#  Lepton anomaly from QED diagrams with vacuum polarization
insertions within the Mellin-Barnes representation

O.P. Solovtsova<EMAIL_ADDRESS><EMAIL_ADDRESS>Bogoliubov Lab.
Theor. Phys., JINR, Dubna, 141980, Russia  Gomel State Technical University,
Gomel, 246746, Belarus  V.I. Lashkevich  Gomel State Technical University,
Gomel, 246746, Belarus  L.P. Kaptari<EMAIL_ADDRESS>Bogoliubov Lab.
Theor. Phys., JINR, Dubna, 141980, Russia

###### Abstract

The contributions to the anomalous magnetic moment of the lepton
$\displaystyle L$ ($\displaystyle L=e\ ,\mu$ or $\displaystyle\tau$) generated
by a specific class of QED diagrams are evaluated analytically up to the
eighth order of the electromagnetic coupling constant. The considered class of
the Feynman diagrams involves the vacuum polarization insertions into the
electromagnetic vertex of the lepton $\displaystyle L$ up to three closed
lepton loops. The corresponding analytical expressions are obtained as
functions of the mass ratios $\displaystyle r=m_{l}/m_{L}$ in the whole region
$\displaystyle 0<r<\infty$. Our consideration is based on a combined use of
the dispersion relations for the polarization operators and the Mellin-Barnes
integral transform for the Feynman parametric integrals. This method is widely
used in the literature in multi-loop calculations in relativistic quantum
field theories. For each order of the radiative correction, we derive
analytical expressions as functions of $\displaystyle r$, separately at
$\displaystyle r<1$ and $\displaystyle r>1$. We argue that in spite of the
obtained explicit expressions in these intervals which are quite different, at
first glance, they represent two branches of the same analytical function.
Consequently, for each order of corrections there is a unique analytical
function defined in the whole range of $\displaystyle r\in(0,\infty)$.

The results of numerical calculations of the $\displaystyle 4th$,
$\displaystyle 6th$ and $\displaystyle 8th$ order corrections to the anomalous
magnetic moments of leptons ($\displaystyle L=e,\mu,\tau$) with all possible
vacuum polarization insertions are represented as functions of the ratio
$\displaystyle r=m_{l}/m_{L}$. Whenever pertinent, we compare our analytical
expressions and the corresponding asymptotical expansions with the known
results available in the literature.

anomalous magnetic moment of leptons, QED polarization operators, bubble-type
diagrams, higher order electromagnetic corrections

###### pacs:

13.40.Em, 12.20.Ds, 14.60.Ef

Some preliminary notes on the theory of the anomalous magnetic moment of a
lepton: the electro-magnetic contribution.

##  I Introduction

The giromagnetic factor $\displaystyle g$ is an important physical quantity
relating the magnetic-dipole moment of a particle to its spin. According to
Dirac’s theory dirac , the electron has $\displaystyle g=2$. The interaction
with photons shifts $\displaystyle g$, resulting in the famous notion of the
electron anomaly, $\displaystyle a_{e}=(g_{e}-2)/2\neq 0$, which can be
considered as a measure of the magnetic field surrounding the electron. The
effect of shifts in $\displaystyle g$ is also inherent in muons and tau
leptons. Albeit this anomaly is rather small, its study is of great importance
since the systematic discrepancy between the model-based theoretical
calculations and the experimentally measured giromagnetic factor may indicate
possible limitations of the Standard Model (SM) or possible existence of as
yet undiscovered particles belonging to SM, however giving rise to some “new
physics”. Besides interactions with virtual photons, the lepton giromagnetic
factor can acquire corrections from hadronic vacuum polarization (HVP) and
light by light and electroweak scattering review-2021 ; Jegerlehner:2017gek .
Nowadays, experimental measurements of electron and muon anomalies
Parker:2018vye ; E989 ; E821  are performed with an impressive precision,
which allows for meticulous comparison of data with theoretical predictions,
cf. Refs. Davoudiasl:2018fbb ; Nomura ; Keshavarzi:2020bfy ; Malaescu .

The pure QED contribution is by far the largest and can be calculated
numerically with an accuracy compatible with the errors of experimental
measurements. The HVP mechanism, due to the well-known difficulties of the
nonperturbative quantum chromodynamics (QCD), is much more complicated for
theoretical analysis and hence represents a leading contributor to
uncertainties in theoretical predictions. Electroweak corrections are
suppressed by the $\displaystyle Z-$bozon mass as $\displaystyle m_{L}/M_{Z}$
and can contribute only at the seventh significant digit in calculations and
are of the same order as corrections from the diagrams with light by light
scattering. A comprehensive analysis of the role of different mechanisms in to
the lepton anomaly can be found, e.g. in Ref. review-2021 . The net result is
that with adding together all the mentioned mechanisms, the deviation of
theoretical calculations from experimental data for electrons
$\displaystyle\Delta a_{e}\equiv
a_{e}^{\mathrm{exp}}-a_{e}^{\mathrm{SM}}\simeq(-87\pm 36)\times 10^{-14}$ is
as large as $\displaystyle\sim 2.5\,$ standard deviations
$\displaystyle\sigma$, see Refs. Parker:2018vye ; Davoudiasl:2018fbb ,
whereas, according to the most recent measurements of the E989 experiment at
Fermilab E989  together with previous measurements of the muon anomaly of the
E821 experiment at BNL E821 , the muon deviation is found to be
$\displaystyle\Delta
a_{\mu}=a_{\mu}^{\mathrm{exp}}-a_{\mu}^{\mathrm{SM}}=(251\pm 59)\times
10^{-11}\sim 4.2\,\sigma\,$. It should be stressed that the contribution of
the HVP mechanism is usually estimated either using the dispersive technique
combined with “$\displaystyle e^{+}e^{-}\to hadrons$” cross section data, see
Refs. Malaescu ; Nomura ; HJEP ; Kubis , or using cumbersome and lengthy
lattice QCD calculations, cf., Ref. lattice  (for more detailed discussions,
see Ref. Jul2022  and references therein). If we rely on the latest lattice
calculations of the HVP corrections, the discrepancy $\displaystyle\Delta
a_{\mu}$ becomes reduced up to $\displaystyle\sim 1.5\ \sigma$. Nevertheless,
it is essential that this result be confirmed by independent lattice
calculations prospects ; Kuberski . Note, that the systematic theoretical
results overestimates of the electron anomaly and underestimates of the muon
anomaly. These circumstances motivate future experimental investigations of
lepton magnetic moments, viz., at Fermilab Grange:2015fou  and J-PARC
Iinuma:2011zz  and renew interest in improving the accuracy of theoretical
calculations within the mentioned mechanisms.

In the present paper, we consider corrections solely from the pure QED
contributions from a subset of Feynman diagrams that allow one to obtain close
analytical expressions for the lowest order, up to the fourth, in the fine
structure constant $\displaystyle\alpha$. The very first calculations of the
radiative corrections to the electron giromagnetic factor were performed by J.
S. Schwinger Schwinger1948  who obtained the electron anomaly to be
$\displaystyle\alpha/(2\pi)\simeq 0.00116$ in excellent agreement with the
experimental data, available at that time. Further increase in measurement
accuracy requires more refined theoretical calculations of the QED
contribution, e.g., corrections of the eighth-, tenth- and higher order w.r.t.
electromagnetic coupling constant. So far, higher order analysis has been
based mainly on either approximate asymptotic expansion of the corresponding
Feynman diagrams or more accurate but cumbersome and computer time consuming
numerical calculations, cf. Refs. Jegerlehner:2017gek ; Kinoshita-1990 ;
Laporta:1993ju ; Laporta:1993ds ; Kinoshita:2005zr ; Laporta:2017okg ;
Aguilar:2008qj ; Kurz:2013exa ; Kurz:2016bau ; Baikov ; Marquard:2017iib  and
references therein. However, as far as we know, the corresponding higher order
exact analytic expressions, which can serve as serious tests for both
asymptotic formulas and numerical results, have not been presented in the
literature. In this context, it is rather appealing to separate from the full
set of diagrams of a given order at least a subset that permits one to perform
calculations in an explicit analytical form. This type subset consists
exclusively of diagrams with insertions of the photon polarization operator
with at least three/four closed lepton loops, the so-called bubble-like
diagrams. Certainly, the explicit expressions are more attractive since they
allow performing calculations with any desired accuracy and also testing the
reliability of asymptotic expansions and numerical evaluation of the
corresponding integrals.

In this paper, we consider this type of QED Feynman diagrams with insertions
of the photon polarization operator and derive analytical expressions for the
radiative corrections up to the eighth order to anomalous magnetic moments of
leptons. In this sense, the paper can be viewed as a generalization of the
results previously reported in the literature concerning mainly muons, cf.
Refs. Laporta:1993ju ; Laporta:1993ds ; Aguilar:2008qj ; Friot:2005cu , to all
types of leptons, $\displaystyle e,\mu$ and $\displaystyle\tau$ and to the
whole region of the mass ratio, $\displaystyle 0<\,m_{\ell}/m_{L}\,<\,\infty$,
where $\displaystyle m_{L}$ and $\displaystyle m_{\ell}$ denote the mass of
the considered lepton $\displaystyle L$ and the mass of the loop leptons,
respectively. The considered diagrams refer to any lepton and include all
possible combinations of leptons in the polarization operator with maximum
three loops formed by two different or three identical leptons.

The paper is organized as follows. In order to facilitate the reading of the
paper, in Section II we briefly recall the main definitions relevant to
calculations of the lepton anomaly. The general relation between the anomaly
induced by the polarization operator of the virtual photon with an arbitrary
number of closed lepton loops and the anomaly due to the exchange of a single
but massive photon is established. Section III is dedicated to calculations of
the Feynman bubble-like diagrams of any order and applications of dispersion
relations and the Mellin-Barnes transform to the corresponding $\displaystyle
x-$parametrizations of Feynman integrals. The theoretical approach is
basically the one developed by E. de Rafael and coauthors Friot:2005cu ;
Aguilar:2008qj  for investigations of the muon anomaly. Previously, a similar
approach was applied to calculate analytically the anomaly of muons up to the
sixth order w.r.t. the electron charge in Ref. Laporta:1993ju . Here we
generalize the method to any type of leptons with all possible insertions in
the polarization operator to determine analytically radiative corrections up
to the eighth and, possibly, higher order. Sections IV-VI are entirely devoted
to establishing analytical expressions for radiative corrections from diagrams
with one, two and three loop insertions, respectively. The radiative
corrections are expressed in terms of the mass-ratios $\displaystyle
r=m_{\ell}/m_{L}$ of the internal to external masses of the concerned leptons.
Particular attention is paid to the determination, for each type of diagrams,
of the generic analytical function valid in the whole interval of
$\displaystyle r$ which, as shown below, represents an analytical continuation
of the corresponding correction derived separately for $\displaystyle r<1$ and
$\displaystyle r>1$. Once such functions are determined explicitly, one can
perform numerical calculations for the corrections of a given order with any
desired precision. In Section VII, we present a qualitative numerical analysis
of the obtained analytical expressions for the corrections up to the eighth
order. We investigate the dependence of corrections originating from all
possible types of loop insertions as a function of the variable $\displaystyle
r$. We argue that in the Feynman diagram the contribution of the lepton loops
decreases with increasing mass of the internal leptons. The hierarchy of three
loop diagrams is determined (numerically) for each type of the considered
leptons. In Subsection VII.1, we compare our results with the known
asymptotical expansions at $\displaystyle r\ll 1$ reported in the literature
and augment them by the asymptotics of $\displaystyle A_{2}^{(8)}(r)$ for
$\displaystyle r\gg 1$, not yet considered hitherto. Summary and conclusions
are collected in Section VIII. Eventually, some useful relations among the
relevant special functions which can facilitate comparisons with previous
results obtained by different authors are presented in Appendix A.

##  II Lepton anomaly and radiative corrections

In order to investigate the magnetic property of the lepton $\displaystyle L$
(electron, muon or tau-lepton), one considers the scattering of the lepton
$\displaystyle L$ in an external magnetic field $\displaystyle
A_{\mu}^{(ext)}(q^{2})$. The corresponding scattering amplitude is stipulated
by only two scalar functions $\displaystyle F_{1}(q^{2})$ and $\displaystyle
F_{2}(q^{2})$:

$\displaystyle\displaystyle
T(p_{1},p_{2})=e\bar{u}(p_{2})\Gamma_{\mu}(p_{1},p_{2})u(p_{1})A^{\mu(ext)}(q^{2})=e\bar{u}(p_{2})\left[\gamma_{\mu}F_{1}(q^{2})+i\frac{\sigma_{\mu\nu}q^{\nu}}{2m_{L}}F_{2}(q^{2})\right]u(p_{1})A^{\mu(ext)}(q^{2}),$
(1)

where the initial and final lepton momenta are $\displaystyle p_{1}$ and
$\displaystyle p_{2}$, respectively, $\displaystyle q=p_{2}-p_{1}$,
$\displaystyle A^{(ext)}=A^{(ext)}(0,{\bf A})$ with $\displaystyle{\bf B}={\rm
rot}\ {\bf A}$ and the electromagnetic vertex $\displaystyle\Gamma_{\mu}$ is

$\displaystyle\displaystyle\Gamma_{\mu}=\left[\gamma_{\mu}F_{1}(q^{2})+i\frac{\sigma_{\mu\nu}q^{\nu}}{2m_{L}}F_{2}(q^{2})\right].$
(2)

The Gordon identity

$\displaystyle\displaystyle\bar{u}(p_{2})\gamma^{\mu}u(p_{1})=\frac{1}{2m_{L}}\bar{u}(p_{2})\left[(p_{1}+p_{2})^{\mu}+i\sigma^{\mu\nu}q_{\nu}\right]u(p_{1})$
(3)

allows one to rewrite the scattering amplitude (1) as

$\displaystyle\displaystyle
T(p_{1},p_{2})=e\bar{u}(p_{2})\Gamma_{\mu}u(p_{1})A^{\mu(ext)}(q^{2}).$ (4)

with

$\displaystyle\displaystyle\Gamma_{\mu}=\left[\frac{(p_{1}+p_{2})^{\mu}}{2m_{L}}F_{1}(q^{2})+i\frac{\sigma_{\mu\nu}q^{\nu}}{2m_{L}}\left(F_{1}(q^{2})+F_{2}(q^{2})\right)\right].$
(5)

Obviously, for the on shell leptons, Eqs. (2) and (5) are completely
equivalent. In the non-relativistic and static limit of $\displaystyle q\to
0$, only $\displaystyle\sigma_{ij}$, ($\displaystyle ij=1,2,3$ ) contributes
to $\displaystyle\Gamma_{\mu}$ and the second term in (5) reduces to

$\displaystyle\displaystyle-\frac{e}{2m}(1+F_{2}(0)){{\psi}^{+}}{\mathchar
28955\relax}\cdot{\bf B}\;{\psi}\to-g_{L}{\mathchar 28950\relax}\,{\bf B},$
(6)

where $\displaystyle{\mathchar
28950\relax}=\left(\displaystyle\frac{e}{2m}{{\bf s}}\right)$ is the magnetic
moment of an elementary particle with spin $\displaystyle{\bf s}={\mathchar
28955\relax}/2$ and the gyromagnetic ratio $\displaystyle g_{L}=2(1+F_{2}(0))$
is the measure of the magnetic anomaly of the lepton. In practice, it is more
convenient to consider another quantity,

$\displaystyle\displaystyle a_{L}=\displaystyle\frac{g_{L}-2}{2}=F_{2}(0).$
(7)

It is therefore clear that to find the lepton magnetic anomaly $\displaystyle
a_{L}$ one should express the calculated (fully dressed) electromagnetic
vertex $\displaystyle\Gamma_{\mu}$ in the form of Eq. (2) or (5), take the
limit $\displaystyle q\to 0$ and find the coefficient in front of the operator
$\displaystyle i{\sigma_{\mu\nu}q^{\nu}}/{2m_{L}}$. An alternative and rather
elegant method for determining $\displaystyle a_{L}$ consists in employing a
properly defined projection operator $\displaystyle{\cal P}_{\mu}$ that,
acting on $\displaystyle\Gamma_{\mu}$, separates $\displaystyle a_{L}$ and,
consequently, the gyromagnetic factor $\displaystyle g_{L}$. For instance, one
can define the following projection operator proj :

$\displaystyle\displaystyle{\cal
P}_{\mu}=\frac{1}{Q^{2}}(\gamma_{\mu}\hat{q}+p_{\mu}\hat{q}/m_{L}-q_{\mu})(\hat{p}+m_{L})+\frac{1}{3}\gamma_{\mu}-\frac{p_{\mu}}{m_{L}}-\frac{4}{3}p_{\mu}\frac{\hat{p}}{m_{L}^{2}},$
(8)

where $\displaystyle p=(p_{1}+p_{2})/2$ and $\displaystyle q=(p_{2}-p_{1})$
with $\displaystyle(p\cdot q)=0$. Then

$\displaystyle\displaystyle a_{L}=\lim_{q\to 0}\left[\frac{1}{4}Tr\left({\cal
P}^{\mu}\Gamma_{\mu}\right)\right].$ (9)

Theoretically, to calculate the anomalous magnetic moment of a lepton up to
the desired order, one should consider radiative corrections to the
electromagnetic vertex $\displaystyle\Gamma_{\mu}(p_{1},p_{2})$ and fold the
obtained expression with the projection operator Eq. (8) and use Eq. (9). In
the present work, we consider the radiative corrections due to Feynman
diagrams with insertions in to the virtual photon propagator of the vacuum
polarization operator. The corresponding diagram is depicted in Fig. 1, left
panel, where the photon propagator is

Figure 1: Radiative corrections to the lepton electromagnetic vertex due to
vacuum polarization insertions, left panel, and the equivalent diagram with
vacuum polarization of a massive photon, right panel.

