11institutetext: IFT (UAM/CSIC), Universidad Autónoma de Madrid Cantoblanco,
8048, Spain 22institutetext: Campus of International Excellence UAM+CSIC,
Cantoblanco, 28049, Madrid, Spain 33institutetext: Instituto de Física de
Cantabria (CSIC-UC), 39005 Santander, Spain 44institutetext: Institute of
Nuclear Physics Polish Academy of Sciences, Krakow, Poland 55institutetext:
DESY, Theory Group, Notkestr. 85, 22607 Hamburg, Germany

# Theory requirements for SM Higgs and EW precision physics at the FCC-ee

S. Heinemeyer 112233 S. Jadach 44 J. Reuter 55

(Received: date / Revised version: date)

###### Abstract

High precision experimental measurements of the properties of the Higgs boson
at $\sim 125$ GeV as well as electroweak precision observables such as the
$W$-boson mass or the effective weak leptonic mixing angle are expected at
future $e^{+}e^{-}$ colliders such as the FCC-ee. This high anticipated
precision has to be matched with theory predictions for the measured
quantities at the same level of accuracy. We briefly summarize the status of
these predictions within the Standard Model (SM) and of the tools that are
used for their determination. We outline how the theory predictions will have
to be improved in order to reach the required accuracy, and also comment on
the simulation frameworks for the Higgs and EW precision program.

###### pacs:

PACS-keydiscribing text of that key and PACS-keydiscribing text of that key

##  1 Introduction

With the discovery of the Higgs boson, all possible elements of the Standard
Model (SM) have been experimentally confirmed and tested in great depth. On
the other hand, observational evidence for dark matter and the matter-
antimatter asymmetry require physics beyond the SM. One promising way to probe
such new physics is through precision measurements of the properties of the
Higgs boson. A complementary way is to measure electroweak precision
(pseudo-)observables (EWPO) with highest precision. These are the avenues
pursued by several proposals for a future $e^{+}e^{-}$ collider. In
particular, the FCC-ee concept is designed to run at
$\sqrt{s}=250\,\,\mathrm{GeV}$ as a Higgs factory, and at $\sqrt{s}\sim
M_{Z},2M_{W}$ for high-precision EWPO measurements. In this way it can improve
(indirect) probes for new physics by several orders of magnitude compared to
existing bounds Gomez-Ceballos:2013zzn ; Abada:2019zxq ; Abada:2019lih .

The anticipated experimental accuracy of an observable has to be matched with
a theory prediction of at least the same level of accuracy to make maximum use
of the experimental data. For the accurate study of the properties of the
Higgs boson, precise predictions for the various partial decay widths, the
branching ratios (BRs) and the Higgs-boson production cross sections along
with their theoretical uncertainties are indispensable. For the EWPO on the
one hand the SM prediction has to reach the level of the experimental
accuracy. Similarly, the extraction of these quantities from experimental data
must be equally well under control. All theses types of uncertainties (various
theoretical ones and experimental) must be taken into account when deriving
constraints on new physics from the data.

Several sources of theory uncertainties have to be distinguished. The
intrinsic uncertainties are due to missing higher orders in the perturbative
expansion of the SM (or BSM) prediction for an observable. The parametric
uncertainties are due to the imperfect experimental knowledge of the SM input
parameters (as well as theory uncertainties induced in their extraction from
data). The extraction of a quantity from a cross section or an asymmetry
requires the theory prediction of this cross section or asymmetry to at least
the same order of precision111Theoretical predictions for the past electron
collider experiments (LEP/SLC) were at least a factor 3 more precise than
experimental data..

In this paper the current status and future implications of theory
uncertainties on (SM) Higgs-boson physics and EWPO’s will be summarized. While
we focus on the FCC-ee, they are valid for all future high precision
$e^{+}e^{-}$ colliders running above the $HZ$ threshold (such as ILC, CLIC, or
CEPC). We will use anticipated FCC-ee precisions to illustrate the impact of
theory uncertainties. We build on Freitas:2019bre , where also a discussion of
the determination of the SM input parameters can be found. In addition, we
review Monte Carlo (MC) event generator frameworks which are indispensable for
simulations in the perturbative and non-perturbative regime, building upon the
experience gained from the LEP era Kobel:2000aw ; Jadach:1996gu ;
Beenakker:1996kt  and the extensive simulations for CEPC, ILC and CLIC. Work
on dedicated FCC-ee MC samples has started.

##  2 Pseudo-observables versus realistic observables

The quantities that can be directly measured in experiments are cross
sections, line shape observables, forward–backward asymmetries etc., called
“realistic observables” Bardin:1999gt ; ALEPH:2005ab . The obtained results
depend on the specific set of experimental cuts, however detector
inefficiencies are removed and kinematic cuts are brought to simplified form
using MC event generators. In order to determine idealized quantities like
masses, partial widths or couplings, or the effective electroweak mixing
angle, from the primarily measured realistic observables, a “QED unfolding”
procedure is applied. This procedure involves manipulations like subtracting
photon-exchange or interference terms, subtracting box-diagram contributions,
unfolding higher-order QED or QCD corrections, etc. These secondary quantities
are therefore called “pseudo-observables”. The Higgs-boson observables and
EWPOs on which we focus here are pseudo-observables. It should be stressed
that the role of EWPOs is to represent and encapsulate experimental data in
the model-independent way222For instance $\sin^{2}\theta_{\rm eff}^{f}$ is not
a SM parameter – it parametrizes data on the ratio of the vector to axial
couplings in $e^{+}e^{-}\to f\bar{f}$.. Fitting EWPOs to the SM and its
extensions comes as an independent next step. In order to reach the required
theoretical accuracy special care has to be taken in the application of these
unfolding procedures – in particular the procedure of the extraction of EWPOs
from data causes certain bias, partial loss of information on physics in the
realistic data. The unfolding procedure for the LEP experiments is described
in Refs. Bardin:1999gt ; ALEPH:2005ab . A discussion of the additional
problems in this respect arising in beyond the SM (BSM) models can be found in
Heinemeyer:2004gx . In Refs. Bardin:1999gt ; ALEPH:2005ab  it was proven that
the bias induced in the extraction of EWPOs (QED deconvolution) was smaller
than the LEP experimental errors. This will not be true in the FCC-ee era
where experimental errors will be smaller by up to two orders. The very
survival of the technique of the model independent EWPOs as the bridge between
the experiment and the theory at FCC-ee will require significant improvement
of the precision of the trivial but large QED effects. Ref.Jadach:2019bye
covers this issue in a great detail, listing challenges and outlining
solutions. In addition, the definition of EWPOs and the algorithm of their
extraction will have to be modified. This is elaborated in Sect. 3 in Ref.
Blondel:2018mad . Two major changes are anticipated. One is that we may be
forced to remove from EWPOs not only QED but also first order pure electroweak
corrections, in order to achieve sharper resolution in the search of the BSM
effects in the new EWPOs. Another important change will be the increased role
of Monte Carlo generators in extracting EWPOs from data – this is because
semianalytic programs like ZFITTER and TOPAZ0 of LEP era will not be able to
handle precisely enough detector geometry (cut-offs).

