetric 14 polynomial. Therefore, Π mat has poles only when the difference of two rapidities equals ±i or ±2i.
Actually, for the present proof we need only to know the behaviour of Π (2n) mat when two arbitrary rapidities u p , u q get large in the same way, i.e. c p = c q and c i = 0 for i = p, q: ]}) and the notation u k means the omission of the rapidity u k . Then, we remember that Π (2n) mat (u 1 , . . . , u 2n ) depends only on the differences u ij , with i < j, i, j = 1, . . . , 2n, and, consequently, may show singularities when u ij pick particular values. Of course, any singular values of Π (2n) mat for the particular difference u pq are left unchanged by the shifts in the left hand side (LHS) of (3.19), whose RHS (in its second factor) tells us where they occur: u pq = ±i, ±2i. Repeating this reasoning for all the possible differences of rapidities, we obtain the structure (3.18).
What is left unknown in (3.18) are the polynomials P 2n . The simplest ones, corresponding to n = 1, 2, are reported in Appendix C ; for n ≥ 3 expressions for P 2n get rapidly unwieldy. The residue formula (2.41) allows us to relate the polynomial P 2n , evaluated in a specific configuration, to a smaller polynomial. Other general properties of these polynomials, which can be found without much ado, are their degree and their highest degree monomial. The highest degree monomial will be discussed in Appendix C. Instead, the degree of the polynomial P 2n (u 1 , . . . , u 2n ) may be found here by comparing (3.18) to (2.11). The degree of Π (2n) mat (u 1 , . . . , u 2n ) is found to be equal to −4n 2 by using integral representation (2.11) and the fact that the degree of δ 2n (u 1 , . . . , u 2n ) is 2n(n − 1). It then follows that the degree of P 2n (u 1 , . . . , u 2n ) is −4n 2 + 4 2n(2n−1) 2 = 4n(n − 1). It is worth to remark that the two polynomials P 2n and δ 2 2n have both degree 4n(n − 1) and, as discussed in appendix C, their highest degrees share the same structure.

The expansion in the strong coupling regime
As a preliminary remark, we will show that, since scalars decouple from the rest of the particles for large values of λ, the hexagonal Wilson loop can be decomposed into the product of a factor accounting for the minimal surfaces on AdS 5 , multiplied per an O(6) factor ascribable to scalars. We will carefully show how the scalar factor can be isolated, by initially mixing scalars with a single kind of particles at once, then by considering all the species together. As a notation remark, in what follows W α 1 ,...,α k stands for the expectation value of a hexagonal Wilson loop, taking into account scalars, gluons, fermions and antifermions as excitations (i.e. α 1 , . . . , α k ∈ {s, g, f,f }), while W (N 1 α 1 +···+N k α k ) denotes the contribution to W α 1 ,...,α k , brought by an intermediate state made of N 1 particles of type α 1 , N 2 of type α 2 , and so on. In order to keep in touch with the previous notation, the 2N-scalar contribution to the hexagonal Wilson loop and the 2N-scalar matrix factor are shortly denoted as W (2N ) and Π (2N ) mat (instead of W (2N s) and Π (2N s) mat ), while in the strong coupling limit µ s (u) reduces to µ (3.2).
• Scalars and gluons: When considering 2N scalars of rapidities u k along with M gluons with rapidities v k , their contribution to the hexagonal Wilson loop reads Since gluons behave as singlets under SU(4), the matrix factor Π (2N s) mat takes into account only the 2N scalars, arranged into the singlet configuration, whereas the dynamical factor enjoys a pairwise decomposition [60] Π (2N s+M g) dyn 2) being E g , p g , µ g the expressions for energy, momentum and measure of gluons, whereas P gg , P sg stand for the gluon-gluon and scalar-gluon amplitudes. In the strong coupling regime considered, the gluon rapidities get rescaled, i.e. v k = √ λ 2πv k , while holding the scalar rapidities fixed, resulting in a decoupling at the level of pentagon amplitudes [61] and as a by-product the scalar and gluon contributions become clearly distinguishable From the last line of (4.4) we can infer the strong coupling factorisation of the hexagonal Wilson loop into a scalar part W (2.1) and a gluon part W g : • Scalars, fermions and antifermions: When studying the contribution to the hexagonal Wilson loop from M fermions (with rapidities u k ), M antifermions (v k ) and 2N scalars (w k ), the computation is more involved, since the matrix factor needs to take into account that fermions and antifermions are not SU(4)-singlets, where the number of a-roots K a equals the number of c-roots, the number of b-roots is K b , and they are related to the number of particles by In the strong coupling limit, we take finite scalar rapidities, w k = O(1), whereas fermion/antifermion rapidities get rescaled, u k = (4.10) The pairwise decomposition of the dynamical factor [61] 15 , together with the strong coupling 15 Here portrayed for scalars and fermions only, for simplicity.
