HU-EP-23/52-RTG

Brick wall diagrams as a completely integrable system

Moritz Kade, Matthias Staudacher

<EMAIL_ADDRESS>
Institut für Mathematik und Institut für Physik,
Humboldt-Universität zu Berlin,
Zum Großen Windkanal 2, 12489 Berlin, Germany

We study the free energy of an integrable, planar, chiral and non-unitary
four-dimensional Yukawa theory, the bi-fermion fishnet theory discovered by
Pittelli and Preti. The typical Feynman-diagrams of this model are of regular
“brick-wall”-type, replacing the regular square lattices of standard fishnet
theory. We adapt A. B. Zamolodchikov’s powerful classic computation of the
thermodynamic free energy of fishnet graphs to the brick-wall case in a
transparent fashion, and find the result in closed form. Finally, we briefly
discuss two further candidate integrable models in three and six dimensions
related to the brick wall model.

##  1 Introduction and results

The discovery and use of integrability of quantum field theories (QFT) is
often strongly related to techniques initially established for exactly
solvable models of statistical physics, see the prime example of planar
$\mathcal{N}=4$ super Yang-Mills theory (SYM) [1]. Concerning QFTs in more
than two dimensions, A. B. Zamolodchikov showed in 1980 [2] that massless
propagators of lattice-like vacuum Feynman diagrams akin to “fishing-net”
diagrams can be interpreted as certain weights analogous to Boltzmann weights
appearing in integrable lattice models. More concretely, a star-triangle
relation [3, 4], sometimes also called method of uniqueness, for the
propagators results in a Yang-Baxter equation, thereby leading to the manifest
appearance of quantum integrability. To demonstrate the power of his
observation, Zamolodchikov calculated the free energy of his system in the
thermodynamic limit. It corresponds to the critical coupling of a then still
unknown QFT, producing the asymptotic growth of the latter’s vacuum graphs. To
be more precise, he applied the method of inversion relations [5, 6, 7, 8, 9,
10], which had been developed for integrable statistical models, to the QFT’s
propagator-weights.

Many years later (2015) the conformal, non-unitary QFT that actually produces
the examined graphs (and essentially nothing else!) in the limit
$\mathrm{N}\rightarrow\infty$ was constructed by Gürdoğan and Kazakov [11] and
christened “bi-scalar fishnet theory”. In fact, it turns out to be a special
case of an even more general class of such theories obtained by performing a
double-scaling limit of $\gamma$-deformed planar $\mathcal{N}=4$ SYM [12],
yielding the so-called $\chi$-CFT [11] with three effective couplings
corresponding to the Cartan elements of the R-symmetry algebra
$\mathfrak{so}(6)\cong\mathfrak{su}(4)$ [13]. Here all of the gauge fields of
$\mathcal{N}=4$ SYM disappear, along with one of its four fermions. Setting
two out of these three effective couplings to zero yields the special case of
Zamolodchikov’s bi-scalar fishnet theory. The authors of [11] also briefly
discussed an intermediate version of these models named $\chi_{0}$-CFT that
still contained all of the three scalars of the $\chi$-CFT and two out of its
three fermions. A further simplified version of this “intermediate” model,
called bi-fermion fishnet theory by them, was then proposed by Pittelli and
Preti [14]. It contains only one scalar, but keeps the two fermions and will
be the focus of the present paper. We will call this theory brick wall model,
according to the shape of its Feynman diagrams. This name was proposed for the
these types of diagrams in [15]. Extending and adapting Zamolodchikov’s
approach to a larger class of diagrams including fermions, we show that the
asymptotic scaling of the model’s vacuum graphs can once again be determined
by a known star-triangle relation [16] and calculate its critical coupling by
deriving and solving suitable inversion relations. Interestingly, while the
brick wall model cannot be obtained by a double-scaling limit of planar
$\gamma$-deformed $\mathcal{N}=4$ SYM, the authors of [14] derived it from a
strong, imaginary Lunin-Maldacena deformation of $\mathcal{N}=2$
superconformal field theory (SCFT).

Along with Zamolodchikov’s free energy, other observables such as certain
anomalous dimensions and correlation functions [17] could be determined
exactly for the bi-scalar fishnet model, which may be interpreted as a
large-$\mathrm{N}$ (planar) matrix field toy model version of four-dimensional
$\phi^{4}$-theory. Unlike its textbook relative, it is, albeit non-unitary,
integrable [18] and conformal [19, 20]. Moreover, increasing the complexity of
bi-scalar fishnet theory back to the direction of full-fledged planar
$\mathcal{N}=4$ SYM showed that integrability continues to provide exact
answers for anomalous dimensions and correlators [21, 15, 22]. More precisely,
the $\chi_{0}$\- and $\chi$-models stayed integrable and conformal even after
including fermions with their characteristic Yukawa interactions. Some of this
work was also adapted to the brick wall model, see [14], which may be
interpreted as a large-N (planar) matrix field toy model version of four-
dimensional Yukawa theory.

Our paper is organized as follows: In section 2 we discuss the Lagrangian of
the brick wall model, explain its graphical motivation from non-planar
$\mathcal{N}=4$ SYM, and very briefly discuss its quantization. In section 3
we review the star-triangle relation (STR) and its consequences for the
manipulation of Feynman integrals using integrability. Section 4 is devoted to
the vacuum graphs in the large-$\mathrm{N}$ limit and to their description by
generalized graph building operators in similarity to partition functions of
statistical models. Section 5 then proceeds with the calculation of the
critical coupling by the method of inversion relations. Finally section 6
gives an outlook on future applications of the brick wall theory and exhibits
two related candidate integrable models in three and six dimensions. There are
two appendices: appendix A proves the main relation used to derive the
inversion relations. In appendix B we transparently review Zamolodchikov’s
famous but extremely concise results [2] for the asymptotics of vacuum fishnet
graphs in $D=3,4,6$ dimensions.

##  2 Graphical motivation of the brick wall model

The existence of the brick wall model, apart from its derivation in[14], is in
some graphical sense also motivated by the relation betweeen the $\chi$-CFT to
fishnet theory. We start by recalling the derivation of the former from the
$\mathcal{N}=4$ model [11]. As a first step, one performs a three-parameter
$\gamma$-deformation of $\mathcal{N}=4$ by modifying the product between two
fields $A\cdot B$ to a star-product

$A\cdot
B\rightarrow\mathrm{e}^{-\frac{\mathrm{i}}{2}\,\mathrm{det}\left(\mathbf{q}_{A}|\mathbf{q}_{B}|\mathbf{\gamma}\right)}\,A\cdot
B\;,$ (2.1)

where $\mathbf{q}_{A}$ and $\mathbf{q}_{B}$ are the $\mathfrak{su}(4)$
$R$-symmetry weight vectors of $A$ and $B$. This introduces factors
$q_{i}:=\mathrm{e}^{-\frac{\mathrm{i}}{2}\gamma_{i}}$ into the $\mathcal{N}=4$
Lagrangian, where $i=1,2,3$. Secondly, the double-scaling limit consists of
taking the ’t Hooft coupling $\lambda\rightarrow 0$ and the twist
parameters111A more general double-scaling procedure, where each $q_{i}$ is
allowed to either go to zero or to infinity, was suggested in [13, 23].
However, up to redefining couplings and permuting fields, it does not lead to
any new models. $q_{i}\rightarrow\infty$ (by considering imaginary angles
$\gamma_{i}\rightarrow\mathrm{i}\infty$). The two limits are balanced such
that $\xi_{i}:=q_{i}\cdot\lambda$ is finite. Note that the limit taking
renders the theory non-hermitian, with only chiral222Chiral, not in the usual
particle physics sense, but meaning that each vertex defines the same
orientation on the curve each graph in the genus expansion is embedded into.
interaction vertices remaining. The new finite, effective coupling constants
$\xi_{i}$ can now be tuned to describe different models. In the case of
generic three parameters $\xi_{i}$, this theory has been christened $\chi$-CFT
[11, 13, 21] by Gürdoğan and Kazakov, with the Lagrangian

$\begin{split}&\mathcal{L}^{\chi}~{}=~{}\mathrm{N}\cdot\mathrm{tr}\left[\sum_{j=1}^{3}\left(-\frac{1}{2}\partial^{\mu}\phi_{j}^{\dagger}\partial_{\mu}\phi_{j}^{\phantom{\dagger}}+\mathrm{i}\bar{\psi}_{j}\not{\partial}\psi_{j}\right)+\xi_{1}^{2}\phi_{2}^{\dagger}\phi_{3}^{\dagger}\phi_{2}^{\phantom{\dagger}}\phi_{3}^{\phantom{\dagger}}+\xi_{2}^{2}\phi_{3}^{\dagger}\phi_{1}^{\dagger}\phi_{3}^{\phantom{\dagger}}\phi_{1}^{\phantom{\dagger}}+\xi_{3}^{2}\phi_{1}^{\dagger}\phi_{2}^{\dagger}\phi_{1}^{\phantom{\dagger}}\phi_{2}^{\phantom{\dagger}}\right.\\\
&\Biggl{.}+\mathrm{i}\sqrt{\xi_{2}\xi_{3}}\left(\psi_{3}\phi_{1}\psi_{2}+\bar{\psi}_{3}^{\phantom{\dagger}}\phi_{1}^{\dagger}\bar{\psi}_{2}^{\phantom{\dagger}}\right)+\mathrm{i}\sqrt{\xi_{1}\xi_{3}}\left(\psi_{1}\phi_{2}\psi_{3}+\bar{\psi}_{1}^{\phantom{\dagger}}\phi_{2}^{\dagger}\bar{\psi}_{3}^{\phantom{\dagger}}\right)+\mathrm{i}\sqrt{\xi_{1}\xi_{2}}\left(\psi_{2}\phi_{3}\psi_{1}+\bar{\psi}_{2}^{\phantom{\dagger}}\phi_{3}^{\dagger}\bar{\psi}_{1}^{\phantom{\dagger}}\right)\Biggr{]}.\end{split}$
(2.2)

All fields here and in the following transform in the adjoint representation
of the “gauge group” $\mathrm{SU}(\mathrm{N})$, which has become a global
symmetry: The gauge field interactions of $\mathcal{N}=4$ are suppressed in
the limit, and we may thus completely project out the gauge fields. As is
common, we omit all the Lorentz indices related to the fermions, e. g.
$\psi_{i}^{\alpha}\psi_{j,\alpha}$ and
$\bar{\psi}_{i,\dot{\alpha}}\bar{\psi}_{j}^{\dot{\alpha}}$ become,
respectively, $\psi_{i}\psi_{j}$ and $\bar{\psi}_{i}\bar{\psi}_{j}$, and so
on. The spacetime derivative $\partial_{\mu}$ is contracted with the Weyl
matrices $\sigma_{\mu}$. Our convention in Euclidean space is
$\sigma_{\mu}=(-\mathrm{i}\vec{\boldsymbol{\sigma}},\mathbbm{1}_{2})$, such
that
$\not{\partial}_{\dot{\alpha}}^{\alpha}=(\sigma^{\mu})_{\dot{\alpha}}^{\alpha}\partial_{\mu}$,
where $\sigma_{i}$ are the Pauli matrices. Actually, (2.2) it is not quite
complete, since quantum corrections generate a number of double-trace terms
required to render the model fully conformal[24, 19]. These however only
appear in a very restricted class of quantities, typically those that involve
length-two operators. This is also true for all the models in the remainder of
this chapter; we will only explicitly exhibit the double-trace terms for the
brick wall model to follow below, cf. (2.6).