##  3 Electroweak precision observables

The most important electroweak precision observables (EWPOs) are related to
properties of the $Z$ and $W$ bosons. The $Z$-boson properties are determined
from measurements of $e^{+}e^{-}\to f\bar{f}$ on the $Z$-pole. To isolate the
physics of the $Z$-boson, the typical set of pseudo-observables is defined in
terms of the de-convoluted cross-section $\sigma_{f}(s)$, where the effect of
initial- and final-state photon radiation and from $s$-channel photon and
double-boson (box) exchange has been removed. The customary set of pseudo-
observables are total and partial $Z$-boson cross sections with their
subsequent decay to quarks or leptons, forward-backward and left-right
asymmetries, from which the effective weak leptonic mixing angle,
$\sin^{2}\theta_{\rm eff}^{f}$, is extracted, as well as the $W$-boson mass,
$M_{W}$, see Ref. Schael:2013ita . The expected (pure) experimental precision
at FCC-ee of $M_{W}$ and $\sin^{2}\theta_{\rm eff}^{\ell}$ is summarized in
Tab. 1 based on Refs. Abada:2019zxq ; Abada:2019lih .

Quantity | FCC-ee | Current intrinsic unc. | Projected intrinsic unc.
---|---|---|---
$M_{W}$ [MeV] | 0.5 | 4 | ($\alpha^{3},\alpha^{2}\alpha_{s}$) | 1
$\sin^{2}\theta_{\rm eff}^{\ell}$ [$10^{-5}$] | 0.6 | 4.5 | ($\alpha^{3},\alpha^{2}\alpha_{s})$ | 1.5
Table 1: Estimated precision for the direct determination of $M_{W}$ and
$\sin^{2}\theta_{\rm eff}^{\ell}$ at FCC-ee Abada:2019zxq ; Abada:2019lih
(column two), current intrinsic uncertainty (column three) and projected
future uncertainty when leading 3-loop corrections become available (column
4).

The quantities listed in Tab. 1 can be predicted within the SM in terms of the
input parameters $G_{F}$, $\alpha(M_{Z})$, $\alpha_{s}(M_{Z})$, $M_{Z}$,
$M_{H}$ and $m_{t}$. The radiative corrections in these predictions are
currently known including complete two-loop corrections In addition,
approximate three- and four-loop corrections of ${\cal O}(\and^{3})$, ${\cal
O}(\and^{2}\alpha_{s})$, ${\cal O}(\and\alpha_{s}^{2})$ and ${\cal
O}(\and\alpha_{s}^{3})$ are available (with $\and=y_{t}^{2}/(4\pi)$ and
$y_{t}$ being the top Yukawa coupling), see Freitas:2016sty  for a review. The
theory uncertainties from missing higher-order corrections are given in the
third column of Tab. 1 Awramik:2003rn ; Awramik:2004ge ; Freitas:2014hra ;
Dubovyk:2018rlg . Also indicated are the main sources for the respective
uncertainties. In order to match the FCC-ee precision the theory predictions
will have to be improved substantially. It was estimated in Freitas:2014owa ;
Freitas:2019bre ; Blondel:2018mad  that the intrinsic uncertainties will be
reduced by complete ${\cal O}(\alpha\alpha_{s}^{2})$ corrections, fermionic
${\cal O}(\alpha^{2}\alpha_{s})$ corrections, double-fermionic ${\cal
O}(\alpha^{3})$ corrections, and leading four-loop corrections enhanced by the
top Yukawa coupling, resulting in the projected intrinsic uncertainties shown
in column four of Tab. 1.

A crucial point here is that the determination of the pseudo-observables in
Tab. 1 from experimental data also requires theory input for the removal of
initial-state and final-state photon radiation and $s$-channel photon exchange
and box contributions. The theory uncertainty from missing higher QED orders
is estimated to amount to a few times 0.01% Bardin:1999gt ; Jadach:2000ir ;
Kobel:2000aw ; ALEPH:2005ab  for the $Z$-peak cross-section and total width
measurements. In order not to be dominated by this uncertainty, it will need
to be reduced by about a factor 10 for the FCC-ee. This will require the
calculation of non-leading log contributions to ${\cal O}(\alpha^{3})$
corrections, ${\cal O}(\alpha^{3}L^{2})$ and ${\cal O}(\alpha^{4}L^{4})$
contributions (with $L\equiv\log s/m_{e}^{2}$) as well as an improved
treatment of fermion pair production from off-shell photons.

The $W$-boson mass will be determined from a threshold scan near the $W$-pair
threshold, $\sqrt{s}\approx 161\,\,\mathrm{GeV}$. It is foreseen that the
experimental uncertainty at FCC-ee for this measurement is about
$0.5\,\,\mathrm{MeV}$ Gomez-Ceballos:2013zzn ; Abada:2019zxq ; Abada:2019lih .
At the point of highest sensitivity, an uncertainty of the cross-section
measurement of $0.1\%$ translates to an uncertainty of $\sim
1.5\,\,\mathrm{MeV}$ on $M_{W}$ Azzi:2017iih . Therefore a theoretical
prediction for the process $e^{+}e^{-}\to 4f$ with an accuracy of
$\Delta\sigma\sim 0.01\%$ is desirable, including effects of off-shell $W$
bosons, which become important near threshold.

