?

A Large Twist Limit for Any Operator

Gwenaël Ferrandoa, Amit Severa, Adar Sharonb, and Elior Urismana

a School of Physics and Astronomy, Tel Aviv University, Ramat Aviv 69978,
Israel
b Simons Center for Geometry and Physics, SUNY, Stony Brook, NY 11794, USA

Abstract

We argue that for any single-trace operator in ${\cal N}=4$ SYM theory there
is a large twist double-scaling limit in which the Feynman graphs have an
iterative structure. Such structure can be recast using a graph-building
operator. Generically, this operator mixes between single trace operators with
different scaling limits. The mixing captures both the finite coupling
spectrum and corrections away from the large twist limit. We first consider a
class of short operators with gluons and fermions for which such mixing
problems do not arise, and derive their finite coupling spectra. We then focus
on a class of long operators with gluons that do mix. We invert their graph-
building operator and prove its integrability. The picture that emerges from
this work opens the door to a systematic expansion of ${\cal N}=4$ SYM theory
around the large twist limit.

###### Contents

  1. 1 Introduction
  2. 2 Short Operators
    1. 2.1 The Operator ${\rm tr\,}(XX^{\dagger}Z)$
    2. 2.2 The Operator ${\rm tr\,}(F_{\mu\nu}Z)$
      1. 2.2.1 Spectrum
  3. 3 Mixing Between Operators and Between Scaling Limits
    1. 3.1 Integrability
    2. 3.2 Higher-Order Corrections
  4. 4 More Examples Without Mixing
    1. 4.1 The Operators ${\rm tr\,}(Z\psi_{4})$ and ${\rm tr\,}(Z\psi_{1}^{\dagger})$
    2. 4.2 The Operators ${\rm tr\,}(Z\psi_{2})$ and ${\rm tr\,}(Z\psi_{3}^{\dagger})$
    3. 4.3 The Operators ${\rm tr\,}(\psi_{4}Z^{J})$
    4. 4.4 The Operator ${\rm tr\,}(\psi_{4}\psi_{1}^{\dagger}Z)$
  5. 5 Holography and the Fishchain Model
  6. 6 Conclusion and Discussion
  7. A Notation and Conventions
    1. A.1 Metric and Sigma Matrices
      1. A.1.1 4 Dimensions
      2. A.1.2 6 Dimensions
    2. A.2 $\gamma$-Deformed $\mathcal{N}=4$ SYM
    3. A.3 Feynman Rules
  8. B Star-Triangle Relations
  9. C Correlators
    1. C.1 The Correlator of Two Length-Two Non-local Operators
    2. C.2 The Operator ${\rm tr\,}(XX^{\dagger}Z)$
    3. C.3 The Operator ${\rm tr\,}(Z\psi_{4})$
  10. D Integrability
    1. D.1 General Construction
    2. D.2 Principal Series Representations
    3. D.3 Lax Matrices for Operators Without Gluons
    4. D.4 Integrability for ${\rm tr\,}(F_{\mu\nu}Z)$
  11. E One-Loop Checks of the Spectrum
    1. E.1 The operators ${\rm tr\,}(ZXX^{\dagger})$ and ${\rm tr\,}(ZX^{\dagger}X)$
    2. E.2 The operators ${\rm tr\,}(Z^{\dagger}X^{\dagger}\psi_{1\alpha})$ and ${\rm tr\,}(Z^{\dagger}\psi_{1\alpha}X^{\dagger})$
  12. F Embedding Formalism and the Spectrum of ${\rm tr\,}(F_{\mu\nu}Z)$
    1. F.1 Embedding Formalism
    2. F.2 Spectrum
  13. G Graph-Building Operator for Section 3

##  1 Introduction

Due to the enormous attention the holographic principle has received in the
last two decades, we have a fairly complete dictionary between the bulk and
boundary observables, at least in the large $N$ limit. Yet knowing and testing
the holographic map is not the same as understanding it. Even in the planar
limit, we are still lacking a first-principle derivation of the duality. Such
a derivation, however, was achieved for a simplifying limit of
$\gamma$-deformed ${\cal N}=4$ SYM theory, called the fishnet theory [1].
Taking such a limit, with simplified dynamics, comes at a price – it only
captures a subset of the observables and has more restrictive physical
phenomena. Hence, the question remains how to extend this derivation to the
full theory. In this paper, we take the first step in this direction.
Explicitly, we show that the introduction of a so-called “twist” into the
planar diagrams allows us to extend the results from the fishnet theory to any
operator in ${\cal N}=4$ SYM theory.

A twist is defined using a symmetry generator and a continuous parameter,
$e^{\operatorname{i}\theta}$, [2]. It has the effect of giving different
weights to different diagrams in a physically meaningful way, see appendix A.2
and [1, 2, 3] for details. Taking a large twist double-scaling limit, in
particular, makes it possible to project to a certain subclass of diagrams.
This limit is the limit in which we simultaneously take the twist parameter to
be large and the ’t Hooft coupling $g^{2}=g_{\text{YM}}^{2}N/(4\pi)^{2}$ to be
small as

$e^{\operatorname{i}\theta}\to\infty\,,\qquad\text{with}\quad\xi_{n}^{2}\equiv{g^{2}\,e^{\operatorname{i}\frac{\theta}{n}}\over
8\pi^{2}}\quad\text{fixed}\,,$ (1.1)

where $n>0$ depends on the operator. The remaining diagrams enter in a
systematic expansion around this double-scaling limit.

The case of $n=1$ with a certain $R$-symmetry twist is the so-called “fishnet”
limit that has been extensively studied in the literature [1, 4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. It was found to preserve the
integrability of the planar theory, [20, 21, 22, 23, 24]. Moreover, it turned
out to have a holographic dual description in terms of a chain of point
particles in $AdS_{5}$, [25, 26, 27, 28]. Notably, both of these properties
have been derived from first principles.

To extend these derivations to the mother ${\cal N}=4$ SYM theory, we need to
gradually turn on the ’t Hooft coupling $g^{2}$ while holding $\xi_{1}^{2}$
fixed. However, we then have to face the following problem. Most of the
operators of the ${\cal N}=4$ SYM theory remain trivial in the fishnet limit.
The perturbative (in $g$) expansion around the fishnet limit is analogous to
the perturbative expansion around the free theory for them. Hence, we did not
gain much by first taking the limit (1.1). Here, we suggest an alternative
constructive path towards ${\cal N}=4$ SYM theory. It involves a
reorganisation of the perturbative expansion in an operator-dependent way. If
all orders in this expansion are resumed, the full (twisted) ${\cal N}=4$ SYM
result is reconstructed.

Concretely, we consider two-point functions of single-trace operators. We
argue that, for any operator ${\cal O}$, there is some twist generator and
some $n_{\cal O}>0$ such that in the corresponding double-scaling limit (1.1),
the diagrams that survive have an iterative structure. For most of the
operators, taking their double-scaling limit involves a mixing problem with
sub-leading loop corrections to operators with the same twist generator, but
lower $n_{\cal O}$.

Diagrams with an iterative structure can be generated by a so-called graph-
building operator. Such an operator is expected to exist even for the full
${\cal N}=4$ SYM mother theory. However, in that case, it is an infinite by
infinite matrix, and therefore of very little practical use. Nevertheless,
turning on a twist gives a grading on this infinite matrix, such that at any
fixed order only a finite matrix remains. This property then allows us to
analyse the matrix in steps. At each step we have to deal with a finite block
of this matrix, which corresponds to a subset of operators. Gradually, more
and more operators are added. Each time an operator ${\cal O}$ is added, it
receives all loop corrections in $\xi_{n_{\cal O}}$ and has a corresponding
holographic description. At the same time, we capture higher loop corrections
in $g$ to operators ${\cal O}^{\prime}$ that were already included to all
orders in their $\xi_{n_{{\cal O}^{\prime}}}$.

To demonstrate this structure, consider a toy example of an $SU(N)$ gauge
theory consisting of two complex scalars, $Z$ and $X$, transforming in the
adjoint representation:

${\cal L}=-N\,{\rm
tr\,}\Big{[}\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+(D^{\mu}Z)^{\dagger}D_{\mu}Z+(D^{\mu}X)^{\dagger}D_{\mu}X-2g^{2}(X^{\dagger}Z^{\dagger}XZ+Z^{\dagger}X^{\dagger}ZX)\Big{]}\,,$
(1.2)

where $D_{\mu}=\partial_{\mu}+\operatorname{i}g[A_{\mu},\cdot]$ is the
covariant derivative. In the planar limit, the rank $N\to\infty$ while the ’t
Hooft coupling $g^{2}$ is held fixed. In general, twisting a single-trace
operator by a symmetry transformation is defined at the diagrammatic level,
see [2] for details. Here, we choose to twist by an internal symmetry under
which the $X$ and $Z$ have charge one. For this choice of symmetry, it can be
shown that twisting is equivalent to deforming the interaction term in the
Lagrangian (1.2) as

$2g^{2}(X^{\dagger}Z^{\dagger}XZ+Z^{\dagger}X^{\dagger}ZX)\quad\rightarrow\quad
2g^{2}(e^{\operatorname{i}\theta}X^{\dagger}Z^{\dagger}XZ+e^{-\operatorname{i}\theta}Z^{\dagger}X^{\dagger}ZX)\,,$
(1.3)

where $\theta$ is the twist angle. Consider for example the diagrams in figure
1 that contribute to the two-point function of the operator ${\rm
tr\,}\\!\\!\left(Z^{2}\right)$. The diagram in figure 1.a scales as
$g^{4}e^{2\operatorname{i}\theta}$ whereas the diagram in 1.b scales as
$g^{2}$. Hence, only the first survives the fishnet double-scaling limit,
(1.1) with $n=1$. At higher loop orders, the only diagrams that survive in the
fishnet limit are the wheel-type diagrams in figure 1.c . They are all
generated by iteration of the fishnet graph-building operator. This operator,
that acts on two scalars, adds one more wheel to the diagram. The gauge field,
on the other hand, decouples from any other correlator and we remain with the
two complex scalars and the first quartic interaction term in (1.2) only. The
corresponding non-trivial local single-trace operators are made from these two
scalars, with no gauge fields.

Figure 1: In the fishnet limit, diagrams that contribute to the two-point
function of the operator ${\rm tr\,}(Z^{2})$ have wheel, or fishnet [1, 4],
structure that can be resummed. The diagram in (a) has one wheel and is of
order $\xi_{1}^{4}$. The diagram in (b), on the other hand, is of order
$g^{2}$ and does not survive the fishnet limit. A typical diagram that does
survive and has many wheels is plotted in (c). Here, the black lines denote
$Z$-propagators and red lines denote $X$-propagators. The arrows indicate the
flow of the $U(1)$ charge of these operators and are pointing towards
$X^{\dagger}$, $Z^{\dagger}$.

Suppose that instead of the operator ${\rm tr\,}\\!\\!\left(Z^{2}\right)$ we
start with the operator

${\cal O}_{\mu\nu}(x)={\rm tr\,}\\!\big{(}F_{\mu\nu}(x)Z(x)\big{)}\,.$ (1.4)

This is an example of an operator that decouples in the fishnet limit because
it includes gluons. However, if we take the limit (1.1) with $n=2$ instead of
the fishnet limit, the gauge field no longer decouples and the operator (1.4)
receives loop corrections in $\xi_{2}$. The corresponding diagrams that
survive in this limit are of wheel-type with gluons, see figure 2. In section
2.2 we study the two-point function between operators of this type in detail
and use the corresponding graph-building operator to resum them exactly. In
particular, we find that in the limit (1.1) with $n=2$, the exact conformal
dimension of ${\cal O}_{\mu\nu}$ is

$\Delta_{\cal O}=2+\sqrt{5-4\sqrt{1+\xi_{2}^{4}}}\,.$ (1.5)

It is worth noting that the operator ${\cal O}_{\mu\nu}$ in (1.4) is in an
antisymmetric representation of the Lorentz group. Consequently, it has zero
overlap with the trace of two $Z$-scalars at arbitrary positions. As a result,
the fishnet wheel diagrams that contribute to the two-point function of ${\rm
tr\,}\\!\\!\left(Z^{2}\right)$ and are associated with the limit (1.1) with
$n=1$, decouple from the two-point function of ${\cal O}_{\mu\nu}$. This is no
longer the case for operators of the form ${\rm
tr\,}\\!\\!\left(F_{\mu\nu}Z^{J}\right)$ with $J>1$. For them, we have to deal
with a mixing problem between operators with different scaling limits. This
mixing is studied in section 3 and involves the operators ${\rm
tr\,}\\!\\!\left(Z^{J}\right)$ and ${\rm
tr\,}\\!\\!\left(XX^{\dagger}Z^{J}\right)$.111These two operators have
different tree-level dimension. Therefore, they do not mix with each other at
the level of the dilatation operator. Instead, the mixing alluded to here
happens at the level of the graph-building operator. It represents a mixing
between these operators with any number of derivatives acting on the fields in
the trace.

Figure 2: Example of a high-loop diagram that contributes to the two-point
function of the operator ${\rm tr\,}(F_{\mu\nu}Z)$ in the double-scaling limit
(1.1) with $n=2$.

In this paper, we focus on a relatively simple subset of single trace
operators, with scalars, gluons, and fermions. The strategy we follow for
studying their two-point functions involves two steps. We first twist them and
then take the appropriate large twist double-scaling limit. We expect that the
structure we find is general. That is, a single-trace local operator can be
twisted by any internal or space-time global symmetry that leaves it
invariant. Using the internal $SU(4)$ $R$-symmetry, an appropriate double-
scaling limit can be applied to any operator. For simplicity, we will only
consider a subset of twists for which the computation of the two-point
function of the twisted operators in the ${\cal N}=4$ SYM theory is the same
as the computation of the untwisted correlator in the $\gamma$-deformed
theory. We refer the reader to [2] for a detailed definition of the colour-
twist method and proof of its equivalence to a $\gamma$-deformation for the
cases we study here.

In section 2, we present three examples of short operators whose scaling limit
is different from the fishnet one. In section 3, we tackle the issue of mixing
alluded to above. In section 4, we study another infinite family of operators
with fermions and a different scaling limit. Lastly, in section 5, we briefly
discuss the holographic dual description of these operators. Various
appendices contain additional details.

##  2 Short Operators

For the examples studied in this paper, it is enough to consider correlators
of operators in the $\gamma$-deformed ${\cal N}=4$ SYM theory, whose
Lagrangian we recall in appendix A.2. In this section, we will focus on the
operators

${\rm tr\,}(XX^{\dagger}Z)\,,\qquad{\rm
tr\,}(X^{\dagger}XZ)\,,\qquad\text{and}\qquad{\rm tr\,}(F_{\mu\nu}Z)\,,$ (2.1)

as well as their generalisations with derivatives. In the fishnet limit ((1.1)
with $n=1$) all of the loop corrections to their conformal dimensions vanish.
Hence, these are examples of operators that do not survive the fishnet limit.

Instead of the fishnet limit, we consider the limit (1.1) with $n=2$. More
explicitly, in terms of the $\gamma$-deformation parameters that we review in
appendix A.2, the limit we consider is

$\gamma_{1}=\gamma_{2}=0\,,\qquad
e^{-\operatorname{i}\gamma_{3}}\to\infty\,,\qquad\text{with}\qquad\xi_{2}^{2}\equiv{g^{2}\,e^{-\operatorname{i}\frac{\gamma_{3}}{2}}\over
8\pi^{2}}\quad\text{fixed}\,.$ (2.2)

In this limit, the correlators involving the operators in (2.1) receive loop
corrections of arbitrarily high order. As in the fishnet theory, these loop
corrections have an iterative structure. In this section, we use the
corresponding graph-building operator to compute the operators’ conformal
dimensions at finite $\xi_{2}$.

###  2.1 The Operator ${\rm tr\,}(XX^{\dagger}Z)$

Single-trace operators composed of a single $Z$, $X$, and an $X^{\dagger}$
field are the simplest examples of operators with a double-scaling limit
different from that of the fishnet. For every spin, we find two twist three
primary operators with a non-trivial anomalous dimension, while the other
operators of this form remain trivial. In order to analyse all non-trivial
operators at once, it is sufficient to study an operator that has some overlap
with all of them. This is achieved by choosing one ordering of the fields in
the trace and placing them at two different spacetime points. The result does
not depend on these choices, as explained later. We choose to use the
operators

${\rm tr\,}\big{(}Z(x)(XX^{\dagger})(y)\big{)}\qquad\text{and}\qquad{\rm
tr\,}\big{(}(Z^{\dagger}X)(x)X^{\dagger}(y)\big{)}\,.$ (2.3)

At finite coupling and twist, operators of this form are not gauge invariant.
Instead, similar operators with Wilson lines between the fields in the trace
can be thought of. However, in the limit (2.2), the Wilson lines decouple, and
one remains with the operators in (2.3).

To compute the non-trivial conformal dimensions of the local operators in this
sector, we study the two-point function of the non-local ones in (2.3)

$G(x_{1},x_{2}|x_{3},x_{4})\equiv\langle{\rm
tr\,}\big{(}Z(x_{1})(XX^{\dagger})(x_{2})\big{)}\,{\rm
tr\,}\big{(}(Z^{\dagger}X)(x_{3})X^{\dagger}(x_{4})\big{)}\rangle=\sum_{L=0}^{\infty}\xi_{2}^{4L}G^{(L)}(x_{1},x_{2}|x_{3},x_{4})\,.$
(2.4)

We will sometimes abuse the notation and refer to this correlator as a “four-
point function”, even though it is really a two-point function of two non-
local single-trace operators.

The only interaction terms in the Lagrangian (A.2) that contribute to this
correlator in the limit (2.2) are

${\cal L}_{\text{int}}=Ng^{2}\,{\rm
tr\,}(X^{\dagger}X^{\dagger}XX)+2Ng^{2}e^{-\operatorname{i}\gamma_{3}}\,{\rm
tr\,}(X^{\dagger}Z^{\dagger}XZ)+\text{non-relevant for }G\,.$ (2.5)

Figure 3: The first few diagrams that contribute to the correlator (2.4) in
the double-scaling limit (2.2). Here, the red lines denote $X$ propagators and
black lines denote $Z$ propagators. The intersection of a red line and a black
line represents the vertex ${\rm tr\,}(X^{\dagger}Z^{\dagger}XZ)$, and the
intersection of two red lines represents the vertex ${\rm
tr\,}(X^{\dagger}X^{\dagger}XX)$.

In figure 3 we have plotted some of the first few diagrams contributing to the
correlator (2.4). At tree-level, (order $\xi_{2}^{0}$), the correlator is
plotted in figure 3. and is given by the product of three propagators

$G^{(0)}(x_{1},x_{2}|x_{3},x_{4})=\frac{(4\pi^{2})^{-3}}{x_{13}^{2}x_{23}^{2}x_{24}^{2}}\,,$
(2.6)

where $x_{ij}=x_{i}-x_{j}$. At the next order in perturbation theory (order
$\xi_{2}^{4}$), we have the two diagrams of figure 3. They give exactly the
same contribution. This phenomenon repeats itself at each order in
perturbation theory. Namely, at each order, one can add a ring to the diagrams
of the previous order using either of the drawings in figure 4, both resulting
in the same kernel. After taking this factor of $2$ into account in the
definition of $\xi_{2}$ in (2.2), we arrive at

$G^{(2)}(x_{1},x_{2}|x_{3},x_{4})=\int\frac{\differential^{4}y_{1}\differential^{4}y_{2}}{\pi^{4}}\frac{G_{1}^{(0)}(y_{1},y_{2}|x_{3},x_{4})}{(x_{1}-y_{1})^{2}(x_{2}-y_{1})^{2}(x_{2}-y_{2})^{2}y_{12}^{2}}\,.$
(2.7)

Figure 4: The graph-building operator $\widehat{B}$ in (2.9). The two drawings
represent the two ways of adding a wheel to a diagram. The corresponding
overall factor of two is part of our definition of the coupling $\xi_{2}^{2}$
in (1.1).

At order $\xi_{2}^{4}$, we also have the two diagrams in figure 5, where a
gluon is exchanged between the $X$ scalars. These two contributions, however,
cancel against each other and do not affect the correlator (2.4). This
cancellation persists at all orders in perturbation theory. Consequently, the
only graphs that survive are of the type of those drawn in figure 3. They have
an iterative structure that is generated by the graph-building operator
$\widehat{B}$. It is defined by its action on the correlator (2.4) as

$G^{(L+2)}(x_{1},x_{2}|x_{3},x_{4})=\int\frac{\differential^{4}y_{1}\differential^{4}y_{2}}{\pi^{4}}B(x_{1},x_{2}|y_{1},y_{2})\,G^{(L)}(y_{1},y_{2}|x_{3},x_{4})\,,$
(2.8)

or in shorthand notation $G^{(L+2)}=\widehat{B}G^{(L)}$, with the kernel

$B(x_{1},x_{2}|y_{1},y_{2})\equiv\frac{1}{(x_{1}-y_{1})^{2}(x_{2}-y_{1})^{2}(x_{2}-y_{2})^{2}y_{12}^{2}}\,.$
(2.9)

The graph-building operator can be visualised diagrammatically as in figure
4.222Note that it is equal to the square of the graph-building operator for
the one-magnon case of [5].

Figure 5: Cancellation of diagrams where a gluon appears.

The resummation of the perturbative expansion can therefore be represented as
the geometrical series

$G=\frac{1}{1-\xi_{2}^{4}\widehat{B}}G^{(0)}\,.$ (2.10)

If, for example, we had chosen to use a non-local operator with the other
ordering of the fields in the trace, such as ${\rm
tr\,}\big{(}(Z^{\dagger}X^{\dagger})(x_{3})X(x_{4})\big{)}$ instead of ${\rm
tr\,}\big{(}(Z^{\dagger}X)(x_{3})X^{\dagger}(x_{4})\big{)}$, then the only
difference would have been in the definition of ${G}^{(0)}$, but $\widehat{B}$
would remain the same.

