#  Calculation of lepton magnetic moments in quantum electrodynamics: a
justification of the flexible divergence elimination method

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

###### Abstract

The flexible method of reduction to finite integrals, briefly described in
earlier publications of the author, is described in detail. The method is
suitable for the calculation of all quantum electrodynamical contributions to
the magnetic moments of leptons. It includes mass-dependent contributions.

The method removes all divergences (UV, IR and mixed) point-by-point in
Feynman parametric space without any usage of limit-like regularizations. It
yields a finite integral for each individual Feynman graph. The subtraction
procedure is based on the use of linear operators applied to the Feynman
amplitudes of UV-divergent subgraphs; a placement of all terms in the same
Feynman parametric space is implied. The final result is simply the sum of the
individual graph contributions; no residual renormalization is required. The
method also allows us to split the total contribution into the contributions
of small gauge-invariant classes.

The procedure offers a great freedom in the choice of the linear operators.
This freedom can be used for improving the computation speed and for a
reliability check.

The mechanism of divergence elimination is explained, as well as the
equivalence of the method and the on-shell renormalization.

For illustrative purposes, all 4-loop contributions to the anomalous magnetic
moments of the electron and muon are given for each small gauge-invariant
class, as well as their comparison with previously known results. This also
includes the contributions that depend on the ratios of the tau-lepton mass to
the electron and muon mass.

##  I Introduction

###  I.1 Quantum field theory, divergences and renormalization

It is well known that the first formulations of quantum field theory suffered
from ultraviolet (UV) divergences. In 1947-1949 the first results appeared
showing that the UV divergences in the physical observables are cancelled if
we define the physical parameters properly. This demonstration was the result
of the efforts of R. Feynman, J. Schwinger, H. Bethe, S. Tomonaga and others.
A procedure for dealing with UV divergences in quantum electrodynamics (QED)
at any order of the perturbation series was developed by F. J. Dyson [1] and
A. Salam [2]. A further development and a mathematical understanding of these
ideas has led to N. Bogoliubov’s $\mathcal{R}$-operation. This operation can
be represented as a subtraction procedure that deals with the Feynman
amplitudes of the UV-divergent subgraphs in each Feynman graph that
contributes to the needed probability amplitude; it removes all UV divergences
in each Feynman graph. The application of the $\mathcal{R}$-operation to
Feynman graphs is equivalent to an introduction of counterterms into the
Lagrangian; it can be treated as a renormalization of the theory parameters.
The corresponding theorem was proved in 1956 by N. Bogoliubov and O. Parasiuk
[3]. This proof contains some inaccuracies, but nevertheless, the Bogoliubov-
Parasiuk theorem set a standard of quality and raised the hope that quantum
field theory has a meaning and can be rigorously examined. The shortcomings of
the proof were corrected by K. Hepp in 1964 [4].

Not only the Bogoliubov-Parasiuk-Hepp theorem is of interest, but also its
proof, because a stronger statement was proved: The UV subtractions lead
directly to finite integrals in Schwinger parametric space111Usually, these
statements are referred to as the BPHZ approach. The last letter “Z” refers to
W. Zimmermann’s paper [5]. In that work, the convergence theorem was proved
directly in Minkowsky momentum space. Moreover, an explicit solution of the
recurrence relations given by the $\mathcal{R}$-operation was provided, also
known as Zimmermann’s forest formula (more precisely, the solution had been
provided four years earlier by O. I. Zavialov and B. M. Stepanov [6], but this
publication was only in Russian and was not noticed).. In QED we have Feynman
graphs with the propagators

$\frac{i(\not{q}+m)}{q^{2}-m^{2}+i\varepsilon},\quad\frac{-g_{\mu\nu}}{q^{2}+i\varepsilon}$
(1)

for the lepton and photon lines correspondingly, where
$\not{q}=q^{\nu}\gamma_{\nu}$ stands for the Dirac gamma matrices
$\gamma_{\nu}$. If we transfer to the Schwinger parameters by using the
formula

$\frac{i}{x}=\int_{0}^{+\infty}e^{ixz}dz,$

we express the Feynman amplitude as an integral

$\int_{z_{1},\ldots,z_{L}>0}I(z_{1},\ldots,z_{L},p_{1},\ldots,p_{n},\varepsilon)dz_{1}\ldots
dz_{L},$ (2)

where $p=(p_{1},\ldots,p_{n})$ is the vector of external momenta,
$z=(z_{1},\ldots,z_{L})$ is the vector of Schwinger parameters, $z_{j}$
corresponds to the $j$-th internal line of the graph. If we apply the
$\mathcal{R}$-operation,
$I(z_{1},\ldots,z_{L},p_{1},\ldots,p_{n},\varepsilon)$ is expressed as a
linear combination of terms; each term is a product of some functions, each of
which corresponds to a Feynman amplitude that uses a subset of the set of
internal lines; if the same mapping is used between $z_{j}$ and the internal
lines, all the counterterms are placed in the same Schwinger parametric space.
A byproduct of the Bogoliubov-Parasiuk-Hepp theorem is that (2) is finite for
any $\varepsilon>0$.

Unfortunately, these subtractions in Schwinger’s parametric space are useless
for calculations. The reason is the inability to handle the limit
$\varepsilon\rightarrow 0$. Infrared (IR) divergences and their cancellation
have been widely discussed in literature. However, a rigorous examination for
the general case has not yet been done. At higher orders IR and UV divergences
mix with each other222See, e.g., the explicit formulas in [7]. In the case of
magnetic moments of leptons, all IR divergences are removed by the physical
on-shell renormalization as well as the UV and mixed ones333See, for example,
[8] and Appendix C of [9].. A general scaterring process requires
consideration of a finite photon detector sensitivity in addition to
renormalization. The on-shell renormalization can be performed in place in
Feynman graphs by a simple modification of the $\mathcal{R}$-operation. In the
case of the lepton magnetic moments, it removes all divergences in the final
result. However, the individual integrals remain IR divergent.

The combinatorics of the IR and mixed divergences in Feynman graphs is very
complicated. For this reason, their subtraction before integration is very
rarely used in modern quantum field theory computations. Nowadays, most
calculations use dimensional regularization, which allows us to work directly
with infinite components without exact mappings for cancellation. However, the
use of dimensional regularization requires an enormous amount of symbolic
manipulation at higher orders. An experience with calculations of the lepton
magnetic moment shows that the methods that substract divergences before
integration work much faster and allow us to obtain high-order corrections
that are not achievable with other methods.

###  I.2 Anomalous magnetic moments of leptons and their calculations

The anomalous magnetic moments (AMM) of the electron and muon are known with a
very high precision. A recent measurement [10] gave the result

$a_{e}=0.00115965218059(13)$ (3)

Standard Model predictions for the electron AMM $a_{e}$ 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, muon, and tau-
lepton, respectively, $\alpha$ is the fine-structure constant; a similar
expression is used for the muon AMM $a_{\mu}$.

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

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

was obtained in 1948 by J. Schwinger [11, 12]. The 2-loop contribution
$A_{1}^{(4)}$ was calculated mainly by R. Karplus and N. Kroll [13]. However,
this calculation had an error; the correct value

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

was independently presented by A. Petermann [14] and C. Sommerfield [15] in
1957. The value of $A_{1}^{(6)}$ was being calculated in 1970-x by various
research groups using numerical integration ([16, 17]; [18]; [19]); each of
these groups used its own method of divergence elimination at the integrand
level; the most accurate value $A_{1}^{(6)}=1.195\pm 0.026$ for those times
was obtained in 1974 by T. Kinoshita and P. Cvitanović; the uncertainty is
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 obtained by S. Laporta and E. Remiddi in 1996 [20]. This value was the
result of the efforts of many researchers (see, for example, [21, 22, 23, 24,
25, 26] ). First numerical estimates for $A_{1}^{(8)}$ were obtained by T.
Kinoshita and W. B. Lindquist in 1981 [27]. The most accurate value presented
by T. Kinoshita’s team, $A_{1}^{(8)}=-1.91298(84)$, was published in 2015
[28]. This value was obtained by Monte Carlo integration. The semianalytic
result of S. Laporta

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

was obtained in 2017 [29]. These two calculations of $A_{1}^{(8)}$ agree well,
as do another independent calculations [30, 31] and our calculation of Feynman
graphs without lepton loops [32]. First overall calculation results for
$A_{1}^{(10)}$ were published in 2012 by T. Aoyama, M. Hayakawa, T. Kinoshita,
M. Nio in [33]444And another subtraction procedure that removes all
divergences before integration was developed for these calculations; it is
different from those used for 6th and 8th order calculations.. The last value
obtained by this group in 2019 [34] is

$A_{1}^{(10)}[\text{AHKN}]=6.737(159).$ (4)

This coefficient has not yet been verified, and a significant computational
error could be apparent in experiments. We recalculated the total contribution
of the graphs without lepton loops to $A_{1}^{(10)}$ and presented [35] in
2019 the value

$A_{1}^{(10)}[\text{no lepton loops,Volkov}]=6.793(90)$

that leads to

$A_{1}^{(10)}[\text{Volkov+AHKN}]=5.862(90)$ (5)

with a discrepancy of $4.8\sigma$ with (4). At the moment the discrepancy is
unresolved, but independent calculations are coming [36]. The values (4) and
(5) in combination with the experimental value (3) and another known
contributions [34] lead to

$\alpha^{-1}[a_{e},\text{AHKN}]=137.0359991663(155)$ (6)

and

$\alpha^{-1}[a_{e},\text{Volkov+AHKN}]=137.0359991593(155),$ (7)

correspondingly. The experimental values

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

come from [37], [38], and [39], respectively555Note that there is a
discrepancy of $5.4\sigma$ between the last two measurements.. The tensions
with (6) are $1.97\sigma$, $3.86\sigma$, $2.09\sigma$; the corresponding
tensions with (7) are $1.89\sigma$, $3.64\sigma$, $2.46\sigma$.

Other results exist for small classes of 5-loop and higher order graphs [40,
41, 42]; they are in good agreement with the ones mentioned above.

An analytic formula for $A_{2}^{(4)}(x)$ was first established by H. H. Elend
[43] in 1966. Numerical values for $A_{2}^{(6)}$ appeared about 1969 [44].
Most attention was paid to $A_{2}^{(6)}(m_{\mu}/m_{e})$, because its value
$A_{2}^{(6)}(m_{\mu}/m_{e})\approx 20$ became unexpectedly large. These values
were also being calculated analytically in the form of an expansion in the
mass ratio. These expansions are being published since 1975 with an increasing
number of terms [45, 46, 47, 48, 49]. The values $A_{3}^{(6)}$ were also
evaluated numerically [50], semianalytically [46] and analytically [51] as
expansions in the mass ratios. First numerical values of
$A_{2}^{(8)}(m_{\mu}/m_{e})$ appeared in 1990 [50]. The recent values of
$A_{2}^{(8)}$ and $A_{3}^{(8)}$ for the electron and muon obtained by T.
Aoyama, M. Hayakawa, T. Kinoshita, M. Nio with the help of Monte Carlo
integration are presented in [52, 28]. Semianalytic calculations using
expansions in mass ratios confirmed these results [53, 54]. Other results for
certain classes of graphs are also in good agreement with them [55, 56, 57,
58, 59, 60, 61]. The only known values of $A_{2}^{(10)}$ and $A_{3}^{(10)}$
for the electron and muon were presented in 2012 by T. Aoyama, M. Hayakawa. T.
Kinoshita, M. Nio [33, 52]. Partial calculations of small graph classes
confirm these results [46, 57, 62, 42]. However, a very important contributing
to $a_{\mu}$ value

$A_{2}^{(10)}(m_{\mu}/m_{e})=742.18(87)$

has not been double-checked yet. It is much larger (in absolute value) than
the corresponding electron value. A significant error in this value could be
noticeable in experiments, and the shift would be comparable to the hadronic
uncertainty. In addition, rapid growth of $A_{2}^{(2n)}(m_{\mu}/m_{e})$ with
$n$ may cause the higher-order terms to significantly affect the result.
However, estimates based on known lower-order values and renormgroup-inspired
arguments [50, 63, 64, 65, 52], as well as non-relativistic calculations [66],
show that both possibilities seem unlikely.