The currently best calculations are based on complete one-loop results for
$e^{+}e^{-}\to 4f$ Denner:2005fg  and partial higher-order effects for the
total cross section from an effective field theory framework Beneke:2007zg ;
Actis:2008rb . The resulting theory uncertainty on $M_{W}$ is estimated to be
about 3 MeV Actis:2008rb . This result must be improved by complete 2-loop
corrections to $e^{+}e^{-}\to W^{+}W^{-}$ and to $W\to f\bar{f}^{\prime}$
(based on the effective field theory framework). In addition, a more accurate
description of initial-state radiation will be important, which includes
universal contributions from soft and collinear photon radiation (see
Bardin:1999gt ; Jadach:2000ir ; Kobel:2000aw  for a review), as well as hard
photon radiation. For the latter, a proper matching and merging procedure
needs to be employed to avoid double counting Denner:2000bj ; Beneke:2007zg ,
see Freitas:2019bre  for more details. There it was estimated that a theory
induced systematic uncertainty of $\Delta
M_{W}\;\raisebox{-3.00003pt}{$\stackrel{{\scriptstyle\displaystyle<}}{{\sim}}$}\;0.60\,\,\mathrm{MeV}$
can be feasible.

##  4 The SM Higgs boson

###  4.1 Higgs-boson production

The very narrow width of the Higgs boson allows for a factorization of all
cross-sections with resonant Higgs bosons into production and decay parts to
very high precision if the Higgs boson can be fully reconstructed. In this
case, finite-width effects and off-shell contributions are of relative size
$\Gamma_{H}/M_{H}\sim 0.00003$ and thus not relevant. If the Higgs boson is
not fully reconstructable (e.g. in $H\to W^{(*)}W^{(*)}\to 2\ell 2\nu$) Higgs
off-shell contributions have to be taken into account (which is
straightforward at NLO).

At the FCC-ee with $\sqrt{s}=240$ GeV (or other $e^{+}e^{-}$ machines near
this center-of-mass energy), the Higgs-boson production cross-section is
strongly dominated by $e^{+}e^{-}\to ZH$, and $e^{+}e^{-}\to\nu\bar{\nu}H$
contributes less than 20% Gomez-Ceballos:2013zzn ; Moortgat-Picka:2015yla .
For these two processes full one-loop corrections in the SM are available
Belanger:2002ik ; Denner:2003yg . For the dominating $ZH$ production mode they
are found at the level of $\sim 5{-}10\%$. Leading two-loop corrections to
this cross section were evaluated at ${\cal O}(\alpha\alpha_{s})$ Gong:2016jys
; Sun:2016bel , which were found to amount to $\sim 1-2\%$. This number has to
be compared to the anticipated experimental accuracy of 0.4% Gomez-
Ceballos:2013zzn ; Abada:2019zxq ; Abada:2019lih . It becomes clear that with
a full two-loop calculation of $e^{+}e^{-}\to ZH$ the intrinsic uncertainty
will be sufficiently small. Calculational techniques for $2\to 2$ processes at
the two-loop level exist, and it is reasonable to assume that, if required,
this calculation within the SM can be incorporated for the FCC-ee Higgs
precision studies (see also Song:2021vru  for recent progress in this
direction.)

For WBF production, the calculation of the full two-loop corrections will be
significantly more difficult, since this is a $2\to 3$ process. However, one
may assume that a partial result based on diagrams with closed light-fermion
loops and top-quark loops (in a large-$m_{t}$ approximation) can be achieved,
which should reduce the intrinsic theory uncertainty to below the 1% level.
Given the fact that the WBF process is less crucial than the $HZ$ channel for
the Higgs physics program FCC-ee with $\sqrt{s}=240$ GeV, this will probably
be adequate for most practical purposes. It would be desirable to have a
complete $2\to 4$ calculation at two loops, including both processes with
their interference, but very likely factorized processes with on-shell
projections will be sufficient.

###  4.2 Higgs-boson decay

The current intrinsic and parametric uncertainties for the various Higgs-boson
decay widths are given in Tab. 2 (see Freitas:2019bre ). The status of the
intrinsic uncertainties was evaluated following Lepage:2014fla ;
deFlorian:2016spz ; lhchxswg-br ; Denner:2011mq .

Partial width | intr. QCD | intr. electroweak | total | para. $m_{q}$ | para. $\alpha_{s}$ | para. $M_{H}$
---|---|---|---|---|---|---
$H\to b\bar{b}$ | $\sim 0.2\%$ | ${<0.3}\%$ | ${<0.4}\%$ | ${1.4\%}$ | ${0.4\%}$ | –
$H\to c\bar{c}$ | $\sim 0.2\%$ | ${<0.3}\%$ | ${<0.4}\%$ | ${4.0\%}$ | ${0.4\%}$ | –
$H\to\tau^{+}\tau^{-}$ | – | ${<0.3}\%$ | ${<0.3}\%$ | – | – | –
$H\to\mu^{+}\mu^{-}$ | – | ${<0.3}\%$ | ${<0.3}\%$ | – | – | –
$H\to gg$ | $\sim 3\%$ | $\sim 1\%$ | $\sim 3.2\%$ | ${<0.2\%}$ | $3.7\%$ | –
$H\to\gamma\gamma$ | $<{0.1}\%$ | $<1\%$ | ${<}1\%$ | ${<0.2\%}$ | – | –
$H\to Z\gamma$ | ${\;\raisebox{-3.00003pt}{$\stackrel{{\scriptstyle\displaystyle<}}{{\sim}}$}\;0.1}\%$ | $\sim 5\%$ | $\sim 5\%$ | – | – | ${2.1\%}$
$H\to WW\to 4$f | $<0.5\%$ | ${<0.3\%}$ | $\sim 0.5\%$ | – | – | ${2.6\%}$
$H\to ZZ\to 4$f | $<0.5\%$ | ${<0.3\%}$ | $\sim 0.5\%$ | – | – | ${3.0\%}$
Table 2:  Current intrinsinc and parametric uncertainties in the various
Higgs-boson decay width calculations, see text and Refs. Lepage:2014fla ;
deFlorian:2016spz ; lhchxswg-br ; Denner:2011mq . “–” indicates a negligible
source of uncertainty.

