where the estimate is computed. This gives the estimate and error bar as

$\langle x\rangle=\frac{x_{1}+\ldots+x_{n}}{n},\qquad\Delta
x=t_{0.95,n}\sqrt{\frac{(\langle x\rangle-x_{1})^{2}+\ldots+(\langle x\rangle-
x_{n})^{2}}{n(n-1)}}\;.$ (50)

where $t_{0.95,n}$ is the $t$-distribution with $p=0.95$ confidence level. In
the situation where fewer than 3 approximants survive, we do not provide an
error bar since usually this error bar is unreliable. We consider Padé
approximants as marginal if the denominator in eq. (48) has a root in the
interval $[0,2\varepsilon_{\text{phys}}]$.

The standard Padé technique also suffers from problems when the argument
becomes large because power law asymptotics $\varepsilon^{L-M}$ become
dominant. In this case the estimates have little predictive value. Thus we
limit ourselves to approximants with $|L-M|<3$ for which the asymptotics are
not so strong and become dominant at much larger values. The double-sided Padé
is free of this problem because the asymptotic growth at physical values of
the expansion parameter is limited by the constraint imposed at $d=2$ or
$d=1$.

#### Padé–Borel–Leroy.

By contrast the Padé–Borel–Leroy method is one of the simplest methods of
Borel resummation. The series under consideration is assumed to be Borel
summable with factorially growing and sign-alternating coefficients as in eq.
(47). In this case details of the large order behaviour such as the parameters
$a$ and $b$ are not relevant. To obtain an estimate and error bar we also
follow the technique suggested in [70] which is almost similar to the above
Padé strategy.

#### Borel resummation with conformal mapping.

An extension of the previous approach is Borel resummation with conformal
mapping which has proved itself as one of the most reliable and precise
resummation methods for the $\varepsilon$ expansion [71, 72, 73, 74, 30]. It
allows one to utilize information about the large order asymptotics and other
properties of the series under consideration. Meanwhile there are plenty of
realizations of this approach. For instance, in this paper we use the KP17
procedure which was developed in [30] for studying the $\varepsilon$ expansion
of $\phi^{4}$ theory critical exponents at six loops. It is very reliable and
well-documented with the technical details provided in Section V of [30].

#### Constrained resummations.

The idea of the constrained Borel resummation was proposed by R. Guida and J.
Zinn-Justin in [74] where it was called resummation with boundary condition.
We have also applied this idea not only to the KP17 procedure, which is very
similar to the method used in [74], but also to the above Padé and PBL
methods. It should be noted that Padé approximants are not Borel resummable.
It is more natural to apply constraints in the way described for double-sided
Padé, but we retain Padé approximants here as it provides an additional view
on constrained resummation. Also results from constrained and double-sided
Padé are in fact different because Padé approximants apply to different
series. Constrained resummation is based on the following series
transformation

$f(\varepsilon)=\sum_{n=0}^{\infty}f_{n}\varepsilon^{n}=f(\varepsilon_{\text{bc}})+(\varepsilon_{\text{bc}}-\varepsilon)\sum_{n=0}^{\infty}h_{n}\varepsilon^{n},$
(51)

where $f(\varepsilon_{\text{bc}})$ is the known value of the expanded exponent
at $\varepsilon_{\text{bc}}$ with bc standing for _boundary condition_. The
resummation is then applied to the series
$h(\varepsilon)=\sum_{n=0}^{\infty}h_{n}\varepsilon^{n}$ before being
substituted into (51) to obtain an estimate for $f(\varepsilon)$.
Uncertainties are computed for $h(\varepsilon)$ as described earlier for each
method before being transformed to an uncertainty for $f(\varepsilon)$ by
standard algebraic rules.

It should be noted that this approach allows one to apply only one constraint.
So for the Lee–Yang problem we will provide two constrained resummations for
each method. There will be one using the value of the exponent in $d=1$ and
another for $d=2$. We will distinguish these two constrained procedures as
<<c{Method}-{$d_{bc}$}>>. So for instance cPadé-1 is the constrained Padé with
the $d$ $=$ $1$ constraint.

#### Overall estimate.

Given that we will have a large number of estimates for each exponent from
different methods we must present an overall final estimate to summarize all
our results. One of the problems is that there is no method which provides the
smallest uncertainty for every exponent. Here we introduce an automatic
algorithm that gives higher weight to results from methods which have smaller
uncertainty. So we will compute final estimates as a weighted average of all
estimates with weights $w_{i}$ proportional to the inverse uncertainties.
Specifically we define

$A_{\text{final}}=\frac{\sum_{i}A_{i}w_{i}}{\sum_{i}w_{i}}$ (52)

which allows us to discard almost all estimates with very large errors. If
instead we were to perform a simple averaging that would significantly shift
the final estimate to an incorrect value.

The error estimate is computed from two parts which are the weighted standard
deviation and weighted average of the uncertainties given by

$E_{\text{final}}=\sqrt{E_{1}^{2}+E_{2}^{2}},\quad\textrm{where}\quad
E_{1}=\sqrt{\frac{\sum_{i}(A_{\text{final}}-A_{i})^{2}w_{i}}{\sum_{i}w_{i}}},\qquad
E_{2}=\frac{1}{\sum_{i}w_{i}}~{}.$ (53)

This strategy results in the following:

  1. 1.

methods with very large error bars almost always do not contribute to an
estimate and only slightly increase the error bar;

  2. 2.

if methods provide significantly different results with comparable error bars,
then the overall error bar increases;

  3. 3.

if all methods provide almost the same value then the error decreases.

In parallel to analysing the estimates from the five loop exponents we have
also repeated this exercise for Monte Carlo and series data in each integer
dimension. In this way we can compare data for exponents from perturbation
theory and simulations on the same level.

##  6 Resummation analysis

Having outlined the technical description of each of the resummation methods
we have used to extract critical exponents as well as how we arrive at our
error estimates, we devote this section to recording the actual values for a
wide range of exponents. For both Lee–Yang edge singularity and percolation
theory a table is provided for each exponent. In addition we illustrate that
data with an associated figure. Each figure shows the exponent estimates with
errors for dimensions 2, 3, 4 and 5 together with our overall Monte Carlo
summary values where these are available. The horizontal axis in each figure
corresponds to dimension $d$.

###  6.1 Lee–Yang edge singularity

For our Lee–Yang singularity analysis we focus on a small set of exponents. As
the main work of others has centred primarily around the exponent $\sigma$ we
have compiled as comprehensive an amount of independent results as possible in
order to draw comparisons. For $\eta$ and $\nu$, aside from the three loop
work of [5], we were only able to find one study of these exponents which was
[11]. That used the functional renormalization group approach and determined
$\eta$ and $\sigma$ directly. An additional exponent termed $\nu_{c}$ was also
computed. From the hyperscaling relations given in [11], it can be related to
our $\eta$ and $\nu$ via the expression $\nu$ $=$ $\nu_{c}/\sigma$. We have
used the values for $\eta$ and $\nu_{c}$ of [11] to produce an independent
estimate for $\nu$ to benchmark our results against. Each of the tables
follows a common theme. The top part of each is a compilation of data from
numerical methods with sources. This is followed by our resummation analysis
for both four and five loops. The former is provided to gauge convergence. In
each case three lines summarize the unconstrained and constrained estimates as
well as the combination of both. This can be compared to a similar estimate
from Monte Carlo (MC) and series data. To save space in the presentation of
the results in the tables the use of MC in the first column includes results
from series too.

| Reference |  $d$ $=$ $1$ |  $d$ $=$ $2$ |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---|---|---|---
| exact | -1.0 | -0.8 |  |  |
MC |  [11] (2016) |  |  | -0.586(29) | -0.316(16) | -0.126(6)
overall |  |  | -0.586(29) | -0.316(16) | -0.126(6)
4 loops | Padé | -1.177 | -0.8655 | -0.5893 | -0.38(11) | -0.153(12)
KP17 | -1.2(9) | -0.9(5) | -0.6(3) | -0.45(10) | -0.161(8)
cPadé-1 |  | -0.77(4) | -0.55(4) | -0.34(2) | -0.149(4)
cPadé-2 |  |  | -0.56(2) | -0.344(15) | -0.149(4)
cPBL-1 |  | -0.78(2) | -0.56(3) | -0.345(14) | -0.150(3)
cPBL-2 |  |  | -0.570(13) | -0.349(10) | -0.151(2)
cKP17-1 |  | -0.763(8) | -0.539(11) | -0.332(9) | -0.147(3)
cKP17-2 |  |  | -0.556(9) | -0.338(8) | -0.148(3)
Constr. Padé |  |  | -0.581(11) | -0.356(10) | -0.152(3)
non constr. | -1.2(9) | -0.9(5) | -0.6(3) | -0.42(11) | -0.158(10)
constr. |  | -0.77(2) | -0.56(2) | -0.343(14) | -0.149(3)
all | -1.2(9) | -0.77(3) | -0.56(2) | -0.35(2) | -0.150(5)
5 loops | Padé | -1.249 | -0.9148 | -0.6185 | -0.39(9) | -0.154(7)
PBL | -0.9(5) | -0.8(3) | -0.55(12) | -0.34(4) | -0.150(3)
KP17 | -1.2(2) | -0.91(11) | -0.61(4) | -0.36(7) | -0.152(11)
cPadé-1 |  | -0.78(3) | -0.56(2) | -0.346(11) | -0.151(2)
cPadé-2 |  |  | -0.570(13) | -0.349(8) | -0.1509(14)
cPBL-1 |  | -0.785(11) | -0.564(11) | -0.348(6) | -0.1508(11)
cPBL-2 |  |  | -0.571(6) | -0.350(4) | -0.1511(8)
cKP17-1 |  | -0.765(7) | -0.542(9) | -0.334(8) | -0.148(4)
cKP17-2 |  |  | -0.557(9) | -0.338(8) | -0.148(3)
Constr. Padé |  |  | -0.580(7) | -0.356(6) | -0.1521(13)
non constr. | -1.2(3) | -0.9(2) | -0.60(7) | -0.36(6) | -0.152(6)
constr. |  | -0.774(15) | -0.565(15) | -0.347(10) | -0.151(2)
all | -1.2(3) | -0.78(3) | -0.57(2) | -0.347(12) | -0.151(2)
Table 1: Estimates for Lee–Yang $\eta$. Figure 2: Plot of estimates for
Lee–Yang $\eta$.

