# A way of fast calculating lepton magnetic moments in quantum electrodynamics

###### Abstract

A new method of divergence subtraction in Feynman parametric integrals is
presented. The method is suitable for calculating the lepton anomalous
magnetic moments (AMM) in quantum electrodynamics (QED). The subtraction
procedure eliminates all divergences before integration and leads to a finite
Feynman parametric integral for each individual Feynman diagram. It is based
on a forest formula with linear operators applied to the Feynman amplitudes of
ultraviolet-divergent subdiagrams. The formula is similar to BPHZ; the
difference is only in the linear operators used and in the way of combining
them. The subtraction is equivalent to the on-shell renormalization from the
beginning: for obtaining the final result we should only sum up the
contributions of all Feynman diagrams after subtraction. The developed method
is an improvement of the method presented by the author in 2016. The
modification is specifically designed for calculating the contributions
dependent on the relations of particle masses. In comparison with the old
version, the new subtraction formula does not contain redundant terms and
possesses some flexibility that can be used for improving the precision of
calculations. Numerical test results are presented up to four loops.

a Skobeltsyn Institute of Nuclear Physics at Lomonosov Moscow State University
(Moscow, Russia) b Dzhelepov Laboratory of Nuclear Problems at Joint Institute
of Nuclear Research (Dubna, Russia)

## Introduction

The electron AMM $a_{e}$ is measured with a very high precision [1] as well as
the muon AMM $a_{\mu}$ [2]111the value is the statistical average of the new
experimental value and the old one [3]:

$\displaystyle a_{e}[\text{expt.}]$
$\displaystyle=0.001\,159\,652\,180\,73(28),$ (1) $\displaystyle
a_{\mu}[\text{expt.}]$ $\displaystyle=0.001\,165\,920\,61(41).$ (2)

The Standard Model prediction of $a_{e}$ uses the following representation:

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

where $m_{e},m_{\mu},m_{\tau}$ are the masses of the electron, muon and tau-
lepton, respectively. Different terms of this expression were calculated by
different researchers. A similar expression is used for $a_{\mu}$. The QED
part forms the most significant contribution to both $a_{e}$ and $a_{\mu}$.

$A_{1}^{(2n)}$, $n=1,2,3,4$ have reliable double-checked values. However,
$A_{1}^{(10)}$ is still sensitive in $a_{e}$ experiments, but is not double-
checked yet. The value

$A_{1}^{(10)}[\text{AHKN}]=6.737(159)$ (3)

was presented by T. Aoyama, M. Hayakawa, T. Kinoshita, M. Nio in 2019 [4].
This value in combination with the remaining terms [4] and with the measured
values of the fine-structure constant $\alpha$

$\displaystyle\alpha^{-1}(\text{Rb-2011})$
$\displaystyle=137.035\,998\,996(85),$ (4)
$\displaystyle\alpha^{-1}(\text{Cs-2018})$
$\displaystyle=137.035\,999\,046(27),$ (5)
$\displaystyle\alpha^{-1}(\text{Rb-2020})$
$\displaystyle=137.035\,999\,206(11)$ (6)

from [5, 6], [7], [8] gives

$a_{e}[\text{theory},\text{Rb-2011},\text{AHKN}]=0.001\,159\,652\,182\,037(11)(12)(720),$
$a_{e}[\text{theory},\text{Cs-2018},\text{AHKN}]=0.001\,159\,652\,181\,606(11)(12)(229),$
$a_{e}[\text{theory},\text{Rb-2020},\text{AHKN}]=0.001\,159\,652\,180\,254(11)(12)(94),$

respectively. Each of these values has three uncertainties: from
$A_{1}^{(10)}$ (statistical uncertainty of the Monte Carlo integration), the
hadronic and electroweak corrections, the uncertainty of $\alpha$,
respectively. Since the $\alpha$ uncertainty dominates, this calculation can
be used for improving the precision of $\alpha$: (1) in combination with (3)
gives

$\alpha^{-1}[a_{e},\text{AHKN}]=137.035\,999\,149\,6(331).$

The tension between it and (4), (5), (6) amounts $1.69\sigma$, $2.43\sigma$,
$1.61\sigma$.

