aainstitutetext: School of Physics and Astronomy, Tel Aviv University, Ramat
Aviv 69978, Israelbbinstitutetext: Max-Planck-Institut für Physik, Werner-
Heisenberg-Institut, Föhringer Ring 6, 80805 München, Germanyccinstitutetext:
Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100
Copenhagen Ø, Denmark

# An Operator Product Expansion for Form Factors III.
Finite Coupling and Multi-Particle Contributions

Amit Sever b Alexander G. Tumanov c Matthias Wilhelm

###### Abstract

Form factors in planar $\mathcal{N}=4$ super-Yang-Mills theory have a dual
description in terms of periodic Wilson loops. This duality maps the multi-
collinear expansion of the former to an operator product expansion of the
latter. The coefficients of this expansion are decomposed in terms of several
elementary building blocks and can be determined at finite ’t Hooft coupling
using bootstrap and integrability techniques. Some of these building blocks
are known from an analogous expansion of scattering amplitudes. In addition to
these, the expansion for form factors includes a new type of building block,
called form factor transitions, that encode information about the local
operator. In the present paper, we consider the form factor of the chiral part
of the stress-tensor supermultiplet. We bootstrap the corresponding form
factor transitions of two-particle flux-tube states and use them to predict
the leading term in the collinear expansion at finite coupling. The
transitions we find can be expressed in terms of a quantity that previously
appeared in a seemingly unrelated context, namely the octagon kernel. Lastly,
we use a factorized ansatz to determine the multi-particle form factor
transitions at finite coupling, which we use to predict the first subleading
term in the collinear expansion. A perfect match is found between our
predictions and the available perturbative data.

††preprint: MPP-2021-213

##  1 Introduction

Form Factors (FFs) are fundamental quantities in Quantum Field Theory (QFT)
that exhibit traits of both scattering amplitudes and correlation functions.
They describe the amplitude of a state created by a local operator to decay
into an $n$-point asymptotic state. In the maximally supersymmetric Yang-Mills
theory in four dimensions ($\mathcal{N}=4$ SYM theory), they have been studied
using modern on-shell methods (see, for instance, the review Yang:2019vag ) as
well as integrability at weak coupling Frassek:2015rka  and strong coupling
Maldacena:2010kp ; Gao:2013dza .

In $\mathcal{N}=4$ SYM theory, planar $n$-point form factors have a dual
description in terms of a certain type of periodic null polygonal Wilson loops
${\cal W}_{n}$ with $n$ unique edges, which we also refer to as wrapped
polygons Alday:2007he ; Maldacena:2010kp ; Brandhuber:2010ad ; Ben-
Israel:2018ckc ; Bianchi:2018rrj . Based on this duality, in Sever:2020jjx ;
Sever:2021nsq  we have developed the form factor operator product expansion
(FFOPE), used to compute FFs in planar ${\cal N}=4$ SYM theory non-
perturbatively. In this approach, a generic wrapped polygon is broken into a
sequence of pentagon transitions ${\cal P}$ Basso:2013vsa ; Basso:2013aha ;
Basso:2014koa ; Basso:2014nra ; Basso:2014hfa ; Basso:2015rta ; Basso:2015uxa
; Belitsky:2014sla ; Belitsky:2014lta ; Belitsky:2016vyq . The sequence ends
with a form factor transition ${\cal F}$ that encodes the information about
the local operator,

${\cal W}_{n}=\langle{\cal
F}|e^{-H\tau_{n-2}+iP\sigma_{n-2}+iJ\phi_{n-2}}{\cal P}\dots\cal
Pe^{-H\tau_{1}+iP\sigma_{1}+iJ\phi_{1}}{\cal P}|0\rangle\,.$ (1)

Here, $H,P,J$ are the Gubser-Klebanov-Polyakov (GKP) flux-tube Hamiltonian,
momentum and angular momentum operators, respectively, and
$\tau_{i},\sigma_{i},\phi_{i}$ are the cross-ratios that parametrize the
polygon, with $\phi_{n-2}=0$; see Sever:2020jjx  for more details.

In this paper, we determine the FF transition of the chiral part of the
stress-tensor supermultiplet at any value of the ’t Hooft coupling. Together
with the known spectrum of the GKP excitations Basso:2010in  and the pentagon
transitions, this enables one to compute form factors at finite ’t Hooft
coupling as an expansion around the large-$\tau_{i}$ multi-collinear limit.

We determined the FF transition by solving a set of bootstrap axioms these
objects satisfy Sever:2020jjx . To uniquely fix the solution of the bootstrap
axioms for the two-particle transitions, we use the Born-level two-particle
transitions that we constructed in Sever:2021nsq  as well as a fruitful
interplay with the perturbative bootstrap Dixon:2020bbt ;
PerturbativeBootstrap2 . Interestingly, we find that the FF transition for
scalars is defined in terms of the so-called octagon kernel. This kernel has
previously occurred in the study of the origin of the six-gluon amplitude
Basso:2020xts , as well as for the octagon form factor Coronado:2018ypq ;
Coronado:2018cxj ; Kostov:2019stn ; Kostov:2019auq ; Belitsky:2019fan ;
Bargheer:2019exp . The solutions for multi-particle GKP states are determined
from the two-particle ones via a simple factorized ansatz, up to a coupling-
independent matrix part that we explicitly construct for four particles.

This paper is structured as follows. In section 2, we review the set of axioms
that the FF transitions obey, as well as discuss some immediate consequences
of them. Based on these axioms, in section 3 we bootstrap the finite-coupling
expressions for the FF transitions of two particles. In section 4, we
bootstrap the FF transitions for more than two particles. We end in section 5
with conclusions and an outlook. Six appendices include some calculations
needed to demonstrate the validity of our claims.

##  2 Bootstrap axioms

Figure 2.1: The FF transition is defined as the amplitude of the two-sided
wrapped polygon to absorb a GKP state. It is normalized in such a way that the
FF transition for the GKP vacuum is equal to one.

The FF transitions are subject to a set axioms that follow from their
definition in figure 2.1, see Sever:2020jjx  for further details. To set the
ground for the FF transition bootstrap, we summarize them here.

Watson

Reordering two adjacent excitations within a GKP eigenstate is equivalent to
acting on it with the S-matrix, which is known at any value of the ’t Hooft
coupling $\lambda$ Basso:2013pxa . This property is inherited by the FF
transition:

$F_{\dots\,a_{j}a_{j+1}\,\dots}(\dots,u_{j},u_{j+1},\dots)=\pm\,S(u_{j},u_{j+1})_{a_{j}a_{j+1}}^{b_{j+1}b_{j}}F_{\dots\,b_{j+1}b_{j}\,\dots}(\dots,u_{j+1},u_{j},\dots)\,,$
(2)

where the indices $b_{j},b_{j+1}$ run over all flux-tube excitations, and the
minus sign corresponds to the case where the two excitations are fermions,
otherwise it is plus.

Singlet

The two-sided wrapped polygon preserves the $U(1)_{\phi}\times SU(4)_{R}$
subgroup of the superconformal symmetry. Here, $U(1)_{\phi}$ parametrizes the
rotations in the plane transverse to the polygon, while $SU(4)_{R}$ is the
R-symmetry group. This implies that

$F_{a_{1}\,\dots\,a_{n}}(u_{1},\ldots,u_{n})={\cal M}_{a_{1}}^{b_{1}}\dots\cal
M_{a_{n}}^{b_{n}}\,F_{b_{1}\,\dots\,b_{n}}(u_{1},\ldots,u_{n})\,,$ (3)

with ${\cal M}\in U(1)_{\phi}\times SU(4)_{R}$. As a result, FF transitions of
charged states are equal to zero.

The $U(1)_{\phi}\times SU(4)_{R}$ singlet states are easy to classify. All of
the single-particle states are charged under at least one of the two
symmetries. Therefore, the simplest singlet states are two-particle ones,
consisting of two conjugate scalars $\phi\bar{\phi}$, two conjugate fermions
$\psi\bar{\psi}$, or two conjugate gluon bound states $F_{n}F_{-n}$. All
singlet states with more than two excitations are built from products of these
elemental two-particle ones.

Reflection

The two-sided wrapped polygon is invariant under spacetime reflections. As a
result, flipping the order of the GKP excitations and the signs of their
momenta is a symmetry of the FF transition:

$F_{\bf a}({\bf u})=F_{\bar{\bf a}}(\bar{\bf u})\,.$ (4)

Here, we use the abbreviation ${\bf a}=\\{a_{1},\dots,a_{n}\\}$, ${\bf
u}=\\{u_{1},\ldots,u_{n}\\}$, $\bar{\bf a}=\\{a_{n},\ldots,a_{1}\\}$ and
$\bar{\textbf{u}}=\\{-u_{n},\dots,-u_{1}\\}$.

Crossing

Applying two mirror transformations to the first excitation results in it
being transferred to the last position:

$F_{a_{1}a_{2}\dots a_{n}}(u_{1}^{2\gamma},u_{2},\dots,u_{n})=F_{a_{2}\dots
a_{n}a_{1}}(u_{2},\dots,u_{n},u_{1})\,.$ (5)

Note that this transformation involves an analytic continuation which crosses
certain cuts that shrink to zero in the perturbative regime. Its path depends
on whether the $u_{1}$ excitation is a scalar, fermion or gluon; see
Basso:2010in  for details.

Square limit

The FF transition has a factorization pole at the kinematical point in which
the momenta of the first and last excitations become equal,

$\displaystyle\lim_{u_{1}\to u_{n}}F_{\bf a}({\bf u})$
$\displaystyle=\frac{-\,i\delta_{a_{n}\bar{a}_{1}}}{\mu_{a_{1}}(u_{1})}\frac{F_{a_{2}\,\dots\,a_{n-1}}(u_{2},\dots,u_{n-1})}{u_{1}-u_{n}-i\epsilon}$
(6)
$\displaystyle\pm\frac{-\,i\delta_{b_{n}\bar{b}_{1}}}{\mu_{b_{1}}(u_{1})}\frac{F_{b_{2}\,\dots\,b_{n-1}}(u_{2},\dots,u_{n-1})}{u_{n}-u_{1}-i\epsilon}\times$
$\displaystyle\qquad\quad\times\Big{(}\big{[}S_{12}\,S_{13}\ldots
S_{1n-1}\big{]}\cdot S_{1n}\cdot\big{[}S_{n-1n}\dots
S_{3n}\,S_{2n}\big{]}\Big{)}_{\bf a}{}^{\bf b}\,,$

where $S_{ij}\equiv S(u_{i},u_{j})$ and the square measure $\mu_{a}(u)$ is
known at any value of the coupling Basso:2013vsa ; Basso:2013aha ;
Basso:2014koa ; Basso:2014nra . Here, the plus sign corresponds to the case of
a bosonic $u_{1}$ excitation and minus to the case of a fermionic $u_{1}$
excitation. The first (second) term in this equation originates from the part
of the GKP wave function in which the leftmost excitation is $u_{1}$ ($u_{n}$)
and the rightmost excitation is $u_{n}$ ($u_{1}$). The two terms are related
by the S-matrix prefactor, in accordance with the Watson axiom.

##  3 Two-particle form factor transitions

In this section, we bootstrap the two-particle FF transitions at finite
coupling using the axioms presented in the previous section as well as
perturbative data.

###  3.1 Gluons and Fermions

Let us consider the consequences of the square-limit axiom for the FF
transitions of $\psi\bar{\psi}$ and $F_{n}F_{-n}$ first. Because the
S-matrices for these excitations satisfy $S_{F_{n}F_{-m}}(u,u)=\delta_{nm}$
and $S_{\psi_{A}\bar{\psi}^{B}}(u,u)=-\,\delta_{A}^{B}$, the two terms in (6)
combine into a delta function:

$\lim_{u\to
v}F_{\psi_{A}\bar{\psi}^{B}}(u,v)=\frac{2\pi\delta_{A}^{B}}{\mu_{\psi}(u)}\,\delta(u-v)\,,\qquad\lim_{u\to
v}F_{F_{n}F_{-n}}(u,v)=\frac{2\pi}{\mu_{F_{n}}(u)}\,\delta(u-v)\,.$ (7)

In Sever:2021nsq  we have computed the transitions for $\psi\bar{\psi}$ and
$F_{1}F_{-1}$ at Born level and found that they are given solely by their
square-limit expressions. It is therefore natural to assume that the
transitions for $\psi\bar{\psi}$ and $F_{n}F_{-n}$ are supported only on their
square limit. In other words, these excitations cannot tell the difference
between the wrapped two-sided polygon and the square:

$F_{\psi_{A}\bar{\psi}^{B}}(u,v)=\frac{2\pi\delta_{A}^{B}}{\mu_{\psi}(u)}\,\delta(u-v)\,,\qquad
F_{F_{n}F_{-n}}(u,v)=\frac{2\pi}{\mu_{F_{n}}(u)}\,\delta(u-v)\,.$ (8)

It is easy to see that the transitions in (8) are consistent with the Watson
and reflection axioms.111The measures are symmetric under reflection and
crossing and are given in Basso:2013vsa ; Basso:2013aha ; Basso:2014koa ;
Basso:2014nra . The remaining crossing axiom is a little delicate to check.
The FF transitions are distributions, that should be inserted under an
integral over the rapidities. The crossing analytic continuation, $2\gamma$,
should thus be understood as taking place inside an integral. To be concrete,
we can realize a delta function as the difference between two poles, one above
and one below the contour of integration. Its contribution to the integral is
invariant under any analytic continuation of the path, and under $2\gamma$ in
particular.

Another supporting argument for the solution (8) comes from considering
transitions with more than two excitations. One of the celebrated
manifestations of integrability is the factorization of multi-particle
fundamental quantities such as the S-matrix and the pentagon transitions into
products of the two-particle ones. Likewise, we also expect the FF transitions
to factorize. The square-limit axiom, however, contains a product of
S-matrices in one of the two terms in (6), but not in the other. If this
product is not equal to $\pm 1$, then this fact would be inconsistent with a
factorized solution. The solution (8) forces the gluons and fermions into
pairs of conjugate states with the same rapidity. As explained in section 4,
this results in these pairs dropping out from the product of S-matrices in the
square limit of any other two excitations.222For the full consistency of a
factorized ansatz, the scalars should also drop out of the product of
S-matrices in (6). In section 4, we show that this is indeed the case.

Lastly, let us remark that the FF transition for the bound states $F_{n}$ and
$F_{-n}$ follow from those of $F_{1},F_{-1}$ and the factorized ansatz for
multi-particle transitions via so-called fusion, as we will elaborate on in
section 4.

###  3.2 Scalars

Let us now turn to the scalars. An immediate consequence of the singlet axiom
(3) is that only the singlet $SO(6)$ tensor structure is present333Note that
in Sever:2020jjx ; Sever:2021nsq  we do not write the flavor indices, as the
sum over them is already explicitly performed in all the expressions.