For the exponent $\eta$ our results are in Table 1. We see that the estimates
for the three key dimensions are stable from four to five loops. However for
the only available study that we could locate [11] there is some overlap
agreement for the 3 dimensional estimate in contrast to those for higher
dimensions. Although this exponent is ordinarily regarded as one of the more
difficult ones to reconcile between different methods since it is very close
to zero. This is not the case here.

| Reference |  $d$ $=$ $1$ |  $d$ $=$ $2$ |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---|---|---|---
MC |  [11] (2016) |  |  | (4.826146) | (1.187477) | (0.696008)
overall |  |  |  |  |
4 loops | Padé |  |  |  | 1.088 | 0.701(4)
KP17 | 1.6(1.2) | 1.3(9) | 1.1(6) | 0.9(3) | 0.69(3)
non constr. | 1.6(1.2) | 1.3(9) | 1.1(6) | 0.9(3) | 0.699(8)
all | 1.6(1.2) | 1.3(9) | 1.1(6) | 0.9(3) | 0.699(8)
5 loops | Padé |  |  |  |  | 0.7022(12)
PBL | 2(2) | 1.6(1.2) | 1.3(5) | 0.98(10) | 0.696(3)
KP17 | 1.6(1.3) | 1.4(1.0) | 1.1(7) | 0.9(2) | 0.70(2)
non constr. | 2(2) | 1.5(1.1) | 1.2(6) | 0.95(14) | 0.700(4)
| all | 2(2) | 1.5(1.1) | 1.2(6) | 0.95(14) | 0.700(4)
Table 2: Estimates for Lee–Yang $\nu$. | Reference |  $d$ $=$ $1$ |  $d$ $=$ $2$ |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---|---|---|---
| exact | -1.0 | -2.5 |  |  |
MC |  [11] (2016) |  |  | (4.826146) | (1.187477) | (0.696008)
overall |  |  |  |  |
4 loops | Padé | -1.1(1.1) | -3(5) | 4(3) | 1.18(10) | 0.701(4)
PBL | -1(2) | -4(11) | 3(3) | 1.17(14) | 0.700(6)
KP17 | -0.8(3) | -2.1(1.1) | 6(4) | 1.23(5) | 0.702(3)
cPadé-1 |  | -2.58(11) | 4.5(3) | 1.205(14) | 0.7015(10)
cPadé-2 |  |  | 4.6(2) | 1.208(10) | 0.7016(8)
cPBL-1 |  | -2.56(8) | 4.5(3) | 1.208(10) | 0.7017(6)
cPBL-2 |  |  | 4.64(14) | 1.211(7) | 0.7019(5)
cKP17-1 |  | -2.61(7) | 4.4(3) | 1.20(2) | 0.701(2)
cKP17-2 |  |  | 4.58(12) | 1.206(8) | 0.7017(11)
Constr. Padé |  |  | 4.74(2) | 1.215(2) | 0.7022(2)
non constr. | -1.0(7) | -2(3) | 4(3) | 1.20(9) | 0.701(4)
constr. |  | -2.58(9) | 4.67(13) | 1.211(8) | 0.7019(6)
all | -1.0(7) | -2.6(2) | 4.7(2) | 1.211(11) | 0.7019(9)
5 loops | Padé | -1.0(4) | -2.4(1.3) | 5(2) | 1.22(4) | 0.7022(12)
PBL | -2(2) | -5(11) | 3(2) | 1.17(8) | 0.701(2)
KP17 | -0.92(15) | -2.3(5) | 5.0(1.2) | 1.22(2) | 0.7021(11)
cPadé-1 |  | -2.55(8) | 4.6(2) | 1.209(8) | 0.7019(4)
cPadé-2 |  |  | 4.65(14) | 1.211(6) | 0.7020(3)
cPBL-1 |  | -2.54(4) | 4.60(11) | 1.210(4) | 0.7019(2)
cPBL-2 |  |  | 4.66(6) | 1.212(2) | 0.70199(14)
cKP17-1 |  | -2.61(9) | 4.4(3) | 1.211(15) | 0.7020(5)
cKP17-2 |  |  | 4.58(14) | 1.212(10) | 0.7020(7)
Constr. Padé |  |  | 4.71(7) | 1.214(4) | 0.7021(2)
non constr. | -1.0(3) | -2.4(1.2) | 4(2) | 1.21(4) | 0.7019(14)
constr. |  | -2.56(6) | 4.61(14) | 1.211(5) | 0.7020(3)
all | -1.0(3) | -2.6(2) | 4.6(2) | 1.211(8) | 0.7020(4)
Table 3: Estimates for Lee–Yang $\nu$ using the series for $1/\nu$. Figure 3:
Plot of estimates (from $1/\nu$) for Lee–Yang $\nu$.

Our direct estimation of $\nu$ suffered from singularity issues. To illustrate
this situation we have included Table 2 where data from those few techniques
that did render a reliable estimate are presented. However the resultant large
error bars mean that these values cannot be regarded as reliable. Instead we
took a different tack and applied our resummation to the series for $1/\nu$
before inverting the final numerical value. This gave problem-free data for
all our approaches which is recorded in Table 3. The behaviour of $\nu$ over
the dimensions indicates that there is a maximum. Clearly $\nu$ is positive
and increases as $d$ decreases to a large value in $3$ dimensions before
becoming negative in two dimensions where the exact values are known. The
functional renormalization group results have a similar behaviour and we are
in qualitative accord at the very least.

| Reference |  $d$ $=$ $1$ |  $d$ $=$ $2$ |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---|---|---|---
| exact | -0.5 | -0.1667 |  |  |
MC |  [5] (1981) |  | -0.146 – -0.166 | 0.079–0.091 | 0.262–0.266 | 0.399–0.400
[75] (1995) |  | -0.165(6) | 0.080(6) | 0.261(12) | 0.40(2)
(hdsc) [25] (2012) |  | -0.1662(5) | 0.077(2) | 0.258(5) | 0.401(9)
(hdbcc) [25] (2012) |  | -0.1662(5) | 0.076(2) | 0.261(4) | 0.402(2)
[76] (2014) |  | -0.161(8) | 0.0877(25) | 0.2648(16) | 0.402(5)
[10] (2014) |  | -0.1664(5) | 0.085(1) | 0.2685(1) | 0.4105(5)
[11] (2016) |  |  | 0.0742(56) | 0.2667(32) | 0.4033(12)
(FRG) [12] (2017) |  |  |  | 0.2648(6) | 0.40166(2)
(LPA) [12] (2017) |  | -0.193 | 0.0588 | 0.245 | 0.394
| overall |  | -0.1661(11) | 0.081(5) | 0.267(2) | 0.402(2)/0.407(4)
4 loops | Padé | -0.1(4) | 0.0(2) | 0.16(12) | 0.27(4) | 0.400(4)
PBL | -0.1(4) | 0.0(2) | 0.16(12) | 0.28(4) | 0.401(4)
KP17 | 0.1(6) | -0.3(4) | 0.0(2) | 0.24(2) | 0.397(3)
cPadé-1 |  | -0.18(2) | 0.07(2) | 0.258(7) | 0.3983(12)
cPadé-2 |  |  | 0.073(10) | 0.259(6) | 0.3983(10)
cKP17-1 |  | -0.17(3) | 0.07(3) | 0.258(11) | 0.398(2)
cKP17-2 |  |  | 0.074(12) | 0.258(8) | 0.398(2)
Constr. Padé |  |  | 0.077(2) | 0.260(2) | 0.3985(5)
non constr. | -0.1(5) | -0.1(3) | 0.1(2) | 0.26(4) | 0.399(4)
constr. |  | -0.18(3) | 0.075(7) | 0.259(5) | 0.3984(7)
all | -0.1(5) | -0.16(9) | 0.08(2) | 0.259(9) | 0.3985(11)
5 loops | Padé | -0.3113 | -0.0923 | 0.1002 | 0.26(3) | 0.398(2)
PBL | -0.2(2) | -0.02(12) | 0.13(5) | 0.272(13) | 0.3991(8)
KP17 |  | -0.21(10) | 0.06(6) | 0.257(14) | 0.3982(12)
cPadé-1 |  | -0.18(2) | 0.068(12) | 0.257(4) | 0.3983(8)
cPadé-2 |  |  | 0.072(6) | 0.258(3) | 0.3981(4)
cPBL-1 |  | -0.18(3) | 0.07(2) | 0.257(7) | 0.3981(7)
cPBL-2 |  |  | 0.072(11) | 0.258(5) | 0.3981(6)
cKP17-1 |  | -0.17(2) | 0.072(12) | 0.258(4) | 0.3983(5)
cKP17-2 |  |  | 0.075(6) | 0.259(2) | 0.3984(4)
Constr. Padé |  |  | 0.078(2) | 0.2602(14) | 0.3984(2)
non constr. | -0.2(2) | -0.12(14) | 0.10(6) | 0.26(2) | 0.3986(13)
constr. |  | -0.18(2) | 0.075(7) | 0.259(3) | 0.3983(5)
all | -0.2(2) | -0.17(5) | 0.075(11) | 0.259(5) | 0.3983(6)
Table 4: Estimates for Lee–Yang $\sigma$. Figure 4: Plot of estimates for
Lee–Yang $\sigma$.

