backscattering, where the initial lepton beam sources highly energetic photons
through Compton scattering Badelek  _et al._ (2004a); Żarnecki (2003);
Archibald  _et al._ (2008), or the equivalent photon approximation
(Weizsäcker–Williams method), where the beam particles act as quasi-classical
sources of collinear quasi-real photons von Weizsacker (1934); Williams
(1934); Budnev  _et al._ (1974). An interface to CIRCE Ohl (1997), which
parameterizes the $e^{\pm}$, and $\gamma$ beam-spectra based on the collider
geometry, is anticipated (cf. Secs. 8.1.2 and 8.1.2). The thus generated
electrons or photons can then either directly enter the hard interaction or be
further resolved.

To model the momentum distribution of incident electrons, SHERPA offers two
different approaches. The first approach uses LL+LO QED PDFs for the
electrons, that are solutions of the DGLAP evolution equations, cf. Sec. 8.2.
It can be combined with a traditional QED parton shower Höche  _et al._ (2010)
to generate exclusive kinematic distributions for the collinear photons. In
the second approach, the emission of soft photons is resummed to all orders
using the Yennie-Frautschi-Suura (YFS) formalism Yennie  _et al._ (1961). In
this method, photon emissions are considered in fully differential form, get
generated explicitly, and the treatment of their phase space is exact. At the
same time, the substructure of photons can be resolved using, e.g., the
dedicated photon PDFs of Glück  _et al._ (1992a, b), upon which the produced
partons enter a traditional QCD and QED evolution through a parton shower.

The hard scattering interaction is then described at LO, NLO and, in some
cases, at NNLO in QCD Höche  _et al._ (2012); Hoeche  _et al._ (2013);
Gehrmann  _et al._ (2013); Höche  _et al._ (2015b) accuracy, matched to the
parton shower Schumann and Krauss (2008); Höche and Prestel (2015). NLO EW
will be offered in the near future within the YFS soft-photon resummation.

####  8.4.3 WHIZARD

WHIZARD is a multi-purpose event generator framework for very general high-
energy physics simulations for all types of colliders and fixed-target
experiments, that allows a high level of flexibility Kilian  _et al._ (2011).

For hard-scattering processes, WHIZARD provides its own (tree-level) matrix-
element generator, OMEGA Moretti  _et al._ (2001), which is based on recursion
relations and the color-flow formalism Kilian  _et al._ (2012a) for QCD. This
framework supports as hard-coded models the SM in different variants and the
most studied BSM models. It has a general interface to FEYNRULES Christensen
_et al._ (2012) and UFO Degrande  _et al._ (2012), cf. Sec. 2.8 and supports
Les Houches accord input files Allanach  _et al._ (2009). Loop-matrix elements
for next-to-leading (NLO) QCD and EW processes can be accessed through
dedicated interfaces to one-loop providers (OLP) like OPENLOOPS, GOSAM and
RECOLA, cf. Sec. 4.1.

For all processes within all models, Feynman diagram selection is possible,
however should be treated with care due to possible gauge-invariance
violations. Processes can be factorized into production and decays by using
the decay features of WHIZARD, also consecutively in decay chains. All decays
of a resonance at a given final-state multiplicity can be auto-generated. In
resonant processes, intermediate resonances can be specified as polarized,
e.g. for the study of specific longitudinal or transversal EW gauge bosons
like in multi-EW boson production and vector-boson scattering Brass  _et al._
(2018); Alboteanu  _et al._ (2008); Kilian  _et al._ (2015); Fleper  _et al._
(2017).

WHIZARD ships with its own scripting language, Sindarin, which allows
arbitrary cuts to be defined on the hard matrix elements, as well as
selections for the event generation. The hard matrix elements can be
convoluted with a large number of parton distribution functions, structure
functions or beam spectra: these include hadron-collider proton PDFs via
LHAPDF externally or from a selected set of internally shipped PDFs, effective
$W/Z$ approximation for the content of EW bosons inside quarks or leptons, the
effective-photon approximation (EPA), also known as Weizsäcker-Williams
approximation. The latter is provided in different variants, depending on the
application for low-energy $\gamma\gamma\to$ hadron background simulations or
for high-energy particle production.