behaviour of the mixed pentagons [52] Π allow us to the separate the scalar contribution from the fermion contribution • Scalars, gluons, fermions and antifermions: Finally, we consider scalars, gluons, fermions and antifermions altogether, each kind of particle being labelled by a Greek index α ∈ {s, g, f,f} (for scalars, gluons, fermions and antifermions respectively), while the Latin label distinguishes the particle of a given type α (i = 1, . . . , N α ); since the system carries no overall SU(4)-charge, we need N f = Nf . Once the multiparticle pentagon factorisation is assumed [60] , (4.14) the hexagonal Wilson loop can be split into the product of the minimal area contribution W AdS and a scalar contribution As a consequence, in order to compute the same-order correction to the minimal area contribution, it is sufficient to consider scalars alone, ignoring their interactions with other particles. It is worth remarking that the two contributions, AdS 5 and S 5 , behave very differently in the collinear limit τ → +∞. The classical contribution W AdS ≃ e − √ λ 2n A 6 becomes trivial, since the area is exponentially suppressed ) in the regime considered. On the other hand, the effect of the five sphere S 5 , or scalars from the OPE point of view, may be finite and remains the only contribution to the hexagonal Wilson loop in the strong coupling limit. As it will be clearer in the following, the necessary condition is that the combination z ≃ e −

The importance of being connected
A Leitmotiv (cf. the explicit Young tableaux computation of subsection 2.1 and the parallel with N = 2 theories [57]) is to consider the WL (2.1), in general, as a partition function. Therefore, we find convenient to compute its logarithm in terms of the 'connected' functions g (2n) : As well known the 'non-connected' function G (2n) can be expressed in terms of the connected g (2l) , with l ≤ n; here we list the first few examples, upon introducing the shorthand notation g i 1 ...in ≡ g (n) (u i 1 , . . . , u in ): G 1234 = g 1234 + g 12 g 34 + g 13 g 24 + g 14 g 23 = g 1234 + (g 12 g 34 + 2 perm.) G 123456 = g 123456 + (g 12 g 3456 + 14 perm.) + (g 12 g 34 g 56 + 14 perm.) . (4.17) The relations above can be inverted to gain the desired g (2n) in terms of the G (2l) , l ≤ n: These formulae can easily be made fully general (arbitrary n) as explained in Appendix D.
As well established in field theory, the connected functions g (2n) enjoy a plethora of computational advantages with respect to the G (2m) quite in general. In the present case, for instance, they make possible the large coupling expansion by allowing this limit inside the series F on the F (2n) : this exchange is not possible on the original (2.1) because of the asymptotic divergence of the G (2m) . In physical words, the connected functions re-sum many infinities to finite contributions. Therefore, as we will extensively see later, it is crucial that the functions g (2n) (differently from the G (2m) ) are integrable over the 2n − 1 variables they depend on. To this aim, it is sufficient to prove that g (2n) belongs to the class L 1 (R 2n−1 ), which is a stronger condition since it involves the modulus |g (2n) | inside the integral. To ensure g (2n) ∈ L 1 (R 2n−1 ) we need to address all the possible asymptotic behaviours in the integration space. The most general situation concerns l subsets composed of k i (i = 1, . . . , l) variables going to infinity by the shifts Λ i = c i R, i = 1, . . . , l, where R is large and the coefficients c i = c j ( i = j) are finite. Sufficient condition for a connected function to be integrable at infinity is the behaviour (4.19) This condition is the generalisation of the one dimensional case R a≤−2 , once we take into account the integration volume which grows as R l−1 . A rigorous proof of g (2n) ∈ L 1 (R 2n−1 ) is not easy, as the number of regions grows very rapidly with n. However, there are many indications and explicit computations that all the functions belong to L 1 (R 2n−1 ). In particular, a thorough discussion of the condition (4.19) for the first cases g (4) and g (6) can be found in Appendix D. In conclusion, we can assume that all the multi-integrals are finite. Eventually, we shall not forget that also numerical calculations here on the g (2n) are much easier that those on the G (2m) [42,52].