$F_{\phi^{i}\phi^{j}}(u,v)=F_{\phi\bar{\phi}}(u,v)\times\delta^{ij}\,.$ (9)

Consider now the square-limit axiom. Because $S_{\phi\bar{\phi}}(u,u)=-1$, it
takes the form

$\lim_{u\to
v}F_{\phi\bar{\phi}}(u,v)=\frac{2i}{\mu_{\phi}(u)}\frac{1}{u-v}\,,$ (10)

where the $i\epsilon$ in the square-limit axiom leads to a principle-part
prescription for the pole. Hence, unlike for gluons and fermions, the scalar
transition cannot be localized at equal rapidities. Indeed, in Sever:2020jjx ;
Sever:2021nsq , we found that at Born level it is given by

$F_{\phi\bar{\phi}}(u,v)=-\,\frac{4}{g^{2}\,(u-v-2i)\,(u-v-i)}\,\frac{\Gamma\left(iu-
iv\right)}{\Gamma\left(\frac{1}{2}+iu\right)\Gamma\left(\frac{1}{2}-iv\right)}+\mathcal{O}(g^{0})\,,$
(11)

where $g^{2}=\frac{\lambda}{16\pi^{2}}$. In this subsection, we construct a
minimal solution to the bootstrap axioms of $F_{\phi\bar{\phi}}$, following
similar steps to those taken for the bootstrap of the scalar pentagon
transition in Basso:2013aha .

The Watson axiom (2) for $F_{\phi\bar{\phi}}$ takes the form

$F_{\phi\bar{\phi}}(u,v)=S_{\phi\bar{\phi}}(u,v)F_{\phi\bar{\phi}}(v,u)\,,$
(12)

where $S_{\phi\bar{\phi}}$ is the S-matrix in the singlet channel, see
Basso:2010in  and appendix A for a review. This relation can be rewritten as

$\frac{F_{\phi\bar{\phi}}(u,v)^{2}}{F_{\phi\bar{\phi}}(v,u)^{2}}=S_{\phi\bar{\phi}}^{2}(u,v)=\frac{S_{\phi\bar{\phi}}(u,v)}{S_{\phi\bar{\phi}}(v,u)}\,,$
(13)

where in the last step we used the unitarity of the S-matrix,
$S_{\phi\bar{\phi}}(u,v)S_{\phi\bar{\phi}}(v,u)=1$. It can be solved by

$F_{\phi\bar{\phi}}(u,v)^{2}=z(u,v)\,S_{\phi\bar{\phi}}(u,v)=\frac{(u-v+2i)(u-v+i)}{(u-v-2i)(u-v-i)}\,z(u,v)\,S_{\phi\phi}(u,v)\,,$
(14)

where $z(u,v)=z(v,u)$ is a symmetric function and in the last step we have
used the relation (75) between the scalar S-matrices in the symmetric channel
($S_{\phi\phi}$) and the singlet channel ($S_{\phi\bar{\phi}}$).

Inserting this form of the FF transition into the crossing axiom (5) yields a
crossing relation for the symmetric function $z(u,v)$:

$\frac{z(u,v)}{z(u^{2\gamma},v)}=\frac{(u-v+4i)(u-v+3i)(u-v+2i)}{(u-v-2i)(u-v-i)(u-v)}\frac{S_{\phi\phi}(u^{2\gamma},v)}{S_{\phi\phi}(v,u)}\,,$
(15)

where we used that for scalars the mirror analytic continuation is simply
$u^{\gamma}=u+i$. Next, we use crossing symmetry of the scalar S-matrix
Basso:2013aha ,

$S_{\phi\phi}(u^{2\gamma},v)\,S_{\phi\phi}(u,v)=\frac{u-v}{u-v+2i}\,,$ (16)

and unitarity of the S-matrix to find

$\frac{z(u,v)}{z(u^{2\gamma},v)}=\frac{(u-v+4i)(u-v+3i)}{(u-v-2i)(u-v-i)}\,.$
(17)

This can be solved via

$z(u,v)=w(u,v)\times\frac{1}{(u-v-2i)(u-v-i)(u-v)(u-v+i)(u-v+2i)}\,,$ (18)

where

$w(u,v)=w(u^{2\gamma},v)\,,\qquad w(u,v)=-\,w(v,u)\,.$ (19)

In terms of the function $w(u,v)$, the square-limit axiom reads

$\lim_{u\to v}w(u,v)=\frac{16}{\mu_{\phi}(u)^{2}}\frac{1}{u-v}\,.$ (20)

At leading order in the coupling, we find $w(u,v)$ by comparison with (11):

$w(u,v)=\frac{16}{\pi g^{4}}\,\frac{\cosh(\pi u)\cosh(\pi
v)}{\sinh(\pi(u-v))}+\mathcal{O}(g^{6})\,.$ (21)

Hence, a minimal finite-coupling solution for $w(u,v)$ that is consistent with
(19), (20), and (21) is

$w_{\text{min}}(u,v)=\frac{16\pi}{\mu_{\phi}(u)\mu_{\phi}(v)}\frac{1}{\sinh(\pi(u-v))}\,.$
(22)

As we will see later, this ansatz (22) does not reproduce the next-to-leading
order (NLO) data, though. We can parametrize the mismatch using a crossing-
invariant symmetric Castillejo-Dalitz-Dyson (CDD) factor

$w(u,v)=w_{\text{min}}(u,v)\times G(u,v)^{2}\,.$ (23)

It is subject to the constraints

$G(u,v)^{2}=G(v,u)^{2}\,,\qquad G(u,v)=1+{\cal O}(g)\,,\qquad
G(u^{2\gamma},v)^{2}=G(u,v)^{2}\,.$ (24)

In terms of the CCD factor $G$, the FF transition reads

$F_{\phi\bar{\phi}}(u,v)^{2}=\frac{S_{\phi\bar{\phi}}(u,v)}{((u-v)^{2}+4)((u-v)^{2}+1)}\,\frac{16\pi}{\mu_{\phi}(u)\mu_{\phi}(v)}\,\frac{G(u,v)^{2}}{(u-v)\sinh(\pi(u-v))}\,,$
(25)

with the finite-coupling expressions for the measure and the S-matrix given in
(78) and (76), respectively. Combining the constraints $G(u,v)^{2}=G(v,u)^{2}$
and $G(u,v)=1+{\cal O}(g)$, we see that $G(u,v)=G(v,u)$. We will later find
that $G(u^{2\gamma},v)=-\,G(u,v)$.444In contrast to the Watson relation,
crossing does not commute with taking the weak-coupling limit.

###  3.3 Determining the scalar CDD factor with perturbative data

We now use perturbative data to find a finite-coupling expression for the CCD
factor $G(u,v)=1+\mathcal{O}(g^{2})$ that remained undetermined in the
previous subsection. This perturbative data originates from the remainder
function of the three-point form factor, which has been calculated up to two-
loop order via unitarity Brandhuber:2012vm  and was subsequently bootstrapped
up to eight-loop order Dixon:2020bbt ; PerturbativeBootstrap2 .

While the two-loop unitarity calculation Brandhuber:2012vm  is naturally
independent, the perturbative bootstrap Dixon:2020bbt ; PerturbativeBootstrap2
made significant use of the FFOPE data, which in turn depends on the FF
transitions and thus the function $G$ that is to be determined in this
subsection. Before diving into the details of determining $G$, let us briefly
explain this interplay between the FFOPE and the perturbative bootstrap.

The NLO correction to $G$ can be determined by requiring it to reproduce the
$e^{-2\tau}\tau^{0}$ contribution to the large-$\tau$ expansion of the two-
loop remainder. As elaborated on in Sever:2020jjx ; Sever:2021nsq , the
operator product expansion of the three-point form factor consists of terms
proportional to $e^{-\tau E}$, with $E$ being the energy of the corresponding
GKP state, that is known at any value of the coupling Basso:2010in . The loop
corrections to $E$ are the only source of polynomial terms in $\tau$, which
means that the term proportional to $e^{-2\tau}\tau^{k}$ is guaranteed to have
at least $k$ powers of $g^{2}$. Because of this, the $\ell$-loop correction to
the $e^{-2\tau}\tau^{k}$ term only receives contributions from the FF
transition up to Nℓ-k-1LO. Thus, knowing $G$ at NkLO fully determines the
$e^{-2\tau}\tau^{\ell-k-1}$ corrections to the form factor at $\ell$-loop
order. This data can then be fed into the perturbative bootstrap to fully
determine the remainder function at $(k+1)$-loop order, from which the Nk+1LO
correction to $G$ can be fixed. This sequence is repeated to obtain the next
order correction, and so on. This interplay between the FFOPE and the
perturbative bootstrap is summarized in figure 3.1.

$\displaystyle G\text{ at
N}^{k}\text{LO}\xrightarrow{\text{FFOPE}}\begin{array}[]{c}e^{-2\tau}\tau^{\ell-k-1}\text{
term}\\\ \text{at $\ell$
loops}\end{array}\xrightarrow{\tiny\begin{array}[]{c}\text{perturbative}\\\
\text{bootstrap}\end{array}}\begin{array}[]{c}\text{$(k+1)$-loop}\\\
\text{remainder}\end{array}\longrightarrow G\text{ at N}^{k+1}\text{LO}$
Figure 3.1: Interplay between the FF transition bootstrap and the perturbative
bootstrap.

To determine the function $G(u,v)$, we first write an ansatz for it that is
built from $\Psi^{(n)}(\frac{1}{2}\pm iu)$ and $\Psi^{(n)}(\frac{1}{2}\pm iv)$
as well as products thereof, where
$\Psi^{(n)}(z)=\frac{d^{n+1}}{dz^{n+1}}\log\Gamma(z)$ is the polygamma
function of order $n$. These functions are the building blocks occurring in
the loop corrections to the pentagon transitions, the measures, the energy and
the momentum of the scalar, and it is thus natural to expect them to occur
also in the loop corrections to the FF transitions. Moreover, we observe that
when we assign weight $n+1$ to $\Psi^{(n)}$, only functions up to weight $m$
occur at order $g^{m}$ for quantities that start at order $g^{0}$. We thus
assume this property in the ansatz, since $G(u,v)=1+\mathcal{O}(g^{2})$. Next,
we impose $G(u,v)=G(v,u)$ due to the Watson axiom, $G(u,u)=1$ due to the
square-limit axiom and $G(u,v)=G(-v,-u)$ due to the reflection axiom. Note
that we cannot impose crossing at the perturbative level, since the branch cut
through which we are supposed to analytically continue closes at weak
coupling. The remaining coefficients in this ansatz can be fixed by comparing
the FFOPE prediction in terms of this ansatz with the data from the
perturbative bootstrap. This way, we found

$\displaystyle\log[$ $\displaystyle
G(u,v)]=\frac{\pi^{2}g^{2}}{2}\left(t(u)-t(v)\right)^{2}$ (26)
$\displaystyle+\frac{\pi^{4}g^{4}}{12}\left(t(u)-t(v)\right)^{2}\left[15\,t(u)^{2}+15\,t(v)^{2}+18\,t(u)t(v)-16\right]$
$\displaystyle+\frac{\pi^{6}g^{6}}{45}\left(t(u)-t(v)\right)^{2}\left[165\,t(u)^{4}+165\,t(v)^{4}+240\,t(u)^{3}t(v)+240\,t(u)t(v)^{3}\right.$
$\displaystyle\left.+\,270\,t(u)^{2}t(v)^{2}-300\,t(u)^{2}-300\,t(v)^{2}-360\,t(u)t(v)+136\right]+\mathcal{O}(g^{8})\,.$

where we used the shorthand notation $t(u)=\tanh(\pi u)$. Note that the
logarithm of $G(u,v)$ is a symmetric function of $\tanh(\pi u)$ and $\tanh(\pi
u)$, with an explicit factor of $(\tanh(\pi u)-\tanh(\pi u))^{2}$ at every
loop order. As such, it is not only satisfying the aforementioned constraints
but also $G(u+2i,v)=G(u,v)$.555This property looks like a perturbative
counterpart of crossing, but we are not aware of a relation between these
analytic continuations. We assume that this property continues to hold at any
order in perturbation theory.

In fact, we can make a finite-coupling conjecture for $G$ that reproduces (26)
by introducing building blocks quite similar to those occurring in the
S-matrices, pentagon transitions and measures.

Let us begin by reviewing the key building blocks occurring in the
construction of the aforementioned quantities, namely the Beisert-Eden-
Staudacher (BES) kernel, the source terms $\kappa_{n}$ and
$\tilde{\kappa}_{n}$ as well as the functions $f_{i}$ for $i=1,2,3,4$; see
also appendix A. The BES kernel can be written as Beisert:2006ez

$\mathbb{K}_{ij}=2j(-1)^{ij+j}\int\limits_{0}^{\infty}\frac{dt}{t}\frac{J_{i}(2gt)J_{j}(2gt)}{e^{t}-1}\,,$
(27)

where $J_{i}$ denotes a Bessel function. The functions $f_{1,2,3,4}$,
introduced in Basso:2013pxa , are constructed by contracting the source terms
$\kappa$ and $\tilde{\kappa}$,

$\displaystyle\kappa_{n}^{v}=-\int\limits_{0}^{\infty}\frac{dt}{t}J_{n}(2gt)\frac{\cos(vt)e^{t/2}-J_{0}(2gt)}{e^{t}-1}\,,\qquad\tilde{\kappa}_{n}^{v}=-\int\limits_{0}^{\infty}\frac{dt}{t}J_{n}(2gt)\frac{\sin(vt)e^{t/2}}{e^{t}-1}\,,$
(28)

with the BES kernel (27) in the following way:

$\displaystyle
f_{1}(u,v)=2\,\tilde{\kappa}^{u}\,\mathbb{Q}\,\frac{1}{1+\mathbb{K}}\,\kappa^{v}\,,\qquad
f_{2}(u,v)=2\,\kappa^{u}\,\mathbb{Q}\,\frac{1}{1+\mathbb{K}}\,\tilde{\kappa}^{v}\,,$
(29) $\displaystyle
f_{3}(u,v)=2\,\tilde{\kappa}^{u}\,\mathbb{Q}\,\frac{1}{1+\mathbb{K}}\,\tilde{\kappa}^{v}\,,\qquad
f_{4}(u,v)=2\,\kappa^{u}\,\mathbb{Q}\,\frac{1}{1+\mathbb{K}}\,\kappa^{v}\,,$

where $\mathbb{Q}_{ij}=(-1)^{i+1}\,\delta_{ij}\,i$. Note that at finite
coupling the source terms $\kappa$ and $\tilde{\kappa}$ are vectors with
infinitely many components, and $\mathbb{K}$ is an infinite matrix acting on
them. At any loop order at weak coupling, only finitely many of these
components are non-vanishing, making this representation particularly suitable
for a weak-coupling analysis.

