rather large uncertainties, but it can nevertheless be observed rather
clearly. In contrast, we can see a somewhat larger $Q_{0}$-dependence of the
same four parameters for the default model in Fig. 8. This is visible most
prominently for $\mathtt{PwtDIquark}$. Since these parameters directly affect
the Baryon production and thus the charge particle multiplicities generated in
the simulation, we can conclude that in the default model the shower cutoff
value $Q_{0}$ affects these multiplicities so that the tuned value of
$\mathtt{PwtDIquark}$ needs to compensate in a significant way. In the novel
dynamical hadronization model this rather unnatural feature is not visible.
Overall, the results shown in Figs. 8 and 9 again confirm that the novel
dynamical hadronization model achieves a better separation of the low-scale
hadronization processes from the parton level description provided by the
parton shower. This separation is essentially the MC simulation analogue of
factorization in analytic QCD studies and an important prerequisite for the
parton shower cutoff to be interpreted as a factorization scale.

###  7.3 Rescaled Migration Matrix Function

Figure 10: Rescaled parton-to-hadron migration matrix funtions for the default
hadronization model for the 2-jettiness distribution for the shower cutoff
$Q_{0}=1.25$ GeV extracted for c.m. energy $Q=91.2$ GeV. The left panel shows
$\tilde{S}^{\rm MC}(k,\hat{k},\\{Q,Q_{0}\\})$ and the right panel $S^{\rm
MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$, which is analogous to the shape function.

Figure 11: Rescaled parton-to-hadron migration matrix funtions for the novel
dynamical hadronization model for the 2-jettiness distribution for the shower
cutoff $Q_{0}=1.25$ GeV extracted for c.m. energy $Q=91.2$ GeV. The left panel
shows $\tilde{S}^{\rm MC}(k,\hat{k},\\{Q,Q_{0}\\})$ and the right panel
$S^{\rm MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$, which is analogous to the shape
function.

Let us now start the discussion on the MC results for the rescaled migration
matrix functions $\tilde{S}^{\rm MC}(k,\hat{k},\\{Q,Q_{0}\\})$ and $S^{\rm
MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ defined in Eqs. (18) and (19), respectively,
for the default and the dynamic hadronization models obtained from the
2-jettiness $\tau$ distributions. For a given (true) parton level soft
momentum $\hat{k}=Q\hat{\tau}$ and shower cutoff $Q_{0}$, the migration
function $\tilde{S}^{\rm MC}(k,\hat{k},\\{Q,Q_{0}\\})$ gives the distribution
of hadron level momenta $k=Q\tau$, while $S^{\rm
MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ gives the distribution of the non-
perturbative soft momenta $\ell=k-\hat{k}$ that the hadronization adds to the
parton level configuration. We remind the reader that $S^{\rm
MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ is the MC analogue of the infrared cutoff-
dependent shape function $S_{\rm had}(\ell,Q_{0})$ in the dijet 2-jettiness
QCD factorization formula discussed in Sec. 3.2. Details concerning the
extraction of the migration matrix functions from the Herwig simulations have
been explained in Sec. 6.3.

In the left panel of Fig. 10 the MC migration function $\tilde{S}^{\rm
MC}(k,\hat{k},\\{Q,Q_{0}\\})$ is shown for the Herwig default cluster
hadronization model. It has been extracted at the c.m. energy $Q=91.2$ GeV for
the reference $Q_{0}=1.25$ GeV tune (explained in more detail in Sec. 6.1) in
the range $0\leq k,\hat{k}\leq 8$ GeV. For $Q=91.2$ GeV this corresponds to
thrust values between $0$ and $0.088$. The corresponding shifted migration
function $S^{\rm MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ is shown in the right panel.
In Fig. 11 the analogous migration functions are shown for the novel dynamical
model. The shape and structures shown in both figures are representative for
the migration matrix functions for all cases we obtain in our analyses

We see that for $\hat{k}>2$ GeV the shape of the migration matrix functions
for both hadronization models are very well consistent with the expectations
from a shape function, which states that the migration function $S^{\rm
MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ should be independent of the partonic
momentum $\hat{k}$. We can spot deviations from this expectation for
$\hat{k}<2$ GeV for the default as well as for the novel dynamical
hadronization model. While for the default hadronization model the migration
function $S^{\rm MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ for $\hat{k}<2$ GeV is
considerably flatter than for $\hat{k}>2$ GeV, for the dynamic hadronization
model it is much peakier. These features play an essential role in the more
important quantitative analyses we carry out in Sec. 7.4.