Small mass behaviour
In this subsection we provide analytical evidence that, when the strong coupling limit is considered, the scalar partition function (2.1) yields an exponentially large contribution to the Wilson loop, which happens to be of the same order as the one from the classical area [47]. In fact, the energy and momentum are in general complicated coupling dependent dispersion relations (in terms of the rapidity u) [36], but reduce to the relativistic ones in the non perturbative regime λ → +∞ with the characteristic, for the scalar sector, of the dynamically generated mass [47,48,49,50,51] m gap (λ) = 2 1/4 Γ(5/4)  in the (Euclidean) relativistic invariant O(6) NLSM of a specific twist operatorV . In this picture G (2n) is the square modulus, summed over the internal O(6) indices, of the form factor 0|V (0)|Φ a 1 (u 1 ) · · · Φ a 2n (u 2n ) , where Φ a (u) represents a scalar with rapidity u and a as O(6) degree of freedom. The two cross ratios are just the coordinate of the difference z 1 − z 2 ≡ z 12 = (τ, σ) and rotational invariance (we have rotated into the euclidean space) imposes that everything must depend only on the distance (modulus) |z 12 | ≡ √ σ 2 + τ 2 . In fact, we just need to insert the identities inside (2.1):

23) and then define the natural variable
and shift the integration variables as this does not affect the functions G 2n (u 1 , . . . , u 2n ), depending only on the differences u i − u j . For the same reason we can safely perform a shift back of the contours to the real axis 16 : The final expression (4.26) depends on the cross ratios (better: on the modulus) and on λ only through the 'adimensional' variable (4.24) z = m gap (λ)|z 12 | . In this (non-scaling) regime the pentagonal amplitude P ss (u i |u j ) depends at LO only on u i − u j and not on the coupling, while the measure becomes a constant (3.2). The function Π(u) has simple poles when u = ±(4m + 2)i, u = ±(4m + 3)i, with m a positive or null integer. In addition, the function Π(u) (3.2) has a double zero for u = 0 and simple zeroes when u = ±(4m + 1)i, u = ±(4m + 4)i with m a positive or null integer.
The asymptotic behaviour of the connected functions assumes a paramount importance when studying the logarithm of the Wilson loop at strong coupling, or small z, as conceived by [56], 16 As a proof, we can make a change of variables in (4.25), integrating in u 1 on Imu 1 = 2 arctan σ/τ π and in the differences u i =1 − u 1 on the real line; since G (2n) depends only on u i =1 − u 1 , the shift of the u 1 contour to Imu 1 = 0 does not produce any additional terms, since exp{−z cosh π 2 u 1 } is analytic.
from which we generalise to the asymptotically free case. The physical reason for their efficiency resides in the fact that they re-sum an infinite number of particle contributions from the original series (2.1). In order to highlight how the connected functions depend only on 2n − 1 independent differences of the rapidities, let the reader allow us for a slight abuse of notation: upon introducing the rescaled rapidities θ i = π 2 u i , we denote by g (2n) (θ 2 − θ 1 , θ 3 − θ 1 , . . . , θ 2n − θ 1 ) the function (2/π) 2n g (2n) (u 1 , u 2 , . . . , u 2n ). Hence, the generic term F (2n) of the series (4.16) for the logarithm of the Wilson loop reads (4.27) The new set of variables α i = θ i+1 − θ 1 for i = 1, . . . , 2n − 1, allows us to recast the integral I (2n) It turns out convenient to define a = 1 + in terms of a real parameter η, depending on the α i but not on θ 1 , which can be thus integrated away: We stress that the result (4.30) for I (2n) holds for any z and does not rely on any properties of the functions g (2n) , but their dependence on the rapidities only through their differences θ ij . Motivated by the findings from Section 4.1, we claim that the integral (4.30) is finite regardless of the damping factor K 0 (zξ). On the contrary, the functions G (2n) are not integrable with respect to the 2n − 1 variables α i : indeed, when an even number m < 2n of θ i get shifted by the same quantity Λ ≫ 1 (i.e. θ i → θ i + Λ ∀i ≤ m), the factorisation G (2n) −→ G (2n−m) G (m) prevents G (2n) from decreasing to zero, since both the factors are order O(1) as a result of their dependence on differences of rapidities. As already observed, the connected functions g (2n) (contrarily to the G (2n) ) are of class L 1 (R 2n−1 ) and this fact allow us to expand the Bessel function (inside the integral (4.30)) at any order of small z ≪ 1: where the second equality follows by expanding the definition (4.24) at large coupling ln z = − √ λ 4 + O(ln λ). A caveat arises, though, when putting forward a systematic expansion: in fact, the term ln ξ in the asymptotic series (4.31) grows linearly for large rapidities, making the integral diverging at infinity. We can overcome this difficulty by introducing a cutoff zξ < 1 and splitting (4.30) accordingly While the cutoff zξ < 1 can be safely removed from the first line of (4.35) 19 , the second line needs a regularisation, which entails the peculiar form ln ln(1/z) of the subleading term. In conclusion, the small z (e.g. strong coupling) expansion of the logarithm of the hexagonal WL (4.16) enjoys the form F (z) ≃ J ln(1/z) + s ln ln(1/z) + t (4.36) which reveals its peculiar double logarithmic behaviour, ascribable to the asymptotic freedom of the O(6) NLSM. Correspondingly, the Wilson loop can be rewritten by means of (4.21) and (4.24), so to highlight its dependence on the coupling λ and the cross ratios Now, it is evident that the leading term does not depend on the cross ratios and can be comparable to (if not bigger than) the classical minimal area contribution [16] W AdS ≃ C AdS e − √ λ A 6 2π (arising from the contribution of gluons and fermions in [30]). Moreover, from (4.38) one can directly read the subleading correction λ B , brought by scalars: it is the only contribution of that type, as the one-loop string corrections give a constant contribution C AdS (τ, σ, φ) of the same kind of C(τ, σ). In the collinear limit τ → +∞, though, the one-loop corrections become negligible and the prefactor is fully given by C(τ, σ).
The small z expression (4.37) proves the proposal of [42] coming from associating the pentagonal amplitudes to O(6) twist fields, whose scaling properties suggest the values J = 1/36 and s = −1/24 [20] (also confirmed by numerical computations [42,52], which are much easier here thanks to the employment of the connected g (2n) ). Differently, we obtained formula (4.37) directly from the OPE series and thus can fruitfully decompose so that the 2n-particle connected contribution to F is parametrised, in the limit z → 0, according to 20 F (2n) ≃ δJ (2n) ln(1/z) + δs (2n) ln ln(1/z) + δt (2n) . . In fact, we wish to determine analytically these coefficients for n = 1, 2 and provide some considerations for arbitrary n.
We emphasise that this is not the original number of particles contributing to W (4.26), as any connected function re-sums (in ln W ) an infinite number of particle contributions from the non-connected ones in W (4.26). This simple fact entails a great improvement in the accuracy; in fact, on one side, the functional form of the expansion is the right one even for the lowest n = 1 (cf. also below), on the other the numerical values of the coefficients are quite precise yet for lower n.
• Two scalars: When considering (4.30) for n = 1, there is only one integration variable where the rescaled function g (2) (α) = 4 π 2 g (2) (u 1 , u 2 ) reads and enjoys the asymptotic behaviour g(α) = Cα −2 + O(α −4 ) with C = 6π Γ 2 (3/4) Γ 2 (1/4) . We want to determine the coefficients in the expansion (4.40) According to (4.34), we divide the integral in two parts 2 goes to zero because K 0 is bounded within the integration support and the function g (2) behaves as Cα −2 + O(α −4 ) for large rapidity, resulting in an O(1/ ln z) contribution. As far as F (2) 1 is concerned, in order to estimate the diverging and the finite contributions for z → 0, we are allowed to expand K 0 (2z cosh α 2 ) for small argument (4.31). Renaming the function h(α) ≡ 1 2π 2 g (2) (α), we get (4.47) In order to disentangle the O(ln ln(1/z)) contribution from the constant ones in we split the integration domain into two intervals the latter housing the divergence ln ln(1/z): to extract it, we add and subtract a counterterm The piece stays finite in the limit 2 ln(1/z) → ∞ while the second yields the subleading ln ln(1/z), up to an additive constant (4.51) Keeping track of all the divergent and finite pieces, we obtain (4.44) with: Our numerical estimate for t (2) amounts to t (2) ≃ −0.00819, in agreement with the Montecarlo evaluation by [42,52].
