Spinning Hexagons

Carlos Bercini${}^{\text{\tiny 1}}$, Vasco Gonçalves${}^{\text{\tiny 1,\tiny
2}}$, Alexandre Homrich${}^{\text{\tiny 3}}$, Pedro Vieira${}^{\text{\tiny
1,\tiny 3}}$
††#@gmail.com&/@{carlos.bercini,vasco.dfg,alexandre.homrich,pedrogvieira}

${}^{\text{\tiny 1}}$ICTP South American Institute for Fundamental Research,
IFT-UNESP, São Paulo, SP Brazil 01440-070
${}^{\text{\tiny 2}}$Centro de Fisica do Porto e Departamento de Fisica e
Astronomia, Faculdade de Ciencias da Universidade do Porto, Porto 4169-007,
Portugal
${}^{\text{\tiny 3}}$Perimeter Institute for Theoretical Physics, Waterloo,
Ontario N2L 2Y5, Canada

###### Abstract

We reduce the computation of three point function of three spinning operators
with arbitrary polarizations to a statistical mechanics problem via the
hexagon formalism. The central building block of these correlation functions
is the hexagon partition function. We explore its analytic structure and use
it to generate perturbative data for spinning three point functions. For
certain polarizations and any coupling, we express the full asymptotic three
point function in determinant form. With the integrability approach
established we open the ground to study the large spin limit where dualities
with null Wilson loops and integrable pentagons must appear.

###### Contents

  1. 1 Introduction
  2. 2 The Hexagon Partition Function
    1. 2.1 The Partition Function
    2. 2.2 Properties of the partition function
    3. 2.3 Tree Level
    4. 2.4 All Loop Abelian Simplifications
  3. 3 Spinning hexagons
    1. 3.1 Polarizing the Hexagon OPE
    2. 3.2 Perturbative checks
    3. 3.3 Abelian
    4. 3.4 A two loop check
  4. 4 Discussion
  5. A Spinors
  6. B Frames, Vertex, and Markers
  7. C $\sigma$ crossing
  8. D Equal spins operators
  9. E From C to H’s
  10. F Abelian $C^{\bullet\bullet\bullet}$ and Pfaffians

##  1 Introduction

Three point functions of single trace operators in planar $\mathcal{N}=4$ SYM
describe the scattering of three closed strings in AdS and are thus given by a
pairs of pants. Pairs of pants can be obtained by stitching two hexagons
together. That is how tailors make pants and it is also how one computes three
point functions in this gauge theory using integrability [1]. This paper is
about spinning pair of pants where two or more operators have spin. In this
case the 3pt function is given by a sum of conformal invariant tensor
structures and we need to explain how the hexagons extract the coefficient
multiplying each such structure. For three twist-two operators with spin 2, 4
and 6, for instance, we have the perturbative result [2]

$\displaystyle C(2,4,6)$ $\displaystyle=\frac{1}{84\sqrt{55}}\Bigg{(}\langle
11\rangle\langle 13\rangle\langle 22\rangle^{3}\langle 23\rangle\langle
31\rangle\langle 32\rangle\langle
33\rangle^{4}\left({\color[rgb]{0,0,1}960}-{\color[rgb]{0,0,0}\frac{2400756}{385}}g^{2}\right)+\langle
12\rangle^{2}\langle 21\rangle^{2}\langle 22\rangle^{2}\langle
33\rangle^{6}\left({\color[rgb]{0,0,1}15}-{\color[rgb]{0,0,0}\frac{88797}{770}}g^{2}\right)+$
$\displaystyle+\langle 11\rangle\langle 13\rangle\langle 23\rangle^{4}\langle
31\rangle\langle 32\rangle^{4}\langle
33\rangle\left(-{\color[rgb]{0,0,1}2007}-{\color[rgb]{0,0,0}\frac{822990467}{34650}}g^{2}\right)+\langle
13\rangle^{2}\langle 22\rangle\langle 23\rangle^{3}\langle
31\rangle^{2}\langle 32\rangle^{3}\langle
33\rangle\left(-{\color[rgb]{0,0,1}7680}-{\color[rgb]{0,0,0}\frac{3113024}{231}}g^{2}\right)+$
$\displaystyle+\langle 12\rangle\langle 13\rangle\langle 21\rangle\langle
23\rangle^{3}\langle 31\rangle\langle 32\rangle^{3}\langle
33\rangle^{2}\left(-{\color[rgb]{0,0,1}15360}-{\color[rgb]{0,0,0}\frac{13877312}{1155}}g^{2}\right)+\langle
13\rangle^{2}\langle 22\rangle^{4}\langle 31\rangle^{2}\langle
33\rangle^{4}\left({\color[rgb]{0,0,1}23}-{\color[rgb]{0,0,0}\frac{7780337}{34650}}g^{2}\right)+$
$\displaystyle+\langle 12\rangle\langle 13\rangle\langle 21\rangle\langle
22\rangle\langle 23\rangle^{2}\langle 31\rangle\langle 32\rangle^{2}\langle
33\rangle^{3}\left({\color[rgb]{0,0,1}23040}-{\color[rgb]{0,0,0}\frac{12990112}{1155}}g^{2}\right)+\langle
11\rangle^{2}\langle 22\rangle^{4}\langle
33\rangle^{6}\left({\color[rgb]{0,0,1}1}-{\color[rgb]{0,0,0}\frac{814939}{34650}}g^{2}\right)+$
$\displaystyle+\langle 11\rangle\langle 12\rangle\langle 21\rangle\langle
22\rangle^{2}\langle 23\rangle\langle 32\rangle\langle
33\rangle^{5}\left({\color[rgb]{0,0,1}576}-{\color[rgb]{0,0,0}\frac{5341948}{1925}}g^{2}\right)+\langle
11\rangle^{2}\langle 23\rangle^{4}\langle 32\rangle^{4}\langle
33\rangle^{2}\left({\color[rgb]{0,0,1}327}+{\color[rgb]{0,0,0}\frac{128149187}{34650}}g^{2}\right)+$
$\displaystyle+\langle 13\rangle^{2}\langle 22\rangle^{3}\langle
23\rangle\langle 31\rangle^{2}\langle 32\rangle\langle
33\rangle^{3}\left(-{\color[rgb]{0,0,1}1883}-{\color[rgb]{0,0,0}\frac{14853709}{4950}}g^{2}\right)+\langle
11\rangle^{2}\langle 22\rangle^{2}\langle 23\rangle^{2}\langle
32\rangle^{2}\langle
33\rangle^{4}\left({\color[rgb]{0,0,1}360}-{\color[rgb]{0,0,0}\frac{3289708}{1155}}g^{2}\right)+$
$\displaystyle+\langle 12\rangle^{2}\langle 21\rangle^{2}\langle
22\rangle\langle 23\rangle\langle 32\rangle\langle
33\rangle^{5}\left(-{\color[rgb]{0,0,1}567}-{\color[rgb]{0,0,0}\frac{3484307}{1650}}g^{2}\right)+\langle
11\rangle^{2}\langle 22\rangle\langle 23\rangle^{3}\langle
32\rangle^{3}\langle
33\rangle^{3}\left(-{\color[rgb]{0,0,1}640}-{\color[rgb]{0,0,0}\frac{5365984}{3465}}g^{2}\right)+$
$\displaystyle+\langle 12\rangle\langle 13\rangle\langle 21\rangle\langle
22\rangle^{3}\langle 31\rangle\langle
33\rangle^{5}\left({\color[rgb]{0,0,1}192}+{\color[rgb]{0,0,0}\frac{145072}{5775}}g^{2}\right)+\langle
11\rangle\langle 12\rangle\langle 21\rangle\langle 22\rangle^{3}\langle
33\rangle^{6}\left(-{\color[rgb]{0,0,1}16}+{\color[rgb]{0,0,0}\frac{2405402}{17325}}g^{2}\right)+$
$\displaystyle+\langle 11\rangle\langle 13\rangle\langle 22\rangle^{4}\langle
31\rangle\langle
33\rangle^{5}\left(-{\color[rgb]{0,0,1}24}+{\color[rgb]{0,0,0}\frac{1432546}{5775}}g^{2}\right)+\langle
11\rangle\langle 13\rangle\langle 22\rangle\langle 23\rangle^{3}\langle
31\rangle\langle 32\rangle^{3}\langle
33\rangle^{2}\left({\color[rgb]{0,0,1}7680}+{\color[rgb]{0,0,0}\frac{20817136}{1155}}g^{2}\right)+$
$\displaystyle+\langle 13\rangle^{2}\langle 23\rangle^{4}\langle
31\rangle^{2}\langle
32\rangle^{4}\left({\color[rgb]{0,0,1}960}+{\color[rgb]{0,0,0}\frac{272704}{231}}g^{2}\right)+\langle
11\rangle\langle 12\rangle\langle 21\rangle\langle 22\rangle\langle
23\rangle^{2}\langle 32\rangle^{2}\langle
33\rangle^{4}\left(-{\color[rgb]{0,0,1}2880}+{\color[rgb]{0,0,0}\frac{3467144}{1155}}g^{2}\right)+$
$\displaystyle+\langle 12\rangle\langle 13\rangle\langle 21\rangle\langle
22\rangle^{2}\langle 23\rangle\langle 31\rangle\langle 32\rangle\langle
33\rangle^{4}\left(-{\color[rgb]{0,0,1}5760}+{\color[rgb]{0,0,0}\frac{1434656}{385}}g^{2}\right)+\langle
12\rangle^{2}\langle 21\rangle^{2}\langle 23\rangle^{2}\langle
32\rangle^{2}\langle
33\rangle^{4}\left({\color[rgb]{0,0,1}1440}+{\color[rgb]{0,0,0}\frac{6762608}{1155}}g^{2}\right)+$
$\displaystyle+\langle 13\rangle^{2}\langle 22\rangle^{2}\langle
23\rangle^{2}\langle 31\rangle^{2}\langle 32\rangle^{2}\langle
33\rangle^{2}\left({\color[rgb]{0,0,1}8640}+{\color[rgb]{0,0,0}\frac{852608}{77}}g^{2}\right)+\langle
11\rangle\langle 12\rangle\langle 21\rangle\langle 23\rangle^{3}\langle
32\rangle^{3}\langle
33\rangle^{3}\left({\color[rgb]{0,0,1}2560}+{\color[rgb]{0,0,0}\frac{46393456}{3465}}g^{2}\right)+$
$\displaystyle+\langle 11\rangle\langle 13\rangle\langle 22\rangle^{2}\langle
23\rangle^{2}\langle 31\rangle\langle 32\rangle^{2}\langle
33\rangle^{3}\left(-{\color[rgb]{0,0,1}5760}+{\color[rgb]{0,0,0}\frac{19842568}{1155}}g^{2}\right)+\langle
11\rangle^{2}\langle 22\rangle^{3}\langle 23\rangle\langle 32\rangle\langle
33\rangle^{5}\left(-{\color[rgb]{0,0,1}48}+{\color[rgb]{0,0,0}\frac{1389824}{1925}}g^{2}\right)+$
$\displaystyle+\left(\langle 11\rangle^{2}\langle 23\rangle^{4}\langle
32\rangle^{4}\langle 33\rangle^{2}-\langle 11\rangle\langle 13\rangle\langle
23\rangle^{4}\langle 31\rangle\langle 32\rangle^{4}\langle
33\rangle\right)\left(-\textbf{{\color[rgb]{0,0,1}87}}+{\color[rgb]{0,1,1}\frac{\textbf{121910653}}{\textbf{34650}}}g^{2}\right)+$
$\displaystyle+\left(\langle 13\rangle^{2}\langle 22\rangle^{4}\langle
31\rangle^{2}\langle 33\rangle^{4}-\langle 13\rangle^{2}\langle
22\rangle^{3}\langle 23\rangle\langle 31\rangle^{2}\langle 32\rangle\langle
33\rangle^{3}\right)\left({\color[rgb]{0,0,1}\textbf{37}}+{\color[rgb]{0,1,1}\frac{\textbf{15349637}}{\textbf{34650}}}g^{2}\right)+$
$\displaystyle+\left(\langle 12\rangle^{2}\langle 21\rangle^{2}\langle
22\rangle^{2}\langle 33\rangle^{6}-\langle 12\rangle^{2}\langle
21\rangle^{2}\langle 22\rangle\langle 23\rangle\langle 32\rangle\langle
33\rangle^{5}\right)\left({\color[rgb]{0,0,1}\textbf{9}}+{\color[rgb]{0,1,1}\frac{\textbf{3091483}}{\textbf{11550}}}g^{2}\right)\Bigg{)}+\dots$
(1)

where $\langle ij\rangle$ is a scalar product involving a left spinor
parametrizing operator $i$ and a right spinor parametrizing operator
$j$.111These spinor variables render the various conformal invariant tensor
structures into simple scalar products, recalled in appendix A. A main goal of
this paper is to develop the formalism to reproduce such results from
integrability.

Not all terms are equally easy to get. The terms in blue, for instance, are
the tree level contributions; we will develop an efficient recursion algorithm
which will allow us to determine them all (and produce a plethora of new
predictions for structure constants of larger twist operators). The boldfaced
terms are what we call the abelian terms; these structure constants are very
integrability friendly as they lack a complicated so-called hexagon matrix
part; these abelian terms we can actually compute easily at one loop (in cyan
here) or even at higher loops. The remaining non boldfaced orange terms are
non-abelian one loop terms; we can also get them but it is quite painful to do
so specially for operators of larger spin.

At the center of all these integrability based computations is a beautiful
partition function represented in figure 1. We call it the hexagon partition
function. The vertex in this partition function is Beisert’s centrally
extended $SU(2|2)$ R-matrix [3] while the open boundary conditions are given
by contracting each boundary with a fixed two dimensional spinor. The various
possible choices of such spinors parametrize the various tensor structures
described above. (For some choice of boundary conditions this partition
function trivializes – leading to the much simpler abelian contributions
described above.) This hexagon partition function is the building block of all
spinning correlators. It is an interesting object on its own right which we
want to advertise in this paper. It has a beautiful very rich integrable
structure, the surface of which we are barely starting to scratch.

This paper is naturally split into two main sections.

In section 2 we study the hexagon partition function mentioned above. This
section might be interesting for hardcore integrabilists, even those with no
interest in three point functions in SYM. We will study this partition
function at weak coupling when it reduces to a rational vertex model partition
function and at finite coupling where it is richer, of Hubbard type. Recursion
relations, analytic continuations and several other tricks will play a key
role in this analysis.

In section 3 we describe how to introduce polarizations into the hexagon
formalism to compute spinning structure constants. We will do it mostly in the
so called asymptotic regime where mirror particles can be ignored except in
section 3.4 where we perform some checks involving mirror particles at two
loops. We will show how the triangle partition function naturally shows up and
the fundamental building block and we will test the hexagon construction
against perturbative data for a few simple examples with low spins.

##  2 The Hexagon Partition Function

###  2.1 The Partition Function

Figure 1: (a) The hexagon partition function $Z_{J_{1},J_{2},J_{3}}$,
illustrated here in the case $Z_{3,3,3}$, describe the scattering of three
sets of fermions, labelled by their rapidities $v_{i},u_{j},w_{k}$, in the
$\mathcal{N}=4$ SYM spin-chain. Each set starts polarized in a fixed direction
labeled by the spinors $R_{2},L_{1},R_{3}$ respectively. The particles then
scatter in all possible pairings according to Beisert’s $PSU(2|2)$ vertex. The
final state is then projected into fermions of definite polarization spinors
$L_{2},R_{1},L_{3}$. In the gauge theory, the boundary conditions are set by
the spacetime polarizations of spinning operators whose structure constants
are governed by the hexagon partition function. Because the vertex is
proportional to the identity when the incoming or outgoing particles are
fermions with identical polarizations, the outer parts of the graph are
trivial and result in a simple abelian factor. We represent these trivial
scattering by the green red and blue colours. They can be factored out
resulting in the equivalent representation (b). The interesting dynamics
happens in the pink region in which particles from different sets interact.

The central object in this paper is dubbed the hexagon partition function or
simply the hexagon. It is defined in figure 1a. The name hexagon becomes clear
when we realize it is given by a simpler partition function depicted in figure
1b.

As usual, the partition function can be thought of as 2d classical statistical
mechanical model where we sum over statistical weights at each vertex (more
precisely the vertex is the Shastry R-matrix of the Hubbard model) and where
at the edges we impose appropriate boundary conditions. For the hexagon
partition function there are six different boundary conditions to impose each
parametrized by its own two dimensional spinor.

Alternatively we can think of it as an integrable 1d quantum mechanical model.
In this picture we start with three sets of fermions in an in-state and let
them scatter – in a purely factorized fashion – into a final state also with
three sets of fermions. (At intermediate time steps these fermions can scatter
into bosons as well but in the initial and final states we only consider
fermions.) These fermions have an $SU(2)$ flavour index and we contract each
of the six sets of fermions (three in plus three out) with the same spinor; in
other words, we scatter three groups of identical fermions. Because these
fermions are identical scattering among the fermions of the same type will be
trivial and this is why the big partition function in 1a ends up simplifying
to the hexagon shape in figure 1b which gives the name to the partition
function. In fact, this simplified shape is far from unique. Since the
underlying model is integrable the order by which the particles scatter is
imaterial so we can alternatively cast this partition function in a myriad of
equivalent ways as illustrated in figure 2.