Also the parametric uncertainties can play a non-negligible role for the
evaluation of the partial widths. The most important parameters are the bottom
quark mass and the strong coupling constant. In Ref. deFlorian:2016spz  the
current uncertainties of $\alpha_{s}$ and $m_{b}$ have been assumed to be
$\delta\alpha_{s}=0.0015$ and $\delta m_{b}=0.03\,\,\mathrm{GeV}$.
Additionally, $\delta m_{c}=0.025\,\,\mathrm{GeV}$, $\delta
m_{t}=0.85\,\,\mathrm{GeV}$ and $\delta M_{H}=0.24\,\,\mathrm{GeV}$
Zyla:2020zbs  have been taken into account. The effect on the various partial
widths has been evaluated as in Ref. Lepage:2014fla  and is shown in the three
right columns of Tab. 2.

When comparing the combined intrinsic and parametric uncertainties with the
target precision of FCC-ee Gomez-Ceballos:2013zzn ; Abada:2019zxq ;
Abada:2019lih , see the last column in Tab. 3, it is clear that improvements
are necessary. Concerning the intrinsic theory uncertainty, the available
predictions for the $f\bar{f}$ and $\gamma\gamma$ channels are already
sufficiently precise to match the expected FCC-ee experimental uncertainty.
With available calculational techniques, the evaluation of complete two-loop
corrections to $H\to f\bar{f}$ can be achieved. This would reduce the
uncertainty of the electroweak contributions to less than 0.1%. Similarly, the
complete NLO corrections to $H\to Z\gamma$ can be carried out with existing
methods, resulting in an estimated precision of about 1%. More theoretical
work is needed for $H\to WW,ZZ,gg$, which are currently limited by QCD
uncertainties. For $H\to gg$ the calculation of massless four-loop QCD
diagrams, will be required, which may be within reach Baikov:2005rw , reducing
the intrinsic uncertainties to the level of about 1%. For $H\to WW,ZZ$, the
required QCD corrections are essentially identical to those for $e^{+}e^{-}\to
WW$, and it is straightforward to improve them to a practically negligible
level. Further significant progress would require the calculation of two-loop
electroweak corrections, which for a $1\to 4$ process is beyond reach for the
forseeable future. Here it should be noted, however, that the $HZZ$ coupling
will be mostly constrained by the measurement of the $e^{+}e^{-}\to HZ$
production process at FCC-ee with $\sqrt{s}=240$ GeV, rather than the decay
$H\to ZZ^{*}$, see the discussion in section 4.1, leading to a remaining
intrinsic uncertainty of less than 0.3%.

decay | projected intr. | para. $m_{q}$ | para. $\alpha_{s}$ | para. $M_{H}$ | FCC-ee prec. on ${g^{2}_{HXX}}$
---|---|---|---|---|---
$H\to b\bar{b}$ | $\sim 0.2\%$ | $0.6\%$ | $<0.1\%$ | – | $\sim 0.8\%$
$H\to c\bar{c}$ | $\sim 0.2\%$ | $\sim 1\%$ | $<0.1\%$ | – | $\sim 1.4\%$
$H\to\tau^{+}\tau^{-}$ | $<0.1\%$ | – | – | – | $\sim 1.1\%$
$H\to\mu^{+}\mu^{-}$ | $<0.1\%$ | – | – | – | $\sim 12\%$
$H\to gg$ | $\sim 1\%$ |  |  $0.5\%$ (0.3%) | – | $\sim 1.6\%$
$H\to\gamma\gamma$ | $<1\%$ | – | – | – | $\sim 3.0\%$
$H\to Z\gamma$ | $\sim 1\%$ | – | – | $\sim 0.1\%$ |
$H\to WW$ | $\;\raisebox{-3.00003pt}{$\stackrel{{\scriptstyle\displaystyle<}}{{\sim}}$}\;0.3\%$ | – | – | $\sim 0.1\%$ | $\sim 0.4\%$
$H\to ZZ$ | $\;\raisebox{-3.00003pt}{$\stackrel{{\scriptstyle\displaystyle<}}{{\sim}}$}\;0.3\%^{\dagger}$ | – | – | $\sim 0.1\%$ | $\sim 0.3\%$
$\Gamma_{\rm tot}$ | $\sim 0.3\%$ | $\sim 0.4\%$ | $<0.1\%$ | $<0.1\%$ | $\sim 1\%$
† From $e^{+}e^{-}\to HZ$ production |  |
Table 3:  Projected intrinsic and parametric uncertainties for the partial and
total Higgs-boson decay width predictions (see text). The last column shows
the target of FCC-ee precisions on the respective coupling squared.

Also shown in Tab. 3 are the projected parametric uncertainties, assuming FCC-
ee precisions. For inputs, we use $\delta\alpha_{s}=0.0002$ and $\delta
m_{t}=50\,\,\mathrm{MeV}$ 333This assumes a precision from an $e^{+}e^{-}$ top
threshold scan. At the highest FCC-ee energy, 365 GeV, the top Yukawa coupling
can also be inferred from the top threshold dependence on electroweak loop
corrections., $\delta M_{H}\sim 10\,\,\mathrm{MeV}$, and $\delta m_{b}\sim
13\,\,\mathrm{MeV}$ and $\delta m_{\rm c}\sim 7\,\,\mathrm{MeV}$, see
Freitas:2019bre  for details. 444 Note that the numbers in Tab. 3 do not take
into account correlations between the uncertainties in the Higgs production
and decay processes or between different decay processes, in particular
entering via $\Gamma_{\rm tot}$. Their impact can only be evaluated when the
full experimental correlation matrix is known, see the discussion in
Freitas:2019bre . It becomes clear that the intrinsic and parametric
uncertainties will be able to match the high anticipated experimental accuracy
at FCC-ee.