Next, we define the even and odd parts of the source terms $\kappa_{n}$ and
$\tilde{\kappa}_{n}$ (28):

$\displaystyle\kappa_{n}^{v}$
$\displaystyle=\kappa_{+,n}^{v}+\kappa_{-,n}^{v}\,,$
$\displaystyle\tilde{\kappa}_{n}^{v}$
$\displaystyle=\tilde{\kappa}_{+,n}^{v}+\tilde{\kappa}_{-,n}^{v}\,,$ (30)

where the odd (even) part vanishes for even (odd) $n$. Since the weak-coupling
expansion of the source terms $\kappa_{+,n}^{v}$ and
$\tilde{\kappa}_{-,n}^{v}$ is periodic under $v\to v+2i$ and can be expressed
in terms of $\tanh(v\pi)$, $\kappa_{+,n}^{v}$ and $\tilde{\kappa}_{-,n}^{v}$
are natural building blocks for $G$, while $\kappa_{-,n}^{v}$ and
$\tilde{\kappa}_{+,n}^{v}$ are not. Moreover, the BES kernel (27) can be
generalized to a tilted BES kernel by splitting it into four blocks as
Basso:2020xts

$\mathbb{K}=\mathbb{K}(\pi/4)\,,\qquad\mathbb{K}(\alpha)=2\cos(\alpha)\begin{pmatrix}\cos(\alpha)\mathbb{K}_{\circ\circ}&\sin(\alpha)\mathbb{K}_{\circ\bullet}\\\
\sin(\alpha)\mathbb{K}_{\bullet\circ}&\cos(\alpha)\mathbb{K}_{\bullet\bullet}\end{pmatrix}\,.$
(31)

Here, $\mathbb{K}_{\circ\circ}$ is the overlap of odd Bessel functions with
odd Bessel functions, $\mathbb{K}_{\circ\bullet}$ is the overlap of odd Bessel
functions with even Bessel functions, etc. For $\alpha=\pi/4$, one recovers
the original BES kernel. For $\alpha=0$, the kernel does not mix between even
and odd parts and is therefore the natural kernel to consider for our problem.
It has previously occurred for the origin of the hexagon Basso:2020xts  as
well as the octagon Coronado:2018ypq ; Coronado:2018cxj ; Kostov:2019stn ;
Kostov:2019auq ; Belitsky:2019fan ; Bargheer:2019exp , and is referred to as
the octagon kernel. Using it, we now define functions $f_{5}$ and $f_{6}$ in
analogy to (29):666One might also define
$f_{7}(u,v)=2\,\kappa_{-}^{u}\,\mathbb{Q}\,\frac{1}{1+\mathbb{K}(0)}\,\kappa_{-}^{v}\,,\qquad
f_{8}(u,v)=2\,\tilde{\kappa}_{+}^{u}\,\mathbb{Q}\,\frac{1}{1+\mathbb{K}(0)}\,\tilde{\kappa}_{+}^{v}\,.$
As previously mentioned, they are not symmetric under $u\to u+2i$ and $v\to
v+2i$, and hence they do not play a role for the FF transition under
consideration. However, it is tempting to speculate that they could occur in
other quantities.

$f_{5}(u,v)=2\,\kappa_{+}^{u}\,\mathbb{Q}\,\frac{1}{1+\mathbb{K}(0)}\,\kappa_{+}^{v}\,,\qquad
f_{6}(u,v)=2\,\tilde{\kappa}_{-}^{u}\,\mathbb{Q}\,\frac{1}{1+\mathbb{K}(0)}\,\tilde{\kappa}_{-}^{v}\,.$
(32)

These functions are symmetric in $u$ and $v$ and vanish for $g=0$.

We can use the functions $f_{5}$ and $f_{6}$ to build $G(u,v)$ that matches
(26) as well as the weak-coupling data up to eight-loop order, while
satisfying all the constraints imposed on it:

$G(u,v)=\exp\left[2f_{5}(u,v)-f_{5}(u,u)-f_{5}(v,v)-2f_{6}(u,v)+f_{6}(u,u)+f_{6}(v,v)\right]\,.$
(33)

This solution is manifestly symmetric in $u$ and $v$ and equal to $1$ for
$g=0$ and for $u=v$. To fully prove its validity, in appendix C we show that
it also has the right behaviour under crossing, namely
$G(u^{2\gamma},v)=-\,G(u,v)$.

Inserting the various finite-coupling expressions into (25) and taking the
square root, we find the following expression for the scalar FF transition:

$\displaystyle F_{\phi\bar{\phi}}(u,v)=-\,$
$\displaystyle\frac{4\,\Gamma\left(i(u-v)\right)}{g^{2}\,(u-v-i)\,(u-v-2i)\,\Gamma\left(\frac{1}{2}+iu\right)\Gamma\left(\frac{1}{2}-iv\right)}$
$\displaystyle\times\exp\Big{[}$
$\displaystyle\int\limits_{0}^{\infty}\frac{dt}{t}\frac{J_{0}(2gt)^{2}-e^{t/2}[J_{0}(2gt)-1]\left(e^{-itu}+e^{itv}\right)-1}{e^{t}-1}$
(34) $\displaystyle-
if_{1}(u,v)+if_{2}(u,v)-\tfrac{1}{2}f_{3}(u,u)-\tfrac{1}{2}f_{3}(v,v)+\tfrac{1}{2}f_{4}(u,u)+\tfrac{1}{2}f_{4}(v,v)$
$\displaystyle+2f_{5}(u,v)-f_{5}(u,u)-f_{5}(v,v)-2f_{6}(u,v)+f_{6}(u,u)+f_{6}(v,v)\Big{]}\,,$

where the first line reproduces (11) at Born level and the second and third
line stem from the S-matrix (76) and the measure (78).

###  3.4 Strong coupling

The solution (33) matches highly non-trivial data up to eight loops. We now
show that it also agrees with the minimal solution to the bootstrap axioms at
strong coupling.777We thank B. Basso for valuable discussions and for sharing
his note on twist operators in the $O(N)$ $\sigma$-model.

At strong coupling, the scalars are the lightest excitations. They describe
the fluctuations of the GKP string on the sphere, and their dynamics are
governed by the $O(6)$ non-linear $\sigma$-model. This leads us to interpret
the two-scalar FF transition at strong coupling as a two-dimensional form
factor of an operator in the $O(6)$ $\sigma$-model (see Basso:2014jfa  for a
similar identification of the pentagon transition):

$\langle\Phi(0)|\theta_{1},\theta_{2}\rangle_{i_{1}i_{2}}\equiv\lim_{g\to\infty}\left.F_{\phi_{i}\phi_{j}}(u_{1},u_{2})\right|_{u_{i}=\frac{2}{\pi}\theta_{i}}\,.$
(35)

Here, $\theta_{i}$ are the hyperbolic rapidities parametrizing the incoming
scalars’ relativistic dispersion relation. The scalar indices are mapped to
the $O(6)$ polarizations of the two incoming scalar excitations.

Through (35), the bootstrap axioms of the FF transition are translated into
defining properties of the operator $\Phi$. Firstly, the singlet axiom
together with the relativistic symmetry of the $O(6)$ $\sigma$-model implies
that

$\langle\Phi(0)|\theta_{1},\theta_{2}\rangle_{i_{1}i_{2}}={\mathbb{F}}(\theta_{1}-\theta_{2})\,\delta_{i_{1}i_{2}}\,,$
(36)

where ${\mathbb{F}}(\theta)$ is the strong-coupling limit of
$\left.F_{\phi\bar{\phi}}(u,v)\right|_{u-v={2\over\pi}\theta}$. The crossing
axiom (5) now takes the form

${\mathbb{F}}(\theta+i\pi)={\mathbb{F}}(-\theta)\,.$ (37)

This implies that $\Phi(0)$ is a twist field which generates a conical defect
angle of $\pi$ around the origin.888This picture is also supported by the
minimal area solution dual to the $n$-gluon form factor Maldacena:2010kp ;
Gao:2013dza . In the collinear limit, the induced metric on the worldsheet
approaches $ds^{2}\simeq\left[P(z)\bar{P}(z)\right]^{1\over 4}dz\,d\bar{z}$,
with $P(z)={1\over z^{2}}\prod_{i=1}^{n-2}(z-z_{i})$. Due to the $1/z^{2}$
behavior near the origin, there is a conical deficit angle of $\pi$ that
corresponds to the FF transition. Similarly, around the points $z_{i}$ there
is a conical excess angle of $\pi/2$ that corresponds to the $n-2$ pentagon
transitions in (1). Secondly, the form factor ${\mathbb{F}}(\theta)$ has to
satisfy the Watson equation

${\mathbb{F}}(\theta)=S_{\textrm{singlet}}(\theta)\times{\mathbb{F}}(-\theta)\,,$
(38)

where the singlet S-matrix is given by Zamolodchikov:1977nu

$S_{\textrm{singlet}}(\theta)=\frac{\Gamma\left({\textstyle\frac{3}{2}}-i{\textstyle\frac{\theta}{2\pi}}\right)\Gamma\left({\textstyle\frac{5}{4}}-i{\textstyle\frac{\theta}{2\pi}}\right)\Gamma\left({\textstyle\frac{3}{4}}+i{\textstyle\frac{\theta}{2\pi}}\right)\Gamma\left(+\,i{\textstyle\frac{\theta}{2\pi}}\right)}{\Gamma\left({\textstyle\frac{3}{2}}+i{\textstyle\frac{\theta}{2\pi}}\right)\Gamma\left({\textstyle\frac{5}{4}}+i{\textstyle\frac{\theta}{2\pi}}\right)\Gamma\left({\textstyle\frac{3}{4}}-i{\textstyle\frac{\theta}{2\pi}}\right)\Gamma\left(-\,i{\textstyle\frac{\theta}{2\pi}}\right)}\,.$
(39)

It can also be obtained by taking the strong-coupling limit of
$S_{\phi\bar{\phi}}(u,v)$ from (76), with
$\theta={\textstyle\frac{\pi}{2}}(u-v)$.

Lastly, the square-limit pole (10) takes the form

$\lim_{\theta\to 0}{\mathbb{F}}(\theta)={i\pi\over\mu}\times{1\over\theta}\,,$
(40)

where $\mu=g\,{\sqrt{\pi}\,\Gamma(3/4)\over\Gamma(1/4)}$ is the strong-
coupling limit of the scalar measure. Under crossing (37), this square-limit
pole maps into the standard kinematical pole of two-dimensional relativistic
form factors:

${\mathbb{F}}(\theta+i\pi)\sim\frac{1}{i\theta}\,.$ (41)

Following Karowski:1978vz ; Cardy:2007mb  we construct a minimal solution to
(37)-(40) that is analytic in the physical strip $0<\Im m(\theta)<2\pi$,
except for the square-limit pole (40) and its images under crossing (37). Note
first that the scattering phase (39) has the integral representation
Karowski:1978vz

$S_{\textrm{singlet}}(\theta)=-\exp\left[-i\int\limits_{0}^{\infty}{dt\over
t}g(t)\sin(t{\textstyle\frac{\theta}{\pi}})\right]\,,\qquad
g(t)=2\,\frac{e^{-t/2}+e^{-t}}{e^{t}+1}\,.$ (42)

Using this representation, the minimal solution to the twist field matrix
elements takes the form Cardy:2007mb

${\mathbb{F}}(\theta)=\frac{2i\pi}{g\sinh\theta}\times\exp{\left[\int\limits_{0}^{\infty}{dt\over
t}\,g(t){\sin^{2}\left(t({\textstyle\frac{\theta}{2\pi}}-{\textstyle\frac{i}{4}})\right)\over\sinh({\textstyle\frac{t}{2}})}\right]}\,.$
(43)

Here, the factor of $1/\sinh\theta$ accounts for the kinematical and square-
limit poles, as well as the minus sign in the right-hand side of (42), while
the exponential factor produces the singlet S-matrix in (38) without
introducing any other singularity. By performing the integral explicitly, we
arrive at

${\mathbb{F}}(\theta)=\frac{i\pi\,\Gamma\left({1\over
4}\right)\Gamma\left({5\over
4}\right)\,\text{csch}(\theta)}{2g\,\Gamma\left(\frac{3}{4}-\frac{i\theta}{2\pi}\right)\Gamma\left(1-\frac{i\theta}{2\pi}\right)\Gamma\left(\frac{5}{4}+\frac{i\theta}{2\pi}\right)\Gamma\left(\frac{3}{2}+\frac{i\theta}{2\pi}\right)}\,.$
(44)

By inserting this expression into (25), together with the strong-coupling
expressions for the S-matrix (39) and the measure $\mu$, we find

$\lim_{g\to\infty}G(u,v)=\cosh\left({\textstyle\frac{\pi}{2}}(u-v)\right)\,.$
(45)

In appendix D, we confirm that this is indeed the strong-coupling limit of our
finite-coupling ansatz (33). Hence, we are confident that this ansatz is
correct and, in particular, that it does not miss any non-perturbative
contributions to the CDD factor $G(u,v)$.

Using (45), it is easy to see that $G(u^{2\gamma},v)=-\,G(u,v)$. It implies
that the minimal solution (with $G(u,v)=1$) also acquires a minus sign under
crossing and therefore was not a true solution to crossing.

##  4 Multi-particle form factor transitions

As briefly discussed above, the multi-particle FF transitions are expected to
factorize into products of two-particle FF transitions. Here, we provide a
factorized ansatz that is consistent with all the axioms and test it against
the perturbative data.

###  4.1 Scalars

Let us first consider transitions involving only scalars. According to the
singlet axiom, the scalars must form an $SU(4)_{R}$ singlet. Hence, it is only
possible to have an even number of them, $2n$. As will soon become apparent, a
single factorized term is not consistent with both the Watson and the square-
limit axioms. Instead, we are forced to consider a sum of factorized terms.
Each of them captures a subset of the square limits and is invariant under a
subset of Watson relations. Different terms in the sum are related to each
other by Watson equations (2) that do not leave them invariant. Concretely,
the sum takes the form

$\displaystyle F_{\phi_{i_{1}}\dots\,\phi_{i_{2n}}}(u_{1},\ldots,u_{2n})=$
$\displaystyle\,\,F_{(\phi\bar{\phi})^{n}}(u_{1},\dots,u_{2n})\times\Pi_{i_{1}\,\dots\,i_{2n}}(u_{1},\ldots,u_{2n})$
(46) $\displaystyle+$
$\displaystyle\sum\limits_{\sigma}F_{(\phi\bar{\phi})^{n}}^{\sigma}(u_{1},\dots,u_{2n})\times\Pi^{\sigma}_{i_{1}\,\dots\,i_{2n}}(u_{1},\ldots,u_{2n})\,,$

where $\phi_{i_{1}}\ldots\phi_{i_{2n}}$ are the $2n$ ordered scalars. Here,
the terms are factorized into a dynamical part, $F_{(\phi\bar{\phi})^{n}}$,
and a matrix part, $\Pi$, that encodes the $SU(4)_{R}$ structure and is
independent of the coupling. The sum in the second line goes over all
inequivalent structures that are generated from $F_{(\phi\bar{\phi})^{n}}$ via
Watson relations, and which can hence be labelled by permutations.999What
exact permutations leave $F_{(\phi\bar{\phi})^{n}}$ invariant depends, of
course, on $F_{(\phi\bar{\phi})^{n}}$, which is constructed below. For
example, consider the permutation of the first two scalars, $\sigma_{12}$. The
corresponding term in (46) is

$\displaystyle
F_{(\phi\bar{\phi})^{n}}^{\sigma_{12}}(u_{1},u_{2},\dots,u_{2n})\times\Pi^{\sigma_{12}}_{i_{1}i_{2}\,\dots\,i_{2n}}(u_{1},u_{2},\ldots,u_{2n})$
(47) $\displaystyle\equiv$
$\displaystyle\left[F_{(\phi\bar{\phi})^{n}}(u_{2},u_{1},\dots,u_{2n})\times\Pi_{j_{1}j_{2}\,\dots\,i_{2n}}(u_{2},u_{1},\ldots,u_{2n})\right]\times
S^{j_{2}j_{1}}_{i_{1}i_{2}}(u_{2},u_{1})$ $\displaystyle=$
$\displaystyle\left[F_{(\phi\bar{\phi})^{n}}(u_{2},u_{1},\dots,u_{2n})\,S_{\phi\bar{\phi}}(u_{2},u_{1})\right]\times\left[\Pi_{j_{1}\,j_{2}\,\dots\,i_{2n}}(u_{2},u_{1},\ldots,u_{2n})R^{j_{2}j_{1}}_{i_{1}i_{2}}(u_{2},u_{1})\right]\,,$

where we factorized the S-matrix into its coupling-dependent (singlet) part
and the R-matrix,

$S_{ij}^{kl}(u,v)=S_{\phi\bar{\phi}}(u,v)\,R_{ij}^{kl}(u,v)\,.$ (48)

The explicit expressions for these factors are given in appendix A. Note that
(47) is not equivalent to just flipping the rapidities and indices in the
first term of (46).

