# Photoproduction in general-purpose event generators

I<EMAIL_ADDRESS>University of Jyvaskyla, Department of
Physics, P.O. Box 35, FI-40014 University of Jyvaskyla, Finland  Helsinki
Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland
P<EMAIL_ADDRESS>S. Plätzer444simon.plaetzer@uni-
graz.at Institut für Physik, NAWI, Universität Graz, Universitätsplatz 5, 8010
Graz, AT  Teilchenphysik, Fakultät für Physik, Universität Wien,
Boltzmanngasse 5, 1030 Wien, AT  P<EMAIL_ADDRESS>Institute for Particle Physics Phenomenology, Durham University, Durham DH1
3LE, UK

(12 June 2024)

###### Abstract

We present a comparison of three different general-purpose Monte Carlo event
generators, H  ERWIG , P  YTHIA , and S  HERPA , with respect to the
simulation of photoproduction. We outline the default inputs, implementation
differences and compare the results at different stages of the event
generation. We find that, despite a similar starting point, the final cross
sections do have some differences related to different non-perturbative
inputs. We compare the simulations with experimental data for jet production
in LEP and HERA and find that all generators provide a decent desription of
the data within the considered uncertainties. We also present predictions for
the upcoming EIC for jet observables and event shapes and conclude that
accurate simulations will require further phenomenological advances.

##  1 Introduction

General-purpose Monte Carlo (MC) event generators [1] provide computational
tools for complete simulations of high-energy collisions with different beam
configurations. They model broad ranges of physics and energy scales starting
from a hard partonic scattering down to hadronization and decays of unstable
hadrons. Thus these generators can be applied to quantify what kind of
particle distributions a specific partonic scattering would result in a
particle detector and use to connect a theoretical first-principle calculation
to an experimental measurement. Accurate and realistic simulations are crucial
for planning and performing experimental analyses with a prime example being
the detector collaborations at the LHC applying these programs virtually in
all analyses.

The main players in the field of general-purpose event generators are H  ERWIG
[2, 3], P  YTHIA [4] and S  HERPA [5, 6], each including a long history of
development over several decades. The main theoretical framework that the
generators built upon consists of the collinear factorization and perturbative
Quantum Chromodynamics (pQCD) [7] that can be used to calculate differential
cross section of varying perturbative order for relevant partonic scatterings.
The radiation of the initial and final partons participating to the hard
scattering can be modelled using DGLAP evolution equations [8, 9, 10, 11] and
parton-showers algorithms [12] that turn the inclusive evolution equations
into exclusive emission probabilities using Sudakov factors. In case of beam
particles with partonic structure a complete partonic final state require also
modelling multiparton interactions (MPIs) [13], that generates the underlying
event present in addition to hard scattering of interest, and beam remnants to
account for conservation of energy and quantum numbers. Finally the partons
carrying colour charge are turned into colour-neutral particles applying non-
pertubative hadronization [14, 15] and colour reconnection models [16, 17].

The development of the current event generators have mainly been driven by the
studies of proton-proton (pp) collisions at the LHC. The provided luminosities
have enabled measurements to reach accuracies of a few percent-level,
necessitating persisting theoretical computations of the differential cross-
sections to matching or higher precision. As an example of the latter,
matching parton shower consistently with matrix-element computations at next-
to-leading order (NLO) [18, 19] have been developed and implemented in all
major event generators and for some particluar cases even beyond NLO [20, 21,
22]. Another avenue where recent progress has been made is to extend the
modern generators to handle other beam configurations including heavy-ion
collisions [23] and electron-proton (ep) collisions [24, 21, 25, 26, 27]. For
the latter the main motivation is provided by the electron-ion collider (E  IC
) that is the next large high-energy collider project pushed forward at BNL in
US [28]. The ep collisions can be classified in terms of $Q^{2}$, the
virtuality of intermediate photon, which is related to the kinematics of the
scattered electron. In deep inelastic scattering processes (DIS) the exchanged
photon is considered as a virtual particle scattering with a single quark
inside the proton. In the photoproduction limit, $Q^{2}\rightarrow 0$, the
photon may, however, fluctuate into a hadronic state where the underlying hard
scattering takes place between two coloured partons and are thus similar to a
collision between two hadrons. In fact this resolved component dominates the
total cross section of photon-proton ($\gamma$p) system and has to be
accounted for in addition to contributions of a direct photon scattering.
Similarly one can study photon-photon ($\gamma\gamma$) collisions in electron-
positron (e+e-) annihilation.