In 2019 the author recalculated a part of $A_{1}^{(10)}$ and discovered a
discrepancy with (3) [9]. The calculation gave the value

$A_{1}^{(10)}[\text{Volkov+AHKN}]=5.862(90).$ (7)

The value leads to

$a_{e}[\text{theory},\text{Rb-2011},\text{Volkov+AHKN}]=0.001\,159\,652\,181\,969(6)(12)(720),$
$a_{e}[\text{theory},\text{Cs-2018},\text{Volkov+AHKN}]=0.001\,159\,652\,181\,547(6)(12)(229),$
$a_{e}[\text{theory},\text{Rb-2020},\text{Volkov+AHKN}]=0.001\,159\,652\,180\,195(6)(12)(94),$
$\alpha^{-1}[a_{e},\text{Volkov+AHKN}]=137.035\,999\,142\,7(331)$

with the same order of uncertainties. The last value has the tension
$1.61\sigma$, $2.26\sigma$, $1.81\sigma$ with (4), (5), (6), respectively. The
shift is relatively small, but a significant error in $A_{1}^{(10)}$ would be
sensitive in experiments.

A recent theoretical prediction for $a_{\mu}$ [10] gives

$a_{\mu}=0.001\,165\,918\,095\,3(4386)(100)(10),$

where the uncertainties came from the hadronic, electroweak and QED
contributions, respectively. It has the discrepancy $4.2\sigma$ with (2). The
QED uncertainty is relatively small, but it is based on the assumption that
all the coefficient values are correct. $A_{2}$ and $A_{3}$ for the muon
suffer from powered large logarithms of $(m_{\mu}/m_{e})$. For example, in
[11] the value

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

was published. It is more than 100 times larger than $A_{1}^{(10)}$ and is not
double-checked yet. A significant error in it would give a shift in $a_{\mu}$
comparable with the hadronic uncertainty. Moreover, the higher-order terms are
still important [11]. Thus, the development of a method of high-order QED
calculcations is still actual. A possibility to reduce the influence of large
logarithms in intermediate values is very important for improving the
precision.

## Old and new method

Let $G$ be a QED Feynman diagram contributing to the lepton AMM (it contains
one external photon line and two external lepton lines). The old method [12]
uses the linear operators $A$, $L$, $U$, where $A$ is the AMM projector
multiplied by the Dirac matrix $\gamma_{\mu}$, $L$ is the standard on-shell
renormalization operator for vertex-like diagrams, $U$ is an intermediate
operator222See the definitions in [12]..

Let us examine the diagram from Fig. LABEL:fig_example contributing to
$a_{\mu}$, where all lepton loops correspond to electrons. The old method
gives the expression

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

where $G_{c}=aa_{1}a_{2}b_{1}b_{2}c_{1}c_{2}c_{3}c_{4}$,
$G_{e}=aa_{1}a_{2}b_{1}b_{2}c_{1}c_{2}c_{3}c_{4}d_{1}d_{2}d_{3}e_{1}e_{2}e_{3}$,
$G_{d}=aa_{1}a_{2}b_{1}b_{2}c_{1}c_{2}c_{3}c_{4}d_{1}d_{2}d_{3}$, the operator
subscript means the subdiagram to which the operator is applied (encoded by
the set of vertexes). Here all brackets should be expanded, each
multiplication should be considered as a sequential transformation of the
Feynman amplitudes from smaller to larger subdiagrams, all terms should be
written in the same Feynman parametric space. In this example, the expression,
that gives a contribution to $a_{\mu}$, contains $A$ applied to an electron
subdiagram $G_{c}$. This looks like a nonsense, but the proof of the
equivalence to the on-shell renormalization in [12] does not allow to avoid
these nonsense terms.

Figure 1: Example of a Feynman diagram for lepton anomalous magnetic moments.

fig_example

Also, an ability to preserve minimal gauge-invariant classes is important for
divergence subtraction methods. Such a class is obtained from one diagram by
moving internal photons along lepton loops and paths without jumping over the
external photon (and without violating one-particle irreducibility). The old
method preserves the classes without lepton loops [9]. However, the proof in
[12] requires swapping the layers between subdiagrams to which $A$ is applied
(like $G$, $G_{c}$, $G_{e}$); this can lead to exit from the class.

It surprisingly turned out that the drawbacks described above are fictitious:
the nonsense terms are cancelled, and all gauge-invariant classes are
preserved due to Ward identities of some kind. However, the old method remains
redundand and nonflexible.