For lepton collisions, electron/muon PDFs are provided at LL accuracy and are
currently being extended to NLL. WHIZARD supports several options for
structured lepton beams: Gaussian beam spreads, individually adjustable for
each beam, beam spectrum files read in as tables of pairs of energy values and
an approximation to GUINEA-PIG spectra via its beamstrahlung generators CIRCE
Ohl (1997) (cf. Sec. 8.1.2) and CIRCE2 (cf. Sec. 8.1.2). WHIZARD allows
initial-state particle of all kinds to be polarized, both for scattering as
well as for decay processes. For scattering processes, the polarization of
both beams can be arbitrarily correlated by specifying a spin density matrix.
In addition, polarization fractions can be given. Initial-state beams can be
asymmetric (like in flavor factories or HERA), and crossing angles can be
defined.

As the program has always had a strong focus on weak production processes of
multi-fermion final states, at lepton colliders it contains a phase-space
parameterization that is flexible enough to adapt to many different
(interfering) resonant production channels. This builds upon an adaptive
multi-channel Monte Carlo integration encoded in the VAMP subpackage Ohl
(1999). In recent years, there have been several attempts to make the MC
integration much more efficient by using highly parallelized computing
architectures: using Multi-Processing Interface (MPI), speed-ups of several
tens up to roughly a hundred can be achieved Brass  _et al._ (2019). Also,
OMEGA can produce very small expressions for the matrix elements in the form
of a so-called virtual machine, which are as efficient as compiled code
Chokoufe Nejad  _et al._ (2015b).

Turning to high-precision calculations at next-to-leading order in the SM,
WHIZARD has a complete implementation for NLO QCD and electroweak (as well as
mixed) corrections for hadron and lepton colliders, based on its
implementation of the FKS subtraction scheme, standard and resonance-aware. In
the same setup as for scattering processes, NLO decays can be calculated. The
automated implementation has been validated for hadron and lepton colliders
and allows to calculate processes at NLO (QCD) for rather high final-state
multiplicities, e.g. $e^{+}e^{-}\to\ell\ell\nu\nu b\bar{b}H$ or $e^{+}e^{-}\to
jjjjjj$ Rothe (2021); Brass  _et al._ (2022). The program provides a default
framework to produce arbitrary differential distributions, using the concept
of event groups with an analysis like e.g. RIVET Bierlich  _et al._ (2020);
Buckley  _et al._ (2013). Electroweak corrections for lepton colliders can be
calculated already for massive leptons, while the infrastructure for
corrections for massless leptons is being validated right now. The NLL
electron PDFs are being implemented, and first results for this are expected
within 2022. To define infrared-safe quantities, WHIZARD provides jet
clustering with a FastJet interface as well as photon recombination. To define
fixed-order NLO cross sections in a 4- or 3-flavor scheme, the user can
specify $b$-jet, $c$-jet and light jet clustering.

In order to generate complete events, WHIZARD provides its own parton shower,
specifically a $k_{T}$-ordered parton shower, as well as, an analytic,
virtuality-order (QCD) shower Kilian  _et al._ (2012b). In addition, WHIZARD
ships with the final Fortran version of PYTHIA (6.427) Sjöstrand  _et al._
(2006) which is specifically tuned to LEP2 hadron data, and with a dedicated
interface to PYTHIA 8 Sjöstrand  _et al._ (2015); the latter allows to
directly communicate between the event records of WHIZARD and PYTHIA.

For inclusive jet samples at LO, WHIZARD can apply the MLM merging algorithm,
while for NLO matching between fixed-order and the parton shower, a general,
process-independent version of POWHEG matching is available for QCD Chokoufe
Nejad  _et al._ (2015a). POWHEG matching for EW corrections is in preparation,
while for mid-term planning also alternative matching and merging schemes are
envisioned. In order to reconcile multi-fermion final states, e.g.
$e^{+}e^{-}\to jjjj$ with parton showering and hadronization, WHIZARD is able
to find and quantify contributions from underlying resonant subprocesses, and
then provide pseudo-shower histories to the final state weighted by the
relative cross sections of these underlying processes. This is important to
correctly model hadronic correlations in the final states, and also inclusive
observables like the total number of final-state neutral and charged hadrons
as well as photons.