For $\sigma$ a similar picture emerges in Table 4 where $hdsc$ and $hdbcc$
indicate results in $d$ dimensions from separate simulations. Also FRG denotes
the functional renormalization group and LPA indicates the use of the local
potential approximation. In compiling the overall estimate and error for the
first bank of the table we have excluded the results of [5] and the LPA data
of [12] due to the absence of errors. We provide two estimates in 5
dimensions. The first includes the FRG result of [12] while the second omits
it. Clearly with the increase in the order of $\varepsilon$ the individual
constrained estimates are in good agreement for $4$ and $5$ dimensions but
less so for $3$ dimensions. This is primarily because the monotonic decrease
in the value of $\sigma$ from $6$ down to $2$ dimensions where it is positive
in the critical dimension but negative in $1$ and $2$ means that at some non-
integer dimension it will vanish. This appears to be in the neighbourhood of
$3$ dimensions as indicated by the relatively small value in the hundredths.
Indeed this is similar to the situation alluded to for $\eta$ in certain
problems. Moreover the same feature is present in $3$ dimensional estimates
from the other methods we have provided in Table 4. This is ultimately
reflected in our final five loop estimate and in particular in the error. This
should not overlook however the very good agreement for $4$ and $5$
dimensions.

| Reference |  $d$ $=$ $1$ |  $d$ $=$ $2$ |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---|---|---|---
4 loops | Padé | 1(4) | 1(3) | 1.3(1.4) | 1.1(5) | 0.69(10)
PBL | 2.4(5) | 2.1(3) | 1.7(2) | 1.22(6) | 0.704(11)
KP17 | 2.3(6) | 1.9(4) | 1.6(3) | 1.18(14) | 0.69(3)
non constr. | 2.3(8) | 2.0(5) | 1.6(3) | 1.20(13) | 0.70(2)
all | 2.3(8) | 2.0(5) | 1.6(3) | 1.20(13) | 0.70(2)
5 loops | Padé | 1(4) | 1(2) | 1.2(1.1) | 1.0(4) | 0.69(6)
PBL | 1.7(3) | 1.6(2) | 1.42(8) | 1.12(3) | 0.689(3)
KP17 | 2.3(4) | 2.2(3) | 1.70(15) | 1.22(6) | 0.701(11)
non constr. | 1.9(6) | 1.8(4) | 1.5(2) | 1.15(7) | 0.691(8)
all | 1.9(6) | 1.8(4) | 1.5(2) | 1.15(7) | 0.691(8)
Table 5: Estimates for Lee–Yang $\omega$. Figure 5: Plot of estimates for
Lee–Yang $\omega$.

Finally the situation with the exponent $\omega$ is less clear. This is
primarily due to the lack of an exact value for this exponent in two
dimensions. Therefore we are only able to record the results of unconstrained
resummations which are given in Table 5. Equally we were unable to compare the
estimates with Monte Carlo studies. So no definite conclusions should be drawn
for $\omega$ in the Lee–Yang study.

###  6.2 Percolation theory

For percolation theory we recall the exact two dimensional values of the
various exponents in Table 6 that were used when constraints were implemented
as indicated in Section 5. We will use the 2 dimensional estimate for the
unconstrained resummations as a benchmark to gauge whether our extrapolation
is consistent with the exact value. In several instances there was either a
significant undershoot or overshoot for both the unconstrained and constrained
case. This gave a strong indication that the higher dimensional estimates
could be unreliable. Consequently we examined the $\varepsilon$ expansion of
various functions of the exponent, such as its reciprocal. Some of these gave
improved estimates and a projection to 2 dimensions that was more in keeping
with the exact values. This is illustrated in Table 6 which also summarizes
our five loop two dimensional estimates from the various resummation methods.
The table also records the summary of [26]; see also [77, Table 2]. While the
exact values derive from two dimensional conformal field theory we note that a
Monte Carlo study [78] which predated [24] gave estimates that were in
remarkably good agreement. For instance, the values of $\alpha$ $=$ $-$
$0.708$ $\pm$ $0.030$, $\beta$ $=$ $0.138(+0.006,-0.005)$, $\gamma$ $=$
$2.432$ $\pm$ $0.035$, $\delta$ $=$ $18.6$ $\pm$ $0.6$, $\nu$ $=$ $1.354$
$\pm$ $0.015$ and $\eta$ $=$ $0.204$ $\pm$ $0.006$ were determined in [78].

We note that here we use the value of $3/2$ for $\omega$ and $72/91$ for
$\Omega$. This is in contrast with the value of $2$ for $\omega$ used in [17]
based on [79] which also gave an argument that $\Omega$ was $96/91$. The
discrepancy between the exact 2 dimensional values for $\omega$ and $\Omega$
was discussed at length in [80]. In both cases the ratio $\omega/\Omega$ is
the same and corresponds to the fractal dimension of the system. Further a
summary is given in [80] of independent calculations of both exponents. These
appear to be more consistent with the respective exact values of $3/2$ and
$72/91$. One test is the value of the exponent $\Delta_{1}$ $=$
$\Omega/\sigma$ which is $2$ for $\Omega$ $=$ $72/91$ but $8/3$ for [79]. We
note that naively taking our central values gives $2.04545$ for $\Delta_{1}$.

exponent | exact | numerical | 4 loop | 5 loop
---|---|---|---|---
overall | Padé | PBL | KP17 | overall
$\alpha$ | $\nicefrac{{-2}}{{3}}$ | -0.667 | -0.56(9) | -0.57(6) | -0.49(13) | -0.41(12) | -0.51(11)
$\beta$ | $\nicefrac{{5}}{{36}}$ | 0.139 | 0.2(3) | 0.3(3) | 0.3(3) | 0.04(40) | 0.2(3)
$\gamma$ | $\nicefrac{{43}}{{18}}$ | 2.39 | 1.9(4) | 2.0(4) | 1.8(3) | 2.2(3) | 2.0(4)
$\delta$ | $\nicefrac{{91}}{{5}}$ | 18.2 | 4(2) |  | 3(5) | 4(2) | 4(3)
$\delta$ $(1/\delta)$ | 7(15) | 16.78 | 10(6) | -25(75) | 7(14)
$\eta$ | $\nicefrac{{5}}{{24}}$ | 0.208 | -0.32(11) | -0.32(12) | -0.1(2) | -0.31(10) | -0.28(15)
$\nu$ | $\nicefrac{{4}}{{3}}$ | 1.33 | 0.87(15) | 0.9315 | 0.86(13) | 0.8(3) | 0.9(2)
$\nu$ $(1/\nu)$ | 1.1(4) | 1.1(2) | 0.99(13) | 1.17(12) | 1.1(2)
$\sigma$ | $\nicefrac{{36}}{{91}}$ | 0.396 | 0.46(4) | 0.44(4) | 0.446(13) | 0.44(5) | 0.44(2)
$\tau$ | $\nicefrac{{197}}{{91}}$ | 2.06 | 2.08(10) | 2.07(4) | 2.08(2) | 2.00(13) | 2.07(4)
$\Omega$ | $\nicefrac{{72}}{{91}}$ | 0.791 | 0.76(2) | 0.6(3) | 0.96(5) | 0.7(2) | 0.9(2)
$\omega$ | $\nicefrac{{3}}{{2}}$ | 1.5 | 2.6(2) | 2(2) | 2.37(5) | 2.8(7) | 2.4(2)
Table 6: Exact values of exponents in $d$ $=$ $2$ together with our estimates.

We now discuss the results for each exponent in alphabetical order. In Tables
7–18 we have summarized the estimates for the various exponents in the
literature. One excellent source we benefited from in this respect is the
table of [26] which is regularly updated. Included in this are our results for
the various resummations. In several of our tables we also have a line
designated as $\varepsilon^{4}$. This records the results of the constrained
Padé method at four loops given in [17].