We use two types of Ward identities for individual lepton self-energy and
vertex-like diagrams333See examples of applying Ward identities for individual
diagrams in [13].. The first type is
$\Gamma_{\mu}(p,0)=-\partial\Sigma(p)/\partial p^{\mu}$, where
$\Gamma_{\mu}(p,q)$ is a vertex-like Feynman amplitude, $\Sigma(p)$ is a
lepton self-energy amplitude; the $\Gamma_{\mu}$-diagrams are obtained from
the $\Sigma$-diagram by inserting the external photon into all possible places
on the main lepton path444The main path is a lepton path between external
lepton lines.. The second type is $\Gamma_{\mu}(p,0)=0$; here the
$\Gamma_{\mu}$-diagrams are obtained from some diagram by inserting the
external photon into one lepton cycle.

Let us describe the new method. We will use four operators
$U_{0},U_{1},U_{2},U_{3}$ instead of $U$. $U_{0}$ is applied to photon self-
energy and photon-photon scattering subdiagrams and works as in the standard
renormalization. $U_{i},i=1,2,3$ are not fixed, but given by requirements
(tentative):

  * •

$U_{i}$ extracts the overall UV-divergent part completely;

  * •

$U_{i}$ preserves the Ward identity: if $\Gamma_{\mu}$ and $\Sigma$ (or
$\Gamma_{\mu}$ itself) satisfy the Ward identity described above, then
$U_{i}\Gamma_{\mu}$ and $U_{i}\Sigma$ (or $U_{i}\Gamma_{\mu}$ itself) satisfy
it too;

  * •

$U_{i}$ cancels IR divergences of the subdiagram;

  * •

$U_{i}$ extracts the mass part completely:

$U_{i}[a(p^{2})+b(p^{2})\not{p}]=a(m^{2})+b(m^{2})m+(\not{p}-m)(\ldots).$

The definitions may differ for different particles. It is possible to put
$U_{3}=L$.

By $\mathfrak{I}^{\prime}[G]$ we denote the set of all vertex-like subgraphs
of $G$ (including $G$) lying on the main path of $G$ and having the external
photon of $G$. By $\mathfrak{F}[G]$ we denote the set of all forests of UV-
divergent subdiagrams of $G$ containing $G$. The expression is

$\sum_{\begin{subarray}{c}F=\\{G_{1},\ldots,G_{n}\\}\in\mathfrak{F}[G]\\\
G^{\prime}\in\mathfrak{I}^{\prime}[G]\cap
F\end{subarray}}(-1)^{n-1}M^{G^{\prime}}_{G_{1}}M^{G^{\prime}}_{G_{2}}\ldots
M^{G^{\prime}}_{G_{n}}.$

Here $M^{G^{\prime}}_{G^{\prime\prime}}$ equals $A_{G^{\prime\prime}}$, if
$G^{\prime\prime}=G^{\prime}$,
$L_{G^{\prime\prime}}-(U_{1})_{G^{\prime\prime}}$, if $G^{\prime\prime}=G\neq
G^{\prime}$, $L_{G^{\prime\prime}}$, if $G^{\prime}\subset
G^{\prime\prime}\subset G$, $(U_{0})_{G^{\prime\prime}}$, if
$G^{\prime\prime}$ is a photon self-energy or a light-by-light subgraph. In
the remaining cases it equals $(U_{2})_{G^{\prime\prime}}$, if
$G^{\prime\prime}$ lies on a lepton loop, $(U_{w})_{G^{\prime\prime}}$, if
$G^{\prime\prime}\in\mathfrak{I}^{\prime}[G]$ and $G^{\prime\prime}\subset
G^{\prime}$, $(U_{1})_{G^{\prime\prime}}$ in the other cases; here $w=3$, if
$G$ has its external photon on a lepton loop, $w=1$ otherwise.

For the example from Fig. LABEL:fig_example we have
$\mathfrak{I}^{\prime}[G]=\\{G,G_{e}\\}$ and the expression

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

Figure 2: 3-loop gauge-invariant classes for the muon $g-2$. Each class is
given by one generating diagram. Electron and tau-lepton loops are marked with
$e$ and $\tau$.

fig_3loops

Figure 3: 4-loop gauge-invariant classes IV(b) (left) and IV(c) (right) from
[11].

