#  Calculation of the total 10-th order QED contribution to the lepton
magnetic moments

Sergey Volkov<EMAIL_ADDRESS><EMAIL_ADDRESS>Institute
for Theoretical Physics, Karlsruhe Institute of Technology (KIT),
76128 Karlsruhe, Germany

###### Abstract

The total 5-loop quantum electrodynamics universal contribution to the
anomalous magnetic moments of the leptons was calculated by the author. The
obtained value $A_{1}^{(10)}=5.891(61)$ provides the first complete
verification of the previously known value obtained by T. Aoyama, M. Hayakawa,
T. Kinoshita, M. Nio (AHKN). The discrepancy is $5\sigma$. The computation
includes the recalculation of the part that the author calculated in 2019
using a slightly different method and the calculation of the remaining part of
the coefficient.

A comparison with the AHKN values in 32 gauge-invariant classes is provided.
In addition, the results are divided into 95 small gauge-invariant classes
that subdivide the former ones. Such a detailzation is provided for the first
time.

The method described in previous works of the author was used in general.
However, the Monte Carlo integration method is new and is described in detail.

Some useful technical information and information on numerical cancellations
is also given.

##  I INTRODUCTION

The electron anomalous magnetic moment $a_{e}$ is known with very high
accuracy. The measurement Fan  _et al._ (2023) provided the result

$a_{e}=0.00115965218059(13)$ (1)

Standard Model predictions for $a_{e}$ use the following expression:

$a_{e}=a_{e}(\text{QED})+a_{e}(\text{hadronic})+a_{e}(\text{electroweak}),$
$a_{e}(\text{QED})=\sum_{n\geq
1}\left(\frac{\alpha}{\pi}\right)^{n}a_{e}^{2n},$
$a_{e}^{2n}=A_{1}^{(2n)}+A_{2}^{(2n)}(m_{e}/m_{\mu})+A_{2}^{(2n)}(m_{e}/m_{\tau})+A_{3}^{(2n)}(m_{e}/m_{\mu},m_{e}/m_{\tau}),$

where $m_{e},m_{\mu},m_{\tau}$ are the masses of the electron, the muon and
the tau-lepton respectively, $\alpha$ is the fine-structure constant.

The universal QED terms $A_{1}^{(2n)}(\alpha/\pi)^{n}$ form the main
contribution to $a_{e}$. The value

$A_{1}^{(2)}=0.5$

was determined by J. Schwinger in 1948 Schwinger (1948, 1949). The 2-loop
contribution $A_{1}^{(4)}$ was mainly calculated by R. Karplus and N. Kroll
Karplus and Kroll (1950). However, this calculation had an error; the correct
value

$A_{1}^{(4)}=-0.328478965579\ldots$

was presented independently in 1957 by A. Petermann Petermann (1957) and C.
Sommerfield Sommerfield (1957). The value of $A_{1}^{(6)}$ was being
calculated in the 1970s by several research groups using numerical integration
(Carroll and Yao (1974); Carroll (1975); Levine and Wright (1973); Cvitanović
and Kinoshita (1974a)); the most accurate value $A_{1}^{(6)}=1.195\pm 0.026$
for that era was determined in 1974 by T. Kinoshita and P. Cvitanović; the
uncertainty was caused by the statistical error of the Monte Carlo
integration. At the same time, an analytical calculation of $A_{1}^{(6)}$
using computers was started. The final value

$A_{1}^{(6)}=1.181241456\ldots$

was presented in 1996 by S. Laporta and E. Remiddi Laporta and Remiddi (1996).
This value was the result of the efforts of many researchers (e.g., Mignaco
and Remiddi (1969); Billi  _et al._ (1972); L. L. DeRaad  _et al._ (1974);
Levine  _et al._ (1976); Barbieri  _et al._ (1978); Laporta (1993) ). The
first numerical estimates for $A_{1}^{(8)}$ were given in 1981 by T. Kinoshita
and W. B. Lindquist Kinoshita and Lindquist (1981). The most precise value
presented by T. Kinoshita’s team, $A_{1}^{(8)}=-1.91298(84)$, was published in
2015 Aoyama  _et al._ (2015). This value was obtained by Monte Carlo
integration. The semianalytic result of S. Laporta

$A_{1}^{(8)}=-1.9122457649\ldots$

was presented in 2017 Laporta (2017). These two calculations of $A_{1}^{(8)}$
agree well, as do another independent calculations Marquard  _et al._ (2019);
Rappl (2016); S. Volkov (2024). First overall computation results for
$A_{1}^{(10)}$ were published in 2012 by T. Aoyama, M. Hayakawa, T. Kinoshita,
M. Nio (AHKN) in Aoyama  _et al._ (2012). The last value presented by these
researchers in 2019 Aoyama  _et al._ (2019) is

$A_{1}^{(10)}[\text{AHKN}]=6.737(159).$ (2)

Up to this moment, the coefficient had not been verified and a significant
computational error could be noticeable in experiments. In 2019, we
recalculated the total contribution of the graphs without lepton loops to
$A_{1}^{(10)}$ Volkov (2017); the value disagreed with the AHKN one.

In this paper we present new calculation results for $A_{1}^{(10)}$. Our value
for the remaining part is

$A_{1}^{(10)}[\text{with lepton loops,Volkov}]=-0.9377(35).$

It agrees with the value

$A_{1}^{(10)}[\text{with lepton loops,AHKN}]=-0.933(17)$

from Aoyama  _et al._ (2012).

This value was obtained using a method slightly different from the one used in
2019 (see Sections III and IV); the programs for the integrand code generation
were also different. For this reason, we recalculated the contribution of the
graphs without lepton loops with the aim of resolving the discrepancy. Our new
value

$A_{1}^{(10)}[\text{no lepton loops,Volkov,2024 only}]=6.857(81)$

agrees with our number published in S. Volkov (2019), but does not agree with
that of the AHKN. The combined value is

$A_{1}^{(10)}[\text{no lepton loops,Volkov}]=6.828(60).$

Our total value

$A_{1}^{(10)}[\text{Volkov}]=5.891(61)$ (3)

comes from our new calculation in combination with the old one. The
discrepancy with (2) is $5\sigma$.

At the moment the discrepancy in $A_{1}^{(10)}[\text{no lepton loops}]$ is
unresolved, but independent calculations are coming Kitano and Takaura (2022).
The values (2) and (3) in combination with the experimental value (1) and
another known contributions Aoyama  _et al._ (2019) lead to

$\alpha^{-1}[a_{e},\text{AHKN}]=137.0359991663(155)$ (4)

and

$\alpha^{-1}[a_{e},\text{Volkov}]=137.0359991595(155),$ (5)

respectively. The values obtained from the measured ratios of the atomic
masses and the Planck constant

$\alpha^{-1}[\text{Rb-2011}]=137.035998996(85),$
$\alpha^{-1}[\text{Cs-2018}]=137.035999046(27),$
$\alpha^{-1}[\text{Rb-2020}]=137.035999206(11)$

come from Bouchendira  _et al._ (2011), R. H. Parker and C. Yu and W. Zhong
and B. Estey and H. Müller (2018), and Morel  _et al._ (2020), respectively.
Note that $\alpha^{-1}[\text{Rb-2020}]$ is the largest among these three
values and has a discrepancy of $5.4\sigma$ relative to
$\alpha^{-1}[\text{Cs-2018}]$. The tensions with (4) are $1.97\sigma$,
$3.86\sigma$, $2.09\sigma$; the corresponding tensions with (5) are
$1.89\sigma$, $3.65\sigma$, $2.45\sigma$, respectively.

Other results exist for small classes of 5-loop and higher order graphs
Lautrup (1977); Caffo  _et al._ (1978); Baikov  _et al._ (2013); they are in
good agreement with the ones mentioned above.

The values for the individual classes and their comparison with the AHKN
values as well as the hardware used are described in Section II. The method of
reduction to finite integrals used was described in detail in S. Volkov (2024)
and briefly recapitulated in Section III; the chosen modification of the
procedure was also described in this section. The integrals were evaluated
numerically using Monte Carlo integration. We used a hand-made Monte Carlo.
The general ideas of our Monte Carlo were described in Volkov (2017). It is
based on predefined probability density functions obtained from the
combinatorics of Feynman graphs. Since the procedure was developed in the
previous works only for graphs without lepton loops, a serious modification
and improvement is needed. This is described in Section IV. The information
that can be used for the resolution of the discrepancy, another calculations,
and theoretical investigations, is presented in Section V.

##  II RESULTS IN DETAIL

We extract $a_{e}$ from QED Feynman graphs with $N_{l}=2$, $N_{\gamma}=1$,
where we denote by $N_{l}$ and $N_{\gamma}$ the number of external lepton and
photon lines in the graph; each graph may contain electron, muon, tau-lepton
lines. We also assume that all graphs are one-particle irreducible and have no
odd lepton loops (Furry’s theorem). _Undirected_ graphs are used everywhere in
the calculations.

We say that a vertex $v$ is _incident_ to a line $l$, if $v$ is one of the
ends of $l$.

Our contributions to $A_{1}^{(10)}$ are split into 95 gauge-invariant classes
depicted in FIG. 1. One class is the set of all Feynman graphs that can be
obtained from one picture by moving internal photon lines along lepton paths
and loops, but without jumping over the vertex incident to the external photon
line. The individual class contributions has to do with the in-place on-shell
renormalization; the exact definition is given in S. Volkov (2024). The proof
of their gauge invariance is given in Cvitanović (1977)111The proof was only
given for graphs without lepton loops. It looks the whole proof has not been
published, but the fact is widely used; we believe that the ideas of
Cvitanović (1977) can be extended to the general case..

Figure 1: Gauge-invariant classes contributing to $A_{1}^{(10)}$.

The newly calculated contributions to $A_{1}^{(10)}[\text{with lepton loops}]$
with the corresponding technical information can be found in Table
LABEL:table_calculations. The calculation was performed on the supercomputer
“HoreKa” (Karlsruhe, Germany). The GPUs NVidia A100 were used for the Monte
Carlo integration (most of the time is taken by the integrand evaluation); The
CPUs Intel Xeon Platinum 8368 were used for the control. To avoid biases
caused by the pseuderandom number generators, two generators from the NVidia
library were used: MRG32k3a and Philox_4x32_10. The separate results as well
as the statistical averages are listed in the table. The number of the
undirected graphs, of the Monte Carlo samples, the calculation time are
presented in the last three columns. The corresponding contributions to
$A_{1}^{(10)}[\text{no lepton loops}]$ are given in Table
LABEL:table_calculations_setv. A comparison with the old values from S. Volkov
(2019) as well as the statistical mean values are presented in Table
LABEL:table_comparison_with_old_setv. The new Monte Carlo algorithm described
in Section IV takes approximately by $1.6$ times fewer samples to achieve the
needed accuracy than the previous one S. Volkov (2019). However, this does not
affect the computation time as much, since more samples require increased
precisions in this case; see Section V.