As we review in appendix C.1, the spectrum of local operators,
$\Delta_{\ell,\bar{\ell}}(\xi_{2})$, can be read from the locations of the
poles in (2.10). Namely, it is given by the solutions to the equation

$\xi_{2}^{4}\,E(\Delta_{*},\ell,\bar{\ell})=1\,,$ (2.11)

where $E(\Delta,\ell,\bar{\ell})$ is the eigenvalue of the graph-building
operator $\widehat{B}$ in the principal series representations of the
conformal group $(\Delta,\ell,\bar{\ell})$. The corresponding eigenfunctions
are fixed by the conformal invariance of the graph-building operator (2.9).
They take the form of conformal three-point functions involving a scalar of
dimension 1 and a scalar of dimension 2. The third operator, at $x_{0}$, can
then only be a symmetric traceless tensor representation
($\bar{\ell}=\ell\equiv S$) of arbitrary rank. If we contract the index
structure of that operator with a null vector $\zeta$, this eigenfunction
reads

$\Psi^{(\Delta,S,S)}_{x_{0},\zeta}(x_{1},x_{2})=\frac{\left(\frac{\zeta\cdot
x_{10}}{x_{10}^{2}}-\frac{\zeta\cdot
x_{20}}{x_{20}^{2}}\right)^{S}}{|x_{12}|^{S-\Delta+3}|x_{10}|^{\Delta-S-1}|x_{20}|^{\Delta-S+1}}\,.$
(2.12)

To compute the eigenvalue $E(\Delta,S,S)$ we can either use the star-triangle
identity (B.2) to act with $\widehat{B}$ on $\Psi$, or directly act with the
inverse

$\widehat{B}^{-1}=\frac{1}{16}x_{12}^{2}\Box_{x_{2}}x_{12}^{2}\Box_{x_{1}}\,.$
(2.13)

The corresponding eigenvalue is

$E(\Delta,S,S)=\frac{16}{(\Delta+S-1)^{2}((4-\Delta)+S-1)^{2}}\,.$ (2.14)

Equation (2.11) has two pairs of solutions related by the shadow transform,
$\Delta\to 4-\Delta$. The two solutions, which in the free theory have
dimension above the unitarity bound, $\Delta_{0}\geqslant 2$, are

$\Delta_{\pm}(S,\xi_{2})=2+\sqrt{\left(S+1\right)^{2}\pm 4{\xi}^{2}_{2}}\,.$
(2.15)

The weak coupling expansion of (2.15) in small ${\xi}^{2}_{2}$ reads

$\Delta_{\pm}(S,\xi_{2})=3+S\pm\frac{2{\xi}^{2}_{2}}{(S+1)}-\frac{2{\xi}^{4}_{2}}{(S+1)^{3}}\pm\frac{4{\xi}^{6}_{2}}{(S+1)^{5}}+O\left({\xi}^{8}_{2}\right)\,.$
(2.16)

For $S=0$, these two dimensions are those of linear combinations of ${\rm
tr\,}(ZXX^{\dagger})$ and ${\rm tr\,}(ZX^{\dagger}X)$. An interesting feature
of the weak coupling expansion in (2.16) is that it is an expansion in
${\xi}^{2}_{2}$, while the graph-building operator appears in (2.4) with a
factor of ${\xi}^{4}_{2}$. This has to do with the fact that the two operators
${\rm tr\,}(ZXX^{\dagger})$ and ${\rm tr\,}(ZX^{\dagger}X)$ can mix with each
other in perturbation theory. In appendix E.1, we explicitly demonstrate how
this happens at one-loop order in perturbation theory.

At strong coupling the dispersion relation (2.15) becomes that of a
holographic dual classical fishchain of length two [25]

$\Delta_{\text{cl}}^{2}=\ell_{\text{cl}}^{2}\pm 4\xi_{2}^{2}+O(\xi_{2})\,.$
(2.17)

Finally, following the procedure reviewed in appendix C.1, we have collected
in appendix C.2 the other quantities required for a complete computation of
the correlator (2.4).

###  2.2 The Operator ${\rm tr\,}(F_{\mu\nu}Z)$

The second example of a family of operators that becomes non-trivial in the
limit (2.2) is that of operators involving one $Z$-field and an $F_{\mu\nu}$,
with any distribution of derivatives. To study these operators of abritrary
spin, in analogy with (2.3), we consider the non-local operators

${\rm tr\,}\big{(}Z(x)F^{\mu\nu}(y)\big{)}\,,\qquad{\rm
tr\,}\big{(}Z^{\dagger}(x)F_{\mu\nu}(y)\big{)}\,.$ (2.18)

Their corresponding two-point function now takes the form

$G_{F,\rho\sigma}^{\mu\nu}(x_{1},x_{2}|x_{3},x_{4})\equiv\langle{\rm
tr\,}\big{(}Z(x_{1})F^{\mu\nu}(x_{2})\big{)}\,{\rm
tr\,}\big{(}Z^{\dagger}(x_{3})F_{\rho\sigma}(x_{4})\big{)}\rangle\,.$ (2.19)

As before, this correlator is gauge invariant in the limit (2.2). We have
found it useful to also consider the non-gauge invariant correlator

$G_{A,\nu}^{\mu}(x_{1},x_{2}|x_{3},x_{4})\equiv\langle{\rm
tr\,}\big{(}Z(x_{1})A^{\mu}(x_{2})\big{)}\,{\rm
tr\,}\big{(}Z^{\dagger}(x_{3})A_{\nu}(x_{4})\big{)}\rangle\,.$ (2.20)

It is related to the correlator in (2.19) through

$G_{F}^{\mu\nu,\rho\sigma}(x_{1},x_{2}|x_{3},x_{4})=\partial_{x_{2}}^{[\mu}\partial_{x_{4}}^{[\rho}G_{A}^{\nu]\,\sigma]}(x_{1},x_{2}|x_{3},x_{4})\,.$
(2.21)

where we have used that $g\rightarrow 0$ in the scaling limit (2.2), and hence
we can replace $F_{\mu\nu}$ by $\partial_{[\mu}A_{\nu]}$.

The interaction terms in the Lagrangian (A.9) that contribute to this double
scaled correlator are

${\cal L}_{\text{int}}=2Ng^{2}{\rm
tr\,}\big{(}X^{\dagger}A_{\mu}XA^{\mu}+e^{-\operatorname{i}\gamma_{3}}X^{\dagger}Z^{\dagger}XZ\big{)}-\operatorname{i}Ng\,{\rm
tr\,}\big{(}[A^{\mu},X]\partial_{\mu}X^{\dagger}-\partial_{\mu}X[X^{\dagger},A^{\mu}]\big{)}+\dots\,.$
(2.22)

Figure 6: The first few diagrams in the perturbative expansion in $\xi_{2}$.
The black line denotes the Z field propagator and the red line denotes the X
field propagator. With each wheel, the gluon can either continue straight
through or annihilate and immediately reappear. For the latter, the
reappearing gluon can appear on either side of the annihilated gluon, but we
have suppressed such diagrams for clarity.

We have drawn some of the first few diagrams that contribute to the
correlators (2.19) and (2.20) in figure 6. These diagrams have an iterative
structure in which an $X$-wheel absorbs and emits the $Z$ scalar and the
gluon. For this picture to hold true, it is important that an $X$-wheel cannot
absorb the gluon without emitting one on its other side. Such a process would
leave us with the trace of a single intermediate $Z$ field, see figure 7.a.
This trace vanishes in the $SU(N)$ theory and is $1/N$-suppressed in the
$U(N)$ theory.

Another type of process that is not supressed in the limit (2.2) is that of a
${\rm tr\,}(XXX^{\dagger}X^{\dagger})$ interaction between two $X$-wheels, see
figure 7.b. These types of interactions come in pairs, one on each side of the
incoming gluon. They cancel out due to the anti-symmetry of the
$A\\!\\!-\\!\\!X\\!\\!-\\!\\!X^{\dagger}$ interaction vertex, (2.22). All
other processes, such as fermion wheels or ghost loops in figure 7.c, are
suppressed in the limit (2.2).

Figure 7: Examples of diagrams which do not contribute in the double-scaling
limit. Solid lines denote scalars and dashed lines denote fermions.

In what follows we work in the $R_{\alpha}$ gauge, where the gluon propagator
takes the form

$\Delta^{\mu}_{\nu}(x)={1\over
4\pi^{2}\alpha\,x^{2}}\left[\frac{1}{2}(\alpha+1)\delta^{\mu}_{\nu}+(\alpha-1)\frac{x^{\mu}x_{\nu}}{x^{2}}\right]\,.$
(2.23)

The tree-level correlator is plotted in figure 6.a and is given by the product
of the gluon and scalar propagators

$G_{A,\mu\nu}^{(0)}=\frac{\Delta_{\mu\nu}(x_{42})}{4\pi^{2}x_{13}^{2}}\,.$
(2.24)

The higher loops are generated iteratively by the graph-building operator as

$\displaystyle{G^{(L+2)}_{A}}^{\mu}_{\nu}(x_{1},x_{2}|x_{3},x_{4})=\,$
$\displaystyle\left(B^{\mu\rho}_{A}{G^{(L)}_{A}}_{\rho\nu}\right)(x_{1},x_{2}|x_{3},x_{4})$
(2.25) $\displaystyle\equiv\,$
$\displaystyle\int\frac{\differential^{4}y_{1}\differential^{4}y_{2}}{\pi^{4}}B^{\mu\rho}_{A}(x_{1},x_{2}|y_{1},y_{2})\,G^{(L)}_{A,\rho\nu}(y_{1},y_{2}|x_{3},x_{4})\,.$

Unlike the previous cases and the fishnet, this operator is not equal to a
single diagram. Instead, it is given by the sum of the three diagrams in
figure 8

$B^{\mu\nu}_{A}(x_{1},x_{2}|y_{1},y_{2})=\frac{4\pi^{2}\Delta^{\mu\nu}(x_{2}-y_{2})}{(x_{1}-y_{1})^{2}y_{12}^{4}}+\int\\!\differential^{4}z\,\frac{\Delta^{\mu}_{\sigma}(x_{2}-z)}{(x_{1}-y_{1})^{2}}\Bigg{[}\frac{1}{(y_{1}-z)^{2}}\overleftrightarrow{\partial}_{z}^{\sigma}\frac{1}{(y_{2}-z)^{2}}\overleftrightarrow{\partial}_{y_{2}}^{\nu}\frac{1}{y_{12}^{2}}\Bigg{]}\,,$
(2.26)

where $f\overleftrightarrow{\partial}^{\mu}g\equiv
f(\partial^{\mu}g)-(\partial^{\mu}f)g$. Here, the first term result from the
quartic interaction in (2.22), and the second term comes from the cubic ones.

The graph-building operator $\widehat{B}_{A}$ depends on our gauge choice.
This fact is manifest in the $\alpha$-dependence of the gluon propagator
(2.23) in (2.26). We now use it to derive a gauge-invariant graph-building
operator $\widehat{B}_{F}$, that generates the perturbative expansion of
$G_{F}$ as

${G_{F}^{(L+2)}}^{\mu\nu}_{\rho\sigma}(x_{1},x_{2}|x_{3},x_{4})=({B^{\mu\nu}_{F,\tau\chi}}{G^{(L)}_{F}}^{\tau\chi}_{\rho\sigma})(x_{1},x_{2}|x_{3},x_{4})\,,$
(2.27)

where $G^{(L)}_{F}$ is related to $G^{(L)}_{A}$ as in (2.21).

To obtain $\widehat{B}_{F}$, we notice that the action of $\widehat{B}_{A}$ on
an arbitrary vector function $\Phi^{\mu}$ takes the form

$B_{A,\tau}^{\nu}\Phi^{\tau}={H}_{\tau\rho}^{\nu}\left(\partial_{2}^{\tau}\Phi^{\rho}-\partial_{2}^{\rho}\Phi^{\tau}\right)\,,$
(2.28)

where the explicit form of ${H}_{\tau\upsilon}^{\nu}$ is given below. By
plugging (2.25) into (2.21) and using this relation we conclude that

$B^{\mu\nu}_{F,\rho\sigma}=\partial_{2}^{\mu}H_{\rho\sigma}^{\nu}-\partial_{2}^{\nu}H_{\rho\sigma}^{\mu}\,.$
(2.29)

Note that $\widehat{B}_{F}$ is indeed independent of the gauge-fixing
parameter $\alpha$ in (2.23). To show this we use that the dependence of
$\widehat{B}_{A}$ on $x_{2}$ only enters through a gluon propagator connecting
to $A(x_{2})$. The dependence of this propagator on $\alpha$ drops out from
$\partial_{\mu}\Delta^{\nu\rho}-\partial_{\nu}\Delta^{\mu\rho}$. Hence, when
plugging (2.25) into (2.21) the dependence of $\widehat{B}_{A}$ on $\alpha$
drops out. The $\alpha$-independence of $\widehat{B}_{F}$ then follows using
(2.28) and (2.29).

Figure 8: The graph-building operator $\widehat{B}_{A}$ in (2.26). The point
$y_{2}$ marks the position of the gluon in the wave function $\Phi$.

To prove (2.28), we first notice that the action of $\widehat{B}_{A}$ in
(2.26) on an arbitrary vector function $\Phi$ takes the form

$\displaystyle\left[\widehat{B}_{A}\Phi\right]^{\mu}(x_{1},x_{2})$ (2.30)
$\displaystyle=4\int\frac{\differential^{4}y_{1}\,\differential^{4}y_{3}}{\pi^{2}}\frac{\Delta^{\mu}_{\tau}(x_{2}-y_{3})}{(x_{1}-y_{1})^{2}y_{13}^{4}}\Bigg{[}\Phi^{\tau}(y_{1},y_{3})+\partial_{y_{3}}^{\tau}y_{13}^{2}\int\frac{\differential^{4}y_{2}}{4\pi^{2}}\,\Phi_{\rho}(y_{1},y_{2})\left(\frac{1}{y_{23}^{2}}\overleftrightarrow{\partial}_{y_{2}}^{\rho}\frac{1}{y_{12}^{2}}\right)\Bigg{]}\,,$

where the three terms correspond to the three diagrams in figure 8
respectively. Using the relation $\Box\,x^{-2}=-4\pi^{2}\delta^{(4)}(x)$, the
first term can be expressed as

$\Phi^{\tau}(y_{1},y_{3})=-\int\frac{\differential^{4}y_{2}}{4\pi^{2}}\left[{\Psi_{\rho}}^{\tau}(y_{1},y_{2})+\Phi_{\rho}(y_{1},y_{2})\partial_{y_{3}}^{\tau}\right]\partial_{y_{3}}^{\rho}\frac{1}{y_{32}^{2}}\,,$
(2.31)

where

$\Psi^{\mu\nu}(x,y)\equiv\partial_{y}^{\mu}\Phi^{\nu}(x,y)-\partial_{y}^{\nu}\Phi^{\mu}(x,y)\,.$
(2.32)

For the other terms in (2.30) we use that under the $y_{2}$-integration we
have

$\Phi_{\rho}(y_{1},y_{2})y_{13}^{2}\left[\frac{1}{y_{23}^{2}}\overleftrightarrow{\partial}_{y_{2}}^{\rho}\frac{1}{y_{12}^{2}}\right]=2\Psi_{\rho\sigma}(y_{1},y_{2})\frac{y_{32}^{\rho}y_{12}^{\sigma}}{y_{32}^{2}\,y_{12}^{2}}+\Phi_{\rho}(y_{1},y_{2})\partial_{y_{2}}^{\rho}\left[\frac{1}{y_{12}^{2}}-\frac{1}{y_{32}^{2}}\right]\,.$
(2.33)

The term $\Phi_{\rho}(y_{1},y_{2})\partial_{y_{2}}^{\rho}1/y_{12}^{2}$ on the
right-hand side does not depend on $y_{3}$ and therefore drops out when we
plug (2.33) into (2.30). We are left with

$\displaystyle\left[\widehat{B}_{A}\Phi\right]^{\mu}(x_{1},x_{2})$
$\displaystyle=2\int\frac{\differential^{4}y_{1}\differential^{4}y_{2}\differential^{4}y_{3}}{\pi^{4}}\frac{\Delta^{\mu}_{\tau}(x_{2}-y_{3})}{(x_{1}-y_{1})^{2}y_{13}^{4}}\Bigg{[}\frac{y_{32}^{\rho}}{y_{32}^{4}}\delta^{\tau\sigma}+\partial_{y_{3}}^{\tau}\frac{y_{32}^{\rho}\,y_{12}^{\sigma}}{y_{32}^{2}\,y_{12}^{2}}\Bigg{]}\Psi_{\rho\sigma}(y_{1},y_{2})$
$\displaystyle\equiv\left[\widehat{H}\Psi\right]^{\mu}(x_{1},x_{2})\,.$ (2.34)

This concludes the proof of (2.28).

The resumation of the perturbative expansion is again a geometrical series:

$G_{F}=\frac{1}{1-\xi_{2}^{4}\widehat{B}_{F}}G^{(0)}_{F}\,.$ (2.35)

To obtain the spectrum, we need to diagonalise $\widehat{B}_{F}$. It is
simpler to diagonalise the inverse $\widehat{B}_{F}^{-1}$ rather than
$\widehat{B}_{F}$ itself because the former is a differential operator. For
this aim, we first notice that the image of $\widehat{B}_{F}$ is always of the
form $\partial_{2}^{\mu}\Phi^{\nu}-\partial_{2}^{\nu}\Phi^{\mu}$ for some
$\Phi$ (that is only defined up to a gauge transformation), see (2.29).
Moreover, in (2.35) $\widehat{B}_{F}$ only acts within the subspace of
antisymmetric tensor functions of this type. Therefore, we should only invert
$\widehat{B}_{F}$ inside this subspace.

Next, we observe that, for all antisymmetric tensor function $\Psi$, the
vector function $\widehat{H}\Psi$ in (2.2) is divergence-less:

${\partial\over\partial x_{2}^{\mu}}\left[\widehat{H}\Psi\right]^{\mu}=0\,.$
(2.36)

This non-trivial identity is straightforward to check, but we do not have a
physical understanding of it. By combining (2.29) with (2.36) we conclude that

${\partial\over\partial
x_{2}^{\mu}}\left[\widehat{B}_{F}\Psi\right]^{\mu\nu}=\Box_{x_{2}}\left[\widehat{H}\Psi\right]^{\nu}\,.$
(2.37)

Because the left-hand side is gauge invariant, so is the right-hand side. To
proceed, we consider the right-hand side of this equation in Feynman gauge,
$\Delta^{\mu}_{\nu}(x)=\delta^{\mu}_{\nu}/(4\pi^{2}x^{2})$. The action of the
Laplacian on $x_{2}$ then produces a $\delta$-function that localises the
$y_{3}$-integral in (2.2). To localise the $y_{1}$-integral we act with
$\Box_{x_{1}}$. This leaves us with

$\displaystyle
x_{12}^{4}\Box_{x_{1}}\partial_{x_{2}}^{\mu}{\big{[}\widehat{B}_{F}\Psi\big{]}_{\mu}}^{\nu}$
$\displaystyle(x_{1},x_{2})=x_{12}^{4}\Box_{x_{1}}\Box_{x_{2}}\big{[}\widehat{H}\Psi\big{]}^{\nu}(x_{1},x_{2})$
(2.38)
$\displaystyle=16\Phi^{\nu}(x_{1},x_{2})+4\partial_{x_{2}}^{\nu}\left[x_{12}^{2}\int\frac{\differential^{4}y_{2}}{\pi^{2}}\Phi_{\rho}(x_{1},y_{2})\Big{(}\frac{1}{(y_{2}-x_{2})^{2}}\overleftrightarrow{\partial}_{y_{2}}^{\rho}\frac{1}{(x_{1}-y_{2})^{2}}\Big{)}\right]\,,$

where
$\Psi^{\mu\nu}=\partial_{2}^{\mu}\Phi^{\nu}-\partial_{2}^{\nu}\Phi^{\mu}$ for
some $\Phi$. It therefore follows that

$\partial_{x_{2}}^{[\rho}x_{12}^{4}\Box_{x_{1}}\partial_{x_{2}}^{\mu}{\big{[}\widehat{B}_{F}\Psi\big{]}_{\mu}}^{\nu]}=16\Psi^{\rho\nu}\,.$
(2.39)

####  2.2.1 Spectrum

We denote the eigenfunctions of $\widehat{B}_{F}$ by
$\Psi^{\mu\nu}_{E}=\partial_{x_{2}}^{\mu}\Phi^{\nu}_{E}-\partial_{x_{2}}^{\nu}\Phi^{\mu}_{E}$,
where $E$ is the eigenvalue and the vector-potential $\Phi_{E}^{\mu}$ is
defined up to a total derivative. To represent these eigenfunctions it is
useful to introduce an auxiliary polarisation vector $\theta$, whose
components anticommute among themselves, $\\{\theta^{\mu},\theta^{\nu}\\}=0$.
In terms of these auxiliary variables, the functions $\Psi$ and $\Phi$ take
the form

$\Phi(\theta)=\theta^{\mu}\Phi_{\mu}\,,\qquad\Psi(\theta)=\frac{1}{2}\theta^{\mu}\theta^{\nu}\Psi_{\mu\nu}=(\theta\cdot\partial_{x_{2}})\Phi(\theta)\,,$
(2.40)

and the eigenfunction equation becomes

$\widehat{B}_{F}^{-1}\Psi_{E}={1\over
16}(\theta\cdot\partial_{x_{2}})\,x_{12}^{4}\,(\partial_{x_{2}}\cdot\partial_{\theta})\,\Box_{x_{1}}\Psi_{E}=\frac{1}{E}\Psi_{E}\,.$
(2.41)

Equivalently, in terms of the vector-potential, this equation reads

$\theta\cdot\partial_{x_{2}}\left[x_{12}^{4}\,(\partial_{x_{2}}\cdot\partial_{\theta})(\theta\cdot\partial_{x_{2}})\Box_{x_{1}}\Phi_{E}-\frac{16}{E}\Phi_{E}\right]=0\,,\qquad\text{with}\qquad\Phi_{E}\simeq\Phi_{E}+(\theta\cdot\partial_{x_{2}})f\,,$
(2.42)

for arbitrary function $f=f(x_{1},x_{2})$.

