Nuclear Inst. and Methods in Physics Research, A 1025 (2022) 166164

10.1016/j.nima.2021.166164

MPP-2021-174

aainstitutetext: Department of Physics and Astronomy, University of
California, Riverside, CA 92521, USAbbinstitutetext: Thomas Jefferson National
Accelerator Facility, Newport News, VA 23606, USAccinstitutetext: Max-Planck-
Institut für Physik, Föhringer Ring 6, 80805 München, Germanyddinstitutetext:
Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USAeeinstitutetext: Berkeley Institute for Data Science, University of
California, Berkeley, CA 94720, USA

# Reconstructing the Kinematics of Deep Inelastic Scattering with Deep
Learning

Miguel Arratia c Daniel Britzger a Owen Long,***Corresponding author. d,e and
Benjamin Nachman

###### Abstract

We introduce a method to reconstruct the kinematics of neutral-current deep
inelastic scattering (DIS) using a deep neural network (DNN). Unlike
traditional methods, it exploits the full kinematic information of both the
scattered electron and the hadronic-final state, and it accounts for QED
radiation by identifying events with radiated photons and event-level momentum
imbalance. The method is studied with simulated events at HERA and the future
Electron-Ion Collider (EIC). We show that the DNN method outperforms all the
traditional methods over the full phase space, improving resolution and
reducing bias. Our method has the potential to extend the kinematic reach of
future experiments at the EIC, and thus their discovery potential in polarized
and nuclear DIS.

##  1 Introduction

The process of deep-inelastic scattering (DIS) is governed by the four-
momentum transfer squared of the exchanged boson $Q^{2}$, the inelasticity
$y$, and the Bjorken scaling variable $x$ ellis_stirling_webber_1996 ;
Devenish:2004pb ; ParticleDataGroup:2020ssz . These kinematic variables are
related via the relation $Q^{2}=sxy$, where $s$ is the square of the center-
of-mass energy.

Conservation of momentum and energy over constrain the DIS kinematics and
leads to a freedom to calculate $Q^{2}$, $y$ and $x$ from measured quantities
JB:1979 ; Blumlein:1990dj ; Hoeger:1991wj ; Bentvelsen:1992fu ;
Bassler:H1intnote93 ; Bassler:1994uq ; ZEUS:1996uid ; Bassler:1997tv . Each of
these methods has advantages and disadvantages and no single approach is
optimal over the entire phase space. In addition, each method exhibits
different sensitivity to QED radiative effects, which further complicates the
choice of an optimal approach. It is a critical time to re-examine
reconstruction techniques given ongoing analyses of data from HERA and the
future electron-ion colliders in the USA (EIC) Accardi:2012qut ;
AbdulKhalek:2021gbh  and China (EicC) Anderle:2021wcy , as well as the
proposed Large Hadron electron Collider (LHeC) at CERN LHeCStudyGroup:2012zhm
; LHeC:2020van .

Machine learning is a promising tool for kinematic reconstruction in DIS
because of its potential to automatically synthesize many dimensions at once.
Deep learning has been proposed for a variety of tasks in hadronic final-state
(HFS, $h$) reconstruction, including particle identification
deOliveira:2018lqd ; Belayneh:2019vyx ; ATL-PHYS-PUB-2020-018 ;
MicroBooNE:2020hho ; Aurisano:2016jvx , jet-energy reconstruction CMS:2019uxx
; ATL-PHYS-PUB-2018-013 ; ATL-PHYS-PUB-2020-001 ; Baldi:2020hjm ;
Kasieczka:2020vlh ; 1910.03773 , jet tagging deOliveira:2015xxd ; CMS:2019dqq
; ATLAS:2018wis ; Kasieczka:2019dbj , unfolding Andreassen:2019cjw ;
Datta:2018mwd ; Bellagente:2019uyp ; Glazov:2017vni ; Vandegar:2020yvw ;
Howard:2021pos ; Baron:2021vvl ; Andreassen:2021zzk ; H1:2021wkz , and more
2102.02770 ; Larkoski:2017jix ; Guest:2018yhq ; Albertsson:2018maf ;
Carleo:2019ptp ; Bourilkov:2019yoi .

We develop a method to reconstruct DIS kinematic variables and account for QED
radiation that relies on deep-neural networks (DNNs). This paper is organized
as follows. Section 2 briefly reviews the current methods for kinematic
reconstruction in DIS. A DNN trained using fast simulations of the proposed
ATHENA detector at the future EIC is studied in Section 3 and the same
approach is demonstrated in full simulations of the H1 detector at HERA in
Section 4. Section 5 explores the impact of additional acceptance and
resolution effects on a fast simulation in comparison with a full detector
simulation. The paper ends with conclusions and outlook in Section 6.
Concurrently to our proposal, M. Diefenthaler et al. Diefenthaler:2021rdj
studied the application of DNNs for the combination of the input and output
variables of three reconstruction methods for $Q^{2}$ and $x$ in NC DIS.

##  2 Basic kinematic reconstruction in DIS

Figure 1 illustrates the process of lepton-proton scattering†††In the
following, the lepton can be an $e^{-}$ or $e^{+}$ but will generically just
be referred to as an electron., which is defined by the incoming electron
four-vector $k$, the outgoing electron four-vector $k^{\prime}$, the incoming
proton four-vector $P$, and the four-vector of the HFS defined as the sum of
all four-vectors that originate from the hadron vertex. Using the photon four-
vector $q=k-k^{\prime}$, the QED Born-level kinematics are described by:

$s=(k+P)^{2}\,,~{}~{}~{}~{}Q^{2}=-q^{2}\,,~{}~{}~{}~{}y=\frac{q\cdot P}{k\cdot
P}\,,~{}~{}~{}~{}\text{and}~{}~{}~{}~{}x=Q^{2}/(sy)\,.$ (1)

Figure 1: Illustration of $ep$ scattering. Left: (QED) Born-level diagram.
Middle: generic illustration of a radiative leptonic tensor including higher-
order QED corrections at the lepton vertex (the box-diagrams are additive and
not shown). Right: practical implementation of QED higher-order corrections in
MC event generators in terms of initial state radiation (ISR, green), final
state radiation (FSR, red), and an effective coupling (orange).

