Nikhef-2021-006, LTH-1254

# Five loop renormalization of $\phi^{3}$ theory with applications to the
Lee–Yang edge singularity and percolation theory

M. Borinsky,a J. A. Gracey,b M. V. Kompanietsc and O. Schnetzd

(aNikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands
bTheoretical Physics Division, Department of Mathematical Sciences,
University of Liverpool, P.O. Box 147, Liverpool, L69 3BX, United Kingdom
cSt. Petersburg State University, 7/9 Universitetskaya nab.,
St. Petersburg 199034, Russia
dDepartment Mathematik, Friedrich Alexander Universität Erlangen-Nürnberg,
Cauerstraße 11, 91058 Erlangen, Germany)

###### Abstract

We apply the method of graphical functions that was recently extended to six
dimensions for scalar theories, to $\phi^{3}$ theory and compute the $\beta$
function, the wave function anomalous dimension as well as the mass anomalous
dimension in the $\overline{\mbox{MS}}$ scheme to five loops. From the results
we derive the corresponding renormalization group functions for the Lee–Yang
edge singularity problem and percolation theory. After determining the
$\varepsilon$ expansions of the respective critical exponents to
$\mathcal{O}(\varepsilon^{5})$ we apply recent resummation technology to
obtain improved exponent estimates in $3$, $4$ and $5$ dimensions. These
compare favourably with estimates from fixed dimension numerical techniques
and refine the four loop results. To assist with this comparison we collated a
substantial amount of data from numerical techniques which are included in
tables for each exponent.

##  1 Introduction

One of the core quantum field theories with applications to various areas of
physics is that of a scalar field with a cubic self-interaction. It has many
interesting properties. For instance, it is known to be asymptotically free in
six dimensions [1, 2] which is also its critical dimension. As such it offered
a much simpler forum to study this property rather than in the more
complicated non-abelian gauge field theory underlying the strong interactions
which also has this property. Another major area where scalar $\phi^{3}$
theory has important consequences is that of condensed matter physics. For
example, the non-unitary version of the model, [3], describes the phase
transitions of the _Lee–Yang edge singularity problem_. In particular the
critical exponents computed in the cubic theory produce estimates which are
not out of line with those of other methods, [4, 5]. Having an accurate
estimate for the exponent $\sigma$ for the Lee–Yang edge singularity is
important in lattice gauge theory studies of Quantum Chromodynamics (QCD), [6,
7]. Specifically it governs the analytic behaviour of the partition function
in the smooth chiral symmetry crossover when there is a non-zero chemical
potential. The latter is included in QCD as an purely imaginary parameter that
leads to loss of unitarity. Though for the region of application in [6, 7]
physical meaningful results can still be extracted. In addition endowing the
cubic model with specific symmetries means it can also describe phase
transitions in _percolation problems_. This follows from taking the replica
limit of the $(N+1)$-state Potts model [8]. The critical dynamics of the phase
transition in percolation has been widely studied using Monte Carlo or series
methods, on discrete or spin systems. References for this comprehensive body
of work will be provided in later sections. Indeed such analyses proceed over
a range of integer dimensions from $2$ to $6$ inclusive where the latter would
correspond to the mean field approximation given its relation to the critical
dimension. The relation of the discrete percolation theory models to that of a
continuum field theory resides in the fact that at the phase transition both
scalar $\phi^{3}$ theory and the spin models lie in the same universality
class. What differs of course in both approaches are the techniques used to
estimate the physically measurable critical exponents. On the continuum
quantum field theory side these are _renormalization group invariants_ that
are determined from high perturbative loop order renormalization group
functions. While these functions are scheme dependent, the critical exponents
at the $d$ dimensional Wilson–Fisher fixed point, [9], are scheme independent.
In more recent years other continuum field theory techniques have been
developed. Two of the main ones are the functional renormalization group and
the modern manifestation of the conformal bootstrap and applied to the
Lee–Yang and percolation problem for example in [10, 11, 12] and [13, 14, 15]
respectively.

In terms of basic scalar $\phi^{3}$ theory the multiloop renormalization of
the model has proceeded in stages over the last half century or so. The one
and two loop renormalization group functions were determined in [1]. This was
extended to the $\operatorname{O}(N)$ group in [2] where the leading order
value of $N$ for the existence of the conformal window was determined. The
extension to three loops for both the Lee–Yang edge singularity and
percolation problems was carried out in [4, 5]. From the point of view of
hindsight that computation was well ahead of its time given the difficulty of
several of the three loop vertex graphs that needed to be evaluated. Moreover,
estimates for the exponents in the percolation and Lee–Yang problems were
extracted in dimensions $d$ in the range $2$ $<$ $d$ $<$ $6$ that were
competitive with other results available then. To achieve a high degree of
accuracy the analysis benefited from improved resummation techniques such as
Padé approximants and Borel transformations, where in the latter conformal
mappings were applied and the behaviour of the $\varepsilon$ expansion was
incorporated. Thereafter progress in systematically renormalizing theories to
higher loop order was hindered in general by a lack of technology to push to
four loops. With the development of Laporta’s integration by parts algorithm,
[16], and its implementation within a variety of publicly available packages,
the four loop renormalization of $\phi^{3}$ was carried out in [17]. That
article covered a range of applications to various problems that had emerged
in the interim. For instance, in recent years it has been shown that there is
a connection of $\phi^{3}$ theory with dualities in higher spin AdS/CFTs, [18,
19]. This generated an interest in understanding the conformal window of
$\phi^{3}$ theories for various symmetry groups such as $\operatorname{O}(N)$
and $\operatorname{Sp}(N)$, [20, 21, 22]. One highlight of the four loop
result of [17] was the improvement in estimates for the Lee–Yang and
percolation theory exponents. What this analysis benefited immensely from was
the progress in classifying two dimensional conformal field theories in the
years after [4, 5]. Specifically the values of the exponents of each problem
in two dimensions were found exactly. For instance, the exponent $\sigma$ in
one and two dimensions was determined exactly in [23]. For percolation theory
the two dimensional values can be derived from the unitary minimal conformal
field theories of [24] when $m$ $=$ $2$ and the central charge is $c$ $=$ $0$.
Therefore it was possible to use that data, together with hyperscaling
relations, to construct constrained Padé approximants motivated by the
application of this idea given in [25]. The upshot was that exponent estimates
based on four loop results for three dimensions were within one standard
deviation of Monte Carlo and series results. This is all the more remarkable
when one recalls this is the resummation of a series that is more reliable
near six dimensions down to three dimensions. We note that for brevity we will
refer to results from non-continuum field theories as Monte Carlo but this
will cover those from series methods too. We note that in this respect we have
compiled exponent estimates from as many sources as we could. These will also
include strong coupling methods, functional renormalization group techniques
and several specialized approaches. An excellent live source for percolation
exponents is [26].

While the Laporta approach has revolutionized our ability to extend results
for many theories to loop orders that what would have been impossible to
conceive of a decade ago, one is always interested in going beyond even those
orders. In the short term any such developments have to proceed in simpler
theories. Indeed this has been the case for scalar $\phi^{4}$ theory which was
renormalized at five loops in the $90$’s in [27, 28] but not extended to six
loops until around a quarter of a century later, [29, 30]. Moreover the latter
article [30] contained a most comprehensive analysis of the resummation of
exponents using the asymptotic properties of the series. Such an analysis was
much-needed after the huge jump in precision. Indeed it revisited the
assumptions made in earlier approaches in the literature. Subsequent to [29,
30] a novel method was applied to scalar $\phi^{4}$ theory in [31] which is
termed graphical functions. This extended the renormalization group functions
to the staggeringly high seven loop order111The eight loop evaluation of the
field anomalous dimension is in progress [32].. One highlight was the
appearance of a new period in the $\beta$ function which is conjectured to not
be expressible in terms of a multiple zeta value (MZV), [33, 31]. En route
expectations concerning the non-MZV content predicted in [34] at high loop
order were confirmed.

One lesson from [31] was the potential usefulness of the graphical function
technique to extend the renormalization group functions. An obstruction that
limited the technique to four dimensional problems was overcome by the first
and fourth author, who extended the method to arbitrary even dimensions. The
details of this extension will be given elsewhere [35]. This availability of
graphical functions in arbitrary even dimensions immediately opened up new
possibilities for the computation of the renormalization group functions in
$\phi^{3}$ theory in six dimensions. In this article we make use of this new
tool and provide the renormalization group functions of $\phi^{3}$ theory up
to five loops. Consequently we will find the next term in the $\varepsilon$
expansion of the critical exponents for the Lee–Yang singularity and
percolation problems. This will form the first part of the article. The second
part will be devoted to a comprehensive resummation analysis of the respective
critical exponents from the $\varepsilon$ expansion in the region between $2$
and $6$ dimensions. This resummation analysis will be based on technology from
[30] but combined with constraints from two dimensional theories as in [17].
Thanks to the additional perturbative order and the more advanced resummation
technology we are able to improve on the estimates obtained in [17]. In broad
terms our estimates of the critical exponents are consistent with Monte Carlo
and series results which have been similarly refined in recent years. In order
to make this comparison of our five loop estimates with numerical data we have
carried out an analysis on all exponent estimates in the literature that had
error bars and produced a global average.

The paper is organized as follows. In Section 2 we introduce the scalar cubic
theory for the cases where there is one scalar field and the Lagrangian which
is relevant for the percolation problem. The core machinery of the graphical
function formalism used to compute the five loop renormalization group
functions is discussed in depth in Section 3. The outcome of this mammoth task
is provided via the explicit five loop expressions for the $\beta$ function,
field anomalous dimension and the mass anomalous dimension in Section 4 for
both the Lee–Yang and percolation problems. As a corollary the $\varepsilon$
expansion of the three corresponding critical exponents are determined to
$\mathcal{O}(\varepsilon^{5})$ too. In the subsequent section we introduce and
discuss aspects of the different resummation methods we use to extract
exponent estimates. Then Section 6 will provide the full consequences of that
analysis for both the Lee–Yang problem and percolation theory. A central
element of this section is the compilation of tables for each exponent for
both problems. Each table provides comprehensive data of fixed dimension
exponent estimates as well as the outcome of each of our resummations. An
overall average is provided for each dimension for both Monte Carlo data and
our five loop results in order to have a common comparison point. Concluding
remarks are provided in Section 7. Throughout the article we will in general
follow the notation of [36].

