DESY 22–024

DO-TH 22/04

TTP 22–011

SAGEX 22–19

High Precision QED Initial State Corrections for

$e^{+}e^{-}\rightarrow\gamma^{*}/Z^{*}$ Annihilation

J. Blümleina and K. Schönwalda,b

a Deutsches Elektronen–Synchrotron DESY,
Platanenallee 6, 15738 Zeuthen, Germany

b Institut für Theoretische Teilchenphysik,
Karlsruher Institut für Technologie (KIT), D-76128 Karlsruhe, Germany

###### Abstract

The precise knowledge of the QED initial state corrections is of instrumental
importance in studying high luminosity measurements in $e^{+}e^{-}$
annihilation at facilities like LEP, the International Linear Collider ILC,
CLIC, a Giga-$Z$ facility, and the planned FCC_ee. Logarithmic corrections of
up to $O(\alpha^{6}L^{5})$ are necessary with various subleading terms taken
into account. This applies to both the inclusive measurement of processes like
$e^{+}e^{-}\rightarrow\gamma^{*}/Z^{*}$ and also the forward–backward
asymmetry. As has been shown recently, techniques from massive QCD, such as
the computation of massive on–shell operator matrix elements, can be used for
these calculations. We give an introduction to this topic and present both the
calculation methods and the numerical corrections having been reached so far.

##  1 Introduction

The precision physics for the process $e^{+}e^{-}\rightarrow\gamma^{*}/Z^{*}$
started at LEP-1 in the early 1990ies leading to a highly precise measurement
of the $Z$–peak [1]. Among other radiative corrections the QED initial state
corrections are of high importance in these measurements, since they change
the shape–parameters of the $Z$–peak significantly. The most far–reaching
calculation of the initial state QED corrections (ISR) of $O(\alpha^{2})$,
with $\alpha=4\pi a$ the fine structure constant, had been performed in Ref.
[2] at that time, which had been a theoretical challenge and milestone. This
calculation has never been checked until 2011, some time after LEP had already
been ended, when the same result was derived using massive operator matrix
elements (OMEs) in Ref. [3]. Massive operator elements were known to present
all contributions except power corrections in QCD in the cases with external
massless partons [4]. Therefore, one could try to apply this method also in
the case of massive fermions, the electrons. However, the present case is one
with also massive external lines. After correcting typographical errors in [2]
all logarithmic contributions of both calculations [2, 3] and the constant
term at $O(\alpha)$ agreed, but not the result for the constant term of
$O(\alpha^{2})$. Furthermore, in one of the subprocesses Ref. [2] agreed with
[5]. Due to this, one option to explain the observed differences has been that
the method of massive OMEs does not work in the case of massive external
lines, or, otherwise, that the results in Refs. [2, 5] need corrections.

One way to unambiguously resolve these discrepancies has been to repeat the
complete calculation to $O(\alpha^{2})$ without neglecting any term at
intermediate steps and carry out the limit $m_{e}^{2}/s\rightarrow 0$ for the
power corrections only in the final results. Such a calculation has not been
fully possible in 2011, but became accessible in 2019, again working on
complete QCD calculation in which specific phase space integrals occurred. The
organization of these integrals as iterated integrals over various root–valued
letters, cf. e.g. [6, 7], and their reduction using shuffle algebras [8],
brought the solution to also organize the phase–space integrals in the case of
QED. In this way, we firstly found the correct results for the fermionic
non–singlet corrections of [2, 5], if the radiated particles are electrons,
but not muons, [9]. While our result agreed with [2, 5] for muons, it
disagreed for electron radiation, since there electron mass terms had been
neglected too early in Refs. [2, 5]. For Ref. [2] this also applied to the
pure singlet channel, which has been calculated earlier in the massless case
in QCD in Ref. [10], and been used as such in [2]. However, also here electron
mass terms cannot be neglected. We also newly calculated the bosonic
contributions in [11, 12], and fully confirmed the results given in Ref. [3]
by the phase space integration method intended to be used in Ref. [2]. Through
this we proofed that the method of massive OMEs can also be used in the case
of external massive fermionic lines.

The technique of massive OMEs now also allows to access logarithmically
enhanced terms through the renormalization group. With the known pieces it is
possible to obtain three consecutive logarithmic orders in all higher orders
in $\alpha$, as e.g. $O(\alpha^{3}L^{3}),O(\alpha^{3}L^{2}),O(\alpha^{3}L)$,
etc., with $L=\ln(s/m_{e}^{2})$ and $s$ the $cms$ energy. We applied this
technique to all corrections up to $O(\alpha^{6}L^{5})$ in Ref. [13] and to
higher order corrections to the forward–backward asymmetry [14], although in
this case new contributions have to be calculated. We also note that the
universal QED corrections $O(\alpha^{k}L^{k})$ are known to at least fifth
order $k=5$, cf. [15, 16, 17, 18, 19, 20, 21, 22]. Highly precise QED
radiative corrections are of importance for the analysis of the precision
measurements at LEP, [1], the planned international linear collider, ILC, [23,
24, 25] CLIC [26], the FCC_ee [27] and muon colliders [28].

The paper is organized as follows. In Section 2 we describe the direct
calculation of the $O(\alpha^{2})$ corrections due to initial state radiation
and the factorized approach by using massive OMEs, which lead to the same
results. For the $O(\alpha^{2})$ ISR corrections we present numerical results
on the $Z^{0}$-peak and width, for the process $e^{+}e^{-}\rightarrow Z^{0}H$
and $t\bar{t}$ production in the threshold region. The method of massive OMEs
is then extended to higher orders for the leading and two subleading
logarithmic terms in Section 3. In Section 4 we present precision results for
the forward–backward asymmetry and Section 5 contains the conclusions. We also
present numerical results on the calculated radiative corrections for
different key observables.

##  2 Two–loop Corrections

After observing the differences in the results of Refs. [3] and [2], the only
way to clarify this consisted in a direct calculation of the complete
$O(a^{2})$ initial state corrections to
$e^{+}e^{-}\rightarrow\gamma^{*}/Z^{*}$ annihilation. Its result confirmed the
results of Ref. [3]. Due to this the explicit high energy factorization could
be shown to $O(a^{2})$ also for massive external fermions. Assuming this
factorization to even higher orders, allows to calculate the first three
logarithmic correction terms to even higher order. These results are reported
in Section 3. In the following we first summarize the complete $O(a^{2})$
calculation in Section 2.1 and turn then to the calculation based on massive
OMEs in Section 2.2. Finally, we derive numerical results on the $O(a^{2})$
corrections for a series of key processes to be measured at future facilities.

###  2.1 Two–loop Corrections: the Direct Calculation

Upon neglecting power corrections of $O(m_{e}^{2}/s)$ the initial state QED
corrections can be written in terms of the following functions

$\displaystyle H\left(z,\alpha,\frac{s}{m^{2}}\right)$ $\displaystyle=$
$\displaystyle\delta(1-z)+\sum_{k=1}^{\infty}\left(\frac{\alpha}{4\pi}\right)^{k}C_{k}\left(z,\frac{s}{m^{2}}\right)$
(1) $\displaystyle C_{k}\left(z,\frac{s}{m^{2}}\right)$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{k}\ln^{k-l}\left(\frac{s}{m^{2}}\right)c_{k,l}(z),$
(2)

which yield the respective differential cross sections by

$\displaystyle\frac{d\sigma_{e^{+}e^{-}}}{ds^{\prime}}=\frac{1}{s}\sigma_{e^{+}e^{-}}(s^{\prime})H\left(z,\alpha,\frac{s}{m^{2}}\right),$
(3)

with $\sigma_{e^{+}e^{-}}(s^{\prime})$ the scattering cross section without
the ISR corrections, $\alpha\equiv\alpha(s)$ the fine structure constant and
$z=s^{\prime}/s$, where $s^{\prime}$ is the invariant mass of the produced
(off-shell) $\gamma/Z$ boson.

The Born cross section is given by

$\displaystyle\sigma^{(0)}_{e^{+}e^{-}}(s)$ $\displaystyle=$
$\displaystyle\frac{4\pi\alpha^{2}}{3s}N_{C,f}\sqrt{1-\frac{4m_{f}}{s}}\left[\left(1+\frac{2m_{f}^{2}}{s}\right)G_{1}(s)-6\frac{m_{f}^{2}}{s}G_{2}(s)\right],$
(4)

see e.g. [29], where $N_{C,f}$ denotes the color factor for the final state
fermions, i.e. $N_{C}=3$ for quarks and $N_{C}=1$ for leptons. The effective
couplings $G_{i}(s)|_{i=1...2}$ read

$\displaystyle G_{1}(s)$ $\displaystyle=$ $\displaystyle
Q_{e}^{2}Q_{f}^{2}+2Q_{e}Q_{f}v_{e}v_{f}{\sf
Re}[\chi_{Z}(s)]+(v_{e}^{2}+a_{e}^{2})(v_{f}^{2}+a_{f}^{2})|\chi_{Z}(s)|^{2},$
(5) $\displaystyle G_{2}(s)$ $\displaystyle=$
$\displaystyle(v_{e}^{2}+a_{e}^{2})a_{f}^{2}|\chi_{Z}(s)|^{2},$ (6)

where the reduced $Z$–propagator is given by

$\displaystyle\chi_{Z}(s)=\frac{s}{s-M_{Z}^{2}+iM_{Z}\Gamma_{Z}},$ (7)

with $M_{Z}$ and $\Gamma_{Z}$ the mass and the with of the $Z$–boson and
$m_{f}$ is the mass of the final state fermion. $Q_{e,f}$ are the
electromagnetic charges of the electron $(Q_{e}=-1)$ and the final state
fermion, respectively, and the electro–weak couplings $v_{i}$ and $a_{i}$ are
given by

$\displaystyle v_{e}$ $\displaystyle=$
$\displaystyle\frac{1}{\sin\theta_{w}\cos\theta_{w}}\left[I^{3}_{w,e}-2Q_{e}\sin^{2}\theta_{w}\right],$
(8) $\displaystyle a_{e}$ $\displaystyle=$
$\displaystyle\frac{1}{\sin\theta_{w}\cos\theta_{w}}I^{3}_{w,e},$ (9)
$\displaystyle v_{f}$ $\displaystyle=$
$\displaystyle\frac{1}{\sin\theta_{w}\cos\theta_{w}}\left[I^{3}_{w,f}-2Q_{f}\sin^{2}\theta_{w}\right],$
(10) $\displaystyle a_{f}$ $\displaystyle=$
$\displaystyle\frac{1}{\sin\theta_{w}\cos\theta_{w}}I^{3}_{w,f}\leavevmode\nobreak\
,$ (11)

where $\theta_{w}$ is the weak mixing angle, and $I^{3}_{w,i}=\pm 1/2$ the
third component of the weak isospin for up and down particles, respectively.