Due to azimuthal symmetry and ignoring mass effects, only two variables of the
outgoing four-vectors are of relevance. The usual choice is

  * •

the scattered electron energy $E$ and its polar angle $\theta$, and

  * •

the energy-momentum balance of the HFS, $\Sigma=E_{h}-p_{z,h}$, and the
inclusive angle of the HFS $\gamma$.

The HFS quantity $\Sigma$ can be calculated as the sum of all HFS particles
$\Sigma=\textstyle{\sum_{i}^{\text{HFS}}}(E_{i}-p_{z,i})$, and $\gamma$ is
defined using the transverse momentum of the HFS, $T$, through
$\tan\frac{\gamma}{2}=\frac{\Sigma}{T}$. Together with the electron-beam
energy $E_{0}$, five observables are known, while three of them suffice to
define a _basic reconstruction method_ for $Q^{2}$, $y$ and $x$. Only for $x$
one further needs the proton beam energy $E_{p}$.

Table 1 summarizes some of the most common reconstruction methods, which use
derived quantities from the scattered electron and HFS including
$p_{T,e}^{2}=E^{2}\sin^{2}\theta$, $p_{z,e}=E\cos\theta$,
$\Sigma_{e}=E-p_{z,e}=E(1-\cos\theta)$,
$\tan\frac{\theta}{2}=\frac{\Sigma_{e}}{p_{T,e}}$ and
$T^{2}=p_{x,h}^{2}+p_{y,h}^{2}=(\textstyle{\sum_{i}^{\text{HFS}}}p_{x,i})^{2}+(\textstyle{\sum_{i}^{\text{HFS}}}p_{y,i})^{2}=E^{2}_{h}\sin^{2}\gamma=\Sigma^{2}\cot^{2}\frac{\gamma}{2}=\Sigma(2E_{h}-\Sigma)$.

Method name | Observables | $y$ | $Q^{2}$ | $x\cdot E_{p}$
---|---|---|---|---
Electron ($e$) |  $[E_{0}$,$E$,$\theta]$ | $1-\frac{\Sigma_{e}}{2E_{0}}$ | $\frac{E^{2}\sin^{2}\theta}{1-y}$ | $\frac{E(1+\cos\theta)}{2y}$
Double angle (DA) Hoeger:1991wj ; Bentvelsen:1992fu  |  $[E_{0}$,$\theta$,$\gamma]$ | $\frac{\tan\tfrac{\gamma}{2}}{\tan\tfrac{\gamma}{2}+\tan\tfrac{\theta}{2}}$ | $4E_{0}^{2}\cot^{2}\tfrac{\theta}{2}(1-y)$ | $\frac{Q^{2}}{4E_{0}y}$
Hadron ($h$, JB) JB:1979  |  $[E_{0}$,$\Sigma$,$\gamma]$ | $\frac{\Sigma}{2E_{0}}$ | $\frac{T^{2}}{1-y}$ | $\frac{Q^{2}}{2\Sigma}$
ISigma (I$\Sigma$) Bassler:1994uq  |  $[E$,$\theta$,$\Sigma]$ | $\frac{\Sigma}{\Sigma+\Sigma_{e}}$ | $\frac{E^{2}\sin^{2}\theta}{1-y}$ | $\frac{E(1+\cos\theta)}{2y}$
IDA Bentvelsen:1992fu  |  $[E$,$\theta$,$\gamma]$ | $y_{\text{DA}}$ | $\frac{E^{2}\sin^{2}\theta}{1-y}$ | $\frac{E(1+\cos\theta)}{2y}$
$E_{0}E\Sigma$ |  $[E_{0}$,$E$,$\Sigma]$ | $y_{h}$ | $4E_{0}E-4E_{0}^{2}(1-y)$ | $\frac{Q^{2}}{2\Sigma}$
$E_{0}\theta\Sigma$ |  $[E_{0}$,$\theta$,$\Sigma]$ | $y_{h}$ | $4E_{0}^{2}\cot^{2}\tfrac{\theta}{2}(1-y)$ | $\frac{Q^{2}}{2\Sigma}$
$\theta\Sigma\gamma$ Bassler:H1intnote93  |  $[\theta$,$\Sigma$,$\gamma]$ | $y_{\text{DA}}$ | $\frac{T^{2}}{1-y}$ | $\frac{Q^{2}}{2\Sigma}$
Double energy (A4) Bentvelsen:1992fu  |  $[E_{0}$,$E$,$E_{h}]$ | $\frac{E-E_{0}}{(xE_{p})-E_{0}}$ | $4E_{0}y(xE_{p})$ | $E+E_{h}-E_{0}$
$E\Sigma T$ |  $[E$,$\Sigma$,$T]$ | $\frac{\Sigma}{\Sigma+E\pm\sqrt{E^{2}+T^{2}}}$ | $\frac{T^{2}}{1-y}$ | $\frac{Q^{2}}{2\Sigma}$
$E_{0}ET$ |  $[E_{0}$,$E$,$T]$ | $\frac{2E_{0}-E\mp\sqrt{E^{2}-T^{2}}}{2E_{0}}$ | $\frac{T^{2}}{1-y}$ | $\frac{Q^{2}}{4E_{0}y}$
Sigma ($\Sigma$) Bassler:1994uq  |  $[E_{0}$,$E$,$\Sigma$,$\theta]$ | $y_{\text{I}\Sigma}$ | $Q^{2}_{\text{I}\Sigma}$ | $\frac{Q^{2}}{4E_{0}y}$
$e$Sigma ($e\Sigma$) Bassler:1994uq  |  $[E_{0}$,$E$,$\Sigma$,$\theta]$ | $\frac{2E_{0}\Sigma}{(\Sigma+\Sigma_{e})^{2}}$ | $2E_{0}E(1+\cos\theta)$ | $\frac{E(1+\cos\theta)(\Sigma+\Sigma_{e})}{2\Sigma}$
Table 1:  Summary of basic reconstruction methods that employ only three out
of five quantities: $E_{0}$ (electron-beam energy), $E$ and $\theta$
(scattered electron energy and polar angle), $\Sigma$ and $\gamma$
(longitudinal energy-momentum balance,
$\Sigma=\textstyle\sum_{\text{HFS}}(E_{i}-p_{z,i})$, and the inclusive angle
of the HFS). Alternatively, the A4 method makes use of the HFS total energy
$E_{h}$. Shorthand notations are used for the longitudinal energy-momentum
balance of the electron, $\Sigma_{e}$, and for the transverse momentum of the
HFS, $T$. The $E\Sigma T$ and $E_{0}ET$ methods are under-constrained and have
two solutions, referring to two possible electron polar angles, and several
more are existent when using $E_{h}$ (see Ref. Bentvelsen:1992fu  for two
examples). The two bottom rows provide the equations of the $\Sigma$ and
$e\Sigma$-methods, which combine quantities of different basic reconstruction
methods, while further methods (like the PT (rD$\Sigma$), D$\Sigma$,
r$e\Sigma$ or mixed method) are found, e.g., in Ref. Bassler:1994uq ;
Bassler:1997tv .