Each particle has a physical momenta which is conserved in each elastic
scattering event; it is thus associated to each line in the figure. We
parametrize it by a physical rapidity. Since we have three sets of momenta we
will have three sets of rapidities which we label as $v_{i},u_{j},w_{k}$. What
was before a valance four statistical mechanical weight is in this picture the
two-to-two scattering matrix of two of the 1d particles:

$\displaystyle S_{ab}^{cd}(z,z^{\prime})\equiv\vbox{
\includegraphics[width=140.92792pt]{S-matrix.pdf}}$ (2)

where each index can take four values (a fermion doublet and a boson doublet
combined into an $SU(2|2)$ fundamental; for the external states we have
fermions only but in intermediate states we will of course produce everyone in
the multiplet.) This S-matrix is Beisert’s PSU(2|2) S-matrix [3, 4] explicitly
given by a simple combination of ten terms

$\displaystyle
S_{ab}^{cd}(z,z^{\prime})=h(z,z^{\prime}){\Bigg{(}}\mathcal{A}(z,z^{\prime})\frac{\Delta_{a}^{c}\Delta_{b}^{d}+\Delta_{a}^{d}\Delta_{b}^{c}}{2}+\mathcal{B}(z,z^{\prime})\frac{\Delta_{a}^{c}\Delta_{b}^{d}-\Delta_{a}^{d}\Delta_{b}^{c}}{2}+\frac{1}{2}\mathcal{C}(z,z^{\prime})\phi_{\mathcal{Z}}^{-1}(z,z^{\prime})E_{ab}\epsilon^{cd}$
$\displaystyle+\mathcal{D}(z,z^{\prime})\frac{\delta_{a}^{c}\delta_{b}^{d}+\delta_{a}^{d}\delta_{b}^{c}}{2}+\mathcal{E}(z,z^{\prime})\frac{\delta_{a}^{c}\delta_{b}^{d}-\delta_{a}^{d}\delta_{b}^{c}}{2}+\frac{1}{2}\mathcal{F}(z,z^{\prime})\phi_{\mathcal{Z}}(z,z^{\prime})\epsilon_{ab}E^{cd}+\mathcal{G}(z,z^{\prime})\delta_{b}^{c}\Delta_{a}^{d}$
(3)
$\displaystyle+\mathcal{L}(z,z^{\prime})\delta_{a}^{d}\Delta_{b}^{c}+\mathcal{K}(z,z^{\prime})\delta_{b}^{d}\Delta_{a}^{c}+\mathcal{H}(z,z^{\prime})\delta_{a}^{c}\Delta_{b}^{d}\Bigg{)}.$

The Beisert matrix elements
$\mathcal{A},\mathcal{B},\mathcal{C},\mathcal{D},\mathcal{E},\mathcal{F},\mathcal{G},\mathcal{H},\mathcal{L},\mathcal{K}$
as well as the non-local markers $\phi_{\mathcal{Z}}$ are defined in appendix
B and depend on the rapidities $z$,$z^{\prime}$ through the Zhukovsky
variables $x^{\pm}$ only. These are defined through

$\frac{z+i/2}{g}=x^{+}(z)+\frac{1}{x^{+}(z)},\qquad\frac{z-i/2}{g}=x^{-}(z)+\frac{1}{x^{-}(z)}.$
(4)

where $g=\lambda/(4\pi)^{2}$ is the coupling.

The three particle sets are not on equal footing. In our conventions, $u_{j}$
are physical rapidities while $v_{i}$ and $w_{k}$ are crossed kinematics. We
can think of the corresponding excitations as anti-particles. Crossed
parameters in the matrix elements are to be understood as analytically
continued. The result of this analytic continuation is simple in this case: we
should pick monodromies around the branch points of the Zhukovsky variables.
This amounts to

$x^{\pm}(v^{\circlearrowright})\rightarrow 1/x^{\pm}(v),\qquad
x^{\pm}(w^{\circlearrowleft})\rightarrow 1/x^{\pm}(w)$ (5)

so that when we write, e.g.,

$\mathcal{A}(z,z^{\prime})\equiv\frac{x^{+}(z^{\prime})-x^{-}(z)}{x^{-}(z^{\prime})-x^{+}(z)},$

we are simultaneously defining

$\mathcal{A}(v,w)=\frac{1/x^{+}(w)-1/x^{-}(v)}{1/x^{-}(w)-1/x^{+}(v)},\qquad\mathcal{A}(v,u)=\frac{x^{+}(u)-1/x^{-}(v)}{x^{-}(u)-1/x^{+}(v)},\qquad\mathcal{A}(u,w)=\frac{1/x^{+}(w)-x^{-}(u)}{1/x^{-}(w)-x^{+}(u)}.$

All other factors are treated in the same simple fashion; the exception is the
overall factor $h(x,y)$ given by [1]

$\displaystyle h(z,z^{\prime})$
$\displaystyle\equiv\frac{x^{-}(z)-x^{-}(z^{\prime})}{x^{-}(z)-x^{+}(z^{\prime})}\frac{1-\frac{1}{x^{-}(z)x^{+}(z^{\prime})}}{1-\frac{1}{x^{+}(z)x^{+}(z^{\prime})}}\sigma(z,z^{\prime})^{-1},$
(6)

with $\sigma$ the BES dressing phase [5] and which transforms nontrivial due
to the non-trivial crossing transformation of $\sigma$ under the
$v^{\circlearrowright}$ and $w^{\circlearrowleft}$ monodromies, see appendix
C. In any case, this overall factor will lead to an overall factorized product
sitting outside as an overall normalization; it is irrelevant for most of our
discussion.

Figure 2: The PSU(2|2) vertex satisfies the Yang-Baxter relation illustrated
in the first line. Simply put, the partition function is invariant under
translating lines across vertices. These allows for many possible rewritings
of the partition function, as expressed in the second line. The term on the
left-hand side corresponds to the second line of (7). The next two terms give
alternative representations whose explicit form we hope is clear from the
picture. In particular, the middle shape relates the hexagon partition
function to a Hubbard-type model in a Kagome-type lattice.

As emphasized above, each line segment in the partition function is a
PSU$(2|2)$ state $|x\rangle$ parametrized by two component polarizations
$c_{\alpha}$ and $d_{a}$ as

$|x\rangle=\underbrace{c_{\alpha}|\psi^{\alpha}\rangle}_{\text{fermions}}+\underbrace{d_{a}|\phi^{a}\rangle}_{\text{bosons}}\equiv(c_{1},c_{2},d_{1},d_{2})^{T}.$

In this basis, the matrices $\delta,\Delta,\epsilon,E$ act as

$\delta=\begin{pmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
\end{pmatrix},\qquad\Delta=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&1&0\\\
0&0&0&1\\\ \end{pmatrix},\qquad\epsilon=\begin{pmatrix}0&1&0&0\\\ -1&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\\\ \end{pmatrix},\qquad E=\begin{pmatrix}0&0&0&0\\\
0&0&0&0\\\ 0&0&0&1\\\ 0&0&-1&0\\\ \end{pmatrix}.$

The boundary conditions are so that initial and final states are purely
fermionic. Particles in the sets $u,v,w$ have, respectively, incoming
polarization spinors $L_{1},R_{2},R_{3}$ and outgoing polarization spinors
$R_{1},L_{2},L_{3}$222This weird choice of labels is made to simplify formulas
in section 3, in which $L_{i}$ and $R_{i}$ will correspond to physical left
and right space-time polarization spinors of local operators in
$\mathcal{N}=4$ SYM. Particle sets $v_{i}$ and $w_{k}$, being crossed, have
their right and left handed spinning components exchanged. It will also
simplify some of the formulas to follow, see e.g. equation (16).. In the bulk
of the partition function, coloured in pink in figure 1, the particles can
transmute into scalars and explore the full PSU$(2|2)$ dynamics.

Having defined all these elements, we can translate picture 1 into an equation
for the partition function. This is equation (7). Note the vertex simplify
dramatically for identical fermionic outgoing states, the matrix part being
reduced to the element $\mathcal{D}=-1$. This was used to write the last three
factors in equation (7).

$\displaystyle Z_{J_{1},J_{2},J_{3}}$
$\displaystyle\equiv\prod_{i=1}^{J_{2}}\prod_{j=1}^{J_{1}}\prod_{k=1}^{J_{3}}\underbrace{{L_{1}}^{b_{j}^{1}}{R_{2}}^{a_{i}^{1}}{R_{3}}^{c_{k}^{1}}}_{\text{incoming
b.c.}}\times\underbrace{{R_{1}}_{b_{j}^{J_{2}+J_{3}}}{L_{2}}_{a_{i}^{J_{1}+J_{3}}}{L_{3}}_{c_{k}^{J_{1}+J_{2}}}}_{\text{outgoing
b.c.}}\times\underbrace{(-1)^{J_{2}+J_{3}}}_{\text{crossing factor}}\times$
(7)
$\displaystyle{\underbrace{S_{a_{i}^{j}b_{j}^{i}}^{a_{i}^{j+1}b_{j}^{i+1}}(v_{J_{2}+1-i},u_{j})}_{{\color[rgb]{0,0,0}{\text{$v_{i}$,
$u_{j}$
scattering}}}}}\times\underbrace{S_{a_{i}^{J_{1}+k}c_{k}^{i}}^{a_{i}^{J_{1}+k+1}c_{k}^{i+1}}(v_{J_{2}+1-i},w_{k})}_{{\color[rgb]{0,0,0}{\text{$v_{i}$,
$w_{k}$
scattering}}}}\times\underbrace{S_{b_{j}^{J_{2}+k}c_{k}^{J_{2}+j}}^{b_{j}^{J_{2}+k+1}c_{k}^{J_{2}+j+1}}(u_{J_{1}+1-j},w_{k})}_{{\color[rgb]{0,0,0}{\text{$u_{j}$,
$w_{k}$ scattering}}}}\times$
$\displaystyle\prod_{i^{\prime}<i}\underbrace{(-1)^{J_{2}(J_{2}-1)/2}h(v_{i^{\prime}},v_{i})}_{{\color[rgb]{0,0,0}{\text{$v_{i^{\prime}},v_{i}$
scattering}}}}\prod_{j^{\prime}<j}\underbrace{(-1)^{J_{1}(J_{1}-1)/2}h(u_{j^{\prime}},u_{j})}_{{\color[rgb]{0,0,0}{\text{$u_{j^{\prime}},u_{j}$
scattering}}}}\times\prod_{{k^{\prime}}<k}\underbrace{(-1)^{J_{3}(J_{3}-1)/2}h(w_{k^{\prime}},w_{k})}_{{\color[rgb]{0,0,0}{\text{$w_{k^{\prime}},w_{k}$
scattering}}}}.$

###  2.2 Properties of the partition function

The $PSU(2|2)$ vertex enjoys four important properties which we will make use
in the next few sections. First, it satisfies the Yang-Baxter relation

$S_{bc}^{\alpha\beta}(z^{\prime},z^{\prime\prime})S_{a\alpha}^{d\gamma}(z,z^{\prime\prime})S_{\gamma\beta}^{ef}(z,z^{\prime})=S_{ab}^{\beta\gamma}(z,z^{\prime})S_{\gamma
c}^{\alpha
f}(z,z^{\prime\prime})S_{\beta\alpha}^{de}(z^{\prime},z^{\prime\prime})$ (8)

which allows for many possible rewritings of the partition formula (7) as
illustrated in figure 2. Second, it satisfies the unitarity relation

$S_{ab}^{\alpha\beta}(z,z^{\prime})S_{\alpha\beta}^{cd}(z^{\prime},z)=h(z,z^{\prime})h(z^{\prime},z),$
(9)

as follows from the unitarity of the Beisert PSU$(2|2)$ S-matrix.

Note that unitarity immediately allows us to show that, up to a trivial
overall factor, the hexagon partition function is an invariant function under
swapping of any fermions of the same type as illustrated in figure 3. More
precisely we can decompose any permutation of fermions within the same group
as a sequence of neighbouring swaps, each of which simply generates an abelian
S-matrix factor $S_{0}(z_{i},z_{i-1})\equiv
h(z_{i},z_{i-1})/h(z_{i-1},z_{i})$,

$Z_{J_{1},J_{2},J_{3}}\Bigr{|}_{z_{i}\leftrightarrow
z_{i-1}}=S_{0}(x_{i},x_{i-1})Z_{J_{1},J_{2},J_{3}}$ (10)

This is the so-called Watson relation.333Note that we could have redefined the
partition function by a trivial product of $h$ factors so that the symmetry
relation (10) would simplify to full invariance under any swap of rapidities.
We found it better not to do so to preserve some nice analytic properties.

Figure 3: The Watson axiom imply that particles in the same external set can
be exchanged through the action of the $SL(2,R)$ S-matrix $S_{0}(x,y)\equiv
h(x,y)/h(y,x)$. This follows explicitly from the fact that for identical
outgoing polarizations the vertex (3) projects into the $\mathcal{D}$ element.
In equations,
${R_{1}}_{c}{R_{1}}_{d}S^{cd}_{ab}(u_{i},u_{i+1})=-{R_{1}}_{a}{R_{1}}_{b}h(u_{i},u_{i+1})=S_{0}(u_{i},u_{i+1}){R_{1}}_{c}{R_{1}}_{d}S^{cd}_{ab}(u_{i+1},u_{i})$.

The two properties above allow one to move lines around (Yang-Baxter) and
collapse bubbles (Unitarity). The next two properties lead to situations which
prepare the lines to be moved.

First, at equal rapidities, the scattering vertex degenerates. For example,
when $v_{i}\rightarrow w_{k}$444Here we mean to take $w_{k}$ and $v_{i}$ to be
equal after performing the crossing monodromies $v_{i}^{\circlearrowright}$
and $w_{k}^{\circlearrowleft}$. we have, up to regular terms,

$\displaystyle
S_{ab}^{cd}(v_{i},w_{k})\sim\frac{i/\mu(w_{k})}{v_{i}-w_{k}}\left(-\delta_{a}^{c}\delta_{b}^{d}-\delta_{a}^{c}\Delta_{b}^{d}-\Delta_{a}^{c}\delta_{b}^{d}-\Delta_{a}^{c}\Delta_{b}^{d}\right)=\vbox{
\includegraphics[width=140.92792pt]{decouplingmatrix.pdf}}.$ (11)

In the limit, the vertex develops a pole proportional to the index permutation
operator, the residue of which defines the measure $\mu(w_{k})$. Physically we
can interpret this singularity as a a clustering property (two far away
particles disconnect from the rest). As illustrated in the figure, at this
point the two lines meeting at a vertex are now disentangled; they are thus
ready to be moved around with unitarity and Yang-baxter in a sequence of moves
which can dramatically simplify the partition function.

Figure 4: Decoupling limit in the $Z_{2,2,2}$ case. Top: lines can be
disentangled through the basic Yang-Baxter and Unitarity moves. Bottom: When
$v_{1}\rightarrow w_{2}$ the vertex degenerate into the permutation operator,
see equation (11). One can then use the basic moves (8, 9) to completely
factorize the depende on $v_{1}$ and $w_{2}$, leading to equation (14). In
doing so one makes use of the unitarity identities
$h(v^{\circlearrowright}_{i},v^{\circlearrowright}_{j})h(v^{\circlearrowright}_{j},v^{\circlearrowleft}_{i})=h(w^{\circlearrowleft}_{j},w^{\circlearrowleft}_{j})h(w^{\circlearrowright}_{i},w^{\circlearrowleft}_{j})=h(v^{\circlearrowright}_{i},u_{j})h(u_{j},v^{\circlearrowleft}_{i})=1,$.

Finally, the vertex also simplifies in the infinite rapidity limit. We have,
up to $1/u$ corrections

$S_{ab}^{cd}(v,u)=S_{ba}^{dc}(u,w)\sim\left(-\delta_{a}^{d}\delta_{b}^{c}+\delta_{a}^{d}\Delta_{b}^{c}+\Delta_{a}^{d}\delta_{b}^{c}+\Delta_{a}^{d}\Delta_{b}^{c}\right)+O(1/u)=\vbox{
\includegraphics[width=140.92792pt]{infty.pdf}},$ (12)

the minus sign in the first term corresponding to the trivial scattering of
anticommuting fermions. Physically, when the rapidity is sent to infinity the
particle has zero momenta and became a goldstone excitation which scatters
trivially with other particles. After this limit the two lines also became two
disentangled pairs, albeit a different choice of pairs compared to the
decoupling example. In this case the particle whose root is sent to infinity
scatters trivially with all other particles so the simplification of the
partition function implies right away that in the $u_{1}\rightarrow\infty$
limit, we simply have

$Z_{J_{1},J_{2},J_{3}}\sim(-1)^{J_{1}+J_{2}+J_{3}+1}\langle L_{1},R_{1}\rangle
Z_{J_{1}-1,J_{2},J_{3}}^{u_{1}},\qquad{u_{1}\to\infty}$ (13)

which is depicted in figure 6. Here and below, rapidities in a superscript in
a partition function $Z$ indicates that the rapidities in the superscript are
absent from the partition function.

These four properties can be used to derive recursion formulas for the
partition function. When a vertex degenerates into a permutation operator we
can sequentially apply the Yang-baxter and unitarity properties to completely
factorize the two decoupling particles from the rest of the partition
function, as explained in figure 4. The result is that the singular term in
the $v_{1}\rightarrow w_{J_{3}}$ limit is

$Z_{J_{1},J_{2},J_{3}}\sim-\frac{i}{\mu(w_{J})}\frac{\langle
L_{2},R_{3}\rangle\langle
L_{3},R_{2}\rangle}{v_{1}-w_{J_{3}}}{Z_{J_{1},J_{2}-1,J_{3}-1}\Bigr{|}}_{v_{a}\rightarrow
v_{a+1}},$ (14)

with $\langle L_{a},R_{b}\rangle\equiv{L_{a}}_{i}R_{b}^{i}$. This relation was
particularly easy to visualize graphically because we decoupled neighboring
particles, namely the last particle of type $w$ with the first particle of
type $v$, see kets and bras in figure (1) where that is even more manifest.
However, since we can rearrange any particles within any set using (10) we can
immediately write down the decoupling between any $v$ and any $w$ particle,

$Z_{J_{1},J_{2},J_{3}}\sim-\frac{i}{\mu(w_{j})}\frac{\langle
L_{2},R_{3}\rangle\langle
L_{3},R_{2}\rangle}{v_{i}-w_{k}}\prod_{i^{\prime}=1}^{i-1}S_{0}(v_{i^{\prime}},v_{i})\prod_{k^{\prime}=k+1}^{J_{3}}S_{0}(w_{k},w_{k^{\prime}}){Z_{J_{1},J_{2}-1,J_{3}-1}\Bigr{|}}_{v_{a}\rightarrow
v_{a+1}},$ (15)

Decouplings for other pairs $v_{i}$, $w_{k}$ can be obtained by moving the
particle $v_{i}$ to the position of $v_{1}$, and similar for $w_{k}$ and
$w_{J_{3}}$, by means of Watson equation, (10) .