$\displaystyle\displaystyle
D_{\alpha\beta}(k^{2})=-ig_{\alpha\beta}\frac{1}{k^{2}}\frac{1}{1+\Pi(k^{2})}=$
$\displaystyle\displaystyle-
ig_{\alpha\beta}\frac{1}{k^{2}}\left[1-\left(\Pi(k^{2})+\Pi^{2}(k^{2})-\Pi^{3}(k^{2})+\cdots\right)\right]\equiv-
ig_{\alpha\beta}\frac{1}{k^{2}}\left[1-\widetilde{\Pi}(k^{2})\right],$ (10)

where $\displaystyle\widetilde{\Pi}(k^{2})$ is the full polarization operator
of the virtual photon. Explicitly, the sought vertex function
$\displaystyle\Gamma_{\mu}(p_{1},p_{2})$ is

$\displaystyle\displaystyle\Gamma_{\mu}(p_{1},p_{2})=-ie\frac{e^{2}}{(2\pi)^{4}}\int
d^{4}k\gamma_{\alpha}\frac{(\hat{p_{2}}-\hat{k}+m_{L})\gamma_{\mu}(\hat{p_{1}}-\hat{k}+m_{L})}{(k^{2}-2p_{2}k)(k^{2}-2p_{1}k)}\gamma_{\alpha}\frac{\widetilde{\Pi}(k^{2})}{k^{2}}.$
(11)

The next step is to apply the dispersion relations to the operator
$\displaystyle-\widetilde{\Pi}(k^{2})/k^{2}$,

$\displaystyle\displaystyle\Gamma_{\mu}(p_{1},p_{2})=-ie\frac{e^{2}}{(2\pi)^{4}}\int\frac{dt}{t}\frac{1}{\pi}\frac{\mbox{\rm
Im}\widetilde{\Pi}(t)}{k^{2}-t}\int
d^{4}k\gamma_{\alpha}\frac{(\hat{p_{2}}-\hat{k}+m_{L})\gamma_{\mu}(\hat{p_{1}}-\hat{k}+m_{L})}{(k^{2}-2p_{2}k)(k^{2}-2p_{1}k)}=$
$\displaystyle\displaystyle\frac{1}{\pi}\int\frac{dt}{t}{\mbox{\rm
Im}}\widetilde{\Pi}(t)\left[-ie\frac{e^{2}}{(2\pi)^{4}}\int
d^{4}k\gamma_{\alpha}\frac{(\hat{p_{2}}-\hat{k}+m_{L})\gamma_{\mu}(\hat{p_{1}}-\hat{k}+m_{L})}{(k^{2}-2p_{2}k)(k^{2}-2p_{1}k)}\gamma_{\alpha}\frac{1}{k^{2}-t}\right],$
(12)

where the expression in square brackets is nothing but the second order
correction to the vertex $\displaystyle\Gamma_{\mu}^{(2)}(p_{1},p_{2},t)$ from
diagrams with the exchange of one massive photon with the mass $\displaystyle
m_{\gamma}^{2}=t$, see right panel in Fig. 1. Then

$\displaystyle\displaystyle\Gamma_{\mu}(p_{1},p_{2})=\frac{1}{\pi}\int\frac{dt}{t}\mbox{\rm
Im}\widetilde{\Pi}(t)\Gamma_{\mu}^{(2)}(p_{1},p_{2},t).$ (13)

In such a way, the corrections to the electromagnetic vertex
$\displaystyle\Gamma_{\mu}$ of an arbitrary order can be expressed via the
polarization operator $\displaystyle\tilde{\Pi}(t)$ and the second order
electromagnetic vertex of a massive photon. This implies that the anomalous
magnetic moment $\displaystyle a_{L}$ is also determined by the second order
anomalous magnetic moment $\displaystyle a_{L}(t)$ of a massive
($\displaystyle m_{\gamma}^{2}=t$) photon folded with the polarization
operator $\displaystyle\mbox{\rm Im}\,\tilde{\Pi}(t)$. The former is known
explicitly berestetski ; brodski  and one can write

$\displaystyle\displaystyle a_{L}$
$\displaystyle\displaystyle=\frac{1}{\pi}\int\frac{dt}{t}\mbox{\rm
Im}\widetilde{\Pi}(t)F_{2}^{(2)}(t)=\frac{1}{\pi}\int\frac{dt}{t}\mbox{\rm
Im}\widetilde{\Pi}(k^{2})\frac{\alpha}{\pi}\int
dx\frac{x^{2}(1-x)}{x^{2}+(1-x)t/m_{L}^{2}}=$ (14)
$\displaystyle\displaystyle-\frac{\alpha}{\pi}\int
dx(1-x)\widetilde{\Pi}(q_{eff}^{2}),$

where $\displaystyle\alpha=e^{2}/4\pi$ is the fine structure constant and the
effective momentum $\displaystyle q_{eff}$ is defined as

$\displaystyle\displaystyle q_{eff}^{2}=-\frac{x^{2}}{1-x}m_{L}^{2}.$ (15)

Note the Euclidean nature of $\displaystyle q_{eff}^{2}<0$ and that

$\displaystyle\widetilde{\Pi}(k^{2})=\Pi(k^{2})-\Pi^{2}(k^{2})+\Pi^{3}(k^{2})-\cdots\,.$

##  III Basic formalism: Mellin-Barnes integral representations

Although Eq. (14) can be considered as the final expression for numerical
calculations of $\displaystyle a_{L}$ up to the desired order, in this paper
we focus on revealing the prerequisites for explicit analytical expressions
for $\displaystyle a_{L}$. The general method of analytical consideration of
$\displaystyle a_{L}$ was presented in Refs. Friot:2005cu ; Aguilar:2008qj ;
Friot:2011ic ; Rafael-HVP  where the sixth-order corrections to the muon
anomaly were examined in some detail. Below we use the method for any type of
lepton and apply it to find analytically the corresponding corrections up to
the eighth order. As in Ref. Aguilar:2008qj , we consider diagrams with lepton
loops of the polarization operator consisting of two different leptons, i.e.
diagrams for which
$\displaystyle\Pi(k^{2})=\left(\Pi^{(l_{1})}(k^{2})+\Pi^{(l_{2})}(k^{2})\right)$.
Moreover, one of the internal leptons is chosen to be of the same kind as the
external one. The case of identical internal loops is included as well. The
case of diagrams depending on three masses, $\displaystyle m_{e}$,
$\displaystyle m_{\mu}$ and $\displaystyle m_{\tau}$, or, equivalently, on two
mass ratios $\displaystyle r_{1}$ and $\displaystyle r_{2}$, are more
complicated for analytical calculations. The first exact expressions for this
case were obtained only for the sixth order diagrams Ananthanarayan:2020acj ,
i.e. for diagrams with two different internal loops other than the external
lepton. An analytical analysis of such corrections will be represented
elsewhere.

Generally, the polarization operator for Feynman diagrams with insertions of
$\displaystyle n=p+j$ closed loops, where $\displaystyle p$ and $\displaystyle
j$ denote the number of leptons loops of $\displaystyle\ell_{1}$ and
$\displaystyle\ell_{2}$ kinds, can be presented as

$\displaystyle\displaystyle\Pi^{n}(k^{2})=\sum\limits_{p=0}^{n}(-1)^{n+1}C_{n}^{p}\left[\Pi^{(\ell_{1})}(k^{2})\right]^{p}\left[\Pi^{(\ell_{2})}(k^{2})\right]^{j=n-p}\equiv\sum\limits_{p=0}^{n}F_{(p,j)}\left[\Pi^{(\ell_{1})}(k^{2})\right]^{p}\left[\Pi^{(\ell_{2})}(k^{2})\right]^{j=n-p},$
(16)

where $\displaystyle C_{n}^{p}$ are the combinatorial coefficients and the
quantity $\displaystyle F_{(p,j)}=(-1)^{p+j+1}C_{p+j}^{p}$ has been introduced
as to reconcile our formulae with the commonly adopted notation, cf. Ref.
Aguilar:2008qj . Note that $\displaystyle p$ and $\displaystyle j$ enter
symmetrically in the polarization operator $\displaystyle\Pi^{n}$, i.e. one
can interchange $\displaystyle\ell_{1}\leftrightarrow\ell_{2}$ in Eq. (16). In
what follows we choose the leptons $\displaystyle\ell_{1}$ to be of the same
kind as the external one. Consequently, for the sake of brevity, below we
release the labels $\displaystyle\ell_{1}$, $\displaystyle\ell_{2}$ for the
loop leptons and use merely the notation $\displaystyle\ell_{1}=L$, and
$\displaystyle\ell_{2}=\ell$. Then the contribution to the anomalous magnetic
moment from diagrams with $\displaystyle p$ leptons of type $\displaystyle L$
and $\displaystyle j$ leptons of type $\displaystyle\ell$ (c.f. Eqs. (14) and
(16)) acquires the form

$\displaystyle\displaystyle
a_{L}^{(p,j)}=\frac{\alpha}{\pi}F_{(p,j)}\int_{0}^{1}dx\;x^{2}(1-x)\left[\Pi^{(L)}(q_{eff}^{2})\right]^{p}\frac{1}{\pi}\int_{0}^{\infty}\frac{dt}{t}\frac{\mbox{\rm
Im}\left[\Pi^{(\ell)}(t)\right]^{j}}{x^{2}+(1-x)t/m_{L}^{2}}=$
$\displaystyle\displaystyle
F_{(p,j)}\frac{\alpha}{\pi}\int\limits_{0}^{\infty}\frac{dt}{t}\int\limits_{0}^{1}dx\frac{x^{2}(1-x)}{x^{2}+(1-x)t/m_{L}^{2}}\left[\Pi^{(L)}\left(-\frac{x^{2}}{1-x)}m_{L}^{2}\right)\right]^{p}\frac{1}{\pi}\;\mbox{\rm
Im}\left[\Pi^{(\ell)}(t)\right]^{j},$ (17)

where, as mentioned, the external lepton is labeled as $\displaystyle L$. This
is the well known expression obtained by E. de Rafael and coauthors
Aguilar:2008qj  for the muon anomaly governed by the polarization operator
with $\displaystyle n$ closed loops of two different leptons. Usually one
introduces a new notation

$\displaystyle\displaystyle\rho_{j}\left(\frac{4m_{\ell}^{2}}{t}\right)=\frac{1}{\pi}\mbox{\rm
Im}\left[\Pi^{(\ell)}(t)\right]^{j}\,,$ (18)

which is inspired by the fact that actually the polarization operator depends
rather on the combination $\displaystyle 4m^{2}/t$ than solely on
$\displaystyle t$.

Equation (17) is the main expression suitable for further analytical
calculations of $\displaystyle a_{L}$ which will allow one to obtain
$\displaystyle a_{L}$ with the precision as high as permitted by the knowledge
of the fundamental constants entering into $\displaystyle a_{L}$, viz., the
fine structure constant and lepton masses $\displaystyle m_{e}$,
$\displaystyle m_{\mu}$ and $\displaystyle m_{\tau}$. To this end, let us
consider the known Mellin-Barnes integral representation for propagator-like
functions of a massive scalar particle, cf. Ref. boos . One starts with the
known integral representation of the Euler beta-function

$B(s,\xi)=\int\limits_{0}^{\infty}dx\;\frac{x^{s-1}}{(1+x)^{s+\xi}}\,,$ (19)

which, for a particular choice of the variables $\displaystyle s$ and
$\displaystyle\xi$, namely for $\displaystyle\xi=\beta-s$, reads as

$B(s,\beta-s)=\int\limits_{0}^{\infty}dx\;\frac{x^{s-1}}{(1+x)^{\beta}}=\int\limits_{0}^{\infty}dx\;x^{s-1}F(x,\beta),$
(20)

i.e., it can be considered as the Mellin transform of the function

$\displaystyle\displaystyle F(x,\beta)=\displaystyle\frac{1}{(1+x)^{\beta}}.$
(21)

Then, the inverse Mellin transform of $\displaystyle F(x,\beta)$ is

$F(x,\beta)=\displaystyle\frac{1}{(1+x)^{\beta}}=\frac{1}{2\pi
i}\int\limits_{c-i\infty}^{c+i\infty}ds\ x^{-s}\ B(s,\beta-s)=\frac{1}{2\pi
i}\int\limits_{c-i\infty}^{c+i\infty}ds\ x^{-s}\ \frac{\Gamma(s)\
\Gamma(\beta-s)}{\Gamma(\beta)}\,,$ (22)

where $\displaystyle 0<c<\beta$. Equation (22) is known as the Mellin-Barnes
representation for propagator functions. Coming back to Eq. (17), let us apply
it to the integrand in integration over $\displaystyle x$

$\displaystyle\displaystyle\frac{x^{2}(1-x)}{x^{2}+(1-x)t/m_{L}^{2}}=\frac{1}{2\pi
i}\int\limits_{c-i\infty}^{c+i\infty}ds\;\left(\frac{4m_{\ell}^{2}}{t}\right)^{s}\left(\frac{4m_{\ell}^{2}}{m_{L}^{2}}\right)^{-s}x^{2s}(1-x)^{1-s}\;\Gamma(s)\Gamma(1-s),$
(23)

where the quantity $\displaystyle 4m_{\ell}^{2}$ has been introduced for
further convenience. With this representation, Eq. (17) reads as