One of the most important tasks for precision physics at future Higgs
factories will be a modeling of QED and photon radiation as precise as
possible. For the normalization of QED cross section, a resummation based on
collinear factorization gives very precise results. As mentioned above, this
is possible in WHIZARD at LL level and will be available at NLL level soon.
Specifically for the assessment of experimental/systematic uncertainties, an
exclusive simulation of photon radiation is necessary. WHIZARD can of course
produce explicit photons from matrix elements; on the other hand, the energy
loss from LL electron PDFs in the collinear approximation is collected in a
single photon per beam. A heuristic approximation is possible to generate a
$p_{T}$ distribution for these photons, which can also be applied to recoiling
charged leptons from Weizsäcker-Williams/EPA Fermi (1932); von Weizsacker
(1934); Williams (1934). The photon distribution is generated using a
logarithmic scaling, while the event of the hard scattering (with subsequent
parton shower and hadronization) is boosted accordingly. It has been shown
that for both signal and background in monophoton dark sector searches (where
this distribution plays a key role), this heuristic approach agrees very well
with exact matrix-element calculations Kalinowski  _et al._ (2020); Sopicki
_et al._ (2021). While collinear resummation results in very accurate
predictions for cross sections of (inclusive) processes with initial-state
radiation, combining the highest possible precision of resummation with
exclusive photon radiation is achieved using the YFS formalism Yennie  _et
al._ (1961). The WHIZARD team is working on the implementation of this
formalism for general processes. Furthermore, WHIZARD will also get its own
QED parton shower, which will be available as a relatively slim stand-alone
shower for pure QED, and potentially also as an interleaved shower together
with the QCD shower.

WHIZARD has special support for the top threshold to model the process
$e^{+}e^{-}\to t\bar{t}\to bW^{-}\bar{b}W^{+}$. This process allows for the
determination of the top mass with the highest possible theoretical precision.
Analytic calculations using an NRQCD EFT approach can include NNNLO
corrections, which reduces the error to 30-70 MeV. This calculation, however,
is completely inclusive. To study experimental efficiencies and systematic
uncertainties, Monte Carlo samples resembling event weights close to the true
top threshold cross sections are needed. WHIZARD includes a special treatment
of a matched calculation between the continuum NLO off-shell fixed-order
calculation and the NLL-threshold resummed calculation avoiding double
counting Bach  _et al._ (2018), fully off-shell and exclusive in the top decay
products. This allows to study fully exclusive distributions.

In a similar framework, work has started to match resummed QED-Coulomb
corrections to the QED NLO/NNLO fixed-order calculation for the $WW$ threshold
Beneke  _et al._ (2008); Actis  _et al._ (2009). Like the top threshold, this
will be available as a dedicated process within a specialized model. These
simulations allow experimental studies with the desired accuracy to support
measurements of the $W$ mass from a threshold scan with a precision of 1-2 MeV
or below.

####  8.4.4 BABAYAGA@NLO

BABAYAGA@NLO Balossini  _et al._ (2006, 2008) is a Monte Carlo event generator
that provides precise predictions for the processes $e^{+}e^{-}\to
e^{+}e^{-}$, $e^{+}e^{-}\to\mu^{+}\mu^{-}$ and $e^{+}e^{-}\to\gamma\gamma$ at
NLOPS accuracy in QED. It is a reference tool for luminosity measurements and
other physics studies at $e^{+}e^{-}$ colliders with center of mass energy at
the GeV scale (flavour factories) Actis  _et al._ (2010). We will first
discuss the QED fixed-order corrections implemented in the code that are
necessary to describe processes at low energies, and then mention recent and
future upgrades of the generator that are necessary to make it suitable for
simulations at high-energy colliders.

In a nutshell, the theoretical formulation inherent to BABAYAGA@NLO is based
on the matching of next-to-leading order (NLO) corrections with a QED Parton
Shower (PS) algorithm modeling multiple photon emission exclusively Carloni
Calame  _et al._ (2000); Carloni Calame (2001).

The master formula for the cross section calculation reads as follows444 For
the sake of simplicity, we limit the notation for our matching procedure to
the case of one radiating particle. The extension of Eq. (29) to the realistic
case, where every charged particle radiates photons, is almost
straightforward.:

$\mathrm{d}\sigma=F_{\textrm{SV}}\,\Pi\left(Q^{2},\epsilon\right)\sum_{n=0}^{\infty}\dfrac{1}{n!}\left(\prod_{i=0}^{n}F_{\textrm{H},i}\right)\left|{\cal{M}}_{n,\textrm{LL}}\right|^{2}\mathrm{d}\Phi_{n}$
(29)