Each of these methods has pros and cons, and yield good performance in limited
kinematic ranges Bentvelsen:1992fu ; Bassler:1994uq ; Bassler:1997tv . For
example, the methods that mostly rely on the scattered electron yield the best
resolution in events with large $y$, but their resolution on $x$ quickly
diverges at low $y$. In contrast, the methods that rely mostly on the HFS
variables yield better performance at low $y$, but are rather limited at high
$y$. Consequently, the H1 and ZEUS collaborations have used different methods
in different kinematic ranges (see Refs. Klein:2008di ; H1:2015ubc  and
references therein). For example, in Refs. H1:2009bcq ; H1:2012qti ;
H1:2013ktq , the electron method is used for $y\gtrsim 0.19$, while the
$\Sigma$ or $e\Sigma$ method are used at lower $y$, and DA method is employed
for calibration.

In a massless, Born-level calculation, all methods yield equivalent results
because of momentum and energy conservation ($2E_{0}=\Sigma+\Sigma_{e}$,
$p_{T,e}=T$ and $Q^{2}=sxy$). However, once (real) higher-order QED effects
are considered, the various methods yield different results and the calculated
quantities for $Q^{2}$, $y$ and $x$ are not representative for the
$\gamma/Z+p$ scattering process at the hadronic vertex.

Higher-order QED effects at the lepton vertex are generically represented as a
correction to the leptonic tensor, as displayed in the middle diagram of
Figure 1. Such radiative corrections include QED bremsstrahlung off the
lepton, photonic lepton-vertex corrections, self-energy contributions at the
external lepton lines, and fermionic contributions to the running of the fine-
structure constant, and additional box-diagrams representing multi-boson
exchange. The complete first-order corrections are calculable semi-
analytically Spiesberger:237380 ; Kwiatkowski:1990cx ; Kwiatkowski:1990es ;
Blumlein:1994ii ; Arbuzov:1995id . For an implementation in Monte-Carlo (MC)
event generators, these calculations are split by partial-fraction
decomposition into initial-state and final-state photon radiation (ISR and
FSR) and using effective couplings, as displayed in the right diagram of
Figure 1.

Two techniques are commonly applied to reduce sensitivity to QED radiation.
The first technique is to merge the FSR photons with the electron, thus
providing the four-vector linked to the exchanged boson, and the second
technique is to take the ISR radiation to be collinear, which implies that
$(\Sigma+\Sigma_{e})/2$ provides the incoming electron beam energy that
contributes in the interaction. For soft and collinear FSR, the first is done
implicitly also at detector level, e.g. when the photon is measured in the
same calorimeter cell as the electron. ISR photons often escape undetected
through the beam hole of the detector.

In cross-section measurements, the radiative particle-level is commonly just
an intermediate step, and additionally requires the application of well-
defined QED correction factors. For precision measurements, these factors
should be small. Common definitions for cross sections in DIS are:

  * •

Structure-function measurements are made as a function of $Q^{2}$ and $x$ and
are quoted at the ‘Born-level’ and thus have to correct for all higher-order
QED effects, such that the fine-structure constant factorizes from the
calculation of the structure functions.

  * •

At HERA, measurements of the HFS were quoted as non-radiative $\gamma^{*}p$
cross sections, which are corrected for first-order QED and electroweak
effects.

A _radiative cross section_ can be defined by merging any radiated photon with
the scattered electron that is closer to the scattered electron than to the
electron beam. By specifying a single reconstruction method, a well-defined,
meaningful and almost model-independent definition of $Q^{2}$, $y$ and $x$ is
obtained.

In the following, we will deal carefully with radiated photons at the particle
level and the detector level to determine kinematic quantities $Q^{2}$, $y$
and $x$, in an optimal way. These observables can then be used for subsequent
cross-section measurement with small and well-defined QED corrections.

##  3 Method

We use TensorFlow tensorflow  to construct and train a DNN to estimate the DIS
kinematics using both the scattered electron and the HFS. To demonstrate our
methodology, we use a fast simulation of the proposed EIC experiment ATHENA
using the Delphes package deFavereau:2013fsa ; miguel_arratia_2021_4592887 .
After presenting results for ATHENA, we will show results applying the same
methodology to a full simulation of the H1 experiment Brun:1987ma ; H1:1996prr
; H1:1996jzy . Both studies use the Rapgap MC generator Jung:1993gf , which
employs routines from Refs. Bengtsson:1987kr ; Bentvelsen:1992fu ;
Ingelman:1996mq ; Dobbs:2001ck . For H1, Rapgap version 3.1 is used, while
verion 3.3 is used for the ATHENA studies. In addition, the MC generator
Djangoh 1.4 Charchula:1994kf  is used to test prior dependence. Both MC
generators employ the Heracles routines Spiesberger:237380 ;
Kwiatkowski:1990cx ; Kwiatkowski:1990es  for the simulation of higher-order
QED effects.

We restrict our study to events with $Q^{2}>200$ $\mathrm{GeV}^{2}$. This
kinematic region is well measured, since the electron is scattered into the
central regions of the detector. However, no single reconstruction method
gives optimal performance over the full phase space Bassler:1994uq .