$\displaystyle\displaystyle a_{L}(p,j)$
$\displaystyle\displaystyle=\frac{\alpha}{\pi}\frac{1}{2\pi
i}F_{(p,j)}\int\limits_{c-i\infty}^{c+i\infty}ds\;\left(\frac{4m_{\ell}^{2}}{m_{L}^{2}}\right)^{-s}\Gamma(s)\Gamma(1-s)\int_{0}^{1}dx\;x^{2s}(1-x)^{1-s}\times$
(24)
$\displaystyle\displaystyle\left[\Pi^{(L}\left(-\frac{x^{2}}{1-x)}m_{L}^{2}\right)\right]^{p}\int_{0}^{\infty}\frac{dt}{t}\left(\frac{4m_{\ell}^{2}}{t}\right)^{s}\rho_{j}\left(\frac{4m_{\ell}^{2}}{t}\right),$

where $\displaystyle 0<c<1$. It can be seen that the Mellin-Barnes transform
made it possible to present the contribution to the lepton anomaly from
different kinds of lepton loops in the following factorized form of two Mellin
momenta

$\displaystyle\displaystyle a_{L}(p,j)=\frac{\alpha}{\pi}\frac{1}{2\pi
i}F_{(p,j)}\int\limits_{c-i\infty}^{c+i\infty}ds\;\left(\frac{4m_{\ell}^{2}}{m_{L}^{2}}\right)^{-s}\Gamma(s)\Gamma(1-s)\;\left(\frac{\alpha}{\pi}\right)^{p}\Omega_{p}(s)\left(\frac{\alpha}{\pi}\right)^{j}R_{j}(s),$
(25)

where

$\displaystyle\displaystyle\left(\frac{\alpha}{\pi}\right)^{p}\Omega_{p}(s)=\int_{0}^{1}dx\;x^{2s}(1-x)^{1-s}\left[\Pi^{(L)}\left(-\frac{x^{2}}{1-x}m_{L}^{2}\right)\right]^{p}$
(26)
$\displaystyle\displaystyle\left(\frac{\alpha}{\pi}\right)^{j}R_{j}(s)=\int_{0}^{\infty}\frac{dt}{t}\left(\frac{4m_{\ell}^{2}}{t}\right)^{s}\rho_{j}\left(\frac{4m_{\ell}^{2}}{t}\right).$
(27)

It is seen from (25)-(27), that to compute $\displaystyle a_{L}$ up to the
$\displaystyle 2(p+j+1)$ order it suffices to calculate separately the
polarization operators $\displaystyle\Pi^{(L)}(q^{2})$ and
$\displaystyle\mbox{\rm Im}\,\Pi^{(\ell)}(q^{2})$. Recall that the generic
variable for the $\displaystyle q^{2}$-dependence of the operator
$\displaystyle\Pi(q^{2})$ is the dimensionless combination $\displaystyle
y=\frac{4m^{2}}{q^{2}}$, i.e. $\displaystyle\Pi^{(L)}(q^{2})=\Pi^{(L)}(y)$. It
implies that the quantity $\displaystyle\Omega_{p}(s)$, for which
$\displaystyle y=-4(1-x)/x^{2}$, does not depend on the lepton mass
$\displaystyle m_{L}$. Furthermore, the term relevant to the second lepton,
$\displaystyle R_{j}(s)$

$\displaystyle\displaystyle\left(\frac{\alpha}{\pi}\right)^{j}R_{j}(s)=\int_{0}^{\infty}\frac{dt}{t}\left(\frac{4m_{\ell}^{2}}{t}\right)^{s}\rho_{j}\left(\frac{4m_{\ell}^{2}}{t}\right)=\int_{0}^{\infty}d\xi\;\xi^{s-1}\rho_{j}(\xi)$
(28)

is mass independent as well. Hence, the mass-dependence in Eq. (25) enters
only via the ratio $\displaystyle r=\frac{m_{\ell}}{m_{L}}.$ Therefore, it is
commonly adopted to classify the contribution to the anomalous magnetic moment
$\displaystyle a_{L}$ by this ratio

$a_{L}=~{}A_{1}\left(\frac{m_{L}}{m_{L}}\right)+~{}A_{2}\left(\frac{m_{\ell}}{m_{L}}\right)+A_{3}\left(\frac{m_{\ell}}{m_{L}},\frac{m_{\ell_{2}}}{m_{L}}\right),$
(29)

where $\displaystyle A_{1}$ corresponds to diagrams for which all the internal
loops are formed by the same leptons as the external one. It also includes the
case of diagrams without lepton loops, i.e. the diagrams of the second order
with respect to the electromagnetic coupling $\displaystyle e$ with exchange
of only one virtual photon. Clearly, the coefficients $\displaystyle A_{1}$
are universal for all kinds of leptons. The coefficients $\displaystyle A_{2}$
correspond to diagrams with either two different types of internal loops with
at least one coinciding with the external lepton $\displaystyle L$ and the
other $\displaystyle\ell\neq L$, or with loops all formed by identical leptons
of type $\displaystyle\ell$, $\displaystyle\ell\neq L$. Eventually,
$\displaystyle A_{3}$ correspond to diagrams with loops formed by maximum
three kinds of leptons with at least two non identical and different from the
external lepton, $\displaystyle l\neq l_{2}\neq L$. Consequently, in
dependence of the number of insertions of the polarization operators, each
coefficient $\displaystyle A_{i}$ can be classified by the corresponding
number of bubble-like loops $\displaystyle n$ ($\displaystyle n=0,1...$) or,
equivalently, by the $\displaystyle(n+1)$-th power of $\displaystyle\alpha$
and computed order by order.

$\displaystyle\displaystyle
A_{1}(m_{L}/m_{L})=A_{1}^{(2)}\left(\frac{\alpha}{\pi}\right)^{1}+A_{1}^{(4)}\left(\frac{\alpha}{\pi}\right)^{2}+A_{1}^{(6)}\left(\frac{\alpha}{\pi}\right)^{3}+\cdots,$
(30) $\displaystyle\displaystyle
A_{2}\left(r={m_{\ell}}/{m_{L}}\right)=A_{2}^{(4)}(r)\left(\frac{\alpha}{\pi}\right)^{2}+A_{2}^{(6)}(r)\left(\frac{\alpha}{\pi}\right)^{3}+A_{2}^{(8)}(r)\left(\frac{\alpha}{\pi}\right)^{4}+\cdots,$
(31) $\displaystyle\displaystyle
A_{3}\left(r_{1},r_{2}\right)=A_{3}^{(6)}(r_{1},r_{2})\left(\frac{\alpha}{\pi}\right)^{3}+A_{3}^{(8)}(r_{1},r_{2})\left(\frac{\alpha}{\pi}\right)^{4}+A_{3}^{(10)}(r_{1},r_{2})\left(\frac{\alpha}{\pi}\right)^{5}+\cdots,~{}~{}~{}$
(32)

where $\displaystyle
r_{1}={m_{\ell_{1}}}/{M_{L}},~{}r_{2}={m_{\ell_{2}}}/{M_{L}}$ and the
superscripts of the coefficient in the r.h.s. indicate the corresponding order
of the radiative corrections, while the powers of $\displaystyle\alpha^{n+1}$
correspond to the number $\displaystyle n$ of loops in the bubble-like Feynman
diagrams. N.B: The diagram without loops ($\displaystyle n=0$), as mentioned,
was calculated explicitly by Schwinger Schwinger1948  and found to be equal to
$\displaystyle\alpha/2\pi$, i.e., in our notation $\displaystyle
A_{1}^{(2)}=1/2$. Moreover, it can be seen form Eqs. (17), (26) and (30) that
each of the coefficients $\displaystyle A_{1}$ in (30) is uniquely determined
by the corresponding value of $\displaystyle\Omega_{p}(s=0)$. For instance,
for $\displaystyle n=0$ one has
$\displaystyle\Omega_{0}(s=0)=A_{1}^{(2)}=\frac{1}{2}$ which is exactly the
normalization of (29) to the Schwinger term $\displaystyle{\alpha}/{2\pi}$.
The mass-independent coefficients $\displaystyle A_{1}$ are less complicated
and have been thoroughly investigated previously; their general explicit
analytical expressions can be found in Refs. Samuel-n-bubble ;
Jegerlehner:2017gek ; Petermann:1957hs ; Sommerfield:1957zz ; Laporta:1996mq ;
Laporta:2018ngl .

In the present paper, we consider the radiative corrections up to the eighth
order, i.e. the coefficients $\displaystyle A_{1,2}^{(4)-(8)}(r)$ in Eqs.
(30)-(31). They include diagrams with one, two and three lepton loops. Since
calculations of the coefficients $\displaystyle A_{2}(r)$ presuppose also
explicit calculations of $\displaystyle\Omega_{p}(s)$, the coefficients
$\displaystyle A_{1}$ for which $\displaystyle r=1$ can be easily inferred
from $\displaystyle\Omega_{p}(s\to 0)$ or from $\displaystyle A_{2}(r\to 1)$.

##  IV One lepton loop corrections

Here we present in some detail the derivation of the explicit expression for
the one loop $\displaystyle l\neq L$ corrections. The corresponding diagram is
depicted in Fig. 2, left panel.

Figure 2: One and two loop corrections to the lepton ($\displaystyle L$)
anomalous magnetic moment of the fourth, left panel, and sixth, central and
right panels, order determined by the coefficients $\displaystyle
A^{(4,6)}_{2}(r=m_{l}/m_{L})$ in Eq. (31). Left panel: one loop diagram with
the internal lepton $\displaystyle(\ell)$ different from $\displaystyle L$ for
which $\displaystyle\Omega_{p}(s)=\Omega_{0}(s)$ and $\displaystyle
R_{j}(s)=R_{1}(s)$. Central panel: diagrams with two different
$\displaystyle(Ll)$ leptons for which
$\displaystyle\Omega_{p}(s)=\Omega_{1}(s)$ and $\displaystyle
R_{j}(s)=R_{1}(s)$. Right panel: diagrams with two identical
$\displaystyle(ll)$ leptons ($\displaystyle l\neq L$) for which
$\displaystyle\Omega_{p}(s)=\Omega_{0}(s)$ and $\displaystyle
R_{j}(s)=R_{2}(s)$, cf. Eqs. (26) and (27).

In this case in Eq. (25) $\displaystyle p=0$, $\displaystyle j=1$ and the
corresponding anomalous magnetic moment is

$\displaystyle\displaystyle a_{L}(0,1)=\frac{\alpha}{\pi}\frac{1}{2\pi
i}\int\limits_{c-i\infty}^{c+i\infty}ds\left(4r^{2}\right)^{-s}\Gamma(s)\Gamma(1-s)\;\Omega_{0}(s)\;\left(\frac{\alpha}{\pi}\right)R_{1}(s),$
(33)

where, according to Eqs. (26) and (27)

$\displaystyle\displaystyle\Omega_{0}(s)=\int_{0}^{1}dx\;x^{2s}(1-x)^{1-s}=\frac{\Gamma(2-s)\Gamma(1+2s)}{\Gamma(3+s)},$
(34)
$\displaystyle\displaystyle\left(\frac{\alpha}{\pi}\right)R_{1}(s)=\int_{0}^{\infty}\frac{dt}{t}\left(\frac{4m_{l}^{2}}{t}\right)^{s}\rho_{1}\left(\frac{4m_{l}^{2}}{t}\right).$
(35)

The polarization operator of photons $\displaystyle\Pi^{(\ell)}(q^{2})$ in QED
is known explicitly and reported in a series of publications, see e.g. Refs.
Jegerlehner:2017gek ; Aguilar:2008qj . With
$\displaystyle\delta=\sqrt{1-{4m_{\ell}^{2}}/{q^{2}}}$, it reads as

$\displaystyle\displaystyle\mbox{\rm
Re}\;\Pi^{(\ell)}(q^{2})=\left(\frac{\alpha}{\pi}\right)\left[\frac{8}{9}-\frac{\delta^{2}}{3}+\delta\left(\frac{1}{2}-\frac{\delta^{2}}{6}\right)\ln\frac{|1-\delta|}{1+\delta}\right],$
(36) $\displaystyle\displaystyle\frac{1}{\pi}\mbox{\rm
Im}\;\Pi^{(\ell)}(q^{2})=\left(\frac{\alpha}{\pi}\right)\delta\left(\frac{1}{2}-\frac{1}{6}\delta^{2}\right)\theta\left(q^{2}-4m_{\ell}^{2}\right).$
(37)

Evidently, since the Euclidean nature of the effective momentum $\displaystyle
q_{eff}$, c.f. Eq. (15), and due to the presence of
$\displaystyle\theta\left(q^{2}-4m_{\ell}^{2}\right)$ in (37), the
polarization operator $\displaystyle\Pi(q_{eff}^{2})$ in (26) is pure real and
can be written as Lautrup77 ,

$\displaystyle\displaystyle\Pi^{(L)}(q_{eff}^{2})=\frac{\alpha}{\pi}f(x),\,\,{\rm
with}\,\,f(x)=\frac{5}{9}+\frac{4}{3x}-\frac{4}{3x^{2}}+\left(-\frac{1}{3}+\frac{2}{x^{2}}-\frac{4}{3x^{3}}\right)\ln(1-x).$
(38)

Consequently $\displaystyle\Omega_{p}(s)$ in (26) is also pure real. Yet, the
function $\displaystyle\theta\left(q^{2}-4m_{\ell}^{2}\right)$ in (37)
restricts integrations over $\displaystyle t$ in Eq. (28) to the interval
$\displaystyle[4m_{\ell}^{2}\ ,\ \infty]$, which is converted in to
$\displaystyle\xi\in[0\ ,\ 1]$ in Eqs. (28) and (35). Then, direct calculation
of the integral (35) provides

$\displaystyle\displaystyle
R_{1}(s)=\frac{\sqrt{\pi}}{~{}4}\frac{1}{s}\frac{~{}\Gamma(2+s)}{\Gamma(5/2+s)}.$
(39)

Inserting Eqs. (34) and (39) into Eq. (33), the coefficient $\displaystyle
A_{2}^{(4)}(r)$ reads as

$\displaystyle\displaystyle A_{2}^{(4)}(r)=\frac{1}{2\pi
i}\int\limits_{c-i\infty}^{c+i\infty}ds\;\left(4r^{2}\right)^{-s}\Gamma(s)\Gamma(1-s)\frac{\Gamma(2-s)\Gamma(1+2s)}{\Gamma(3+s)}\frac{\sqrt{\pi}}{4}\frac{1}{s}\frac{\Gamma(2+s)}{\Gamma(5/2+s)}.$
(40)

It is worth emphasizing that in Eq (33) the ratio $\displaystyle r$ acts as an
external parameter so that the integral, if exists, defines $\displaystyle
A_{2}^{(4)}(r)$ as an analytical function of $\displaystyle r$ in the whole
interval of $\displaystyle r\in(0,\infty)$. Its explicit expression can be
found directly by integrating over $\displaystyle s$ in (40), which can be
done, e.g. by the Cauchy residue theorem. Depending on the value of
$\displaystyle r$, one can close the integration contour to the left
$\displaystyle r<1$ or to the right $\displaystyle r>1$ semiplane,
respectively, where the integrand (40) exhibits explicitly pole-like
singularities with known residues. The Cauchy residue theorem yields