Thirdly, as an intermediate step, still more general than bi-scalar fishnet
theory, we obtain the $\chi_{0}$-CFT, see [13, 21], by setting $\xi_{1}=0$.
The theory has the Lagrangian

$\begin{split}\mathcal{L}^{\chi_{0}}~{}=~{}\mathrm{N}\cdot\mathrm{tr}\Biggl{[}\Biggr{.}&-\frac{1}{2}\sum_{j=1}^{3}\partial^{\mu}\phi_{j}^{\dagger}\partial_{\mu}\phi_{j}^{\phantom{\dagger}}+\sum_{k=2}^{3}\mathrm{i}\bar{\psi}_{k}\not{\partial}\psi_{k}\\\
&\Biggl{.}+\xi_{2}^{2}\phi_{3}^{\dagger}\phi_{1}^{\dagger}\phi_{3}^{\phantom{\dagger}}\phi_{1}^{\phantom{\dagger}}+\xi_{3}^{2}\phi_{1}^{\dagger}\phi_{2}^{\dagger}\phi_{1}^{\phantom{\dagger}}\phi_{2}^{\phantom{\dagger}}+\mathrm{i}\sqrt{\xi_{2}\xi_{3}}\left(\psi_{3}\phi_{1}\psi_{2}+\bar{\psi}_{3}^{\phantom{\dagger}}\phi_{1}^{\dagger}\bar{\psi}_{2}^{\phantom{\dagger}}\right)\Biggr{]}.\end{split}$
(2.3)

Finally, to obtain the bi-scalar fishnet theory (FN), one puts also
$\xi_{2}=0$ and defines for simplicity $\xi:=\xi_{3}$. This results in

$\begin{split}&\mathcal{L}^{\mathrm{FN}}~{}=~{}\mathrm{N}\cdot\mathrm{tr}\left[\frac{1}{2}\sum_{j=1}^{2}\partial^{\mu}\phi_{j}^{\dagger}\partial_{\mu}\phi_{j}^{\phantom{\dagger}}+\xi^{2}\phi_{1}^{\dagger}\phi_{2}^{\dagger}\phi_{1}^{\phantom{\dagger}}\phi_{2}^{\phantom{\dagger}}\right].\end{split}$
(2.4)

Now the theory contains only one chiral interaction term, which in the limit
$\mathrm{N}\rightarrow\infty$ generates fishnet-like Feynman diagrams. The
more general $\chi_{0}$-CFT (2.3) contains two vertices of this type,
alongside two Yukawa vertices that couple fermions and bosons.

We now contemplate exclusively focussing on the Yukawa interaction vertices,
ad-hoc entirely eliminating the fields $\phi_{2}$ and $\phi_{3}$ from (2.3).
This results in the Lagrangian of the brick wall model, first obtained from a
double-scaling limit of $\mathcal{N}=2$ SCFT in [14]:

$\begin{split}&\mathcal{L}^{\mathrm{BW}}~{}=~{}\mathrm{N}\cdot\mathrm{tr}\left[-\frac{1}{2}\partial^{\mu}\phi^{\dagger}\partial_{\mu}\phi+\mathrm{i}\sum_{k=1}^{2}\bar{\psi}_{k}\not{\partial}\psi_{k}+\rho\cdot\left(\psi_{1}\phi\psi_{2}+\bar{\psi}_{1}\phi^{\dagger}\bar{\psi}_{2}\right)\right].\end{split}$
(2.5)

It is important to note that the brick model model cannot be obtained by a
special choice of the parameters $\xi_{i}$ from (2.3) or (2.2). Its vertices
are still chiral, since the hermitian conjugates of the Yukawa interaction
terms, for which the fields are ordered in reverse, are missing. This
chirality, in combination with the fact that the two vertices have to
alternate when following a fermion line, result in the typical “brick wall”
structure of the generated Feynman diagrams, see figure 1 below. A similar set
of fishnet-inspired integrable QFTs, the so-called loom models, have recently
been proposed in [25].

As already mentioned above, just like the $\chi$-CFT model, the brick wall
model (2.5) requires a double-trace (dt) counter-term for full consistency as
a conformal field theory [14]. It may be derived just like for
$\mathcal{L}^{\chi}$ in (2.2), cf. [24, 21], and reads

$\mathcal{L}^{\mathrm{BW}}_{\mathrm{dt}}~{}=~{}\alpha\cdot\mathrm{tr}\bigl{[}\phi\phi\bigr{]}\mathrm{tr}\bigl{[}\phi^{\dagger}\phi^{\dagger}\bigr{]}.$
(2.6)

In order to reach the conformal fixpoint, the parameter $\alpha$ has to be
finetuned as a suitable function $\alpha(\rho)$ of $\rho$ in (2.5). The value
of the fixpoint is a zero of the quadratic beta function of $\alpha$ [26]. It
is however not needed for any of the computations performed in the paper,
since no ultraviolet divergences appear in asymptotic vacuum diagrams of the
brick wall theory (2.5).

##  3 Star-triangle relation for scalars and fermions

In this section, we first review the star-triangle relation and its
consequences for bosons and fermions [16, 27, 28]. This sets the notation and
points out parallels to integrable two-dimensional lattice models. For the
latter, see also the interesting early work [29]. We then proceed to derive
the x-unity relation, which is a manipulation of a particular Feynman graph.
This is the key ingredient needed in later chapters to establish the inversion
relations.

In a Wick-rotated $D$-dimensional conformal QFT ($D\in\mathbbm{N}$), the
fields depend on coordinates $x^{\mu}\in\mathbbm{R}^{D}$ of the Euclidean
spacetime. For the norm of a vector we write
$\left|x\right|=\sqrt{x^{2}}=\sqrt{(x^{1})^{2}+\cdots+(x^{D})^{2}}$ and we use
the shorthand $x_{ij}:=x_{i}-x_{j}$. In a planar QFT we can make use of
integrable lattice model techniques as follows. The planar Feynman diagrams
form the lattice. Each vertex carries an infinite dimensional function space,
where the “spin components” are not labelled by a discrete index, but by the
coordinates $x^{\mu}$. Every summation over the discrete index is replaced by
an integration over $x^{\mu}$. The lattice model’s weights are assigned to the
edges connecting the vertices, and given by the generalized coordinate-space
propagators

$W_{u}^{\ell}(x_{10})~{}:=~{}\frac{1}{\left(x_{10}^{2}\right)^{u}}\left[\frac{\not{x}_{10}}{\left|x_{10}\right|}\right]^{2\ell}~{}=~{}\scalebox{1.0}[1.0]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}$
(3.1)

as in [2, 30]. They depend on a spectral parameter $u\in\mathbbm{C}$, which
eventually is to be tuned to the conformal dimension of the field under
consideration, thereby recovering the QFT’s standard massless propagator of
that field. Concretely, we have $u=\frac{D-2}{2}$ for scalars and
$u=\frac{D-1}{2}$ for fermions. The label
$\ell\in\left\\{0,\frac{1}{2}\right\\}$ is an a-priori undetermined spin
label, whose physical values are $\ell=0$ for scalars and $\ell=\frac{1}{2}$
for fermions. Inside a Feynman graph, we will indicate $\ell=0$ by a straight
solid line and and $\ell>0$ by a wavy line. If $\ell$ is left undetermined, we
will draw both on top of each other, as done above. In even dimensions, we
have $\not{x}=\sigma_{\mu}x^{\mu}$ for chiral fermions and
$\not{\bar{x}}=\bar{\sigma}_{\mu}x^{\mu}$ for anti-chiral ones, where the
$\sigma_{\mu}$, $\bar{\sigma}_{\mu}$ are now333In the preceding chapter we
have already used the notation $\sigma_{\mu}$ for the special case $D=4$,
which should not cause any confusion. the standard $(2^{D/2-1})$-dimensional
$\sigma$-matrices for massless chiral and anti-chiral spinors in $D$
dimensions, satisfying the $D$-dimensional Clifford algebra. However, in all
the diagrams considered in this work the two chiralities always appear in
alternation. Therefore, we will not distinguish them graphically. For odd
dimensions, the slash denotes contraction with the $2^{(D-1)/2}$-dimensional
gamma matrices, once again satisfying the associated $D$-dimensional Clifford
algebra, i. e. we write $\not{x}=\gamma_{\mu}x^{\mu}$. Finally, the
propagators in the theories considered here correspond to complex scalars and
complex fermions, which are usually denoted with an arrow on their
propagators. For the sake of simplicity, we will drop these arrows, too.

The spin structure also plays a role when one considers the so called merging
rules (see e. g. [28]). The spectral parameters of two propagators with
coinciding spin $\ell$ and the same initial and end points get added, which
translates into

$W_{u}^{\ell}(x)W_{-u}^{\ell}(x)~{}=~{}\mathbbm{I}^{(\ell)}\,,$ (3.2)

where the unit matrix $\mathbbm{I}^{(\ell)}$ is the one appropriate to the
spin structure on the right-hand side of the $D$-dimensional Clifford algebra:
One has $\mathbbm{I}_{2^{D/2-1}}$ for even $D$ and $\mathbbm{I}_{2^{(D-1)/2}}$
for odd $D$ if $\ell=\frac{1}{2}$. In the scalar case $\ell=0$, the unit
matrix becomes a scalar factor, $\mathbbm{I}^{(0)}=1$.

The following calculations are all based on integrability, since we employ the
star-triangle relation (STR) for bosons and fermions [4, 16] (see also [28,
27] for a review and generalizations). Expressed in terms of the generalized
propagators $W_{u}^{\ell}(x_{jk})$ in (3.1), it reads

$\begin{split}\int\mathrm{d}^{2\eta}x_{0}~{}~{}&W_{u_{1}}^{0}(x_{10})W_{u_{2}}^{\ell}(x_{20})W_{u_{3}}^{\ell}(x_{30})\\\
&\stackrel{{\scriptstyle\sum_{i}u_{i}=2\eta}}{{~{}=~{}}}r_{\ell}(u_{1},u_{2},u_{3})\cdot
W_{\eta-u_{1}}^{0}(x_{23})W_{\eta-u_{2}}^{\ell}(x_{13})W_{\eta-
u_{3}}^{\ell}(x_{12}),\end{split}$ (3.3)

where we identified the dimension $D$ with twice the so-called crossing
parameter $\eta$ [30], i.e.