In Eq. (29), $\Pi\left(Q^{2},\epsilon\right)$ is the Sudakov form factor,
which accounts for the exponentiation of leading logarithmic (LL)
contributions due to soft and virtual corrections, $Q^{2}$ and $\epsilon$
being the hard scale of the process555For charged fermion pair production, we
set in the code $Q^{2}=s\,t/u$ in order to exponentiate also the dominant
contribution due to box diagrams and initial-final state interference, in
addition to the leading logarithms from initial and final state radiation. For
$e^{+}e^{-}\to\gamma\gamma$, only the choice $Q^{2}=s$ describing initial
state radiation is physically meaningful. and a soft-hard photon separator,
respectively. ${\left|{\cal{M}}_{n,\textrm{LL}}\right|}^{2}$ is the squared
matrix element describing the emission in LL approximation of $n$ hard
photons, i.e. with energy larger than $\epsilon$. The factor
$\mathrm{d}\Phi_{n}$ is the exact phase space element of the underlying
process accompanied by photon radiation.

In Eq. (29) the matching of the above PS ingredients with the NLO QED
corrections is realized by the fixed-order factors $F_{\textrm{SV}}$ and
$F_{\textrm{H}}$, which are process dependent and whose definitions can be
found in Balossini  _et al._ (2006). They are infrared/collinear safe
correction factors that account for those $O(\alpha)$ non-logarithmic terms
entering the NLO calculation and absent in the universal PS approach. The
factor $F_{\textrm{SV}}$ takes into account soft+virtual corrections, while
$F_{\textrm{H}}$ is a hard Bremsstrahlung contribution including the exact
matrix element of the $2\to 3$ radiative process of the underlying $2\to 2$
process under consideration. It is worth stressing that both $F_{\textrm{SV}}$
and $F_{\textrm{H}}$ are applied at differential level on an event-by-event
basis.

By construction, the matching procedure as in Eq. (29) is such that its
$O(\alpha)$ expansion reproduces the NLO cross section and exponentiation of
LL contribution is preserved as in a pure PS approach. Moreover, as a by-
product of its factorized structure, the bulk of the photonic sub-leading
contributions at NNLO, i.e those of the order of $\alpha^{2}L$, is
automatically included by means of terms of the type
$F_{{\textrm{SV}}~{}|~{}{\textrm{H}}}~{}\otimes$ LL corrections Montagna  _et
al._ (1996).

Concerning the treatment of the angular degrees of freedom in the PS approach,
the generation of the transverse momentum of fermions and photons at each
branching is achieved in BABAYAGA@NLO according to the following function

$I(k)~{}=~{}\sum_{i,j=1}^{N}~{}\eta_{i}\eta_{j}~{}\frac{p_{i}\cdot
p_{j}}{(p_{i}\cdot k)(p_{j}\cdot k)}~{}E_{\gamma}^{2}$ (30)

where $N$ is the number of involved charged particles, $p_{\ell}$ is the
momentum of the external fermion $\ell$, $\eta_{\ell}$ is a charge factor, see
e.g. Carloni Calame (2001), $k$ is the photon momentum, $E_{\gamma}$ is its
energy and the sum runs over all the external fermions. It is worth noting
that Eq. (30) is inspired to the expression of the differential cross section
for a generic process with emission of additional soft photons and accounts
for the interference of radiation coming from different particles.

For QED processes at flavour factories, the theoretical accuracy of
BABAYAGA@NLO is at the 0.1% level. This conclusion follows from a number of
detailed tuned comparisons with the results of independent codes, such as
BHWIDE Jadach  _et al._ (1997b) and MCGPJ Arbuzov  _et al._ (2006), as well as
with NNLO predictions and the estimate of $O(\alpha^{2}L)$ contributions
according to realistic event selection criteria.

For the high-precision requirements of high-energy $e^{+}e^{-}$ colliders,
possible improvements of BABAYAGA@NLO are given by

  1. 1.

the inclusion of the full set of NLO electroweak corrections. A first step
towards this goal has been taken in Ref. Carloni Calame  _et al._ (2019) for
the study of the process $e^{+}e^{-}\to\gamma\gamma$ at FCC-ee energies;

  2. 2.

the implementation into a QED PS algorithm of the parton evolution at next-to-
leading-logarithmic (NLL) accuracy, in order to improve the description of
higher-order corrections;

  3. 3.

the generalization of the matching algorithm to include also exact NNLO in
QED.