###  3.1 Fast simulation of the ATHENA experiment

Neutral-current DIS events are generated with the Rapgap 3.3 event generator
for electron-proton scattering with beam energies of $E_{0}=18\,\mathrm{GeV}$
and $E_{p}=275\,\mathrm{GeV}$ and processed with the Delphes fast simulation
of the ATHENA detector at the EIC. The scattered electron is selected as the
highest-$p_{T}$ track that satisfies the following criteria: correct charge,
$p_{T}>5$ GeV, electromagnetic fraction in the calorimeter $>0.80$, and
isolation $<0.20$, where isolation is defined as the scalar $p_{T}$ sum of all
other tracks and neutral hadrons within a cone of $\Delta R<0.5$ around the
electron direction divided by the electron $p_{T}$. The HFS is reconstructed
from the sum of all Energy-Flow candidates (tracks, photons, and neutral
hadrons), excluding the scattered electron and any photon candidates that is
within a cone of $\Delta R<0.4$ around the scattered electron. We require
$\Sigma+\Sigma_{e}$ to be within $\pm 4\,\mathrm{GeV}$ of $2E_{0}$ to suppress
ISR events.

###  3.2 QED radiation and categorization of events

We introduce a practical categorization of events, which is closely related to
our proposed radiative cross-section definition above:

  * •

if the radiated photon is closer to the electron-beam direction, it is an
Initial-State Radiation event (ISR);

  * •

if the radiated photon is closer to the scattered-electron direction, it is a
Final-State Radiation event (FSR);

  * •

if no photon radiation is emitted by the generator, it is a non-radiative
event (NoR).

Within Rapgap, which implements first-order QED corrections, the radiated
photon either branches off from the beam electron before it interacts with the
proton, or it branches off of the scattered electron after interacting with
the proton. This leads to the natural interpretation of the former as ISR and
the latter as FSR, which agrees with our practical definitions in 94% of QED
radiation events.

To define the target values of $Q^{2}$, $y$ and $x$ for events with QED
radiation, we use the generated beam electron after radiation for ISR events
and the generated scattered electron prior to radiation for FSR events. While
this definition can be considered as the _true_ kinematic quantities‡‡‡Note
that learning the true value from detector-level quantities introduces a prior
dependence ATL-PHYS-PUB-2018-013 ., alternative definitions in terms of
particle-level observables are also possible. For example, by applying FSR
merging and using the equations of ISR insensitive reconstructions methods for
calculating $Q^{2}$, $y$ and $x$. In our studies, these different definitions
yield indistinguishable results. The generator-level quantities are
generically denoted with a subscript ‘gen’ in the following.

###  3.3 DNN inputs

For our main goal of determining $Q^{2}$, $y$ and $x$, the task of the DNN is
in fact two-fold. In the absence of QED radiation, the task of the DNN would
be to learn to compute $Q^{2}$, $y$ and $x$ from the input quantities subject
to finite-resolution and acceptance effects. In events with QED radiation, the
DNN needs to learn to quantify the extent of QED radiation and account for it
while calculating $Q^{2}$, $y$ and $x$. Hence, the regression DNN must learn
to treat ISR, FSR, and NoR events separately in order to achieve optimal
performance. This is a key feature of our DNN method, which is absent in
traditional methods.

We define the following variables to characterize the strength of QED
radiation in the event:

$p_{T}^{\rm bal}\ =\ 1-\frac{p_{T,e}}{T}\ =\
1-\frac{\Sigma_{e}\tan\frac{\gamma}{2}}{\Sigma\tan\frac{\theta}{2}}\ \ \
~{}~{}\text{and}~{}~{}\ \ \ p_{z}^{\rm bal}\ =\ 1-\frac{\Sigma_{e}+\Sigma}{2\
E_{0}}\,.$ (2)

When calculating them at particle level, both quantities $p_{T}^{\rm bal}$ and
$p_{z}^{\rm bal}$, are zero for events with no QED radiation, but positive for
events with FSR and ISR, respectively. The $p_{T}^{\rm bal}$ ($p_{z}^{\rm
bal}$ ) value indicates the strength of FSR (ISR).

The following observables that help indicate QED radiation in the event are
included as inputs to the regression DNN for $Q^{2}$, $y$ and $x$:

  * •

The values of $p_{T}^{\rm bal}$ and $p_{z}^{\rm bal}$.

  * •

The energy, $\eta$, and $\Delta\phi$ of the reconstructed photon in the event
that is closest to the electron-beam direction, where $\Delta\phi$ is with
respect to the scattered electron.

  * •

The sum ECAL energy within a cone of $\Delta R<0.4$ around the scattered
electron divided by the scattered-electron track momentum.

  * •

The number of ECAL clusters within a cone of $\Delta R<0.4$ around the
scattered electron.

These seven observables are combined with the following eight:

  * •

Scattered-electron quantities $p_{T,e}$, $p_{z,e}$ and $E$.

  * •

HFS four-vector quantities $T$, $p_{z,h}$ and $E_{h}$.

  * •

$\Delta\phi(e,h)$ between the scattered electron and the HFS momentum vector.

  * •

The difference $\Sigma_{e}-\Sigma$.

The transverse momenta of the scattered electron and the HFS are highly
correlated. We replace the pair of $p_{T}$ values with the difference and the
sum of the $p_{T}$ values, which removes the correlation and aids the DNN
training. The sum $\Sigma_{e}+\Sigma$ appears in the definition of $p_{z}^{\rm
bal}$ and is sharply peaked at twice the electron-beam energy, making the
$\Sigma$ values anti-correlated; hence, we include the orthogonal combination
$\Sigma_{e}-\Sigma$ in addition.

###  3.4 DNNs for the classification or quantification of QED radiation

To determine the ability of the DNN to identify and quantify QED radiation, we
investigated two approaches: a classification network and a regression network
for $p_{T}^{\rm bal}$ and $p_{z}^{\rm bal}$. Both DNNs differ only in the
activation function for the final layer, the learning targets, and the loss
function for the training.