Crossing symmetry relates different expressions for the partition function
that exchange the various sets of particles. We illustrate this in figure 5.
In particular, the expression (1) is invariant under $u\rightarrow
v,v\rightarrow w,w\rightarrow u$ with $L_{i}\rightarrow
L_{i+1},R_{i},\rightarrow R_{i+1}$:

$\qquad\qquad\qquad
Z_{J_{1},J_{2},J_{3}}=Z_{J_{2},J_{3},J_{1}}\bigg{\rvert}_{\begin{subarray}{c}u\rightarrow
v,v\rightarrow w,w\rightarrow u\\\
(L_{i},R_{i})\rightarrow(L_{i+1},R_{i+1})\end{subarray}}.$ (16)

Combining permutation invariance (16) with the $v\rightarrow w$ decoupling
(14) provides $u\rightarrow v$ and $u\rightarrow w$ decoupling formulas

$\displaystyle Z_{J_{1},J_{2},J_{3}}\sim$
$\displaystyle-\frac{i}{\mu(v_{i})}\frac{\langle L_{1},R_{2}\rangle\langle
L_{2},R_{1}\rangle}{u_{j}-v_{i}}\prod_{j^{\prime}=1}^{j-1}S_{0}(u_{j^{\prime}},u_{j})\prod_{i^{\prime}=i+1}^{J_{2}}S_{0}(v_{i},v_{i^{\prime}})\times
Z_{J_{1}-1,J_{2}-1,J_{3}}^{u_{j},v_{i}},$ (17) $\displaystyle
Z_{J_{1},J_{2},J_{3}}\sim$ $\displaystyle\frac{i}{\mu(w_{k})}\frac{\langle
L_{1},R_{3}\rangle\langle
L_{3},R_{1}\rangle}{u_{j}-w_{k}}\prod_{j^{\prime}=j+1}^{J_{1}}S_{0}(u_{j},u_{j^{\prime}})\prod_{k^{\prime}=1}^{k-1}S_{0}(w_{k^{\prime}},w_{k})\times
Z_{J_{1}-1,J_{2},J_{3}-1}^{u_{j},w_{k}}.$ (18)

Once again we write only the singular terms in the $u_{j}\rightarrow v_{i}$
and $u_{j}\rightarrow w_{k}$ limits respectively.

Figure 5: The $\mathcal{N}=4$ hexagon form factor [1] from which the hexagon
partition function originates is symmetric in its three physical edges,
represented here by solid lines. The expression (7) is obtained by moving all
particles from the $v$ and $w$ edges to the top $u$ edge, left picture. Moving
particles to non-adjacent edges of the hexagon correspond to crossing
transformations. Alternatively, the hexagon form factor could be evaluated
through the prescription of the right figure, in which $v$ particles remain
physical while $w$ is crossed clockwise and $u$ crossed anticlockwise. The
result is equation (16). An equivalent statement is that the hexagon form
factor is invariant under crossing all particles in the same direction. Figure
6: Decoupling when $u_{1}\rightarrow\infty$ for $Z_{2,2,1}$. When one of the
rapidities is sent to infinity the vertex reduces to the free S-matrix. All
scatterings of particle $u_{1}$ are of the fermion-fermion type, resulting in
the $(-1)^{J_{1}+J_{2}+J_{3}-1}$ factor of equation (13).

The equations (13) and (17, 18) in boxes are powerful constraints. They relate
partition functions with a certain number of particles (lhs’s) with partition
functions with less particles (rhs’s). An obvious question is whether we can
turn them into recursion relations which completely determine the partition
function in terms of trivial tiny partition functions with a vertex or two. We
could not find a way to do it at finite coupling. The basic obstacle is that
the partition function is not a simple rational function of the rapidities
whose residues are simple, but instead a multi-sheeted object since the
Zhukosky variables naturally take values in tori [6]. Before crossing cuts,
all poles are of the type described in (13, 17, 18). However, in the second
sheet there are extra poles capturing the fusion of the fundamental fermions
into bound states. Correspondingly, the residues are no longer simple
decoupled partition functions but instead partition functions with external
bound-states. One could hope to bootstrap the complete system of partition
functions including all possible external bound-states in a recursive way, but
we did not achieve that. On the other hand, in perturbation theory the
coupling goes to zero and these tori degenerate and simplify into simple
(punctured) spheres. In more pedestrian terms, the square roots in the
Zhukosky variables disappear and these become simply proportional to the Bethe
roots so that the partition function becomes a nice rational function of the
Bethe roots whose poles correspond to simple decouplings.555Even this is
actually not totally trivial because of the non-local $\phi$ Z-markers with
its own square roots. It turns out that these square roots can be simplify
away. In that case we can indeed convert the above relations into powerful
recursion relations. It is what we will do in section 2.3. After that we will
consider a particular example where we can evaluate the full hexagon partition
function at any value of the coupling.

###  2.3 Tree Level

When $g\rightarrow 0$ the Zhukovsky variables (4) become rational functions of
the rapidities. More surprisingly, the full hexagon partition function becomes
rational, the various square roots from the matrix elements (63 \- 64)
cancelling out. In this section we use rationality to derive recursion
relations that efficiently compute the hexagon partition function for any
$J_{i}$.

Rational functions are completely fixed by their singularities and
asymptotics. As a function of $u_{1}$, $Z(J_{1},J_{2},J_{3})$ has two types of
poles both of which with clear physical meaning. The first are poles
corresponding to the fusion of two physical particles, $u_{1}$ and $u_{j}$,
into a bound state. These are the singularities contained in the
$h(u_{1},u_{j})$ factors in (7). All other singularities at tree level666At
$g\neq 0$ the physical particles may form bound states with the crossed
particles, resulting in new singularities.  are decoupling poles (17,18). We
are immediately led to the recursion relation

$\displaystyle\hat{Z}_{J_{1},J_{2},J_{3}}=$
$\displaystyle(-1)^{J_{1}+J_{2}+J_{3}+1}\langle
L_{1},R_{1}\rangle\hat{Z}_{J_{1}-1,J_{2},J_{3}}^{u_{1}}+$ (19)
$\displaystyle-\sum_{i=1}^{J_{2}}\frac{i}{\mu(v_{i})}\frac{\langle
L_{1},R_{2}\rangle\langle
L_{2},R_{1}\rangle}{u_{1}-v_{i}}\prod_{i^{\prime}=i+1}^{J_{2}}S_{0}(v_{i},v_{i^{\prime}})\prod_{j=2}^{J_{1}}\hat{h}(v_{i},u_{j})^{-1}\
\times\hat{Z}_{J_{1}-1,J_{2}-1,J_{3}}^{u_{1},v_{i}}+$
$\displaystyle+\sum_{k=1}^{J_{3}}\frac{i}{\mu(w_{k})}\frac{\langle
L_{1},R_{3}\rangle\langle
L_{3},R_{1}\rangle}{u_{1}-w_{k}}\prod_{j^{\prime}=2}^{J_{1}}S_{0}(u_{1},u_{j^{\prime}})\prod_{k^{\prime}=1}^{k-1}S_{0}(w_{k^{\prime}},w_{k})\prod_{j=2}^{J_{1}}\hat{h}(w_{k},u_{j})^{-1}\times\hat{Z}_{J_{1}-1,J_{2},J_{3}-1}^{u_{1},w_{k}}$

with
$\hat{Z}_{J_{1},J_{2},J_{3}}={Z}_{J_{1},J_{2},J_{3}}\prod_{j<j^{\prime}}h(u_{j},u_{j}^{\prime})^{-1}$.
Here $\hat{h}=h$ computed without performing crossing monodromies in the
arguments, since in the decoupling limit $u_{j}$ approaches $v_{i}$ and
$w_{k}$ without going around Zhukovsky branch points777These branch points
disappear at tree level, the Zhukovsky variables becoming rational. This is an
important point. The crossing transformations do not commute with the
perturbative expansion. Crossing can only be performed at finite coupling. In
a sense, the simplicity of the tree level result encode some of the finite
coupling structure through the many properties derived from crossing and used
to bootstrap (19).. Explicitly, at tree level

$\hat{h}(x,y)=\frac{x-y}{x-y-i},\qquad\mu(x)=1,\qquad
S_{0}(x,y)=\frac{x-y+i}{x-y-i}.$ (20)

Repeated use of (19) reduces the computation of $Z_{J_{1},J_{2},J_{3}}$ to
that of ${Z}_{0,J_{2},J_{3}}$. One could then use rationality in $v_{i}$ and
$w_{k}$ to write further recursions that reduce the formula all the way to
$Z_{0,0,0}=1$. One can skip this step, in practice, since the recursions for
${Z}_{0,J_{2},J_{3}}$ can be solved explicitly. Through induction one finds
the determinant formula

${Z}_{0,J_{2},J_{3}}=(-1)^{A_{J_{2}J_{3}}}\frac{\prod_{i=1}^{J_{2}}\prod_{k=1}^{J_{3}}(v_{i}-w_{k}+i)}{(\prod_{i<i^{\prime}}v_{i^{\prime}}-v_{i}+i)(\prod_{k<k^{\prime}}w_{k}-w_{k^{\prime}}-i)}\det\text{M},$

where M is a square matrix of size $\max(J_{2},J_{3})$ whose elements are
given by

$M_{ij}=\begin{cases}\frac{\langle 23\rangle\langle
32\rangle}{v_{i}-w_{j}}+\frac{\langle 22\rangle\langle
33\rangle}{v_{i}-w_{j}+i}-\frac{\langle 23\rangle\langle
32\rangle}{v_{i}-w_{j}+i}\,,&\text{ if }i\leq J_{2}\land j\leq J_{3}\\\ &\\\
-\frac{\langle 23\rangle\langle 32\rangle}{\langle 33\rangle
v_{i}^{J_{3}-j+1}}-\frac{\langle
22\rangle}{(v_{i}+i)^{J_{3}-j+1}}+\frac{\langle 23\rangle\langle
32\rangle}{\langle 33\rangle(v_{i}+i)^{J_{3}-j+1}}\,,&\text{ if $j>J_{3}$}\\\
&\\\ \frac{\langle 23\rangle\langle 32\rangle}{\langle 22\rangle
w_{j}^{J_{2}-i+1}}+\frac{\langle
33\rangle}{(w_{j}-i)^{J_{2}-i+1}}-\frac{\langle 23\rangle\langle
32\rangle}{\langle 22\rangle(w_{j}-i)^{J_{2}-i+1}}\,,&\text{ if
$i>J_{2}$}\end{cases},$

with $\langle ij\rangle\equiv\langle L_{i},R_{j}\rangle$ and

$A_{J_{2}J_{3}}=(J_{2}+1)(J_{3}+1)+\sum_{i=2}^{3}(J_{i}-1)J_{i}/2+\begin{cases}\frac{(J_{3}-J_{2})(1+J_{3}-J_{2})}{2}\,,&\text{if
$J_{2}<J_{3}$}\\\ 0\,,&\text{otherwise}\end{cases}.$

In section 3.2 the recursive formulas (19 ,2.3) will be used to produce three
point structures data for operators of arbitrary twist at tree level. For
convenience, we attach a ready to copy-and-paste Mathematica implementation of
the tree level recursion. The SL$(2)$ S-matrix and dynamical factor are ` `
` S0[z_, zp_] = (z - zp + I)/(z - zp - I);`
` h[z_, zp_] = (z - zp)/(z - zp - I);` ` `

We then write the recursive formulas for the partition function. As explained
above, when applying the recursion to one of the three particle sets, it is
useful to factor out the dynamical factors. We denote this procedure through
“hats" which must be removed to recover $Z$. First we do recursions for “v’s"
starting with the trivial one-set answer:

``
` Zhathat[0,J3_] := V[3]^J3 Product[(-1) h[w[k], w[kk]], {kk, J3}, {k, kk -
1}];`
` Zhathat[J2_,J3_]:= Zhathat[J2,J3] = (-1)^(J2+J3+1) V[2] (Zhathat[J2-1,J3]
/.v[i_]:>v[i+1])-` ` Sum[I
H[2,3]/(v[1]-w[k])Product[S0[w[k],w[kk]],{kk,k+1,J3}]
Product[h[w[k],v[i]]^-1,{i,2,J2}]`
` (Zhathat[J2-1,J3-1] /.{v[i_]:>v[i+1],w[kkk_]/;kkk>=k:>w[kkk+1]}),{k,J3}];`
` Zhat[0,J2_,J3_]:=Zhathat[J2, J3]Product[h[v[i], v[ii]],{ii,J2},{i,ii-1}];`

Next, we perform recursions for the “u" particles to recover the full
partition function,

``
` Zhat[J1_,J2_,J3_]:= Zhat[J1,J2,J3]=(-1)^(J1+J2+J3+1)V[1](Zhat[J1-1,J2,J3]
/.u[j_]:>u[j+1])-` ` Sum[I
H[1,2]/(u[1]-v[i])Product[S0[v[i],v[ii]],{ii,i+1,J2}]Product[h[v[i],u[j]]^-1,{j,2,J1}]`
` (Zhat[J1-1,J2-1,J3]/.{u[j_]:>u[j+1],v[iii_]/;iii>=i:>v[iii+1]}),{i,J2}]+`
` Sum[I
H[3,1]/(u[1]-w[k])Product[S0[u[1],u[jj]],{jj,2,J1}]Product[S0[w[kk],w[k]], `
` {kk,1,k-1}] Product[h[w[k],u[j]]^-1,{j,2,J1}](Zhat[J1-1,
J2,J3-1]/.{u[j_]:>u[j+1],`
` w[kkk_]/;kkk>=k:>w[kkk+1]}),{k,J3}];`
`Z[J1_,J2_,J3_]:=Zhat[J1,J2,J3]Product[h[u[j],u[jj]],{jj,J1},{j,jj-1}];`

as promised. In the code we denoted $V[i]\equiv\langle L_{i},R_{i}\rangle$ and
$H[i,j]\equiv\langle L_{i},R_{j}\rangle\langle R_{i},L_{j}\rangle$.

###  2.4 All Loop Abelian Simplifications

Figure 7: When the incoming state are fermions with parallel polarizations,
the vertex becomes proportional to the identity, e.g.
${R_{2}}^{a}{L_{1}}^{b}S^{cd}_{ab}(u_{i},u_{i+1})=-{R_{2}}^{c}{L_{1}}^{d}h(v_{2},u_{1})$.
This simplification happens at all vertices if $L_{1}=R_{2}=R_{3}$, the
partition function reducing to the product of abelian factors times the inner
product between incoming and outgoing polarizations, equation (21). Of course,
the same simplification holds if instead we have $R_{1}=L_{2}=L_{3}$.

When $L_{1}=R_{2}=R_{3}$ the triangle partition function dramatically
simplifies. In that case, only the terms proportional to the $\mathcal{D}$
matrix element in (3) survives. As shown in figure 7, the result is

$\displaystyle Z^{\text{Abelian}}_{J_{1},J_{2},J_{3}}=$ $\displaystyle\langle
L_{1},R_{1}\rangle^{J_{1}}\langle L_{2},R_{2}\rangle^{J_{2}}\langle
L_{3},R_{3}\rangle^{J_{3}}\times(-1)^{J_{2}+J_{3}+J(J-1)/2}$ (21)
$\displaystyle\prod_{i=1}^{J_{2}}\prod_{j=1}^{J_{1}}\prod_{k=1}^{J_{3}}h(v_{i},u_{j})h(u_{j},w_{k})h(v_{i},w_{k})\prod_{\begin{subarray}{c}{i^{\prime}<i}\\\
{j^{\prime}<j}\\\
{k^{\prime}<k}\end{subarray}}h(v_{i^{\prime}},v_{i})h(u_{j^{\prime}},u_{j})h(w_{k^{\prime}},w_{k}),$

with $J=J_{1}+J_{2}+J_{3}$. We emphasize this result holds at finite coupling
$g$.

Of course, from permutation invariance, the result also simplifies for
$L_{2}=R_{1}=R_{3}$ and $L_{3}=R_{1}=R_{2}$. This is non-trivial from the
expression (7) but can be made manifest through crossing transformations. The
vertices are also projected into the $\mathcal{D}$ element if
$R_{1}=L_{2}=L_{3}$ or permutations, as can be seen from (3). The partition
function for these polarizations also reduces to formula (21).

##  3 Spinning hexagons

###  3.1 Polarizing the Hexagon OPE

Now we shift gears by focusing on the computation of three point functions of
primary operators in the SL$(2|\mathbb{R})$ sector of $\mathcal{N}=4$ SYM. One
important ingredient in our analysis, as we shall see below, is the hexagon
partition function analysed in the previous section. Before delving into those
details let us define the precise form of the operators that are being
considered and summarize how the hexagon formalism can be used to compute
three point functions.