####  4.1.1 Dynamical part

We would like to construct an ansatz in which the dynamical parts of all the
terms in the sum in (46) factorize into products of two-particle scalar
transitions, $F_{\phi\bar{\phi}}(u,v)$ in (9). To ensure the closure of such a
factorized ansatz under the Watson relations, each term in (46) has to contain
either $F_{\phi\bar{\phi}}(u_{i},u_{j})$ or
$1/F_{\phi\bar{\phi}}(u_{j},u_{i})$ for each pair of excitations $i<j$.
Because
$F_{\phi\bar{\phi}}(u,v)=S_{\phi\bar{\phi}}(u,v)\,F_{\phi\bar{\phi}}(v,u)$ and
$S_{\phi\bar{\phi}}(u,v)\,S_{\phi\bar{\phi}}(v,u)=1$, when two neighbouring
excitation are swapped, these factors produce the dynamical part of the
S-matrix appearing in (47). It means that in total, each scalar appears in
$(2n-1)$ two-particle factors, one for each other scalar.

Next, we consider the square-limit axiom (6). It consists of two terms, one
with and the other without a product of S-matrices. This might naively
contradict a factorized ansatz. It turns out, however, that the product of
S-matrices is trivial and the right-hand side of the square-limit axiom does
factorize. The reason for this simplification is that the S-matrix between two
scalars of the same rapidity is a permutation,

$S_{ij}^{kl}(v,v)=\delta_{i}^{l}\delta_{j}^{k}\,.$ (49)

As a result, the S-matrices in the second term in (6) cancel in pairs, see
figure 4.1. We conclude that, as for two scalars, the two terms in the square-
limit axiom combine into a principle-part prescription for the factorization
pole, times the measure and the $(2n-2)$-particle FF transition.

Figure 4.1: The scalar S-matrix at equal rapidities is a permutation (49). As
a result, the S-matrices in the second term in the square-limit axiom (6)
cancel in pairs, $S(u_{i},v)S(v,u_{i})=1\hskip-4.55254pt1$, where
$u_{1}=u_{n}=v$.

Let us now use the Watson relation to move excitation $u_{i}$ to the first
position, excitation $u_{j}$ to the last, and then take the $u_{i}\to u_{j}$
square limit. To correctly reproduce the square-limit pole, there must be a
term in (46) with a factor of $F_{\phi\bar{\phi}}(u_{i},u_{j})$ in the
numerator. For the rest of the FF transition to factor out, all other two-
particle transitions involving $u_{i}$ and $u_{j}$ must cancel. Hence, among
the $2n-1$ two-particle transitions involving $u_{i}$, $n$ factors are in the
numerator and $(n-1)$ factors are in the denominator.

One immediate consequence of this is that a single factorized term cannot be
permutation invariant, as claimed above. Namely, because some of the
transitions are in the numerator of $F_{(\phi\bar{\phi})^{n}}$ and some are in
the denominator, reordering the scalars using the Watson relation generically
leads to a different function of the ordered scalars. This is the reason the
ansatz (46) involves a sum of factorized terms instead of a single one, which
would have required the two-particle transitions to be all together in the
numerator or denominator. This structure is more complicated than the one for
the pentagon transition, for which a single factorized term is consistent with
all the axioms Basso:2013vsa .101010The reason is that for the pentagon
transition, the square-limit axiom involves excitations in two different
groups, the ones on the top and the ones on the bottom of the pentagon. On the
other hand, for the FF transition a single group of excitations is subject to
an analogous square-limit constraint.

Let us illustrate this construction for the case of the four-scalar FF
transition. In this case, there are three different factorized dynamical
structures that are consistent with the Watson and the square-limit axioms,
see figure 4.2. Correspondingly, equation (46) becomes

$\displaystyle
F_{\phi_{i_{1}}\phi_{i_{2}}\phi_{i_{3}}\phi_{4}}(u_{1},u_{2},u_{3},u_{4})$
$\displaystyle=F_{(\phi\bar{\phi})^{2}}(u_{1},u_{2},u_{3},u_{4})\times\Pi_{i_{1}i_{2}i_{3}i_{4}}(u_{1},u_{2},u_{3},u_{4})$
(50)
$\displaystyle+F_{(\phi\bar{\phi})^{2}}^{\sigma_{34}}(u_{1},u_{2},u_{3},u_{4})\times\Pi^{\sigma_{34}}_{i_{1}i_{2}i_{3}i_{4}}(u_{1},u_{2},u_{3},u_{4})$
$\displaystyle+F_{(\phi\bar{\phi})^{2}}^{\sigma_{23}}(u_{1},u_{2},u_{3},u_{4})\times\Pi^{\sigma_{23}}_{i_{1}i_{2}i_{3}i_{4}}(u_{1},u_{2},u_{3},u_{4})\,,$

Figure 4.2: The three dynamical structures entering the four-scalar FF
transition, (50). Each of them factorizes into six two-particle FF
transitions. A blue line from $u_{i}$ to $u_{j}$ represents a factor of
$F_{\phi\bar{\phi}}(u_{i},u_{j})$, while a red one represents a factor of
$1/F_{\phi\bar{\phi}}(u_{i},u_{j})$. The first structure on the left is
invariant under crossing, while the other two are interchanged.

where

$F_{(\phi\bar{\phi})^{2}}(u_{1},u_{2},u_{3},u_{4})=\frac{F_{\phi\bar{\phi}}(u_{1},u_{2})F_{\phi\bar{\phi}}(u_{2},u_{3})F_{\phi\bar{\phi}}(u_{3},u_{4})F_{\phi\bar{\phi}}(u_{1},u_{4})}{F_{\phi\bar{\phi}}(u_{3},u_{1})F_{\phi\bar{\phi}}(u_{4},u_{2})}\,.$
(51)

These three structures form a closed set under Watson relations. We see that
the $u_{1}\to u_{4}$ square limit is captured by the terms in the first and
last lines of (50). In this limit, the factor
$F_{\phi\bar{\phi}}(u_{1},u_{4})$ produces the factorization pole times the
scalar measure, while the rest of the factors involving $u_{1}$ and $u_{4}$
cancel out. Similarly, the factorization pole involving $u_{1}$ and $u_{3}$ is
reproduced by the second and third lines.

For the transition of $2n$ scalars we take the dynamical part of the first
term in (46) to be the ratio between the product of two-particle transitions
of scalars that are separated by an odd number of sites and the product of the
ones between scalars that are separated by an even number of sites

$F_{(\phi\bar{\phi})^{n}}(u_{1},\dots,u_{2n})=\prod\limits_{i<j}^{2n}\frac{F_{\phi\bar{\phi}}(u_{i},u_{j})^{\frac{1-(-1)^{i-j}}{2}}}{F_{\phi\bar{\phi}}(u_{j},u_{i})^{\frac{1+(-1)^{i-j}}{2}}}\,.$
(52)

This term is then multiplied by the matrix part and summed over the ${1\over
2}\binom{2n}{n}$ permutations that produce the inequivalent structures
$F_{(\phi\bar{\phi})^{n}}^{\sigma}$, see (46).

Lastly, we consider the crossing axiom. Let us focus on (52) first. The
product of the two-particle FF transition in the numerator of (52) is
manifestly crossing symmetric. This is because
$F_{\phi\bar{\phi}}(u_{1}^{2\gamma},u_{j})=F_{\phi\bar{\phi}}(u_{j},u_{1})$
and two scalars that are separated an odd (even) number of sites remain
separated an odd (even) number of sites after crossing. Up to a simple
rational factor, the denominator is also crossing invariant:

$\displaystyle
F_{\phi\bar{\phi}}(u_{j},u_{1}^{2\gamma})=S_{\phi\bar{\phi}}(u_{j},u_{1}^{2\gamma})\,F_{\phi\bar{\phi}}(u_{1}^{2\gamma},u_{j})=S_{\phi\bar{\phi}}(u_{j},u_{1}^{2\gamma})\,F_{\phi\bar{\phi}}(u_{j},u_{1})$
(53)
$\displaystyle\qquad=S_{\phi\bar{\phi}}(u_{j},u_{1}^{2\gamma})\,S_{\phi\bar{\phi}}(u_{j},u_{1})\,F_{\phi\bar{\phi}}(u_{1},u_{j})=\frac{(u_{1}-u_{j}+3i)\,(u_{1}-u_{j}+4i)}{(u_{1}-u_{j}-i)\,(u_{1}-u_{j}-2i)}\,F_{\phi\bar{\phi}}(u_{1},u_{j})\,,$

where we used (75) and (16). The rational coupling-independent pre-factor on
the right-hand side of (53) can be absorbed in the crossing transformation of
the matrix part, which we turn to next. It is not to hard to see that crossing
of any of the terms in (46) is equivalent to a cyclic permutation and, hence,
the sum is invariant.

####  4.1.2 Matrix part

The bootstrap axioms for the FF transition reduce to simplified constraints on
the matrix part. We list them here and solve them explicitly for the case of
four scalars.

##### Watson

The dynamical part satisfies the Watson relation with the dynamical part of
the S-matrix, see (48) and (12). Hence, the matrix part should satisfy the
same relation, but with the R-matrix instead:

$\Pi^{\sigma_{ii+1}\cdot\tilde{\sigma}}_{\ldots\,ii+1\,\ldots}(\ldots,u_{i},u_{i+1},\ldots)=\Pi^{\tilde{\sigma}}_{\ldots\,k+1k\,\ldots}(\ldots,u_{i+1},u_{i},\ldots)\times
R^{kk+1}_{ii+1}(u_{i+1},u_{i})\,.$ (54)

Using this relation, the matrix parts that correspond to terms with non-
trivial permutations in (46) can be expressed in terms of the matrix part that
multiplies the unpermuted dynamical part (52).

##### Reflection

The dynamical factor (52) is invariant under reflection (4). Hence, so is the
matrix part

$\Pi_{i_{1}\,\dots\,i_{2n}}(-u_{1},\dots,-u_{2n})=\Pi_{i_{2n}\,\dots\,i_{1}}(u_{2n},\dots,u_{1})\,.$
(55)

##### Crossing

The crossing transformation of the factorized dynamical part (52) translates
into the following crossing transformation of the corresponding matrix part

${\Pi_{j_{1}\,\dots\,j_{2n}}(u_{1}+2i,\dots,u_{2n})\over\Pi_{j_{2}\,\dots
j_{2n}j_{1}}(u_{2},\dots,u_{2n},u_{1})}=\prod_{i=1}^{n-1}\frac{(u_{1}-u_{2i+1}+3i)\,(u_{1}-u_{2i+1}+4i)}{(u_{1}-u_{2i+1}-i)\,(u_{1}-u_{2i+1}-2i)}\,,$
(56)

where we used that $u^{2\gamma}=u+2i$ and (53).

##### Square limit

In the square limit, the matrix part must factorize as

$\Pi_{i_{1}\,\dots\,i_{2n}}(u,u_{2},\dots,u_{2n-1},u)={1\over
2}\,\delta_{i_{1}i_{2n}}\times\Pi_{i_{2}\,\dots\,i_{2n-1}}(u_{2},\dots,u_{2n-1})\,,$
(57)

where the factor of 2 in the denominator stands for the two terms in (46) that
reduce to the same term after removing the first and last excitations.

##### Singlet

The matrix part $\Pi_{i_{1}\dots i_{2n}}$ projects $2n$ scalars into an
$SU(4)_{R}$ singlet. It consists of ${(2n-1)!!}$ tensors structures, which are
factorized products of $\delta_{ij}$’s. Each one of them is multiplied by a
function of the rapidities $\pi_{I}(u_{1},\dots,u_{n})$, with
$I=1,\dots,(2n-1)!!$. Namely, we have

$\Pi_{i_{1}\,\dots\,i_{2n}}(u_{1},\dots,u_{2n})=\pi_{1}(u_{1},\dots,u_{n})\prod_{k=1}^{n}\delta_{kk+n}+\dots\,.$
(58)

#### Four scalars

We now solve for the four-scalars matrix part explicitly, starting from (50).
The first line of (50), with the three singlet tensor structures inserted,
reads

$\displaystyle
F_{(\phi\bar{\phi})^{2}}(u_{1},u_{2},u_{3},u_{4})\,\Pi_{i_{1}i_{2}i_{3}i_{4}}(u_{1},u_{2},u_{3},u_{4})$
$\displaystyle=\frac{F_{\phi\bar{\phi}}(u_{1},u_{2})F_{\phi\bar{\phi}}(u_{2},u_{3})F_{\phi\bar{\phi}}(u_{3},u_{4})F_{\phi\bar{\phi}}(u_{1},u_{4})}{F_{\phi\bar{\phi}}(u_{3},u_{1})F_{\phi\bar{\phi}}(u_{4},u_{2})}$
(59)
$\displaystyle\times\left(\pi_{1}\,\delta_{i_{1}i_{4}}\delta_{i_{2}i_{3}}+\pi_{2}\,\delta_{i_{1}i_{3}}\delta_{i_{2}i_{4}}+\pi_{3}\,\delta_{i_{1}i_{2}}\delta_{i_{3}i_{4}}\right)\,.$

In general, the Watson constraint (54) mixes the $\Pi$’s of the three
different dynamical structures in (50). Among all possible permutations of the
four scalars, which are generated by multiple uses of the Watson relation, it
is sufficient to consider the ones that take $\Pi$ in (50) back to itself. The
rest can then be used to generate the two other matrix parts in (50) from
$\Pi$. The permutations that preserve $\Pi$ are cyclic permutations,
reflection, as well as any combination of these.