$\eta~{}=~{}\frac{D}{2}\;.$ (3.4)

It is straightforward to check that (3.3) is invariant under $D$-dimensional
conformal transformations: Invariance under translations and rotations is
manifest. Scale invariance is ensured through the constraint
$\sum_{i}u_{i}=u_{1}\\!+\\!u_{2}\\!+\\!u_{3}=2\eta$. Finally, under inversion
$x_{\mu}\rightarrow x_{\mu}/x^{2}$, one has $x_{jk}^{2}\rightarrow
x_{jk}^{2}/(x_{j}^{2}x_{k}^{2})$ and
$\mathrm{d}^{2\eta}x_{0}\rightarrow\mathrm{d}^{2\eta}x_{0}/(x_{0}^{2})^{2\eta}$,
which also leaves (3.3) invariant if and only if the constraint holds. In
terms of Feynman diagrams, (3.3) becomes

$\scalebox{0.9}[0.9]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\stackrel{{\scriptstyle\sum_{i}u_{i}=2\eta}}{{=}}r_{\ell}(u_{1},u_{2},u_{3})\cdot\scalebox{0.9}[0.9]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\;,$
(3.5)

fully justifying the name “star-triangle relation”. Here and below, we use
filled, black circles within Feynman diagrams as internal integrated vertices,
whereas unfilled circles indicate fixed external points. Notice the crucial
normalization factor

$\displaystyle r_{\ell}(u_{1},u_{2},u_{3})~{}$
$\displaystyle:=~{}\pi^{\eta}\cdot
a_{0}(u_{1})\,a_{\ell}(u_{2})\,a_{\ell}(u_{3})$ (3.6a)
$\displaystyle\mathrm{with}~{}~{}a_{\ell}(u)~{}$
$\displaystyle:=~{}\frac{\Gamma(\eta-u+\ell)}{\Gamma(u+\ell)}\;,$ (3.6b)

where one easily observes that $a_{\ell}(u)a_{\ell}(\eta-u)=1$. From the STR
(3.3) one may derive important lemmas, which we will then use to derive the
inversion relations.

  * •

First, thanks to the just discussed conformal covariance of the STR (3.3), one
may derive so-called chain relations by sending one of the three external
points $x_{i}\rightarrow\infty$. One easily sees that essentially three
different limits exist, resulting in three distinct chain rules. While the
resulting three “chain integrals” are evaluated in position space, they are
formally equivalent to bubble integrals in a dual momentum space
interpretation. One finds

$\displaystyle\int\mathrm{d}^{2\eta}x_{0}~{}~{}W_{u_{2}}^{0}(x_{20})W_{u_{3}}^{0}(x_{30})~{}=~{}$
$\displaystyle r_{0}(2\eta-u_{2}-u_{3},u_{2},u_{3})\cdot
W_{u_{2}+u_{3}-\eta}^{0}(x_{23}),$ (3.7a)
$\displaystyle\scalebox{0.65}[0.65]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}~{}=~{}$
$\displaystyle r_{0}(2\eta-
u_{2}-u_{3},u_{2},u_{3})\cdot\scalebox{0.65}[0.65]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\;,$
(3.7b)
$\displaystyle\int\mathrm{d}^{2\eta}x_{0}~{}~{}W_{u_{1}}^{0}(x_{10})W_{u_{2}}^{\frac{1}{2}}(x_{20})~{}=~{}$
$\displaystyle r_{\frac{1}{2}}(u_{1},u_{2},2\eta-u_{1}-u_{2})\cdot
W_{u_{1}+u_{2}-\eta}^{\frac{1}{2}}(x_{12}),$ (3.8a)
$\displaystyle\scalebox{0.65}[0.65]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}~{}=~{}$
$\displaystyle r_{\frac{1}{2}}(u_{1},u_{2},2\eta-
u_{1}-u_{2})\cdot\scalebox{0.65}[0.65]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\;,$
(3.8b)
$\displaystyle\int\mathrm{d}^{2\eta}x_{0}~{}~{}W_{u_{2}}^{\frac{1}{2}}(x_{20})W_{u_{3}}^{\frac{1}{2}}(x_{30})\;$
$\displaystyle=-\,r_{\frac{1}{2}}(2\eta-
u_{2}-u_{3},u_{2},u_{3})\cdot\mathbbm{I}^{(\frac{1}{2})}\cdot
W_{u_{2}+u_{3}-\eta}^{0}(x_{23}),$ (3.9a)

$\displaystyle=\,-\,r_{\frac{1}{2}}(2\eta-
u_{2}-u_{3},u_{2},u_{3})\cdot\mathbbm{I}^{(\frac{1}{2})}\cdot\scalebox{0.5}[0.5]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\;.$
(3.9b)

  * •

Second, by setting the spectral parameter of the scalar line to
$u_{1}=\eta-\varepsilon$, the scalar side of the triangle on the RHS of (3.3)
effectively disappears and on the LHS of the remaining equation (3.3) the
weight $W_{u_{1}}^{0}(x_{10})$ divided by $r_{\ell}(u_{1},u_{2},u_{3})$ acts
like killing the integral in the limit $\varepsilon\rightarrow 0$.
Accordingly, one interprets this combination as a representation of the
$D$-dimensional delta function

$\delta^{(2\eta)}\left(x_{10}\right)~{}=~{}\lim_{\varepsilon\rightarrow
0}~{}\pi^{-\eta}a_{0}(\varepsilon)\cdot
W_{\eta-\varepsilon}^{0}(x_{10})~{}=~{}\scalebox{0.7}[0.7]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}},$
(3.10)

which is denoted by a dotted line in a Feynman graph. Furthermore, observe
that setting $u_{2}=\eta-u-\varepsilon$ and $u_{3}=\eta+u$ in (3.7) and (3.9),
subsequently multiplying both sides with
$\pi^{D}a_{\ell}(u+\varepsilon)\,a_{\ell}(-u)$ and then taking the limit
$\varepsilon\rightarrow 0$ gives the prescription of the delta-distribution
(3.10) on the RHS. Here and in the following derivations, we find it very
convenient to define444We introduced the factor $\pi^{-2u}$ for convenience.
It drops out in the product $f_{\ell}(u)f_{\ell}(-u)$, but turns out to be
useful below when constructing the relevant solution of the functional
inversion relations (5.4).

$f_{\ell}(u)~{}:=~{}\pi^{\eta}\pi^{-2u}a_{\ell}(u)^{-1}~{}=~{}\pi^{\eta}\pi^{-2u}\,\frac{\Gamma(u+\ell)}{\Gamma(\eta-u+\ell)}.$
(3.11)

We may then compactly write for $\ell=0$

$\lim_{\varepsilon\rightarrow
0}\int\mathrm{d}^{2\eta}x_{0}~{}~{}W_{\eta-u-\varepsilon}^{0}(x_{20})W_{\eta+u}^{0}(x_{30})~{}=~{}f_{0}(u)f_{0}(-u)\cdot\delta^{(2\eta)}(x_{23}).$
(3.12)

We now employ the STR and the chain relations to establish useful identities
for certain x-shaped Feynman diagrams which we will need later in section 5:

$\displaystyle=~{}(-1)^{\delta_{m,\frac{1}{2}}}\cdot
f_{\ell}(\eta-u)f_{\ell}(u-\eta)\cdot~{}$
$\displaystyle\scalebox{0.6}[0.6]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\cdot\mathbbm{I}^{(\ell)}\mathbbm{I}^{(m)}\;,$
(3.13a)

$\displaystyle=~{}(-1)^{\delta_{l,\frac{1}{2}}}\cdot
f_{m}(\eta-v)f_{m}(v-\eta)\cdot~{}$
$\displaystyle\scalebox{0.6}[0.6]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\cdot\mathbbm{I}^{(\ell)}\mathbbm{I}^{(m)}\,.$
(3.13b)

We will call them x-unity relations and derive them in appendix A. They are
closely related to the so-called exchange relations [22]. The second relation
(3.13b) may be obtained by relabeling the external points of (3.13a).

##  4 Vacuum graphs in the thermodynamic limit

Figure 1: The toroidal vacuum graph $Z^{\mathrm{BW}}_{3,4}$ is shown as an
example for a contribution to the brick wall model’s free energy. It contains
three rows, each consisting of four graph building kernels (4.1). The faces of
the graph generate the typical brick wall pattern of this theory. In the limit
$N\rightarrow\infty$, the leading order results in toroidal diagrams. This is
impemented into the figure: top and bottom lines are identified, and so are
the left and right boundary lines.

The vacuum diagrams of the brick wall theory form a regular lattice. One
easily checks that, due to the chirality of the vertices
$\mathrm{N}\rho\cdot\mathrm{tr}\left(\psi_{1}\phi\psi_{2}\right)$ and
$\mathrm{N}\rho\cdot\mathrm{tr}\left(\bar{\psi}_{1}\phi^{\dagger}\bar{\psi}_{2}\right)$,
no diagrams of spherical topology, i.e. ${\cal O}(\mathrm{N}^{2})$ in the
large-N expansion, exist. Accordingly, the expansion of the free energy, which
is a generating function of all of the model’s vacuum graphs, starts at
$\mathrm{N}^{0}$, corresponding to graphs of toroidal topology. Thus, this is
the leading contribution in the planar limit $\mathrm{N}\rightarrow\infty$
that we consider.

Next, we would like to explain the appearance of the brick wall shape,
referring to figure 1 as an illustrating example of a vacuum graph. The
vertices change the flavor of the fermions, therefore no two vertices of the
same kind can be adjacent. As a consequence, the two flavors of fermions form
sub-graphs, which wrap one cycle of the torus. A generic vacuum diagram
$Z^{\mathrm{BW}}_{MN}$ contains $N$ vertical fermion lines, each of them
consisting of $M$ $\psi_{1}$ and $M$ $\psi_{2}$ propagators. The bosonic
propagators then connect the fermionic ones such that the faces of the Feynman
graphs are hexagonal. This gives the characteristic brick wall shape of the
diagram.