See reviews [67] on the muon (2021) and [34] on the electron (2019) for
details.

###  I.3 Methods of numerical calculation and gauge-invariant classes

Methods based on subtraction of divergences before integration have been
widely used for calculations of QED contributions to magnetic moments of
leptons [18, 19, 16, 34] and for other quantum field theory calculations [68].

In 2016, we presented a method [69] suitable for computing $A_{1}^{(2n)}$ for
arbitrary $n$. The difference of this method from the previously known methods
is that it is based on linear operators applied to the Feynman amplitudes of
UV-divergent subgraphs, and the final result is obtained directly by summing
the contributions of the Feynman graphs (no residual renormalization is
required).

A distribution of the final result over Feynman graphs is a matter of free
choice, but it is very untrivial to realize this freedom in the context of
methods that subtract divergences before integration and yield the final
result without residual renormalization, because the integrals obtained must
be finite for each individual Feynman graph, but the whole procedure must
conform to the physical on-shell renormalization. In a sense, this subtraction
procedure must contain an evidence that the on-shell renormalization removes
all divergences, including infrared and mixed divergences. In 2021, we
presented a modification [70] of the developed method; it allows us to choose
3 linear subtraction operators independently666The number is not related to
the number of leptons; the number of leptons can be arbirtarily large..
Moreover, the new method is suitable for the calculation of mass-dependent
contributions777More precisely, the old method is also suitable for this, but
the argument in [69] does not prove this and is not extensible to prove this.
$A_{2}^{(2n)}$, $A_{3}^{(2n)}$.

Some classes of Feynman graphs form gauge-invariant classes. For example, any
class of graphs closed under a motion of the internal photon lines along the
lepton paths and loops, but without jumping over the external photon line, is
invariant in the class of Lorentz-invariant gauges with the photon propagators
of the form $-(g_{\mu\nu}+k_{\mu}k_{\nu}f(k^{2}))/(k^{2}+i0)$, if we properly
define the on-shell renormalization for subclasses of graphs888The gauge
invariance was proved in [8], but only for graphs without lepton loops. It
looks the whole proof is not published, but the fact is widely used; we
believe that the ideas of [8] can be extended to the general case. 999The
gauge invariance implies not only the possibility of using different photon
propagators, but also the ability to renormalize photon self-energies
differently., see Section IV.1.

Examples of the use of the new method, as well as a comparison with the old
method, were given in [70, 71]. In this work we show that the method yields
the exact value for each gauge-invariant class mentioned above, including
those contributing to $A_{2}^{(2n)}$ and $A_{3}^{(2n)}$, as well as a finite
integral for each Feynman graph.

##  II The method formulation

We work in the unit system where $\hbar=c=1$, 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 satisfy the
condition $\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=2g_{\mu\nu}$.

We extract the lepton AMM from QED Feynman graphs with $N_{l}=2$,
$N_{\gamma}=1$, where by $N_{l}$ and $N_{\gamma}$ we denote 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).

We work in the Feynman gauge with the propagators (1) for leptons and photons,
respectively, where $m$ is the lepton mass.

There are the following types of UV-divergent subgraphs101010We 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_ subgraphs111111Photon-photon scattering subgraph
divergences cancel in the final result without subtraction, but they remain in
the individual graphs. ($N_{l}=0$, $N_{\gamma}=4$).

Two subgraphs are said to overlap 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 $G\in F$. The lepton path of a graph $G$ connecting its external
lepton lines is called the _main path_ of $G$. By $\mathcal{I}[G]$ we
denote121212The definition differs from that in [69]. the set of all
vertexlike subgraphs of $G$ (including $G$) which have the external photon of
$G$ and at least one vertex of the main path of $G$.

To define the subtraction procedure, we use the linear operators labeled $A$,
$U_{0},U_{1},U_{2},U_{3}$, $L$; sometimes the same names are used for
operators applied to the Feynman amplitudes of subgraphs of different types.
The definitions are:

  * •

$A$ is applied to vertexlike Feynman amplitudes131313$p-\frac{q}{2}$,
$p+\frac{q}{2}$ are incoming and outgoing lepton momenta, $q$ is the photon
momentum. $\Gamma_{\mu}(p,q)$ and is defined as

$A\Gamma_{\mu}(p,q)=[A^{\prime}\Gamma_{\mu}(p,q)]\gamma_{\mu},$

where $A^{\prime}$ is the projector of the AMM. See the definition of the
projector in [69, 35].

  * •

$L$ is the standard on-shell renormalization operator for vertexlike graphs;
it is defined as

$L\Gamma_{\mu}(p,q)=e[L^{\prime}\Gamma_{\mu}(p,q)]\gamma_{\mu},\quad
L^{\prime}\Gamma_{\mu}=\frac{1}{e}[a(m^{2})+mb(m^{2})+m^{2}c(m^{2})],$

where

$\Gamma_{\mu}(p,0)=a(p^{2})\gamma_{\mu}+b(p^{2})p_{\mu}+c(p^{2})\not{p}p_{\mu}+d(p^{2})(\not{p}\gamma_{\mu}-\gamma_{\mu}\not{p}),$
(8)

$e$ is the lepton charge.

  * •

$U_{0}$ is applied to photon self-energy and photon-photon scattering
subgraphs and works as in standard renormalization. For the Feynman amplitude
$\Pi_{\mu\nu}(p^{2})$ of the photon self-energy, we can take, for example,

$U_{0}\Pi_{\mu\nu}(p^{2})=e[U_{0}^{\prime}\Pi_{\mu\nu}(p^{2})][g_{\mu\nu},p^{2}g_{\mu\nu},p_{\mu}p_{\nu}],$

where

$U_{0}^{\prime}\Pi_{\mu\nu}(p^{2})=\frac{1}{e}\left[a(0),\left.\frac{\partial
a(p^{2})}{\partial p^{2}}\right|_{p^{2}=0},b(0)\right],$

a scalar product is implied, and

$\Pi_{\mu\nu}(p^{2})=a(p^{2})g_{\mu\nu}+b(p^{2})p_{\mu}p_{\nu}.$

Optimizations as described in [72] can also be used for photon self-energy
graphs. For the Feynman amplitudes of photon-photon scattering, we can take

$U_{0}\Pi_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(p_{1},p_{2},p_{3},p_{4})=\Pi_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(0).$

We can write this as

$U_{0}\Pi_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(p_{1},p_{2},p_{3},p_{4})=e[U_{0}^{\prime}\Pi_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(p_{1},p_{2},p_{3},p_{4})][g_{\mu_{1}\mu_{2}}g_{\mu_{3}\mu_{4}},g_{\mu_{1}\mu_{3}}g_{\mu_{2}\mu_{4}},g_{\mu_{1}\mu_{4}}g_{\mu_{2}\mu_{3}}],$

where

$U_{0}^{\prime}\Pi_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(p_{1},p_{2},p_{3},p_{4})=\frac{1}{e}\Pi_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(0)$
$\times\left(\left[\frac{5}{72},-\frac{1}{72},-\frac{1}{72}\right]g_{\mu_{1}\mu_{2}}g_{\mu_{3}\mu_{4}}+\left[-\frac{1}{72},\frac{5}{72},-\frac{1}{72}\right]g_{\mu_{1}\mu_{3}}g_{\mu_{2}\mu_{4}}+\left[-\frac{1}{72},-\frac{1}{72},\frac{5}{72}\right]g_{\mu_{1}\mu_{4}}g_{\mu_{2}\mu_{3}}\right).$

These vector representations with $U_{0}^{\prime}$ are used in Section IV.

  * •

$U_{j}$, $j=1,2$ are applied to vertexlike and lepton self-energy Feynman
amplitudes; $U_{3}$ is applied only to vertexlike amplitudes. They are defined
as

$U_{j}\Gamma_{\mu}(p,q)=e[U_{j}^{\prime}\Gamma_{\mu}(p,q)]\gamma_{\mu},$
$U_{j}\Sigma(p)=e([U_{j}^{\prime}\Sigma](\not{p}-m)+M^{\prime}\Sigma(p))$ (9)

where $\Gamma_{\mu}(p,q)$ and $\Sigma(p)$ are vertexlike and lepton self-
energy Feynman amplitudes, $U_{j}^{\prime}$ are number-valued linear
operators,

$\Sigma(p)=r(p^{2})+s(p^{2})\not{p}.$ (10)
$M^{\prime}\Sigma(p)=\frac{1}{e}[r(m^{2})+s(m^{2})m],$ (11)

We require that:

    1. 1.

$U_{j}$ ($j=1,2$) _preserve the Ward identity_ : if $\Gamma_{\mu}$ and
$\Sigma$ satisfy

$\Gamma_{\mu}(p,0)=-e\frac{\partial\Sigma(p)}{p^{\mu}},$

then

$[U_{j}\Gamma_{\mu}](p,0)=-e\frac{\partial[U_{j}\Sigma](p)}{p^{\mu}}$

is also satisfied. The latter can be rewritten as

$U_{j}^{\prime}\Gamma_{\mu}=-eU_{j}^{\prime}\Sigma.$

    2. 2.

If $\Gamma_{\mu}(p,0)=0$, then $U^{\prime}_{j}\Gamma_{\mu}=0$ ($j=1,2,3$).

These conditions are sufficient to prove combinatorially that the subtraction
procedure defined below is equivalent to on-shell renormalization, provided
that all regularization issues are resolved.

To obtain finite integrals for each graph, more stringent requirements are
needed. We take

$U_{j}^{\prime}\Gamma_{\mu}=\frac{1}{e}a(M^{2}),\quad
U_{j}^{\prime}\Sigma=-\frac{1}{e}s(M^{2}),\quad j=1,2,$

where (8), (10) are satisfied, $M^{2}$ is an arbitrary real number (it may be
different for different operators). As $U_{3}^{\prime}\Gamma_{\mu}$ we can
take $\frac{1}{e}a(M^{2})$ or $L^{\prime}\Gamma_{\mu}$.

$U_{j}$ can be chosen independently for different $j$. $U_{1}$ and $U_{3}$ are
defined only for the lepton whose AMM is calculated; $U_{2}$ are defined for
all leptons occuring in graphs and can be chosen independently.