To open with, our estimates for $\alpha$ are presented in Table 7. One aspect
that is evident is the close agreement of the estimates from both the
constrained and unconstrained approaches for both 4 and 5 dimensions and to a
lesser extent for $3$ dimensions. The latter is in essence due to the
overshoot of the two dimensional exact value for the unconstrained methods.
Indeed the five loop value has a more pronounced discrepancy than at four
loops. Given the slower decrease in the constrained estimate as $d$ decreases,
we would take the position that those estimates are probably more reliable.
Independent Monte Carlo data for each dimension would be useful to compare
with.

| Reference |  $d$ $=$ $2$ |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---|---|---
| exact | -0.6667 |  |  |
4 loops | Padé | -0.58(6) | -0.67(4) | -0.76(2) | -0.873(3)
PBL | -0.57(6) | -0.66(4) | -0.76(2) | -0.872(3)
KP17 | -0.46(15) | -0.60(7) | -0.74(3) | -0.869(4)
cPadé-2 |  | -0.708(11) | -0.778(9) | -0.875(3)
cPBL-2 |  | -0.710(13) | -0.780(11) | -0.876(4)
cKP17-2 |  | -0.72(2) | -0.79(2) | -0.874(6)
Constr. Padé |  | -0.7212(4) | -0.782(14) | -0.877(5)
non constr. | -0.56(9) | -0.65(5) | -0.76(2) | -0.871(4)
constr. |  | -0.720(3) | -0.781(13) | -0.876(4)
all | -0.56(9) | -0.719(12) | -0.77(2) | -0.874(5)
5 loops | Padé | -0.57(6) | -0.66(3) | -0.759(12) | -0.872(2)
PBL | -0.49(13) | -0.62(6) | -0.75(2) | -0.871(2)
KP17 | -0.41(12) | -0.59(4) | -0.736(7) | -0.8690(6)
cPadé-2 |  | -0.703(14) | -0.773(9) | -0.8739(13)
cPBL-2 |  | -0.69(4) | -0.77(2) | -0.872(4)
cKP17-2 |  | -0.72(2) | -0.78(2) | -0.870(6)
Constr. Padé |  | -0.723(2) | -0.78(2) | -0.880(10)
non constr. | -0.51(11) | -0.62(5) | -0.745(15) | -0.870(2)
constr. |  | -0.720(11) | -0.77(3) | -0.873(6)
| all | -0.51(11) | -0.71(3) | -0.76(2) | -0.870(3)
Table 7: Estimates for $\alpha$ in percolation. Figure 6: Plot of estimates
for $\alpha$ in percolation.

For $\beta$ the situation is much clearer in that we have several independent
results provided in Table 8. Again there is a large degree of stability within
each of the comparable methods and loop order although the unconstrained Padé
clearly overshoots the two dimensional exact result. From four to five loops
the final estimates are remarkably stable. Comparing, however, with [81, 82]
the loop estimates are all slightly larger even for 5 dimensions.

| Reference |  $d$ $=$ $2$ |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---|---|---
| exact | 0.1389 |  |  |
$\varepsilon^{4}$ |  [17] (2015) |  | 0.4273 | 0.6590 | 0.8457
MC |  [82] (1976) |  | 0.41(1) |  |
[81] (1990) |  | 0.405(25) | 0.639(20) | 0.835(5)
overall |  | 0.409(14) | 0.64(2) | 0.835(5)
4 loops | Padé | 0.3(3) | 0.49(13) | 0.68(4) | 0.847(4)
KP17 | 0.0(4) | 0.40(9) | 0.65(2) | 0.845(4)
cPadé-2 |  | 0.426(8) | 0.657(5) | 0.8453(12)
cPBL-2 |  | 0.419(7) | 0.653(5) | 0.8444(12)
cKP17-2 |  | 0.424(11) | 0.656(7) | 0.845(2)
Constr. Padé |  | 0.4303 | 0.6599(2) | 0.8458
non constr. | 0.2(3) | 0.43(11) | 0.66(3) | 0.846(4)
constr. |  | 0.423(9) | 0.656(6) | 0.8449(14)
all | 0.2(3) | 0.42(2) | 0.656(10) | 0.845(2)
5 loops | Padé | 0.3(3) | 0.49(13) | 0.68(4) | 0.845(4)
PBL | 0.3(3) | 0.48(13) | 0.67(4) | 0.847(4)
KP17 | 0.0(4) | 0.39(6) | 0.650(11) | 0.8447(10)
cPadé-2 |  | 0.427(6) | 0.658(4) | 0.8455(6)
cPBL-2 |  | 0.417(5) | 0.653(2) | 0.8448(2)
cKP17-2 |  | 0.422(5) | 0.655(3) | 0.8450(5)
Constr. Padé |  | 0.42(2) | 0.656(12) | 0.8453(15)
non constr. | 0.2(3) | 0.44(10) | 0.66(2) | 0.845(2)
constr. |  | 0.420(8) | 0.655(4) | 0.8449(5)
| all | 0.2(3) | 0.42(2) | 0.655(7) | 0.8449(7)
Table 8: Estimates for $\beta$ in percolation. Figure 7: Plot of estimates for $\beta$ in percolation. | Reference |  $d$ $=$ $2$ |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---|---|---
| exact | 2.3889 |  |  |
$\varepsilon^{4}$ |  [17] (2015) |  | 1.8357 | 1.4500 | 1.1817
MC |  [82] (1976) |  | 1.6 |  |
[81] (1990) |  | 1.805(20) | 1.435(15) | 1.185(5)
overall |  | 1.805(20) | 1.435(15) | 1.185(5)
4 loops | Padé | 1.946 | 1.7(3) | 1.44(6) | 1.179(7)
PBL | 1.9(3) | 1.65(14) | 1.40(5) | 1.175(7)
KP17 | 1.9(8) | 1.8(2) | 1.44(6) | 1.181(4)
cPadé-2 |  | 1.83(3) | 1.45(2) | 1.181(3)
cKP17-2 |  | 1.82(6) | 1.44(4) | 1.180(6)
Constr. Padé |  | 1.82(4) | 1.44(2) | 1.180(3)
non constr. | 1.9(4) | 1.7(2) | 1.42(6) | 1.179(6)
constr. |  | 1.82(4) | 1.44(2) | 1.180(4)
all | 1.9(4) | 1.81(8) | 1.44(3) | 1.180(5)
5 loops | Padé | 2.0(4) | 1.8(2) | 1.45(4) | 1.181(4)
PBL | 1.8(3) | 1.6(2) | 1.38(4) | 1.176(3)
KP17 | 2.2(3) | 1.77(13) | 1.43(2) | 1.179(2)
cPadé-2 |  | 1.83(3) | 1.448(13) | 1.181(2)
cPBL-2 |  | 1.83(4) | 1.45(2) | 1.181(3)
cKP17-2 |  | 1.81(5) | 1.437(13) | 1.1797(14)
Constr. Padé |  | 1.83(3) | 1.444(14) | 1.181(2)
non constr. | 2.0(4) | 1.7(2) | 1.42(4) | 1.179(3)
constr. |  | 1.82(4) | 1.44(2) | 1.180(2)
all | 2.0(4) | 1.81(8) | 1.44(3) | 1.180(3)
Table 9: Estimates for $\gamma$ in percolation. Figure 8: Plot of estimates
for $\gamma$ in percolation.

In the case of $\gamma$ the independent Monte Carlo data was only available
from the same articles as our comparison for $\beta$. There was a similar
picture for our estimates in that there was remarkable stability between four
and five loops for the three dimensions of interest as is clear from Table 9.
However the check estimate for two dimensions was much closer to the exact
value than that for $\beta$. This may be one of the reasons why our final
values for five loops are in remarkably good agreement with the Monte Carlo
results of [81]. There was no error bar for the 3 dimensional result of [82].
So it is not clear how to interpret the central value that is 12% lower than
both [81] and our estimate at five loops.

| Reference |  $d$ $=$ $2$ |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---|---|---
| exact | 18.2 |  |  |
MC |  [82] (1976) |  |  | 3.198(6) |
[83] (1980) |  | 5.3 | 3.9 | 3.0
[84] (1998) |  | 5.29(6) |  |
overall |  | 5.29(6) | 3.198(6) |
4 loops | Padé | 4.756 | 3.742 | 3.1(2) | 2.391(14)
KP17 | 4(2) | 3.3(1.3) | 3.1(4) | 2.39(3)
cPadé-2 |  | 7.831 | 3.2(3) | 2.40(2)
cPBL-2 |  | 11(4) | 6(4) | 2.6(4)
cKP17-2 |  | 10(4) | 3(2) | 2.40(11)
Constr. Padé |  | 5.1(2) | 3.15(5) | 2.393(5)
non constr. | 4(2) | 3.3(1.3) | 3.1(3) | 2.39(2)
constr. |  | 6(2) | 3.2(3) | 2.40(2)
all | 4(2) | 5(2) | 3.2(3) | 2.40(2)
5 loops | Padé |  |  | 3.15(8) | 2.396(5)
PBL | 3(5) | 3(3) | 2.5(1.5) | 2.3(2)
KP17 | 4(2) | 3.3(1.5) | 3.2(2) | 2.396(6)
cPadé-2 |  | 7.831 | 3.3(3) | 2.40(2)
cPBL-2 |  | 8.92(5) | 3.84(2) | 2.4061(12)
cKP17-2 |  | 10(4) | 3.3(1.3) | 2.40(5)
Constr. Padé |  | 5.16(7) | 3.175(14) | 2.3954(11)
non constr. | 4(3) | 3(2) | 3.1(2) | 2.395(12)
constr. |  | 7(2) | 3.4(3) | 2.401(6)
| overall | 4(3) | 7(2) | 3.4(3) | 2.400(7)
Table 10: Estimates for $\delta$ in percolation.