• Four scalars: As far as the leading order J ln(1/z) is concerned, it is not difficult to evaluate the correction δJ (4) coming from the explicit expression of the four scalar connected function g (4) . Specializing (4.33) for n = 2 we get (4.55) We simply integrate it with Mathematica R and obtain a correction to J by an amount δJ (4) = (−3.44 ± 0.01) · 10 −3 , i.e. J (2) + δJ (4) ≡ J (4) ≃ 0.02765: this value differs from the 2D-CFT prediction J = 1 36 = 0.027 [42,52] by just 0.5%. The correction δs (4) to the subleading coefficient s is more involved, as it depends on the asymptotic behaviour of the connected function g (4) and there are many different regions to take into account. We remind from formula (4.35) that the divergence ln ln(1/z) comes from the combined action of the cutoff zξ < 1 and the piece g (4) ln ξ due to the expansion of K 0 (zξ). More precisely, it is contained in the integral (4.56) When one or more variables are large we have, from (4.28), ln ξ ≃ |α i | 2 , where α i is the largest of them, and the cutoff condition translates into |α i | < 2 ln(1/z). The linearity in α i tells us that the only region where the integral becomes divergent corresponds to the split 4 → 3 + 1, where g (4) goes to zero with the minimum power required by convergence, see Appendix D. This region has multiplicity four 21 and they are all physically equivalent, so we choose to send α 1 → ∞ and keep the other variables finite. We define the asymptotic function g  where we considered only the upper integration limit, which contains the divergent part. In addition, a factor 4 · 2 due to the number of regions (a particle can be sent either to +∞ or −∞) shows up. The coefficient in (4.56) is then , (4.60) where in the last piece we used the variables u 3,4 = 2 π α 2,3 for brevity. The numerical integration yields δs (4) ≃ 0.017650 and the four particles prediction sums up to s (4) = s (2) + δs (4) ≃ −0.036894: the discrepancy with respect to the expected value [42] s = −1/24 = −0.0416 is about 11%.
The finite part δt (4) of the four particles integral has three different contributions δt (4) = δt (4) 1 + δt (4) 2 + δt (4) 3 . They can be obtained from once we subtract both the divergent terms, δJ (4) ln(1/z) and δs (4) ln ln(1/z), previously analysed. Referring to formula (4.35), we immediately see that a finite contribution comes from the constant term in the expansion of the Bessel function ln 2 − γ, then (4) . Another one appears when we remove the cutoff zξ < 1 (see the first line of (4.35)) in the computation of δJ (4) δt Repeating the same argument as for the subleading δs (4) , only the regions 4 → 3 + 1 matter and their contribution is exactly the same δt (4) as (α 2 , α 3 ) = δs (4) . The last one, δt 3 , must be extracted from the integral (4.56) − 1 12(2π) 4 zξ<1 dα 1 dα 2 dα 3 g (4) (α 1 , α 2 , α 3 ) ln ξ ≃ δs (4) ln ln(1/z) + δt which, besides the ln ln(1/z) contribution obtained in (4.59), contains also a finite piece. As in the two particle case, we regulate the infinity by subtracting the asymptotic behaviours and get the finite integral (it is allowed to remove the cutoff zξ < 1) where we introduced a > 0 to prevent the singularities on the axes α i = 0. This parameter does not affect the large α i limit and we can take any finite value we want. On the contrary of the two particle case we do not split the integration in parts, for it turns out to be rather involved; the insertion of a parameter a which avoid the singularity in α i = 0 is more effective. 22 The divergence δs (4) ln ln(1/z) is isolated in g (4) as (α 2 , α 3 ) + 1 2(|α 2 | + a) g (4) as (α 1 , α 3 ) + 22 Alternatively, we could have used this procedure also for the n = 1 case.
g (4) as (α 1 , α 2 ) + 1 2(|α 1 | + a) g (4) as (α 2 − α 1 , α 3 − α 1 ) , (4.67) which also contains δt 3 . The four terms contribute the same thanks to the invariance of the cutoff zξ < 1 under permutation of variables and (α 1 , Disregarding the vanishing terms, the integral simplifies to as the divergence appears only where |α 1 | is large and we can safely remove the cutoff in the other directions. The integral over α 1 yields which reproduces (4.59) plus a finite correction proportional to δs (4) and eventually we get where the dependence on a drops out thanks to To simplify the result we choose a = 2 and sum up everything to get the final answer (4) .