The expression for the subtraction and extraction of the AMM corresponding to
a Feynman graph $G$ is

$\sum_{\begin{subarray}{c}F=\\{G_{1},\ldots,G_{n}\\}\in\mathfrak{F}[G]\\\
G^{\prime}\in\mathcal{I}[G]\cap
F\end{subarray}}(-1)^{n-1}S^{G^{\prime}}_{G_{1}}\ldots
S^{G^{\prime}}_{G_{n}},$

where

$S^{G^{\prime}}_{G^{\prime\prime}}=\begin{cases}A_{G^{\prime\prime}},\text{ if
}G^{\prime\prime}=G^{\prime},\\\
L_{G^{\prime\prime}}-(U_{1})_{G^{\prime\prime}},\text{ if
}G^{\prime\prime}=G\neq G^{\prime},\\\ L_{G^{\prime\prime}},\text{ if
}G^{\prime}\subset G^{\prime\prime}\subset G,\\\
(U_{\xi})_{G^{\prime\prime}},\text{ if
}G^{\prime\prime}\in\mathcal{I}[G]\text{ and }G^{\prime\prime}\subset
G^{\prime},\\\ (U_{1})_{G^{\prime\prime}},\text{ if
}G^{\prime\prime}\notin\mathcal{I}[G]\text{ and }G^{\prime\prime}\text{ has
its external leptons on the main path of }G,\\\
(U_{2})_{G^{\prime\prime}},\text{ if }G^{\prime\prime}\text{ has its external
leptons on a lepton loop of }G,\\\ (U_{0})_{G^{\prime\prime}},\text{ if
}G^{\prime\prime}\text{ is a photon self-energy or photon-photon scattering
subgraph};\end{cases}$

here

$\xi=\begin{cases}3,\text{ if the external photon of }G\text{ lies on a lepton
loop},\\\ 1\text{ otherwise};\end{cases}$ (12)

the index of an operator denotes the subgraph to whose Feynman amplitude it is
applied; $G_{1}\subset G_{2}$ or $G_{1}\subseteq G_{2}$ means $V(G_{1})\subset
V(G_{2})$ or $V(G_{1})\subseteq V(G_{2})$, where $V(G)$ denotes the set of
vertices of $G$. At the level of Feynman amplitudes, each term of the
expression sequentially transforms the Feynman amplitudes of subgraphs using
corresponding linear operators (the subgraphs should be ordered by inclusion
in this sequence, from smaller to larger). Finally, the coefficient before
$\gamma_{\mu}$ should be taken.

For example, for the graph $G$ from Figure 1 the expression is

$\displaystyle\left[A_{G}\left(1-(U_{3})_{G_{e}}\right)-\left(L_{G}-(U_{1})_{G}\right)A_{G_{e}}\right]\times\left(1-(U_{2})_{G_{c}}\right)\times\left(1-(U_{1})_{e_{2}e_{4}e_{5}}\right)$
$\displaystyle\times\left(1-(U_{0})_{G_{d}}\right)\times\left(1-(U_{0})_{c_{1}c_{2}c_{3}c_{4}}\right)\times\left(1-(U_{2})_{c_{1}c_{2}c_{3}}-(U_{2})_{c_{1}c_{3}c_{4}}\right)\times\left(1-(U_{2})_{a_{1}a_{2}}\right),$

where
$G_{e}=aa_{1}a_{2}b_{1}b_{2}c_{1}c_{2}c_{3}c_{4}d_{1}d_{2}d_{3}e_{1}e_{2}e_{3}e_{4}e_{5}$,
$G_{d}=\\{aa_{1}a_{2}b_{1}b_{2}c_{1}c_{2}c_{3}c_{4}d_{1}d_{2}d_{3}\\}$,
$G_{c}=aa_{1}a_{2}b_{1}b_{2}c_{1}c_{2}c_{3}c_{4}$,
$\mathcal{I}[G]=\\{G_{e},G\\}$, subdiagrams are denoted by enumeration of
their internal vertices; we should expand the parentheses to obtain the forest
formula. The expression for this example does not depend on the type of
leptons on the lepton loops ($U_{2}$ may be defined differently for different
types of leptons, however).

Figure 1: An example of a Feynman diagram contributing to AMM.

To obtain finite Feynman parametric integrals, all terms should be placed in
the same Feynman parametric space. The algorithm for obtaining integrals from
expressions is briefly explained in [71] and in detail in [69]. The needed
value of $A_{l}^{(2n)}$ is simply the sum of the integrals corresponding to
the Feynman graphs involved.

##  III Divergence elimination

The principle of divergence elimination in the method is similar to that
explained in [69] for the old method. Here we describe schematically how this
choice of linear operators allows us to separate divergences and how it leads
to divergence cancellation.

Let us consider the 1-loop unrenormalized vertexlike Feynman amplitude

$\Gamma_{1,\mu}(p,q)=\int\gamma^{\nu}\frac{\not{p}+\frac{\not{q}}{2}+\not{k}+m}{\left(p+\frac{q}{2}+k\right)^{2}-m^{2}+i0}\gamma_{\mu}\frac{\not{p}-\frac{\not{q}}{2}+\not{k}+m}{\left(p-\frac{q}{2}+k\right)^{2}-m^{2}+i0}\gamma_{\nu}\frac{1}{k^{2}+i0}d^{4}k.$

Its IR-divergent part at $p^{2}=m^{2}$ is equal to

$\Gamma_{1,\mu}(p,0)_{\text{IR}}=\left[\int\gamma^{\nu}\frac{\not{p}+m}{\left(p+k\right)^{2}-m^{2}+i0}\gamma_{\mu}\frac{\not{p}+m}{\left(p+k\right)^{2}-m^{2}+i0}\gamma_{\nu}\frac{1}{k^{2}+i0}d^{4}k\right]_{\text{IR}}.$

With

$(\not{p}+m)\gamma_{\mu}(\not{p}+m)=2mp_{\mu}+2p_{\mu}\not{p}$

and the fact that $p_{\mu}$ commutes with all other multipliers that may
appear in the expression, we obtain

$\Gamma_{1,\mu}(p,0)_{\text{IR}}=Ap_{\mu}+Bp_{\mu}\not{p},$

where $A$ and $B$ some IR-divergent values141414We mean that a regularization
is introduced, but we omit it for simplicity.. Thus, $U_{1}\Gamma_{1,\mu}$ and
$U_{2}\Gamma_{1,\mu}$ are IR-finite. On the other hand, the UV-divergent part
of $\Gamma_{1,\mu}(p,q)$ does not depend on $p,q$ and is therefore
proportional to $\gamma_{\mu}$. Thus, $(L-U_{j})\Gamma_{1,\mu}$, $j=1,2$ are
UV-finite.

There is a general observation:

  * •

$(L-U_{j})\Gamma$, $j=1,2,3$ has no _overall_ UV divergence for any vertexlike
Feynman amplitude $\Gamma$ (of any order);

  * •

$U_{j}\Gamma$, $j=1,2$ does not have IR divergences at all, provided that the
mass renormalization has been done properly (see below);

  * •

this is also true for $U_{j}\Sigma$, $j=1,2$, where $\Sigma$ is a lepton self-
energy Feynman amplitude.

In particular, this means that $U_{1},U_{2}$ can be used to subtract UV
divergences without generating additional IR divergences (unlike $L$).

Figure 2: 2-loop examples

Consider the 2-loop ladder graph $G$ in Fig. 2,left151515Since changing the
lepton line directions on a path or loop does not change the contribution, we
will sometimes work with undirected Feynman graphs.. The corresponding
expression is

$A_{G}-A_{G}(U_{1})_{abc}-(L-U_{1})_{G}A_{abc}.$

In this case, we have only logarithmic divergences. The overall UV divergence
corresponds to $k_{1},k_{2}\rightarrow\infty$; it is proportional to
$\gamma_{\mu}$ and is cancelled in each term by $A$ or $L-U_{1}$. The overall
IR divergence corresponds to $k_{1},k_{2}\rightarrow 0$. In all terms it is
cancelled by $A$ since it is proportional to the Born amplitude
$A\gamma_{\mu}=0$; this can be shown using standard methods for studying IR
divergences in QED. There is also a mixed UV-IR divergence in this graph:
$k_{1}\rightarrow 0,k_{2}\rightarrow\infty$; in this case, the UV divergence
corresponding to $abc$ is IR-infinitely enhanced by the remaining propagators.
In the first 2 terms it is cancelled by $A$, because the UV-divergent part of
$abc$ is proportional to $\gamma_{\mu}$, so the “enhanced” value is
proportional to $A\gamma_{\mu}=0$. In the remaining term, it is cancelled in
$A_{abc}$ for the same reason. The remaining divergences161616This
investigation is intended to demonstrate ideas only; a rigorous proof is not
the goal in this case. are the UV divergence $k_{2}\rightarrow\infty$ and the
IR divergence $k_{1}\rightarrow 0$. The first is cancelled in the sum of two
terms by $1-U_{1}$ applied to the amplitude of $abc$ and in the last term by
$A_{abc}$. The second does not exist in $A_{G}(U_{1})_{abc}$, in $A_{G}$ it
enhances the amplitude generated by $A_{abc}$ by an IR-infinite multiplier,
and it is cancelled by $(L-U_{1})_{G}A_{abc}$, because $L-U_{1}$ completely
extracts the IR-divergent part of the enhancing multiplier.

Let us examine the graph $G$ on Fig. 2,right. The expression is

$A_{G}-A_{G}(U_{1})_{bc}.$

The difficulty in this case is the power-type IR-divergence corresponding to
$k_{1}\rightarrow 0$ and a fixed $k_{2}\neq 0$. However, it is eliminated by
the mass subtraction part in $U_{1}$. To make this possible, we give no
freedom to the definition (11) of $M^{\prime}$ in (9). After the mass
subtraction, the IR divergence becomes logarithmic and is removed by $A$, as
is the overall UV divergence. The UV subdivergence of $bc$ is removed by
$1-U_{1}$, and no additional IR divergence is generated.

Figure 3: An example with $U_{3}$ in the expression

Let us consider the graph $G$ on Fig. 3 . The expression is

$\left[A_{G}-A_{G}(U_{3})_{bcdfghi}-(L-U_{1})_{G}A_{bcdfghi}\right]\left(1-(U_{0})_{fghi}\right).$

The last multiplier eliminates the UV divergence corresponding to $fghi$; in
this case there are no problems with IR divergences. The last term of the
first multiplier removes the IR divergence corresponding to $k_{ae}\rightarrow
0$ (the subscript denotes the graph line by its ends). The term
$A_{G}(U_{3})_{bcdfghi}$ removes the UV divergence corresponding to $bcdfghi$.
It is important to note that it does not create any additional IR divergence,
although $U_{3}$ has more freedom in definition than $U_{1}$ and $U_{2}$ (we
can take $U_{3}=L$, for example). The reason is that the on-shell Feynman
amplitude of $bcdfghi$ has no IR divergence, because its external photon is on
a lepton loop.

The cancellation of divergences in Feynman parametric space for an arbitrary
graph $G$ can be shown in a similar way as in [69], but $U$ should be replaced
everywhere by the corresponding $U_{j}$. Let us note one more small change.
Following [69], Appendix C, we assume that $G_{1}\subset\ldots\subset
G_{n}\subset G_{n+1}=G$ are all elements of $\mathcal{I}[G]$. If $\xi=1$,
where (12) is implied, we also assume that $G_{0}$ is the graph consisting of
only one vertex of $G$ joining the external photon (if $\xi=3$, we use $G_{0}$
as the empty graph). By $z_{l}$ we denote the Feynman parameter corresponding
to the internal line $l$. With the help of power counting, we investigate the
case

$z_{l}\asymp\delta^{\beta_{l}},\quad\delta\rightarrow 0,$

where $\beta_{l}\geq 0$ are some numbers corresponding to the internal lines.
By $P_{j}$ we denote171717In [69] it is denoted by $P_{j}^{\prime}\cup
P_{j}^{\prime\prime}$. the set of all lines $l$ on the main path of $G$ such
that $l\in G_{j+1}$ and $l\notin G_{j}$. The cancellation of UV divergences
and UV parts of mixed divergences is demonstrated in the same way181818The
expression is organized in a very similar way to Zimmermann’s forest formula;
this makes it possible to handle UV divergences. as in [69]. If all
divergences not belonging to $\mathcal{I}[G]$ are properly subtracted, a
potentially IR divergent case corresponds to a _vector of divergences_
$[v_{0},\ldots,v_{n};w_{0},\ldots,w_{n}]$, where $v_{j}\geq w_{j}\geq 0$,
$2v_{0}-w_{0}=2v_{1}-w_{1}=\ldots=2v_{n}-w_{n}>0$, there exists $j$ such that
$w_{j}=0$. This vector gives the values

$\beta_{l}=\begin{cases}v_{j},\text{ if }l\in P_{j},\\\ w_{j},\text{ if }l\in
G_{j+1},\ j\notin G_{j},\ j\notin P_{j}.\end{cases}$

We will refer to the index $j$, $1\leq j\leq n$ as a _transition index_ for
the divergence vector $[v,w]$ if $v_{j}>v_{j-1}$. Correspondingly, the index
$j$ is called an _inverse transition index_ if $v_{j}<v_{j-1}$. The difference
with the investigation in [69] is the following:

  1. 1.