The primary operators in the SL$(2|\mathbb{R})$ sector are traceless symmetric
single traces made out of covariant derivative excitations
$\mathcal{D}^{\alpha\dot{\alpha}}$ on top of a BPS “vaccum" of fundamental
scalars, $\text{Tr}(\vec{Y}\cdot\vec{\phi})^{\tau}$. To denote operators of
spin $J_{i}$ we use the index free notation in terms of polarization spinors

$\mathcal{O}_{J_{i}}(x)\equiv\mathcal{O}_{J_{i}}(x,L_{i},R_{i})=\left(\prod_{j=1}^{J_{i}}{L_{i}}_{\alpha_{j}}{R_{i}}_{\dot{\alpha}_{j}}\right)\mathcal{O}_{J_{i}}^{\alpha_{1}\dots\alpha_{J_{i}}\dot{\alpha}_{1}\dots\dot{\alpha_{J_{i}}}}(x).$
(22)

These operators are the same type of spinning operators we consider in the
conformal bootstrap [7], being the only difference here that we do not need to
consider leading twist, in fact, here they can have any twist $\tau$, which we
leave implicit.

By conformal symmetry, the full three point function can be recovered from its
values in the conformal frame888More generally, by superconformal symmetry, we
can also fix the R-symmetry polarizations $Y_{i}$ to point at fixed directions
in $SO(6)$. The dependence on them is completely fixed by the Ward identities.
In practice we use the twisted translated polarizations of [1]. The R-symmetry
factors will not be important in our discussion and we mostly omit them.. We
place operators along the $x_{2}$ axis at $0$, $1$ and $\infty$. The rescaled
correlator

$C(J_{1},J_{2},J_{3})\equiv\lim_{\mathcal{L}\rightarrow\infty}\mathcal{L}^{2\Delta_{3}}\langle\mathcal{O}_{J_{1}}(0)\rangle\mathcal{O}_{J_{2}}(1)\mathcal{O}_{J_{3}}(\mathcal{L})$
(23)

can then be expressed as a sum over tensor structures invariant under the
residual unfixed symmetry [8, 9]. As defined in appendix A, these tensor
structures can be written as inner products of the polarization spinors
$L_{i}$ and $R_{i}$, as follows

$C(J_{1},J_{2},J_{3})=\sum_{\ell_{1},\ell_{2},\ell_{3}}C^{J_{1},J_{2},J_{3}}_{\ell_{1},\ell_{2},\ell_{3}}\frac{\langle
1,1\rangle^{J_{1}-\ell_{2}-\ell_{3}}\langle
2,2\rangle^{J_{2}-\ell_{1}-\ell_{3}}\langle
3,3\rangle^{J_{3}-\ell_{1}-\ell_{2}}}{\langle 2,3\rangle^{-\ell_{1}}\langle
3,2\rangle^{-\ell_{1}}\langle 1,3\rangle^{-\ell_{2}}\langle
3,1\rangle^{-\ell_{2}}\langle 1,2\rangle^{-\ell_{3}}\langle
2,1\rangle^{-\ell_{3}}}.$ (24)

where $\langle i,j\rangle=\langle L_{i},R_{j}\rangle$. The goal is to
reproduce this three point function – which at once captures all structure
constants for any tensor structures – from the hexagon formalism.

The hexagon formalism has already been developed and tested thoroughly for
spinless operators to very high perturbative orders (as well as for spinning
correlators without non-trivial tensor structures) so the main novelty here is
how to deal with the various spinor polarizations leading to the various terms
summed in (24). Before presenting the relevant expressions with these spinor
boundary conditions accounted for let us recall that the hexagon formalism
entails two main components as illustrated in figure 8:

  * •

One is a sum over partitions of physical rapidities for each external
operator. When we cut the three point function pair of pants into two hexagons
the excitations on each operator can end up on either hexagon so we must sum
over ways of partitioning such excitations. These sums are often referred to
as the asymptotic sums.

  * •

The other are integrals over mirror rapidities along the three seams of the
pair of pants. When we glue back the hexagons into a pair of pants we must sum
over all possible quantum states along each edge where we glue. These mirror
integrals are often referred to as wrapping effects.

In this work we completely ignore the second effect. Except for section 3.4.

Figure 8: Each closed spin chain operator is split into two open chain
operators. We sum over all the ways its excitations can end up in either one
of the chains. Gluing the hexagons together amounts to integrating over all
possible mirror states.

There are two simple reasons for doing so. The first is that wrapping
corrections are delayed in perturbation theory. At tree level and one loop
they do not show up even for the smallest twist two external operators for
instance and for larger twist operators they are delayed even more in
perturbation theory. So we better clean up the asymptotic part first – where
all the subtleties related to the spinors already shows up – before addressing
these finite size mirror corrections. The second reason is that even at loop
level, sometimes we can drop the wrapping corrections altogether if the
distances travelled by the mirror particles are very large. This happens for
large operators with lots of R-charge but also for operators of small R-charge
and very large spin. The intuition is that when the spin is very large the
centripetal force effectively opens up the operator making it effectively very
large; in this case sometimes these finite size corrections can be ignored. We
thus suspect this to be the case for the case of three twist two operators
with very large spin and generic polarizations. Evaluating this kind of
correlator should beautifully connect to the remarkable Wilson loop dualities
[7, 10, 11]. So if our goal is to eventually prove these dualities from
integrability perhaps we can ignore wrapping for now.

With wrapping out of the window, we can then present our main formula for the
spinning three point function. It is a long formula but fortunately most
ingredients are self-explanatory and have appeared before in several spinless
studies. It reads

$C(J_{1},J_{2},J_{3})=\underbrace{\mathcal{N}(\textbf{u})\mathcal{N}(\textbf{v})\mathcal{N}(\textbf{w})}_{\texttt{normalizations
}}\times\sum_{\begin{subarray}{c}{a\,\cup\,\bar{a}=\textbf{u}}\\\
{b\,\cup\,\bar{b}=\textbf{v}}\\\
{c\,\cup\,\bar{c}=\textbf{w}}\end{subarray}}\underbrace{\omega_{\ell_{13}}(a,\bar{a})\omega_{\ell_{12}}(b,\bar{b})\omega_{\ell_{23}}(c,\bar{c})}_{\texttt{splitting
factors}}\times\underbrace{\mathcal{H}(a,b,c)\times\mathcal{H}(\bar{a},{\bar{c}},{\bar{b}})}_{\texttt{hexagons}}$
(25)

where the normalization and splitting factors are given by

$\displaystyle\mathcal{N}(\mathbf{u})^{2}=\frac{\prod\limits\limits_{i=1}^{J}\mu(u_{i})}{\prod\limits_{i\neq
j}^{J}S(u_{i},u_{j})\det{\left(\partial_{u_{i}}\phi_{j}\right)}}\,,\qquad\omega_{\ell}(a,\bar{a})=\prod\limits_{u_{j}\in\bar{a}}(-e^{ip(u_{j})\ell})\prod\limits_{\begin{subarray}{c}{u_{i}\in
a}\\\ {i>j}\end{subarray}}S(u_{i},u_{j})\,.$ (26)

These are the same factors arising already in [1]; the notation and physical
meaning of these factors is the same as there.

Figure 9: To obtain a correlator with all excitations on top we perform one
crossing transformation in the left ($\circlearrowright$) and another in the
right ($\circlearrowleft$). These monodromies will change the Zhukovsky
variables as well as the indices of the excitations.

The most important objects in (25) – and where important novelty lies – are
the hexagons form factors $\mathcal{H}$. We can relate either of them to a
purely creation amplitude with all excitations in the top by applying a
sequence of crossing transformations, see figure 9. This leads to a
representation of the hexagon as

$\displaystyle\mathcal{H}(a,b,c)$
$\displaystyle=(-1)^{\frac{(a+b+c)(a+b+c-1)}{2}}\prod_{\begin{subarray}{c}{u_{i}\in
a}\\\ {v_{j}\in b}\\\ {w_{k}\in c}\end{subarray}}L_{1,\alpha_{1}}\dots
L_{2,\alpha_{2,j}}L_{3,\alpha_{3,k}}R_{1,\beta_{1,i}}R_{2,\beta_{2,j}}R_{3,\beta_{3,k}}\times$
$\displaystyle\times\langle\mathfrak{h}|\mathcal{D}^{\alpha_{2,j}\beta_{2,j}}(v_{j}^{\circlearrowright})\mathcal{D}^{\alpha_{1,i}\beta_{1,i}}(u_{i})\mathcal{D}^{\alpha_{3,k}\beta_{3,k}}(w_{k}^{\circlearrowleft})\rangle$
(27)

where the minus factor in front is the grading associated with the fermionic
nature of the excitations.

This creation amplitude describes the scattering of three sets of fermions,
labelled by their rapidities u, v and w in the $\mathcal{N}=4$ SYM spin-chain.
Each set starts polarized in a fixed direction labeled by their correspondent
left-hand spinors. The particles then scatter in all possible pairings
according to Beisert’s $PSU(2|2)$ vertex, with the final state being the
projection into fermions of definite polarization defined by its right-hand
spinors. The object that accounts for all these scatterings is the hexagon
partition function ($Z$), defined as

$\mathcal{H}(u_{a},v_{b},w_{c})=(-1)^{\frac{(a+b+c)(a+b+c-1)}{2}}Z_{a,b,c}\,.$
(28)

The particles entering this partition function are not on equal footing, since
we used crossing transformations for the rapidities v and w. The first effect
of these transformations is the exchange their left and right $PSU(2|2)$
indices

$\mathcal{D}^{\alpha\beta}\xrightarrow{\circlearrowleft}-\mathcal{D}^{\beta\alpha}\;\;\text{and}\;\;\mathcal{D}^{\alpha\beta}\xrightarrow{\circlearrowright}-\mathcal{D}^{\beta\alpha}\,,$
(29)

this will swap incoming and outgoing indices for particles v and w. More
precisely, it will set $L_{1}$, $R_{2}$, $R_{3}$ to be the incoming boundary
conditions and $R_{1}$, $L_{2}$, $L_{3}$ to be the outgoing boundary
conditions for the scattering, see figure 1a.

Another implication of crossing is the introduction of crossed parameters in
the S-matrix elements. This amounts to analytically continue these matrix
elements when considering the crossed rapidities v and w. The effects of these
analytical continuations are simple: we must pick monodromies around the
branch point of the Zhukovsky variables (5) and mind the non-trivial factors
introduced by the BES dressing phase, see appendix C.

In the next sections we use the properties of the hexagon partition function
(7) discussed in section 2 to generate spinning three point functions data and
compare it with the perturbative results such as (1).

###  3.2 Perturbative checks

As derived in (19) the tree-level hexagon partition function can be written as
a recursion relation. The minus signs introduced by gradding slightly change
the expression (28), but the physics behind it is still the same. One can
fully determine the tree-level partition function by considering its
decoupling poles and its behavior at infinity.

Both the recursion relation and its sum over partitions can be easily
evaluated using for example Mathematica. Replacing the u, v and w by the Bethe
roots for twist-2 operators with spins $J_{1}=2$, $J_{2}=4$ and $J_{3}=6$ we
reproduce all the blue numbers in (1).

One nice aspect of the integrability formalism is that, even though the amount
of terms in the recursion relation and in the sum over partitions grows
exponentially with the spins of the operators they do not grow with the twist
of these operators.

Therefore, the recursion relation above is a powerful tool to explore the
corner of low spin and large twist three point functions. For example, the
data for twist-10 operators with spins $J_{1}=2$, $J_{2}=4$ and $J_{3}=6$ can
be easily evaluated. In particular for equal bridge lengths ($\ell_{ij}=5$)
and Bethe roots

$\displaystyle\mathbf{u}$ $\displaystyle=\\{-0.0718891,0.0718891\\}\,,$
$\displaystyle\mathbf{v}$
$\displaystyle=\\{0.164766,0.327921,0.844103,1.31232\\}\,,$
$\displaystyle\mathbf{w}$
$\displaystyle=\\{0.296976,0.491824,0.756996,1.13713,2.62857,3.91077\\}\,,$

it reads

$\displaystyle C(2,4,6)$ $\displaystyle=\langle 11\rangle\langle
13\rangle\langle 22\rangle^{3}\langle 23\rangle\langle 31\rangle\langle
32\rangle\langle
33\rangle^{4}\left({\color[rgb]{0,0,1}-0.0000297465}\right)+\langle
12\rangle^{2}\langle 21\rangle^{2}\langle 22\rangle^{2}\langle
33\rangle^{6}\left({\color[rgb]{0,0,1}1.9812\times 10^{-6}}\right)+$
$\displaystyle+\langle 11\rangle\langle 13\rangle\langle 23\rangle^{4}\langle
31\rangle\langle 32\rangle^{4}\langle
33\rangle\left({\color[rgb]{0,0,1}0.00491993}\right)+\langle
13\rangle^{2}\langle 22\rangle\langle 23\rangle^{3}\langle
31\rangle^{2}\langle 32\rangle^{3}\langle
33\rangle\left({\color[rgb]{0,0,1}-0.00216325}\right)+$ $\displaystyle+\langle
12\rangle\langle 13\rangle\langle 21\rangle\langle 23\rangle^{3}\langle
31\rangle\langle 32\rangle^{3}\langle
33\rangle^{2}\left({\color[rgb]{0,0,1}0.00999526}\right)+\langle
13\rangle^{2}\langle 22\rangle^{4}\langle 31\rangle^{2}\langle
33\rangle^{4}\left({\color[rgb]{0,0,1}1.1861\times 10^{-6}}\right)+$
$\displaystyle+\langle 12\rangle\langle 13\rangle\langle 21\rangle\langle
22\rangle\langle 23\rangle^{2}\langle 31\rangle\langle 32\rangle^{2}\langle
33\rangle^{3}\left({\color[rgb]{0,0,1}-0.00073364}\right)+\langle
11\rangle^{2}\langle 22\rangle^{4}\langle
33\rangle^{6}\left({\color[rgb]{0,0,1}1.30467\times 10^{-7}}\right)+$
$\displaystyle+\langle 11\rangle\langle 12\rangle\langle 21\rangle\langle
22\rangle^{2}\langle 23\rangle\langle 32\rangle\langle
33\rangle^{5}\left({\color[rgb]{0,0,1}-0.000071986}\right)+\langle
11\rangle^{2}\langle 23\rangle^{4}\langle 32\rangle^{4}\langle
33\rangle^{2}\left({\color[rgb]{0,0,1}-0.00144007}\right)+$
$\displaystyle+\langle 13\rangle^{2}\langle 22\rangle^{3}\langle
23\rangle\langle 31\rangle^{2}\langle 32\rangle\langle
33\rangle^{3}\left({\color[rgb]{0,0,1}-0.000092995}\right)+\langle
11\rangle^{2}\langle 22\rangle^{2}\langle 23\rangle^{2}\langle
32\rangle^{2}\langle
33\rangle^{4}\left({\color[rgb]{0,0,1}0.000522219}\right)+$
$\displaystyle+\langle 12\rangle^{2}\langle 21\rangle^{2}\langle
22\rangle\langle 23\rangle\langle 32\rangle\langle
33\rangle^{5}\left({\color[rgb]{0,0,1}0.0000727415}\right)+\langle
11\rangle^{2}\langle 22\rangle\langle 23\rangle^{3}\langle
32\rangle^{3}\langle
33\rangle^{3}\left({\color[rgb]{0,0,1}0.000908203}\right)+$
$\displaystyle+\langle 12\rangle\langle 13\rangle\langle 21\rangle\langle
22\rangle^{3}\langle 31\rangle\langle
33\rangle^{5}\left({\color[rgb]{0,0,1}-2.52289\times 10^{-6}}\right)+\langle
11\rangle\langle 12\rangle\langle 21\rangle\langle 22\rangle^{3}\langle
33\rangle^{6}\left({\color[rgb]{0,0,1}-2.11167\times 10^{-6}}\right)+$
$\displaystyle+\langle 11\rangle\langle 13\rangle\langle 22\rangle^{4}\langle
31\rangle\langle 33\rangle^{5}\left({\color[rgb]{0,0,1}-1.31656\times
10^{-6}}\right)+\langle 11\rangle\langle 13\rangle\langle 22\rangle\langle
23\rangle^{3}\langle 31\rangle\langle 32\rangle^{3}\langle
33\rangle^{2}\left({\color[rgb]{0,0,1}-0.00342357}\right)+$
$\displaystyle+\langle 13\rangle^{2}\langle 23\rangle^{4}\langle
31\rangle^{2}\langle
32\rangle^{4}\left({\color[rgb]{0,0,1}0.000683143}\right)+\langle
11\rangle\langle 12\rangle\langle 21\rangle\langle 22\rangle\langle
23\rangle^{2}\langle 32\rangle^{2}\langle
33\rangle^{4}\left({\color[rgb]{0,0,1}-0.00146526}\right)+$
$\displaystyle+\langle 12\rangle\langle 13\rangle\langle 21\rangle\langle
22\rangle^{2}\langle 23\rangle\langle 31\rangle\langle 32\rangle\langle
33\rangle^{4}\left({\color[rgb]{0,0,1}0.000215642}\right)+\langle
12\rangle^{2}\langle 21\rangle^{2}\langle 23\rangle^{2}\langle
32\rangle^{2}\langle
33\rangle^{4}\left({\color[rgb]{0,0,1}0.00255232}\right)+$
$\displaystyle+\langle 13\rangle^{2}\langle 22\rangle^{2}\langle
23\rangle^{2}\langle 31\rangle^{2}\langle 32\rangle^{2}\langle
33\rangle^{2}\left({\color[rgb]{0,0,1}0.00212987}\right)+\langle
11\rangle\langle 12\rangle\langle 21\rangle\langle 23\rangle^{3}\langle
32\rangle^{3}\langle
33\rangle^{3}\left({\color[rgb]{0,0,1}0.000540332}\right)+$
$\displaystyle+\langle 11\rangle\langle 13\rangle\langle 22\rangle^{2}\langle
23\rangle^{2}\langle 31\rangle\langle 32\rangle^{2}\langle
33\rangle^{3}\left({\color[rgb]{0,0,1}-0.00103067}\right)+\langle
11\rangle^{2}\langle 22\rangle^{3}\langle 23\rangle\langle 32\rangle\langle
33\rangle^{5}\left({\color[rgb]{0,0,1}9.51785\times 10^{-6}}\right)+$
$\displaystyle+\left(\langle 11\rangle^{2}\langle 23\rangle^{4}\langle
32\rangle^{4}\langle 33\rangle^{2}-\langle 11\rangle\langle 13\rangle\langle
23\rangle^{4}\langle 31\rangle\langle 32\rangle^{4}\langle
33\rangle\right)\left({\color[rgb]{0,0,1}0.0020167}\right)+$
$\displaystyle+\left(\langle 13\rangle^{2}\langle 22\rangle^{4}\langle
31\rangle^{2}\langle 33\rangle^{4}-\langle 13\rangle^{2}\langle
22\rangle^{3}\langle 23\rangle\langle 31\rangle^{2}\langle 32\rangle\langle
33\rangle^{3}\right)\left({\color[rgb]{0,0,1}4.14448\times 10^{-6}}\right)+$
$\displaystyle+\left(\langle 12\rangle^{2}\langle 21\rangle^{2}\langle
22\rangle^{2}\langle 33\rangle^{6}-\langle 12\rangle^{2}\langle
21\rangle^{2}\langle 22\rangle\langle 23\rangle\langle 32\rangle\langle
33\rangle^{5}\right)\left({\color[rgb]{0,0,1}6.5287\times
10^{-6}}\right)+\dots\,,$ (30)

Beyond tree-level, the hexagon partition function is not a simple rational
function of the rapidities. Lack of control over the singularities in other
Riemann sheets prevent us from writing a recursion relation akin to (19) at
loop order. We are forced to evaluate the scattering of the hexagon form
factor via the explicit form of the hexagon partition function (7).