####  8.4.5 BHLUMI/BHWIDE

BHWIDE Jadach  _et al._ (1997b) is a Monte Carlo (MC) event generator for
large-angle Bhabha scattering with multiphoton emission. It is based on the
Yennie–Frautschi–Suura exclusive exponentiation Yennie  _et al._ (1961) in
which all the infrared singularities are summed-up to the infinite order and
cancelled properly in the YFS form factor. The remaining non-IR residuals,
$\bar{\beta}_{n}^{(l)}$, corresponding to the emission of $n$-real photons,
are calculated perturbatively up to a given order $l$, where $l\geq n$, and
$(l-n)$ corresponds to a number of loops accounted for in
$\bar{\beta}_{n}^{(l)}$. In BHWIDE, an arbitrary number $n$ of real photons
with non-zero transverse momenta and energies above some soft-photon cut-off
are generated according to the YFS MC algorithm Jadach  _et al._ (1992b). This
cut-off can be set to an arbitrary low value without any danger of negative
probability distributions, as it can happen in standard fixed-order
calculations. The non-IR residuals are calculated up to NLO:
$\bar{\beta}_{0}^{(1)}$, corresponding no real-photon emission, includes one-
loop electroweak (EW) radiative corrections, and $\bar{\beta}_{1}^{(1)}$,
corresponding to one-real photon emission, includes a tree-level hard-
bremsstrahlung matrix element. In $\bar{\beta}_{0}^{(1)}$, two libraries of
the ${\cal O}(\alpha)$ virtual EW corrections are used: (1) the older one of
Refs. Bohm  _et al._ (1988); Berends  _et al._ (1988), and (2) the newer of
Ref. Beenakker  _et al._ (1991) (recommended). In $\bar{\beta}_{0}^{(1)}$, two
independent matrix elements for single hard-photon radiation are implemented:
(1) the one expressed in terms of helicity amplitudes Jadach  _et al._ (1997b)
(recommended), and (2) the formula of CALKUL Berends  _et al._ (1982) for the
matrix element squared. In both cases, finite electron/positron mass effects
are taken into account. The above two implementations agree numerically up to
at least $6$ digits event-by-event.

The MC algorithm of BHWIDE is based on that of the event generator BHLUMI for
small-angle Bhabha scattering Jadach  _et al._ (1992b) with three important
extensions: (1) QED interferences between the electron and positron lines
(“up-down” interferences) are added, (2) the full YFS form factor for the
$e^{+}e^{-}\rightarrow e^{+}e^{-}$ process is implemented, and (3) the exact
NLO matrix element for the full Bhabha process (all channels and
contributions) is included. At a low-level MC stage, the multiphoton radiation
is generated for a $t$-channel scattering process, while an $s$-channel as
well as all interferences are reintroduced through appropriate MC weights.
This means that the program is more efficient when the $t$-channel
contribution is dominant, although it proved to work well also at the
$Z$-peak.

BHWIDE offers two modes of event generation: (1) variable-weight events
(faster, several MC weights can be computed in a single run), and (2) unit-
weight events (useful for experimental analyses where events need to be
processed through time-consuming detector simulations). Various input
parameter options, to be set by the user, allow to choose between different
contributions/corrections to the cross section, such as EW corrections (two
libraries), vacuum-polarization parameterization, etc. Other input parameters
allow to specify the necessary ingredients for event generation and cross-
section calculations, such as collision energy, physical parameters (masses,
widths, etc.), the soft-photon cut-off, phase-space cuts, etc. For each
generated event, information on four-momenta of the final-state electron,
positron, and all emitted photons is provided through a dedicated event
record. In the variable-weight mode, it is supplemented with a value of the
nominal event weight and a vector of weight values corresponding to various
models or approximations. In the final stage of a program run, values of the
total cross section and its statistical error corresponding to the generated
event sample are computed and provided together with other useful information.

Several cross-checks and comparisons of BHWIDE with other calculations were
done for the LEP experiments Jadach  _et al._ (1996); Placzek  _et al._ (1999)
which were its first main users. Based on these tests, its overall theoretical
precision was estimated at the level of $\sim 1\%$ for the LEP energies.
BHWIDE has also been used by some experiments at low-energy $e^{+}e^{-}$
colliders Actis  _et al._ (2010). Recently, it was helpful in studies of
theoretical precision of luminosity measurements at future lepton colliders
(FCC-ee, ILC, CLIC) Jadach  _et al._ (2019b, 2021), where it can be
complementary to BHLUMI in providing predictions for contributions to the
small-angle Bhabha process other than a pure $t$-channel $\gamma$-exchange. In
order to match the expected experimental precision of the future lepton
colliders, BHWIDE needs further improvements, such as interfaces to modern NLO
EW libraries, inclusion of the NNLO QED radiative corrections, and for FCC-ee,
possibly also the NNLO EW corrections.