• 2n scalars: Referring to the notation and recalling the expansion (4.39), we get the following expressions of the 2n particle contributions to J, s and t: the leading divergence J gets corrected by while the subleading δs (2n) is contained in the integral which also yields the finite piece δt (2n) 3 . As for t, we have three contributions δt (2n) = δt , where the first is simply due to the constant term in the expansion of K 0 while the second comes from the removal of the cutoff in the computation of δJ (2n) and reads As in the n = 1, 2 cases, it can be shown to equal 23 δs (2n) . Collecting all the contributions, we get δt (2n) = (ln 2 − γ)δJ (2n) + δs (2n) + δt (2n) 3 .

(4.78)
In summary, we provided many explicit formulae for the coefficients J, s and t, which parametrise the small z limit of W (4.37). Thanks to the expansion of ln W , we have been able to represent them as a series (4.39), which is a very effective procedure, as their contributions δJ (2n) , δs (2n) and δt (2n) can be easily extracted from the integral (4.35), in most cases analytically. Already for n = 2, the expected values [42] for J and s are reproduced with a good accuracy. A deeper numerical analysis of (4.35) could confirm their values with even more precision, and would allow to compute the constant contribution t very precisely. On the other hand, by means of (4.38), the coefficients J, s and t can be used to parametrise the scalar contribution W O (6) in terms of the coupling constant λ.

Conclusions and perspectives
For our purposes, the strong coupling behaviour of the quantum GKP dynamics shows at least two different regimes depending on the value of the rapidities. In the first, the strong coupling non-perturbative regime (fixed rapidity), the scalars of the hexagonal WL (OPE) series yield a dominant contribution, W O(6) , and decouple from the other particles (gluons and fermions) whose effect is negligible. Yet, in the perturbative regime (rapidity scaling like ∼ √ λ), only gluons and fermions yield a contribution, W AdS , and it is nothing but that of the classical AdS string [44,30]. The two contributions are comparable and compete one with the other depending on the values of the cross ratios. Naturally, this scenario admits a generalisation for the other polygons, and the analysis has been restricted here only to the simplest polygon for sake of simplicity.
In fact, the matrix factors appearing in the hexagonal WL are the simplest ones and they have been recast into a shape recalling the Nekrasov instanton partition function of N = 2 SYM. Thus, they have allowed us an efficient and elegant treatment through Young tableaux, which culminated in the calculation (2.23) in terms of rational functions. Since the starting formula inherits its structure from the SO(6) symmetry of the scalars, we would like to think that the entire procedure may be generalised to the more complicated matrix parts appearing in the other polygons. To support this idea and elucidate the method, we have explicitly performed the computations for two and four scalars and eventually obtained final elegant expressions. In fact, this rather holds in general as we have provided explicit closed expressions for the polynomials δ 2n entering the integral expression for the matrix factor, (2.11), and we have introduced and partially disentangled the polynomials P 2n (3.18), which completely define the explicit expressions of the matrix factors. Eventually, this Young tableaux approach naturally applies as well to the SU(4) matrix part of the fermion/anti-fermion contributions, which give rise at strong coupling to the so-called meson excitation [30,31]. This is actually the topic of an upcoming paper [62].
On the contrary, it is suitable that we recall why the dynamical parts of the pentagonal transitions are more explicit: they are always products of two-body components. Besides, they become relativistic when the 't Hooft coupling grows.
