aainstitutetext: II. Institut für Theoretische Physik, Universität Hamburg,
22761 Hamburg, Germanybbinstitutetext: Pittsburgh Particle Physics,
Astrophysics & Cosmology Center (PITT PACC), Department of Physics &
Astronomy, University of Pittsburgh, Pittsburgh, PA 15260, USAccinstitutetext:
Institute of Physics, University of Silesia, Katowice, Poland ddinstitutetext:
Faculty of Science, University of Hradec Králové, Czech
Republiceeinstitutetext: Deutsches Elektronen–Synchrotron, DESY, 15738
Zeuthen, Germanyffinstitutetext: School of Mathematics, Trinity College
Dublin, University of Dublin, Dublin 2, Ireland

#  DESY 19-103
KW 19-005
Electroweak pseudo-observables and Z-boson form factors at two-loop accuracy

Ievgen Dubovyk b Ayres Freitas c,d Janusz Gluza c,e Tord Riemann f Johann
Usovitsch<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(today)

###### Abstract

We present Standard Model predictions for the complete set of
phenomenologically relevant electroweak precision pseudo-observables related
to the $Z$-boson: the leptonic and bottom-quark effective weak mixing angles
$\sin^{2}\theta_{\rm eff}^{\ell}$, $\sin^{2}\theta_{\rm eff}^{b}$, the
$Z$-boson partial decay widths $\Gamma_{f}$, where $f$ indicates any charged
lepton, neutrino and quark flavor (except for the top quark), as well as the
total $Z$ decay width $\Gamma_{Z}$, the branching ratios $R_{\ell}$, $R_{c}$,
$R_{b}$, and the hadronic cross section $\sigma_{\rm had}^{0}$. The input
parameters are the masses $M_{\mathrm{Z}}$, $M_{\mathrm{H}}$ and
$m_{\mathrm{t}}$, and the couplings $\alpha_{\mathrm{s}}$, $\alpha$. The
scheme dependence due to the choice of $M_{\mathrm{W}}$ or its alternative
$G_{\mu}$ as a last input parameter is also discussed. Recent substantial
technical progress in the calculation of Minkowskian massive higher-order
Feynman integrals allows the calculation of the complete electroweak two-loop
radiative corrections to all the observables mentioned. QCD contributions are
included appropriately. Results are provided in terms of simple and convenient
parameterization formulae whose coefficients have been determined from the
full numerical multi-loop calculation. The size of the missing electroweak
three-loop or QCD higher-order corrections is estimated. We briefly comment on
the prospects for their calculation. Finally, direct predictions for the
$Z{\bar{f}}f$ vector and axial-vector form-factors are given, including a
discussion of separate order-by-order contributions.

##  1 Introduction

In 2018 we celebrated 50 years of the Standard Model of elementary particles.
The basics of the model were formulated and experimentally validated in the
1960s/70s. The next decade brought an intensive development of the calculation
of quantum field theoretical radiative corrections in that model and in its
alternatives. An experimental highlight in this context was the
$e^{+}e^{-}$-collider LEP, which enabled us to check the Standard Model at an
accuracy of better than the per-cent level, which corresponds to effects from
more than one electroweak and two QCD loop orders. This proved, for the first
time in a systematic way, the Standard Model as a quantum field theory. LEP 1
was running, from Summer 1989 to 1995, at and around the $Z$-boson peak. The
expectation for the experimental precision of $M_{\mathrm{Z}}$ and
$\Gamma_{\mathrm{Z}}$ was 20 MeV in 1986 Blondel:1986kj  and reached finally 2
MeV Arduini:1996pp . This precision tag was extremely important because
$M_{\mathrm{Z}}$ is one of the Standard Model input parameters to the commonly
used on-mass-shell renormalization scheme. Indeed, the experimental accuracy
of $M_{\mathrm{Z}}$ triggered much of the precision loop calculations,
including the prediction of the top quark and Higgs masses prior to their
discoveries from loop corrections to LEP observables in the Standard Model,
see Refs. Ellis:1986jba ; Altarelli:1989YR ; Bardin:1995-YR03  (as well as
Refs. Blondel:2018mad ; Blondel:2019vdq  for an overview of the current state
of the art). Data from the $Z$ peak and the $Z$ resonance curve (the $Z$ line
shape) allow to measure a large variety of observables, such as
$M_{\mathrm{Z}}$, $\Gamma_{\mathrm{Z}}$, cross-sections for different two-
fermion final states and their ratios and angular asymmetries, together with
radiation of (sufficiently soft) photons, gluons, etc. From the real
observables, the so-called electroweak pseudo-observables (EWPOs) are
extracted by means of a de-convolution of initial-state radiation and
subtraction of backgrounds. The fine details of relating EWPOs to real cross-
sections at LEP 1 precision are described in detail in Ref. Gluza:inYR2018
and references quoted therein.

On occasion of the 50th anniversary of the Standard Model, many of the related
crucial developments were remembered at the conference ”SM@50” smat50:2019 .

This article is devoted to the state-of-the art calculation of the Standard
Model (SM) corrections to the $Z{\bar{f}}f$-vertex and their inclusion into
the predictions for the various EWPOs. We will mostly focus on recent advances
in the calculation of the electroweak two-loop terms. QCD contributions, which
are known up to four-loop order, have also been taken into account in the
results presented here, but we will refer to the literature for further
details.

The $Z$ resonance curve can be described theoretically by writing the S-matrix
elements as a Laurent series in the center-of-mass energy squared $s$ (also
called S-matrix ansatz). This Laurent series ansatz is worked out up to two
loops Consoli:1986kr ; Sirlin:1991fd ; Willenbrock:1991hu ; Stuart:1991xk ;
Sirlin:1991rt ; Stuart:1991cc ; Stuart:1992jf ; Veltman:1992tm ;
Passera:1998uj . The coefficients of the leading series term contain the $Z$
vertex form factors. Their one-loop corrections were studied in the 1980s;
first with massless fermions Bardin:1980fe ; Bardin:1981sv ; Marciano:1980pb ;
Marciano:1983wwa , and slightly later with the full mass dependence of the
Standard Model Akhundov:1985fc ; Bernabeu:1987me ; Jegerlehner:1988ak ;
Beenakker:1988pv . After several papers on approximate/partial higher-order
corrections, the complete two-loop weak corrections were determined in a
series of papers from 2004 to 2018 Awramik:2004ge ; Awramik:2006ar ;
Awramik:2006uz ; Hollik:2005va ; Hollik:2006ma ; Awramik:2008gi ;
Freitas:2013dpa ; Freitas:2014owa ; Freitas:2014hra ; Dubovyk:2016aqv ;
Dubovyk:2018rlg . The correct formulation of the interplay of the 2$\to$2 loop
corrections with higher order real QED corrections in the S-matrix approach,
also called un-folding of the effective 2$\to$2 Born terms from the realistic
2$\to$n observables, is a topic on its own. It was first studied in Refs.
Leike:1991pq ; Riemann:1992gv ; Kirsch:1994cf ; Riemann:2015wpn , but its
extension beyond two-loop level will require more work Gluza:inYR2018-C3 ;
Gluza:inYR2018 ; Jadach:2019bye . The numerically relevant two-loop and
partial higher-order corrections were included in the final analysis of LEP 1
data ALEPH:2005ab .111The EWPOs at LEP 1 were determined order by order
without a Laurent expansion. This was based on the ZFITTER software
Bardin:1999yd ; Arbuzov:2005ma , for both the Standard Model loop calculation
and the unfolding of cross-sections. The relevant higher-order corrections to
the input $W$ mass Awramik:2003rn  and to the leptonic weak mixing angle
Awramik:2004ge  are implemented in ZFITTER v.6.42. While ZFITTER v.6.44beta
Akhundov:2013ons  also contains the QCD four-loop corrections of Baikov:2012er
, they are of no experimental relevance so far.

The theoretical advances described here go beyond the Standard Model theory
used for physics at LEP 1 Bardin:1999yd ; Arbuzov:2005ma  but will be needed
for the FCC-ee Tera-Z project Jadach:April2018 ; Blondel:2018mad ;
Mangano:2018mur ; Blondel:2019vdq ; Dubovyk:inYR2018  whose unique
experimental precision calls for perturbative predictions at three electroweak
loops together with corresponding QCD terms.

In this work, the following pseudo-EWPOs are discussed: The partial widths
$\Gamma_{f}$ for $Z$-boson decay into $f\bar{f}$ final states; the total $Z$
width $\Gamma_{\mathrm{Z}}$; the branching fractions
$\Gamma_{f}/\Gamma_{\mathrm{Z}}$; the total hadronic $Z$-pole quark-pair
production cross-section $\sigma^{0}_{\rm had}$; and the effective weak mixing
angles $\sin^{2}\theta_{\rm eff}^{f}$, defined from the ratio of vector- and
axial $Z$-boson couplings. The precise definition of all these pseudo-
observables will be given below. Pseudo-observables differ from real
observables by removing from the former the effects of initial-state and
initial-final QED radiation, as well as non-resonant photon-exchange, box and
$t$-channel contributions ALEPH:2005ab ; Blondel:2018mad .