##  5 Monte-Carlo and exclusive predictions

To determine systematic uncertainties, to properly extract parameters from the
Higgs measurements and to find deviations from SM predictions, it is
indispensable to have Monte Carlo (MC) simulations. The challenge compared to
LEP/SLC times is the strongly increased experimental precision and much higher
luminosities which demands much better simulations. The precise description of
QED effects, both regarding the correct normalization of cross sections as
well as exclusive multi-photon radiation will be one of the highest
priorities. In the recent years, there has been steady progress in dealing
with the calculation of higher logarithmic and finite orders in the QED
radiation, enhanced by the very small electron mass, see Refs.
Ablinger:2020qvo ; Blumlein:2020jrf ; Blumlein:2021jdl ; Frixione:2019lga ;
Bertone:2019hks . These very advanced calculations are only partially
applicable for the exclusive description of photon radiation which is
necessary if cuts are to be applied on those photons. At the level of LO
matrix elements, there is a matching procedure between explicit matrix element
photons and the electron PDFs in collinear factorization in the context of
WHIZARD (cf. below) which has been recently shown to work unexpectedly well
Kalinowski:2020lhp ; 1859727 .

From the LEP era there are many dedicated MC programs which feature soft
photon resummation in the exclusive form. Most of them YFS2/KORALZ
Jadach:1999tr , BHLUMI Jadach:1996is , BHWIDE Jadach:1995nk , YFSWW
Placzek:2003zg  are based directly on the Yennie-Frautschi-Suura (YSF)
resummation scheme Yennie:1961ad . A more powerful coherent exclusive
exponentiation (CEEX) at the amplitude level was introduced in Jadach:2000ir
and is so far implemented only in KKMC Jadach:1999vf  program for two-fermion
processes $e^{+}e^{-}\to f\bar{f}$, $f=\mu,\tau,\nu,q$. A generic semi-
automatic implementation of CEEX for a wider class of processes would be a
very desirable development, and might pave the path towards a consistent
inclusion of the non-soft higher order contributions over the complete photon
phase space, merging together genuine electroweak corrections with QED
correction calculated to much higher orders, inclusion of spin correlations,
initial-final state interferences and more. The ultimate goal is to match non-
soft QED corrections up to forth order and genuine SM electroweak corrections
up to second order.

Multi-purpose event generators allow to simulate the full spectrum of SM (and
BSM) processes at FCC-ee. They provide (for the moment) a lower level of
precision for the two-fermion processes, but allow to simulate them in the
same framework as four-, six- and event eight-fermion processes. Due to the
much higher precision and better detectors than at SLC/LEP these processes
have to be included already at the $250\,\,\mathrm{GeV}$ stage, and not only
if triple electroweak resonances or top pairs are kinematically accessible. In
this context a lot of experience has been gained from the large full SM event
samples generated for ILC, CLIC and CEPC using WHIZARD Moretti:2001zz ;
Kilian:2007gr . This supports leading-logarithmic QED collinear factorization
for QED photons, automated NLO QCD corrections for arbitrary processes, while
NLO electroweak corrections are under way. For experimental studies, crucial
observables are the correct numbers of neutral and charged hadrons. These can
only be correctly simulated if the color-flow assignments are correctly made,
which includes that the invariant masses of the shower systems from the most
prominent processes, $e^{+}e^{-}\to W^{+}W^{-},ZZ$ are preserved; in a full
electroweak matrix element they are, however, quantum mechanically entangled.
The MC hence needs to determine the correct probability of the resonant
subprocesses and hand them over to parton shower and hadronization
Sjostrand:2006za ; Sjostrand:2014zea . It is important to note that the
solution of the long-standing problem of the soft photon resummation to narrow
width semi-stable charged resonances like $W^{\pm}$ was recently outlined in
Ref. Jadach:2019wol . It is also very important to take photon-induced
backgrounds from Weizsäcker-Williams QED splittings into account, particularly
$\gamma\gamma\to$ hadrons. WHIZARD has a special treatment for the top
threshold run Bach:2017ggt  that is used by the FCC-ee collaboration. It also
supports all different kinds of beam spectra, from simple parameterized files,
over Gaussian beam spreads to a sophisticated beamstrahlung setup for linear
and circular lepton machines Ohl:1996fi .

To search for deviations in the Higgs sector from the SM, WHIZARD supports a
large setup of BSM models like supersymmetry, extended Higgs models like 2HDM,
Higgs singlet extensions, SMEFT, composite and Little Higgs models, which can
be generically extended to its interface to automated Lagrangian model tools,
cf. e.g. Christensen:2010wz ; Degrande:2011ua

##  6 Outlook and deliverables

The most important steps to be taken in the future to match the high
anticipated accuracy of FCC-ee in the context of Higgs physics and EWPO with
SM theory predictions are the following.

  * •

Improved unfolding techniques to go from observables to pseudo-observables.

  * •

Calculations for the EWPO: complete ${\cal O}(\alpha\alpha_{s}^{2})$
corrections, fermionic ${\cal O}(\alpha^{2}\alpha_{s})$ corrections, double-
fermionic ${\cal O}(\alpha^{3})$ corrections, and leading four-loop
corrections enhanced by the top Yukawa coupling.

  * •

To extract the $W$-boson mass: complete 2-loop corrections to $e^{+}e^{-}\to
W^{+}W^{-}$ and to $W\to f\bar{f}^{\prime}$ (based on the effective field
theory framework).

  * •

Development on matching scheme between EW corrections and radiated/resummend
photons.

  * •

Full two-loop corrections for $e^{+}e^{-}\to ZH$ and $e^{+}e^{-}\to VVH$

  * •

Full two-loop corrections for $H\to ff(+\gamma)$.

  * •

(Mostly) automatized MC generation including NLO QCD and EW corrections.

  * •

Improvements by roughly one order of magnitude in the determination of the
most important SM parameters ($m_{t}$, $m_{b}$, $\alpha_{s}$,
$\Delta\alpha_{\rm had}$, $M_{H}$, …)

  * •

Complete simulations of relevant SM processes at different energy stages in
full detector simulation

Finally, the following should be kept in mind. The SM constitutes the model in
which highest theoretical precision for the predictions of EWPO and Higgs-
boson observables can be obtained. This concerns their prediction as well as
their extraction. However, as soon as BSM physics will be discovered, an
evaluation of the predictions of the EWPO and the Higgs-boson observables in
any preferred BSM model will be necessary. The corresponding theory
uncertainties, both intrinsic and parametric, can then be larger (see, e.g.,
Freitas:2013xga ; Heinemeyer:2004gx  for the Minimal Supersymmetric SM). A
dedicated theory effort (beyond the SM) would be needed in this case.