The situation with $\delta$ was much less straightforward and we include two
tables with data. In Table 10 we have applied our various resummation
techniques directly to the $\varepsilon$ expansion of $\delta$. This
illustrates one immediate difficulty in obtaining reliable values for low $d$.
The value for $\delta$ in exactly 6 dimensions is $2$ whereas it is exactly
$18.2$ in two dimensions. The latter places a big restriction on the ability
of the $\varepsilon$ expansion to manufacture large corrections as $d$
decreases. This is clear in both our four and five loop estimates from both
our unconstrained and constrained resummations. Moreover the skew that is
necessary to meet the two dimensional constraint clearly makes the convergence
between loops in 3 dimensions impossible. Equally there is no agreement with
the few independent estimates which both have more precise values. Although
the direct $\varepsilon$ expansion approach is problematic we have provided it
here partly to be transparent in the application of our methods but also to
highlight the hidden pitfalls in naively applying various resummation methods.
Equally to address this problem we have instead tried a second approach which
was to sum the $\varepsilon$ expansion of $1/\delta$, determine a numerical
value and then invert this. The results of this exercise are given in Table
11. Many of the failings with the direct evaluation are absent now. There is a
degree of stability between the various four and five loop results especially
in 3 dimensions. Equally and more importantly we find that this and the 4
dimensional results are certainly not out of line with the results of [84, 82]
although our 3 dimensional result has large errors. While the extrapolation to
2 dimensions has larger errors it does accommodate the exact value. In some
sense this much more consistent agreement across Table 11 is an a posteriori
justification of following this second strategy.

| Reference |  $d$ $=$ $2$ |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---|---|---
| exact | 18.2 |  |  |
MC |  [82] (1976) |  |  | 3.198(6) |
[83] (1980) |  | 5.3 | 3.9 | 3.0
[84] (1998) |  | 5.29(6) |  |
overall |  | 5.29(6) | 3.198(6) |
4 loops | Padé | 8(10) | 5(2) | 3.1(2) | 2.391(14)
PBL | 8(10) | 4.4(1.4) | 3.1(2) | 2.39(2)
KP17 | -20(120) | 6.3(1.5) | 3.25(9) | 2.399(4)
cPadé-2 |  | 5.13(11) | 3.17(3) | 2.395(3)
cPBL-2 |  | 5.19(8) | 3.18(2) | 2.396(2)
cKP17-2 |  | 5.15(12) | 3.17(3) | 2.395(5)
Constr. Padé |  | 5.2(2) | 3.17(6) | 2.394(6)
non constr. | 7(15) | 5(2) | 3.2(2) | 2.395(9)
constr. |  | 5.17(12) | 3.18(3) | 2.395(3)
| all | 7(15) | 5.2(3) | 3.18(5) | 2.395(5)
5 loops | Padé | 16.78 | 6(2) | 3.22(15) | 2.397(7)
PBL | 10(6) | 4.7(6) | 3.13(7) | 2.393(3)
KP17 | -25(75) | 6.1(1.5) | 3.25(9) | 2.398(4)
cPadé-2 |  | 5.16(5) | 3.179(14) | 2.3956(12)
cPBL-2 |  | 5.18(7) | 3.179(14) | 2.3956(11)
cKP17-2 |  | 5.16(3) | 3.175(8) | 2.3952(10)
Constr. Padé |  | 5.1(2) | 3.17(3) | 2.395(2)
non constr. | 7(14) | 5.2(1.2) | 3.19(11) | 2.396(5)
constr. |  | 5.16(6) | 3.176(14) | 2.3954(12)
| all | 7(14) | 5.16(15) | 3.18(3) | 2.396(2)
Table 11: Estimates for $\delta$ in percolation using the series for $1/\delta$. Figure 9: Plot of estimates (from $1/\delta$) for $\delta$ in percolation. | Reference |  $d$ $=$ $3$ |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---|---|---
| exact | 0.2083 |  |  |
$\varepsilon^{4}$ |  [17] (2015) |  | -0.0470 | -0.0954 | -0.0565
MC |  [81] (1990) |  | -0.07(5) | -0.12(4) | -0.075(20)
[85] (1996) |  |  | -0.0944(28) |
[84] (1998) |  | -0.046(8) |  |
[86] (1998) |  | -0.059(9) |  |
[87] (2001) |  |  | -0.0929(9) |
overall |  | -0.054(14) | -0.094(4) | -0.075(20)
4 loops | Padé | -0.32(13) | -0.21(6) | -0.13(2) | -0.059(3)
KP17 | -0.33(10) | -0.23(5) | -0.14(2) | -0.061(3)
cPadé-2 |  | 0.01143 | -0.06287 | -0.055(7)
cPBL-2 |  | 0.04(9) | -0.06(3) | -0.051(6)
cKP17-2 |  | -0.09(12) | -0.10(3) | -0.057(3)
non constr. | -0.32(11) | -0.22(5) | -0.13(2) | -0.060(3)
constr. |  | -0.02(12) | -0.08(4) | -0.055(5)
all | -0.32(11) | -0.15(13) | -0.12(4) | -0.057(5)
5 loops | Padé | -0.32(12) | -0.22(5) | -0.13(2) | -0.059(2)
PBL | -0.1(2) | -0.14(10) | -0.11(3) | -0.057(3)
KP17 | -0.31(10) | -0.23(5) | -0.14(2) | -0.061(3)
cPadé-2 |  | -0.003(13) | -0.074(7) | -0.056(4)
cPBL-2 |  | 0.06(7) | -0.03(5) | -0.049(5)
cKP17-2 |  | -0.04(2) | -0.091(8) | -0.0553(9)
Scaling Padé |  | (-0.0566) | (-0.0833) | (-0.0545)
Hyperscaling Padé |  | (-0.0176) | (-0.0790) | (-0.0540)
non constr. | -0.28(15) | -0.20(7) | -0.13(2) | -0.059(3)
constr. |  | -0.01(4) | -0.08(2) | -0.055(3)
all | -0.28(15) | -0.06(10) | -0.10(3) | -0.056(3)
Table 12: Estimates for $\eta$ in percolation. Figure 10: Plot of estimates
for $\eta$ in percolation.

We now turn to one of the more widely measured exponents which is $\eta$ with
data recorded in Table 12. As is the case with $\eta$ in other problems its
value in percolation is relatively small and close to zero being associated
with the wave function anomalous dimension. Consequently it is difficult to
obtain an accurate value for it that is in consistent agreement with other
estimates. This is evident in the numerical values of [81, 84, 86, 87, 85] in
3 and 4 dimensions although those of [87, 85] are the most accurate and do
overlap. On the $\varepsilon$ expansion side the Padé resummations suffered
from the presence of at least one singularity in $2$ $<$ $d$ $<$ $6$ or were
not monotonic. For this case we indirectly estimated $\eta$ using the
respective scaling and hyperscaling relations

$\eta~{}=~{}2~{}-~{}\frac{\gamma}{\nu}~{},\quad\eta~{}=~{}\frac{d+2-(d-2)\delta}{1+\delta}~{}.$
(54)

The numerical values for each of these are bracketed to indicate that they
were not derived directly. A more difficult issue to circumvent for $\eta$ in
the constrained approach is the relatively large and positive exact value in 2
dimensions. This means that $\eta$ would have to be identically zero somewhere
between $3$ and $2$ dimensions as $d$ decreases. So while our final 3
dimensional estimate of $\eta$ has shown an improvement from four to five
loops this is to be balanced against a necessarily large error.