The crossing constraint (56) now takes the form

$\displaystyle\pi_{1}(u_{1}+2i,u_{2},u_{3},u_{4})=\frac{(u_{1}-u_{3}-i)\,(u_{1}-u_{3}-2i)}{(u_{1}-u_{3}+3i)\,(u_{1}-u_{3}+4i)}\,\pi_{3}(u_{2},u_{3},u_{4},u_{1})\,,$
(60)
$\displaystyle\pi_{3}(u_{1}+2i,u_{2},u_{3},u_{4})=\frac{(u_{1}-u_{3}-i)\,(u_{1}-u_{3}-2i)}{(u_{1}-u_{3}+3i)\,(u_{1}-u_{3}+4i)}\,\pi_{1}(u_{2},u_{3},u_{4},u_{1})\,,$
$\displaystyle\pi_{2}(u_{1}+2i,u_{2},u_{3},u_{4})=\frac{(u_{1}-u_{3}-i)\,(u_{1}-u_{3}-2i)}{(u_{1}-u_{3}+3i)\,(u_{1}-u_{3}+4i)}\,\pi_{2}(u_{2},u_{3},u_{4},u_{1})\,.$

The square-limit condition (57) reads

$\displaystyle\left.\pi_{1}\right|_{u_{1}=u_{4}}=\frac{1}{2}\,,\qquad\left.\pi_{2}\right|_{u_{1}=u_{4}}=\left.\pi_{3}\right|_{u_{1}=u_{4}}=0\,,$
(61)

and the reflection symmetry of $\Pi$ (55) simply becomes a symmetry of the
$\pi$’s

$\pi_{i}(-u_{4},-u_{3},-u_{2},-u_{1})=\pi_{i}(u_{1},u_{2},u_{3},u_{4})\,.$
(62)

These constraints are easily seen to be satisfied by the following solution:

$\displaystyle\pi_{2}(u_{1},u_{2},u_{3},u_{4})$
$\displaystyle=\frac{(u_{1}-u_{4})\,(u_{1}-u_{2}-2i)\,(u_{2}-u_{3}-2i)\,(u_{3}-u_{4}-2i)}{4\,(u_{1}-u_{3}+i)\,(u_{1}-u_{3}+2i)\,(u_{2}-u_{4}+i)\,(u_{2}-u_{4}+2i)}\,,$
(63) $\displaystyle\pi_{1}(u_{1},u_{2},u_{3},u_{4})$
$\displaystyle=-\,\frac{\pi_{2}(u_{2},u_{1},u_{4},u_{3})-\sum\limits_{i=1}^{2}r^{(i)}(u_{2},u_{1})\,r^{(i)}(u_{4},u_{3})\,\pi_{2}(u_{1},u_{2},u_{3},u_{4})}{i\left(u_{1}-u_{2}+u_{3}-u_{4}\right)r^{(2)}(u_{2},u_{1})\,r^{(2)}(u_{4},u_{3})}\,,$
$\displaystyle\pi_{3}(u_{1},u_{2},u_{3},u_{4})$
$\displaystyle=\frac{(u_{1}-u_{3}-3i)\,(u_{1}-u_{3}-4i)}{(u_{1}-u_{3}+i)\,(u_{1}-u_{3}+2i)}\,\pi_{1}(u_{2},u_{3},u_{4},u_{1}-2i)\,,$

where the rational functions $r^{(i=1,2,3)}(u,v)$ represent the coefficients
of the three tensor structures of the R-matrix and are given in (A).

In subsection 4.4, we will verify that this solution is in agreement with the
perturbative data. We leave the generalization of this solution to matrix
parts with more than four scalars to future work.

###  4.2 Gluons and Fermions

For gluons and fermions, a factorized ansatz means that they only appear in
singlet pairs of two conjugate excitations with the same momenta, (8). Such a
gluon-gluon or fermion anti-fermion pair behaves as an effective single-
particle excitation that is an $SU(4)_{R}\times U(1)_{\phi}$ singlet. Hence,
constructing a factorized ansatz involving gluons and fermions is trivial. We
simply divide them into pairs and take products of two-particle transitions
for all the pairs. To cover all the different multi-square limits, we have to
sum over all possible divisions. We consider the ordering of the excitations
in which the square limits of the gluons and fermions can be taken directly.
Other orderings can be obtained using the Watson equation.

For example, the four-particle FF transitions involving only gluons and
fermions are

$\displaystyle F_{F\psi_{A}\bar{\psi}^{B}\bar{F}}(u,u^{\prime},v^{\prime},v)$
$\displaystyle=\delta_{A}^{B}\,F_{F\bar{F}}(u,v)\,F_{\psi\bar{\psi}}(u^{\prime},v^{\prime})\,,$
(64) $\displaystyle F_{FF\bar{F}\bar{F}}(u,u^{\prime},v^{\prime},v)$
$\displaystyle=F_{F\bar{F}}(u,v)\,F_{F\bar{F}}(u^{\prime},v^{\prime})+S_{FF}(u,u^{\prime})\,F_{F\bar{F}}(u,v^{\prime})\,F_{F\bar{F}}(u^{\prime},v)\,,$
$\displaystyle
F_{\psi_{A}\psi_{B}\bar{\psi}^{C}\bar{\psi}^{D}}(u,u^{\prime},v^{\prime},v)$
$\displaystyle=\delta_{A}^{D}\delta_{B}^{C}\,F_{\psi\bar{\psi}}(u,v)\,F_{\psi\bar{\psi}}(u^{\prime},v^{\prime})-S_{AB}^{CD}(u,u^{\prime})\,F_{\psi\bar{\psi}}(u,v^{\prime})\,F_{\psi\bar{\psi}}(u^{\prime},v)\,,$

where $F\equiv F_{1}$ and $\bar{F}\equiv F_{-1}$. Note that in the mixed
transitions given in the first line there is a unique square-limit pairing in
which each type of excitation decouples in pairs. In these cases, we have
chosen one of the four orderings where this limit can be taken directly,
without reordering of the excitations ($F\psi_{A}\bar{\psi}^{B}\bar{F}$). The
other orderings of this type (such as $\psi_{A}F\bar{F}\bar{\psi}^{B}$) give
the same answer because the S-matrices that result from the permutations
cancel each other. In the last two lines, transitions involve four excitations
of the same type. In these cases, there are two different possible pairings of
conjugate excitations and, correspondingly, two terms in the FF transition,
which are related by the Watson equation.

Such a factorized ansatz is consistent with the square-limit axiom because, as
in the scalar case, the product of S-matrices in the second term of (6) is
trivial. The reason for this is, however, somewhat different and applies only
to our factorized ansatz. Consider first the case in which the $u_{1}$ and
$u_{n}$ excitations in (6) are two conjugate gluons or gluon bound states.
Because these are abelian excitations, their scattering with any other
excitation $\chi\in\\{F_{m},\psi_{A},\bar{\psi}^{A},\phi_{i}\\}$ is a phase
instead of a matrix. As a result of unitarity and charge conjugation symmetry,
the terms in the product of S-matrices in (6) cancel in pairs,

$S_{\chi F_{m}}(v,u)S_{F_{-m}\bar{\chi}}(u,v)=S_{\chi
F_{m}}(v,u)S_{F_{m}\chi}(u,v)=1\,,\qquad S_{F_{m}F_{-m}}(u,u)=1\,,$ (65)

where $u=u_{1}=u_{n}$ in (6). The same cancellation also applies for the
square limit of $\chi$ with $\bar{\chi}$, in which $v=u_{1}=u_{n}$. Hence, for
any state the gluon and gluon-bound-state S-matrices in (6) cancel in pairs.

Figure 4.3: The square limit axiom (6) for fermions is consistent with
factorization because the product of S-matrices in the second term becomes
trivial when evaluated on a factorized ansatz. To see this, we note that two
conjugate fermions of equal rapidity scatter trivially with a scalar (${\bf
b}$) and with another pair of conjugate fermions (${\bf a}$). Here, the dashed
directed lines represent the contraction of the $SU(4)_{R}$ index, the solid
arrow represents the flow of $U(1)_{\phi}$ charge in $1\over 2$ units and the
black dots represents the R-matrices in (126) and (128).

Now consider the case in which the $u_{1}$ and $u_{n}$ excitations in (6) are
two conjugate fermions. Because the square-limit axiom is consistent with the
Watson relation, it is sufficient to consider any particular ordering of the
excitations. To show that the product of S-matrices in the second term in (6)
is trivial for any state, it is therefore enough to show that we can freely
pass the $u_{1}$ or $u_{n}$ fermionic excitation through a single scalar or
through another pair of conjugate fermions with the same rapidity. The
S-matrices factorize into a dynamical and the R-matrix parts, see (125),
(128). The dynamical parts are abelian factors. They cancel due to unitarity
and charge conjugation symmetry, similarly to (65). We can therefore focus on
the matrix part, which is captured by the R-matrices. Using their explicit
form given in (126) and (128), we have checked that indeed, the $u_{1}$ and
$u_{n}$ excitations pass freely through a scalar or another pair of fermions,
see figure 4.3.

Note also that unlike in the scalar case, in which each dynamical part in (46)
factorized in terms of the two-particle transitions only, in (64) the S-matrix
explicitly appears in the last two lines. The two terms in
$F_{FF\bar{F}\bar{F}}$ and $F_{\psi\psi\bar{\psi}\bar{\psi}}$ are supported on
different kinematical limits, which describe two pairs of decoupled
excitations that effectively propagate on the square. Hence, no additional
interaction between the pairs is present.

#### Fusion

Lastly, we note that the multi-gluon FF transitions fuse properly into the
gluon-bound-state FF transitions. To see this, the FF transitions have to be
contracted with the pentagon creation amplitudes and the measures, in the way
they enter the FFOPE (1), see Sever:2020jjx ; Sever:2021nsq  for details. For
example, we can start with the four-gluon contribution to the OPE for the
three-point form factor (1):111111Compared to Sever:2020jjx ; Sever:2021nsq ,
here we have used the reflection symmetry to replace
$F_{\bar{F}\bar{F}FF}(-u_{4},-u_{3},-u_{2},-u_{1})$ with
$F_{FF\bar{F}\bar{F}}(u_{1},u_{2},u_{3},u_{4})$.

$\displaystyle{\cal W}_{3,F^{4}}=\left.\langle{\cal
F}|e^{iP\sigma-H\tau}|0\rangle\right|_{4-\rm gluons}$ (66)
$\displaystyle={1\over 4}\int
P_{FF\bar{F}\bar{F}}(0|u_{1},u_{2},u_{3},u_{4})F_{FF\bar{F}\bar{F}}(u_{1},u_{2},u_{3},u_{4})\prod_{i=1}^{4}\mu_{F}(u_{i})e^{ip_{F}(u_{i})\sigma-
E_{F}(u_{i})\tau}{du_{i}\over 2\pi}$ $\displaystyle=\frac{1}{2}\int{1\over
P_{F\bar{F}}(u_{1}|u_{1})P_{F\bar{F}}(u_{2}|u_{2})P_{F\bar{F}}(u_{1}|u_{2})P_{F\bar{F}}(u_{2}|u_{1})}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\times\left[{\mu_{F}(u_{1})\mu_{F}(u_{2})\over
P_{FF}(u_{1}|u_{2})P_{FF}(u_{2}|u_{1})}\right]\prod_{i=1}^{2}e^{2ip_{F}(u_{i})\sigma-2E_{F}(u_{i})\tau}{du_{i}\over
2\pi}\,.$

As detailed in Basso:2014koa ; Basso:2014nra , gluons are fused into bound
states by evaluating the integrand at $u_{1}=u^{+}$, $u_{2}=v^{-}$ with
$u^{\pm}=u\pm{i\over 2}$, and then taking the $v\to u$ limit. In terms of the
original four rapidities in the second line, this corresponds to the limit
$(u_{1},u_{2},u_{3},u_{4})\to(u^{+},u^{-},u^{-},u^{+})$. The term in the
brackets in the last line is the two-gluon contribution to the hexagon. It has
a pole at $v=u$, whose residue is the bound-state measure $\mu_{F_{2}}(u)$,
Basso:2014koa ; Basso:2014nra . At the same time, the first factor becomes the
creation amplitude for two conjugate bound states. Hence, in this limit we get

$\displaystyle i\,\underset{v=u}{\operatorname{{\rm
residue}}}\,{\mu_{F}(u^{+})\mu_{F}(v^{-})\over
P_{FF}(u^{+}|v^{-})P_{FF}(v^{-}|u^{+})}{1\over
P_{F\bar{F}}(u^{+}|u^{+})P_{F\bar{F}}(v^{-}|v^{-})P_{F\bar{F}}(u^{+}|v^{-})P_{F\bar{F}}(v^{-}|u^{+})}{du\over
2\pi}$ $\displaystyle=\,$ $\displaystyle
P_{F_{2}F_{-2}}(0|u,u)\,\mu_{F_{2}}(u)\,{du\over
2\pi}=P_{F_{2}F_{-2}}(0|u,w)F_{F_{2}F_{-2}}(u,w)\,\mu_{F_{2}}(u)\,\mu_{F_{2}}(w)\,{du\,dw\over(2\pi)^{2}}\,,$
(67)

which is indeed the way the two bound states contribute to the FFOPE.

###  4.3 Mixed states

Finally, let us consider states containing all possible combinations of
excitations. We find that they likewise factorize, following essentially the
same rules as discussed above.

For example, for two scalars and two gluons or fermions we have

$\displaystyle F_{F\phi_{i}\phi_{j}\bar{F}}(u,u^{\prime},v^{\prime},v)$
$\displaystyle=F_{\phi\bar{\phi}}(u^{\prime},v^{\prime})F_{F\bar{F}}(u,v)\,\delta_{ij}\,,$
(68) $\displaystyle
F_{\psi_{A}\phi_{i}\phi_{j}\bar{\psi}^{B}}(u,u^{\prime},v^{\prime},v)$
$\displaystyle=F_{\phi\bar{\phi}}(u^{\prime},v^{\prime})F_{\psi\bar{\psi}}(u,v)\,\delta_{ij}\delta_{A}^{B}\,.$

Similarly, for FF transitions involving $2m$ scalars, the scalar FF
transitions themselves form the building blocks of the factorization. For
example,

$F_{F\phi_{i_{1}}\ldots\phi_{i_{2m}}\bar{F}}(v_{1},u_{1},\dots,u_{2m},v_{2})=F_{\phi_{i_{1}}\ldots\phi_{i_{2m}}}(u_{1},\dots,u_{2m})\times
F_{F\bar{F}}(v_{1},v_{2})\,.$ (69)

To summarize, we see that for any paring of the gluons, gluon bound states and
fermions, they propagate as if they where on the square, with no reference to
the form factor. In particular, that means that the matrix structure of a FF
transition involving scalars and any number of gluons and fermions is the same
as the matrix structure of the FF transition involving only the scalars. This
is a great simplification compared to the case of the pentagon transitions,
which include effective transitions between different type of excitations and
in which fermions also contribute to the matrix part.

###  4.4 Matching with data

As discussed in section 3.3, perturbative data is by now available up to
eight-loop order Dixon:2020bbt ; PerturbativeBootstrap2 . We find that the OPE
results which follow from our conjectured multi-particle FF transitions
involving four particles are in perfect agreement with the perturbative data
at order $e^{-4\tau}$ in the collinear limit.

In Sever:2021nsq , we already reported on a full match at order $e^{-4\tau}$
up to three loops, where only two-particle states and four-particle states
with so-called small fermions contribute. We now extend this to contributions
from generic four-particle states.