An index $j$ is called _separating_ in a term of the expression if there is an
operator $A_{G_{j}}$, $L_{G_{j}}$ or $(U_{3})_{G_{j}}$ ($U_{3}=L$) in this
term. The difference with [69] is the possibility to take $U_{3}$ (if it is
equal to $L$).

  2. 2.

If $\xi=3$, then no consideration of the case $v_{0}>w_{0}$ is necessary; this
case cannot produce divergence in a term, because $G_{1}$ has its external
photon on a lepton loop.

  3. 3.

If $\xi=3$ and $U_{3}=L$, we rewrite the expression differently for a given
index $H$ of the inverse transition in $[v,w]$:

$-\sum_{1\leq
l<H}\left(L_{G}-(U_{1})_{G})(1-L_{G_{n}})\ldots(1-L_{G_{l+1}})A_{G_{l}}(1-L_{G_{l-1}})\ldots(1-L_{G_{1}})\right)$
$+\sum_{H<l\leq
n}\left((L_{G}-(U_{1})_{G})(1-L_{G_{n}})\ldots(1-L_{G_{l+1}})[L_{G_{l}}A_{G_{H}}-A_{G_{l}}](1-L_{G_{H-1}})\ldots(1-L_{G_{1}})\right)$
$+[A_{G}-(L_{G}-(U_{1})_{G})A_{G_{H}}](1-L_{G_{H-1}})\ldots(1-L_{G_{1}})$
$+\sum_{H<l\leq
n}\left[\left(A_{G}(1-L_{G_{n}})\ldots(1-L_{G_{l+1}})-(L_{G}-(U_{1})_{G})\sum_{l<k\leq
n}\left(A_{G_{k}}\prod_{\begin{subarray}{c}l<r\leq n\\\ r\neq
k\end{subarray}}(1-L_{G_{r}})\right)\right)\right.$
$\times\left.\vphantom{\prod_{\begin{subarray}{c}l<r\leq n\\\ r\neq
k\end{subarray}}\left[\left(A_{G}(1-L_{G_{n}})\right)\right]}[L_{G_{l}}L_{G_{H}}-L_{G_{l}}](1-L_{G_{H-1}})\ldots(1-L_{G_{1}})\right].$

The last term was added to (C.7) in [69], inappropriate $(1-U)$ multipliers
were removed. The factor $L_{G_{l}}L_{G_{H}}-L_{G_{l}}$ cancels the divergence
in this term, analogous to the other terms.

This argument shows how it works, but a full rigorous examination seems
difficult and cumbersome; we rely on a numerical check (see Section V).

##  IV Equivalence to on-shell renormalization

###  IV.1 Definition

The definition of on-shell renormalization is well known for the coefficients
of the expansion in $\alpha$, but it requires a definition for subsets of
Feynman graphs. We define it as the sum of the individual graph contributions
obtained by the in-place on-shell renormalization. It can be written in terms
of expressions such as those used above. The expression for a graph $G$ is

$\sum_{F=\\{G,G_{1},\ldots,G_{n}\\}\in\mathfrak{F}[G]}(-1)^{n}A_{G}S_{G_{1}}\ldots
S_{G_{n}},$

where

$S_{G^{\prime}}=\begin{cases}L_{G^{\prime}},\text{ if }G^{\prime}\text{ is a
vertexlike graph},\\\ B_{G^{\prime}},\text{ if }G^{\prime}\text{ is a lepton
self-energy graph},\\\ (U_{0})_{G^{\prime}},\text{ if }G^{\prime}\text{ is a
photon self-energy or photon-photon scattering graph};\end{cases}$

here

$B\Sigma(p)=e([B^{\prime}\Sigma(p)](\not{p}-m)+M^{\prime}\Sigma(p)),\quad
B^{\prime}\Sigma(p)=\frac{1}{e}\left(s(m^{2})+2m\left.\frac{\partial
r(x)}{\partial x}\right|_{x=m^{2}}+2m^{2}\left.\frac{\partial s(x)}{\partial
x}\right|_{x=m^{2}}\right),$

where (10) and (11) are satisfied.

The multipliers $S_{G^{\prime}}$ for photon-photon scattering graphs
$G^{\prime}$ can be omitted in gauge-invariant classes as well as
transformations like described in [72] can be applied for photon self-energy
graphs.

###  IV.2 Examples

2-loop examples are given in [69] for the old version of the method. In the
new version each operator $U$ should be replaced by $U_{1}$ or $U_{0}$.

The Ward identities for individual Feynman graphs play an important role in
proving the equivalence. We will illustrate this by the gauge-invariant class
shown in Fig. 4. The investigation applies to both the new and old methods
(the difference is only in the name of the $U$-operator).

Figure 4: An example of a 3-loop gauge-invariant class

Table 1 provides the parts of the contributions, obtained with our method,
that contain $U$-operators. The lower-order Feynman amplitudes used in these
expressions are shown in Fig. 5. The boxed dots represent the mass vertices
(the vertices that give $1$ in the Feynman rules). By $\Gamma_{1}$ and
$\Sigma_{1}$ we also denote the 1-loop vertexlike and lepton self-energy
amplitudes (without special vertices), respectively. In this case,
$\mathcal{I}[G]=\emptyset$, and we have no IR subtractions; therefore, the
remaining contributions are the same as those obtained by the in-place on-
shell renormalization.

Table 1: Contributions with $U$-operators of the graphs from Fig. 4 obtained with our method Set number | Contribution
---|---
1 | $-(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Gamma_{\lambda})-(A^{\prime}\Gamma_{\lambda})(U_{1}^{\prime}\Gamma_{1})+(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Gamma_{1})^{2}$
2 | $-(A^{\prime}\Gamma_{\rho})(U_{1}^{\prime}\Gamma_{1})-e(A^{\prime}\Gamma_{\lambda})(U_{1}^{\prime}\Sigma_{1})+e(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Gamma_{1})(U_{1}^{\prime}\Sigma_{1})+(A^{\prime}\Gamma_{M})(U_{1}^{\prime}\Gamma_{1})(M^{\prime}\Sigma_{1})$
3 | $-(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Gamma_{\times})$
4 | $-(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Gamma_{\lambda})-(A^{\prime}\Gamma_{\lambda})(U_{1}^{\prime}\Gamma_{1})+(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Gamma_{1})^{2}$
5 | $-(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Gamma_{\rho})-e(A^{\prime}\Gamma_{\lambda})(U_{1}^{\prime}\Sigma_{1})+e(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Gamma_{1})(U_{1}^{\prime}\Sigma_{1})+(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Gamma_{M})(M^{\prime}\Sigma_{1})$
6 | $-(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Gamma_{\approx})-(A^{\prime}\Gamma_{\lambda})(U_{1}^{\prime}\Gamma_{1})+(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Gamma_{1})^{2}$
7 | $-(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Gamma_{\rho})-e(A^{\prime}\Gamma_{\lambda})(U_{1}^{\prime}\Sigma_{1})+e(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Gamma_{1})(U_{1}^{\prime}\Sigma_{1})+(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Gamma_{M})(M^{\prime}\Sigma_{1})$
8 | $-2e(A^{\prime}\Gamma_{\rho})(U_{1}^{\prime}\Sigma_{1})+e^{2}(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Sigma_{1})^{2}+2e(A^{\prime}\Gamma_{M})(U_{1}^{\prime}\Sigma_{1})(M^{\prime}\Sigma_{1})$
9 | $-e(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Sigma_{\times})-2(A^{\prime}\Gamma_{\rho})(U_{1}^{\prime}\Gamma_{1})+2e(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Sigma_{1})(U_{1}^{\prime}\Gamma_{1})+2(A^{\prime}\Gamma_{M})(M^{\prime}\Sigma_{1})(U_{1}^{\prime}\Gamma_{1})$
10 | $-e(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Sigma_{\approx})-e(A^{\prime}\Gamma_{\rho})(U_{1}^{\prime}\Sigma_{1})+e^{2}(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Sigma_{1})^{2}+e(A^{\prime}\Gamma_{M})(M^{\prime}\Sigma_{1})(U_{1}^{\prime}\Sigma_{1})+e(A^{\prime}\Gamma_{1})(U_{1}^{\prime}\Sigma_{M})(M^{\prime}\Sigma_{1})$
Figure 5: Feynman amplitudes participating in the expressions for the set in
Fig. 4 and the corresponding Feynman graphs

After summation, we obtain

$(A^{\prime}\Gamma_{1})(-[2U_{1}^{\prime}\Gamma_{\lambda}+U_{1}^{\prime}\Gamma_{\times}+eU_{1}^{\prime}\Sigma_{\times}]-[2U_{1}^{\prime}\Gamma_{\rho}+U_{1}^{\prime}\Gamma_{\approx}+eU_{1}^{\prime}\Sigma_{\approx}])$
$+[U_{1}^{\prime}\Gamma_{1}+eU_{1}^{\prime}\Sigma_{1}](-3A^{\prime}\Gamma_{\lambda}-3A^{\prime}\Gamma_{\rho}+(A^{\prime}\Gamma_{1})(3U_{1}^{\prime}\Gamma_{1}+2eU_{1}^{\prime}\Sigma_{1}))+(A^{\prime}\Gamma_{1})(M^{\prime}\Sigma_{1})[2U_{1}^{\prime}\Gamma_{M}+eU_{1}^{\prime}\Sigma_{M}]$

We use the following Ward identities for individual graphs:

$(\Gamma_{1})_{\mu}(p,0)+e\frac{\partial\Sigma_{1}(p)}{\partial
p^{\mu}}=0,\quad
2(\Gamma_{M})_{\mu}(p,0)+e\frac{\partial\Sigma_{M}(p)}{\partial p^{\mu}}=0,$
$2(\Gamma_{\lambda})_{\mu}(p,0)+(\Gamma_{\times})_{\mu}(p,0)+e\frac{\partial\Sigma_{\times}(p)}{\partial
p^{\mu}}=0,\quad
2(\Gamma_{\rho})_{\mu}(p,0)+(\Gamma_{\approx})_{\mu}(p,0)+e\frac{\partial\Sigma_{\approx}(p)}{\partial
p^{\mu}}=0$

Each identity contains one lepton self-energy graph and all vertexlike graphs
obtained by inserting an external photon into a line on the main path. Each
coefficient equals $1$, but some of our coefficients equal $2$, because we are
working with undirected Feynman graphs. The identities can be proved by
standard methods used in QED191919See [73].. The corollary is

$U_{1}^{\prime}\Gamma_{1}+eU_{1}^{\prime}\Sigma_{1}=0,\quad
2U_{1}^{\prime}\Gamma_{M}+eU_{1}^{\prime}\Sigma_{M}=0,\quad
2U_{1}^{\prime}\Gamma_{\lambda}+U_{1}^{\prime}\Gamma_{\times}+eU_{1}^{\prime}\Sigma_{\times}=0,\quad
2U_{1}^{\prime}\Gamma_{\rho}+U_{1}^{\prime}\Gamma_{\approx}+eU_{1}^{\prime}\Sigma_{\approx}=0.$

From this it follows that all terms with $U_{1}^{\prime}$ are cancelled.
Similarly, we can prove that all terms with $L^{\prime}$ and $B^{\prime}$ are
lifted in the on-shell expression, which implies the equivalence202020provided
that all regularization issues are resolved; dimensional regularization can be
used in most cases..

Let us examine examples where the new and old methods work differently. We
start with the 4-loop class IV(b) from Fig. 6,left. We suppose that the graphs
contribute to the muon $g-2$ and contain an electron loop.