Before moving on, let us comment on a general feature of the migration
functions. It concerns that $S^{\rm MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ is finite
for $\ell$ values down to $-1$ GeV. We can see this feature clearly in the
right panels of Figs. 10 and Fig. 11 for partonic values $\hat{k}>1$ GeV. Even
though the major portion of $S^{\rm MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ is
located at positiv $\ell$, which means that the bulk of hadronization
corrections shift the thrust distribution towards larger thrust values, this
negative tail shows that hadronization can sometimes also decrease the thrust
value. Since this feature arises from the tuning to actual LEP data, it means
that at least for the Herwig simulation implementation this particular feature
of the hadronization corrections is demanded by data. For $\hat{k}\geq 1.5$
GeV this behaviour arises consistently and becomes independent of the value of
$\hat{k}$ also for other $Q$ and $Q_{0}$. This feature is per se not at all
problematic and also consistent with a shape function within QCD
factorization. However, for partonic $\hat{k}$ values below $1.5$ GeV the
migration function is not capable to build up such a negative tail, simply
because the physical thrust values are restricted to be positive. This entails
that the first $\ell$-moment of the rescaled migration function $S^{\rm
MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ (see Eq. (13)) for $\hat{k}$ values below
$1.5$ GeV is always larger than for values larger than $2$ GeV, where the
first moment stabilizes. This feature is absent in the shape function $S_{\rm
had}(\ell,Q_{0})$, which is strictly $\hat{k}$-independent. We come back to
this feature in our discussion below.

###  7.4 Shower Cutoff and Energy Dependence of the Migration Matrix Function

Figure 12: Rescaled parton-to-hadron migration matrix function $S^{\rm
MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ as a function of $\ell$ for the default
hadronization model for $Q_{0}=1.25$ and a number of different partonic
$\hat{k}=\hat{\tau}/Q$ soft momenta. The upper panels show the results for
some $\hat{k}<2$ GeV and the lower panels for larger $\hat{k}$ values. The
left, middle and right panels have been extracted from the 2-jettiness
distributions for c.m. energies $Q=45$, $91.2$ and $200$ GeV, respectively.
Figure 13: Rescaled parton-to-hadron migration matrix function $S^{\rm
MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ as a function of $\ell$ for the novel
dynamical hadronization model for $Q_{0}=1.25$ and a number of different
partonic $\hat{k}=\hat{\tau}/Q$ soft momenta. The upper panels show the
results for some $\hat{k}<2$ GeV and the lower panels for larger $\hat{k}$
values. The left, middle and right panels have been extracted from the
2-jettiness distributions for c.m. energies $Q=45$, $91.2$ and $200$ GeV,
respectively.

Finally, let us now discuss at the more quantitative level the properties of
the migration matrix functions $S^{\rm MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ we
have obtained from our $Q_{0}$-dependent tuning analyses for the Herwig
default and the novel dynamical hadronization model.