Table 1: The contributions of all gauge-invariant classes containing lepton loops to $A_{1}^{(10)}$, calculated with different random number generators Class | Calc_MRG | Calc_Philox | Average | $N_{\text{graph}}$ | $N_{\text{samples}}$ | Time (GPU-hours)
---|---|---|---|---|---|---
I(a) | 0.00047108(23) | 0.00047102(23) | 0.00047105(16) | 1 | $15\times 10^{11}$ | 30.9
I(b1) | 0.0046717(16) | 0.0046715(16) | 0.0046715(11) | 2 | $12\times 10^{11}$ | 25.8
I(b2) | 0.00233645(75) | 0.00233679(78) | 0.00233656(54) | 2 | $12\times 10^{11}$ | 26.3
I(c) | 0.0234651(35) | 0.0234645(41) | 0.0234643(26) | 3 | $81\times 10^{10}$ | 22.8
I(d) | 0.00380375(84) | 0.00380366(91) | 0.00380370(61) | 2 | $10^{12}$ | 22.8
I(e) | 0.010293(16) | 0.010282(17) | 0.010289(11) | 9 | $29\times 10^{11}$ | 70.5
I(f) | 0.00757078(67) | 0.00757131(66) | 0.00757106(47) | 2 | $84\times 10^{10}$ | 17.6
I(g) | 0.0285715(26) | 0.0285678(24) | 0.0285696(18) | 4 | $15\times 10^{11}$ | 39.8
I(h) | 0.001682(10) | 0.0016807(91) | 0.0016826(63) | 11 | $50\times 10^{11}$ | 114.5
I(i) | 0.01727(43) | 0.01718(41) | 0.01726(29) | 39 | $85\times 10^{12}$ | 1416.2
I(j) | 0.000401(11) | 0.0004044(79) | 0.0004038(63) | 2 | $74\times 10^{10}$ | 50.0
II(ap1) | 0.0092489(11) | 0.0092448(14) | 0.00924758(84) | 3 | $79\times 10^{10}$ | 23.9
II(ap2) | -0.0297767(17) | -0.0297768(16) | -0.0297767(11) | 2 | $85\times 10^{10}$ | 23.2
II(ap3) | -0.0262168(17) | -0.0262195(14) | -0.0262183(11) | 2 | $49\times 10^{10}$ | 21.0
II(as1) | -0.0466165(58) | -0.0466065(61) | -0.0466111(41) | 2 | $57\times 10^{10}$ | 19.3
II(as2) | 0.0137932(21) | 0.0137901(23) | 0.0137916(15) | 3 | $13\times 10^{11}$ | 25.9
II(as3) | -0.0299263(27) | -0.0299284(24) | -0.0299276(18) | 2 | $87\times 10^{10}$ | 13.4
II(bp1) | 0.0326220(40) | 0.0326252(60) | 0.0326223(31) | 6 | $86\times 10^{10}$ | 28.7
II(bp2) | -0.080010(10) | -0.0800158(91) | -0.0800129(67) | 4 | $56\times 10^{10}$ | 16.3
II(bp3) | -0.083589(11) | -0.083617(13) | -0.0836017(85) | 4 | $29\times 10^{10}$ | 9.6
II(bs1) | -0.125046(16) | -0.125060(15) | -0.125057(11) | 4 | $80\times 10^{10}$ | 34.0
II(bs2) | -0.125034(21) | -0.125080(19) | -0.125056(14) | 4 | $67\times 10^{10}$ | 16.9
II(bs3) | 0.088110(14) | 0.088118(19) | 0.088110(10) | 8 | $20\times 10^{11}$ | 40.4
II(bs4) | -0.180651(20) | -0.180611(19) | -0.180629(14) | 8 | $13\times 10^{11}$ | 26.2
II(c1) | -0.086935(12) | -0.086943(14) | -0.0869389(86) | 4 | $63\times 10^{10}$ | 16.2
II(c2) | 0.0375586(58) | 0.0375556(85) | 0.0375573(46) | 6 | $10^{12}$ | 24.8
II(c3) | -0.067120(11) | -0.067129(11) | -0.0671244(72) | 4 | $10^{12}$ | 20.9
II(d1) | -0.18984(16) | -0.18988(16) | -0.18988(11) | 18 | $47\times 10^{11}$ | 110.9
II(d2) | 0.11133(11) | 0.11137(10) | 0.111341(75) | 25 | $50\times 10^{11}$ | 100.1
II(d3) | -0.16439(12) | -0.16438(12) | -0.164378(83) | 18 | $43\times 10^{11}$ | 76.1
II(e1) | -1.13784(67) | -1.13735(64) | -1.13757(46) | 30 | $20\times 10^{11}$ | 730.1
II(e2) | -0.20445(41) | -0.20506(39) | -0.20478(28) | 20 | $69\times 10^{10}$ | 306.6
II(f1) | -0.23991(13) | -0.23990(12) | -0.239896(88) | 3 | $13\times 10^{10}$ | 43.9
II(f2) | -1.91548(33) | -1.91474(32) | -1.91510(23) | 9 | $54\times 10^{10}$ | 186.6
II(f3) | -0.28090(25) | -0.28019(24) | -0.28054(18) | 6 | $17\times 10^{10}$ | 118.1
III(ap1) | 0.056150(65) | 0.056165(57) | 0.056161(42) | 10 | $30\times 10^{10}$ | 30.6
III(ap2) | 0.251177(35) | 0.251187(41) | 0.251184(26) | 12 | $30\times 10^{10}$ | 39.5
III(ap3) | 0.227933(70) | 0.227875(67) | 0.227894(47) | 20 | $82\times 10^{10}$ | 45.2
III(ap4) | 0.143148(25) | 0.143129(25) | 0.143139(17) | 4 | $28\times 10^{10}$ | 26.4
III(ap5) | 0.071158(55) | 0.071180(51) | 0.071170(37) | 20 | $69\times 10^{10}$ | 39.0
III(ap6) | 0.164803(28) | 0.164778(32) | 0.164795(20) | 10 | $92\times 10^{10}$ | 28.8
III(ap7) | 0.0275440(43) | 0.0275462(35) | 0.0275451(27) | 3 | $64\times 10^{10}$ | 34.3
III(as1) | 0.054626(81) | 0.054572(76) | 0.054599(55) | 10 | $96\times 10^{10}$ | 31.3
III(as2) | 0.225735(25) | 0.225727(25) | 0.225730(17) | 3 | $38\times 10^{10}$ | 24.4
III(as3) | 0.05552(10) | 0.055647(92) | 0.055588(66) | 20 | $12\times 10^{11}$ | 52.9
III(as4) | 0.289459(50) | 0.289388(50) | 0.289427(35) | 12 | $60\times 10^{10}$ | 26.1
III(as5) | 0.439958(81) | 0.440098(78) | 0.440024(55) | 20 | $16\times 10^{11}$ | 48.2
III(as6) | 0.102922(30) | 0.103027(28) | 0.102979(21) | 4 | $93\times 10^{10}$ | 26.6
III(as7) | 0.017024(52) | 0.017036(49) | 0.017022(35) | 10 | $81\times 10^{10}$ | 22.2
III(b1) | 0.35844(19) | 0.35866(18) | 0.35850(13) | 20 | $22\times 10^{11}$ | 104.5
III(b2) | 0.55074(11) | 0.55070(11) | 0.550730(74) | 6 | $51\times 10^{10}$ | 28.9
III(b3) | -0.21853(25) | -0.21798(23) | -0.21825(16) | 40 | $36\times 10^{11}$ | 164.2
III(b4) | 0.91605(13) | 0.91600(13) | 0.916024(91) | 24 | $17\times 10^{11}$ | 69.0
III(b5) | 0.43822(15) | 0.43817(15) | 0.43819(10) | 8 | $13\times 10^{11}$ | 48.3
III(b6) | 1.35328(23) | 1.35317(22) | 1.35323(15) | 40 | $42\times 10^{11}$ | 146.5
III(b7) | -0.07136(15) | -0.07093(14) | -0.07113(10) | 20 | $21\times 10^{11}$ | 65.4
III(c1) | 1.86016(70) | 1.86020(67) | 1.86018(48) | 12 | $20\times 10^{11}$ | 749.5
III(c2) | 3.2810(13) | 3.2801(13) | 3.28050(91) | 27 | $36\times 10^{11}$ | 2590.2
III(c3) | 0.27186(70) | 0.27149(66) | 0.27166(48) | 6 | $12\times 10^{11}$ | 705.1
III(c4) | -0.7592(11) | -0.7596(11) | -0.75943(79) | 24 | $23\times 10^{11}$ | 1982.5
III(c5) | 0.22273(75) | 0.22339(71) | 0.22308(51) | 15 | $13\times 10^{11}$ | 858.0
III(c6) | 0.04462(66) | 0.04334(64) | 0.04392(46) | 18 | $10^{12}$ | 744.0
IV(1) | -0.51577(63) | -0.51599(59) | -0.51586(43) | 74 | $53\times 10^{11}$ | 734.9
IV(2) | -0.64897(40) | -0.64858(38) | -0.64878(28) | 20 | $18\times 10^{11}$ | 276.2
IV(3) | -1.14814(88) | -1.14833(84) | -1.14824(60) | 148 | $78\times 10^{11}$ | 1544.6
IV(4) | 1.19585(56) | 1.19608(53) | 1.19593(38) | 55 | $32\times 10^{11}$ | 594.6
IV(5) | -1.52733(88) | -1.52833(83) | -1.52785(60) | 162 | $74\times 10^{11}$ | 1605.4
IV(6) | 0.50510(44) | 0.50550(42) | 0.50531(30) | 56 | $19\times 10^{11}$ | 444.5
IV(7) | -0.19297(38) | -0.19292(36) | -0.19295(26) | 20 | $26\times 10^{11}$ | 254.9
IV(8) | -0.78534(55) | -0.78358(53) | -0.78444(38) | 40 | $61\times 10^{11}$ | 526.5
IV(9) | -4.4900(11) | -4.4904(10) | -4.49029(73) | 222 | $21\times 10^{12}$ | 2075.6
IV(10) | 0.19705(47) | 0.19693(44) | 0.19698(32) | 50 | $40\times 10^{11}$ | 432.6
IV(11) | 0.05155(81) | 0.05307(76) | 0.05233(55) | 148 | $12\times 10^{12}$ | 1317.3
IV(12) | -0.37235(48) | -0.37267(44) | -0.37250(31) | 54 | $35\times 10^{11}$ | 476.4
VI(ap) | 0.482912(86) | 0.483009(83) | 0.482955(59) | 5 | $14\times 10^{10}$ | 41.4
VI(as) | 0.558642(83) | 0.558526(80) | 0.558582(57) | 5 | $27\times 10^{10}$ | 37.4
VI(b) | 1.34680(17) | 1.34710(16) | 1.34697(11) | 10 | $10^{12}$ | 64.7
VI(c) | -2.53393(71) | -2.53241(67) | -2.53312(49) | 37 | $30\times 10^{11}$ | 839.9
VI(d) | 1.8426(32) | 1.8506(30) | 1.8468(22) | 127 | $41\times 10^{12}$ | 14465.2
VI(e) | -0.43170(25) | -0.43094(24) | -0.43129(17) | 13 | $55\times 10^{10}$ | 125.5
VI(f) | 0.77156(34) | 0.77158(32) | 0.77154(23) | 46 | $12\times 10^{11}$ | 295.7
VI(g) | -1.5962(14) | -1.5968(14) | -1.5965(10) | 122 | $10^{13}$ | 3221.2
VI(h) | 0.1855(10) | 0.18576(93) | 0.18554(68) | 162 | $12\times 10^{12}$ | 1913.8
VI(i) | -0.04398(15) | -0.04392(14) | -0.04396(10) | 16 | $62\times 10^{10}$ | 68.3
VI(j) | -0.22921(63) | -0.22917(60) | -0.22920(43) | 9 | $63\times 10^{10}$ | 644.1
VI(k) | 0.67989(58) | 0.67970(56) | 0.67974(39) | 32 | $32\times 10^{11}$ | 611.5
TOTAL | -0.9438(51) | -0.9314(49) | -0.9377(35) | 2323 | $32\times 10^{13}$ | 45214.1
Table 2: The contributions of all gauge-invariant classes without lepton loops to $A_{1}^{(10)}$, calculated with different random number generators Class | Calc_MRG_V | Calc_Philox_V | Average | $N_{\text{graph}}$ | $N_{\text{samples}}$ | Time (GPU-hours)
---|---|---|---|---|---|---
V(1) | 6.175(30) | 6.167(30) | 6.169(20) | 706 | $28\times 10^{12}$ | 4428.0
V(2) | 0.962(44) | 0.966(45) | 0.964(31) | 148 | $40\times 10^{12}$ | 5436.2
V(3) | 0.342(29) | 0.326(29) | 0.334(20) | 55 | $13\times 10^{12}$ | 1942.6
V(4) | -0.832(53) | -0.748(53) | -0.789(37) | 706 | $54\times 10^{12}$ | 9497.4
V(5) | -2.139(49) | -2.174(49) | -2.157(34) | 370 | $42\times 10^{12}$ | 8028.9
V(6) | -0.402(51) | -0.426(51) | -0.415(35) | 558 | $32\times 10^{12}$ | 9676.1
V(7) | 2.677(33) | 2.612(33) | 2.644(23) | 261 | $18\times 10^{12}$ | 4360.5
V(8) | -1.005(31) | -0.942(31) | -0.975(21) | 336 | $14\times 10^{12}$ | 4300.8
V(9) | 1.0807(92) | 1.0795(92) | 1.0807(63) | 73 | $21\times 10^{11}$ | 560.8
TOTAL | 6.86(12) | 6.86(12) | 6.857(81) | 3213 | $24\times 10^{13}$ | 48231.4
Table 3: Comparison of the contributions of the classes without lepton loops to $A_{1}^{(10)}$ with the values from S. Volkov (2019), statistical averages Class | New value | Old value (2019) | Average
---|---|---|---
V(1) | 6.169(20) | 6.157(33) | 6.166(17)
V(2) | 0.964(31) | 0.970(33) | 0.967(23)
V(3) | 0.334(20) | 0.315(20) | 0.325(15)
V(4) | -0.789(37) | -0.754(42) | -0.774(28)
V(5) | -2.157(34) | -2.165(36) | -2.161(25)
V(6) | -0.415(35) | -0.403(34) | -0.409(24)
V(7) | 2.644(23) | 2.625(24) | 2.635(17)
V(8) | -0.975(21) | -1.011(21) | -0.993(15)
V(9) | 1.0807(63) | 1.0902(62) | 1.0855(44)
TOTAL | 6.857(81) | 6.793(90) | 6.828(60)

The comparison of the averaged values with the AHKN values Aoyama  _et al._
(2012, 2019) is given in Table LABEL:table_comparison_ahkn. Each class is the
union of the classes from FIG. 1 with the corresponding names; the value is
the sum of the corresponding numbers. For example, II(b) consists of II(bp1),
II(bp2), II(bp3), II(bs1), II(bs2), II(bs3), II(bs4); here “p” means
“parallel”, “s” means “sequential” (it concerns the placement of the photon
self-energy insertions). The names are taken from Aoyama  _et al._ (2012). The
results agree very well in all sets except Set V. In the latter, a discrepancy
of $5\sigma$ remains.