fig_IV_b_c

Table 1: Contributions of the classes from Fig. LABEL:fig_3loops to $a_{\mu}^{6}$ and their comparison with known values obtained by different researchers with the help of analytical expressions. Diagram | Our value | Analyt. value | Ref.
---|---|---|---
1 | $0.448\,703(35)$ | $0.448\,70$ |  [14, 15, 16, 17]
2 | $-0.498\,224(67)$ | $-0.498\,25$ |  [18, 19, 14, 15, 17, 20]
3 | $0.533\,289(54)$ | $0.533\,36$ |  [20, 18, 16, 19, 15, 14, 17]
4 | $0.421\,080(43)$ | $0.421\,17$ |  [20, 18, 19]
5 | $0.050\,178(16)$ | $0.050\,148\,7$ |  [21, 22]
6 | $-0.112\,324(21)$ | $-0.112\,336$ |  [21, 23]
7 | $-0.087\,987(12)$ | $-0.087\,984\,7$ |  [21, 24]
8 | $0.002\,559\,8(15)$ | $0.002\,558\,5$ |  [25]
9 | $0.052\,865(11)$ | $0.052\,87$ |  [25]
10 | $0.370\,94(15)$ | $0.371\,005$ |  [26]
11 | $1.617\,52(28)$ | - | -
12 | $-2.061\,83(39)$ | - | -
13 | $-1.948\,80(28)$ | - | -
11–13 | $-2.393\,11(56)$ | $-2.392\,391\,81(7)$ |  [27]
14 | $2.718\,85(62)$ | $2.718\,655\,7(2)$ |  [27]
15 | $0.100\,38(12)$ | $0.100\,519\,296(3)$ |  [27]
16 | $1.495\,45(92)$ | $1.493\,671\,80(4)$ |  [27]
17 | $20.947\,5(13)$ | $20.947\,1(29)$ |  [28]
18 | $0.000\,544\,10(20)$ | - | -
19 | $-0.001\,600\,08(30)$ | - | -
20 | $-0.001\,061\,37(17)$ | - | -
18–20 | $-0.002\,117\,35(40)$ | $-0.002\,117\,13$ |  [27]
21 | $0.000\,000\,276\,6(30)$ | $0.000\,000\,277\,833$ |  [27]
22 | $0.000\,038\,704(63)$ | $0.000\,038\,687\,5$ |  [27]
23 | $0.000\,295\,496(73)$ | $0.000\,295\,557$ |  [27]
24 | $0.002\,144\,3(12)$ | $0.002\,143\,31$ |  [28]
25 | $0.000\,524\,5(70)$ | $0.000\,527\,761$ |  [29]
Table 2: Contributions of the diagrams from the class IV(b) with a tau-lepton loop (Fig. LABEL:fig_IV_b_c, left) to $a_{\mu}^{8}$ obtained by the old method [12] and by the new developed method (New 2). Diagram |  Old [12] | New 2 | Diagram |  Old [12] | New 2
---|---|---|---|---|---
1 | $-0.045\,644(18)$ | $-0.045\,642(19)$ | 10 | $0.008\,448(22)$ | $0.016\,543(25)$
2 | $-0.001\,059(16)$ | $-0.009\,095(15)$ | 11 | $0.016\,191(23)$ | $0.016\,173(25)$
3 | $0.027\,752(20)$ | $0.027\,771(20)$ | 12 | $0.020\,202(21)$ | $0.020\,204(25)$
4 | $-0.024\,477(15)$ | $-0.024\,472(15)$ | 13 | $0.002\,047(29)$ | $0.002\,011(32)$
5 | $0.030\,956(24)$ | $0.030\,970(24)$ | 14 | $-0.011\,533(28)$ | $-0.011\,572(34)$
6 | $-0.014\,374(24)$ | $-0.014\,395(24)$ | 15 | $-0.004\,808(25)$ | $-0.004\,823(23)$
7 | $0.005\,023(17)$ | $0.005\,006(17)$ | 16 | $0.018\,435(28)$ | $0.018\,450(30)$
8 | $-0.041\,921(27)$ | $-0.041\,973(28)$ | $\sum$ | $0.006\,117(92)$ | $0.006\,058(96)$
9 | $0.020\,880(24)$ | $0.020\,904(22)$ |  |  |
Table 3: Contributions of the diagrams from the class IV(c) with an electron loop (Fig. LABEL:fig_IV_b_c, right) to $a_{\mu}^{8}$ obtained by the old method [12] and by the new developed method (New 1). Diagram |  Old [12] | New 1 | Diagram |  Old [12] | New 1
---|---|---|---|---|---
1 | $-118.181(23)$ | $-118.179(22)$ | 8 | $4.972(33)$ | $20.775(32)$
2 | $91.815(22)$ | $91.841(21)$ | 9 | $100.498(24)$ | $100.475(24)$
3 | $-80.851(20)$ | $-80.820(20)$ | 10 | $-76.896(32)$ | $-76.924(32)$
4 | $2.607(21)$ | $2.620(22)$ | 11 | $89.117(31)$ | $89.140(31)$
5 | $-74.660(27)$ | $-74.730(26)$ | 12 | $-26.755(13)$ | $-26.766(12)$
6 | $37.012(23)$ | $37.016(22)$ | 13 | $-10.547(19)$ | $-10.543(21)$
7 | $64.808(19)$ | $48.934(19)$ | $\sum$ | $2.940(87)$ | $2.840(86)$