In Fig. 12 we show $S^{\rm MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ obtained from the
default hadronization model over $\ell$ for different $\hat{k}$ for the
reference shower cut scale $Q_{0,\mathrm{ref}}=$1.25\text{\,}\mathrm{GeV}$$
for the hard scales $Q=45$ GeV (left panels), $91.2$ GeV (middle panels) and
$200$ GeV (right panels). In Fig. 13 the analogous results are displayed
obtained from the dynamical hadronization model. The upper panels show $S^{\rm
MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ for several small $\hat{k}$ values below $2$
GeV, which is the range where $S^{\rm MC}$ is still strongly depending on
$\hat{k}$. The lower panels show $S^{\rm MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ for
larger $\hat{k}$ values between $5$ and $8$ GeV where it is rather
$\hat{k}$-independent. We remind the reader that $\hat{k}=8$ GeV corresponds
to partonic 2-jettiness values of $\hat{\tau}=(0.178,0.088,0.040)$ for
$Q=\\{45,91.2,200\\}$ GeV, so all rescaled transfer functions which are shown
are well within the dijet region, see our discssion of Fig. 1. It is one of
the predictions of QCD factorization in the dijet region that the shape
function $S_{\rm had}(\ell,Q_{0})$ is independent of the hard scale $Q$ and
the partonic $\hat{k}$ value. As we can see in the lower panels of Figs. 12
and 13, this is indeed realized quite well for $S^{\rm
MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ obtained from both hadronization models at
large $\hat{k}$ values. For the dynamic model the shape of $S^{\rm
MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ is somewhat broader than for the default
model but for both models the results are very stable concerning the values of
$Q$ and $\hat{k}$. For small $\hat{k}$ the situation is, however, quite
different. We see that for the default model the $S^{\rm
MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ functions broaden considerably for decreasing
$\hat{k}$. This behavior is also visible in the 3D plots of Fig. 10 and
furthermore depends significantly on the c.m. energy $Q$, as can be seen most
clearly from the blue curves for $\hat{k}=0.025$ GeV in the upper panels of
Fig. 12. This feature of the default hadronization model causes very large
positive hadronization corrections for events with no partonic branching or
small partonic $\hat{\tau}$ values. Since the no-branching events still
constitute a considerable fraction of the events ($(16.1,6.2,1.9)\%$ for
$Q=(45,91.2,200)$ GeV and $Q_{0}=$1.25\text{\,}\mathrm{GeV}$$ for the Herwig’s
shower), this broadening effect affects the hadron level $\tau$ distribution
in a notable way. The important aspect of this effect is, that it is
incompatible with QCD factorization. For the dynamical model there is still a
visible dependence on $\hat{k}$, but it is substantially milder. In particular
there is no broadening for decreasing values of $\hat{k}$ and the $Q$
dependence is significantly smaller as well, which can be seen by again
comparing the blue curves in the upper panel of Fig. 13 for $\hat{k}=0.025$
GeV and the three $Q$ values. Instead, decreasing $\hat{k}$ the shape of
$S^{\rm MC}(\ell)$ in the dynamical hadronization model becomes more peaky.
This behavior is an attempt of the dynamical hadronization model to compensate
for the principle inability to describe the negative hadronization corrections
for $\hat{k}\leq 1.5$ GeV, which we already mentioned in the previous
subsection, while at the same time avoiding the QCD-incompatible broadening
effects of the default model.

Figure 14: First moment $\Omega_{1}^{\rm MC}(\hat{k},Q,Q_{0})$ of the rescaled
parton-to-hadron migration function over $\hat{k}$ for the default (left
panel) and the novel dynamical hadronization model (right panel) extracted
from the 2-jettiness distribution simulated for shower cutoff $Q_{0}=1.25$ GeV
at c.m. energies $Q=45$ GeV (blue), $91.2$ (orange) and $200$ GeV (green
dots).

To gain a more quantitative insight into the $Q_{0}$-, $\hat{k}$\- and
$Q$-dependence of the migration function $S^{\rm
MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ it is instructive to have a close look on its
first moment defined by

$\displaystyle\Omega_{1}^{\rm
MC}(\hat{k},Q,Q_{0})\,\equiv\,\frac{1}{2}\int\\!\mathrm{d}\ell\,\ell\,S^{\rm
MC}(\ell,\\{\hat{k},Q,Q_{0}\\})\,,$ (61)

in analogy to the QCD factorization’s shape function first moment
$\Omega_{1}(Q_{0})$ given in Eq. (13). The latter only depends on the shower
cut $Q_{0}$, but is independent of $\hat{k}$ or $Q$. In Fig. 14 the value of
$\Omega_{1}^{\rm MC}(\hat{k},Q,Q_{0})$ is displayed for $0<\hat{k}<3$ GeV and
$Q=45$ GeV (blue), $91.2$ GeV (orange) and $200$ GeV (green) for $Q_{0}=1.25$
GeV. The left panel shows $\Omega_{1}^{\rm MC}$ for the default hadronization
model and the right panel that for the novel dynamical model. For both we only
see a mild $Q$\- and $\hat{k}$-dependence for $\hat{k}\gtrsim 1.5$ GeV, where
the dependence is slightly smaller for the dynamical model. For small
$\hat{k}<1.5$ GeV, however, the dependence on $\hat{k}$ and $Q$ is enormous
for the default model. For the smallest $\hat{k}$ value we have
$\Omega_{1}^{\rm MC}\approx 2.1$ GeV for all $Q$ values for the dynamical
model. On the other hand, $\Omega_{1}^{\rm MC}$ varies wildly with $Q$ and
even reaches $5.5$ GeV for $Q=200$ GeV. It is this uncontrolled behavior for
the default hadronization model that is responsible for the stronger $Q_{0}$
dependence visible in Fig. 5 in comparison to Fig. 6.