One may distinguish four processes: i) photon radiation, ii) $e^{+}e^{-}$ pair
emission, iii) fermionic pure singlet corrections, and iv) the interference
terms between the diagrams contributing to ii) and iii). Furthermore, there is
also $\mu^{+}\mu^{-}$ initial state radiation and that of light quarks. The
latter contribution belongs to the QCD corrections, and we will label the one
of $\mu^{+}\mu^{-}$ pair emission as process v). It has been described in
Refs. [2, 5] correctly, as has been confirmed in Ref. [9, 11]. We will
consider the processes i)–iv) in the following.

The phase–space integrals to be performed are organized as iterated integrals
over a set of letters, which are found by solving associated differential
equations, cf. [30]. The alphabet consists of letters forming generalized
harmonic polylogarithms [31], but also of elliptic letters. Since all
integrations are indefinite, still iterative integrals are obtained. We
therefore call the corresponding integrals Kummer–elliptic integrals. The
respective iterative integrals have the structure

$\displaystyle H_{b,\vec{a}}(z)=\int_{0}^{z}dyf_{b}(y)H_{\vec{a}}(y).$ (12)

Examples for these letters are, cf. [11] (using the abbreviation
$f_{b}(t;z,\rho)\equiv f_{b}$),

$\displaystyle d_{24}$ $\displaystyle=$
$\displaystyle\frac{1}{\big{(}t^{2}(1-z)^{2}-8\rho(1+z)t+4tz+16\rho^{2}\big{)}\sqrt{t^{2}(1-z)^{2}-8\rho
t(1+z)+16\rho^{2}}},$ (13) $\displaystyle d_{26}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1-t}\big{(}t^{2}(1-z)^{2}-8\rho(1+z)t+4tz+16\rho^{2}\big{)}\sqrt{t^{2}(1-z)^{2}-8\rho
t(1+z)+16\rho^{2}}}.$ (14)

These integrals depend on the tiny parameter $\rho=m_{e}^{2}/s$ and one may
now expand in $\rho$ controlling the accuracy by high precision numerics for
the unexpanded integrals at each step. One finally arrives at iterated
integrals which integrate in terms of harmonic polylogarithms or even
classical polylogarithms and Nielsen integrals [32, 33, 34, 35, 36, 37].

The size of the amplitudes to be dealt with amounts to 10 Gb (process i)), 25
kb (process ii)), 56 kb (process iii)) and 124 kb (process iv)), requiring
several months for code design and 30h of computation time. The calculation
could only be done by using computer–algorithmic methods, which have been
developed very recently.

The 2–loop corrections for process i) consist out of soft (S), virtual (V) and
hard (H) corrections and their interference terms [11] which can be written
schematically by

$\displaystyle R^{\gamma\gamma}_{2}=T_{2}^{\rm S_{2}}+T_{2}^{\rm
V_{2}}+T_{2}^{\rm S_{1}V_{1}}+T_{2}^{\rm S_{1}H_{1}}+T_{2}^{\rm
V_{1}H_{1}}+T_{2}^{\rm H_{2}}.$ (15)

We agree with [2] for all but the virtual–hard contributions, i.e. $T_{2}^{\rm
V_{1}H_{1}}$. The radiator $R_{2}^{\gamma\gamma}$ is given by

$\displaystyle R^{\it i)}_{2}(z)$ $\displaystyle=$ $\displaystyle
R_{2}^{\gamma\gamma}(z)$ (16) $\displaystyle=$
$\displaystyle\delta(1-z)\Bigl{\\{}32(L-1)^{2}\ln^{2}(\varepsilon)+\big{(}48L^{2}-(112-64\zeta_{2})L+64-64\zeta_{2}\big{)}\ln(\varepsilon)$
$\displaystyle+(18-32\zeta_{2})L^{2}-(45-88\zeta_{2}-48\zeta_{3})L+76+(6-96\ln(2))\zeta_{2}-72\zeta_{3}-\frac{96}{5}\zeta_{2}^{2}\Bigr{\\}}$
$\displaystyle+\theta(1-z-\varepsilon)\Bigl{\\{}64(L-1)^{2}{\cal
D}_{1}+\big{(}48L^{2}-(112-64\zeta_{2})L+64-64\zeta_{2}\big{)}{\cal D}_{0}$
$\displaystyle-L^{2}\biggl{(}8(5+z)+32(1+z)\ln(1-z)+\frac{8\big{(}1+3z^{2}\big{)}}{1-z}\ln(z)\biggr{)}+L\biggl{(}8(14+z)$
$\displaystyle+\frac{8\big{(}5+2z+7z^{2}\big{)}}{1-z}\ln(z)-\frac{4\big{(}1+3z^{2}\big{)}}{1-z}\ln^{2}(z)+\frac{16\big{(}1+z^{2}\big{)}}{1-z}\text{Li}_{2}(1-z)$
$\displaystyle-32(1+z)\zeta_{2}+\biggl{[}64(1+z)+\frac{16\big{(}1+z^{2}\big{)}}{1-z}\ln(z)\biggr{]}\ln(1-z)\biggr{)}-\frac{8\big{(}18+z-15z^{2}\big{)}}{3(1-z)}$
$\displaystyle-\frac{8\big{(}1+z^{2}\big{)}}{3(1-z)}\ln^{3}(z)+\frac{4}{3(1-z)^{3}}\big{(}12-33z+51z^{2}-51z^{3}+13z^{4}\big{)}\ln^{2}(z)$
$\displaystyle+\frac{32}{3}(3+8z)\zeta_{2}-32(1+z)\zeta_{3}-\biggl{(}16(1+3z)+\frac{8\big{(}2+6z-3z^{2}\big{)}}{1-z}\ln(z)$
$\displaystyle-\frac{8\big{(}3-z^{2}\big{)}}{1-z}\ln^{2}(z)\biggr{)}\ln(1-z)+16z\ln^{2}(1-z)-\biggl{(}\frac{8\big{(}6+3z+26z^{2}-27z^{3}\big{)}}{3(1-z)^{2}}$
$\displaystyle+\frac{16\big{(}1-3z^{2}\big{)}}{1-z}\zeta_{2}\biggr{)}\ln(z)+\biggl{(}-\frac{8\big{(}9+19z-13z^{2}\big{)}}{3(1-z)}-16(1+z)\ln(1-z)$
$\displaystyle+\frac{8\big{(}5-3z^{2}\big{)}}{1-z}\ln(z)\biggr{)}\text{Li}_{2}(1-z)+24(1+z)\text{Li}_{3}(1-z)+32(1+z)\text{Li}_{3}(z)\Bigr{\\}},$

with

$\displaystyle{\cal D}_{k}(z)=\left(\frac{\ln^{k}(1-z)}{1-z}\right)_{+}.$ (17)

Here the separation parameter $\varepsilon$ has been introduced to realize the
$+$-distributions in terms of functions. It disappears after the $z$-integral
has been performed.

