# DEMPgen: Physics event generator for Deep Exclusive Meson Production
at Jefferson Lab and the EIC

Z. Ahmed  R. S. Evans  I. Goel  0000-0002-2553-4100 G. M. Huber
0000-0002-5658-1065 S. J. D. Kay  0000-0002-8855-3034 W. B. Li
0000-0002-8108-8045 L. Preet  0009-0004-7667-077X A. Usman  University of
Regina, Regina, SK S4S 0A2, Canada  Stony Brook University, Stony Brook NY
11794, USA  University of York, Heslington, York, Y010 5DD, UK  Center for
Frontiers in Nuclear Science, Stony Brook University, Stony Brook NY 11794,
USA  College of William and Mary, Williamsburg, VA 23187, USA

###### Abstract

There is increasing interest in deep exclusive meson production (DEMP)
reactions, as they provide access to Generalized Parton Distributions over a
broad kinematic range, and are the only means of measuring pion and kaon
charged electric form factors at high $Q^{2}$. Such investigations are a
particularly useful tool in the study of hadronic structure in QCD’s
transition regime from long-distance interactions described in terms of meson-
nucleon degrees of freedom, to short-distance interactions governed by hard
quark-gluon degrees of freedom. To assist the planning of future experimental
investigations of DEMP reactions in this transition regime, such as at
Jefferson Lab and the Electron-Ion Collider (EIC), we have written a special
purpose event generator, DEMPgen. Several types of DEMP reactions can be
generated: $t$-channel $p(e,e^{\prime}\pi^{+})n$,
$p(e,e^{\prime}K^{+})\Lambda[\Sigma^{0}]$, and $\vec{n}(e,e^{\prime}\pi^{-})p$
from a polarized 3He target.

DEMPgen is modular in form, so that additional reactions can be added over
time. The generator produces kinematically-complete reaction events which are
absolutely-normalized, so that projected event rates can be predicted, and
detector resolution requirements studied. The event normalization is based on
parameterizations of theoretical models, appropriate to the kinematic regime
under study. Both fixed target modes and collider beam modes are supported.
This paper presents the structure of the generator, the model
parameterizations used for absolute event weighting, the kinematic
distributions of the generated particles, some initial results using the
generator, and instructions for its use.

††journal: Computer Physics Communications

##  1 Introduction

We have written a Deep Exclusive Meson Production (DEMP) event generator
(DEMPgen) [1], which is modular in form, so that the variety of reactions it
simulates can be expanded over time. The motivation for the writing of the
event generator is to evaluate the feasibility of hadron structure studies
with polarized targets at Jefferson Lab (JLab), and with colliding beams at
the Electron-Ion Collider (EIC).

Figure 1: DEMP handbag diagram for the colinear factorization regime at large
$Q^{2}$. The portion above the dashed line represents the hard scattering with
a parton of momentum fraction $x+\xi$, which can be treated perturbatively.
The portion below the line contains the soft contributions, where the
Generalized Parton Distributions (GPD) encode the response of the nucleon to
the exchange of a parton $x+\xi$ with one with fraction $x-\xi$.

The process of interest is deep inelastic scattering of an electron and
nucleon. The value of $Q^{2}$ is high enough to probe the parton structure via
deep inelastic scattering. Deep Exclusive Meson Production is a kind of
inelastic scattering in which a target nucleon is split into a meson and
recoil hadron, where either all three outgoing particles are detected, or two
are detected with sufficient resolution to construct the missing mass and
momentum of the third particle. Generically, a DEMP reaction involves an
electron ($e$) interacting with a nucleon ($N$) yielding a scattered electron
($e^{\prime}$), an ejectile ($X_{Ej}$, the produced meson) and a recoil hadron
($X_{Rec}$) which can be written as

$\displaystyle N\left(e,e^{\prime}X_{Ej}X_{Rec}\right).$ (1)

An example $p(e,e^{\prime}\pi^{+}n)$ DEMP reaction is shown in Fig. 1. At
moderate $Q^{2}$, DEMP reactions have significant higher twist contributions
at the amplitude level, but these contributions are expected to largely cancel
in some asymmetries [2].

At present, two modules are available in DEMPgen:

  1. 1.

A colliding beam kinematics module for the EIC. In this module, the ejectile
is emitted at small $-t$, at forward angles in the center of mass frame. In
this case, the recoil hadron takes most of the incident nucleon (protons
typically) beam energy and is scattered at very small angles ($<1^{\circ}$).
In this module, the following reactions are currently available:

     * (a)

$p(e,e^{\prime}\pi^{+}n)$

     * (b)

$p(e,e^{\prime}K^{+}\Lambda[\Sigma^{0}])$.

  2. 2.

A fixed target kinematics module for polarized targets at Jefferson Lab. This
module includes optional corrections for Fermi momentum and other nuclear
effects. This module is primarily designed to generate exclusive
${}^{3}He(e,e^{\prime}\pi^{-}p)(pp)_{sp}$ events from a transversely polarized
${}^{3}He$ target.

This paper is divided into sections as follows. In Sec. 2, we will briefly
describe the scientific motivation for our studies, so the structure and
kinematic ranges of applicability of the generator can be better understood.
In Sec. 3, we summarize the coding structure of the two modules of the
generator. The cross-section parameterizations of the currently implemented
physics processes are also discussed. In Sec. 4, we present some results
obtained with the generator, to display some of the ways in which it can be
used. Sec. 5 presents a summary of our work to date, and an outlook of some
extensions to the generator that are being considered.

##  2 Scientific Motivation for our Studies

###  2.1 Motivation for EIC $\pi^{+}$ and $K^{+}$ studies

The Electron-Ion Collider is a next generation collider to be constructed at
Brookhaven National Laboratory to study the structure of nucleons in detail.
According to the National Academies of Science assessment [3], the EIC will
address three major questions.

  1. 1.

How does the mass of the nucleon arise? The problem is that while gluons have
no mass, and $u$, $d$ quarks are nearly massless, the nucleons that contain
them are heavy; the total mass of a nucleon is significantly larger than the
mass of the valence quarks it contains. The largest contribution to the mass
of the proton originates from the gluon field energy. In this sense, the
source of the vast majority of the visible mass in the universe is not the
Higgs field, but the gluon field. Measurements of deep exclusive $\pi^{+}$
production from the nucleon at the EIC allow the quark-gluon energy
contributions to the nucleon mass budget to be studied.

  2. 2.

How does the spin of the nucleon arise? How the angular momentum, both
intrinsic as well as orbital, of the internal quarks and gluons gives rise to
the known nucleon spin is not well understood. The quark polarization
contribution to the nucleon spin is only about 30%. The remainder of the spin
must reside in the orbital angular momenta of quarks and gluons or gluon
polarization. A central goal of the EIC program is to provide a determination
of the gluon spin contribution and its orbital angular momentum.

  3. 3.

What are the emergent properties of dense systems of gluons? The nature of
gluons in matter, i.e. their arrangements or states, and the details of how
they hold matter together are not well known. Gluons in matter are somewhat
like dark matter in the universe, unseen but playing a crucial role. The EIC
would be able to study the gluons that bind quarks and antiquarks into
nucleons and nuclei with unprecedented precision. A central goal of such
studies is to explore the limit of low parton momentum fraction $x$, where the
number of gluons in the target is very large. The EIC would also be able to
explore modifications of the quark distributions in nuclei.

Answers to these questions are essential for understanding the nature of
visible matter in the universe.

As for our specific investigation, the pion and kaon are two of the simplest
systems available to study the structure of hadrons. The elastic
electromagnetic form factors of the charged pion and kaon, $F_{\pi}(Q^{2})$
and $F_{K}(Q^{2})$, are a rich source of insights into basic features of
hadron structure. For example, $F_{\pi}(Q^{2})$ and $F_{K}(Q^{2})$ can provide
insight on the roles played by confinement and Dynamical Chiral Symmetry
Breaking (DCSB) in fixing the hadron’s size, determining its mass, and
defining the transition from the strong- to perturbative-QCD domains [4].

The experimental determination of the $\pi^{+}$ electric form factor
($F_{\pi}$) is challenging. The best way to determine $F_{\pi}$ would be
electron-pion elastic scattering. However, the lifetime of the $\pi^{+}$ is
only 26.0 ns. Since $\pi^{+}$ targets are not possible, and $\pi^{+}$ beams
with the required properties are not yet available, one of the experimentally
feasible approaches is high-energy exclusive electroproduction of a $\pi^{+}$
from a proton at low Mandelstam four-momentum transfer to the target proton,
$-t=-(p_{p}-p_{n})^{2}$. This is best described as quasi-elastic ($t$-channel)
scattering of the electron from the virtual $\pi^{+}$ cloud of the proton.
Scattering from the $\pi^{+}$ cloud dominates the longitudinal photon cross
section ($d\sigma_{L}/dt$), when $|t|\ll M_{p}^{2}$ [5]. To reduce background
contributions, normally one separates the components of the cross-section due
to longitudinal (L) and transverse (T) virtual photons (and the LT, TT
interference contributions), via a Rosenbluth separation (Eqn. 24). The value
of $F_{\pi}(Q^{2})$ is determined by comparing the measured $d\sigma_{L}/dt$
values at small $-t$ to the best available electroproduction model. The
obtained $F_{\pi}$ values are in principle dependent upon the model used, but
one anticipates this dependence to be reduced at sufficiently small $-t$.

Using this approach, the charged pion form factor, $F_{\pi}(Q^{2})$, has been
measured in Jefferson Lab (JLab) Hall C via $\pi^{+}$ electroproduction up to
$Q^{2}=-(p_{e}-p_{e^{\prime}})^{2}=2.45$ GeV2 with high precision [6]. This
result generated confidence in the reliability of $\pi^{+}$ electroproduction
as a tool for pion form factor extraction. JLab experiment E12-19-006 [7], one
of the flagships of the 12 GeV upgrade, will extend the high precision studies
to $Q^{2}=6.0$ GeV2, and with lower precision to $Q^{2}=8.5$ GeV2. This
experiment is expected to deliver pion form factor data bridging the region
where QCD transitions from the strong (color confinement, long-distance) to
perturbative (asymptotic freedom, short-distance) domains. The measurement of
${F}_{\pi}(Q^{2})$ at the EIC is a continuation of the study of the pion form
factor at higher $Q^{2}$ kinematics. The comparison of the pion and kaon form
factors in this regime would provide vital information for the understanding
of the role of DCSB in the generation of hadronic mass.

The reliability of the electroproduction method to determine the $K^{+}$ form
factor, $F_{K}(Q^{2}),$ is not yet established. JLab E12-09-011 [8] has
acquired data for the $p(e,e^{\prime}K^{+})\Lambda$,
$p(e,e^{\prime}K^{+})\Sigma^{0}$ reactions at hadronic invariant mass
$W=\sqrt{(p_{K}+p_{\Lambda,\Sigma})^{2}}>2.5$ GeV, to search for evidence of
scattering from the proton’s “kaon cloud”. The data are still being analyzed,
with L/T-separated cross-sections expected in the near future. If they confirm
that the scattering from the virtual $K^{+}$ in the nucleon contributes
significantly to $d\sigma_{L}/dt$ at low four-momentum transfer to the target
$|t|\ll M_{p}^{2}$, the experiment will yield the world’s first quality data
for $F_{K}$ above $Q^{2}>0.2$ GeV2. This would then open up the possibility of
using the same exclusive reactions to determine the kaon form factor over a
wide range of $Q^{2}$ at the EIC. DEMPgen includes two modules of the
$p(e,e^{\prime}\pi^{+}n)$ and $p(e,e^{\prime}K^{+}\Lambda[\Sigma^{0}])$
reactions in colliding beam mode, to enable feasibility studies for these
measurements at the EIC. These are discussed further in Secs. 4.2 and 4.3.