The regular pattern of the vacuum graphs relies on the fact that it can be
generated by a fundamental piece of the diagram, the graph building kernel
[21]

$\mathbbm{R}^{\mathrm{BW}}~{}=~{}\scalebox{0.6}[0.6]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\;.$
(4.1)

It should be understood as an integration kernel. In order to form the graph
$Z^{\mathrm{BW}}_{MN}$, we chain together $N$ graph building kernels to form
the row-matrix

$\begin{split}\mathbbm{T}_{N}^{\mathrm{BW}}~{}&=~{}\mathrm{tr}\overbrace{\left[\mathbbm{R}^{\mathrm{BW}}\circ\cdots\circ\mathbbm{R}^{\mathrm{BW}}\right]}^{N-\mathrm{times}}\\\
~{}&=~{}\scalebox{0.4}[0.4]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\;.\end{split}$
(4.2)

Here, the thick black lines should be identified horizontally, as we are
computing a trace. The final step to obtain a brick wall vacuum graph is to
convolute $M$ row-matrices on top of each other and to take a vertical trace
over the resulting diagram. The vertical trace also includes the indices of
the elements of the Clifford algebra. It yields

$\begin{split}&Z^{\mathrm{BW}}_{MN}~{}=~{}\mathrm{Tr}\left[\left(\mathbbm{T}_{N}^{\mathrm{BW}}\right)^{M}\right]\\\
&~{}=~{}\int\left[\prod_{m,n=1}^{M,N}\frac{\mathrm{d}^{4}x_{mn}\mathrm{d}^{4}y_{mn}}{(x_{m,n-1}-y_{mn})^{2}}\frac{(x_{mn}-y_{mn})^{\mu}(\sigma_{\mu})^{\alpha_{mn}}_{\dot{\alpha}_{mn}}}{(x_{mn}-y_{mn})^{4}}\frac{(x_{mn}-y_{m+1,n})^{\nu}(\bar{\sigma}_{\nu})^{\dot{\alpha}_{mn}}_{\alpha_{m+1,n}}}{(x_{mn}-y_{m+1,n})^{4}}\right],\end{split}$
(4.3)

a generic vacuum graph of the brick wall theory, see figure 1 for
$Z^{\mathrm{BW}}_{3,4}$ as an example. The double-periodic boundary conditions
are built into (4.3) by the index identifications $M+1\equiv 1$ and $0\equiv
N$.

The free energy is then the infinite sum of all vacuum diagrams

$Z~{}=~{}\sum_{M,N=1}^{\infty}Z^{\mathrm{BW}}_{MN}\rho^{2MN}\;,$ (4.4)

where the two in the exponent of $\rho$ comes from the fact that
$\mathbbm{R}^{\mathrm{BW}}$ has four vertices, each shared with one other
graph building kernel. Note that this exponent might differ for graphs of
other theories if some of the weights in their graph building kernel are
required to turn into delta functions, see appendix B. The critical coupling
$\rho_{\mathrm{cr}}$ of the brick wall model is defined as the radius of
convergence of the series (4.4), which we find to be
$\rho_{\mathrm{cr}}=\left(\mathbbm{K}^{\mathrm{BW}}\right)^{-\frac{1}{2}}$
with the quantity

$\mathbbm{K}^{\mathrm{BW}}~{}:=~{}\lim_{M,N\rightarrow\infty}\left|Z^{\mathrm{BW}}_{MN}\right|^{\frac{1}{MN}}\;.$
(4.5)

It happens to be equal to the free energy in the thermodynamic limit of the
brick wall vacuum diagrams $Z^{\mathrm{BW}}_{MN}$. This is the quantity we
will calculate with the help of integrability by using the STR (3.3).

To proceed, we first generalize the relevant graphs by using the more general
graph building kernel

$\mathbbm{R}_{\ell m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)~{}=~{}\scalebox{0.6}[0.6]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\;.$
(4.6)

The propagators are the ones in (3.1), and their weights are $u_{+}$, $u_{-}$,
$v_{-}$ and $v_{+}$ (going counterclockwise, starting at the top right
propagator) and $\ell$ and $m$ for the ones on the left and on the right,
respectively. All of them need to be finetuned in order to obtain the desired
vacuum graphs of the theory under consideration. For the brick wall model this
means $\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)=\left(\begin{smallmatrix}0&\frac{3}{2}\\\
1&\frac{3}{2}\end{smallmatrix}\right)$ and $\ell=0$, as well as
$m=\frac{1}{2}$.

In the literature [16, 22, 27], the parameters $u_{+}$, $u_{-}$, $v_{+}$ and
$v_{-}$ of the graph building kernel (4.6) are to be restricted due to the
requirement of it being a R-matrix as an element of
$\mathrm{End}(V_{\Delta,\ell,\dot{\ell}}\otimes V_{\Delta,\ell,\dot{\ell}})$
of the principle series representation $V_{\Delta,\ell,\dot{\ell}}$ of the
Euclidean conformal group. However, this specification is not necessary in
what follows. This is the reason why we neither use the term R-matrix for
(4.6) nor transfer matrix for the row-matrices.

The graph building kernel gets composed to a row-matrix

$\begin{split}\mathbbm{T}_{N\;\ell m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)~{}&=~{}\mathrm{tr}\overbrace{\left[\mathbbm{R}_{\ell
m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)\circ\cdots\circ\mathbbm{R}_{\ell
m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)\right]}^{N-\mathrm{times}}\\\
~{}&=~{}\scalebox{0.4}[0.4]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\;,\end{split}$
(4.7)

such that the generalized graphs are again obtained by stacking up $M$ rows
and identifying the vertical cycle of the torus. This is denoted by taking the
vertical trace, including any potential spinorial indices. We find

$Z_{MN\;\ell m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)~{}=~{}\mathrm{Tr}\left[\mathbbm{T}_{N\;\ell
m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)^{M}\right]\;,$ (4.8)

and similar to (4.5), the generalized graph’s free energy in the thermodynamic
limit is

$\mathbbm{K}_{\ell m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)~{}=~{}\lim_{M,N\rightarrow\infty}\left|Z_{MN\;\ell
m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)\right|^{\frac{1}{MN}}\;.$ (4.9)

Finally, we assume that the fact that the trace of a matrix equals the sum of
its eigenvalues also holds for the integral operators considered here. In the
limit $M\rightarrow\infty$ all but the highest eigenvalue are suppressed, so
we can write the free energy in the thermodynamic limit as

$\mathbbm{K}_{\ell m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)~{}=~{}\lim_{N\rightarrow\infty}\left|\Lambda_{\mathrm{max},N\;\ell
m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)\right|^{\frac{1}{N}}\;.$ (4.10)

We have denoted the highest eigenvalue of $\mathbbm{T}_{N\;\ell
m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\ u_{-}&v_{-}\end{smallmatrix}\right)$
as $\Lambda_{\mathrm{max},N\;\ell m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)$. The quantity $\mathbbm{K}_{\ell
m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\ u_{-}&v_{-}\end{smallmatrix}\right)$
will be derived in the next section in generality, and then specified to the
brick wall model.

##  5 Inversion relations of the free energy by integrability

The strategy for deriving the inversion relations for the free energy (4.9) is
based on finding an integration kernel that trivializes the generalized
transfer matrix (4.7) to the identity matrix, up to a convolution factor. We
will construct such an “inverse” by usage of the x-unity relations (3.13a) and
(3.13b). It will turn out that there are four of these kernels, leading to
four inversion relations for the free energy. They are sufficiently
constraining to allow for the construction of the desired solution. By tuning
the spectral parameters to the ones of the brick wall model, we obtain its
free energy in the thermodynamic limit. To check our procedure, we verified
that the method also reproduces Zamolodchikov’s original result [2] in a
transparent fashion, see appendix B.

To illustrate the inversion up to a factor, consider the convolution of the
transfer matrix with its transpose at a different spectral parameter. As an
example, we choose $\left(\begin{smallmatrix}-u_{+}&2\eta-v_{+}\\\ 2\eta-
u_{-}&-v_{-}\end{smallmatrix}\right)$. After applying (3.13a) a total of $2N$
times, we get the relation

$\displaystyle\mathbbm{T}$ ${}_{N\;\ell
m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)\circ\mathbbm{T}_{N\;\ell
m}^{\mathrm{T}}\left(\begin{smallmatrix}-u_{+}&2\eta-v_{+}\\\ 2\eta-
u_{-}&-v_{-}\end{smallmatrix}\right)$ (5.1a)
$\displaystyle\stackrel{{\scriptstyle\phantom{\eqref{eq:XUnity1}}}}{{=}}~{}\scalebox{0.5}[0.5]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}$
(5.1b)
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:XUnity1}}}{{=}}~{}\scalebox{0.5}[0.5]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\cdot\mathbbm{I}^{(\ell)}\mathbbm{I}^{(m)}$
(5.1c) $\displaystyle\hskip
42.67912pt\cdot\left[(-1)^{\delta_{m,\frac{1}{2}}}f_{\ell}(\eta-
u_{-})f_{\ell}(u_{-}-\eta)\right]^{N}$ (5.1d)
$\displaystyle\stackrel{{\scriptstyle\phantom{\eqref{eq:XUnity1}}}}{{=}}~{}\scalebox{0.5}[0.5]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\cdot\mathbbm{I}^{(\ell)}\mathbbm{I}^{(m)}$
(5.1e) $\displaystyle\hskip
42.67912pt\cdot\left[(-1)^{\delta_{m,\frac{1}{2}}}f_{\ell}(\eta-
u_{-})f_{\ell}(u_{-}-\eta)\right]^{N}$ (5.1f)
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:XUnity1}}}{{=}}~{}\scalebox{0.5}[0.5]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\cdot\mathbbm{I}^{(\ell)}\mathbbm{I}^{(m)}$
(5.1g) $\displaystyle\hskip
42.67912pt\cdot\left[(-1)^{\delta_{m,\frac{1}{2}}+\delta_{\ell,\frac{1}{2}}}f_{\ell}(\eta-
u_{-})f_{\ell}(u_{-}-\eta)f_{m}(\eta-v_{+})f_{m}(v_{+}-\eta)\right]^{N}$
(5.1h)
$\displaystyle\stackrel{{\scriptstyle\phantom{\eqref{eq:XUnity1}}}}{{=}}\left[(-1)^{\delta_{m,\frac{1}{2}}+\delta_{\ell,\frac{1}{2}}}f_{\ell}(\eta-
u_{-})f_{\ell}(u_{-}-\eta)f_{m}(\eta-
v_{+})f_{m}(v_{+}-\eta)\right]^{N}\cdot\mathbbm{1}_{N}\cdot\mathbbm{I}^{(\ell)}\mathbbm{I}^{(m)}\;.$
(5.1i)