An important complication over the two-particle case is that both the multi-
particle pentagon creation amplitudes and the multi-scalar FF transitions
include non-trivial matrix parts. For example, to evaluate the four-scalar
contribution to the three-gluon form factor, the two matrix structures are
contracted together as

${\cal
W}_{3,\phi^{4}}=\frac{1}{4!}\int\\!P_{\phi^{i}\phi^{j}\phi^{k}\phi^{l}}(0|u_{1},u_{2},u_{3},u_{4})F_{\phi^{i}\phi^{j}\phi^{k}\phi^{l}}(u_{1},u_{2},u_{3},u_{4})\prod_{i=1}^{4}\mu_{\phi}(u_{i})e^{ip_{\phi}(u_{i})\sigma-E(u_{i})\tau}du_{i}.$
(70)

To date, these matrix parts have not been worked out in full generality, not
even for the pentagon transitions (see Belitsky:2016vyq  for some of them).
Fortunately, all matrix parts for four particles are known; we summarize them
in appendix E. For the four-scalar example (70),
$F_{\phi^{i}\phi^{j}\phi^{k}\phi^{l}}$ is the four-scalar FF transition in
(50), (63), while the matrix part of the pentagon transition is given in
(118). Using the technique detailed in Sever:2021nsq , the integral above
evaluates to

$\displaystyle{\cal W}_{3,\phi^{4}}=\,\,$ $\displaystyle
g^{8}e^{-4\tau}\biggl{[}18e^{4\sigma}+\left(40\sigma^{3}+24\sigma^{2}+10\pi^{2}\sigma+36\sigma+2\pi^{2}-\frac{847}{36}\right)e^{6\sigma}+{\cal
O}(e^{8\sigma})\biggr{]}$ $\displaystyle+\,\,$ $\displaystyle
g^{10}e^{-4\tau}\biggl{[}\biggl{\\{}\left(6\pi^{2}-\frac{1323}{4}\right)e^{4\sigma}+\biggl{(}60\zeta_{3}\sigma^{2}-72\zeta_{3}\sigma-96\sigma^{5}-256\sigma^{4}-\frac{320}{3}\pi^{2}\sigma^{3}$
$\displaystyle\hphantom{+g^{10}e^{-4\tau}\biggl{[}\biggl{\\{}}-362\sigma^{3}-116\pi^{2}\sigma^{2}-\frac{765\sigma^{2}}{2}-15\pi^{4}\sigma-\frac{185}{2}\pi^{2}\sigma-\frac{9463\sigma}{18}+5\pi^{2}\zeta_{3}$
$\displaystyle\hphantom{+g^{10}e^{-4\tau}\biggl{[}\biggl{\\{}}-30\zeta_{5}-\frac{17\pi^{4}}{5}+\frac{13300}{27}-\frac{6851\pi^{2}}{216}\biggr{)}e^{6\sigma}+{\cal
O}(e^{8\sigma})\biggr{\\}}$ (71)
$\displaystyle\hphantom{+g^{10}e^{-4\tau}\biggl{[}\biggl{\\{}}-\tau\biggl{\\{}144\sigma
e^{4\sigma}+\biggl{(}320\sigma^{4}+352\sigma^{3}+120\pi^{2}\sigma^{2}+264\sigma^{2}+72\pi^{2}\sigma$
$\displaystyle\hphantom{+g^{10}e^{-4\tau}\biggl{[}\biggl{\\{}-\tau\,\biggl{\\{}}+\frac{34\sigma}{9}+96\zeta_{3}+\frac{8\pi^{4}}{3}+10\pi^{2}-\frac{1006}{9}\biggr{)}e^{6\sigma}+{\cal
O}(e^{8\sigma})\biggr{\\}}\biggr{]}$ $\displaystyle+\,\,$ $\displaystyle{\cal
O}(g^{12})\,.$

Adding all contributions, we performed a full check up to five-loop order,
finding a perfect match with the perturbative data given in the ancillary
files of Dixon:2020bbt . Moreover, we have done selective checks up to eight-
loop order, again finding perfect agreement with the perturbative data given
in the ancillary files of PerturbativeBootstrap2 .

It would be interesting to determine the matrix parts of the FF transitions
and the pentagon transitions for states with more particles and to use them
for confronting our conjectures with data also at higher orders in
$e^{-2\tau}$.

##  5 Conclusions

In this paper, we have bootstrapped the FF transitions of the chiral part of
the stress-tensor supermultiplet in the planar maximally supersymmetric Yang-
Mills theory at finite value of the ’t Hooft coupling.

The form factor transitions of a GKP state vanish unless the excitations are
pairwise conjugate. For transitions involving fermions, gluons, and gluon
bound states, we found that the conjugate pairs decouple and move together as
if they were propagating on the square. On the other hand, the transitions
involving scalars are non-trivial. They are given by a certain sum of
factorized ratios of two-scalar transitions, which we have determined up to a
group-theoretical coupling-independent matrix part. We found that the finite-
coupling behavior of the two-scalar transition is governed by the so-called
octagon kernel – a quantity that has previously occurred for other
observables. The determination of the two-scalar FF transition has benefited
from a very fruitful interplay with the perturbative form factor bootstrap
Dixon:2020bbt ; PerturbativeBootstrap2 , and our predictions at order
$e^{-2\tau}$ and $e^{-4\tau}$ perfectly match with the eight-loop perturbative
data. Moreover, the strong-coupling limit of the two-scalar FF transition
agrees with the minimal solution to the form factor axioms for a twist
operator in the $O(6)$ non-linear sigma model.

We conclude by listing some of the many future directions.

  * •

Both the pentagon transitions and the FF transitions factorize into a known
dynamical part that encodes all dependence on the coupling and a matrix part
that encodes the group-theoretic dependence on the fermions’ and scalars’
$SU(4)_{R}$ indices. The matrix part has not been determined in full
generality, neither for form factors nor for pentagons. Fixing the matrix
parts for the FF transition seems simpler than for the pentagon transitions,
because the former only depend on the scalars, not on the fermions. Moreover,
instead of fixing these matrix parts individually, it may be simpler to find a
closed-form integral representation for the full contraction of the matrix
parts that contribute to a particular $n$-point form factor or $n$-point
amplitude, as was done for the case of the six-point amplitude in
Basso:2015uxa .

The missing matrix parts would allow us to perform more tests of the FFOPE and
produce more high-loop predictions.

  * •

It would be interesting to see how the minimal surface area of
Maldacena:2010kp ; Gao:2013dza  emerges from the FFOPE at large coupling. In
particular, it would be important to compute the scalar correction to it,
which is expected to be of the same order as the minimal area Basso:2014jfa .

  * •

The octagon kernel has previously occurred in the description of four-point
functions Coronado:2018ypq ; Coronado:2018cxj ; Kostov:2019stn ;
Kostov:2019auq ; Belitsky:2019fan ; Bargheer:2019exp  and the origin of the
six-point amplitude Basso:2020xts ; see also Arkani-Hamed:2021iya . Very
recently, the three-point form factor was related to a particular subspace of
the six-point amplitude in an intricate way Dixon:2021tdw . It would be
interesting to see whether this relation can be used to connect the
occurrences of the octagon kernel in both cases. Moreover, it would be
interesting to understand a potential connection with the octagon.

  * •

While we used the three-point MHV form factor to guide and check our
derivation of the FF transitions, they equally determine the $n$-point MHV
form factor. Adding more than three gluons in MHV configuration is achieved by
simply gluing more OPE channels, connected by the pentagon transitions, see
(1). This should provide valuable data for a perturbative bootstrap at $n>3$.

  * •

It would be interesting to consider the NkMHV configurations with $k\geq 1$,
which would require the known charged pentagons Basso:2014hfa ; Basso:2015rta
.

  * •

The chiral part of the stress-tensor supermultiplet is in a sense the simplest
operator multiplet, having BPS protected dimension and trivial higher
integrability charges. The form factors of local operators with non-trivial
charges can equally be incorporated into the FFOPE. One has to twist the
crossing axiom (5) by (T-dual of) the corresponding charges, see
Cavaglia:2020hdb . It would be fascinating to bootstrap the corresponding FF
transitions and try to express them in terms of the operators’ Q-functions.
Corresponding perturbative data is available in Engelund:2012re ;
Brandhuber:2014ica ; Wilhelm:2014qua ; Nandan:2014oga ; Loebbert:2015ova ;
Brandhuber:2016fni ; Loebbert:2016xkw ; Caron-Huot:2016cwu ; Banerjee:2016kri
; Ahmed:2016vgl ; Brandhuber:2018xzk ; Lin:2020dyj ; Guo:2021bym ; Lin:2021qol
.

  * •

By inserting a complete basis of states, form factors can be used to decompose
other observables in terms of sums of product of form factors. In particular,
it would be interesting to decompose the double-trace corrections to
scattering amplitudes. These are dual to correlation functions between two
periodic Wilson loops Ben-Israel:2018ckc , such as the ones we studied in the
present paper. The expansion of this correlator around the limit where the two
Wilson loops are far apart is controlled by the FF transition of the low-lying
operators, the leading contribution being given by the FF transitions of the
chiral part of the stress-tensor supermultiplet, which we studied in this
paper.

##### Acknowledgments:

We are very grateful to B. Basso for many valuable discussions and comments on
the manuscript. MW thanks L. Dixon, Ö. Gürdoğan and A. McLeod for
collaboration on Dixon:2020bbt ; PerturbativeBootstrap2 . AT and MW are
grateful to CERN for hospitality. AS is grateful to NBI for hospitality. AS
was supported by the Israel Science Foundation (grant number 1197/20). AT
received funding from the European Research Council (ERC) under the European
Unions Horizon 2020 research and innovation programme, Novel structures in
scattering amplitudes (grant agreement No. 725110). MW was supported in part
by the ERC starting grant 757978 and the research grants 00015369 and 00025445
from Villum Fonden.

##  Appendix A Scalar S-matrix and measure

In this appendix, we review the scalar S-matrix Basso:2013pxa  as well as the
scalar measure, in a form that is convenient for our needs.

The S-matrix between two scalar GKP excitations takes the form

$S_{ij}^{kl}(u,v)=S_{\phi\bar{\phi}}(u,v)\,R_{ij}^{kl}(u,v)\,,$ (72)

where

$R_{ij}^{kl}(u,v)=r^{(1)}(u,v)\,\delta_{i}^{k}\delta_{j}^{l}+r^{(2)}(u,v)\,\delta_{i}^{l}\delta^{k}_{j}+r^{(3)}(u,v)\,\delta_{ij}\delta^{kl}\,,$
(73)

with

$\displaystyle r^{(1)}(u,v)=\frac{(u-v)(u-v-2i)}{(u-v+i)(u-v+2i)}\,,$
$\displaystyle r^{(2)}(u,v)=-\,\frac{i(u-v-2i)}{(u-v+i)(u-v+2i)}\,,$ (74)
$\displaystyle r^{(3)}(u,v)=\frac{i(u-v)}{(u-v+i)(u-v+2i)}\,.$

The scattering phase in different channels are related by a simple phase
factor. In particular, the phase in the singlet and symmetric channels are
related as

$S_{\phi\bar{\phi}}(u,v)\equiv{1\over
6}S_{ij}^{kl}(u,v)\delta^{ij}\delta_{kl}=\frac{(u-v+2i)\,(u-v+i)}{(u-v-2i)\,(u-v-i)}\,S_{\phi\phi}(u,v)\,.$
(75)

The S-matrix in the symmetric channel can be expressed as

$S_{\phi\phi}(u,v)=\frac{\Gamma\left(\tfrac{1}{2}-iu\right)\Gamma\left(\tfrac{1}{2}+iv\right)\Gamma\left(iu-
iv\right)}{\Gamma\left(\tfrac{1}{2}+iu\right)\Gamma\left(\tfrac{1}{2}-iv\right)\Gamma\left(iv-
iu\right)}\,\mathcal{G}(u,v)\,,$ (76)

where the function $\mathcal{G}(u,v)$ captures the entire dependence on the
coupling constant. It is given by

$\mathcal{G}(u,v)=\exp\biggl{[}2i\int_{0}^{\infty}\frac{dt}{t}(J_{0}(2gt)-1)\frac{e^{t/2}(\sin(ut)-\sin(vt))}{e^{t}-1}-2if_{1}(u,v)+2if_{2}(u,v)\biggr{]}\,,$
(77)

where $f_{1}$ and $f_{2}$ were defined in (29).

The scalar measure is given by Basso:2013aha

$\displaystyle\mu(u)=\frac{\pi g^{2}}{\cosh(\pi
u)}\exp\biggl{[}\int_{0}^{\infty}\frac{dt}{t}(J_{0}(2gt){-}1)\frac{2e^{t/2}\cos(ut)-J_{0}(2gt)-1}{e^{t}-1}+f_{3}(u,u)-f_{4}(u,u)\biggr{]},$
(78)

where $f_{3}$ and $f_{4}$ were defined in (29).

##  Appendix B Integral representations for the finite-coupling solution

In this appendix, we derive an integral representation for the functions
$f_{5}$ and $f_{6}$ that enter our solution for the scalar two-particle FF
transitions at finite coupling. The integral representations are more
tractable for the analysis of crossing symmetry and strong coupling than the
matrix notation, which is more advantageous at weak coupling.

Let us define, using matrix notation,

$\gamma_{o,+}^{v}=\frac{1}{1+\mathbb{K}(0)}\kappa_{+}^{v}\,,\qquad\tilde{\gamma}_{o,-}^{v}=\frac{1}{1+\mathbb{K}(0)}\tilde{\kappa}_{-}^{v}\,.$
(79)

These objects satisfy

$\gamma_{o,+}^{v}+\mathbb{K}(0)\,\gamma_{o,+}^{v}=\kappa_{+}^{v}\,,\qquad\tilde{\gamma}_{o,-}^{v}+\mathbb{K}(0)\,\tilde{\gamma}_{o,-}^{v}=\tilde{\kappa}_{-}^{v}\,,$
(80)

or, in components,

$\displaystyle\gamma_{o,+,2n}^{v}+2\sum\limits_{m=1}^{\infty}\mathbb{K}_{2n,2m}\gamma_{o,+,2m}^{v}$
$\displaystyle=\kappa_{+,2n}^{v}\,,$ (81)
$\displaystyle\tilde{\gamma}_{o,-,2n-1}^{v}+2\sum\limits_{m=1}^{\infty}\mathbb{K}_{2n-1,2m-1}\tilde{\gamma}_{o,-,2m-1}^{v}$
$\displaystyle=\tilde{\kappa}_{-,2n-1}^{v}\,,$

where, similarly to (28), we defined

$\gamma_{o,+,2n}=\int_{0}^{\infty}\frac{dt}{t}J_{2n}(2gt)\gamma_{o,+}^{v}(2gt)\,,\qquad\tilde{\gamma}_{o,-,2n-1}=\int_{0}^{\infty}\frac{dt}{t}J_{2n-1}(2gt)\tilde{\gamma}_{o,-}^{v}(2gt)\,.$
(82)

The odd components of $\gamma_{o,+}$ and the even components of
$\tilde{\gamma}_{o,-}$ are absent as a direct consequence of the fact that the
tilted BES kernel (31) is diagonal for $\alpha=0$ and therefore does not mix
even and odd components. The above relations can be inverted using the
orthogonality of the Bessel functions,

$\int_{0}^{\infty}\frac{dt}{t}J_{2n}(t)J_{2m}(t)=\frac{\delta_{n,m}}{4n}\,,\qquad\int_{0}^{\infty}\frac{dt}{t}J_{2n-1}(t)J_{2m-1}(t)=\frac{\delta_{n,m}}{2(2n-1)}\,,$
(83)

which results in

$\gamma_{o,+}^{v}(2gt)=\sum_{n=1}^{\infty}2(2n)J_{2n}(2gt)\gamma_{o,+,2n}\,,\qquad\tilde{\gamma}_{o,-}^{v}(2gt)=\sum_{n=1}^{\infty}2(2n-1)J_{2n-1}(2gt)\tilde{\gamma}_{o,-,2n-1}\,.$
(84)