##  2 Background

We begin by outlining the essential background to the problem of extracting
critical exponents for both the Lee–Yang and percolation problems. First the
action for the basic cubic theory is

$S(\phi_{0})~{}=~{}-~{}\int\text{d}^{d}x\left[\frac{1}{2}(\partial\phi_{0})^{2}+\frac{1}{2}m_{0}^{2}\phi_{0}^{2}+\frac{g_{0}}{3!}\phi_{0}^{3}\right]~{},$
(1)

where $\phi_{0}$, $m_{0}$ and $g_{0}$ are the bare field, mass and coupling
constant. It is this version of the theory that was shown to be asymptotically
free in six dimensions, [1]. The connection to the action underlying the
Lee–Yang edge singularity problem is given by the coupling constant mapping
$g_{0}$ $\to$ $ig_{0}$, which yields a _non-unitary theory_ , [3]. The
analogous action for percolation theory requires $N$ fields $\phi_{i}$ prior
to taking the replica limit defined formally by $N$ $\to$ $0$. First we note
that the most general renormalizable $\phi^{3}$ theory type action in six
dimensions is

$S(\phi_{0})~{}=~{}-~{}\int\text{d}^{d}x\left[\frac{1}{2}(\partial\phi_{0i})^{2}+\frac{1}{2}m_{0}^{2}\phi_{0i}^{2}+\frac{g_{0}}{3!}d_{ijk}\phi_{0i}\phi_{0j}\phi_{0k}\right]~{},$
(2)

where the indices run from $1$ to $N$. Specific values for the fully symmetric
tensor $d_{ijk}$ correspond to different problems and eventually lead to
different prefactors which multiply the individual scalar Feynman integrals in
the respective perturbative expansions. For percolation theory the $d_{ijk}$
tensor is related to the vertices of an $N$-dimensional tetrahedron, [37], and
corresponds to the $(N+1)$-state Potts model, [38]. In particular $d_{ijk}$ is
given by

$d_{ijk}=\sum_{\alpha=1}^{N+1}e_{\alpha}^{i}e_{\alpha}^{j}e_{\alpha}^{k}~{},$
(3)

where the $N$-dimensional vectors $e_{1}^{i},\ldots,e_{N+1}^{i}$ satisfy the
algebra, [37],

$\sum_{\alpha=1}^{N+1}e_{\alpha}^{i}=0~{},\quad\sum_{\alpha=1}^{N+1}e_{\alpha}^{i}e_{\alpha}^{j}=(N+1)\delta^{ij}~{},\quad\sum_{i=1}^{N}e_{\alpha}^{i}e_{\beta}^{i}=(N+1)\delta_{\alpha\beta}-1~{}.$
(4)

In previous calculations [5, 17] these algebraic rules were used to compute
the $N$ dependence of each individual graph. For instance in [5] the
renormalization group functions were written in terms of invariants which
corresponded with the primitively divergent Feynman integrals. Therefore the
algebra was used to determine the $N$ dependence of the invariants as well as
their subsequent value in the replica limit. Here we use a presumably new
diagrammatic approach which is based on a similar one given in [5] to
calculate the necessary prefactors for each individual graph. The details of
this approach will be given in Section 3.3.

For both the Lee–Yang and the percolation problem we renormalize in the
modified minimal subtraction ($\overline{\mbox{MS}}$) scheme in the
dimensionally regularized theory in $d$ $=$ $6$ $-$ $\varepsilon$ which
retains multiplicative renormalizability222The three loop renormalization
group calculations [4, 5] were also performed in $d=6-\varepsilon$, while the
most recent four loop computations [17] used $d=6-2\varepsilon$, which is more
common in high energy physics.. This means that the number of independent
renormalization constants equates to the number of terms in each action. Thus
the renormalized action in terms of renormalized entities for the general
theory is

$S^{R}(\phi)~{}=~{}-~{}\int\text{d}^{d}x\left[\frac{1}{2}Z_{1}(\partial\phi_{i})^{2}+\frac{1}{2}Z_{2}m^{2}\phi_{i}^{2}+\frac{g\mu^{\varepsilon/2}}{3!}Z_{3}d_{ijk}\phi_{i}\phi_{j}\phi_{k}\right]~{},$
(5)

where

$\displaystyle\begin{aligned}
\phi_{0i}&=\phi_{i}Z_{\phi}~{},&&&m_{0}^{2}&=m^{2}Z_{m^{2}}~{},&&&g_{0}&=g\mu^{\varepsilon/2}Z_{g},\\\
Z_{1}&=Z_{\phi}^{2}~{},&&&Z_{2}&=Z_{m^{2}}Z_{\phi}^{2}~{},&&&Z_{3}&=Z_{g}Z_{\phi}^{3}~{},\end{aligned}$
(6)

and we have defined the field renormalization constant in the way which is
more common in statistical physics. We highlight this explicitly in
contradistinction to the high energy physics convention where it is usually
defined as “$\phi_{0i}$ $=$ $\phi_{i}Z_{\phi}^{1/2}\,$”. As we use the
$\overline{\mbox{MS}}$ scheme we recall that the coupling constant and
$\varepsilon$ dependence of the renormalization constants $Z_{n}$ is given by

$Z_{n}=Z_{n}(g,\varepsilon)=1+\sum_{k=1}^{\infty}\frac{Z_{n,k}(g)}{\varepsilon^{k}}~{}.$
(7)

These are determined perturbatively by requiring the finiteness of the
renormalized 1-PI Green’s functions
$\Gamma^{R}_{n}(g,m,\mu)=\left(Z_{\phi}\right)^{n}\Gamma_{n}(g_{0},m_{0})$.
The conventional method to ensure finiteness is to use the Bogoliubov–Parasiuk
$R$-operation [39, 40], as well as the $R^{*}$-operation [41, 42, 43, 44].
These allow for the use of infrared regularization or infrared rearrangement
for Feynman diagrams on an individual basis. This significantly simplifies the
calculations. In this article however we will use the much more powerful
technique of graphical functions to calculate the divergences of all the
diagrams. This will be discussed in the next section. One major advantage of
that approach is that it enables us to compute all diagrams in a
straightforward way without any infrared rearrangement and
$R/R^{*}$-operations. The only simplification we use is to consider purely
massless graphs throughout. To extract the mass renormalization constant we
evaluate a $2$-point Green’s function with the mass operator inserted at zero
momentum.

Once we have extracted the renormalization constants to five loops the next
stage is to produce the corresponding renormalization group functions
$\beta(g)$, $\gamma_{\phi}(g)$ and $\gamma_{m^{2}}(g)$ which are defined by

$\displaystyle\begin{gathered}\beta(g):=\mu\frac{\partial
g}{\partial\mu}\bigg{|}_{g_{0}}=\frac{-\varepsilon
g/2}{1+g{\partial_{g}}\ln{(Z_{g})}}~{},\quad\gamma_{\phi}(g):=\mu\frac{\partial}{\partial\mu}\ln(Z_{\phi})\bigg{|}_{g_{0}}=\beta(g)\frac{\partial}{\partial
g}\ln(Z_{\phi})~{},\\\
\gamma_{m^{2}}(g):=\mu\frac{\partial}{\partial\mu}\ln(Z_{m^{2}})\bigg{|}_{g_{0}}=\beta(g)\frac{\partial}{\partial
g}\ln(Z_{m^{2}})~{}.\end{gathered}$ (10)

These ensure that after renormalization all finite renormalized $n$-point 1-PI
Green’s functions $\Gamma_{n}^{R}$ will satisfy the renormalization group
equation

$\left[\mu\partial_{\mu}+\beta(g)\partial_{g}-\gamma_{m^{2}}(g)m^{2}\partial_{m^{2}}-n\gamma_{\phi}(g)\right]\Gamma_{n}^{R}=0~{},$
(11)

where $\partial_{g}$ $=$ $\frac{\partial}{\partial g}$ for instance. Once the
renormalization group functions have been established, they lay the foundation
for our application to critical exponents. In general if there is a non-
trivial infrared fixed point of the $\beta$ function at $g^{\ast}$ with
$\beta(g^{\ast})$ $=$ $0$ then in the limit $g$ $\to$ $g^{\ast}$ eq. (11)
transforms to an equation describing critical scaling with exponents
$\gamma_{\phi}^{\ast}$ $=$ $\gamma_{\phi}(g^{\ast})$ and
$\gamma_{m^{2}}^{\ast}$ $=$ $\gamma_{m^{2}}(g^{\ast})$. In addition the
correction to scaling exponent $\beta^{\prime\,\ast}$ $=$
$\partial_{g}\beta(g)|_{g=g^{*}}$ corresponding to the $\beta$ function slope
at criticality will be of interest. In terms of the notation used in
statistical physics the connection between these critical point
renormalization group functions and the critical exponents is

$\displaystyle\begin{gathered}\eta~{}=~{}2\gamma_{\phi}^{*}~{},\quad
1/\nu~{}=~{}2+\gamma_{m^{2}}^{*}~{}=~{}\eta_{O}-\eta+2~{},\quad\omega~{}=~{}\beta^{\prime\,\ast}~{},\end{gathered}$
(13)

where $\eta_{O}$ is the anomalous dimension derived from $Z_{2}$ evaluated at
$g^{\ast}$ and $\nu$ corresponds to the correlation length exponent. Knowledge
of the two basic exponents $\eta$ and $\nu$ means that other critical
exponents can be accessed via hyperscaling relations. These are given by

$\displaystyle\begin{gathered}\alpha~{}=~{}2~{}-~{}d\nu~{},\quad\beta~{}=~{}\frac{1}{2}(d-2+\eta)\nu~{},\quad\gamma~{}=~{}(2-\eta)\nu~{},\quad\delta~{}=~{}\frac{d+2-\eta}{d-2+\eta}~{},\\\
\sigma~{}=~{}\frac{2}{(d+2-\eta)\nu}~{},\quad\tau~{}=~{}1~{}+~{}\frac{2d}{d+2-\eta}~{},\quad\Omega~{}=~{}\frac{2\omega}{d+2-\eta}~{}.\end{gathered}$
(16)

These will be our main focus for the percolation problem. For the Lee–Yang
problem we will concentrate on $\eta$, $\nu$, $\sigma$ and $\omega$
specifically.

##  3 Computational technique

###  3.1 Graphical Functions

For the evaluation of the required Feynman integrals, we made heavy use of the
graphical function technique that has been introduced in $d=4$ by the fourth
author in [45], extended to $d=4-\varepsilon$ in [31] and recently generalized
to all even dimensions $\geq 4$ by the first and the fourth author [35].

Recall that the massless scalar propagator in $d$-dimensional Euclidean space
time is the Green’s function for the respective massless Klein–Gordon
equation,

$\displaystyle\Box_{\boldsymbol{x}}\frac{1}{|\boldsymbol{x}-\boldsymbol{y}|^{d-2}}=-\frac{4}{\Gamma(d/2-1)}\delta^{(d)}(\boldsymbol{x}-\boldsymbol{y}),$
(17)

where $\Gamma(x)=\int_{0}^{\infty}t^{x-1}\exp(-t)\text{d}t$ is the gamma
function. Up to a rescaling and a reparametrization which will be specified
later, a graphical function is an Euclidean massless position space three-
point correlation function which can be written as an integral over a product
of such propagators,

$\displaystyle
G_{\Gamma}(\boldsymbol{x}_{a},\boldsymbol{x}_{b},\boldsymbol{x}_{c})=\left(\prod_{v\in
V_{\Gamma}^{\text{int}}}\int_{\mathbb{R}^{d}}\frac{\text{d}\boldsymbol{x}_{v}}{\pi^{d/2}}\right)\prod_{\\{v,w\\}\in
E_{\Gamma}}\frac{1}{|\boldsymbol{x}_{v}-\boldsymbol{x}_{w}|^{d-2}}.$ (18)

It is determined by a graph $\Gamma$ with edges $E_{\Gamma}$, internal
vertices in $V_{\Gamma}^{\text{int}}$ and external vertices
$\\{a,b,c\\}=V_{\Gamma}^{\text{ext}}$ such that $V_{\Gamma}^{\text{int}}\cap
V_{\Gamma}^{\text{ext}}=\emptyset$. Note that in our position space setting,
external vertices can have any number of incident edges. Such a three-point
function $G_{\Gamma}$ has translation, rotation and scaling symmetries, such
that for all
$\boldsymbol{x}_{a},\boldsymbol{x}_{b},\boldsymbol{x}_{c}\in\mathbb{R}^{d}$,