As before, these functions are fixed by the conformal symmetry of the graph-
building operator (2.39). The vector-potential eigenfunction $\Phi_{E}^{\mu}$
is a conformal three-point functions between a dimension one primary scalar at
$x_{1}$, a dimension one vector at $x_{2}$, and a primary operator at some
arbitrary point $x_{0}$, that we take to transform in a representation
$\rho=(\Delta,\ell,\bar{\ell})$ of the conformal group

$\Phi_{E}=\langle{\cal O}_{(\Delta,\ell,\bar{\ell})}(x_{0}){\cal
O}_{(1,0,0)}(x_{1}){\cal
O}_{(1,1,1)}(x_{2})\rangle+(\theta\cdot\partial_{x_{2}})f\,.$ (2.43)

Correspondingly, the eigenfunction $\Psi_{E}$ is a conformal three-point
function involving the same operators at $x_{0}$ and $x_{1}$ and a dimension
two self-dual or anti-self-dual rank-2 anti-symmetric tensor at $x_{2}$.

The simplest way of constructing these three point functions is using
embedding coordinates. We relegate the details to appendix F, and we merely
present here the results. We find that there are three families of such non-
zero three point functions, with

$(\ell,\bar{\ell})\in\\{(S,S),(S-1,S+1),(S+1,S-1)\\}\qquad\text{for}\qquad
S\geqslant 1\,,$ (2.44)

where $S$ is the total spin. The first family of operators are symmetric
traceless tensors of rank $S$. Their corresponding three-point function takes
the form

$\Phi^{(\Delta,S,S)}_{x_{0},\zeta}(x_{1},x_{2},\theta)=\frac{\left(\frac{\theta\cdot
x_{12}}{x_{12}^{2}}-\frac{\theta\cdot
x_{02}}{x_{02}^{2}}\right)\left(\frac{\zeta\cdot
x_{10}}{x_{10}^{2}}-\frac{\zeta\cdot
x_{20}}{x_{20}^{2}}\right)^{S}}{|x_{12}|^{1+S-\Delta}|x_{10}|^{1+\Delta-S}|x_{20}|^{\Delta-S-1}}\,,$
(2.45)

where $\zeta$ is a polarisation vector for the operator at $x_{0}$. By
plugging this function into (2.42) we find the eigenvalue

$E(\Delta,S,S)=\frac{16}{(\Delta-S-3)^{2}((4-\Delta)-S-3)^{2}}\,.$ (2.46)

The dimensions of the corresponding operators are determined by (2.11). The
physical solutions, having $\Delta(0,S,S)\geqslant 2$, are

$\Delta_{\pm}(\xi_{2},S,S)=2+\sqrt{(S+1)^{2}\pm 4\xi_{2}^{2}}\,.$ (2.47)

Note that this is the same spectrum as the one we have found before in (2.15).
The operators are, however, different. For example, for $S=1$ the two non-
trivial operators we have found here and the two in the previous section are
different combinations of the operators333Here, we have used the equation of
motion, $dF=J$, to trade field strength for a scalar bi-linear. We have also
used integration by parts to remove the derivative from the $Z$-field.

${\rm tr\,}(ZXD^{\mu}X^{\dagger})\,,\qquad{\rm
tr\,}(ZD^{\mu}XX^{\dagger})\,,\qquad{\rm
tr\,}(ZX^{\dagger}D^{\mu}X)\,,\qquad\text{and}\qquad{\rm
tr\,}(ZD^{\mu}X^{\dagger}X)\,.$ (2.48)

We expect the degeneracy between (2.47) and (2.15) to be lifted at the next
order in the $g$ expansion around the large twist limit.

The other two families have $(\ell,\bar{\ell})=(S-1,S+1)$ and
$(\ell,\bar{\ell})=(S+1,S-1)$. For $S=1$, they correspond to self dual and
anti-self dual field strength. They can be combined into a single (reducible)
representation $\rho=(\Delta,S-1,S+1)\oplus(\Delta,S+1,S-1)$. The index
structure is encoded using two polarisation vectors, $\eta$ and $\zeta$, see
appendix F.2 for details. Two linearly independent wave functions are
constructed in the appendix and are given by

$\Phi^{\rho,1}_{x_{0},\zeta,\eta}(x_{1},x_{2},\theta)=\frac{x_{20}^{2}(\theta\cdot\eta)-2(\theta\cdot
x_{20})(\eta\cdot
x_{20})}{|x_{12}|^{2+S-\Delta}|x_{10}|^{\Delta-S}|x_{20}|^{2+\Delta-S}}\,(\eta\cdot\partial_{\zeta})\left(\frac{\zeta\cdot
x_{10}}{x_{10}^{2}}-\frac{\zeta\cdot x_{20}}{x_{20}^{2}}\right)^{S}\,,$ (2.49)

and

$\Phi^{\rho,2}_{x_{0},\zeta,\eta}(x_{1},x_{2},\theta)=\epsilon_{\mu\nu\rho\sigma}\frac{(x_{20}^{2}x_{10}^{\rho}-x_{10}^{2}x_{20}^{\rho})\theta^{\sigma}-2(\theta\cdot
x_{20})x_{10}^{\rho}x_{20}^{\sigma}}{|x_{12}|^{2+S-\Delta}|x_{10}|^{2+\Delta-S}|x_{20}|^{2+\Delta-S}}\eta^{\nu}(\eta\cdot\partial_{\zeta})\zeta^{\mu}\left(\frac{\zeta\cdot
x_{10}}{x_{10}^{2}}-\frac{\zeta\cdot x_{20}}{x_{20}^{2}}\right)^{S-1}\,.$
(2.50)

For $S=1$ these two functions are related by the duality map $*{\cal
O}_{\mu\nu}(x_{0})=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}{\cal
O}^{\rho\sigma}(x_{0})$. For $S>1$ they are related in an analogous way given
in (F.36) and (F.37), such that
$\Phi^{\rho,1}_{x_{0},\zeta,\eta}/S+\Phi^{\rho,2}_{x_{0},\zeta,\eta}/(S+1)$ is
self-dual, while
$\Phi^{\rho,1}_{x_{0},\zeta,\eta}/S-\Phi^{\rho,2}_{x_{0},\zeta,\eta}/(S+1)$ is
anti-self-dual.

These relations imply that the wave functions (2.49) and (2.50) correspond to
two different representations of the same operator. Hence, they give rise to
the same spectrum. In the appendix, we use (2.49) to obtain

$E(\Delta,S\pm 1,S\mp
1)=\frac{16}{(\Delta-S-4)((4-\Delta)-S-4)(\Delta-S-2)((4-\Delta)-S-2)}\,.$
(2.51)

The physical solutions of (2.11), namely with tree-level dimensions
$\Delta_{0}\geqslant 2$, have twist two and four. They are given by444Note
that under the replacement $\xi_{2}\to\xi_{1}$, the spectrum in (2.52) exactly
coincides with that of the fishnet operators ${\rm tr\,}(Z\partial^{S}Z)$ and
${\rm tr\,}(Z\partial^{S}\Box Z)$. In the fishnet case, the perturbative
expansion of the twist-two dimension, $\Delta_{-}(\xi_{1},-1,1)$, starts at
order $\xi_{1}^{2}$ and is related to the presence of double-trace
interactions. Here, $S\geqslant 1$, and no such issue arises.

$\Delta_{\pm}(\xi_{2},S+1,S-1)=\Delta_{\pm}(\xi_{2},S-1,S+1)=2+\sqrt{1+(S+1)^{2}\pm
2\sqrt{(S+1)^{2}+4\xi_{2}^{4}}}\,.$ (2.52)

At small coupling, the dimensions behave as

$\Delta_{\pm}=\left(S+3\pm 1\right)\pm\frac{2\,\xi_{2}^{4}}{(S+1)(S+1\pm
1)}-\frac{2\,\xi_{2}^{8}(S+1\pm(S+1\pm 1)^{2})}{(S+1)^{3}(S+1\pm
1)^{3}}+O\left(\xi_{2}^{12}\right)\,.$ (2.53)

When $S=1$, these solutions correspond to the operators ${\rm
tr\,}(F_{\mu\nu}Z)$ and ${\rm tr\,}(F_{\mu\nu}\Box Z)$ respectively.

##  3 Mixing Between Operators and Between Scaling Limits

Consider the extension of the operators discussed in the previous section to
operators with more $Z$-fields, such as ${\rm tr\,}(XX^{\dagger}Z^{J})$ with
$J>1$. These cases come with a new complication – after adding more $Z$-lines
to the diagrams in figure 5, they no longer cancel. As a result, the graph-
building operator mixes between wave functions with different number and type
of fields, all having the same $R$-charge. In addition, diagrams where the $X$
and $X^{\dagger}$ annihilate or where the gluon in ${\rm
tr\,}(F(x_{1})Z(x_{2})Z(x_{3})\dots)$ is absorbed by the $X$-wheel no longer
vanish. As a result, the graph-building operator also mixes between operators
with different double-scaling limits (1.1). Such a graph-building operator can
be represented by a matrix. Importantly for us, this matrix is always finite.
Different elements of this matrix have different dependence on the ’t Hooft
coupling and the twist. By analysing it we can obtain both the spectrum of
operators with different double-scaling limits, as well as the systematic
corrections away from the large twist double-scaling limit. Depending on the
operator we are interested in, we consider the same matrix-graph-building
operator expanded around different double-scaling limits.

Our main focus in this section is the generalisation of the operators we have
studied so far to operators with more $Z$-fields. Namely, the dimensions of
the operators ${\rm tr\,}(FZ^{J})$ and ${\rm tr\,}(XX^{\dagger}Z^{J})$ with
$J>1$. Their corresponding double-scaling limit is (1.1) with $n=1+1/J$. In
this limit, only the mixing between these two non-local operators and with the
fishnet operator ${\rm tr\,}(Z^{J})$ is relevant. Hence, the graph-building
operator is a $3\times 3$ matrix acting on the vector of wave functions

$\vec{\Psi}=\begin{pmatrix}\Psi_{\emptyset}\\\ \Psi_{A}\\\
\Psi_{X}\end{pmatrix}\equiv\left(\begin{array}[]{r}\langle{\rm
tr\,}(Z(x_{1})Z(x_{2})\dots Z(x_{J}))\,{\cal O}\rangle\\\ \langle{\rm
tr\,}(A(x_{0})Z(x_{1})Z(x_{2})\dots Z(x_{J}))\,{\cal O}\rangle\\\ \langle{\rm
tr\,}(XX^{\dagger}(x_{0})Z(x_{1})Z(x_{2})\dots Z(x_{J}))\,{\cal
O}\rangle\end{array}\right)\,.$ (3.1)

Here, like in (2.3), we have chosen to use the operators with the fields $X$
and $X^{\dagger}$ placed at the same position. As before, we first consider
the correlator with a gauge field insertion, $\Psi_{A}$, and then construct
from it the correlator with a field strength insertion as

$\Psi_{F}^{\mu\nu}=\partial_{x_{0}}^{[\mu}\Psi_{A}^{\nu]}\,,\qquad\text{or}\qquad\Psi_{F}=\theta\cdot\partial_{x_{0}}\Psi_{A}\,.$
(3.2)

The graph-building operator takes the following form

$\widehat{B}=\begin{pmatrix}g^{-2}{\mathcal{\widehat{B}}}_{\emptyset\emptyset}&g^{-1}{\mathcal{\widehat{B}}}_{\emptyset
A}&g^{-1}{\mathcal{\widehat{B}}}_{\emptyset X}\\\
g^{-1}{\mathcal{\widehat{B}}}_{A\emptyset}&{\mathcal{\widehat{B}}}_{AA}&{\mathcal{\widehat{B}}}_{AX}\\\
g^{-1}{\mathcal{\widehat{B}}}_{X\emptyset}&{\mathcal{\widehat{B}}}_{XA}&{\mathcal{\widehat{B}}}_{XX}\end{pmatrix}\,,$
(3.3)

where ${\mathcal{\widehat{B}}}_{\emptyset\emptyset}$ is the fishnet graph-
building operator. The corresponding eigenvalue equation reads

$\begin{pmatrix}1&0&0\\\ 0&\theta\cdot\partial_{0}&0\\\
0&0&1\end{pmatrix}\left[\widehat{B}-E\,\boldsymbol{\mathbbm{1}}\right]\vec{\Psi}=0\,.$
(3.4)

Finally, we read the spectrum by solving

$\xi^{2(J+1)}_{1+1/J}\,E(\Delta_{*},\ell,\bar{\ell})=1\,.$ (3.5)

We look for a solution to (3.4) for which the eigenvalue $E$ stays finite when
$g\rightarrow 0$ with fixed $\xi^{2}_{1+1/J}$. Due to the scaling of the
($\emptyset,j$) and ($i,\emptyset$) components of $\widehat{B}$ in (3.3) it
follows that

${\mathcal{\widehat{B}}}_{\emptyset\emptyset}\Psi_{\emptyset}=0\,,\qquad\text{and
hence}\qquad\Psi_{\emptyset}=0\,.$ (3.6)

Similarly, it follows that the second and third components of $\Psi$ have to
satisfy the reduced eigenvalue equation

$\begin{pmatrix}\theta\cdot\partial_{0}&0\\\
0&1\end{pmatrix}\left[{\mathfrak{B}}-E\,\boldsymbol{\mathbbm{1}}\right]\begin{pmatrix}\Psi_{A}\\\
\Psi_{X}\end{pmatrix}=0\,,$ (3.7)

where

${\mathfrak{B}}_{ij}={\mathcal{\widehat{B}}}_{ij}-{\mathcal{\widehat{B}}}_{i\emptyset}{\mathcal{\widehat{B}}}_{\emptyset\emptyset}^{-1}{\mathcal{\widehat{B}}}_{\emptyset
j}\qquad\text{for}\qquad i,j\in\\{A,X\\}\,.$ (3.8)

All the diagrams that contribute to $\widehat{B}$ in (3.3) have an $X$ wheel
crossing all the $Z$ fields, that result in the $\xi^{2(J+1)}_{1+1/J}$ factor
in (3.5). Among these diagrams, the only ones that contribute to
${\mathfrak{B}}_{ij}$ are those where a single $X$-propagator from that wheel
absorbs and emits an $XX^{\dagger}$ or an $A$ field. All other diagrams, where
an $XX^{\dagger}$ or an $A$ field is attached to different propagators, cancel
out between the two terms in (3.8). This cancellation is demonstrated in
figure 9.

Figure 9: a) The combination
${\mathcal{\widehat{B}}}_{A\emptyset}{\mathcal{\widehat{B}}}_{\emptyset\emptyset}^{-1}{\mathcal{\widehat{B}}}_{\emptyset
X}$ has the effect of joining diagrams from
${\mathcal{\widehat{B}}}_{A\emptyset}$ and ${\mathcal{\widehat{B}}}_{\emptyset
X}$ where the gluon and the $XX^{\dagger}$ fields do not connect to the same
$X$-propagator into a diagram of ${\mathcal{\widehat{B}}}_{AX}$. b) As a
result, the only diagrams that contributes to
${\mathcal{\widehat{B}}}_{AX}-{\mathcal{\widehat{B}}}_{A\emptyset}{\mathcal{\widehat{B}}}_{\emptyset\emptyset}^{-1}{\mathcal{\widehat{B}}}_{\emptyset
X}$ are those with the interaction with the gluon and the $XX^{\dagger}$
fields is local on the chain of $Z$’s.

The remaining diagrams have the $XX^{\dagger}$ and the $A$ fields interact
locally on the chain of $Z$’s. They are all drawn in figures 9.b, 10, and are
evaluated in appendix G. The result is

Figure 10: The diagrams that contribute to the reduced graph-building operator
${\mathfrak{B}}$ in (3.8). The interactions with the gluon and the
$XX^{\dagger}$ fields take place locally on the chain of $Z$’s. This is in
contrast to the diagrams that contribute to $B$ in (3.27).

$\left[{\mathfrak{B}}_{FF}\Psi_{F}\right](\theta,x_{0},\dots,x_{J})=\theta\cdot\partial_{0}\int\prod_{j=0}^{J}\frac{\differential^{4}y_{j}}{\pi^{2}}\frac{\Psi_{F}^{\mu\nu}(y_{0},\dots,y_{J})}{\prod_{i=1}^{J}(x_{i}-y_{i})^{2}y_{i,i+1}^{2}}\\\
\times\int{\differential^{4}z\over\pi^{2}}{y_{J0}^{2}\over(x_{0}-z)^{2}(y_{1}-z)^{2}(z-y_{J})^{2}}\Bigg{[}\frac{(z-y_{0})_{\mu}}{(z-y_{0})^{4}}\theta_{\nu}+{1\over
2}\theta\cdot\partial_{z}\frac{(y_{0}-z)_{\mu}}{(y_{0}-z)^{2}}\left(\frac{y_{01,\nu}}{y_{01}^{2}}+\frac{y_{0J,\nu}}{y_{0J}^{2}}\right)\\\
+\left(\frac{\theta\cdot(y_{J}-z)}{(y_{J}-z)^{2}}-\frac{\theta\cdot(y_{1}-z)}{(y_{1}-z)^{2}}\right)\bigg{(}\frac{(y_{0}-z)_{\mu}y_{0J,\nu}}{(y_{0}-z)^{2}y_{0J}^{2}}+\frac{y_{0J,\mu}y_{01,\nu}}{y_{0J}^{2}y_{01}^{2}}+\frac{y_{01,\mu}(y_{0}-z)_{\nu}}{y_{01}^{2}(y_{0}-z)^{2}}\bigg{)}\Bigg{]}\,,$
(3.9)

where $y_{J+1}\equiv y_{0}$ and we have used (3.2) to convert $\Psi_{A}$ into
$\Psi_{F}$ as

$\theta\cdot\partial_{x_{0}}\,{\mathfrak{B}}_{AA}\Psi_{A}={\mathfrak{B}}_{FF}\Psi_{F}\,,\qquad\theta\cdot\partial_{x_{0}}\,{\mathfrak{B}}_{AX}\Psi_{X}={\mathfrak{B}}_{FX}\Psi_{X}\,.$
(3.10)

Similarly,

$\left[{\mathfrak{B}}_{FX}\Psi_{X}\right](\theta,x_{0},\dots,x_{J})=\frac{1}{4}\theta\cdot\partial_{0}\int\prod_{j=0}^{J}\frac{\differential^{4}y_{j}}{\pi^{2}}\frac{\Psi_{X}(y_{0},\dots,y_{J})}{\prod_{i=0}^{J}(x_{i}-y_{i})^{2}y_{i,i+1}^{2}}\\\
\times\int{\differential^{4}z\over\pi^{2}}{(x_{0}-y_{0})^{2}\over(x_{0}-z)^{2}}\Bigg{[}\frac{1}{2(z-y_{1})^{2}(z-y_{J})^{2}}\theta\cdot\partial_{z}\left(\frac{y_{10}^{2}(z-y_{J})^{2}}{(z-y_{0})^{2}}-\frac{y_{J0}^{2}(z-y_{1})^{2}}{(z-y_{0})^{2}}\right)\\\
+\left(\frac{\theta\cdot(y_{J}-z)}{(y_{J}-z)^{2}}-\frac{\theta\cdot(y_{1}-z)}{(y_{1}-z)^{2}}\right)\bigg{(}\frac{y_{10}^{2}}{(z-y_{1})^{2}(z-y_{0})^{2}}+\frac{y_{J0}^{2}}{(z-y_{J})^{2}(z-y_{0})^{2}}-\frac{y_{1J}^{2}}{(z-y_{1})^{2}(z-y_{J})^{2}}\bigg{)}\Bigg{]}\,,$
(3.11)

and

$\left[{\mathfrak{B}}_{XF}\Psi_{F}\right](x_{0},\dots,x_{J})=2\int\prod_{j=0}^{J}\frac{\differential^{4}y_{j}}{\pi^{2}}\frac{\Psi_{F}^{\mu\nu}(y_{0},\dots,y_{J})}{\prod_{i=1}^{J}(x_{i}-y_{i})^{2}y_{i,i+1}^{2}}{y_{J0}^{2}\over(x_{0}-y_{1})^{2}(x_{0}-y_{J})^{2}}\\\
\times\bigg{[}\frac{(y_{0}-x_{0})_{\mu}y_{0J,\nu}}{(y_{0}-x_{0})^{2}y_{0J}^{2}}+\frac{y_{0J,\mu}y_{01,\nu}}{y_{0J}^{2}y_{01}^{2}}+\frac{y_{01,\mu}(y_{0}-x_{0})_{\nu}}{y_{01}^{2}(y_{0}-x_{0})^{2}}\bigg{]}\,.$
(3.12)

Finally,

$\left[{\mathfrak{B}}_{XX}\Psi_{X}\right](x_{0},\dots,x_{J})=\frac{1}{2}\int\prod_{j=0}^{J}\frac{\differential^{4}y_{j}}{\pi^{2}}\frac{\Psi_{X}(y_{0},y_{1},\dots,y_{J})}{\prod_{i=0}^{J}(x_{i}-y_{i})^{2}y_{i,i+1}^{2}}\\\
\times\bigg{[}\frac{y_{10}^{2}}{(x_{0}-y_{1})^{2}}+\frac{y_{J0}^{2}}{(x_{0}-y_{J})^{2}}-\frac{y_{1J}^{2}(x_{0}-y_{0})^{2}}{(x_{0}-y_{J})^{2}(x_{0}-y_{1})^{2}}\bigg{]}\,.$
(3.13)

Surprisingly, we find that ${\mathfrak{B}}$ has a simple inverse

$\displaystyle{\mathfrak{B}}^{-1}=$
$\displaystyle(-4)^{-J-1}\begin{pmatrix}(\theta\cdot\partial_{x_{0}})\,x_{J0}^{2}x_{10}^{2}\,(\partial_{x_{0}}\cdot\partial_{\theta})&2(\theta\cdot\partial_{x_{0}})\left[x_{10}^{2}(\theta\cdot
x_{J0})-x_{J0}^{2}(\theta\cdot x_{10})\right]\\\\[6.0pt]
2\left[x_{J0}^{2}(x_{10}\cdot\partial_{\theta})-x_{10}^{2}(x_{J0}\cdot\partial_{\theta})\right](\partial_{x_{0}}\cdot\partial_{\theta})&\partial_{x_{0},\mu}\,x_{J0}^{2}x_{10}^{2}\,\partial_{x_{0}}^{\mu}+8(x_{10}\cdot
x_{J0})\end{pmatrix}$ (3.14)
$\displaystyle\times\prod_{i=1}^{J-1}x_{i,i+1}^{2}\prod_{j=1}^{J}\Box_{x_{j}}\,.$

For $J=1$, this operator becomes diagonal with its elements exactly coinciding
with (2.41) and (2.13) respectively. For $J>1$, this operator is conformally
invariant, but conformal symmetry is no longer sufficient to fix its
eigenfunctions. Instead, we now prove that it is part of the commuting charges
of an integrable model.