Another radiator is the one of the non–singlet process for $e^{+}e^{-}$
emission, $R^{\it ii)}_{2}$. The correction to the scattering cross section
for electron pair emission is given by

$\displaystyle R_{2}^{\it ii)}(z)$
$\displaystyle=\Biggl{\\{}\frac{64}{3}z(1-z)(1+z-4\rho){\rm
H}^{*}_{v_{4},d_{7}}+\frac{256}{3}z\rho(1+z-4\rho){\rm H}^{*}_{v_{4},d_{6}}$
$\displaystyle+\frac{128z(1-4\rho^{2})(1-z+2\rho)(1-z-4\rho)}{3(1-z)^{2}}{\rm
H}^{*}_{d_{8},d_{7}}$
$\displaystyle+\frac{512z\rho(1-4\rho^{2})(1-z+2\rho)(1-z-4\rho)}{3(1-z)^{3}}{\rm
H}^{*}_{d_{8},d_{6}}$
$\displaystyle+\frac{16}{9(1-z)^{2}}\Bigl{[}(1+z)^{2}\big{(}4-9z+4z^{2}\big{)}+2\big{(}9-16z+13z^{2}-2z^{3}\big{)}\rho+32\rho^{2}\Bigr{]}{\rm
H}^{*}_{d_{2}}$
$\displaystyle+\frac{512z\rho}{9(1-z)^{4}}\Bigl{[}3(1-z)^{4}z-(1-z)^{3}\big{(}4+z^{2}\big{)}\rho-2\big{(}9-29z+38z^{2}-17z^{3}+3z^{4}\big{)}\rho^{2}$
$\displaystyle-4(2-z)\big{(}3+6z-5z^{2}\big{)}\rho^{3}+16\big{(}7-8z+9z^{2}\big{)}\rho^{4}+128(3-z)\rho^{5}\Bigr{]}{\rm
H}^{*}_{d_{4}}$
$\displaystyle-\frac{16}{9(1-z)^{4}}\Bigl{[}3-34z+129z^{2}-212z^{3}+129z^{4}-34z^{5}+3z^{6}+8\big{(}2-16z+9z^{2}$
$\displaystyle+4z^{3}-5z^{4}+2z^{5}\big{)}\rho+16z\big{(}12-13z+18z^{2}-z^{3}\big{)}\rho^{2}+32\big{(}1+22z-7z^{2}\big{)}\rho^{3}\Bigr{]}{\rm
H}^{*}_{d_{1}}$
$\displaystyle-\frac{128z}{9(1-z)^{4}}\Bigl{[}1+7z-47z^{2}+86z^{3}-47z^{4}+7z^{5}+z^{6}-2\big{(}7-55z+54z^{2}$
$\displaystyle+16z^{3}-17z^{4}+3z^{5}\big{)}\rho-4\big{(}39-16z+16z^{2}+4z^{3}+5z^{4}\big{)}\rho^{2}$
$\displaystyle+16\big{(}8-23z+22z^{2}+9z^{3}\big{)}\rho^{3}+128\big{(}7+2z-z^{2}\big{)}\rho^{4}\Bigr{]}{\rm
H}^{*}_{d_{5}}-\frac{64}{3}(2z+(1-z)\rho){\rm H}^{*}_{d_{3}}$
$\displaystyle+\biggl{[}\frac{16}{3\sqrt{1-4\rho}}(1+z-4\rho){\rm
H}^{*}_{v_{4}}+\frac{32(1-4\rho^{2})(1-z+2\rho)(1-z-4\rho)}{3(1-z)^{3}\sqrt{1-4\rho}}{\rm
H}^{*}_{d_{8}}\biggr{]}$
$\displaystyle\times\ln\left(\frac{1-z-4\rho-\sqrt{1-4\rho}\sqrt{(1-z)^{2}-8(1+z)\rho+16\rho^{2}}}{1-z-4\rho+\sqrt{1-4\rho}\sqrt{(1-z)^{2}-8(1+z)\rho+16\rho^{2}}}\right)\Biggr{\\}}.$
(18)

It can even be obtained without expanding in the parameter $\rho$. Here the
iterative integrals ${\rm H}^{*}$ read

$\displaystyle{\rm H}^{*}_{b,\vec{a}}(z)=\int_{z}^{1}dyf_{b}(y){\rm
H}^{*}_{\vec{a}}(y)\leavevmode\nobreak\ $ (19)

and the respective letters are given in Ref. [11].

Figure 1: Relative deviations of the results of Ref. [2] from the exact result
in % for the $O(\alpha^{2})$ corrections. The non–singlet contribution
(process II): dash-dotted line; the pure singlet contribution (process III):
dashed; the interference term between both contributions (process IV): dots;
for $s=M_{Z}^{2}$, $M_{Z}=91.1879$ GeV; from [9].

After the expansion in $\rho$ one obtains

$\displaystyle R_{2}^{\it ii)}(z)$
$\displaystyle=\frac{8\big{(}1+z^{2}\big{)}}{3(1-z)}L^{2}+\biggl{[}-\frac{16\big{(}11-12z+11z^{2}\big{)}}{9(1-z)}+\frac{32\big{(}1+z^{2}\big{)}}{3(1-z)}\ln(1-z)$
$\displaystyle-\frac{16\big{(}1+z^{2}\big{)}}{3(1-z)}\ln(z)\biggr{]}L+\frac{32\big{(}1+z^{2}\big{)}}{3(1-z)}\ln^{2}(1-z)-\frac{16z^{2}}{3(1-z)}\text{Li}_{2}(1-z)$
$\displaystyle+\frac{32}{9(1-z)^{3}}\big{(}7-13z+8z^{2}-13z^{3}+7z^{4}\big{)}-\frac{16z}{9(1-z)^{4}}\big{(}3-36z+94z^{2}$
$\displaystyle-72z^{3}+19z^{4}\big{)}\ln(z)-\frac{32\big{(}1+z^{2}\big{)}}{3(1-z)}\zeta_{2}-\biggl{(}\frac{32\big{(}11-12z+11z^{2}\big{)}}{9(1-z)}$
$\displaystyle+\frac{32\big{(}1+z^{2}\big{)}}{3(1-z)}\ln(z)\biggr{)}\ln(1-z)-\frac{8z^{2}}{3(1-z)}\ln^{2}(z)+\mathcal{O}\left(\rho
L^{2}\right).$ (20)

This expression differs from those given in [2, 5]. The other radiators are
given in Ref. [11].

Let us compare the fermionic corrections to the non–logarithmic contributions
at $O(a^{2})$ for processes ii)–iv) of Refs. [2] and [9, 11]. In Figure 1 we
show the relative deviation of the former result from the latter.

Both the non–singlet and pure singlet terms grow towards small values of $z$.
In the latter case this is caused also by a deviation in the $1/z$
contributions in [2]. The difference in the interference term, showing various
structural differences, first grows and then drops in the large $z$ region,
taking negative values.

Figure 2 shows the non–logarithmic contributions to the radiators at
$O(a^{2})$, weighted by the function $z(1-z)$ for convenience, to illustrate
the relative impact of the contributions from processes ii)–iv).

Figure 2: The initial state $O(\alpha^{2})$ corrections to $\gamma^{*}/Z^{*}$
production due to $e^{+}e^{-}$ pair production multiplied by $z(1-z)$. The
non–singlet contribution (process II): dash-dotted line; the pure singlet
contribution (process III): dashes; the interference term between both
contributions (process IV) $\times 10$: dotted; the vector contributions
implied by process B, Ref. [38], and interferences $\times 100$: long dash-
dotted; all contributions: full line for $s=M_{Z}^{2}$; from [9].

In the small $z$ region up to about $z\sim 0.2$ the pure singlet contribution
dominates. For still larger values, the non–singlet contribution dominates and
at intermediate values there is a negative contribution due to the
interference terms of both. There is another contribution at $O(a^{2})$ having
no logarithmic parts, which has not been considered in [2] but in [38], which
is rather small. We would also like to mention that the radiative corrections
are different for the vector–axial vector interference contributions if
compared to the pure vector and pure axial vector contributions, which has
been considered in Refs. [9, 11].

In this and the subsequent calculations reported in this survey we have
extensively used the packages FORM [39] Sigma [40, 41], EvaluateMultiSum [42],
HarmonicSums [43, 8, 49, 31, 44, 46, 47, 48, 45], HolonomicFunctions [50], and
private implementations [30] to calculate the respective integrals
analytically.

###  2.2 The Method of Massive Operator Matrix Elements

The method of massive operator matrix elements has been introduced in QCD to
calculate heavy flavor corrections to deep–inelastic scattering at two–loop
order for all but contributions due to power corrections of
$O((m^{2}_{Q}/Q^{2})^{k}),k\geq 1$, with $Q^{2}$ the virtuality of the
exchanged gauge boson and $m_{Q}$ the heavy quark mass [4]. It is also
applicable to massive Drell–Yan processes as studied in the present paper and
has been applied in [2], showing that it works at one–loop order and for all
logarithmic contributions at two–loop order111Upon correcting typographical
errors in [2] in Ref. [3].. However, there has not been a clear conclusion in
[2], whether it works also for the non–logarithmic contribution at $O(a^{2})$.
It has been found in [3], that the $O(a^{2})$ terms given in [2] are indeed
not reproduced. For us it initially caused doubts on its inapplicability,
because this factorization might not be present in the massive Drell–Yan
process, although it is expected. The complete result obtained in Ref. [6],
see Section 2.1, has then fully confirmed our earlier result in the factorized
approach [3]. In the following we will outline the different steps in this
approach in detail, also for later use at even higher orders [13], see Section
3.

The general decomposition of the scattering cross section in Mellin space is
given by, cf. [3]222In the massless case the principle solution of the
renormalization group equations (RGEs) to general orders has been known for
long, see [52, 53].