Nevertheless, also for the dynamic model $\Omega_{1}^{\rm
MC}(\hat{k},Q,Q_{0})$ still exhibits a visible $\hat{k}$ dependence for
$\hat{k}<1.5$ GeV which is in principle not compatible with the shape
functions first moment. This is again related to the fact already mentioned at
the end of Sec. 3.2, namely that for the Herwig MC setup the hadronization
effects can sometimes lead to a decrease of the hadron level 2-jettiness value
$\tau$ with respect to its parton level value $\hat{\tau}$. For values
$\hat{k}\lesssim 1$ GeV this behavior cannot be realized any longer because
the physical 2-jettiness values cannot become negative. Since the
hadronization model’s capability to increase the hadron level 2-jettiness
value remains unchanged for these low $\hat{k}$ values, this feature leads to
the more peaky shape of the $S^{\rm MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ just
mentioned in the discussion of Fig. 6 above, which unavoidably leads to an
increasing first moment for $\hat{k}\lesssim 1$ GeV.

Figure 15:  First moment difference $\Omega_{1}^{\rm
MC}(\hat{k},Q,Q_{0})-\Omega_{1}^{\rm
MC}(\hat{k},Q,Q_{0,\mathrm{ref}}=$1.25\text{\,}\mathrm{GeV}$)$ of the rescaled
parton-to-hadron migration matrix function over $\hat{\tau}=\hat{k}/Q$ for the
default (upper panel) and the novel dynamical hadronization model (lower
panels) extracted from the 2-jettiness distribution simulated for shower
cutoff $Q_{0}=1$ (blue), $1.25$ (orange), $1.5$ (green) and $1.75$ GeV (red)
at c.m. energies $Q=45$ GeV (left), $91.2$ (middle) and $200$ GeV (right
panels). The dashed lines show the results expected from QCD factorization.

Finally, let us examine the dependence of $\Omega_{1}^{\rm
MC}(\hat{k},Q,Q_{0})$ on the shower cutoff $Q_{0}$, which we have not yet
covered in our analysis so far. Given the previous observations concerning the
$\hat{k}$ dependence of $\Omega_{1}^{\rm MC}$ for $\hat{k}\lesssim 1.5$ GeV,
the consistency of $\Omega_{1}^{\rm MC}(\hat{k},Q,Q_{0})$ for the dynamic
model with the $Q_{0}$-dependence of the shape function’s first moment
$\Omega_{1}$ in Eq. (13) should at least be realized for $\hat{k}>1$ GeV,
which corresponds to $\hat{\tau}>(0.022,0.011,0.005)$ for $Q=(45,91.2,200)$
GeV. In Fig. 15 the values for $\Omega_{1}^{\rm
MC}(\hat{k},Q,Q_{0})-\Omega_{1}^{\rm
MC}(\hat{k},Q,Q_{0,\mathrm{ref}}=$1.25\text{\,}\mathrm{GeV}$)$ are displayed
for $0<\hat{\tau}=\hat{k}/Q<0.15$ GeV for $Q=45$ GeV (left panels), $91.2$ GeV
(middle panels) and $Q=200$ GeV (right panels), where we have averaged the
corresponding moment results in bins of width $\Delta\hat{\tau}=0.01$ GeV. The
upper panels show the results for the default hadronization model and the
lower panels for the novel dynamical model. The results for the moment
difference are shown for $Q_{0}=1.0$ (blue), $1.25$ (orange), $1.5$ (green)
and $1.75$ (red). The horizontal colored dashed lines represent the value of
the corresponding shape function moment differences from Eq. (14) as expected
from QCD factorization:

$\displaystyle\Omega_{1}(Q_{0})-\Omega_{1}(Q_{0,\mathrm{ref}})\,=\,\frac{1}{2}\,\Delta_{\rm
soft}(Q_{0},Q_{0,\mathrm{ref}})\,,$ (62)

where $\Delta_{\rm soft}(Q_{0},Q_{0,\mathrm{ref}})$ is determined from the
$R$-evolution equation in Eq. (10).