The product of the dynamical and matrix parts makes the (modulus) square of the full pentagonal transition, G (2n) (for 2n particles). This enters the OPE series (multiplied by the exponential of the free propagation) and, for large rapidities, enjoys a factorisation property in terms of products of squared transitions G (2m) with less particles, m < n, up to corrections decreasing as inverse powers (of the rapidities). This feature constitutes a crucial issue of our paper in itself and for future studies, but also because it has led us to interesting achievements. For instance, the structure of the matrix factor, i.e. formula (3.18): the poles of the matrix factor have been completely identified and all the unknowns are the polynomials P 2n , whose structure is studied and discussed in Appendix C. Another consequence is the power-like decay of the connected counterparts g (2n) , which in their turn characterise the series expansion of the logarithm, ln W . This passage has allowed us the strong coupling expansion despite the ambiguous behaviour caused by an exponentially small mass gap m gap ∼ e − √ λ 4 . In perspective, this manoeuvre should be very efficient and fruitful for future studies. For instance for performing a numerical summation of the OPE series, also for other polygons. This seems quite evident if we look at the expressions for the 2n scalar leading terms at the end of Subsection 4.2. We have proved explicitly the two and four particles contributions to J, s and t, which give an estimate for A, B and C(τ, σ), parameters of the strong coupling behaviour of the scalar contribution (4.38). This line of research should allow us a precise numerical determination of the coefficients A, B, C of the aforementioned formulae from the OPE series. On the contrary, all this does not apply to the series for W , because of the single term behaviour W (2n) ∼ (1/z) n at LO. Physically, the fundamental difference is due to the asymptotic behaviour of the connected functions g (2n) with respect to the G (2n) .
Enlarging our point of view, if, as written above, it is natural to apply these ideas to arbitrary null polygonal Wilson loops with n sides, it is also very important to understand how our approach compares with FFs for twist fields [21] and how much of it survives for other operators in SO(6) (SU(4)) symmetric (relativistic) theories.

A Properties of the δ 2n polynomials
In this Appendix we list some properties of the polynomials δ 2n (b 1 , . . . , b 2n ) (2.10) which appear in the integrand (2.11), giving the matrix factor Π (2n) mat (u 1 , . . . , u 2n ). For convenience, we recall their expression as sums over partitions In the first place, the function δ 2n (b 1 , . . . , b 2n ) is invariant under the exchange of its arguments, i.e.
and vanishes whenever three or more variables lie aligned (i.e. spaced by i) in the complex plane From (A.1) a more compact representation to δ 2n can be obtained, by borrowing some results from the Quantum Hall effect: indeed, one can recognise the Moore-Read wave function [63,64] in the highest degree 2n(n − 1) of the δ-polynomials so that a more elegant expression in terms of a Pfaffian 24 follows We can extend (A.5) to the full δ 2n by means of the substitution 25 b ij , to find the compact formula [59] in terms of the Pfaffian of the 2n × 2n matrix D. The Pfaffian representation (A.6) also allows for a recursive description of the δ-polynomials: (for any arbitrarily chosen k ∈ {1, . . . , 2n}), where the notation b k means that b k does not appear as a variable of the function δ 2n−2 .

B Factorisations
In this Appendix, we report the main calculations which prove the asymptotic factorisation of the 2n-point function when some of the rapidities u i get large. Following the main text (Section 3), we first discuss the four point functions, then the general 2n-point case.

B.1 Four point functions
Starting from the integral representation (2.4) for Π (4) mat (u 1 + Λ 1 , u 2 + Λ 2 , u 3 , u 4 ) and performing the shifts in the isotopic variables described in the main text, we get Π (4) where we defined .
The reason to shift the isotopic variables is that we get an expression, (B.1), in which one is allowed to perform the limit Λ i → ∞ inside the integrals. We have The possible values for the string of shifts {Λ b 1 , Λ b 2 , Λ a 1 , Λ c 1 } are ten: Summing over the ten possible shifts, we eventually obtain Π (4) where Π (2) mat (u 1 + Λ 1 , u 2 + Λ 2 ) = 6/Λ 4 12 + . . .. In order to evaluate the subleading terms in the square bracket of (B.3), we use the following formulae: Formulae (B.4) are necessary to reconstruct the term 6/Λ 4 12 , the leading one in Π (2) mat (u 1 +Λ 1 , u 2 +Λ 2 ), after summing over the ten possible shifts.