$\displaystyle\frac{d\sigma_{e^{+}e^{-}}}{ds^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{1}{s}\sigma^{(0)}(s^{\prime})\Biggl{[}\Gamma_{e^{+}e^{+}}\left(N,\frac{\mu^{2}}{m_{e}^{2}}\right)\tilde{\sigma}_{e^{+}e^{-}}\left(N,\frac{s^{\prime}}{\mu^{2}}\right)\Gamma_{e^{-}e^{-}}\left(N,\frac{\mu^{2}}{m_{e}^{2}}\right)$
(21) $\displaystyle+\Gamma_{\gamma
e^{+}}\left(N,\frac{\mu^{2}}{m_{e}^{2}}\right)\tilde{\sigma}_{e^{-}\gamma}\left(N,\frac{s^{\prime}}{\mu^{2}}\right)\Gamma_{e^{-}e^{-}}\left(N,\frac{\mu^{2}}{m_{e}^{2}}\right)$
$\displaystyle+\Gamma_{e^{+}e^{+}}\left(N,\frac{\mu^{2}}{m_{e}^{2}}\right)\tilde{\sigma}_{e^{+}\gamma}\left(N,\frac{s^{\prime}}{\mu^{2}}\right)\Gamma_{\gamma
e^{-}}\left(N,\frac{\mu^{2}}{m_{e}^{2}}\right)$ $\displaystyle+\Gamma_{\gamma
e^{+}}\left(N,\frac{\mu^{2}}{m_{e}^{2}}\right)\tilde{\sigma}_{\gamma\gamma}\left(N,\frac{s^{\prime}}{\mu^{2}}\right)\Gamma_{\gamma
e^{-}}\left(N,\frac{\mu^{2}}{m_{e}^{2}}\right)\Biggr{]}.$

The terms in the brackets $[...]$ are Mellin–convoluted. Only massive OMEs of
the kind $\Gamma_{e^{\pm}e^{\pm}}$ and $\Gamma_{\gamma e^{\pm}}$ contribute
because the process considered has electron–positron initial states and the
$\tilde{\sigma}_{\gamma\gamma}$ terms appear only from $O(a^{4})$ onward. In
the factorization also the massless Wilson coefficients for the Drell-Yan
process $\tilde{\sigma}_{ij}$, $i,j=e^{\pm},\gamma$ contribute. The
factorization scale $\mu$ cancels after expanding in $a_{0}=a(m_{e}^{2})$. The
following expansions hold for the massive OMEs and massless Wilson
coefficients

$\displaystyle\Gamma_{li}\left(N,\frac{\mu^{2}}{m_{e}^{2}}\right)$
$\displaystyle=$
$\displaystyle\delta_{li}+\sum_{r=1}^{\infty}a^{r}(\mu^{2})\sum_{n=0}^{r}a_{li;nr}(N)\Lambda^{n}$
(22)
$\displaystyle\tilde{\sigma}_{lk}\left(N,\frac{s^{\prime}}{\mu^{2}}\right)$
$\displaystyle=$
$\displaystyle\delta_{lk}+\sum_{r=1}^{\infty}a^{r}(\mu^{2})\sum_{n=0}^{r}b_{lk;nr}(N)\lambda^{n},$
(23)

with the logarithms $\Lambda$ and $\lambda$ given by

$\displaystyle\Lambda=\ln\left(\frac{\mu^{2}}{m_{e}^{2}}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\lambda=\ln\left(\frac{s^{\prime}}{\mu^{2}}\right).$ (24)

The massive OMEs $\Gamma_{ij}$ and massless Wilson coefficients
$\tilde{\sigma}_{kl}$ fulfill the following renormalization group equations,
cf. [51],

$\displaystyle\left[\left(\frac{\partial}{\partial\Lambda}+\beta(a)\frac{\partial}{\partial
a}\right)\delta_{kl}+\frac{1}{2}\gamma_{kl}(N,a)\right]\Gamma_{li}\left(N,a,\frac{\mu^{2}}{m_{e}^{2}}\right)$
$\displaystyle=$ $\displaystyle 0$ (25)
$\displaystyle\left[\left(\frac{\partial}{\partial\lambda}-\beta(a)\frac{\partial}{\partial
a}\right)\delta_{kl}\delta_{jm}+\frac{1}{2}\gamma_{kl}(N,a)\delta_{jm}+\frac{1}{2}\gamma_{jm}(N,a)\delta_{kl}\right]\tilde{\sigma}_{lj}\left(N,a,\frac{s^{\prime}}{\mu^{2}}\right)$
$\displaystyle=$ $\displaystyle 0\leavevmode\nobreak\ ,$ (26)

and the QED $\beta$ function has the representation

$\displaystyle\beta(a)=-\sum_{k=0}^{\infty}\beta_{k}a^{k+2},$ (27)

with $\beta_{0}=-4/3,\leavevmode\nobreak\ \beta_{1}=-4$.

The calculation of the scattering cross section (21) to $O(a_{0}^{2})$
requires the knowledge of the expansion coefficients of the massless Drell–Yan
cross section [38, 54, 9] to $O(a_{0}^{2})$, the one– and two–loop anomalous
dimensions $\gamma_{ij}^{(k)}$ [55], and the massive OMEs
$\Gamma_{ee}^{(0)},\Gamma_{e\gamma}^{(0)}$ and $\Gamma_{ee}^{(1)}$. The latter
quantity needs to be known in its contributions to processes i) + iv), ii) and
iii). The major task in Ref. [3] has been to calculate $\Gamma_{ee}^{(1)}$. At
that time, many of the automated methods and function spaces found in later
massive calculations have been not yet known, cf. [56], and majorly
hypergeometric integration techniques [57] were used. Checks have been
performed using the package tarcer [58] and detailed lists of special
integrals had to be calculated, cf. [3].

As an example, we show the massive OME $\hat{\hat{A}}_{ee}^{\rm III}$, from
which $\Gamma_{ee}^{\rm III}$ is obtained,

$\displaystyle\hat{\hat{A}}_{ee}^{(2),\rm III}$ $\displaystyle=$
$\displaystyle
S_{\varepsilon}^{2}\int^{1}_{0}dx\,\,x^{N}\frac{1+(-1)^{N}}{2}\left\\{\phantom{\rule{0.28453pt}{16.21828pt}}\\!\right.\frac{8}{\varepsilon^{2}}\left[\frac{1-x}{3x}(4x^{2}+7x+4)+2(1+x)\ln(x)\right]$
(28)
$\displaystyle\phantom{\int^{1}_{0}dx\,\,x^{n}\left\\{\phantom{\rule{0.28453pt}{16.21828pt}}\\!\right.}+\frac{4}{\varepsilon}\left[\phantom{\rule{0.28453pt}{15.93347pt}}\\!\right.5(1+x)\ln^{2}(x)-\frac{1+x}{3x}(8x^{2}-17x-16)\ln(x)+\frac{4(1-x)}{9x}(5x^{2}$
$\displaystyle\phantom{\int^{1}_{0}dx\,\,x^{n}\left\\{\phantom{\rule{0.28453pt}{16.21828pt}}\\!\right.}+23x+14)\left.\\!\phantom{\rule{0.28453pt}{15.93347pt}}\right]+\frac{2}{x}(1-x)(4x^{2}+13x+4)\zeta_{2}+\frac{1}{3x}(8x^{3}+135x^{2}+75x$
$\displaystyle\phantom{\int^{1}_{0}dx\,\,x^{n}\left\\{\phantom{\rule{0.28453pt}{16.21828pt}}\\!\right.}+32)\ln^{2}(x)+\left[\frac{304}{9x}-\frac{80}{9}x^{2}-\frac{32}{3}x+108-\frac{32}{1+x}-\frac{64(1+2x)}{3(1+x)^{3}}\right]\ln(x)$
$\displaystyle\phantom{\int^{1}_{0}dx\,\,x^{n}\left\\{\phantom{\rule{0.28453pt}{16.21828pt}}\\!\right.}-\frac{224}{27}x^{2}+16\frac{1-x}{3x}(x^{2}+4x+1)\left[2\ln(x)\ln(1+x)-{\rm
Li}_{2}(1-x)\right.$
$\displaystyle\phantom{\int^{1}_{0}dx\,\,x^{n}\left\\{\phantom{\rule{0.28453pt}{16.21828pt}}\\!\right.}\left.+2{\rm
Li}_{2}(-x)\right]+(1+x)\left[\phantom{\rule{0.28453pt}{15.93347pt}}\\!\right.4\zeta_{2}\ln(x)+\frac{14}{3}\ln^{3}(x)-32\ln(x){\rm
Li}_{2}(-x)$
$\displaystyle\phantom{\int^{1}_{0}dx\,\,x^{n}\left\\{\phantom{\rule{0.28453pt}{16.21828pt}}\\!\right.}-16\ln(x){\rm
Li}_{2}(x)+64{\rm Li}_{3}(-x)+32{\rm
Li}_{3}(x)+16\zeta_{3}\left.\\!\phantom{\rule{0.28453pt}{15.93347pt}}\right]-\frac{182}{3}x+50$
$\displaystyle\phantom{\int^{1}_{0}dx\,\,x^{n}\left\\{\phantom{\rule{0.28453pt}{16.21828pt}}\\!\right.}-\frac{32}{1+x}+\frac{800}{27x}+\frac{64}{3(1+x)^{2}}\left.\\!\phantom{\rule{0.28453pt}{15.93347pt}}\right\\}m_{e}^{2\varepsilon}(\Delta\cdot
p)^{N}\leavevmode\nobreak\ .$

Here $\Delta$ is a light–like vector, $p$ the external momentum of the OME,
$N$ labels the Mellin moment, $\varepsilon=D-4$ is the dimensional parameter,
and $S_{\varepsilon}=\exp[(\varepsilon/2)(\gamma_{E}-\ln(4\pi))]$ is the
spherical factor, with $\gamma_{E}$ the Euler–Mascheroni constant.

After all building blocks which enter (21) have been calculated, one can
perform the corresponding Mellin transforms analytically and obtain all the
radiators, which were also obtained in the direct calculation, presented in
Section 2.1.

###  2.3 Numerical Results at $O(\alpha^{2})$

The initial state QED corrections can be written in terms of the radiator
functions introduced in Section 2.1, which we use in the following to
calculate the ISR corrections to different processes.