The Standard Model predictions for $Z$-pole pseudo-observables can be
constructed in terms of the following three theoretical building blocks
Awramik:2006uz :

$\displaystyle v_{f}(s)$ $\displaystyle\equiv
v_{f}^{\mathrm{Z}}(s)-v_{f}^{\gamma}(s)\,\frac{\Sigma_{\rm\gamma
Z}(s)}{s+\Sigma_{\gamma\gamma}(s)}\,,$ (1) $\displaystyle a_{f}(s)$
$\displaystyle\equiv
a_{f}^{\mathrm{Z}}(s)-a_{f}^{\gamma}(s)\,\frac{\Sigma_{\rm\gamma
Z}(s)}{s+\Sigma_{\gamma\gamma}(s)}\,,$ (2)
$\displaystyle\Sigma_{\mathrm{Z}}(s)$ $\displaystyle\equiv\Sigma_{\rm
ZZ}(s)-\frac{[\Sigma_{\rm\gamma Z}(s)]^{2}}{s+\Sigma_{\gamma\gamma}(s)}\,,$
(3)

where $v_{f}^{\mathrm{Z}}$ and $a_{f}^{\mathrm{Z}}$ are the one-particle
irreducible $Zf\bar{f}$ vector- and axial-vector vertex contributions,
respectively, whereas $v_{f}^{\gamma}$ and $a_{f}^{\gamma}$ are their
counterparts for the $\gamma f\bar{f}$ vertex. The $\Sigma_{V_{1}V_{2}}$
denotes the one-particle irreducible $V_{1}$–$V_{2}$ self-energy. At tree
level,

$\displaystyle v^{\mathrm{Z}}_{f(0)}$
$\displaystyle=e\frac{I^{3}_{f}-2Q_{f}s_{\scriptscriptstyle{\mathrm{W}}}^{2}}{2s_{\scriptscriptstyle{\mathrm{W}}}c_{\scriptscriptstyle{\mathrm{W}}}},$
$\displaystyle v^{\gamma}_{f(0)}$ $\displaystyle=eQ_{f},$ (4) $\displaystyle
a^{\mathrm{Z}}_{f(0)}$
$\displaystyle=e\frac{I^{3}_{f}}{2s_{\scriptscriptstyle{\mathrm{W}}}c_{\scriptscriptstyle{\mathrm{W}}}},$
$\displaystyle a^{\gamma}_{f(0)}$ $\displaystyle=0.$ (5)

Here $I^{3}_{f}$ and $Q_{f}$ are the weak isospin and electric charge (in
units of the elementary charge $e>0$) of the fermion $f$, respectively.
$s_{\scriptscriptstyle{\mathrm{W}}}$ and $c_{\scriptscriptstyle{\mathrm{W}}}$
are the sine and cosine of the weak mixing angle, respectively, and the
subscript $(0)$ is used to denote tree-level order.

For the theory calculations, these building blocks must be evaluated at the
complex $Z$ pole Willenbrock:1991hu ; Sirlin:1991fd ; Stuart:1991xk ;
Veltman:1992tm ,
$s_{0}\equiv\overline{M}_{\mathrm{Z}}^{2}-i\overline{M}_{\mathrm{Z}}\overline{\Gamma}_{\mathrm{Z}}$,
where $\overline{M}_{\mathrm{Z}}$ and $\overline{\Gamma}_{\mathrm{Z}}$ are the
on-shell mass and width of the $Z$-boson, respectively. Note that
$\overline{M}_{\mathrm{Z}}$ and $\overline{\Gamma}_{\mathrm{Z}}$ differ from
the mass $M_{\mathrm{Z}}$ and width $\Gamma_{\mathrm{Z}}$ reported in
publications of LEP, Tevatron and LHC experiments by a fixed factor
Bardin:1988xt ; Leike:1991pq :

$\displaystyle\textstyle\overline{M}_{\mathrm{Z}}$
$\displaystyle=M_{\mathrm{Z}}\big{/}\sqrt{1+\Gamma_{\mathrm{Z}}^{2}/M_{\mathrm{Z}}^{2}}\,,$
$\displaystyle\overline{\Gamma}_{\mathrm{Z}}$
$\displaystyle=\Gamma_{\mathrm{Z}}\big{/}\sqrt{1+\Gamma_{\mathrm{Z}}^{2}/M_{\mathrm{Z}}^{2}}\,.$
(6)

Similar expressions hold for $\overline{M}_{\mathrm{W}}$ and
$\overline{\Gamma}_{\mathrm{W}}$ Bardin:1986fi ; Denner:1990tx .

##  2 $Z$-boson decay width, branching ratios and cross-sections

The width of the $Z$ boson, $\overline{\Gamma}_{\mathrm{Z}}$, is related to
the imaginary part of the $Z$ self-energy. Using the optical theorem, one can
derive the following expression for $\overline{\Gamma}_{\mathrm{Z}}$
Freitas:2013dpa ; Freitas:2014hra :

$\displaystyle\overline{\Gamma}_{\mathrm{Z}}$
$\displaystyle=\sum_{f}\overline{\Gamma}_{f},~{}~{}~{}f=e,\mu,\tau,\nu_{e},\nu_{\mu},\nu_{\tau},u,c,s,d,b,$
(7) $\displaystyle\overline{\Gamma}_{f}$
$\displaystyle=\frac{N_{c}^{f}\overline{M}_{\mathrm{Z}}}{12\pi}\Bigl{[}{\cal
R}_{\rm V}^{f}F_{\rm V}^{f}+{\cal R}_{\rm A}^{f}F_{\rm
A}^{f}\Bigr{]}_{s=\overline{M}_{\mathrm{Z}}^{2}}\;.$ (8)

Here $N_{c}^{f}$ is the color factor and ${\cal R}_{\rm V,A}$ are radiator
functions that capture final-state QCD and QED corrections, see section 7 in
Ref. Chetyrkin:1994js , whereas the remaining electroweak and mixed
electroweak–QCD corrections are contained in the form factors $F_{\rm
V,A}^{f}$. Up to two-loop accuracy, the form factors can be written as follows
Freitas:2014hra :

$\displaystyle F_{\rm
V}^{f}=\frac{|v_{f}|^{2}}{1+\text{Re}\\{\Sigma^{\prime}_{\mathrm{Z}}-\frac{i}{2}\overline{M}_{\mathrm{Z}}\overline{\Gamma}_{\mathrm{Z}}\Sigma^{\prime\prime}_{\mathrm{Z}}\\}}\bigg{|}_{s=\overline{M}_{\mathrm{Z}}^{2}}\,,$
(9) $\displaystyle F_{\rm
A}^{f}=\frac{|a_{f}|^{2}}{1+\text{Re}\\{\Sigma^{\prime}_{\mathrm{Z}}-\frac{i}{2}\overline{M}_{\mathrm{Z}}\overline{\Gamma}_{\mathrm{Z}}\Sigma^{\prime\prime}_{\mathrm{Z}}\\}}\bigg{|}_{s=\overline{M}_{\mathrm{Z}}^{2}}\,,$
(10)

where $\Sigma^{\prime}_{\mathrm{Z}}$ and $\Sigma^{\prime\prime}_{\mathrm{Z}}$
are shorthand expressions for $d\Sigma_{\mathrm{Z}}/ds$ and
$d^{2}\Sigma_{\mathrm{Z}}/ds^{2}$, respectively.

In addition to the partial widths, certain branching ratios are of
phenomenological importance:

$\displaystyle R_{\ell}$ $\displaystyle=\overline{\Gamma}_{\rm
had}/\overline{\Gamma}_{\ell},$ $\displaystyle R_{c}$
$\displaystyle=\overline{\Gamma}_{c}/\overline{\Gamma}_{\rm had},$
$\displaystyle R_{b}$
$\displaystyle=\overline{\Gamma}_{b}/\overline{\Gamma}_{\rm had}.$ (11)

Here $\overline{\Gamma}_{\rm had}=\sum_{f=u,c,d,s,b}\overline{\Gamma}_{f}$.
Further, the cross-section for $e^{+}e^{-}\to\text{hadrons}$ at the $Z$ peak
can be expressed in terms of partial widths Freitas:2014hra ,

$\displaystyle\sigma_{\rm had}$ $\displaystyle=\sigma^{0}_{\rm
had}+\sigma_{\rm had,non-res}\,,$ $\displaystyle\sigma^{0}_{\rm had}$
$\displaystyle=\sum_{f=u,d,c,s,b}\frac{12\pi}{\overline{M}_{\mathrm{Z}}^{2}}\,\frac{\overline{\Gamma}_{e}\overline{\Gamma}_{f}}{\overline{\Gamma}_{\mathrm{Z}}^{2}}(1+\delta
X)\,.$ (12)

Here $\sigma^{0}_{\rm had,non-res}$ accounts for non-resonant photon-exchange,
box and $t$-channel contributions. Furthermore, $\delta X$ occurs from higher-
order terms of the Laurent expansion of the full amplitude around the complex
pole $s_{0}$. At two-loop order, $\delta X$ can be written as $\delta
X_{(2)}=-(\text{Im}\,\Sigma^{\prime}_{{\mathrm{Z}}(1)})^{2}-2\overline{\Gamma}_{\mathrm{Z}}\overline{M}_{\mathrm{Z}}\;\text{Im}\,\Sigma^{\prime\prime}_{{\mathrm{Z}}(1)}$,
where subscripts $(n)$ indicate the loop order. In the limit $m_{f}\ll
M_{\mathrm{W}}$ ($f\neq t$), it is given by

$\displaystyle\delta X_{(2)}$
$\displaystyle=-\biggl{[}\frac{\alpha}{s_{\scriptscriptstyle{\mathrm{W}}}^{2}c_{\scriptscriptstyle{\mathrm{W}}}^{2}}\Bigl{(}\frac{7}{8}-\frac{5}{3}s_{\scriptscriptstyle{\mathrm{W}}}^{2}+\frac{20}{9}s_{\scriptscriptstyle{\mathrm{W}}}^{4}\Bigr{)}\biggr{]}^{2}.$
(13)

It is important to note that eq. (12) assumes that initial-state photon
radiation effects have been removed by means of a de-convolution procedure,
see e.g. Ref. Schael:2013ita .