$\displaystyle\displaystyle
A_{2}^{(4)}(r<1)=\frac{1}{2}\,r\,\left(5\,{r}^{2}-1\right)\left[{\rm
Li_{2}}\left(r\right)-{\rm Li_{2}}(-r)\right]-\,r^{4}{\rm Li_{2}}(r^{2})\
+2\,{r}^{4}\ln^{2}(r)+$ (41)
$\displaystyle\displaystyle\left\\{\frac{1}{2}\,r\,\left(5\,{r}^{2}-1\right)\left[\phantom{\\!\\!\\!\\!\\!\frac{1}{1}}\ln\left(1-r\right)-\ln\left(1+r\right)\right]-\frac{1}{3}-2\,{r}^{4}\ln\left(1-{r}^{2}\right)+3\,{r}^{2}\right\\}\ln\left(r\right)-$
$\displaystyle\displaystyle~{}{\frac{25}{36}}+\frac{1}{4}\,{\pi}^{2}r-\frac{5}{4}\,{\pi}^{2}{r}^{3}+4\,{r}^{2}+\frac{1}{3}\,{r}^{4}{\pi}^{2}\,,$
$\displaystyle\displaystyle~{}~{}~{}~{}A_{2}^{(4)}(r>1)=\frac{1}{2}\,r\,\left(-5\,{r}^{2}+1\right)\left[{\rm
Li_{2}}\left(1/r\right)-{\rm
Li_{2}}\left(-1/r\right)\right]-\frac{25}{36}+r^{4}{\rm
Li_{2}}\left(1/r^{2}\right)+$ (42)
$\displaystyle\displaystyle~{}~{}~{}~{}\left\\{\frac{1}{2}\,r\,\left(-5\,r^{2}+1\right)\left[\phantom{\\!\\!\\!\\!\\!\frac{1}{1}}\ln(r+1)-\ln(r-1)\right]-\frac{1}{3}+3\,{r}^{2}-2\,{r}^{4}\ln\left(1-1/r^{2}\right)\right\\}\ln\left(r\right)+4\,{r}^{2},$

where $\displaystyle{\rm Li_{2}}$ denotes the dilogarithm function. As seen,
Eqs. (41) and (42) are quite different. However, using the known properties of
complex logarithms and dilogarithms (see Appendix A), one can show that both
equations are identical in the whole region of the parameter $\displaystyle
r\in(0,\infty)$. It means that in practice one can use, for numerical
calculations, any expression (41) or (42) regardless of the value of
$\displaystyle r$. Nevertheless, one shall do it with some caution, noting
that the logarithms in the square brackets in (41) and (42) should not be
converted into one logarithm of the ratios of their arguments. This is because
when using (41) at $\displaystyle r>1$, the complex logarithms
$\displaystyle\ln\left[(1+r)/(1-r)\right]$ and
$\displaystyle\ln(1+r)-\ln(1-r)$ differ by a factor of $\displaystyle 2\pi i$.
It also can be shown that using the known relations among the special
functions (see Appendix A) the analytical results for $\displaystyle
A_{2}^{(4)}(r)$ reported in Ref. Friot:2011ic  and ours, Eqs. (41)-(42), are
identical. The analytical continuation of Eqs. (41) and (42) can be written as
one function valid in the whole region $\displaystyle r\in(0,\infty)$:

$\displaystyle\displaystyle
A_{2}^{(4),(l)}(r)=-\frac{1}{3}\;\ln(r){-\frac{25}{36}}+4\,{r}^{2}+3\;r^{2}\ln(r)+\frac{1}{2}\,r\,\left(1-5\,{r}^{2}\right)\bigg{[}\;{\rm
Li_{2}}\left(\frac{1-r}{1+r}\right)-$ (43) $\displaystyle\displaystyle{\rm
Li_{2}}\left(-\frac{1-r}{1+r}\right)+\frac{1}{4}\pi^{2}\;\bigg{]}+2\left[-{\rm
Li_{2}}\left(\frac{1-r}{1+r}\right)+{\rm
Li_{2}}\left(-\frac{1-r}{1+r}\right)+\frac{1}{12}\pi^{2}\;-\;2\;{\rm
Li_{2}}\left(1-\frac{1}{r}\right)\right]r^{4}\,.$

An analogous expression for the coefficient $\displaystyle A_{2}^{(4)}(r)$, in
a form slightly different from Eq. (43) form, was also reported in Ref.
eidelman .

Furthermore, we complementary checked that Eqs. (41) and (42), as well as Eq.
(43), reproduce the well-known asymptotic expansions 111Notice that the
leading asymptotic terms for $\displaystyle A_{2}^{(4)}(r)$ werte firstly
reported in Refs. suura ; Lautrup:1969fr .

$\displaystyle\displaystyle A_{2}^{(4),(\ell)}(r)=$
$\displaystyle\displaystyle-\frac{1}{3}\;\ln(r)-\frac{25}{36}+\frac{1}{4}\pi^{2}r+\big{[}3+4\ln(r)\big{]}r^{2}-\frac{5}{4}\pi^{2}r^{3}+O(r^{4});\quad
r\ll 1$ (44) $\displaystyle\displaystyle A_{2}^{(4),(\ell)}(r)=$
$\displaystyle\displaystyle\frac{1}{45\;r^{2}}+\left[\frac{9}{19600}-\frac{1}{70}\ln(r)\right]\frac{1}{r^{4}}+O\left(\frac{1}{r^{6}}\right);\quad
r\gg 1.$ (45)

and also the limit

$\displaystyle\displaystyle\lim_{r\to
1}A_{2}^{(4)}(r)=A_{1}^{(4)}=\frac{119}{36}-\frac{\pi^{2}}{3}\,,$ (46)

which exactly coincides with the known result obtained before, see e.g., Ref.
Li:1992xf ; Czarnecki:1998rc .

##  V Two lepton loops

In this section we present calculations of corrections due to two loop
diagrams, as depicted in Fig. 2, central and right panels, according to which
one has two kinds of contributions.

###  V.1 Two identical leptons $\displaystyle(ll)$, $\displaystyle l\neq L$:
Fig. 2, right panel

As follows from expression (25), in this case $\displaystyle p=0$,
$\displaystyle j=2$. Hence the anomaly reads as

$\displaystyle\displaystyle a_{L}(0,2)=-\frac{\alpha}{\pi}\frac{1}{2\pi
i}\int\limits_{c-i\infty}^{c+i\infty}\
ds\left(4r^{2}\right)^{-s}\Gamma(s)\Gamma(1-s)\;\Omega_{0}(s)\;\left(\frac{\alpha}{\pi}\right)^{2}R_{2}(s),$
(47)

where $\displaystyle\Omega_{0}(s)$ is defined by Eq. (34) and $\displaystyle
R_{2}(s)$ is given by Eq. (27) with

$\displaystyle\displaystyle\rho_{2}\left(\frac{4m_{l}^{2}}{t}\right)=\frac{1}{\pi}\left[2\mbox{\rm
Re}\Pi^{(l)}(t)\mbox{\rm Im}\Pi^{(\ell)}(t)\right].$ (48)

Inserting (48), (36) and (37) in to Eq. (27) we get

$\displaystyle\displaystyle
R_{2}(s)=\frac{\sqrt{\pi}}{9}\;\frac{(s-1)(6+13s+4s^{2})}{s^{2}(2+s)(3+s)}\;\frac{\Gamma\left(1+s\right)}{\Gamma\left(\frac{3}{2}+s\right)}\,.$
(49)

Applying in Eqs. (34) and (49) the known properties of the Euler gamma-
functions, e.g. the Euler reflection and the Legendre duplication formulae,
the sixth order contribution to $\displaystyle a_{L}$, Eq. (47), due to
diagrams of the type $\displaystyle(ll)$ reads as

$A_{2}^{(6),(ll)}(r)=\frac{1}{2\pi
i}\int\limits_{c-i\infty}^{c+i\infty}ds\;r^{-2s}\left[\frac{2(1-s)^{2}(6+13s+4s^{2})}{9s(2s+1)(1+s)(2+s)^{2}(3+s)}\right]\frac{\pi^{2}}{\sin^{2}(\pi
s)}\,,$ (50)

which manifests explicitly all singularities of the integrand in the complex
plane of the Mellin variable $\displaystyle s$. Consequently, integration in
(50) can be straightforwardly performed by closing the integration contour in
the left ($\displaystyle r<1$) or right ($\displaystyle r>1$) semiplane. The
analytical expressions for $\displaystyle A_{2}^{(6),(ll)}(r<1)$ and
$\displaystyle A_{2}^{(6),(ll)}(r>1)$ turn to be two branches of one
analytical function $\displaystyle A_{2}^{(6),(ll)}(r)$ valid in the whole
interval $\displaystyle r\in(0,\infty)$,

$\displaystyle\displaystyle
A_{2}^{(6),(ll)}(r)=\frac{2}{3}\;\left({\frac{1}{3}}-4\,r^{2}+5\,r^{4}-{\frac{16}{15}}\,r^{6}\right)\left[-{\rm
Li_{2}}\left(\frac{1-r}{1+r}\right)+{\rm
Li_{2}}\left(-\frac{1-r}{1+r}\right)+\right.$
$\displaystyle\displaystyle\left.\frac{1}{12}\;\pi^{2}-2\;{\rm
Li_{2}}\left(1-\frac{1}{r}\right)\right]\,+\frac{8}{3}\,\left[{\rm
Li_{3}}\,({r}^{2})-\bigg{(}{\rm
Li_{2}}\,({r}^{2})+\frac{1}{3}\;\pi^{2}\bigg{)}\ln(r)-\frac{2}{3}\;\ln^{3}(r)\right]r^{4}-$
$\displaystyle\displaystyle\frac{8}{45}\left[{\rm
Li_{2}}\left(\frac{1-r}{1+r}\right)-{\rm
Li_{2}}\left(-\frac{1-r}{1+r}\right)+\frac{1}{4}\;\pi^{2}\right]r+\frac{317}{324}+\frac{25}{27}\,\ln(r)-\frac{191}{45}\;r^{2}-\frac{254}{45}\,r^{2}\ln\,(r)+$
$\displaystyle\displaystyle\frac{16}{45}\,r^{4}+\frac{32}{45}\,r^{4}\ln(r).$
(51)

Notice that Eq. (51) determines also the mass-independent coefficient
$\displaystyle A_{1}^{(6)}$ in (30) as

$\displaystyle\displaystyle A_{1}^{(6)}=\lim_{r\to
1}A_{2}^{(6),(ll)}(r)=-\frac{943}{324}-\frac{4\pi^{2}}{135}+\frac{8}{3}\;\zeta(3)$
(52)

in agreement with the well-known result (see Refs. Jegerlehner:2017gek ;
Samuel-n-bubble  for more details). Above, in Eq. (52),
$\displaystyle\zeta(3)$ denotes the Euler-Riemann zeta function.

###  V.2 Two different loops $\displaystyle(L\ell)$, Fig. 2, left panel

For diagrams of type $\displaystyle(L\ell)$ one has $\displaystyle p=1$,
$\displaystyle j=1$ and the corresponding Mellin–Barnes representation looks
like

$\displaystyle\displaystyle a_{L}(1,1)=-\frac{\alpha}{\pi}\frac{2}{2\pi
i}\int\limits_{c-i\infty}^{c+i\infty}ds\left(4r^{2}\right)^{-s}\Gamma(s)\Gamma(1-s)\left(\frac{\alpha}{\pi}\right)\Omega_{1}(s)\,\left(\frac{\alpha}{\pi}\right)R_{1}(s),$
(53)

where the quantity $\displaystyle R_{1}(s)$ has already been computed, cf. Eq.
(39), whereas $\displaystyle\Omega_{1}(s)$, Eq. (26), with the polarization
operator (38), reads as

$\displaystyle\displaystyle\Omega_{1}(s)=\int_{0}^{1}dx\;x^{2s}(1-x)^{1-s}f(x)=\bigg{[}\frac{(1-2s)(1+2s)\left(36+54s-29s^{2}-34s^{3}+5s^{4}+4s^{5}\right)}{9(1-s)s(1+s)(2+s)}$
$\displaystyle\displaystyle-2\pi\left(1+s-s^{2}\right)\text{cot}(\pi
s)\bigg{]}\bigg{(}\frac{2^{2s-1}}{\sqrt{\pi}}\bigg{)}{\Gamma\bigg{(}-2-s\bigg{)}}{\Gamma\left(-\frac{1}{2}+s\right)}.$
(54)

It is worth noting that for $\displaystyle\Omega_{1}(s=0)$ one has

$\displaystyle\displaystyle-\Omega_{1}(0)=-\lim_{s\to
0}\Omega_{1}(s)=\frac{119}{36}-\frac{\pi^{2}}{3}\,,$ (55)

which is exactly the limit (46) for the coefficient $\displaystyle
A_{1}^{(4)}$ in (30).

By taking into account the singularities in Eqs. (39) and (54 ) in the complex
plane of the variable $\displaystyle s$, the Cauchy residue theorem provides
the following explicit expression for $\displaystyle A_{2}^{(6),(L\ell)}(r)$

$\displaystyle\displaystyle
A_{2}^{(6),(L\ell)}(r)=\left(\frac{4}{45\,{r}^{2}}+\frac{1}{9}+\frac{4}{3}\,{r}^{2}-\frac{13}{9}\,r^{4}\right)\bigg{[}{\rm
Li_{2}}\left(\frac{1}{{r}^{2}}\right)-2\ln\left(1-\frac{1}{{r}^{2}}\right)\ln(r)\bigg{]}-2\left(1+{r}^{4}\right)\times$
$\displaystyle\displaystyle{\rm
Li_{3}}\left(\frac{1}{{r}^{2}}\right)+\bigg{(}\frac{1}{r}+\frac{2}{3}\,r+\frac{11}{3}\,r^{3}\bigg{)}\bigg{\\{}{\rm
Li}_{3}\left(\frac{1}{r}\right)-{\rm
Li}_{3}\left(-\frac{1}{r}\right)+\bigg{[}{\rm
Li}_{2}\left(\frac{1}{r}\right)-{\rm
Li}_{2}\left(-\frac{1}{r}\right)\bigg{]}\ln(r)+$
$\displaystyle\displaystyle\frac{1}{2}\big{[}\ln(1+r)-\ln(1-r)\big{]}\ln^{2}(r)\bigg{\\}}-\frac{8}{3}\;(1+r^{4})\bigg{[}{\rm
Li_{2}}\left(\frac{1}{{r^{2}}}\right)-\frac{1}{2}\ln\left(1-\frac{1}{r^{2}}\right)\ln(r)\bigg{]}\ln(r)-$
$\displaystyle\displaystyle\frac{8}{45}\;r^{3}\bigg{[}{\rm
Li_{2}}\left(\frac{1}{r}\right)-{\rm
Li_{2}}\left(-\frac{1}{r}\right)+\big{[}\ln\left(1+r\right)-\ln\left(1-r\right)\big{]}\ln(r)\bigg{]}-\bigg{(}\frac{8}{45r^{2}}+\frac{11}{9}+\frac{7}{3}r^{2}\bigg{)}\times$
$\displaystyle\displaystyle\ln^{2}(r)-\frac{1853}{810}-\frac{349}{135}\ln(r)-\frac{53}{15}\,r^{2}-\frac{64}{45}\,r^{2}\ln(r)-{\frac{4\,{\pi}^{2}}{135\,{r}^{2}}}\,,$
(56)

which defines the single analytical function $\displaystyle
A_{2}^{(6),(lL)}(r)$ valid for all $\displaystyle r\in(0,\infty)$. As in the
case of Eqs. (41) and (42), one should treat with caution the complex
logarithms in the square brackets in (56).

##  VI Three lepton loops

In this section, we present the details of calculations of the 8-th order
corrections from the bubble-type diagrams, i.e., calculations of the
coefficients $\displaystyle A_{2}^{(8)}(r)$ in (31). The corresponding
diagrams are depicted in Fig. 3 for which the ingredients
$\displaystyle\Omega_{p}(s)$ and $\displaystyle R_{j}(s)$, Eqs.(26) and (27)
are: $\displaystyle p=1,\ j=2$ (left panel), $\displaystyle p=2,\ j=1$
(central panel) and $\displaystyle p=0,\ j=3$ (right panel). The quantities
$\displaystyle\Omega_{0}(s)$, $\displaystyle\Omega_{1}(s)$, $\displaystyle
R_{1}(s)$ and $\displaystyle R_{2}(s)$ have been already calculated in Eqs.
(34), (54), (39) and (49), respectively. The remaining
$\displaystyle\Omega_{2}(s)$ and $\displaystyle R_{3}(s)$ will be presented
explicitly as they appear in calculations of the corresponding diagrams.