Using the definition of the BES kernel (27) and (84), we can rewrite (81) in
the following way

$\displaystyle\gamma_{o,+,2n}^{v}+2\int_{0}^{\infty}\frac{dt}{t}\frac{J_{2n}(2gt)\gamma_{o,+}^{v}}{e^{t}-1}$
$\displaystyle=\kappa_{+,2n}^{v}\,,$ (85)
$\displaystyle\tilde{\gamma}_{o,-,2n-1}^{v}+2\int_{0}^{\infty}\frac{dt}{t}\frac{J_{2n-1}(2gt)\tilde{\gamma}_{o,-}^{v}}{e^{t}-1}$
$\displaystyle=\tilde{\kappa}_{-,2n-1}^{v}\,.$

These equations can be brought into the form

$\int_{0}^{\infty}\frac{dt}{t}(\cos(ut)-J_{0}(2gt))\left[\gamma_{o,+}^{v}(2gt)\,\frac{e^{t}+1}{e^{t}-1}+\frac{\cos(vt)e^{t/2}-J_{0}(2gt)}{e^{t}-1}\right]=0$
(86)

and

$\int_{0}^{\infty}\frac{dt}{t}\sin(ut)\left[\tilde{\gamma}_{o,-}^{v}(2gt)\,\frac{e^{t}+1}{e^{t}-1}+\frac{\sin(vt)e^{t/2}}{e^{t}-1}\right]=0\,.$
(87)

Both equations only hold for $u\leq(2g)^{2}$. These relations will be later
used to find the strong-coupling behavior of $\gamma_{o,+}^{v}$ and
$\tilde{\gamma}_{o,-}^{v}$, as well as to determine how the mirror
transformation acts on $f_{5}$ and $f_{6}$ (32), which we can now rewrite in
the following way

$f_{5}(u,v)=\int_{0}^{\infty}\frac{dt}{t}\frac{\cos(ut)\,e^{t/2}-J_{0}(2gt)}{e^{t}-1}\,\gamma_{o,+}^{v}(2gt)\,,$
(88)

and similarly

$f_{6}(u,v)=-\,\int_{0}^{\infty}\frac{dt}{t}\frac{\sin(ut)\,e^{t/2}}{e^{t}-1}\,\tilde{\gamma}_{o,-}^{v}(2gt)\,.$
(89)

##  Appendix C Crossing symmetry

In this appendix, we check that the finite-coupling solution (34) we found for
the scalar FF transition indeed satisfies the crossing axiom (5).121212A
similar calculation for the S-matrix can be found in appendix B.2 of
Basso:2013pxa .

The mirror transformation maps the rapidity $u$ of a scalar excitation to
$u^{\pm\gamma}=u\pm i$. While this operation seems rather straightforward, one
has to keep in mind that it involves passing through cuts connecting the
points $-2g\pm i/2$ and $2g\pm i/2$, which would not be visible from within
perturbation theory. To perform this delicate operation, we split the mirror
transformation into three distinct steps: shifting $u$ all the way up to the
cut, actually passing through the cut, and the remaining $\pm\frac{i}{2}$
shift that completes the mirror transformation.

We first follow this prescription to compute the mirror transformation acting
on $f_{6}$, using its integral representation (89) in terms of
$\tilde{\gamma}_{o,-}^{v}$. We define $u^{\pm}=u\pm\frac{i}{2}\mp i0$ and
using

$\sin(u^{\pm}t)=\sin(ut)e^{-t/2}\pm ie^{\mp iut}\sinh(t/2)$ (90)

find

$f_{6}(u^{\pm},v)=-\int_{0}^{\infty}\frac{dt}{t}\frac{\sin(ut)\tilde{\gamma}_{o,-}^{v}(2gt)}{e^{t}-1}\mp\frac{i}{2}\int_{0}^{\infty}\frac{dt}{t}e^{\mp
iut}\tilde{\gamma}_{o,-}^{v}(2gt)\,.$ (91)

Next, we move through the cut using

$\mp\frac{i}{2}\int_{0}^{\infty}\frac{dt}{t}e^{\mp
iut}\tilde{\gamma}_{o,-}^{v}(2gt)=\mp\frac{i}{2}\int_{0}^{\infty}\frac{dt}{t}e^{\pm
iut}\tilde{\gamma}_{o,-}^{v}(2gt)-\int_{0}^{\infty}\frac{dt}{t}\sin(ut)\tilde{\gamma}_{o,-}^{v}(2gt)\,,$
(92)

leading to

$\displaystyle f_{6}(u^{\pm},v)$
$\displaystyle=-\int_{0}^{\infty}\frac{dt}{t}\frac{\sin(ut)\tilde{\gamma}_{o,-}^{v}(2gt)}{e^{t}-1}\mp\frac{i}{2}\int_{0}^{\infty}\frac{dt}{t}e^{\pm
iut}\tilde{\gamma}_{o,-}^{v}(2gt)-\int_{0}^{\infty}\frac{dt}{t}\sin(ut)\tilde{\gamma}_{o,-}^{v}(2gt)$
$\displaystyle=\int_{0}^{\infty}\frac{dt}{t}\frac{\sin(u^{\mp}t)e^{t/2}\tilde{\gamma}_{o,-}^{v}(2gt)}{e^{t}-1}-\int_{0}^{\infty}\frac{dt}{t}\sin(ut)\frac{e^{t}+1}{e^{t}-1}\tilde{\gamma}_{o,-}^{v}(2gt)$
$\displaystyle=\int_{0}^{\infty}\frac{dt}{t}\frac{\sin(u^{\mp}t)e^{t/2}\tilde{\gamma}_{o,-}^{v}(2gt)}{e^{t}-1}+\int_{0}^{\infty}\frac{dt}{t}\sin(ut)\frac{\sin(vt)e^{t/2}}{e^{t}-1}\,,$
(93)

where in the last step we used (87). Performing the final shift by
$\pm\frac{i}{2}$, we notice that the first term in the last line gives us back
the original $f_{6}$. As a result, we find

$\displaystyle f_{6}(u^{\pm\gamma},v)$
$\displaystyle=-f_{6}(u,v)+\int_{0}^{\infty}\frac{dt}{t}\sin(u^{\pm}t)\frac{\sin(vt)e^{t/2}}{e^{t}-1}\,.$
(94)

Taking the difference between the $+$ and $-$ versions of this equation gives
us

$f_{6}(u^{\gamma},v)-f_{6}(u^{-\gamma},v)=i\int_{0}^{\infty}\frac{dt}{t}\cos(ut)\sin(vt)\,.$
(95)

We now repeat the same derivation for $f_{5}$ using its integral
representation (88) in terms of $\gamma_{o,+}^{v}$. We have

$\cos(u^{\pm}t)=\cos(ut)e^{-t/2}+e^{\mp iut}\sinh(t/2)\,.$ (96)

Using it, we find

$f_{5}(u^{\pm},v)=\int_{0}^{\infty}\frac{dt}{t}\frac{\cos(ut)-J_{0}(2gt)}{e^{t}-1}\gamma_{o,+}^{v}(2gt)+\frac{1}{2}\int_{0}^{\infty}\frac{dt}{t}e^{\mp
iut}\gamma_{o,+}^{v}(2gt)\,.$ (97)

To cross through the cut, we use

$\frac{1}{2}\int_{0}^{\infty}\frac{dt}{t}e^{\mp
iut}\gamma_{o,+}^{v}(2gt)=-\frac{1}{2}\int_{0}^{\infty}\frac{dt}{t}e^{\pm
iut}\gamma_{o,+}^{v}(2gt)+\int_{0}^{\infty}\frac{dt}{t}\cos(ut)\gamma_{o,+}^{v}(2gt)\,.$
(98)

We thus have

$\displaystyle f_{5}(u^{\pm},v)$
$\displaystyle=\int_{0}^{\infty}\frac{dt}{t}\frac{\cos(ut)-J_{0}(2gt)}{e^{t}-1}\gamma_{o,+}^{v}(2gt)$
(99)
$\displaystyle\phantom{{}=}-\frac{1}{2}\int_{0}^{\infty}\frac{dt}{t}e^{\pm
iut}\gamma_{o,+}^{v}(2gt)+\int_{0}^{\infty}\frac{dt}{t}\cos(ut)\gamma_{o,+}^{v}(2gt)$
$\displaystyle=-\int_{0}^{\infty}\frac{dt}{t}\frac{\cos(u^{\mp}t)e^{t/2}-J_{0}(2gt)}{e^{t}-1}\gamma_{o,+}^{v}(2gt)$
$\displaystyle\phantom{{}=}+\int_{0}^{\infty}\frac{dt}{t}\left(\cos(ut)-J_{0}(2gt)\right)\frac{e^{t}+1}{e^{t}-1}\gamma_{o,+}^{v}(2gt)$
$\displaystyle=-\int_{0}^{\infty}\frac{dt}{t}\frac{\cos(u^{\mp}t)e^{t/2}-J_{0}(2gt)}{e^{t}-1}\gamma_{o,+}^{v}(2gt)$
$\displaystyle\phantom{{}=}-\int_{0}^{\infty}\frac{dt}{t}\left(\cos(ut)-J_{0}(2gt)\right)\frac{\cos(vt)e^{t/2}-J_{0}(2gt)}{e^{t}-1}\,,$

where in the second line we used that
$\int_{0}^{\infty}\frac{dt}{t}J_{0}(2gt)\gamma_{o,+}^{v}(2gt)=0$ due to the
orthogonality of the Bessel functions, and in the last line we have used (86).
The final shift results in

$f_{5}(u^{\pm\gamma},v)=-f_{5}(u,v)-\int_{0}^{\infty}\frac{dt}{t}\left(\cos(u^{\pm}t)-J_{0}(2gt)\right)\frac{\cos(vt)e^{t/2}-J_{0}(2gt)}{e^{t}-1}\,.$
(100)

This implies

$f_{5}(u^{\gamma},v)-f_{5}(u^{-\gamma},v)=i\int_{0}^{\infty}\frac{dt}{t}\sin(ut)\cos(vt)-i\int_{0}^{\infty}\frac{dt}{t}e^{-t/2}J_{0}(2gt)\sin(ut)\,.$
(101)

Combining $f_{5}$ and $f_{6}$, we find

$\displaystyle f_{5}(u^{\gamma},v)$ $\displaystyle-
f_{5}(u^{-\gamma},v)-f_{6}(u^{\gamma},v)+f_{6}(u^{-\gamma},v)$ (102)
$\displaystyle=i\int_{0}^{\infty}\frac{dt}{t}\sin((u-v)t)-i\int_{0}^{\infty}\frac{dt}{t}e^{-t/2}J_{0}(2gt)\sin(ut)$
$\displaystyle=\frac{i\pi}{2}\,{\rm
sign}(u-v)-i\int_{0}^{\infty}\frac{dt}{t}e^{-t/2}J_{0}(2gt)\sin(ut)\,.$

Analytic continuation of $f_{5}(u,u)$ and $f_{6}(u,u)$ can be done in steps by
first performing the mirror transformation in $u$, then in $v$ and then
setting $v=u$. For $f_{6}$, this results in

$f_{6}(u^{\pm\gamma},v^{\pm\gamma})-f_{6}(u,v)=\pm\,\frac{i}{2}\int_{0}^{\infty}\frac{dt}{t}\sin((u+v\pm
i)t)\,,$ (103)

which leads to

$f_{6}(u^{\gamma},v^{\gamma})-f_{6}(u^{-\gamma},v^{-\gamma})=i\int_{0}^{\infty}\frac{dt}{t}\cosh(t)\sin((u+v)t)\,.$
(104)

Similarly,

$\displaystyle f_{5}(u^{\pm\gamma},v^{\pm\gamma})-f_{5}(u,v)=$
$\displaystyle\pm\frac{i}{2}\int_{0}^{\infty}\frac{dt}{t}\sin((u+v\pm
i)t)+\frac{1}{2}\log{g^{2}}$ (105)
$\displaystyle-\int_{0}^{\infty}\frac{dt}{t}\left(\frac{1}{2}\,J_{0}(2gt)\left(e^{\pm
iu^{\pm}t}+e^{\pm iv^{\pm}t}\right)-e^{-t}\right)\,.$

Deriving this equation involves a number of subtle steps, which we briefly
outline here.131313We thank B. Basso for useful discussions of these steps.
After the mirror transformation in $u$ is performed, the second mirror
transformation in $v$ acts on both terms in (100). In the second term, simply
replacing $v$ with $v\pm i$ would not lead to the correct result, due to this
integral having a cut between $-2g$ and $2g$. One can see, however, that $v\to
v\pm i$ gives the correct result for almost all terms in this integral, with
an exception of the one term that comes from integrating $J_{0}(2gt)$ with
$e^{\pm ivt}$. This integral by itself is divergent at $t=0$, so we can add a
$v$-independent regulator $e^{-t}$ to it to cancel the divergence, while
subtracting the same regulator from the rest of the terms. The analytic
continuation through the cut can then be performed using the following
relation:

$\displaystyle\int\limits_{0}^{\infty}\frac{dt}{t}\left(J_{0}(2gt)\,e^{\pm
i(v\pm
i0)t}-e^{-t}\right)=-\log{g^{2}}+\int\limits_{0}^{\infty}\frac{dt}{t}\left(-\,J_{0}(2gt)\,e^{\mp
i(v\mp i0)t}+e^{-t}\right).$ (106)

This then leads to (105). The difference between the positive and negative
transformations is given by

$\displaystyle f_{5}(u^{\gamma},v^{\gamma})-f_{5}(u^{-\gamma},v^{-\gamma})$
$\displaystyle=i\int_{0}^{\infty}\frac{dt}{t}\cosh(t)\sin((u+v)t)$ (107)
$\displaystyle\phantom{{}=}-i\int_{0}^{\infty}\frac{dt}{t}e^{-t/2}J_{0}(2gt)\left(\sin(ut)+\sin(vt)\right)\,.$

Combining (104) with (107) and setting $v$ equal to $u$, we find

$\displaystyle f_{5}(u^{\gamma},u^{\gamma})-f_{5}(u^{-\gamma},u^{-\gamma})$
$\displaystyle-
f_{6}(u^{\gamma},u^{\gamma})+f_{6}(u^{-\gamma},u^{-\gamma})=-\,2i\int_{0}^{\infty}\frac{dt}{t}e^{-t/2}J_{0}(2gt)\sin(ut)\,.$
(108)

After inserting (102) and (108) into (33), we see that the remaining integrals
cancel and conclude that

$G(u^{2\gamma},v)^{2}=\exp(2\pi i\,{\rm sign}(u-v))\,G(u,v)^{2}=G(u,v)^{2}\,,$
(109)

but

$G(u^{2\gamma},v)=\exp(\pi i\,{\rm sign}(u-v))\,G(u,v)=-\,G(u,v)\,.$ (110)

This relation is exactly what we expect from the minimal solution at strong
coupling, which is presented in section 3.4.

##  Appendix D Strong-coupling limit

In this appendix, we find the strong-coupling limit of our finite-coupling
solution for the scalar two-particle FF transition.