In this study we present comparisons of the three general purpose event
generators, P  YTHIA , H  ERWIG and S  HERPA , with respect to
photoproduction. In Sec. 2 we present the frameworks for each of the
generators and discuss their differences. In Sec. 3 and 4 we present cross
sections for dijet photoproduction and compare the results with the L  EP and
H  ERA data. We also show predictions for dijet photoproduction at kinematics
relevant to the EIC including comparisons to event-shape and inclusive QCD
observables in Sec. 5. We conclude our findings and provide a brief outlook in
Sec. 6.

##  2 Event Generators

For photons resolved into a hadronic state, the partonic structure can be
described with DGLAP-evolved parton distribution functions (PDFs),
$f^{\,\gamma}_{i}(x_{\gamma},\mu_{\mathrm{F}}^{2})$, where $x_{\gamma}$ is the
momentum fraction of a given parton wrt. the photon and $\mu_{\mathrm{F}}^{2}$
is the factorization scale at which the PDFs are evaluated at. Similarly as
the PDFs for protons,
$f^{\,\mathrm{p}}_{i}(x_{\mathrm{p}},\mu_{\mathrm{F}}^{2})$, the photon PDFs
are obtained in a global analysis where the parameters of a non-perturbative
input are fitted into experimental data. In addition to a fitted hadron-like
part, the photon PDFs contain also a point-like (anomalous) contribution from
perturbative $\gamma\rightarrow q\overline{q}$ splittings that has to be
included in the evolution equations when comparing to experimental data. The
structure of the resolved photons were an active topic around 20 years ago
when LEP was producing experimental data for such analyses but, since then,
activities have been sparse and the available fits are lacking the recent
methodological developments, e.g. for the uncertainty estimation.

Knowing the structure of the incoming states, one can factorize hard-process
generation for photoproduction in ep collisions as

$\mathrm{d}\sigma^{\mathrm{ep}}_{\mathrm{hard}}=f^{\,\mathrm{e}}_{\gamma}(x,Q^{2})\otimes
f_{i}^{\gamma}(x_{\gamma},\mu_{\mathrm{F}}^{2})\otimes
f_{j}^{\mathrm{p}}(x_{\mathrm{p}},\mu_{\mathrm{F}}^{2})\otimes\mathrm{d}\sigma^{ij}_{\mathrm{hard}},$
(2.1)

where $f^{\,\mathrm{e}}_{\gamma}(x,Q^{2})$ is the photon flux from an electron
at a given momentum fraction $x$ and photon virtuality $Q^{2}$, and
$\sigma^{ij}_{\mathrm{hard}}$ perturbatively calculable hard coefficient
function for a given hard process with initiating particles $i$ and $j$. In
case of direct-photon scattering the photon will act as an initiating particle
and there one can replace the $f_{i}^{\gamma}$ with a delta function. Thus,
there are two contributions that should be accounted: direct and resolved. At
leading order (LO) these contributions can be taken as separate but at higher
orders one has to make sure that no double counting takes place when combining
these two event classes. The photon flux can be computed using equivalent
photon approximation (EPA) [29]. For photons from a charged lepton $l$ it
gives in the leading-log (LL) approximation

$f^{\,l}_{\gamma}(x)=\frac{\alpha_{\mathrm{em}}}{2\pi}\frac{1+(1-x)^{2}}{x}\log\left[\frac{Q^{2}_{\mathrm{max}}}{Q^{2}_{\mathrm{min}}(x,m_{l})}\right].$
(2.2)

The lower limit for the virtuality follows from kinematics and the upper limit
is adjusted according to the experimental configuration.

In case of $\gamma\gamma$ collisions in e+e- the corresponding factorization
reads

$\mathrm{d}\sigma^{\mathrm{ee}}_{\mathrm{hard}}=f^{\,\mathrm{e}}_{\gamma}(x_{1},Q_{1}^{2})\otimes
f_{i}^{\gamma}(x^{+}_{\gamma},\mu_{\mathrm{F}}^{2})\otimes
f^{\,\mathrm{e}}_{\gamma}(x_{2},Q_{2}^{2})\otimes
f_{j}^{\gamma}(x^{-}_{\gamma},\mu_{\mathrm{F}}^{2})\otimes\mathrm{d}\sigma^{ij}_{\mathrm{hard}}.$
(2.3)

In this case there are four different contributions, direct-direct, direct-
resolved, resolved-direct and resolved-resolved that need to be accounted for.
This factorized approach provides the starting point for all considered event
generators over which further modelling of parton showers, beam remnants, MPIs
and hadronization are included. Thus, even though the starting point is the
same for all generators, the results will depend on the applied inputs and
particular choices made in this modelling. Before comparing the different
event generators to data, we will therefore describe the main features of the
generators and the default inputs, and highlight some significant differences
in the following subsections.