Here, $\mathbbm{1}_{N}$ is the identity kernel consisting of delta functions.
One could have applied similar steps to the products $\mathbbm{T}_{N\;\ell
m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)\circ\mathbbm{T}_{N\;\ell
m}^{\mathrm{T}}\left(\begin{smallmatrix}2\eta-u_{+}&-v_{+}\\\ 2\eta-
u_{-}&-v_{-}\end{smallmatrix}\right)$, $\mathbbm{T}_{N\;\ell
m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)\circ\mathbbm{T}_{N\;\ell
m}^{\mathrm{T}}\left(\begin{smallmatrix}-u_{+}&2\eta-v_{+}\\\ -u_{-}&2\eta-
v_{-}\end{smallmatrix}\right)$ and $\mathbbm{T}_{N\;\ell
m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)\circ\mathbbm{T}_{N\;\ell
m}^{\mathrm{T}}\left(\begin{smallmatrix}2\eta-u_{+}&-v_{+}\\\ -u_{-}&2\eta-
v_{-}\end{smallmatrix}\right)$. Consequently, one would have used (3.13a) and
(3.13b), (3.13b) and (3.13a) and (3.13b) and (3.13b). In total, this leads to
four relations similar to (5.1). We proceed the same way in all four cases. We
let the product of transfer matrices act on the eigenvector corresponding to
the highest eigenvalue, take their magnitude to the power $\frac{1}{N}$ and
take the limit $N\rightarrow\infty$ as in (4.10). Since the eigenvalues are
insensitive to transposition, the LHS of (5.1b) turns into a product of free
energy contributions at different spectral parameters. The RHS (5.1i) yields
its scalar factor by these steps. Applying these manipulations to the four
relations yields the four inversion relations for the free energy
$\mathbbm{K}_{\ell m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)$:

$\displaystyle\mathbbm{K}_{\ell m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)\mathbbm{K}_{\ell
m}\left(\begin{smallmatrix}-u_{+}&2\eta-v_{+}\\\ 2\eta-
u_{-}&-v_{-}\end{smallmatrix}\right)~{}$ $\displaystyle=~{}f_{\ell}(\eta-
u_{-})f_{\ell}(u_{-}-\eta)f_{m}(\eta-v_{+})f_{m}(v_{+}-\eta),$ (5.2a)
$\displaystyle\mathbbm{K}_{\ell m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)\mathbbm{K}_{\ell
m}\left(\begin{smallmatrix}2\eta-u_{+}&-v_{+}\\\ 2\eta-
u_{-}&-v_{-}\end{smallmatrix}\right)~{}$ $\displaystyle=~{}f_{\ell}(\eta-
u_{-})f_{\ell}(u_{-}-\eta)f_{\ell}(\eta-u_{+})f_{\ell}(u_{+}-\eta),$ (5.2b)
$\displaystyle\mathbbm{K}_{\ell m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)\mathbbm{K}_{\ell
m}\left(\begin{smallmatrix}-u_{+}&2\eta-v_{+}\\\ -u_{-}&2\eta-
v_{-}\end{smallmatrix}\right)~{}$ $\displaystyle=~{}f_{m}(\eta-
v_{-})f_{m}(v_{-}-\eta)f_{m}(\eta-v_{+})f_{m}(v_{+}-\eta),$ (5.2c)
$\displaystyle\mathbbm{K}_{\ell m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)\mathbbm{K}_{\ell
m}\left(\begin{smallmatrix}2\eta-u_{+}&-v_{+}\\\ -u_{-}&2\eta-
v_{-}\end{smallmatrix}\right)~{}$ $\displaystyle=~{}f_{m}(\eta-
v_{-})f_{m}(v_{-}-\eta)f_{\ell}(\eta-u_{+})f_{\ell}(u_{+}-\eta).$ (5.2d)

Next, to construct a solution for (5.2), we make the ansatz

$\mathbbm{K}_{\ell m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)~{}=~{}\kappa^{1}_{\ell}(u_{+})\kappa^{2}_{\ell}(u_{-})\kappa^{3}_{m}(v_{+})\kappa^{4}_{m}(v_{-})$
(5.3)

with a-priori four functions $\kappa^{1}_{\ell}(u)$, $\kappa^{2}_{\ell}(u)$,
$\kappa^{3}_{\ell}(u)$ and $\kappa^{4}_{\ell}(u)$ yet to be determined.
Plugging the ansatz (5.3) into the four inversion relations (5.2), we find
that all four functions are identical:
$\kappa^{1}_{\ell}(u)=\kappa^{2}_{\ell}(u)=\kappa^{3}_{\ell}(u)=\kappa^{4}_{\ell}(u)=:\kappa_{\ell}(u)$.
It has to satisfy the functional relations

$\displaystyle\kappa_{\ell}(u)\kappa_{\ell}(-u)~{}$ $\displaystyle=~{}1\,,$
(5.4a) $\displaystyle\frac{\kappa_{\ell}(\eta-u)}{\kappa_{\ell}(u)}~{}$
$\displaystyle=~{}f_{\ell}(u)\;,$ (5.4b)

in slight generalization of a result in [30]. We can construct a solution
satisfying (5.4) by using the explicit form of $f_{\ell}(u)$ and alternately
requiring the function $\kappa_{\ell}(u)$ to satisfy (5.4a) and (5.4b) [31,
32, 33]. This yields the solution

$\kappa_{\ell}(u)~{}=~{}\pi^{u}\frac{\Gamma(\eta-u+\ell)}{\Gamma(\eta+\ell)}\prod_{k=1}^{\infty}\frac{\Gamma(2\eta
k+\eta-u+\ell)\,\Gamma(2\eta k+u+\ell)\,\Gamma(2\eta
k-\eta+\ell)}{\Gamma(2\eta k-\eta+u+\ell)\,\Gamma(2\eta
k-u+\ell)\,\Gamma(2\eta k+\eta+\ell)}\;,$ (5.5)

which reduces to the findings of [30] after setting $\ell=0$. We may now apply
the general solution

$\mathbbm{K}_{\ell m}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)~{}=~{}\kappa_{\ell}(u_{+})\kappa_{\ell}(u_{-})\kappa_{m}(v_{+})\kappa_{m}(v_{-})$
(5.6)

to the brick wall graphs by setting the spectral parameters to
$\left(\begin{smallmatrix}0&\frac{3}{2}\\\
1&\frac{3}{2}\end{smallmatrix}\right)$ and $\ell=0$, as well as
$m=\frac{1}{2}$. Putting everything together, the free energy of the brick
wall graphs reads

$\begin{split}\mathbbm{K}^{\mathrm{BW}}~{}&=~{}\mathbbm{K}_{0,\frac{1}{2}}\left(\begin{smallmatrix}0&\frac{3}{2}\\\
1&\frac{3}{2}\end{smallmatrix}\right)~{}=~{}\kappa_{0}(0)\kappa_{0}(1)\kappa_{\frac{1}{2}}\left(\tfrac{3}{2}\right)\kappa_{\frac{1}{2}}\left(\tfrac{3}{2}\right)\\\
~{}&\stackrel{{\scriptstyle
2\eta=D=4}}{{=}}~{}1\cdot\pi^{\frac{3}{2}}\frac{\Gamma\left(\frac{5}{4}\right)}{\Gamma\left(\frac{3}{4}\right)}\cdot\frac{\pi^{2}}{2}\cdot\frac{\pi^{2}}{2}~{}=~{}\frac{\pi^{\frac{11}{2}}}{16}\frac{\Gamma\left(\frac{1}{4}\right)}{\Gamma\left(\frac{3}{4}\right)}\;.\end{split}$
(5.7)

The resulting critical coupling of the brick wall theory (2.5) is then
$\rho_{\mathrm{cr}}=\left[\frac{\pi^{\frac{11}{2}}}{16}\frac{\Gamma\left(\frac{1}{4}\right)}{\Gamma\left(\frac{3}{4}\right)}\right]^{-\frac{1}{2}}$.

Finally, we make the important observation that $\mathbbm{K}$ factorizes into
contributions $\kappa$ of the individual propagators contained in the graph
building kernel, see (5.6). The connectivity of the vacuum diagrams depends on
how many of the propagators in the graph building kernel are either tuned such
that they vanish or else turn into a delta function. In both cases the
contribution to the free energy is trivial, i.e. $\kappa_{\ell}(0)=1$,
$\lim_{\varepsilon\rightarrow
0}\pi^{-\eta}a_{0}(\varepsilon)\kappa_{0}(\eta-\varepsilon)=1$ (cf. appendix
B). Thus, in the thermodynamic limit, the topology of the vacuum graphs is
solely reflected in the exponent of the free energy. However, its value is
sensitive to the different values of $u$ and $\ell$ in the evaluation of
(5.5). For $\ell=0$ and $u=\frac{D-2}{2}$, i.e. for a bosonic field in $D$
dimensions, we find the compact result

$\kappa_{0}\left(\tfrac{D-2}{2}\right)~{}\stackrel{{\scriptstyle
2\eta=D}}{{=}}~{}\frac{\pi^{\frac{D-1}{2}}\Gamma\left(\frac{3}{2}-\frac{1}{D}\right)}{\Gamma\left(\frac{D}{2}\right)\Gamma\left(1-\frac{1}{D}\right)}\;.$
(5.8)

For the fermionic case $\ell=\frac{1}{2}$ and $u=\frac{D-1}{2}$, we
unfortunately were not able to deduce a similarly compact form. Accordingly,
here we only present some specific results for
$\kappa_{\frac{1}{2}}\left(\tfrac{D-1}{2}\right)$: for, respectively,
$D=3,4,6$ one has $\frac{2\pi}{\sqrt{3}}$, $\frac{\pi^{2}}{2}$ and
$\frac{\pi^{3}}{2\sqrt{6}}$.

##  6 Conclusions and outlook

Let us summarize our work. Our focus is on large vacuum energy diagrams of the
brick wall model, whose Lagrangian is given in (2.5) with (2.6). This is an
integrable, planar, chiral and non-unitary four-dimensional matrix quantum
field theory, and was first proposed in [14]. It may be considered an
integrable relative of the standard non-integrable Yukawa theory, in close
analogy with bi-scalar fishnet theory, which in the same vein may be
considered to be an integrable relative of standard non-integrable
$\phi^{4}_{4}$-theory. The typical Feynman-diagrams of this bi-fermionic model
are of regular “brick-wall”-type, replacing the regular square lattices of
standard fishnet theory. Curiously, in contradistinction to the fishnet
theory, and more generally to the $\chi_{0}$\- and $\chi$-CFTs, the model may
not be obtained as a double-scaling limit of planar, twisted $\mathcal{N}=4$
SYM; instead, one instead needs to consider twisted $\mathcal{N}=2$ SCFT [14].
Our key result is then the adaptation of A. B. Zamolodchikov’s powerful
computation of the thermodynamic free energy of fishnet graphs to the brick-
wall case. This allows us to find this quantity in the closed form (5.7).