Figure 3: The 8-th order diagrams considered in the present paper. From left
to right: one lepton loop as the external lepton and two loops with
$\displaystyle\ell\neq L$ ($\displaystyle(L\ell\ell)$); two lepton loops with
the same leptons as the external one and one loop $\displaystyle\ell\neq L$
($\displaystyle(LL\ell)$); three identical lepton lops $\displaystyle\ell\neq
L$, ($\displaystyle(\ell\ell\ell)$). The relevant quantities
$\displaystyle\Omega_{p}(s)$ and $\displaystyle R_{j}(s)$, Eqs.(26) and (27)
are: $\displaystyle p=1,\ j=2$ (left panel), $\displaystyle p=2,\ j=1$
(central panel) and $\displaystyle p=0,\ j=3$ (right panel).

###  VI.1  Two loops with leptons $\displaystyle(\ell\ell)\neq L$ and one loop
with lepton $\displaystyle L$: $\displaystyle p=1$, $\displaystyle j=2$, left
panel in Fig. 3

The Mellin-Barnes integral representation for the contribution to the lepton
anomaly from the diagram shown in Fig. 3, left panel, reads as

$\displaystyle\displaystyle a_{L}(1,2)=\frac{\alpha}{\pi}\frac{3}{2\pi
i}\int\limits_{c-i\infty}^{c+i\infty}\
ds\left(4r^{2}\right)^{-s}\Gamma(s)\Gamma(1-s)\left(\frac{\alpha}{\pi}\right)\Omega_{1}(s)\
\left(\frac{\alpha}{\pi}\right)^{2}R_{2}(s),$ (57)

where $\displaystyle R_{2}(s)$ and $\displaystyle\Omega_{1}(s)$ are given by
Eqs. (49) and (54), respectively. From these expressions one can infer
explicitly the locations of the singularities of the integrand (57)
contributing to the Cauchy residue theorem in the complex plane of the Mellin
variable $\displaystyle s$. Closing the integration contour in the left
semiplane $\displaystyle(r<1)$ and computing the corresponding residues, we
obtain

$\displaystyle\displaystyle
A_{2}^{(8),(L\ell\ell)}(r<1)=\frac{16}{5r}\left(-{\frac{3}{7}}+\frac{1}{9}\;r^{2}\right)\bigg{[}\big{[}{\rm
Li}_{2}\left(r\right)-{\rm
Li}_{2}\left(-r\right)\big{]}\ln(r)-\frac{1}{2}\big{[}\,\ln\left(1+r\right)-\,\ln\left(1-r\right)\big{]}$
$\displaystyle\displaystyle\times\ln^{2}(r)+{\rm Li}_{3}\left(-r\right)-{\rm
Li}_{3}\left(r\right)\bigg{]}-\left(\frac{13}{9}-4\,{r}^{2}+\frac{67\,}{27}\,{r}^{4}+\frac{2}{27}\,{r}^{6}\right)\bigg{[}{\rm
Li}_{2}\left({r}^{2}\right)+2\,\ln(1-r^{2})\ln\left(r\right)$
$\displaystyle\displaystyle-\frac{1}{3}\,{\pi}^{2}\bigg{]}+\left(\frac{1}{3}+4\,{r}^{2}-\frac{73\,}{15}\,{r}^{4}+\frac{88}{315}\,{r}^{6}\right){\rm
Li}_{3}({r}^{2})-\frac{2}{3}\bigg{[}\left(\frac{2}{3}+8{r}^{2}-\frac{154}{15}\,{r}^{4}+\frac{88}{105}\,{r}^{6}\right){\rm
Li}_{2}({r}^{2})$
$\displaystyle\displaystyle+\left(\frac{1}{9}+\frac{4}{3}{r}^{2}-{\frac{13}{9}}\,{r}^{4}\right)\pi^{2}\bigg{]}\ln(r)+4\left(1+2r^{4}\right)\bigg{[}{\rm
Li}_{4}\left({r}^{2}\right)-{\rm
Li}_{3}\left({r}^{2}\right)\ln(r)+\frac{1}{3}\bigg{(}{\rm
Li}_{2}\left({r}^{2}\right)-$
$\displaystyle\displaystyle\frac{1}{3}\pi^{2}\bigg{)}\ln^{2}(r)\bigg{]}+\left({\frac{9911}{2835}}+{\frac{9136}{945}{r}^{2}}-{\frac{668}{945}\,{r}^{4}}\right)\ln\left(r\right)+\bigg{[}\frac{2869}{945}-\frac{58}{35}\,{r}^{2}+\frac{4162}{945}\,{r}^{4}+\frac{4}{27}\,{r}^{6}-$
$\displaystyle\displaystyle\frac{2}{3}\left(\frac{1}{3}+4\,{r}^{2}-\frac{89}{15}{r}^{4}+\frac{88\,}{105}{r}^{6}\right)\ln(1-r^{2})\bigg{]}\ln^{2}(r)-\frac{32}{45}\left(1-\frac{11}{41}\,{r}^{2}\right){r}^{4}\ln^{3}(r)+$
$\displaystyle\displaystyle{\frac{80321}{68040}}+\frac{6509}{630}\,{r}^{2}-\frac{334}{945}\,{r}^{4}-\frac{4}{45}\bigg{(}1+2r^{4}\bigg{)}{\pi}^{4},$
(58)

where, regardless that the integral (57) has been calculated by the Cachy’s
theorem in the left semiplane, the obtained expression represents the sought
analytical function $\displaystyle A_{2}^{(8),(L\ell\ell)}(r)$ valid in the
whole interval $\displaystyle r\in(0,\infty)$. This assertion has been checked
by calculating the integral closing the contour in the right semiplane and
also by direct numerical calculations of the integral (14) with the
polarization operator (38).

###  VI.2  Two leptons $\displaystyle(LL)$ as the external one and one loop
with $\displaystyle l\neq L$: $\displaystyle p=2$, $\displaystyle j=1$,
central panel in Fig. 3

For these diagrams the Mellin-Barnes representation provides

$\displaystyle\displaystyle
a_{L}^{LL\ell}(2,1)=\frac{\alpha}{\pi}\frac{3}{2\pi
i}\int\limits_{c-i\infty}^{c+i\infty}\
ds\left(4r^{2}\right)^{-s}\Gamma(s)\Gamma(1-s)\left(\frac{\alpha}{\pi}\right)^{2}\Omega_{2}(s)\,\left(\frac{\alpha}{\pi}\right)R_{1}(s),$
(59)

where $\displaystyle R_{1}(s)$ is given by Eq. (39). Due to presence of the
squared polarization operator and logarithmic functions in Eqs. (26) and (36),
calculations of $\displaystyle\Omega_{2}(s)$ turn to be rather cumbersome. To
present the results in a more or less compact form, we introduce several
auxiliary functions containing integrals with powers of the logarithm
$\displaystyle\ln^{k}(1-x)$, $\displaystyle k=0,~{}1,~{}2$:

$X_{k}(s,n)=\int\limits_{0}^{1}dx\;x^{2s+n}(1-x)^{1-s}\ln^{k}(1-x).$ (60)

Then we obtain

$\displaystyle\displaystyle
X_{0}(s,n)=\frac{\Gamma(2-s)\Gamma(1+n+2s)}{\Gamma(3+n+s)}\,,\quad
X_{1}(s,n)=X_{0}(s,n)\bigg{(}\psi\left(2-s\right)-\psi\left(3+n+s\right)\bigg{)},\quad$
(61) $\displaystyle\displaystyle
X_{2}(s,n)=X_{0}(s,n)\left[\bigg{(}\psi\left(2-s\right)-\psi\left(3+n+s\right)\bigg{)}^{2}+\;\psi^{(1)}\left(2-s\right)-\;\psi^{(1)}\left(3+n+s\right)\right],\quad$

where $\displaystyle\psi(x)$ and $\displaystyle\psi^{(1)}(x)$ are the
polygamma functions of the order $\displaystyle 0$ and $\displaystyle 1$,
respectively. Then, with (61), equation (26) casts the form

$\displaystyle\displaystyle\Omega_{2}(s)\;=\;\frac{25}{81}X_{0}(s,0)+\frac{16}{9}X_{0}(s,-4)-\frac{32}{9}X_{0}(s,-3)+\frac{8}{27}X_{0}(s,-2)+\frac{40}{27}X_{0}(s,-1)-$
$\displaystyle\displaystyle\frac{10}{27}X_{1}(s,0)+\frac{32}{9}X_{1}(s,-5)-\frac{80}{9}X_{1}(s,-4)+\frac{104}{27}X_{1}(s,-3)+\frac{28}{9}X_{1}(s,-2)-\frac{8}{9}X_{1}(s,-1)+$
$\displaystyle\displaystyle\frac{1}{9}X_{2}(s,0)+\frac{16}{9}X_{2}(s,-6)-\frac{16}{3}X_{2}(s,-5)+4X_{2}(s,-4)+\frac{8}{9}X_{2}(s,-3)-\frac{4}{3}X_{2}(s,-2).$
(62)

It immediately follows from Eq. (62) that

$\displaystyle\displaystyle
A_{1}^{(6)}=\Omega_{2}(0)=-\frac{943}{324}-\frac{4\pi^{2}}{135}+\frac{8}{3}\,\zeta(3).$
(63)

Having calculated explicitly $\displaystyle\Omega_{2}(s)$ and $\displaystyle
R_{1}(s)$, further integration in (59) is performed by the Cauchy residue
theorem. Closing the integration contour in the left semiplane ($\displaystyle
r<1$) or right ($\displaystyle r>1$) semiplanes in (59), we obtain the
coefficient corresponding to the anomaly $\displaystyle a_{L}^{LL\ell}~{}$.

####  VI.2.1  Closing the integration contour in the left semiplane of
$\displaystyle s$: $\displaystyle A_{2}^{(8),(LL\ell)}(r<1)$

The result of integration for $\displaystyle A_{2}^{(8),(LLl)}(r<1)$ can be
represented in the following form:

$\displaystyle\displaystyle
A_{2}^{(8),(LL\ell)}(r<1)=\frac{4}{3}\;\bigg{\\{}{\frac{34}{105}}+{\frac{4}{35\,{r}^{2}}}+{\frac{323}{210}}\,{r}^{2}+{\frac{89}{36}}\,{r}^{4}+\frac{2}{105}(21r-r^{3})\big{[}\ln\left(1+r\right)-\ln\left(1-r\right)\big{]}$
$\displaystyle\displaystyle+\left(\frac{1}{6}+{\frac{4}{35\,{r}^{4}}}+{\frac{4}{15\,{r}^{2}}}+2\,{r}^{2}-{\frac{13}{6}}\,{r}^{4}\right)\ln\left(1-{r}^{2}\right)-\left(2+{r}^{4}\right){\rm
Li}_{2}(r^{2})\bigg{\\}}\ln^{2}(r)+2\;\bigg{\\{}{\frac{1813}{43200\,{r}^{4}}}-$
$\displaystyle\displaystyle{\frac{8731477}{4536000}}-{\frac{371429}{907200\,{r}^{2}}}+{\frac{33907}{60480}}\,{r}^{2}+{\frac{590}{189}}\,{r}^{4}+\left(\frac{1}{5}+{\frac{8}{105\,{r}^{4}}}+{\frac{8}{45\,{r}^{2}}}+\frac{8}{3}\,{r}^{2}-{\frac{26\,{r}^{4}}{9}}\right){\rm
Li}_{2}(r^{2})+$
$\displaystyle\displaystyle\left(\frac{8}{3}+2\,{r}^{4}\right){\rm
Li}_{3}(r^{2})+\left(-{\frac{37939}{69120}}+{\frac{7}{1536\,{r}^{4}}}-{\frac{173}{4608\,{r}^{2}}}+{\frac{55}{1536}}\,{r}^{2}-{\frac{442}{567}}\,{r}^{4}\right){\pi}^{2}-\frac{4}{3}\,\left(2+{r}^{4}\right)\times$
$\displaystyle\displaystyle\zeta\left(3\right)\bigg{\\}}\ln(r)-\bigg{[}\;{\frac{1813}{43200{r}^{5}}}\,-{\frac{25}{72\,{r}^{3}}}-{\frac{145}{32\,r}}+{\frac{1481}{192}}\,{r}^{3}+\left({\frac{7}{1536\,{r}^{5}}}-{\frac{5}{128\,{r}^{3}}}-{\frac{145}{256\,r}}-{\frac{167}{384}}\,r-\right.$
$\displaystyle\displaystyle\left.{\frac{1481}{1536}}\,{r}^{3}\right){\pi}^{2}\bigg{]}\bigg{[}{\rm
Li}_{2}(-r)-{\rm
Li}_{2}(r)+\big{[}\ln\left(1+r\right)-\ln\left(1-r\right)\big{]}\ln(r)\bigg{]}+{\frac{16}{15}}\left(r-{\frac{{r}^{3}}{21}}\right)\bigg{[}{\rm
Li}_{3}(r)-$ $\displaystyle\displaystyle{\rm Li}_{3}(-r)-\big{(}{\rm
Li}_{2}(r)-{\rm
Li}_{2}(-r)\big{)}\ln(r)\bigg{]}-\left({\frac{13}{45}}+{\frac{8}{105\,{r}^{4}}}+{\frac{8}{45\,{r}^{2}}}+4\,{r}^{2}-\frac{13}{3}\,{r}^{4}\right){\rm
Li}_{3}\left({r}^{2}\right)+$
$\displaystyle\displaystyle\left({\frac{5383}{1350}}-{\frac{28}{9}}\,{r}^{2}-{\frac{125}{54}}\,{r}^{4}+\frac{2}{3}\,{\pi}^{2}+\frac{1}{3}\,{\pi}^{2}{r}^{4}\right)\bigg{[}{\rm
Li}_{2}\left({r}^{2}\right)+2\ln(1-r^{2})\ln(r)\bigg{]}-4\left(1+{r}^{4}\right){\rm
Li}_{4}\left({r}^{2}\right)$
$\displaystyle\displaystyle-{\frac{1813}{21600\,{r}^{4}}}+{\frac{1035989}{~{}1360800\,{r}^{2}}}+{\frac{519835531}{34020000}}+\,{\frac{147587}{30240}}\,{r}^{2}-{\frac{~{}558857}{59535}}\,{r}^{4}+\left({\frac{211546\,{r}^{4}}{178605}}+{\frac{6799\,{r}^{2}}{80640}}\right.$
$\displaystyle\displaystyle\left.-{\frac{97463}{518400}}+{\frac{533}{6912}}\,{r}^{2}-{\frac{7}{768}}\,{r}^{4}\right){\pi}^{2}+\frac{\pi^{4}}{27}(2+{r}^{4})-\left({\frac{2}{45}}+\frac{8}{3}\,{r}^{2}-{\frac{676}{189}}\,{r}^{4}\right)\zeta(3)+S_{1}(r)$
(64)

with

$\displaystyle\displaystyle
S_{1}(r)=2\sum_{n=3}^{\infty}\bigg{[}C_{1}(n)\;\bigg{(}\psi^{(2)}({n})+\psi^{(1)}({n})\ln(r^{2})\bigg{)}-C_{2}(n)\;\psi^{(1)}({n})\bigg{]}r^{2n}\,,$
(65) $\displaystyle\displaystyle
C_{1}(n)=(-60+5n+86n^{2}+36n^{3}+3n^{4})/Y_{1}(n)\,,$
$\displaystyle\displaystyle
C_{2}(n)=\left(5400-1080n-67455n^{2}+20260n^{3}+107834n^{4}-17494n^{5}\right.$
$\displaystyle\displaystyle\left.~{}~{}~{}~{}~{}~{}~{}~{}-77654n^{6}-10816n^{7}+14768n^{8}+4704n^{9}+288n^{10}\right)/[Y_{1}(n)]^{2}\,,$
$\displaystyle\displaystyle Y_{1}(n)=n(n-2)(2n+5)(4n^{2}-9)(4n^{2}-1)\,.$

A scrupulous analysis of the sum $\displaystyle S_{1}(r<1)$ demonstrated that
it converges very fast in the region $\displaystyle r<1$ and can be calculated
numerically with any desired accuracy. However, we have not succeeded in
finding explicitly its convergency function and hence presenting
$\displaystyle A_{2}^{(8),(LL\ell)}(r<1)$ in terms of a single complex
function, similar to Eq. (58). Moreover, as it can be easily seen, the sum
$\displaystyle S_{1}(r)$ diverges for $\displaystyle r>1$. Consequently,
$\displaystyle A_{2}^{(8),(LL\ell)}(r<1)$, Eq. (64), cannot be
straightforwardly continued analytically to the right semiplane, i.e., the
resulting expressions must be considered strictly as only the left semiplane
branch of the analytical function $\displaystyle A_{2}^{(8),(LL\ell)}(r)$.