###  3.1 Integrability

Let us consider a conformal spin chain of length $J+1$ consisting of the
reducible representation
$\rho_{0}=\left[\theta\cdot\partial_{x}\left(1,1,1\right)\right]\oplus(2,0,0)$
at site $0$ and of (irreducible) scalar representations of dimension $1$ at
all the other sites.555Note that $\theta\cdot\partial_{x}\left(1,1,1\right)$
is included in the usual dimension-$2$ antisymmetric rank-$2$ tensor
representation. The fact that it is obtained by taking an anti-symmetric
derivative of the vector representation is an additional constraint. It
implies that the tensor is annihilated by $\theta\cdot\partial_{x}$, and is
therefore conserved. We also consider a six-dimensional auxiliary space
carrying the defining representation $\mathbf{6}$ of the conformal group. In
order to simplify some of the computations, we choose to work in the embedding
space realisation of the principal series representations, see appendix F.1
for details. We denote by $Y^{M}$ the coordinates and by $\Theta^{M}$ the
polarisation vectors in embedding space. The relevant Lax matrix at site $0$
takes the form

$L^{(\rho_{0};\mathbf{6})}_{Y}(u)=u^{2}-u\,q_{MN}^{(\rho_{0})}\otimes
e^{MN}+\mathcal{L}^{MN}_{Y}\otimes e_{MN}\,,$ (3.15)

where $?e_{M}^{N}?$ is the $6\times 6$ matrix with the $(M,N)$ element equal
to one and the others are equal to zero. Here, $q_{MN}$ are the conformal
generators. In the representation $\rho_{0}$, they take the form

$q_{MN}^{(\rho_{0})}=\begin{pmatrix}Y_{M}\partial_{Y^{N}}-Y_{N}\partial_{Y^{M}}+\Theta_{M}\partial_{\Theta^{N}}-\Theta_{N}\partial_{\Theta^{M}}&0\\\
0&Y_{M}\partial_{Y^{N}}-Y_{N}\partial_{Y^{M}}\end{pmatrix}\,.$ (3.16)

Finally, the operator $\mathcal{L}_{Y}$ is

$\mathcal{L}^{MN}_{Y}=-\frac{1}{2}\begin{pmatrix}(\Theta\cdot\partial_{Y})\,Y^{M}Y^{N}\,(\partial_{Y}\cdot\partial_{\Theta})&(\Theta\cdot\partial_{Y})\left[Y^{M}\Theta^{N}-Y^{N}\Theta^{M}\right]\\\
\left[Y^{N}\partial_{\Theta}^{M}-Y^{M}\partial_{\Theta}^{N}\right](\partial_{Y}\cdot\partial_{\Theta})&\frac{1}{2}\left[Y^{M}\Box_{Y}Y^{N}+Y^{N}\Box_{Y}Y^{M}\right]+2\eta^{MN}\end{pmatrix}\,.$
(3.17)

The first column of these operators acts on weight-two functions
$\Psi(\Theta,\lambda\,Y)=\lambda^{-2}\Theta^{M}\Theta^{N}\partial_{Y^{M}}\Phi_{N}(Y)$,
which also satisfy the transversality condition
$Y\cdot\partial_{\Theta}\Psi=0$.666In order to simplify the computations, we
also impose the relations $\Psi=\Theta\cdot\partial_{Y}\Phi$ and
$Y\cdot\partial_{\Theta}\Psi=0$ away from $Y^{2}=\Theta\cdot Y=0$. This is not
a restriction; it simply means that we have fixed the gauge associated to
$\Phi\rightarrow\Phi+(\Theta\cdot Y)F$. Otherwise, we would have to add
$\frac{1}{2}(\partial_{Y}^{M}\Theta^{N}+\partial_{Y}^{M}\Theta^{N})Y\cdot\partial_{\Theta}$
to the $FF$ coefficient of $\mathcal{L}^{MN}_{Y}$ and replace the $XF$
coefficient by
$\frac{1}{4}\left[Y^{N}\partial_{\Theta}^{M}-Y^{M}\partial_{\Theta}^{N}\right](Y\cdot\partial_{\Theta})\Box_{Y}$.
The second column acts on weight-two functions of $Y$ only. Introducing a
second copy of the auxiliary space, we have checked that

$R_{12}^{(\mathbf{6},\mathbf{6})}(u-v)L^{(\rho_{0};\mathbf{6})}_{Y,1}(u)L^{(\rho_{0};\mathbf{6})}_{Y,2}(v)=L^{(\rho_{0};\mathbf{6})}_{Y,2}(v)L^{(\rho_{0};\mathbf{6})}_{Y,1}(u)R_{12}^{(\mathbf{6},\mathbf{6})}(u-v)\,,$
(3.18)

where the indices $1$ and $2$ indicate in which of the auxiliary spaces the
operators are acting non-trivially, and the Zamolodchikovs’ R-matrix is [29]

$R_{12}^{(\mathbf{6},\mathbf{6})}(u)=u(u+2)+(u+2)\left(e_{1,MN}\otimes
e_{2}^{NM}\right)-u\left(e_{1,MN}\otimes e_{2}^{MN}\right)\,.$ (3.19)

At sites $1,\ldots,J$ we have a dimension $1$ scalar. The corresponding
fishnet Lax matrix, which satisfies the RLL-relation (3.18), reads

$L^{(1,0,0;\mathbf{6})}_{Y}(u)=u^{2}-u\left(Y^{M}\partial_{Y}^{N}-Y^{N}\partial_{Y}^{M}\right)\otimes
e_{MN}-{1\over 2}Y^{M}Y^{N}\Box_{Y}\otimes e_{MN}\,.$ (3.20)

It acts on weight-one functions $\Phi(\lambda\,Y)=\lambda^{-1}\Phi(Y)$. The
transfer matrix

$T_{\mathbf{6}}(u)\equiv{\rm
tr\,}_{\mathbf{6}}\left(L^{(\rho_{0};\mathbf{6})}_{Y_{0}}(u)L_{Y_{1}}^{(1,0,0;\mathbf{6})}(u)\cdots
L_{Y_{J}}^{(1,0,0;\mathbf{6})}(u)\right)$ (3.21)

thus commutes with itself for arbitrary values of the spectral parameter,

$\left[T_{\mathbf{6}}(u),T_{\mathbf{6}}(v)\right]=0\,.$ (3.22)

Finally, we claim that

$T_{\mathbf{6}}(0)=(-1)^{J+1}\mathfrak{B}^{-1}\,.$ (3.23)

Hence, the inverse graph-building operator is part of the conserved charges of
an integrable model. The proof of (3.23) uses the map to embedding coordinates
that we review in appendix F.1.777To that end, one can also use the fact that
the diagonal elements of $\mathcal{L}_{Y,MN}$ in (3.17) are equal to
$\frac{1}{2}?q^{(\rho_{0})}_{M}{}^{P}?q^{(\rho_{0})}_{PN}-q^{(\rho_{0})}_{MN}$.
Eventually, it turns out that the reduction
$\mathcal{L}_{Y}\rightarrow\mathcal{L}_{x}$ is equivalent to performing the
replacement
$(\Theta\cdot\partial_{Y},\partial_{Y}\cdot\partial_{\Theta},\Box_{Y})\rightarrow(\theta\cdot\partial_{x},\partial_{x}\cdot\partial_{\theta},\Box_{x})$,
and plugging the following expressions in $L^{(\rho_{0};\mathbf{6})}_{Y}(0)$:
$Y^{\mu}=x^{\mu}\,,\quad Y^{+}=1\,,\quad
Y^{-}=x^{2}\,,\quad\Theta^{\mu}=\theta^{\mu}\,,\quad\Theta^{+}=0\,,\quad\Theta^{-}=2x\cdot\theta\,,$
(3.24) and
$\partial_{\Theta^{\mu}}=\partial_{\theta^{\mu}}\,,\quad\partial_{\Theta^{-}}=0\,,\quad\partial_{\Theta^{+}}=-x\cdot\partial_{\theta}\,.$
(3.25)

We point out that the Lax matrix (3.15) is different from the naïve guess
associated to the reducibility of the representation $\rho_{0}$. This guess
would correspond to a direct sum of the two standard Lax matrices constructed
from the fusion procedure, recalled in appendix D.1. Moreover, because this
direct sum contains an epsilon-tensor, the Lax matrix (3.15) does not seem to
be related to it by a change of basis. This situation is in contrast to all
other examples studied in this paper, where the standard fusion procedure
produces the relevant integrable models. It would therefore be interesting to
understand how the Lax matrix (3.15) fits inside the known classification of
integrable models.

###  3.2 Higher-Order Corrections

The _same_ graph-building operator (3.27) can also be used to compute
perturbative corrections in $g$ to the fishnet dimensions

$\Delta_{\text{f.n.}}(\xi_{1},g)=\Delta_{\text{f.n.}}^{(0)}(\xi_{1})+g^{2}\Delta_{\text{f.n.}}^{(2)}(\xi_{1})+O(g^{3})\,.$
(3.26)

In order to compute $\Delta_{\text{f.n.}}^{(2)}$, we consider the limit (1.1)
with $n=1$ and expand around it in $g$. At order $g^{2}$ we must take into
account elements of the graph-building operator that mix between the zeroth
order fishnet wave function, $\Psi_{\emptyset}$ in (3.1), and $\Psi_{A}$,
$\Psi_{X}$, as well as wave functions with other intermediate fields such as
fermions. Using the relation $\xi_{1+1/J}^{2(J+1)}=g^{2}\xi_{1}^{2J}$, the
corresponding graph-building operator takes the form

$B_{\text{f.n.}}=\begin{pmatrix}{\mathcal{B}}_{\emptyset\emptyset}&g{\mathcal{B}}_{\emptyset
A}&g{\mathcal{B}}_{\emptyset X}&g{\mathcal{B}}_{\emptyset,\text{other}}\\\
g{\mathcal{B}}_{A\emptyset}&g^{2}{\mathcal{B}}_{AA}&g^{2}{\mathcal{B}}_{AX}&0\\\
g{\mathcal{B}}_{X\emptyset}&g^{2}{\mathcal{B}}_{XA}&g^{2}{\mathcal{B}}_{XX}&0\\\
g{\mathcal{B}}_{\text{other},\emptyset}&0&0&g^{2}{\mathcal{B}}_{\text{other},\text{other}}\\\
\end{pmatrix}\,.$ (3.27)

Our main point in this subsection is that at the upper $3\times 3$ corner we
have the graph-building operator $g^{2}B$ from (3.27). Similarly, the elements
with ${\mathcal{B}}_{\text{other}}$ enter the computation of other type of
operators in their corresponding scaling limit (1.1) with $n>1$. As we explain
below, using these elements, one can compute
$\Delta_{\text{f.n.}}^{(2)}(\xi_{1})$. Carrying out this computation
explicitly is beyond the scope of this paper, and we leave it to a future
work.

We are interested in eigenvalues of $B_{\text{f.n.}}$ of the form

$E=E^{(0)}+gE^{(1)}+g^{2}E^{(2)}+O(g^{3})\,,$ (3.28)

where the coefficients $E^{(j)}$’s are independent of $\xi_{1}$. The
corresponding correction to the scaling dimension in (3.26) is obtained by
solving the equation $\xi_{1}^{2J}E(\Delta_{\text{f.n.}},\ell,\bar{\ell})=1$.

At leading order we have the fishnet equation

${\mathcal{B}}_{\emptyset\emptyset}\Psi_{\emptyset}^{(0)}=E^{(0)}\Psi_{\emptyset}^{(0)}\,.$
(3.29)

The following orders are determined iteratively from $\Psi_{\emptyset}^{(0)}$
in a standard perturbative fashion. At the next order, we obtain

$\Psi_{i}^{(1)}=\frac{1}{E^{(0)}}{\mathcal{B}}_{i\emptyset}\Psi_{\emptyset}^{(0)}\qquad\text{for}\qquad
i\in\\{A,X,\text{other}\\}\,,$ (3.30)

and

$\left({\mathcal{B}}_{\emptyset\emptyset}-E^{(0)}\right)\Psi_{\emptyset}^{(1)}=E^{(1)}\Psi_{\emptyset}^{(0)}\,.$
(3.31)

Because of (3.29), the only finite solution of this equation is $E^{(1)}=0$,
and therefore $\Psi_{\emptyset}^{(1)}\propto\Psi_{\emptyset}^{(0)}$. This
solution represents our freedom in choosing a normalisation for $\vec{\Psi}$.
We choose it to be $g$-independent, so that $\Psi_{\emptyset}^{(1)}=0$.

At second order, there is only one non-trivial equation

$\left({\mathcal{B}}_{\emptyset\emptyset}-E^{(0)}\right)\Psi_{\emptyset}^{(2)}=\Big{(}E^{(2)}-\frac{1}{E^{(0)}}\left[{\mathcal{B}}_{\emptyset
A}{\mathcal{B}}_{A\emptyset}+{\mathcal{B}}_{\emptyset
X}{\mathcal{B}}_{X\emptyset}+{\mathcal{B}}_{\emptyset,\text{other}}{\mathcal{B}}_{\text{other},\emptyset}\right]\Big{)}\Psi_{\emptyset}^{(0)}\,.$
(3.32)

As in the previous order, the right-hand side needs to be orthogonal to
$\Psi_{\emptyset}^{(0)}$ for $\Psi_{\emptyset}^{(2)}$ to be well defined.
Hence,888To compute this ratio one has to introduce a UV regulator, see (C.6).

$E^{(2)}=\frac{\langle\Psi_{\emptyset}^{(0)}|{\mathcal{B}}_{\emptyset
A}{\mathcal{B}}_{A\emptyset}+{\mathcal{B}}_{\emptyset
X}{\mathcal{B}}_{X\emptyset}+{\mathcal{B}}_{\emptyset,\text{other}}{\mathcal{B}}_{\text{other},\emptyset}|\Psi_{\emptyset}^{(0)}\rangle}{E^{(0)}\langle\Psi_{\emptyset}^{(0)}|\Psi_{\emptyset}^{(0)}\rangle}\,.$
(3.33)

Once this condition is satisfied, equation (3.32) is trivially solved by
expanding $\Psi_{\emptyset}^{(2)}$ over a complete basis of eigenstates of
${\mathcal{B}}_{\emptyset\emptyset}$.

The spectrum is obtained as a solution to
$\xi_{1}^{2J}E(\Delta_{\text{f.n.}},\ell,\bar{\ell})=1$. This gives the
fishnet condition
$\xi_{1}^{2J}E^{(0)}(\Delta^{(0)}_{\text{f.n.}},\ell,\bar{\ell})=1$ as well as
the first correction to the anomalous dimension:

$\Delta^{(2)}_{\text{f.n.}}=-\left.\frac{E^{(2)}(\Delta,\ell,\bar{\ell})}{\frac{\differential}{\differential\Delta}E^{(0)}(\Delta,\ell,\bar{\ell})}\right|_{\Delta=\Delta^{(0)}_{\text{f.n.}}}\,.$
(3.34)

Note that beyond leading order, the basis of wave functions we work with is no
longer gauge invariant, and neither are the matrix elements of the graph-
building operator. However, the eigenvectors of the graph-building operator
are gauge invariant. For example, the order $g$ mixing between the fishnet
state and the state with one more gluon in (3.30) is responsible for
converting the $Z$-derivatives into covariant ones.

Note also that in (3.27) we did not include order $g^{2}$ correction to
$\mathcal{B}_{\emptyset\emptyset}$. The absence of such corrections would
imply that all the loop corrections to the twist 2 and twist 4 fishnet states
(3.26) can be obtained through the mixing structure in (3.33). Further
research will be needed to determine whether or not this simple structure is
valid.

##  4 More Examples Without Mixing

By adjusting the twist, the construction of the previous section can be
extended to any operator.999For a generic operator one has to use the twisting
method of [2] instead of the $\gamma$-deformed action (A.2). We now present
more examples of operators with different double-scaling limits and no mixing.
The operators we study have one or two fermions and any number of $Z$-scalars.
For operators with more than two fields in the trace, conformal symmetry is
not sufficient to fix eigenfunctions of the corresponding graph-building
operator, as before. Instead, we prove that the graph-building operator is
integrable.

###  4.1 The Operators ${\rm tr\,}(Z\psi_{4})$ and ${\rm
tr\,}(Z\psi_{1}^{\dagger})$

A relatively simple family of operators that have a closed non-trivial double
scaling limit are those made of one $Z$ field, one $\psi_{4}$ or
$\psi_{1}^{\dagger}$ field, and at most one $X$ field. The corresponding limit
is (1.1) with $\theta=-\gamma_{3}$ and $n=4/3$. Namely, the limit we consider
is

$e^{-\operatorname{i}\gamma_{3}}\rightarrow\infty\,,\qquad\text{with}\qquad\xi_{4/3}^{2}=\frac{g^{2}\,e^{-\operatorname{i}\frac{3}{4}\gamma_{3}}}{8\pi^{2}}\quad\text{fixed}\,.$
(4.1)

In this case, we find four non-trivial operators at any half-integer spin, one
twist 2, one twist 4, and two of twist 3. In analogy with (2.4), here it is
sufficient to consider the correlator

$G_{2,\alpha\dot{\alpha}}(x_{1},x_{2}|x_{3},x_{4})=\langle{\rm
tr\,}\big{(}Z(x_{1})\psi_{4\alpha}(x_{2})\big{)}\,{\rm
tr\,}\big{(}Z^{\dagger}(x_{3})\psi_{4\dot{\alpha}}^{\dagger}(x_{4})\big{)}\rangle\,,$
(4.2)

where $\alpha$ and $\dot{\alpha}$ are spinor indices.

The relevant interaction terms that contribute to this correlator in the limit
(2.2) are

${\cal
L}_{\text{int}}=\sqrt{2}Nge^{\frac{\operatorname{i}}{2}\gamma_{1}^{-}}\,{\rm
tr\,}\big{(}\psi^{\dagger}_{4}X\psi^{\dagger}_{1}\big{)}-\sqrt{2}Nge^{\frac{\operatorname{i}}{2}\gamma_{1}^{-}}\,{\rm
tr\,}\big{(}\psi_{4}X^{\dagger}\psi_{1}\big{)}+2Ng^{2}e^{-\operatorname{i}\gamma_{3}}\,{\rm
tr\,}\big{(}X^{\dagger}Z^{\dagger}XZ\big{)}+\dots\,.$ (4.3)

Figure 11: The first few diagrams that contributes to the correlator (4.2).
Black dashed lines denote $\psi_{4}$ propagators and green dashed line denote
$\psi_{1}$ ones.

The first few diagrams contributing to the correlator
$G_{2,\alpha\dot{\alpha}}$ are plotted in figure 11. At tree level, we have

$G_{2}^{(0)}(x_{1},x_{2}|x_{3},x_{4})=\frac{\cancel{x}_{24}}{8\pi^{4}x_{13}^{2}x_{24}^{4}}\,,$
(4.4)

where $\cancel{x}=x^{\mu}\sigma_{\mu}$ and
$\cancel{\bar{x}}=x^{\mu}\bar{\sigma}_{\mu}$ are $2\times 2$ matrices. As
before, the diagrams have an iterative structure with
$G^{(L+2)}=\xi_{4/3}^{4}\,\widehat{B}G^{(L)}$. The corresponding graph-
building operator $\widehat{B}$ now also acts on the spinor index of $G$.

Figure 12: The graph-building operator $B$ in (4.5).

The kernel of this integral operator is plotted in figure 12. It takes the
form

$\displaystyle B(x_{1},x_{2}|y_{1},y_{2})\equiv$
$\displaystyle\int\frac{\differential^{4}z}{\pi^{2}}\frac{-(\cancel{x}_{2}-\cancel{z})(\cancel{\bar{z}}-\cancel{\bar{y}}_{2})}{(x_{1}-y_{1})^{2}(x_{2}-z)^{4}(z-y_{2})^{4}(z-y_{1})^{2}y_{12}^{2}}$
(4.5) $\displaystyle=$
$\displaystyle\frac{(\cancel{y}_{1}-\cancel{x}_{2})\cancel{\bar{y}}_{12}}{(x_{1}-y_{1})^{2}(x_{2}-y_{1})^{2}(x_{2}-y_{2})^{2}y_{12}^{4}}\,,$

where in the second step we have preformed the integration using the star-
triangle relation (B.2).