###  2.2 Motivation for Jefferson Lab $\vec{n}(e,e^{\prime}\pi^{-})p$ studies

The development of the Generalized Parton Distribution (GPD) formalism in the
last 20 years is a notable advance in our understanding of the structure of
the nucleon. GPDs unify the concepts of parton distributions and hadronic form
factors, and are “universal objects” that provide a comprehensive framework
for describing the quark and gluon structure of the nucleon. GPDs are probed
through Deep Exclusive reactions, and their knowledge would allow a
tomographic 3D understanding of the nucleon to be built up [9, 10, 11]. A
special kinematic regime is probed in DEMP, where the initial hadron emits a
quark-antiquark or gluon pair. This has no counterpart in the usual parton
distributions and carries information about $q\bar{q}$ and $gg$-components in
the hadron wavefunction.

The four lowest-order GPDs are parameterized in terms of quark chirality. At
leading twist, there are four GPDs: $H(x,\xi,t)$, $E(x,\xi,t)$,
$\tilde{H}(x,\xi,t)$ and $\tilde{E}(x,\xi,t)$ associated with each quark
flavor as well as gluons. $H$ and $\tilde{H}$ GPDs conserve helicity, while
the $E$ and $\tilde{E}$ GPDs are associated with a helicity flip of the
nucleon. Each GPD depends upon three variables, the four-momentum transferred,
$Q^{2}$, the average longitudinal momentum of the struck quark $x$ (in the
high $Q^{2}$ regime $x=x_{B}$) and the skewness,
$\xi=(p_{1}^{+}-p_{2}^{+})/(p_{1}^{+}+p_{2}^{+})$, where $p_{1}$, $p_{2}$
refer to the light-cone plus components of the initial and final nucleon
momenta in Fig. 1.

GPDs also describe the correlation between partons in a nucleon. By utilizing
a Fourier transform, one can access the longitudinal momentum fraction of
quarks and their position in the transverse plane simultaneously [12]. The
first moments of GPDs are related to the elastic form factors of the nucleon.
The GPDs integrals over $x$ give [13]:

$\displaystyle\int^{1}_{-1}dx\quad H^{q}(x,\xi,t)=F^{q}_{1}(t),$ (2)
$\displaystyle\int^{1}_{-1}dx\quad E^{q}(x,\xi,t)=F^{q}_{2}(t),$ (3)
$\displaystyle\int^{1}_{-1}dx\quad\tilde{H}^{q}(x,\xi,t)=g^{q}_{A}(t),$ (4)
$\displaystyle\int^{1}_{-1}dx\quad\tilde{E}^{q}(x,\xi,t)=h^{q}_{A}(t),$ (5)

where $F_{1}$, $F_{2}$, $g_{A}$ and $h_{A}$ are the Dirac, Pauli, pseudoscalar
and axialvector form factors respectively.

GPDs provide information about the nucleon in a manner that is independent of
the probing reaction. One way to determine GPDs is via DEMP reactions. Because
quark helicity is conserved in the hard scattering regime, the produced meson
acts as a helicity filter [11]. In particular, leading order perturbative QCD
predicts that longitudinally polarized vector meson production (e.g.
$\rho^{0,\pm}_{L},\omega_{L}$) is sensitive only to the unpolarized GPDs ($H$
and $E$), whereas pseudoscalar mesons (e.g. $\pi$, $\eta$) produced via
longitudinally polarized virtual photons are sensitive only to the polarized
GPDs ($\tilde{H}$ and $\tilde{E}$). Thus, DEMP reactions are complementary to
the Deeply Virtual Compton Scattering (DVCS) process, as they provide the
additional data needed to disentangle the different GPDs.

The GPD $\tilde{E}$ is particularly poorly known [14]. It is related to the
pseudoscalar nucleon form factor $G_{P}(t)$, which is itself highly uncertain
because it is negligible at the momentum transfer of nucleon $\beta$-decay.
$\tilde{E}$ is believed to contain an important pion pole contribution and
hence is optimally studied in DEMP. $\tilde{E}$ cannot be related to already
known parton distributions, and so experimental information about it can
provide new information on nucleon structure, which is unlikely to be
available from any other source. Furthermore, this observable has been noted
as being important for the reliable extraction of $F_{\pi}$ from pion
electroproduction [15], due to the significant $\pi$ pole contribution.

Figure 2:  Scattering and hadronic reaction planes for exclusive the
$\vec{N}(e,e^{\prime}\pi)N^{\prime}$ reaction. $\theta_{q}$ is the scattering
angle of the virtual photon with respect to the incident electron beam, and
$\theta_{\pi}$ is the scattering angle of the pion with respect to the
$q$-vector. $\phi_{\pi}$ is the azimuthal angle between the hadronic reaction
plane and the electron scattering plane. $\phi_{S}$ is the azimuthal angle
between the target nucleon polarization and the scattering plane, and
$\beta=(\phi-\phi_{s})$ is the angle between the target nucleon polarization
vector and the reaction plane.

The most sensitive observable to probe $\tilde{E}$ is the transverse single
spin asymmetry in exclusive $\pi^{\pm}$ production:

$A^{\perp}_{L}=\frac{\int^{\pi}_{0}d\beta\frac{d\sigma^{\pi}_{L}}{d\beta}-\int^{2\pi}_{\pi}d\beta\frac{d\sigma^{\pi}_{L}}{d\beta}}{\int^{2\pi}_{0}d\beta\frac{d\sigma^{\pi}_{L}}{d\beta}},$
(6)

where d$\sigma^{L}_{\pi}$ is the exclusive $\pi$ cross-section for
longitudinal virtual photons and $\beta$ is the angle between the transversely
polarized target vector and the reaction plane (Fig. 2). Frankfurt et al. have
shown that $A^{\perp}_{L}$ vanishes if $\tilde{E}$ is zero [2]. If
$\tilde{E}\neq 0$, the asymmetry will produce a $\sin(\beta)$ dependence.
Refs. [2, 16] note that “precocious scaling” is likely to set in at moderate
$Q^{2}\sim 2-4$ GeV2 for this observable, as opposed to the absolute cross
section, where scaling is not expected until $Q^{2}>10$ GeV2.

So that all final state particles are charged, and hence relatively easily
detectable experimentally, the fixed target DEMPgen module simulates the
reaction ${}^{3}He(e,e^{\prime}\pi^{-}p)(pp_{sp})$ from a transversely
polarized target, including Fermi momentum and final state interaction (FSI)
effects. This is described in more detail in Sec. 3.3.6.

##  3 DEMP Event Generator

This section describes the structure of the event generator, as well as the
model parameterization, scattering cross-section and kinematic ranges for each
of the DEMP processes. DEMPgen has two distinct modules, one for colliding
beam event generation and one for beam-on-fixed-target event generation. Both
modules use a common .json input file, the options for which are detailed in
A.

The colliding beams module has initially been developed to enable the
simulation of various DEMP channels at the upcoming EIC. As such, this is
referred to as the EIC module.

The fixed target module is primarily focused on the generation of
${}^{3}He(e,e^{\prime}\pi^{-}p)(pp)_{sp}$ events from a polarized ${}^{3}He$
target at the upcoming Solenoidal Large Intensity Device (SoLID) experiment at
JLab [17]. Consequently, this module is referred to as the SoLID module.

The EIC and SoLID modules are detailed in the following subsections, including
the structure of each module and the relevant event kinematics. The physics
models utilized in the generator are described in Sec. 3.3. Results from
studies using both modules are presented in Sec. 4. In the following
discussion, reaction kinematics will be referred to in generic terms as much
as possible using the nomenclature of scattered electron, ejectile (the
produced meson), and recoil hadron.

###  3.1 EIC Module

The EIC module reads in the input .json file and calculates events in
colliding beam kinematics. Input parameters specified in this file include the
incoming beam energies, random number generator seed, number of events to
generate, the output file type, generation ranges, and the reaction to
generate. Details on the names of the input parameters, and where relevant
their default values, are included in A.

Currently, DEMPgen is capable of generating events for the following
reactions:

  1. 1.

$p(e,e^{\prime}\pi^{+}n)$

  2. 2.

$p(e,e^{\prime}K^{+}\Lambda)$

  3. 3.

$p(e,e^{\prime}K^{+}\Sigma^{0})$.

In the future, DEMPgen will be extended to handle deep exclusive meson
production from $t$-channel $e$-A and $u$-channel $e$-$p$ collisions.

The EIC module considers electrons and nucleons as the incoming particles at
different beam energies. For electron-proton collisions, four of the proposed
beam energy combinations are $5(e)\times 41(p)$, $5(e)\times 100(p)$,
$10(e)\times 100(p)$, and $18(e)\times 275(p)$. DEMPgen can produce events
with any arbitrary electron-proton beam energies, however, only a few
“standard” beam energy combinations (including those defined above) have
projected luminosity values. The luminosity is required for event weight
calculations, see Sec. 3.1.2 for further details on this, and B for details on
the luminosity values. DEMPgen generates three outgoing particles in the
inertial frame of the EIC detector (collider frame) based on the pion or kaon
electroproduction reactions.

####  3.1.1 Event Generation

After reading in the .json input file, DEMPgen initialises several parameters
and begins generating events. The main event processing loop generates
$N_{Requested}$ events (specified in the input .json file). Each pass through
the main event processing loop increments $N_{Gen}$ by 1, effectively this is
the number of “tries” so far. Generally, the majority of the generated events
are discarded by various event selection cuts. If an event fails a cut, no
further processing is carried out for this event. Another event is generated,
so long as $N_{Gen}$ $<$ $N_{Requested}$. The general sequence and flow of
events through the EIC module of DEMPgen are summarized in a flowchart, Fig.
3. As shown in this figure, cuts are conducted in a specific order and
conducted sequentially to reject events as soon as possible and prevent any
redundant calculations being conducted. Cut values for
$p(e,e^{\prime}\pi^{+}n)$ reactions are shown in Tab. 1. As an example, an
event generated with $Q^{2}=0.1~{}GeV^{2}$ would be rejected and no further
processing would occur for this event.

Figure 3: Flowchart of the basic operation of the EIC module in simplified steps. Key process and steps are described in the rounded boxes. Cuts on the generated events are represented by the elliptical boxes. Beginning from the initialisation box, the chart then flows counter-clockwise. Many of the cuts are on kinematic quantities that are detailed further in Sec. 3.3.1. The cut values vary depending upon the requested reaction, cut values for the $p(e,e^{\prime}\pi^{+}n)$ reaction are shown in Table 1. This table also indicates which cuts are reaction dependent. Cut |  Values |  Reaction Dependent?
---|---|---
$Q^{2}$ |  $Q^{2}<3$GeV2 & $Q^{2}>35$ GeV2 |  Yes
$W$ |  $W<2$ GeV & $W>10.2$ GeV |  Yes
Unphysical |  $E_{X}$ returns NaN |  No
4-Momentum Conservation |  $0.00001$ GeV |  No
$-t$ |  $-t>1.3$ GeV2 |  Yes
Table 1: Table of event selection cuts and their values in the EIC module.
Some cut ranges depend upon the reaction specified, as indicated in the table,
the values shown are the values for $p(e,e^{\prime}\pi^{+}n)$ reactions. Note
that for the 4-momentum conservation cut, the difference between the initial
and final 4-momenta sum is calculated. All components of this resulting
4-vector are checked against this value. The cut will fail upon any component
exceeding the specified value.