Figure 6: 4-loop gauge-invariant classes IV(b) (left) and IV(c) (right) from
[52].

The difference is that the old method yields additional subtractions in graphs
2,10 because each of them has a vertexlike subgraph joining the external
photon and not lying on the main path212121These subgraphs correspond to
fictitious IR divergences, but the subtraction was necessary in [69] to make
the procedure equivalent to on-shell renormalization.. Also, the operators
$U_{2}$ are used everywhere instead of $U$. The contribution of IV(b) is equal
to

The first term is cancelled because its second multiplier equals $0$ due to
the Ward identity; the light-by-light term is similar and is cancelled for the
same reason. The expression for the on-shell renormalization has the same
form, but the occurrences of $U_{2}^{\prime}$ are replaced with $L^{\prime}$,
$B^{\prime}$; its terms already considered are cancelled due to the same
reason. The remaining terms are identical.

Consider the class IV(c) from Fig. 6,right. In this case, we also assume that
the graphs contribute to the muon $g-2$ and have an electron loop. Graphs 7
and 8 give terms with $U_{3}$. The total contribution of these graphs is

We ignore the terms with $U_{0}^{\prime}$ and $M^{\prime}$ in this
investigation, because they do not cause any problems. To show that the first
term equals $0$, we need the Ward identities of the form $\Gamma_{\mu}(p,0)=0$
for single graphs. It holds for sets of vertexlike graphs that have the
external photon on a _lepton loop_ that are closed under the movement of it
along the loop. The validity can be proved in the usual way. Since $U_{3}$
preserves this kind of Ward identities, its contributions are cancelled. The
remaining graphs yield

With $U_{1}^{\prime}\Gamma_{1}=-eU_{1}^{\prime}\Sigma_{1}$, we get that all
terms containing $U_{1}^{\prime}$ are cancelled after summation. The total
contribution obtained by the in-place on-shell renormalization is

The difference between our and on-shell contributions equals

It equals $0$, because $L^{\prime}\Gamma_{1}=-eB^{\prime}\Sigma_{1}$.

Let us summarize. To prove the equivalence, we need two kinds of the “Ward
identities” for single graphs:

  * •

$\Gamma_{\mu}(p,0)=-e\times\partial\Sigma(p)/p^{\mu}$; the set includes one
lepton self-energy graph and all vertexlike graphs obtained from it by
inserting an external photon into a line on the main path.

  * •

$\Gamma_{\mu}(p,0)=0$; the set consists of vertexlike graphs having the
external photon on a lepton loop; it is closed under the movement of the
external photon along the loop.

Graphs in the sets may contain special vertices. See also examples of the use
of the Ward identities for individual graphs in Section IV.H of [32].

###  IV.3 Complete proof

Suppose we have a class $\mathcal{K}$ of _1-particle irreducible_ vertexlike
Feynman graphs that is closed under the movement of internal photons along
lepton loops and paths, but without jumping over the external photon. We prove
that its contribution obtained with our method is equal to the contribution
obtained with the in-place on-shell renormalization. Since the ideas used here
are similar to those widely used in QED, the explanation is just a framework.

Let us give some definitions.

Suppose we have one term $X$ of the expression with linear operators
constructed for some graph $G$ of this class. The subgraphs to which the
operators in $X$ are applied form a rooted tree; in this tree, $G$ is a root,
and $G^{\prime}$ is a descendant of $G^{\prime\prime}$ if $G^{\prime}\subset
G^{\prime\prime}$. The contribution of $X$ can be expressed with $A^{\prime}$,
$U_{j}^{\prime}$, $L^{\prime}$, $L^{\prime}-U_{j}^{\prime}$, $M^{\prime}$,
$B^{\prime}$ applied to some graphs with special vertices. This expression can
have several terms $Y_{1},\ldots,Y_{n}$, because $U_{j}\Sigma$ is split into
$U_{j}^{\prime}$-part and $M^{\prime}$-part. Each term $Y_{j}$ can be
represented as a _layer tree_. A _layer_ is a Feynman graph obtained from one
graph to which an operator in $X$ is applied by schrinking all its childs to
special vertices (corresponding to operators in $Y$). A layer tree is a rooted
tree, each node of it is $(l,O,r)$, where $l$ is a layer; $O$ is one of the
operators $A^{\prime}$, $U_{j}^{\prime}$, $L^{\prime}$,
$L^{\prime}-U_{j}^{\prime}$, $M^{\prime}$, $B^{\prime}$; $r$ is the reference
to the corresponding vertex of the parent (it does not exist for the root).
The root of the layer tree corresponds to $G$ in $X$. Vertices and inputs of
one vertex that have the same type are enumerated222222The enumeration is
needed to avoid symmetry problems with 4-photon vertices. If we have an
enumeration, the coefficients should be properly accounted for; this can be
done with a standard technique; we do not do this in our consideration. An
examination of the coefficients is only necessary for symmetry issues; no
reduction of similar terms takes place in our consideration.. The contribution
of the layer tree (without coefficient) is the product of the node
values232323Strictly speaking, tensor contractions are needed when we have
$U_{0}^{\prime}$ layers, because for simplicity we decided to use one label
for all kinds of special vertices that can be generated., where the value of
the node $(l,O,r)$ is $O\Gamma_{l}$, where $\Gamma_{l}$ is the Feynman
amplitude corresponding to the layer $l$. For simplicity, we will sometimes
interchange the layer tree nodes with the layers.

For example, the term $A_{G}(U_{0})_{defghi}(U_{2})_{gh}$ for the graph $G$
from Fig. 7,left gives the layer tree depicted in Fig. 7,right.

Figure 7: An example of a Feynman graph (left) and a layer tree (right) that
is possible for it (with $U_{0}^{\prime}$ and $U_{2}^{\prime}$ for lepton
self-energy graphs)

The following markers are used for vertices on images:

  1. 1.

a round dot is an ordinary $\gamma_{\nu}$ vertex;

  2. 2.

a boxed dot is a mass vertex (which gives $1$);

  3. 3.

a cross-dot is a $(\not{p}-m)$-vertex;

  4. 4.

a cross in a circle is a vertex coming from photon self energy subgraphs;

  5. 5.

a diamond dot is a vertex originating from photon-photon scattering subgraphs.

Another example is the term

$A_{G}B_{bcdefghi}L_{cdefg}B_{de}B_{klno}(U_{0})_{no}$

for $G$ from Fig. 8,left, where $M^{\prime}$ parts are taken for each $B$ (the
term is part of the direct on-shell renormalization expression). Its layer
tree is depicted in Fig. 8,right. The root is layer 1.

The sequence of layer tree nodes $Y_{1},\ldots,Y_{n}$ of vertexlike type is
called the _main branch_ if the following conditions are satisfied:

  * •

$Y_{1}$ is the root of the layer tree;

  * •

$Y_{j+1}$ is the child of $Y_{j}$ corresponding to the vertex of $Y_{j}$
joining its external photon;

  * •

$Y_{n}$ has no vertexlike children corresponding to its vertex joining its
external photon;

the _type_ of a layer is its multiset of the external line types.

Figure 8: An example of a Feynman graph (left) and a possible layer tree for
it (right) with $M^{\prime}$-operators

We say that the node $X$ of a layer tree is _locally on a path_ , if the
corresponding to $X$ parent’s vertex lies on parent’s main path; we also
assume that the root of the layer tree is locally on a path. If the node is
not locally on a path, it may be on a loop or on a special 2-photon or
4-photon vertex of the parent.

The maximal initial segment $Y_{1},Y_{2},\ldots,Y_{k}$ of the main branch
$Y_{1},Y_{2},\ldots,Y_{n}$, all elements of that are locally on a path, is
called the _$\mathcal{I}$ -branch_.

Let us proceed with the proof. We represent the contribution of $\mathcal{K}$
obtained by our method as the sum of the layer tree contributions (with
coefficients). Each layer tree has one node $X$ on the $\mathcal{I}$-branch
that has the operator $A^{\prime}$. We call it the _A-node_. The operators
corresponding to vertexlike or lepton self-energy nodes $X$ satisfy the
following rules:

  * •

if $X$ is the A-node, then the corresponding operator is $A^{\prime}$;

  * •

if $X$ is the root but not A-node, then the operator is
$L^{\prime}-U^{\prime}_{1}$;

  * •

if $X$ is an ascendant of the A-node but not the root, then the operator is
$L$;

  * •

if $X$ is a descendant of the A-node and belongs to the $\mathcal{I}$-branch,
then the operator is $U_{\xi}$, where (12) is satisfied for graphs $G$ in
$\mathcal{K}$;

  * •

if $X$ is not on the $\mathcal{I}$-branch and all ascendants of $X$ and $X$
itself are locally on a path, then the operator is $U_{1}^{\prime}$ or
$M^{\prime}$;

  * •

if $X$ is not on the $\mathcal{I}$-branch and at least one ascendant of $X$ or
$X$ itself is not locally on a path, then the operator is $U_{2}^{\prime}$ or
$M^{\prime}$.

Let us perform the following operations with this layer tree sum:

  1. 1.

_Remove all terms containing layers of photon-photon scattering type_. In
fact, they are cancelled by the identities for individual photon-photon
scattering graphs of the form

$\Gamma_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}(0)=0,$

where $\Gamma$ is the corresponding Feynman amplitude, and the external
momenta are in the parentheses. The identity is satisfied for a given set of
graphs, if the set is closed under the movement of external photons along
lepton loops, and _every graph in this set contains at least one external
photon line joining a lepton loop_ (we consider graphs with special vertices;
thus an external photon can fall into a 4-photon vertex). The identities can
be proved with a standard technique that is used in QED242424See [73].. Let us
consider the elimination in detail. We say that the layer trees $T_{1}$ and
$T_{2}$ are equivalent if one is obtained from the other by the movement of
external photons along lepton loops in some layers of photon-photon scattering
types _which have no descendants of photon-photon scattering type_ (the
absence of the descendants guarantees that all moved photons are on lepton
loops252525They cannot fall to 2-photon vertices due to 1-particle
irreducibility.); we suppose that the structure of the descendants is
preserved when a photon is shifted; the children referring to the vertex
joining the moved photon change their reference to the new photon location on
a lepton loop. It is easy to see that if $T_{1}$ is obtained from a Feynman
graph of $\mathcal{K}$, this is also true for $T_{2}$. Therefore, all terms
are split into equivalence classes. In the contribution of each of these
classes, the identities mentioned above are factorized. This leaves only the
classes that have no photon-photon scattering layers at all. Note that the
movement of photons (both external and internal) along lepton loops does not
change the 1-particle irreducibility.

  2. 2.

_Remove all terms containing vertexlike layers with a non- $A^{\prime}$
operator and the external photon on a loop_262626We consider here only the
internal structure of the layer, not its reference to its parent.. The
operator is $L^{\prime}$, $U_{1}^{\prime}$, $U_{2}^{\prime}$,
$U_{3}^{\prime}$. All these operators yield $0$ for the Feynman amplitudes
$\Gamma_{\mu}(p,q)$ satisfying $\Gamma_{\mu}(p,0)=0$. Thus all these terms are
cancelled by the identities of this form mentioned in Section IV.2. This can
be proved in a similar way as for the case with photon-photon scattering
layers. After this step, all occurrences of $U_{3}^{\prime}$ are removed,
since at least the $U_{3}^{\prime}$ corresponding to the end of the
$\mathcal{I}$-branch is applied to a layer with the external photon on a
lepton loop.

  3. 3.