Our graphical view of the brick wall model immediately suggests the
formulation of two further planar QFTs, thereby continuing the quest for
integrable QFTs that are not derivable from deformations of maximally
supersymmetric $\mathcal{N}=4$ SYM. A similar motivation drove, perhaps, the
recent construction of the “loom” models of [25]. First, replacing the
fermions of the brick wall model (2.5) by two different bosons and thus making
it a tri-scalar theory does not fundamentally change the diagrammatics in the
limit $\mathrm{N}\rightarrow\infty$. The vacuum Feynman diagrams are still of
brick wall, or, equivalently, honeycomb form. However, now the interaction
terms are exactly marginal in six as opposed to four dimensions. This leads us
to write down the following non-unitary, but integrable cousin of
$\phi^{3}_{6}$-theory with Lagrangian

$\begin{split}&\mathcal{L}^{\phi^{3}}_{D=6}~{}=~{}\mathrm{N}\cdot\mathrm{tr}\left[\frac{1}{2}\sum_{j=1}^{3}\partial^{\mu}\phi_{j}^{\dagger}\partial_{\mu}\phi_{j}+\rho\cdot\left(\phi_{1}\phi_{2}\phi_{3}+\phi^{\dagger}_{1}\phi_{2}^{\dagger}\phi_{3}^{\dagger}\right)\right]\;.\end{split}$
(6.1)

It is equivalent to one of the models formulated in [34], where in addition
the anomalous dimension of this theory’s length-two operators was calculated
in a non-perturbative fashion. Clearly this theory generates honeycomb graphs
with propagators living in six dimensions. Here it is totally unclear whether
the model is derivable from a unitary supersymmetric strongly twisted SYM
theory, as no suitable candidates are known in dimension six. However, the
corresponding free energy in the thermodynamic limit was also calculated as
the third of the three cases considered in [2]. As a check, we successfully
reproduce his result by our methods in appendix B in (B.14). Second, our ad-
hoc graphical view of the the brick wall model (2.5), inspired (but not
derived) from deformed $\mathcal{N}=4$ SYM may also be applied to the
$\gamma$-deformation of ABJM theory [13]. Keeping only the boson-fermion
interactions in the analogue of the $\chi_{0}$-theory (2.3) then yields the
following three-dimensional Lagrangian

$\begin{split}&\mathcal{L}^{\mathrm{ABJM~{}FN}}_{D=3}~{}=~{}-\mathrm{N}\cdot\mathrm{tr}\left[\sum_{j=1}^{2}\partial^{\mu}Y_{j}^{\dagger}\partial_{\mu}Y_{j}^{\phantom{\dagger}}+\mathrm{i}\sum_{k=1}^{2}\Psi^{\dagger}_{k}\not{\partial}\Psi_{k}^{\phantom{\dagger}}+\rho\cdot\left(Y_{1}^{\phantom{\dagger}}Y_{2}^{\dagger}\Psi_{1}^{\phantom{\dagger}}\Psi_{2}^{\dagger}-Y_{1}^{\dagger}Y_{2}^{\phantom{\dagger}}\Psi_{1}^{\dagger}\Psi_{2}^{\phantom{\dagger}}\right)\right]\;.\end{split}$
(6.2)

In contrast to the four-dimensional case, the vacuum diagrams do not follow a
brick wall pattern but have the appearance of square-lattice fishnets. The
difference to the vacuum graphs of the bi-scalar fishnet theory (2.4) is that
now there are zig-zag lines formed by fermionic propagators in alternation
with bosonic zig-zag lines. Once again, we are able to model these graphs by
suitably composing the generalized graph building kernel (4.6), evaluated as
$\mathbbm{R}_{0,\frac{1}{2}}\left(\begin{smallmatrix}\frac{1}{2}&1\\\
\frac{1}{2}&1\end{smallmatrix}\right)$. Tracing out the steps of section 5, we
find the free energy of the theory (6.2) in the thermodynamic limit to be

$\mathbbm{K}_{0,\frac{1}{2}}\left(\begin{smallmatrix}\frac{1}{2}&1\\\
\frac{1}{2}&1\end{smallmatrix}\right)~{}=~{}\kappa_{0}(\tfrac{1}{2})^{2}\kappa_{\frac{1}{2}}(1)^{2}~{}\stackrel{{\scriptstyle
2\eta=D=3}}{{=}}~{}\frac{4\pi^{3}}{27}\frac{\Gamma\left(\frac{1}{6}\right)^{2}}{\Gamma\left(\frac{2}{3}\right)^{2}}\;.$
(6.3)

Clearly it would again be interesting to see whether the model (6.2) may be
derived as a double scaling limit from a supersymmetric strongly twisted SCFT
in three dimensions.

The brick-wall model (2.5), as well as the theories (6.1) and (6.2), should
allow for an adaption of the calculations of a large number of quantities that
have been computed exactly for the fishnet model. For the brick wall model,
some of these have already been performed in [14], such as the leading order
of $\alpha(\rho)$ in (2.5) and the anomalous dimension of the operator
$\mathrm{tr}\left[\phi^{L}\right]$. The diagonalization of the brick wall
graph building kernel (4.1) was already examined in [22] and would allow to
get a closed form for Basso-Dixon [35] brick wall diagrams. Conceptually, the
construction of a “brickchain” similar to the fishchain, a quantum mechanical
model based on the graph building kernel [36, 37], would give hints on a dual
AdS description of the brick wall model. The fact that the brick wall theory
cannot be obtained from a double-scaling limit of $\mathcal{N}=4$ SYM makes a
holographic dual description very intriguing.

The method of inversion relations does not yet account for twisted toroidal
diagrams, where the fermion lines wrap the cycle of the torus multiple times
before closing. It would be interesting to implement them in the framework,
however the result for the fishnet theory should not be altered, since it was
confirmed by TBA [38]. A confirmation by TBA for the brick wall result would
be an equally good sanity check. Finally, the brick wall model, similar to the
fishnet model, should serve as a stepping stone to uncover the integrable
structure of more evolved theories. For this reason, translating methods from
those two simplest toy models to the $\chi_{0}$-, $\chi$-CFT and eventually to
$\mathcal{N}=4$ SYM is of high priority.

## Acknowledgments

We thank Matthias Volk for initial collaboration on this project. We are
thankful to Changrim Ahn, Zoltan Bajnok and Lance Dixon for very useful
discussions. Special thanks to Lance Dixon, Enrico Olivucci and Volodya
Kazakov for very careful readings of our manuskript, and several valuable
suggestions on references. This work is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) \- Projektnummer
417533893/GRK2575 “Rethinking Quantum Field Theory”.

##  Appendix A Derivation of x-unity

In this appendix, we will prove the auxiliary relations stated in (3.13) by
using the STR and the chain rules. In detail, the calculation goes as follows:

$\displaystyle\scalebox{0.57}[0.57]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}~{}=~{}\lim_{\varepsilon\rightarrow
0}\scalebox{0.57}[0.57]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}$
(A.1a)
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:Inversion1Weights}}}{{=}}~{}\lim_{\varepsilon\rightarrow
0}\scalebox{0.57}[0.57]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\cdot\frac{1}{f_{0}(\eta-2\varepsilon)f_{0}(2\varepsilon-\eta)}$
(A.1b)
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:STRgenFormula}}}{{=}}~{}\lim_{\varepsilon\rightarrow
0}\scalebox{0.57}[0.57]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\cdot\frac{r_{\ell}(\varepsilon,u,2\eta-u-\varepsilon)}{f_{0}(\eta-2\varepsilon)f_{0}(2\varepsilon-\eta)}$
(A.1c)
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:DeltaDefi},\eqref{eq:STRgenFormula}}}{{=}}~{}\lim_{\varepsilon\rightarrow
0}\scalebox{0.57}[0.57]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\cdot\frac{r_{\ell}(\varepsilon,u,2\eta-u-\varepsilon)}{f_{0}(\eta-2\varepsilon)f_{0}(2\varepsilon-\eta)}\frac{\pi^{\eta}}{a_{0}(\varepsilon)}r_{m}(2\eta-2\varepsilon,v,2\varepsilon-v)$
(A.1d)
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:Inversion2Weights}}}{{=}}~{}\scalebox{0.57}[0.57]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\cdot\lim_{\varepsilon\rightarrow
0}\frac{r_{\ell}(\varepsilon,u,2\eta-u-\varepsilon)}{f_{0}(\eta-2\varepsilon)f_{0}(2\varepsilon-\eta)}\frac{\pi^{\eta}}{a_{0}(\varepsilon)}r_{m}(2\eta-2\varepsilon,v,2\varepsilon-v)\cdot\mathbbm{I}^{(\ell)}$
(A.1e)
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:ChainRuleEllHalf}\;\mathrm{or}\;\eqref{eq:ChainRuleEll0}}}{{=}}~{}\scalebox{0.57}[0.57]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\cdot(-1)^{\delta_{m,\frac{1}{2}}}r_{m}(0,\eta+v,\eta-v)$
(A.1f) $\displaystyle\hskip 56.9055pt\cdot\lim_{\varepsilon\rightarrow
0}\frac{r_{\ell}(\varepsilon,u,2\eta-u-\varepsilon)}{f_{0}(\eta-2\varepsilon)f_{0}(2\varepsilon-\eta)}\frac{\pi^{\eta}}{a_{0}(\varepsilon)}r_{m}(2\eta-2\varepsilon,v,2\varepsilon-v)\cdot\mathbbm{I}^{(\ell)}\mathbbm{I}^{(m)}$
(A.1g)
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:Inversion2Weights}}}{{=}}~{}(-1)^{\delta_{m,\frac{1}{2}}}\cdot
f_{\ell}(\eta-u)f_{\ell}(u-\eta)\cdot\scalebox{0.57}[0.57]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}~{}\cdot\mathbbm{I}^{(\ell)}\mathbbm{I}^{(m)}$
(A.1h)

In the last step, we simplified the overall factor by extensive use of
$a_{m}(u)a_{m}(\eta-u)=1$ and the definition of $f_{\ell}(u)$ in (3.11).

##  Appendix B Derivation of scalar results

In this appendix, we will carefully re-derive Zamolodchikov’s expression for
the free energy per propagator [2, eq.(18)] from our general solution (5.6)
and (5.5). This appears to be a worthwhile exercise, given the extremely
concise arguments and derivations in [2].