The exclusive nature of DEMP reactions pose some constraints on event
generation. Specifically, the relevant differential cross section is 5-fold
($\theta_{e},\phi_{e},E_{e},\theta_{Ej},\phi_{Ej}$), rather than 6-fold
($\theta_{e},\phi_{e},E_{e},\theta_{Ej},\phi_{Ej},E_{Ej}$), as the final
states are discrete for the considered reactions. To calculate the cross
section, the outgoing ejectile momentum magnitude is uniquely determined from
four-momentum conservation at the photon-ejectile vertex once the scattered
electron energy and angles, and the outgoing ejectile $(\theta,\phi)$ angles
are chosen. The energy of the scattered electron is selected from a uniform
random distribution in a configurable range specified in the DEMPgen .json
input card, given in A. The direction of the scattered electron and the
produced ejectile is selected using sphere point picking [18]. The angular
range over which the scattered electron and ejectile are distributed is also a
configurable parameter in the DEMPgen .json input card. The energy of the
ejectile is left to be solved for.

These variables provide all the information necessary to uniquely solve for
all remaining kinematic variables. Applying conservation of energy and
momentum yields the following equation:

$\nu+E_{N}-\sqrt{m_{Ej}^{2}+|\vec{X_{Ej}}|^{2}}-\sqrt{m_{Rec}^{2}+|\vec{q}+\vec{N}-\vec{X_{Ej}}|^{2}}=0,$
(7)

where $\nu$ and $E_{N}$ represent the energy of the virtual photon and the
nucleon, respectively. The vectors represents the three-momenta of the
respective particles. The only unknown in this equation is the momentum vector
of the ejectile, $\vec{X_{Ej}}$. Since the direction of the ejectile has
already been specified, Eqn. 7 can be further reduced to a single-valued
unknown: the magnitude of the ejectile momentum.

The energy of the ejectile is determined analytically. Eqn. 7 is modified in
terms of the energy of the ejectile in the collider frame and then solved to
get a quadratic equation. Finally, the coefficients of the quadratic equation
are defined and calculated,

$a[E^{2}_{Ej}]+b[E_{Ej}]+c=0.$ (8)

Here, $a$, $b$, and $c$ depend on the known quantities, such as the four-
momenta of other particles and direction of ejectile and are specified in C.
Using the quadratic formula, the solutions of Eqn. 8 are determined. The
direction and momentum of the recoil hadron are generated by applying
conservation of energy and momentum at the physics reaction vertex.

Kinematic quantities are determined as soon as the relevant information is
available to calculate them. E.g., as soon as the ejectile 4-vector is
determined, $t$ and $u$ can be calculated. Selection cuts are applied to the
events, as shown in Fig 3, and detailed in Tab. 1. As a consequence of these
cuts, the actual number of successfully generated events will be different (or
smaller) than the number of events tried. If an event passes all selection
cuts, a cross section value is determined from a parameterized model as
described in Sec. 3.3.4. As a validation check of this determination, events
that return negative cross section values or NaN are removed. This cross
section is then used to determine a weight for the event, as described in the
next section, and the event information is saved to the output file.

####  3.1.2 Event Weighting

DEMPgen produces events with variable weight, corresponding to the rate of the
given reaction at the input luminosity. Following the generation of events as
presented in Sec. 3.1.1, every event that passes all of the selection cuts is
assigned a weight value as follows,

$\mathrm{Weight}=\frac{\sigma\times PSF\times
CF\times\mathcal{L}}{N_{Requested}},$ (9)

where $\sigma$ is the 5-fold differential cross-section in the collider frame,
$PSF$ is the phase space factor (see D for more details), $CF$ is a conversion
factor to convert $\mu$b to cm2, $\mathcal{L}$ is the luminosity (in units of
cm-2s-1), and $N_{Requested}$ is the total number of events that the generator
tried to produce. $N_{Requested}$ includes events that were discarded due to
either falling outside of acceptable parameters, or having no valid solutions
in the kinematics solver. The value of $\sigma$ is determined for the
generated kinematics from a parameterized model calculation, further details
on the models used and the parameterization for each physics process are
provided in Sec. 3.3.

The resulting weight value is in units of Hz. The output of the generator is a
CERN-ROOT file and a LUND format file for the SoLID module. For the EIC
module, only a text file of events is produced, this can be in LUND, Pythia6
or HEPMC3 format depending upon what the user specifies in the control card
(see A for more info).

###  3.2 SoLID Module

Figure 4: Flowchart describing the flow of data through the program, and the
structure of the SoLID module of the event generator. The rounded boxes
signify the most important named variables that appear in the main file of the
program. Their values are pointers to an instance of the given class. Arrows
indicate how data is moved between these objects. The four open boxes indicate
static functions. The “FSI Weights” node represents a simple collection of
doubles. The “Output” box represents the destination for data to be saved into
the output CERN-ROOT tree.

Fig. 4 demonstrates the flow of data in the event generator for the SoLID
module, starting from the random number generators, and leading to the output
file. The rounded boxes in this chart each indicate the main named variables
whose values are pointers to instances of the indicated class. For example,
“VertTargNeut” is a pointer to an object of class “Particle”. These variables,
the classes, and their place in the structure of the event generator, are
discussed in the following paragraphs.

####  3.2.1 Particle Class

Instances of the particle class contain all of the pieces of information about
a single particle, in a single frame of reference, that are relevant to the
event generator. This includes the four-momentum, rest mass, charge, etc. The
particle inherits from CERN-ROOT’s TLorentzVector class, which allows for the
creation and manipulation of general four-vectors. The TLorentzVector class
includes methods to calculate components, angles, and magnitudes of a four-
momentum, as well as perform Lorentz boosts and rotations. It also defines
algebraic operators for four-vectors. Implementation of this class
significantly simplifies calculations within the event generator, and
eliminates a large number of messy algorithms that would otherwise need to be
present.

####  3.2.2 DEMPEvent class

Instances of the DEMPEvent class represent the event viewed from a single
reference frame. The class stores seven particle objects: the incident
electron, target nucleon, virtual photon, scattered electron, produced meson,
and recoiled nucleon. The class has methods to calculate Mandelstam variables
$s$, $t$, $u$, and functions to perform coordinate transformations on the
event. There are five DEMPEvent objects initialized in the event generator, as
seen in Fig. 4. VertEvent contains the particles as viewed at the vertex of
the interaction, in the laboratory rest frame. Once the kinematics
calculations have been completed, this object is no longer modified. All other
DEMPEvent objects are calculated from this object by copying and then
transforming them. CofMEvent is the event viewed at the vertex in the center
of the momentum reference frame. RestEvent is the event viewed at the vertex
in the rest frame of the target neutron. TConEvent is the event viewed at the
vertex in the coordinate system defined by the Trento Conventions [19].

###  3.3 Physics Models

As discussed in Sec. 3.1.2, generated events are assigned a weight calculated
from various quantities. One of these quantities is the 5-fold differential
cross-section,
$d^{5}\sigma/dE{e^{\prime}}d\Omega_{e^{\prime}}d\Omega^{*}_{Ej}$. This cross
section is determined based on the kinematics of the generated event using
parameterized physics models. The utilized models and their parameterizations
are described in this section.

####  3.3.1 Kinematic definitions

In general, the following Lorentz invariants are used to define the kinematics
of $p\left(e,e^{\prime}X_{Ej}X_{Rec}\right)$ events:

$\displaystyle-Q^{2}=(p_{e}-p_{e^{\prime}})^{2}$ (10) $\displaystyle
W^{2}=(p_{\gamma^{*}}+p_{p})^{2}=(p_{Ej}+p_{Rec})^{2}$ (11) $\displaystyle
t=(p_{\gamma^{*}}-p_{Ej})^{2}=(p_{p}-p_{Rec})^{2}$ (12) $\displaystyle
u=(p_{\gamma^{*}}-p_{Rec})^{2}=(p_{p}-p_{Ej})^{2},$ (13)

where for the EIC module, $p_{e}$, $p_{e^{\prime}}$, $p_{\gamma^{*}}$,
$p_{p}$, $p_{Ej}$, and $p_{Rec}$ represent four vectors for the electron beam,
scattered electron, virtual photon, proton beam, ejectile, and recoil hadron,
respectively. For the SoLID module with a quasi-free 3He target configuration,
substitute the neutron four momentum ($p_{n}$) for the proton four momentum
($p_{p}$) in the above equations.

In the fixed target frame, $t$ can be expressed as

$t=(E_{Ej}-\nu)^{2}-|\vec{p_{Ej}}|^{2}-|\vec{q}|^{2}+2|\vec{p_{Ej}}||\vec{q}|\cos\theta_{Ejq},$
(14)

where $\nu,\vec{q}$ are the energy and three-momentum of the virtual photon,
and $\theta_{Ejq}$ is the angle between the ejectile and the $q$-vector shown
in Fig. 2 (as $\theta_{\pi}$). For spacelike DEMP reactions, $t$ is always
negative, so the variable $-t$ is used throughout the paper. The minimal value
of $-t$ (or $-t_{min}$) is obtained when $\theta_{Ejq}=0$, i.e. when the
ejectile is emitted in the direction of the virtual photon (referred to as
parallel kinematics). $-t_{max}$ is the maximum value of $-t$, corresponding
to anti-parallel kinematics, $\theta_{Ejq}=\pi$. $u$ is not single valued,
taking its most negative value $-u_{max}$ at $\theta_{Ejq}=0$, passing through
zero just before anti-parallel kinematics, and ending slightly positive at
$\theta_{Ejq}=\pi$ (taken as $-u_{min}$). For convenience,
$t^{\prime}=t-t_{min}$ and $u^{\prime}=u-u_{min}$ are defined.

We define the missing mass and momentum in DEMP as follows:

$\displaystyle\centering\vec{p}_{miss}\@add@centering$
$\displaystyle=\vec{p}_{e}+\vec{p}_{N}-\vec{p}_{e^{\prime}}-\vec{p}_{Ej}$ (15)
$\displaystyle E_{miss}$ $\displaystyle=E_{e}+E_{N}-E_{e^{\prime}}-E_{Ej}$
(16) $\displaystyle m_{miss}^{2}$ $\displaystyle=E_{miss}^{2}-p_{miss}^{2},$
(17)

where $N$ refers to the quasi-free target nucleon (proton or neutron, e.g.
from 3He). For DEMP reactions, $m_{miss}$ must equal the mass of the recoil
particle.

####  3.3.2 Cross Section Formalism

In the one-photon exchange approximation, the reduced five-fold differential
cross section for DEMP in terms of virtual photon flux factor, $\Gamma_{V}$,
and a virtual photon cross-section, $\frac{d{{}^{2}}\sigma}{d\Omega^{*}_{Ej}}$
is given by Eqn. 18, with incoming and outgoing particles described by plane
waves

$\frac{d^{5}\sigma}{dE_{e^{\prime}}d\Omega_{e^{\prime}}d\Omega^{*}_{Ej}}=\Gamma_{V}\frac{d{{}^{2}}\sigma}{d\Omega^{*}_{Ej}},$
(18)

where $E_{e^{\prime}}$ and $\Omega_{e^{\prime}}$ are the scattered electron’s
lab energy and solid angle, respectively, and $\Omega^{*}_{Ej}$ is the
ejectile solid angle in the center-of-mass frame. The virtual photon flux
factor, $\Gamma_{V}$, can be written as

$\Gamma_{V}=\frac{\alpha}{2\pi^{2}}\frac{E_{e^{\prime}}}{E_{e}}\frac{K}{Q^{2}}\frac{1}{1-\epsilon},$
(19)

where $\alpha$ is the fine structure constant, the factor
$K=(W^{2}-M_{N}^{2})/(2M_{N})$ is the equivalent real photon energy [20],
i.e., the laboratory energy required by a real photon to excite a target of
mass, $M_{N}$, and create a system with invariant mass equal to $W$.
$\epsilon$ is the polarization of the virtual photon given by

$\epsilon=\left(1+\frac{2|\mathbf{q}|^{2}}{Q^{2}}\tan^{2}\frac{\theta_{e}}{2}\right)^{-1}.$
(20)

For colliding beams, $\epsilon$ can be expressed in terms of the fractional
energy loss of the collision, $y$, where

$\displaystyle y=\frac{Q^{2}}{x_{B}(s_{tot}-M_{N}^{2})},$ (21)

and

$\displaystyle\epsilon=\frac{2(1-y)}{1+(1-y)^{2}},$ (22)

where $x_{B}$ is the Bjorken scaling variable and $s_{tot}$ is the square of
the center of mass energy of the system.