####  VI.2.2 Closing the integration contour in the right semiplane of
$\displaystyle s$: $\displaystyle A_{2}^{(8),(LL\ell)}(r>1)$

By doing as before, we get

$\displaystyle\displaystyle
A_{2}^{(8),(LL\ell)}(r>1)=\frac{16}{9}\big{(}2+r^{4}\big{)}\,\ln^{4}(r)+\bigg{(}\frac{1949}{1728}-\frac{701\,}{192}{r}^{2}+{\frac{52\,}{9}}{r}^{4}-\frac{1447}{6720\,{r}^{4}}-{\frac{359}{960\,{r}^{2}}}\bigg{)}\ln^{3}(r)+$
$\displaystyle\displaystyle\bigg{[}\left(\frac{2}{9}+\frac{8}{3}\,{r}^{2}-\frac{26}{9}r^{4}+\frac{16}{105\,{r}^{4}}+\frac{16}{45\,{r}^{2}}\right)\ln\left({r}^{2}-1\right)+\frac{4}{9}\big{(}2+r^{4}\big{)}\pi^{2}+\frac{11261}{2240}+\frac{295409}{20160}\,{r}^{2}+$
$\displaystyle\displaystyle{\frac{268}{27}}\,{r}^{4}+{\frac{7}{192\,{r}^{4}}}-{\frac{9439}{60480\,{r}^{2}}}\bigg{]}\ln^{2}(r)+\bigg{\\{}\bigg{[}-\frac{2}{9}\big{(}2+r^{4}\big{)}\pi^{2}-{\frac{56\,{r}^{2}}{9}}-{\frac{134\,{r}^{4}}{27}}-{\frac{16}{27}}\bigg{]}\ln\left({r}^{2}-1\right)-$
$\displaystyle\displaystyle\left(\frac{4}{9}+\frac{16}{105\,{r}^{4}}+{\frac{16}{45\,{r}^{2}}}+\frac{16}{3}\,{r}^{2}-{\frac{52\,{r}^{4}}{9}}\right){\rm
Li_{2}}\left(\frac{1}{{r}^{2}}\right)+\left({\frac{2717}{6912}}+{\frac{173}{6912\,{r}^{2}}}+{\frac{323}{768}}\,{r}^{2}-{\frac{7}{2304{r}^{4}}\,}\right){\pi}^{2}$
$\displaystyle\displaystyle+{\frac{7}{864{r}^{4}}\,}-{\frac{1291}{10080\,{r}^{2}}}+{\frac{501857}{30240}}\,{r}^{2}+{\frac{380971}{30240}}\bigg{\\}}\ln(r)+\bigg{[}\frac{\pi^{2}}{9}\;\big{(}2+r^{4}\big{)}+{\frac{8}{27}}+{\frac{28}{9}}\,{r}^{2}+{\frac{67}{27}}\,{r}^{4}\bigg{]}$
$\displaystyle\displaystyle\times\,{\rm
Li_{2}}\left(\frac{1}{{r}^{2}}\right)-\bigg{(}\frac{1}{3}+{\frac{8}{105\,{r}^{4}}}+{\frac{8}{45\,{r}^{2}}}+4\,{r}^{2}-\frac{13}{3}\,{r}^{4}\bigg{)}{\rm
Li_{3}}\left(\frac{1}{{r}^{2}}\right)+\frac{14}{3}\;\big{(}2+{r}^{4}\big{)}{\rm
Li_{4}}\left(\frac{1}{{r}^{2}}\right)+$
$\displaystyle\displaystyle\bigg{[}{\frac{7405\,{r}^{3}}{864}}+{\frac{167}{48}}\,r-{\frac{5}{16\,{r}^{3}}}+{\frac{35}{864\,{r}^{5}}}+\bigg{(}{\frac{5}{384\,{r}^{3}}}+{\frac{145}{768\,r}}-{\frac{7}{4608\,{r}^{5}}}+{\frac{1481}{4608}}\,{r}^{3}+{\frac{167}{1152}}\,r\bigg{)}\pi^{2}\bigg{]}\times$
$\displaystyle\displaystyle\bigg{[}{\rm
Li_{2}}\left(-\frac{1}{{r}}\right)-{\rm
Li_{2}}\left(\frac{1}{{r}}\right)+\bigg{(}\ln\left(1-\frac{1}{{r}}\right)-\ln\left(1+\frac{1}{{r}}\right)\bigg{)}\ln(r)\bigg{]}+\bigg{(}{\frac{7}{1152\,{r}^{5}}}-{\frac{5}{96\,{r}^{3}}}-{\frac{145}{192\,r}}-$
$\displaystyle\displaystyle{\frac{167}{288}}\,r-{\frac{1481}{1152}}\,{r}^{3}\bigg{)}\bigg{\\{}\bigg{[}\ln\left(1+\frac{1}{{r}}\right)-\ln\left(1-\frac{1}{{r}}\right)\bigg{]}\ln^{3}(r)-3\;\bigg{[}{\rm
Li_{2}}\left(-\frac{1}{{r}}\right)-{\rm
Li_{2}}\left(\frac{1}{{r}}\right)\bigg{]}\ln^{2}(r)-$
$\displaystyle\displaystyle 6\;\bigg{[}{\rm
Li_{3}}\left(-\frac{1}{{r}}\right)-{\rm
Li_{3}}\left(\frac{1}{{r}}\right)\bigg{]}\ln(r)-6\;{\rm
Li_{4}}\left(-\frac{1}{{r}}\right)+6\;{\rm
Li_{4}}\left(\frac{1}{{r}}\right)\bigg{\\}}+\bigg{(}{\frac{8}{15}}\,r-{\frac{8}{315}}\,{r}^{3}\bigg{)}\times$
(66)
$\displaystyle\displaystyle\bigg{\\{}\bigg{[}\ln\left(1+\frac{1}{{r}}\right)-\ln\left(1-\frac{1}{{r}}\right)\bigg{]}\ln^{2}(r)-2\;\bigg{[}{\rm
Li_{2}}\left(-\frac{1}{{r}}\right)-{\rm
Li_{2}}\left(\frac{1}{{r}}\right)\bigg{]}\ln(r)-2\;{\rm
Li_{3}}\left(-\frac{1}{{r}}\right)+$ $\displaystyle\displaystyle 2\;{\rm
Li_{3}}\left(\frac{1}{{r}}\right)\bigg{\\}}+\frac{8}{3}\,\big{(}2+r^{4}\big{)}\,\bigg{\\{}2\;\bigg{[}{\rm
Li_{3}}\left(\frac{1}{{r^{2}}}\right)+{\rm
Li_{2}}\left(\frac{1}{{r^{2}}}\right)\bigg{]}\ln(r)-\frac{1}{3}\ln(r^{2}-1)\ln^{2}(r)\bigg{\\}}\ln(r)+$
$\displaystyle\displaystyle\bigg{(}{\frac{6917}{20736}}-{\frac{533}{20736\,{r}^{2}}}+{\frac{7}{2304\,{r}^{4}}}+{\frac{1225}{2304}}\,{r}^{2}\bigg{)}\pi^{2}-{\frac{7}{864\,{r}^{4}}}+{\frac{2059}{30240\,{r}^{2}}}+{\frac{23731\,{r}^{2}}{1120}}+{\frac{1208147}{90720}}+S_{2}(r),$

where

$S_{2}(r)=2\sum_{n=1}^{\infty}\bigg{[}C_{1}(-n)\;\bigg{(}\psi^{(2)}({n})+\psi^{(1)}({n})\ln(r^{2})\bigg{)}-C_{2}(-n)\;\psi^{(1)}({n})\bigg{]}r^{-2n}\,.$
(67)

As in the case $\displaystyle r<1$, the sum (67) converges fast in the
interval $\displaystyle r\in[1,\infty)$ and diverges at $\displaystyle r<1$.
Also, we have not found explicitly the convergency function for $\displaystyle
S_{2}(r)$, so that a direct analytical continuation to the left semiplane
remains hindered. Correspondingly, Eqs. (VI.2.2)-(67) define only the right
semiplane branch of the analytical function $\displaystyle
A_{2}^{(8),(LLl)}(r)$.

###  VI.3  Three identical leptons $\displaystyle(lll)$, $\displaystyle l\neq
L$: right panel of Fig. 3, $\displaystyle p=0$, $\displaystyle j=3$

For these diagrams the lepton anomaly reads as

$\displaystyle\displaystyle
a_{L}^{\ell\ell\ell}(0,3)=\frac{\alpha}{\pi}\frac{1}{2\pi
i}\int\limits_{c-i\infty}^{c+i\infty}\
ds\left(4r^{2}\right)^{-s}\Gamma(s)\Gamma(1-s)\Omega_{0}(s)\,\left(\frac{\alpha}{\pi}\right)^{3}R_{3}(s),,$
(68)

where $\displaystyle\Omega_{0}(s)$ is defined in Eq. (34). As for
$\displaystyle R_{3}(s)$, direct calculations by Eqs. (28) and (36) provide

$R_{3}(s)=\frac{\sqrt{\pi}}{864}\frac{\Gamma(s)}{\Gamma\left(\frac{11}{2}+s\right)}\left\\{\frac{P(s)}{s(1+s)(2+s)}-27(1+s)(35+21s+3s^{2})\left[\pi^{2}-6\
\psi^{(2)}(s)\right]\right\\},$ (69)

where $\displaystyle
P(s)=3492-8748s-26575s^{2}-9214s^{3}+18395s^{4}+17018s^{5}+5120s^{6}+512s^{7}\,$.

Explicitly, the coefficient $\displaystyle A_{2}^{(8),\ell\ell\ell}(r)$ is

$\displaystyle\displaystyle A_{2}^{(8),(\ell\ell\ell)}(r)=\frac{1}{2\pi
i}\int\limits_{c-i\infty}^{c+i\infty}ds\;r^{-2s}\left[\frac{(1-s)}{(s+2)(2s+1)(2s+3)(2s+5)(2s+7)(2s+9)}\right]\times$
$\displaystyle\displaystyle\bigg{[}\frac{-(1-s)(-3492+5256s+31831s^{2}+41045s^{3}+22650s^{4}+5632s^{5}+512s^{6})}{27s(s+1)^{2}(s+2)}~{}~{}~{}$
$\displaystyle\displaystyle-\;{\pi^{2}(35+21s+3s^{2})}+\;{6\psi^{(1)}(s)(35+21s+3s^{2})}\bigg{]}\frac{\pi^{2}}{\sin^{2}(\pi
s)}\;.$ (70)

Equations (34), (69) and (VI.3) define explicitly all the singularities in the
complex plane of the variable $\displaystyle s$. Accordingly, the integration
contour can be closed in the left ($\displaystyle r<1$) or right
($\displaystyle r>1$) semiplane of $\displaystyle s$:

####  VI.3.1 Integration contour in the left semiplane: $\displaystyle
A_{2}^{(8),(\ell\ell\ell)}(r<1)$

$\displaystyle\displaystyle
A_{2}^{(8),(\ell\ell\ell)}(r<1)=\bigg{[}-{\frac{8609}{5832}}+\frac{1760}{567}{r}^{2}+\frac{7899407}{396900}\,{r}^{4}-\frac{692549}{1786050}\,{r}^{6}-\frac{50171}{793800}\,{r}^{8}+$
$\displaystyle\displaystyle\frac{1}{3}\,\left(-\frac{25}{54}+\frac{4237}{1152}\,{r}^{2}-\frac{13709}{3456}\,{r}^{4}+\frac{767}{1152}\,{r}^{6}+\frac{11}{128}\,{r}^{8}\right){\pi}^{2}+\frac{1}{5}\,{\pi}^{4}{r}^{4}-\frac{1}{1536}{\pi}^{4}r{\it
V_{1}(r)}+$
$\displaystyle\displaystyle\frac{r}{4}\,\left(\frac{101}{16}-\frac{1025}{18}\,{r}^{2}-\frac{1813}{120}\,{r}^{4}-\frac{3229}{525}\,{r}^{6}-\frac{1292159}{1587600}\,{r}^{8}+{\frac{{\pi}^{2}}{192}}{\it
V_{1}(r)}\right)\bigg{(}{\rm Li_{2}}\left(r\right)-{\rm
Li_{2}}\left(-r\right)\bigg{)}+$ $\displaystyle\displaystyle\frac{r}{64}{\it
V_{1}(r)}\bigg{(}{\rm Li_{4}}\left(r\right)-{\rm
Li_{4}}\left(-r\right)\bigg{)}+{\it V_{2}(r)}\,{\rm
Li_{2}}\left({r}^{2}\right)-\frac{32}{315}\,{r}^{2}\left(7\,{r}^{2}-12\right){\rm
Li_{3}}\left({r}^{2}\right)-2\,r^{4}{\rm Li_{4}}\left(r^{2}\right)+$
$\displaystyle\displaystyle\left(-\frac{2}{9}+\frac{136}{35}\,{r}^{2}-\frac{64}{15}\,{r}^{4}+\frac{16}{15}\,{r}^{6}\right)\zeta\left(3\right)\bigg{]}+2\,\bigg{[}-\frac{317}{324}+\frac{15863}{2835}\,{r}^{2}-\frac{12963943}{1587600}\,{r}^{4}+\frac{1046261}{2381400}\,{r}^{6}-$
$\displaystyle\displaystyle\frac{50171}{1587600}\,{r}^{8}+\frac{1}{3}\,{\pi}^{2}\left(-\frac{1}{9}+\frac{721}{768}\,{r}^{2}+\frac{2981}{2304}\,{r}^{4}+\frac{263}{768}\,{r}^{6}+\frac{11}{256}\,{r}^{8}\right)+\frac{1}{2}\,r\left({\it
V_{3}(r)}-\frac{{\pi}^{2}}{768}{\it V_{1}(r)}\right)\times$
$\displaystyle\displaystyle\bigg{(}\ln\left(r+1\right)-\ln\left(1-r\right)\bigg{)}-\frac{1}{128}{\it
V_{1}(r)}\,r\bigg{(}{\rm Li_{3}}\left(r\right)-{\rm
Li_{3}}\left(-r\right)\bigg{)}+{\it
V_{2}(r)}\,\ln(1-{r}^{2})+\frac{16}{15}\times$
$\displaystyle\displaystyle\bigg{(}-\frac{4}{7}+\frac{1}{3}\,r^{2}\bigg{)}r^{2}{\rm
Li_{2}}({r}^{2})+2\,{r}^{4}{\rm
Li_{3}}\left({r}^{2}\right)+4\,\zeta\left(3\right){r}^{4}\bigg{]}\ln(r)+4\,\bigg{[}-\frac{25}{108}+\frac{4237}{2304}\,{r}^{2}-\frac{13709}{6912}\,{r}^{4}+$
$\displaystyle\displaystyle\frac{1}{3}\,{\pi}^{2}{r}^{4}+\frac{767}{2304}\,{r}^{6}+\frac{11}{256}\,{r}^{8}+\frac{r}{512}{\it
V_{1}(r)}\bigg{(}{\rm Li_{2}}(r)-{\rm Li_{2}}(-r)\bigg{)}-{r}^{4}{\rm
Li_{2}}\left({r}^{2}\right)\bigg{]}\ln^{2}(r)+$
$\displaystyle\displaystyle\bigg{[}-\frac{4}{27}+\frac{721}{576}\,{r}^{2}+\frac{2981}{1728}\,{r}^{4}+\frac{263}{576}\,{r}^{6}+\frac{11}{192}\,{r}^{8}-\frac{r}{384}{\it
V_{1}(r)}\bigg{(}\,\ln(r+1)-\,\ln(1-r)\bigg{)}-\,$
$\displaystyle\displaystyle\frac{8}{3}\,{r}^{4}\ln(1-{r}^{2})\bigg{]}\ln^{3}(r)+\frac{4}{3}\,{r}^{4}\ln^{4}(r)+S_{3}(r)$
(71)

with

$\displaystyle\displaystyle
S_{3}(r)=6\sum\limits_{n=4}^{\infty}\bigg{[}C_{3}(n)H^{(2)}(n)+C_{4}(n)\psi^{(2)}({n+1})-C_{4}(n)H^{(2)}(n)\ln(r^{2})\bigg{]}r^{2n},$
(72)

where $\displaystyle H^{(2)}(n)$ is the $\displaystyle n$-th generalized
harmonic number and the shorthand notation is