As before, the eigenfunctions of the graph-building operator (4.5) are fixed
by its conformal symmetry. They take the form of conformal three-point
functions involving a scalar of dimension 1 and a fermion of dimension
$\frac{3}{2}$. The third operator can be in either the $(\Delta,S-1/2,S+1/2)$
or the $(\Delta,S+1/2,S-1/2)$ representation with half-integer spin $S$. Using
[30] and the accompanying Mathematica notebook, we find that the
eigenfunctions are given respectively by

$\Psi_{x_{0},s,\bar{s}}^{(\Delta,S-{1\over 2},S+{1\over
2})}(x_{1},x_{2})=\frac{\cancel{x}_{20}\bar{s}\left[s\left(\frac{\cancel{x}_{10}}{x_{10}^{2}}-\frac{\cancel{x}_{20}}{x_{20}^{2}}\right)\bar{s}\right]^{S-{1\over
2}}}{|x_{12}|^{S-\Delta+2}|x_{10}|^{\Delta-S}|x_{20}|^{\Delta-S+2}}\,,$ (4.6)

and

$\Psi_{x_{0},s,\bar{s}}^{(\Delta,S+{1\over 2},S-{1\over
2})}(x_{1},x_{2})=\frac{\cancel{x}_{21}\cancel{\bar{x}}_{10}\epsilon
s\left[s\left(\frac{\cancel{x}_{10}}{x_{10}^{2}}-\frac{\cancel{x}_{20}}{x_{20}^{2}}\right)\bar{s}\right]^{S-{1\over
2}}}{|x_{12}|^{S-\Delta+3}|x_{10}|^{\Delta-S+1}|x_{20}|^{\Delta-S+1}}\,.$
(4.7)

Here, $s$ and $\bar{s}$ are the two spinor polarisations of the third operator
is located at $x_{0}$.

The corresponding eigenvalues can be obtained by either acting with
$\widehat{B}$ on the eigenfunctions (4.6), (4.7) and performing the
integrations using the star triangle identity (B.2),101010The first integral
is of the form (B.2) with $S=1$, $\zeta^{\mu}=s_{0}\sigma^{\mu}\bar{s}_{0}$,
and $A_{\mu\nu}=\bar{\sigma}_{\nu}\sigma_{\mu}\bar{s}_{0}$. The second
integral is of the form (B.2) with $S=0$. or by acting on them with the
inverse

$\widehat{B}^{-1}=\frac{1}{16}x_{12}^{2}\,\cancel{\partial}_{x_{2}}\,x_{12}^{2}\,\cancel{\bar{\partial}}_{x_{2}}\Box_{x_{1}}=\frac{1}{16}x_{12}^{2}\,\cancel{x}_{21}\,\Box_{x_{2}}\,\cancel{\bar{x}}_{21}\Box_{x_{1}}\,.$
(4.8)

The eigenvalues we find are

$\displaystyle E(\Delta,S-1/2,S+1/2)=$
$\displaystyle\frac{16}{\left(\Delta+S-1\right)^{2}\left(\Delta-S-2\right)\left(\Delta-S-4\right)}\,,$
(4.9) $\displaystyle E(\Delta,S+1/2,S-1/2)=$
$\displaystyle\frac{16}{\left(\Delta-S-3\right)^{2}\left(\Delta+S-2\right)\left(\Delta+S\right)}\,.$
(4.10)

Note that the shadow transform maps the $(\Delta,S-1/2,S+1/2)$ representation
to the $(4-\Delta,S+1/2,S-1/2)$ one. Correspondingly, the eigenfunctions of
the graph-building operators are interchanged under this transformation.

There are two solutions to $\xi_{4/3}^{4}E(\Delta,S-1/2,S+1/2)=1$ with
positive tree-level dimension, one with twist two and the other with twist
four. They can be written in closed form, but these are not very illuminating.
Instead, we only give the first few terms in their perturbative expansion

$\displaystyle\Delta_{2}(\xi_{4/3},S-1/2,S+1/2)$
$\displaystyle=2+S-\frac{2}{(S+1/2)^{2}}\xi_{4/3}^{4}+\frac{2(S-3/2)}{(S+1/2)^{5}}\xi_{4/3}^{8}+O(\xi_{4/3}^{12})\,,$
(4.11) $\displaystyle\Delta_{4}(\xi_{4/3},S-1/2,S+1/2)$
$\displaystyle=4+S+\frac{2}{(S+3/2)^{2}}\xi_{4/3}^{4}-\frac{2(S+7/2)}{(S+3/2)^{5}}\xi_{4/3}^{8}+O(\xi_{4/3}^{12})\,.$
(4.12)

For $S=1/2$ and $\xi_{4/3}=0$, they correspond to two sets of degenerate
primary operators, ${\rm tr\,}(Z\psi_{4})$ and ${\rm
tr\,}(Z\Box\psi_{4})+\dots$ or ${\rm tr\,}(Z\psi_{1}^{\dagger})$ and ${\rm
tr\,}(Z\Box\psi_{1}^{\dagger})+\dots$ respectively.

Similarly, there are two solutions to $\xi_{4/3}^{4}E(\Delta,S+1/2,S-1/2)=1$
with $\real(\Delta)>2$. Perturbatively, they take the form

$\Delta_{\frac{7}{2},\pm}=3+S\pm\frac{2}{\sqrt{(S+1/2)(S+3/2)}}\xi_{4/3}^{2}-\frac{2(S+1)}{(S+1/2)^{2}(S+3/2)^{2}}\xi_{4/3}^{4}+O(\xi_{4/3}^{6})\,.$
(4.13)

For $S=1/2$ and $\xi_{4/3}=0$, they correspond to linear combinations of ${\rm
tr\,}(Z\psi_{1}^{\dagger}X)$ and ${\rm tr\,}(ZX\psi_{1}^{\dagger})$ or ${\rm
tr\,}(Z\psi_{4}X)$ and ${\rm tr\,}(ZX\psi_{4})$. As before, the perturbative
expansion is in terms of $\xi_{4/3}^{2}$ (and not $\xi_{4/3}^{4}$) because of
the mixing between the two sets, see appendix E.2.

Finally, following the procedure reviewed in appendix C.1, we have collected
in appendix C.3 the other quantities required for the complete computation of
the correlator (4.2).

###  4.2 The Operators ${\rm tr\,}(Z\psi_{2})$ and ${\rm
tr\,}(Z\psi_{3}^{\dagger})$

Instead of the fermion $\psi_{4}$ and $\psi_{1}^{\dagger}$, we can consider
operators that are composed from one $\psi_{2}$ or $\psi_{3}^{\dagger}$ field,
(together with one $Z$ field and at most one $X$ field). The suitable double-
scaling limit in this case is (1.1) with $n=4$. The relevant interaction terms
are now

$\mathcal{L}_{\text{int}}=\operatorname{i}\sqrt{2}Nge^{\frac{\operatorname{i}}{2}\gamma_{1}^{+}}\,{\rm
tr\,}\big{(}\psi_{3}X\psi_{2}\big{)}+\operatorname{i}\sqrt{2}Nge^{\frac{\operatorname{i}}{2}\gamma_{1}^{+}}\,{\rm
tr\,}\big{(}\psi^{\dagger}_{3}X^{\dagger}\psi^{\dagger}_{2}\big{)}+2Ng^{2}e^{-\operatorname{i}\gamma_{3}}\,{\rm
tr\,}\big{(}X^{\dagger}Z^{\dagger}XZ\big{)}+\dots\,.$ (4.14)

The diagrams that survive in this limit look the same as the ones before, see
figure 11. The only difference is in the labeling of the fermions. Hence, all
the results above also apply here, with $\xi_{4/3}\to\xi_{4}$.

###  4.3 The Operators ${\rm tr\,}(\psi_{4}Z^{J})$

One can generalise the operator from section 4.1 and study ${\rm
tr\,}(\psi_{4}Z^{J})$ with $J>1$ perturbatively under the limit of (1.1) with
$n=\frac{2J+2}{2J+1}$.

We take the wave function in this case to be the correlator

$\psi=\langle{\rm tr\,}(\psi_{4}(x_{0})Z(x_{1})\dots Z(x_{J}))\,{\cal
O}\rangle\,.$ (4.15)

The inverse of the corresponding graph-building operator takes the form

$\widehat{B}^{-1}=x^{2}_{0J}\cancel{\partial}_{x_{0}}\prod_{i=1}^{J}x^{2}_{i,i-1}\cancel{\bar{\partial}}_{x_{0}}\prod_{j=1}^{J}\Box_{x_{j}}\,.$
(4.16)

When $J=1$, this naturally coincides with (4.8), up to normalisation and
relabeling of the points.

In the integrability picture, we construct a spin chain with one fermion at
position $x_{0}$ and $J$ scalars at positions $x_{1},\dots,x_{J}$. The inverse
of the graph-building operator (4.16) is proportional to the corresponding
transfer matrix in the $\bf 6$ representation, evaluated at $u=0$:

$\widehat{B}^{-1}\ \propto\
\widehat{T}_{\mathbf{6}}\big{(}0;-1/4,0,..,0\big{)}={\rm
tr\,}_{\mathbf{6}}\left(L_{x_{0}}^{\left(3/2,1,0;\boldsymbol{6}\right)}\left(-1/4\right)\cdot
L_{x_{1}}^{\left(1,0,0;\boldsymbol{6}\right)}(0)\cdot\ldots\cdot
L_{x_{J}}^{\left(1,0,0;\boldsymbol{6}\right)}(0)\right)\,.$ (4.17)

The relevant Lax matrices are given in equations (D.13) and (D.15) of appendix
D.3.

###  4.4 The Operator ${\rm tr\,}(\psi_{4}\psi_{1}^{\dagger}Z)$

There are also operators with more than one fermion that do not mix. Consider
for example the operator ${\rm tr\,}(\psi_{4}\psi_{1}^{\dagger}Z)$ in the
double-scaling limit (1.1) with $n=\frac{3}{2}$.

We take the wave function in this case to be the correlator

$\psi=s_{1}^{\alpha}s_{2}^{\beta}\,\langle{\rm
tr\,}(\psi_{4,\alpha}(x_{1})\psi_{1,\beta}^{\dagger}(x_{2})Z(x_{3}))\,{\cal
O}\rangle\,.$ (4.18)

where $s_{1}$ and $s_{2}$ are the (independent) spinor polarisations.

The bulk diagrams contributing in perturbation theory appear in figure 13. The
corresponding graph-building operator is plotted in figure 14. Its inverse
reads

$\widehat{B}^{-1}=x_{13}^{2}\,x_{23}^{2}\,s_{1}^{\alpha}\,\bar{s}_{2,\dot{\alpha}}\,\cancel{\partial}_{x_{1},\alpha\dot{\beta}}\,\bar{\cancel{\partial}}_{x_{2}}^{\,\dot{\alpha}\beta}x_{12}^{2}\,\bar{\cancel{\partial}}_{x_{1}}^{\,\dot{\beta}\gamma}\,\cancel{\partial}_{x_{2},\beta\dot{\gamma}}\,\partial_{s^{\gamma}_{1}}\partial_{\bar{s}_{2}}^{\dot{\gamma}}\,\Box_{x_{3}}\,.$
(4.19)

Figure 13: The first few diagrams in the perturbative expansion that
contribute to the correlator $\langle{\rm
tr\,}(\psi_{4}(x_{1})\psi_{1}^{\dagger}(x_{2})Z(x_{3}))\,{\rm
tr\,}(Z^{\dagger}(x_{6})\psi_{1}(x_{5})\psi_{4}^{\dagger}(x_{4}))\rangle$.
Figure 14: The graph-building operator for the correlator $\psi$ in (4.18).

The associated integrable spin chain contains a fermion and an anti-fermion of
dimension $\frac{3}{2}$, and a scalar of dimension $1$. The inverse of the
graph-building operator (4.19) is proportional to the corresponding transfer
matrix in the $\bf 6$ representation, evaluated at $u=0$:

$\widehat{B}^{-1}\ \propto\
\widehat{T}_{\mathbf{6}}\left(0;-1/4,1/4,0\right)={\rm
tr\,}_{\mathbf{6}}\left(L_{x_{1}}^{\left(3/2,1,0;\boldsymbol{6}\right)}\left(-1/4\right)\cdot
L_{x_{2}}^{\left(3/2,0,1;\boldsymbol{6}\right)}\left(1/4\right)\cdot\text{$L_{x_{3}}^{\left(1,0,0;\boldsymbol{6}\right)}$}\left(0\right)\right)\,.$
(4.20)

The relevant Lax matrices are given in equations (D.13), (D.15), and (D.16) of
appendix D.3.

Similarly, one could consider operators with more scalars like ${\rm
tr\,}(\psi_{4}Z^{m}\psi_{1}^{\dagger}Z^{n})$, or operators with more fermions
of the same flavour like ${\rm tr\,}(\psi_{4}\psi_{4}Z^{m}\psi_{4}Z^{n})$.

##  5 Holography and the Fishchain Model

One of the main motivations for this work is to extend our understanding of
holography at large $N$. For the fishnet model, a first-principles derivation
of a holographic dual was put forward in [25, 26, 27]. Ideally, one would like
to extend this derivation to ${\cal N}=4$ SYM theory. The dual “fishchain
model” however, only captures a tiny set of operators of the mother ${\cal
N}=4$ SYM theory. In this section we show that the fishchain model can be
extended to include all the operators that we have studied here. They all have
the same strong coupling classical description as that of the fishnet
operators. They differ at the quantum level, and may include internal degrees
of freedom at the fishchain sites.

The basic idea of the holographic derivation is to identify the equation

$\mathcal{H}\,|\psi\rangle\equiv\left(\xi^{-2M}\widehat{B}^{-1}-1\right)|\psi\rangle=0\,,$
(5.1)

with a time reparametrisation constraint for a chain of particles. This
identification then results in a chain of particles with nearest-neighbour
interactions moving in $AdS_{5}$.

In the fishchain model, quantum corrections are controlled by $\hbar=1/\xi$.
Because $\partial={\operatorname{i}\over\hbar}\hat{p}$, at large $|\xi|$ the
only terms in $\widehat{B}^{-1}$ that remain are the ones with two derivatives
at each site. For all the cases we have studied these terms turn out to be the
same as the fishnet ones. Therefore, at strong coupling they are all described
by the classical fishchain model [25].

For example, consider the inverse of the graph-building operator in (4.8) for
the case of $\text{Tr}(Z\psi_{4\alpha})$. It can be written as

$\widehat{B}^{-1}=x^{4}_{12}\,\Box_{x_{2}}\,\Box_{x_{1}}+2x^{2}_{12}\,\cancel{x}_{21}\cancel{\bar{\partial}}_{x_{2}}\,\Box_{x_{1}}\;.$
(5.2)

Here the first term is the inverse of the fishnet graph-building operator
while the second has only one derivative at site $2$. It is therefore
suppressed at large $|\xi_{4/3}|$. Moreover, at this site we have a fermionic
degree of freedom on which the second term acts non-trivially.

The quantisation of the classical fishchain model results in a chain of
particles moving in $AdS_{5}$. These particles may now have internal degrees
of freedom. The map between the boundary wave function $\Phi_{\cal O}$ and the
fishchain bulk wave function $\Psi_{\cal O}$ is defined as111111To include
fermions, a different bulk-to-boundary propagator is required and also a
different embedding formalism, see [31].

$\Psi\left(\\{Z_{i}\\}_{i=0}^{M};\\{\Omega_{i}\\}_{i=0}^{M}\right)\equiv\int\prod_{j=1}^{M}\left(D^{4}Y_{j}\,G^{\Delta_{j},[p_{j}]}\left(Z_{j},\Omega_{j}|Y_{j},\partial_{\Theta_{j}}\right)\right)\Phi\left(\\{Y_{j}\\}_{i=0}^{M};\\{\Theta_{i}\\}_{i=0}^{M}\right)\,,$
(5.3)

where the $Y_{i}$’s and the $\Theta_{i}$’s are the boundary embedding
coordinates and polarisation vectors. Similarly, the $Z_{i}$’s and
$\Omega_{i}$’s are the bulk AdS embedding coordinates and polarisation
vectors. These coordinates are subject to the constraints $Z\cdot Z=-1$ and
$\Omega\cdot Z=0$. The definition of the integration measure can be found in
Section 5.1 of [26]. Finally, $G^{\Delta,[p]}$ is the $p$-form bulk-to-
boundary propagator. It is given by [32]

$G^{\Delta,[p]}\left(Z,\Omega|Y,\partial_{\Theta}\right)=\frac{\left[\left(\partial_{\Theta}\cdot\Omega\right)\left(Y\cdot
Z\right)-\left(Z\cdot\partial_{\Theta}\right)\left(Y\cdot\Omega\right)\right]^{p}}{-4\pi^{2}\left(Z\cdot
Y\right)^{4-\Delta+p}}\,.$ (5.4)

The map (5.3) can be extended to the case with mixing that we have studied in
the previous section. In that case, $M=J$ and the wave function is a direct
sum of two spaces

$\Phi\left(\\{Y_{j}\\}_{i=0}^{J};\Theta_{0}\right)=\left(\\!\\!\begin{array}[]{c}\Phi_{F}\left(\\{Y_{j}\\}_{i=0}^{J};\Theta_{0}\right)\\\
\Phi_{X}\left(\\{Y_{j}\\}_{i=0}^{J}\right)\end{array}\\!\\!\right)\,,\qquad\Psi\left(\\{Z_{j}\\}_{i=0}^{J};\Omega_{0}\right)=\left(\\!\\!\begin{array}[]{c}\Psi_{F}\left(\\{Z_{j}\\}_{i=0}^{J};\Omega_{0}\right)\\\
\Psi_{X}\left(\\{Z_{j}\\}_{i=0}^{J}\right)\end{array}\\!\\!\right)\,.$ (5.5)

At sites $i=1,\ldots,J$ we have $(\Delta_{i},p_{i})=(1,0)$. At site $i=0$ we
have the direct sum of $(\Delta,p)=(2,2)\bigoplus(2,0)$. Correspondingly, in
(5.3) we have

$G^{\Delta_{0},[p_{0}]}\,\Phi=\left(\\!\\!\begin{array}[]{c}G^{2,[2]}\,\Phi_{F}\\\
G^{2,[0]}\,\Phi_{X}\end{array}\\!\\!\right)\,.$ (5.6)

It follows from this map that $(Z\cdot\partial_{\Omega})\Psi_{F}(Z,\Omega)=0$.
Moreover, the constraint $(Y\cdot\partial_{\Theta})\Phi_{F}=0$ implies that
$(\partial_{Z}\cdot\partial_{\Omega})\Psi_{F}(Z,\Omega)=0$. Using (3.23),
(5.1), and (5.3), the inverse of the graph-building operator maps into the
following quantum fishchain Hamiltonian

$\mathcal{H}=\xi^{-2(J+1)}_{1+1/J}\,{\rm tr\,}_{\bf
6}\left[\left(\\!\\!\begin{array}[]{cc}{\mathbb{L}}_{FF}(Z_{0},\Omega_{0})&{\mathbb{L}}_{FX}(Z_{0},\Omega_{0})\\\
{\mathbb{L}}_{XF}(Z_{0},\Omega_{0})&{\mathbb{L}}_{XX}(Z_{0},\Omega_{0})\end{array}\\!\\!\right)\cdot\mathbb{L}\left(Z_{1}\right)\cdot\ldots\cdot\mathbb{L}\left(Z_{J}\right)\right]-1\,,$
(5.7)

with

$\displaystyle{\mathbb{L}}(Z)$
$\displaystyle=-\frac{1}{2}\left(Z^{M}Z^{N}\Box_{Z}+2Z^{(M}\partial_{Z}^{N)}-\partial_{Z}^{M}\partial_{Z}^{N}+2\eta^{MN}\right),$
$\displaystyle{\mathbb{L}}_{FF}(Z,\Omega)$
$\displaystyle=-\frac{1}{2}\left(\left(\partial_{Z}\cdot\partial_{\Omega}\right)Z^{M}Z^{N}\left(\Omega\cdot\partial_{Z}\right)-\partial_{Z}^{M}\partial_{Z}^{N}\right),$
$\displaystyle{\mathbb{L}}_{XX}(Z,\Omega)$
$\displaystyle=-\frac{1}{2}\left(\frac{1}{2}Z^{(M}\Box_{Z}Z^{N)}-\partial_{Z}^{M}\partial_{Z}^{N}+2\eta^{MN}\right),$
(5.8) $\displaystyle{\mathbb{L}}_{FX}(Z,\Omega)$
$\displaystyle=\frac{1}{2}\left(\Omega\cdot\partial_{Z}\right)\Omega^{[M}\partial_{Z}^{N]},$
$\displaystyle{\mathbb{L}}_{XF}(Z,\Omega)$
$\displaystyle=-\frac{1}{4}\partial^{[N}_{\Omega}\partial_{\Omega}^{M]}\,.$

##  6 Conclusion and Discussion

In this paper, we have argued that any single-trace operator in the planar
limit of ${\cal N}=4$ SYM theory has a large twist double-scaling limit. In
this operator-dependent limit, the loop corrections to the operator two-point
function have an iterative structure. They can therefore be analysed to all
loop order using the graph-building operator technique. Correspondingly, they
all have a dual fishchain description. This description is discrete at any
order in the expansion around the large twist limit. The picture that emerges
is that of operators breaking into sectors with different double-scaling
limits. At the level of the spectrum, these sectors decouple from each other.

We have considered several types of examples. The case of mixing between
operators with different double-scaling limits is both interesting and
generic. The corresponding graph-building operator is a matrix, which can be
used both to study the large twist operator’s dimensions and to compute
corrections away from this limit. Finally, we have proven the integrability of
all the cases that we have considered.

There are many future directions to pursue, some of which we list below.