Table 4: Comparison of the contributions to $A_{1}^{(10)}$ with the AHKN values Aoyama  _et al._ (2012, 2019) Class | This value | AHKN value
---|---|---
I(a) | 0.00047105(16) | 0.000470940(60)
I(b) | 0.0070081(12) | 0.00701080(70)
I(c) | 0.0234643(26) | 0.0234680(20)
I(d) | 0.00380370(61) | 0.00380170(50)
I(e) | 0.010289(11) | 0.0102960(40)
I(f) | 0.00757106(47) | 0.0075684(20)
I(g) | 0.0285696(18) | 0.0285690(60)
I(h) | 0.0016826(63) | 0.001696(13)
I(i) | 0.01726(29) | 0.01747(11)
I(j) | 0.0004038(63) | 0.0003975(18)
II(a) | -0.1094945(50) | -0.109495(23)
II(b) | -0.473625(27) | -0.473559(84)
II(c) | -0.116506(12) | -0.116489(32)
II(d) | -0.24291(15) | -0.24300(29)
II(e) | -1.34235(54) | -1.3449(10)
II(f) | -2.43553(30) | -2.4336(15)
III(a) | 2.12726(14) | 2.12733(17)
III(b) | 3.32730(32) | 3.32712(45)
III(c) | 4.9199(15) | 4.921(11)
IV | -7.7303(16) | -7.7296(48)
V | 6.828(60) | 7.670(159)
VI(a) | 1.041537(82) | 1.04132(19)
VI(b) | 1.34697(11) | 1.34699(28)
VI(c) | -2.53312(49) | -2.5289(28)
VI(d) | 1.8468(22) | 1.8467(70)
VI(e) | -0.43129(17) | -0.43120(70)
VI(f) | 0.77154(23) | 0.7703(22)
VI(g) | -1.5965(10) | -1.5904(63)
VI(h) | 0.18554(68) | 0.1792(39)
VI(i) | -0.04396(10) | -0.0438(12)
VI(j) | -0.22920(43) | -0.2288(18)
VI(k) | 0.67974(39) | 0.6802(38)

##  III REDUCTION TO FINITE INTEGRALS

We work in the unit system with $\hbar=c=1$, the factors of $4\pi$ appear in
the fine-structure constant: $\alpha=e^{2}/(4\pi)$, the tensor $g_{\mu\nu}$
corresponds to the signature $(+,-,-,-)$, the Dirac matrices fulfill the
condition $\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=2g_{\mu\nu}$.

We work in the Feynman gauge with the propagators

$\frac{i(\not{q}+m)}{q^{2}-m^{2}+i\varepsilon},\quad\frac{-g_{\mu\nu}}{q^{2}+i\varepsilon}$

for leptons and photons, where $m$ is the lepton mass.

The method of reduction of $A_{1}^{(2n)}$ to finite integrals is described in
detail in S. Volkov (2024). It gives one Feynman parametric integral for each
Feynman graph. The final value is simply the sum of these integrals. To make
each integral finite, a subtraction procedure with linear operators applied to
the Feynman amplitudes of UV-divergent subgraphs is utilized before arriving
at the Feynman parameters. We focus here only on the aspects that are
important for the current implementation.

There are the following types of UV-divergent subgraphs222We consider only
those subgraphs which are one-particle irreducible and contain all lines
connecting the vertexes of the given subgraph; since odd lepton loops are
forbidden, a UV-divergent subgraph is one-particle irreducible if and only if
it is amputated. in QED Feynman graphs: _lepton self-energy_ subgraphs
($N_{l}=2$, $N_{\gamma}=0$), _vertexlike_ subgraphs ($N_{l}=2$,
$N_{\gamma}=1$), _photon self-energy_ subgraphs ($N_{l}=0$, $N_{\gamma}=2$),
_photon-photon scattering_ subgraphs333Photon-photon scattering subgraph
divergences cancel out in the final result without subtraction, but they
remain in the individual graphs. ($N_{l}=0$, $N_{\gamma}=4$).

We use standard operators like the anomalous magnetic moment projector and
those that define the on-shell renormalization, but also special operators $U$
that are applied to the Feynman amplitudes of vertexlike and lepton self-
energy graphs. In terms of S. Volkov (2024), we put $U_{1}=U_{2}=U_{3}=U$.

If $\Gamma_{\mu}(p,q)$ is a vertexlike Feynman amplitude, $p-\frac{q}{2}$,
$p+\frac{q}{2}$ are incoming and outgoing lepton momenta, $q$ is the photon
momentum,

$\Gamma_{\mu}(p,0)=a(p^{2})\gamma_{\mu}+b(p^{2})p_{\mu}+c(p^{2})\not{p}p_{\mu}+d(p^{2})(\not{p}\gamma_{\mu}-\gamma_{\mu}\not{p})$

is satisfied, then put by definition

$(U\Gamma)_{\mu}(p,q)=a(M^{2})\gamma_{\mu},$ (6)

where $M^{2}$ is an arbitrary number. Similarly, for a lepton self-energy
Feynman amplitude $\Sigma(p)=r(p^{2})+s(p^{2})\not{p}$ we define

$(U\Sigma)(p)=r(m^{2})+s(m^{2})m+s(M^{2})(\not{p}-m).$ (7)

In old calculations we used $M^{2}=m^{2}$. Here we use $M^{2}=-m^{2}$; this
slightly reduces inter-graph cancellations; see Section V and Volkov (2023).

For photon self-energy Feynman amplitudes
$\Pi_{\mu\nu}(p^{2})=h_{1}(p^{2})g_{\mu\nu}+h_{2}(p^{2})p_{\mu}p_{\nu}$ we
utilize the substitution

$\Pi_{\mu\nu}(p^{2})\rightarrow p^{2}g_{\mu\nu}[h_{2}(0)-h_{2}(p^{2})]$

instead of subtractions. This trick makes the expressions shorter and has been
widely used in literature; see, for example, Cvitanović and Kinoshita (1974b).

##  IV MONTE CARLO INTEGRATION

###  IV.1 General idea

Suppose we have a Feynman parametric integral

$\int_{z_{1},\ldots,z_{K}>0}I(z_{1},\ldots,z_{K})\delta(z_{1}+\ldots+z_{K}-1)dz_{1}\ldots
dz_{K}$

corresponding to a graph of $A_{1}^{(2n)}$. Here $K=3n-1$: we use a trick to
reduce the number from $3n$ to $3n-1$ (see Volkov (2017)); each $z_{i}$
corresponds to an internal graph line, except one corresponding to two lepton
lines adjoining the external photon line.

For the integration we use the predefined probability density functions of the
form

$g(\underline{z})=C_{\text{main}}\times
g_{0}(\underline{z})+C_{\text{min}}\times
g_{\text{min}}(\underline{z})+C_{\text{uniform}}\times
g_{\text{uniform}}(\underline{z})+C_{\text{modify}}\times\frac{1}{N}\sum_{i=1}^{N}g_{i}(\underline{z}),$

where $\underline{z}=(z_{1},\ldots,z_{K})$;
$C_{\text{main}},C_{\text{min}},C_{\text{uniform}},C_{\text{modify}}$ are
arbitrary nonnegative numbers,

$C_{\text{main}}+C_{\text{min}}+C_{\text{uniform}}+C_{\text{modify}}=1;$

$N$ is an arbitrary natural number.

The function $g_{0}$ has the form

$g_{0}(\underline{z})=C\times\frac{\prod_{l=2}^{K}\left(z_{j_{l}}/z_{j_{l-1}}\right)^{\mathrm{Deg}_{0}(\\{j_{l},j_{l+1},\ldots,j_{K}\\})}}{z_{1}z_{2}\ldots
z_{K}},$

where

  * •

$(j_{1},\ldots,j_{K})$ is a permutation of $1,\ldots,K$ such that
$z_{j_{1}}\geq z_{j_{2}}\geq\ldots\geq z_{j_{K}}$ (this splitting of the
integration area is called the Hepp sectors);

  * •

$\mathrm{Deg}_{0}(s)$ are arbitrary positive real numbers defined for any set
$s\subseteq\\{1,2,\ldots,K\\}$ (except the empty and full sets);

  * •

$C$ is a constant such that

$\int_{z_{1},\ldots,z_{K}>0}g_{0}(\underline{z})\delta(z_{1}+\ldots+z_{K}-1)dz_{1}\ldots
dz_{K}=1.$

The algorithm for determining $\mathrm{Deg}_{0}$ for each set of internal
graph lines is described in Section IV.2; each set
$s\subseteq\\{1,\ldots,K\\}$ can easily be mapped to the set of internal graph
lines.

The functions $g_{i}$, $1\leq i\leq N$, and $g_{\text{min}}$ have the same
form as $g_{0}$, but with the numbers $\mathrm{Deg}_{i}(s)$ and
$\mathrm{Deg}_{\text{min}}(s)$ instead of $\mathrm{Deg}_{0}(s)$ (and with
different $C$). We use

$\mathrm{Deg}_{i}(s)=\max(S^{\text{sat}}_{i},S^{\text{mul}}_{i}\mathrm{Deg}_{0}(s)-S^{\text{sub}}_{i}),$
$\mathrm{Deg}_{\text{min}}(s)=D,$

where $D$, $S^{\text{sat}}_{i}$, $S^{\text{mul}}_{i}$, $S^{\text{sub}}_{i}$
($1\leq i\leq N$) are arbitrary positive real numbers. The function
$g_{\text{uniform}}$ corresponds to the uniform distribution (constant).

We use the following constant values:

$C_{\text{min}}=0.002,\quad C_{\text{uniform}}=0.06,\quad
C_{\text{modify}}=0.02,$
$C_{\text{main}}=1-C_{\text{min}}-C_{\text{uniform}}-C_{\text{modify}}=0.918,$
$D=0.75,\quad N=2,$ $S^{\text{sat}}_{1}=0.6,\quad S^{\text{mul}}_{1}=1.0,\quad
S^{\text{sub}}_{1}=1.3,$ $S^{\text{sat}}_{2}=1.0,\quad
S^{\text{mul}}_{2}=0.4,\quad S^{\text{sub}}_{2}=0.6.$

The algorithm for generating the random samples with the probability density
functions of this form is described in Volkov (2017). The function
$g_{0}(\underline{z})$ is designed in such a way to be “near” to
$C|I(\underline{z})|$. The remaining terms of $g(\underline{z})$ are
stabilization terms; they prevent from significant underestimation of
$|I(\underline{z})|$, which can lead to poor Monte Carlo convergence.

The choice of $S^{\text{sat}}_{i}$, $S^{\text{mul}}_{i}$, $S^{\text{sub}}_{i}$
is the result of experiments after the discovery of poor convergence for some
Feynman graphs from Set I(i).

We have several integrals in our calculation. They are evaluated
simultaneously by blocks of samples in random order. The probabilities of
choosing each integral are adjusted in real time to make the overall
convergence as fast as possible. The algorithm is described in S. Volkov
(2019) (Section IV.A).

###  IV.2 Obtaining $\mathrm{Deg}_{0}(s)$

####  IV.2.1 Preliminaries

This construction is based on S. Volkov (2020) (ideologically444In fact, a
serious improvement of the theory is needed to justify this procedure.) and on
numerical experiments with 4-loop graphs. It would be great if the integral

$\int_{z_{1},\ldots,z_{K}>0}\frac{|I(\underline{z})|^{2}}{g_{0}(\underline{z})}\delta(z_{1}+\ldots+z_{K}-1)dz_{1}\ldots
dz_{K}$

was finite and not so large (see Volkov (2017)). However, we can not guarantee
this for all orders; moreover, examples of high order are known that result in
an infinite integral.

To write a formula for $\mathrm{Deg}_{0}(s)$, we need some additional
definitions.

Two subgraphs are called _overlapping_ if they are not contained in each other
and the intersection of their line sets is not empty.

A set of subgraphs of a graph is called a _forest_ if any two elements of this
set do not overlap.

For a vertexlike graph $G$, we denote by $\mathfrak{F}[G]$ the _set_ of all
forests $F$ consisting of UV-divergent subgraphs of $G$ and satisfying the
condition $G\in F$.

If $s$ is some set of internal graph lines, by $\mathrm{Loop}(s)$ we denote
the number of _independent loops_ in $s$; by $\mathrm{Lept}(s)$ we denote the
set of all _lepton lines_ in $s$; by $\mathrm{Ph}(s)$ we denote the set of all
_photon lines_ in $s$; by $\mathrm{V}(s)$ we denote the set of all _vertices_
incident to at least one line in $s$.

If $G$ is an arbitrary Feynman graph, we denote by $\mathrm{E}[G]$ the set of
all _internal lines_ of $G$, by $\mathrm{V}[G]$ we denote the set of all
vertices of $G$. If additionally $G$ has two external leptons, by
$\mathrm{LPath}[G]$ we denote the _lepton path_ connecting the external lepton
lines of $G$ (as a set of lines).

A subgraph $G^{\prime}$ of a graph $G$ is said to lie on $\mathrm{LPath}[G]$,
if $G^{\prime}$ has external lepton lines, and each of these lines is in
$\mathrm{LPath}[G]$ or coincides with an external line of $G$.

A set $s\subseteq\mathrm{E}[G]$ is called _cyclo-complete_ , if

$(\mathrm{Lept}(\mathrm{E}[G])\backslash\mathrm{LPath}[G])\subseteq s.$

####  IV.2.2 Auxiliary functions

For convenience, we mark each new definition of global scope with bold
lettering.