For general boundary conditions the one-loop evaluation is computationally
demanding. One of the problems is that structure constants are given by linear
combinations of multiple hexagons, which can be made explicit by combining
(24) and (25) to write

$\displaystyle\sum_{\ell_{1},\ell_{2},\ell_{3}}C^{J_{1},J_{2},J_{3}}_{\ell_{1},\ell_{2},\ell_{3}}\frac{\langle
1,1\rangle^{J_{1}-\ell_{2}-\ell_{3}}\langle
2,2\rangle^{J_{2}-\ell_{1}-\ell_{3}}\langle
3,3\rangle^{J_{3}-\ell_{1}-\ell_{2}}}{\langle 2,3\rangle^{-\ell_{1}}\langle
3,2\rangle^{-\ell_{1}}\langle 1,3\rangle^{-\ell_{2}}\langle
3,1\rangle^{-\ell_{2}}\langle 1,2\rangle^{-\ell_{3}}\langle
2,1\rangle^{-\ell_{3}}}=$
$\displaystyle=\mathcal{N}(\textbf{u})\mathcal{N}(\textbf{v})\mathcal{N}(\textbf{w})\times\sum_{\begin{subarray}{c}{a\,\cup\,\bar{a}=\textbf{u}}\\\
{b\,\cup\,\bar{b}=\textbf{v}}\\\
{c\,\cup\,\bar{c}=\textbf{w}}\end{subarray}}\omega_{\ell_{13}}(a,\bar{a})\omega_{\ell_{12}}(b,\bar{b})\omega_{\ell_{23}}(c,\bar{c})\times\mathcal{H}(a,b,c)\times\mathcal{H}(\bar{a},{\bar{c}},{\bar{b}})\,.$
(31)

When we fix the external polarizations $L_{i}$ and $R_{i}$ we must evaluate
all the scaterrings, sum over partitions and in the end we obtain that a
single glued hexagon is given by a combination of structure constants
$C^{J_{1},J_{2},J_{3}}_{\ell_{1},\ell_{2},\ell_{3}}$. By considering several
polarizations one can invert these relations obtaining single structure
constants in terms of combinations of glued hexagons, see appendix E.

For simplicity we will consider a particular polarization where the
computation of a single hexagon gives us a simple combination of structure
constants. This is the configuration where all the operators have the same
polarizations (i.e all the excitations are chosen to be the longitudinal modes
$\mathcal{D}^{1\dot{2}}$). In terms of spinors, these means
$L_{1}=L_{2}=L_{3}$ and $R_{1}=R_{2}=R_{3}$, making the inner products of (25)
equal. The tensor structure factors out and we get that

$\langle
1,1\rangle^{J_{1}+J_{2}+J_{3}}\sum_{\ell_{1},\ell_{2},\ell_{3}}C^{J_{1},J_{2},J_{3}}_{\ell_{1},\ell_{2},\ell_{3}}$
(32)

is given by a single glued hexagon where all the excitations have the same
polarization. Note that, this is not an abelian configuration, once we cross
the rapidities v and w to the top operator we end up with distinct excitations
that do not scatter trivially.

For twist-2 operators we were able to evaluate this sum for arbitrary values
of spins in a close formula

$\displaystyle\sum_{\ell_{1},\ell_{2},\ell_{3}}C^{J_{1},J_{2},J_{3}}_{\ell_{1},\ell_{2},\ell_{3}}$
$\displaystyle=\frac{\Gamma\left(1+J_{1}+J_{2}\right)\Gamma\left(1+J_{1}+J_{3}\right)\Gamma\left(1+J_{2}+J_{3}\right)}{\Gamma\left(1+J_{1}\right)\Gamma\left(1+J_{2}\right)\Gamma\left(1+J_{3}\right)\sqrt{\Gamma\left(1+2J_{1}\right)\Gamma\left(1+2J_{2}\right)\Gamma\left(1+2J_{3}\right)}}\times$
$\displaystyle\times{}_{4}F_{3}{\left[\genfrac{.}{.}{0.0pt}{}{-J_{1}\text{,}\mskip
8.0mu-J_{2}\text{,}\mskip 8.0mu-J_{3}\text{,}\mskip
8.0mu-1-J_{1}-J_{2}-J_{3}}{-J_{1}-J_{2}\text{,}\mskip 8.0mu-
J_{1}-J_{3}\text{,}\mskip 8.0mu-J_{2}-J_{3}};1\right]}$ (33)

By computing the equal polarizations hexagon for twist-2 and spin $J_{1}=2$,
$J_{2}=4$ and $J_{3}=6$ we recovered this tree-level value from the hexagon
partition function (7) and also obtained the one-loop contribution

$\sum_{\ell_{1},\ell_{2},\ell_{3}}C^{2,4,6}_{\ell_{1},\ell_{2},\ell_{3}}=\frac{75}{\sqrt{55}}-\frac{59077}{4620\sqrt{55}}g^{2}+\dots\,.$
(34)

which is reproduces the sum over all structure constants (all orange and cyan
terms) appearing in the three point function (1).

###  3.3 Abelian

As advertised in 2.4 there some polarizations which completely trivialize the
scattering of the excitations. Since they are given by the abelian part of the
hexagon partition function (7), we denote them by abelian components. They are
associated with three-point functions where one operator has orthogonal
polarization to the other two, see figure (10). By crossing these two
operators we are left with three sets of excitations with identical
polarizations in the same physical edge, which in turn scatter trivially.

One of such configurations can be obtained by considering $R_{1}=L_{2}=L_{3}$
and $L_{1}=R_{2}=R_{3}$, which sets the inner products of (24) to zero
$\langle 12\rangle=\langle 13\rangle=0$. Moreover, $\langle 23\rangle\langle
32\rangle=\langle 22\rangle\langle 33\rangle$, so the tensor structure factors
out, resulting that the three point function

$\langle 1,1\rangle^{J_{1}}\langle 2,2\rangle^{J_{2}}\langle
3,3\rangle^{J_{3}}\sum_{\ell_{1},\ell_{2},\ell_{3}}C^{J_{1},J_{2},J_{3}}_{\ell_{1},0,0}$
(35)

is given by an abelian hexagon. By permutation of the indices $1,2,3$ we have
the other components

$\mathbb{A}_{1}=\sum_{\ell_{1}}C^{J_{1},J_{2},J_{3}}_{\ell_{1},0,0}\,,\quad\quad\mathbb{A}_{2}=\sum_{\ell_{2}}C^{J_{1},J_{2},J_{3}}_{0,\ell_{2},0}\,,\quad\quad\mathbb{A}_{3}=\sum_{\ell_{3}}C^{J_{1},J_{2},J_{3}}_{0,0,\ell_{3}}$
(36)

which are structure constants given by hexagon form factors with purely
abelian factors.

Figure 10: The first operator is orthogonal to the other two operators, once
we perform crossing we end up with identical excitations in the physical edge,
which scatter trivially. This configuration corresponds to the abelian
configuration $\mathbb{A}_{1}$. By considering the other configurations where
the operators two and three are the ones orthogonal to the other operators we
get the abelian structures $\mathbb{A}_{2}$ and $\mathbb{A}_{3}$,
respectively.

The scattering with these boundary conditions forces a trivial matrix part in
(25) and yields the totally factorized contribution for the hexagon partition
function shown in (21) and reviewed here

$\displaystyle\mathcal{H}(u_{a},v_{b},w_{c})$ $\displaystyle=\langle
1,1\rangle^{a}\langle 2,2\rangle^{b}\langle
3,3\rangle^{c}\times(-1)^{b+c}\prod_{v_{i}\in b}\prod_{u_{j}\in
a}\prod_{w_{k}\in c}h(v_{i},u_{j})h(u_{j},w_{k})h(v_{i},w_{k})\times$
$\displaystyle\times\prod_{\begin{subarray}{c}{v_{l}\in b}\\\
{l<i}\end{subarray}}\prod_{\begin{subarray}{c}{u_{m}\in a}\\\
{m<j}\end{subarray}}\prod_{\begin{subarray}{c}{w_{n}\in c}\\\
{n<k}\end{subarray}}h(v_{l},v_{i})h(u_{m},u_{j})h(w_{n},w_{k})\,,$ (37)

where the $h^{\prime}s$ above must be crossed according with their arguments,
following (6).

These abelian form factors can be easily evaluated at any order in
perturbation theory, in particular at one-loop and for twist-2 operators with
spins $J_{1}=2$, $J_{2}=4$ and $J_{3}=6$ their sum over partitions reproduce
the one-loop structure constants (colored in cyan) in expression (1).

It turns out that in the abelian case the sum over partitions can be performed
analytically. The result (42) is a determinant formula for the three point
function in the asymptotic regime valid for operators with arbitrary twist,
which dependence enters only implicitly through the bridge lengths and the
Bethe roots. This vastly reduces the computational effort required to evaluate
the structure constants, and opens avenues to explore via integrability the
large spin limit of the three point functions.

In this section we go over the result and derivation whose details and
notation can be found in appendix F. The result for the case of one spinning
operator was obtained in [12]. The key observation was that the sum over
partitions of products of hexagon dynamical factors,

$H(u_{\bar{a}},u_{\bar{a}})\equiv\prod_{i<j\in\bar{a}}h(u_{i},u_{j})h(u_{j},u_{i}),$

could be interpreted as the expansion of a Fredholm pfaffian,

$\sum_{\bar{a}\subset\mathbf{u}}(-1)^{|\bar{a}|}w^{\prime}(u_{\bar{a}})H(u_{\bar{a}},u_{\bar{a}})=pf(I-\mathcal{K})_{2J_{1}\times
2J_{1}}.$ (38)

We define the matrix $\mathcal{K}$ in appendix F. Crucial for our purposes is
that this holds for any factorized function of the rapidities
$w^{\prime}(u_{\bar{a}})=\prod_{i\in\bar{a}}w^{\prime}(u_{i})$.

The asymptotic structure constant for three spinning operators (31) can be
rewritten in the abelian case as999For the rest of this section we will be
explicit about the crossing kinematics $\circlearrowleft$,
$\circlearrowright$.

$\displaystyle\mathbb{A}_{1}(J_{1},J_{2},J_{3})=$ $\displaystyle\langle
11\rangle^{J_{1}}\langle 22\rangle^{J_{2}}\langle
33\rangle^{J_{3}}{\mathcal{N}(J_{1})\mathcal{N}(J_{2})\mathcal{N}(J_{3})}h(\hat{u},\hat{u})h(\hat{v},\hat{v})h(\hat{w},\hat{w})$
(39)
$\displaystyle\times\sum_{\begin{subarray}{c}{\color[rgb]{1,0,0}{a\cup\bar{a}\subset\mathbf{u}}}\\\
{\color[rgb]{0,0,1}{b\cup\bar{b}\subset\mathbf{v}}}\\\
{\color[rgb]{0,0,1}{c\cup\bar{c}\subset\mathbf{w}}}\end{subarray}}(-1)^{{\color[rgb]{1,0,0}{|\bar{a}|}}{\color[rgb]{0,0,1}{+|\bar{b}|+|\bar{c}|}}}{\color[rgb]{1,0,0}{e^{u}_{\bar{a}}}}{\color[rgb]{0,0,1}{e^{v}_{\bar{b}}e^{w}_{\bar{c}}}}{\color[rgb]{1,0,0}{H(u_{\bar{a}},u_{\bar{a}})}}{\color[rgb]{0,0,1}{H(v_{\bar{b}},v_{\bar{b}})H(w_{\bar{c}},w_{\bar{c}})}}$
$\displaystyle\times{\color[rgb]{1,0,0}{h(v^{\circlearrowright}_{b},u_{a})h(u_{a},w^{\circlearrowleft}_{c})}}{\color[rgb]{1,0,0}{h(w^{\circlearrowright}_{\bar{c}},u_{\bar{a}},)h(u_{\bar{a}},v^{\circlearrowleft}_{\bar{b}})}}{\color[rgb]{0,0,1}{h(v^{\circlearrowright}_{b},w^{\circlearrowleft}_{c})h(w^{\circlearrowright}_{\bar{c}},v^{\circlearrowleft}_{\bar{b}})}},$

with $h(\hat{u},\hat{u})\equiv\prod_{i<j}h(u_{i},u_{j})$,

$e^{u}_{\bar{a}}\equiv\prod_{i\in\bar{a}}\frac{e^{iP(u_{i})\ell_{13}}}{h(\hat{u},u_{i})},\qquad
h(\hat{u},u_{i})\equiv\prod_{j\neq i}h(u_{j},u_{i}),\qquad
h(v^{\circlearrowright}_{b},u_{a})\equiv\prod_{\begin{subarray}{c}i\in a\\\
j\in b\end{subarray}}h(v^{\circlearrowright}_{j},u_{i}),$

and similar for the other terms. The red terms depend explicitly on both sets
$a$, $\bar{a}$. However, by completing the factors
$h(v^{\circlearrowright}_{b},u_{a})h(u_{a},w^{\circlearrowleft}_{c})$ to
$h(v^{\circlearrowright}_{b},\hat{u})h(\hat{u},w^{\circlearrowleft}_{c})$ and
using the unitarity relation
$h(x^{\circlearrowright},y)h(y,x^{\circlearrowleft})=1$, it can be rewritten
as

red
$\displaystyle={\color[rgb]{0,0,1}{h(v^{\circlearrowright}_{b},\hat{u})h(\hat{u},w^{\circlearrowleft}_{c})}}\sum_{a\cup\bar{a}\subset\mathbf{u}}(-1)^{|\bar{a}|}e^{u}_{\bar{a}}H(u_{\bar{a}},u_{\bar{a}})\frac{h(w^{\circlearrowright}_{\bar{c}},u_{\bar{a}})h(u_{\bar{a}},v^{\circlearrowleft}_{\bar{b}})}{h(v^{\circlearrowright}_{b},u_{\bar{a}})h(u_{\bar{a}},w^{\circlearrowleft}_{c})}$
$\displaystyle={\color[rgb]{0,0,1}{h(v^{\circlearrowright}_{b},\hat{u})h(\hat{u},w^{\circlearrowleft}_{c})}}\sum_{\bar{a}\subset\mathbf{u}}(-1)^{|\bar{a}|}\omega^{\prime}(u_{\bar{a}})H(u_{\bar{a}},u_{\bar{a}}),$

with
$\omega^{\prime}(u_{\bar{a}})=e^{u}_{\bar{a}}/\left(h(v^{\circlearrowright},u_{\bar{a}})h(u_{\bar{a}},w^{\circlearrowleft})\right)$,
for which the pfaffian identity (38) applies.

Physically, from the point of view of $O_{1}$, the effect of the orthogonal
operators $O_{2}$, $O_{3}$ is to create an effective background redefining the
propagation factor $e^{u}_{\bar{a}}$ but leaving the interaction term
$H(u_{\bar{a}},u_{\bar{a}})$ untouched. As we will see next, the same is true
from the point of view of $O_{2}$, $O_{3}$ provided we interpret it as a
larger operator with $J_{2}+J_{3}$ excitations.