We followed a heuristic approach in designing the DNN, guided by prior
experience. We chose to have the number of nodes per hidden layer grow in the
first layers, reach a peak size, and then fall at a rate that is symmetric
about the peak layer. Several trial configurations were tested, each with a
different number of hidden layers and/or number of nodes per layer, though we
did not perform a thorough scan. We also tried a few sets of optimizer
hyperparameters before finding values that gave good results in a reasonable
amount of training time. Various standard activation functions were tested
with no appreciable differences in the results. This basic DNN design is
effective for a variety of tasks, including classification and regression, as
we explain below. The same design works well for both the ATHENA fast
simulation and the H1 full simulation. Our design explorations were terminated
once we achieved DIS reconstruction performance that exceeded the conventional
methods. We have not carried out thorough hyperparameter scans, so further
improvements may be possible.

The QED-classification DNN consists of a sequential network with 8 layers. The
15 DNN inputs, defined in the previous section, are transformed prior to
training to have zero mean and unit RMS using the sklearn StandardScaler
scikit-learn . The learning targets are three binary (0 or 1) state variables
to tag the events as ISR, FSR, or NoR. The activation function is a rectified
linear unit (relu) for the first layer, scaled exponential linear unit (selu)
NIPS2017_5d44ee6f  for the middle layers, and the softmax function for the
final output layer. The numbers of nodes per layer are 64, 128, 512, 1024,
512, 128, and 64, with 3 outputs in the final layer. The outputs are three
numbers, each between 0 and 1, that sum up to 1. The loss function for the
training is categorical cross entropy. The training is performed with the Adam
optimizer kingma2017adam  with a learning rate of $10^{-4}$. The total number
of parameters of the DNN model is 1,199,555. The training and validation are
performed using over 28 million simulated events with half for training and
half for validation. The batch size for the training is 128. The training
terminates after 38 epochs, finding no further improvement in the validation
loss function.

Figure 2 shows distributions for the three outputs of the QED-classification
DNN. Some events have clear evidence of QED radiation and are strongly
identified. Some degree of mis-classification is to be expected, since neither
soft ISR/FSR, nor collinear FSR induce a measurable signal in the detector.

Figure 2: Distributions of the QED-classification DNN predictions for the
ATHENA fast simulation. The distributions are normalized to equal area.

In the second study, the QED regression for $p_{T}^{\rm bal}$ and $p_{z}^{\rm
bal}$, the DNN has the same structure as the QED-classification DNN except for
the following differences. The activation function is the linear function for
the final output layer, which has 2 nodes instead of 3. The learning targets
are the particle-level values of $p_{T}^{\rm bal}$ and $p_{z}^{\rm bal}$,
which are transformed to have zero mean and unit RMS prior to training. The
loss function for the training is Huber loss 10.1214/aoms/1177703732  with a
transition between quadratic and linear loss at $\pm 0.01$. The batch size for
the training is 1024. The training and validation are done using the same
sample as for the QED-classification DNN.

Figure 3 shows distributions of the DNN predictions for $p_{T}^{\rm bal}$ and
$p_{z}^{\rm bal}$, separately for ISR, FSR, and NoR events, as well as scatter
plots of the predicted values vs the particle-level values. The predicted
$p_{T}^{\rm bal}$ distribution for FSR events is shifted to positive values
for FSR events, while the predicted $p_{z}^{\rm bal}$ distribution is shifted
to positive values for ISR events, as expected. For many events, the DNN is
able to accurately estimate both $p_{T}^{\rm bal}$ and $p_{z}^{\rm bal}$.
There are also QED-radiation events where the prediction is zero, which
correspond to cases where the radiated photon is either out of acceptance or
not clearly identified.

Figure 3: Distributions of the QED-regression DNN predictions and particle-
level values of $p_{T}^{\rm bal}$ and $p_{z}^{\rm bal}$ for the ATHENA fast
simulation.

###  3.5 Regression DNN for DIS kinematic variables $Q^{2}$, $y$ and $x$

We estimate $Q^{2}$, $y$ and $x$ using a regression DNN that has a similar
structure as the QED regression DNN described above, except for the final
output layer that has three nodes corresponding to the target variables
$Q^{2}$, $y$ and $x$. The learning rate is $10^{-5}$ and the batch size is
1024. Since the distributions of $Q^{2}$, $y$ and $x$ are approximately
exponential, the DNN is trained to predict the logarithm of each variable. The
training and validation are performed using over 28 million events, using half
for validation and half for training.

As a pilot study, we consider three different choices for the input variables
to the DNN:

  * •

add the three QED-classification DNN outputs (FSR, ISR, NoR) as inputs, in
addition to the 15 variables.

  * •

add the QED-regression DNN predictions $p_{T}^{\rm bal}$ and $p_{z}^{\rm bal}$
as inputs, in addition to the 15 variables.

  * •

use the same 15 inputs as in the QED-classification and regression DNNs.

We found that the results of these three approaches are essentially the same.
This suggests that the QED-classification and the QED-regression DNNs do not
provide any additional information beyond what the regression DNN for $Q^{2}$,
$y$ and $x$ learned. In light of this, we chose the simplest (third) option,
which uses only the original 15 variables as inputs. The total number of
parameters for the DNN model is 1,199,619. The training terminates after 103
epochs, finding no further improvement in the validation loss function.

###  3.6 Benchmark of the DNN reconstruction vs. standard methods

Figure 4 shows the results for the DNN and traditional methods using events
with $Q^{2}>200~{}{\rm GeV}^{2}$ and two $y$ regions, obtained with the ATHENA
fast simulation.

Figure 4: Resolution for $Q^{2}$ (left), $y$ (middle), and $x$ (right) for the
DNN, electron, and double-angle (DA) methods for the fast simulation of the
ATHENA experiment. The top (bottom) row is for events with $y_{\rm gen}>0.15$
($y_{\rm gen}<0.15$). All distributions are normalized to the same area.

The resolutions from the DNN exhibit a peak at unity and mostly Gaussian-like
tails; in contrast, classical methods yield a peak but larger tails caused by
their limited use of the reconstructed quantities and the presence of ISR or
FSR.