$\displaystyle
G_{\Gamma}(\boldsymbol{x}_{a},\boldsymbol{x}_{b},\boldsymbol{x}_{c})$
$\displaystyle=G_{\Gamma}(\boldsymbol{x}_{a}+\boldsymbol{v},\boldsymbol{x}_{b}+\boldsymbol{v},\boldsymbol{x}_{c}+\boldsymbol{v})$
$\displaystyle\text{for all }\boldsymbol{v}\in\mathbb{R}^{d}$
$\displaystyle=G_{\Gamma}(\Lambda\boldsymbol{x}_{a},\Lambda\boldsymbol{x}_{b},\Lambda\boldsymbol{x}_{c})$
$\displaystyle\text{for all }\Lambda\in\operatorname{SO}(d)$ (19)
$\displaystyle=\xi^{-\Delta_{\Gamma}}G_{\Gamma}(\xi\boldsymbol{x}_{a},\xi\boldsymbol{x}_{b},\xi\boldsymbol{x}_{c})$
$\displaystyle\text{for all }\xi\in\mathbb{R}_{>0},$

with the _superficial degree of divergence_ given by

$\Delta_{\Gamma}=d|V_{\Gamma}^{\text{int}}|-(d-2)|E_{\Gamma}|.$

It follows that two degrees of freedom together with $\Delta_{\Gamma}$ are
sufficient to parameterize the three-point function $G_{\Gamma}$. A convenient
parameterization is given by a single complex variable $z\in\mathbb{C}$,

$\displaystyle z{\overline{z}}$
$\displaystyle=\frac{\boldsymbol{x}_{ac}^{2}}{\boldsymbol{x}_{ab}^{2}},$
$\displaystyle(1-z)(1-{\overline{z}})$
$\displaystyle=\frac{\boldsymbol{x}_{bc}^{2}}{\boldsymbol{x}_{ab}^{2}},$ (20)

where $\boldsymbol{x}_{ij}=\boldsymbol{x}_{j}-\boldsymbol{x}_{i}$. Using this
parameterization, we can write $G_{\Gamma}$ as

$\displaystyle
G_{\Gamma}(\boldsymbol{x}_{a},\boldsymbol{x}_{b},\boldsymbol{x}_{c})=|\boldsymbol{x}_{ab}|^{\Delta_{\Gamma}}f_{\Gamma}(z),$
(21)

where $f_{\Gamma}:\mathbb{C}\rightarrow\mathbb{R}$, the _graphical function_ ,
is independent of the overall scale.

An important feature is that $f_{\Gamma}(z)$ is a _single-valued_ real
analytic function on $\mathbb{C}\backslash\\{0,1\\}$, [46]. Furthermore, a
graphical function admits expansions of $\log$-Laurent type at the singular
points $0$, $1$ and $\infty$, [45, 35],

$\displaystyle
f_{\Gamma}(z)=\sum_{\ell=0}^{L_{a}}\sum_{m,\overline{m}=M_{a}}^{\infty}c^{a}_{\ell,m,\overline{m}}[\log(z-a)({\overline{z}}-a)]^{\ell}(z-a)^{m}({\overline{z}}-a)^{\overline{m}},\quad|z-a|<1,\quad
a\in\\{0,1\\}$ (22) $\displaystyle
f_{\Gamma}(z)=\sum_{\ell=0}^{L_{\infty}}\sum_{m,\overline{m}=-\infty}^{M_{\infty}}c^{\infty}_{\ell,m,\overline{m}}[\log(z{\overline{z}})]^{\ell}z^{m}{\overline{z}}^{\overline{m}},\quad|z|>1.$
(23)

Due to the existence of the expansion at infinity in eq. (23) the graphical
function naturally lives on the Riemann sphere $\mathbb{C}\cup\\{\infty\\}$.

These fundamental structures of graphical functions are vital for the
efficiency of the graphical function method. After going from the
$\boldsymbol{x}_{a},\boldsymbol{x}_{b},\boldsymbol{x}_{c}\in\mathbb{R}^{d}$
coordinates to $z,{\overline{z}}$ via the parameterization in eq. (20), it is
convenient to rename the external vertices $a,b$ and $c$ to $0,1$ and $z$. The
reason for this is that eq. (20) effectively identifies the plane that is
spanned in $d$-dimensional space by the points
$\boldsymbol{x}_{a},\boldsymbol{x}_{b},\boldsymbol{x}_{c}$ with the complex
plane $\mathbb{C}$ such that $0$ is mapped to $\boldsymbol{x}_{a}$, $1$ is
mapped to $\boldsymbol{x}_{b}$ and $z$ to $\boldsymbol{x}_{c}$.

Graphical functions fulfill a large number of _combinatorial_ identities. We
can depict a general graph with the three labeled external vertices as

$\Gamma=\leavevmode\hbox to42.51pt{\vbox
to37.39pt{\pgfpicture\makeatletter\hbox{\hskip
12.81747pt\lower-18.69687pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.48447pt}{-2.15277pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$z$}}
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{21.358pt}{-15.36386pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$0$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{8.61108pt}{0.0pt}{11.10825pt}{6.02773pt}{17.22217pt}{12.14165pt}\pgfsys@curveto{11.10825pt}{6.02773pt}{11.10825pt}{-6.02773pt}{17.22217pt}{-12.14165pt}\pgfsys@curveto{11.10825pt}{-6.02773pt}{8.61108pt}{0.0pt}{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{0.55229pt}{0.55229pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-0.55229pt}{1.0pt}{-1.0pt}{0.55229pt}{-1.0pt}{0.0pt}\pgfsys@curveto{-1.0pt}{-0.55229pt}{-0.55229pt}{-1.0pt}{0.0pt}{-1.0pt}\pgfsys@curveto{0.55229pt}{-1.0pt}{1.0pt}{-0.55229pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{17.22217pt}{12.14165pt}\pgfsys@moveto{18.22217pt}{12.14165pt}\pgfsys@curveto{18.22217pt}{12.69394pt}{17.77446pt}{13.14165pt}{17.22217pt}{13.14165pt}\pgfsys@curveto{16.66988pt}{13.14165pt}{16.22217pt}{12.69394pt}{16.22217pt}{12.14165pt}\pgfsys@curveto{16.22217pt}{11.58936pt}{16.66988pt}{11.14165pt}{17.22217pt}{11.14165pt}\pgfsys@curveto{17.77446pt}{11.14165pt}{18.22217pt}{11.58936pt}{18.22217pt}{12.14165pt}\pgfsys@closepath\pgfsys@moveto{17.22217pt}{12.14165pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{17.22217pt}{-12.14165pt}\pgfsys@moveto{18.22217pt}{-12.14165pt}\pgfsys@curveto{18.22217pt}{-11.58936pt}{17.77446pt}{-11.14165pt}{17.22217pt}{-11.14165pt}\pgfsys@curveto{16.66988pt}{-11.14165pt}{16.22217pt}{-11.58936pt}{16.22217pt}{-12.14165pt}\pgfsys@curveto{16.22217pt}{-12.69394pt}{16.66988pt}{-13.14165pt}{17.22217pt}{-13.14165pt}\pgfsys@curveto{17.77446pt}{-13.14165pt}{18.22217pt}{-12.69394pt}{18.22217pt}{-12.14165pt}\pgfsys@closepath\pgfsys@moveto{17.22217pt}{-12.14165pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{},$

and identify the graph with its associated graphical function (as long as no
confusion is possible),

$f_{\Gamma}(z)=\left[\leavevmode\hbox to37.84pt{\vbox
to31.32pt{\pgfpicture\makeatletter\hbox{\hskip
12.60217pt\lower-15.66145pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.26917pt}{-2.15277pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$z$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.90175pt}{5.88402pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.90175pt}{-12.32845pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$0$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{6.45831pt}{0.0pt}{8.33119pt}{4.5208pt}{12.91663pt}{9.10623pt}\pgfsys@curveto{8.33119pt}{4.5208pt}{8.33119pt}{-4.5208pt}{12.91663pt}{-9.10623pt}\pgfsys@curveto{8.33119pt}{-4.5208pt}{6.45831pt}{0.0pt}{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{0.55229pt}{0.55229pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-0.55229pt}{1.0pt}{-1.0pt}{0.55229pt}{-1.0pt}{0.0pt}\pgfsys@curveto{-1.0pt}{-0.55229pt}{-0.55229pt}{-1.0pt}{0.0pt}{-1.0pt}\pgfsys@curveto{0.55229pt}{-1.0pt}{1.0pt}{-0.55229pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{12.91663pt}{9.10623pt}\pgfsys@moveto{13.91663pt}{9.10623pt}\pgfsys@curveto{13.91663pt}{9.65852pt}{13.46892pt}{10.10623pt}{12.91663pt}{10.10623pt}\pgfsys@curveto{12.36433pt}{10.10623pt}{11.91663pt}{9.65852pt}{11.91663pt}{9.10623pt}\pgfsys@curveto{11.91663pt}{8.55394pt}{12.36433pt}{8.10623pt}{12.91663pt}{8.10623pt}\pgfsys@curveto{13.46892pt}{8.10623pt}{13.91663pt}{8.55394pt}{13.91663pt}{9.10623pt}\pgfsys@closepath\pgfsys@moveto{12.91663pt}{9.10623pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{12.91663pt}{-9.10623pt}\pgfsys@moveto{13.91663pt}{-9.10623pt}\pgfsys@curveto{13.91663pt}{-8.55394pt}{13.46892pt}{-8.10623pt}{12.91663pt}{-8.10623pt}\pgfsys@curveto{12.36433pt}{-8.10623pt}{11.91663pt}{-8.55394pt}{11.91663pt}{-9.10623pt}\pgfsys@curveto{11.91663pt}{-9.65852pt}{12.36433pt}{-10.10623pt}{12.91663pt}{-10.10623pt}\pgfsys@curveto{13.46892pt}{-10.10623pt}{13.91663pt}{-9.65852pt}{13.91663pt}{-9.10623pt}\pgfsys@closepath\pgfsys@moveto{12.91663pt}{-9.10623pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
In this notation we have the following identities which can be used to add and
remove edges between external vertices:

$\displaystyle\left[\leavevmode\hbox to37.84pt{\vbox
to31.32pt{\pgfpicture\makeatletter\hbox{\hskip
12.60217pt\lower-15.66145pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.26917pt}{-2.15277pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$z$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.90175pt}{5.88402pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.90175pt}{-12.32845pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$0$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{6.45831pt}{0.0pt}{8.33119pt}{4.5208pt}{12.91663pt}{9.10623pt}\pgfsys@curveto{8.33119pt}{4.5208pt}{8.33119pt}{-4.5208pt}{12.91663pt}{-9.10623pt}\pgfsys@curveto{8.33119pt}{-4.5208pt}{6.45831pt}{0.0pt}{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{0.55229pt}{0.55229pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-0.55229pt}{1.0pt}{-1.0pt}{0.55229pt}{-1.0pt}{0.0pt}\pgfsys@curveto{-1.0pt}{-0.55229pt}{-0.55229pt}{-1.0pt}{0.0pt}{-1.0pt}\pgfsys@curveto{0.55229pt}{-1.0pt}{1.0pt}{-0.55229pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{12.91663pt}{9.10623pt}\pgfsys@moveto{13.91663pt}{9.10623pt}\pgfsys@curveto{13.91663pt}{9.65852pt}{13.46892pt}{10.10623pt}{12.91663pt}{10.10623pt}\pgfsys@curveto{12.36433pt}{10.10623pt}{11.91663pt}{9.65852pt}{11.91663pt}{9.10623pt}\pgfsys@curveto{11.91663pt}{8.55394pt}{12.36433pt}{8.10623pt}{12.91663pt}{8.10623pt}\pgfsys@curveto{13.46892pt}{8.10623pt}{13.91663pt}{8.55394pt}{13.91663pt}{9.10623pt}\pgfsys@closepath\pgfsys@moveto{12.91663pt}{9.10623pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{12.91663pt}{-9.10623pt}\pgfsys@moveto{13.91663pt}{-9.10623pt}\pgfsys@curveto{13.91663pt}{-8.55394pt}{13.46892pt}{-8.10623pt}{12.91663pt}{-8.10623pt}\pgfsys@curveto{12.36433pt}{-8.10623pt}{11.91663pt}{-8.55394pt}{11.91663pt}{-9.10623pt}\pgfsys@curveto{11.91663pt}{-9.65852pt}{12.36433pt}{-10.10623pt}{12.91663pt}{-10.10623pt}\pgfsys@curveto{13.46892pt}{-10.10623pt}{13.91663pt}{-9.65852pt}{13.91663pt}{-9.10623pt}\pgfsys@closepath\pgfsys@moveto{12.91663pt}{-9.10623pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right]$
$\displaystyle=\left[\leavevmode\hbox to37.84pt{\vbox
to31.32pt{\pgfpicture\makeatletter\hbox{\hskip
12.60217pt\lower-15.66145pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-9.26917pt}{-2.15277pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$z$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.90175pt}{5.88402pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.90175pt}{-12.32845pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$0$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{6.45831pt}{0.0pt}{8.33119pt}{4.5208pt}{12.91663pt}{9.10623pt}\pgfsys@curveto{8.33119pt}{4.5208pt}{8.33119pt}{-4.5208pt}{12.91663pt}{-9.10623pt}\pgfsys@curveto{8.33119pt}{-4.5208pt}{6.45831pt}{0.0pt}{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}{}{{}}{}{{}}{{{{}}{{}}{{}}{{}}{{}}}{{{{}}{}{}{}}}
}{{}{}}{{}} {}{}{}{{{}}{{}}{{}}} {{{}}{{}}{{}}}
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A permutation of the external vertices $0,1$ and $z$ corresponds to a specific
Möbius transformation of the $z$ variables together with an overall conformal
rescaling,

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{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{12.91663pt}{-9.10623pt}\pgfsys@moveto{13.91663pt}{-9.10623pt}\pgfsys@curveto{13.91663pt}{-8.55394pt}{13.46892pt}{-8.10623pt}{12.91663pt}{-8.10623pt}\pgfsys@curveto{12.36433pt}{-8.10623pt}{11.91663pt}{-8.55394pt}{11.91663pt}{-9.10623pt}\pgfsys@curveto{11.91663pt}{-9.65852pt}{12.36433pt}{-10.10623pt}{12.91663pt}{-10.10623pt}\pgfsys@curveto{13.46892pt}{-10.10623pt}{13.91663pt}{-9.65852pt}{13.91663pt}{-9.10623pt}\pgfsys@closepath\pgfsys@moveto{12.91663pt}{-9.10623pt}\pgfsys@fillstroke\pgfsys@invoke{
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{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(25)

These two identities generate the full permutation group of the external
vertices. The factorization rules in eq. (24) and the permutation identities
in eq. (25) follow immediately from eq. (21), the definition of a position
space three-point function in eq. (18) and the parametrization in eq. (20).

Up to this point, all statements on graphical functions are valid in even
dimensions $\geq 4$. To handle QFTs with subdivergences we need to use a
regulator. It is convenient to use dimensional regularization because the
concept of graphical functions is based on the complex plane which is
independent of the ambient space. Hence, it is stable under the deformation of
the dimension to real numbers. In fact, all the previous identities work for
general dimensions. We use the notation

$d=2n+4-\varepsilon,\quad
n\in\\{0,1,2,3,\ldots\\},\quad\varepsilon\in\mathbb{R}~{}.$

The general idea is to consider graphical functions in $d$ dimensions as
Laurent series in $\varepsilon$. The coefficient of every power in
$\varepsilon$ conjecturally reflects the structure of graphical functions in
fixed even dimensions. In particular, every coefficient is a single-valued
real analytic function on $\mathbb{C}\backslash\\{0,1\\}$ and admits
$\log$-Laurent expansions (eqs. (22) and (23)) at $0$, $1$, and $\infty$.

The most important identity for the graphical function technique follows from
the definition of the propagator, eq. (17), combined with eqs. (18) and (20).
The intuition behind this identity is that, according to eq. (17), the box
operator can be used to ‘amputate’ single external edges of a position space
Feynman diagram: Two position space three-point functions $G_{\tilde{\Gamma}}$
and $G_{\Gamma}$ whose underlying Feynman graphs $\widetilde{\Gamma}$ and
$\Gamma$ only differ by an appended edge along the external vertex $c$,

$\displaystyle\widetilde{\Gamma}=\leavevmode\hbox to56.74pt{\vbox
to30.5pt{\pgfpicture\makeatletter\hbox{\hskip 31.21436pt\lower-14.59201pt\hbox
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}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{-19.37494pt}{0.0pt}\pgfsys@moveto{-18.37494pt}{0.0pt}\pgfsys@curveto{-18.37494pt}{0.55229pt}{-18.82265pt}{1.0pt}{-19.37494pt}{1.0pt}\pgfsys@curveto{-19.92723pt}{1.0pt}{-20.37494pt}{0.55229pt}{-20.37494pt}{0.0pt}\pgfsys@curveto{-20.37494pt}{-0.55229pt}{-19.92723pt}{-1.0pt}{-19.37494pt}{-1.0pt}\pgfsys@curveto{-18.82265pt}{-1.0pt}{-18.37494pt}{-0.55229pt}{-18.37494pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-19.37494pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
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{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{0.55229pt}{0.55229pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-0.55229pt}{1.0pt}{-1.0pt}{0.55229pt}{-1.0pt}{0.0pt}\pgfsys@curveto{-1.0pt}{-0.55229pt}{-0.55229pt}{-1.0pt}{0.0pt}{-1.0pt}\pgfsys@curveto{0.55229pt}{-1.0pt}{1.0pt}{-0.55229pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
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{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{12.91663pt}{9.10623pt}\pgfsys@moveto{13.91663pt}{9.10623pt}\pgfsys@curveto{13.91663pt}{9.65852pt}{13.46892pt}{10.10623pt}{12.91663pt}{10.10623pt}\pgfsys@curveto{12.36433pt}{10.10623pt}{11.91663pt}{9.65852pt}{11.91663pt}{9.10623pt}\pgfsys@curveto{11.91663pt}{8.55394pt}{12.36433pt}{8.10623pt}{12.91663pt}{8.10623pt}\pgfsys@curveto{13.46892pt}{8.10623pt}{13.91663pt}{8.55394pt}{13.91663pt}{9.10623pt}\pgfsys@closepath\pgfsys@moveto{12.91663pt}{9.10623pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{12.91663pt}{-9.10623pt}\pgfsys@moveto{13.91663pt}{-9.10623pt}\pgfsys@curveto{13.91663pt}{-8.55394pt}{13.46892pt}{-8.10623pt}{12.91663pt}{-8.10623pt}\pgfsys@curveto{12.36433pt}{-8.10623pt}{11.91663pt}{-8.55394pt}{11.91663pt}{-9.10623pt}\pgfsys@curveto{11.91663pt}{-9.65852pt}{12.36433pt}{-10.10623pt}{12.91663pt}{-10.10623pt}\pgfsys@curveto{13.46892pt}{-10.10623pt}{13.91663pt}{-9.65852pt}{13.91663pt}{-9.10623pt}\pgfsys@closepath\pgfsys@moveto{12.91663pt}{-9.10623pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}~{},\qquad\Gamma=\leavevmode\hbox
to37.36pt{\vbox to30.5pt{\pgfpicture\makeatletter\hbox{\hskip
11.83942pt\lower-14.59201pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$c$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.90175pt}{-11.259pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$a$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{6.45831pt}{0.0pt}{8.33119pt}{4.5208pt}{12.91663pt}{9.10623pt}\pgfsys@curveto{8.33119pt}{4.5208pt}{8.33119pt}{-4.5208pt}{12.91663pt}{-9.10623pt}\pgfsys@curveto{8.33119pt}{-4.5208pt}{6.45831pt}{0.0pt}{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{0.55229pt}{0.55229pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-0.55229pt}{1.0pt}{-1.0pt}{0.55229pt}{-1.0pt}{0.0pt}\pgfsys@curveto{-1.0pt}{-0.55229pt}{-0.55229pt}{-1.0pt}{0.0pt}{-1.0pt}\pgfsys@curveto{0.55229pt}{-1.0pt}{1.0pt}{-0.55229pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{12.91663pt}{9.10623pt}\pgfsys@moveto{13.91663pt}{9.10623pt}\pgfsys@curveto{13.91663pt}{9.65852pt}{13.46892pt}{10.10623pt}{12.91663pt}{10.10623pt}\pgfsys@curveto{12.36433pt}{10.10623pt}{11.91663pt}{9.65852pt}{11.91663pt}{9.10623pt}\pgfsys@curveto{11.91663pt}{8.55394pt}{12.36433pt}{8.10623pt}{12.91663pt}{8.10623pt}\pgfsys@curveto{13.46892pt}{8.10623pt}{13.91663pt}{8.55394pt}{13.91663pt}{9.10623pt}\pgfsys@closepath\pgfsys@moveto{12.91663pt}{9.10623pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{12.91663pt}{-9.10623pt}\pgfsys@moveto{13.91663pt}{-9.10623pt}\pgfsys@curveto{13.91663pt}{-8.55394pt}{13.46892pt}{-8.10623pt}{12.91663pt}{-8.10623pt}\pgfsys@curveto{12.36433pt}{-8.10623pt}{11.91663pt}{-8.55394pt}{11.91663pt}{-9.10623pt}\pgfsys@curveto{11.91663pt}{-9.65852pt}{12.36433pt}{-10.10623pt}{12.91663pt}{-10.10623pt}\pgfsys@curveto{13.46892pt}{-10.10623pt}{13.91663pt}{-9.65852pt}{13.91663pt}{-9.10623pt}\pgfsys@closepath\pgfsys@moveto{12.91663pt}{-9.10623pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

fulfill the partial differential equation

$\displaystyle\Box_{\boldsymbol{x}_{c}}G_{\tilde{\Gamma}}(\boldsymbol{x}_{a},\boldsymbol{x}_{b},\boldsymbol{x}_{c})=-\frac{4}{\Gamma(d/2-1)}G_{\Gamma}(\boldsymbol{x}_{a},\boldsymbol{x}_{b},\boldsymbol{x}_{c})~{}.$
(26)