The _ultraviolet degree of divergence_ is defined as

$\omega_{G}(s)=2\times\mathrm{Loop}(s)-|s|+\frac{1}{2}|\mathrm{Lept}(s)|,$

where $s\subseteq\mathrm{E}[G]$. For example, for the graph $G$ in FIG. 2 we
have

$\omega_{G}(\\{1,2,3,4,5,6,7,8,9,10,11\\})=6-11+4=-1,$
$\omega_{G}(\\{1,3,4,5,8,9,10,11,12\\})=6-9+2.5=-0.5.$

Figure 2: Example of a Feynman graph without lepton loops

A graph $G^{\prime\prime}$ is called a _child_ of $G^{\prime}$ in a forest $F$
($G^{\prime}\in F$), if $G^{\prime\prime}$ is a maximal (with respect to
inclusion) element of $F$ properly contained in $G^{\prime}$.

If $f\in\mathfrak{F}[G]$ and $G^{\prime}\in F$, then by $G^{\prime}/F$ we
denote the graph that is obtained from $G^{\prime}$ by shrinking all children
of $G^{\prime}$ in $F$ to points. We will also use the symbols like $\omega$
for graphs like $G^{\prime}/F$ and for sets $s\subseteq\mathrm{E}[G]$. This
means that we really apply it to the set $s^{\prime}$ that is the intersection
of $s$ and the set of all internal lines of $G^{\prime}/F$.

Let us consider an example. Suppose $G$ is the graph from FIG. 2, the UV-
divergent subgraphs are

$G_{1}=\mathrm{bcdefgh},\ G_{2}=\mathrm{cdef},\ G_{3}=\mathrm{cde},\
G_{4}=\mathrm{def}$

(each subgraph is given by vertex enumeration). Suppose

$F=\\{G,G_{1},G_{2},G_{3}\\}.$

In this case,

$\omega_{G_{1}/F}(\\{1,2,3,4,5,6,7,8,9,10,11\\})=\omega_{G_{1}/F}(\\{2,6,7,11\\})$
$=2-4+1.5=-0.5$

($G_{2}$ is the only child of $G_{1}$ in $F$).

Let us define $\overline{\omega}_{G}(s)$, $s\subseteq\mathrm{E}[G]$. It is
defined only for $G$ with two external fermion lines (vertexlike or lepton
self-energy graphs). The set $s\cap\mathrm{LPath}[G]$ can be considered as a
union of nonintersecting paths. Suppose $I_{1},\ldots,I_{l}$ are the sets of
vertices corresponding to these paths (it is obvious that $|I_{j}|\geq 2$ for
any $j$). The empty set gives $l=0$. By definition, put

$I_{0}=\mathrm{LPath}[G]\backslash(I_{1}\cup\ldots\cup I_{l}).$

We say that two vertices $v_{1},v_{2}$ from $\mathrm{V}(\mathrm{LPath}[G])$
are 1-equivalent, if there is a path in
$\mathrm{E}[G]\backslash\mathrm{LPath}[G]$ from $v_{1}$ to $v_{2}$ (we
consider all lines as undirected). Suppose $B_{1},B_{2},\ldots,B_{r}$ are
classes of equivalence of $\mathrm{V}(\mathrm{LPath}[G])$ with respect to
1-equivalence. We say that $v_{1}$ and $v_{2}$ are 2-equivalent, if there is a
path in $s\backslash\mathrm{LPath}[G]$ from $v_{1}$ to $v_{2}$ (we consider
all lines as undirected). By $N_{jb}$ we denote the number of equivalence
classes in $I_{j}\cap B_{b}$ with respect to 2-equivalence (the number is $0$
if the set is empty). By definition, put

$a_{jb}=\begin{cases}1,\text{ if }0<N_{jb}<\sum_{i=0}^{l}N_{ib},\\\
\frac{1}{2},\text{ if }1=N_{jb}=\sum_{i=0}^{l}N_{ib},\\\ 0,\text{ in the other
cases}.\end{cases}$

The function $\overline{\omega}_{G}$ is defined as

$\overline{\omega}_{G}(s)=2\times\mathrm{Loop}(s)-|s|+\frac{1}{2}|\mathrm{Lept}(s)\backslash\mathrm{LPath}[G]|+\frac{1}{2}\sum_{j=1}^{l}|I_{j}|+\frac{1}{2}\sum_{j=1}^{l}\sum_{b=1}^{r}N_{jb}-\sum_{j=1}^{l}\sum_{b=1}^{r}a_{jb}-\frac{1}{2}.$

Let us examine several examples. If $G$ is the graph from FIG. 3, we have

$r=2,\quad B_{1}=\\{\mathrm{a,b,d,e}\\},\quad B_{2}=\\{\mathrm{c}\\}.$

Figure 3: Example of a Feynman graph with a lepton loop of length 4

For $s=\\{1,2,3,9,10,11,12\\}$, we have

$l=1,\quad I_{1}=\\{\mathrm{a,b,c,d}\\},\quad I_{0}=\\{\mathrm{e}\\},$
$I_{0}\cap B_{1}=\\{\mathrm{e}\\},\ N_{01}=1,\quad I_{1}\cap
B_{1}=\\{\mathrm{a,b,d}\\},\ N_{11}=3,$ $I_{0}\cap B_{2}=\emptyset,\
N_{02}=0,\quad I_{1}\cap B_{2}=\\{\mathrm{c}\\},\ N_{12}=1,$ $a_{11}=1,\quad
a_{12}=\frac{1}{2},$ $\overline{\omega}_{G}(s)=0-7+0+2+2-1.5-0.5=-5$

For $s=\\{1,2,3,5,6,7,8,9,10,11,12\\}$ we have the same $l,I_{0},I_{1}$,

$N_{01}=1,\quad N_{11}=1,\quad N_{02}=0,\quad N_{12}=1,$

$a_{jb}$ are the same,

$\overline{\omega}_{G}(s)=6-11+2+2+1-1.5-0.5=-2$

For $s=\\{1,2,3,6,9,10,11,12\\}$ we have the same $l,I_{0},I_{1}$,

$N_{01}=1,\quad N_{11}=2,\quad N_{02}=0,\quad N_{12}=1,$

$a_{jb}$ are the same,

$\overline{\omega}_{G}(s)=2-8+0.5+2+1.5-1.5-0.5=-4.$

For $s=\\{1,2,3,4,5,6,7,8\\}$ we have

$l=1,\quad I_{1}=\\{\mathrm{a,b,c,d,e}\\},\quad I_{0}=\emptyset,$
$N_{01}=0,\quad N_{11}=4,\quad N_{02}=0,\quad N_{12}=1,$ $a_{11}=0,\quad
a_{12}=\frac{1}{2},$ $\overline{\omega}_{G}(s)=2-8+2+2.5+2.5-0.5-0.5=0.$

For $s=\\{1,2,3,4,5,6,7,8,9,10,11\\}$ we have the same $l,I_{0},I_{1}$,

$N_{01}=0,\quad N_{11}=2,\quad N_{02}=0,\quad N_{12}=1,$

$a_{jb}$ are the same,

$\overline{\omega}_{G}(s)=6-11+2+2.5+1.5-0.5-0.5=0.$

For $s=\\{1,2,3,4,5,6,7,8,9,10,11,12\\}$ we have the same $l,I_{0},I_{1}$

$N_{01}=0,\quad N_{11}=1,\quad N_{02}=0,\quad N_{12}=1,$
$a_{11}=\frac{1}{2},\quad a_{12}=\frac{1}{2},$
$\overline{\omega}_{G}(s)=8-12+2+2.5+1-1-0.5=0.$

If $G$ is the graph from FIG. 4 containing special vertices, we have

$r=2,\quad B_{1}=\\{\mathrm{a,c}\\},\quad B_{2}=\\{\mathrm{b}\\}.$

Figure 4: Example of a Feynman graph with special vertices

For $s=\emptyset$, we have

$l=0,\quad I_{0}=\\{\mathrm{a,b,c}\\},$
$\overline{\omega}_{G}(s)=0-0+0+0+0-0-0.5=-0.5.$

For $s=\\{3,4,5\\}$, we have the same $l=0$,

$\overline{\omega}_{G}(s)=0-3+0+0+0-0-0.5=-3.5.$

For $s=\\{1,2\\}$, we have

$l=1,\quad I_{1}=\\{\mathrm{a,b,c}\\},\quad I_{0}=\emptyset,$ $N_{01}=0,\quad
N_{11}=2,\quad N_{02}=0,\quad N_{12}=1,$ $a_{11}=0,\quad a_{12}=\frac{1}{2},$
$\overline{\omega}_{G}(s)=0-2+0+1.5+1.5-0.5-0.5=0.$

For $s=\\{1,2,3,4,5\\}$, we have the same $l,I_{0},I_{1}$,

$N_{01}=0,\quad N_{11}=1,\quad N_{02}=0,\quad N_{12}=1,$
$a_{11}=\frac{1}{2},\quad a_{12}=\frac{1}{2},$
$\overline{\omega}_{G}(s)=2-5+0+1.5+1-1-0.5=-2.$

If $G$ is the graph from FIG. 5, we have

$r=2,\quad B_{1}=\\{\mathrm{a,c,d}\\},\quad B_{2}=\\{\mathrm{b,e}\\}.$

Figure 5: Example of a Feynman graph with an external photon on a lepton loop

If $s=\\{1,2,4,5,6,7,8,9,10,11,12\\}$, we have

$l=2,\quad I_{1}=\\{\mathrm{a,b,c}\\},\quad I_{2}=\\{\mathrm{d,e}\\},\quad
I_{0}=\emptyset,$ $N_{01}=0,\ N_{11}=1,\ N_{21}=1,\ N_{02}=0,\ N_{12}=1,\
N_{22}=1,$ $a_{11}=a_{21}=a_{12}=a_{22}=1,$
$\overline{\omega}_{G}(s)=6-11+2+2.5+2-4-0.5=-3.$

If $s=\\{1,2,4,5,6,7,8,9,11,12\\}$, we have the same $l,I_{0},I_{1},I_{2}$,

$N_{01}=0,\ N_{11}=2,\ N_{21}=1,\ N_{02}=0,\ N_{12}=1,\ N_{22}=1,$

$a_{jb}$ are the same,

$\overline{\omega}_{G}(s)=4-10+2+2.5+2.5-4-0.5=-3.5.$

If $s=\\{2,3,4,12\\}$, we have

$l=1,\quad I_{1}=\\{\mathrm{b,c,d,e}\\},\quad I_{0}=\\{\mathrm{a}\\},$
$N_{01}=1,\ N_{11}=2,\ N_{02}=0,\ N_{12}=1,$ $a_{11}=1,\quad
a_{12}=\frac{1}{2},$ $\overline{\omega}_{G}(s)=2-4+0+2+1.5-1.5-0.5=-0.5.$

If $s=\\{2,3,4,6,10,11\\}$, we have the same $l,I_{0},I_{1}$,

$N_{01}=1,\ N_{11}=1,\ N_{02}=0,\ N_{12}=2,$ $a_{11}=1,\quad a_{12}=0,$
$\overline{\omega}_{G}(s)=2-6+0.5+2+1.5-1-0.5=-1.5.$

If $s=\\{2,3,4,5,6,9,11\\}$, we have the same $l,I_{0},I_{1}$,

$N_{01}=1,\ N_{11}=2,\ N_{02}=0,\ N_{12}=2,$

the same $a_{jb}$,

$\overline{\omega}_{G}(s)=0-7+1+2+2-1-0.5=-3.5.$

If $s=\\{1,2,3,4,5,6,7,8\\}$, we have

$l=1,\quad,I_{1}=\\{\mathrm{a,b,c,d,e}\\},\quad I_{0}=\emptyset,$ $N_{01}=0,\
N_{11}=3,\ N_{02}=0,\ N_{12}=2,$ $a_{11}=a_{12}=0,$
$\overline{\omega}_{G}(s)=2-8+2+2.5+2.5-0-0.5=0.5.$

If $s=\\{1,2,3,4,5,6,7,8,12\\}$, we have the same $l,I_{0},I_{1}$,

$N_{01}=0,\ N_{11}=3,\ N_{02}=0,\ N_{12}=1,$ $a_{11}=0,\quad
a_{12}=\frac{1}{2},$ $\overline{\omega}_{G}(s)=4-9+2+2.5+2-0.5-0.5=0.5.$

If $G$ is the graph from FIG. 6, we have

$r=2,\quad B_{1}=\\{\mathrm{a,b,c,e,f,g}\\},\quad B_{2}=\\{\mathrm{d}\\}.$

Figure 6: Example of a Feynman graph with a lepton loop of length 6

If $s=\\{1,2,3,4,5,6,7,8,9,10,11,12\\}$, we have

$l=1,\quad I_{1}=\\{\mathrm{a,b,c,d,e,f,g}\\},\quad I_{0}=\emptyset,$
$N_{01}=0,\ N_{11}=6,\ N_{02}=0,\ N_{12}=1,$ $a_{11}=0,\quad
a_{12}=\frac{1}{2},$ $\overline{\omega}_{G}(s)=2-12+3+3.5+3.5-0.5-0.5=-1.$

The sequence $(l_{1},G_{1},l_{2},\ldots,G_{n},l_{n+1})$, where $n\geq 1$,
$G_{1},\ldots,G_{n}$ are self-energy subgraphs of $G$,
$l_{1},\ldots,l_{n+1}\in\mathrm{E}[G]$, is called a _chain_ in $G$, if the
following conditions are satisfied:

  * •

$\mathrm{V}[G_{1}],\ldots,\mathrm{V}[G_{n}]$ do not intersect;

  * •

$l_{1},\ldots,l_{n+1}$ are pairwise different;

  * •

for all $i$ ($1\leq i\leq n$), each of the lines $l_{i}$, $l_{i+1}$ is
incident to at least one vertex of $\mathrm{V}[G_{i}]$;

  * •

the set $\\{l_{1},\ldots,l_{n+1}\\}$ is maximal with respect to inclusion
(among all the sets obtained from the objects satisfying the conditions
above).