  * •

In section 3.1 we have found that the case with mixing is an integrable spin-
chain model. It would be interesting to understand whether it fits inside the
standard construction of integrable models or whether it is new.

  * •

In section 3.2 we have described how certain corrections away from the large
twist double-scaling limit arise from mixing between operators with a
different double-scaling limit. It would be interesting to understand if this
construction can be extended such that all the corrections are of this type.
If true, it would imply that the elements of the corresponding graph-building
operator do not receive loop corrections.

  * •

To solve the theory at large twist we also need to compute the structure
constants. Three point functions between operators with the same double-
scaling limit are expected to take a similar form to the ones in the fishnet
theory. On top of these, it would be interesting to explore three point
functions between operators with different double-scaling limits.

  * •

A similar fishnet-like limit for $\mathcal{N}=2$ supersymmetric gauge theories
was introduced in [33]. It would be interesting to study this structure in
such theories using similar generalised double-scaling limits.

## Acknowledgements

We thank S. Chester and D.-l. Zhong for invaluable discussions. We thank D.-l.
Zhong for comments on the draft. AS, GF and EO are supported by the Israel
Science Foundation (grant number 1197/20). GF is grateful to the Azrieli
Foundation for the award of an Azrieli Fellowship.

##  Appendix A Notation and Conventions

In this appendix, we detail the conventions and notation we use throughout the
paper.

###  A.1 Metric and Sigma Matrices

####  A.1.1 4 Dimensions

We work in Euclidean signature and use Greek indices to denote the flat four
dimensional coordinates as $x^{\mu}$, with $\mu=1,2,3,4$. The sigma matrices
(or chiral projections of the gamma matrices) are

$\sigma^{\mu}=\left(\operatorname{i}\sigma_{x},\operatorname{i}\sigma_{y},\operatorname{i}\sigma_{z},\mathrm{I}_{2}\right)\quad\text{and}\quad\bar{\sigma}^{\mu}=\left(-\operatorname{i}\sigma_{x},-\operatorname{i}\sigma_{y},-\operatorname{i}\sigma_{z},\mathrm{I}_{2}\right)\,,$
(A.1)

where $\sigma_{x},\sigma_{y},$ and $\sigma_{z}$ are the usual Pauli matrices
while $\mathrm{I}_{2}$ is the $2\times 2$ identity matrix.

The sigma matrices satisfy

$\sigma^{\mu}\bar{\sigma}^{\nu}+\sigma^{\nu}\bar{\sigma}^{\mu}=\bar{\sigma}^{\mu}\sigma^{\nu}+\bar{\sigma}^{\nu}\sigma^{\mu}=2\,\delta^{\mu\nu}\mathrm{I}_{2}\,,$
(A.2)
$\sigma^{\mu}=(\bar{\sigma}^{\mu})^{\dagger}\,,\qquad\epsilon\sigma^{\mu}\epsilon=-^{t}\\!\bar{\sigma}^{\mu}\,,$
(A.3)

where $\epsilon=\begin{pmatrix}0&1\\\ -1&0\\\ \end{pmatrix}$ and
${}^{t}\\!\bar{\sigma}^{\mu}$ is the transpose of $\bar{\sigma}^{\mu}$.
Moreover,

$\sigma^{\mu}\otimes\bar{\sigma}_{\mu}=2\,\mathbb{P}\,,\quad\sigma^{\mu}\otimes\sigma_{\mu}=\bar{\sigma}^{\mu}\otimes\bar{\sigma}_{\mu}=2\left(\mathrm{I}_{4}-\mathbb{P}\right)\,,$
(A.4)

where $\mathrm{I}_{4}$ is the $4\times 4$ identity matrix and $\mathbb{P}$ is
the permutation operator on $\mathbb{C}^{2}\otimes\mathbb{C}^{2}$. In explicit
index notations, the previous relations read

$(\sigma^{\mu})_{\alpha\dot{\alpha}}(\bar{\sigma}_{\mu})^{\dot{\beta}\beta}=2\,\delta^{\beta}_{\alpha}\delta^{\dot{\beta}}_{\dot{\alpha}}\,,\quad(\sigma^{\mu})_{\alpha\dot{\alpha}}(\sigma_{\mu})_{\beta\dot{\beta}}=2\,\epsilon_{\alpha\beta}\epsilon_{\dot{\alpha}\dot{\beta}}\,,\quad(\bar{\sigma}^{\mu})^{\dot{\alpha}\alpha}(\bar{\sigma}_{\mu})^{\dot{\beta}\beta}=2\,\epsilon^{\alpha\beta}\epsilon^{\dot{\alpha}\dot{\beta}}\,,$
(A.5)

with
$\epsilon_{\alpha\beta},\epsilon_{\dot{\alpha}\dot{\beta}},\epsilon^{\alpha\beta},\epsilon^{\dot{\alpha}\dot{\beta}}$
all antisymmetric in their indices and such that
$\epsilon_{12}=\epsilon_{\dot{1}\dot{2}}=\epsilon^{12}=\epsilon^{\dot{1}\dot{2}}=1$.

The matrices

$\sigma_{\mu\nu}=\frac{\sigma_{\mu}\bar{\sigma}_{\nu}-\sigma_{\nu}\bar{\sigma}_{\mu}}{4}\quad\text{and}\quad\bar{\sigma}_{\mu\nu}=\frac{\bar{\sigma}_{\mu}\sigma_{\nu}-\bar{\sigma}_{\nu}\sigma_{\mu}}{4}$
(A.6)

are the generators of the two inequivalent two-dimensional representations of
$\mathfrak{so}(4,\mathbb{C})$.

####  A.1.2 6 Dimensions

We use upper-case Latin indices $M,N,P$, etc. to denote indices ranging from
$1$ to $6$. The indices are lowered using the tensor
$\eta_{MN}=\operatorname{Diag}(1,1,1,1,1,-1)_{MN}$. For an arbitrary tensor
$F^{\cdots MNP\cdots}$ it is also convenient to define

$F^{\cdots M\pm P\cdots}=F^{\cdots M6P\cdots}\pm F^{\cdots M5P\cdots}\,.$
(A.7)

A contraction of two tensors can then be written

$A^{M}B_{M}=A^{M}B^{N}\eta_{MN}=A^{\mu}B_{\mu}+A^{5}B^{5}-A^{6}B^{6}=A^{\mu}B_{\mu}-\frac{A^{+}B^{-}+A^{-}B^{+}}{2}\,.$
(A.8)

###  A.2 $\gamma$-Deformed $\mathcal{N}=4$ SYM

The $\gamma$-deformed theory is a family of deformations of the interaction
terms in the ${\cal N}=4$ SYM Euclidean Lagrangian. It is parameterised by
$\gamma_{i=1,2,3}$, and is defined as

$\mathcal{L}=-N\,\text{Tr}\left[\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+D^{\mu}\phi_{i}^{\dagger}D_{\mu}\phi^{i}+\psi^{\dagger}\overline{\cancel{D}}\psi\right]+\mathcal{L}_{int}\,,$
(A.9)

with

$\displaystyle\mathcal{L}_{\text{int}}=\ \ g^{2}N\,{\rm tr\,}$
$\displaystyle\Big{[}2\,e^{-\operatorname{i}\epsilon^{ijk}\gamma_{k}}\phi_{i}^{\dagger}\phi_{j}^{\dagger}\phi^{i}\phi^{j}-\frac{1}{2}\\{\phi_{i}^{\dagger},\phi^{i}\\}\\{\phi_{j}^{\dagger},\phi^{j}\\}\Big{]}$
$\displaystyle+\sqrt{2}gN\,{\rm tr\,}$
$\displaystyle\Big{[}e^{\frac{\operatorname{i}}{2}\gamma_{j}^{-}}\left(\psi^{\dagger}_{4}\phi^{j}\psi^{\dagger}_{j}-\psi^{4}\phi_{j}^{\dagger}\psi^{j}\right)+e^{-\frac{\operatorname{i}}{2}\gamma_{j}^{-}}\left(\psi^{j}\phi_{j}^{\dagger}\psi^{4}-\psi^{\dagger}_{j}\phi^{j}\psi^{\dagger}_{4}\right)\Big{]}$
(A.10) $\displaystyle+\sqrt{2}gN\,{\rm tr\,}$
$\displaystyle\Big{[}\operatorname{i}e^{\frac{\operatorname{i}}{2}\epsilon_{imk}\gamma_{m}^{+}}\left(\epsilon_{ijk}\psi^{i}\phi^{j}\psi^{k}+\epsilon^{ijk}\psi^{\dagger}_{i}\phi_{j}^{\dagger}\psi^{\dagger}_{k}\right)\Big{]}\,.$

Here,

$\gamma_{1}^{\pm}=-\frac{\gamma_{3}\pm\gamma_{2}}{2},\qquad\gamma_{2}^{\pm}=-\frac{\gamma_{1}\pm\gamma_{3}}{2},\qquad\gamma_{3}^{\pm}=-\frac{\gamma_{2}\pm\gamma_{1}}{2}\,,$
(A.11)

and we have suppressed the spinor indices, assuming the contractions
$\epsilon^{\beta\alpha}\psi^{i}_{\alpha}\psi^{j}_{\beta}$ and
$\epsilon^{\dot{\alpha}\dot{\beta}}\psi^{\dagger}_{i,\dot{\alpha}}\psi^{\dagger}_{j,\dot{\beta}}$
for instance. The covariant derivative and the field strength are

$D_{\mu}=\partial_{\mu}+\operatorname{i}g[A_{\mu},\cdot]$ (A.12)

and

$F_{\mu\nu}=-\frac{\operatorname{i}}{g}[D_{\mu},D_{\nu}]=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}+\operatorname{i}g[A_{\mu},A_{\nu}]\,,$
(A.13)

And we use $\overline{\cancel{D}}=D_{\mu}\bar{\sigma}^{\mu}$. In (A.9),
moreover, summation over doubly and triply repeated indices is assumed.
Throughout this paper we use either Feynman gauge, or more generally, the
$R_{\alpha}$ gauge. The Feynman rules associated to the propagators are
recalled in appendix A.3. In other parts of this paper, we also use notation
where the three complex scalars are $X=\phi_{1}$, $Z=\phi_{2}$, and
$Y=\phi_{3}$.

In terms of the three $\gamma$-parameters, the double-scaling limit in all the
examples we consider are of the type (1.1) with $\theta=-\gamma_{3}$, and
$\gamma_{1}=\gamma_{2}=0$. In particular, the fishnet limit is that limit with
$n=1$. Namely,121212Note that in (A.14) $\gamma_{3}$ is complex. Hence, the
$\gamma$-deformed CFT is not unitary. Instead of twisting the theory, as in
(A.2), one can take the viewpoint where we stay with ${\cal N}=4$ SYM theory
and twist the operators, see [2].

$\gamma_{1}=\gamma_{2}=0\,,\qquad
e^{-\operatorname{i}\gamma_{3}}\to\infty\,,\qquad\text{with}\quad\xi_{1}^{2}\equiv{g^{2}\,e^{-\operatorname{i}\gamma_{3}}\over
8\pi^{2}}\quad\text{fixed}\,.$ (A.14)

The twists we consider can be obtained by replacing the product of fields in
the Lagrangian with a star-product (before integrating out the auxiliary
fields), [34]. This star-product is defined as

$A\star B=e^{\frac{\operatorname{i}}{2}q_{A}\wedge q_{B}}AB$ (A.15)

where

$q_{A}\wedge q_{B}=\left(q_{A}\right)^{\mathrm{T}}Cq_{B},\quad
C=\left(\begin{array}[]{ccc}0&-\gamma_{3}&\gamma_{2}\\\
\gamma_{3}&0&-\gamma_{1}\\\ -\gamma_{2}&\gamma_{1}&0\end{array}\right)\;,$
(A.16)

and the $q_{A}$’s are the $SU(4)$ R-symmetry Cartan charges. Explicitly, the
charges are given by

$\begin{array}[]{c|cccc|c|ccc}&\psi_{\alpha}^{1}&\psi_{\alpha}^{2}&\psi_{\alpha}^{3}&\psi_{\alpha}^{4}&A_{\mu}&\phi_{1}&\phi_{2}&\phi_{3}\\\
\hline\cr
q_{B}^{1}&+\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&+\frac{1}{2}&0&1&0&0\\\
q_{B}^{2}&-\frac{1}{2}&+\frac{1}{2}&-\frac{1}{2}&+\frac{1}{2}&0&0&1&0\\\
q_{B}^{3}&-\frac{1}{2}&-\frac{1}{2}&+\frac{1}{2}&+\frac{1}{2}&0&0&0&1\end{array}$
(A.17)

Note that the Lagrangian (A.9) needs to be supplemented with ghost fields.
They are not relevant in the examples we study.

###  A.3 Feynman Rules

The free propagators in the $R_{\alpha}$ gauge are given by

$\displaystyle\langle(\phi_{i}^{\dagger})_{ab}(x)\phi^{j}_{cd}(y)\rangle_{0}$
$\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}N_{c}(x-y)^{2}}\delta_{i}^{j}\left(\delta_{ad}\delta_{bc}-\frac{\delta_{ab}\delta_{cd}}{N_{c}}\right)\,,$
(A.18) $\displaystyle\langle A^{\mu}_{ab}(x)A^{\nu}_{cd}(y)\rangle_{0}$
$\displaystyle=$
$\displaystyle\frac{\frac{\alpha+1}{2\alpha}\delta^{\mu\nu}+\frac{\alpha-1}{\alpha}\frac{(x-y)^{\mu}(x-y)^{\nu}}{(x-y)^{2}}}{4\pi^{2}N_{c}(x-y)^{2}}\left(\delta_{ad}\delta_{bc}-\frac{\delta_{ab}\delta_{cd}}{N_{c}}\right)\,,$
(A.19) $\displaystyle\langle\psi^{A}_{\alpha
cd}(x)(\psi_{\dot{\alpha}B}^{\dagger})_{ab}(y)\rangle_{0}$ $\displaystyle=$
$\displaystyle\frac{(\cancel{x}-\cancel{y})_{\alpha\dot{\alpha}}}{2\pi^{2}N_{c}(x-y)^{4}}\delta^{A}_{B}\left(\delta_{ad}\delta_{bc}-\frac{\delta_{ab}\delta_{cd}}{N_{c}}\right)\,,$
(A.20)

where $\cancel{x}=x_{\mu}\sigma^{\mu}$, and, in this equation only, the upper-
case indices $A$ and $B$ range from 1 to 4.

##  Appendix B Star-Triangle Relations

We collect here some integral identities, useful for the eigenvalue
computations of section 2.1 and section 4.1. Such identities appeared for the
first time in [35] (see also [6]). In this section, $\ell$ and $S$ are non-
negative integers, $\zeta$ is a null vector ($\zeta^{\mu}\zeta_{\mu}=0$), and
$A$ is a 2-tensor such that

$A_{\mu\nu}+A_{\nu\mu}=\frac{\delta_{\mu\nu}}{2}?A^{\rho}_{\rho}?\,,\quad?A^{\mu}_{\nu}??A_{\mu\rho}?=?A_{\mu\nu}??A_{\rho}^{\nu}?=0\,.$
(B.1)

Provided that $\zeta^{\mu}A_{\mu\nu}=0$ and $\alpha+\beta+\gamma=4+S$, the
following integral identities hold true

$\int\frac{\left[(x_{1}-y)^{\mu}A_{\mu\nu}(y-x_{2})^{\nu}\right]^{S}\left[\zeta\cdot\left(\frac{y-x_{1}}{(y-x_{1})^{2}}-\frac{x_{21}}{x_{21}^{2}}\right)\right]^{\ell}}{(x_{1}-y)^{2\alpha}(x_{2}-y)^{2\beta}(x_{3}-y)^{2\gamma}}\frac{\differential^{4}y}{\pi^{2}}\\\
=\frac{\Gamma(2+S-\alpha)\Gamma(2+S+\ell-\beta)\Gamma(2-\gamma)}{\Gamma(\alpha+\ell)\Gamma(\beta)\Gamma(\gamma)}\frac{\left[x_{13}^{\mu}A_{\mu\nu}x_{32}^{\nu}\right]^{S}\left[\zeta\cdot\left(\frac{x_{31}}{x_{31}^{2}}-\frac{x_{21}}{x_{21}^{2}}\right)\right]^{\ell}}{x_{12}^{2(2-\gamma)}x_{23}^{2(2+S-\alpha)}x_{31}^{2(2+S-\beta)}}\,,$
(B.2)

and

$\int\frac{\left[(x_{1}-y)^{\mu}A_{\mu\nu}(y-x_{2})^{\nu}\right]^{S}\left[\zeta\cdot\left(\frac{y-x_{1}}{(y-x_{1})^{2}}-\frac{x_{31}}{x_{31}^{2}}\right)\right]^{\ell}}{(x_{1}-y)^{2\alpha}(x_{2}-y)^{2\beta}(x_{3}-y)^{2\gamma}}\frac{\differential^{4}y}{\pi^{2}}\\\
=\frac{\Gamma(2+S-\alpha)\Gamma(2+S-\beta)\Gamma(2+\ell-\gamma)}{\Gamma(\alpha+\ell)\Gamma(\beta)\Gamma(\gamma)}\frac{\left[x_{13}^{\mu}A_{\mu\nu}x_{32}^{\nu}\right]^{S}\left[\zeta\cdot\left(\frac{x_{21}}{x_{21}^{2}}-\frac{x_{31}}{x_{31}^{2}}\right)\right]^{\ell}}{x_{12}^{2(2-\gamma)}x_{23}^{2(2+S-\alpha)}x_{31}^{2(2+S-\beta)}}\,.$
(B.3)

If $\zeta_{\rho}\partial_{3}^{\rho}A_{\mu\nu}=0$, $\alpha+\beta+\gamma=4+S$,
and there exists another null vector $\eta$ such that

$A_{\mu\nu}x_{32}^{\nu}=x_{23}^{2}\eta_{\mu}\,,\qquad\zeta^{\mu}\eta_{\mu}=0\,,$
(B.4)

then

$\int\frac{\left[(x_{1}-y)^{\mu}A_{\mu\nu}(y-x_{2})^{\nu}\right]^{S}\left[\zeta\cdot\left(\frac{y-x_{3}}{(y-x_{3})^{2}}-\frac{x_{23}}{x_{23}^{2}}\right)\right]^{\ell}}{(x_{1}-y)^{2\alpha}(x_{2}-y)^{2\beta}(x_{3}-y)^{2\gamma}}\frac{\differential^{4}y}{\pi^{2}}\\\
=\frac{\Gamma(2+S-\alpha)\Gamma(2+S+\ell-\beta)\Gamma(2-\gamma)}{\Gamma(\alpha)\Gamma(\beta)\Gamma(\gamma+\ell)}\frac{\left[x_{13}\cdot\eta\right]^{S}\left[\zeta\cdot\left(\frac{x_{13}}{x_{13}^{2}}-\frac{x_{23}}{x_{23}^{2}}\right)\right]^{\ell}}{x_{12}^{2(2-\gamma)}x_{23}^{2(2-\alpha)}x_{31}^{2(2+S-\beta)}}\,.$
(B.5)

##  Appendix C Correlators

In this appendix, we compute some of the correlators of short non-local
operators studied in the main text.

###  C.1 The Correlator of Two Length-Two Non-local Operators

We begin with a review of the method for computing the two-point function
between non-local operators made from two fields. We loosely call it a “four-
point function”. The procedure has been widely developed in the literature,
see for example [5] for the case of scalars. Since we will also be interested
in fermionic representations, we keep the discussion very general; we merely
assume that the correlator has an iterative structure encoded in a graph-
building operator. In line with this, we will heavily rely on the general
results about harmonic analysis on the conformal group from [36].

Consider some four-point function

$G(x_{1},x_{2}|x_{3},x_{4})\equiv\langle\text{Tr}(\mathcal{O}_{1}(x_{1})\mathcal{O}_{2}(x_{2}))\text{Tr}(\mathcal{O}_{3}(x_{3})\mathcal{O}_{4}(x_{4}))\rangle\,,$
(C.1)

where $\mathcal{O}$ is any combination of the fields in the theory and their
derivatives. We assume that all of the diagrams contributing to $G$ have an
iterative structure. Namely, that we can write the correlator as the geometric
series

$\widehat{G}=\sum_{L=0}^{\infty}\chi^{L}\widehat{B}^{L}\widehat{G}^{(0)}=\frac{1}{1-\chi\widehat{B}}\widehat{G}^{(0)}\;,$
(C.2)

where $\chi$ is some coupling constant, $\widehat{B}$ is the graph-building
operator, and $\widehat{G}^{(0)}$ is the first contribution. All these
operators are integral operators whose kernels we denote by
$B(x_{1},x_{2}|y_{1},y_{2})$, $G^{(0)}(x_{1},x_{2}|y_{1},y_{2})$, etc. In
order to simplify the expressions, we include factors of $\pi^{-2}$ in the
integration measure. The action of $\widehat{B}$ on an arbitrary function
$\Phi$ is thus given by

$\widehat{B}\Phi(x_{1},x_{2})=\int\frac{\differential^{4}y_{1}\differential^{4}y_{2}}{\pi^{4}}B(x_{1},x_{2}|y_{1},y_{2})\Phi(y_{1},y_{2})\,.$
(C.3)

And, for example, the kernel of $\widehat{B}^{2}=\widehat{B}\widehat{B}$ is

$B^{2}(x_{1},x_{2}|y_{1},y_{2})=\int
B(x_{1},x_{2}|z_{1},z_{2})B(z_{1},z_{2}|y_{1},y_{2})\frac{\differential^{4}z_{1}\differential^{4}z_{2}}{\pi^{4}}\,.$
(C.4)

Finding a complete basis of eigenvectors of $\widehat{B}$ will allow us to
compute the four-point function (C.2). We now turn to the diagonalisation of
$\widehat{B}$.