In fact, such combinatorial differential equations hold for arbitrary
$n$-point functions. In our case of the three-point function we can translate
the Laplacian $\Box_{\boldsymbol{x}_{c}}$ via eq. (20) into an operator on the
space of complex functions in $z$ and ${\overline{z}}$ to get an effective
Laplacian, which operates on graphical functions,

$\displaystyle\begin{gathered}\left(\Delta_{n}+\frac{\varepsilon/2}{z-{\overline{z}}}(\partial_{z}-\partial_{{\overline{z}}})\right)\left[\leavevmode\hbox
to57.21pt{\vbox to31.32pt{\pgfpicture\makeatletter\hbox{\hskip
31.97711pt\lower-15.66145pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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{{}}{}{{}}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}}
{}{}{}\pgfsys@moveto{-19.37494pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}
{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{-19.37494pt}{0.0pt}\pgfsys@moveto{-18.37494pt}{0.0pt}\pgfsys@curveto{-18.37494pt}{0.55229pt}{-18.82265pt}{1.0pt}{-19.37494pt}{1.0pt}\pgfsys@curveto{-19.92723pt}{1.0pt}{-20.37494pt}{0.55229pt}{-20.37494pt}{0.0pt}\pgfsys@curveto{-20.37494pt}{-0.55229pt}{-19.92723pt}{-1.0pt}{-19.37494pt}{-1.0pt}\pgfsys@curveto{-18.82265pt}{-1.0pt}{-18.37494pt}{-0.55229pt}{-18.37494pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-19.37494pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{0.55229pt}{0.55229pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-0.55229pt}{1.0pt}{-1.0pt}{0.55229pt}{-1.0pt}{0.0pt}\pgfsys@curveto{-1.0pt}{-0.55229pt}{-0.55229pt}{-1.0pt}{0.0pt}{-1.0pt}\pgfsys@curveto{0.55229pt}{-1.0pt}{1.0pt}{-0.55229pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{12.91663pt}{9.10623pt}\pgfsys@moveto{13.91663pt}{9.10623pt}\pgfsys@curveto{13.91663pt}{9.65852pt}{13.46892pt}{10.10623pt}{12.91663pt}{10.10623pt}\pgfsys@curveto{12.36433pt}{10.10623pt}{11.91663pt}{9.65852pt}{11.91663pt}{9.10623pt}\pgfsys@curveto{11.91663pt}{8.55394pt}{12.36433pt}{8.10623pt}{12.91663pt}{8.10623pt}\pgfsys@curveto{13.46892pt}{8.10623pt}{13.91663pt}{8.55394pt}{13.91663pt}{9.10623pt}\pgfsys@closepath\pgfsys@moveto{12.91663pt}{9.10623pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{12.91663pt}{-9.10623pt}\pgfsys@moveto{13.91663pt}{-9.10623pt}\pgfsys@curveto{13.91663pt}{-8.55394pt}{13.46892pt}{-8.10623pt}{12.91663pt}{-8.10623pt}\pgfsys@curveto{12.36433pt}{-8.10623pt}{11.91663pt}{-8.55394pt}{11.91663pt}{-9.10623pt}\pgfsys@curveto{11.91663pt}{-9.65852pt}{12.36433pt}{-10.10623pt}{12.91663pt}{-10.10623pt}\pgfsys@curveto{13.46892pt}{-10.10623pt}{13.91663pt}{-9.65852pt}{13.91663pt}{-9.10623pt}\pgfsys@closepath\pgfsys@moveto{12.91663pt}{-9.10623pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right]=-\frac{1}{\Gamma(n+1-\varepsilon/2)}\left[\leavevmode\hbox
to37.84pt{\vbox to31.32pt{\pgfpicture\makeatletter\hbox{\hskip
12.60217pt\lower-15.66145pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{{}{}}}{{}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
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{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}{{}}{{}}{{}}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@fill{.5}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@curveto{6.45831pt}{0.0pt}{8.33119pt}{4.5208pt}{12.91663pt}{9.10623pt}\pgfsys@curveto{8.33119pt}{4.5208pt}{8.33119pt}{-4.5208pt}{12.91663pt}{-9.10623pt}\pgfsys@curveto{8.33119pt}{-4.5208pt}{6.45831pt}{0.0pt}{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{0.55229pt}{0.55229pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-0.55229pt}{1.0pt}{-1.0pt}{0.55229pt}{-1.0pt}{0.0pt}\pgfsys@curveto{-1.0pt}{-0.55229pt}{-0.55229pt}{-1.0pt}{0.0pt}{-1.0pt}\pgfsys@curveto{0.55229pt}{-1.0pt}{1.0pt}{-0.55229pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{12.91663pt}{9.10623pt}\pgfsys@moveto{13.91663pt}{9.10623pt}\pgfsys@curveto{13.91663pt}{9.65852pt}{13.46892pt}{10.10623pt}{12.91663pt}{10.10623pt}\pgfsys@curveto{12.36433pt}{10.10623pt}{11.91663pt}{9.65852pt}{11.91663pt}{9.10623pt}\pgfsys@curveto{11.91663pt}{8.55394pt}{12.36433pt}{8.10623pt}{12.91663pt}{8.10623pt}\pgfsys@curveto{13.46892pt}{8.10623pt}{13.91663pt}{8.55394pt}{13.91663pt}{9.10623pt}\pgfsys@closepath\pgfsys@moveto{12.91663pt}{9.10623pt}\pgfsys@fillstroke\pgfsys@invoke{
}
{{}}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{12.91663pt}{-9.10623pt}\pgfsys@moveto{13.91663pt}{-9.10623pt}\pgfsys@curveto{13.91663pt}{-8.55394pt}{13.46892pt}{-8.10623pt}{12.91663pt}{-8.10623pt}\pgfsys@curveto{12.36433pt}{-8.10623pt}{11.91663pt}{-8.55394pt}{11.91663pt}{-9.10623pt}\pgfsys@curveto{11.91663pt}{-9.65852pt}{12.36433pt}{-10.10623pt}{12.91663pt}{-10.10623pt}\pgfsys@curveto{13.46892pt}{-10.10623pt}{13.91663pt}{-9.65852pt}{13.91663pt}{-9.10623pt}\pgfsys@closepath\pgfsys@moveto{12.91663pt}{-9.10623pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right]\\\ \text{with
}\Delta_{n}=\frac{1}{(z-{\overline{z}})^{n+1}}\partial_{z}\partial_{{\overline{z}}}(z-{\overline{z}})^{n+1}+\frac{n(n+1)}{(z-{\overline{z}})^{2}}.\end{gathered}$
(29)

To obtain a graphical function of higher complexity from a graphical function
of lower complexity, we would like to solve this partial differential equation
for the graphical function with the appended edge. To do this, we need to
invert the effective Laplacian. This inversion can be roughly separated into
two related problems: First, we have to find a general solution of the
differential equation. Second, we need to specify the solution that gives the
wanted graphical function.

The first problem is reasonably easy in four dimensions, i.e. $n=0$. As long
as $\varepsilon=0$ the partial differential operator $\Delta_{0}$ factorizes
into $\partial_{z}$ and $\partial_{\overline{z}}$. Even though, naively
integrating with respect to $z$ and ${\overline{z}}$ results in undetermined
integration constants which may be arbitrary functions of ${\overline{z}}$ and
$z$ respectively, these integration constants are restricted if we require the
result to be single-valued. Single-valued integration needs to be performed on
a suitable class of functions. Because of the denominator $z-{\overline{z}}$
the class of single-valued multiple polylogarithms as studied in [47, 48] is
not enough. However, there exists a generalization by the fourth author to
_generalized single-valued hyperlogarithms_ (GSVHs) which exactly accommodates
the situation [49]. This theory comes with general and conveniently fast
algorithms for single-valued integration in $z$ and ${\overline{z}}$. For
$\varepsilon>0$ the generalization is straightforward: We are not interested
in a full result but rather in a Laurent series in $\varepsilon$. This allows
us to treat the $\varepsilon$ term in eq. (29) as a perturbation and to solve
the differential equation by iteratively inverting $\Delta_{0}$.

The situation is much more complicated for $n\geq 1$ (i.e. $d\geq 6$). Note
that the effective Laplacian $\Delta_{n}$ is a partial differential operator
in $z$ and ${\overline{z}}$ which in general does not admit a simple solution
by integration. However, somewhat surprisingly, a general solution of eq. (29)
for all $n=0,1,2,3,\ldots$ in the case $\varepsilon=0$ was found by the first
and the fourth author [35]. The function space of GSVHs is perfectly suitable
for general $n$. The case $\varepsilon>0$ is again treated as a perturbation.

The second problem—specifying the exact solution—is solved for $\varepsilon=0$
by a theorem [45, 35]: the structures of graphical functions—single-valuedness
and $\log$-Laurent expansions—are so restrictive that they fully specify the
solution. That means there is always only one special solution among the
family of general solutions that behaves like a graphical function. Finding
this special solution is solved by an algorithm given in [35]. For
$\varepsilon>0$, handling subdivergences in this context is a bit subtle but
always possible. Altogether we obtain a general and surprisingly efficient
algorithm to append an edge to the external label $z$ of a graphical function.

Using the three basic operations—adding edges between external vertices,
permuting external vertices, and appending an edge—allows one to construct a
wide class of graphical functions from the empty graphical function (the
graphical function with neither edges nor internal vertices), which is the
constant $1$. See Fig. 1 for a non-trivial example of a two-point function in
$\phi^{4}$ theory that can be constructed from these basic operations.

In practice, there exist a few more elementary operations (products, factors,
completion) that provide combinatorial relations between different graphical
functions. A graphical function that can be expressed as a sequence these
elementary operation applied to the trivial graphical function is called
_constructible_ and can be calculated to any reasonable order in $\varepsilon$
(limited by time and memory demands). We want to emphasize that this concept
of constructible graphical functions can easily be applied to any massless QFT
in even dimensions.

Still, starting at some loop order, there exist graphical functions which
cannot be constructed. Some of these graphical functions may be amendable to
some reductions by elementary operations, but eventually a non-empty
irreducible graphical function will be reached that has to be calculated by
other means. Beyond constructability there exists a toolbox of additional
identities which has some resemblance to standard momentum space techniques
(e.g. momentum space IBPs). Explicitly we have special identities,
approximations in $\varepsilon$ and position space IBPs that can be used to
further simplify the calculations. However this toolbox is still in its
infancy and small additions may lead to dramatic increases in the
applicability of the overall graphical function technique.

Eventually, some (typically small) set of (typically small) graphical
functions remains that has to be integrated by different means. A brute force
approach is to write the respective graphical functions as parametric
integrals [46] which in many cases can be integrated using an algorithm by F.
Brown [50] which is implemented as HyperInt by E. Panzer [51]. In four
dimensions this parametric integration often works quite well, even though it
is always very slow compared to operations on graphical functions. In six or
higher dimensions the use of parametric integration is limited by the
existence on squares and higher powers in the denominators of the parametric
integrals as this causes problems in the implementation of HyperInt and
results in large time and memory consumption. For our purpose this is not
severe as only six graphical functions with at most two internal vertices were
needed to solve the five loop renormalization problem.