###  2.1 H  ERWIG

H  ERWIG [2, 3, 30] is a multi-purpose event generator, which has
traditionally been focusing on accurate QCD simulations. It includes two
parton shower modules, an angular ordered one based on coherent branching
[31], and more recently a dipole parton shower [32, 33]. Both parton shower
modules include full mass effects, spin correlations, and initial and final
state radiation [34, 35]. Hard processes can be simulated using a wide range
of built-in matrix elements at leading and next to leading order, and via the
Matchbox module [33, 3] using external matrix element providers and an
automated matching to NLO QCD either within the MC@NLO or POWHEG approaches.
Multi-jet merging [36, 37] is available using the dipole shower.
Photoproduction processes can be simulated in $e^{+}e^{-}$, $ep$ and $pp$
collisions including all direct and resolved components. The PDF sets and
fluxes we currently provide are the SaS photon PDFs [38] with the 1D, 1M, 2D
and 2M schemes, and photon fluxes following a Weizsäcker-Williams
parametrization for electrons, and the Budnev approach [39] for protons.
Hadronization is modelled using the cluster hadronization model [2], and we
provide an eikonal MPI model, which currently however is limited to $pp$
collisions and unfortunately cannot yet treat resolved photons due to
technical issues. Similar restrictions currently apply to NLO matching using
Matchbox, though both of these issues are currently being worked on and should
be available with one of the next H  ERWIG releases. In order to assess
uncertainties, we can vary the scales of the hard process as well as the hard
veto scales of the parton showers, which we perform in this study,
additionally to varying the photon PDF sets. All of these variations turn out
not to be significant, with the PDF set variation contributing the biggest
uncertainty with similar findings as reported in . The H  ERWIG parton showers
always need to terminate on a valence quark of the incoming parton, and thus
create a remnant which complements the parton which had been extracted. In
case of the photon we will therefore always generate an anti-quark remnant,
which might cause additional (collinear) shower activity even if radiation in
the showers is switched off entirely.

###  2.2 P  YTHIA

Simulations for photoproduction in ep and $\gamma\gamma$ collisions in e+e-
have been fully enabled from release 8.226 onwards and the current setup is
described in the 8.3 manual (Ref. [4]). Both hard and soft QCD processes can
be generated, the former including processes like inclusive and diffractive
jet production, and the latter non-diffractive and diffractive events and also
elastic scattering. The implementation of resolved processes is based on CJKL
photon PDFs [40] which includes hadron-like part with non-perturbative input
separately from the perturbatively calculated anomalous contribution. In the
generation of resolved processes these are not considered separately but the
parton-shower algorithm includes the $\gamma\rightarrow q\overline{q}$
splitting that may collapse the resolved photon into an unresolved state at a
perturbative scale corresponding to the anomalous part of the evolution
equation.

The simulations include generation of MPIs as long as the photons are in a
resolved state and the MPI framework, where the QCD cross sections are
regulated with the screening parameter $p_{\mathrm{T,0}}$, can be applied also
for non-diffractive soft QCD processes without any phase-space cuts. The cross
section for the different soft processes are obtained from the SaS
parametrizations in Ref. [38]. For proton PDFs the current default
NNPDF23_lo_as_0130_qed has been applied and the value of $\alpha_{\mathrm{S}}$
has been fixed accordingly to 0.130 at
$\mu_{\mathrm{R}}^{2}=M_{\mathrm{Z}}^{2}$. Here we have used P  YTHIA version
8.310 released in July 2023.

In this study we have applied the default setup for photoproduction in P
YTHIA 8.3. In case of $\gamma\gamma$ a specific tune for $p_{\mathrm{T,0}}$
has been applied based on single-inclusive charged-particle production data
from OPAL at LEP [41] and in case $\gamma$p the
$p_{\mathrm{T,0}}^{\mathrm{ref}}$ in the standard parametrization has been
adjusted to 3.0 GeV based on comparisons to single-inclusive charged particle
production in photoproduction at HERA [42] which is in line also with the
multiplicity distributions measured recently by ZEUS in Ref. [43]. Heavy quark
masses are included apart from the $gQ\rightarrow gQ$ subprocess where
massless expressions for the matrix element are applied also for charm and
bottom quarks. The remnants are constructed by adding a minimal number of
partons such that momentum and colour are conserved. In case of resolved
photons the flavour for “valence” quark-antiquark pair is sampled according to
relative fractions of the PDFs at the specific kinematics and part of the
phase-space is cut out to allow room for the remnant partons. The scale
variations have been performed by varying renormalization and factorization
scales independently in the hard process generation by a factor of two,
excluding the variations where the relative difference would be 4.