The two-fold differential cross-section in Eqn. 18 can be expressed in terms
of the invariant cross-section as

$\frac{d^{2}\sigma}{d\Omega^{*}_{Ej}}=J\frac{d^{2}\sigma}{dtd\phi},\\\ $ (23)

where $J$ is the Jacobian factor to transform $\Omega^{*}_{Ej}$ to $t,\phi$.

The general two-fold differential cross section in Eqn. 23 can be expressed in
terms of four structure functions:

$\begin{split}2\pi\frac{d^{2}\sigma}{dtd\phi}&=\epsilon\frac{d\sigma_{\mathrm{L}}}{dt}+\frac{d\sigma_{\mathrm{T}}}{dt}\\\
&+\sqrt{2\epsilon(\epsilon+1)}\frac{d\sigma_{\mathrm{LT}}}{dt}\cos{\phi}+\epsilon\frac{d\sigma_{\mathrm{TT}}}{dt}\cos{2\phi},\end{split}$
(24)

where subscripts $L$ and $T$ represent the longitudinal and transverse
polarizations of the virtual photon. For brevity, we refer to $d\sigma_{L}/dt$
as $\sigma_{L}$, and so on.

To study DEMP at the EIC, the cross terms $\sigma_{LT}$ and $\sigma_{TT}$,
which arise from longitudinal transverse and transverse transverse
interference states of the virtual photon, are ignored as they are expected to
be small, and even more highly uncertain than $\sigma_{L}$ and $\sigma_{T}$.

Finally, for the EIC module, the five-fold cross-section is transformed into
the collider frame with the help of the following Jacobians:

$\displaystyle J=A\times J^{col}_{rf},$ (25) $\displaystyle
J^{col}_{rf}=\frac{|\vec{p}_{Ej}|^{2}\times
W}{|\left(|\vec{p}^{cm}_{Ej}|(M_{p}+E_{\gamma})|\vec{p}_{Ej}|-E_{Ej}|\vec{p}_{\gamma}|cos(\theta_{Ej})|\right)|},$
(26) $\displaystyle A=J_{cm}\frac{|\vec{p}^{cm}_{Ej}|}{\pi},$ (27)
$\displaystyle
J_{cm}=\frac{\vec{p}^{rf}_{\gamma}-\beta^{rf}_{cm}E^{rf}_{\gamma}}{\gamma^{rf}_{cm}(1-(\beta^{rf}_{cm})^{2})}.$
(28)

In Eqn. 26, $\vec{p}_{Ej}$ and $\vec{p}^{cm}_{Ej}$ are the three-momentum
vectors of the ejectile in the collider and center-of-mass frame,
respectively. $E_{\gamma^{*}}$ is the energy of the virtual photon in the
proton’s rest frame, $E_{Ej}$ is the ejectile’s energy in the collider frame,
$\vec{p}_{\gamma^{*}}$ is the three momentum vector of the virtual photon in
the collider frame, and $\theta_{Ej}$ is the ejectile angle in the collider
frame. In Eqn 28, $\vec{p}^{rf}_{\gamma^{*}}$ is the three momentum vector of
the virtual photon in the proton’s rest frame, $\beta^{rf}_{cm}$ is $\beta$ of
the center of mass frame relative to proton’s rest frame, $\gamma^{rf}_{cm}$
is the Lorentz $\gamma$ of the center of mass frame relative to proton’s rest
frame, and $E^{rf}_{\gamma^{*}}$ is the energy of the virtual photon in the
proton’s rest frame.

Figure 5: Comparison of the VR (red circles) and CKY (blue squares) models for
$\sigma_{L}$ (left) and $\sigma_{T}$ (right) of the EIC pion module for
selected kinematics.

Figure 6: Parameterization of CKY $\sigma_{L}$ for the EIC pion module with
CKY model values (black circles) plotted along with the landau (red line) and
exponential fits (blue and green lines) in Eqn. 29.

Figure 7: Parameterization of CKY $\sigma_{T}$ for the EIC pion module with
CKY model values (black circles) plotted along with the second order
polynomial (red line) and exponential fits (blue line) in Eqn. 30.

####  3.3.3 Cross Section Parameterization

The following sections provide details of selecting the appropriate
theoretical model, parameterizing various cross section components using the
model, and implementing the parameterizations into DEMPgen to determine the
total differential cross-section, along with assigning the suitable weights to
the events to enable the study of different reactions.

####  3.3.4 Exclusive $p(e,e^{\prime}\pi^{+}n)$ Physics Model

##### VR and CKY models

The VR model [21] by Tom Vrancx and Jan Ryckebusch introduces a strong
hadronic form factor in the Reggeized background amplitudes to improve the
description of $\sigma_{T}$, while retaining good agreement with $\sigma_{L}$
data, at low $-t$ and $W>2$ GeV. The VR model for pion electroproduction is
fine-tuned with L/T-separated data up to $-t\lesssim 0.5$ GeV2 and
$0.7<Q^{2}<4.35$ GeV2.

The CKY [22] model by Tae Keun Choi, Kook Jin Kong and Byung Geel Yu is also a
Regge-based model. The CKY model accounts for the importance of the roles of
the pion and proton form factors in DEMP to provide a good description of
separated $W>2$ GeV DESY and JLab data for $-t<0.7$ GeV2, and unseparated data
for $-t<5$ GeV2, both up to $Q^{2}\approx 5$ GeV2 [22, 23].

##### Comparison of two models

For EIC kinematics, a detailed comparison of the VR and CKY models was
undertaken for $\sigma_{L}$ and $\sigma_{T}$. A typical graph of each
$\sigma_{L,T}$ is shown in Fig. 5. As shown in Fig. 5, the VR and CKY models
are in generally good agreement with each other at low $-t$. However, at
higher $-t$, the two models differ from each other in some cases and it was
observed that the CKY model behaves more consistently than the VR model.

Therefore, DEMPgen utilizes the CKY model to determine the cross section and
assign each event an appropriate weight. The cutoff mass parameter values for
$\pi$, $\rho$, and $p$ (proton) trajectories were chosen to be $\Lambda_{\pi}$
= 0.65 GeV, $\Lambda_{\rho}$ = 0.782 GeV, and $\Lambda_{p}$ = 1.55 GeV to fit
high the $Q^{2}$ region. We also have performed simulations demonstrating the
feasibility of pion electric form factor, $F_{\pi}$, measurements at the EIC
using this event generator, as presented in Sec. 4.2.

##### Model Implementation in DEMPgen

For the cross-section parameterization of the pion module, the following
ranges are chosen, $W$ from 2 to 10.2 GeV, $Q^{2}$ from 3 to 35 GeV2, and $-t$
up to 1.3 GeV2. There are 22 bins of $W$, each of 0.2 GeV width. For each $W$
bin, there are 33 $Q^{2}$ bins, each of 1 GeV2 width. For each unique bin of
$W$ and $Q^{2}$, $\sigma_{L}$ and $\sigma_{T}$ are parameterized against $-t$
from 0 GeV2 to 1.3 GeV2, as shown in Figs. 6, 7.

In order to make the event generator more efficient and save CPU time, some
hard cuts are applied, as discussed in Sec. 3.1.1. Events with $Q^{2}<5$ GeV2,
$W<3$ GeV, $W>10.6$ GeV are ignored. If the FF (Form Factor) generator option
is used, then $-t>0.6$ GeV2 events are removed, or if the TSSA (Transverse
Single Spin Asymmetry) generator option is used, then $-t>1.3$ GeV2 events are
removed.

When an event is generated with specific values of $Q^{2}$ and $W$, the
generator looks for the closest parameterization combination bin and uses the
relevant function to determine the cross-section value.

##### Parameterization of $\sigma_{L}$

$\sigma_{L}$ is parameterized with a Landau function, $\mathcal{L}_{Landau}$,
and two exponential functions as described below:

$\sigma_{L}(Q^{2}_{bin},-t,W_{bin})=\left\\{\begin{array}[]{l
l}\mathcal{L}_{Landau},\quad 0\leq-t<0.15\\\ exp(c_{1}+c_{2}|-t|),\quad
0.15\leq-t<0.5\\\ exp(c_{3}+c_{4}|-t|),\quad 0.5\leq-t<1.3\end{array}\right.$
(29)

For an analytic expression of $\mathcal{L}_{Landau}$, please see the relevant
CERN-ROOT documentation [24].

##### Parameterization of $\sigma_{T}$

$\sigma_{T}$ is parameterized with a polynomial and an exponential function as
described by Eqn. 30

$\sigma_{T}(Q^{2}_{bin},-t,W_{bin})=\left\\{\begin{array}[]{l
l}c_{0}+c_{1}|-t|+c_{2}|-t|^{2},\quad 0\leq-t<0.2\\\
exp(c_{3}+c_{4}|-t|),\quad 0.2\leq-t<1.3\end{array}\right.$ (30)

Figure 8: Parameterization of VGL $\sigma_{L}$ for the EIC kaon module with
$\Lambda$ channel values (black circles) and $\Sigma^{0}$ channel values
(black squares) plotted along with the polynomial (red line) and exponential
fits (magenta, blue, and green lines) in Eqns. 32, 33.

Figure 9: Parameterization of VGL $\sigma_{T}$ for the EIC kaon module with
$\Lambda$ channel values (black circles) and $\Sigma^{0}$ channel values
(black squares) plotted along with the polynomial (red and blue lines) and
exponential fits (green line) in Eqn. 34. The $\Sigma^{0}$ channel is scaled
down by a factor of 10 to make the two channels visible.

####  3.3.5 Exclusive $p(e,e^{\prime}K^{+}\Lambda[\Sigma^{0}]$) Physics Model

The VGL model [25, 26, 27, 28] by Vanderhaeghen, Guidal, and Laget describes
exclusive hadronic reactions above the resonance region (above $W\approx 2$
GeV) and at low four-momentum transfer ($-t<2$ GeV2). The framework of this
model is based on the exchange of one or more meson Regge trajectories ($K$
and $K^{*}$ trajectories for kaon production) in the $t$-channel. The model
cutoff mass parameter values for these trajectories are taken as
$\Lambda_{K}^{2}$ = $\Lambda_{K^{*}}^{2}$ = 1.5 GeV2 to fit the high $Q^{2}$
behavior of $\sigma_{L}$ and $\sigma_{T}$. Similar to the pion module, a
detailed comparison of the VR [29] and VGL models was undertaken for
$\sigma_{L}$ and $\sigma_{T}$ over a wide range of EIC kinematics. In this
case, the CKY $K^{+}$ model was not yet available for comparison. It was found
that at higher $-t$, the VGL model behaves more consistently than the VR
model, therefore, DEMPgen uses the VGL model to determine the event cross
section (and eventually the weight) for the $\Lambda$ and $\Sigma^{0}$
channels of the kaon module.

##### Model Implementation in DEMPgen

For the cross section parameterization of the $\Lambda$ and $\Sigma^{0}$
channels, the following ranges are chosen, $W$ from 2 to 10 GeV, $Q^{2}$ from
1 to 35 GeV2, and $-t$ up to 2.0 GeV2. There are 9 bins of $W$, each of 1 GeV
width. For each $W$ bin, there are 35 $Q^{2}$ bins, each of 1 GeV2 width. For
each unique bin of $W$ and $Q^{2}$, $\sigma_{L}$ and $\sigma_{T}$ are
parameterized against $-t$ from 0 GeV2 to 2.0 GeV2 as shown in Figs. 8, 9.