To begin, we restrict ourself to scalar propagator weights in this section.
Furthermore, we notice the different normalization of the weights
$G_{\alpha}(x)$ in [2, eq.(8)] compared to the ones considered here,
$W_{u}^{0}(x)$ , in (3.1). We observe that we can relate them as

$W_{u}^{0}(x)~{}=~{}\pi^{u}\frac{\Gamma(\eta-u)}{\Gamma(\eta)}G_{\pi-\pi\frac{u}{\eta}}(x)~{}=:~{}h(u)G_{\pi-\pi\frac{u}{\eta}}(x)\;.$
(B.1)

The graph building kernel in terms of the weight $G_{\alpha}(x)$,
$\mathbbm{R}^{G}$, is related to the generalized one (4.6) by

$\mathbbm{R}_{0,0}\left(\begin{smallmatrix}u_{+}&v_{+}\\\
u_{-}&v_{-}\end{smallmatrix}\right)~{}=~{}h\left(\eta-\eta\tfrac{\alpha_{+}}{\pi}\right)h\left(\eta-\eta\tfrac{\alpha_{-}}{\pi}\right)h\left(\eta-\eta\tfrac{\beta_{+}}{\pi}\right)h\left(\eta-\eta\tfrac{\beta_{-}}{\pi}\right)\cdot\mathbbm{R}^{G}\left(\begin{smallmatrix}\alpha_{+}&\beta_{+}\\\
\alpha_{-}&\beta_{-}\end{smallmatrix}\right)$ (B.2)

and we introduced Zamolodchikov’s angles on the medial lattice
$\frac{\alpha_{\pm}}{\pi}:=1-\frac{u_{\pm}}{\eta}$ and
$\frac{\beta_{\pm}}{\pi}:=1-\frac{v_{\pm}}{\eta}$ [25]. According to (4.9),
the normalized free energy is

$\begin{split}\mathbbm{K}^{G}\left(\begin{smallmatrix}\alpha_{+}&\beta_{+}\\\
\alpha_{-}&\beta_{-}\end{smallmatrix}\right)~{}&=~{}\frac{\mathbbm{K}_{0,0}\left(\begin{smallmatrix}\eta-\eta\frac{\alpha_{+}}{\pi}&\eta-\eta\frac{\beta_{+}}{\pi}\\\
\eta-\eta\frac{\alpha_{-}}{\pi}&\eta-\eta\frac{\beta_{-}}{\pi}\end{smallmatrix}\right)}{h\left(\eta-\eta\frac{\alpha_{+}}{\pi}\right)h\left(\eta-\eta\frac{\alpha_{-}}{\pi}\right)h\left(\eta-\eta\frac{\beta_{+}}{\pi}\right)h\left(\eta-\eta\frac{\beta_{-}}{\pi}\right)}\\\
~{}&=:~{}\kappa^{G}(\alpha_{+})\kappa^{G}(\alpha_{-})\kappa^{G}(\beta_{+})\kappa^{G}(\beta_{-})\;,\end{split}$
(B.3)

which implies for the factors

$\begin{split}\kappa^{G}(\alpha)~{}&=~{}\frac{\kappa_{0}\left(\frac{D}{2}-\frac{D\alpha}{2\pi}\right)}{h\left(\frac{D}{2}-\frac{D\alpha}{2\pi}\right)}\\\
~{}&=~{}\prod_{k=1}^{\infty}\frac{\Gamma(Dk+D/2-D\alpha/2\pi)\,\Gamma(Dk+D\alpha/2\pi)\,\Gamma(Dk-D/2)}{\Gamma(Dk-D/2+D\alpha/2\pi)\,\Gamma(Dk-D\alpha/2\pi)\,\Gamma(2\eta
k+D/2)}~{}=:~{}\frac{1}{Z(\alpha)}\;.\end{split}$ (B.4)

$Z(\alpha)$ is Zamolodchikov’s result for the free energy per propagator in
the thermodynamic limit, so is the factor $\kappa^{G}(\alpha)$. We obtain the
inverse of $Z(\alpha)$, since he defines $Z(\alpha):=\mathrm{e}^{-f(\alpha)}$
with $f(\alpha)=\mathrm{log}\,\kappa^{G}(\alpha)$.

Next, we derive the expressions for the critical coupling of regular
triangular, quadratic and honeycomb lattices in, respectively, $D=3,4,6$
dimensions from the general free energy per weight (B.4), as considered in
[2]. However, therein the critical coupling was computed in relation to yet
another weight, called $G\mathcal{D}(x):=s^{-1}\cdot G_{\frac{2\pi}{D}}(x)$
[2, eq.(1)], and the factor of proportionality is $s=2\pi,4\pi,8\pi$,
respectively to $D=3,4,6$. The motivation for the new weight is that
$G\mathcal{D}(x)$ is the $D$-dimensional Fourier transform of the scalar
momentum space propagator. Under the assumption that the Lagrangian contains
an interaction term $g\cdot\mathcal{L}_{\mathrm{int}}$, we will demonstrate
how to calculate
$g_{\mathrm{cr}}=\left(\mathbbm{K}^{G\mathcal{D}}\right)^{-b}$ in the three
cases. The quantity $b$ denotes the ratio of the number of graph building
kernels to the number of vertices in the vacuum diagrams and will be
determined below.

  * •

In three dimensions, the Feynman graphs can have a triangular shape, which is
due to a sixtic interaction, as provided by e. g. the ABJM fishnet theory
[13]. In order to obtain a triangular graph building kernel from (4.6), we
have to turn one propagator into a delta-distribution. We pick the north-west
propagator to undergo the metamorphosis and by (3.10) we can construct the
triangular graph building kernel as

$\mathbbm{R}^{D=3}~{}:=~{}\lim_{\varepsilon\rightarrow
0}\pi^{-\eta}a_{0}(\varepsilon)\cdot\mathbbm{R}_{0,0}\left(\begin{smallmatrix}\eta-\varepsilon&\frac{1}{2}\\\
\frac{1}{2}&\frac{1}{2}\end{smallmatrix}\right)~{}=~{}\scalebox{0.6}[0.6]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\;.$
(B.5)

We observe that, due to the delta-distribution, the number of vertices equals
the number of graph building kernels in a asymptotic vacuum diagram, hence
$b=1$. The corresponding free energy related to the canonical weight function
$W_{u}^{0}(x)$ is

$\begin{split}\mathbbm{K}^{D=3}~{}:&=~{}\lim_{\varepsilon\rightarrow
0}\pi^{-\eta}a_{0}(\varepsilon)\cdot\mathbbm{K}_{0,0}\left(\begin{smallmatrix}\eta-\varepsilon&\frac{1}{2}\\\
\frac{1}{2}&\frac{1}{2}\end{smallmatrix}\right)~{}=~{}\kappa_{0}\left(\tfrac{1}{2}\right)^{3}\cdot\lim_{\varepsilon\rightarrow
0}\pi^{-\eta}a_{0}(\varepsilon)\kappa_{0}(\eta-\varepsilon)\\\
~{}&=~{}\left[\frac{\sqrt{\pi}}{3}\frac{\Gamma(\frac{1}{6})}{\Gamma(\frac{2}{3})}\right]^{3}\;,\end{split}$
(B.6)

which is obtained by (5.6) and (5.5). Regarding the weight $G_{\alpha}(x)$,
the correcting factor yielding a delta function is provided by $h(u)$ in
(B.1). By noting that $u=\eta$ and $u=\frac{1}{2}$ correspond to $\alpha=0$
and $\alpha=\frac{2\pi}{3}$ for $D=3$, respectively, one can calculate
directly by (B.3) and (B.4)

$\mathbbm{K}^{G,D=3}~{}=~{}\frac{1}{Z(0)Z(\frac{2\pi}{3})^{3}}~{}=~{}\left[\frac{\sqrt{\pi}}{6}\frac{\Gamma(\frac{1}{6})}{\Gamma(\frac{2}{3})}\right]^{3}\;.$
(B.7)

Changing to the weight $G\mathcal{D}(x)$ via dividing by $s=2\pi$ for each of
the three remaining propagators, see the explanation in the paragraph
following (B.4), gives the critical coupling

$g_{\mathrm{cr}}^{D=3}~{}=~{}\left(\mathbbm{K}^{G\mathcal{D},D=3}\right)^{-1}~{}=~{}\left(\frac{\mathbbm{K}^{G,D=3}}{s^{3}}\right)^{-1}~{}=~{}\left[12\sqrt{\pi}\;\frac{\Gamma(\frac{2}{3})}{\Gamma(\frac{1}{6})}\right]^{3}~{}=~{}\frac{1}{c_{3}}\;.$
(B.8)

This is the original result by Zamolodchikov [2], which was reproduced by TBA
in [38].

  * •

In four dimensions, the Feynman graphs under consideration have four-valent
vertices. A theory producing them is the bi-scalar fishnet theory [11]. The
corresponding graph building kernel solely consists of scalar propagators in
four dimensions with weight $u=1$ and takes the form

$\mathbbm{R}^{D=4}~{}:=~{}\mathbbm{R}_{0,0}\left(\begin{smallmatrix}1&1\\\
1&1\end{smallmatrix}\right)~{}=~{}\scalebox{0.6}[0.6]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\;,$
(B.9)

and the ratio of graph building kernels to vertices is $b=\frac{1}{2}$. The
free energy corresponding to the scalar canonical weight (3.1) is

$\mathbbm{K}^{D=4}~{}:=~{}\mathbbm{K}_{0,0}\left(\begin{smallmatrix}1&1\\\
1&1\end{smallmatrix}\right)~{}=~{}\kappa_{0}\left(1\right)^{4}~{}=~{}\left[\frac{\pi^{3/2}}{4}\frac{\Gamma\left(\frac{1}{4}\right)}{\Gamma\left(\frac{3}{4}\right)}\right]^{4}\;.$
(B.10)

Transitioning to the weights $G_{\alpha}(x)$ by (B.3), we note that
$\alpha=\frac{\pi}{2}$ for $D=4$ such that we find

$\mathbbm{K}^{G,D=4}~{}=~{}\frac{1}{Z(\frac{\pi}{2})^{4}}~{}=~{}\left[\frac{\sqrt{\pi}}{4}\frac{\Gamma\left(\frac{1}{4}\right)}{\Gamma\left(\frac{3}{4}\right)}\right]^{4}\;.$
(B.11)

In order to recover Zamolodchikov’s famous result [2], we need to once again
properly normalize the weight, see above. Choosing $s=4\pi$, we indeed find
the critical coupling to be

$g_{\mathrm{cr}}^{D=4}~{}=~{}\left(\mathbbm{K}^{G\mathcal{D},D=4}\right)^{-\frac{1}{2}}~{}=~{}\left(\frac{\mathbbm{K}^{G,D=4}}{s^{4}}\right)^{-\frac{1}{2}}~{}=~{}\left[16\sqrt{\pi}\;\frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}\right]^{2}~{}=~{}\frac{1}{c_{4}}\;.$
(B.12)

This result was also obtained by different techniques in [38].