_Represent the layer trees as trees of M-blocks_. An _M-block_ includes a node
(vertexlike or lepton self-energy-like) with an operator $A^{\prime}$,
$L^{\prime}$, $U_{j}^{\prime}$ ($j=1,2$) of the layer tree, all its children
with operators $M^{\prime}$, $U_{0}^{\prime}$, their children with these
operators and so on272727We do not use here that $M^{\prime}$ properly
extracts the mass part, but it is used for the proof of gauge invariance [8]..
For example, there are two M-blocks in 8,right: $\\{1,2,3,5\\}$ and
$\\{4,6\\}$; the latter is a child of the former and refers to the vertex of
layer 2 to which layer 4 refers. M-blocks can also be represented as layers.
The layer is obtained from the corresponding subgraph by shrinking the
subgraphs corresponding to its children (in the tree of M-blocks) to special
vertices (in a sense, it is obtained by gluing the layer tree layers belonging
to the M-block). An M-block can also be viewed as a tree of nodes. Let us
define the _main vertex_ of the M-block; if the layer has its external photon
on its main path, the main vertex is layer’s vertex joining its external
photon; otherwise, it is the _last vertex of the main path_ of the layer.

  4. 4.

_Single out the $\mathcal{S}$-branch of M-blocks in each layer tree_. We
define the $\mathcal{S}$-branch of M-blocks
$Y_{1}^{\prime},\ldots,Y_{n}^{\prime}$ in a tree of M-blocks by the following
conditions:

     * •

$Y_{1}^{\prime}$ is the root of the M-block tree;

     * •

$Y^{\prime}_{j}$, $j\geq 2$, is a child of $Y^{\prime}_{j-1}$ and refers to
the main vertex of $Y^{\prime}_{j-1}$;

     * •

there are no children of $Y_{n}^{\prime}$ that refer to the main vertex of
$Y_{n}^{\prime}$.

An initial segment $Y_{1}^{\prime},\ldots,Y_{k}^{\prime}$ of the
$\mathcal{S}$-branch corresponds to the $\mathcal{I}$-branch
$Y_{1},\ldots,Y_{k}$. Due to the 1-particle irreducibility of each subgraph to
which an operator is applied, all elements of the $\mathcal{S}$-branch are of
vertexlike type.

  5. 5.

_For each M-block take a one-to-one correspondence between the layer vertices
(except the main vertex) and its $\Sigma$-chains lying on the same lepton path
part or loop._ A _$\Sigma$ -chain_ is a peace of a lepton path or loop, which
ends are ordinary $\gamma_{\mu}$ vertices, but all intermediate vertices are
of $(\not{p}-m)$ type (the situation when there are no intermediate vertices
is also possible). We require that the $\Sigma$-chain lies on the same side of
the M-block external photon as the corresponding vertex (if it is applicable).
We should take this correspondence _deterministically_ (it should depend only
on the topology of the $\Sigma$-chains and external lines). For example, for
each $v$ on a lepton loop, we can take the outgoing from $v$ chain; for $v$ on
the main path we can take the $\Sigma$-chain joining $v$ and having another
end closer to the main vertex of the M-block.

  6. 6.

_Remove all terms containing M-blocks outside the $\mathcal{S}$-branch_. Let
us explain how they are cancelled. An M-block is called _factorizable_ if it
has at least one child that does not refer to its main vertex. We say that the
M-block trees $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ are equivalent if one is
obtained from the other by a sequence of the following transformations:

     * •

suppose $Y^{\prime}$ is a factorizable M-block that _has no factorizable
children_ , $v$ is an ordinary $\gamma_{\mu}$-vertex in $Y^{\prime}$, the main
vertex of $Y^{\prime}$ is not equal $v$, the $\Sigma$-chain $\sigma$
corresponds to $v$, the child M-block referring to $v$ exists and is denoted
by $Y^{\prime\prime}$; the transformation is: remove the external photon in
the _root layer_ of $Y^{\prime\prime}$, move the reference of
$Y^{\prime\prime}$ to the new $(\not{p}-m)$ vertex inserted at the beginning
of $\sigma$; the operator applied is $U_{j}^{\prime}$, $j=1,2$ and remains the
same after the transformation; the structure of the descendants and their
references are preserved, _the reference to the main vertex of
$Y^{\prime\prime}$ is moved to the main vertex of the transformed M-block_
(there are no references to non-main vertices, since $Y^{\prime}$ has no
factorizable children).

     * •

if under the same conditions there are no children referring to $v$, and the
intermediate vertexes of $\sigma$ are $w_{1},\ldots,w_{r}$, $r\geq 1$ (ordered
along the line direction), $Y^{\prime\prime}$ is the child M-block referring
to $w_{1}$, then insert an external photon into an arbitrary line on the main
path of the _root layer_ of $Y^{\prime\prime}$, remove $w_{1}$ and move the
reference of $Y^{\prime\prime}$ to $v$. the applied operator is
$U_{j}^{\prime}$, $j=1,2$ and remains the same after the transformation; the
structure of the descendants and their references are preserved (as in the
previous case).

These transformations are inverse to each other. They are illustrated in Fig.
9; here $v$ is the round dot, $\sigma$ starts from $v$, $T_{0}$ is a subtree
referring to $v$ in the layer tree, the subtrees $T_{1},\ldots,T_{n}$ refer to
the intermediate vertexes of $\sigma$; on the right side, $T_{0}$ is replaced
by $T_{0}^{\prime}$ (by removing the external photon in the root layer) and is
connected to the begin of $\sigma$.

Figure 9: Transformations leading to the cancellation of all terms with
M-blocks outside the $\mathcal{S}$-branch.

This equivalence relation splits the terms into equivalence classes. If a term
belongs to the expression, all equivalent terms also belong to the expression;
it is important here that these transformations do not change the 1-particle
irreducibility of the whole graph as well as of the subgraphs to which the
operators are applied282828This can be proved using three facts: 1) a Feynman
graph and all its subgraphs to which the operators are applied are 1-particle
irreducible if and only if each layer of the corresponding layer tree is
1-particle irreducible; 2) adding and removing $(\not{p}-m)$ vertices in
internal lines does not change the 1-particle irreducibility; 3) inserting the
external photon into a lepton line (or removing it) does not change the
1-particle irreducibility.; the absence of odd lepton loops in the whole graph
is also preserved. In the sum of contributions belonging to one equivalence
class, the multipliers of the form
$U^{\prime}_{j}\Gamma+eU^{\prime}_{j}\Sigma$ (as mentioned in Section IV.2)
are factorized. If the number of these multipliers is at least 1, the
contribution equals $0$.

The presence of M-blocks outside the $\mathcal{S}$-branch means that there is
at least one factorizable M-block in the $\mathcal{S}$-branch. If it has
factorizable children, take one of them; and repeat the operation until it has
no factorizable children. So there is at least one factorizable M-block that
has no factorizable children. The cancellation of these terms follows from
this.

Note: On the one hand, the correspondence between $v$ and $\sigma$ may cross
the boundaries of the layers constituting the M-block; on the other hand, the
procedure of inserting and removing the external photon is always performed in
the root layer of the M-block, and the structure of $\Sigma$-chains is ignored
in this layer (the main path may contain special vertices); but when we move
the reference, we use the main vertex of the whole M-block.

  7. 7.

_Combine the terms with different placements of the A-nodes._ After the
previous step, the contribution can be written as a sum of terms like

$(-1)^{n-1}(O_{1}^{\prime}\Gamma_{1})(O_{2}^{\prime}\Gamma_{2})\ldots(O_{n}^{\prime}\Gamma_{n}),$
(13)

where $\Gamma_{1},\ldots,\Gamma_{n}$ are the vertexlike Feynman amplitudes
(after the appropriate application of $-M^{\prime}$ and $-U_{0}^{\prime}$ for
subgraphs) corresponding to the $\mathcal{S}$-branch elements
$Y^{\prime}_{1},\ldots,Y_{n}^{\prime}$, the corresponding linear operators are
$O^{\prime}_{1},\ldots,O^{\prime}_{n}$.

The Feynman graph $G$ for which the expression is constructed is uniquely
determined from the layers $Y_{1}^{\prime},\ldots,Y_{n}^{\prime}$ by a
sequential insertion of the layers into a point; if
$Y_{1}^{\prime},\ldots,Y^{\prime}_{j-1}$ have their external photons on their
main paths, then $Y_{j}^{\prime}$ is inserted into the vertex joining the
external photon; otherwise, it is inserted into the end of the main path.

The term (13) corresponding to $Y_{1}^{\prime},\ldots,Y_{n}^{\prime}$,
$O_{1}^{\prime},\ldots,O_{n}^{\prime}$ exists in the expression for
$\mathcal{K}$ if the following conditions are satisfied:

     * •

$Y_{j}^{\prime}$ are correct vertexlike M-blocks (taking into account their
internal layer structure);

     * •

$G\in\mathcal{K}$, where $G$ is reconstructed by the procedure described
above;

     * •

the vector $(O_{1}^{\prime},\ldots,O_{n}^{\prime})$ is of the form
$(A^{\prime},U_{1}^{\prime},\ldots,U_{1}^{\prime})$ or
$(L^{\prime}-U_{1}^{\prime},L^{\prime},\ldots,L^{\prime},A^{\prime},U_{1}^{\prime},\ldots,U_{1}^{\prime})$,
where $x,\ldots,x$ may have $0$ elements;

     * •

for all $i$ such that $O_{i}^{\prime}\neq A^{\prime}$ the layer
$Y_{i}^{\prime}$ has its external photon on its main path.

Any permutation of $Y_{1}^{\prime},\ldots,Y_{n}^{\prime}$ does not change the
validity of $G\in\mathcal{K}$ for the reconstructed graph $G$292929To prove
this, consider two cases: 1) each $Y_{j}^{\prime}$ has its external photon on
its main path (in this case the inability to jump over the external photon
should be taken into account); 2) at least one $Y_{j}^{\prime}$ has its
external photon on a lepton loop (in this case the external photon is on a
lepton loop).. Thus, the sum can be represented as a sum of terms (13) with
the same conditions, but with vectors $(O_{1}^{\prime},\ldots,O_{n}^{\prime})$
of the form $(A^{\prime},U_{1}^{\prime},\ldots,U_{1}^{\prime})$ or
$(A^{\prime},L^{\prime},\ldots,L^{\prime},L^{\prime}-U_{1}^{\prime},U_{1}^{\prime},\ldots,U_{1}^{\prime})$.

We can prove by induction that

$(U_{1}^{\prime},U_{1}^{\prime},U_{1}^{\prime},\ldots,U_{1}^{\prime},U_{1}^{\prime})$
$+(L-U_{1}^{\prime},U_{1}^{\prime},U_{1}^{\prime},\ldots,U_{1}^{\prime},U_{1}^{\prime})$
$+(L^{\prime},L^{\prime}-U_{1}^{\prime},U_{1}^{\prime},\ldots,U_{1}^{\prime},U_{1}^{\prime})$
$+\ldots$
$+(L^{\prime},L^{\prime},L^{\prime},\ldots,L^{\prime},L^{\prime}-U_{1}^{\prime})$
$=(L^{\prime},L^{\prime},L^{\prime},\ldots,L^{\prime},L^{\prime}).$

Thus, we arrive at the sum of terms

$(-1)^{n-1}(A^{\prime}\Gamma_{1})(L^{\prime}\Gamma_{2})\ldots(L^{\prime}\Gamma_{n})$
(14)

with the same conditions.

With the contribution of $\mathcal{K}$ obtained by the in-place on-shell
renormalization303030Note that the definition of the contribution of the
gauge-invariant class in [8] already corresponds our final representation;
however, it looks like it becomes cumbersome if graphs may have lepton loops.
we perform the same operations, but without the last step and using the
identities for $L^{\prime}$ and $B^{\prime}$. These operations result in the
same sum of terms (14) with the same conditions.