Unlike the pion module, when an event is generated with specific values of
$Q^{2}$ and $W$ in the kaon module, the generator looks for the lower and
upper bound values for $Q^{2}$ and $W$ based on the parameterization ranges
and gets the relevant cross-section values. After the values are computed, the
generator uses a truncated Taylor series to determine the cross-section value
at the desired point. The first-degree Taylor polynomial in two variables, $x$
and $y$, for a function $f(x,y)$ about the point $(a,b)$ is given by Eqn. 31.

$f(x,y)=f(a,b)+f_{x}(a,b)(x-a)+f_{y}(a,b)(x-b),$ (31)

where the subscripts represent the respective partial derivatives. This
feature will be implemented for the EIC pion module in a future patch.

##### Parameterization of $\sigma_{L}$

The functional form of the $\sigma_{L}$ parameterization depends upon the $W$
range to effectively describe the VGL model points. For lower $W$ values,
$2<W$ (GeV) $<3$, three exponential functions are utilized. For
$4<W$(GeV)$<10$, a polynomial and two exponential functions are used for the
parameterization, as in Eqns. 32, 33.

For $2<W$ (GeV) $<3$,

$\sigma_{L}(Q^{2}_{bin},-t,W_{bin})=\left\\{\begin{array}[]{l
l}exp(c_{0}+c_{1}|-t|),&t_{1}\leq-t<t_{2}\\\
exp(c_{2}+c_{3}|-t|),&t_{2}\leq-t<t_{3}\\\
exp(c_{4}+c_{5}|-t|),&t_{3}\leq-t<2.0\end{array}\right.$ (32)

and for $4<W$ (GeV) $<10$,

$\sigma_{L}(Q^{2}_{bin},-t,W_{bin})=\left\\{\begin{array}[]{l
l}c_{0}+c_{1}|-t|+c_{2}|-t|^{2},&t_{1}\leq-t<t_{2}\\\
exp(c_{3}+c_{4}|-t|),&t_{2}\leq-t<t_{3}\\\
exp(c_{5}+c_{6}|-t|),&t_{3}\leq-t<2.0\end{array}\right.$ (33)

##### Parameterization of $\sigma_{T}$

Unlike $\sigma_{L}$, $\sigma_{T}$ is parameterized with a common set of
functions over the full $W$ range, two polynomials and an exponential function
as described by Eqn. 34,

$\sigma_{T}(Q^{2}_{bin},-t,W_{bin})=\left\\{\begin{array}[]{l
l}c_{0}+c_{1}|-t|+c_{2}|-t|^{2},&t_{1}\leq-t<t_{2}\\\
c_{3}+c_{4}|-t|+c_{5}|-t|^{2},&t_{2}\leq-t<t_{3}\\\
exp(c_{6}+c_{7}|-t|),&t_{3}\leq-t<2.0\end{array}\right.$ (34)

$t_{1}$ is the minimum value of $-t$ (or $-t_{min}$) for a given set of values
for $Q^{2}$ and $W$. The cutoff points, $t_{2}$ and $t_{3}$, between the
different parameterizations are chosen according to the point where each pair
of parameterizations intersect, $f_{1}(Q^{2}_{bin},-t,W_{bin})$ =
$f_{2}(Q^{2}_{bin},-t,W_{bin})$ and $f_{2}(Q^{2}_{bin},-t,W_{bin})$ =
$f_{3}(Q^{2}_{bin},-t,W_{bin})$ [30]. It is important to note that the
$\sigma_{L}$ and $\sigma_{T}$ are parameterized similarly for the $\Lambda$
and $\Sigma^{0}$ channels.

####  3.3.6 Exclusive ${}^{3}He(e,e^{\prime}\pi^{-}p)(pp)_{sp}$ Physics Model

Many of the elements for this module are the same as the
$p(e,e^{\prime}\pi^{+}n)$ module detailed in Section 3.3.4, except that the
Jacobians from target rest frame to collider frame are not needed, and the
Fermi momentum of the quasi-free struck neutron in 3He must be included. For
brevity, only the differences from the previous module are described.

##### Parameterization of $\sigma_{UU}$

The unpolarized differential cross section (Eqn 18), shorthanded as
$\sigma_{UU}$, and its components are parameterized from the phenomenological
Vrancx-Ryckebusch (VR) model [21] (same model as in Fig. 5 except for
different kinematic range). Model data are generated in the kinematic region
of $Q^{2}$ from 4.0 to 7.5 GeV2, $-t$ from $-t_{min}$ to $-1.0$ GeV2, at a
fixed $W=3.0$ GeV [21, 29, 31], which is within the region of validity of the
VR model. The $W$ dependence is then taken as $(W^{2}-M_{p}^{2})^{-2}$, where
$M_{p}$ is the proton mass [32].

These data were parameterized to fit the following functions:

$\displaystyle\sigma_{L}=\exp{(P_{1}(Q^{2})+|t|P^{\prime}_{1}(Q^{2}))}+\exp{(P_{2}(Q^{2})+|t|P^{\prime}_{2}(Q^{2}))},$
(35)
$\displaystyle\sigma_{T}=\frac{\exp{(P_{1}(Q^{2})+|t|P^{\prime}_{1}(Q^{2}))}}{P_{1}(|t|)},$
(36) $\displaystyle\sigma_{LT}=P_{5}(t(Q^{2})),$ (37)
$\displaystyle\sigma_{TT}=P_{5}(t(Q^{2})).$ (38)

The results of this parameterization are accessed by the SigmaCalc class.

Figure 10: $\vec{n}(e,e^{\prime}\pi^{-}p)$ asymmetry amplitudes vs
$t^{\prime}$ for different values of $Q^{2}$ and $W$. Data points are the raw
model data provided by Goloskokov and Kroll [33]. The lines are the
parameterized fit for each $Q^{2},W$ pair (Eqn. 41).

##### Parameterization of Azimuthal Modulations

The transversely polarized neutron target gives rise to a variety of azimuthal
modulations,

$A(\phi,\phi_{s})=\frac{d\sigma_{UT}(\phi,\phi_{s})}{d\sigma_{UU}(\phi)}=-\sum_{k}A_{UT}^{\sin(\mu\phi+\lambda\phi_{s})_{k}}\sin(\mu\phi+\lambda\phi_{s})_{k},$
(39)

where $d\sigma_{UU}$ is the unpolarized nucleon cross section in terms of the
well-known L, T, LT and TT response functions described above. Six different
azimuthal angular modulations contribute to $A_{UT}$ [34]:

$\displaystyle A_{UT}(\phi,\phi_{S})$ $\displaystyle=$ $\displaystyle
A_{UT}^{\sin(\phi-\phi_{S})}\sin(\phi-\phi_{S})$ (40) $\displaystyle+$
$\displaystyle A_{UT}^{\sin(\phi+\phi_{S})}\sin(\phi+\phi_{S})$
$\displaystyle+$ $\displaystyle A_{UT}^{\sin(\phi_{S})}\sin(\phi_{S})$
$\displaystyle+$ $\displaystyle
A_{UT}^{\sin(2\phi-\phi_{S})}\sin(2\phi-\phi_{S})$ $\displaystyle+$
$\displaystyle A_{UT}^{\sin(3\phi-\phi_{S})}\sin(3\phi-\phi_{S})$
$\displaystyle+$ $\displaystyle
A_{UT}^{\sin(2\phi+\phi_{S})}\sin(2\phi+\phi_{S}).$

The main physics goal of our measurement with SoLID [35] is to measure the
$k=1$ asymmetry amplitude, $A_{UT}^{\sin(\phi-\phi_{s})}$. In addition, the
$k=3$ asymmetry amplitude, $A_{UT}^{\sin(\phi_{s})}$, is also accessible
through the SoLID experiment, and gives information on higher order
transversity GPDs [36, 37].

S. V. Goloskokov and P. Kroll (GK) have provided model data for the first five
asymmetry amplitudes [33]. These data are at discrete values of $Q^{2}$ from
4.107 to 7.167 GeV2, $W$ from 2.362 to 3.191 GeV, and $t^{\prime}=t-t_{min}$
from 0 to 0.5 GeV2. The GK model data are shown in Fig. 10. The sixth
asymmetry amplitude is expected to be much smaller, and is taken to be zero.

The fit functions were chosen only to closely match the shape of the GK model
data, and were not based on any physical principle. They are as follows:

$\displaystyle
A_{UT}^{\sin(\mu\phi+\lambda\phi_{s})_{k}}=\begin{cases}ae^{bt^{\prime}}-(a+c)e^{dt^{\prime}}+c,&\quad
k=1\\\ ae^{bt^{\prime}}+c,&\quad k=2,3,4,5\end{cases}$ (41)

where $a$, $b$, $c$, and $d$ are fit parameters. These fits are carried out
independently for each $Q^{2},W$ pair. The parameterized functions are
displayed alongside the model data in Fig. 10. The $k=1$ fit function
originally had an additional, independent parameter in place of $(a+c)$, but
the fit did not converge reliably. As such, it was constrained to pass through
the origin, justified by the requirement that all asymmetries dependent on
$\phi$ must vanish at $t=t_{min}$, as $\phi$ is undefined in parallel
kinematics. This also applies to all but the $k=3$ asymmetry. However, the
fits were satisfactory, and the additional constraint was not deemed
necessary.

The parameters are stored in a file and read at runtime into instances of the
Asymmetry class. Each Asymmetry object corresponds to one of the five
asymmetries and contains each of the parameterized functions for that
asymmetry, one for each $Q^{2}$, $W$ pair. The Asymmetry class implements a
function to retrieve the asymmetry amplitude given $Q^{2}$ and $t^{\prime}$ as
arguments. The value is calculated by selecting the two functions with the
associated values of $Q^{2}$ closest to the input. These two functions are
each evaluated at the input $t^{\prime}$ value, resulting in two data points,
$(Q^{2}_{1},A_{1})$ and $(Q^{2}_{2},A_{2})$. A line is then drawn between
these two points to interpolate a value for the asymmetry amplitude at the
input $Q^{2}$ value.

The asymmetries are accessed by the SigmaCalc class and used to calculate the
cross section components $\sigma_{UT}$ according to Eqn. 43.

$\displaystyle\Sigma_{k}$
$\displaystyle=d\sigma_{UU}(\phi)A_{UT}^{\sin(\mu\phi+\lambda\phi_{s})_{k}}$
(42) $\displaystyle d\sigma_{UT}$
$\displaystyle=-\frac{P_{T}}{\sqrt{1-\sin^{2}\theta\sin^{2}\phi_{s}}}\sum_{k=1}^{6}\sin(\mu\phi+\lambda\phi_{s})_{k}\Sigma_{k}.$
(43)

#####  $\sigma$ and Event Weight

The cross section components $\sigma_{UU}$ and $\sigma_{UT}$ are summed to
give the overall cross section $\sigma$, as shown by Eqn. 44,

$d\sigma=d\sigma_{UU}+d\sigma_{UT},$ (44)

where $\sigma_{UT}$ is determined from Eqns 39, 40, 41.

Figure 11: 3He spectral function generated according to the Argonne Nuclear
Potential [38]. Figure 12: Comparison of $Q^{2}$ (left) and $t$ (right)
weighted distributions for the 3He$(e,e^{\prime}\pi^{-}p)(pp)_{SP}$ reaction
with Fermi momentum disabled (blue) and enabled (red). Figure 13: Missing mass
and momentum distributions for the 3He$(e,e^{\prime}\pi^{-})p(pp)_{SP}$
reaction with only energy loss effects enabled (red squares), and with no
corrective effect enabled (blue circles).