In what follows, we assume that $\widehat{B}$ is conformally invariant.
Namely, we assume that $\widehat{B}$ commutes with the action of two principal
series representations of the conformal algebra. One
$\rho_{1}=(\Delta_{1},\ell_{1},\bar{\ell}_{1})$ acts on $x_{1}$, and the other
$\rho_{2}=(\Delta_{2},\ell_{2},\bar{\ell}_{2})$ acts on $x_{2}$. The
diagonalisation is thus reduced to decomposing the tensor product into
irreducible representations. Consequently, the eigenvectors of $\widehat{B}$
take the form of conformal three-point functions between the operators at
$x_{1}$, $x_{2}$, and a third operator in a representation $\rho$ at $x_{0}$,
$\Psi_{x_{0}}^{\rho}$. For brevity, we do not display the $\mathfrak{so}(4)$
indices of $\Psi_{x_{0}}^{\rho}$ associated with each representation. The
eigenvalue equation takes the form

$\widehat{B}\Psi^{\rho}_{x_{0}}(x_{1},x_{2})=E(\rho)\,\Psi^{\rho}_{x_{0}}\left(x_{1},x_{2}\right)\,.$
(C.5)

In this equation, for instance, there is an implicit contraction of the
indices associated with representations $\rho_{1}$ and $\rho_{2}$ between
$\widehat{B}$ and $\Psi^{\rho}_{x_{0}}$, whereas the indices associated with
$\rho$ remain arbitrary. In order to have a complete basis of eigenvectors, it
will be enough to consider $\rho=(\Delta,\ell,\bar{\ell})$ such that
$\Delta=2+\operatorname{i}\nu$ with $\nu$ real and positive. We introduce the
left eigenvector $\widetilde{\Psi}_{x_{0}}^{\rho}$ of $\widehat{B}$ associated
to the eigenvalue $E(\rho)$. It has the form of a conformal three-point
function of operators transforming according to the representations
$\tilde{\rho}\equiv(4-\Delta,\ell,\bar{\ell})=(2-\operatorname{i}\nu,\ell,\bar{\ell})$,
$\tilde{\rho}_{1}$, and $\tilde{\rho}_{2}$. The orthogonality relation will
read

$\int\frac{\differential^{4}x_{1}\differential^{4}x_{2}}{\pi^{4}}\widetilde{\Psi}_{x_{0}}^{\rho}(x_{1},x_{2})\Psi_{x^{\prime}_{0}}^{\rho^{\prime}}(x_{1},x_{2})=\frac{2\pi^{3}}{\mu(\rho)}\delta_{\rho,\rho^{\prime}}\delta^{(4)}(x_{0}-x^{\prime}_{0})\operatorname{Id}_{\ell,\bar{\ell}}\,,$
(C.6)

where $\Delta=2+\operatorname{i}\nu$ and
$\Delta^{\prime}=2+\operatorname{i}\nu^{\prime}$ with $\nu,\nu^{\prime}>0$,
the indices for $\rho_{1}$ and $\rho_{2}$ have been contracted in the left-
hand side, and

$\delta_{\rho,\rho^{\prime}}=\delta(\nu-\nu^{\prime})\delta_{\ell,\ell^{\prime}}\delta_{\bar{\ell},\bar{\ell}^{\prime}}\,.$
(C.7)

Here, $\operatorname{Id}_{\ell,\bar{\ell}}$ denotes the unit operator in the
representation $(\ell,\bar{\ell})$ of $\mathfrak{so}(4)$. In practice, we will
always contract the indices associated to this space with auxiliary
polarisation vectors $s,\bar{s},s^{\prime},\bar{s}^{\prime}$ so that
$\operatorname{Id}_{\ell,\bar{\ell}}$ would be replaced with $(s\cdot
s^{\prime})^{\ell}(\bar{s}\cdot\bar{s}^{\prime})^{\bar{\ell}}$. The
coefficient $\mu(\rho)$ can be found in [36], and the completeness relation
reads [37]131313We only consider representations of the complementary series
such that $|2-\Delta_{1}|+|2-\Delta_{2}|\leqslant 2$. We can thus use the
results of section 10.D of [37].

$\int\limits_{0}^{\infty}\frac{\differential\nu}{2\pi}\sum_{\begin{subarray}{c}\ell,\bar{\ell}\\\
\rho\subset\rho_{1}\otimes\rho_{2}\end{subarray}}\mu(\rho)\int\frac{\differential^{4}x_{0}}{\pi^{2}}\,\Psi_{x_{0}}^{(2+\operatorname{i}\nu,\ell_{1},\bar{\ell}_{1})}(x_{1},x_{2})\widetilde{\Psi}_{x_{0}}^{(2+\operatorname{i}\nu,\ell_{2},\bar{\ell}_{2})}(y_{1},y_{2})\\\
=\pi^{4}\delta^{(4)}(x_{1}-y_{1})\delta^{(4)}(x_{2}-y_{2})\operatorname{Id}_{\ell_{1},\bar{\ell}_{1}}\operatorname{Id}_{\ell_{2},\bar{\ell}_{2}}\,.$
(C.8)

Inserting this completeness relation in the expression for the four-point
function yields

$G(x_{1},x_{2}|x_{3},x_{4})=\int\limits_{0}^{\infty}\frac{\differential\nu}{2\pi}\sum_{\begin{subarray}{c}\ell,\bar{\ell}\\\
\rho\subset\rho_{1}\otimes\rho_{2}\end{subarray}}\frac{\mu(\rho)}{1-\chi
E(\rho)}\int\frac{\differential^{4}x_{0}}{\pi^{2}}\,\Psi_{x_{0}}^{(2+\operatorname{i}\nu,\ell,\bar{\ell})}(x_{1},x_{2})\left[\widetilde{\Psi}_{x_{0}}^{(2+\operatorname{i}\nu,\ell,\bar{\ell})}\widehat{G}^{(0)}\right](x_{3},x_{4})\,.$
(C.9)

The function
$\widetilde{\Psi}_{x_{0}}^{(2+\operatorname{i}\nu,\ell,\bar{\ell})}\widehat{G}^{(0)}$
will also be of the form of conformal three-point function—up to some power of
$x_{34}^{2}$. The integral over the auxiliary point $x_{0}$ can therefore be
expressed in terms of four-dimensional conformal blocks $\mathfrak{G}$ as
[36]141414Here, the conformal blocks are defined as in subsection 2.7.1 of
[36].

$\int\frac{\differential^{4}x_{0}}{\pi^{2}}\Psi_{x_{0}}^{(2+\operatorname{i}\nu,\ell,\bar{\ell})}(x_{1},x_{2})\left[\widetilde{\Psi}_{x_{0}}^{(2+\operatorname{i}\nu,\ell,\bar{\ell})}\widehat{G}^{(0)}\right](x_{3},x_{4})\\\
=\widetilde{G}_{0}(x_{3},x_{4})C(\rho)\left[S(\rho_{3},\rho_{4};\tilde{\rho})\mathfrak{G}(x_{1},x_{2}|x_{3},x_{4})+S(\rho_{1},\rho_{2};\rho)\widetilde{\mathfrak{G}}(x_{1},x_{2}|x_{3},x_{4})\right]\,.$
(C.10)

In the equation above, $\widetilde{G}_{0}(x_{3},x_{4})C(\rho)$ comes from
$\widetilde{\Psi}_{x_{0}}^{(2+\operatorname{i}\nu,\ell,\bar{\ell})}\widehat{G}^{(0)}$,
whereas the terms between brackets result from the integral over $x_{0}$. The
second term in the brackets can be absorbed into that of the first one upon
extension of the integration domain to the whole real line:

$G(x_{1},x_{2}|x_{3},x_{4})=\widetilde{G}_{0}(x_{3},x_{4})\int\limits_{-\infty}^{\infty}\frac{\differential\nu}{2\pi}\sum_{\begin{subarray}{c}\ell,\bar{\ell}\\\
\rho\subset\rho_{1}\otimes\rho_{2}\end{subarray}}\frac{C(\rho)\mu(\rho)S(\rho_{3},\rho_{4};\tilde{\rho})}{1-\chi
E(\rho)}\mathfrak{G}(x_{1},x_{2}|x_{3},x_{4})\,.$ (C.11)

The block $\mathfrak{G}$ decays exponentially in the $\nu$ lower half-plane,
so we can deform the integration contour in that direction and write the
integral as the sum over the residues:

$G=\widetilde{G}_{0}\sum_{\rho^{*}}P\left(\rho^{*}\right)\mathfrak{G}_{\mathcal{O}_{*}}\,,$
(C.12)

where

$P(\rho^{*})=-\text{Res}_{\Delta=\Delta^{*}}\left[\frac{C(\rho)\mu(\rho)S(\rho_{3},\rho_{4};\tilde{\rho})}{1-\chi
E(\rho)}\right],$ (C.13)

are the conformal block coefficients. The locations of the associated physical
poles are obtained by solving

$\chi\,E(\Delta^{*},\ell,\bar{\ell})=1\,.$ (C.14)

Note that when deforming the contour away from the principal series, there are
also poles arising from the conformal block $\mathfrak{G}$ and the numerator.
These are expected to cancel against each other, but we did not attempt to
verify it.

###  C.2 The Operator ${\rm tr\,}(XX^{\dagger}Z)$

We give here the results for the correlator (2.4) of subsection 2.1. The left
eigenvectors of $\widehat{B}$ are

$\widetilde{\Psi}^{(\Delta,S,S)}_{x_{0},\zeta}(x_{1},x_{2})=\frac{\left(\frac{\zeta\cdot
x_{10}}{x_{10}^{2}}-\frac{\zeta\cdot
x_{20}}{x_{20}^{2}}\right)^{S}}{|x_{12}|^{1+S+\Delta}|x_{10}|^{5-\Delta-S}|x_{20}|^{3-\Delta-S}}\,.$
(C.15)

In particular, they satisfy

$\left[\widetilde{\Psi}^{(\Delta,S,S)}_{x_{0},\zeta}\widehat{G}^{(0)}\right](x_{3},x_{4})=\frac{16\,x_{34}^{2}}{(4\pi^{2})^{3}(3+S-\Delta)^{2}(-1+S+\Delta)^{2}}\widetilde{\Psi}^{(\Delta,S,S)}_{x_{0},\zeta}(x_{3},x_{4})\,,$
(C.16)

where $\widehat{G}^{(0)}$ is given by (2.6). Thus, we conclude that in (C.10)
we have

$C(\Delta,S)=\frac{16}{(4\pi^{2})^{3}(3+S-\Delta)^{2}(-1+S+\Delta)^{2}}\qquad\text{and}\qquad\widetilde{G}_{0}(x_{3},x_{4})=x_{34}^{2}\,.$
(C.17)

The kinematic factors are

$\displaystyle\mu(\Delta,S,S)=2^{S-1}(S+1)(\Delta-2)^{2}(\Delta-S-3)(\Delta+S-1)\,,$
(C.18)

and

$\displaystyle
S((3,0,0),(2,0,0);(4-\Delta,S,S))=\frac{\pi^{2}\Gamma(S-\Delta+3)\Gamma\left(\frac{1}{2}(S+\Delta-1)\right)\Gamma\left(\frac{1}{2}(S+\Delta+1)\right)}{(2-\Delta)\Gamma(S+\Delta)\Gamma\left(\frac{1}{2}(S-\Delta+3)\right)\Gamma\left(\frac{1}{2}(S-\Delta+5)\right)}\,.$
(C.19)

Putting everything together, we obtain

$\displaystyle
P(\Delta_{\pm},S,S)=\mp\frac{(S+1)2^{S-2\Delta_{\pm}}\Gamma\left(\frac{S+4-\Delta_{\pm}}{2}\right)\Gamma\left(\frac{S+1+\Delta_{\pm}}{2}\right)}{4\pi^{4}\,\xi^{2}_{2}\,\Gamma\left(\frac{S+3-\Delta_{\pm}}{2}\right)\Gamma\left(\frac{S+\Delta_{\pm}}{2}\right)}\,,$
(C.20)

where $\Delta_{\pm}$ is given in (2.15).

###  C.3 The Operator ${\rm tr\,}(Z\psi_{4})$

We give here the results for the correlator (4.2) of subsection 4.1. The first
family of left eigenvectors of $\widehat{B}$ is

$\widetilde{\Psi}^{(\Delta,S-\frac{1}{2},S+\frac{1}{2})}_{x_{0},s,\bar{s}}(x_{1},x_{2})=\frac{\bar{s}\,\cancel{\bar{x}}_{02}\left[s\left(\frac{\cancel{x}_{10}}{x_{10}^{2}}-\frac{\cancel{x}_{20}}{x_{20}^{2}}\right)\bar{s}_{0}\right]^{S-\frac{1}{2}}}{|x_{12}|^{\Delta+S+1}|x_{10}|^{-\Delta-S+5}|x_{20}|^{-\Delta-S+5}}\,.$
(C.21)

In particular, using the formulae from appendix B, they satisfy

$\left[\widetilde{\Psi}^{(\Delta,S-\frac{1}{2},S+\frac{1}{2})}_{x_{0},s,\bar{s}}\widehat{G}_{2}^{(0)}\right](x_{3},x_{4})=\frac{1}{\pi^{4}\left(\Delta+S-1\right)\left(\Delta-S-2\right)\left(\Delta-S-4\right)}\\\
\times\frac{\bar{s}\,\cancel{\bar{x}}_{03}\,\cancel{x}_{34}\left[s\left(\frac{\cancel{x}_{30}}{x_{30}^{2}}-\frac{\cancel{x}_{40}}{x_{40}^{2}}\right)\bar{s}\right]^{S-\frac{1}{2}}}{|x_{34}|^{\Delta+S-1}|x_{30}|^{-\Delta-S+5}|x_{40}|^{-\Delta-S+5}}\,,$
(C.22)
$\left[\widetilde{\Psi}^{(\Delta,S-\frac{1}{2},S+\frac{1}{2})}_{x_{0},s,\bar{s}}\widehat{G}_{2}^{(0)}\right](x_{3},x_{4})=\frac{x_{34}^{2}\,\widetilde{\Psi}^{(\Delta,S-\frac{1}{2},S+\frac{1}{2})}_{x_{0},s,\bar{s}}(x_{3},x_{4})\cdot\cancel{x}_{04}\cancel{x}_{03}\cancel{x}_{34}}{\pi^{4}\left(\Delta+S-1\right)\left(\Delta-S-2\right)\left(\Delta-S-4\right)x_{04}^{2}}\,,$
(C.23)

where $\widehat{G}_{2}^{(0)}$ is given in (4.4). Since $\widehat{G}_{2}^{(0)}$
is not proportional to $\widehat{B}$, we are not getting back the same
eigenvector. Instead, the resulting function is the shadow transform with
respect to point $x_{1}$ and point $x_{2}$. From the previous equation, we
read off

$C(\Delta,S-\tfrac{1}{2},S+\tfrac{1}{2})=\frac{1}{\pi^{4}\left(\Delta+S-1\right)\left(\Delta-S-2\right)\left(\Delta-S-4\right)}\quad\text{and}\quad\widetilde{G}_{0}(x_{3},x_{4})=1\,.$
(C.24)

The kinematic factors are

$\displaystyle\mu(\Delta,S-\tfrac{1}{2},S+\tfrac{1}{2})=\frac{1}{2}(S+\tfrac{1}{2})\left(\Delta-\tfrac{5}{2}\right)\left(\Delta-\tfrac{3}{2}\right)\left(\Delta-S-3\right)\left(\Delta+S-1\right)\,,$
(C.25)

and

$\displaystyle
S\left((1,0,0),\left(\tfrac{3}{2},0,1\right);(4-\Delta,S-\tfrac{1}{2},S+\tfrac{1}{2})\right)=\frac{\operatorname{i}\pi^{2}\Gamma\left(S-\Delta+3\right)\Gamma\left(\frac{1}{2}\left(S+\Delta-1\right)\right)\Gamma\left(\frac{1}{2}\left(S+\Delta+1\right)\right)}{\left(\Delta-\frac{5}{2}\right)\Gamma\left(\frac{1}{2}\left(S-\Delta+4\right)\right)^{2}\Gamma\left(S+\Delta\right)}\,.$
(C.26)

Putting everything together, we obtain

$P\left(\Delta,S-\tfrac{1}{2},S+\tfrac{1}{2}\right)=\frac{-\operatorname{i}(S+\frac{1}{2})\left(\Delta-\frac{3}{2}\right)\left(\Delta+S-1\right)^{3}\left(\Delta-S-4\right)}{\xi^{4}_{4/3}2^{2(\Delta+1)}\pi^{2}\left(\Delta^{2}-\left(S+5\right)\Delta+2S+\frac{11}{2}\right)}\\\
\times\frac{\Gamma\left(\frac{1}{2}\left(S-\Delta+5\right)\right)\Gamma\left(\frac{1}{2}\left(S+\Delta-1\right)\right)}{\Gamma\left(\frac{1}{2}\left(S-\Delta+2\right)\right)\Gamma\left(\frac{1}{2}\left(S+\Delta\right)\right)}\,,$
(C.27)

where $\Delta=\Delta_{2}$ or $\Delta=\Delta_{4}$, are the dimensions given in
(4.11) and (4.12).

The second family of left eigenvectors is

$\widetilde{\Psi}_{x_{0},s,\bar{s}}^{\left(\Delta,S+\frac{1}{2},S-\frac{1}{2}\right)}\left(x_{1},x_{2}\right)=\frac{s\,\cancel{x}_{01}\,\cancel{\bar{x}}_{12}\left[s\left(\frac{\cancel{x}_{10}}{x_{10}^{2}}-\frac{\cancel{x}_{20}}{x_{20}^{2}}\right)\bar{s}\right]^{S-\frac{1}{2}}}{|x_{12}|^{\Delta+S+2}|x_{10}|^{-\Delta-S+6}|x_{20}|^{-\Delta-S+4}}\,.$
(C.28)

They satisfy the relation

$\left[\widetilde{\Psi}_{x_{0},s,\bar{s}}^{\left(\Delta,S+\frac{1}{2},S-\frac{1}{2}\right)}\widehat{G}_{2}^{\left(0\right)}\right]\left(x_{3},x_{4}\right)=\frac{1}{\pi^{4}\left(-\Delta+S+3\right)\left(\Delta+S-2\right)\left(\Delta+S\right)}\\\
\times\frac{s\,\cancel{x}_{40}\left[s\left(\frac{\cancel{x}_{30}}{x_{30}^{2}}-\frac{\cancel{x}_{40}}{x_{40}^{2}}\right)\bar{s}\right]^{S-\frac{1}{2}}}{|x_{34}|^{\Delta+S-2}|x_{30}|^{-\Delta-S+4}|x_{40}|^{-\Delta-S+6}}\,.$
(C.29)

The resulting function is again the shadow transform with respect to $x_{1}$
and $x_{2}$. We have

$\displaystyle
C(\Delta,S+\tfrac{1}{2},S-\tfrac{1}{2})=\frac{1}{\pi^{4}\left(-\Delta+S+3\right)\left(\Delta+S-2\right)\left(\Delta+S\right)}\qquad\text{and}\qquad\widetilde{G}_{0}(x_{3},x_{4})=1\,.$
(C.30)

Regarding the kinematic factors, one has
$\mu(\Delta,S+\frac{1}{2},S-\frac{1}{2})=\mu(\Delta,S-\frac{1}{2},S+\frac{1}{2})$,
but

$\displaystyle
S\left((1,0,0),\big{(}\tfrac{3}{2},0,1\big{)};\big{(}4-\Delta,S+\tfrac{1}{2},S-\tfrac{1}{2}\big{)}\right)=\frac{\operatorname{i}\pi^{2}\Gamma\left(S-\Delta+3\right)\Gamma\left(\frac{1}{2}\left(S+\Delta\right)\right)^{2}}{\left(\Delta-\frac{5}{2}\right)\Gamma\left(\frac{1}{2}\left(S-\Delta+3\right)\right)\Gamma\left(\frac{1}{2}\left(S-\Delta+5\right)\right)\Gamma\left(S+\Delta\right)}\,.$
(C.31)

Putting everything together, we obtain

$P\left(\Delta_{\frac{7}{2},\pm},S+\tfrac{1}{2},S-\tfrac{1}{2}\right)=\frac{-\operatorname{i}(S+\frac{1}{2})2^{-2\Delta_{\frac{7}{2},\pm}-2}\left(\Delta_{\frac{7}{2},\pm}-\frac{3}{2}\right)\left(\Delta_{\frac{7}{2},\pm}-S-3\right)^{3}\left(\Delta_{\frac{7}{2},\pm}+S-2\right)}{\xi^{4}_{4/3}\pi^{2}\left(\Delta_{\frac{7}{2},\pm}^{2}+\Delta_{\frac{7}{2},\pm}\left(S-3\right)-2S+\frac{3}{2}\right)}\\\
\times\frac{\Gamma\left(\frac{1}{2}\left(S-\Delta_{\frac{7}{2},\pm}+4\right)\right)\Gamma\left(\frac{1}{2}\left(S+\Delta_{\frac{7}{2},\pm}+2\right)\right)}{\Gamma\left(\frac{1}{2}\left(S-\Delta_{\frac{7}{2},\pm}+5\right)\right)\Gamma\left(\frac{1}{2}\left(S+\Delta_{\frac{7}{2},\pm}-1\right)\right)}\,,$
(C.32)

where $\Delta_{\frac{7}{2},\pm}$ is given in (4.13).

##  Appendix D Integrability

In this appendix we review the construction of integrable spin chains with
four-dimensional conformal symmetry. We then apply it to demonstrate the
integrability of the graph-building operators for all the cases without
mixing.