$\displaystyle\displaystyle
C_{3}(n)=(-295995+836500\,n-787336\,n^{2}+206366\,n^{3}+131386\,n^{4}-114304\,n^{5}+$
$\displaystyle\displaystyle\phantom{P_{4}(n)=(}35216\,n^{6}-5088\,n^{7}+288\,n^{8})/[Y_{2}(n)]^{2},$
$\displaystyle\displaystyle C_{4}(n)=(1+n)(35-21\,n+3\,n^{2})/Y_{2}(n),,$
$\displaystyle\displaystyle
Y_{2}(n)=(n-2)(2\,n-9)(2\,n-7)(2\,n-5)(2\,n-3)(2\,n-1),$

and

$\displaystyle\displaystyle
V_{1}(r)=-101+820\,r^{2}+210\,r^{4}+84\,r^{6}+11\,r^{8}$ (73)
$\displaystyle\displaystyle
V_{2}(r)=-\frac{97}{2835}+\frac{10348}{11025}\,r^{2}+\frac{5524}{675}\,r^{4}-\frac{2}{3}\,\pi^{2}\,r^{4}$
(74) $\displaystyle\displaystyle
V_{3}(r)=-\frac{101}{64}+\frac{1025}{72}\,r^{2}+\frac{1813}{480}\,r^{4}+\frac{3229}{2100}\,r^{6}+\frac{1292159}{6350400}\,r^{8},$
(75)

####  VI.3.2 Integration contour in the right semiplane: $\displaystyle
A_{2}^{(8),(\ell\ell\ell)}(r>1)$

$\displaystyle\displaystyle
A_{2}^{(8),(\ell\ell\ell)}(r>1)=\left[\frac{307987}{893025}-\frac{1620853897}{222264000}r^{2}-\frac{137496269}{31752000}r^{4}-\frac{30585647}{19051200}r^{6}-\frac{1292159}{6350400}r^{8}+\right.$
$\displaystyle\displaystyle\left.\left(-\frac{97}{2835}+\frac{10348}{11025}r^{2}+\frac{5524}{675}r^{4}\right)\ln\left(1-\frac{1}{r^{2}}\right)+\left(\frac{64}{105}-\frac{16}{45}r^{2}\right)r^{2}\;{\rm
Li}_{2}\left(\frac{1}{r^{2}}\right)\right]\ln(r^{2})+$
$\displaystyle\displaystyle\left(-\frac{101}{64}r+\frac{1025}{72}r^{3}+\frac{1813}{480}r^{5}+\frac{3229}{2100}r^{7}+\frac{1292159}{6350400}r^{9}\right)\bigg{[}{\rm
Li}_{2}\left(\frac{1}{r}\right)-{\rm
Li}_{2}\left(-\frac{1}{r}\right)+\ln(r)\times$
$\displaystyle\displaystyle\bigg{(}\ln\left(r+1\right)-\ln\left(r-1\right)\bigg{)}\bigg{]}+\left(\frac{97}{2835}-\frac{10348}{11025}r^{2}-\frac{5524}{675}r^{4}\right){\rm
Li}_{2}\left(\frac{1}{r^{2}}\right)+\left(\frac{128}{105}-\frac{32}{45}r^{2}\right)\times$
$\displaystyle\displaystyle~{}~{}r^{2}{\rm
Li}_{3}\left(\frac{1}{r^{2}}\right)+\frac{415506937}{281302875}-\frac{79924199429}{3889620000}r^{2}-\frac{628065509}{79380000}r^{4}-\frac{89172623}{28576800}r^{6}-\frac{1292159}{3175200}r^{8}+$
$\displaystyle\displaystyle~{}~{}S_{4}(r)\,,$ (76)

where

$\displaystyle\displaystyle
S_{4}(r)=6\sum\limits_{n=2}^{\infty}\bigg{[}C_{3}(-n)H^{(2)}_{n-1}-C_{4}(-n)\psi^{(2)}({n})-C_{4}(-n)H^{(2)}_{n-1}\ln(r^{2})\bigg{]}r^{-2n}.$
(77)

Note that the sums $\displaystyle S_{1}(r)$, Eq. (65), and $\displaystyle
S_{3}(r)$, Eq. (72), converge rapidly at $\displaystyle r<1$ and diverge at
$\displaystyle r>1$. Contrarily, the sums $\displaystyle S_{2}(r)$, Eq. (67),
and $\displaystyle S_{4}(r)$, Eq. (77), converge at $\displaystyle r>1$ and
diverge at $\displaystyle r<1$. It means that these sums can be applied
strictly in the region of their convergence. However, an important comment is
in line here. Namely, we observed that despite the fact that the coefficient
$\displaystyle A_{2}^{(8),(\ell\ell\ell)}(r>1)$ has been obtained exclusively
for $\displaystyle r>1$, its formal employment in the region $\displaystyle
r<1$, from $\displaystyle r\ll 1$ up to $\displaystyle r=0.5\div 0.55$, shows
that (VI.3.2) provides absolutely the same numerical results as the
coefficient $\displaystyle A_{2}^{(8),(\ell\ell\ell)}(r<1)$, Eq. (71). It
means that if one finds explicitly the convergence function for $\displaystyle
S_{4}(r)$, it would be possible to analytically continue $\displaystyle
A_{2}^{(8),(\ell\ell\ell)}(r)$ by a single function valid in the whole
interval $\displaystyle r\in(0,\infty)$. In the case when the results of
integration do not contain infinite sums, the analytical functions for one,
two and three loops are determined by Eqs. (42),(51), (56) and (58),
respectively.

##  VII Discussions

With the above results we conclude our analytical consideration of the
radiative corrections to the lepton anomaly from QED diagrams with insertions
of the photon polarization operators up to the eighth order. The obtained
analytical expressions allowing find numerically the corresponding corrections
with any predetermined accuracy avoiding lengthy and computer time consuming
calculations. With these analytical expressions in hand, precision of further
numerical calculations is restricted only by the knowledge of the involved
fundamental constants, viz. lepton masses and the fine structure constant. In
the present paper, we give only the qualitative peculiarities of the radiative
corrections to the anomalies of any type of leptons from diagrams with
insertions of one, two and three lepton loops in the polarization operator. An
interested reader can easily use the presented analytical expressions to find
numerically the corresponding corrections with the desired precision.

As mentioned above, in the literature only corrections up to the sixth order
Laporta:1993ju  were reported in details focusing mainly on the muon anomaly
with insertions of electron and muon loops, i.e. on the sixth order
coefficients $\displaystyle A_{2}^{(6)}(r)$ for $\displaystyle r\,\leq\,1$, in
our notation. Analytically, the radiative corrections for tauons within the
considered technique have not yet been considered. Moreover, even for
electrons and muons, higher order corrections have been investigated solely as
asymptotic expansions. Here we qualitatively analyze the fourth, sixth and
eighth order coefficients (30) and (31) for all possible combinations of the
existing three leptons with one, two and three internal lepton loops, cf. Eqs.
(43), (51), (56), (64), (VI.2.2), (71) and (VI.3.2).

Figure 4: The fourth and sixth order coefficients $\displaystyle
A_{2}^{(4)}(r)$ and $\displaystyle A_{2}^{(6)}(r)$, Eq. (31), as functions of
the mass ratio $\displaystyle r=m_{\ell}/m_{L}$, for an external lepton
$\displaystyle L$ with insertions of the polarization operators with one loop,
left panel, and two loops, central and right panels. The notation corresponds
to $\displaystyle A_{L}^{(\ell)}(r)$, $\displaystyle A_{L}^{(L\ell)}$, and
$\displaystyle A_{L}^{(\ell\ell)}$, Eqs. (43), (56) and (50), respectively,
where $\displaystyle L$ and $\displaystyle\ell=e,\mu,\tau$. The horizontal
dashed lines indicate the values of the coefficients at $\displaystyle r=1$,
i.e., the values $\displaystyle A_{1}^{(4)}$, Eq. (46), and $\displaystyle
A_{1}^{(6)}$, Eq. (63), which are universal for any kind of the considered
leptons. The central and right panels illustrate the same dependence of
$\displaystyle A_{2}^{(6)}(r)$ but for the combinations
$\displaystyle(L\ell)$, Eq. (56), and $\displaystyle(\ell\ell)$, Eqs. (50),
respectively. The open circles, as well as the associated with them labels,
point to physical values of the ratio $\displaystyle r$ and to the
corresponding coefficients $\displaystyle A_{2}^{(\ell)}(r)$ or $\displaystyle
A_{2}^{(L\ell)}(r)$.

In Fig. 4, we present the results of calculations of the fourth,
$\displaystyle A_{2}^{(4)}(r)$, and sixth order corrections, $\displaystyle
A_{2}^{(6)}(r)$, for diagrams with one and two lepton loops, as depicted in
Fig. 2. The variable $\displaystyle r$ is defined as $\displaystyle
r=m_{\ell}/m_{L}$ and varies in the interval $\displaystyle r\in(0,\infty)$.
To ease the visibility of the results, the coefficients $\displaystyle
A_{2}^{(4,6)}(r)$ for physical values of $\displaystyle r$, i.e., when
$\displaystyle L$ and $\displaystyle\ell$ denote the real existing leptons
$\displaystyle e,\mu$ and $\displaystyle\tau$, are marked in Fig. 4 by open
circles. Moreover, for a better illustration of the combinations of
$\displaystyle L$ and $\displaystyle\ell$, each physical value of
$\displaystyle A_{2}^{(4,6)}(r)$ is additionally labeled by $\displaystyle
A_{L}^{(\ell)}$ (left panel), $\displaystyle A_{L}^{(L\ell)}$ (central panel)
and and $\displaystyle A_{L}^{(\ell\ell)}$ (right panel). Analogously, in Fig.
5 we present the results of calculations of the eighth order contributions
from diagrams of three types: (i) insertions of three loops where one loop
corresponds to the external (under consideration) lepton and two loops with
lepton pairs different from the first one, i.e., diagrams of type
$\displaystyle(L\ell\ell)$, left panel in Figs. 3 and 5 (ii) one lepton loop
different from the external lepton, two other correspond to the external one,
i.e., diagrams of the type $\displaystyle(LL\ell)$, central panel in Figs. 3
and 5, (iii) eventually, the diagrams with all three loops with leptons of the
same type $\displaystyle(\ell\ell\ell)$, including also the case when
$\displaystyle L=\ell$, right panel in Figs. 3 and 5.

Figure 5: The eighth order coefficients $\displaystyle A_{2}^{(8)}(r)$, Eq.
(31), as functions of the mass ratio $\displaystyle r=m_{\ell}/m_{L}$, for an
external lepton $\displaystyle L=e,\mu,\tau$ with insertions of the
polarization operators with three loops formed by leptons of different types.
Left panel: the coefficients $\displaystyle A_{2}^{(8)}(r)$ determined by
diagrams with three loops with one lepton as the external $\displaystyle L$
and two other different from $\displaystyle L$. The notation corresponds to
$\displaystyle A_{L}^{(L\ell\ell)}(r)$, Eq. (58), where $\displaystyle L$ and
$\displaystyle\ell=e,\mu,\tau$. The horizontal dashed line indicates the value
of the coefficients at $\displaystyle r=1$, i.e., the coefficints
$\displaystyle A_{1}^{(8)}$, Eq. (30) which are universal for any kind of the
considered leptons. The central and right panels illustrate the same
dependence of $\displaystyle A_{L}^{(L\ell\ell)}(r)$ but for the combinations
$\displaystyle(LL\ell)$, Eqs. (64)-(67), and $\displaystyle(\ell\ell\ell)$,
Eqs. (71)-(VI.3.2), respectively.

From Figs. 4 and 5 one infers that the radiative corrections decrease rather
fast, by about 10-15 orders of magnitude, with increase of $\displaystyle r$
from its smallest to highest physical values. It means that the contribution
of the heaviest $\displaystyle\tau$-lepton to the anomaly of the lighter ones
is much smaller than the contribution from properly the muon and electron
loops. This effect increases with increasing of order of the radiative
corrections from $\displaystyle\alpha^{2}$ to $\displaystyle\alpha^{4}$ and
depends on the combination of the internal and external leptons. However, due
to the extremely high precision achieved in the measurements of the lepton
anomalies Parker:2018vye ; Malaescu ; Davoudiasl:2018fbb ; E989 ; E821 , the
corrections from $\displaystyle\tau$-leptons cannot be neglected. The
horizontal dashed lines in Figs. 4 and 5 correspond to the values of the
coefficients $\displaystyle A_{2}^{(4)-(8)}(r)$ at $\displaystyle r=1$, i.e.,
to the universal coefficients $\displaystyle A_{1}^{(4)-(8)}$, Eq. (30).

Another interesting circumstance to be stressed is that, according to Eq.
(25), the radiative corrections are governed not only by the mass ratio
$\displaystyle r$ but, at the same value of $\displaystyle r$, they are also
quite sensitive to the combinations of $\displaystyle\Omega_{p}$ and
$\displaystyle R_{j}$ in (25), i.e., to the concrete form of the Feynman
diagram. Recall that $\displaystyle p+j=n$ and that $\displaystyle\Omega_{p}$
is determined by the $\displaystyle p-th$ power of the lepton polarization
operator, while $\displaystyle R_{j}$ contains the imaginary part of the sum
$\displaystyle\big{[}\Pi_{l}(t)+\Pi_{L}(t)\big{]}^{j}$, cf. Eqs. (26) and
(18). So for the corrections to the electron anomaly at, e.g. $\displaystyle
r=m_{\mu}/m_{e}$, one has $\displaystyle A_{e}^{ee\mu}(r)\gg
A_{e}^{e\mu\mu}(r)\gg A_{e}^{\mu\mu\mu}(r)$ which vary from $\displaystyle
A_{e}^{ee\mu}(r)\sim 2\cdot 10^{-7}$ to $\displaystyle
A_{e}^{\mu\mu\mu}(r)\sim 2\cdot 10^{-12}$. The same situation occurs for
$\displaystyle r=m_{e}/m_{\tau}$ as well as for muons and tauons.
Qualitatively, the relative contributions of the three loop diagrams to the
$\displaystyle e,\mu$ and $\displaystyle\tau$-leptons is illustrated in Fig.
6, where the corrections from different types of insertions decrease from left
to right.