Figure 3: The $Z$-resonance in $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$. Dotted
line: Born cross section; Dashed line: $O(\alpha)$ ISR corrections; Full line:
$O(\alpha^{2})$ \+ soft resummation ISR corrections, with
$s_{0}=4m_{\tau}^{2}$; from Ref. [12]. Figure 4: Relative contributions of the
ISR QED corrections to the cross section for $e^{+}e^{-}\rightarrow ZH$ in %.
Dotted line: $O(\alpha^{0})$; Dashed line: $O(\alpha)$; Dash-dotted line:
$O(\alpha^{2})$; Full line: soft resummation beyond $O(\alpha^{2})$, with
$s_{0}=4m_{\tau}^{2}$; from Ref. [12].

Here we consider the contribution to $O(\alpha^{2})$ only and turn to higher
order corrections in Section 3. We discuss the ISR corrections for the $Z$
peak and its surrounding, the process $e^{+}e^{-}\rightarrow ZH$, and
$t\bar{t}$ production in the threshold region, cf. Ref. [12]. The ISR
corrections depend on the experimental cut $s_{0}$. In the LEP1 analysis it
has been chosen $s_{0}=4m_{\tau}^{2}$ or $s_{0}=0.01M_{Z}^{2}$ [1].
Furthermore, one considers the cases of a fixed $Z$-width or the $s$-dependent
$Z$-width, leading to a peak shift of $34.2$ MeV and a shift of the width of 1
MeV, in accordance with Refs. [59]. We illustrate the different QED ISR
corrections to $e^{+}e^{-}\rightarrow Z^{*}/\gamma^{*}$ around the $Z$ peak in
Figure 3. The ISR corrections change the profile of the resonance, i.e. the
peak position, height and the half width. The effect of soft resummation
beyond $O(\alpha^{2})$ are nearly invisible.

More detailed effects, also due to higher order corrections, are discussed in
Table 1 in Section 3. The new results, compared with [2], imply a relative
shift in the $Z$-width by $\sim$4 MeV for $s_{0}=4m_{\tau}^{2}$, being larger
than the current value of $\Delta\Gamma_{Z}=\pm 2.3$ MeV [60], which may
require a reanalysis of the LEP data to obtain consistent results.

For the study of the radiative corrections to the process
$e^{+}e^{-}\rightarrow ZH$ we refer to the Born cross section in [61]. The
anticipated experimental accuracy for this process should reach 1% [62] at
future colliders like the ILC, CLIC, and even $0.4\%$ at the FCC_ee [63].
Figure 4 shows the relative corrections of ISR corrections.

Figure 5: The QED ISR corrections to $e^{+}e^{-}\rightarrow t\overline{t}$
($s$-channel photon exchange) in the threshold region for a PS-mass of
$m_{t}=172\leavevmode\nobreak\ \mathrm{GeV}$. Dotted line $O(\alpha^{0})$;
Dashed line $O(\alpha)$; Dash-dotted line $O(\alpha^{2})$; Full line
$O(\alpha^{2})$ \+ soft resummation; from Ref. [12].

The NNLO corrections vary between +4.8% and $-1\%$ and are thus larger or of
the size of the expected experimental errors, which might imply the need of
higher order corrections. On the other hand, the soft resummation
contributions are of $O(\pm 0.2\%)$ and reach half of the projected accuracy.

The QED corrections to $t\bar{t}$ production in the threshold region are shown
in Figure 5. For the cross section at threshold we use the N3LO QCD
corrections implemented in the code QQbar_threshold [64, 65, 66] without QED
corrections. Figure 5 illustrates the convergence of the corrections from the
uncorrected cross section to the one with the $O(\alpha^{2})$ corrections and
higher order soft resummation, which is stable up to the peak region an
displays still some shift towards the continuum region. Accuracy studies for
this process have been made in Refs. [67, 68].

The full two–loop QED corrections are mandatory down to the constant
contribution to match the anticipated experimental accuracy for key processes
to be measured at future $e^{+}e^{-}$ colliders and they were mandatory in the
analysis of the LEP experiments. Given the luminosity projected for future
experiments, even higher order corrections will become necessary, to which we
turn now.

##  3 The Higher Order Corrections

In the previous section it has been shown, that the method of massive OMEs
provides an equivalent way to calculate the initial state corrections in the
limit $m_{e}^{2}/s\rightarrow 0$. This is fully justified in particular for
inclusive calculations, where the power corrections being neglected do not
play any role as e.g. in high energy electron-positron collisions. However,
the constant part is still important. One arrives at hierarchies of the kind
$O(\alpha^{k}L^{k}),O(\alpha^{k}L^{k-1}),O(\alpha^{k}L^{k-2}),...,O(\alpha^{k})$.
By using the renormalization group equation one can now derive three
consecutive corrections, for which all massive OMEs are available together
with the two–loop corrections to the massless Drell–Yan cross section [38, 54,
9]. In Ref. [13] we have calculated all the corresponding corrections up to
$O(\alpha^{6}L^{5})$.

To obtain the leading logarithmic contributions to the radiators we solve (21)
to the desired order and calculate the coefficients of the expansion of the
inclusive radiator given in Eqs. (1,2) in Mellin space. While at lower orders
all terms are obtained, at three–loop order we get the coefficients
$c_{3,3},c_{3,2}$ and $c_{3,1}$ only, etc. For example we obtain

$\displaystyle c_{3,1}$ $\displaystyle=$
$\displaystyle-\gamma_{ee}^{(2)}-2\Gamma_{ee}^{(0)}\gamma_{ee}^{(1)}-\Gamma_{ee}^{(0)}\gamma_{e\gamma}^{(0)}\Gamma_{\gamma
e}^{(0)}-\gamma_{e\gamma}^{(1)}\Gamma_{\gamma
e}^{(0)}-\gamma_{e\gamma}^{(0)}\Gamma_{\gamma
e}^{(1)}-\beta_{1}\sigma_{ee}^{(0)}-\gamma_{ee}^{(1)}\sigma_{ee}^{(0)}$ (29)
$\displaystyle-\gamma_{e\gamma}^{(0)}\Gamma_{\gamma
e}^{(0)}\sigma_{ee}^{(0)}-2\Gamma_{ee}^{(0)}\gamma_{\gamma
e}^{(0)}\sigma_{e\gamma}^{(0)}-\gamma_{\gamma
e}^{(1)}\sigma_{e\gamma}^{(0)}-\Gamma_{\gamma
e}^{(0)}\gamma_{\gamma\gamma}^{(0)}\sigma_{e\gamma}^{(0)}-\gamma_{\gamma
e}^{(0)}\sigma_{\gamma
e}^{(1)}+\beta_{0}\Bigl{[}-2\Gamma_{ee}^{(0)}\sigma_{ee}^{(0)}$
$\displaystyle-2\sigma_{ee}^{(1)}-2\Gamma_{\gamma
e}^{(0)}\sigma_{e\gamma}^{(0)}\Bigr{]}-\gamma_{ee}^{(0)}\left[{\Gamma_{ee}^{(0)}}^{2}+2\Gamma_{ee}^{(1)}+2\Gamma_{ee}^{(0)}\sigma_{ee}^{(0)}+\sigma_{ee}^{(1)}+\Gamma_{\gamma
e}^{(0)}\sigma_{e\gamma}^{(0)}\right].$

The calculation is best carried out in Mellin space and the $z$–space
representation is finally obtained by an analytic Mellin inversion using the
tools of the package HarmonicSums.

As can be seen from Eq. (29), starting with three–loop order the expansion
coefficient $\Gamma_{\gamma e}^{(1)}$ contributes. In Mellin space it is
obtained from the massive OME $A_{\gamma e}^{(2)}$, cf. [13],

$\displaystyle A_{\gamma e}^{(2)}(N)$ $\displaystyle=$
$\displaystyle\Biggl{[}\frac{\big{(}N^{2}+N+2\big{)}\big{(}N^{2}+N+6\big{)}}{3(N-1)N^{2}(N+1)^{2}}-\frac{4\big{(}N^{2}+N+2\big{)}}{(N-1)N(N+1)}S_{1}\Biggr{]}L^{2}$
(30)
$\displaystyle-\Biggl{[}\frac{2P_{2}}{9(N-1)^{2}N^{3}(N+1)^{3}}-\frac{4P_{1}}{3(N-1)N^{2}(N+1)^{2}}S_{1}+\frac{12\big{(}N^{2}+N+2\big{)}}{(N-1)N(N+1)}S_{1}^{2}$
$\displaystyle+\frac{12\big{(}N^{2}+N+2\big{)}}{(N-1)N(N+1)}S_{2}\Biggr{]}L+\frac{P_{8}}{27(N-4)(N-3)(N-2)(N-1)N^{4}(N+1)^{4}}$
$\displaystyle+\biggl{(}\frac{2P_{7}}{9(N-4)(N-3)(N-2)(N-1)N^{3}(N+1)^{3}}+\frac{2\big{(}N^{2}+N+2\big{)}}{(N-1)N(N+1)}S_{2}\biggr{)}S_{1}$
$\displaystyle+\frac{P_{3}}{3(N-2)(N-1)N(N+1)^{2}}S_{1}^{2}+\frac{2\big{(}N^{2}+N+2\big{)}}{3(N-1)N(N+1)}S_{1}^{3}$
$\displaystyle+\frac{P_{6}}{3(N-2)(N-1)N^{2}(N+1)^{2}}S_{2}+\frac{4\big{(}N^{2}+N+2\big{)}}{3(N-1)N(N+1)}S_{3}$
$\displaystyle-\frac{48\big{(}N^{2}+N+2\big{)}}{(N-1)N(N+1)}S_{2,1}+\frac{3\cdot
2^{6+N}}{(N-2)(N+1)^{2}}S_{1,1}\biggl{(}\frac{1}{2},1\biggr{)}$
$\displaystyle+\frac{2^{6-N}P_{5}}{3(N-3)(N-2)(N-1)^{2}N^{2}}\biggl{(}S_{2}(2)+S_{1}S_{1}(2)-S_{1,1}(1,2)-S_{1,1}(2,1)\biggr{)}$
$\displaystyle-\frac{32\big{(}N^{2}+N+2\big{)}}{(N-1)N(N+1)}\biggl{[}S_{1}(2)S_{1,1}\biggl{(}\frac{1}{2},1\biggr{)}+S_{1,2}\biggl{(}\frac{1}{2},2\biggr{)}-S_{1,1,1}\biggl{(}\frac{1}{2},1,2\biggr{)}$
$\displaystyle-
S_{1,1,1}\biggl{(}\frac{1}{2},2,1\biggr{)}-\frac{\zeta_{2}}{2}S_{1}(2)\biggr{]}+\frac{4P_{4}}{(N-2)(N-1)N^{2}(N+1)^{2}}\zeta_{2},$