Observable | $X_{0}$ | $a_{1}$ | $a_{2}$ | $a_{3}$ | $a_{4}$ | $a_{5}$ | $a_{6}$ | $a_{7}$ | $a_{8}$
---|---|---|---|---|---|---|---|---|---
$\Gamma_{e,\mu}$ [MeV] | 83.983 |  $-$0.1202 |  $-$0.06919 | 0.00383 | 0.0597 | 0.8037 |  $-$0.015 |  $-$0.0195 | 0.0032
$\Gamma_{\tau}$ [MeV] | 83.793 |  $-$0.1200 |  $-$0.06905 | 0.00382 | 0.0596 | 0.8023 |  $-$0.015 |  $-$0.0195 | 0.0032
$\Gamma_{\nu}$ [MeV] | 167.176 |  $-$0.1752 |  $-$0.1249 | 0.00595 | 0.1046 | 1.253 |  $-$0.110 |  $-$0.0232 | 0.0064
$\Gamma_{u}$ [MeV] | 299.994 |  $-$0.6152 |  $-$0.2771 | 0.0174 | 0.2341 | 4.051 |  $-$0.467 |  $-$0.0676 | 0.017
$\Gamma_{c}$ [MeV] | 299.918 |  $-$0.6152 |  $-$0.2771 | 0.0174 | 0.2340 | 4.051 |  $-$0.467 |  $-$0.0676 | 0.017
$\Gamma_{d,s}$ [MeV] | 382.829 |  $-$0.6685 |  $-$0.3322 | 0.0193 | 0.2792 | 3.792 |  $-$0.18 |  $-$0.0706 | 0.020
$\Gamma_{b}$ [MeV] | 375.890 |  $-$0.6017 |  $-$0.3158 | 0.0190 | 0.227 |  $-$2.174 | 0.042 |  $-$0.027 | 0.021
$\Gamma_{\mathrm{Z}}$ [MeV] | 2494.75 |  $-$4.055 |  $-$2.117 | 0.122 | 1.746 | 19.68 |  $-$1.63 |  $-$0.432 | 0.12
$R_{\ell}$ [$10^{-3}$] | 20751.6 |  $-$8.112 |  $-$1.174 | 0.155 | 0.16 |  $-$37.59 |  $-$10.9 | 1.27 | 0.29
$R_{c}$ [$10^{-5}$] | 17222.2 |  $-$4.049 |  $-$0.749 | 0.0832 | 1.08 | 98.956 |  $-$15.1 |  $-$0.761 | 0.080
$R_{b}$ [$10^{-5}$] | 21585.0 | 4.904 | 0.9149 |  $-$0.0535 |  $-$2.676 |  $-$292.21 | 20.0 | 1.97 |  $-$0.11
$\sigma^{0}_{\rm had}$ [pb] | 41489.6 | 0.408 |  $-$0.320 | 0.0424 | 1.32 | 60.17 | 16.3 |  $-$2.31 |  $-$0.19

Observable | $a_{9}$ | $a_{10}$ | $a_{11}$ | $a_{12}$ | $a_{13}$ | $a_{14}$ | $a_{15}$ | $a_{16}$ | max. dev.
---|---|---|---|---|---|---|---|---|---
$\Gamma_{e,\mu}$ [MeV] |  $-$0.0956 |  $-$0.0078 |  $-$0.0095 | 0.25 |  $-$1.08 | 0.056 |  $-$0.37 | 286 | $<0.0015$
$\Gamma_{\tau}$ [MeV] |  $-$0.0954 |  $-$0.0078 |  $-$0.0094 | 0.25 |  $-$1.08 | 0.056 |  $-$0.37 | 285 | $<0.0015$
$\Gamma_{\nu}$ [MeV] |  $-$0.187 |  $-$0.014 |  $-$0.014 | 0.37 |  $-$0.085 | 0.054 |  $-$0.30 | 503 | $<0.002$
$\Gamma_{u}$ [MeV] | 14.26 | 1.6 |  $-$0.046 | 1.82 |  $-$11.1 | 0.16 |  $-$1.0 | 1253 | $<0.006$
$\Gamma_{c}$ [MeV] | 14.26 | 1.6 |  $-$0.046 | 1.82 |  $-$11.1 | 0.16 |  $-$1.0 | 1252 | $<0.006$
$\Gamma_{d,s}$ [MeV] | 10.20 |  $-$2.4 |  $-$0.052 | 0.71 |  $-$10.1 | 0.16 |  $-$0.92 | 1469 | $<0.007$
$\Gamma_{b}$ [MeV] | 10.53 |  $-$2.4 |  $-$0.056 | 1.2 |  $-$10.1 | 0.15 |  $-$0.95 | 1458 | $<0.007$
$\Gamma_{\mathrm{Z}}$ [MeV] | 58.61 |  $-$4.0 |  $-$0.32 | 8.1 |  $-$56.1 | 1.1 |  $-$6.8 | 9267 | $<0.04$
$R_{\ell}$ [$10^{-3}$] | 732.30 |  $-$44 |  $-$0.61 | 5.7 |  $-$358 |  $-$4.7 | 37 | 11649 | $<0.12$
$R_{c}$ [$10^{-5}$] | 230.9 | 125 | 0.045 | 36.9 |  $-$120 | 1.2 |  $-$6.2 | 3667 | $<0.1$
$R_{b}$ [$10^{-5}$] |  $-$131.9 |  $-$84 |  $-$0.27 | 4.4 | 71.9 |  $-$0.77 |  $-$4.4 |  $-$1790 | $<0.12$
$\sigma^{0}_{\rm had}$ [pb] |  $-$579.58 | 38 | 0.010 | 7.5 | 85.2 | 9.1 |  $-$68 |  $\\!\\!\\!-$85957 | $<0.15$

Table 1: Coefficients for the parameterization formula (15) for various
observables. Within the ranges
$25\,\,\mathrm{GeV}<M_{\mathrm{H}}<225\,\,\mathrm{GeV}$,
$155\,\,\mathrm{GeV}<m_{\mathrm{t}}<195\,\,\mathrm{GeV}$,
$\alpha_{\mathrm{s}}=0.1184\pm 0.0050$, $\Delta\alpha=0.0590\pm 0.0005$ and
$M_{\mathrm{Z}}=91.1876\pm 0.0084\,\,\mathrm{GeV}$, the formulae approximate
the full result with maximal deviations given in the last column.

Results for the partial and total $Z$ widths, branching ratios and
$\sigma^{0}_{\rm had}$ including the full two-loop corrections have first been
published in Ref. Dubovyk:2018rlg . They can be expressed in simple
parameterization formulae, which are adequate for most phenomenological
applications. Here, we present slightly more complicated formulae that cover
an extended numerical range of input parameters:222This extended input
parameter range is useful for determining indirect constraints on various SM
parameters from electroweak precision observables (see e.g. section 10 in Ref.
Patrignani:2016xqp ), since these indirect bounds often extend over larger
intervals than the corresponding direct measurements.

$\displaystyle 25\,\,\mathrm{GeV}<M_{\mathrm{H}}<225\,\,\mathrm{GeV},$
$\displaystyle 155\,\,\mathrm{GeV}<m_{\mathrm{t}}<192\,\,\mathrm{GeV},$
$\displaystyle M_{\mathrm{Z}}=91.1876\pm 0.0084\,\,\mathrm{GeV},$
$\displaystyle\alpha_{\mathrm{s}}=0.1184\pm 0.0050,$
$\displaystyle\Delta\alpha=0.0590\pm 0.0005.$ (14)

Here $\Delta\alpha$ is the shift in the running electromagnetic coupling
$\alpha(q^{2})$ from $q^{2}=0$ to $M_{\mathrm{Z}}^{2}$, defined by
$\alpha(M_{\mathrm{Z}}^{2})=\alpha(0)/(1-\Delta\alpha)$. It can be divided
into a leptonic and a hadronic part, $\Delta\alpha=\Delta\alpha_{\rm
lept}+\Delta\alpha_{\rm had}$. $\Delta\alpha_{\rm lept}$ has been computed to
three-loop order Steinhauser:1998rq , whereas $\Delta\alpha_{\rm had}$
contains non-perturbative hadronic contributions, which are commonly extracted
from data Davier:2017zfy ; Jegerlehner:2017zsb ; Keshavarzi:2018mgv . We
neglect the light fermion masses $m_{f}$, $f\neq t$, everywhere besides in
$\Delta\alpha$ and (at leading power) in the radiator functions ${\cal
R}^{f}_{\rm V/A}$. The $W$ boson mass $M_{\mathrm{W}}$ can be computed from
the Fermi constant $G_{\mu}$ (see eqs. (6)–(8) in the arXiv version of Ref.
Awramik:2003rn ) and thus is not listed as an independent input parameter.
Both $G_{\mu}$ and $\alpha$, the electromagnetic fine structure constant in
the Thomson limit, are known with very small uncertainties, and thus we use
their central experimental values Patrignani:2016xqp  without any uncertainty
interval.