Chains can be _leptonic_ or _photonic_ (according to the type of
$l_{1},\ldots,l_{n+1}$). Analogously, the chain lies on $\mathrm{LPath}[G]$,
if $l_{1},\ldots,l_{n+1}\in\mathrm{LPath}[G]$. The chains that are obtained
from each other by reversing the order are considered as the same.

For example, we have the following chains in the graph $G$ from FIG. 7:

$(6,G_{2},7,G_{4},8),\quad(15,G_{3},16),\quad(1,G_{1},3),\quad(18,G_{5},20),$

where

$G_{1}=\mathrm{bc},\ G_{2}=\mathrm{fghijk},\ G_{3}=\mathrm{jk},\
G_{4}=\mathrm{lmno},\ G_{5}=\mathrm{no}.$ (8)

Figure 7: Example of a Feynman graph with chains

If $G$ is a Feynman graph, $F\in\mathfrak{F}[G]$, $G^{\prime}\in F$, by
$\mathrm{Chains}[G^{\prime},F]$ we denote the set of all chains
$(l_{1},G_{1},\ldots,l_{n+1})$ in $G$ such that each $G_{i}$ is a child of
$G^{\prime}$ in $F$.

For example, for the graph $G$ in FIG. 7 and (8), for the forest

$F=\\{G,G_{1},G_{2},G_{3},G_{4},G_{4}\\},$ (9)

we have

$\mathrm{Chains}[G,F]=\\{(1,G_{1},3),\ (6,G_{2},7,G_{4},8)\\},$
$\mathrm{Chains}[G_{2},F]=\\{(15,G_{3},16)\\},$
$\mathrm{Chains}[G_{4},F]=\\{(18,G_{5},20)\\}.$

For a fixed Feynman graph $G$, a forest $F\in\mathfrak{F}[G]$, and
$G^{\prime}\in F$, we define the function $\mathrm{Ch}_{G^{\prime},F}(s)$ for
all $s\in\mathrm{E}[G]$ in the following way:

$\mathrm{Ch}_{G^{\prime},F}(s)=\sum_{c\in\mathrm{Chains}[G^{\prime},F]}g(c,s),$

where

$g(c,s)=\begin{cases}f(c,s),\text{ if $c$ is a fermionic chain, $c$ does not
lie on $\mathrm{LPath}[G]$, $s$ is cyclo-complete in $G$},\\\
\overline{f}(c,s),\text{ if $c$ is a fermionic chain, $c$ lies on
$\mathrm{LPath}[G]$, $s$ is cyclo-complete in $G$},\\\ \tilde{f}(c,s),\text{
if $c$ is a fermionic chain, $s$ is not cyclo-complete in $G$},\\\
p(c,s),\text{ if $c$ is a photonic chain, $s$ is cyclo-complete in $G$},\\\
\tilde{p}(c,s),\text{ if $c$ is a photonic chain, $s$ is not cyclo-complete in
$G$},\end{cases}$
$f((l_{1},G_{1},\ldots,l_{n+1}),s)=\begin{cases}-\frac{n}{2},\text{ if
}l_{1},\ldots,l_{n+1}\in s,\\\ \frac{1}{2}(h(G_{1},s)+\ldots+h(G_{n},s))\text{
otherwise},\end{cases}$ $h(G^{\prime},s)=\begin{cases}1,\text{ if
}\mathrm{E}[G^{\prime}]\subseteq s,\\\ 0\text{ otherwise},\end{cases}$
$\overline{f}((l_{1},G_{1},\ldots,l_{n+1}),s)=\begin{cases}-\frac{n}{2},\text{
if }l_{1},\ldots,l_{n+1}\in s,\\\ 0\text{ otherwise},\end{cases}$
$\tilde{f}((l_{1},G_{1},\ldots,l_{n+1}),s)=\begin{cases}-\frac{n}{2},\text{ if
}l_{1},\ldots,l_{n+1}\in s,\\\ \frac{1}{2}|\\{l_{1},\ldots,l_{n+1}\\}\cap
s|+h(G_{1},s)+\ldots+h(G_{n},s)\text{ otherwise},\end{cases}$
$p((l_{1},G_{1},\ldots,l_{n+1}),s)=\begin{cases}-n,\text{ if
}l_{1},\ldots,l_{n+1}\in s,\\\ 0\text{ otherwise},\end{cases}$
$\tilde{p}((l_{1},G_{1},\ldots,l_{n+1}),s)=\begin{cases}-n,\text{ if
}l_{1},\ldots,l_{n+1}\in s,\\\ h(G_{1},s)+\ldots+h(G_{n},s)\text{
otherwise}.\end{cases}$

The auxiliary functions $f,\overline{f},\tilde{f},p,\tilde{p}$ are illustrated
by the following examples concerning the graph $G$ from FIG. 7 and the
subgraphs 8:

$f((1,G_{1},3),\\{1,2,3\\})=-0.5,\quad f((1,G_{1},3),\\{1,2\\})=0,$
$f((1,G_{1},3),\\{1,2,5\\})=0.5,$
$\overline{f}((1,G_{1},3),\\{1,2,3\\})=-0.5,\quad\overline{f}((1,G_{1},3),\\{1,2\\})=0,$
$\overline{f}((1,G_{1},3),\\{1,2,5\\})=0,$
$\tilde{f}((1,G_{1},3),\\{1,2,3\\})=-0.5,\quad\tilde{f}((1,G_{1},3),\\{1,2\\})=0.5,$
$\tilde{f}((1,G_{1},3),\\{1,2,5\\})=1.5,$
$p((6,G_{2},7,G_{4},8),\\{6,7,9,10,11,12,15,16\\})=0,$
$\tilde{p}((6,G_{2},7,G_{4},8),\\{6,7,9,10,11,12,15,16\\})=0,$
$p((6,G_{2},7,G_{4},8),\\{6,7,8,9,10,11,12,15,16\\})=-2,$
$\tilde{p}((6,G_{2},7,G_{4},8),\\{6,7,8,9,10,11,12,15,16\\})=-2,$
$p((6,G_{2},7,G_{4},8),\\{6,7,9,10,11,12,13,14,15,16\\})=0,$
$\tilde{p}((6,G_{2},7,G_{4},8),\\{6,7,9,10,11,12,13,14,15,16\\})=1.$

The set $s=\\{1,3,9,10,\ldots,20\\}$ is cyclo-complete in $G$. If (9) is
satisfied, we have

$\mathrm{Ch}_{G,F}(s)=\overline{f}((1,G_{1},3),s)+p((6,G_{2},7,G_{4},8),s)$
$=-0.5+0=-0.5,$ $\mathrm{Ch}_{G_{2},F}(s)=p((15,G_{3},16),s)=-1,$
$\mathrm{Ch}_{G_{4},F}(s)=f((18,G_{5},20),s)=-0.5.$

For the not cyclo-complete set $s=\\{13,14,19,20\\}$, we have

$\mathrm{Ch}_{G,F}(s)=\tilde{f}((1,G_{1},3),s)+\tilde{p}((6,G_{2},7,G_{4},8),s)$
$=0+0=0,$ $\mathrm{Ch}_{G_{2},F}(s)=\tilde{p}((15,G_{3},16),s)=1,$
$\mathrm{Ch}_{G_{4},F}(s)=\tilde{f}((18,G_{5},20),s)=0.5.$

For another not cyclo-complete set $s=\\{2,5,6,8,9,10,11,12,17,18,19,20\\}$,
we have

$\mathrm{Ch}_{G,F}(s)=\tilde{f}((1,G_{1},3),s)+\tilde{p}((6,G_{2},7,G_{4},8),s)$
$=1+0=1.$

Suppose the Feynman graph $G$ is fixed. Let us define the functions
$\tilde{\omega}_{G^{\prime},F}(s)$,
$\tilde{\omega}^{\prime}_{G^{\prime},F}(s)$, where $F\in\mathfrak{F}[G]$,
$G^{\prime}\in F$, $s\subseteq\mathrm{E}[G]$:

$\tilde{\omega}_{G^{\prime},F}(s)=\begin{cases}\max(\overline{\omega}_{G^{\prime}/F}(s),\omega_{G^{\prime}/F}(s))-\mathrm{Ch}_{G^{\prime},F}(s),\text{
if $s$ is cyclo-complete in $G$, and $G^{\prime}$ lies on
$\mathrm{LPath}[G]$},\\\
\omega_{G^{\prime}/F}(s)-\mathrm{Ch}_{G^{\prime},F}(s)\text{
otherwise},\end{cases}$
$\tilde{\omega}^{\prime}_{G^{\prime},F}(s)=\min(0,\tilde{\omega}_{G^{\prime},F}(s)+C_{\mathrm{Sub}}[G^{\prime},s]),$

where

$C_{\mathrm{Sub}}[G^{\prime},s]=\begin{cases}C_{\mathrm{SubLSEOP}}(s),\text{
if $G^{\prime}$ is a lepton self-energy subgraph, $G^{\prime}$ lies on
$\mathrm{LPath}[G]$},\\\ C_{\mathrm{SubLSEOL}}(s),\text{ if $G^{\prime}$ is a
lepton self-energy subgraph, $G^{\prime}$ does not lie on
$\mathrm{LPath}[G]$},\\\ C_{\mathrm{SubI}}(s),\text{ if $G^{\prime}$ is
vertexlike, $G^{\prime}$ lies on $\mathrm{LPath}[G]$,}\\\
\quad\quad\quad\text{the external photon of $G$ is incident to some
$v\in\mathrm{V}[G^{\prime}]$},\\\ C_{\mathrm{SubVOP}}(s),\text{ if
$G^{\prime}$ is vertexlike, $G^{\prime}$ lies on $\mathrm{LPath}[G]$},\\\
\quad\quad\quad\text{there is no $v\in\mathrm{V}[G^{\prime}]$ incident to the
external photon of $G$ },\\\ C_{\mathrm{SubVOL}}(s)\text{ if $G^{\prime}$ is
vertexlike, $G^{\prime}$ does not lie on $\mathrm{LPath}[G]$},\\\ 0,\text{ in
the other cases},\end{cases}$

the values $C_{\mathrm{SubLSEOP}}(s)$, $C_{\mathrm{SubLSEOL}}(s)$,
$C_{\mathrm{SubI}}(s)$, $C_{\mathrm{SubVOP}}(s)$, $C_{\mathrm{SubVOL}}(s)$ are
taken from Table 5 depending on whether $s$ is cyclo-complete in $G$ and
whether the external photon of $G$ is incident to $v\in\mathrm{LPath}[G]$. The
table constants were obtained by numerical experiments with the Monte Carlo
convergence speed at the 4-loop level.