Figure 6: Qualitative illustration of the contributions to the eighth order
coefficients $\displaystyle A_{2}^{(8)}(r)$, Eq. (31), of different types of
Feynman diagrams, Fig. 3, in descending order: the upper row refers to
electrons, the middle row to muons and the lower row to taons. The notation
corresponds to $\displaystyle A_{L}^{(\ell\ell\ell)}(r)$, $\displaystyle
A_{L}^{(LL\ell)}(r)$ and $\displaystyle A_{L}^{(L\ell\ell)}(r)$, where
$\displaystyle L=e$ (upper row), $\displaystyle L=\mu$ (middle row),
$\displaystyle L=\tau$ (lower row) and $\displaystyle\ell=e,\mu,\tau$.

Here it is appropriate to reiterate that the presented results demonstrate the
qualitative analysis of the contribution of one, two and three loop diagrams
to the lepton anomaly. For a scrupulous quantitative investigation of the
considered radiative corrections one can use the exact analytical expressions
reported above.

###  VII.1 The asymptotic expansions

The performed qualitative analysis can be be essentially relieved if, instead
of the exact analytical formulae, one employs their asymptotic expansions.
Such analyzes have been widely used in the literature Aguilar:2008qj  by
approximate calculations of the corresponding integrals (25) with
preliminarily found asymptotic expansions of $\displaystyle\Omega_{p}(s)$ and
$\displaystyle R_{j}(s)$. Explicitly, the corresponding expansions can be
found in Refs. Aguilar:2008qj . In our case, as an additional check of the
obtained analytical expressions, we compare our asymptotical expansions with
the known results reported before by taking the limits $\displaystyle r\ll 1$
and $\displaystyle r\gg 1$ for the coefficients (31). We have found that
$\displaystyle A_{2}^{(8),(L\ell\ell)}(r\ll 1)$, Eq. (58), $\displaystyle
A_{2}^{(8),(LL\ell)}(r\ll 1)$, Eq. (64), and $\displaystyle
A_{2}^{(8),(\ell\ell\ell)}(r\ll 1)$, Eq. (71), are entirely consistent with
the corresponding expressions reported in Refs. Aguilar:2008qj ; Kurz:2016bau
. For the sake of brevity, we do not present them here, mentioning only that
such expansions work surprisingly well in a large interval of $\displaystyle
r<1$, from $\displaystyle r\sim 0$ up to $\displaystyle r\sim 0.3-0.4$.

As for $\displaystyle r\gg 1$, the asymptotic expansion for $\displaystyle
A_{2}^{(8)}(r)$ has not been considered insofar. For this reason below we
write out explicitly our asymptotics, $\displaystyle r\gg 1$, of the
coefficients $\displaystyle A_{2}^{(8),(L\ell\ell)}(r\gg 1)$, $\displaystyle
A_{2}^{(8),(LL\ell)}(r\gg 1)$ and $\displaystyle
A_{2}^{(8),(\ell\ell\ell)}(r\gg 1)$, cf. Eqs. (58), (VI.2.2) and (VI.3.2):

$\displaystyle\displaystyle A_{2}^{(8),(L\ell\ell)}(r\gg
1)=\left[\frac{5809}{1080000}-\frac{61}{27000}\ln(r)+\frac{2}{225}\ln^{2}(r)\right]\frac{1}{r^{4}}+$
(78)
$\displaystyle\displaystyle\left[\frac{1862387}{277830000}-\frac{6073}{496125}\ln(r)+\frac{2}{175}\ln^{2}(r)\right]\frac{1}{r^{6}}+~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\displaystyle\left[\frac{12916049}{9001692000}-\frac{1940611}{100018800}\ln(r)+\frac{671}{7938}\ln^{2}(r)\right]\frac{1}{r^{8}}+{\cal
O}\left(\frac{1}{r^{10}}\right)\,,$

$\displaystyle\displaystyle A_{2}^{(8),(LL\ell)}(r\gg
1)=\bigg{[}\frac{16}{45}\,\zeta(3)-\frac{203}{486}\bigg{]}\frac{1}{r^{2}}+\bigg{[}\frac{17}{105}\zeta(3)-\frac{40783}{2315250}\ln(r)-\frac{1023526159}{5186160000}$
(79)
$\displaystyle\displaystyle-\frac{37}{11025}\ln^{2}(r)-\frac{2}{315}\ln^{3}(r)\bigg{]}\frac{1}{r^{4}}+\bigg{[}-{\frac{1243103\,}{93767625}}\ln\left(r\right)-{\frac{4744350631}{472588830000}}+{\frac{8}{945}}\,\zeta\left(3\right)+$
$\displaystyle\displaystyle{\frac{2122\,}{297675}}\ln^{2}\left(r\right)-{\frac{16\,}{2835}}\ln^{3}\left(r\right)\bigg{]}\frac{1}{{r}^{6}}+\bigg{[}\frac{8}{2079}\zeta(3)-\frac{1013327141}{99843767100}\ln(r)-$
$\displaystyle\displaystyle{\frac{8721404003611}{2767669224012000}}+{\frac{166657}{14407470}}\ln^{2}(r)-{\frac{16}{6237}}\ln^{3}(r)\bigg{]}\frac{1}{r^{8}}+{\cal
O}\left(\frac{1}{r^{10}}\right)\,,$ $\displaystyle\displaystyle
A_{2}^{(8),(\ell\ell\ell)}$ $\displaystyle\displaystyle(r\gg
1)=\bigg{[}\frac{87709}{9729720}-\frac{89}{15015}\zeta(3)\bigg{]}\frac{1}{r^{4}}+\bigg{[}\frac{12204667}{1824322500}-\frac{4}{1125}\ln(r)-\frac{40}{9009}\zeta(3)\bigg{]}\frac{1}{r^{6}}$
(80)
$\displaystyle\displaystyle+\bigg{[}\frac{73879547}{17721990000}-\frac{2}{375}\ln(r)-\frac{334}{109395}\zeta(3)\bigg{]}\frac{1}{r^{8}}+{\cal
O}\left(\frac{1}{r^{10}}\right)\,.$

NB: the asymptotics of the infinite sums $\displaystyle S_{2}(r)$, Eq. (67),
and $\displaystyle S_{4}(r)$, Eq. (77), which, as mentioned, converge rapidly
as $\displaystyle n\gg 1$, have been obtained by restricting the summation
index $\displaystyle n$ to a finite value $\displaystyle N\sim 5000$ and then
by taking the asymptotics $\displaystyle r\gg 1$.

Comparing the asymptotic expansion (78)-(80) with their corresponding exact
expressions one concludes that (78)-(80) provide basically the same
(numerical) results as the exact ones already starting from $\displaystyle
r\sim 2\div 2.5\,$.

It is worth mentioning that in computing integrals of type (59), one obtains,
depending on the method of integration used in dependence of the used,
analytical results in terms of various special functions, viz. polygammas
$\displaystyle\psi^{(n)}(r)$, polylogarithms $\displaystyle{\rm Li}_{n}(r)$,
harmonic functions $\displaystyle H(n,r)$, Hurwits-Lerch transcendent
$\displaystyle\Phi(r,n,a)$ etc. To reconcile different analytical results to
each other, the number of special functions involved in integration must be
maximally reduced by using the known relations among these special functions
(see Appendix A). In such a way, the asymptotical expressions for
$\displaystyle r\ll 1$ have been identically reduced to the previously
reported expansions. This further persuades us of the correctness of our
analytical analysis of the coefficients (30)-(31) and, consequently, that for
each coefficient in Eq (31) there is a corresponding analytical function valid
in the whole interval $\displaystyle r\in(0,\infty)$.

##  VIII Summary

In summary, we have presented an investigation of the contributions to the
anomalous magnetic moment of leptons $\displaystyle L$ ($\displaystyle
L=e\,,\,\mu$ or $\displaystyle\tau$) generated by QED diagrams with insertions
of one, two and three loops in the photon vacuum polarization operator. We
considered all possible combinations of external and internal leptons in the
bubble-like diagrams. The radiative corrections for each diagram are obtained
in close analytical forms. We argued that each coefficient $\displaystyle
A_{2}(r)$ determining the corresponding $\displaystyle 4$th, $\displaystyle
6$th and $\displaystyle 8$th order of the radiative corrections can be
represented explicitly by analytic functions which depend on the mass ratio
$\displaystyle r=m_{l}/m_{L}$, being different for different combinations of
the external $\displaystyle(L)$ and internal ($\displaystyle\ell$) leptons in
the Feynman diagrams of the corresponding order. The generic variable
$\displaystyle r$ of these functions is defined in the whole region
$\displaystyle 0<r<\infty$.

Our consideration is based on a combined use of the dispersion relations for
the vacuum polarization operators and the Mellin-Barnes integral transform for
the Feynman parametric integrals. This technique is widely used in the
literature in multi-loop calculations in relativistic quantum field theories,
c.f. Refs. Friot:2005cu ; Rafael-HVP ; Kotikov:2018wxe . The ultimate
integrations have been performed by the Cauchy residue theorem in the left
($\displaystyle r<1$) and right ($\displaystyle r>1$) semiplanes of the
complex Mellin variable $\displaystyle s$. We demonstrate that the results in
these two regions complement each other and determine the two branches of a
common, for each considered Feynman diagram, analytical function. We
investigated numerically the behaviour of these functions in the whole
interval $\displaystyle 0<r<\infty$ and classified for each lepton the
contribution of various diagrams in descending order of their significance. We
also showed that the diagrams with $\displaystyle r>1$, i.e. diagrams with
loops formed by heavier leptons, play a minuscule role in comparison with the
role of corrections in the interval $\displaystyle r<1$. However, in high
precision calculations such contributions can not be neglected.

Whenever pertinent, we compared our analytical expressions and the
corresponding asymptotical expansions with well-known results available in the
literature and found that they are fully compatible with the early known
calculations.

The present paper can be considered as further developments of the efforts
aimed at a better understanding of the role of bubble-like diagrams in the
lepton anomaly and as an extension of previously reported analysis to the all
three muons in the whole interval of the mass ratio $\displaystyle r$,
$\displaystyle(0\,<\,m_{l}/m_{L}\,<\,\infty)$. The performed analysis
persuades us that the Mellin-Barnes approach probably can be successfully
applied in calculating analogous diagrams with insertions, this time, of
hadron loops in the polarization operator.

## Acknowledgments

We gratefully acknowledge helpful discussions with A. L. Kataev and O. V.
Teryaev and their support of the present activity. We also thank A. V. Sidorov
for discussions and cooperation in the earlier stages of this work.

##  Appendix A Some useful relations

Direct employment of the Cauchy residue theorem to the integrals, Eqs. (40),
(50), (53), (57), (59), and (68) results in expressions containing a variety
of special functions, viz. polygammas $\displaystyle\psi^{(m)}(n)$,
polylogarithms $\displaystyle{\rm Li}_{n}(r)$, generalized harmonic functions
$\displaystyle H_{n}^{(m)}$, Hurwits-Lerch transcendent
$\displaystyle\Phi(r,n,a)$ etc. Using the herebelow relations, the number of
necessary special functions can be essentially reduced. This allows one to
reconcile explicitly our results to the ones known in the literature and
reported in different forms with different special functions, cf. Refs.
Friot:2005cu ; Laporta:1993ju ; Sidorov:2019 . Also, by using the appropriate
properties of the remaining functions for $\displaystyle r<1$ and
$\displaystyle r>1$, one can express the corresponding coefficients
$\displaystyle A_{2}(r<1)$ through the coefficients $\displaystyle
A_{2}(r<1)$, which allows one to assert that there exist, for each Feynman
diagram, a common analytical function valid in the whole interval
$\displaystyle(0\,<\,r\,\infty)$.

$\displaystyle\displaystyle\begin{array}[]{lr}\Phi(r,2,1/2)=\frac{2}{\sqrt{r}}\left[{\rm{Li}_{2}}(\sqrt{r})-{\rm{Li}_{2}}(-\sqrt{r})\right];\\\\[8.53581pt]
\Phi(r,2,3/2)=-\frac{4}{r}+\frac{2}{r\sqrt{r}}\left[{\rm{Li}_{2}}(\sqrt{r})-{\rm{Li}_{2}}(-\sqrt{r})\right];\\\\[8.53581pt]
\Phi(r,2,5/2)=-\frac{4}{9r}-\frac{4}{r^{2}}+\frac{2}{r^{2}\sqrt{r}}\left[{\rm{Li}_{2}}(\sqrt{r})-{\rm{Li}_{2}}(-\sqrt{r})\right];\\\\[8.53581pt]
{\rm{Li}_{2}}(1-r)+{\rm{Li}_{2}}\left(1-\frac{1}{r}\right)=-\frac{1}{2}\ln^{2}(r);\qquad\qquad{\rm(r>0)};\\\\[8.53581pt]
{\rm{Li}_{2}}(r)+{\rm{Li}_{2}}(1/r)=-\frac{\pi^{2}}{6}-\frac{1}{2}\ln^{2}(-r);\qquad\qquad{\rm(r>1)};\\\\[8.53581pt]
{\rm{Li}_{3}}(r)-{\rm{Li}_{3}}(1/r)=-\frac{\pi^{2}}{6}\ln(-r)-\frac{1}{6}\ln^{3}(-r);\qquad{\rm(r>1)}\\\\[8.53581pt]
{\rm{Li}_{3}}(r)-{\rm{Li}_{3}}(1/r)=\frac{\pi^{2}}{6}\ln(-1/r)+\frac{1}{6}\ln^{3}(-1/r);\qquad{\rm(r<1)}\\\\[8.53581pt]
{\rm{Li}_{4}}(r)+{\rm{Li}_{4}}(1/r)=-\frac{7\pi^{4}}{360}-\frac{\pi^{2}}{12}(\ln(-r))^{2}-\frac{1}{24}\ln^{4}(-r);\qquad\qquad{\rm(r>1)}\\\\[8.53581pt]
{\rm Li_{2}}\left(\frac{1-r}{1+r}\right)-{\rm
Li_{2}}\left(-\frac{1-r}{1+r}\right)={\rm Li_{2}}(-r)-{\rm
Li_{2}}(r)+\bigg{(}\ln(1+r)\ln(r)-\ln(1-r)\bigg{)}\ln(r)+\frac{\pi^{2}}{4};\\\\[14.22636pt]
{\rm Li_{n}}(r)+{\rm Li_{n}}(-r)=\frac{1}{2^{n-1}}{\rm
Li_{n}}(r^{2}).\\\\[8.53581pt] \end{array}$ (91)
$\displaystyle\displaystyle{\rm
arctanh(r)}=\frac{1}{2}\bigg{[}\ln(1+r)-\ln(1-r)\bigg{]};$ (92)
$\displaystyle\displaystyle{\rm H_{n}^{(1)}}=\psi^{(1)}(n+1)+\gamma;\quad{\rm
H}_{n}^{(2)}=\frac{\pi^{2}}{6}-\psi^{(1)}(n+1),$ (93)

where $\displaystyle{\rm Li_{n}}(r)$, $\displaystyle\Phi(r,s,a)$,
$\displaystyle{\rm H_{n}^{(m)}}$ and $\displaystyle\psi^{(m)}(n)$ are the
polylogaritm, Lerch transcendent, generalized harmonic number and Euler
polygamma functions, respectively.

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