### Acknowledgements

The work of S.H. is supported in part by the MEINCOP Spain under contract
PID2019-110058GB-C21 and in part by the AEI through the grant IFT Centro de
Excelencia Severo Ochoa SEV-2016-0597. S.J. acknowledges funding from the
European Union’s Horizon 2020 research and innovation programme under under
grant agreement No 951754 and support of the National Science Centre, Poland,
Grant No. 2019/34/E/ST2/00457. JRR acknowledges the support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Association) under Germany’s
Excellence Strategy-EXC 2121 ”Quantum Universe”-39083330. The authors would
like to thank Tord Riemann, who passed away as this report was written, for
his decade-long work on EW precision physics for $e^{+}e^{-}$ colliders.

## References

  * (1) M. Bicer, et al., First Look at the Physics Case of TLEP, JHEP 01 (2014) 164 (2014).  arXiv:1308.6176, doi:10.1007/JHEP01(2014)164.
  * (2) A. Abada, et al., FCC-ee: The Lepton Collider, Eur. Phys. J. ST 228 (2) (2019) 261–623 (2019).  doi:10.1140/epjst/e2019-900045-4.
  * (3) A. Abada, et al., FCC Physics Opportunities, Eur. Phys. J. C79 (6) (2019) 474 (2019).  doi:10.1140/epjc/s10052-019-6904-3.
  * (4) A. Freitas, et al., Theoretical uncertainties for electroweak and Higgs-boson precision measurements at FCC-ee, (2019).  arXiv:1906.05379.
  * (5) M. Kobel, et al., Two-Fermion Production in Electron-Positron Collisions, in: Proceedings, Monte Carlo Workshop: Report of the working groups on precision calculation for LEP-2 physics: CERN, Geneva, Switzerland, March 12-13, June 25-26, October 12-13 Oct 1999, 2000, p. (2000).  arXiv:hep-ph/0007180, doi:10.5170/CERN-2000-009.269.
URL http://weblib.cern.ch/abstract?CERN-2000-09-D