For six loops approximately $80\%$ of the Feynman integrals are calculated.
Because of the vast number and the higher complexity of graphs at six loops it
is necessary to make better use of some techniques in the theory of graphical
functions. Most prominently, making full use of a generalized $\Delta-Y$
identity and position space integration by parts (IBP) requires the
(notoriously tedious) solution of large linear systems. In the primitive case
these identities (with some others) were implemented by the first author. This
solved all primitive six loop integrals in $\phi^{3}$ theory [52]. Even at $7$
loops $92\%$ of the primitive integrals could be calculated [52]. The
implementation for subdivergent graphs is straight forward but still lacking.
We expect that the implementation of position space IBP will solve $\phi^{3}$
theory up to six loops in perturbation theory. It might be necessary to use
some minor additions from other techniques such as $R/R^{*}$ [43, 42, 41] or
its Hopf algebraic version [53] that make it possible to exclusively work with
finite expressions. A hypothetical extension to $7$ loops might already
require the addition of numerical methods such as tropical parametric
integration [54] to evaluate a small number of primitive graphs that cannot be
evaluated via GSVHs. Such graphs are known to appear in $\phi^{4}$ theory at
$8$ loops [55].

###  3.2 Periods and renormalization group functions

$0$$1$$0$$1$$z$$\int\text{d}z\text{d}{\overline{z}}$$z$$1$$0$$0$$0\leftrightarrow
z$$z$$1$$0$$0$$0$$0$$\Box_{z}^{-1}$$z$$1$$0$$0$$0$$0$ext.
edges$1$$z$$0$$0$$0$$0$$1\leftrightarrow
z$$1$$z$$0$$0$$0$$0$$\Box_{z}^{-1}$$1$$z$$0$$0$$0$$0$ext.
edges$1$$z$$0$$0$$0$$0$$\Box_{z}^{-1}$$1$$z$$0$$0$$0$$0$ext. edges Figure 1:
Example of the graphical reduction of a two-point function in $\phi^{4}$
theory. The explicit expression can be obtained by following the arrows which
start from the trivial graphical function which is equal to $1$.

To determine renormalization functions it is sufficient to calculate
($\varepsilon$-dependent) periods, i.e. the Feynman integrals of graphs with
two external vertices. If a graph has two external vertices
$\boldsymbol{x}_{a},\boldsymbol{x}_{b}$ then translation and scale-invariance
(eq. (19)) forces the respective position space Feynman integral to evaluate
to $|\boldsymbol{x}_{ab}|^{\Delta_{\Gamma}}P_{\Gamma}(\varepsilon)$ where the
_period_ , $P_{\Gamma}(\varepsilon)$, is only a function of $\varepsilon$.

The calculation of periods is much simpler than the calculation of graphical
functions: one can _complete_ a period by adding an edge between
$\boldsymbol{x}_{a}$ and $\boldsymbol{x}_{b}$ with a specific weight so that
one is free to choose an arbitrary set of three external vertices $0$, $1$,
$z$ to obtain a graphical function. If this graphical function can be
calculated it is always possible to efficiently integrate over the third
vertex $z$ to obtain the period. This freedom is of great benefit for
calculating periods. Often there exists a particularly convenient choice which
facilitates the calculation. In a certain sense graphical functions were
originally invented to use them for the calculation of periods with exactly
this concept. Fig. 1 shows an example of a full reduction chain of a period in
$\phi^{4}$ theory.

The graphical function technique is powerful enough to take a naive approach
to renormalization. We calculate each amplitude to the necessary order in
$\varepsilon$. Instead of calculating the three-point function directly in
terms of graphical functions, we set one external vertex to infinity and
calculate the simpler period of the resulting truncated three-point function
as an effective two-point function. We add all amplitudes to get regularized
(effective) two-point functions from which we read off the $Z$ factors by the
condition that renormalization renders the (effective) two-point functions
finite, see eq. (5). From the $Z$ factors we read off the renormalization
functions, see eq. (6).

A detailed explanation of the graphical function method will be in [32]. All
algorithms are implemented as Maple procedures in HyperlogProcedures by the
fourth author [56]. The calculation of the fifth order result in six
dimensional $\phi^{3}$ theory is fully automated in HyperlogProcedures and
takes about two days on a single core consuming 38 GB of memory. It can easily
be parallelized.

###  3.3 Percolation theory prefactors

Our diagrammatic approach to compute the combinatorial prefactors
$\mathscr{C}_{\Gamma}$ for percolation theory follows from eqs. (3) and (4)
via the diagrammatic discussion in [5, eq. (3.21) and the paragraph that
follows]. For a graph $\Gamma$, we write the associated _completed graph_ as
$\overline{\Gamma}$. A graph is completed by attaching all external legs to an
additional vertex [57]. Completion utilizes the fact that the tensor structure
of two- and three-point graphs is fixed up to a constant. Adapting the
notation of [30], for the percolation theory problem we find the following
relations of the combinatorial factors $\mathscr{C}_{\Gamma}$, which follow
from the algebraic rules above

$\mathscr{C}_{\Gamma}(\alpha)=\mathscr{C}_{\overline{\Gamma}}\cdot\begin{cases}\frac{(N+1)\delta_{\alpha_{1},\alpha_{2}}-1}{(N+1)N}&\text{
if $\Gamma$ is a $2$-point function,}\\\
\frac{(N+1)^{2}\delta_{\alpha_{1},\alpha_{2}}\delta_{\alpha_{1},\alpha_{3}}-(N+1)(\delta_{\alpha_{1},\alpha_{2}}+\delta_{\alpha_{1},\alpha_{3}}+\delta_{\alpha_{2},\alpha_{3}})+2}{(N+1)N(N-1)}&\text{
if $\Gamma$ is a $3$-point function,}\end{cases}$ (30)

where $\alpha_{1},\alpha_{2}$ and $\alpha_{3}$ are associated with the two or
three external tensor indices of the respective $n$-point functions. The
combinatorial factors of the completed graphs are readily computed by
_contraction-deletion_.

$\begin{array}[]{rll}\mathscr{C}_{\overline{\Gamma}}&=\;\;(N+1)\mathscr{C}_{\overline{\Gamma}/e}-\mathscr{C}_{\overline{\Gamma}\setminus
e}&\text{ for each edge }e\in\overline{\Gamma},\\\
\mathscr{C}_{\overline{\Gamma}_{1}\,\overline{\Gamma}_{2}}&=\;\;\mathscr{C}_{\overline{\Gamma}_{1}}\cdot\mathscr{C}_{\overline{\Gamma}_{2}}&\text{
for the disjoint union $\overline{\Gamma}_{1}\,\overline{\Gamma}_{2}$,}\\\
\mathscr{C}_{\bullet}&=\;\;N+1&\text{ for the single vertex
$\bullet$,}\end{array}$ (31)

where $\overline{\Gamma}/e$ and $\overline{\Gamma}\setminus e$ denote the
contraction and deletion of the edge $e$ in the graph $\overline{\Gamma}$. The
contraction of a self-loop is the deletion of the loop in this context.

The percolation theory problem is described by subsequently taking the
$N\rightarrow 0$ limit for each prefactor.

##  4 Five loop renormalization group functions

With the major task of calculating all the integrals and carrying out the
renormalization, we devote this section solely to recording the results of the
full five loop renormalization in the context of the two critical systems that
we are interested in.

###  4.1 Lee–Yang edge singularity

For the Lee–Yang (LY) edge singularity problem the five loop
$\overline{\mbox{MS}}$ renormalization group functions are

$\displaystyle\begin{aligned}
\beta^{(\mathrm{LY})}(g)=&-\frac{\varepsilon}{2}\,g+\frac{3}{4}\,{g}^{3}-{\frac{125\,{g}^{5}}{144}}+\left({\frac{33085}{20736}}+\frac{5\,\zeta_{{3}}}{8}\right){g}^{7}\\\
&+\left({\frac{15\,\zeta_{4}}{32}}-{\frac{3404365}{746496}}-{\frac{4891\,\zeta_{{3}}}{864}}+\frac{5\,\zeta_{{5}}}{3}\right){g}^{9}\\\
&+\left({\frac{75\,\zeta_{6}}{32}}-{\frac{46519\,\zeta_{4}}{9216}}+{\frac{102052031}{6718464}}+{\frac{99\,{\zeta_{{3}}}^{2}}{16}}+{\frac{366647\,\zeta_{{3}}}{6912}}+{\frac{151795\,\zeta_{{5}}}{3456}}-{\frac{5495\,\zeta_{{7}}}{64}}\\!\right){g}^{11}\\\
&+\mathcal{O}(g^{13})\end{aligned}$ (32) $\displaystyle\begin{aligned}
\gamma^{(\mathrm{LY})}_{\phi}(g)=&-\frac{1}{12}\,{g}^{2}+{\frac{13}{432}}\,{g}^{4}+\left(-{\frac{5195}{62208}}+\frac{1}{24}\,\zeta_{{3}}\right){g}^{6}+\left({\frac{53449}{248832}}+{\frac{7\,\zeta_{{4}}}{96}}+{\frac{35\,\zeta_{{3}}}{864}}-{\frac{5\,\zeta_{{5}}}{18}}\right){g}^{8}\\\
&+\left(-{\frac{125\,\zeta_{{6}}}{288}}-{\frac{5651\,\zeta_{{4}}}{27648}}-{\frac{16492987}{20155392}}-{\frac{25\,{\zeta_{{3}}}^{2}}{144}}-{\frac{56693\,\zeta_{{3}}}{62208}}+{\frac{4471\,\zeta_{{5}}}{10368}}+{\frac{147\,\zeta_{{7}}}{64}}\right){g}^{10}\\\
&+\mathcal{O}(g^{12})\end{aligned}$ (33) $\displaystyle\begin{aligned}
\gamma^{(\mathrm{LY})}_{m^{2}}(g)=&-\frac{5}{6}\,{g}^{2}+{\frac{97}{108}}\,{g}^{4}+\left(-{\frac{52225}{31104}}-{\frac{7\,\zeta_{{3}}}{12}}\right){g}^{6}+\left(-{\frac{19\,\zeta_{{4}}}{48}}+{\frac{445589}{93312}}+{\frac{821\,\zeta_{{3}}}{144}}-{\frac{35\,\zeta_{{5}}}{18}}\right){g}^{8}\\\
&+\left(-{\frac{25\,\zeta_{{6}}}{9}}+{\frac{66953\,\zeta_{{4}}}{13824}}-{\frac{40331135}{2519424}}-{\frac{229\,{\zeta_{{3}}}^{2}}{36}}-{\frac{839129\,\zeta_{{3}}}{15552}}-{\frac{225457\,\zeta_{{5}}}{5184}}\right.\\\
&\left.~{}~{}~{}~{}~{}+{\frac{2821\,\zeta_{{7}}}{32}}\right){g}^{10}+\mathcal{O}(g^{12})\end{aligned}$
(34)

where $\zeta_{z}$ is the Riemann zeta function. Each expression agrees with
the earlier expressions given in [1, 4, 5, 17] up to four loops after the
coupling constant mapping is undone. We note that the five loop expression
also agrees with the recent calculation of [58]. Equipped with these it is
straightforward to derive the respective critical exponents which are