Table 5: The constants that are used for obtaining $\mathrm{Deg}_{0}(s)$. The values depend on whether $s$ is cyclo-complete in the whole graph $G$ and whether the external photon of $G$ is incident to $v\in\mathrm{LPath}[G]$ (“on path”) or not (“on loop”). Constant | cyclo-complete | not cyclo-complete
---|---|---
| on path | on loop | on path | on loop
$C_{\mathrm{SubLSEOP}}(s)$ | 0 | 0 | 0.293 | 0
$C_{\mathrm{SubLSEOL}}(s)$ | 0.021 | 0 | 0 | 0
$C_{\mathrm{SubI}}(s)$ | 0 | 0.011 | 0.228 | 0.622
$C_{\mathrm{SubVOP}}(s)$ | 0.085 | 0 | 0.041 | 0.394
$C_{\mathrm{SubVOL}}(s)$ | 0.185 | 0 | 0 | 0
$C_{\mathrm{BigZ}}$ | 0.695 | 1.079 | … | …
$C_{\mathrm{BigF}}$ | 0.285 | 0.38 | … | …
$C_{\mathrm{Add}}(s)$ | 0.199 | -0.122 | 0 | 0.187
$C_{\mathrm{Sat}}(s)$ | 0.46 | 0.716 | 0.785 | 0.684

Let us examine several examples. Suppose the graph $G$ is taken from FIG. 7,
the conditions (8) and (9) are satisfied. If $s=\\{13,14,19,20\\}$, we have
for

$\tilde{\omega}_{G,F}(s)=\omega_{G,F}(s)-\mathrm{Ch}_{G,F}(s)=0-0=0,$
$\tilde{\omega}^{\prime}_{G,F}(s)=\min(0,0+C_{\mathrm{SubI}}(s))=\min(0,C_{\mathrm{SubI}}(s)),$
$\tilde{\omega}_{G_{1},F}(s)=\omega_{G_{1},F}(s)-\mathrm{Ch}_{G_{1},F}(s)=0-0=0,$
$\tilde{\omega}^{\prime}_{G_{1},F}(s)=\min(0,C_{\mathrm{SubI}}(s))=\min(0,C_{\mathrm{SubLSEOP}}(s)),$
$\tilde{\omega}_{G_{2},F}(s)=\omega_{G_{2},F}(s)-\mathrm{Ch}_{G_{2},F}(s)=0-1=-1,$
$\tilde{\omega}^{\prime}_{G_{2},F}(s)=\min(0,-1)=-1,$
$\tilde{\omega}_{G_{4},F}(s)=\omega_{G_{4},F}(s)-\mathrm{Ch}_{G_{4},F}(s)=-0.5-0.5=-1,$
$\tilde{\omega}^{\prime}_{G_{4},F}(s)=\min(0,-1)=-1,$
$\tilde{\omega}_{G_{5},F}(s)=\omega_{G_{5},F}(s)-\mathrm{Ch}_{G_{5},F}(s)=-0.5-0=-0.5,$
$\tilde{\omega}^{\prime}_{G_{5},F}(s)=\min(0,-0.5+C_{\mathrm{SubLSEOL}}(s)).$

If $F=\\{G,G_{1},G_{2},G_{3},G_{4},G_{5},G_{6}\\}$ with $G_{1},\ldots,G_{5}$
from (8) and $G_{6}=\mathrm{ghijk}$, $s=\\{9,12,15,16\\}$, we have

$\tilde{\omega}_{G_{6},F}(s)=\omega_{G_{6},F}(s)-\mathrm{Ch}_{G_{6},F}(s)=-2-(-1)=-1,$
$\tilde{\omega}^{\prime}_{G_{6},F}(s)=\min(0,-1+C_{\mathrm{SubVOL}}(s)).$

Suppose $G$ is from FIG. 2,

$F_{1}=\\{G,G_{1},G_{2},G_{3}\\},\quad F_{2}=\\{G,G_{1},G_{2},G_{4}\\},$ (10)
$G_{1}=\mathrm{bcdefgh},\ G_{2}=\mathrm{cdef},\ G_{3}=\mathrm{cde},\
G_{4}=\mathrm{def}.$ (11)

If $s=\\{4,5,10,11\\}$, we have

$\tilde{\omega}_{G,F_{1}}(s)=\tilde{\omega}_{G,F_{2}}(s)=0,$
$\tilde{\omega}^{\prime}_{G,F_{1}}(s)=\tilde{\omega}^{\prime}_{G,F_{2}}(s)=\min(0,C_{\mathrm{SubI}}(s)),$
$\tilde{\omega}_{G_{1},F_{1}}(s)=\tilde{\omega}_{G_{1},F_{2}}(s)=\max(\overline{\omega}_{G_{1},F_{1}}(s),\omega_{G_{1},F_{1}}(s))$
$=\max(-1.5,-1)=-1,$
$\tilde{\omega}^{\prime}_{G_{1},F_{1}}(s)=\tilde{\omega}^{\prime}_{G_{1},F_{2}}(s)=\min(0,-1+C_{\mathrm{SubI}}(s)),$
$\tilde{\omega}_{G_{2},F_{1}}(s)=\max(\overline{\omega}_{G_{2},F_{1}}(s),\omega_{G_{2},F_{1}}(s))=\max(0.5,-0.5)=0.5,$
$\tilde{\omega}^{\prime}_{G_{2},F_{1}}(s)=\min(0,0.5+C_{\mathrm{SubLSEOP}}(s)),$
$\tilde{\omega}_{G_{2},F_{2}}(s)=\max(\overline{\omega}_{G_{2},F_{2}}(s),\omega_{G_{2},F_{2}}(s))=\max(-0.5,0)=0,$
$\tilde{\omega}^{\prime}_{G_{2},F_{2}}(s)=\min(0,C_{\mathrm{SubLSEOP}}(s)),$
$\tilde{\omega}_{G_{3},F_{1}}(s)=\max(\overline{\omega}_{G_{3},F_{1}}(s),\omega_{G_{3},F_{1}}(s))=\max(-1,-0.5)=-0.5,$
$\tilde{\omega}^{\prime}_{G_{3},F_{1}}(s)=\min(0,-0.5+C_{\mathrm{SubVOP}}(s)),$
$\tilde{\omega}_{G_{4},F_{2}}(s)=\max(\overline{\omega}_{G_{4},F_{2}}(s),\omega_{G_{4},F_{2}}(s))=\max(0,0)=0,$
$\tilde{\omega}^{\prime}_{G_{4},F_{2}}(s)=\min(0,C_{\mathrm{SubVOP}}(s)).$

Suppose $G$ is taken from FIG. 6,

$F=\\{G\\},\quad s=\\{1,2,\ldots,12\\}.$

We have

$\tilde{\omega}_{G,F}(s)=\max(\overline{\omega}_{G,F}(s),\omega_{G,F}(s))=\max(-1,-4)=-1,$
$\tilde{\omega}^{\prime}_{G,F}(s)=\min(0,-1+C_{\mathrm{SubI}}(s)).$

####  IV.2.3 The formula

Suppose $G$ is a vertexlike Feynman graph.

By $\mathfrak{F}_{\mathrm{max}}[G]$ we denote the set of all maximal sets from
$\mathfrak{F}[G]$ (with respect to inclusion).

By definition, put

$\mathrm{Deg}_{0}(s)=\begin{cases}C_{\mathrm{BigZ}}+(C_{\mathrm{BigF}}-C_{\mathrm{BigZ}})\frac{|\mathrm{Ph}(s)|}{|\mathrm{Ph}(\mathrm{E}[G])|},\text{
if $\mathrm{Lept}(\mathrm{E}[G])\subseteq s$, and}\\\
\quad\quad\quad\text{there exists $F\in\mathfrak{F}_{\mathrm{max}}[G]$ such
that $\tilde{\omega}_{G^{\prime},F}(s)\geq 0$ for all $G^{\prime}\in F$},\\\
\min_{F\in\mathfrak{F}_{\mathrm{max}}[G]}D_{F}(s)\text{
otherwise},\end{cases}$

where

$D_{F}(s)=\tau\left(-\sum_{G^{\prime}\in
F}\tilde{\omega}^{\prime}_{G^{\prime},F}(s),\ C_{\mathrm{Add}}(s),\
C_{\mathrm{Sat}}(s)\right),$

$C_{\mathrm{BigZ}}$, $C_{\mathrm{BigF}}$, $C_{\mathrm{Add}}(s)$,
$C_{\mathrm{Sat}}(s)$ are taken from Table 5,

$\tau(x,C_{\mathrm{a}},C_{\mathrm{s}})=\frac{1}{2}(C_{\mathrm{a}}+C_{\mathrm{s}})\times\mu\left(\frac{x-C_{\mathrm{s}}}{C_{\mathrm{a}}+C_{\mathrm{s}}}\right),$
$\mu(t)=2+t+\sqrt{t^{2}+0.25}.$

Here, the table constants were determined by numerical experiments with the
Monte Carlo convergence speed at the 4-loop level; $\mu(t)$ is a smooth
approximation for $2+\max(0,2t)$; the function
$\tau(x,C_{\mathrm{a}},C_{\mathrm{s}})$ is a smooth approximation for
$\max(x,C_{\mathrm{s}})+C_{\mathrm{a}}$.

Let us examine several examples. Suppose $G$ is Feynman graph from FIG. 2. In
this case,

$\mathfrak{F}_{\mathrm{max}}[G]=\\{F_{1},F_{2}\\},$

where (10) and (11) are satisfied. If $s=\\{4,5,10,11\\}$, we have

$\mathrm{Deg}_{0}(s)=\min\left(\tau(-\tilde{\omega}^{\prime}_{G,F_{1}}(s)-\tilde{\omega}^{\prime}_{G_{1},F_{1}}(s)-\tilde{\omega}^{\prime}_{G_{2},F_{1}}(s)-\tilde{\omega}^{\prime}_{G_{3},F_{1}}(s),C_{\mathrm{Add}}(s),C_{\mathrm{Sat}}(s)),\right.$
$\left.\tau(-\tilde{\omega}^{\prime}_{G,F_{2}}(s)-\tilde{\omega}^{\prime}_{G_{1},F_{2}}(s)-\tilde{\omega}^{\prime}_{G_{2},F_{2}}(s)-\tilde{\omega}^{\prime}_{G_{4},F_{2}}(s),C_{\mathrm{Add}}(s),C_{\mathrm{Sat}}(s))\right)$

Since all the constants $C_{\mathrm{SubLSEOP}}$, $C_{\mathrm{SubLSEOL}}$,
$C_{\mathrm{SubI}}$, $C_{\mathrm{SubVOP}}$, $C_{\mathrm{SubVOL}}$ are
nonnegative, we have

$\mathrm{Deg}_{0}(s)=\tau(\max(0,1-C_{\mathrm{SubI}}(s)),C_{\mathrm{Add}}(s),C_{\mathrm{Sat}}(s)).$

If $s=\\{4,5,10\\}$, we have

$\tilde{\omega}_{G,F_{2}}(s)=\tilde{\omega}_{G_{1},F_{2}}(s)=\tilde{\omega}_{G_{2},F_{2}}(s)=\tilde{\omega}_{G_{4},F_{2}}(s)=0,$

but we still use $D_{F}$, because $s$ does not contain
$\mathrm{Lept}(\mathrm{E}[G])$:

$\mathrm{Deg}_{0}(s)=D_{F_{2}}(s)=\tau(0,C_{\mathrm{Add}}(s),C_{\mathrm{Sat}}(s)).$

If $s=\\{1,2,3,4,5,6,7,8,10\\}$, we have

$\tilde{w}_{G,F_{2}}(s)=0,\ \tilde{w}_{G_{1},F_{2}}(s)=0,\
\tilde{w}_{G_{2},F_{2}}(s)=0.5,\ \tilde{w}_{G_{4},F_{2}}(s)=0,$
$\mathrm{Deg}_{0}(s)=\frac{1}{4}C_{\mathrm{BigF}}+\frac{3}{4}C_{\mathrm{BigZ}}.$

Suppose $G$ is Feynman graph from FIG. 6. In this case, we have

$\mathfrak{F}_{\mathrm{max}}[G]=\\{F\\},\quad F=\\{G\\}.$

If $s=\\{1,2,\ldots,12\\}$, we have $\mathrm{Lept}(\mathrm{E}[G])\subseteq s$,
but we use $D_{F}$, because $\tilde{\omega}_{G,F}(s)=-1<0$:

$\mathrm{Deg}_{0}(s)=\tau(\max(0,C_{\mathrm{SubI}}(s)-1),C_{\mathrm{Add}}(s),C_{\mathrm{Sat}}(s)).$

###  IV.3 Additional stabilization techniques

During the integration, we perform the summation of the values
$I(\underline{z})/g(\underline{z})$, where $I$ is the integrand, $g$ is the
probability density function. Since these values are unbounded, it is very
important to prevent from occasional emergence of very large values, but to
allow their systematic occurence; we used the technique described in Section
III.D of Volkov (2017) adapted to GPUs. The presence of acute peaks in the
integrands can lead to an error underestimation; to prevent from this, we used
the heuristic described in Section IV.F of Volkov (2018).

##  V TECHNICAL INFORMATION AND NUMERICAL CANCELLATIONS

In our approach, all divergences are numerically cancelled at the level of the
integrands. This cancellation leads to round-off errors. We evaluate the
integrand values using interval arithmetic to control the errors. The samples
that result in too large intervals are evaluated with increased precisions.

The integrand expressions at the 5-loop level are huge. Special tricks are
required to evaluate them on GPUs. The general scheme of the realization as
well as the interval arithmetic are described in S. Volkov (2019). We have
generated the integrand code for different precisions separately:

  * •

“Eliminated Interval Arithmetic” (EIA) is an approach that works with the
machine double precision, but significantly faster than the usual interval
arithmetic (the price for this is larger intervals). It is described in
Section IV.C of Volkov (2018).

  * •

The usual machine double precision interval arithmetic.

  * •

Arbitrary-precision interval arithmetic with the mantissa size as a parameter.

The codegenerator for the integrands was written in the D programming
language. The generation took a couple of months on ITP KIT office computers.
The code was generated in C++555Formally it is C++, but for the most part the
constructions from C were used. with CUDA, the size is 136 GB and 294 GB for
graphs with lepton loops and without lepton loops, respectively. The size in
the compiled form is 191 GB and 393 GB, respectively. The compilation with
nvcc was performed on the supercomputer “HoreKa” (Karlsruhe, Germany) with the
help of the processors Intel Xeon Platinum 8368 (38 cores) and took about 1
core-month.

Table LABEL:table_round_off_contr contains the information about the
contributions of the Monte Carlo samples that required increased precisions.
$\triangle^{\text{fail}}_{\text{EIA}}$, $\triangle^{\text{fail}}_{\text{IA}}$,
$\triangle^{\text{fail}}_{\text{128}}$, $\triangle^{\text{fail}}_{\text{192}}$
are the contributions of the samples for which the following precision failed:
Eliminated Interval Arithmetic, the usual mashine double precision interval
arithmetic, 128-bit-mantissa interval arithmetic, 192-bit-mantissa interval
arithmetic, respectively. The integrands and the probability density functions
tend very quickly towards infinity at the boundary; sometimes the situation
arises that the probability density value is too large to be stored in the
mashine double precision. This is not a problem for our integration program,
but the contribution $\triangle^{\text{dens}}_{>\text{double}}$ of these
samples is also given in the table.