##  V Computational results

The new method was tested by numerical calculations. The contributions of 25
gauge-invariant classes contributing to $A_{1}^{(6)}$, $A_{2}^{(6)}$,
$A_{3}^{(6)}$ for the muon were presented in [70]. Here we present all 4-loop
results for the electron and the muon for all gauge-invariant classes (with
electron, muon and tau-lepton loops). They are summarized in Table 2. We used
the values

$m_{\mu}/m_{e}=206.76828103,\quad m_{\tau}/m_{\mu}=16.81665.$

in our calculations. The uncertainty of the masses was not taken into account.
The aim of the calculation was to verify the method, not to obtain precise
physical results (the mass uncertainty does not play a significant role at
this level of precision). Some of the previously known values for comparison
that are given in the tables were calculated with other values of the mass
ratios (and the uncertainties may be inaccurate). The Monte Carlo integration
was performed on the ITP/TTP KIT computing cluster with the help of the GPU
NVidia A100 and took about three GPU-weeks; a method similar to that described
in [35, 32] was used.

Table 2: The calculated total 4-loop contributions (for the electron and the muon) and their comparison with previously known values; the values for comparison are from the first reference in the lists. Contribution | Our value | Value for comparison | Ref.
---|---|---|---
$A_{1}^{(8)}$ | -1.9118(41) | -1.91224576 |  [29]; [28, 30, 31]
$A_{2}^{(8)}(m_{e}/m_{\mu})$ | 0.000924(11) | 0.0009141970703(372) |  [54]; [33]
$A_{2}^{(8)}(m_{e}/m_{\tau})$ | 0.00000710(60) | 0.00000742924(118) |  [54]; [33]
$A_{3}^{(8)}(m_{e}/m_{\mu},m_{e}/m_{\tau})$ | 0.000000745(24) | 0.00000074687(28) |  [54]; [33]
$A_{2}^{(8)}(m_{\mu}/m_{e})$ | 132.673(84) | 132.6852(60) |  [52]; [53]
$A_{2}^{(8)}(m_{\mu}/m_{\tau})$ | 0.04252(11) | 0.0424941(53) |  [54]; [52]
$A_{3}^{(8)}(m_{\mu}/m_{e},m_{\mu}/m_{\tau})$ | 0.0622(33) | 0.062722(10) |  [53]; [52]

Table 3 presents the computed contributions of the gauge-invariant classes
from Fig. 10 to $A_{1}^{(8)}$ and a comparison with S. Laporta’s results [29].
Table 4 contains all the contributions of the classes from Fig. 11 for
different particles $x$ and $y$. Since the contribution does not depend on the
line directions, all graphs on the images are undirected. Also, permutting the
elements on a photon does not change the contribution, but we calculated these
contributions separately, except $A_{2}^{(8)}[\text{II(c2,xy)}]$,
$A_{2}^{(8)}[\text{II(c2,yx)}]$ and $A_{3}^{(8)}[\text{II(c2,yt)}]$,
$A_{3}^{(8)}[\text{II(c2,ty)}]$. Tables 5 and 6 contain a comparison with
known values for $A_{2}^{(8)}[\text{electron}]$ and
$A_{2}^{(8)}[\text{muon}]$, respectively. Table 7 contains the contributions
of the classes from Fig. 12 to $A_{3}^{(8)}$ for electron and muon, Table 8
contains their comparison with known results.

Throughout the calculations, the operators

$U_{j}\Gamma_{\mu}(p,q)=a(-m^{2})\gamma_{\mu},\quad
U_{j}\Sigma(p)=s(-m^{2})(\not{p}-m)+r(m^{2})+s(m^{2})m$

are used for $j=1,2,3$ and all leptons, where (8) and (10) are satisfied;
therefore, $U$-subtractions are performed at space-like momenta ($p^{2}<0$);
this is a general observation: the subtractions at space-like points make the
oscillations slightly smaller and the convergence of the Monte Carlo
integration slightly faster [71].