###  2.3 S  HERPA

In the S  HERPA event generator [5, 6], photoproduction of jets at M  C @N  LO
accuracy has been recently implemented and validated against data [26] and
will be published under version 3.0. Resolved processes can be calculated
through interfaces to various different photon PDFs, where the default is the
SaS PDF library [38].

We generated events at Leading Order and M  C @N  LO accuracy as described in
[26]. We used A  MEGIC [44] and C  OMIX [45] for tree-level matrix elements
and subtraction terms [46], and O  PEN L  OOPS [47] for one-loop matrix
elements. For the calculation of the matrix element, we treated the $b$-quarks
for Z  EUS , and additionally the $c$-quark for O  PAL and E  IC runs, as
massive and included them in the final state. While at LO all subprocesses are
available, the initial state subtraction for massive quarks has not been
implemented yet. The cssparton shower [48] was used, combined with the M  C @N
LO method [18, 49] as implemented in S  HERPA [50].

Multiple-parton interactions were modelled through an implementation of the
Sjostrand-van Zijl model [51, 52] in which the total hadronic cross-section is
calculated using Regge theory and the parametrizations for the photon are
obtained as a superposition of light neutral vector mesons as proposed and
parametrised in [52]. The tuning of the rewritten MPI modelling is currently
work-in-progress as part of the new release. Particles were hadronized through
the cluster fragmentation model in A  HADIC [53] and has already been tuned
against data. The photon flux was modelled as computed in [54], which includes
a correction for $x\to 1$ relevant for lepton-hadron colliders. For the PDFs,
we used the SAS2M [55, 56] set for the photon and the PDF4LHC21_40_pdfas [57]
set for the proton, and the value of $\alpha_{\mathrm{S}}$ was kept at its
default value of 0.118 with three-loop running. Factorisation and
renormalisation scale were set to
$\mu_{\mathrm{F}}=\mu_{\mathrm{R}}=H_{\mathrm{T}}/2$ and a 7-point scale
variation in the matrix element and the shower was done as an uncertainty
estimation.

###  2.4 Differences between generators

As a baseline for a broader comparison between the generators, we compared
events of L  EP -like setups with beam energies of 99 GeV from P  YTHIA and S
HERPA at the bare cross-section-level for each of the components. Here we used
the same photon flux, PDF, and a setting of $\alpha_{S}=0.130$ with 1-loop
running and computed only processes of light partons. In Fig. 1 we can see
that indeed the results are, up to statistical fluctuations, in a perfect
agreement. The considered observables include the energy and pseudorapidity
distribution, and transverse momentum spectra of the outgoing partons for
direct, single-resolved and double resolved contributions separately. In case
of single-resolved contribution the direct photon has a positive $p_{z}$.

In Tab. 1, we summarise the differences between the generators with respect to
the simulation of photoproduction. Other differences are for example the
fragmentation model and QED corrections, however, these will not be discussed
here and can be found in the manuals [2, 58, 4].

Property | P  YTHIA | S  HERPA | H  ERWIG
---|---|---|---
Flux | LL | NLL | LL
$\alpha_{S}(M_{\mathrm{Z}}^{2})$ | 0.130, 1-loop running | 0.118, 3-loop running | 0.118, 2-loop running (hard process)
PDFs | CJKL | SAS2M | SAS2M
Remnants | forced splittings/PS rejection | PS rejection | forced splitting
Photon Splitting | yes | no | no
MPI tuning | preliminary $\gamma\mathrm{p}$/$\gamma\gamma$ tune | untuned | untuned
Table 1: Differences between the generators of default settings, specifically for photoproduction. |  |
---|---|---
Figure 1: Comparison of fixed-order parton level events between P  YTHIA and S
HERPA for a L  EP -like setup.

Based on this, we changed the beam configuration on the previous setup to an E
IC -like setup with electron-proton beams with 18 and 275 GeV, and looked at
photoproduction events with resolved photons. We used the same settings as
before, however, now also modelling the Initial State Radiation (ISR) and the
beam remnants, including charm and bottom quarks in the processes and using
the NNPDF23_lo_as_0130_qed PDF for the proton. The total cross-section in P
YTHIA is larger than in S  HERPA due to the way the remnants are created: in P
YTHIA , forced splittings are combined with phase space rejection to ensure
the correct creation of the valence quark content of the beam particle, which
allows for an iterative procedure. In S  HERPA , the remnant creation is
purely based on phase space rejection which naturally will lead to more events
being rejected, especially for charm and bottom quarks. In the subsequent
modelling of the beam breakup, we can see that S  HERPA creates more activity
in the low-$E_{T}$ and low-$p_{T}$ region, whereas P  YTHIA tends to have
higher transverse momenta and energies in its particle spectrum.

|  |
---|---|---
Figure 2: Comparison of fixed-order parton level events including ISR parton
shower and beam remnants between P  YTHIA and S  HERPA for a E  IC -like
setup.