with the polynomials

$\displaystyle P_{1}$ $\displaystyle=$ $\displaystyle
25N^{4}+44N^{3}+87N^{2}+56N+12,$ (31) $\displaystyle P_{2}$ $\displaystyle=$
$\displaystyle 112N^{7}+194N^{6}+347N^{5}+339N^{4}+93N^{3}-293N^{2}-60N+36,$
(32) $\displaystyle P_{3}$ $\displaystyle=$ $\displaystyle
17N^{4}-66N^{3}-179N^{2}-272N-212,$ (33) $\displaystyle P_{4}$
$\displaystyle=$ $\displaystyle N^{5}+4N^{4}+25N^{3}+14N^{2}+12N+8-3\cdot
2^{N+3}N^{2}\bigl{(}N-1\bigr{)},$ (34) $\displaystyle P_{5}$ $\displaystyle=$
$\displaystyle 9N^{5}-24N^{4}+8N^{3}+4N^{2}+33N-18,$ (35) $\displaystyle
P_{6}$ $\displaystyle=$ $\displaystyle
11N^{5}-90N^{4}-329N^{3}-356N^{2}-284N-48,$ (36) $\displaystyle P_{7}$
$\displaystyle=$ $\displaystyle
17N^{9}+213N^{8}-1729N^{7}+2329N^{6}-5196N^{5}+7898N^{4}+16196N^{3}$ (37)
$\displaystyle+12528N^{2}-4896N-3456,$ $\displaystyle P_{8}$ $\displaystyle=$
$\displaystyle-509N^{11}+2365N^{10}+2797N^{9}-13158N^{8}+31274N^{7}-4694N^{6}-64636N^{5}$
(38) $\displaystyle-107861N^{4}-14622N^{3}+6588N^{2}-2376N-2592.$

The radiators are obtained in terms of harmonic sums [46, 47] and generalized
harmonic sums [31] or harmonic polylogarithms [48] $H_{\vec{a}}(z)$ and
$H_{\vec{a}}(1-z)$ weighted by rational coefficients. The choice of arguments
has been made to avoid the occurrence of more general iterated integrals. As
usual, the radiators in $z$-space are distribution valued. We show the
coefficient $c_{3,1}$ in Mellin space as an example,

$\displaystyle c_{3,1}$ $\displaystyle=$
$\displaystyle\Biggl{\\{}\frac{\big{(}2+N+N^{2}\big{)}2^{8-N}[S_{2}(2)-S_{1,1}(1,2)-S_{1,1}(2,1)]Q_{36}}{3(N-3)(N-2)(N-1)^{2}N^{3}(N+1)(N+2)}$
(39)
$\displaystyle+\frac{16S_{2,1}Q_{54}}{9(N-1)N^{2}(N+1)^{2}(N+2)}+\frac{8S_{3}Q_{91}}{27(N-1)N^{2}(N+1)^{2}(N+2)}$
$\displaystyle+\frac{Q_{206}}{81(N-4)(N-3)(N-2)(N-1)^{2}N^{5}(N+1)^{5}(N+2)^{3}}$
$\displaystyle+\Biggl{[}\Biggl{[}-\frac{64S_{1}Q_{119}}{3(N-1)N^{2}(N+1)^{2}(N+2)^{2}}+\frac{32Q_{144}}{9(N-1)N^{3}(N+1)^{3}(N+2)^{2}}$
$\displaystyle+\frac{768(N+3)\big{(}2+N^{2}\big{)}S_{1}^{2}}{(N-1)N(N+1)(N+2)}\Biggr{]}S_{-2}+\Biggl{[}-\frac{32(N+3)\big{(}2+N^{2}\big{)}\big{(}6+13N+13N^{2}\big{)}}{3(N-1)N^{2}(N+1)^{2}(N+2)}$
$\displaystyle+\frac{128(N+3)\big{(}2+N^{2}\big{)}S_{1}}{(N-1)N(N+1)(N+2)}\Biggr{]}S_{-3}+\frac{64(N+3)\big{(}2+N^{2}\big{)}\big{(}6+13N+13N^{2}\big{)}S_{-2,1}}{(N-1)N^{2}(N+1)^{2}(N+2)}$
$\displaystyle-\frac{768(N+3)\big{(}2+N^{2}\big{)}S_{1}S_{-2,1}}{(N-1)N(N+1)(N+2)}+\Biggl{[}-\frac{32S_{1}Q_{119}}{3(N-1)N^{2}(N+1)^{2}(N+2)^{2}}$
$\displaystyle+\frac{16Q_{144}}{9(N-1)N^{3}(N+1)^{3}(N+2)^{2}}+\frac{384(N+3)\big{(}2+N^{2}\big{)}S_{1}^{2}}{(N-1)N(N+1)(N+2)}\Biggr{]}\zeta_{2}$
$\displaystyle+\Biggl{[}\frac{32(N+3)\big{(}2+N^{2}\big{)}\big{(}6+13N+13N^{2}\big{)}}{(N-1)N^{2}(N+1)^{2}(N+2)}-\frac{384(N+3)\big{(}2+N^{2}\big{)}S_{1}}{(N-1)N(N+1)(N+2)}\Biggr{]}\zeta_{3}\Biggr{]}(-1)^{N}$
$\displaystyle+\Biggl{[}-\frac{64S_{2,1}Q_{6}}{3(N-1)N(N+1)}+\frac{\big{(}2+N+N^{2}\big{)}2^{8-N}S_{1}(2)Q_{36}}{3(N-3)(N-2)(N-1)^{2}N^{3}(N+1)(N+2)}$
$\displaystyle-\frac{64S_{3}Q_{20}}{9(N-1)N}-\frac{16S_{2}Q_{132}}{27(N-1)N^{2}(N+1)^{2}(N+2)^{2}}$
$\displaystyle-\frac{8Q_{201}}{81(N-4)(N-3)(N-2)(N-1)^{2}N^{4}(N+1)^{4}(N+2)^{3}}-1152S_{2}^{2}-384S_{4}$
$\displaystyle-1280S_{3,1}-1024S_{-2,-1}+768S_{2,1,1}\Biggr{]}S_{1}+\Biggl{[}\frac{32S_{2}Q_{8}}{(N-1)N(N+1)}-256S_{2,1}$
$\displaystyle+\frac{16Q_{150}}{27(N-2)(N-1)^{2}N^{3}(N+1)^{3}(N+2)^{2}}\Biggr{]}S_{1}^{2}+\Biggl{[}-\frac{64Q_{40}}{27(N-1)N^{2}(N+1)^{2}(N+2)}$
$\displaystyle-128S_{2}\Biggr{]}S_{1}^{3}+\frac{512}{9}S_{1}^{4}+\Biggl{[}\frac{4Q_{175}}{27(N-2)(N-1)^{2}N^{3}(N+1)^{3}(N+2)^{2}}-384S_{3}\Biggr{]}S_{2}$
$\displaystyle+\frac{32\big{(}54+97N+97N^{2}\big{)}S_{2}^{2}}{3N(N+1)}+\frac{32(2+5N)(3+5N)S_{4}}{N(N+1)}+384S_{5}$
$\displaystyle+\Biggl{[}\Biggl{[}-\frac{256\big{(}2+3N+3N^{2}\big{)}}{N(N+1)}+1024S_{1}\Biggr{]}S_{-1}-256S_{-2,1}\Biggr{]}S_{-2}+640S_{-3}S_{-2}+192S_{-2}^{2}$
$\displaystyle+\frac{64\big{(}30+11N+11N^{2}\big{)}S_{3,1}}{3N(N+1)}-384S_{3,2}-384S_{4,1}+\frac{256\big{(}2+3N+3N^{2}\big{)}S_{-2,-1}}{N(N+1)}$
$\displaystyle-\frac{64\big{(}6+11N+11N^{2}\big{)}S_{2,1,1}}{N(N+1)}+768S_{3,1,1}-1024S_{-2,1,-2}$
$\displaystyle+\frac{3\big{(}2+N+N^{2}\big{)}2^{8+N}S_{1,1}\big{(}\frac{1}{2},1\big{)}}{(N-2)N(N+1)^{3}(N+2)}-\frac{128\big{(}2+N+N^{2}\big{)}^{2}S_{1}(2)S_{1,1}\big{(}\frac{1}{2},1\big{)}}{(N-1)N^{2}(N+1)^{2}(N+2)}$
$\displaystyle-\frac{128\big{(}2+N+N^{2}\big{)}^{2}(S_{1,2}\big{(}\frac{1}{2},2\big{)}-S_{1,1,1}\big{(}\frac{1}{2},1,2\big{)}-S_{1,1,1}\big{(}\frac{1}{2},2,1\big{)})}{(N-1)N^{2}(N+1)^{2}(N+2)}$
$\displaystyle+\Biggl{[}-\frac{32S_{1}^{2}Q_{12}}{3(N-1)N(N+1)}-\frac{4Q_{174}}{27(N-2)(N-1)^{2}N^{3}(N+1)^{3}(N+2)^{2}}$
$\displaystyle+\ln(2)\Biggl{[}-\frac{576\big{(}2+3N+3N^{2}\big{)}}{N(N+1)}+2304S_{1}\Biggr{]}$
$\displaystyle+128S_{1}^{3}+\Biggl{[}\frac{16Q_{133}}{27(N-1)N^{2}(N+1)^{2}(N+2)^{2}}+2432S_{2}\Biggr{]}S_{1}-\frac{64\big{(}19+30N+30N^{2}\big{)}S_{2}}{N(N+1)}$
$\displaystyle+576S_{3}+\Biggl{[}-\frac{128\big{(}2+3N+3N^{2}\big{)}}{N(N+1)}+512S_{1}\Biggr{]}S_{-1}+192S_{-2}+320S_{-3}$
$\displaystyle-640S_{-2,1}+\frac{64\big{(}2+N+N^{2}\big{)}^{2}S_{1}(2)}{(N-1)N^{2}(N+1)^{2}(N+2)}+160\zeta_{3}\Biggr{]}\zeta_{2}$
$\displaystyle+\Biggl{[}\frac{16\big{(}588+1171N+1171N^{2}\big{)}}{15N(N+1)}-\frac{6272}{5}S_{1}\Biggr{]}\zeta_{2}^{2}+\Biggl{[}\frac{64S_{1}Q_{34}}{3(N-1)N(N+1)}+512S_{1}^{2}$
$\displaystyle-\frac{16Q_{85}}{9(N-1)N^{2}(N+1)^{2}(N+2)}-192S_{2}+192S_{-2}\Biggr{]}\zeta_{3}-416\zeta_{5}\Biggr{\\}},$