$\displaystyle\begin{aligned}
\eta^{(\mathrm{LY})}(\varepsilon)=&-\frac{1}{9}\,\varepsilon-{\frac{43}{729}}\,{\varepsilon}^{2}+\left(-{\frac{8375}{236196}}+{\frac{16\,\zeta_{{3}}}{243}}\right){\varepsilon}^{3}+\left(-{\frac{3883409}{76527504}}+{\frac{4\,\zeta_{4}}{81}}-{\frac{716\,\zeta_{{3}}}{19683}}-{\frac{80\,\zeta_{{5}}}{2187}}\right){\varepsilon}^{4}\\\
&+\left(-{\frac{100\,\zeta_{6}}{2187}}-{\frac{179\,\zeta_{4}}{6561}}-{\frac{1545362585}{24794911296}}+{\frac{136\,{\zeta_{{3}}}^{2}}{2187}}+{\frac{37643\,\zeta_{{3}}}{59049}}+{\frac{80524\,\zeta_{{5}}}{59049}}\right.\\\
&\left.~{}~{}~{}~{}~{}-{\frac{4172\,\zeta_{{7}}}{2187}}\right){\varepsilon}^{5}+\mathcal{O}(\varepsilon^{6})\\\
=&-0.11111\,\varepsilon-0.058985\,{\varepsilon}^{2}+0.043693\,{\varepsilon}^{3}-0.078951\,{\varepsilon}^{4}+0.20843\,{\varepsilon}^{5}+\mathcal{O}(\varepsilon^{6})\end{aligned}$
(35) $\displaystyle\begin{aligned}
\eta_{O}^{(\mathrm{LY})}(\varepsilon)=&-\frac{2}{3}\,\varepsilon-{\frac{43}{486}}\,{\varepsilon}^{2}+\left(-{\frac{8375}{157464}}+{\frac{8\,\zeta_{{3}}}{81}}\right){\varepsilon}^{3}+\left({\frac{2\,\zeta_{4}}{27}}-{\frac{3883409}{51018336}}-{\frac{358\,\zeta_{{3}}}{6561}}-{\frac{40\,\zeta_{{5}}}{729}}\right){\varepsilon}^{4}\\\
&+\left(-{\frac{50\,\zeta_{6}}{729}}-{\frac{179\,\zeta_{4}}{4374}}-{\frac{1545362585}{16529940864}}+{\frac{68\,{\zeta_{{3}}}^{2}}{729}}+{\frac{37643\,\zeta_{{3}}}{39366}}+{\frac{40262\,\zeta_{{5}}}{19683}}\right.\\\
&\left.~{}~{}~{}~{}~{}-{\frac{2086\,\zeta_{{7}}}{729}}\right){\varepsilon}^{5}+\mathcal{O}(\varepsilon^{6})\\\
=&-0.66667\,\varepsilon-0.088477\,{\varepsilon}^{2}+0.065543\,{\varepsilon}^{3}-0.11843\,{\varepsilon}^{4}+0.31241\,{\varepsilon}^{5}+\mathcal{O}(\varepsilon^{6})\end{aligned}$
(36) $\displaystyle\begin{aligned}
1/\nu^{(\mathrm{LY})}(\varepsilon)=&~{}2-{\frac{5}{9}}\varepsilon-{\frac{43}{1458}}{\varepsilon}^{2}+\left(-{\frac{8375}{472392}}+{\frac{8\,\zeta_{{3}}}{243}}\right){\varepsilon}^{3}+\\\
&+\left({\frac{2\,\zeta_{{4}}}{81}}-{\frac{3883409}{153055008}}-{\frac{358\,\zeta_{{3}}}{19683}}-{\frac{40\,\zeta_{{5}}}{2187}}\right){\varepsilon}^{4}+\\\
&+\left(-{\frac{1545362585}{49589822592}}-{\frac{50\,\zeta_{{6}}}{2187}}-{\frac{179\,\zeta_{{4}}}{13122}}+{\frac{37643\,\zeta_{{3}}}{118098}}+\right.\\\
&\left.~{}~{}~{}~{}~{}~{}+{\frac{40262\,\zeta_{{5}}}{59049}}+{\frac{68\,{\zeta_{{3}}}^{2}}{2187}}-{\frac{2086\,\zeta_{{7}}}{2187}}\right){\varepsilon}^{5}+O\left({\varepsilon}^{6}\right)\\\
=&~{}2-0.55556\,\varepsilon-0.029493\,\varepsilon^{2}+0.021845\,\varepsilon^{3}-0.039477\,\varepsilon^{4}+0.10413\,\varepsilon^{5}\\\
&+\mathcal{O}(\varepsilon^{6})\end{aligned}$ (37)

$\displaystyle\begin{aligned}
\omega^{(\mathrm{LY})}(\varepsilon)=&~{}\varepsilon-{\frac{125}{162}}\,{\varepsilon}^{2}+\left({\frac{36755}{52488}}+{\frac{20\,\zeta_{{3}}}{27}}\right){\varepsilon}^{3}+\left(\frac{5}{9}\,\zeta_{4}-{\frac{31725355}{17006112}}-{\frac{9673\,\zeta_{{3}}}{2187}}+{\frac{160\,\zeta_{{5}}}{81}}\right){\varepsilon}^{4}\\\
&+\left({\frac{200\,\zeta_{6}}{81}}-{\frac{9673\,\zeta_{4}}{2916}}+{\frac{17088604709}{5509980288}}+{\frac{1384\,{\zeta_{{3}}}^{2}}{243}}+{\frac{12094613\,\zeta_{{3}}}{354294}}\right.\\\
&\left.~{}~{}~{}~{}~{}+{\frac{1050770\,\zeta_{{5}}}{19683}}-{\frac{21980\,\zeta_{{7}}}{243}}\right){\varepsilon}^{5}+\mathcal{O}(\varepsilon^{6})\\\
=&~{}\varepsilon-0.77160\,{\varepsilon}^{2}+1.5907\,{\varepsilon}^{3}-4.5329\,{\varepsilon}^{4}+15.440\,{\varepsilon}^{5}+\mathcal{O}(\varepsilon^{6})\end{aligned}$
(38) $\displaystyle\begin{aligned}
\sigma^{(\mathrm{LY})}(\varepsilon)=&~{}\frac{1}{2}-\frac{1}{12}\,\varepsilon-{\frac{79}{3888}}\,{\varepsilon}^{2}+\left(-{\frac{10445}{1259712}}+{\frac{\zeta_{{3}}}{81}}\right){\varepsilon}^{3}\\\
&+\left({\frac{\zeta_{4}}{108}}-{\frac{4047533}{408146688}}-{\frac{161\,\zeta_{{3}}}{26244}}-{\frac{5\,\zeta_{{5}}}{729}}\right){\varepsilon}^{4}\\\
&+\left(-{\frac{25\,\zeta_{6}}{2916}}-{\frac{161\,\zeta_{4}}{34992}}-{\frac{1601178731}{132239526912}}+{\frac{17\,{\zeta_{{3}}}^{2}}{1458}}+{\frac{112399\,\zeta_{{3}}}{944784}}+{\frac{20101\,\zeta_{{5}}}{78732}}\right.\\\
&\left.~{}~{}~{}~{}~{}-{\frac{1043\,\zeta_{{7}}}{2916}}\right){\varepsilon}^{5}+\mathcal{O}(\varepsilon^{6})\\\
=&~{}0.50000-0.083333\,\varepsilon-0.020319\,{\varepsilon}^{2}+0.0065494\,{\varepsilon}^{3}-0.014381\,{\varepsilon}^{4}+0.038131\,{\varepsilon}^{5}\\\
&+\mathcal{O}(\varepsilon^{6})\end{aligned}$ (39)

where we have also recorded the numerical value of each coefficient in the
$\varepsilon$ expansion.

###  4.2 Percolation theory

For the percolation problem, denoted by $\mathrm{P}$, the analogous five loop
renormalization group functions after taking the replica limit as described in
Section 3.3 are

$\displaystyle\begin{aligned}
\beta^{(\mathrm{P})}(g)=&-\frac{\varepsilon}{2}\,g-\frac{7}{4}\,{g}^{3}-{\frac{671}{144}}\,{g}^{5}+\left(-{\frac{414031}{20736}}-{\frac{93\,\zeta_{{3}}}{8}}\right){g}^{7}\\\
&+\left({\frac{651\,\zeta_{4}}{32}}-{\frac{84156383}{746496}}-{\frac{121109\,\zeta_{{3}}}{864}}-{\frac{595\,\zeta_{{5}}}{6}}\right){g}^{9}\\\
&+\left(-{\frac{74773124579}{107495424}}+{\frac{930967\,\zeta_{4}}{3072}}+{\frac{20825\,\zeta_{6}}{64}}-{\frac{7411\,{\zeta_{{3}}}^{2}}{32}}-{\frac{56477573\,\zeta_{{3}}}{27648}}\right.\\\
&\left.~{}~{}~{}~{}~{}-{\frac{11846549\,\zeta_{{5}}}{3456}}+{\frac{97293\,\zeta_{{7}}}{64}}\right){g}^{11}+\mathcal{O}(g^{13})\end{aligned}$
(40) $\displaystyle\begin{aligned}
\gamma^{(\mathrm{P})}_{\phi}(g)=&~{}\frac{1}{12}\,{g}^{2}+{\frac{37}{432}}\,{g}^{4}+\left({\frac{29297}{62208}}-{\frac{5\,\zeta_{{3}}}{24}}\right){g}^{6}+\left({\frac{225455}{82944}}+{\frac{33\,\zeta_{4}}{32}}+{\frac{233\,\zeta_{{3}}}{864}}-{\frac{55\,\zeta_{{5}}}{18}}\right){g}^{8}\\\
&+\left({\frac{5907303973}{322486272}}+{\frac{169325\,\zeta_{4}}{27648}}+{\frac{10675\,\zeta_{6}}{576}}+{\frac{719\,{\zeta_{{3}}}^{2}}{288}}+{\frac{3841369\,\zeta_{{3}}}{248832}}+{\frac{5443\,\zeta_{{5}}}{10368}}\right.\\\
&\left.~{}~{}~{}~{}~{}-{\frac{3969\,\zeta_{{7}}}{64}}\right){g}^{10}+\mathcal{O}(g^{12})\end{aligned}$
(41) $\displaystyle\begin{aligned}
\gamma^{(\mathrm{P})}_{m^{2}}(g)=&~{}\frac{5}{6}\,{g}^{2}+{\frac{193}{108}}\,{g}^{4}+\left({\frac{41\,\zeta_{{3}}}{12}}+{\frac{237751}{31104}}\right){g}^{6}+\left({\frac{3299\,\zeta_{{3}}}{72}}+{\frac{190\,\zeta_{{5}}}{9}}-{\frac{33\,\zeta_{{4}}}{8}}+{\frac{4114259}{93312}}\right){g}^{8}\\\
&+\left({\frac{11167\,{\zeta_{{3}}}^{2}}{144}}+{\frac{91152569\,\zeta_{{3}}}{124416}}+{\frac{5839991\,\zeta_{{5}}}{5184}}-{\frac{26117\,\zeta_{{7}}}{32}}-{\frac{1100099\,\zeta_{{4}}}{13824}}-{\frac{11725\,\zeta_{{6}}}{288}}\right.\\\
&\left.~{}~{}~{}~{}~{}+{\frac{44535597533}{161243136}}\right){g}^{10}+\mathcal{O}(g^{12})~{}.\end{aligned}$
(42)

Again the four loop expressions are in agreement with previous
$\overline{\mbox{MS}}$ computations, [1, 4, 5, 17]. Consequently the critical
exponents are