B.2 2n point functions
Starting from the integral representation (2.4) for Π (2n) mat (u 1 + Λ 1 , . . . , u m + Λ m , u m+1 , . . . , u 2n ) and performing the shifts in the isotopic variables described in the main text, we get Π (2n) where we used the shorthand notations Λ a ij = Λ a i − Λ a j , Λ ab ij = Λ a i − Λ b j and where we multiplied the previous expression by a combinatorial factor 2n m n k 2 which takes into account the different choices of isotopic roots that after shifting give the same result. We are now allowed to send Λ i → +∞ inside the integrals in (B.5). We have Summing over all the choices for the shifts on the auxiliary variables, the second and the third line of (B.7) cancel and we are left with Π (2n) where, however, Π mat (u 1 + Λ 1 , . . . , u 2k + Λ 2k ) goes to zero like Π (2k) C The polynomials P 2n In Section 3 we proved the polar structure (3.18), where P 2n is a polynomial of degree 4n(n − 1), symmetric under permutations of its variables. In this Appendix we list some properties of these functions and give explicit expressions for P 2 and P 4 . In addition, by means of the factorisation, we derive a simple formula for the highest degree P 2n for any n and relate it to that of δ 2n .

C.1 Explicit expressions
We now provide the polynomials appearing in (3.18) in the cases n = 1, n = 2: P 2 (u 1 , u 2 ) = 6 (C.6) a free boson with propagator u −2 ij . The generalization of this formula to the 2n goes through an expression that, in the factorisation limit, reproduces exactly the property (C.13) for P (0) 2n , with any n, k. We thus conjecture where the sum is restricted 26 over the (inequivalent) pairings, such that the total number of terms is (2n − 1)!!, as in the Wick expansion. A careful analysis shows that (C.9) is the only polynomial solution satisfying factorisation (C.13) and the required symmetry under u i ↔ u j . Formula (C.9) is confirmed for n = 3, directly from the sum over Young tableaux (2.23), by taking the leading order in the large rapidities limit of the general contribution (2.22). As anticipated in the main text, there is an interesting link with the polynomials δ 2n : we use the identity 27 for the special 2n × 2n antisymmetric matrix to relate the highest degrees of the polynomials P 2n and δ 2n . Combining (A.5),(C.9) and (C.10) we can express the highest degree in terms of a determinant as This remarkable equality does not survive when we consider the full polynomial P 2n , as we can verify for n = 2 with the explicit formula (C.7). We do not know if a determinant representation of the full P 2n exists: however, it is an interesting idea to pursue since it would allow to find a nice representation of W .

D Connected functions
This Appendix focuses on the connected functions g (2n) . We first analyse the relation with the 'Green' functions G (2n) , that we sketched for the first few cases n = 2, 3 in the main text (Section 4.1). We also mentioned the importance of the property g (2n) ∈ L 1 (R 2n−1 ): here we tackle this issue and give evidence of its validity. Sometimes, concerning the asymptotic behaviour of G (2n) , we will refer to the results obtained in Section 3. The expansion of G (2n) in terms of the connected parts is well-known, it involves a sum over all the possible arrangements of 2n particles in subgroups of even particles 28 containing also an oscillating sign. In an equivalent manner, it is possible to sum over all the permutations and account for the overcounting with the specific prefactor, and rewrite (D.1) and (D.2) as [56] G (n) (u 1 , . . . , u n ) = n q=1 1 q!
which actually holds also for n odd.
Now we turn our attention to the asymptotic properties of g (2n) . In Section 4.1 we stated that the connected functions must be integrable over the 2n − 1 variables α i on which they depend: a sufficient condition for that is g (2n) ∈ L 1 (R 2n−1 ), which involves the integral of |g (2n) |, being then a stronger requirement. The general condition to be fulfilled is (4.19), which covers all the possible asymptotic regions in the integration space. Here we will address some of them for any n, giving hints for g (2n) ∈ L 1 (R 2n−1 ). Moreover, we perform a complete study of the functions g (4) and g (6) and show that they belong respectively to L 1 (R 3 ) and L 1 (R 5 ).
In conclusion, despite the lack of a general proof, we collected much evidence for g (2n) ∈ L 1 (R 2n−1 ). First, both functions g (4) and g (6) satisfy the requirement and their decay is even faster in some regions. The extension of the factorisation analysed in Section 3 to the region 6 → 2 + 2 + 2 (D.19) is needed for g (6) , which actually holds. In addition, the constraint (D. 19) is very physical and we can imagine it holds true, conveniently extended, for any splitting of the type 2n → 2k 1 + · · · + 2k l , which would guarantee the integrability of our connected functions.