The fitting formulae for the EWPOs have the form

$\displaystyle\begin{aligned}
X=&X_{0}+a_{1}L_{\mathrm{H}}+a_{2}L_{\mathrm{H}}^{2}+a_{3}L_{\mathrm{H}}^{4}+a_{4}\Delta_{\mathrm{H}}+a_{5}\Delta_{\mathrm{t}}+a_{6}\Delta_{\mathrm{t}}^{2}+a_{7}\Delta_{\mathrm{t}}L_{\mathrm{H}}+a_{8}\Delta_{\mathrm{t}}L_{\mathrm{H}}^{2}\\\
&+a_{9}\Delta_{\alpha_{\mathrm{s}}}+a_{10}\Delta_{\alpha_{\mathrm{s}}}^{2}+a_{11}\Delta_{\alpha_{\mathrm{s}}}\Delta_{\mathrm{H}}+a_{12}\Delta_{\alpha_{\mathrm{s}}}\Delta_{\mathrm{t}}+a_{13}\Delta_{\alpha}+a_{14}\Delta_{\alpha}\Delta_{\mathrm{H}}\\\
&+a_{15}\Delta_{\alpha}\Delta_{t}+a_{16}\Delta_{\mathrm{Z}},\end{aligned}$
(15) $\displaystyle
L_{\mathrm{H}}=\log\frac{M_{\mathrm{H}}}{125.7\,\,\mathrm{GeV}},\quad\Delta_{\mathrm{H}}=\frac{M_{\mathrm{H}}}{125.7\,\,\mathrm{GeV}}-1,\quad\Delta_{\mathrm{t}}=\Bigl{(}\frac{m_{\mathrm{t}}}{173.2\,\,\mathrm{GeV}}\Bigr{)}^{2}-1,\quad$
$\displaystyle\Delta_{\alpha_{\mathrm{s}}}=\frac{\alpha_{\mathrm{s}}(M_{\mathrm{Z}})}{0.1184}-1,\quad\Delta_{\alpha}=\frac{\Delta\alpha}{0.059}-1,\quad\Delta_{\mathrm{Z}}=\frac{M_{\mathrm{Z}}}{91.1876\,\,\mathrm{GeV}}-1.$

The coefficients $X_{0}$ and $a_{1},\ldots a_{16}$ are obtained from fits to a
grid of 8750 data points of the full computation. The latter includes

  * •

Complete one-loop corrections Akhundov:1985fc , which have been re-computed
for this work, and full two-loop Freitas:2013dpa ; Freitas:2014hra ;
Dubovyk:2018rlg  electroweak corrections;

  * •

Corrections of order ${\mathcal{O}}(\alpha\alpha_{\mathrm{s}})$ to vector-
boson self-energies Djouadi:1987gn ; Djouadi:1987di ; Kniehl:1989yc ;
Kniehl:1991gu ; Djouadi:1993ss , which have been re-evaluated for this work;

  * •

Non-factorizable ${\mathcal{O}}(\alpha\alpha_{\mathrm{s}})$ $Zq\bar{q}$ vertex
contributions Czarnecki:1996ei ; Harlander:1997zb ; Fleischer:1992fq ;
Buchalla:1992zm ; Degrassi:1993ij ; Chetyrkin:1993jp , which are not captured
in the products ${\cal R}_{i}^{f}F_{i}^{f}$ ($i=$ V,A);

  * •

Higher-loop QCD corrections in the limit of a large top Yukawa coupling
$y_{\mathrm{t}}$, of orders ${\mathcal{O}}(\and\alpha_{\mathrm{s}}^{2})$
Avdeev:1994db ; Chetyrkin:1995ix ,
${\mathcal{O}}(\and^{2}\alpha_{\mathrm{s}})$, ${\mathcal{O}}(\and^{3})$
vanderBij:2000cg ; Faisst:2003px , and
${\mathcal{O}}(\and\alpha_{\mathrm{s}}^{3})$ Schroder:2005db ;
Chetyrkin:2006bj ; Boughezal:2006xk , where $\and\equiv
y_{\mathrm{t}}^{2}/(4\pi)$.

  * •

Final-state QED and QCD radiation effects, which enter through the radiator
functions ${\cal R}_{\rm V,A}$, up to the orders ${\mathcal{O}}(\alpha^{2})$,
${\mathcal{O}}(\alpha\alpha_{\mathrm{s}})$ and
${\mathcal{O}}(\alpha_{\mathrm{s}}^{4})$ Chetyrkin:1994js ; Baikov:2008jh ;
Kataev:1992dg .

Both the $Z$ vertex corrections as well as the prediction of $M_{\mathrm{W}}$
from $G_{\mu}$ have been computed to this same level of perturbation theory.

Numerical values for the coefficients are given in Tab. 1. Some of the numbers
for $X_{0}$ deviate slightly in the last digit from those in Ref.
Dubovyk:2018rlg . This is due to the larger grid of input parameters used
here, which can exert a pull on the fit parameters. The differences are well
within the accuracy quoted in the last column of Tab. 1.

##  3 Asymmetries and effective weak mixing angles

The effective weak mixing angle for the $Zf\bar{f}$ vertex is defined, from
the theory side, as

$\displaystyle\sin^{2}\theta_{\rm
eff}^{f}\equiv\frac{1}{4|Q_{f}|}\biggl{(}1-\text{Re}\biggl{\\{}\frac{v_{f}(\overline{M}_{\mathrm{Z}}^{2})}{a_{f}(\overline{M}_{\mathrm{Z}}^{2})}\biggr{\\}}\biggr{)}.$
(16)

Here $\overline{M}_{\mathrm{W}}^{2}$ and $\overline{M}_{\mathrm{Z}}^{2}$ are
the real parts of the complex pole of the $W$ and $Z$ propagators,
respectively. They are related to the masses commonly reported by experiments
at LEP, Tevatron, LHC according to eq (6). Moreover, $Q_{f}$ denotes the
electric charge of the fermion $f$.

The effective weak mixing angles can be extracted from a range of asymmetries
Gluza:inYR2018 , defined from effective Born two-particle cross-sections,
including the left-right asymmetry

$\displaystyle A_{\rm LR}$ $\displaystyle=\frac{\sigma_{e_{\rm
L}}-\sigma_{e_{\rm R}}}{\sigma_{e_{\rm L}}+\sigma_{e_{\rm R}}}=A_{e}+A_{\rm
LR}^{\rm non-res}$ (17) and the forward-backward asymmetry $\displaystyle
A_{\rm FB}^{f}$
$\displaystyle=\frac{\sigma_{\cos\theta>0}-\sigma_{\cos\theta<0}}{\sigma_{\cos\theta>0}+\sigma_{\cos\theta<0}}=\tfrac{3}{4}A_{e}A_{f}+A_{\rm
FB}^{f,\rm non-res}\,,$ (18) where $\displaystyle A_{f}$
$\displaystyle\equiv\frac{2\,\text{Re}\\{v_{f}/a_{f}\\}}{1+(\text{Re}\\{v_{f}/a_{f}\\})^{2}}=\frac{1-4|Q_{f}|\sin^{2}\theta_{\rm
eff}^{f}}{1-4|Q_{f}|\sin^{2}\theta_{\rm eff}^{f}+8(|Q_{f}|\sin^{2}\theta_{\rm
eff}^{f})^{2}}\,.$ (19)

Here $\sigma_{e_{\rm L}}$ and $\sigma_{e_{\rm R}}$ are the cross-sections for
$e^{+}e^{-}\to f\bar{f}$ for left- and right-handed polarized electron beams,
respectively, whereas $\sigma_{\cos\theta>0}$ and $\sigma_{\cos\theta<0}$
denote the cross-section for $f$ restricted to the forward and backward
hemisphere, respectively. Furthermore, $A_{X}^{\rm non-res}$ accounts for the
non-resonant photon-exchange, box and $t$-channel contributions.

The most precisely measured effective weak mixing angles are the leptonic
effective weak mixing angle $\sin^{2}\theta_{\rm eff}^{\ell}$ (extracted from
$A_{\rm LR}$) and the bottom-quark one, $\sin^{2}\theta_{\rm eff}^{b}$
(extracted from $A_{\rm FB}^{b}$) Schael:2013ita .

Standard Model predictions for $\sin^{2}\theta_{\rm eff}^{\ell}$ including the
full two-loop corrections have been presented originally in Ref.
Awramik:2006ar ; Awramik:2006uz ; Hollik:2006ma . We reproduced by an
independent calculation the contribution of the bosonic electroweak two-loop
corrections using the methods of Ref. Dubovyk:2018rlg . The corrections can be
expressed in terms of a weak form factor $\Delta\kappa_{\ell}^{(\alpha^{2},\rm
bos)}$, where

$\displaystyle\Delta\kappa_{f}=\biggl{(}1-\frac{\overline{M}_{\mathrm{W}}}{\overline{M}_{\mathrm{Z}}}\biggr{)}^{-1}\sin^{2}\theta_{\rm
eff}^{f}-1\,.$ (20)

The comparison with Ref. Awramik:2006ar  is shown in Tab. 2, which
demonstrates that the two calculations agree to an accuracy of
${\mathcal{O}}(10^{-7})$, which implies an accuracy of better than $10^{-7}$
for $\sin^{2}\theta_{\rm eff}^{\ell}$. The full two-loop corrections for
$\sin^{2}\theta_{\rm eff}^{b}$ have been presented first in Ref.
Dubovyk:2016aqv .

$M_{\mathrm{H}}$ [GeV] | Result of Ref. Awramik:2006ar  | Our result
---|---|---
100 | $-0.733989\times 10^{-4}$ | $-0.733955\times 10^{-4}$
200 | $-0.469470\times 10^{-4}$ | $-0.471273\times 10^{-4}$
Table 2: Comparison of numerical results for
$\Delta\kappa_{\ell}^{(\alpha^{2},\rm bos)}$ from Ref. Awramik:2006ar  with
our calculation of $Z$ vertex corrections from Ref. Dubovyk:2018rlg . We use
$\overline{M}_{\mathrm{Z}}=91.1876\,\,\mathrm{GeV}$ and
$\overline{M}_{\mathrm{W}}=80.385\,\,\mathrm{GeV}$.