  * (6) S. Jadach, et al., Event generators for Bhabha scattering, in: CERN Workshop on LEP2 Physics (followed by 2nd meeting, 15-16 Jun 1995 and 3rd meeting 2-3 Nov 1995) Geneva, Switzerland, February 2-3, 1995, 1996, pp. 229–298 (1996).  arXiv:hep-ph/9602393.
  * (7) W. Beenakker, et al., $WW$ cross-sections and distributions, in: CERN Workshop on LEP2 Physics (followed by 2nd meeting, 15-16 Jun 1995 and 3rd meeting 2-3 Nov 1995) Geneva, Switzerland, February 2-3, 1995, 1996, pp. 79–139 (1996).  arXiv:hep-ph/9602351.
  * (8) D. Yu. Bardin, M. Grunewald, G. Passarino, Precision calculation project report, (1999).  arXiv:hep-ph/9902452.
  * (9) S. Schael, et al., Precision electroweak measurements on the $Z$ resonance, Phys. Rept. 427 (2006) 257–454 (2006).  arXiv:hep-ex/0509008, doi:10.1016/j.physrep.2005.12.006.
  * (10) S. Heinemeyer, W. Hollik, G. Weiglein, Electroweak precision observables in the minimal supersymmetric standard model, Phys. Rept. 425 (2006) 265–368 (2006).  arXiv:hep-ph/0412214, doi:10.1016/j.physrep.2005.12.002.
  * (11) S. Jadach, M. Skrzypek, QED challenges at FCC-ee precision measurements, Eur. Phys. J. C79 (9) (2019) 756 (2019).  arXiv:1903.09895, doi:10.1140/epjc/s10052-019-7255-9.
  * (12) A. Blondel, et al., Standard model theory for the FCC-ee Tera-Z stage, in: Mini Workshop on Precision EW and QCD Calculations for the FCC Studies : Methods and Techniques CERN, Geneva, Switzerland, January 12-13, 2018, CERN, CERN, Geneva, 2019, p. (2019).  arXiv:1809.01830, doi:10.23731/CYRM-2019-003.
  * (13) S. Schael, et al., Electroweak Measurements in Electron-Positron Collisions at W-Boson-Pair Energies at LEP, Phys. Rept. 532 (2013) 119–244 (2013).  arXiv:1302.3415, doi:10.1016/j.physrep.2013.07.004.
  * (14) A. Freitas, Numerical multi-loop integrals and applications, Prog. Part. Nucl. Phys. 90 (2016) 201–240 (2016).  arXiv:1604.00406, doi:10.1016/j.ppnp.2016.06.004.
  * (15) M. Awramik, M. Czakon, A. Freitas, G. Weiglein, Precise prediction for the W boson mass in the standard model, Phys. Rev. D 69 (2004) 053006 (2004).  arXiv:hep-ph/0311148, doi:10.1103/PhysRevD.69.053006.
  * (16) M. Awramik, M. Czakon, A. Freitas, G. Weiglein, Complete two-loop electroweak fermionic corrections to $\sin^{2}\theta^{\rm lept}_{\rm eff}$ and indirect determination of the Higgs boson mass, Phys. Rev. Lett. 93 (2004) 201805 (2004).  arXiv:hep-ph/0407317, doi:10.1103/PhysRevLett.93.201805.
  * (17) A. Freitas, Higher-order electroweak corrections to the partial widths and branching ratios of the Z boson, JHEP 04 (2014) 070 (2014).  arXiv:1401.2447, doi:10.1007/JHEP04(2014)070.
  * (18) I. Dubovyk, A. Freitas, J. Gluza, T. Riemann, J. Usovitsch, Complete electroweak two-loop corrections to Z boson production and decay, Phys. Lett. B 783 (2018) 86–94 (2018).  arXiv:1804.10236, doi:10.1016/j.physletb.2018.06.037.
  * (19) A. Freitas, Electroweak precision tests in the LHC era and Z-decay form factors at two-loop level, in: 12th DESY Workshop on Elementary Particle Physics: Loops and Legs in Quantum Field Theory, 2014, p. (6 2014).  arXiv:1406.6980.
  * (20) S. Jadach, B. F. L. Ward, Z. Was, Coherent exclusive exponentiation for precision Monte Carlo calculations, Phys. Rev. D63 (2001) 113009 (2001).  arXiv:hep-ph/0006359, doi:10.1103/PhysRevD.63.113009.
  * (21) P. Azzurri, et al., Physics Behind Precision, (3 2017).  arXiv:1703.01626.
  * (22) A. Denner, S. Dittmaier, M. Roth, L. H. Wieders, Electroweak corrections to charged-current e+ e- —$>$ 4 fermion processes: Technical details and further results, Nucl. Phys. B 724 (2005) 247–294, [Erratum: Nucl.Phys.B 854, 504–507 (2012)] (2005).  arXiv:hep-ph/0505042, doi:10.1016/j.nuclphysb.2011.09.001.
  * (23) M. Beneke, P. Falgari, C. Schwinn, A. Signer, G. Zanderighi, Four-fermion production near the W pair production threshold, Nucl. Phys. B 792 (2008) 89–135 (2008).  arXiv:0707.0773, doi:10.1016/j.nuclphysb.2007.09.030.
  * (24) S. Actis, M. Beneke, P. Falgari, C. Schwinn, Dominant NNLO corrections to four-fermion production near the W-pair production threshold, Nucl. Phys. B 807 (2009) 1–32 (2009).  arXiv:0807.0102, doi:10.1016/j.nuclphysb.2008.08.006.
  * (25) A. Denner, S. Dittmaier, M. Roth, D. Wackeroth, Electroweak radiative corrections to e+ e- —$>$ W W —$>$ 4 fermions in double pole approximation: The RACOONWW approach, Nucl. Phys. B 587 (2000) 67–117 (2000).  arXiv:hep-ph/0006307, doi:10.1016/S0550-3213(00)00511-3.
  * (26) A. Arbey, et al., Physics at the e+ e- Linear Collider, Eur. Phys. J. C 75 (8) (2015) 371 (2015).  arXiv:1504.01726, doi:10.1140/epjc/s10052-015-3511-9.
  * (27) G. Belanger, F. Boudjema, J. Fujimoto, T. Ishikawa, T. Kaneko, K. Kato, Y. Shimizu, Full one loop electroweak radiative corrections to single Higgs production in e+ e-, Phys. Lett. B 559 (2003) 252–262 (2003).  arXiv:hep-ph/0212261, doi:10.1016/S0370-2693(03)00339-3.
  * (28) A. Denner, S. Dittmaier, M. Roth, M. M. Weber, Electroweak radiative corrections to single Higgs boson production in e+ e- annihilation, Phys. Lett. B 560 (2003) 196–203 (2003).  arXiv:hep-ph/0301189, doi:10.1016/S0370-2693(03)00370-8.
  * (29) Y. Gong, Z. Li, X. Xu, L. L. Yang, X. Zhao, Mixed QCD-EW corrections for Higgs boson production at $e^{+}e^{-}$ colliders, Phys. Rev. D 95 (9) (2017) 093003 (2017).  arXiv:1609.03955, doi:10.1103/PhysRevD.95.093003.
  * (30) Q.-F. Sun, F. Feng, Y. Jia, W.-L. Sang, Mixed electroweak-QCD corrections to $e+e-\to{}HZ$ at Higgs factories, Phys. Rev. D 96 (5) (2017) 051301 (2017).  arXiv:1609.03995, doi:10.1103/PhysRevD.96.051301.
  * (31) Q. Song, A. Freitas, On the evaluation of two-loop electroweak box diagrams for $e^{+}e^{-}\to HZ$ production, JHEP 04 (2021) 179 (2021).  arXiv:2101.00308, doi:10.1007/JHEP04(2021)179.
  * (32) G. P. Lepage, P. B. Mackenzie, M. E. Peskin, Expected Precision of Higgs Boson Partial Widths within the Standard Model, (4 2014).  arXiv:1404.0319.
  * (33) D. de Florian, et al., Handbook of LHC Higgs Cross Sections: 4. Deciphering the Nature of the Higgs Sector, 2/2017 (10 2016).  arXiv:1610.07922, doi:10.23731/CYRM-2017-002.
  * (34) D. de Florian, et al., LHC Higgs Cross Section Working Group, (2016).
URL https://twiki.cern.ch/twiki/bin/view/LHCPhysics/LHCHXSWGBRs