ATHENA fast simulation (Rapgap+Delphes)

Figure 5:  Resolution on the reconstruction of $Q^{2}$ (left), $y$ (middle),
and $x$ (right) as a function of the generated $y$ for the fast simulation of
ATHENA. The top (bottom) row shows the RMS (mean) of the measured-over-
generated distribution as a function of generated $y_{\text{gen}}$. The RMS
and mean are calculated using events with the measured-over-generated ratio
within the interval 0 to 2.

The properties of the resolutions, their mean and RMS, are displayed in Figure
5 for small intervals of $y$ §§§More detailed representations of the
resolutions for the different methods are shown in Appendix A.. The DNN
reconstruction has the smallest RMS among all methods, for all three kinematic
variables and all $y$ intervals. Also, the mean distributions are unbiased for
$Q^{2}$, $y$ and $x$ for all $y$ intervals, while the classical methods
exhibit large biases.

We examine more closely the resolution and bias for events with and without
QED radiation in Figures 6 and 7, where we use the definition of NoR, ISR and
FSR events from Section 3.2. The RMS for events with no QED radiation gives a
measure of the core resolution, free from the tails that are visible in the
distributions of Figure 4 for the conventional methods. All methods, except
the hadron method, show no bias in NoR events. For $x$ and $y$, the electron
method has a better core resolution than the DNN for $y>0.15$; however, it
suffers from poorer resolution and a strong bias in events with QED radiation.
The DNN reconstruction has some loss in performance in QED-radiation events,
but it is about a factor of two better than the electron method in these
events, and shows no bias. We conclude that the DNN has successfully learned
to mitigate the effects of QED radiation that spoil the resolution and bias
the calculations of the conventional methods.

ATHENA fast simulation (Rapgap+Delphes)

Figure 6:  Comparison of the RMS of the resolution of $Q^{2}$, $y$ and $x$
(from left to right) for NoR (top), ISR (middle), and FSR (bottom) events. The
RMS is calculated using events with the measured-over-generated ratio within
the interval 0 to 2.

ATHENA fast simulation (Rapgap+Delphes)

Figure 7:  Comparison of the mean of the resolution distributions of $Q^{2}$,
$y$ and $x$ (from left to right) for NoR (top), ISR (middle), and FSR (bottom)
events. The mean is calculated using events with the measured-over-generated
ratio within the interval 0 to 2.

##  4 Demonstration using the full simulation of the H1 experiment

We apply our DNN methodology to simulated events of the H1 experiment at HERA.
The events were simulated by the H1 Collaboration using the Rapgap 3.1
Jung:1993gf  and Djangoh 1.4 Charchula:1994kf  generators for the beam
energies $E_{0}=27.6\,\mathrm{GeV}$ and $E_{p}=920\,\mathrm{GeV}$. The
generators employ the Heracles routines Spiesberger:237380 ;
Kwiatkowski:1990cx ; Kwiatkowski:1990es  for QED radiation, the CTEQ6L PDF set
Pumplin:2002vw , and the Lund hadronization model Andersson:1983ia  with
parameters fitted by the ALEPH Collaboration Schael:2004ux . The simulation of
the H1 experiment H1:1996prr ; H1:1996jzy  employs the Geant 3 package
Brun:1987ma  and includes real calorimeter noise and fast shower simulations
Fesefeldt:1985yw ; Grindhammer:1989zg ; Gayler:1991cr ; Kuhlen:1992ey ;
Grindhammer:1993kw ; Glazov:2010zza . The simulation includes time-dependent
properties (‘run-specific’), where the detector state and beam properties
correspond altogether to the HERA-II data taking periods.

The simulated events are reconstructed just like data, in particular, an
energy-flow algorithm energyflowthesis ; energyflowthesis2 ; energyflowthesis3
is used to define objects whose sum yields the HFS four-vector, and the
scattered electron candidate are defined using the same approach as Refs.
H1:2012qti ; Andreev:2014wwa ; H1:2021wkz . The simulated events also undergo
the same (in-situ) calibration procedure as real data, using the latest
calibration by the H1 Collaboration H1:2012qti ; Kogler:2011zz ;
Andreev:2014wwa . Some technical selections and fiducial cuts are applied as
it would be done similarly to real data. In particular, events are required to
have $45<\Sigma+\Sigma_{e}<65$ GeV to suppress ISR events; a veto on QED
Compton events is imposed; and since a trigger simulation is included, our
study is limited to $E\gtrsim 11\,\mathrm{GeV}$ Andreev:2014wwa . The
simulated events are processed within H1’s computing environment
Britzger:2021xcx  and altogether several $10^{8}$ events were simulated, and
after ‘run’-selection, acceptance effects and our selection of
$Q^{2}>200\,\mathrm{GeV}^{2}$ yield about $7\cdot 10^{7}$ simulated and
reconstructed events for our DNN studies.

We use the same regression DNN structure, input variables, and training
methods for the simulated H1 events as previously for the ATHENA events. We
take a sample of over 12 million Rapgap events and use half of the events for
training and half for validation. The training terminates after 125 epochs,
finding no further improvement in the validation loss function.

Figure 8: Resolution for $Q^{2}$ (left), $y$ (middle), and $x$ (right) for the
DNN, electron, and double-angle (DA) reconstruction methods for the full
simulation of the H1 experiment. The top (bottom) row is for events with
$y_{\rm gen}>0.15$ ($y_{\rm gen}<0.15$). All distributions are normalized to
the same area.

Figure 8 shows the resolutions for the DNN and two classical methods in two
intervals of $y$. Similarly to the results obtained with the ATHENA
simulation, the DNN yields a peak at unity and no asymmetric tails for all
three quantities, in contrast with classical methods.

H1 full simulation

Figure 9:  Resolution on the reconstruction of $Q^{2}$ (left), $y$ (middle),
and $x$ (right) as a function of the generated $y$ for the full simulation of
H1. The top (bottom) row shows the RMS (mean) of the measured-over-generated
distribution as a function of the generated $y$. The RMS and mean are
calculated using events with the measured-over-generated ratio within the
interval 0 to 2.