Coming back to the modelling of the perturbative part, the event generators
model the flux to different accuracies where the difference is given by

$f_{\mathrm{NLL}}(x)-f_{\mathrm{LL}}(x)=-\frac{\alpha_{\mathrm{em}}}{\pi}m_{e}^{2}x\left(\frac{1}{Q^{2}_{\mathrm{min}}}-\frac{1}{Q^{2}_{\mathrm{max}}}\right)\
.$ (2.4)

The correction comes with an overall negative sign and hence will lead to
smaller total cross-sections. In differential observables, the correction will
however only be sizeable in regions where $x\to 1$. Even though not specific
to photoproduction we point out that the settings of $\alpha_{\mathrm{S}}$ are
different: while in S  HERPA the current default is set according to the
default proton PDF set and close to the PDG world average [59], in P  YTHIA it
is being used as a dynamical $K$ factor, also in accordance with its default
proton PDF set. Photon PDFs do not allow the nowadays standard procedure of
setting $\alpha_{\mathrm{S}}$ in accordance with the fit as this information
is often not given in the corresponding publication and would in any case be
in conflict with settings of modern proton PDFs. Hence, strictly speaking the
factorisation of the cross section is not fully consistent due to the
different values of $\alpha_{\mathrm{S}}$ used throughout event generation.

As a last point, we plot the parton distributions of the light partons for the
two PDF libraries SAS2M and CJKL in Fig. 3. The two fits come to vastly
different behaviour at small $x$, where the SAS2M sets reach an almost
constant value and the CJKL sets behave like a power-law, with the most
pronounced difference in the gluon distribution.

It is also worthy to point out that unlike the settings for $\alpha_{S}$ and
the flux, which cover the perturbative region and are independent of any
fitting, the remnant creation and the MPIs interplay with non-perturbative
effects and rely on precise data by experiments like H  ERA and L  EP .

Figure 3: Comparison of the CJKLLO and SAS2M PDF sets for light quarks and the
gluon at $\mu_{\mathrm{F}}^{2}=5\ \mathrm{GeV}^{2}$.

##  3 Comparisons to L  EP

For the comparison to L  EP data, we used a dijet measurement from the O  PAL
collaboration [60]. At lepton colliders, the cross-section for photoproduction
can be separated into $\sigma=\sigma_{\gamma\gamma\to X}+\sigma_{\gamma j\to
X}+\sigma_{j\gamma\to X}+\sigma_{jj\to X}$, where $j$ denotes a photon being
resolved into partons. The analysis used data taken at $\sqrt{s}=198$ GeV and
clustered jets with the $k_{T}$ algorithm with $R=1$ demanding $E_{T}>3$ GeV
and $|\eta|<2$ with at least two jets. To separate resolved from direct
photoproduction processes, the analysis defined

$x_{\gamma}^{\pm}=\frac{\sum_{j=1,2}E^{(j)}\pm
p_{z}^{(j)}}{\sum_{i\in\mathrm{hfs}}E^{(i)}\pm p_{z}^{(i)}}$ (3.1)

and associated values $x_{\gamma}^{\pm}<0.75$ with resolved processes. For a
parton-level $2\rightarrow 2$ scattering these definitions would match with
the momentum fractions in photons going positive and negative $p_{z}$ in Eq.
2.3 but adding parton showers, MPIs and hadronization will smear the
kinematics such that the division will be only approximative. In the numerator
the sum runs over the two jets with the highest transverse momentum,
$p_{\mathrm{T}}$, and in the denominator over the complete hadronic final
state. In Fig. 4 we present the average transverse energy of the di-jets for
different taggings on $x_{\gamma}$. The LO prediction from S  HERPA
undershoots the total cross-section, however, the effect is more pronounced
for resolved photons than for unresolved ones. Going to NLO accuracy, the
simulation describes the data well within the errors. The large $K$ factors
hint at the real corrections and the filled-up phase space as the drivers of
the improved description. P  YTHIA tends to overshoot the data for events with
direct contribution but agree well with the data for the resolved-resolved
case. The H  ERWIG simulation typically lies between the S  HERPA and P  YTHIA
results. In all cases the data is within the large uncertainties from scale
variations. As discussed in Sect. 2.4, the large difference between the LO
results from H  ERWIG , P  YTHIA and S  HERPA builds up from varying inputs.

|  |
---|---|---
Figure 4: Distribution of average transverse energy of di-jets, $\bar{E}_{T}$,
from O  PAL [60] for all $x^{\pm}_{\gamma}$ (left), $x^{+}_{\gamma}$ or
$x^{-}_{\gamma}<0.75$ (middle) and $x^{\pm}_{\gamma}<0.75$ (right), compared
to Leading Order simulations by H  ERWIG , P  YTHIA and S  HERPA and M  C @N
LO -accurate simulations by S  HERPA .