We start with the two equations (86) and (87). They have the following two
particular solutions:

$\gamma_{o,+,\text{part}}^{v}=-\frac{\cos(vt)e^{t/2}-J_{0}(2gt)}{e^{t}+1}\,,\qquad\tilde{\gamma}_{o,-,\text{part}}^{v}=-\frac{\sin(vt)e^{t/2}}{e^{t}+1}\,.$
(111)

Following Basso:2013pxa , one would expect that the physical solutions differ
from the particular solutions by a homogeneous solution that is exponentially
suppressed at strong coupling and hence does not play a role in the strong-
coupling limit.

For strong coupling, assuming the homogeneous solution to be suppressed, we
can obtain the following simple integrals for the combinations of $f_{5}$ and
$f_{6}$ that enter (33):

$\displaystyle 2f_{5}(u,v)$ $\displaystyle-
f_{5}(u,u)-f_{5}(v,v)=\frac{1}{2}\int\limits_{0}^{\infty}\frac{dt}{t}\frac{\left({\rm
cos}(ut)-{\rm cos}(vt)\right)^{2}}{{\rm sinh}(t)}$
$\displaystyle=\frac{1}{2}\,{\rm log}\,\left[{\rm
cosh}\left(\frac{\pi}{2}\,(u+v)\right){\rm
cosh}\left(\frac{\pi}{2}\,(u-v)\right)\right]-\frac{1}{4}\,{\rm log}\left[{\rm
cosh}(\pi u)\,{\rm cosh}(\pi v)\right],$ (112) $\displaystyle 2f_{6}(u,v)$
$\displaystyle-
f_{6}(u,u)-f_{6}(v,v)=-\,\frac{1}{2}\int\limits_{0}^{\infty}\frac{dt}{t}\frac{\left({\rm
sin}(ut)-{\rm sin}(vt)\right)^{2}}{{\rm sinh}(t)}$
$\displaystyle=\frac{1}{2}\,{\rm log}\left[\frac{{\rm
cosh}\left(\frac{\pi}{2}\,(u+v)\right)}{{\rm
cosh}\left(\frac{\pi}{2}\,(u-v)\right)}\right]-\frac{1}{4}\,{\rm
log}\left[{\rm cosh}(\pi u)\,{\rm cosh}(\pi v)\right].$ (113)

Inserting these expressions into (33) gives

$\displaystyle G(u,v)={\rm cosh}\left(\frac{\pi}{2}\,(u-v)\right).$ (114)

This is exactly what we expected from the minimal solution from the $O(6)$
non-linear sigma model, cf. section 3.4.

##  Appendix E Pentagon matrix parts and contractions

In this appendix, we collect the matrix part of the creation pentagons, which
are required for the matching with data in subsection 4.4. Moreover, we
provide some of the more non-trivial contractions between the matrix parts of
the creation pentagon and the annihilation FF transition. We are considering
order $e^{-4\tau}$, where states of twist four occur. These are the
combinations $F_{2}\bar{F}_{2}$, $FF\bar{F}\bar{F}$, $F\psi\bar{\psi}\bar{F}$,
$F\phi\bar{\phi}\bar{F}$, $\psi\psi\bar{\psi}\bar{\psi}$,
$\psi\phi\bar{\phi}\bar{\psi}$ and $\phi\phi\bar{\phi}\bar{\phi}$. Note that
in addition, the fermions in these expressions can have either large or small
momenta.

###  E.1 Matrix parts

The matrix parts of the pentagon transitions that create the aforementioned
states have been computed in Basso:2013aha ; Belitsky:2016vyq ; Sever:2021nsq
.

The matrix part arises for $SU_{R}(4)$-charged fundamental excitations and is
trivial for pentagons involving gluons. The matrix part for a state involving
two gluons and two non-gluons is identical to the one for the state involving
only the two non-gluons.

The two-particle matrix parts have the following form:

$\displaystyle\Pi_{0|F\bar{F}}(0|v_{1},v_{2})$ $\displaystyle=1,$ (115)
$\displaystyle\Pi_{0|\psi\bar{\psi}}(0|v_{1},v_{2})$
$\displaystyle=\frac{i}{v_{1}-v_{2}+2i},$
$\displaystyle\Pi_{0|\phi\bar{\phi}}(0|v_{1},v_{2})$
$\displaystyle=-\,\frac{1}{(v_{1}-v_{2}+i)\,(v_{1}-v_{2}+2i)}.$

The matrix part for
$\psi^{A_{1}}\psi^{A_{2}}\bar{\psi}_{B_{1}}\bar{\psi}_{B_{2}}$ is
Belitsky:2016vyq

$\displaystyle\Pi_{0|\psi\psi\bar{\psi}\bar{\psi}}(0|u_{1},u_{2},v_{1},v_{2})^{A_{1}A_{2}}_{B_{1}B_{2}}$
$\displaystyle=\delta^{A_{1}}_{B_{1}}\delta^{A_{2}}_{B_{2}}\,\pi_{1}+\delta^{A_{1}}_{B_{2}}\delta^{A_{2}}_{B_{1}}\,\pi_{2},$
(116)

where

$\displaystyle\pi_{1}$
$\displaystyle=-\,\frac{1}{\prod_{j,k=1}^{2}(u_{j}-v_{k}+2i)}\,\frac{(u_{1}-v_{2}+3i)\,(u_{2}-v_{1}+2i)}{(u_{1}-u_{2}+i)\,(v_{1}-v_{2}+i)}\,,$
(117) $\displaystyle\pi_{2}$
$\displaystyle=\frac{1}{\prod_{j,k=1}^{2}(u_{j}-v_{k}+2i)}\,\frac{(u_{1}-v_{1}+3i)\,(u_{2}-v_{2}+3i)-i\,(u_{2}-v_{1})+2}{(u_{1}-u_{2}+i)\,(v_{1}-v_{2}+i)}\,.$

Similarly, for scalars, Basso:2013aha

$\displaystyle\Pi_{0|\phi\phi\phi\phi}(0|u_{1},u_{2},u_{3},u_{4})_{i_{1}i_{2}i_{3}i_{4}}$
$\displaystyle=\delta_{i_{1}i_{2}}\delta_{i_{3}i_{4}}\,\theta_{1}+\delta_{i_{1}i_{3}}\delta_{i_{2}i_{4}}\,\theta_{2}+\delta_{i_{1}i_{4}}\delta_{i_{2}i_{3}}\,\theta_{3},$
(118)

where

$\displaystyle\theta_{1}$
$\displaystyle=\frac{(u_{1}-u_{4}+3i)\,(u_{2}-u_{3}+2i)\left[(u_{1}-u_{3}+2i)\,(u_{2}-u_{4}+3i)+i\,(u_{3}-u_{4})-2\right]}{\prod_{1\leq
j<k\leq 4}(u_{j}-u_{k}+i)(u_{j}-u_{k}+2i)},$ $\displaystyle\theta_{2}$
$\displaystyle=-\,\frac{(u_{1}-u_{2}+2i)\,(u_{2}-u_{3}+2i)\,(u_{3}-u_{4}+2i)\,(u_{1}-u_{4}+3i)}{\prod_{1\leq
j<k\leq 4}(u_{j}-u_{k}+i)(u_{j}-u_{k}+2i)},$ (119) $\displaystyle\theta_{3}$
$\displaystyle=\frac{(u_{1}-u_{2}+2i)\,(u_{3}-u_{4}+2i)\left[(u_{1}-u_{3}+2i)\,(u_{2}-u_{4}+3i)+i\,(u_{3}-u_{4})-1\right]}{\prod_{1\leq
j<k\leq 4}(u_{j}-u_{k}+i)(u_{j}-u_{k}+2i)}.$

Lastly, the matrix part for creating the mixed scalar-fermion state is
Sever:2021nsq

$\displaystyle\Pi_{0|\psi\phi\phi\bar{\psi}}(0|v_{1},u_{1},u_{2},v_{2})=$
$\displaystyle-\frac{3i}{4}\,\frac{1}{\left(u_{1}-v_{1}-\frac{3i}{2}\right)\left(u_{2}-v_{1}-\frac{3i}{2}\right)\left(u_{1}-v_{2}+\frac{3i}{2}\right)\left(u_{2}-v_{2}+\frac{3i}{2}\right)}$
$\displaystyle\times\frac{(u_{1}+u_{2})\,(v_{1}+v_{2})-2\,u_{1}u_{2}-2\,v_{1}v_{2}+5i\,(v_{1}-v_{2})-\frac{21}{2}}{(u_{1}-u_{2}+i)\,(u_{1}-u_{2}+2i)\,(v_{1}-v_{2}+3i)}\,.$
(120)

The most non-trivial four-particle matrix parts that arise are thus for four
scalars, two scalars and two fermions, as well as for four fermions.

###  E.2 Non-trivial contractions

Most contractions of the matrix parts of the creation amplitude and the FF
transition are trivial. The only noteworthy cases correspond to four fermions
and four scalars.

For four fermions, contracting the last line of (64) with (116) yields

$\displaystyle\Pi_{0|\psi\psi\bar{\psi}\bar{\psi}}(0|u_{1},u_{2},v_{1},v_{2})^{A_{1}A_{2}}_{B_{1}B_{2}}F_{\psi^{\text{}}_{A_{1}}\psi^{\prime}_{A_{2}}\bar{\psi}^{\prime}_{B_{1}}\bar{\psi}^{\text{}}_{B_{2}}}(u_{1},u_{2},v_{1},v_{2})$
(121)
$\displaystyle=\frac{4\,(u_{1}-u_{2})^{2}+22}{\left((u_{1}-u_{2})^{2}+1\right)\left((u_{1}-u_{2})^{2}+4\right)}$
$\displaystyle\times\frac{\delta(u_{1}-v_{2})\,\delta(u_{2}-v_{1})-S_{\psi\psi}(u_{1},u_{2})\,\delta(u_{1}-v_{2})\,\delta(u_{2}-v_{1})}{\mu_{\psi}(u_{1})\mu_{\psi}(u_{2})}\,,$

where we used that Belitsky:2016vyq

$S_{B_{1}B_{2}}^{A_{1}A_{2}}(u,v)=S_{\psi\psi}(u,v)\left[\delta_{B_{1}}^{A_{1}}\delta_{B_{2}}^{A_{2}}s^{(1)}_{\psi\psi}(u,v)+\delta_{B_{2}}^{A_{1}}\delta_{B_{1}}^{A_{2}}s^{(2)}_{\psi\psi}(u,v)\right]\,,$
(122)

with

$s^{(1)}_{\psi\psi}=\frac{u-v}{u-v-i}\,,\quad
s^{(2)}_{\psi\psi}=\frac{-\,i}{u-v-i}\,.$ (123)

When multiplying (121) with the dynamical part of the pentagon transition, the
two parts in (121) that are related via the S-matrix give identical
contributions, such that one could drop the second term in favour of a factor
of $2$.

For four scalars, we contract
$\Pi_{i_{1}i_{2}i_{3}i_{4}}(u_{1},u_{2},u_{3},u_{4})$ in (59) with (118),
finding

$\displaystyle\Pi_{0|\phi\phi\phi\phi}(0|u_{1},u_{2},u_{3},u_{4})_{i_{1}i_{2}i_{3}i_{4}}\Pi_{i_{1}i_{2}i_{3}i_{4}}(u_{1},u_{2},u_{3},u_{4})$
(124) $\displaystyle=\frac{9}{\prod_{1\leq j<k\leq
4}(u_{j}-u_{k}+i)(u_{j}-u_{k}+2i)}\sum_{j=1}^{4}\biggl{(}2u_{j}^{2}u_{j+1}^{2}-7u_{j}^{2}u_{j+1}u_{j+2}+3u_{j}u_{j+1}^{2}u_{j+2}$
$\displaystyle\qquad-7u_{j}u_{j+1}u_{j+2}^{2}+\frac{7}{2}u_{j}^{2}u_{j+2}^{2}+4u_{j}^{2}+\frac{11}{2}u_{j}u_{j+1}u_{j+2}u_{j+3}+14u_{j}u_{j+1}-18u_{j}u_{j+2}+16\biggr{)}$

Note that this expression is not real, only becoming real after it is
multiplied by the factors $(u_{j}-u_{k}-i)(u_{j}-u_{k}-2i)$ in
$F_{\phi\bar{\phi}}(u_{j},u_{k})$ in the dynamical part. The other two
structures in (50),
$\Pi^{\sigma_{34}}_{i_{1}i_{2}i_{3}i_{4}}(u_{1},u_{2},u_{3},u_{4})$ and
$\Pi^{\sigma_{23}}_{i_{1}i_{2}i_{3}i_{4}}(u_{1},u_{2},u_{3},u_{4})$, are
related to the one above via Watson, and they thus yield identical
contributions after including also the dynamical part. Thus, they effectively
contribute a factor of $3$.

##  Appendix F R-matrices with fermions

In this appendix, we summarize the matrix form of the S-matrices with fermions
that are used in section 4.2. For more details, we refer the reader to
appendix A of Basso:2014koa .

The S-matrix between two fermions or between a fermion and an anti-fermion
factorizes into a coupling-dependent part and the R-matrix141414It is a non-
trivial fact that the coupling-dependent parts of these two are the same.

$S_{\psi\psi}(u,v)_{AB}^{CD}=S_{\psi\psi}(u,v)R_{{\bf
44}}(u-v)_{AB}^{CD}\,,\qquad
S_{\psi\bar{\psi}}(u,v)_{AD}^{CB}=S_{\psi\psi}(u,v)R_{{\bf 4}\bar{\bf
4}}(u-v)_{AD}^{CB}\,,$ (125)

where

$R_{{\bf 44}}(w)_{AB}^{CD}={w\over w-i}\delta_{A}^{C}\delta_{B}^{D}-{i\over
w-i}\delta_{A}^{D}\delta_{B}^{C}\,,\qquad R_{{\bf 4}\bar{\bf
4}}(w)_{AD}^{CB}=\delta_{A}^{C}\delta_{D}^{B}-{i\over
w-2i}\delta_{A}^{B}\delta_{D}^{C}\,.$ (126)

The conjugated R-matrices are trivially related to the ones given above,

$R_{{\bar{\bf 4}\bar{\bf 4}}}(w)^{AB}_{CD}=R_{{\bf 44}}(w)^{AB}_{CD}\,,\qquad
R_{\bar{\bf 4}{\bf 4}}(w)_{CB}^{AD}=R_{{\bf 4}\bar{\bf 4}}(w)_{BC}^{DA}\,,$
(127)

Similarly, for a fermion and a scalar, we have

$S_{\phi\psi}(u,v)_{iA}^{jB}=S_{\phi\psi}(u,v)R_{\textbf{64}}(u-v)_{iA}^{jB}\,,\qquad
R_{\textbf{64}}(w)_{iA}^{jB}=\delta_{i}^{j}\delta_{A}^{B}+\frac{i}{2w-3i}\rho_{iAC}\rho^{jCB}\,,$
(128)

The $\rho$-matrices entering the R-matrices (128) are the off-diagonal
components of the 6D Dirac $\gamma$-matrices in the Weyl representation. They
satisfy the Clifford algebra

$\rho_{iAC}\rho^{jCB}+\rho_{jAC}\rho^{iCB}=2\delta_{i}^{j}\delta_{A}^{B}\,,\qquad\textrm{where}\qquad\rho^{iAB}=-\rho^{iBA}=-\rho^{*}_{iAB}\,,$
(129)

and are usually referred to as ’t Hooft symbols.

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