  * •

The last class of graphs under consideration contain cubic interactions in
$D=6$. We propose the theory (6.1), first introduced in [34], which has these
graphs as their contribution to the free energy. Originally, they were coined
honeycomb graphs, but essentially they are the scalar versions of the brick
wall graphs considered above. Consequently, the graph building kernel has the
same shape as (4.1) but with $D=6$ scalar propagators of weight $u=2$,
therefore we write

$\mathbbm{R}^{D=6}~{}:=~{}\mathbbm{R}_{0,0}\left(\begin{smallmatrix}0&2\\\
2&2\end{smallmatrix}\right)~{}=~{}\scalebox{0.6}[0.6]{\leavevmode\hbox{$\vbox{\hbox{\resizebox{}{}{{\leavevmode\hbox{\set@color{}}}}}}$}}\;.$
(B.13)

We observe that the ratio $b$ is again $\frac{1}{2}$ and the canonical free
energy is found by (5.6) to be

$\mathbbm{K}^{D=6}~{}:=~{}\mathbbm{K}_{0,0}\left(\begin{smallmatrix}0&2\\\
2&2\end{smallmatrix}\right)~{}=~{}\kappa_{0}\left(0\right)\cdot\kappa_{0}\left(2\right)^{3}~{}=~{}\left[\frac{\pi^{5/2}}{6}\frac{\Gamma\left(\frac{1}{3}\right)}{\Gamma\left(\frac{5}{6}\right)}\right]^{3}\;.$
(B.14)

The free energy in terms of the weights $G_{\alpha}(x)$ can be computed via
(B.1) after recalling that the non-zero angle is $\alpha=\frac{\pi}{3}$ in
$D=6$. It yields

$\mathbbm{K}^{G,D=6}~{}=~{}\frac{1}{Z(\pi)Z(\frac{\pi}{3})^{3}}~{}=~{}\left[\frac{\sqrt{\pi}}{3}\frac{\Gamma\left(\frac{1}{3}\right)}{\Gamma\left(\frac{5}{6}\right)}\right]^{3}\;.$
(B.15)

Finally, we again turn to the weight $G\mathcal{D}(x)$ by dividing by $s=8\pi$
for each non-vanishing propagator in the graph building kernel. The critical
coupling related to this weight is then found to be

$g_{\mathrm{cr}}^{D=6}~{}=~{}\left(\mathbbm{K}^{G\mathcal{D},D=6}\right)^{-\frac{1}{2}}~{}=~{}\left(\frac{\mathbbm{K}^{G,D=6}}{s^{3}}\right)^{-\frac{1}{2}}~{}=~{}\left[24\sqrt{\pi}\;\frac{\Gamma(\frac{5}{6})}{\Gamma(\frac{1}{3})}\right]^{\frac{3}{2}}~{}=~{}\frac{1}{c_{6}}\;,$
(B.16)

which completes the reproduction of the results of [2].

## References

  * [1] N. Beisert et al., _Review of AdS/CFT Integrability: An Overview_ , Lett. Math. Phys. 99 (2012), 3–32, arXiv:1012.3982 [hep-th].
  * [2] A. B. Zamolodchikov, _’Fishnet’ diagrams as a completely integrable system_ , Phys. Lett. B 97 (1980), 63–66.
  * [3] M. D’Eramo, G. Parisi, and L. Peliti, _Theoretical predictions for critical exponents at the lambda point of bose liquids_ , Lett. Nuovo Cim. 2 (1971), no. 17, 878–880.
  * [4] D. I. Kazakov, _Calculation of Feynman diagrams by the “uniqueness” method_ , Teoreticheskaya i Matematicheskaya Fizika 58 (1984), no. 3, 343–353, https://link.springer.com/article/10.1007/BF01018044.
  * [5] Y. Stroganov, _A new calculation method for partition functions in some lattice models_ , Physics Letters A 74 (1979), no. 1, 116–118, https://www.sciencedirect.com/science/article/pii/0375960179906017.
  * [6] R. J. Baxter, _The inversion relation method for some two-dimensional exactly solved models in lattice statistics_ , J. Statist. Phys. 28 (1982), 1–41.
  * [7] R. J. Baxter, _Solving models in statistical mechanics_ , Adv. Stud. Pure Math. 19 (1989), 95.
  * [8] R. J. Baxter, _The ’Inversion relation’ method for obtaining the free energy of the chiral Potts model_ , Physica A 322 (2003), 407–431, cond-mat/0212075.
  * [9] S. V. Pokrovsky and Y. A. Bashilov, _Star-triangle relations in the exactly solvable statistical models_ , Communications in Mathematical Physics 84 (1982), no. 1, 103–132, https://link.springer.com/article/10.1007/BF01208374.
  * [10] M. Bousquet-Melou, A. J. Guttmann, W. P. Orrick, and A. Rechnitzer, _Inversion relations, reciprocity and polyominoes_ , 1999.
  * [11] O. Gürdoğan and V. Kazakov, _New Integrable 4D Quantum Field Theories from Strongly Deformed Planar $\mathcal{N}=$ 4 Supersymmetric Yang-Mills Theory_, Phys. Rev. Lett. 117 (2016), no. 20, 201602, arXiv:1512.06704 [hep-th], [Addendum: Phys.Rev.Lett. 117, 259903 (2016)].
  * [12] N. Beisert and R. Roiban, _Beauty and the twist: The Bethe ansatz for twisted N=4 SYM_ , JHEP 08 (2005), 039, hep-th/0505187.
  * [13] J. a. Caetano, O. Gürdoğan, and V. Kazakov, _Chiral limit of $\mathcal{N}$ = 4 SYM and ABJM and integrable Feynman graphs_, JHEP 03 (2018), 077, arXiv:1612.05895 [hep-th].
  * [14] A. Pittelli and M. Preti, _Integrable fishnet from $\gamma$-deformed $\mathcal{N}=2$ quivers_, Phys. Lett. B 798 (2019), 134971, arXiv:1906.03680 [hep-th].
  * [15] D. Chicherin, V. Kazakov, F. Loebbert, D. Müller, and D.-l. Zhong, _Yangian Symmetry for Fishnet Feynman Graphs_ , Phys. Rev. D 96 (2017), no. 12, 121901, arXiv:1708.00007 [hep-th].
  * [16] D. Chicherin, S. Derkachov, and A. P. Isaev, _Conformal group: R-matrix and star-triangle relation_ , JHEP 04 (2013), 020, arXiv:1206.4150 [math-ph].
  * [17] N. Gromov, V. Kazakov, and G. Korchemsky, _Exact Correlation Functions in Conformal Fishnet Theory_ , JHEP 08 (2019), 123, arXiv:1808.02688 [hep-th].
  * [18] N. Gromov, V. Kazakov, G. Korchemsky, S. Negro, and G. Sizov, _Integrability of Conformal Fishnet Theory_ , JHEP 01 (2018), 095, arXiv:1706.04167 [hep-th].
  * [19] C. Sieg and M. Wilhelm, _On a CFT limit of planar $\gamma_{i}$-deformed $\mathcal{N}=4$ SYM theory_, Phys. Lett. B 756 (2016), 118–120, arXiv:1602.05817 [hep-th].
  * [20] D. Grabner, N. Gromov, V. Kazakov, and G. Korchemsky, _Strongly $\gamma$-Deformed $\mathcal{N}=4$ Supersymmetric Yang-Mills Theory as an Integrable Conformal Field Theory_, Phys. Rev. Lett. 120 (2018), no. 11, 111601, arXiv:1711.04786 [hep-th].
  * [21] V. Kazakov, E. Olivucci, and M. Preti, _Generalized fishnets and exact four-point correlators in chiral CFT 4_, JHEP 06 (2019), 078, arXiv:1901.00011 [hep-th].
  * [22] S. Derkachov and E. Olivucci, _Exactly solvable single-trace four point correlators in $\chi$CFT4_, JHEP 02 (2021), 146, arXiv:2007.15049 [hep-th].
  * [23] C. Ahn and M. Staudacher, _The Integrable (Hyper)eclectic Spin Chain_ , JHEP 02 (2021), 019, arXiv:2010.14515 [hep-th].
  * [24] J. Fokken, C. Sieg, and M. Wilhelm, _Non-conformality of ${{\gamma}_{i}}$-deformed N = 4 SYM theory_, J. Phys. A 47 (2014), 455401, arXiv:1308.4420 [hep-th].
  * [25] V. Kazakov and E. Olivucci, _The loom for general fishnet CFTs_ , JHEP 06 (2023), 041, arXiv:2212.09732 [hep-th].
  * [26] E. Pomoni and L. Rastelli, _Large N Field Theory and AdS Tachyons_ , JHEP 04 (2009), 020, arXiv:0805.2261 [hep-th].
  * [27] S. Derkachov and E. Olivucci, _Conformal quantum mechanics & the integrable spinning Fishnet_, JHEP 11 (2021), 060, arXiv:2103.01940 [hep-th].
  * [28] M. Preti, _STR: a Mathematica package for the method of uniqueness_ , Int. J. Mod. Phys. C 31 (2020), no. 10, 2050146, arXiv:1811.04935 [hep-th].
  * [29] H. Au-Yang and J. H. H. Perk, _The Large N limits of the chiral Potts model_ , Physica A 268 (1999), 175–206, math/9906029.
  * [30] V. V. Bazhanov, A. P. Kels, and S. M. Sergeev, _Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs_ , J. Phys. A 49 (2016), no. 46, 464001, arXiv:1602.07076 [math-ph].
  * [31] D. Bombardelli, _S-matrices and integrability_ , J. Phys. A 49 (2016), no. 32, 323003, arXiv:1606.02949 [hep-th].
  * [32] R. Shankar and E. Witten, _The S Matrix of the Supersymmetric Nonlinear Sigma Model_ , Phys. Rev. D 17 (1978), 2134.
  * [33] A. B. Zamolodchikov, _Exact S Matrix of Quantum Sine-Gordon Solitons_ , JETP Lett. 25 (1977), 468.
  * [34] O. Mamroud and G. Torrents, _RG stability of integrable fishnet models_ , JHEP 06 (2017), 012, arXiv:1703.04152 [hep-th].
  * [35] B. Basso and L. J. Dixon, _Gluing Ladder Feynman Diagrams into Fishnets_ , Phys. Rev. Lett. 119 (2017), no. 7, 071601, arXiv:1705.03545 [hep-th].
  * [36] N. Gromov and A. Sever, _Derivation of the Holographic Dual of a Planar Conformal Field Theory in 4D_ , Phys. Rev. Lett. 123 (2019), no. 8, 081602, arXiv:1903.10508 [hep-th].
  * [37] R. M. Iakhibbaev and D. M. Tolkachev, _Generalising holographic fishchain_ , 2023.
  * [38] B. Basso and D.-l. Zhong, _Continuum limit of fishnet graphs and AdS sigma model_ , JHEP 01 (2019), 002, arXiv:1806.04105 [hep-th].