In the following, we present simple parameterization formulae for
$\sin^{2}\theta_{\rm eff}^{\ell}$ and $\sin^{2}\theta_{\rm eff}^{b}$, which
cover the extended range of input parameters of eq. (14). The parameterization
formula

$\displaystyle\sin^{2}\theta_{\rm eff}^{f}=\begin{aligned}
&s_{0}+d_{1}L_{H}+d_{2}L_{H}^{2}+d_{3}L_{H}^{4}+d_{4}\Delta_{\alpha}+d_{5}\Delta_{\mathrm{t}}+d_{6}\Delta_{\mathrm{t}}^{2}+d_{7}\Delta_{\mathrm{t}}L_{H}\\\
&+d_{8}\Delta_{\alpha_{\mathrm{s}}}+d_{9}\Delta_{\alpha_{\mathrm{s}}}\Delta_{\mathrm{t}}+d_{10}\Delta_{\mathrm{Z}}\end{aligned}$
(21) with $\displaystyle
L_{\mathrm{H}}=\log\frac{M_{\mathrm{H}}}{125.7\,\,\mathrm{GeV}},\quad\Delta_{\mathrm{t}}=\Bigl{(}\frac{m_{\mathrm{t}}}{173.2\,\,\mathrm{GeV}}\Bigr{)}^{2}-1,\quad$
$\displaystyle\Delta_{\alpha_{\mathrm{s}}}=\frac{\alpha_{\mathrm{s}}(M_{\mathrm{Z}})}{0.1184}-1,\quad\Delta_{\alpha}=\frac{\Delta\alpha}{0.059}-1,\quad\Delta_{\mathrm{Z}}=\frac{M_{\mathrm{Z}}}{91.1876\,\,\mathrm{GeV}}-1$

provides a good description of the full result in the parameter region (14).
Values for the coefficients are obtained by fitting (21) to a grid of 8750
data points.

Table 3 shows the result of a fit to a calculation that includes all known
corrections:

  * •

Complete one- and two-loop electroweak corrections, (see Refs. Marciano:1980pb
; Akhundov:1985fc ; Awramik:2004ge ; Hollik:2005va ; Awramik:2006ar ;
Hollik:2006ma ; Awramik:2008gi ; Dubovyk:2016aqv  for the original
references);

  * •

Corrections of order ${\mathcal{O}}(\alpha\alpha_{\mathrm{s}})$ to vector-
boson self-energies Djouadi:1987gn ; Djouadi:1987di ; Kniehl:1989yc ;
Kniehl:1991gu ; Djouadi:1993ss , which we have re-evaluated for this work;

  * •

Non-factorizable ${\mathcal{O}}(\alpha\alpha_{\mathrm{s}})$ $Zb\bar{b}$ vertex
contributions Czarnecki:1996ei ; Harlander:1997zb ; Fleischer:1992fq ;
Buchalla:1992zm ; Degrassi:1993ij ; Chetyrkin:1993jp , which do not cancel in
the ratio $v_{b}/a_{b}$;

  * •

Higher-loop corrections in the limit of a large top Yukawa coupling
$y_{\mathrm{t}}$, of orders ${\mathcal{O}}(\and\alpha_{\mathrm{s}}^{2})$
Avdeev:1994db ; Chetyrkin:1995ix ,
${\mathcal{O}}(\and^{2}\alpha_{\mathrm{s}})$, ${\mathcal{O}}(\and^{3})$
vanderBij:2000cg ; Faisst:2003px , and
${\mathcal{O}}(\and\alpha_{\mathrm{s}}^{3})$ Schroder:2005db ;
Chetyrkin:2006bj ; Boughezal:2006xk  where $\and\equiv
y_{\mathrm{t}}^{2}/(4\pi)$.

As indicated by the last column in the table, the largest deviation of the fit
formulae from the full result is ${\mathcal{O}}(\text{few}\times 10^{-6})$,
while for most of the parameter region in (14) the agreement is better than
$10^{-6}$. The careful reader may realize that the parameterization for
$\sin^{2}\theta_{\rm eff}^{b}$ in Table 3 deviates slightly from Eqs. (20,22)
in Dubovyk:2016aqv . The difference is due to the larger grid of data points
used here. A fit formula is, obviously, not able to reproduce the data points
in a grid perfectly. The fitting aims to find the best average agreement
between the data points (which are generated with our full numerical
calculation) and the fit formula. A larger grid therefore can lead to some
shifts of the coefficients. As a consequence, the formula in Dubovyk:2016aqv
will probably be more accurate for input values within the ranges in Tab. 1
there. On the other hand, while the formula here may be a little less accurate
within these ranges, it covers a much larger range of input values.

Observable | $s_{0}$ | $d_{1}$ | $d_{2}$ | $d_{3}$ | $d_{4}$ | $d_{5}$
---|---|---|---|---|---|---
$\sin^{2}\theta_{\rm eff}^{\ell}\times 10^{4}$ | 2314.64 | 4.616 | 0.539 |  $-$0.0737 | 206 |  $-$25.71
$\sin^{2}\theta_{\rm eff}^{b}\times 10^{4}$ | 2327.04 | 4.638 | 0.558 |  $-$0.0700 | 207 |  $-$9.554

Observable | $d_{6}$ | $d_{7}$ | $d_{8}$ | $d_{9}$ | $d_{10}$ | max. dev.
---|---|---|---|---|---|---
$\sin^{2}\theta_{\rm eff}^{\ell}\times 10^{4}$ | 4.00 | 0.288 | 3.88 |  $-$6.49 |  $-$6560 | $<0.056$
$\sin^{2}\theta_{\rm eff}^{b}\times 10^{4}$ | 3.83 | 0.179 | 2.41 |  $-$8.24 |  $-$6630 | $<0.025$

Table 3: Coefficients for the parameterization formula (21) for the leptonic
and bottom-quark effective weak mixing angles. Within the ranges given in eq.
(14), the formula deviates from the full result up to the maximal amount given
in the last column.

It should also be noted that the fit formula for $\sin^{2}\theta_{\rm
eff}^{\ell}$ in Ref. Awramik:2006ar  does not include the
${\mathcal{O}}(\and\alpha_{\mathrm{s}}^{3})$ corrections from Refs.
Schroder:2005db ; Chetyrkin:2006bj ; Boughezal:2006xk , but they are included
in the formula presented here.

In Tab. 4 it is shown that the technical accuracy of our fit formulae is
adequate for the expected experimental precision of several future
$e^{+}e^{-}$ colliders, although it will get modified by anticipated future
three-loop electroweak corrections.

Observable | max. dev. | EXP now | FCC-ee | CEPC | GigaZ
---|---|---|---|---|---
$\Gamma_{Z}$ [MeV] | 0.04 | 2.3 | 0.1 | 0.5 | 0.8
$\sin^{2}\theta_{\rm eff}^{\ell}\times 10^{4}$ | 0.056 | 1.6 | 0.06 | 0.23 | 0.1
$\sin^{2}\theta_{\rm eff}^{b}\times 10^{4}$ | 0.025 | 160 | 9 | 9 | 15

Table 4: Goodness of fit for some chosen EWPOs, compared with the envisaged
precision measurements for $\Gamma_{Z}$ and $\sin^{2}\theta_{\rm eff}^{\ell}$
(statistical errors), and $\sin^{2}\theta_{\rm eff}^{b}$ (systematic errors)
at the collider projects FCC-ee Tera-Z Abada:2019lih , CEPC
CEPCStudyGroup:2018ghi  and ILC/GigaZ Hawkings:1999ac . The values of maximal
deviations are taken from Tabs. 1 and 3. The entry “EXP now” gives the present
experimental precision, as known since LEP 1 ALEPH:2005ab .

##  4 Vector and axial-vector $Z$-boson form factors $F_{\rm V}^{f}$ and
$F_{\rm A}^{f}$

The pseudo-observables discussed in the previous sections aim to be closely
related to actual observables, such as cross-sections, branching ratios, or
asymmetries. On the other hand, for some purposes it is also useful to have
numerical results for the underlying vertex corrections themselves
Freitas:2014owa , for example: (i) Inclusion of selected corrections from
Beyond Standard Model (BSM) physics, (ii) Estimations of magnitudes of
selected single terms, (iii) Partial cross-checks with other calculations. For
such purposes, the form factors $F_{\rm V}^{f}$ and $F_{\rm A}^{f}$ introduced
in eq. (8) are needed explicitly.

Tables 5 and 6 show the numerical contributions of different orders of
perturbation theory to $F_{\rm V}^{f}$ and $F_{\rm A}^{f}$. Here the form
factors are always understood to include the appropriate (on-shell)
counterterms to render them UV-finite. The non-factorizable
${\mathcal{O}}(\alpha\alpha_{\mathrm{s}})$ corrections are shown separately in
the table. Strictly speaking they are not part of the form factors $F_{\rm
V}^{f}$ and $F_{\rm A}^{f}$, but they finally contribute to observables, such
as the decay width, and we incorporate them in the tables and the fit formula
below. In Tab. 5 the form factors are computed using the following input
values:

$\displaystyle M_{\mathrm{Z}}=91.1876\,\,\mathrm{GeV},$
$\displaystyle\Gamma_{\mathrm{Z}}=2.4952\,\,\mathrm{GeV},$
$\displaystyle\Rightarrow\quad\overline{M}_{\mathrm{Z}}=91.1535\,\,\mathrm{GeV}$
(22a) $\displaystyle M_{\mathrm{W}}=80.385\,\,\mathrm{GeV},$
$\displaystyle\Gamma_{\mathrm{W}}=2.085\,\,\mathrm{GeV},$
$\displaystyle\Rightarrow\quad\overline{M}_{\mathrm{W}}=80.358\,\,\mathrm{GeV}$
(22b) $\displaystyle M_{\mathrm{H}}=125.1\,\,\mathrm{GeV},$ $\displaystyle
m_{\mathrm{t}}=173.2\,\,\mathrm{GeV},$ $\displaystyle
m_{\mathrm{b}}^{\overline{\text{MS}}}=4.2\,\,\mathrm{GeV},$
$\displaystyle\Delta\alpha=0.059,$ $\displaystyle\alpha_{\mathrm{s}}=0.1184$
(22c)

For Tab. 6, on the other hand, the Fermi constant $G_{\mu}$ is used as an
input instead of (22b), and $\overline{M}_{\mathrm{W}}$ is computed from

$\displaystyle\overline{M}_{\mathrm{W}}^{2}\biggl{(}1-\frac{\overline{M}_{\mathrm{W}}^{2}}{\overline{M}_{\mathrm{Z}}^{2}}\biggr{)}=\frac{\pi\alpha}{\sqrt{2}G_{\mu}}(1+\Delta
r),$ (23)

where $\Delta r$ has been evaluated to the same orders as given in each column
of the table. More details about the calculation of $\Delta r$ can be found in
Ref. Awramik:2003rn . As before, the dependence of the Standard Model
prediction on various input parameters can be expressed in terms of the simple
parameterization formula eq. (15).