Plugging this result in (39) leaves us with the blue terms,

$\displaystyle{\color[rgb]{0,0,1}{\text{blue}}}=\sum_{\begin{subarray}{c}b\cup\bar{b}\subset\mathbf{v}\\\
c\cup\bar{c}\subset\mathbf{w}\end{subarray}}(-1)^{|\bar{b}|+|\bar{c}|}e^{v}_{\bar{b}}e^{w}_{\bar{c}}H(v_{\bar{b}},v_{\bar{b}})H(w_{\bar{c}},w_{\bar{c}})h(v^{\circlearrowright}_{b},\hat{u})h(\hat{u},w^{\circlearrowleft}_{c})h(v^{\circlearrowright}_{b},w^{\circlearrowleft}_{c})h(w^{\circlearrowright}_{\bar{c}},v^{\circlearrowleft}_{\bar{b}}).$
(40)

Once again completing to the overall factors and using the unitarity identity,
we can reduce101010See appendix (F) for details. the summand to functions of
$\bar{b}$, $\bar{c}$ only,

${\color[rgb]{0,0,1}{\text{blue}}}=h(\hat{v}^{\circlearrowright},\hat{w}^{\circlearrowleft})h(\hat{v}^{\circlearrowright},\hat{u})h(\hat{u},\hat{w}^{\circlearrowleft})\sum_{\begin{subarray}{c}b\cup\bar{b}\subset\mathbf{v}\\\
c\cup\bar{c}\subset\mathbf{w}\end{subarray}}(-1)^{|\bar{b}|+|\bar{c}|}\omega^{\prime}(v_{\bar{b}}^{\circlearrowright})\omega^{\prime}(w_{\bar{c}})H(v_{\bar{b}}^{\circlearrowright},v_{\bar{b}}^{\circlearrowright})H(w_{\bar{c}},w_{\bar{c}})H(v^{\circlearrowright}_{\bar{b}},w_{\bar{c}}),$
(41)

with $\omega^{\prime}(v_{\bar{b}}^{\circlearrowright})$,
$\omega^{\prime}(w_{\bar{c}})$ defined in appendix (F). To conclude, one must
reinterpret the double sum over partition as a single sum over the partition
of the set $\mathbf{z}=\mathbf{v}^{\circlearrowright}\cup\mathbf{w}$. One can
then write

${\color[rgb]{0,0,1}{\text{blue}}}=h(\hat{v}^{\circlearrowright},\hat{w}^{\circlearrowleft})h(\hat{v}^{\circlearrowright},\hat{u})h(\hat{u},\hat{w}^{\circlearrowleft})\sum_{d\cup\bar{d}\subset\mathbf{z}}(-1)^{|\bar{d}|}\omega^{\prime}(z_{\bar{d}})H(z_{\bar{d}},z_{\bar{d}})$

which can be cast as a pfaffian through (38).

The final result is

$\displaystyle\mathbb{A}_{1}(J_{1},J_{2},J_{3})$ $\displaystyle=\langle
11\rangle^{J_{1}}\langle 22\rangle^{J_{2}}\langle
33\rangle^{J_{3}}h(\hat{v}^{\circlearrowright},\hat{u})h(\hat{u},\hat{w}^{\circlearrowleft})h(\hat{v}^{\circlearrowright},\hat{w}^{\circlearrowleft})h(\hat{v},\hat{v})h(\hat{w},\hat{w})h(\hat{u},\hat{u})\times$
$\displaystyle{\mathcal{N}(J_{1})\mathcal{N}(J_{2})\mathcal{N}(J_{3})}\times
pf\left(\textit{I}-\mathcal{K}\right)_{2J_{1}\times
2J_{1}}pf\left(\textit{I}-\mathcal{K}^{\prime}\right)_{2(J_{2}+J_{3})\times
2(J_{2}+J_{3})},$ (42)

with the matrices $\mathcal{K}$ and $\mathcal{K}^{\prime}$ defined in appendix
(F).

As discussed above, these formulas express the abelian three point functions
at finite coupling in the asymptotic regime, that is, ignoring finite volume
corrections coming from excitations wrapping around the seams of the pair of
pants. These mirror corrections can be properly ignored at weak coupling when
we consider three point functions of large twist operators so that all bridge
numbers $\ell_{ij}$ are large. Due to the Boltzmann suppression of propagation
over large distances $\ell_{ij}$, mirror particles only start to contribute at
$(\ell_{ij}+1)$ loops.

For example, when we consider operators of classical twist
$\tau_{1}=\tau_{2}=\tau_{3}=6$ and bridge lengths $\ell_{ij}=3$, equation (42)
provides exact formulas up to three loops. As an example, for spins $J_{1}=6$,
$J_{2}=4$, $J_{3}=2$ and bethe roots (at tree-level)

$\displaystyle\mathbf{u}$
$\displaystyle=\\{-3.34538,-2.2266,-1.45396,-0.894775,-0.500817,-0.22559\\}\,,$
$\displaystyle\mathbf{v}$
$\displaystyle=\\{-2.18949,-1.24366,-0.66142,0.163878\\}\,,$
$\displaystyle\mathbf{w}$ $\displaystyle=\\{-1.03826,1.03826\\}\,,$

we obtain111111We keep the $\zeta(3)$ from the dressing phase explicit.,

$\displaystyle C_{\text{abelian}}(6,4,2)=$ $\displaystyle
0.000087168875373411949253081221196521450200295689674319409$ $\displaystyle-$
$\displaystyle
0.0010941423754946036043705421214308022132833891655992081\,g^{2}$
$\displaystyle+$ $\displaystyle
0.0024639150636611583478721259566761789123679977153044975\,g^{4}$
$\displaystyle+$
$\displaystyle(0.15149466528731340172164338220385987881409857001459463$
$\displaystyle+$ $\displaystyle
0.0099644970255952430277488412973707304997053789074945953\times\zeta(3))g^{6}.$

We hope that the novel data generated through this result will be useful in
future integrability explorations.

###  3.4 A two loop check

An important question is if the asymptotic part is sufficient to capture the
leading contribution, in the large spin limit, of three point functions with
more than one non-BPS operator. The goal of this section is to lay out the
main ingredients that are necessary to compute the first correction to the
asymptotic part of the OPE coefficient in the Abelian polarization. Most of
the discussion is valid for operators with any spin but we compute explicitly
only the finite size correction of the OPE coefficient of two Konishi
operators and one BPS leaving a more general analysis(including the large spin
limit mentioned above) to the future. These new effects are obtained by gluing
the two hexagons along the three seams (from now on referred to as mirror
edges) to form a pair of pants. The gluing procedure is achieved by dressing
the mirror edges with multiparticle states and integrating over their
rapidities. The multiparticle states are labeled by boundstate number $n$ as
explained [1, 13, 14, 15].

Figure 11: Single mirror particle contribution to a three point function with
three spinning operators. The mirror particle would correspond to a bottom
contribution (in the notation of [1]) if the operators $2,3$ were BPS and
adjacent if the the $1$ and $2$ or $3$ were absent.

One crucial but important detail is that the bound state particles live in the
mirror edges. This might seem innocuous but is the reason why their
contribution only starts at two loops. To have a better grasp on this matter
let us look at the integrand of just one boundstate particle in the hexagon
formalism

int $\displaystyle=\sum_{\begin{subarray}{c}{a\,\cup\,\bar{a}=\textbf{u}}\\\
{b\,\cup\,\bar{b}=\textbf{v}}\\\ {c\,\cup\,\bar{c}=\textbf{w}}\\\
X\end{subarray}}\mu(z^{\gamma})e^{ip_{z}^{\gamma}}\omega_{l_{13}}(a,\bar{a})\omega_{l_{23}}(c,\bar{c})\omega_{l_{12}}(b,\bar{b})\mathcal{H}(b^{2\gamma},a,c^{-2\gamma},z^{-3\gamma})\mathcal{H}(\bar{c}^{2\gamma},\bar{a},\bar{b}^{-2\gamma},z^{-3\gamma})$
(43)

where $X$ is a particle in the mirror theory with rapidity $z$ and

$\displaystyle\mu(z^{\gamma})=\frac{n(x^{[+n]}x^{[-n]})^{2}}{g^{2}(x^{[+n]}x^{[-n]}-1)^{2}((x^{[+n]})^{2}-1)(1-(x^{[-n]})^{2})},\
\ \ e^{ip_{u^{\gamma}}}=\frac{1}{x^{[n]}x^{[-n]}},$ (44)

with $n$ being the boundstate number. Since the boundstate particle lives in
the mirror edge we need to do and off number of $\gamma$ rotation. In this
case the expressions for $\mu(z^{\gamma})$ and $e^{ip_{u^{\gamma}}}$ start
with $g^{2}$ at weak coupling. Physically, this is just saying that we pay a
coupling for the mirror particle to travel from the front hexagon to the back.
For this reason the finite size correction for twist two operators only makes
an appearance at the two loop level. Fortunately there is perturbative data
computed using traditional Feynman integrals that we can compare against [16].
The rest of this section is devoted to obtain the finite size correction at
two loops and for two spin two operators and match it with the result obtained
in [16].

The integrand (43) is very similar to the hexagon partition function, as it is
the scattering of three sets of particles in a pairwise fashion, but now with
an additional bound state particle arising from the mirror contributions.

There are several simplifications for the Abelian polarization, for example
the scattering between the three sets of particles u, v and w is simple. If it
was not for the boundstate particles, the whole matrix part would be trivial.
However, as depicted in figure 12, in this abelian configuration the
boundstate scattering can be recasted as a transfer matrix [1],

$T_{a}(z)=\sum_{n=-1}^{1}(3n^{2}-2)\prod_{m=0}^{n}\frac{R^{(+)}(z^{[2m-a]})}{R^{(-)}(z^{[2m-a]})}\sum_{j=\frac{2-a}{2}}^{\frac{a-2n}{2}}\prod_{k=j+n}^{\frac{a-2}{2}}\frac{R^{(+)}(z^{[2n-2k]})B^{(+)}(z^{[-2k]})}{R^{(-)}(z^{[2n-2k]})B^{(-)}(z^{[-2k]})}\,,$
(45)

where

$R^{(\pm)}(z)=\prod_{j}(x(z)-x^{\mp}_{j})\,,\,\,\,B^{(\pm)}(z)=\prod_{j}\left(\frac{1}{x(z)}-x^{\mp}_{j}\right)\,.$
(46)

and the product in $j$ should be taken over all physical particles. Notice
that some of these particles are mirror rotated as can be seen in figure 11
and for this reason the transfer matrix that shows up in the integrand (43) is
not exactly (45). Instead, the integrand will be written in term of
$T_{n}^{\circlearrowright\circlearrowleft}(z)$ which is defined by

$\displaystyle
T_{n}^{\circlearrowright\circlearrowleft}(z)=T_{n}(z)\Big{|}_{x^{[-a]}\to\frac{1}{x^{[-a]}},\,x^{\pm}_{j}\to\frac{1}{x^{\pm}_{j}}}\
\ \ j\in{\bf{v}\cup\bf{w}}.$ (47)

Figure 12:  The sum over the flavour indices of the mirror particle states in
the matrix part of the hexagon form factor is proportional to a transfer
matrix. This is precisely the same mechanism present in the OPE with just one
spinning operator.

The integrand (43) can then be written in a compact form as

$\displaystyle\textrm{int}(z)=\frac{T_{n}^{\circlearrowright\circlearrowleft}(z)\mu(z^{\gamma})e^{ip_{z}^{\gamma}}}{h_{n}(z^{-\gamma},{\bf{v}})h_{n}(z^{\gamma},{\bf{u}})}h_{n}({\bf{w}},z^{-\gamma})\sum_{\begin{subarray}{c}{a\,\cup\,\bar{a}=\textbf{u}}\\\
{b\,\cup\,\bar{b}=\textbf{v}}\\\
{c\,\cup\,\bar{c}=\textbf{w}}\end{subarray}}\omega_{l_{23}}^{\prime}(c,\bar{c})\omega_{l_{13}}(a,\bar{a})\omega_{l_{12}}^{\prime}(b,\bar{b})\times$
$\displaystyle\times
h(b^{2\gamma},a)h(b^{4\gamma},c)h(a,c^{-2\gamma})h(\bar{c}^{2\gamma},\bar{a})h(\bar{c}^{4\gamma},\bar{b}_{2})h(\bar{a},\bar{b}^{-2\gamma})\prod_{i=1}^{3}h(w_{i},w_{i})h(\bar{w}_{i},\bar{w}_{i}),$
(48)

with $\omega^{\prime}$, $p_{n}$ and $h_{n}$ defined by

$\displaystyle\omega^{\prime}_{\ell_{23}}(c,\bar{c})=\omega_{\ell_{23}}(c,c)p_{n}\left(\bar{c}^{4\gamma},z^{-\gamma}\right)\,,\quad\omega^{\prime}_{\ell_{12}}(b,\bar{b})=\omega_{\ell_{12}}(b,b)p_{n}\left(\bar{b},z^{-\gamma}\right)\,,$
(49) $\displaystyle
p_{n}(u,v)=h_{n}(u,v)h_{n}(v,u)=\frac{(u-v)^{2}+\frac{(n-1)^{2}}{4}}{(u-v)^{2}+\frac{(n+1)^{2}}{4}}\left(\frac{1-\frac{1}{y^{-}x^{[+n]}}}{1-\frac{1}{y^{-}x^{[-n]}}}\frac{1-\frac{1}{y^{+}x^{[-n]}}}{1-\frac{1}{y^{+}x^{[+n]}}}\right)^{2}\,,$
(50) $\displaystyle
S_{n}(u,v)=\frac{1}{\sigma_{n}^{2}(u,v)}\frac{(u-v+i\frac{n-1}{2})(u-v+i\frac{n+1}{2})}{(u-v-i\frac{n-1}{2})(u-v-i\frac{n+1}{2})}\prod_{k=-\frac{n-1}{2}}^{\frac{n-1}{2}}\left(\frac{1-\frac{1}{y^{+}x^{[-2k-1]}}}{1-\frac{1}{y^{-}x^{[+2k+1]}}}\right)^{2}$
(51)

and $\sigma_{n}(u,v)$ is the (fused) BES dressing phase [5]. Let us remark
that these formulas should be valid for operators with any spin but only in
the Abelian polarization.

Figure 13: For two non-BPS operators there is one mirror edge that is distinct
from the other two. It turns out that up to two loops only the finite size
corrections (a) contribute while the finite size correction (b) will only
starts at three loop order.

For simplicity we focus on three point functions with two spinning operators
$J_{1}=J_{2}=2$ and one BPS and compare it with perturbative results obtained
in [16]. Obviously, this choice breaks the symmetry between the mirror edges
since one of them will be between the two non-BPS operators, as can be seen in
figure 13. It is straightforward to check by expanding at weak coupling that
we only need to take into consideration the finite size corrections
represented in 13.a.

The building blocks to get this contribution are very explicit and now it is
just a matter of evaluating them at weak coupling after taking the appropriate
monodromies. After doing this we obtain

$\displaystyle\textrm{int}(z)={\scriptstyle{\frac{55296(-1)^{2/3}n^{2}g^{4}\left(12z^{2}-n^{2}\right)\left(3n^{2}+12z^{2}-4\right)}{\left(n^{2}+4z^{2}\right)^{3}\left(216n^{6}\left(6z^{2}-1\right)+432n^{4}\left(18z^{4}-2z^{2}+1\right)+384n^{2}\left(54z^{6}+9z^{4}-15z^{2}-1\right)+81n^{8}+256\left(9z^{4}+3z^{2}+1\right)^{2}\right)}}}$
(52)

where we used the Konishi Bethe roots for the two spinning operators

$\textbf{u}=\textbf{v}=\left(-\frac{1}{2\sqrt{3}},\frac{1}{2\sqrt{3}}\right)\,.$
(53)

The OPE coefficient121212One might be worried about the $(-1)^{\frac{2}{3}}$
in (52) because it can give rise to an imaginary part in the structure
constant. However this does not pose an issue since this factor is removed
with the normalization factor
$\mathcal{N}(2)^{2}=\frac{(-1)^{-\frac{2}{3}}}{54}$. for two spin two
operators is given by at two loops

$\sum_{\ell=0}^{2}C_{2,2,0}^{\ell}=\mathcal{N}(2)\mathcal{N}(2)\left(\textrm{{\bf{asymptotic}
}}_{\textrm{{\bf{Abelian} }}}+2\sum_{n=1}^{\infty}\int
dz\textrm{int}(z)\right).$ (54)

where asymptotic are the abelian structures given by the hexagon partition
function (37) or by the pfaffian (42) and the factor of two in front of the
integral takes into account the two possibilities to add mirror particles as
shown in figure 13.a.

The integral over the rapidity $z$, that can be done by picking residues and
the sum over the boundstate $n$ can be evaluated without much effort. After
adding everything up we obtain

$\displaystyle\sum_{\ell=0}^{2}C_{2,2,0}^{\ell}=\frac{1}{6}\big{(}1-12g^{2}+147g^{4}\big{)}$
(55)

which matches exactly the result obtained in [16].

It should be possible to obtain the finite size correction for other spins (in
the Abelian polarization). The only issue that might occur is that the
integral/sum needs to be regularized. However, we expect that the HPL method
of [13] can be used to regularize efficiently the sums in a similar fashion as
in one spinning case. Along these lines it would be interesting to extend the
results of [29] to our setting.

Obtaining finite size corrections for other polarizations is somewhat harder
but should be doable at least at leading order in the coupling and for small
spins. The main issue is that the sum over mirror particles in (43) is more
involved. It would be interesting to study this further in the future.

##  4 Discussion

This paper reduces the computation of asymptotic three point functions of
three spinning operators in $\mathcal{N}=4$ SYM to the statistical mechanical
problem of computing the partition function of a system of the Hubbard type on
a Kagome-like lattice whose boundary conditions are determined by the
polarizations and quantum numbers of the spinning operators, equations (7, 25,
28). The analytic structure of this partition function is inherited from the
vertex, Beisert’s SU(2|2) extended S-matrix [3], and therefore is extremely
rich. Its singularities are determined by the spectrum of the dual world-sheet
theory and have clear physical interpretations: particles decoupling,
annihilating, fusing into bound states, and others.

In the limit of weak coupling the vertex reduces to the rational type and the
partition function can be solved. We do so in this work by exploiting the
analyticity of the partition function. After stripping out overall factors,
the partition function in the rational limit is a meromorphic function of the
rapidities with no bound-state poles. It can be then solved recursively by
concatenating the various decoupling poles in each of the rapidities. This is
the result (19). From the point of view of the gauge theory, it provides an
efficient way of generating tree-level structure constant data for spinning
operators of arbitrary twist.