The distributions in pseudo-rapidity $\eta$ in Fig. 5 show a similar picture,
however for double-resolved processes, i.e. $x^{\pm}_{\gamma}<0.75$, all
predictions have a slightly steeper fall-off as a function of $\eta$ than seen
in the data, leaning towards an undershoot in the forward region. One
potential reason could be the weakly constrained gluon-content of the photon,
which should be the leading contribution at low-$E_{T}$ and forward jets.

|
---|---
Figure 5: Distribution of pseudo-rapidity $\eta$ of di-jets from O  PAL [60]
for $x^{+}_{\gamma}$ or $x^{-}_{\gamma}<0.75$ (left) and
$x^{\pm}_{\gamma}<0.75$ (right), compared to Leading Order simulations by H
ERWIG , P  YTHIA and S  HERPA and M  C @N  LO -accurate simulations by S
HERPA .

Fig. 6 shows distributions of $x_{\gamma}$ for low and high average jet
transverse energies $\bar{E}_{T}$, respectively. We see good agreement for
both the P  YTHIA and the S  HERPA -M  C @N  LO simulations; however, the
transition from the resolved to the direct processes seems to be poorly
modelled, as can be seen by the consistent undershoot at around
$x_{\gamma}\approx 0.8$ for all predictions. Unlike H  ERWIG and S  HERPA ,
the P  YTHIA simulation does include the correct evolution of the photon PDF,
i.e. the photon splitting $\gamma\to q\bar{q}$ is taken into account, the
distribution still shows this shape in that case too, though. Potential
reasons could be the poor quality of the photon PDF or insufficient tuning of
the fragmentation and multiple-parton interactions modelling. Combined with
the overshoot in the largest-$x$ bin, we would expect these effects to
increase the multiplicity of direct processes, hence shifting cross-section
towards lower values of $x$.

|
---|---
Figure 6: Distribution of $x_{\gamma}$ of di-jets from O  PAL [60] for bins of
average jet transverse energy $\bar{E}_{T}\in[5\,{\rm GeV},7\,{\rm GeV}]$
(left) and $\bar{E}_{T}\in[11\,{\rm GeV},25\,{\rm GeV}]$ (right), compared to
Leading Order simulations by H  ERWIG , P  YTHIA and S  HERPA and M  C @N  LO
-accurate simulations by S  HERPA .

##  4 Comparisons to H  ERA

For this comparison we used data taken by the Z  EUS collaboration [61] during
H  ERA Run 1. Here, the photoproduction cross-section can be decomposed into
two parts, $\sigma=\sigma_{\gamma P\to X}+\sigma_{jP\to X}$, where again $j$
denotes a parton resolved from within the photon. Similar to the previous
analysis, jets were clustered with the $k_{T}$ algorithm with $R=1$ with cuts
of $\eta\in[-1,2.4]$ and $E_{T}>14\ (11)$ GeV for the (sub-)leading jet,
respectively. The analysis used a similar discriminant for the tagging of
direct and resolved processes, defined as

$x_{\gamma}^{\mathrm{obs}}=\frac{\sum_{j=1,2}E_{\mathrm{T}}^{(j)}\mathrm{e}^{-\eta^{(j)}}}{2yE_{e}}\
,$ (4.1)

where the sum runs over two highest $E_{\mathrm{T}}$ jets, $E_{e}$ is the
energy of the incoming electron and $y$ the inelasticity.

|
---|---
Figure 7: Distributions of leading jet transverse energy
$E_{T}^{\mathrm{jet1}}$ for $x_{\gamma}>0.75$ (left) and $x_{\gamma}<0.75$
(right) with jet pseudo-rapdities for the two leading jets in
$1<\eta^{\mathrm{jet1}}<2.4$ and $0<\eta^{\mathrm{jet2}}<1$ from Z  EUS [61],
compared to Leading Order simulations by H  ERWIG , P  YTHIA and S  HERPA and
M  C @N  LO -accurate simulations by S  HERPA .