Results of numerical tests of the methods are presented in Tables 1, 2, and 3.
We use two setups: New 1 means $U_{1}=U_{2}=U$, $U_{3}=L$, New 2 means
$U_{1}=U_{2}=U_{3}=U$. We use fixed masses of particles without taking into
account their uncertainty: $m_{\mu}=105.658\,374\,5\,\text{MeV}$,
$m_{e}=0.510\,998\,95\,\text{MeV}$, $m_{\tau}=1\,776.82\,\text{MeV}$. All
results from Table 1 are in very good agreement with known values from
literature. The difference between the methods used starts from four loops.
The sum values in Table 2 are in good agreement with each other, with the
value $0.006\,106(31)$ from [11], and with [30]. Diagrams 2 and 10 are the
only diagrams with difference between the methods. The sum values in Table 3
are in good agreement with the value $2.907\,2(44)$ from [11]. A difference
between the methods is only in diagrams 7 and 8.

The author thanks Lidia Kalinovskaya, Gudrun Heinrich, Savely Karshenboim,
Andrey Arbuzov for the important assistance, and Andrey Kataev for valuable
consultations. Also, the author thanks the Laboratory of Information
Technologies of JINR (Dubna, Russia) for providing an access to its
computational resources and additionally the organizers of the conference
FFK-2021 for providing a possibility to make a presentation. And beyond that,
the author considers it proper to honor the memory of Fyodor Tkachov.