We see that for $\hat{k}>1$ GeV the moment difference obtained for the
dynamical model (lower panels) are indeed nicely compatible with Eq. (62). The
visible small discrepancies are related to quadratic and higher order effects
in $Q_{0}$ which the linear approximation of the evolution equation does not
capture. They are also compatible with the corresponding discrepancies
concerning the $Q_{0}$ dependence of the partonic cumulant difference already
discussed in Sec. 3.3 for Fig. 1. In stark contrast, for the default model
results (upper panels) there is no sign of a similar agreement with Eq. (62)
or of any stability with respect to the value of $\hat{\tau}$. We see that for
the default model the moment differences are completely unrelated to the
shower cut dependence expected from QCD factorization, and thus essentially
uncontrolled from the QCD perspective. We stress, that this uncontrolled
$Q_{0}$-dependence of the hadronization corrections from the default
hadronization model takes place even for large $\hat{k}$ values where the
migration function $S^{\rm MC}(\ell,\\{\hat{k},Q,Q_{0}\\})$ appears stable
(also with respect to changes of $Q$) as shown in the lower panels of Fig. 12.
There is furthermore no stability concerning the $\hat{k}$ dependence of
$\Omega_{1}^{\rm MC}(\hat{k},Q,Q_{0})-\Omega_{1}^{\rm
MC}(\hat{k},Q,Q_{0,\mathrm{ref}})$ with respect the hard scattering scale $Q$.

It is instructive to discuss this failure of the default model from a the
perspective of overall size of the hadronization corrections. As we can see
from Fig. 14, for $Q_{0,\mathrm{ref}}=1.25$ we have $\Omega_{1}^{\rm MC}$ in
the range between $1.2$ and $1.5$ GeV for partonic momenta $\hat{k}>2$ GeV.
This should be compared to the overall variation of $\Omega_{1}^{\rm MC}$ in
the range between $Q_{0}=1.0$ GeV and $Q_{0}=1.75$ GeV predicted by
$R$-evolution equation (10) obtained from QCD factorization. The latter
amounts to around $0.5$ GeV, as we see in the lower panels of Fig. 15.
Howerver, because the choice of reference shower cut $Q_{0,\mathrm{ref}}$ is
arbitrary, and we do not have any principle means to tell for which value of
$Q_{0}$ the default model may have the best agreement with the expectations
from QCD factorization (which can only be tested by a $Q_{0}$-dependence
consistent with QCD factorization), this implies that the size of the
hadronization corrections to the (true) partonic thrust distribution that is
provided by the default model is inconsistent with QCD factorization in the
range of $40$%. This is substantially worse in comparison to the new default
model, where we have consistency at the level of better than $10$%. In the
context of the shower cut $Q_{0}$ adopting the role of an IR factorization
scale, so that the hadronization effects have a well-defined and controlled
scheme dependence so that they can be assigned field theoretic meaning, the
default model thus fails quite badly. In this respect the novel dynamical
hadronization model performs substantially better.

Overall, the new dynamic hadronization model provides a significant
improvement concerning the control of the shower cut $Q_{0}$ as an IR
factorization scale. This feature is a prerequisite to combine the
hadronization corrections implemented and quantified in the hadronization
model with parton level theoretical calculations in a meaningful and
systematic manner.

##  8 Conclusions and Outlook

In this article we further promote the idea of the parton shower cutoff
$Q_{0}$ for MC simulations being an infrared factorization scale that
separates, in a controlled manner and compatible with QCD, perturbative and
non-perturbative hadronization effects. In this context, features of the MC’s
hadronization model may be given a systematic field theoretic meaning and, at
the same time, QCD parameters appearing in the parton shower may be related to
their renormalization scheme dependent counter parts appearing in analytic QCD
computations. An important prerequisite to achieve this is to have a parton
shower algorithms that has at least NLL precision for the observable
considered. However, the second essential and equally important prerequisite
is to have a hadronization model that can properly match the unavoidable
infrared cutoff $Q_{0}$-dependence of the parton level description that
emerges from the parton shower. This is a highly nontrivial condition on the
hadronization model since it entails that the parton shower cutoff $Q_{0}$ is
not treated as a tuned hadronization parameter. Rather, the hadron level MC
observable description should be equivalent for different $Q_{0}$ values, at
least within some low energy interval, where QCD perturbation theory can still
be trusted.

In this work we have investigated these features using the angular ordered
parton shower and the cluster hadronization model implemented in the Herwig
7.2 MC event generator focusing primarily on the 2-jettiness distribution in
$e^{+}e^{-}$ annihilation for which the angular ordered parton shower is NLL
precise. In the earlier work Hoang:2018zrp  some of us have analyzed the gluon
transverse momentum cutoff $Q_{0}$-dependence that emerges from the angular
order parton shower for the 2-jettiness distribution in detail at the
subleading ${\cal O}(\alpha_{s})$ level using QCD factorization in the dijet
region. It was also shown that the Herwig parton shower implementation
satisfies the $Q_{0}$ evolution equation from QCD factorization very well. In
this work we have now extended our investigations concerning the parton shower
infrared cutoff being a factorization scale from the perspective of the Herwig
cluster hadronization model.