Figure 10: Gauge-invariant classes contributing to $A_{1}^{(8)}$. Each class is obtained from the corresponding picture by moving the internal photon ends along the lepton paths and loops, but without jumping over the external photon. Table 3: Computed contributions of the classes from Fig. 10 to $A_{1}^{(8)}$ and their comparison with the results of [29]. Class | Value | Value from [29] |  | Class | Value | Value from [29]
---|---|---|---|---|---|---
I(a) | 0.00087614(64) | 0.0008768659 |  | III(7) | -0.012485(63) | -
I(b) | 0.0153300(39) | 0.0153252829 |  | IV(a) | 0.598864(69) | 0.598842072
I(c) | 0.0111324(13) | 0.0111309140 |  | IV(b) | 0.82236(20) | 0.8222844858
I(d) | 0.049544(60) | 0.0495132026 |  | IV(c) | -1.13789(52) | -1.1388228765
II(a1) | -0.255082(24) | - |  | IV(d1) | -0.87251(21) | -0.872657392
II(a2) | -0.317402(29) | - |  | IV(d2) | -0.117978(94) | -0.1179498688
II(a3) | 0.151986(16) | 0.1519895997 |  | V(1) | -1.9723(12) | -1.9710756168
II(b1) | -0.0341771(30) | -0.034179376 |  | V(2) | -0.6227(20) | -0.6219210635
II(b2) | 0.0065025(10) | 0.0065041484 |  | V(3) | -0.1412(21) | -0.1424873798
II(c1) | -0.0376103(54) | - |  | V(4) | 1.0865(15) | 1.0866983948
II(c2) | -0.0536815(84) | - |  | V(5) | -1.0397(19) | -1.04054241
II(c3) | 0.0178533(40) | 0.0178536865 |  | V(6) | 0.51153(71) | 0.512462048
III(1) | 0.128258(83) | - |  | II(a1)+II(a2) | -0.572485(38) | -0.5724718621
III(2) | 0.242948(42) | - |  | II(c1)+II(c2) | -0.091292(10) | -0.09130584
III(3) | -0.04388(10) | - |  | III(1)+III(5) | 0.69078(12) | 0.6904483476
III(4) | 0.374317(49) | 0.3743579348 |  | III(2)+III(6) | 0.409178(76) | 0.4092170285
III(5) | 0.562521(92) | - |  | III(3)+III(7) | -0.05637(12) | -0.0563360902
III(6) | 0.166229(64) | - |  |  |  |
Figure 11: Gauge-invariant classes contributing to $A_{2}^{(8)}$ for particle $x$. Each class is obtained from the corresponding image by moving the internal photon ends along the lepton paths and loops, but without jumping over the external photon. $x$ and $y$ may be arbitrary leptons. Figure 12: Gauge-invariant classes contributing to $A_{3}^{(8)}$ for particle $x$. Each class is obtained from the corresponding image by moving the internal photon ends along the lepton paths and loops, but without jumping over the external photon. $x$ and $y$ are electron and muon (or vice versa), $\tau$ is the tau-lepton. Table 4: Calculated contributions of the classes from Fig. 11 to $A_{2}^{(8)}$ for electron and muon. Class |  $\text{Value}\times 10^{3},$ $x=e$, $y=\mu$ |  $\text{Value}\times 10^{5},$ $x=e$, $y=\tau$ | Value, $x=\mu$, $y=e$ | Value, $x=\mu$, $y=\tau$
---|---|---|---|---
I(a,xxy) | 0.000168(35) | 0.00003(37) | 0.01854(19) | 0.00002116(31)
I(a,xyx) | 0.000077(18) | 0.00008(22) | 0.009378(80) | 0.00001057(16)
I(a,yxy) | 0.0000009(22) | -0.000004(13) | 0.16473(32) | 0.000000274(38)
I(a,yyx) | 0.0000034(45) | 0.000012(27) | 0.32986(60) | 0.000000612(77)
I(a,yyy) | 0.000000006(22) | 0.0000000035(78) | 7.2235(18) | 0.0000000180(35)
I(b,xy) | 0.000963(51) | 0.00050(78) | 0.1195(10) | 0.00014580(54)
I(b,yx) | 0.000811(79) | 0.0010(11) | 0.33397(53) | 0.00010541(69)
I(b,yy) | 0.00000038(20) | 0.000000053(79) | 7.1267(62) | 0.000001950(32)
I(c,xy) | 0.004754(57) | 0.00346(41) | 0.16205(25) | 0.00027729(30)
I(c,yx) | 0.005936(32) | 0.00361(37) | 0.02156(40) | 0.00040798(35)
I(c,yy) | 0.0003543(26) | 0.000127(10) | 1.4372(25) | 0.000053149(67)
I(d) | 0.002457(32) | 0.000915(62) | -0.234(15) | 0.0003681(11)
II(a1) | -0.041120(60) | -0.02104(22) | -2.3873(57) | -0.0037194(13)
II(a2) | -0.07041(11) | -0.04147(57) | -2.4684(73) | -0.0054854(21)
II(a3) | 0.025403(57) | 0.01748(23) | 2.0651(80) | 0.0018698(11)
II(b1,xy) | -0.017667(63) | -0.01333(56) | -0.37697(57) | -0.00098634(55)
II(b1,yx) | -0.027301(55) | -0.02399(68) | -0.31493(55) | -0.00126397(53)
II(b1,yy) | -0.00090489(12) | -0.000319911(44) | -6.2229(30) | -0.000136912(20)
II(b2,xy) | 0.006692(24) | 0.00684(22) | 0.16083(31) | 0.00033086(24)
II(b2,yy) | 0.000133460(43) | 0.000047178(15) | 2.1911(19) | 0.0000204864(67)
II(c1,xy) | -0.04082(29) | -0.0306(33) | -0.6564(10) | -0.0022605(23)
II(c1,yy) | -0.0009045(88) | -0.000336(36) | -6.0791(24) | -0.00014068(19)
II(c2,xy) | -0.04159(30) | -0.0373(30) | -0.4278(10) | -0.0019306(26)
II(c2,yx) | -0.04159(30) | -0.0373(30) | -0.4278(10) | -0.0019306(26)
II(c2,yy) | -0.002274(22) | -0.000921(84) | -6.7191(37) | -0.00027518(40)
II(c3,xy) | 0.03193(20) | 0.0337(19) | 0.36033(65) | 0.0011914(16)
II(c3,yy) | 0.0010686(85) | 0.000539(36) | 4.6173(27) | 0.00009622(17)
III(1) | -0.0384(11) | -0.0679(70) | 1.7884(74) | 0.000526(12)
III(2) | 0.23468(44) | 0.2280(34) | 1.7049(78) | 0.0098734(51)
III(3) | 0.0860(23) | 0.073(10) | -2.703(14) | 0.003569(14)
III(4) | 0.2010(17) | 0.1436(61) | 4.375(14) | 0.0111379(75)
III(5) | 0.2768(11) | 0.2027(75) | 4.400(10) | 0.016024(11)
III(6) | 0.05621(29) | 0.0395(15) | 2.565(10) | 0.0035162(48)
III(7) | 0.00390(33) | -0.0013(16) | -1.3382(82) | 0.0005628(56)
IV(a,xy) | 0.1540(17) | 0.054(16) | 2.6961(53) | 0.012611(17)
IV(a,yx) | 0.4639(13) | 0.364(13) | 4.3409(74) | 0.023262(12)
IV(a,yy) | 0.01755(10) | 0.00658(34) | 116.802(27) | 0.0026298(27)
IV(b) | 0.04174(90) | 0.0118(30) | -0.423(52) | 0.006132(20)
IV(c) | -0.1954(50) | -0.114(25) | 2.860(47) | -0.018337(83)
IV(d1) | -0.1693(70) | -0.092(38) | -3.5037(79) | -0.015203(55)
IV(d2) | -0.0051(45) | -0.000(28) | -0.9187(49) | -0.000555(22)
Table 5: Comparison with previously known values for $A_{2}^{(8)}$ of the electron ($x=e$); the values for comparison are from the first reference in the lists. Class |  $\text{Value}\times 10^{3},$ $y=\mu$ | Value for comparison |  $\text{Value}\times 10^{5},$ $y=\tau$ | Value for comparison | Ref.
---|---|---|---|---|---
I(a) | 0.000250(40) | 0.0002264 | 0.00012(43) | 0.0000802 |  [54]; [33, 55, 60, 61]
I(b) | 0.001774(94) | 0.001704139(76) | 0.0015(14) | 0.000602805(26) |  [33]; [55]
I(c) | 0.011044(66) | 0.0110072(15) | 0.00720(55) | 0.0069819(12) | [33]
I(d) | 0.002457(32) | 0.0024727 | 0.000915(62) | 0.0008746 |  [54]; [33]
II(a) | -0.08613(14) | -0.0864460(90) | -0.04503(65) | -0.0456480(70) |  [33]; [55]
II(b) | -0.039048(87) | -0.0390003(27) | -0.03075(90) | -0.0303937(42) | [33]
II(c) | -0.09418(55) | -0.095097(24) | -0.0721(57) | -0.071697(25) | [33]
III | 0.8203(33) | 0.81715 | 0.617(16) | 0.6059 |  [54]; [33]
IV(a) | 0.6354(21) | 0.63579 | 0.425(21) | 0.451 |  [54]; [33]
IV(b) | 0.04174(90) | 0.041574 | 0.0118(30) | 0.01471 |  [54]; [33, 59, 58]
IV(c) | -0.1954(50) | -0.19548262120(20) | -0.114(25) | -0.0979 |  [54]; [33]
IV(d) | -0.1744(83) | -0.1778(12) | -0.092(47) | -0.0927(13) | [33]
I(a,yyy) | 0.000000006(22) | 0.000000001033 | 0.0000000035(78) | 0.000000000001293 |  [60]; [61, 55]
I(a,yxy) +I(a,yyx) | 0.0000043(51) | 0.0000001346 | 0.000008(30) | 0.0000000003954 |  [60]; [61, 55]
I(a,xxy) +I(a,xyx) | 0.000246(40) | 0.0002263 | 0.00011(43) | 0.00008025 |  [60]; [61, 55]
I(b,xy) | 0.000963(51) | 0.0010008 | 0.00050(78) | 0.000354 | [55]
I(b,yx) | 0.000811(79) | 0.00070428 | 0.0010(11) | 0.000249 | [55]
I(b,yy) | 0.00000038(20) | 0.00000017906 | 0.000000053(79) | 0.000000000354 | [55]
II(c1,yy) +II(c2,yy) +II(c3,yy) | -0.002110(25) | -0.0021216 | -0.00072(10) | -0.00075 | [55]
II(c1,xy) +II(c2,xy) +II(c2,yx) +II(c3,xy) | -0.09207(55) | -0.092981 | -0.0714(57) | -0.0710 | [55]
Table 6: Comparison with previously known values for $A_{2}^{(8)}$ of the muon ($x=\mu$); the values for the comparison are from the first reference in the lists. Class |  $\text{Value},$ $y=e$ | Value for comparison | Ref. |  $\text{Value},$ $y=\tau$ | Value for comparison | Ref.
---|---|---|---|---|---|---
I(a) | 7.7460(19) | 7.745136 |  [53]; [52] | 0.00003263(36) | 0.00003242810(20) |  [54]; [52]
I(b) | 7.5802(63) | 7.58201(71) | [52] | 0.00025316(87) | 0.000252 | [52]
I(c) | 1.6208(25) | 1.624307(40) | [52] | 0.00073843(47) | 0.000737 | [52]
I(d) | -0.234(15) | -0.2303620(50) |  [56]; [52, 53] | 0.0003681(11) | 0.0003677960(40) |  [54]; [52]
II(a) | -2.791(12) | -2.77885 |  [53]; [52, 55] | -0.0073350(26) | -0.0073290(10) | [52]
II(b) | -4.5628(37) | -4.55277(30) | [52] | -0.00203588(80) | -0.002036 | [52]
II(c) | -9.3325(55) | -9.34180(83) | [52] | -0.0052501(46) | -0.0052460(10) | [52]
III | 10.792(28) | 10.7934(27) |  [52]; [53] | 0.045209(24) | 0.0452089860(60) |  [54]; [52]
IV(a) | 123.839(29) | 123.78551(44) |  [52]; [53] | 0.038504(20) | 0.038519670(30) |  [54]; [52]
IV(b) | -0.423(52) | -0.4170(37) |  [52]; [53] | 0.006132(20) | 0.006126610(50) |  [54]; [52, 59, 58]
IV(c) | 2.860(47) | 2.9072(44) |  [52]; [53] | -0.018337(83) | -0.01830100(10) |  [54]; [52]
IV(d) | -4.4225(93) | -4.43243(58) |  [52]; [53] | -0.015757(59) | -0.015868(37) | [52]
I(a,yyy) | 7.2235(18) | 7.22307640(80) |  [55]; [53, 57] | 0.000000037(14) | 0.0000000232 |  [60]; [61]
I(b,xy) | 0.1195(10) | 0.1196024600(20) | [55] | - | - | -
I(b,yx) | 0.33397(53) | 0.333664680(10) | [55] | - | - | -
I(b,yy) | 7.1267(62) | 7.12800840(20) | [55] | - | - | -
I(c,xy) | 0.16205(25) | 0.161982(11) | [74] | - | - | -
I(c,yx) | 0.02156(40) | 0.0215830(20) | [74] | - | - | -
I(c,yy) | 1.4372(25) | 1.440744(16) | [74] | - | - | -
IV(a,xy) | 2.6961(53) | 2.69(14) | [53] | - | - | -
IV(a,yx) | 4.3409(74) | 4.33(17) | [53] | - | - | -
IV(a,yy) | 116.802(27) | 116.760(20) | [53] | - | - | -
I(a,yxy) +I(a,yyx) | 0.49459(68) | 0.494072030(30) |  [55]; [53, 57] | 0.000000885(86) | 0.0000008757 |  [60]; [61]
I(a,xxy) +I(a,xyx) | 0.02792(20) | 0.0279883220(70) |  [55]; [53, 57] | 0.00003173(35) | 0.0000315291 |  [60]; [61]
II(c1,yy) +II(c2,yy) +II(c3,yy) | -8.1808(51) | -8.1895 | [55] | - | - | -
II(c1,xy) +II(c2,xy) +II(c2,yx) +II(c3,xy) | -1.1517(18) | -1.1532 | [55] | - | - | -
II(b1,yy) +II(b2,yy) +II(c1,yy) +II(c2,yy) +II(c3,yy) | -12.2126(63) | -12.212631 | [53] | - | - | -
II(b1,xy) +II(b1,yx) +II(b2,xy) +II(c1,xy) +II(c2,xy) +II(c2,yx) +II(c3,xy) | -1.6828(20) | -1.683165(13) | [53] | - | - | -
Table 7: Computed contributions of classes from Fig. 12 to $A_{3}^{(8)}$ for electron and muon. Class |  $\text{Value}\times 10^{7},$ $x=e$ |  $\text{Value},$ $x=\mu$ | Class |  $\text{Value}\times 10^{7},$ $x=e$ |  $\text{Value},$ $x=\mu$
---|---|---|---|---|---
I(a,tty) | -0.000003(17) | -0.0000012(85) | I(c,yt) | 0.0797(60) | 0.000383(24)
I(a,tyt) | 0.0000063(85) | 0.0000038(35) | II(b1,ty) | -0.30252(13) | -0.006434(89)
I(a,xyt) | -0.0113(95) | 0.000155(34) | II(b1,yt) | -0.22691(14) | -0.00457(10)
I(a,ytx) | -0.0033(93) | 0.000133(30) | II(b2,yt) | 0.070470(61) | 0.002062(38)
I(a,yty) | 0.000037(36) | 0.000884(51) | II(c1,yt) | -0.533(33) | -0.00972(23)
I(a,yxt) | 0.008(11) | 0.000147(30) | II(c2,yt) | -0.718(63) | -0.00753(20)
I(a,yyt) | -0.000022(71) | 0.00184(11) | II(c2,ty) | -0.718(63) | -0.00753(20)
I(b,ty) | 0.00025(20) | 0.00069(13) | II(c3,yt) | 0.771(29) | 0.00498(12)
I(b,yt) | 0.00001(14) | 0.001944(81) | IV(a,ty) | 5.94(16) | 0.0482(12)
I(c,ty) | 0.0965(52) | 0.001389(36) | IV(a,yt) | 2.99(15) | 0.0351(20)
Table 8: Comparison with previously known values for $A_{3}^{(8)}$ for the electron and the muon; the comparison values are from the first reference in the lists. Class |  $\text{Value}\times 10^{7},$ $x=e$ | Value for comparison | Ref. |  $\text{Value},$ $x=\mu$ | Value for comparison | Ref.
---|---|---|---|---|---|---
I(a) | -0.007(17) | 0.0000119956 |  [54]; [33] | 0.00316(14) | 0.003209050(10) |  [53]; [52]
I(b) | 0.00025(24) | 0.0000140970(10) | [33] | 0.00264(16) | 0.002611 | [52]
I(c) | 0.1763(79) | 0.172860(21) | [33] | 0.001772(43) | 0.001807 | [52]
II(b) | -0.45897(20) | -0.458968(17) | [33] | -0.00895(14) | -0.0090080(10) | [52]
II(c) | -1.20(10) | -1.18969(67) | [33] | -0.01980(38) | -0.0196420(20) | [52]
IV(a) | 8.93(22) | 8.9432(25) |  [54]; [33] | 0.0834(23) | 0.083747570(90) |  [53]; [52]
I(a,yty)+I(a,yyt) | - | - | - | 0.00273(12) | 0.00274860(90) | [57]

##  VI Conclusion and discussion

The development of techniques plays a very important role in theoretical
physics. Techniques of a certain kind first appear; then they are refined:
they become simpler, less cumbersome, more efficient; and only after that can
the technique be extended to broader classes of tasks and lead to a
breakthrough.

This work is a part of the “refinement” phase. At present, the described
method of divergence subtraction works only for the lepton magnetic moments
and QED. However, its formulation is much simpler and compact than that of the
previously known methods. Unlike the old method described in [69], the new
method has more freedom and does not require useless extra subtractions; but
most importantly, the ideas behind the new method appear extendable to broader
classes of problems.

It was supposed that the old method [69] only worked for $A_{1}^{(n)}$.
However, it turned out to be suitable for the computation of $A_{2}^{(8)}$,
$A_{3}^{(8)}$ as well. This was first discovered numerically and was
surprising, but this can be proved: all the extra subtraction terms that the
old method has have the operator $L^{\prime}$ or $L^{\prime}-U^{\prime}$
applied to a layer with external photon on a lepton loop; thus, all these
occurrences are cancelled, as it was proved in Section IV.3 for
$U^{\prime}_{3}$.

The old method can also lead to the situation where subtracting a fictitious
IR divergence produces a real IR divergence (see Discussion in [69]). The
cancellation of these divergences was conjectured in [69]. We have now checked
this numerically: for the only 5-loop graph in which this situation occurs,
the obtained integral is indeed finite. In addition, the reasoning in Section
IV.3 looks cumbersome and might contain mistakes; therefore, we checked on a
computer that the required combinatorial identities are satisfied, up to 6
loops and 3 particles. However, the fact of divergece cancellation in every
single Feynman graph requires a full rigorous proof.

###### Acknowledgements.

The author would like to thank Gudrun Heinrich, Savely Karshenboim, Andrey
Arbuzov, Lidia Kalinovskaya for valuable assistance and Andrey Kataev for help
in writing the review section. The computations were performed with the help
of the ITP/TTP KIT computing cluster. All images were drawn using AxoDraw and
JaxoDraw. The research is supported by the Alexander von Humboldt Foundation.

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