The corresponding numbers of the Monte Carlo samples are given in Table
LABEL:table_round_off_np. The table also includes the number
$N^{\text{fail}}_{\text{256}}$ of the samples for which the 256-bit-mantissa
precision failed. It is the highest precision in our integration, these
samples are considered as giving $0$.

Table LABEL:table_oscillations contains an information about oscillations in
individual graph contributions and in integrands. This concerns only the new
calculation. The part referring to the graphs without lepton loops can be
compared with the corresponding table in S. Volkov (2019). The use of
$M^{2}=-m^{2}$ instead of $M^{2}=m^{2}$ in (6) and (7) reduced the
oscillations by about $1.5$ times.

Table 6: Contributions of the Monte-Carlo samples with round-off errors and with the probability density out of mashine double Class | Value | $\triangle^{\text{fail}}_{\text{EIA}}$ | $\triangle^{\text{fail}}_{\text{IA}}$ | $\triangle^{\text{fail}}_{\text{128}}$ | $\triangle^{\text{fail}}_{\text{192}}$ | $\triangle^{\text{dens}}_{>\text{double}}$
---|---|---|---|---|---|---
I(a) | 0.00047105(16) | $-0.00069$ | $2.3\times 10^{-5}$ | $5.7\times 10^{-14}$ | $3.0\times 10^{-25}$ | $-2.1\times 10^{-34}$
I(b) | 0.0070081(12) | $-0.0013$ | $0.00058$ | $-9.2\times 10^{-13}$ | $1.0\times 10^{-23}$ | $1.1\times 10^{-34}$
I(c) | 0.0234643(26) | $0.0074$ | $0.0029$ | $7.3\times 10^{-12}$ | $-3.0\times 10^{-24}$ | $-1.4\times 10^{-33}$
I(d) | 0.00380370(61) | $0.00076$ | $0.00071$ | $1.2\times 10^{-9}$ | $-6.6\times 10^{-15}$ | $-2.9\times 10^{-26}$
I(e) | 0.010289(11) | $0.0051$ | $0.0018$ | $6.8\times 10^{-7}$ | $1.1\times 10^{-17}$ | $-7.8\times 10^{-20}$
I(f) | 0.00757106(47) | $0.0045$ | $0.0019$ | $5.8\times 10^{-8}$ | $3.3\times 10^{-14}$ | $-4.0\times 10^{-25}$
I(g) | 0.0285696(18) | $0.014$ | $0.0051$ | $2.4\times 10^{-8}$ | $-1.9\times 10^{-14}$ | $-5.9\times 10^{-20}$
I(h) | 0.0016826(63) | $0.0016$ | $-0.00067$ | $-2.9\times 10^{-7}$ | $2.9\times 10^{-13}$ | $1.8\times 10^{-23}$
I(i) | 0.01726(29) | $-0.010$ | $-0.0050$ | $-9.9\times 10^{-5}$ | $-1.9\times 10^{-8}$ | $9.6\times 10^{-15}$
I(j) | 0.0004038(63) | $0.0037$ | $-0.0014$ | $-1.1\times 10^{-9}$ | $-1.2\times 10^{-17}$ | $2.2\times 10^{-22}$
II(a) | -0.1094945(50) | $-0.0070$ | $-0.0012$ | $1.4\times 10^{-9}$ | $-1.4\times 10^{-21}$ | $5.0\times 10^{-16}$
II(b) | -0.473625(27) | $-0.052$ | $-0.0082$ | $4.5\times 10^{-8}$ | $-5.0\times 10^{-14}$ | $-1.6\times 10^{-12}$
II(c) | -0.116506(12) | $-0.041$ | $-0.0052$ | $6.3\times 10^{-8}$ | $-1.1\times 10^{-19}$ | $3.2\times 10^{-13}$
II(d) | -0.24291(15) | $-0.042$ | $-0.0078$ | $4.9\times 10^{-6}$ | $-3.1\times 10^{-16}$ | $-3.6\times 10^{-14}$
II(e) | -1.34235(54) | $-0.77$ | $-0.014$ | $0.00014$ | $3.9\times 10^{-8}$ | $-6.0\times 10^{-13}$
II(f) | -2.43553(30) | $-1.8$ | $0.097$ | $0.00095$ | $3.4\times 10^{-7}$ | $-2.5\times 10^{-14}$
III(a) | 2.12726(14) | $0.68$ | $-0.11$ | $-0.00038$ | $-1.6\times 10^{-6}$ | $-1.1\times 10^{-6}$
III(b) | 3.32730(32) | $0.99$ | $-0.20$ | $-0.00026$ | $-1.1\times 10^{-6}$ | $-5.5\times 10^{-7}$
III(c) | 4.9199(15) | $4.9$ | $1.3$ | $0.0010$ | $4.6\times 10^{-7}$ | $-1.2\times 10^{-11}$
IV | -7.7303(16) | $-4.1$ | $0.81$ | $0.0056$ | $-3.6\times 10^{-5}$ | $-5.7\times 10^{-6}$
V | 6.857(81) | $11.4$ | $2.7$ | $0.12$ | $-0.00053$ | $1.9\times 10^{-10}$
VI(a) | 1.041537(82) | $0.45$ | $-0.078$ | $-0.00060$ | $5.2\times 10^{-8}$ | $1.9\times 10^{-16}$
VI(b) | 1.34697(11) | $0.26$ | $-0.17$ | $-0.0013$ | $-1.1\times 10^{-6}$ | $1.3\times 10^{-13}$
VI(c) | -2.53312(49) | $-1.2$ | $0.039$ | $0.0032$ | $3.9\times 10^{-6}$ | $-7.2\times 10^{-13}$
VI(d) | 1.8468(22) | $1.5$ | $1.6$ | $-0.013$ | $-1.5\times 10^{-5}$ | $-1.0\times 10^{-11}$
VI(e) | -0.43129(17) | $-0.28$ | $-0.00087$ | $1.1\times 10^{-7}$ | $-3.1\times 10^{-16}$ | $3.2\times 10^{-17}$
VI(f) | 0.77154(23) | $0.59$ | $-0.11$ | $-0.0011$ | $-5.1\times 10^{-7}$ | $-1.6\times 10^{-13}$
VI(g) | -1.5965(10) | $-1.4$ | $-0.018$ | $-0.0056$ | $-1.0\times 10^{-5}$ | $-3.6\times 10^{-12}$
VI(h) | 0.18554(68) | $0.070$ | $-0.0019$ | $0.00017$ | $2.1\times 10^{-7}$ | $-6.6\times 10^{-13}$
VI(i) | -0.04396(10) | $-0.13$ | $-0.038$ | $-3.6\times 10^{-5}$ | $2.2\times 10^{-9}$ | $-5.1\times 10^{-14}$
VI(j) | -0.22920(43) | $0.032$ | $-0.083$ | $-0.00064$ | $-6.0\times 10^{-7}$ | $1.7\times 10^{-12}$
VI(k) | 0.67974(39) | $0.43$ | $0.80$ | $0.00064$ | $7.9\times 10^{-9}$ | $-2.2\times 10^{-13}$
Table 7: Statistics of round-off errors in the integrand evaluations and the Monte Carlo samples with the probability density out of mashine double Class | $N_{\text{total}}$ | $N^{\text{fail}}_{\text{EIA}}$ | $N^{\text{fail}}_{\text{IA}}$ | $N^{\text{fail}}_{\text{128}}$ | $N^{\text{fail}}_{\text{192}}$ | $N^{\text{fail}}_{\text{256}}$ | $N^{\text{dens}}_{>\text{double}}$
---|---|---|---|---|---|---|---
I(a) | $15\times 10^{11}$ | $38\times 10^{10}$ | $15\times 10^{9}$ | $19\times 10^{7}$ | $36\times 10^{4}$ | $3620$ | $5$
I(b) | $24\times 10^{11}$ | $54\times 10^{10}$ | $40\times 10^{9}$ | $25\times 10^{7}$ | $15\times 10^{4}$ | $1171$ | $6$
I(c) | $81\times 10^{10}$ | $18\times 10^{10}$ | $23\times 10^{9}$ | $22\times 10^{6}$ | $3037$ | $8$ | $6$
I(d) | $10^{12}$ | $19\times 10^{10}$ | $31\times 10^{9}$ | $21\times 10^{6}$ | $5452$ | $11$ | $85$
I(e) | $29\times 10^{11}$ | $51\times 10^{10}$ | $69\times 10^{9}$ | $77\times 10^{6}$ | $42129$ | $114$ | $57$
I(f) | $84\times 10^{10}$ | $11\times 10^{10}$ | $19\times 10^{9}$ | $96\times 10^{5}$ | $4829$ | $8$ | $99$
I(g) | $15\times 10^{11}$ | $25\times 10^{10}$ | $46\times 10^{9}$ | $16\times 10^{6}$ | $5962$ | $2$ | $1057$
I(h) | $50\times 10^{11}$ | $61\times 10^{10}$ | $58\times 10^{9}$ | $87\times 10^{5}$ | $3513$ | $3$ | $3040$
I(i) | $85\times 10^{12}$ | $10^{13}$ | $12\times 10^{11}$ | $23\times 10^{7}$ | $13\times 10^{4}$ | $199$ | $19304$
I(j) | $74\times 10^{10}$ | $25\times 10^{10}$ | $46\times 10^{9}$ | $43\times 10^{7}$ | $19\times 10^{5}$ | $25355$ | $3096$
II(a) | $49\times 10^{11}$ | $12\times 10^{11}$ | $95\times 10^{9}$ | $72\times 10^{7}$ | $23\times 10^{5}$ | $55926$ | $6528$
II(b) | $65\times 10^{11}$ | $18\times 10^{11}$ | $13\times 10^{10}$ | $29\times 10^{7}$ | $50\times 10^{4}$ | $3981$ | $22909$
II(c) | $26\times 10^{11}$ | $61\times 10^{10}$ | $54\times 10^{9}$ | $87\times 10^{5}$ | $682$ | $0$ | $23303$
II(d) | $14\times 10^{12}$ | $29\times 10^{11}$ | $11\times 10^{10}$ | $24\times 10^{6}$ | $4433$ | $3$ | $52772$
II(e) | $27\times 10^{11}$ | $13\times 10^{11}$ | $69\times 10^{9}$ | $68\times 10^{7}$ | $27\times 10^{5}$ | $20662$ | $11\times 10^{4}$
II(f) | $85\times 10^{10}$ | $52\times 10^{10}$ | $43\times 10^{9}$ | $28\times 10^{7}$ | $11\times 10^{5}$ | $10002$ | $27199$
III(a) | $10^{13}$ | $44\times 10^{11}$ | $24\times 10^{10}$ | $12\times 10^{8}$ | $42\times 10^{5}$ | $45636$ | $22\times 10^{4}$
III(b) | $15\times 10^{12}$ | $54\times 10^{11}$ | $38\times 10^{10}$ | $17\times 10^{8}$ | $57\times 10^{5}$ | $41232$ | $54\times 10^{4}$
III(c) | $11\times 10^{12}$ | $63\times 10^{11}$ | $56\times 10^{10}$ | $77\times 10^{8}$ | $43\times 10^{6}$ | $49\times 10^{4}$ | $13\times 10^{5}$
IV | $78\times 10^{12}$ | $35\times 10^{12}$ | $29\times 10^{11}$ | $21\times 10^{9}$ | $10^{8}$ | $13\times 10^{5}$ | $63\times 10^{5}$
V | $24\times 10^{13}$ | $51\times 10^{12}$ | $90\times 10^{11}$ | $19\times 10^{10}$ | $10^{9}$ | $50\times 10^{5}$ | $45\times 10^{5}$
VI(a) | $41\times 10^{10}$ | $21\times 10^{10}$ | $25\times 10^{9}$ | $47\times 10^{6}$ | $46527$ | $457$ | $4932$
VI(b) | $10^{12}$ | $44\times 10^{10}$ | $36\times 10^{9}$ | $70\times 10^{6}$ | $13\times 10^{4}$ | $1745$ | $30089$
VI(c) | $30\times 10^{11}$ | $21\times 10^{11}$ | $11\times 10^{10}$ | $20\times 10^{7}$ | $11\times 10^{5}$ | $23583$ | $40\times 10^{4}$
VI(d) | $41\times 10^{12}$ | $26\times 10^{12}$ | $12\times 10^{11}$ | $36\times 10^{8}$ | $28\times 10^{6}$ | $72\times 10^{4}$ | $10^{7}$
VI(e) | $55\times 10^{10}$ | $35\times 10^{10}$ | $20\times 10^{9}$ | $28\times 10^{6}$ | $12\times 10^{4}$ | $2736$ | $37778$
VI(f) | $12\times 10^{11}$ | $69\times 10^{10}$ | $48\times 10^{9}$ | $10^{8}$ | $68\times 10^{4}$ | $16916$ | $31312$
VI(g) | $10^{13}$ | $60\times 10^{11}$ | $26\times 10^{10}$ | $54\times 10^{7}$ | $30\times 10^{5}$ | $65104$ | $11\times 10^{5}$
VI(h) | $12\times 10^{12}$ | $40\times 10^{11}$ | $27\times 10^{10}$ | $57\times 10^{7}$ | $22\times 10^{5}$ | $50017$ | $22\times 10^{4}$
VI(i) | $62\times 10^{10}$ | $18\times 10^{10}$ | $10^{10}$ | $13\times 10^{6}$ | $20159$ | $281$ | $5509$
VI(j) | $63\times 10^{10}$ | $40\times 10^{10}$ | $28\times 10^{9}$ | $99\times 10^{6}$ | $61\times 10^{4}$ | $11317$ | $11062$
VI(k) | $32\times 10^{11}$ | $13\times 10^{11}$ | $20\times 10^{9}$ | $26\times 10^{5}$ | $507$ | $0$ | $361$
Table 8: Oscillations in the individual graph contributions and integrals; here, $a_{i}=\int I_{i}(\underline{z})d\underline{z}$ is the contribution of the $i$-th graph to the value, $I_{i}$ is the corresponding Feynman parametric integrand. Class | Value $=\sum_{i}a_{i}$ | $\sum_{i}|a_{i}|$ | $\max_{i}|a_{i}|$ | $\sum_{i}\int|I_{i}(\underline{z})|d\underline{z}$ | $N_{\text{graph}}$
---|---|---|---|---|---
I(a) | 0.00047105(16) | $0.00047$ | $0.00047$ | $0.021$ | 1
I(b1) | 0.0046715(11) | $0.0047$ | $0.0027$ | $0.12$ | 2
I(b2) | 0.00233656(54) | $0.0023$ | $0.0014$ | $0.061$ | 2
I(c) | 0.0234643(26) | $0.023$ | $0.012$ | $0.23$ | 3
I(d) | 0.00380370(61) | $0.0038$ | $0.0033$ | $0.051$ | 2
I(e) | 0.010289(11) | $0.074$ | $0.022$ | $0.76$ | 9
I(f) | 0.00757106(47) | $0.0076$ | $0.0073$ | $0.037$ | 2
I(g) | 0.0285696(18) | $0.029$ | $0.015$ | $0.15$ | 4
I(h) | 0.0016826(63) | $0.13$ | $0.036$ | $0.63$ | 11
I(i) | 0.01726(29) | $1.5$ | $0.12$ | $5.8$ | 39
I(j) | 0.0004038(63) | $0.12$ | $0.061$ | $0.67$ | 2
II(ap1) | 0.00924758(84) | $0.011$ | $0.0073$ | $0.067$ | 3
II(ap2) | -0.0297767(11) | $0.030$ | $0.024$ | $0.11$ | 2
II(ap3) | -0.0262183(11) | $0.026$ | $0.022$ | $0.096$ | 2
II(as1) | -0.0466111(41) | $0.047$ | $0.031$ | $0.28$ | 2
II(as2) | 0.0137916(15) | $0.017$ | $0.013$ | $0.15$ | 3
II(as3) | -0.0299276(18) | $0.030$ | $0.023$ | $0.17$ | 2
II(bp1) | 0.0326223(31) | $0.033$ | $0.011$ | $0.23$ | 6
II(bp2) | -0.0800129(67) | $0.080$ | $0.045$ | $0.36$ | 4
II(bp3) | -0.0836017(85) | $0.084$ | $0.045$ | $0.36$ | 4
II(bs1) | -0.125057(11) | $0.13$ | $0.063$ | $0.86$ | 4
II(bs2) | -0.125056(14) | $0.13$ | $0.062$ | $0.86$ | 4
II(bs3) | 0.088110(10) | $0.093$ | $0.050$ | $0.96$ | 8
II(bs4) | -0.180629(14) | $0.18$ | $0.048$ | $1.1$ | 8
II(c1) | -0.0869389(86) | $0.087$ | $0.063$ | $0.42$ | 4
II(c2) | 0.0375573(46) | $0.041$ | $0.024$ | $0.25$ | 6
II(c3) | -0.0671244(72) | $0.067$ | $0.054$ | $0.29$ | 4
II(d1) | -0.18988(11) | $1.6$ | $0.43$ | $6.8$ | 18
II(d2) | 0.111341(75) | $0.87$ | $0.16$ | $4.3$ | 25
II(d3) | -0.164378(83) | $1.3$ | $0.37$ | $4.9$ | 18
II(e1) | -1.13757(46) | $20.5$ | $2.0$ | $65.0$ | 30
II(e2) | -0.20478(28) | $24.2$ | $3.0$ | $38.5$ | 20
II(f1) | -0.239896(88) | $0.32$ | $0.27$ | $2.5$ | 3
II(f2) | -1.91510(23) | $3.8$ | $0.99$ | $14.9$ | 9
II(f3) | -0.28054(18) | $8.6$ | $2.8$ | $10.3$ | 6
III(ap1) | 0.056161(42) | $0.51$ | $0.16$ | $1.4$ | 10
III(ap2) | 0.251184(26) | $0.45$ | $0.14$ | $0.89$ | 12
III(ap3) | 0.227894(47) | $1.0$ | $0.12$ | $2.5$ | 20
III(ap4) | 0.143139(17) | $0.36$ | $0.14$ | $0.64$ | 4
III(ap5) | 0.071170(37) | $0.83$ | $0.15$ | $2.1$ | 20
III(ap6) | 0.164795(20) | $0.37$ | $0.067$ | $1.1$ | 10
III(ap7) | 0.0275451(27) | $0.075$ | $0.051$ | $0.15$ | 3
III(as1) | 0.054599(55) | $1.4$ | $0.47$ | $2.6$ | 10
III(as2) | 0.225730(17) | $0.47$ | $0.35$ | $0.79$ | 3
III(as3) | 0.055588(66) | $2.4$ | $0.51$ | $4.9$ | 20
III(as4) | 0.289427(35) | $0.78$ | $0.20$ | $1.7$ | 12
III(as5) | 0.440024(55) | $1.2$ | $0.13$ | $4.0$ | 20
III(as6) | 0.102979(21) | $0.33$ | $0.12$ | $1.1$ | 4
III(as7) | 0.017022(35) | $0.45$ | $0.088$ | $2.0$ | 10
III(b1) | 0.35850(13) | $4.0$ | $0.76$ | $10.7$ | 20
III(b2) | 0.550730(74) | $1.7$ | $0.74$ | $3.4$ | 6
III(b3) | -0.21825(16) | $8.9$ | $0.78$ | $22.4$ | 40
III(b4) | 0.916024(91) | $3.3$ | $0.50$ | $8.1$ | 24
III(b5) | 0.43819(10) | $2.8$ | $0.48$ | $6.3$ | 8
III(b6) | 1.35323(15) | $5.1$ | $0.25$ | $16.7$ | 40
III(b7) | -0.07113(10) | $2.8$ | $0.31$ | $10.1$ | 20
III(c1) | 1.86018(48) | $13.0$ | $2.7$ | $37.4$ | 12
III(c2) | 3.28050(91) | $43.3$ | $5.8$ | $104.7$ | 27
III(c3) | 0.27166(48) | $14.3$ | $3.8$ | $32.7$ | 6
III(c4) | -0.75943(79) | $33.1$ | $3.7$ | $89.8$ | 24
III(c5) | 0.22308(51) | $36.5$ | $6.6$ | $66.4$ | 15
III(c6) | 0.04392(46) | $31.4$ | $4.8$ | $50.3$ | 18
IV(1) | -0.51586(43) | $15.7$ | $1.0$ | $29.9$ | 74
IV(2) | -0.64878(28) | $7.1$ | $0.86$ | $12.5$ | 20
IV(3) | -1.14824(60) | $30.9$ | $1.5$ | $63.8$ | 148
IV(4) | 1.19593(38) | $15.1$ | $1.3$ | $27.5$ | 55
IV(5) | -1.52785(60) | $32.4$ | $1.3$ | $67.4$ | 162
IV(6) | 0.50531(30) | $8.4$ | $0.59$ | $17.4$ | 56
IV(7) | -0.19295(26) | $4.6$ | $0.46$ | $11.0$ | 20
IV(8) | -0.78444(38) | $10.0$ | $0.63$ | $25.6$ | 40
IV(9) | -4.49029(73) | $33.4$ | $0.64$ | $84.4$ | 222
IV(10) | 0.19698(32) | $8.3$ | $0.66$ | $21.9$ | 50
IV(11) | 0.05233(55) | $18.7$ | $0.48$ | $60.7$ | 148
IV(12) | -0.37250(31) | $6.6$ | $0.43$ | $19.0$ | 54
V(1) | 6.169(20) | $599.3$ | $6.9$ | $1208.7$ | 706
V(2) | 0.964(31) | $2077.2$ | $46.2$ | $2454.3$ | 148
V(3) | 0.334(20) | $737.9$ | $38.0$ | $861.2$ | 55
V(4) | -0.789(37) | $2029.2$ | $33.3$ | $3126.1$ | 706
V(5) | -2.157(34) | $2253.3$ | $58.7$ | $3060.7$ | 370
V(6) | -0.415(35) | $1948.0$ | $43.6$ | $2780.9$ | 558
V(7) | 2.644(23) | $1062.3$ | $51.1$ | $1493.4$ | 261
V(8) | -0.975(21) | $1028.0$ | $43.0$ | $1460.4$ | 336
V(9) | 1.0807(63) | $137.5$ | $17.0$ | $200.4$ | 73
VI(ap) | 0.482955(59) | $0.75$ | $0.49$ | $2.9$ | 5
VI(as) | 0.558582(57) | $1.3$ | $0.70$ | $3.9$ | 5
VI(b) | 1.34697(11) | $2.9$ | $1.1$ | $15.4$ | 10
VI(c) | -2.53312(49) | $39.5$ | $3.9$ | $72.0$ | 37
VI(d) | 1.8468(22) | $352.6$ | $12.1$ | $703.1$ | 127
VI(e) | -0.43129(17) | $4.5$ | $1.1$ | $11.5$ | 13
VI(f) | 0.77154(23) | $9.8$ | $0.63$ | $33.7$ | 46
VI(g) | -1.5965(10) | $137.8$ | $3.8$ | $341.1$ | 122
VI(h) | 0.18554(68) | $84.0$ | $2.2$ | $239.3$ | 162
VI(i) | -0.04396(10) | $2.9$ | $0.49$ | $10.0$ | 16
VI(j) | -0.22920(43) | $16.3$ | $4.7$ | $60.0$ | 9
VI(k) | 0.67974(39) | $56.2$ | $5.1$ | $75.4$ | 32
with lepton loops | -0.9377(35) | $1174.7$ | $12.1$ | $2656.6$ | 2323
no lepton loops | 6.857(81) | $11872.6$ | $58.7$ | $16646.1$ | 3213