The mean and RMS of the resolutions are displayed as a function of $y$ in
Figure 9. The DNN yields the smallest RMS. The mean of the DNN reconstruction
is closest to unity over a wide range of $y$. Only for $Q^{2}$, or at highest
$y$, the electron method achieves comparably good RMS and mean. In contrast,
at lower $y$ the DNN provides a significant improvement and results in a bias-
free reconstruction of $y$ and $x$ with superior resolution¶¶¶More detailed
representations of the resolutions for the different methods are collected in
Appendix B, where also further reconstruction methods from Section 2 are
studied with a real detector simulation..

To assess a possible bias in our DNN methodology that may arise from the
details of the MC event generator that is used to train the DNN, we study the
performance of the DNN reconstruction using two different MC event generators,
Rapgap and Djangoh. The two event generators differ in the modelling of
higher-order QCD radiation that results in significant differences in the
prediction of the HFS. Djangoh employs the color-dipole model, while Rapgap
employs a matrix-element plus parton shower model, where the parton shower is
in the leading-logarithmic approximation. In this study, we use H1’s full
simulation and train the DNN with a Rapgap event sample. Subsequently this DNN
is applied to a statistically independent Rapgap event sample and to a
simulated Djangoh sample.

Figure 10 shows the $Q^{2}$, $y$ and $x$ resolutions as a function of the
generated $y$ for both the Rapgap and Djangoh samples when using the same DNN,
where the measured-over-generated ratio is calculated on an event-by-event
basis with the respective generated values. The results from the Djangoh event
sample are nearly indistinguishable from the Rapgap sample. In particular, the
mean of the distribution is unbiased. This result suggests that any generator-
specific systematic errors in the DNN predictions is negligible.

H1 full simulation

Figure 10:  Resolution comparison of the Rapgap and Djangoh generators on the
DNN reconstruction of $Q^{2}$ (left), $y$ (middle), and $x$ (right) as a
function of the generated $y$ for the full simulation of H1. The top (bottom)
row shows the RMS (mean) of the measured-over-generated distribution as a
function of the generated $y$. The red (blue) curves show the results for the
Rapgap (Djangoh) event sample, while the DNN was trained with the Rapgap
sample in both cases.

The resolution (RMS) for $y$ and $x$ increase at lower $y$, even for the DNN
reconstruction. Since this pattern is not present in the the ATHENA fast
simulation results and may be attributed to further acceptance, noise, or
resolution effects that deteriorates the measurement of the HFS energyflow . A
dedicated study using a Delphes fast simulation is presented in the following
section.

##  5 Impact of further acceptance and resolution effects at low $y$

At low $y$, the HFS-based methods perform better than the electron method. The
reason is, that the ratio $E(1-\cos\theta)/2E_{0}$ gets close to one and
cannot be measured accurately because of large values of $E$. Likewise,
however, the HFS momentum balance $\Sigma$ goes to zero as $y$ goes to zero.
Although for kinematic reconstruction at low $y$ the usage of $\Sigma$ is
preferred over $\Sigma_{e}$, the quantity $\Sigma$ is particularly sensitive
to resolution and acceptance effects. In particular HFS components that are
more in the central region of the calorimeter contribute more, such making
$\Sigma$ at low $y$ especially sensitive resolution effects or efficiency
losses in the central part of the detector or fake components from
calorimetric noise.

Figure 11: Resolutions for $x$ in various $y$ ranges and reconstruction
methods obtained with a full Geant simulation of H1 (shaded area) and a fast
simulation based on Delphes  with (full line) and without (dashed line) an
added ad-hoc noise contribution.

The Delphes fast simulation does not include calorimeter noise hits, nor does
it account in a full-fledged manner for single-particle acceptance effects and
efficiency losses as they can be present at the boundaries of calorimeter
stacks or because of insensitive material. To test the hypothesis that such
detector effects can be responsible for the resolution decrease in $x$ and $y$
for hadronic reconstruction methods at low $y$ ($y\lesssim 0.15$), we have
implemented the H1 experiment in Delphes. Figure 11 shows the $x$ resolution
for the standard reconstruction methods for our fast simulation of H1 compared
to the full simulation. The agreement between the fast and full simulation at
high $y$ is fairly good. At low $y$, however, there is a low-side tail for the
$\Sigma$, hadron and DA method for events processed with the full H1
simulation, while that tail is absent in the fast simulation, and also the
mean value is shifted.

We apply an additional additive component with random sign to the HFS to the
fast simulated events, which mimics further detector effects, like acceptance
or efficiency losses, reduced resolution or artificial components from
electronic noise in the calorimeter. The model we use to simulate these
effects is to generate random numbers from TRandom::Landau in Root
citeulike:363715  with mu=0 and sigma=0.05 in units of GeV, and add it with a
random sign to the $p_{x}$, $p_{y}$, and $p_{z}$ components of the
reconstructed HFS four-vector.

The results are also displayed in Figure 11. We find that adding such an
additional detector effect to the HFS in the fast simulation brings the fast
simulation into good agreement with the full detector simulation. Adding these
ad-hoc detector effects produces a low-side tail in the $x$ resolution at low
$y$ but does not affect the $x$ reconstruction at high $y$. The electron
remains naturally unaltered in that procedure.

This study suggests that further detector effects that are otherwise not
included in the Delphes fast simulation impact the precise measurement of the
HFS and reduce the $x$ and $y$ resolution at low $y$. We do not currently have
an estimate for how large that effect in ATHENA will be, what is the actual
correspondence in the full simulation (an acceptance, efficiency, resolution
or noise effect), or to which extent it is impacted by calibration or noise-
suppression algorithms. Though, due to a larger coverage of the calorimeter,
less dead material and newer detector technologies of ATHENA as compared to
H1, we expect this additive component to be significantly smaller for ATHENA
than what our ad-hoc model adds to H1 to bring the fast and full simulation
into agreement.

To place an upper bound on the impact of these effects in the ATHENA results,
we investigated adding our ad-hoc component to the ATHENA fast simulation. The
full analysis is then repeated, including the DNN training. The results are
shown in Figure 12 for both the conventional reconstruction methods and the
DNN reconstruction, where the DNN reconstruction for the unaltered ATHENA
sample is included for comparison. The DNN resolution does get worse below $y$
of around 0.2 and there is a small bias at very low $y$, as we expected based
on the H1 study. The $Q^{2}$ reconstruction is insensitive to that, since it
is dominated by the electron reconstruction. The results in Figure 12 show
that also with a very conservative ad-hoc model, the DNN outperforms all
standard reconstruction methods.