In Fig. 7 we show the leading jet transverse energy $E_{T}^{\mathrm{jet1}}$
for direct and resolved processes. While the S  HERPA -LO prediction stays
below the data, the P  YTHIA predictions agree within the error bars; the S
HERPA -M  C @N  LO predictions describe the data well with the exception of
the low-$E_{T}$ phase space in the direct process, where we see an undershoot
of about 20%. As the cuts on the pseudo-rapidity select the forward region,
this observable is probably sensitive to additional radiation from underlying
events, the used PDFs, and to the photon splitting in the parton shower. Fig.
8 shows a similar situation for the $\eta$-dependence, where for the resolved
processes both, P  YTHIA and S  HERPA -M  C @N  LO , agree with data but for
direct processes we see an undershoot with S  HERPA .

|
---|---
Figure 8: Distributions of sub-leading jet pseudo-rapidity
$\eta^{\mathrm{jet2}}$ for $x_{\gamma}>0.75$ (left) and $x_{\gamma}<0.75$
(right) with leading jet pseudo-rapdity in $0<\eta^{\mathrm{jet1}}<1$ from Z
EUS [61], compared to Leading Order simulations by H  ERWIG , P  YTHIA and S
HERPA and M  C @N  LO -accurate simulations by S  HERPA .

We finish this section with the distributions in $x_{\gamma}$ for low and high
leading jet transverse energy in Fig. 9. Opposed to the modelling for L  EP ,
there is no undershoot visible at the transition from direct to resolved
processes. S  HERPA undershoots the data in the region where both $x_{\gamma}$
and $E_{T}$ are small, however this can be attributed to the missing tuning of
the MPIs as the same region is fairly well described by P  YTHIA . H  ERWIG is
again compatible with the leading order simulation from S  HERPA .

|
---|---
Figure 9: Distributions of $x_{\gamma}$ for leading jet transverse energies in
$14<E_{T}^{\mathrm{jet1}}<17$ GeV (left) and $25<E_{T}^{\mathrm{jet1}}<90$ GeV
(right) from Z  EUS [61], compared to Leading Order simulations by H  ERWIG ,
P  YTHIA and S  HERPA and M  C @N  LO -accurate simulations by S  HERPA .

##  5 Predictions for E  IC

For the planned E  IC we present predictions for electron-proton beams with 18
and 275 GeV beam energies, respectively, similar to the study in [62]. We
cluster jets with the anti-$k_{T}$ algorithm with $R=1.0$ and demand at least
one jet with $E_{\mathrm{T}}>6$ GeV. Looking at inclusive (di-)jet observables
in Fig. 10, we see a similar behaviour in the comparison between the
generators, where S  HERPA -LO yields the smallest cross-section and P  YTHIA
-LO the largest, while H  ERWIG delivers a significantly different shape of
the $x_{\gamma}$ distribution. The $K$ factor in these observables is roughly
50%, again hinting at the real correction and the phase space driving the
correction at NLO. The P  YTHIA LO prediction deviates another 50% from the
NLO-accurate prediction, which can partially be explained by the different PDF
sets, $\alpha_{\mathrm{S}}$ value and the other differences as pointed out in
Subsec. 2.4. This means that, going towards highest possible precision, the
perturbative accuracy needs to be improved further and the non-perturbative
effects need to be constrained by data. Publicly available R  IVET analyses
for the relevant data from H  ERA and L  EP are crucial to reach the latter
goal. While there have beens some recent progress on porting the existing
analyses to R  IVET framework, there still are some shortages related to MPI
constraints and virtuality modelling.

|
---|---
Figure 10: Predictions of transverse jet energy $E_{T}$ (left) and
$x_{\gamma}$ (right) for jet production at the E  IC , comparing to Leading
Order simulations by H  ERWIG , P  YTHIA and S  HERPA and M  C @N  LO
-accurate simulations by S  HERPA .

To study the event shapes in more detail, we present predictions for
transverse thrust $T_{\perp}$ and transverse sphericity $S_{\perp}$, defined
as

$\displaystyle T_{\perp}$
$\displaystyle=\mathrm{max}_{\vec{n}_{\mathrm{T}}}\frac{\sum_{i}\left|\vec{p}_{\mathrm{T},i}\cdot\vec{n}_{\mathrm{T}}\right|}{\sum_{i}\vec{p}_{\mathrm{T},i}}$
(5.1) $\displaystyle S_{\perp}$
$\displaystyle=\frac{2\lambda_{2}}{\lambda_{1}+\lambda_{2}}$ (5.2)

in Fig. 11. Here $n_{\mathrm{T}}$ is the transverse-thrust axis that maximize
the quantity and $\lambda_{1,2}$ are the eigenvalues of the transverse
linearised sphericity tensor $\mathbf{S_{\alpha\beta}}$ defined as