Explicitly, this cross section is a two-fold differential scattering cross
section in the center of mass frame. In order to calculate the event weight,
the five-fold differential cross section in the lab frame is needed. This is
calculated as follows

$d^{5}\sigma=\frac{d^{5}\sigma}{de^{\prime}d\Omega_{e^{\prime}}d\Omega_{\pi}}=\Gamma_{V}J\frac{d^{2}\sigma}{dtd\phi},$
(45)

where $J$ is the Jacobian transformation from the center of mass frame to the
lab frame. $\Gamma_{V}$ is the virtual photon flux factor,

$\Gamma_{V}=\frac{\alpha}{2\pi}\frac{e^{\prime}}{E}\frac{(W^{2}-M_{n}^{2})}{2M_{n}Q^{2}}\frac{1}{1-\epsilon},$
(46)

where $\alpha$ is the fine structure constant, $W$ is the invariant mass of
the final state, and $\epsilon$ is the virtual photon polarization,

$\epsilon=\left(1+\frac{2|\vec{q}|^{2}}{Q^{2}}\tan^{2}\frac{\theta_{e}}{2}\right)^{-1}.$
(47)

The event weight is then given by the following expression:

$\mathrm{Weight}=\frac{\sigma\times PSF\times CF\times\mathcal{L}\times
TF}{N_{Requested}},$ (48)

where $TF$ are target factors and $N_{Requested}$ is the number of events the
generator tried to produce. The other variables given in this equation are the
same as those defined in Eqn. 9. The assumed luminosity for the SoLID DEMP
experiment is $10^{36}$ cm-2s-1 [39]. The target-factors include the $\sim$60%
target polarization and the 85.6% effective polarized neutron [40, 41] of the
Jefferson Lab polarized 3He target.

##### Fermi Momentum Effects

The target neutron in the SoLID experiment is contained within a 3He nucleus.
As such, the neutron has a non-zero momentum in the 3He target frame, known as
Fermi momentum. Fermi momentum is incorporated into the event generator in the
TargetGen class, which generates the target neutron’s momentum before the main
kinematics calculation is performed.

The calculation of particle kinematics begins with random generation of the
energy and momentum of the target neutron, if Fermi momentum is enabled in the
generator. If Fermi momentum is not enabled, it is set to zero momentum, with
energy equal to the neutron rest mass. The direction of the neutron’s Fermi
momentum is chosen uniformly using sphere point picking [18]. The magnitude of
the Fermi momentum follows the distribution shown in Fig. 11, chosen randomly
according to the Argonne Nuclear Potential [38].

The Fermi distribution was originally generated with the following
normalization,

$4\pi\int p^{2}\frac{dF}{d\vec{p}}dp\sin\theta d\theta d\phi=2.$ (49)

The distribution was normalized to two in order to describe the two protons in
the 3He nucleus. For the single neutron, the distribution needs to be
normalized to one, as such the data have simply been divided by two and reused
here.

The resulting momentum distribution is given by a set of 1000 discrete data
points, providing the probability density from 0 to 1 GeV/c. As there is no
clear, theoretically motivated, functional form for this distribution, the
momentum is selected in the generator by a simple Monte Carlo procedure: A
point, ($x\in[0,300],y\in[0,6.03]$), is randomly selected. If this point lies
within the bounded area below the curve on Fig. 11, then it is used, otherwise
the procedure repeats until a point in the bounded area is found. In order to
cut down on computation time, the data is truncated at 300 MeV/c, beyond which
the probabilities are negligible.

Figs. 12, 13 demonstrate the effect of Fermi momentum on the $Q^{2}$ and $t$
missing momentum and mass distributions (Eqns. 15, 17) of the generated data.
These plots indicate that the effect of Fermi momentum upon the data is
minimal for SoLID kinematics.

##### Final State Interaction Effects

Figure 14: Weighted missing momentum distribution in each $t$ bin for quasi-
free $\pi^{-}$ production from the neutron in 3He, with FSI and Fermi momentum
enabled, compared to the distribution with only Fermi momentum enabled, and
with no effects enabled. The FSI distribution uses the Catchen weight in its
weighting. The solid black line indicates the cut point $|\vec{p}_{miss}|>1.2$
GeV/c.

When the target nucleon emits the charged pion in the
3He$(e,e^{\prime}\pi^{-}p)(pp)_{SP}$ reaction, it is possible for the pion to
scatter off of one of the spectator nucleons via the
$\pi^{-}p\rightarrow\pi^{\prime}p^{\prime}$ process as it passes through the
nuclear volume. This secondary reaction is known as a Final State Interaction
(FSI). FSI effects have been estimated by calculating the kinematics using
elastic scattering and the scattering cross section using phase-shift
parameterizations by Rowe, Solomon and Landau [42]. Charge exchange reactions
of the form $\pi^{0}n\rightarrow\pi^{-}p$ are excluded, as the purpose of the
FSI module is to study the relative effects of FSI on the beam target
asymmetries, and both charge conserving and charge exchanging reactions should
have similar kinematic effects.

FSI is implemented in the event generator using another instance of the
TargetGen class to generate a target proton with Fermi Momentum, as described
in Sec. 3.3.6. A random direction is selected with sphere point picking [18]
to determine the direction of the scattered pion in the pion-nucleon center of
mass frame. In the center of mass frame, the total momentum is zero, so:

$\displaystyle|\vec{p_{\pi}}|$ $\displaystyle=|\vec{p_{p}}|=p$ (50)
$\displaystyle|\vec{p^{\prime}_{\pi}}|$
$\displaystyle=|\vec{p^{\prime}_{p}}|=p^{\prime}.$ (51)

The conservation of energy equation then may be expressed as:

$\displaystyle E_{\pi}+E_{p}$ $\displaystyle=E^{\prime}_{\pi}+E^{\prime}_{p}$
(52) $\displaystyle\sqrt{p^{2}+m_{\pi}^{2}}+\sqrt{p^{2}+m_{p}^{2}}$
$\displaystyle=\sqrt{{p^{\prime}}^{2}+m_{\pi}^{2}}+\sqrt{{p^{\prime}}^{2}+m_{p}^{2}}.$
(53)

The only solution to this equation is $p=p^{\prime}$, and so the kinematics of
the outgoing particles are trivial. They are then transformed back to the lab
frame.

The implementation of the FSI cross section calculation, i.e. the calculation
of the $\pi^{-}p$ differential scattering cross section, was written by A.
Shinozaki [43], and further modified by us. This differential cross section is
given in the center of mass frame, and must be transformed via a Jacobian into
a lab frame value which may be used as a correcting factor to the overall
event weight. Three different formulations of the Jacobian are available.

The “William’s Weight” uses the following Jacobian [44]:

$\displaystyle J_{Williams}$
$\displaystyle=\frac{|p_{\pi,lab}|^{2}}{\gamma|p_{\pi,com}|(|p_{\pi,lab}|-\beta
E_{\pi,lab}\theta_{\pi,lab})}$ (54) $\displaystyle\gamma$
$\displaystyle=\frac{E_{\pi,lab}+E_{p,lab}}{|p_{\pi,lab}+p_{p,lab}|}$ (55)
$\displaystyle\beta$
$\displaystyle=\frac{|p_{\pi,lab}|+|p_{p,lab}|}{E_{\pi,lab}+E_{p,lab}}$ (56)

The “Dedrick Weight” uses the following Jacobian [45]:

$\displaystyle J_{Dedrick}$
$\displaystyle=\frac{\left((g+\cos^{2}(\theta_{\pi,com})+(1-\beta^{2})(1-\cos^{2}(\theta_{\pi,com}))\right)^{3/2}}{(1-\beta^{2})|(1+g\cos(\theta_{\pi,com}))|}$
(57) $\displaystyle g$ $\displaystyle=\frac{\beta E_{\pi,com}}{p_{\pi,com}},$
(58)

where $\beta$ is the same as in the William’s Weight.

Finally, the “Catchen Weight” uses the following Jacobian [46]:

$J_{Catchen}=\frac{|p_{\pi,lab}|^{2}E_{\pi,com}}{|p_{\pi,com}|^{2}E_{\pi,lab}}.$
(59)

The effects of FSI on the missing momentum distributions are shown in Fig. 14,
using Catchen Weight in weighting the FSI-enabled data. The figure shows that
a secondary interaction has a much more significant effect on the data than
the other corrective effects. However, they also indicate that events which
undergo FSI occur at a much smaller rate than those that do not. Furthermore,
Fig. 14 indicates that the majority of FSI events can be eliminated by cutting
events with $|\vec{p}_{miss}|>1.2$ GeV/c. Only $~{}4\%$ of the FSI events
remain after the cut.

##  4 Results

|  Kinematic ranges for EIC Module
---|---
$\theta_{e^{\prime}}$ |  User specified, default is 60∘ to 175∘
$E_{e^{\prime}}$ |  User specified, expressed as fraction of electron beam energy ($E_{e}$), default is 0.5$E_{e}$ to 2.5$E_{e}$
$\phi_{e^{\prime}}$ |  0∘ to 360∘
$\theta_{Ej}$ |  User specified, default is 0∘ to 50∘
$\phi_{Ej}$ |  0∘ to 360∘
$Q^{2}$ |  Varies by reaction, 3 GeV2 to 35 GeV2 for $\pi^{+}$, 1 GeV2 to 35 GeV2 for $K^{+}$
$-t$ |  $\pi^{+}$ FF mode: up to 0.5 GeV2, $\pi^{+}$ TSSA mode: up to 1.2 GeV2, $K^{+}$ up to 2.0 GeV2
$W$ |  User specified, default varies by reaction, typically $\approx$ 2 to 10 GeV
$\mathcal{L}$ |  Varies depending upon beam energy combination, see B for details, default is $1\times 10^{33}$ cm-2s-1
Table 2: Kinematic ranges for $p(e,e^{\prime}X_{Ej}X_{Rec})$ event generation
for the EIC. Note that the $\pi^{+}$ TSSA mode for the EIC is currently a work
in progress. For more details on user specified quantities, see A. Figure 15:
Exclusive $p(e,e^{\prime}\pi^{+}n)$ kinematic distributions for $e^{\prime}$
(left),$\pi^{+}$ (center), $n$ (right) at $5\times 41$ (top), $5\times 100$
(middle) and $10\times 100$ (bottom) GeV EIC beam energy combinations with
$5<Q^{2}$(GeV${}^{2})<35$. Due to the EIC beam crossing angle of 25 mrad, the
neutron event distributions are shifted from zero – in reality, they are
centered about a line tangent to the proton beam trajectory at the interaction
point. The $z$-axis (color scale) is logarithmic and shows the rate (in Hz)
per bin.  Figure 16: Weighted distribution of the $p(e,e^{\prime}\pi^{+}n)$
reaction produced by DEMPgen for EIC kinematics ($5\times 100$ beam
combination). The $z$-axis (color scale) is logarithmic and shows the rate (in
Hz) per bin. Figure 17:  Comparison of DEMP $p_{n}$ distributions and SIDIS
$p_{miss}$ distributions for different bins in $Q^{2}$ from EIC simulations
($5\times 100$ beam combination). The DEMP events are generated for the
kinematic ranges listed in Table 1, while the SIDIS events have significantly
broader distributions in $W$ and $t$. The distributions have been arbitrarily
scaled to demonstrate the difference in shape. Shown in green are example cut
values that could be used to distinguish between DEMP and SIDIS events [47].
Figure 18: Weighted $y$ distribution of the $p(e,e^{\prime}\pi^{+}n)$ reaction
produced by DEMPgen for EIC kinematics ($5\times 100$ beam combination) with
$5<Q^{2}$ (GeV2) $<35$ and the $W$, $t$ ranges listed in Table 1.

###  4.1 EIC Kinematic Ranges

In DEMPgen, EIC events are generated within specific kinematic variable
ranges. Many of the variable limits are user defined, and some depend upon the
reaction being generated. The limits for these variables, over which events
are generated for the EIC, are shown in Table 2.

Note that beam energy combinations at the EIC are typically expressed as

$\displaystyle E_{e}\times E_{Pr},$ (60)

where $E_{e}$ is the electron beam energy in GeV and $E_{Pr}$ is the proton
(or ion) beam energy in GeV, i.e. $5\times 100$ represents 5 GeV electrons
colliding with 100 GeV protons.