Form fact. | Born | ${\cal O}(\alpha)$ | ${\cal O}(\alpha\alpha_{\mathrm{s}})$ |  ${\cal O}(\alpha\alpha_{\mathrm{s}})$ non-fact. |  ${\cal O}(\and\alpha_{\mathrm{s}}^{2},\,\and\alpha_{\mathrm{s}}^{3},$ ${}\quad\,\and^{2}\alpha_{\mathrm{s}},\,\and^{3})$ | ${\cal O}(N_{f}^{2}\alpha^{2})$ | ${\cal O}(N_{f}\alpha^{2})$ | ${\cal O}(\alpha^{2}_{\rm bos})$
---|---|---|---|---|---|---|---|---
$F_{V}^{\ell}$ [$10^{-5}$] | 39.07 |  $-$24.86 | 2.41 | – | 0.35 | 1.47 | 2.37 | 0.27
$F_{A}^{\ell}$ [$10^{-5}$] | 3309.44 | 118.59 | 9.46 | – | 1.22 | 8.60 | 2.60 | 0.45
$F_{V,A}^{\nu}$ [$10^{-5}$] | 3309.44 | 127.56 | 9.46 | – | 1.22 | 8.60 | 3.83 | 0.39
$F_{V}^{u,c}$ [$10^{-5}$] | 544.88 |  $-$44.80 | 7.29 |  $-$0.39 | 1.02 |  $-$1.67 | 3.25 | 0.33
$F_{A}^{u,c}$ [$10^{-5}$] | 3309.44 | 120.79 | 9.46 |  $-$0.98 | 1.22 | 8.60 | 3.27 | 0.44
$F_{V}^{d,s}$ [$10^{-5}$] | 1635.01 | 5.84 | 9.64 |  $-$0.80 | 1.32 | 0.71 | 3.45 | 0.37
$F_{A}^{d,s}$ [$10^{-5}$] | 3309.44 | 123.78 | 9.46 |  $-$1.14 | 1.22 | 8.60 | 3.11 | 0.42
$F_{V}^{b}$ [$10^{-5}$] | 1635.01 |  $-$26.16 | 9.64 |  $-$3.13 | 1.32 | 0.71 | 1.77 | 1.05
$F_{A}^{b}$ [$10^{-5}$] | 3309.44 | 78.26 | 9.46 |  $-$4.45 | 1.22 | 8.60 | 0.13 | 1.18
Table 5: Contributions of different perturbative orders to the $Z$ vertex form factors. A fixed value of $M_{\mathrm{W}}$ has been used as input, instead of $G_{\mu}$. $N_{f}^{n}$ refers to corrections with $n$ closed fermions loops, whereas $\alpha_{\rm bos}^{2}$ denotes corrections without closed fermions loops. Furthermore, $\and=y_{\mathrm{t}}/(4\pi)$ where $y_{\mathrm{t}}$ is the top Yukawa coupling.  Form fact. | Born | ${\cal O}(\alpha)$ | ${\cal O}(\alpha\alpha_{\mathrm{s}})$ |  ${\cal O}(\alpha\alpha_{\mathrm{s}})$ non-fact. |  ${\cal O}(\and\alpha_{\mathrm{s}}^{2},\,\and\alpha_{\mathrm{s}}^{3},$ ${}\quad\,\and^{2}\alpha_{\mathrm{s}},\,\and^{3})$ | ${\cal O}(N_{f}^{2}\alpha^{2},\,N_{f}\alpha^{2})$ | ${\cal O}(\alpha^{2}_{\rm bos})$
---|---|---|---|---|---|---|---
$F_{V}^{\ell}$ [$10^{-5}$] | 77.63 |  $-$59.86 | 0.31 | – |  $-$0.09 | 1.88 | 0.24
$F_{A}^{\ell}$ [$10^{-5}$] | 3426.43 | 19.32 |  $-$1.12 | – |  $-$0.92 | 1.62 | 0.27
$F_{V,A}^{\nu}$ [$10^{-5}$] | 3426.43 | 28.36 |  $-$1.16 | – |  $-$0.93 | 2.81 | 0.22
$F_{V}^{u,c}$ [$10^{-5}$] | 644.45 |  $-$129.87 |  $-$1.36 |  $-$0.40 |  $-$0.73 |  $-$6.25 | 0.19
$F_{A}^{u,c}$ [$10^{-5}$] | 3426.43 | 21.54 |  $-$1.13 |  $-$0.99 |  $-$0.92 | 2.28 | 0.27
$F_{V}^{d,s}$ [$10^{-5}$] | 1760.71 |  $-$100.64 |  $-$1.85 |  $-$0.81 |  $-$1.01 |  $-$6.23 | 0.19
$F_{A}^{d,s}$ [$10^{-5}$] | 3426.43 | 24.56 |  $-$1.15 |  $-$1.15 |  $-$0.93 | 2.11 | 0.25
$F_{V}^{b}$ [$10^{-5}$] | 1760.71 |  $-$133.08 |  $-$1.58 |  $-$3.15 |  $-$0.96 |  $-$7.70 | 0.86
$F_{A}^{b}$ [$10^{-5}$] | 3426.43 |  $-$21.45 |  $-$0.85 |  $-$4.47 |  $-$0.88 |  $-$0.64 | 1.01
Table 6: Same as Tab. 5, but with $M_{\mathrm{W}}$ calculated from $G_{\mu}$.

Table 7 shows the numerical values for the coefficients obtained by fitting
this formula to the currently most precise computation, including the same
corrections as in section 2, except for the final-state QED and QCD radiation
effects, i.e.

$\displaystyle F_{\rm V}^{f}$ $\displaystyle=|v_{f(0)}|^{2}+F^{f}_{\rm
V(\alpha)}+F^{f}_{\rm V(\alpha\alpha_{\mathrm{s}})}+F^{f}_{\rm
V(\alpha^{2})}+F^{f}_{\rm
V(\alpha_{\mathrm{t}}\alpha_{\mathrm{s}}^{2})}+F^{f}_{\rm
V(\alpha_{\mathrm{t}}^{2}\alpha_{\mathrm{s}})}+F^{f}_{\rm
V(\alpha_{\mathrm{t}}^{3})}+F^{f}_{\rm
V(\alpha_{\mathrm{t}}\alpha_{\mathrm{s}}^{3})}\,,$ $\displaystyle F_{\rm
A}^{f}$ $\displaystyle=|a_{f(0)}|^{2}+F^{f}_{\rm A(\alpha)}+F^{f}_{\rm
A(\alpha\alpha_{\mathrm{s}})}+F^{f}_{\rm A(\alpha^{2})}+F^{f}_{\rm
A(\alpha_{\mathrm{t}}\alpha_{\mathrm{s}}^{2})}+F^{f}_{\rm
A(\alpha_{\mathrm{t}}^{2}\alpha_{\mathrm{s}})}+F^{f}_{\rm
A(\alpha_{\mathrm{t}}^{3})}+F^{f}_{\rm
A(\alpha_{\mathrm{t}}\alpha_{\mathrm{s}}^{3})}\,.$ (24)

Note that $G_{\mu}$ (rather than $M_{\mathrm{W}}$) has been used as one input
in Tab. 7.

The form factor results presented here can be easily augmented to include the
effects of some new physics model:

$\displaystyle F^{f}_{\rm V,SM{+}NP}$ $\displaystyle\approx F^{f}_{\rm
V,SM}+2\text{Re}\\{v_{f(0)}v_{f,\rm NP}\\}\,,$ (25) $\displaystyle F^{f}_{\rm
A,SM{+}NP}$ $\displaystyle\approx F^{f}_{\rm
A,SM}+2\text{Re}\\{a_{f(0)}a_{f,\rm NP}\\}\,.$ (26)

Here “SM” denotes the SM contributions discussed in the present paper, while
“NP” stands for the new physics correction on top of the SM. Since the
existing experimental constraints imply that any possible new physics effect
is small, it is sufficient to use the tree-level couplings $v_{f(0)}$ and
$a_{f(0)}$ in the interference terms and neglect the $|v_{f,\rm NP}|^{2}$ and
$|a_{f,\rm NP}|^{2}$ terms.