####  8.4.6 PHOTOS

In contrast to the above examples, PHOTOS Barberio  _et al._ (1991); Barberio
and Was (1994); Golonka and Was (2006); Davidson  _et al._ (2016) is not a
complete stand-alone event generator for $e^{+}e^{-}$ collisions. It is,
however, a widely used tool to approximate higher-order QED final state
radiation corrections to an otherwise LO QED simulation. Its FSR resummation
properties can be summarised by examining PHOTOS’ approximation of the all-
orders decay rate $\mathrm{d}\Gamma$ in terms of a given LO decay rate
$\mathrm{d}\Gamma_{0}$,

$\begin{split}\mathrm{d}\Gamma^{\text{P{HOTOS}}}=\;&\mathrm{d}\Gamma_{0}\left\\{1+\sum\limits_{c=1}^{n_{\text{ch}}}\sum\limits_{n_{\gamma}}\frac{\left(\alpha\,L_{c}\right)^{n_{\gamma}}}{n_{\gamma}!}\left[\prod\limits_{i=1}^{n_{\gamma}}\mathrm{d}x_{c}^{i}\right]\left(P_{\epsilon_{\text{cut}}}(x_{c}^{1})\\!\otimes\\!.\;\\!\\!.\;\\!\\!.\\!\otimes\\!P_{\epsilon_{\text{cut}}}(x_{c}^{n_{\gamma}})\right)\right\\}\;.\end{split}$
(31)

Therein, the emission kernels are summed over all charged particles
$n_{\text{ch}}$ involved and $L_{c}$ is the logarithm of the ratio of the
decaying particle’s mass over the mass of the charged particle $c$. Finally,
$x_{c}=\prod x_{c}^{i}$ is the energy energy fraction retained after the
radiation process, with $n_{\gamma}$ photons being emitted. The photons are
distributed in phase space according to the Altarelli-Parisi splitting
functions $P_{\epsilon_{\text{cut}}}(x)$, regulated with an infrared cut-off
$\epsilon_{\text{cut}}$. These emission kernels can be modified by suitable
weights to recover the correct coherent soft-photon limit, or exact higher-
order corrections. Their precise definitions can be found in Barberio and Was
(1994).

###  8.5 Heavy flavor production and decay

The production of excited heavy mesons and baryons, and their subsequent
decay, have always been a particular focus of the $e^{+}e^{-}$ collider
physics program, especially at flavor factories. Heavy-quark fragmentation
differs from its light-quark counterpart, and to some extent is more
straightforward due to the appearance of a high mass scale.

Decay cascades of unstable hadrons down to the stable ground states are
modeled in several event generators Lange (2001); Bellm  _et al._ (2016a);
Bothmann  _et al._ (2019b) using the same two basic ingredients. First,
branching ratios are taken from experimental measurements for known decay
channels or simulated inclusively through parton shower and hadronization
models for the unmeasured ones. Second, decay kinematics are generated
according to matrix elements including form factors, which are typically
fitted from effective field theories or measurements. Dedicated
implementations for particular phenomenological effects are available in some
of the generators, two of which are exemplified in the following.

####  8.5.1 Polarization effects

Figure 9: Efficiency-corrected normalized signals as functions of $\cos\gamma$
in comparison to experimental data from BELLE Balagura  _et al._ (2008). Green
Histograms show default predictions from HERWIG, while red and blue histograms
are respectively predictions with only the implementation of HQEFT and with
HQEFT effects in the presence of improved hadronization and decay models.