with $Q_{i}$ polynomials in $N$, cf. [13].

The numerical effect of the corrections of up to $O(\alpha^{6}L^{5})$ are
illustrated in Table 1 describing the $Z^{0}$-peak both in the case of a fixed
width and the $s$ dependent width. The radiative corrections change both the
peak positions and the width. The corrections due to the different orders may
be positive or negative. We list the individual contributions.

At $O(\alpha^{6}L^{6})$ the corrections to the peak are 7 keV and to the width
27 keV. The next subleading correction serves as a control term. For
measurements at the future FCC_ee statistical accuracies of $\Delta
M_{Z}=50\leavevmode\nobreak\ \mathrm{keV}\leavevmode\nobreak\
{\rm[stat]},\Delta\Gamma_{Z}=8\leavevmode\nobreak\
\mathrm{keV}\leavevmode\nobreak\ {\rm[stat]}$ are estimated [69]. This
precision level is about reached in the case of the present corrections. In
the case a further logarithmic order is needed, one needs to calculated both,
one more order for the Drell–Yan process and one more for the massive OMEs.
Furthermore, the electro–weak corrections and QCD corrections have to be
pushed to higher orders. All of this will meet theoretical and technological
(i.e. mathematical and computer–algebraic) challenges.

| Fixed width |  $s$ dep. width
---|---|---
| Peak | Width | Peak | Width
| (MeV) | (MeV) | (MeV) | (MeV)
$O(\alpha)$ correction | 185.638 | 539.408 | 181.098 | 524.978
$O(\alpha^{2}L^{2})$: | – 96.894 | –177.147 | – 95.342 | –176.235
$O(\alpha^{2}L)$: | 6.982 | 22.695 | 6.841 | 21.896
$O(\alpha^{2})$: | 0.176 | – 2.218 | 0.174 | – 2.001
$O(\alpha^{3}L^{3})$: | 23.265 | 38.560 | 22.968 | 38.081
$O(\alpha^{3}L^{2})$: | – 1.507 | – 1.888 | – 1.491 | – 1.881
$O(\alpha^{3}L)$: | – 0.152 | 0.105 | – 0.151 | – 0.084
$O(\alpha^{4}L^{4})$: | – 1.857 | 0.206 | – 1.858 | 0.146
$O(\alpha^{4}L^{3})$: | 0.131 | – 0.071 | 0.132 | – 0.065
$O(\alpha^{4}L^{2})$: | 0.048 | – 0.001 | 0.048 | 0.001
$O(\alpha^{5}L^{5})$: | 0.142 | – 0.218 | 0.144 | – 0.212
$O(\alpha^{5}L^{4})$: | – 0.000 | 0.020 | – 0.001 | 0.020
$O(\alpha^{5}L^{3})$: | – 0.008 | 0.009 | – 0.008 | 0.008
$O(\alpha^{6}L^{6})$: | – 0.007 | 0.027 | – 0.007 | 0.027
$O(\alpha^{6}L^{5})$: | – 0.001 | 0.000 | – 0.001 | 0.000
Table 1: Shifts in the $Z$-mass and the width due to the different
contributions to the ISR QED radiative corrections for a fixed width of
$\Gamma_{Z}=2.4952\leavevmode\nobreak\ \mathrm{GeV}$ and $s$-dependent width
using $M_{Z}=91.1876\leavevmode\nobreak\ \mathrm{GeV}$ [60] and
$s_{0}=4m_{\tau}^{2}$, cf. [1]; from [13].

The method of massive OMEs has been also applied to calculate the massive
Wilson coefficients for deep inelastic scattering in the region $Q^{2}\gg
m_{Q}^{2}$ in the single and two–mass cases at three–loop order analytically
in Refs. [70].

##  4 The Forward-Backward Asymmetry

A high luminosity measurement of the forward–backward asymmetry provides an
excellent possibility for a precision measurement of the fine structure
constant at high energy [71]. This is important because of its hadronic
contributions [72]. The calculation of the forward–backward asymmetry at
one–loop order has been performed in [73, 79, 78, 80, 74, 75, 76, 77], and the
leading logarithmic two–loop corrections were computed in [79]. More recently
the leading logarithmic contributions to $O(\alpha^{6}L^{6})$ have been
calculated in Ref. [14]. Also the initial–final state interference terms,
final state corrections, electro–weak and QCD corrections are known at lower
orders, cf. [14].

The forward–backward asymmetry is defined as ratio over the difference and the
sum of the angular integrals over the respective hemispheres measuring the
final state muons,

$\displaystyle\sigma_{F}=2\pi\int\limits_{0}^{1}{\rm d}\cos(\theta)\frac{{\rm
d}\sigma}{{\rm d}\Omega},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \sigma_{B}=2\pi\int\limits_{-1}^{0}{\rm
d}\cos(\theta)\frac{{\rm d}\sigma}{{\rm d}\Omega}\,.$ (40)

The angle $\theta$ is defined between the incoming electron $e^{-}$ and the
outgoing muon $\mu^{-}$ from $\gamma^{*}/Z^{*}$ decay, with

$\displaystyle A_{\text{FB}}(s)$ $\displaystyle=$
$\displaystyle\frac{\sigma_{F}(s)-\sigma_{B}(s)}{\sigma_{T}(s)},$ (41)

and $\sigma_{T}(s)=\sigma_{F}(s)+\sigma_{B}(s)$. At Born level this reduces to
[29]

$\displaystyle\sigma_{\rm FB}^{(0)}(s)$ $\displaystyle=$
$\displaystyle\sigma_{F}^{(0)}(s)-\sigma_{B}^{(0)}(s)=\frac{\pi\alpha^{2}}{s}N_{C,f}\left(1-\frac{4m_{f}^{2}}{s}\right)G_{3}(s)\,,$
(42) $\displaystyle\sigma_{T}^{(0)}(s)$ $\displaystyle=$
$\displaystyle\sigma_{F}^{(0)}(s)+\sigma_{B}^{(0)}(s)=\frac{4\pi\alpha^{2}}{3s}N_{C,f}\sqrt{1-\frac{4m_{f}^{2}}{s}}\left[\left(1+\frac{2m_{f}^{2}}{s}\right)G_{1}(s)-6\frac{m_{f}^{2}}{s}G_{2}(s)\right],$

with $m_{f}$ the final state fermion mass, $m_{f}\equiv m_{\mu}$ and

$\displaystyle G_{3}(s)$ $\displaystyle=$ $\displaystyle
2Q_{e}Q_{f}a_{e}a_{f}{\sf
Re}[\chi_{Z}(s)]+4v_{e}v_{f}a_{e}a_{f}|\chi_{Z}(s)|^{2}.$ (44)

The initial state corrections to $A_{\rm FB}$ at leading logarithmic level are
described by two radiator functions $H_{e}^{LL}$ and $H_{FB}^{LL}$ [79], using
the notation in [73],

$\displaystyle A_{\text{FB}}(s)$ $\displaystyle=$
$\displaystyle\frac{1}{\sigma_{T}(s)}\int\limits_{z_{0}}^{1}{\rm
d}z\,\frac{4z}{(1+z)^{2}}\tilde{H}_{e}^{LL}(z)\sigma_{\rm
FB}^{(0)}(zs),\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\sigma_{T}(s)=\int\limits_{z_{0}}^{1}{\rm
d}z\,H_{e}(z)\sigma_{T}^{(0)}(zs)\,,$ (45)

with

$\displaystyle\tilde{H}_{e}^{LL}(z)=\left[H_{e}^{LL}(z)+H_{FB}^{LL}(z)\right],$
(46)

where the parameter $z_{0}$ plays the role of an energy cut and
$z=s^{\prime}/s$. Here $H_{e}^{LL}$ is angular independent, and denotes the
leading logarithmic contributions of the radiators of Section 2, while
$H_{FB}^{LL}$ is angular dependent and obtained by the following integral