##  VI CONCLUSION

The universal QED expansion coefficient $A_{1}^{(10)}$ of the lepton magnetic
moments was successfully calculated and compared with the previously known
value obtained by T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio. The results
agree very well in 31 out of 32 classes, but remain inconsistent in the
remaining class. The deviating class value agrees with the value presented by
the author in 2019, but not with the AHKN one.

Another calculations of $A_{1}^{(10)}$ are either not accurate enough to
resolve the discrepancy or affecting only a very small part of the
contributions.

The results presented are divided into 95 gauge-invariant classes, which
subdivide the 32 classes mentioned above. The results at this level of detail
were presented for the first time.

The author’s method for removing divergences under the integral sign in
Feynman parametric space was used. The method yields a finite integral for
each Feynman graph. However, the contributions of individual graphs remain
large and oscillating relative to the final value. Changing the subtraction
point to the off-shell one reduces the oscillations slightly, but does not
solve the problem completely. These oscillations occur regardless of the
subtraction method used and still require a comprehensive explanation.

A nonadaptive Monte Carlo integration algorithm was used for the numerical
integration. This nonadaptivity means that the probability density function
for each integral is predefined and remains fixed during the integration.
However, this algorithm has 42 real parameters that can be chosen arbitrarily.
32 of them were adjusted during the experiments with the 4-loop calculations.
6 of them were chosen after poor convergence was observed for Set I(i) and
some adjustments were made to make the convergence for this set as good as
possible. Reasonable values were chosen for the remaining 4 parameters.

The definition of the probability density functions used is based on the
combinatorics of the Feynman graphs. The procedure suitable for all Feynman
graphs, including those with lepton loops, was presented for the first time.
The description looks cumbersome at first glance, but it is accompanied by a
number of examples that provide the opportunity to check and improve the
algorithm.

###### Acknowledgements.

The author would like to thank Gudrun Heinrich, Savely Karshenboim, Andrey
Arbuzov, Lidia Kalinovskaya for important help. All preliminary calculations
were performed using the ITP/TTP KIT computing cluster and ITP KIT office
computers. The main work was carried out on the HoreKa supercomputer funded by
the Ministry of Science, Research and the Arts Baden-Württemberg and by the
Federal Ministry of Education and Research. All images were drawn using
AxoDraw and JaxoDraw. The research is supported by the Alexander von Humboldt
Foundation.

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