###  4.2 Exclusive $p(e,e^{\prime}\pi^{+}n)$ Projections for the EIC

Figure 19:  Existing data (green [48, 49]; black circles [50]; black triangles
[6, 51]; blue and yellow [6, 52, 53]) and projected uncertainties for future
$F_{\pi}$ data from JLab (violet [7]) and the EIC (black squares), in
comparison to a variety of models of charged pion structure (black dot [54];
red solid [55]; orange [56]; cyan [57]; and green [58], where Hard is pQCD
with analytic running coupling, and the band is Hard+Soft including non-
perturbative uncertainties). The EIC projections, obtained with the use of
DEMPgen, cover a wide range in $Q^{2}$, providing access to the emergent mass
scale in QCD.

To illustrate the utility of DEMPgen, we briefly describe our EIC measurement
feasibility studies published in Ref. [59]. The EIC is projected to be capable
of delivering proton beam energies of up to 275 GeV and electron beam energies
up to 18 GeV [60]. The beam crossing angle at IP6, the location of the ePIC
detector, is planned to be 25 mrad, i.e. the proton beam will make an angle of
25 mrad with respect to the $z$ axis, with the electron beam propagating in
the $-\hat{z}$ direction.

$p(e,e^{\prime}\pi^{+}n)$ events were generated with DEMPgen assuming an
integrated luminosity of 20 fb-1 for 5$\times$100 GeV electron/proton
collisions. DEMP event kinematic distributions are shown in Fig. 15. All three
outgoing particles, the electron, pion and neutron are required to be detected
to ensure exclusivity. The neutrons take nearly all of the proton beam
momentum and are detected at very forward angles (in the Zero Degree
Calorimeter, ZDC). The scattered electrons and pions have similar momenta,
however, the electrons are distributed over a wider range of angles. For
example, for 5$\times$100 GeV electron/proton collisions, the 5-6 GeV/c
electrons are primarily scattered 25-45∘ from the electron beam, while the
5-12 GeV/c $\pi^{+}$ are 7-30∘ from the proton beam. Similarly, Fig. 16 shows
the $Q^{2}$ versus $-t$ distribution of the generated DEMP events.

Simulation studies demonstrated that event selection cuts were highly
effective in isolating pion DEMP events from background pion SIDIS
($p(e,e^{\prime}\pi^{+})X$) events, as seen in Fig. 17. The selection cuts
included a cut on $y$, the fractional energy loss (defined in Eqn. 21). DEMP
events predominantly have $y>0.01$ (Fig. 18). Further details of the event
selection cuts can be found in [59].

For the projected statistical and systematic uncertainties, the following
assumptions were made:

  * 1.

Integrated luminosity of 20 fb-1 for the 5$\times$100 GeV measurement, as
described above.

  * 2.

Clean identification of exclusive $p(e,e^{\prime}\pi^{+}n)$ events by tagging
the high energy, forward going neutron in the Zero Degree Calorimeter (ZDC),
as determined by passing the DEMPgen events through a Geant4 simulation of the
IP6 detector.

  * 3.

Systematic uncertainties of 2.5% point-to-point, and 12% scale, similar to the
ZEUS leading neutron measurement [61].

  * 4.

As the EIC cannot access the low $\epsilon<0.8$ needed to do a quality
L/T-separation, a model is required to subtract the estimated $\sigma_{T}$
contribution from the measured cross sections to yield $\sigma_{L}$. For the
error propagation, $R=\sigma_{L}/\sigma_{T}=0.013-0.14$ is assumed at the
lowest $-t$, and $\delta R=R$ systematic uncertainty is assumed in the model
[21] subtraction to isolate $\sigma_{L}$.

  * 5.

Pion pole channel dominance at small $-t$ over the measured $Q^{2},W$ range
will need to be confirmed in a separate measurement of exclusive
$\pi^{-}/\pi^{+}$ ratios obtained from electron-deuteron collision data. If
this check is not done, an additional systematic uncertainty in the pion pole
dominance will be required.

Under these conditions, we have concluded that $F_{\pi}$ measurements at the
EIC are feasible up to $Q^{2}\approx 30$ GeV2 (Fig. 19). A consistent and
robust EIC $F_{\pi}$ data set will probe deep into the region where
$F_{\pi}(Q^{2})$ exhibits strong sensitivity to both emergent mass generation
via DCSB and the evolution of this effect with distance scale.

###  4.3 Exclusive $p(e,e^{\prime}K^{+}\Lambda[\Sigma^{0}]$) Projections for
the EIC

Figure 20: Exclusive $p(e,e^{\prime}K^{+})\Lambda$ kinematic distributions for
$e^{\prime}$ (left), $K^{+}$ (center), $\Lambda$ (right) at $5\times 41$
(top), $5\times 100$ (middle) and $10\times 100$ (bottom) GeV EIC beam energy
combinations with $5<Q^{2}$ (GeV2) $<35$. Due to the EIC beam crossing angle
of 25 mrad, the $\Lambda$ event distributions are shifted from zero – in
reality, they are centered about a line tangent to the proton beam trajectory
at the interaction point. The $z$-axis (color scale) is logarithmic and shows
the rate (in Hz) per bin.  Figure 21: Exclusive
$p(e,e^{\prime}K^{+})\Sigma^{0}$ kinematic distributions for $e^{\prime}$
(left), $K^{+}$ (center), $\Sigma^{0}$ (right), analogous to Fig. 20.  Figure
22: Weighted distribution of $p(e,e^{\prime}K^{+})\Lambda$ reaction produced
by DEMPgen for EIC kinematics ($5\times 100$ beam combination). The $z$-axis
(color scale) is logarithmic and shows the rate (in Hz) per bin.  Figure 23:
Weighted $y$ distribution of the $p(e,e^{\prime}K^{+})\Lambda$ reaction
produced by DEMPgen for EIC kinematics ($5\times 100$ beam combination) with
$5<Q^{2}$ (GeV2) $<35$.

DEMP event kinematic distributions for the $\Lambda$ and $\Sigma^{0}$ channels
are shown in the Figs. 20, 21. The $K^{+}$ event distributions are highly
similar to the $\pi^{+}$ distributions shown in Fig. 15, in that the electron
is scattered (mostly) higher in energy in the electron endcap region, the
$K^{+}$ has moderate momentum in the hadron endcap region, while the ejectile
hyperon takes most of the incident proton beam momentum at very forward angle.
Similarly, Fig. 22 shows the $Q^{2}$ versus $-t$, and Fig. 23 presents the $y$
distribution of generated DEMP events for the $\Lambda$ channel.

A significant difference from the $\pi^{+}$ channel at the EIC is that the
identification of the exclusive $K^{+}$ production reaction requires the
efficient reconstruction of the $\Lambda[\Sigma^{0}]$ from its decay products
in the far forward detectors. This is a non-trivial task for which the DEMPgen
events are an essential prerequisite for the necessary detector reconstruction
and acceptance studies. These simulation studies are in advanced progress,
where we are investigating both the charged $\Lambda\rightarrow p\pi^{-}$ and
neutral $\Lambda\rightarrow n\pi^{0}$ decay modes, as well as
$\Sigma^{0}\rightarrow\Lambda\gamma$. When these studies are completed, we
will have a significantly better understanding of the feasibility of $F_{K}$
measurements using the EIC far forward detectors, and will disseminate these
results in a future publication.

###  4.4 Exclusive $\vec{{}^{3}He}(e,e^{\prime}\pi^{-}p)(pp)_{SP}$ Projections
for SoLID at Jefferson Lab

Figure 24: Weighted kinematic coverage of the three final state particles in
the 3He$(e,e^{\prime}\pi^{-}p)(pp)_{SP}$ reaction produced by the DEMP event
generator for SoLID experiment kinematics. The color axis represents the rate
for each bin. Figure 25: Weighted acceptance of
3He$(e,e^{\prime}\pi^{-}p)(pp)_{SP}$ reaction produced by DEMPgen and measured
in SoLID. The color axis represents the expected yield of DEMP events in the
experiment. Figure 26: Missing momentum spectra of recoil protons in DEMP
(blue) and SIDIS (red) processes in the 3He$(e,e^{\prime}\pi^{-}p)(pp)_{SP}$
reaction. The dashed magenta curve is the DEMP missing mass only considering
the Fermi motion, multiple scattering and the energy loss, while the blue and
red curves have further taken into account the detector resolutions. The
light-blue dashed line indicates $P_{miss}<1.2$ GeV/c, beyond which are mostly
SIDIS events. Only generated events with $W>2$ GeV and $Q^{2}>4$ GeV2 are
shown. The normalization of the SIDIS background is approximate.  Figure 27:
3He$(e,e^{\prime}\pi^{-}p)(pp)_{SP}$ $A_{UT}$ asymmetries of Eqn. 40 for a bin
with central kinematics of $<-t>=0.45$ GeV2, $<Q^{2}>=5.77$ GeV2,
$<x_{B}>=0.47$, binned as a 2-dimensional scatter plot for $12\times 12$
$(\phi,\phi_{S})$ bins. Dark (bright) color indicates negative (positive)
transverse target single-spin asymmetry for that bin.

Like the EIC simulations discussed above, for the SoLID
3He$(e,e^{\prime}\pi^{-}p)(pp)_{SP}$ experiment we anticipate the necessity to
detect the scattered electron, $\pi^{-}$ and recoil proton in coincidence to
ensure exclusivity. The SoLID trigger will be configured so that all
electron-$\pi^{-}$ coincidence events will be written to disk, and the event
sample mined to select those events for which an additional proton is
detected.

To study the feasibility of this approach, we generated events with DEMPgen
using the following configuration:

  * 1.

$e^{-}$ Beam energy: 11 GeV

  * 2.

$e^{\prime}$ energy: 1.1 to 9.9 GeV

  * 3.

$e^{\prime}$ $\theta$: $5^{\circ}$ to $27^{\circ}$

  * 4.

$\pi$ $\theta$: $6^{\circ}$ to $18^{\circ}$

In addition, any events meeting the following criteria are discarded due to
being outside the accurate range for the cross section model:

  * 1.

$-t>1.2$ GeV2

  * 2.

$Q^{2}<4$ GeV2

  * 3.

$W<2$ GeV

Fig. 24 shows the weighted momentum and polar angle distribution of particles
generated in this configuration. This is a fixed target experiment. The
scattered electrons and $\pi^{-}$ are emitted at forward angles from 5 to 27∘
and take most of the beam momentum, up to 9 GeV/c. In contrast, the recoil
proton has much lower momentum and is emitted over a much wider angular range
(up to 50∘). The two spectator protons will essentially have only Fermi
momentum and will be ranged out in the target and other detector elements.
Fig. 25 shows the $Q^{2}$ versus $-t$ coverage of the events in SoLID, and
projected event statistics for a 48 day experimental run.

To further separate the DEMP events from competing non-exclusive reactions,
such as SIDIS, missing mass and momentum cuts are planned to be used in the
analysis of the experimental data. The missing momentum is also useful in
identifying events which have undergone a final state interaction (Sec.3.3.6).
The missing mass and momentum distributions produced by DEMPgen with Fermi
momentum and various other corrective effects enabled are shown in Fig. 26.
Similar to what was already indicated in Fig. 17, the DEMP events have unique
kinematics on the edge of the SIDIS missing momentum distribution. With
sufficiently good event reconstruction resolution, this makes it
experimentally possible to cleanly separate the small DEMP cross section from
the orders of magnitude larger SIDIS background.