| Reference |  $d$ $=$ $2$ |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---|---|---
| exact | 1.3333 |  |  |
$\varepsilon^{4}$ |  [17] (2015) |  | 0.8960 | 0.6920 | 0.5746
MC |  [88] (1975) |  | 0.80(5) |  |
[82] (1976) |  | 0.8(1) |  |
[81] (1990) |  | 0.872(70) | 0.678(50) | 0.571(3)
[85] (1997) |  |  | 0.689(10) |
[84] (1998) |  | 0.875(1) |  |
[89] (2000) |  | 0.8765(18) |  |
[90] (2013) |  | 0.8764(12) |  |
[91] (2014) |  | 0.8751(11) |  |
[92] (2014) |  | 0.87619(11) |  |
[93] (2016) |  | 0.8774(13) | 0.6852(28) | 0.5723(18)
[13] (2018) |  |  | 0.693 |
[94] (2020) |  |  | 0.6845(6) | 0.5757(7)
overall |  | 0.876(4) | 0.685(2) | 0.574(2)
4 loops | Padé | 0.8812 | 0.78(8) | 0.68(2) | 0.573(2)
PBL | 0.88(10) | 0.77(5) | 0.66(2) | 0.572(3)
KP17 | 0.8(3) | 0.9(2) | 0.68(3) | 0.574(3)
cPadé-2 |  | 0.9496 | 0.697(15) | 0.575(2)
cKP17-2 |  | 0.95(12) | 0.69(4) | 0.574(3)
Constr. Padé |  | 0.896(3) | 0.6916(12) | 0.5746(2)
non constr. | 0.87(15) | 0.79(9) | 0.67(2) | 0.573(3)
constr. |  | 0.897(10) | 0.692(4) | 0.5746(5)
all | 0.87(15) | 0.89(4) | 0.690(9) | 0.5744(11)
5 loops | Padé | 0.9315 | 0.81(5) | 0.681(13) | 0.5737(10)
PBL | 0.86(13) | 0.78(4) | 0.669(6) | 0.5730(2)
KP17 | 0.8(3) | 0.85(8) | 0.682(11) | 0.5737(7)
cPadé-2 |  | 0.9496 | 0.697(15) | 0.575(2)
cPBL-2 |  | 0.93(3) | 0.703(14) | 0.575(2)
cKP17-2 |  | 0.88(7) | 0.69(2) | 0.5738(12)
Constr. Padé |  | 0.89(2) | 0.689(6) | 0.5743(7)
non constr. | 0.9(2) | 0.80(6) | 0.675(11) | 0.5732(5)
constr. |  | 0.90(4) | 0.692(13) | 0.563(13)
| all | 0.9(2) | 0.87(7) | 0.685(15) | 0.570(9)
Table 13: Estimates for $\nu$ in percolation.

The situation for $\nu$ is similar as can be seen in Tables 13 and 14 where we
note that there has been a significant number of independent measurements of
$\nu$ especially in $3$ dimensions. This has provided us with a very reliable
benchmark to compare with. Again we had to apply our resummation methods to
both $\nu$ and $1/\nu$ since the estimate of the former undershot the exact 2
dimensional value. This was not the case for the latter which we would regard
as our results for $\nu$. From Table 14 it is clear that our 4 and 5
dimensional five loop estimates are in very good accord with the Monte Carlo
values and are an improvement on the four loop ones. The situation for 3
dimensions is good and also closer to numerical values.

| Reference |  $d$ $=$ $2$ |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---|---|---
| exact | 1.3333 |  |  |
$\varepsilon^{4}$ |  [17] (2015) |  | 0.8960 | 0.6920 | 0.5746
MC |  [88] (1975) |  | 0.80(5) |  |
[82] (1976) |  | 0.8(1) |  |
[81] (1990) |  | 0.872(70) | 0.678(50) | 0.571(3)
[85] (1997) |  |  | 0.689(10) |
[84] (1998) |  | 0.875(1) |  |
[89] (2000) |  | 0.8765(18) |  |
[90] (2013) |  | 0.8764(12) |  |
[91] (2014) |  | 0.8751(11) |  |
[92] (2014) |  | 0.87619(11) |  |
[93] (2016) |  | 0.8774(13) | 0.6852(28) | 0.5723(18)
[13] (2018) |  |  | 0.693 |
[94] (2020) |  |  | 0.6845(6) | 0.5757(7)
overall |  | 0.876(4) | 0.685(2) | 0.574(2)
4 loops | Padé | 1.1(3) | 0.82(9) | 0.68(2) | 0.573(3)
PBL | 1.0(3) | 0.80(9) | 0.67(2) | 0.572(3)
KP17 | 1.5(7) | 0.91(10) | 0.69(2) | 0.574(2)
cPadé-2 |  | 0.896(3) | 0.6916(12) | 0.5746(2)
cPBL-2 |  | 0.896(2) | 0.6918(11) | 0.5746(2)
cKP17-2 |  | 0.90(3) | 0.685(10) | 0.5739(9)
Constr. Padé |  | 0.895(5) | 0.691(2) | 0.5745(3)
non constr. | 1.1(4) | 0.84(10) | 0.68(2) | 0.573(2)
constr. |  | 0.896(4) | 0.691(2) | 0.5745(3)
| all | 1.1(4) | 0.894(14) | 0.691(5) | 0.5744(7)
5 loops | Padé | 1.1(2) | 0.84(7) | 0.681(13) | 0.5737(10)
PBL | 0.99(13) | 0.81(4) | 0.675(8) | 0.5733(6)
KP17 | 1.17(12) | 0.86(3) | 0.683(6) | 0.5738(5)
cPadé-2 |  | 0.895(3) | 0.6914(13) | 0.5746(2)
cPBL-2 |  | 0.88(3) | 0.687(11) | 0.5741(11)
cKP17-2 |  | 0.89(3) | 0.685(5) | 0.5738(3)
Constr. Padé |  | 0.899(11) | 0.6913(12) | 0.5744(5)
non constr. | 1.1(2) | 0.84(5) | 0.680(9) | 0.5736(7)
constr. |  | 0.894(9) | 0.690(3) | 0.5743(5)
| all | 1.1(2) | 0.89(3) | 0.689(5) | 0.5741(6)
Table 14: Estimates for $\nu$ in percolation using the series for $1/\nu$. Figure 11: Plot of estimates (from $1/\nu$) for $\nu$ in percolation. | Reference |  $d$ $=$ $2$ |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---|---|---
| exact | 0.3956 |  |  |
$\varepsilon^{4}$ |  [17] (2015) |  | 0.4419 | 0.4642 | 0.4933
MC |  [95] (1976) |  | 0.42(6) |  |
[85] (1996) |  | 0.4522(8) |  |
[84] (1998) |  | 0.445(10) |  |
[92] (2014) |  | 0.45237(8) |  |
overall |  | 0.4523(13) |  |
4 loops | Padé | 0.45(5) | 0.47(3) | 0.483(10) | 0.494(2)
PBL | 0.45(3) | 0.470(14) | 0.484(5) | 0.4946(10)
KP17 | 0.47(6) | 0.48(4) | 0.475(13) | 0.4937(13)
cPadé-2 |  | 0.442(2) | 0.476(6) | 0.4934(10)
cPBL-2 |  | 0.440(6) | 0.473(3) | 0.4930(5)
cKP17-2 |  | 0.448(9) | 0.478(5) | 0.4939(7)
Constr. Padé |  | 0.444(11) | 0.475(6) | 0.4932(13)
non constr. | 0.46(4) | 0.47(2) | 0.482(9) | 0.4943(13)
constr. |  | 0.443(6) | 0.475(5) | 0.4934(8)
| all | 0.46(4) | 0.447(13) | 0.477(7) | 0.4937(11)
5 loops | Padé | 0.44(4) | 0.46(2) | 0.480(6) | 0.4940(8)
PBL | 0.446(13) | 0.465(5) | 0.4814(15) | 0.49409(14)
KP17 | 0.44(5) | 0.45(2) | 0.478(4) | 0.4939(3)
cPadé-2 |  | 0.4425 | 0.476(6) | 0.4934(7)
cPBL-2 |  | 0.443(15) | 0.475(9) | 0.4935(11)
cKP17-2 |  | 0.452(7) | 0.4789(14) | 0.49396(13)
Constr. Padé |  | 0.447(9) | 0.477(5) | 0.4937(8)
non constr. | 0.44(2) | 0.462(12) | 0.480(3) | 0.4940(3)
constr. |  | 0.448(12) | 0.478(4) | 0.4939(4)
| overall | 0.44(2) | 0.454(14) | 0.479(4) | 0.4940(3)
Table 15: Estimates for $\sigma$ in percolation. Figure 12: Plot of estimates
for $\sigma$ in percolation.

Examining Table 15 which summarizes our investigation for $\sigma$, we note
that the only independent estimates are in $3$ dimensions. We have clearly
obtained excellent agreement with these which exceed the situation with other
exponents. This outcome has to be qualified by noting that in strictly 6
dimensions $\sigma$ is $0.5$ with the exact result in 2 dimensions being just
under $0.4$. In other words the behaviour of $\sigma$ in the intervening
dimensions is a shallow decrease as $d$ decreases which underpins the very
good overlap. Consequently even though there have been no measurements of
$\sigma$ in 4 and 5 dimensions that we can compare with, we would expect our
five loop estimates to be in line with any future Monte Carlo studies in those
dimensions.