The default cluster hadronization model has not been designed in a way so that
it systematically matches to the parton shower for a range of $Q_{0}$ values.
This is related to a number of features in the cluster formation and fission
dynamics that yield good data description through the tuning, but are
otherwise ad-hoc and not compatible with the processes that happen in the
parton shower evolution close to the cutoff $Q_{0}$. As a result, the
hadronization effects to the 2-jettiness distribution provided by the default
Herwig cluster model do not satisfy the QCD constraints that emerge from
$Q_{0}$ being promoted to a factorization scale in a satisfactory manner. We
have demonstrated this in a number of tests based on tuning analyses for
different $Q_{0}$ shower cutoff values where $Q_{0}$ is treated as an external
scale and not a tuned parameter. These tests involve analyses of the
$Q_{0}$-dependence of hadron level simulations for 2-jettiness and also other
event-shapes and observables and of the parton-to-hadron migration matrix for
2-jettiness for which QCD factorization provides nontrivial constraints.

To improve the cluster hadronization model we have added a number of features
to the cluster formation and cluster fission processes that mimic the gluon
emission and splitting dynamics that takes place in the parton shower. These
modifications provide a clearer separation of model parameters that are
expected to be correlated to $Q_{0}$, governing the ’hard’ aspects of the
cluster hadronization dynamics, from those which govern ’soft’ hadron
formation aspects that should be rather $Q_{0}$-independent. In our
$Q_{0}$-dependent tuning studies we found that this novel dynamical cluster
hadronization model performs substantially better concerning the
$Q_{0}$-invariance of Herwig’s hadron level predictions as well as concerning
the QCD factorization constraints on the 2-jettiness parton-to-hadron
migration matrix. We emphasize that our analyses also involved hard scattering
energies that are not accounted for in the reference data used for the tuning.
This shows that the novel dynamical hadronization model properly scales the
consistency to QCD factorization to other hard scattering energies.

Even though the novel dynamical hadronization model we have designed in this
article is not perfect, its features provide an important step forward in
promoting the hadronization corrections encoded in MC generators to have a
well-defined scheme in the QCD context, as also discussed previously in Ref.
Platzer:2022jny . This is an essential aspect that should be followed in
parallel to the ongoing developments of subleading order precise parton shower
algorithms such that the parameters of the parton shower as well as the
hadronization model can acquire a systematic QCD field theoretic meaning.
Beyond a more precise and consistent description of experimental data,
important potential applications of such improvements are the determination of
QCD parameters directly from MC studies as well as a systematic quantification
of hadronization corrections to analytic QCD calculations from MC simulations.
In an upcoming article we will investigate the former application from the
perspective of the MC top quark mass parameter.

## Acknowledgments

We acknowledge support by the FWF Austrian Science Fund under the Project No.
P32383-N27 and under the FWF Austrian Science Fund Doctoral Program “Particles
and Interactions” No. W1252-N27 as well as partial support by the COST Action
No. CA16201 Particleface. This work has also been supported in part by 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). We are grateful the Erwin-Schrödinger International Institute for
Mathematics and Physics for partial support during the Thematic Programme
“Quantum Field Theory at the Frontiers of the Strong Interactions”, July 31 -
September 1, 2023.

##  Appendix A Comparison of Simulations with LEP Data

Figure 16: Selected rivet analyses for $e^{+}e^{-}$ data (black) versus Herwig
simulations for the default hadronization model for shower cutoff values
$Q_{0}=1$ (blue), $1.25$ (orange), $1.5$ (green) and $1.75$ GeV (red). The
ratios in the lower panel sections are shown w.r. to the simulations for
$Q_{0}=1.25$ GeV.

Figure 17: Selected rivet analyses for $e^{+}e^{-}$ data (black) versus Herwig
simulations for the novel dynamical hadronization model for shower cutoff
values $Q_{0}=1$ (blue), $1.25$ (orange), $1.5$ (green) and $1.75$ GeV (red).
The ratios in the lower panel sections are shown w.r. to the simulations for
$Q_{0}=1.25$ GeV.

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