It would be fascinating to solve this partition function away from weak
coupling. From the point of view of analyticity, we now have functions on a
multi-sheeted Riemann surface with multiple additional fusion poles whose
residues should be related to smaller partition functions involving bound-
state lines. From the statistical mechanics viewpoint, similar integrable
systems were solved in a more traditional fashion. For example, in [17] Baxter
solves the thermodynamic partition function of a 8-vertex model defined on the
faces of a Kagome lattice by matching it to the computation in the standard
square lattice. His construction is the same behind the integrability of the
fishnet model by Zamolodchikov [18]. Exploring solid-on-solids dualities of
this type might lead to a direct solution in terms of the transfer-matrices
and functional equations.

Once the statistical mechanics problem is solved, one must still resolve the
sum over partitions (25), the complexity of which grows rapidly at large but
finite spin131313In the strict large spin limit we hope to have alternative
techniques, see discussion below.. Turns out this problem can be solved, at
any value of the coupling, for certain boundary conditions of the partition
function. When two operators are parallel and orthogonal to the third the
hexagon becomes a simple abelian factor, the sum over partitions of which can
be cast in pfaffian form. This is the result (42). It provides an efficient
way of generating some high-loop data for structure constants of high twist
spinning operators.141414Some corners of the cross ratios space parametrized
by the null snowflake OPE limit of the 6-pt function should be controlled by
the abelian structure constants. Having complete control over those, we could
therefore generalize [12] to the higher point case and provide valuable
boundary data for an eventual bootstrap approach to fix these correlators [19,
20, 21].  Note: for each generic three spinning operators, the pfaffian
formulas provide three independent structure constants, one for each choice of
orthogonal operator. The three point function must therefore interpolate
between these three highly non-trivial pfaffian formulas involving matrices of
different dimensionality as the polarization vectors vary. It would be great
to identify non-trivial objects linear in the tensors structures (24) that
performs such interpolation, as they might be relevant for describing the full
three point function of spinning operators in $\mathcal{N}=4$ SYM.

Figure 14: Top arrow: this work, decomposition of the spinning three point
functions into polarized hexagons. Left arrow: large spin three-point
functions duality with null Wilson loops [2]. Bottom arrow: decomposition of
null Wilson loops into integrable pentagons [22]. Right arrow: future work,
how the integrable hexagons and integrable pentagons are related in the large
spin limit.

At large spin the asymptotic structure constant should provide the exact three
point function of the gauge theory, finite size corrections being power-law
suppressed in spin. In [7] we considered the null-hexagon limit of six point
functions of $\mathbf{20}^{\prime}$ operators and its respective snowflake OPE
decomposition in terms of three large spin operators to derive a map between
the expectation value of null hexagonal Wilson loops (WL) and the large spin
limit of the three point functions considered in this paper. Combining [7] and
this work, we can write a sharp formula expressing the expectation value of
the null WL in any geometrical configuration to the hexagons expansion:

$\mathbb{W}(U_{1},U_{2},U_{3})=\lim_{J_{i},\ell_{i}\rightarrow\infty}\frac{\mathcal{N_{\lambda}}\mathcal{N}_{\textbf{u}}\mathcal{N}_{\textbf{v}}\mathcal{N}_{\textbf{w}}}{\left(C^{\ell_{1},\ell_{2},\ell_{3}}_{J_{1},J_{2},J_{3}}\right)_{\text{tree}}}\times\prod_{i=1}^{3}\left(\frac{J_{i}\ell_{i}}{2\ell_{i+1}\ell_{i-1}}\right)^{-\frac{\gamma_{i}}{2}}\times\sum_{\begin{subarray}{c}{a\,\cup\,\bar{a}=\textbf{u}}\\\
{b\,\cup\,\bar{b}=\textbf{v}}\\\
{c\,\cup\,\bar{c}=\textbf{w}}\end{subarray}}\omega_{\ell_{13}}\omega_{\ell_{12}}\omega_{\ell_{23}}\times\mathcal{H}_{a,b,c}\mathcal{H}_{\bar{a},{\bar{c}},{\bar{b}}}$
(56)

where the ratios of $\ell_{i}$ and $J_{i}$ are kept fixed and determined in
terms of the null hexagon cross ratios $U_{i}$ through (10) of [7]; the
boundary conditions of the hexagon partition function $\mathcal{H}$ are fixed
in terms of $U_{i}$ through (37) of [7]. Above, $\gamma_{i}$ are the anomalous
dimensions of operator $O_{i}$, while $\mathcal{N}_{\lambda}$ is a spin
independent normalization constant to be determined.

Null WL can be decomposed as an expansion around the collinear limit by the
Pentagon OPE, higher energy excitations of the GKP vacuum controlling the
geometric expansion [22]. Equation (56) can therefore be understood as a map
between hexagons and pentagons, see figure 14. Since the adjoint Wilson loop
$\mathbb{W}$ is the square of the fundamental Wilson loop in the large $N$
limit and since the latter is obtained gluing two pentagons $\mathcal{P}$
together we can cast the sharp equation (56) as the simplified slogan

$\left(\sum\mathcal{P}\times\mathcal{P}\right)^{2}=\lim_{\texttt{large
spin}}\sum\mathcal{H}\times\mathcal{H}\,.$ (57)

Explicitly uncovering this relation is a problem for the future which could
lead to a more universal integrability framework for $\mathcal{N}=4$ SYM. More
generally, one should ask if (56) could lead to an alternative effective way
of computing WL physics and their dual gluon scattering amplitudes [23, 24].
The answer to this question depends on understanding how to simplify the large
spin limit of the computations discussed in this work. The thermodynamic limit
of the hexagon partition function is therefore of special importance. The
matrix-models inspired methods of [25, 26, 27] should also be useful.

It is also possible that instead of simplifying the building blocks arising in
the summand in (56) we should discard all these partition functions and look
for new tricks to compute directly the full sum over partitions. That would be
bitter sweet.151515Even if that turns out to be the case, these (sums over)
hexagon partition functions will always be useful to produce important data to
test such scenario specially in perturbation theory.

## Acknowledgements

We would like to thank Benjamin Basso, Vladimir Kazakov, João Caetano, Gabriel
Lefundes, Matheus Fabri, Tiago Fleury, Shota Komatsu for illuminating
discussions. Research at the Perimeter Institute is supported in part by the
Government of Canada through NSERC and by the Province of Ontario through MRI.
We also thank Porto University for the organization of the 2022 Bootstrap
conference in Porto, during which this project was completed. This work was
additionally supported by a grant from the Simons Foundation (Simons
Collaboration on the Nonperturbative Bootstrap #488661) and ICTP-SAIFR FAPESP
grant 2016/01343-7 and FAPESP grant 2017/03303-1. The work of P.V. was
partially supported by the Sao Paulo Research Foundation (FAPESP) under Grant
No. 2019/24277-8. The work of C.B. was supported by the Sao Paulo Research
Foundation (FAPESP) under Grant No. 2018/25180-5. V.G. is supported by Simons
Foundation grants #488637 (Simons collaboration on the non-perturbative
bootstrap). Centro de Física do Porto is partially funded by Fundação para a
Ciência e Tecnologia (FCT) under the grant UID04650-FCUP. This research was
supported in part by the National Science Foundation under Grant No. NSF
PHY-1748958.

##  Appendix A Spinors

In section 3 we parametrize spinning operators through their left- and right-
handed polarization spinors ${L_{i}}_{\alpha}$ and ${R_{i}}_{\dot{\alpha}}$.
These are related to the polarization vectors by

$\epsilon_{i}^{\mu}={R_{i}}_{\dot{\beta}}\left(\bar{\sigma}^{\mu}\right)^{\dot{\beta}\alpha}{L_{i}}_{\alpha}.$
(58)

In our conventions the sigma matrices $\sigma^{\mu}_{{\alpha}\dot{\alpha}}$,
$\bar{\sigma}^{\mu\dot{\alpha}\alpha}$ are given by

$\sigma^{0}=-\begin{pmatrix}1&0\\\
0&1\end{pmatrix},\qquad\sigma^{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\qquad\sigma^{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\qquad\sigma^{3}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},\qquad$ (59)

$\bar{\sigma}=(\sigma_{0},-\sigma_{1},-\sigma_{2},-\sigma_{3})$. Indices are
raised and lowered with

$\epsilon^{\alpha\beta}=\epsilon^{\dot{\alpha}\dot{\beta}}=-\epsilon_{{\alpha}{\beta}}=-\epsilon_{\dot{\alpha}\dot{\beta}}=\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\,.$ (60)

The structures $\langle i,j\rangle\equiv
i{R_{j}}_{\dot{\alpha}}(\bar{\sigma}_{2})^{\dot{\alpha}\alpha}{L_{i}}_{\alpha}$
are preserved by the residual symmetry of the conformal frame chosen along the
$x_{2}$ direction. Due to this it is useful to define, when working in the
conformal frame, left-handed spinors
$R^{\alpha}=iR_{\dot{\alpha}}(\bar{\sigma}_{2})^{\dot{\alpha}\alpha}=\epsilon^{\alpha\dot{\alpha}}R_{\dot{\alpha}}$.
These can now be straightforwardly contracted with the $L_{i}$ spinors to form
invariants, as is used in section 2 to define the hexagon partition function
(7).

The $\langle i,j\rangle$ structures are related to the canonical covariant
$V_{i}$, $H_{ij}$ structures used in the literature [8] through

$\langle i,j\rangle\langle j,i\rangle\equiv H_{ij},\qquad\langle
i,i\rangle\equiv V_{i}.$ (61)

##  Appendix B Frames, Vertex, and Markers

The integrability description of $\mathcal{N}=4$ SYM requires a choice of a
frame, i.e. a representation for the single-trace operators in terms of
magnons on top of a spin chain. Two choices are common: the string and spin-
chain frame [1]. For most of this paper we use the spin-chain frame, in which
a single trace operator made out of $L$ fundamental fields is assigned to a
spin chain of length $L$. Clearly the number of fields composing an operator
is not a non-perturbative notion. Moreover, it is not preserved by the
dynamics: the scattering between excitations on top of the vacuum can change
the chain length, deleting or introducing additional sites $\mathcal{Z}$ in
the chain. These are represented by the $\mathcal{Z}^{\mp}$ markers. The spin-
chain frame is nevertheless useful to compare with previous weak coupling
results available in the literature.

The S-matrix for excitations in the spin chain frame is given by

$\displaystyle
S|\phi^{a}(x)\phi^{b}(y)\rangle=\mathcal{A}(x,y)|\phi^{\\{a}(y)\phi^{b\\}}(x)\rangle+\mathcal{B}(x,y)|\phi^{[a}(y)\phi^{b]}(x)\rangle+\frac{1}{2}\mathcal{C}(x,y)\Sigma^{ab}\epsilon_{cd}|\mathcal{Z}^{-}\psi^{c}(y)\psi^{d}(x)\rangle$
$\displaystyle
S|\psi^{a}(x)\psi^{b}(y)\rangle=\mathcal{D}(x,y)|\psi^{\\{a}(y)\psi^{b\\}}(x)\rangle+\mathcal{E}(x,y)|\psi^{[a}(y)\psi^{b]}(x)\rangle+\frac{1}{2}\mathcal{F}(x,y)\epsilon^{ab}\Sigma_{cd}|\mathcal{Z}^{+}\phi^{c}(y)\phi^{d}(x)\rangle$
$\displaystyle
S|\phi^{a}(x)\psi^{b}(y)\rangle=\mathcal{G}(x,y)|\psi^{b}(y)\phi^{a}(x)\rangle+\mathcal{H}(x,y)|\phi^{a}(y)\psi^{b}(x)\rangle$
$\displaystyle
S|\psi^{a}(x)\phi^{b}(y)\rangle=\mathcal{K}(x,y)|\psi^{a}(y)\phi^{b}(x)\rangle+\mathcal{L}(x,y)|\phi^{b}(y)\psi^{a}(x)\rangle.$
(62)

The S-matrix elements are expressed in terms of the Zhukovsky variables as [3,
4]

$\displaystyle\mathcal{A}(x,y)$
$\displaystyle=\frac{x^{+}(y)-x^{-}(x)}{x^{-}(y)-x^{+}(x)},$ (63)
$\displaystyle\mathcal{B}(x,y)$
$\displaystyle=-1+\frac{\left(x^{+}(x)-x^{+}(y)\right)\left(x^{-}(x)\left(x^{-}(y)-2x^{+}(y)\right)+x^{+}(x)x^{+}(y)\right)}{\left(-1+x^{-}(x)x^{-}(y)\right)\left(x^{-}(y)-x^{+}(x)\right)x^{+}(x)x^{+}(y)},$
$\displaystyle\mathcal{C}(x,y)$
$\displaystyle=\frac{2\gamma(x)\gamma(y)\left(x^{+}(x)-x^{+}(y)\right)}{\left(1-x^{-}(x)x^{-}(y)\right)\left(x^{-}(y)-x^{+}(x)\right)},$
$\displaystyle\mathcal{D}(x,y)$ $\displaystyle=-1,$
$\displaystyle\mathcal{E}(x,y)$
$\displaystyle=\frac{x^{+}(y)-x^{-}(x)}{x^{-}(y)-x^{+}(x)}+\frac{\left(x^{+}(x)-x^{+}(y)\right)\left(x^{-}(x)x^{-}(y)+x^{+}(x)\left(x^{+}(y)-2x^{-}(y)\right)\right)}{\left(1-x^{-}(x)x^{-}(y)\right)\left(x^{-}(y)-x^{+}(x)\right)x^{+}(x)x^{+}(y)},$
$\displaystyle\mathcal{F}(x,y)$
$\displaystyle=\frac{2x^{-}(x)x^{-}(y)\left(x^{-}(x)-x^{+}(x)\right)\left(x^{-}(y)-x^{+}(y)\right)\left(x^{+}(x)-x^{+}(y)\right)}{\gamma(x)\gamma(y)\left(-1+x^{-}(x)x^{-}(y)\right)\left(x^{-}(y)-x^{+}(x)\right)x^{+}(x)x^{+}(y)},$
$\displaystyle\mathcal{G}(x,y)$
$\displaystyle=\frac{x^{+}(y)-x^{+}(x)}{x^{-}(y)-x^{+}(x)},$
$\displaystyle\mathcal{H}(x,y)$
$\displaystyle=\frac{\gamma(x)\left(x^{+}(y)-x^{-}(y)\right)}{\gamma(y)\left(x^{-}(y)-x^{+}(x)\right)},$
$\displaystyle\mathcal{L}(x,y)$
$\displaystyle=\frac{x^{-}(y)-x^{-}(x)}{x^{-}(y)-x^{+}(x)},$
$\displaystyle\mathcal{K}(x,y)$
$\displaystyle=\frac{\gamma(y)\left(x^{+}(x)-x^{-}(x)\right)}{\gamma(x)\left(x^{-}(y)-x^{+}(x)\right)},$
(64)

where

$\gamma(x)=\sqrt{x^{-}(x)-x^{+}(x)}.$

In computing the hexagon form factors we cross all
$\mathcal{D}^{\alpha\dot{\alpha}}$ excitations to the u edge, figure (9),
break them into left- and right- fermions, and then scatter all left
components with the $PSU(2|2)$ S-matrix described above. Finally we project
the scattering result into the final state of right-fermions. When fermions
scatter they may produce markers according to the $\mathcal{C}$ and
$\mathcal{F}$ elements in (62). Our computation scheme is then to, as soon as
a $\mathcal{Z}$ marker is created in this chain of scatterings, move it
immediately to the extreme left side of the spin chain. This amounts to a
translation by unit on the asymptotic wavefunctions of the excitations [3, 4]

$|\psi(u)\mathcal{Z}\rangle=\frac{\chi^{+}(u)}{\chi^{-}(u)}|\mathcal{Z}\psi(u)\rangle.$
(65)

Once on the left side of the chain, it can be ignored for the rest of the
computation. The result is an extra factor in the matrix elements
corresponding to the product of momentas $\frac{x^{+}(z)}{x^{-}(z)}$ for all
fermions to the left of those being scattered. The result are the cumulative
$\mathcal{Z}$ marker factors

$\displaystyle\phi_{\mathcal{Z}}(v_{i},u_{j})=\prod_{a=1}^{i-1}$
$\displaystyle\prod_{b=j+1}^{J_{1}}\frac{x^{+}(v_{a})}{x^{-}(v_{a})}\frac{x^{-}(u_{b})}{x^{+}(u_{b})},\qquad\phi_{\mathcal{Z}}(v_{i},w_{j})=\prod_{a=1}^{i-1}\prod_{b=k+1}^{J_{3}}\frac{x^{+}(v_{a})}{x^{-}(v_{a})}\frac{x^{+}(w_{b})}{x^{-}(w_{b})},\qquad$
$\displaystyle\phi_{\mathcal{Z}}(u_{j},w_{k})=\prod_{a=j+1}^{J_{1}}\prod_{b=1}^{J_{2}}\prod_{c=k+1}^{J_{3}}\frac{x^{-}(u_{a})}{x^{+}(u_{a})}\frac{x^{+}(v_{b})}{x^{-}(v_{b})}\frac{x^{+}(w_{c})}{x^{-}(w_{c})}.\qquad$

in equation (3).

Once the $\mathcal{Z}$ marker is removed in this way, we proceed with the
remaining scatterings. Note that the $x(v)$, $x(w)$ factors are inverted with
respect to the $\chi(u)$ factors due to the crossed kinematics, equation (5).