Form fact. | $X_{0}$ | $a_{1}$ | $a_{2}$ | $a_{3}$ | $a_{4}$ | $a_{5}$ | $a_{6}$ | $a_{7}$ | $a_{8}$
---|---|---|---|---|---|---|---|---|---
$F_{V}^{\ell}$ [$10^{-5}$] | 20.11 |  $-$1.317 |  $-$0.2614 | 0.0333 | 0.276 | 7.472 | 1.55 |  $-$0.326 | 0.0012
$F_{A}^{\ell}$ [$10^{-5}$] | 3445.60 |  $-$3.640 |  $-$2.592 | 0.124 | 2.186 | 25.67 |  $-$2.17 |  $-$0.480 | 0.13
$F_{V,A}^{\nu}$ [$10^{-5}$] | 3455.73 |  $-$3.618 |  $-$2.580 | 0.123 | 2.161 | 25.89 |  $-$2.27 |  $-$0.480 | 0.13
$F_{V}^{u,c}$ [$10^{-5}$] | 506.025 |  $-$4.483 |  $-$1.062 | 0.105 | 0.912 | 23.20 |  $-$1.54 |  $-$0.409 | 0.085
$F_{A}^{u,c}$ [$10^{-5}$] | 3447.47 |  $-$3.630 |  $-$2.588 | 0.124 | 2.178 | 25.80 |  $-$2.24 |  $-$0.480 | 0.13
$F_{V}^{d,s}$ [$10^{-5}$] | 1650.36 |  $-$5.243 |  $-$1.834 | 0.134 | 1.540 | 29.20 |  $-$2.78 |  $-$0.458 | 0.13
$F_{A}^{d,s}$ [$10^{-5}$] | 3450.12 |  $-$3.612 |  $-$2.581 | 0.123 | 2.167 | 25.64 |  $-$2.21 |  $-$0.476 | 0.13
$F_{V}^{b}$ [$10^{-5}$] | 1621.40 |  $-$4.888 |  $-$1.756 | 0.134 | 1.27 |  $-$2.87 |  $-$1.5 |  $-$0.21 | 0.13
$F_{A}^{b}$ [$10^{-5}$] | 3408.08 |  $-$3.110 |  $-$2.458 | 0.120 | 1.754 |  $-$21.04 |  $-$0.60 |  $-$0.15 | 0.14

Form fact. | $a_{9}$ | $a_{10}$ | $a_{11}$ | $a_{12}$ | $a_{13}$ | $a_{14}$ | $a_{15}$ | $a_{16}$ | max. dev.
---|---|---|---|---|---|---|---|---|---
$F_{V}^{\ell}$ [$10^{-5}$] |  $-$0.13 |  $-$0.037 |  $-$0.11 | 2.78 |  $-$43.15 | 1.2 |  $-$8.9 | 1396 | $<0.02$
$F_{A}^{\ell}$ [$10^{-5}$] |  $-$3.87 |  $-$0.28 |  $-$0.28 | 7.6 |  $-$1.9 | 1.1 |  $-$6.3 | 6912 | $<0.04$
$F_{V,A}^{\nu}$ [$10^{-5}$] |  $-$3.93 |  $-$0.29 |  $-$0.28 | 7.6 |  $-$1.7 | 1.1 |  $-$6.3 | 6939 | $<0.04$
$F_{V}^{u,c}$ [$10^{-5}$] |  $-$3.89 |  $-$0.16 |  $-$0.22 | 6.5 |  $-$145.6 | 0.98 |  $-$6.9 | 5648 | $<0.03$
$F_{A}^{u,c}$ [$10^{-5}$] |  $-$4.86 |  $-$0.27 |  $-$0.27 | 7.5 |  $-$1.9 | 1.1 |  $-$6.3 | 6918 | $<0.04$
$F_{V}^{d,s}$ [$10^{-5}$] |  $-$5.56 |  $-$0.24 |  $-$0.28 | 8.1 |  $-$132 | 0.95 |  $-$6.0 | 7492 | $<0.04$
$F_{A}^{d,s}$ [$10^{-5}$] |  $-$5.03 |  $-$0.27 |  $-$0.27 | 7.5 |  $-$1.6 | 1.1 |  $-$6.2 | 6923 | $<0.04$
$F_{V}^{b}$ [$10^{-5}$] |  $-$1.3 |  $-$0.27 |  $-$0.32 | 12.5 |  $-$131 | 0.95 |  $-$5.7 | 7451 | $<0.04$
$F_{A}^{b}$ [$10^{-5}$] | 0.87 |  $-$0.31 |  $-$0.31 | 13.5 |  $-$2.4 | 1.1 |  $-$7.2 | 6918 | $<0.04$

Table 7: Coefficients for the parameterization formula (15) for various form
factors ($X$). Within the ranges
$25\,\,\mathrm{GeV}<M_{\mathrm{H}}<225\,\,\mathrm{GeV}$,
$155\,\,\mathrm{GeV}<m_{\mathrm{t}}<195\,\,\mathrm{GeV}$,
$\alpha_{\mathrm{s}}=0.1184\pm 0.0050$, $\Delta\alpha=0.0590\pm 0.0005$ and
$M_{\mathrm{Z}}=91.1876\pm 0.0084\,\,\mathrm{GeV}$, the formula approximates
the full result with average and maximal deviations given in the last two
columns.

##  5 Theoretical error estimates for missing higher order corrections

The main theory uncertainty of the results presented in this paper stems from
unknown three- and four-loop corrections. The leading missing orders are
${\mathcal{O}}(\alpha^{3})$, ${\mathcal{O}}(\alpha^{2}\alpha_{\mathrm{s}})$,
${\mathcal{O}}(\alpha\alpha_{\mathrm{s}}^{2})$, and
${\mathcal{O}}(\alpha\alpha_{\mathrm{s}}^{3})$. Partial results for these
contributions, in the limit of a large top Yukawa coupling $y_{\mathrm{t}}$,
have already been computed Avdeev:1994db ; Chetyrkin:1995ix ; vanderBij:2000cg
; Faisst:2003px ; Schroder:2005db ; Chetyrkin:2006bj ; Boughezal:2006xk .
Therefore, when evaluating the impact of theory uncertainties, it is always
implied that we refer to these contributions _beyond_ the
leading-$y_{\mathrm{t}}$ limit.

There are a number of different methods for assessing theory uncertainties
from unknown higher orders, none of which is fully reliable. Rather, they
should be taken as an order-of-magnitude estimate of the size of these terms.
A convenient and widely applicable method is based on the assumption that the
first few orders of the perturbation series approximately follow a geometric
series Freitas:2014hra ; Freitas:2016sty ; Dubovyk:2018rlg . In this way one
obtains as an ansatz

$\displaystyle{\mathcal{O}}(\alpha^{3})-{\mathcal{O}}(\alpha_{\mathrm{t}}^{3})$
$\displaystyle\sim\frac{{\mathcal{O}}(\alpha^{2})-{\mathcal{O}}(\alpha_{\mathrm{t}}^{2})}{{\mathcal{O}}(\alpha)}{\mathcal{O}}(\alpha^{2}),$
(27)
$\displaystyle{\mathcal{O}}(\alpha^{2}\alpha_{\mathrm{s}})-{\mathcal{O}}(\alpha_{\mathrm{t}}^{2}\alpha_{\mathrm{s}})$
$\displaystyle\sim\frac{{\mathcal{O}}(\alpha^{2})-{\mathcal{O}}(\alpha_{\mathrm{t}}^{2})}{{\mathcal{O}}(\alpha)}{\mathcal{O}}(\alpha\alpha_{\mathrm{s}}),$
$\displaystyle{\mathcal{O}}(\alpha\alpha_{\mathrm{s}}^{2})-{\mathcal{O}}(\alpha_{\mathrm{t}}\alpha_{\mathrm{s}}^{2})$
$\displaystyle\sim\frac{{\mathcal{O}}(\alpha\alpha_{\mathrm{s}})-{\mathcal{O}}(\alpha_{\mathrm{t}}\alpha_{\mathrm{s}})}{{\mathcal{O}}(\alpha)}{\mathcal{O}}(\alpha\alpha_{\mathrm{s}}),$
$\displaystyle{\mathcal{O}}(\alpha\alpha_{\mathrm{s}}^{3})-{\mathcal{O}}(\alpha_{\mathrm{t}}\alpha_{\mathrm{s}}^{3})$
$\displaystyle\sim\frac{{\mathcal{O}}(\alpha\alpha_{\mathrm{s}})-{\mathcal{O}}(\alpha_{\mathrm{t}}\alpha_{\mathrm{s}})}{{\mathcal{O}}(\alpha)}{\mathcal{O}}(\alpha\alpha_{\mathrm{s}}^{2}),$

where $\and=y_{\mathrm{t}}^{2}/(4\pi)$. Since we are only interested in the
missing higher orders beyond the leading large-$y_{\mathrm{t}}$ limit, the
same leading large-$y_{\mathrm{t}}$ approximations have been subtracted in the
numerators on the right-hand sides.

The contribution of these estimates to the overall theory error evaluation is
shown in Tab. 8 for various pseudo-observables, and in Tab. 9 for the
$Z$-boson form factors. Note that the error estimate for $\sin^{2}\theta_{\rm
eff}^{\ell}$ is slightly improved compared to Refs. Awramik:2006ar ;
Awramik:2006uz  due to the inclusion of
${\mathcal{O}}(\and\alpha_{\mathrm{s}}^{3})$ corrections from Refs.
Schroder:2005db ; Chetyrkin:2006bj ; Boughezal:2006xk .

Nevertheless, we would also like to remind the reader that any estimate of the
theory error from missing higher orders is not a precise prediction. Therefore
it is generally desirable to ensure that the theory error is sub-dominant in
any phenomenological analysis. Comparing the numbers in Tab. 8 to current
measurement results Schael:2013ita ; Patrignani:2016xqp , this is clearly seen
to be the case.