An interesting question is whether the polarization of the initial heavy quark
may be detected after hadronization, either in a polarization of the final
ground state or in anisotropies in the decay products of the excited hadron.
The result should hinge, in part, on a non-perturbative parameter that can
measure the net transverse alignment of the light degrees of freedom in the
fragmentation process Falk and Peskin (1994). The effect can be simulated in
the HERWIG framework through the combination of various physics models
Masouminia and Richardson :

  * (a)

Passing through the polarization of heavy hadrons at the end of parton shower
by employing a prescription of the heavy-quark effective field theory (HQEFT)
that allows for the detection of a net polarization of the initial heavy-
quark, either in a polarization of the final ground state or in the decay
products of the excited heavy mesons and heavy baryons Falk and Peskin (1994);
Masouminia and Richardson ,

  * (b)

improving the strong and radiative decay modes of the excited heavy mesons,
i.e charm and bottom mesons,

  * (c)

re-tuning HERWIG’s cluster hadronization model by the use of all existing
production rates of heavy hadrons, and

  * (d)

improvement of the kinematic behavior of the cluster hadronization model,
especially with respect to the treatment of light high-energy clusters.

Figure 9 shows first results, demonstrating the immediate effects of these
modifications in the predictions of the experimental data from BELLE Balagura
_et al._ (2008).

####  8.5.2 Mixing

Figure 10: CP violation in the interference between $B^{0}\to J/\Psi K_{S}$
and $\bar{B}^{0}\to J/\Psi K_{S}$ with $y=-0.05$.

Another interesting phenomenological aspect is the mixing of neutral mesons
(e.g. $D^{0}$, $B^{0}$, $B_{s}$) and in particular its connection to CP
violation, which has been studied in detail at flavor factories. The SHERPA
framework allows a modeling of these effects in several ways:

  * (a)

Explicit mixing of neutral mesons in the event record, optionally with CP
violation in the mixing, i.e. $|q/p|^{2}\neq 1$,

  * (b)

Direct CP violation in decays, i.e. rate differences between the decay $i\to
f$ and $\bar{i}\to\bar{f}$, implemented through separate branching ratio
tables for particles and anti-particles,

  * (c)

CP violation in the interference between mixing and decay amplitudes, for
decays into final states common to both particle and anti-particle.

An example of the latter can be seen in Fig. 10 for the rate asymmetry between
$B^{0}\to J/\Psi K_{S}$ and $\overline{B^{0}}\to J/\Psi K_{S}$ decays in
$\Upsilon(4S)\to B^{0}\overline{B^{0}}$ events assuming
$y\equiv\Delta\Gamma/(2\Gamma)=-0.05$.