DEMPgen can be used to study the experimental effects on the measured target
asymmetries for different kinematics. For example, by turning on or off the
FSI or Fermi momentum modules, their effects on the predicted asymmetries can
be identified. Fig. 27 shows the generated asymmetry from a transversely
polarized 3He target for a bin with central kinematics of $<-t>=0.45$ GeV2,
$<Q^{2}>=5.77$ GeV2, $<x_{B}>=0.47$. The axes are the two azimuthal angles in
Fig. 2, the angle with respect to the target polarization ($\phi_{S}$) on the
vertical axis, and the angle between the scattering and reaction planes
($\phi$) on the horizontal axis. For these kinematics, the
$A_{UT}^{sin(\phi_{S})}$ modulation dominates, as evident by the trend from
positive (yellow) asymmetry bins at top to negative (blue) asymmetry bins at
bottom. The other modulations are significantly smaller, and are identified by
a fitting of the full Eqn. 40 to the data. The SoLID projected data are
expected to be a considerable advance over the only existing dataset from
HERMES [36] in terms of both kinematic coverage and statistical precision. For
more details on the SoLID experiment feasibility studies using DEMPgen,
consult Ref. [35].

##  5 Summary and Outlook

We have developed DEMPgen, a powerful tool for future feasibility studies of
proposed Deep Exclusive Meson Production (DEMP) measurements at Jefferson Lab
(JLab) and the Electron-Ion Collider (EIC). Based on the results we have
obtained, we expect exclusive $\pi^{+}$ form factor measurements at the EIC to
be feasible up to about $Q^{2}=30$ GeV2, which would be a considerable
extension in kinematic range over what is possible with 11 GeV beam at JLab.
DEMPgen was also essential for feasibility studies of proposed transverse
target single-spin asymmetry measurements in exclusive $\pi^{-}$ production
from the polarized neutron in 3He with SoLID. Both of these studies have
established the reliability and utility of DEMPgen.

DEMPgen is modular in form so that additional reaction channels can be added
to it over time. We have described a recent extension, namely $t$-channel
$K^{+}$ production leading to the $\Lambda$ and $\Sigma^{0}$ final states.
There is a lot of interest in our $K^{+}$ studies, to see whether it will be
feasible to extract the $K^{+}$ form factor to high $Q^{2}$ at the EIC [60].
We have used DEMPgen to generate predicted kinematic distributions and rates
for the $e^{\prime}$, $K^{+}$, $\Lambda[\Sigma^{0}]$ final state. Detailed
detector simulations to establish whether the reconstruction of the
$\Lambda[\Sigma^{0}]$ from their detected decay products is sufficiently
reliable are ongoing.

Further extensions of DEMPgen to a broader kinematic regime are envisioned.
Particularly for the $K^{+}$ form factor feasibility studies, the lower beam
energy combinations planned for the Electron-Ion Collider in China (EicC) [62]
may prove to be vital for the reliable identification of the $\Lambda$ from
its decay products, as the $\Lambda$ will decay closer to the interaction
region than at the EIC, and the decay products correspondingly easier to
detect.

Similarly, should the JLab 22 GeV upgrade [63] come to fruition, it would be
straightforward to extend the cross section and target asymmetry model
parameterizations to cover the kinematic range enabled by the upgrade. There
is also interest in our adding exclusive $\pi^{\pm}$ reactions from the
deuteron to the generator [60], as the exclusive $\pi^{\pm}$ ratios versus $t$
are needed to determine the extent of non-pion pole contributions to the data
from which the pion charge form factor are extracted (Sec. 4.2).

Finally, we also plan to add $u$-channel exclusive $\pi^{0}$ production to
DEMPgen, optimized for EIC studies. If the reliability of EIC $u$-channel
$\pi^{0}$ studies can be established, it would open up a novel kinematic range
for the study of Transition Distribution Amplitudes (TDA) [64] in the backward
colinear factorization regime.

## Acknowledgements

We thank Nathan Heinrich and Zhihong Ye for their assistance with Figs. 1, 26,
27. This work was supported by the Natural Sciences and Engineering Research
Council of Canada (NSERC) SAPPJ-2023-00041 and SAPIN-2021-00026, the UK
Science and Technology Facilities Council (STFC) grants ST/V001035/1,
ST/W004852/1, and Department of Energy grant DE-FG02-05ER41372 (WL). We
gratefully acknowledge the support from the Stony Brook University Center for
Frontiers of Nuclear Science Director Abhay Deshpande.

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##  Appendix A  json Control Cards

DEMPgen utilizes .json control cards to customise various parameters when
running the event generator. Some parameters are only applicable to EIC or
SoLID simulations and others are applicable in both cases. Table 3 outlines
the options that are common in both cases, Table 4 specifies the EIC module
options and the SoLID module options are listed in Table 5.

Table 3: Common input .json file options Parameter |  Description
---|---
Experiment |  “eic” or “solid”, controls the type of event generated subsequently.
file_name |  Name of the output file created by the generator.
n_events |  Number of events to be thrown by the generator, this is _not_ the number of total events that will be saved in the output. This is equivalent to $N_{Requested}$ as described eslewhere in the text.
generator_seed |  Random number generator seed used by the event generator.
Kinematics_type |  1 for FF (EIC) or 2 for TSSA (SoLID). May include TSSA for EIC in the future.
particle |  Choices are omega, pi+, pi0 or K+. This is the produced ejectile (meson) in the reaction.
Table 4: EIC module only input .json file options Parameter |  Description
---|---
hadron |  Lambda or Sigma0, only used in EIC kaon DEMP event generation.
ebeam |  Incident electron beam energy. Typically 5, 10 or 18, but can be set arbitrarily.
hbeam |  Incident hadron (ion) beam energy, typically, 41, 100 or 275, but can be set arbitrarily.
hbeam_part |  Hadron (ion) beam particle, proton, deut or helium3. Work in progress.
det_location |  Detector location, “ip6” or “ip8”. This option sets the beam crossing angle accordingly for each potential EIC interaction point.
OutputType |  Fixes the output file type options are LUND, Pythia6 or HEPMC3. SoLID output is LUND only and this flag is not used.
Ee_Low |  Minimum scattered electron energy generated, as a multiple of the electron beam energy. Default is 0.5.
Ee_High |  Maximum scattered electron energy generated, as a multiple of the electron beam energy. Default is 2.5.
e_Theta_Low |  Minimum scattered electron $\theta$ that will be generated, in degrees. Default is 60.
e_Theta_High |  Maximum scattered electron $\theta$ that will be generated, in degrees. Default is 175.
EjectileX_Theta_Low |  Minimum ejectile (meson) $\theta$ that will be generated, in degrees. Default is 0.
EjectileX_Theta_High |  Maximum ejectile (meson) $\theta$ that will be generated, in degrees. Default is 50.
Table 5: SoLID module only input .json file options Parameter |  Description
---|---
beam_energy |  Incoming electron beam energy
Targ_dir |  Target direction, 1 for up or 2 for down.
targ_pol_x |  Target polarisation in the $x$ direction.
targ_pol_y |  Target polarisation in the $y$ direction.
targ_pol_z |  Target polarisation in the $z$ direction.
scat_elec_Emin |  Minimum scattered electron energy generated, in MeV.
scat_elec_Emax |  Maximum scattered electron energy generated, in MeV.
scat_elec_thetamin |  Minimum scattered electron $\theta$ generated, in degrees.
scat_elec_thetamax |  Maximum scattered electron $\theta$ generated, in degrees.
prod_pion_thetamin |  Minimum $\pi$ $\theta$ generated, in degrees.
prod_pion_thetamax |  Maximum $\pi$ $\theta$ generated, in degrees.
multiple_scattering |  “true” or “false” to enable or disable multiple scattering effects.
ionisation |  “true” or “false” to enable or disable ionisation effects.
bremmstrahlung |  “true” or “false” to enable or disable bremmstrahlung effects.
final_state_interaction |  “true” or “false” to enable or disable final state interaction effects.
fermi_momentum |  “true” or “false” to enable or disable Fermi momentum effects.
weight_cut |  “true” or “false” to enable or disable a cut on the event weight
w_cut |  “true” or “false” to enable or disable a cut on the $W$ value of generated events.
w_min |  Minimum value of $W$ retained by the generator (if cut is enabled).
Qsq_cut |  “true” or “false” to enable or disable a cut on the $Q^{2}$ value of generated events.
Qsq_min |  Minimum value of $Q^{2}$ retained by the generator (if cut is enabled).
t_cut |  “true” or “false” to enable or disable a cut on the $t$ value of generated events.
t_min |  Minimum value of $t$ retained by the generator (if cut is enabled).

##  Appendix B  EIC Luminosity

In processing events, DEMPgen utilizes the luminosity to determine the event
weight. For EIC event generation, the luminosity is set depending upon the
beam energy combination specified. The luminosity values for each beam energy
combination are specified in Table 6. Note that some sample beam energy
combinations for the Electron-Ion Collider China (EicC) have been included
too[62].

$E_{e}$ (GeV) |  $E_{p}$ (GeV) |  $\mathcal{L}$ ($10^{33}\,cm^{-2}s^{-1}$) |  Note
---|---|---|---
2.8 |  13 |  0.7 |  EicC
3.5 |  20 |  2 |  EicC
5 |  41 |  0.44 |  -
5 |  100 |  3.68 |  -
10 |  100 |  4.48 |  -
18 |  275 |  1.54 |  -
- |  - |  1 |  Default
Table 6: Luminosity values used by DEMPgen for different electron/proton beam
energy combinations. Values for the EIC were taken from [65].

##  Appendix C  Quadratic Equation in Analytical Solution

While solving for the energy of the ejectile using Eqn .7 in the analytical
solution, a quadratic equation can be generated. This is given by:

$a[E^{2}_{Ej}]+b[E_{Ej}]+c=0,$ (61)

where $a$, $b$ and $c$ are constants and depends on the four momentum of
initial particles as well as the direction of ejectile as:

$a=4[\vec{P}^{2}_{i}(\hat{P}_{i}.\hat{P}_{Ej})^{2}-E^{2}_{i}]$ (62)
$b=4[E_{Ej}(E^{2}_{Ej}-\vec{P}^{2}_{i}+M^{2}_{Ej}-M^{2}_{Rec})]$ (63)
$c=-[4(\vec{P}^{2}_{i}(\hat{P}_{i}.\hat{P}_{Ej})^{2})M^{2}_{Ej}+(E^{2}_{Ej}-\vec{P}^{2}_{i}+M^{2}_{Ej}-M^{2}_{Rec})^{2}]$
(64)

Here, $\vec{P}_{i}$ is the net initial three momenta, $\hat{P}_{in}$ and
$\hat{P}_{Ej}$ are the net initial and ejectile unit vectors and $E_{i}$ is
the net initial energy.

##  Appendix D  Phase Space Factor

The phase-space-factor is the fraction of the total kinematically accessible
phase space that is covered by the event generator. This factor is a function
of the incoming electron beam energy and the angles over which the scattered
electron and produced ejectile are generated. This is calculated as -

$PSF=\left(\left(E_{e^{\prime}Max}-E_{e^{\prime}Min}\right)d\Omega_{e^{\prime}}\left(\theta,\phi\right)d\Omega_{Ej}\left(\theta,\phi\right)\right),$
(65)

where $E_{e^{\prime}Max}$ and $E_{e^{\prime}Min}$ are the maximum and minimum
energy that the generated scattered electron can have. $d\Omega_{e^{\prime}}$
and $d\Omega_{Ej}$ are the solid angles over which the scattered electron and
ejectile are generated respectively. The resulting value of the PSF is a
quantity in units of $MeVsr^{2}$ (SoLID) or $GeVsr^{2}$ (EIC).

$E_{e^{\prime}Max}$ and $E_{e^{\prime}Min}$ can be specified by the user in
the .json input file (see Appendix A for more details). The $\theta$ range
over which scattered electrons and ejectiles are generated is also specified
by the user in the same manner, however, the generator is hard coded to
generate both of these particles across $2\pi$ in $\phi$.