| Reference |  $d$ $=$ $2$ |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---|---|---
| exact | 2.0549 |  |  |
$\varepsilon^{4}$ |  [17] (2015) |  | 2.1888 | 2.3124 | 2.4171
MC |  [83] (1980) |  |  | 2.26 | 2.33
[85] (1996) |  | 2.18906(8) | 2.3127(6) |
[86] (1998) |  | 2.186(2) |  |
[84] (1998) |  | 2.189(2) |  |
[96] (2001) |  |  | 2.313(3) | 2.412(4)
[87] (2001) |  | 2.190(2) | 2.313(2) |
[97] (2006) |  | 2.189(1) |  |
[92] (2014) |  | 2.18909(5) |  |
(site) [98] (2018) |  | 2.1892(1) | 2.3142(5) | 2.419(1)
(bond) [98] (2018) |  | 2.1890(2) | 2.311(2) | 2.422(4)
[99] (2019) |  |  | 2.3135(5) |
| overall |  | 2.1891(4) | 2.3133(12) | 2.418(4)
4 loops | Padé | 2.10(7) | 2.20(5) | 2.318(13) | 2.4180(15)
PBL | 2.11(7) | 2.22(3) | 2.323(10) | 2.4186(14)
KP17 | 1.9(2) | 2.16(11) | 2.31(2) | 2.417(2)
cPadé-2 |  | 2.195(4) | 2.315(3) | 2.4176(5)
cPBL-2 |  | 2.193(3) | 2.314(2) | 2.4174(3)
cKP17-2 |  | 2.194(4) | 2.315(3) | 2.4175(9)
Constr. Padé |  | 2.197(9) | 2.317(6) | 2.4179(10)
non constr. | 2.08(10) | 2.20(5) | 2.318(14) | 2.418(2)
constr. |  | 2.194(5) | 2.315(3) | 2.4176(6)
| all | 2.08(10) | 2.194(9) | 2.315(5) | 2.4176(8)
5 loops | Padé | 2.07(4) | 2.19(3) | 2.314(8) | 2.4174(7)
PBL | 2.08(2) | 2.202(8) | 2.317(2) | 2.4176(2)
KP17 | 2.00(13) | 2.17(4) | 2.312(9) | 2.4173(8)
cPadé-2 |  | 2.194(2) | 2.3146(14) | 2.4174(2)
cPBL-2 |  | 2.193(3) | 2.3145(14) | 2.4174(2)
cKP17-2 |  | 2.1938(12) | 2.3150(8) | 2.4175(2)
Constr. Padé |  | 2.194(2) | 2.3148(14) | 2.4174(2)
non constr. | 2.07(4) | 2.20(2) | 2.316(5) | 2.4176(5)
constr. |  | 2.194(2) | 2.3149(12) | 2.4175(2)
| all | 2.07(4) | 2.194(4) | 2.315(2) | 2.4175(3)
Table 16: Estimates for $\tau$ in percolation. Figure 13: Plot of estimates
for $\tau$ in percolation.

There is a similar picture for $\tau$ which is apparent in Table 16. Again the
behaviour of $\tau$ from 2 to 6 dimensions is relatively flat due to the exact
values in these dimensions being of a similar order. In this case there are
independent data in 4 and 5 dimensions. In the absence of these we could have
repeated our final comment for $\sigma$. So it is reassuring that our five
loop 4 and 5 dimensional estimates are indeed in excellent agreement with the
independent studies. The situation with our 3 dimensional result is in similar
accord.

| Reference |  $d$ $=$ $2$ |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---|---|---
| exact | 0.7912 |  |  |
$\varepsilon^{4}$ |  [17] (2015) |  |  | 0.4008 | 0.2304
MC |  [81] (1990) |  | 0.50(9) | 0.31(5) | 0.27(7)
[85] (1996) |  |  | 0.37(4) |
[84] (1998) |  | 0.64(2) |  |
[86] (1998) |  | 0.73(8) |  |
[89] (1999) |  | 0.64(5) |  |
[100] (2000) |  | 0.65(2) |  |
[87] (2001) |  | 0.60(8) | 0.5(1) |
[98] (2018) |  | 0.77(3) |  |
[99] (2019) |  |  | 0.40(4) |
overall |  | 0.66(8) | 0.38(8) | 0.27(7)
4 loops | Padé | 0.6(3) | 0.5(2) | 0.37(7) | 0.201(14)
PBL | 0.768(4) | 0.583(2) | 0.3956(8) | 0.20593(14)
KP17 | 0.7(2) | 0.51(15) | 0.36(7) | 0.205(15)
cPadé-2 |  | 0.58(3) | 0.39(2) | 0.205(6)
cPBL-2 |  | 0.59425(2) | 0.399952(15) | 0.206613(5)
cKP17-2 |  | 0.593(5) | 0.399(5) | 0.206(2)
Constr. Padé |  | 0.599(4) | 0.405(5) | 0.209(3)
non constr. | 0.76(2) | 0.581(13) | 0.395(5) | 0.2059(6)
constr. |  | 0.5943(5) | 0.4000(4) | 0.20661(10)
| all | 0.76(2) | 0.594(2) | 0.3999(10) | 0.2066(2)
5 loops | Padé | 0.6(3) | 0.50(15) | 0.36(6) | 0.21(3)
PBL | 0.96(5) | 0.68(2) | 0.432(5) | 0.2112(3)
KP17 | 0.7(2) | 0.5(2) | 0.47(9) | 0.215(14)
cPadé-2 |  | 0.58(3) | 0.39(2) | 0.205(5)
cPBL-2 |  | 0.608(6) | 0.411(5) | 0.2095(12)
cKP17-2 |  | 0.595(3) | 0.400(3) | 0.2067(12)
Constr. Padé |  | 0.597(2) | 0.403(2) | 0.2079(11)
non constr. | 0.9(2) | 0.65(8) | 0.43(2) | 0.2112(9)
constr. |  | 0.598(7) | 0.404(6) | 0.208(2)
| all | 0.9(2) | 0.60(2) | 0.409(15) | 0.210(2)
Table 17: Estimates for $\Omega$ in percolation. Figure 14: Plot of estimates
for $\Omega$ in percolation.

For the exponent $\Omega$ there are again a large number of independent
measurements that we have shown in Table 17 though mostly in 3 dimensions.
However these and those in 4 dimensions have a wide range of central values as
well as a range of errors that are 10% or more in some cases. By contrast to
$\sigma$ and $\tau$ there is a much larger variation in $\Omega$ across the
dimensions rising from zero in strictly 6 dimensions as $d$ decreases. Despite
this both our constrained and unconstrained resummations as well as from four
to five loops appear to have a degree of stability. Although our check
estimate against the exact two dimensional value is not as good as for other
exponents, it is encouraging that our 3 dimensional five loop estimate has
overlap with independent data. This is clearly more pronounced for 4 and 5
dimensions. For instance, our final five loop 4 dimensional estimate of
$0.41(2)$ for $\Omega$ is consistent with the recent 4 dimensional value of
$0.40(4)$, [99].

Finally we arrive at the correction to scaling exponent $\omega$ with the
situation summarized in Table 18. As mentioned earlier for our constrained
resummations we have taken the exact value in 2 dimensions to be $3/2$. The
independent data is mostly available in 3 dimensions and like $\Omega$ covers
a wide range from around $1.0$ to $1.62$. Again similar to $\Omega$, $\omega$
rises from zero to a large 2 dimensional value. The overall picture is that in
3 dimensions our five loop estimates are close to one standard deviation away
from our calculated global value from the Monte Carlo data. Though the
constrained data in 3 dimensions is in better accord. However the 2
dimensional projected value for our unconstrained resummations is not close to
the exact value. What would be useful are independent numerical studies in 4
and 5 dimensions to compare our results with.

| Reference |  $d$ $=$ $2$ |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---|---|---
| exact | 1.5 |  |  |
$\varepsilon^{4}$ |  [17] (2015) |  | 1.6334 | 1.2198 | 0.7178
MC |  [81] (1990) |  | 1.26(23) | 0.94(15) | 0.96(26)
[85] (1996) |  | 1.13(10) |  |
[84] (1998) |  | 1.61(5) |  |
[89] (1999) |  | 1.62(13) |  |
[101] (2010) |  | 1.0(2) |  |
overall |  | 1.4(3) | 0.94(15) | 1.0(3)
4 loops | Padé | 2(2) | 1.5(1.3) | 1.2(5) | 0.73(10)
PBL | 2.58(4) | 1.99(2) | 1.388(7) | 0.7544(14)
KP17 | 2.2(8) | 1.8(4) | 1.3(2) | 0.77(3)
cPadé-2 |  | 1.36(7) | 1.12(6) | 0.70(2)
cPBL-2 |  | 1.38(6) | 1.13(6) | 0.70(2)
cKP17-2 |  | 1.38(9) | 1.13(9) | 0.70(3)
Constr. Padé |  | 1.35(4) | 1.10(5) | 0.70(2)
non constr. | 2.6(2) | 1.98(9) | 1.38(4) | 0.755(6)
constr. |  | 1.37(6) | 1.12(6) | 0.70(2)
| all | 2.6(2) | 1.6(3) | 1.30(13) | 0.74(2)
5 loops | Padé | 2(2) | 1.5(1.1) | 1.2(4) | 0.75(10)
PBL | 2.37(5) | 1.88(2) | 1.349(6) | 0.7483(6)
KP17 | 2.8(7) | 2.1(4) | 1.43(12) | 0.76(2)
cPadé-2 |  | 1.37(8) | 1.13(7) | 0.71(2)
cPBL-2 |  | 1.44(9) | 1.19(8) | 0.73(2)
cKP17-2 |  | 1.39(9) | 1.14(9) | 0.71(3)
Constr. Padé |  | 1.35(5) | 1.10(6) | 0.69(3)
non constr. | 2.4(2) | 1.89(9) | 1.35(3) | 0.749(3)
constr. |  | 1.39(11) | 1.15(11) | 0.71(4)
| all | 2.4(2) | 1.7(3) | 1.31(9) | 0.746(11)
Table 18: Estimates for $\omega$ in percolation. Figure 15: Plot of estimates for $\omega$ in percolation. exponent |  $d$ $=$ $3$ |  $d$ $=$ $4$ |  $d$ $=$ $5$
---|---|---|---
$\sigma$ | 0.452(7) | 0.4789(14) | 0.49396(13)
$\tau$ | 2.1938(12) | 2.3150(8) | 2.4175(2)
$\alpha$ | -0.64(4) | -0.75(2) | -0.870(1)
$\beta$ | 0.429(4) | 0.658(1) | 0.8452(2)
$\gamma$ | 1.78(3) | 1.430(6) | 1.1792(7)
$\delta$ | 5.16(4) | 3.175(8) | 2.3952(12)
$\eta$ | -0.03(1) | -0.084(4) | -0.0547(10)
$\nu$ | 0.88(2) | 0.686(2) | 0.5739(1)
Table 19: Estimates of percolation exponents from hyperscaling relations using
$O(\varepsilon^{5})$ cKP17-2 estimates of $\sigma$ and $\tau$ as input.