Observable | $\alpha\alpha_{\mathrm{s}}^{2}$ | $\alpha\alpha_{\mathrm{s}}^{3}$ | $\alpha^{2}\alpha_{\mathrm{s}}$ | $\alpha^{3}$ | Total
---|---|---|---|---|---
$\Gamma_{e,\mu,\tau}$ [MeV] | 0.008 | 0.001 | 0.010 | 0.013 | 0.018
$\Gamma_{\nu}$ [MeV] | 0.008 | 0.001 | 0.008 | 0.011 | 0.016
$\Gamma_{u,c}$ [MeV] | 0.025 | 0.004 | 0.08 | 0.07 | 0.11
$\Gamma_{d,s}$ [MeV] | 0.016 | 0.003 | 0.06 | 0.05 | 0.08
$\Gamma_{b}$ [MeV] | 0.11 | 0.02 | 0.13 | 0.06 | 0.18
$\Gamma_{\mathrm{Z}}$ [MeV] | 0.23 | 0.035 | 0.21 | 0.20 | 0.4
$R_{\ell}$ [$10^{-3}$] | 2.5 | 0.4 | 3.6 | 3.9 | 6
$R_{c}$ [$10^{-5}$] | 1.6 | 0.3 | 3.4 | 3.0 | 5
$R_{b}$ [$10^{-5}$] | 5.5 | 0.9 | 6.4 | 3.7 | 10
$\sigma^{0}_{\rm had}$ [pb] | 0.2 | 0.03 | 4.2 | 3.7 | 6
$\sin^{2}\theta_{\rm eff}^{\ell}$ [$10^{-5}$] | — | 0.3 | 3.0 | 3.1 | 4.3
$\sin^{2}\theta_{\rm eff}^{b}$ [$10^{-5}$] | 0.7 | 0.4 | 4.3 | 3.2 | 5.3
Table 8: Leading unknown higher-order corrections and their estimated order of magnitude for various pseudo-observables. The different orders always correspond to missing higher orders beyond the known approximations in the limit of a large top Yukawa coupling. The total theory error is obtained by adding the individual orders in quadrature.  Observable | $\alpha\alpha_{\mathrm{s}}^{2}$ | $\alpha\alpha_{\mathrm{s}}^{3}$ | $\alpha^{2}\alpha_{\mathrm{s}}$ | $\alpha^{3}$ | Total
---|---|---|---|---|---
$F_{V}^{\ell}$ [$10^{-5}$] | 0.03 | 0.004 | 0.2 | 0.4 | 0.5
$F_{A}^{\ell}$ [$10^{-5}$] | 0.17 | 0.03 | 0.3 | 0.3 | 0.4
$F_{V,A}^{\nu}$ [$10^{-5}$] | 0.16 | 0.02 | 0.3 | 0.5 | 0.6
$F_{V}^{u,c}$ [$10^{-5}$] | 0.09 | 0.01 | 0.4 | 0.2 | 0.5
$F_{A}^{u,c}$ [$10^{-5}$] | 0.17 | 0.03 | 0.3 | 0.4 | 0.5
$F_{V}^{d,s}$ [$10^{-5}$] | 0.2 | 0.03 | 0.6 | 0.8 | 1.1
$F_{A}^{d,s}$ [$10^{-3}$] | 0.3 | 0.04 | 0.4 | 0.5 | 0.7
$F_{V}^{b}$ [$10^{-5}$] | 0.2 | 0.03 | 0.8 | 0.7 | 1.1
$F_{A}^{b}$ [$10^{-5}$] | 0.3 | 0.04 | 0.1 | 0.1 | 0.3
Table 9: Same as Tab. 8, but for various form factors.

##  6 Conclusions and outlook

In this study, we have presented some phenomenologically useful applications
of the recently completed electroweak two-loop calculation of $Z$-boson vertex
corrections Dubovyk:2016aqv ; Dubovyk:2018rlg . The work collects multi-year
efforts of several groups for predictions of the EWPOs related to the $Z$ peak
up to electroweak full two-loop accuracy, supplemented by leading QCD higher-
order terms. We have determined the two-loop electroweak contributions with a
net relative numerical accuracy of about four digits. This ensures that these
two-loop results will be known with sufficient accuracy even when adding the
next perturbative order, as it might be needed for applications at the next
generation of $e^{+}e^{-}$ colliders.

For practical applications, the results for the EWPOs, as well as for the
$Z$-boson vertex form factors, have been presented in terms of simple
parameterization formulae, whose coefficients have been fitted to the full
numerical computation. It is planned to include these fitting formulae into a
new version of the weak library DIZET of ZFITTER AkhundovDIZET:2019?? ;
Bardin:1989di ; Bardin:1989tq ; Bardin:1999yd ; Arbuzov:2005ma ;
Akhundov:2013ons .333Private communication by L. Kalinovskaya for the
ZFITTER/DIZET support team. The accuracy of the fitting formulae is less than
our full numerical two-loop calculation, but more than sufficient for present-
day purposes. For the future FCC-ee Tera-Z project, it may be necessary to
provide more precise formulae by including more terms with higher powers of
the input parameters.

Finally, we would like to make a few comments on the prospects for the
calculation of electroweak three-loop corrections, which will be necessary for
the level of precision foreseen for FCC-ee and similar $e^{+}e^{-}$ collider
proposals. This electroweak third order, by itself, will be needed with only
about two digits accuracy Dubovyk:inYR2018B1 . The generation of the
amplitudes for ${\cal O}(10^{4}{-}10^{5})$ diagrams as well as the evaluation
of the Lorentz and Dirac algebra are routine tasks performed by computer
algebra programs, and they should be straightforward with increased computing
power in the future. Potential specific problems related to the treatment of
$\gamma_{5}$ at three-loop level have to be controlled Marquard:inYR2018D1 .

The most challenging problem will certainly be the stable numerical
computation of three-loop Feynman integrals with several different internal
mass scales. At two loops, we did not perform any reduction of the Feynman
integrals to a smaller number of masters and thus had to calculate about 1000
previously unknown numerical integrals. In the next perturbative order, it may
be advantageous to perform such a reduction to masters, given the ever
increasing performance of programs like KIRA Maierhoefer:2017hyi ;
Maierhofer:2019goc , Reduze 2 vonManteuffel:2012np , FIRE Smirnov:2014hma ;
Smirnov:2014hma , and LiteRED Lee:2013mka .

For the calculation of the (master) integrals themselves, it is desirable to
have a procedure to automatically isolate and treat the ultra-violet and
infra-red singularities. Although there is rapid progress in several
analytical approaches to complicated loop integrals Blondel:2018mad ;
Blondel:2019vdq , one has to expect that the more complicated ones will have
to be done numerically. An additional complication is that for integrals with
physical cuts, their stable numerical evaluation becomes more challenging.
There are two kinds of software packages available that address these
problems, based on either sector decomposition (SD) as realized in the SecDec
project Carter:2010hi ; Borowka:2012yc ; Borowka:2015mxa ; Borowka:2017idc ;
Borowka:2018goh ; Jahn:2018zsh  and the FIESTA project Smirnov:2008py ;
Smirnov:2009pb ; Smirnov:2013eza ; Smirnov:2015mct , or based on Mellin-Barnes
(MB) transformations, as implemented in the AMBRE project Gluza:2007rt ;
Gluza:2009mj ; Gluza:2010rn ; Gluza:2010mz ; Blumlein:2014maa ;
Dubovyk:2015yba ; Dubovyk:2016ocz ; Dubovyk:2017cqw ; Dubovyk:inYR2018E4 ;
Dubovyk:PhD2019?? ; Usovitsch:2018shx ; Usovitsch:April2018 ;
Usovitsch:inYR2018 ; Usovitsch:PhD2018 . Sector decomposition is typically
advantageous for integrals with many different mass scales, while the MB
approach is more efficient for integrals with fewer independent parameters.
Both methods certainly have room for crucial improvements. Several other
numerical integration methods, as reviewed e.g. in Refs. Blondel:2018mad ;
Blondel:2019vdq , are useful for certain classes of multi-loop integrals, even
though they are less general than the SD and MB approaches. Overall, numerical
loop integration techniques are well positioned to meet the necessary future
precision demands.

An electroweak three-loop result for the $Z$-peak EWPOs must be accompanied by
improved calculations of the corrections needed to translate EWPOs to real
observables. These include initial-state and final-state QED corrections and
their interference as well as higher-order terms of the Laurent series
expansion about the $Z$ resonance pole Blondel:2018mad . The latter will, for
example, involve massive two-loop box diagrams. A complete accounting of the
required correction terms is still lacking.

To summarize, we have completed the electroweak two-loop predictions for the
EWPOs of the $Z$ resonance and collect here an extensive set of new fitting
formulae for them. On a longer time scale, the calculation of the next
perturbative order for the calculation of the EWPOs will be necessary and,
with proper investments, realistically accessible.

###### Acknowledgements.

The authors are grateful to V. Yermolchyk for checks and useful comments. The
work of I.D. was supported by a research grant of Deutscher Akademischer
Austauschdienst (DAAD) and by Deutsches Elektronensychrotron DESY and CERN.
The work of J.G. was supported by the Polish National Science Centre (NCN)
under the Grant Agreement 2017/25/B/ST2/01987 and by the program International
mobilities for research activities of the University of Hradec Králové,
CZ.02.2.69/0.0/0.0/16_027/0008487. The work of A.F. was supported in part by
the National Science Foundation under grant no. PHY-1820760. The work of T.R.
is funded by Deutsche Rentenversicherung Bund. He is supported in part by a
2015 Alexander von Humboldt Honorary Research Scholarship of the Foundation
for Polish Sciences (FNP) and by the Polish National Science Centre (NCN)
under the Grant Agreement 2017/25/B/ST2/01987. J.U. received funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme under grant agreement no. 647356 (CutLoops).
The project was partly supported by the COST (European Cooperation in Science
and Technology) Action CA16201 PARTICLEFACE. We finally acknowledge kind
support for a working meeting by Prof. Peter Uwer at Humboldt-Universität zu
Berlin.

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