## References

  * [1] _Hanneke D., Fogwell Hoogerheide S., Gabrielse G._ Cavity Control of a Single-Electron Quantum Cyclotron: Measuring the Electron Magnetic Moment // Phys. Rev. A. —- 2011\. —- V. 83, no. 052122. —- arXiv:1009.4831 [physics.atom-ph].
  * [2] _Abi B. et al._ [Muon g-2 Collaboration] Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm // Phys. Rev. Lett. —- 2021\. —- V. 126, no. 141801. —- arXiv:2104.03281.
  * [3] _Bennett G.W. et al._ [Muon g-2 Collaboration] Final report of the E821 muon anomalous magnetic moment measurement at BNL // Phys. Rev. D. —- 2006\. —- V. 73, no. 072003. —- arXiv:hep-ex/0602035.
  * [4] _Aoyama T., Kinoshita T., Nio M._ Theory of the Anomalous Magnetic Moment of the Electron // Atoms. —- 2019\. —- V. 7, no. 1. —- P. 28.
  * [5] _Bouchendira R., Clade P., Guellati-Khélifa S., Nez F., Biraben F._ New determination of the fine structure constant and test of the quantum electrodynamics // Phys. Rev. Lett. —- 2011\. —- V. 106, no. 080801. —- arXiv:1012.3627 [physics.atom-ph].
  * [6] _Mohr P.J., Newell D.B., Taylor B.N._ CODATA Recommended Values of the Fundamental Physical Constants: 2014 // Rev. Mod. Phys. —- 2016\. —- V. 88, no. 3,035009. —- arXiv:1507.07956.
  * [7] _Parker R.H., Yu C., Zhong W., Estey B., Müller H._ Measurement of the fine-structure constant as a test of the Standard Model // Science. —- 2018\. —- V. 360. —- P. 191. —- arXiv:1812.04130.
  * [8] _Morel L., Yao Z., Cladé P., Guellati-Khélifa S._ Determination of the fine-structure constant with an accuracy of 81 parts per trillion // Nature. —- 2020\. —- V. 588, no. 7836. —- P. 61–65.
  * [9] _Volkov S._ Calculating the five-loop QED contribution to the electron anomalous magnetic moment: Graphs without lepton loops // Phys. Rev. D. —- 2019\. —- V. 100, no. 9,096004. —- arXiv:1909.08015.
  * [10] _Aoyama T., Asmussen N., others._ The anomalous magnetic moment of the muon in the Standard Model // Physics Reports. —- 2020\. —- V. 887. —- P. 1–166. —- arXiv:2006.04822.
  * [11] _Aoyama T., Hayakawa M., Kinoshita T., Nio M._ Complete Tenth-Order QED Contribution to the Muon g-2 // Phys. Rev. Lett. —- 2012\. —- V. 109, no. 111808. —- arXiv:1205.5370 [hep-ph].
  * [12] _Volkov S._ Subtractive procedure for calculating the anomalous electron magnetic moment in QED and its application for numerical calculation at the three-loop level // J. Exp. Theor. Phys. —- 2016\. —- V. 122, no. 6. —- P. 1008–1031.
  * [13] _Volkov S._ Numerical calculation of high-order QED contributions to the electron anomalous magnetic moment // Phys. Rev. D. —- 2018\. —- V. 98, no. 7,076018. —- arXiv:1807.05281.
  * [14] _Laporta S., Remiddi E._ The Analytical value of the electron (g-2) at order $\alpha^{3}$ in QED // Phys. Lett. B. —- 1996\. —- V. 379. —- P. 283. —- arXiv:hep-ph/9602417.
  * [15] _Laporta S._ The Analytical value of the corner ladder graphs contribution to the electron (g-2) in QED // Phys. Lett. B. —- 1995\. —- V. 343. —- P. 421–426.
  * [16] _Barbieri R., Caffo M., Remiddi E., Turrini S., Oury D._ The anomalous magnetic moment of the electron in QED: some more sixth order contributions in the dispersive approach // Nuclear Physics B. —- 1978\. —- V. 144. —- P. 329–348.
  * [17] _Levine M.J., Roskies R._ Analytic contribution to the $g$ factor of the electron in sixth order // Phys. Rev. D. —- 1976\. —- V. 14. —- P. 2191.
  * [18] _Levine M.J., Perisho R.C., Roskies R._ Analytic Contributions to the G Factor of the electron // Phys. Rev. D. —- 1976\. —- V. 13. —- P. 997.
  * [19] _Levine M.J., Remiddi E., Roskies R._ Analytic Contributions To The G Factor Of The Electron In Sixth Order // Phys. Rev. D. —- 1979\. —- V. 20. —- P. 2068–2076.
  * [20] _Levine M.J., Roskies R._ Hyperspherical approach to quantum electrodynamics - sixth-order magnetic moment // Phys. Rev. D. —- 1974\. —- V. 9. —- P. 421.
  * [21] _Barbieri R., Remiddi E._ Sixth order electron and muon $(g-2)/2$ from second order vacuum polarization insertion // Physics Letters B. —- 1974\. —- V. 49. —- P. 468.
  * [22] _Barbieri R., Caffo M., Remiddi E._ A Contribution to Sixth Order electron and Muon Anomalies. 3. // Lett. Nuovo Cimento. —- 1974\. —- V. 9. —- P. 690.
  * [23] _Billi D., Caffo M., Remiddi E._ A Contribution to the sixth-Order electron and muon Anomalies // Lettere al Nuovo Cimento. —- 1972\. —- V. 4, no. 14. —- P. 657–660.
  * [24] _Barbieri R., Caffo M., Remiddi E._ A contribution to sixth-order electron and muon anomalies. – II // Lett. Nuovo Cimento. —- 1972\. —- V. 5. —- P. 769.
  * [25] _Mignaco J., Remiddi E._ Fourth-order vacuum polarization contribution to the sixth-order electron magnetic moment // Nuovo Cimento A. —- 1969\. —- V. 60. —- P. 519.
  * [26] _Laporta S., Remiddi E._ The Analytic value of the light-light vertex graph contributions to the electron (g-2) in QED // Phys. Lett. B. —- 1991\. —- V. 265. —- P. 182–184.
  * [27] _Laporta S._ The Analytical contribution of the sixth order graphs with vacuum polarization insertions to the muon (g-2) in QED // Nuovo Cimento A. —- 1993\. —- V. 106, no. 5. —- P. 675–683.
  * [28] _Laporta S., Remiddi E._ The Analytical value of the electron light-light graphs contribution to the muon (g-2) in QED // Physics Letters B. —- 1993\. —- V. 301. —- P. 440–446.
  * [29] _Czarnecki A., Skrzypek M._ The Muon anomalous magnetic moment in QED: Three loop electron and tau contributions // Physics Letters B. —- 1999\. —- V. 449. —- P. 354–360. —- arXiv:hep-ph/9812394.
  * [30] _Kurz A., Liu T., Marquard P., Steinhauser M._ Anomalous magnetic moment with heavy virtual leptons // Nucl. Phys. B. —- 2014\. —- V. 879. —- P. 1–18. —- arXiv:1311.2471 [hep-ph].