For the most part our focus in this section has been on providing estimates
for various exponents. To do this we used the $\varepsilon$ expansion
expressions for $\eta$ and $\nu$ and then derived $O(\varepsilon^{5})$
expansions for the other exponents through the scaling and hyperscaling
relations of (16). We then applied various different summation methods to each
exponent to arrive at estimates for the three dimensions of interest. One
question arises as a consequence of this and concerns whether the individual
estimates are then consistent with the scaling and hyperscaling relations.
Equally another question is which pair of exponents could reasonably be
regarded as the ones from which all the other estimates can be accurately
described and thereby be our final exponent estimates. To this end we have
presented our analysis in Table 19. In particular we have chosen $\sigma$ and
$\tau$ as our two independent exponents and note that the results using the
cKP17-2 technique are the most reasonable to use for final predictions. This
is because the two dimensional constraints have been implemented and these two
exponents are accurate with a tight error bar in each dimension. Using these
cKP17-2 values we have computed the exponents for $\alpha$, $\beta$, $\gamma$,
$\delta$, $\eta$ and $\nu$ using (16). In the Table the independent values for
$\sigma$ and $\tau$ are given above the rule with the values derived from
scaling relations recorded below. For the remaining two exponents $\omega$ and
$\Omega$, their cKP17-2 values have not been included as they are not
accessible by scaling or hyperscaling relations and those estimates are
already in their respective tables.

Comparing these scaling relation values of Table 19 with those from the direct
summation ones given in the earlier tables, none look far out of line. One
exception might possibly be the 3 dimensional value of $\alpha$ as it appears
to be more consistent with the unconstrained analysis. However looking at the
extrapolation to 2 dimensions suggests it is probably on a better trajectory.
The lack of Monte Carlo or series computations for this exponent means that we
have no way of gauging our $\alpha$ estimates against an independent analysis.
For the other exponents the view is that the individual resummations are
consistent with the scaling relations and also with independent data from
other methods where that is substantial. In the case of $\delta$ where there
are only a few such other cases the estimates in Table 19 are not
inconsistent. Our final 3 dimensional estimate for $\eta$ appears to be on the
low side but this exponent is difficult to measure accurately. For $\nu$ our 4
and 5 dimensional values are remarkably close to the global average we
compiled. While that for 3 dimensions is not as accurate, it does lie within
the errors of the global estimate.

In presenting our final estimates in this Table we need to be clear in stating
that this shows our results are consistent with scaling relations both by
constructing the $\varepsilon$ equations directly and resumming them before
comparing the estimates obtained from using two independent summed values in
the (hyper)scaling relations. So in essence it is a self-consistency check. By
contrast it is usually the case that in the Monte Carlo and series approaches
the focus is on one or two specific exponents. Those values are then used to
generate the remaining exponents via (16) rather than make extra measurements.
So until all techniques have achieved a large degree of computational accuracy
it is probably the case that in comparing exponent estimates the overall
picture is still not perfect.

##  7 Conclusions

The main result of this paper is clearly the provision of the five loop
renormalization group functions of $\phi^{3}$ theory in six dimensions in the
$\overline{\mbox{MS}}$ scheme. This level of precision could not have been
achieved without the use of the graphical function formalism developed in [45,
31]. That method was originally pioneered in four dimensions to renormalize
$\phi^{4}$ theory to seven loops in the same scheme [31, 32]. To achieve the
level of five loop accuracy here required the extension of graphical functions
to six dimensions that was provided in [35, 52]. An indication of the
advantage of such new and powerful techniques can be gained from the dates
that the previous loop order results became available for the cubic theory.
The one and two loop renormalization group functions were recorded in the
early 70’s [1] with the three loop extension appearing within seven years, [4,
5]. Similar to the extension of $\phi^{4}$ theory from five to six loops,
there was a quarter of a century lull before $\phi^{3}$ theory was
renormalized at four loops [17]. The relatively quick extension, in terms of
time, to five loops here is suggestive that with suitable investment in the
underlying mathematics of the graphical functions approach, higher loop orders
are potentially within reach in this and other theories.

Such higher order computations are not purely academic exercises since the
second part of our study was to extract improved estimates for critical
exponents in two important problems. Both the Lee–Yang edge singularity and
percolation problems at criticality lie in separate universality classes but
both have a $\phi^{3}$ continuum field theory at their heart. The main
difference is that the respective versions of the cubic theory are decorated
with different symmetry properties. Given that we have renormalized the pure
$\phi^{3}$ Lagrangian at five loops, it was a relatively simple exercise to
include the respective symmetry decorations and determine the two sets of
renormalization group functions. From these the $\varepsilon$ expansion of the
critical exponents were constructed to $\mathcal{O}(\varepsilon^{5})$ before a
variety of resummation techniques were applied to extract numerical estimates.
Moreover, we devoted a significant part of determining estimates to a careful
error analysis using the same formalism provided in [30] for $\phi^{4}$
theory. What ought to be recognized is that on the whole not only do the five
loop results improve upon the four loop results, as well as showing
convergence, but also that there is good agreement with other techniques in
both problems for the dimensions of interest. These include Monte Carlo and
strong coupling methods to name but a few. In making this remark it should not
be overlooked that this includes 3 dimensions where we have summed from $d$
$=$ $6$ $-$ $\varepsilon$ down to $d$ $=$ $3$ with a large value of
$\varepsilon$. In one sense this confirms the role of $\phi^{3}$ theory as
being in the same universality class. What was useful in making this
comparison between the exponents from discrete systems and the continuum field
theory was collating the available data for the former to produce a global
average. The associated error bars were produced with the same routine that we
used for the five loop results from the various resummations. In this respect
we were endeavouring to compare the picture in the discrete and continuum
sides in the same way.

The determination of the five loop renormalization group functions in the core
cubic theory opens up the possibility of studying related six dimensional
cubic theories to the same level of precision. For instance the conformal
bootstrap formalism represents a powerful tool to calculate exponents. It was
used in [102] to determine exponents for $\phi^{3}$ theory in a variety of
representations of the Lie group $F_{4}$. While the three loop comparison in
that article was in reasonable agreement for various exponents calculated with
the bootstrap, the four loop study of [103] gave an improvement towards
convergence. It would therefore be interesting, given the accuracy of results
in [102], to extend to the $F_{4}$ study of [103] to five loops. Aside from
this particular symmetry of the cubic theory, an intriguing property of
$\phi^{3}$ with a bi-adjoint symmetry was observed in [104]. For any Lie group
it transpires that this theory is asymptotically free due to the two loop
coefficient of the $\beta$ function; the one loop term is identically zero.
One feature of the four loop result was the appearance of higher order group
Casimirs at three loops. This is one order earlier than in four dimensional
non-abelian gauge theories. So the bi-adjoint theory offers a window into the
type of group Casimirs that could appear in six loop gauge theory $\beta$
functions such as QCD. Although such problems are worth pursuing, the more
interesting extension of our current work would clearly be to six loops. This
would obviously provide a higher level of precision with which to compare
future numerical studies of the discrete spin models in the same universality
class. The graphical functions method that was used here, and was successful
in extending $\phi^{4}$ theory to seven loops [31, 32] should be regarded as a
starting point to achieve the six loop renormalization of $\phi^{3}$ theory.

#### Acknowledgements.

We warmly thank A.J. McKane for discussions on the resummation of asymptotic
series in quantum field theory. Also JAG thanks R.M. Ziff for encouragement
and discussions, and in particular that surrounding the value of $\omega$ in
two dimensional percolation theory. The work of MB was supported by the NWO
Vidi grant 680-47-551 “Decoding Singularities of Feynman graphs”. The work of
JAG was supported by a DFG Mercator Fellowship and in part through the STFC
Consolidated grant ST/T000988/1. The work of MVK was supported by Foundation
for the Advancement of Theoretical Physics “BASIS” through Grant 18-1-2-43-1.
The work of OS was supported by the DFG grant SCHN 1240.

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