$\mathbf{S_{\alpha\beta}}=\frac{1}{\sum_{i}\left|\vec{p}_{\mathrm{T},i}\right|}\sum_{i}\frac{1}{\left|\vec{p}_{\mathrm{T},i}\right|}\begin{pmatrix}p_{i,x}^{2}&p_{i,x}p_{i,y}\\\
p_{i,y}p_{i,x}&p_{i,y}^{2}\end{pmatrix}$ (5.3)

with $i$ summing over the momenta in the final state and $\alpha$, $\beta$
over ${1,2}$. In both observables, P  YTHIA predicts slightly more isotropic
events than S  HERPA , which might again be caused by a larger number of MPIs
modelled within P  YTHIA . Comparing the two S  HERPA predictions, we again
observe a sizeable $K$ factor and a shift towards more isotropic events. While
using a higher value for $\alpha_{\mathrm{S}}(M_{\mathrm{Z}})$ in P  YTHIA as
a proxy for the $K$ factor seems to work quite well and gives similar cross
sections, the uncertainties from scale variations are considerably smaller at
NLO.

|
---|---
Figure 11: Predictions of transverse thrust $T_{\perp}$ (left) and transverse
sphericity $S_{\perp}$ (right), for jet production at the E  IC , comparing to
Leading Order simulations by H  ERWIG , P  YTHIA and S  HERPA and M  C @N  LO
-accurate simulations by S  HERPA .

As a last observable, in Fig. 12 we look at the charged particle multiplicity
in the detector acceptance range $|\eta|<4$ again for events that contain at
least one jet with $E_{\mathrm{T}}>6~{}\text{GeV}$ and see large disagreement
between P  YTHIA and S  HERPA . Even though MPIs do not play a huge role when
studying high-$p_{\mathrm{T}}$ observables, such as jets, they do come into
play when studying event structure in more detail. The result shows less
hadrons being generated in S  HERPA than in P  YTHIA or H  ERWIG and it means
that a careful study of MPIs and hadronisation is necessary to correctly
simulate these observables. As discussed before, while the perturbative
accuracy is under good control, corrections due to non-perturbative effects
rely on data being made available.

Figure 12: Predictions of charged particle multiplicity $N_{\mathrm{charged}}$
for events with a jet with $E_{\mathrm{T}}<6~{}\text{GeV}$ at the E  IC ,
comparing to Leading Order simulations by H  ERWIG , P  YTHIA and S  HERPA and
M  C @N  LO -accurate simulations by S  HERPA .

##  6 Conclusion

We presented a comparison of the three general purpose event generators, H
ERWIG , P  YTHIA , and S  HERPA , and contrasted the results against
experimental data from LEP and HERA colliders. While the starting point and
theoretical ingredients are similar in each of the generator, the default
inputs and differences in phenomenological modelling do result in significant
differences for the considered observables. In particular we compared to data
for dijet photoproduction analyzed by the O  PAL and Z  EUS collaborations,
seeing an overall good agreement with the data for the S  HERPA -M  C @N  LO
and the P  YTHIA simulations. At leading order, to which the H  ERWIG
simulations are currently limited, and consistent with S  HERPA at leading
order, shapes are roughly consistent with the data, however normalizations
lack a significant $K$-factor. Furthermore, we presented predictions for the
upcoming E  IC for inclusive QCD observables and event shapes for events
containing at least one high-$p_{\mathrm{T}}$ jet. We found significant
differences in observables sensitive to the underlying event modelling and
concluded that by having more experimental data in R  IVET framework would
allow for further tuning of the parameters related to MPI generation such
discrepancies could be largely resolved.

Being the dominant production mechanism for hadronic final states, more work
is needed in preparation for precision phenomenology for the E  IC to
understand the different regimes and a coherent modelling of these, including
development of the relevant infrastructure in all multi-purpose event
generators. Open questions remain like the transition region between DIS and
photoproduction at virtualities of $Q^{2}\approx 1\ \mathrm{GeV}^{2}$ or the
direct and resolved regimes of the photon, and, more importantly, better fits
to the parton content of the photon which are needed as a key ingredient for
precision phenomenology at the E  IC .

## Acknowledgements

We thank the organizers and the staff at the PhysTev 2023 conference, where
this study was started. P.M. is supported by the STFC under grant agreement
ST/P006744/1. I.H. has been supported by the Academy of Finland, project
331545, and funded as a part of the Center of Excellence in Quark Matter of
the Academy of Finland, project 346326. The reported work is associated with
the European Research Council project ERC-2018-ADG-835105 YoctoLHC.

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