## Acknowledgments

The work of S. Alioli, A. Broggio, A. Gavardi, S. Kallweit, G. Marinelli, D.
Napoletano and R. Nagar is supported by the ERC Starting Grant REINVENT
714788. S. Alioli also acknowledges funding from Fondazione Cariplo and
Regione Lombardia, grant 2017-2070 and from MIUR through the FARE grant
R18ZRBEAFC. The work by M. D. Baker was supported by Jefferson Science
Associates, LLC under Contract No. DE-AC05-06OR23177 with the DOE, Office of
Science, Office of Nuclear Physics and by the U.S. Department of Energy under
Contract No. DE-SC0012704. This work is supported by a Royal Society Research
Professorship (RP$\backslash$R1$\backslash$180112: M. van Beekveld, G. P
.Salam, L. Scyboz), by the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement No.
788223, PanScales: M. Dasgupta, S. Ferrario Ravasio, K. Hamilton, A. Karlberg,
R. Medves, G. P .Salam, A. Soto-Ontoso, G. Soyez and R. Verheyen), and by the
Science and Technology Facilities Council (STFC) under grants ST/T000864/1 (M.
van Beekveld, G. P. Salam), ST/T001038/1 (M. Dasgupta, M. H. Seymour),
ST/T000856/1 (K. Hamilton) and ST/T000694/1 (B. R. Webber). The work of J. M.
Campbell, S. Gardiner, T. Hobbs, S. Hoeche, J. Isaacson, S. W. Li, P. Machado,
S. Mrenna, N. Rocco and M. Wagman was supported by the Fermi National
Accelerator Laboratory (Fermilab), a U.S. Department of Energy, Office of
Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance,
LLC (FRA), acting under Contract No. DE–AC02–07CH11359. The work of N.
Darvishi is supported by the National Natural Science Foundation of China
(NSFC) under grants No. 12022514, No. 11875003 and No. 12047503, and CAS
Project for Young Scientists in Basic Research YSBR-006, by the Development
Program of China under Grant No. 2020YFC2201501 (2021/12/28) and by the CAS
President’s International Fellowship Initiative (PIFI) grant. A. Denner, T.
Ohl and G. Pelliccioli are supported by the Federal Ministry of Education and
Research, Germany (BMBF, grant 05H21WWCAA). The work of M. Diefenthaler was
supported by Jefferson Science Associates, LLC under Contract No. DE-
AC05-06OR23177 with the DOE, Office of Science, Office of Nuclear Physics. T.
Engel is supported by the Swiss National Science Foundation (SNF) under
contract 200021_178967. The work of J. R. G is supported by the Royal Society
through Grant URF\R1\201500. I. Helenius acknowledges support from the Academy
of Finland, project number 331545 and has been funded as a part of the CoE in
Quark Matter of the Academy of Finland. S. Jadach partly supported by the
funding from the European Union’s Horizon 2020 research and innovation
programme under Grant agreement no. 951754 and from the National Science
Centre, Poland, Grant no. 2019/34/E/ST2/00457. The work of A. Jentsch and Z.
Tu was supported by the U.S. Department of Energy under Contract No. DE-
SC0012704, and A. Jentsch was also supported by the Program Development
program at Brookhaven National Laboratory. S. Klein, B. Nachman and C.
Wilkinson are supported by the U.S. Department of Energy (DOE), Office of
Science under contract DE-AC02-05CH11231. F. Kling and J.R. Reuter are
supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Association) under Germany’s Excellence Strategy-EXC2121 “Quantum
Universe”-39083330. C. Krause is supported by DOE grant DOE-SC0010008. M.R.
Masouminia is supported by the UK Science and Technology Facilities Council
(grant numbers ST/P001246/1). The work of J. McFayden was supported by the
Royal Society Fellowship Grant URF\R1\201519. L. Pickering is supported by a
Royal Society University Research Fellowship (URF\R1\211661). V. Pandey
acknowledges the support from US DOE under grant DE-SC0009824. C.T. Preuss is
supported by the Swiss National Science Foundation (SNF) under contract
200021-197130. A. C. Price is supported by the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 945422. L. Rottoli is supported by the SNF grant PZ00P2_201878.
This work is supported by the Royal Society through a University Research
Fellowship (URF\R1\180549: M. Schönherr) and an Enhancement Award
(RGF\EA\181033 and CEC19\100349: W. Ju, M. Schönherr) as well as the STFC
under grant agreement ST/P001246/1 (L. Flower, M. Schönherr). S. Schumann
acknowledges support from the German Federal Ministry of Education and
Research (BMBF, grant 05H21MGCAB) and the German Research Foundation (DFG,
project number 456104544). The work of A. Siodmok and J. Whitehead is
supported also by the National Science Centre, Poland, (Grant no.
2019/34/E/ST2/00457) The work of A. Siodmok was also funded by the Priority
Research Area Digiworld under the program Excellence Initiative – Research
University at the Jagiellonian University in Cracow. T. Sjöstrand is supported
by the Swedish Research Council, contract number 2016-05996. P. Skands is
supported by ARC grant DP170100708. J.T. Sobczyk was supported by NCN grant
UMO-2021/41/B/ST2/02778. D. Soldin acknowledges support from the NSF Grant
PHY-1913607. The work of D. Soper was supported by the United States
Department of Energy under grant DE-SC0011640. S. Trojanowski is supported by
the grant “AstroCeNT: Particle Astrophysics Science and Technology Centre”
carried out within the International Research Agendas programme of the
Foundation for Polish Science financed by the European Union under the
European Regional Development Fund and by the Polish Ministry of Science and
Higher Education through its scholarship for young and outstanding scientists
(decision no 1190/E-78/STYP/14/2019). R. Winterhalder is supported by FRS-FNRS
(Belgian National Scientific Research Fund) IISN projects 4.4503.16. The work
by K. Zapp has received funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme (Grant
agreement No. 803183, collectiveQCD). This research used resources of the
Argonne Leadership Computing Facility, which is a DOE Office of Science User
Facility supported under Contract DE-AC02-06CH11357. The work is funded by
LANL’s Laboratory Directed Research and Development (LDRD/PRD) program under
project number 20210968PRD4. Los Alamos National Laboratory is operated by
Triad National Security, LLC, for the National Nuclear Security Administration
of U.S. Department of Energy (Contract No. 89233218CNA000001). This research
is supported in parts by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under grant 396021762-TRR 257. Much of the event
generator development described above was performed in the context of the
MCnet collaboration, which received funding from the European Union’s Horizon
2020 research and innovation programme as part of the Marie Skłodowska-Curie
Innovative Training Network MCnetITN3 (grant agreement no. 722104).

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