$\displaystyle\begin{aligned}
\eta^{(\mathrm{P})}(\varepsilon)=&-\frac{1}{21}\varepsilon-{\frac{206}{9261}}\,{\varepsilon}^{2}+\left(-{\frac{93619}{8168202}}+{\frac{256\,\zeta_{{3}}}{7203}}\right){\varepsilon}^{3}\\\
&+\left(-{\frac{103309103}{14408708328}}+{\frac{64\,\zeta_{4}}{2401}}+{\frac{189376\,\zeta_{{3}}}{9529569}}-{\frac{320\,\zeta_{{5}}}{3087}}\right){\varepsilon}^{4}\\\
&+\left(-{\frac{43137745921}{3630994498656}}+{\frac{47344\,\zeta_{4}}{3176523}}-{\frac{400\,\zeta_{6}}{3087}}-{\frac{187744\,{\zeta_{{3}}}^{2}}{7411887}}+{\frac{77003747\,\zeta_{{3}}}{600362847}}\right.\\\
&\left.~{}~{}~{}~{}~{}+{\frac{2337824\,\zeta_{{5}}}{9529569}}-{\frac{664\,\zeta_{{7}}}{16807}}\right){\varepsilon}^{5}+\mathcal{O}(\varepsilon^{6})\\\
=&-0.047619\,\varepsilon-0.022244\,{\varepsilon}^{2}+0.031263\,{\varepsilon}^{3}-0.061922\,{\varepsilon}^{4}+0.20454\,{\varepsilon}^{5}+\mathcal{O}(\varepsilon^{6})\end{aligned}$
(43) $\displaystyle\begin{aligned}
\eta_{O}^{(\mathrm{P})}(\varepsilon)=&-\frac{2}{7}\,\varepsilon-{\frac{355}{6174}}\,{\varepsilon}^{2}+\left({\frac{204\,\zeta_{{3}}}{2401}}-{\frac{235495}{10890936}}\right){\varepsilon}^{3}\\\
&+\left({\frac{153\,\zeta_{4}}{2401}}-{\frac{157609181}{19211611104}}+{\frac{99865\,\zeta_{{3}}}{3176523}}-{\frac{2000\,\zeta_{{5}}}{7203}}\right){\varepsilon}^{4}\\\
&+\left({\frac{99865\,\zeta_{4}}{4235364}}-{\frac{2500\,\zeta_{6}}{7203}}-{\frac{97373066851}{4841325998208}}-{\frac{107248\,{\zeta_{{3}}}^{2}}{2470629}}+{\frac{202024997\,\zeta_{{3}}}{800483796}}\right.\\\
&\left.~{}~{}~{}~{}~{}+{\frac{1860076\,\zeta_{{5}}}{3176523}}+{\frac{2288\,\zeta_{{7}}}{16807}}\right){\varepsilon}^{5}+\mathcal{O}(\varepsilon^{6})\\\
=&-0.28571\,\varepsilon-0.057499\,{\varepsilon}^{2}+0.080517\,{\varepsilon}^{3}-0.18935\,{\varepsilon}^{4}+0.63749\,{\varepsilon}^{5}+\mathcal{O}(\varepsilon^{6})\end{aligned}$
(44) $\displaystyle\begin{aligned}
1/\nu^{(\mathrm{P})}(\varepsilon)=&~{}2-{\frac{5\,\varepsilon}{21}}-{\frac{653\,{\varepsilon}^{2}}{18522}}+\left(-{\frac{332009}{32672808}}+{\frac{356\,\zeta_{{3}}}{7203}}\right){\varepsilon}^{3}+\\\
&+\left({\frac{89\,\zeta_{4}}{2401}}-{\frac{59591131}{57634833312}}+{\frac{110219\,\zeta_{{3}}}{9529569}}-{\frac{3760\,\zeta_{{5}}}{21609}}\right){\varepsilon}^{4}+\\\
&+\left(-{\frac{119568216869}{14523977994624}}+{\frac{110219\,\zeta_{4}}{12706092}}-{\frac{4700\,\zeta_{6}}{21609}}+{\frac{298060003\,\zeta_{{3}}}{2401451388}}+\right.\\\
&\left.~{}~{}~{}~{}~{}+{\frac{3242404\,\zeta_{{5}}}{9529569}}+{\frac{2952\,\zeta_{{7}}}{16807}}-{\frac{134000\,{\zeta_{{3}}}^{2}}{7411887}}\right){\varepsilon}^{5}+\mathcal{O}(\varepsilon^{6})\\\
=&~{}2-0.238095\,\varepsilon-0.035255\,\varepsilon^{2}+0.049249\,\varepsilon^{3}-0.12744\,\varepsilon^{4}+0.43287\,\varepsilon^{5}+\mathcal{O}(\varepsilon^{6})\end{aligned}$
(45) $\displaystyle\begin{aligned}
\omega^{(\mathrm{P})}(\varepsilon)=&~{}\varepsilon-{\frac{671}{882}}\,{\varepsilon}^{2}+\left({\frac{40639}{57624}}+{\frac{372\,\zeta_{{3}}}{343}}\right){\varepsilon}^{3}\\\
&+\left({\frac{279\,\zeta_{4}}{343}}-{\frac{317288185}{304946208}}-{\frac{348539\,\zeta_{{3}}}{151263}}-{\frac{1360\,\zeta_{{5}}}{343}}\right){\varepsilon}^{4}\\\
&+\left({\frac{601352852897}{691617999744}}-{\frac{348539\,\zeta_{4}}{201684}}-{\frac{1700\,\zeta_{6}}{343}}+{\frac{207440\,{\zeta_{{3}}}^{2}}{117649}}+{\frac{11664257531\,\zeta_{{3}}}{800483796}}\right.\\\
&\left.~{}~{}~{}~{}~{}+{\frac{17305178\,\zeta_{{5}}}{453789}}-{\frac{55596\,\zeta_{{7}}}{2401}}\right){\varepsilon}^{5}+\mathcal{O}(\varepsilon^{6})\\\
=&~{}\varepsilon-0.76077\,{\varepsilon}^{2}+2.0089\,{\varepsilon}^{3}-7.0413\,{\varepsilon}^{4}+30.216\,{\varepsilon}^{5}+\mathcal{O}(\varepsilon^{6})~{}.\end{aligned}$
(46)

Examining the numerical values one can easily see that the series for the
critical exponents has growing alternating coefficients, which is typical for
an asymptotic series. Therefore in order to obtain reliable estimates for the
exponents one needs to apply resummation methods.

##  5 Resummation strategy

While our focus so far has been in relation to renormalizing $\phi^{3}$ theory
in six dimensions, the physical problems of interest are in lower dimensions.
As noted we require a strategy to resum the asymptotic series in $\varepsilon$
for the critical exponents in order to extract meaningful estimates in for
example three dimensions. In this case $\varepsilon$ itself would take the
value $3$ which is clearly not small as $d=6-\varepsilon$. Therefore we devote
this section to discussing the various resummation techniques that we will
apply to both the Lee–Yang and percolation problems.

First we recall aspects of the resummation formalism that is well-established
in this area of quantum field theory. In general, an $\varepsilon$-expansion
series, $f(\varepsilon)$, for a critical exponent is asymptotic with
factorially growing coefficients, [59, 60, 61],

$f(\varepsilon)=\sum_{k=0}^{\infty}f_{k}\varepsilon^{k},\qquad f_{k}\propto
k!(-a)^{k}k^{b}\text{ for }k\rightarrow\infty~{},$ (47)

where the constant $a>0$ is related to the position of the closest singularity
$-1/a$ in the Borel plane. This number is the same for all exponents of the
underlying model. By contrast the parameter $b$ is related to the type of
singularity and may take a different value for different exponents. In the two
cases considered here we will only use the parameter $a$.

For the Lee–Yang problem we take $a=5/18$ [62, 63, 64]. For percolation theory
we use $a$ $=$ $15/28$ [65]. In [66] the slightly lower value of $10/21$ was
obtained, but later on in [67] it was shown that there was a discrepancy
caused by an incorrect integration contour in [66, 68]. If the contour is
corrected then the approach suggested in [66] leads to the same results as
[65]. Given that the precise value of $a$ is still not fully resolved yet, in
our subsequent resummation for percolation we have carried out the analysis
for both cases. It transpires that using either value of $a$ produces
estimates that are almost the same. The resulting error bars are consistently
of similar size and the difference of the two central values is completely
negligible within these error bars. For this reason, we only provide the
results based on the choice $a$ $=$ $15/28$. Finally we note that even though
the value of the implicit proportionality constant in eq. (47) remains
controversial [61, 69], its precise value is not required for the analysis we
have performed for either problem.

In order to obtain reliable estimates for critical exponents we have applied a
number of different resummation techniques. These are Padé, Padé–Borel–Leroy
(PBL), Borel resummation with conformal mapping (KP17) [30], double-sided Padé
and constrained versions of Padé, PBL and KP17. The main purpose of
implementing double-sided Padé and constrained resummations is to produce more
accurate estimates especially in lower dimensions such as $d$ $=$ $3$. For
both problems the constraint arises from the known exact values for exponents
in two dimensions. Those for percolation are derived from a minimal conformal
field theory with $m$ $=$ $2$ and central charge $c$ $=$ $0$. These conformal
field theories have been classified [24]. It turns out that taking into
account the known exact value at $d$ $=$ $2$ significantly improves estimates.
For the Lee–Yang problem exact values for some exponents are also are known in
$d$ $=$ $1$. We will now discuss technical aspects of each of the resummation
methods we used separately.

#### Padé, double-sided Padé.

Applying the method of Padé approximants is regarded as one of the most simple
resummation methods. It does not require any knowledge about the properties of
the series considered. Specifically the approximant is constructed as a
rational function

$P_{[L/M]}(\varepsilon):=\frac{P_{L}(\varepsilon)}{P_{M}(\varepsilon)}$ (48)

where $P_{L}(\varepsilon)$ and $P_{M}(\varepsilon)$ are polynomials of order
$L$ and $M$ respectively. The polynomials are chosen in such a way that the
expansion of the approximant up to $N=L+M+1$ coincides precisely with the
initial series. For a double-sided Padé expansion information from both sides
of the expansion interval is used. For the lower end, which is $d$ $=$ $2$ in
our case, we only know the first term of the expansion from conformal field
theory. So to find an estimate for the series up to $\varepsilon^{N}$ one
needs to consider approximants with $L+M+1=N+1$. Given that $d$ $=$ $1$ data
is available for the Lee–Yang problem too we need to consider approximants
with $L+M+1=N+2$ in that case.

One of the problems with Padé approximants is that different $L/M$
approximants can produce significantly different estimates and one has to
choose a “proper” approximant. Such subjective choices might be ill-conceived
and hence provide an incorrect estimate. In this article we follow the
strategy suggested in [70] which is as follows. In order to obtain an estimate
and error bar (apparent accuracy) of order $N$ we will consider all non-
marginal approximants of order $N$ and $N-1$ and view them as “independent
measurements”:

$x_{k}:=P_{[L_{k}/M_{k}]}(\varepsilon_{\text{phys}})\quad\text{ for }\quad
k=1,\ldots,n$ (49)

where $\varepsilon_{\text{phys}}$ is the value of the expansion parameter