$\displaystyle H_{FB}^{LL}(z)=\int\limits_{0}^{1}{\rm
d}x_{1}\int\limits_{0}^{1}{\rm
d}x_{2}\left(\frac{(1+z)^{2}}{(x_{1}+x_{2})^{2}}-1\right)\Gamma_{ee}^{LL}(x_{2})\Gamma_{ee}^{LL}(x_{1})\delta(x_{1}x_{2}-z).$
(47)

Here $\Gamma_{ee}^{LL}(x_{i})$ denote the massive operator matrix elements and
(47) is expanded in $(\alpha L)$ to the desired order. It is convenient to
consider first the Mellin–transform

$\displaystyle\mathcal{M}[H_{FB}^{LL}(z)](n)$ $\displaystyle=$
$\displaystyle\int\limits_{0}^{1}{\rm
d}zz^{n}H_{FB}^{LL}(z)=\int\limits_{0}^{1}{\rm d}x_{1}\int\limits_{0}^{1}{\rm
d}x_{2}x_{1}^{n}x_{2}^{n}\left(\frac{(1+x_{1}x_{2})^{2}}{(x_{1}+x_{2})^{2}}-1\right)\Gamma_{ee}^{LL}(x_{2})\Gamma_{ee}^{LL}(x_{1})$

and to calculate the generating function

$\displaystyle\mathcal{G}[H_{FB}^{LL}(z)](t)$ $\displaystyle=$
$\displaystyle\sum\limits_{n=0}^{\infty}t^{n}\mathcal{M}[H_{FB}^{LL}(z)](n),$
(49)

from which we then obtain the individual radiators. They can be expressed by
harmonic and cyclotomic harmonic polylogarithms [49] over the alphabet

$\displaystyle\mathfrak{A}=\left\\{f_{0}=\frac{1}{z},f_{1}=\frac{1}{1-z},f_{-1}=\frac{1}{1+z},f_{\\{4,0\\}}=\frac{1}{1+z^{2}},f_{\\{4,1\\}}=\frac{z}{1+z^{2}}\right\\}.$
(50)

$H_{FB}^{LL}$ starts at $O(\alpha^{2}L^{2})$.

| $A_{\rm FB}(s_{-})$ | $A_{\rm FB}(M_{Z}^{2})$ | $A_{\rm FB}(s_{+})$
---|---|---|---
$\mathcal{O}(\alpha^{0})$ | $-0.3564803$ | $0.0225199$ | $0.2052045$
$\mathcal{O}(\alpha L^{1})$ | $-0.2945381$ | $-0.0094232$ | $0.1579347$
$\mathcal{O}(\alpha L^{0})$ | $-0.2994478$ | $-0.0079610$ | $0.1611962$
$\mathcal{O}(\alpha^{2}L^{2})$ | $-0.3088363$ | $0.0014514$ | $0.1616887$
$\mathcal{O}(\alpha^{3}L^{3})$ | $-0.3080578$ | $0.0000198$ | $0.1627252$
$\mathcal{O}(\alpha^{4}L^{4})$ | $-0.3080976$ | $0.0001587$ | $0.1625835$
$\mathcal{O}(\alpha^{5}L^{5})$ | $-0.3080960$ | $0.0001495$ | $0.1625911$
$\mathcal{O}(\alpha^{6}L^{6})$ | $-0.3080960$ | $0.0001499$ | $0.1625911$

Table 2: $A_{\rm FB}$ evaluated at $s_{-}=(87.9\,{\rm GeV})^{2}$, $M_{Z}^{2}$
and $s_{+}=(94.3\,{\rm GeV})^{2}$ for the cut $z>4m_{\tau}^{2}/s$ from Ref.
[14].

;

As an example $H_{FB}^{(3),LL}$ reads

$\displaystyle H_{FB}^{(3),LL}(z)$ $\displaystyle=$
$\displaystyle-\frac{16(1-z)\big{(}4+11z+4z^{2}\big{)}}{3z}-\pi\biggl{[}\frac{4\big{(}2-3z-2z^{2}-3z^{3}+2z^{4}\big{)}}{3z^{3/2}}$
(51)
$\displaystyle+\frac{4(1-z)(1+5z)}{\sqrt{z}}H_{0}+\frac{16(1-z)^{2}}{\sqrt{z}}H_{\\{4,1\\}}\biggr{]}+\biggl{[}\frac{4(1+z)\big{(}5-18z-19z^{2}\big{)}}{3z}$
$\displaystyle-\frac{16(1-z)(1-7z)}{\sqrt{z}}H_{\\{4,0\\}}-96(1+z)H_{\\{4,1\\}}\biggr{]}H_{0}-8(1+z)H_{0}^{2}$
$\displaystyle+\biggl{[}\frac{16(1-z)(1+z)^{2}}{z}-\frac{64(1-z)^{2}}{\sqrt{z}}H_{\\{4,0\\}}\biggr{]}H_{1}+\biggl{[}\frac{16\big{(}2-3z-2z^{2}-3z^{3}+2z^{4}\big{)}}{3z^{3/2}}$
$\displaystyle+\frac{64(1-z)^{2}}{\sqrt{z}}H_{\\{4,1\\}}\biggr{]}H_{\\{4,0\\}}+\biggl{[}-\frac{16(1-z)(1+z)^{2}}{z}+64(1+z)H_{0}$
$\displaystyle+\frac{64(1-z)^{2}}{\sqrt{z}}H_{\\{4,0\\}}\biggr{]}H_{-1}-64(1+z)H_{0,1}+\frac{32(1-z)^{2}}{\sqrt{z}}H_{0,\\{4,0\\}}+96(1+z)H_{0,\\{4,1\\}}$
$\displaystyle+\frac{64(1-z)^{2}}{\sqrt{z}}H_{1,\\{4,0\\}}-\frac{64(1-z)^{2}}{\sqrt{z}}H_{\\{4,0\\},\\{4,1\\}}-64(1+z)H_{-1,0}$
$\displaystyle-\frac{64(1-z)^{2}}{\sqrt{z}}H_{-1,\\{4,0\\}}+20(1+z)\zeta_{2},$

with $H_{\vec{w}}\equiv H_{\vec{w}}(\sqrt{z})$ and
$\zeta_{k},k\in\mathbb{N},k\geq 2$ are the values of Riemann’s $\zeta$
function at integer argument.

Let us give some numerical illustration. We compute $A_{\rm FB}(s)$ for
different values around the $Z^{0}$-peak as suggested in [71] to higher orders
and see the corresponding improvement in Table 2. Comparing the results at
one–loop to the highest order we obtain corrections of $-3$ % for $s_{-}$ and
$-1\%$ for $s_{+}$. More numerical studies are presented in Ref. [14].

##  5 Conclusions

The precision calculations of the QED initial state corrections to the process
$e^{+}e^{-}\rightarrow\gamma^{*}/Z^{*}$ are an essential ingredient for the
planned key measurements at high energy $e^{+}e^{-}$ colliders such as the
ILC, CLIC, FCC_ee, and future muon colliders. Due to the smallness of the
ratio $m_{l}^{2}/s$, with $m_{l}$ the initial charged lepton mass, the
logarithmic and constant term corrections in a massive environment are
sufficient. It therefore has been important to clarify the differences between
Ref. [2] and Ref. [3] at $O(a^{2})$. The differences found do in principle
require to repeat the LEP electro–weak analysis, given the current accuracy of
the $Z^{0}$ peak and width, because of the respective theoretical shifts.
Since the codes TOPAZ0 [81, 80] and ZFITTER [82] contain the results of [2],
they have to be updated for the use in further experimental analyses.

The agreement between Refs. [11] and [3] allowed to use the method of massive
OMEs for even higher order corrections in the fine structure constant. Having
available all OMEs which contribute at two–loop order, the first three
logarithmic expansion coefficients are available through the renormalization
group equations to any order in the fine structure constant, i.e. two further
orders beyond the leading logarithmic approximation. The calculation of one
more order seems to be technically possible. The presently available
corrections reach the projected accuracies at the FCC_ee and need to be
supplemented by corresponding QCD and electro–weak corrections.

For the forward–backward asymmetry, which might allow a precision measurement
of the fine structure constant in the future, the leading logarithmic
corrections have been extended to $O((\alpha L)^{6})$ and one might want to
consider sub–leading corrections. Beyond the corrections in the leading
logarithmic approximation, where harmonic sums are sufficient to express the
radiators in Mellin space, one also finds generalized and cyclotomic harmonic
sums, forming a part of the function spaces having been already revealed in
other analytic massive single–scale calculations [56, 70]. We are still in the
early phase to calculate the necessary radiative corrections for the
high–luminosity measurements at the FCC_ee and will view more theoretical
results during the decades ahead.

Acknowledgments.
We would like to thank A. De Freitas, W.L. van Neerven, and C.G. Raab for
collaboration on the present topic and P. Janot and G. Passarino for
discussions. This project has received funding from the European Union’s
Horizon 2020 research and innovation programme under the Marie Sklodowska-
Curie grant agreement No. 764850, SAGEX.

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