ATHENA fast simulation with additive resolution effect (Rapgap+Delphes)

Figure 12:  Resolution on the reconstruction of $Q^{2}$ (left), $y$ (middle),
and $x$ (right) as a function of the generated $y$ for the fast simulation of
ATHENA with the ad-hoc additive resolution effects model added. The top
(bottom) row shows the RMS (mean) of the measured-over-generated distribution
as a function of the generated $y$. For comparison, the DNN curves for that
unaltered sample are shown in black for comparison.

##  6 Summary and outlook

We have presented a novel method to reconstruct the DIS kinematic variables
($Q^{2}$, $y$ and $x$) using a deep neural network (DNN) that takes as input
the electron and hadronic-final-state measurements as well as observables that
can indicate the presence of QED radiation.

We have introduced our methodology using a Delphes-based fast simulation of
the ATHENA experiment for the EIC and validated our methods using the well-
understood full simulation of the H1 experiment at HERA.

Our method outperforms traditional methods over a wide kinematic range,
improving the resolution, and decreasing the bias. We validated that our
method is independent of the Monte Carlo event generator used to train the
DNN. We further performed a study to validate the fast simulation approach by
comparing a Delphes model of the H1 detector with the H1 full simulation;
these comparisons allowed us to identify key effects for low $y$ events.

Our DNN-based method shows promise for improving the resolution and extending
the kinematic reach of flagship EIC measurements. Given that EIC experiments
are being designed, our method offers a way to benchmark detector performance
against DIS using an optimal combination of electron and HFS measurements.

## Code availability

The code in this work can be found in: https://github.com/owen234/DIS-reco-
paper/

## Acknowledgments

We thank our colleagues from the H1 Collaboration for allowing us to use the
simulated MC event samples and for providing valuable feedback to the analysis
an on the manusript, in particular Günter Grindhammer, Sergey Levonian, Stefan
Schmitt and Zhiqing Zhang. We also thank members of the ATHENA Collaboration,
and in particular Steven Sekula, for help with the ATHENA Delphes model. We
thank Hubert Spiesberger for valuable discussions about QED radiative effects
and insights into the Heracles routines. We thank Hannes Jung for providing
the Rapgap event generator. Thanks to DESY-IT and MPI für Physik for providing
some computing infrastructure and supporting the data preservation project of
the HERA experiments. M.A was supported through DOE Contract No. DE-
AC05-06OR23177 under which JSA operates the Thomas Jefferson National
Accelerator Facility, and by the University of California, Office of the
President award number 00010100. B.N. was supported by the Department of
Energy, Office of Science under contract number DE-AC02-05CH11231.

##  Appendix A Detailed resolution plots for the ATHENA fast simulation

In this section detailed resolution plots for the variables $Q^{2}$ (Figure
13), $y$ (Figure 14) and $x$ (Figure 15) for the ATHENA fast simulation at the
EIC with $\sqrt{s}=141\,\mathrm{GeV}$ are shown. Our DNN-reconstruction method
is compared to four widely used basic reconstruction methods (I$\Sigma$,
hadron, double-angle and electron-method) for $Q^{2}>200\,\mathrm{GeV}^{2}$ in
five kinematic ranges in $y_{\text{gen}}$.

Figure 13: Resolutions for $Q^{2}$ in various $y$ ranges and reconstruction
methods from the Delphes fast simulation of ATHENA for events with
$Q^{2}>200~{}{\rm GeV}^{2}$. The tails in the distributions are from events
with ISR and FSR radiation.  Figure 14: Resolutions for $y$ in various $y$
ranges and reconstruction methods from the Delphes fast simulation of ATHENA
for events with $Q^{2}>200~{}{\rm GeV}^{2}$. The tails in the distributions
are from events with ISR and FSR radiation.  Figure 15: Resolutions for $x$ in
various $y$ ranges and reconstruction methods from the Delphes fast simulation
of ATHENA for events with $Q^{2}>200~{}{\rm GeV}^{2}$. The tails in the
distributions are from events with ISR and FSR radiation.

##  Appendix B Detailed resolution plots for the H1 full simulation

In this section detailed resolution plots for the variables $Q^{2}$ (Figures
16 and 17), $y$ (Figure 18) and $x$ (Figures 19 and 20) for H1’s full
simulation at HERA at $\sqrt{s}=319\,\mathrm{GeV}$ are shown. Our DNN-
reconstruction method is compared to the full set of eight basic
reconstruction methods (I$\Sigma$, hadron, double-angle (DA), electron ($e$),
IDA, $E_{0}E\Sigma$, $E_{0}\theta\Sigma$, and $\theta\Sigma\gamma$-method) for
$Q^{2}>200\,\mathrm{GeV}^{2}$ in five kinematic ranges in $y_{\text{gen}}$.
The double-energy method is omitted since $E_{h}$ is not measureable du to
relevant acceptance losses through the forward beam-hole. For $y$ there are
only four unambiguous independent basic methods.

Figure 16: Resolutions for $Q^{2}$ in various $y$ ranges and reconstruction
methods from the full simulation of H1 for events with $Q^{2}>200~{}{\rm
GeV}^{2}$.  Figure 17: Resolutions for $Q^{2}$ in various $y$ ranges and
reconstruction methods from the full simulation of H1 for events with
$Q^{2}>200~{}{\rm GeV}^{2}$.  Figure 18: Resolutions for $y$ in various $y$
ranges and reconstruction methods from the full simulation of H1 for events
with $Q^{2}>200~{}{\rm GeV}^{2}$.  Figure 19: Resolutions for $x$ in various
$y$ ranges and reconstruction methods from the full simulation of H1 for
events with $Q^{2}>200~{}{\rm GeV}^{2}$.  Figure 20: Resolutions for $x$ in
various $y$ ranges and reconstruction methods from the full simulation of H1
for events with $Q^{2}>200~{}{\rm GeV}^{2}$.

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