  * (35) A. Denner, S. Heinemeyer, I. Puljak, D. Rebuzzi, M. Spira, Standard Model Higgs-Boson Branching Ratios with Uncertainties, Eur. Phys. J. C 71 (2011) 1753 (2011).  arXiv:1107.5909, doi:10.1140/epjc/s10052-011-1753-8.
  * (36) P. A. Zyla, et al., Review of Particle Physics, PTEP 2020 (8) (2020) 083C01 (2020).  doi:10.1093/ptep/ptaa104.
  * (37) P. A. Baikov, K. G. Chetyrkin, J. H. Kuhn, Scalar correlator at O(alpha(s)**4), Higgs decay into b-quarks and bounds on the light quark masses, Phys. Rev. Lett. 96 (2006) 012003 (2006).  arXiv:hep-ph/0511063, doi:10.1103/PhysRevLett.96.012003.
  * (38) J. Ablinger, J. Blümlein, A. De Freitas, K. Schönwald, Subleading Logarithmic QED Initial State Corrections to $e^{+}e^{-}\rightarrow\gamma^{*}/{Z^{0}}^{*}$ to $O(\alpha^{6}L^{5})$, Nucl. Phys. B 955 (2020) 115045 (2020).  arXiv:2004.04287, doi:10.1016/j.nuclphysb.2020.115045.
  * (39) J. Blümlein, A. De Freitas, C. Raab, K. Schönwald, The $O(\alpha^{2})$ initial state QED corrections to $e^{+}e^{-}\rightarrow\gamma^{*}/Z_{0}^{*}$, Nucl. Phys. B 956 (2020) 115055 (2020).  arXiv:2003.14289, doi:10.1016/j.nuclphysb.2020.115055.
  * (40) J. Blümlein, A. De Freitas, K. Schönwald, The QED initial state corrections to the forward-backward asymmetry of $e^{+}e^{-}\to\gamma^{*}/Z^{0*}$ to higher orders, Phys. Lett. B 816 (2021) 136250 (2021).  arXiv:2102.12237, doi:10.1016/j.physletb.2021.136250.
  * (41) S. Frixione, Initial conditions for electron and photon structure and fragmentation functions, JHEP 11 (2019) 158 (2019).  arXiv:1909.03886, doi:10.1007/JHEP11(2019)158.
  * (42) V. Bertone, M. Cacciari, S. Frixione, G. Stagnitto, The partonic structure of the electron at the next-to-leading logarithmic accuracy in QED, JHEP 03 (2020) 135 (2020).  arXiv:1911.12040, doi:10.1007/JHEP03(2020)135.
  * (43) J. Kalinowski, W. Kotlarski, P. Sopicki, A. F. Zarnecki, Simulating hard photon production with WHIZARD, Eur. Phys. J. C 80 (7) (2020) 634 (2020).  arXiv:2004.14486, doi:10.1140/epjc/s10052-020-8149-6.
  * (44) P. Sopicki, J. Kalinowski, W. Kotlarski, K. Mekała, A. F. Zarnecki, Simulating hard photon production with WHIZARD, PoS ICHEP2020 (2021) 285 (2021).  doi:10.22323/1.390.0285.
  * (45) S. Jadach, B. F. L. Ward, Z. Was, The Monte Carlo program KORALZ, for the lepton or quark pair production at LEP / SLC energies: From version 4.0 to version 4.04, Comput. Phys. Commun. 124 (2000) 233–237 (2000).  arXiv:hep-ph/9905205, doi:10.1016/S0010-4655(99)00437-3.
  * (46) S. Jadach, W. Placzek, E. Richter-Was, B. F. L. Ward, Z. Was, Upgrade of the Monte Carlo program BHLUMI for Bhabha scattering at low angles to version 4.04, Comput. Phys. Commun. 102 (1997) 229–251 (1997).  doi:10.1016/S0010-4655(96)00156-7.
  * (47) S. Jadach, W. Placzek, B. F. L. Ward, BHWIDE 1.00: O(alpha) YFS exponentiated Monte Carlo for Bhabha scattering at wide angles for LEP-1 / SLC and LEP-2, Phys. Lett. B390 (1997) 298–308, also hep-ph/9608412; The Monte Carlo program BHWIDE is available from http://cern.ch/placzek (1997).  arXiv:hep-ph/9608412, doi:10.1016/S0370-2693(96)01382-2.
  * (48) W. Placzek, S. Jadach, Multiphoton radiation in leptonic W boson decays, Eur. Phys. J. C29 (2003) 325–339 (2003).  arXiv:hep-ph/0302065, doi:10.1140/epjc/s2003-01223-4.
  * (49) D. R. Yennie, S. C. Frautschi, H. Suura, The infrared divergence phenomena and high-energy processes, Annals Phys. 13 (1961) 379–452 (1961).  doi:10.1016/0003-4916(61)90151-8.
  * (50) S. Jadach, B. F. L. Ward, Z. Was, The Precision Monte Carlo event generator KK for two fermion final states in $e^{+}e^{-}$ collisions, Comput. Phys. Commun. 130 (2000) 260–325 (2000).  arXiv:hep-ph/9912214, doi:10.1016/S0010-4655(00)00048-5.
  * (51) M. Moretti, T. Ohl, J. Reuter, O’Mega: An Optimizing matrix element generator, (2 2001).  arXiv:hep-ph/0102195.
  * (52) W. Kilian, T. Ohl, J. Reuter, WHIZARD: Simulating Multi-Particle Processes at LHC and ILC, Eur. Phys. J. C 71 (2011) 1742 (2011).  arXiv:0708.4233, doi:10.1140/epjc/s10052-011-1742-y.
  * (53) T. Sjostrand, S. Mrenna, P. Z. Skands, PYTHIA 6.4 Physics and Manual, JHEP 05 (2006) 026 (2006).  arXiv:hep-ph/0603175, doi:10.1088/1126-6708/2006/05/026.
  * (54) T. Sjöstrand, S. Ask, J. R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C. O. Rasmussen, P. Z. Skands, An introduction to PYTHIA 8.2, Comput. Phys. Commun. 191 (2015) 159–177 (2015).  arXiv:1410.3012, doi:10.1016/j.cpc.2015.01.024.
  * (55) S. Jadach, W. Płaczek, M. Skrzypek, QED exponentiation for quasi-stable charged particles: the $e^{-}e^{+}\rightarrow W^{-}W^{+}$ process, Eur. Phys. J. C80 (6) (2020) 499 (2020).  arXiv:1906.09071, doi:10.1140/epjc/s10052-020-8034-3.
  * (56) F. Bach, B. C. Nejad, A. Hoang, W. Kilian, J. Reuter, M. Stahlhofen, T. Teubner, C. Weiss, Fully-differential Top-Pair Production at a Lepton Collider: From Threshold to Continuum, JHEP 03 (2018) 184 (2018).  arXiv:1712.02220, doi:10.1007/JHEP03(2018)184.
  * (57) T. Ohl, CIRCE version 1.0: Beam spectra for simulating linear collider physics, Comput. Phys. Commun. 101 (1997) 269–288 (1997).  arXiv:hep-ph/9607454, doi:10.1016/S0010-4655(96)00167-1.
  * (58) N. D. Christensen, C. Duhr, B. Fuks, J. Reuter, C. Speckner, Introducing an interface between WHIZARD and FeynRules, Eur. Phys. J. C 72 (2012) 1990 (2012).  arXiv:1010.3251, doi:10.1140/epjc/s10052-012-1990-5.
  * (59) C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer, T. Reiter, UFO - The Universal FeynRules Output, Comput. Phys. Commun. 183 (2012) 1201–1214 (2012).  arXiv:1108.2040, doi:10.1016/j.cpc.2012.01.022.
  * (60) A. Freitas, K. Hagiwara, S. Heinemeyer, P. Langacker, K. Moenig, M. Tanabashi, G. W. Wilson, Exploring Quantum Physics at the ILC, in: Community Summer Study 2013: Snowmass on the Mississippi, 2013, p. (7 2013).  arXiv:1307.3962.