###  D.1 General Construction

Integrable spin chains based on the four-dimensional conformal algebra
$\mathfrak{so}(5,1)$ can be built for an arbitrary representation $\rho$ at
each site of the chain. This is done using the Lax matrices

$L^{(\rho;\mathbf{4})}(u)=u\operatorname{Id}-\frac{1}{2}q_{MN}^{(\rho)}\otimes\Sigma^{MN}\,.$
(D.1)

This matrix acts on $\mathcal{V}_{\rho}\otimes\mathbb{C}^{4}$, with
$\mathcal{V}_{\rho}$ being the physical space in the representation $\rho$,
and $\mathbb{C}^{4}$ is the auxiliary space transforming in the ${\bf 4}$ of
$\mathfrak{so}(5,1)$. This matrix is constructed such that it satisfies the
Yang–Baxter equation

$R^{(\mathbf{4};\mathbf{4})}(u-v)L^{(\rho;\mathbf{4})}_{1}(u)L^{(\rho;\mathbf{4})}_{2}(v)=L^{(\rho;\mathbf{4})}_{2}(v)L^{(\rho;\mathbf{4})}_{1}(u)R^{(\mathbf{4};\mathbf{4})}(u-v)\,.$
(D.2)

Here, $\Sigma_{MN}=q^{(\mathbf{4})}_{MN}$ are the $\sigma$-matrices in 6D.
They realise one of the two four-dimensional irreducible representations of
the conformal algebra. The generators $q_{MN}^{(\rho)}$ satisfy the
commutation relations

$\left[q_{MN}^{(\rho)},q_{KL}^{(\rho)}\right]=\eta_{NK}q_{ML}^{(\rho)}-\eta_{MK}q_{NL}^{(\rho)}-\eta_{NL}q_{MK}^{(\rho)}+\eta_{ML}q_{NK}^{(\rho)}\,.$
(D.3)

The R-matrix is given by

$R^{(\mathbf{4};\mathbf{4})}(u)=L^{(\mathbf{4};\mathbf{4})}\left(u+\frac{1}{4}\right)=u\operatorname{Id}+\mathbb{P}\,,$
(D.4)

where $\mathbb{P}$ is the permutation operator on
$\mathbb{C}^{4}\otimes\mathbb{C}^{4}$. Since $R^{(\mathbf{4};\mathbf{4})}(-1)$
projects onto the irreducible representation
$\mathbf{6}\subset\mathbf{4}\otimes\mathbf{4}$, one can use the fusion
procedure to compute the Lax matrices acting on
$\mathcal{V}_{\rho}\otimes\mathbb{C}^{6}$, [38, 39]. We checked, using
Mathematica, that they are given by151515The sign in front of the term
involving $\epsilon_{MNABCD}$ is related to our convention for the
$\sigma$-matrices, which satisfy
$\epsilon_{MNABCD}\Sigma^{AB}\Sigma^{CD}=-12\Sigma_{MN}$.

$L^{(\rho;\mathbf{6})}(u)=\Bigg{[}u^{2}\eta_{MN}-uq_{MN}^{(\rho)}+\frac{1}{2}\bigg{(}?q^{(\rho),}_{M}{}^{P}?q^{(\rho)}_{PN}-2?q^{(\rho)}_{MN}?\\\
-\frac{C_{\rho}+2}{4}\eta_{MN}-\frac{1}{8}?\epsilon_{MN}^{ABCD}?q^{(\rho)}_{AB}q^{(\rho)}_{CD}\bigg{)}\Bigg{]}\otimes?e^{MN}?\,.$
(D.5)

Here, $?e_{M}^{N}?$ is the $6\times 6$ matrix with a single non-zero
coefficient, equal to one, at the intersection of row $M$ and column $N$, and
$q^{(\mathbf{6})}_{MN}=e_{MN}-e_{NM}$, the indices being lowered using the
metric $\eta_{MN}$, see appendix A.1. The quadratic Casimir reads

$C_{\rho}=q^{(\rho),MN}q^{(\rho)}_{NM}\,.$ (D.6)

Finally, for the completely antisymmetric tensor in (D.5) we use the
convention where $\epsilon_{123456}=+1$.

As a consequence of the Yang–Baxter equation (D.2), the transfer matrices (or
T-operators)

$T_{\mathbf{6}}\left(u;\theta_{1},\dots,\theta_{N}\right)\equiv{\rm
tr\,}_{\mathbf{6}}\left(L_{N}^{\left(\rho_{N};\mathbf{6}\right)}\left(u-\theta_{N}\right)L_{N-1}^{\left(\rho_{N-1};\mathbf{6}\right)}\left(u-\theta_{N-1}\right)\cdots
L_{1}^{\left(\rho_{1};\mathbf{6}\right)}\left(u-\theta_{1}\right)\right),$
(D.7)

commute with each other for different values of the spectral parameter $u$,
and fixed values of the inhomogeneities $\theta_{1},\dots,\theta_{N}$.

For each correlator in the sections below, we construct the associated Lax
matrices and show that
$T_{\mathbf{6}}\left(0;\theta_{1},\dots,\theta_{N}\right)$ coincides with the
inverse of the graph-building operator—for an appropriate choice of the
inhomogeneities.

###  D.2 Principal Series Representations

We will be interested in representations belonging to the principal series of
$\mathrm{Spin}(5,1)$, (which is a double cover of the connected component of
the identity in the conformal group). These representations are labeled by a
complex number $\Delta$, the conformal dimension, and two non-negative
integers $\ell$ and $\bar{\ell}$, the spins. They may be realised as acting on
functions $f$ on $\mathbb{R}^{4}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$,
such that $f(x,s,\bar{s})$ is homogeneous of degree $\ell$ in
$s\in\mathbb{C}^{2}$ and, separately, of degree $\bar{\ell}$ in
$\bar{s}\in\mathbb{C}^{2}$. The generators of the conformal algebra take the
following form

  * •

rotations
$q^{(\Delta,\ell,\bar{\ell})}_{\mu\nu}=x_{\mu}\partial_{x^{\nu}}-x_{\nu}\partial_{x^{\mu}}+s^{\alpha}\left(\sigma_{\mu\nu}\right)?\\!_{\alpha}^{\beta}?\partial_{s^{\beta}}+\bar{s}_{\dot{\alpha}}\left(\bar{\sigma}_{\mu\nu}\right)?\\!^{\dot{\alpha}}_{\dot{\beta}}?\partial^{\dot{\beta}}_{\bar{s}}\,,$

  * •

translations $q^{(\Delta,\ell,\bar{\ell}),{\mu+}}=-\partial_{x}^{\mu}\,,$

  * •

dilation $q^{(\Delta,\ell,\bar{\ell})}_{56}=-\Delta-x\cdot\partial_{x}\,,$

  * •

special conformal transformations

$q^{(\Delta,\ell,\bar{\ell}),{\mu-}}=2x^{\mu}\left(\Delta+x\cdot\partial_{x}\right)-x^{2}\partial_{x}^{\mu}+2x_{\nu}\left(s^{\alpha}\left(\sigma^{\mu\nu}\right)?\\!_{\alpha}^{\beta}?\partial_{s^{\beta}}+\bar{s}_{\dot{\alpha}}\left(\bar{\sigma}^{\mu\nu}\right)?\\!^{\dot{\alpha}}_{\dot{\beta}}?\partial^{\dot{\beta}}_{\bar{s}}\right)\,.$

Here we have introduced
$\sigma_{\mu\nu}=\frac{\sigma_{\mu}\bar{\sigma}_{\nu}-\sigma_{\nu}\bar{\sigma}_{\mu}}{4}$
and
$\bar{\sigma}_{\mu\nu}=\frac{\bar{\sigma}_{\mu}\sigma_{\nu}-\bar{\sigma}_{\nu}\sigma_{\mu}}{4}$.
Introducing the additional notations

$x^{+}=1\,,\quad
x^{-}=x^{2}\,,\quad\partial_{x,-}=0\,,\quad\partial_{x,+}=-\Delta-x\cdot\partial_{x}\,,$
(D.8)
$\sigma^{+}=\bar{\sigma}^{+}=0\,,\quad\sigma^{-}=2\,\cancel{x},\quad\text{and}\quad\bar{\sigma}^{-}=2\,\cancel{\bar{x}}\,,$
(D.9)

we can write all the generators in the compact form

$q^{(\Delta,\ell,\bar{\ell})}_{MN}=x_{M}\partial_{x,N}-x_{N}\partial_{x,M}+s^{\alpha}\left(\sigma_{MN}\right)?\\!_{\alpha}^{\beta}?\partial_{s^{\beta}}+\bar{s}_{\dot{\alpha}}\left(\bar{\sigma}_{MN}\right)?\\!^{\dot{\alpha}}_{\dot{\beta}}?\partial^{\dot{\beta}}_{\bar{s}}\,.$
(D.10)

In the following, we shall be interested in representations for which either
$\ell$ or $\bar{\ell}$ is zero. In this case, the L-matrices (D.5) become

$L^{(\Delta,\ell,0;\mathbf{6})}_{MN}(u)=\left(u^{2}-\frac{(\Delta-1)^{2}}{4}\right)\eta_{MN}-\left(u+\frac{\ell}{4}\right)q^{(\Delta,\ell,0)}_{MN}\\\
+\frac{1}{2}\left[-x_{M}x_{N}\Box_{x}-x_{N}s\sigma_{M}\cancel{\bar{\partial}}_{x}\partial_{s}+\left(1-\Delta+\frac{\ell}{2}\right)\left(x_{M}\partial_{x,N}+x_{N}\partial_{x,M}+s\sigma_{MN}\partial_{s}+2\ell
V_{M}x_{N}\right)\right]\,,$ (D.11)

and

$L^{(\Delta,0,\bar{\ell};\mathbf{6})}_{MN}(u)=\left(u^{2}-\frac{(\Delta-1)^{2}}{4}\right)\eta_{MN}-\left(u-\frac{\bar{\ell}}{4}\right)q^{(\Delta,0,\bar{\ell})}_{MN}\\\
+\frac{1}{2}\left[-x_{M}x_{N}\Box_{x}-x_{M}\bar{s}\bar{\sigma}_{N}\cancel{\partial}_{x}\partial_{\bar{s}}+\left(1-\Delta+\frac{\bar{\ell}}{2}\right)\left(x_{M}\partial_{x,N}+x_{N}\partial_{x,M}-\bar{s}\bar{\sigma}_{MN}\partial_{\bar{s}}+2\bar{\ell}x_{M}V_{N}\right)\right]\,,$
(D.12)

where $V$ is a constant vector: $V^{\mu}=V^{+}=0$, and $V^{-}=1$. These
computations were also done using Mathematica.

###  D.3 Lax Matrices for Operators Without Gluons

We collect here the various Lax matrices required to reproduce the inverse
graph-building operators (2.13) for ${\rm tr\,}(XX^{\dagger}Z)$, (4.16) for
${\rm tr\,}(\psi_{4}Z^{J})$, and (4.19) for ${\rm
tr\,}(\psi_{4}\psi^{\dagger}Z)$. Applying the formulae (D.11) and (D.12), we
find

$\displaystyle L_{MN}^{\left(1,0,0;\boldsymbol{6}\right)}\left(0\right)$
$\displaystyle=-\frac{1}{2}x_{M}x_{N}\Box_{x}\,,$ (D.13) $\displaystyle
L_{MN}^{\left(2,0,0;\boldsymbol{6}\right)}\left(0\right)$
$\displaystyle=-\frac{1}{4}\eta_{MN}-\frac{1}{2}\left[x_{M}x_{N}\Box_{x}+x_{M}\partial_{x,N}+x_{N}\partial_{x,M}\right]\,,$
(D.14) $\displaystyle
L_{MN}^{\left(\frac{3}{2},1,0;\boldsymbol{6}\right)}\left(-\frac{1}{4}\right)$
$\displaystyle=-\frac{1}{2}\left[x_{M}x_{N}\Box_{x}+x_{N}s\sigma_{M}\cancel{\bar{\partial}}_{x}\partial_{s}\right]\,,$
(D.15) $\displaystyle
L_{MN}^{\left(\frac{3}{2},0,1;\boldsymbol{6}\right)}\left(\frac{1}{4}\right)$
$\displaystyle=-\frac{1}{2}\left[x_{M}x_{N}\Box_{x}+x_{M}\bar{s}\bar{\sigma}_{N}\cancel{\partial}_{x}\partial_{\bar{s}}\right]\,.$
(D.16)

We recall that they are all written in four-dimensional space using the
notation introduced in (D.8) and (D.9).

The inverse graph-building operator (2.13) for instance is given by

$\widehat{B}^{-1}={\rm
tr\,}_{\mathbf{6}}(L_{x_{1}}^{\left(1,0,0;\boldsymbol{6}\right)}\left(0\right)L_{x_{2}}^{\left(2,0,0;\boldsymbol{6}\right)}\left(0\right))\,.$
(D.17)

###  D.4 Integrability for ${\rm tr\,}(F_{\mu\nu}Z)$

In the case of the operator ${\rm
tr\,}\big{(}Z(x_{1})F^{\mu\nu}(x_{2})\big{)}$ in (2.19) we have an
antisymmetric rank-2 tensor of dimension 2 at $x_{2}$. This representation is
reducible, given by the direct sum $(2,2,0)\oplus(2,0,2)$. These
representations correspond to the self-dual and anti-self-dual parts of the
tensor, respectively.161616Note that because the graph-building operator we
got is not expressed in terms of spinors, we cannot directly apply the general
results of section D.2.

In order to lighten the notations, we introduce an auxiliary polarisation
vector $\theta$, whose components anticommute among themselves:
$\\{\theta^{\mu},\theta^{\nu}\\}=0$. We are thus interested in functions
homogeneous of degree $2$ in $\theta$,
$\psi(x,\theta)=\theta^{\mu}\theta^{\nu}\psi_{\mu\nu}(x)$. In this
representation, the generators of the conformal group take the form

$\displaystyle q_{\mu\nu}$
$\displaystyle=x_{\mu}\partial_{x^{\nu}}-x_{\nu}\partial_{x^{\mu}}+\theta_{\mu}\partial_{\theta^{\nu}}-\theta_{\nu}\partial_{\theta^{\mu}}\,,$
(D.18) $\displaystyle q^{\mu+}$ $\displaystyle=-\partial^{\mu}\,,$ (D.19)
$\displaystyle q_{56}$ $\displaystyle=-\Delta-x\cdot\partial_{x}\,,$ (D.20)
$\displaystyle q^{\mu-}$
$\displaystyle=2\,x_{\mu}\left(\Delta+x\cdot\partial_{x}\right)-x^{2}\partial_{x}^{\mu}+2\left(\theta^{\mu}(x\cdot\partial_{\theta})-(\theta\cdot
x)\,\partial^{\mu}_{\theta}\right)\,.$ (D.21)

One can then apply (D.5) to obtain the Lax matrices at each site of a chain of
length $2$ containing a scalar of dimension 1 (site 1) and an antisymmetric
rank-2 tensor of dimension 2 (site 2). The corresponding transfer matrix (D.7)
becomes

$\widehat{T}_{\mathbf{6}}(0;0,0)=\frac{1}{16}\left[(\theta\cdot\partial_{x_{2}})\,x_{12}^{4}\,(\partial_{x_{2}}\cdot\partial_{\theta})+(\partial_{x_{2}}\cdot\partial_{\theta})\,x_{12}^{4}\,(\theta\cdot\partial_{x_{2}})\right]\Box_{x_{1}}\,.$
(D.22)

Clearly, if $\Psi=\theta\cdot\partial_{2}\,\Phi$ for some function
$\Phi(x_{1},x_{2},\theta)=\theta^{\nu}\Phi_{\nu}(x_{1},x_{2})$, then this
operator simplifies to

$\widehat{T}_{\mathbf{6}}(0;0,0)\Psi=\frac{1}{16}(\theta\cdot\partial_{x_{2}})\,x_{12}^{4}\,(\partial_{x_{2}}\cdot\partial_{\theta})\Box_{x_{1}}\Psi\,.$
(D.23)

This is indeed the inverse of (the restriction of) $\widehat{B}_{F}$, see
(2.39).

##  Appendix E One-Loop Checks of the Spectrum

In this appendix we preform a few checks of the spectrum at one-loop order in
perturbation theory.

###  E.1 The operators ${\rm tr\,}(ZXX^{\dagger})$ and ${\rm
tr\,}(ZX^{\dagger}X)$

We would like to check the spectrum of ${\rm tr\,}(ZXX^{\dagger})$, given in
(2.15), at small $\xi_{2}$. Expanding (2.15) at $S=0$ we obtain

$\Delta_{\pm}(0,\xi_{2})=3\pm 2\xi_{2}^{2}+O(\xi_{2}^{4})\,.$ (E.1)

Surprisingly the leading term is $\xi_{2}^{2}$ instead of $\xi_{2}^{4}$, even
though the graph-building operator is of order $\xi_{2}^{4}$. This is an
outcome of the fact that we have mixing between the operators171717In the
original fishnet model, this was due to the presence of double-trace
interactions in the action.

$\mathcal{O}_{1}={\rm
tr\,}(ZXX^{\dagger})\qquad\text{and}\qquad\mathcal{O}_{2}={\rm
tr\,}(ZX^{\dagger}X)\,.$ (E.2)

Let us see explicitly how this spectrum emerges at one-loop order in the
perturbative expansion.

In the case at hand, the dilatation operator is a $2\times 2$ matrix, acting
on the operators $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$. The diagrams that
contribute to the two point functions of these operators at leading order in
$g^{2}$ and $\xi_{2}^{2}$ are drawn in figure 15.

Figure 15: Diagrams contributing to the two-point functions of ${\cal
O}_{1}={\rm tr\,}(ZX^{\dagger}X)$ and ${\cal O}_{2}={\rm
tr\,}(ZXX^{\dagger})$. Black lines denotes $Z$ propagator and red lines denote
$X$ propagator. The interactions are the same as in (2.5).

We found that they take the form

$\begin{pmatrix}\langle{\cal O}^{\dagger}_{1}{\cal O}_{1}\rangle&\langle{\cal
O}^{\dagger}_{1}{\cal O}_{2}\rangle\\\ \langle{\cal O}^{\dagger}_{2}{\cal
O}_{1}\rangle&\langle{\cal O}^{\dagger}_{2}{\cal
O}_{2}\rangle\end{pmatrix}=\frac{1}{(4\pi^{2})^{3}x^{6}}\begin{pmatrix}1&g^{2}K\\\
256\pi^{4}\frac{\xi_{2}^{4}}{g^{2}}K&1\end{pmatrix}\left(1+O(g^{2},\xi_{2}^{2})\right)\,,$
(E.3)

where $x$ is the separation between the two operators and $K$ is the integral

$K=\int{\differential^{4}y\over(4\pi^{2})^{2}}{x^{4}\over
y^{4}(x-y)^{4}}={1\over 4\pi^{2}}\log(x/\epsilon)+\text{finite}\,,$ (E.4)

where $\epsilon$ is some UV cutoff length scale. The logarithmic divergence of
$K$ comes from the region of integration where $y\to 0,x$. The corresponding
eigenvalues of the matrix (E.3) are therefore

$K_{\pm}=\frac{1\mp
4\xi_{2}^{2}\left(\log(x/\epsilon)+\text{finite}\right)}{(4\pi^{2})^{3}x^{6}}\,.$
(E.5)

From (E.5) we read the anomalous dimensions

$\gamma_{\pm}=\pm 2\xi_{2}^{2}+O(\xi_{2}^{4})\,,$ (E.6)

in agreement with (E.1).

###  E.2 The operators ${\rm tr\,}(Z^{\dagger}X^{\dagger}\psi_{1\alpha})$ and
${\rm tr\,}(Z^{\dagger}\psi_{1\alpha}X^{\dagger})$

Similarly to the case considered above, we compute the dimension of the spin
one-half operators

$\mathcal{O}_{1}={\rm
tr\,}(Z^{\dagger}\psi_{1\alpha}X^{\dagger})\qquad\text{and}\qquad\mathcal{O}_{2}={\rm
tr\,}(Z^{\dagger}X^{\dagger}\psi_{1\alpha})\,,$ (E.7)

at one loop order in $\xi_{4/3}$. The diagrams that contribute to these two
point functions are drawn in figure 16.

Figure 16: Diagrams contributing to the two-point functions of
$\mathcal{O}_{1}={\rm tr\,}(Z^{\dagger}\psi_{1\alpha}X^{\dagger})$ and
$\mathcal{O}_{2}={\rm tr\,}(Z^{\dagger}X^{\dagger}\psi_{1\alpha})$. The black
lines represent $Z$ propagator, the red lines are $X$ propagator, and the
dashed black lines are fermion propagators of $\psi_{1}$ (green) and
$\psi_{4}$ (black). The interactions are given in (4.3).

They take the form

$\begin{pmatrix}\langle{\cal O}^{\dagger}_{1}{\cal O}_{1}\rangle&\langle{\cal
O}^{\dagger}_{1}{\cal O}_{2}\rangle\\\ \langle{\cal O}^{\dagger}_{2}{\cal
O}_{1}\rangle&\langle{\cal O}^{\dagger}_{2}{\cal
O}_{2}\rangle\end{pmatrix}=\frac{\cancel{x}}{32\pi^{6}(x^{2})^{4}}\begin{pmatrix}1&g^{-2/3}\xi_{4/3}^{8/3}\,Y_{1}\\\
\,g^{2/3}\xi_{4/3}^{4/3}\,Y_{2}&1\end{pmatrix}\left(1+O(g^{2},\xi_{4/3}^{2})\right)\,,$
(E.8)

where

$Y_{1}=32\pi^{8/3}\,K\,,\qquad
Y_{2}=\frac{1}{2\pi^{2/3}}\int\frac{\differential^{4}z\,\differential^{4}y}{\pi^{4}}\frac{x^{4}\,\cancel{\bar{x}}\,\cancel{z}(\cancel{\bar{z}}-\cancel{\bar{y}})(\cancel{y}-\cancel{x})}{z^{6}(y-z)^{4}(y-x)^{6}}\,.$
(E.9)

Working in dimensional regularisation, with $d=4-\epsilon$, we find