In the string frame the length of the spin-chain is well defined non-
perturbatively, corresponding to the $R$-charge of the dual single-trace
operator. As a consequence, the S-matrix in the string frame does not produce
$\mathcal{Z}$ markers and non-perturbative crossing transformations are
simple. For a PSU$(2|2)^{2}$ bifundamental magnon excitation
$\eta^{A\dot{B}}(z)$ crossing in the string frame simply amounts to

$\eta^{A\dot{B}}(z)\xrightarrow{\circlearrowright}-\eta^{B\dot{A}}(z^{\circlearrowright}),\qquad\eta^{A\dot{B}}(z)\xrightarrow{\circlearrowleft}-\eta^{B\dot{A}}(z^{\circlearrowleft}).$
(66)

Crossing in the spin-chain frame is in general complicated, requiring one to
map the spin-chain frame magnons to the string frame, use (66), and convert
back. For our purposes the procedure is trivial, since in the
SL$(2,\mathbb{R})$ sector we just need to cross covariant derivatives and
$\mathcal{D}_{\text{spin-
chain}}^{\alpha\dot{\alpha}}=\mathcal{D}_{\text{string}}^{\alpha\dot{\alpha}}$.

##  Appendix C $\sigma$ crossing

The crossing transformations are implemented through analytic continuation on
the rapidities around the $u=\pm\frac{i}{2}+2g$ branch points. The S-matrix
elements (63-64) have simple transformation rules under these monodromies, as
follows from (4). Clockwise $z^{\circlearrowright}$ and anti-clockwise
$z^{\circlearrowleft}$ crossings are equivalent for these factors.

The dynamical factor $h(z,z^{\prime})$, on the other hand, transforms non-
trivially [1]. The necessary formulae can be derived as a consequence of the
crossing equation for the BES factor,

$\sigma(z^{\circlearrowright},z^{\prime})\sigma(z,z^{\prime})=\frac{(1-1/x^{+}y^{+})(1-x^{-}/y^{+})}{(1-x^{-}/y^{-})(1-1/x^{+}y^{-})}$
(67)

combined with unitarity, $\sigma(z,z^{\prime})\sigma(z^{\prime},z)=1$. We use
$x$ and $y$ to denote the Zhukovsky variable associated to $z$ and
$z^{\prime}$ respectively. Some of the most useful equations are

$\displaystyle h(z^{\circlearrowright},z^{\prime})$
$\displaystyle=h(z,{z^{\prime}}^{\circlearrowleft})=\frac{1-1/x^{+}y^{-}}{1-1/x^{-}y^{+}}\sigma(z,z^{\prime})$
(68) $\displaystyle h(z^{\circlearrowright},{z^{\prime}}^{\circlearrowleft})$
$\displaystyle=\frac{y^{-}-x^{+}}{y^{-}-x^{-}}\frac{1-1/y^{+}x^{+}}{1-1/y^{-}x^{+}}\frac{1}{\sigma(z,z^{\prime})}$
(69)

which in particular imply the simplified unitarity equations referred to in
figure 4.

##  Appendix D Equal spins operators

When two or more operators have the same spin the hexagon form factors can
have off-shell singularities which display an order of limits issue when going
on-shell [28].

For two spinning operators (consider $J_{1}=J_{2}=J$ and $J_{3}=0$), this can
be easily avoided via two operations. First we cross the $v$ rapidities twice
$v\to v^{\pm 4\gamma}$, which makes the matrix part of the hexagon form factor
invariant and makes the kinematic pole of the diagonal limit appear only in
the dynamical part. Second we use Bethe equations in the splitting factors of
the $u$ particles

$\omega_{\ell_{13}}(a,\bar{a})=(-1)^{\bar{a}}\prod_{u_{j}\in\bar{a}}e^{-ip(u_{j})\ell_{12}}\prod_{\begin{subarray}{c}{u_{i}\in
a}\\\ {i<j}\end{subarray}}S^{-1}(u_{j},u_{i})$ (70)

which eliminates $\ell_{13}$ for $\ell_{12}$. As worked out in [28], after
summing over partitions the diagonal poles cancel resulting in the well-
defined off-shell object.

Turning on the third operator, provided that it is not equal to the other two,
does not introduce new poles, so again we have well-defined off-shell object.
However, for three spinning operators of equal spins ($J_{1}=J_{2}=J_{3}=J$)
we cannot choose crossing and splitting factors in such a way that all the
sets of particles satisfy the two criteria above. In other words, there is no
choice of crossing and weights for the excitations $w$ that make the glued
hexagon free of off-shell poles. However when going on-shell these poles must
cancel with zeros coming from the matrix part and yield the physical three-
point function.

We were not able to come with a correct prescription for the glued hexagons
that deals with the order of limit issue, therefore we avoided equal three
spinning operators such as the seemingly harmless $J_{1}=J_{2}=J_{3}=2$ case.
We believe that understanding the correct prescription is another interesting
question for the hexagon formalism.

##  Appendix E From C to H’s

In this section we explain how to express the structure constant in terms of
hexagon components. Our starting point is (31), which we rewrite here as

$\displaystyle\sum_{\ell_{1},\ell_{2},\ell_{3}}C^{J_{1},J_{2},J_{3}}_{\ell_{1},\ell_{2},\ell_{3}}\left(\frac{\langle
1,1\rangle^{J_{1}-\ell_{2}-\ell_{3}}\langle
2,2\rangle^{J_{2}-\ell_{1}-\ell_{3}}\langle
3,3\rangle^{J_{3}-\ell_{1}-\ell_{2}}}{\langle 2,3\rangle^{-\ell_{1}}\langle
3,2\rangle^{-\ell_{1}}\langle 1,3\rangle^{-\ell_{2}}\langle
3,1\rangle^{-\ell_{2}}\langle 1,2\rangle^{-\ell_{3}}\langle
2,1\rangle^{-\ell_{3}}}\right)=$
$\displaystyle=L_{1,\vec{\alpha}_{1}}L_{2,\vec{\alpha}_{2}}L_{3,\vec{\alpha}_{3}}R_{1,\vec{\beta}_{1}}R_{2,\vec{\beta}_{2}}R_{3,\vec{\beta}_{3}}\mathcal{H}_{G}^{\vec{\alpha}_{1}\vec{\beta}_{1};\vec{\alpha}_{2}\vec{\beta}_{2};\vec{\alpha}_{3}\vec{\beta}_{3}}(\textbf{u},\textbf{v},\textbf{w})$
(71)

where $\mathcal{H}_{G}$ are the glued hexagons.

By expanding in components and taking derivatives with respect to the spinors
in both sides of the expression above we can construct a matrix $M$ that
writes the polarized glued hexagons in terms of structure constants

$\mathcal{H}_{G}^{(\vec{\alpha}_{1}\vec{\beta}_{1});(\vec{\alpha}_{2}\vec{\beta}_{2});(\vec{\alpha}_{3}\vec{\beta}_{3})}=M^{(\vec{\alpha}_{1}\vec{\beta}_{1});(\vec{\alpha}_{2}\vec{\beta}_{2});(\vec{\alpha}_{3}\vec{\beta}_{3})}_{\ell_{1},\ell_{2},\ell_{3}}C^{J_{1},J_{2},J_{3}}_{\ell_{1},\ell_{2},\ell_{3}}\,,$
(72)

where we emphasize that the indices of $\mathcal{H}_{G}$ are completely
symmetrized, in accordance with the symmetric traceless nature of the
operators we are considering.

Notice that the rectangular matrix $M$ has linearly independent columns, so
that $M^{T}M$ is invertible (since it is a grammiam matrix of linearly
independent vectors). Therefore we can act on the LHS of equation (72) with
the left inverse $M^{+}=(M^{T}M)^{-1}M^{T}$ and find the inverted relation

$C_{J_{1},J_{2},J_{3}}^{\ell_{1},\ell_{2},\ell_{3}}=\left(M_{(\vec{\alpha}_{1}\vec{\beta}_{1});(\vec{\alpha}_{2}\vec{\beta}_{2});(\vec{\alpha}_{3}\vec{\beta}_{3})}^{\ell_{1},\ell_{2},\ell_{3}}\right)^{+}\mathcal{H}_{G}^{(\vec{\alpha}_{1}\vec{\beta}_{1});(\vec{\alpha}_{2}\vec{\beta}_{2});(\vec{\alpha}_{3}\vec{\beta}_{3})}$
(73)

which writes the structure constants as combinations of hexagons. Note also
that the $O(3)$ invariance implies several identities between the hexagon
components. These identities are simply the vanishing of the null vectors of
$M^{+}$.

##  Appendix F Abelian $C^{\bullet\bullet\bullet}$ and Pfaffians

In section (3.3) we presented a determinant formula for structure constants of
three non protected spinning operators polarized so that two are parallel and
orthogonal to the third. Here we make explicit some of the formulas and detail
some steps.

First we discuss the pfaffian identity. As derived in [12],

$\sum_{\bar{a}\subset\mathbf{u}}(-1)^{|\bar{a}|}w^{\prime}(u_{\bar{a}})H(u_{\bar{a}},u_{\bar{a}})=pf(I-\mathcal{K})_{2J_{1}\times
2J_{1}}.$

where the matrix $\mathcal{K}$ is defined as

$\mathcal{K}=\begin{pmatrix}K_{11}(u,u)&K_{12}(u,u)\\\
K_{21}(u,u)&K_{22}(u,u)\end{pmatrix}$ (74)

with

$\displaystyle K_{11}(a,b)_{ij}\equiv
g^{a}_{i}/k(x^{+}(a_{i}),x^{-}(b_{j})),\qquad
K_{12}(a,b)_{ij}\equiv-g^{a}_{i}k(x^{+}(a_{i}),x^{+}(b_{j}))/(x^{+}(a_{i})x^{+}(b_{j}))^{2},$
$\displaystyle K_{21}(a,b)_{ij}\equiv
g^{a}_{i}k(x^{-}(a_{i}),x^{-}(b_{j}))/(x^{-}(a_{i})x^{-}(b_{j}))^{2},\qquad
K_{22}(a,b)_{ij}\equiv-g^{a}_{i}/k(x^{-}(a_{i}),x^{+}(b_{j})),$

and where

$g^{u}_{i}=k(x^{+}(u_{i}),x^{-}(u_{i}))\omega^{\prime}(u_{i}),\qquad
k(x,y)=\frac{x-y}{1-1/xy}.$ (75)

As previously emphasized, this formula holds for any factorized function of
the rapidities $w^{\prime}(u_{\bar{a}})=\prod_{i\in\bar{a}}w^{\prime}(u_{i})$.

The next point we would like to clarify in this appendix is the sequence of
manipulations leading to the pfaffian representation of the blue terms. The
starting point is equation (40) which reads

$\displaystyle{\color[rgb]{0,0,1}{\text{blue}}}=\sum_{\begin{subarray}{c}b\cup\bar{b}\subset\mathbf{v}\\\
c\cup\bar{c}\subset\mathbf{w}\end{subarray}}(-1)^{|\bar{b}|+|\bar{c}|}e^{v}_{\bar{b}}e^{w}_{\bar{c}}H(v_{\bar{b}},v_{\bar{b}})H(w_{\bar{c}},w_{\bar{c}})h(v^{\circlearrowright}_{b},\hat{u})h(\hat{u},w^{\circlearrowleft}_{c})h(v^{\circlearrowright}_{b},w^{\circlearrowleft}_{c})h(w^{\circlearrowright}_{\bar{c}},v^{\circlearrowleft}_{\bar{b}}).$

We then write

$h(v^{\circlearrowright}_{b},w^{\circlearrowleft}_{c})=\frac{h(\hat{v}^{\circlearrowright},\hat{w}^{\circlearrowleft})h(v^{\circlearrowright}_{\bar{b}},w^{\circlearrowleft}_{\bar{c}})}{h(v^{\circlearrowright}_{\bar{b}},w^{\circlearrowleft}_{\bar{c}})h(v^{\circlearrowright}_{\bar{b}},w^{\circlearrowleft}_{c})h(v^{\circlearrowright}_{b},w^{\circlearrowleft}_{\bar{c}})h(v^{\circlearrowright}_{\bar{b}},w^{\circlearrowleft}_{\bar{c}})},\hskip
8.5359pth(v^{\circlearrowright}_{b},\hat{u})=\frac{h(\hat{v}^{\circlearrowright},\hat{u})}{h(v^{\circlearrowright}_{\bar{b}},\hat{u})},\hskip
8.5359pth(\hat{u},w^{\circlearrowleft}_{c})=\frac{h(\hat{u},\hat{w}^{\circlearrowleft})}{h(\hat{u},w^{\circlearrowleft}_{\bar{c}})}$

to end up with

$\displaystyle{\color[rgb]{0,0,1}{\text{blue}}}=$ $\displaystyle
h(\hat{v}^{\circlearrowright},\hat{w}^{\circlearrowleft})h(\hat{v}^{\circlearrowright},\hat{u})h(\hat{u},\hat{w}^{\circlearrowleft})\sum_{\begin{subarray}{c}b\cup\bar{b}\subset\mathbf{v}\\\
c\cup\bar{c}\subset\mathbf{w}\end{subarray}}(-1)^{|\bar{b}|+|\bar{c}|}\underbrace{\left(\frac{e^{v}_{\bar{b}}}{h(v^{\circlearrowright}_{\bar{b}},\hat{w}^{\circlearrowleft})h(v^{\circlearrowright}_{\bar{b}},\hat{u})}\right)}_{\equiv\omega^{\prime}(v^{\circlearrowright}_{\bar{b}})}\underbrace{\left(\frac{e^{w}_{\bar{c}}}{h(\hat{u},w^{\circlearrowleft}_{\bar{c}})h(\hat{v}^{\circlearrowright},w^{\circlearrowleft}_{\bar{c}})}\right)}_{\equiv\omega^{\prime}(w_{\bar{c}})}$
$\displaystyle\times
H(v_{\bar{b}},v_{\bar{b}})H(w_{\bar{c}},w_{\bar{c}})h(v^{\circlearrowright}_{\bar{b}},w^{\circlearrowleft}_{\bar{c}})h(w^{\circlearrowright}_{\bar{c}},v^{\circlearrowleft}_{\bar{b}}).$

Here, as before, we interpret the effect of $O_{1}$ on operators $O_{2}$ and
$O_{3}$ as a background that corrects their propagation but does not affect
their interactions. Next, we use the identity
$h(v^{\circlearrowright}_{\bar{b}},w^{\circlearrowleft}_{\bar{c}})h(w^{\circlearrowright}_{\bar{c}},v^{\circlearrowleft}_{\bar{b}})=h(v^{\circlearrowright}_{\bar{b}},w_{\bar{c}})h(w_{\bar{c}},v^{\circlearrowleft}_{\bar{b}})$
to obtain the result from the main text (41),

${\color[rgb]{0,0,1}{\text{blue}}}=h(\hat{v}^{\circlearrowright},\hat{w}^{\circlearrowleft})h(\hat{v}^{\circlearrowright},\hat{u})h(\hat{u},\hat{w}^{\circlearrowleft})\sum_{\begin{subarray}{c}b\cup\bar{b}\subset\mathbf{v}\\\
c\cup\bar{c}\subset\mathbf{w}\end{subarray}}(-1)^{|\bar{b}|+|\bar{c}|}\omega^{\prime}(v_{\bar{b}}^{\circlearrowright})\omega^{\prime}(w_{\bar{c}})H(v_{\bar{b}}^{\circlearrowright},v_{\bar{b}}^{\circlearrowright})H(w_{\bar{c}},w_{\bar{c}})H(v^{\circlearrowright}_{\bar{b}},w_{\bar{c}}).$

The next step is to recognize the sum over partitions for the effective
operator $O_{2}\cup O_{3}$ as a pfaffian,

$\sum_{d\cup\bar{d}\subset\mathbf{z}}(-1)^{|\bar{d}|}\omega^{\prime}(z_{\bar{d}})H(z_{\bar{d}},z_{\bar{d}})=pf(I-\mathcal{K^{\prime}})_{2(J_{2}+J_{3})\times
2(J_{2}+J_{3})}.$

Comparing with the definition $(\ref{Ksdef})$, $\mathcal{K^{\prime}}$ is
defined in terms of the functions $K_{x,y}(a,b)$ as

$\displaystyle\mathcal{K}^{\prime}=-\begin{pmatrix}K_{11}(v^{\circlearrowright},v^{\circlearrowright})&K_{12}(v^{\circlearrowright},v^{\circlearrowright})&K_{11}(v^{\circlearrowright},w)&K_{12}(v^{\circlearrowright},w)\\\
K_{21}(v^{\circlearrowright},v^{\circlearrowright})&K_{22}(v^{\circlearrowright},v^{\circlearrowright})&K_{21}(v^{\circlearrowright},w)&K_{22}(v^{\circlearrowright},w)\\\
K_{11}(w,v^{\circlearrowright})&K_{12}(w,v^{\circlearrowright})&K_{11}(w,w)&K_{12}(w,w)\\\
K_{21}(w,v^{\circlearrowright})&K_{22}(w,v^{\circlearrowright})&K_{21}(w,w)&K_{22}(w,w)\\\
\end{pmatrix}.$

where
$g^{v^{\circlearrowright}}_{i}=k(x^{+}(v^{\circlearrowright}_{i}),x^{-}(v^{\circlearrowright}_{i}))\omega^{\prime}(v_{i}^{\circlearrowright})$
and $g^{w}_{i}=k(x^{+}(w_{i}),x^{-}(w_{i}))\omega^{\prime}(w_{i})$.

A last comment regards the choice of splitting factors in (39). There we chose
to cross all three sets of magnons from the front to the back hexagon through
the right boundary of the cut chain. Alternative pfaffian representations can
be obtained in an analogous manner for all other possible choices161616e.g.
left-left-left, right-right-left, etc. of splitting factor provided one
replaces (39) by the appropriate expression. The only difference is the final
result is the replacement of the original splitting factors $e^{a}_{\bar{x}}$
by their left alternatives in the pfaffian formula. As usual, the expressions
differ off-shell but reproduce the same structure constants.

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