††institutetext: Faculty of Physics and Applied Computer Science,
AGH University of Science and Technology
Al. Mickiewicza 30, 30-059 Krakow, Poland

# Comparison of general-purpose events generators for particle fluence
simulation in LHC environment

A. Oblakowska-Mucha,111Corresponding author and T. Szumlak
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

At the LHC era, the detector systems are operating at the harsh hadronic
environment with the unprecedentedly high particle flux. Position sensitive
silicon devices are usually positioned at the innermost regions of the
experimental setups and must cope with highly non-uniform radiation fields. At
the end of LHC Run II, fluence in silicon trackers reached 1015 neq/cm2.
Initial simulation studies predict that the maximal fluence for the HL-LHC may
be up to two orders of magnitude higher than the one seen in LHC Run I and Run
II. In this paper, two general-purpose physics events generators used for
simulation of proton-proton collisions at LHC energies: Pythia 8.2 and DPMJET
3, are compared. Fluences obtained using these models, with the latest tuning
to the LHC data, in detectors situated close to the point of proton-proton
collisions are determined as well.

##  1 Introduction

Large Hadron Collider (LHC) at CERN was designed to collide bunches of protons
accelerated to centre-of-mass energy, $\sqrt{s}$, of up to 14 TeV. To
accomplish a rich physics programme, covering high precision Standard Model
(SM) measurements and searches of new physics, the LHC is able to provide 2808
bunches with more than 1011 protons each and collide them with a frequency of
40 MHz (at the nominal conditions). LHC can also supply lead ion beams. During
the years 2010-2012 (Run I) and 2015-2018 (Run II) the proton beams collided
with $\sqrt{s}$=6.5-7 TeV and 13 TeV respectively. The Run III data taking
period will start in the year 2021 and, thanks to the new injectors, the
instantaneous luminosity will be doubled and reach approximately
2.2$\times$1034 cm-2s-1. The LHC program will be continued collecting 3000
fb-1 of data around the year 2035 (during Run IV called High Luminosity LHC,
HL-LHC) HL . Run IV will be preceded by the significant upgrade of the
accelerator together with the substantial change in all main detector trackers
because of the radiation damage.

The total cross-section for proton-proton interaction at $\sqrt{s}$=8 TeV is
$\sigma_{tot}$ = (96.07$\pm$0.18$\\\ \pm$0.31) mb sigma_tot , out of this 71.5
mb is related to the inelastic processes. Assuming the nominal LHC luminosity
to be 1034 cm-2s-1, in each second of running the experiments about
96$\times$107 proton-proton ($pp$) interactions occur. At the LHC energies, a
few hundreds of particles may be produced per one $pp$ interaction with
momenta that range over many orders of magnitude. The planned HL-LHC runs will
double this number, making the radiation damage the main concern in any
detector system.

At the end of LHC Run II data taking period the inner parts of silicon
trackers of the LHCb and ATLAS were irradiated of the fluence that reached
1015 neq/cm2 velo_fluence . The increase of the luminosity in Run III will
cause an increase of the particle fluence as well, reaching annually value
compared to the Run I and II together. When the level of 1016 neq/cm2 is
reached, the detectors’ proper performance will be significantly affected rd50
. The precautions, whereas the operating conditions, must be changed or
eventually sensors should be replaced, will be taken based on the simulation
of the particle fluence and monitoring of the actual state of the detector.

The fluence in the LHC environment is highly non-uniform and comes from
particles emerging from the high energy proton-proton collisions and also from
the interactions of these particles with the sub-detectors, supports and
shielding. The radiation field is composed of charged hadrons and leptons,
neutrons and photons. The proportion of the different particle types at a
given point of the radiation field depends on the distance and the angle with
respect to the proton beams and on the detector’s material. The only reliable
way to estimate the properties of such complex radiation filed is through the
Monte Carlo based simulation. Estimation of the particle fluence is therefore
necessarily both for the monitoring of the current state of a detector (to
anticipate the change in operation) and for the prediction whereas all
components would operate until the end of the planned data taking period. Last
but not least, simulation of the fluence is indispensable for the verification
of the models for the radiation damage models rd50 .

The simulation procedure of particle fluence traversing the LHC detectors
contains two components: generation of particles (i.e., the final state
system) originating from a proton-proton collision at the LHC energies, and
the subsequent simulation of particle transport in the detectors’ material.
The former is elaborated in this document. Two models of the high energy
proton-proton collision used in programs for simulation of particle fluence
are compared: Pythia 8.2 and DPMJET 3. Samples are generated without any
selection criteria which could bias the comparison. If any differences are
observed at the generator lever, they will be naturally propagated to the
discrepancies in the particle transport and fluence simulation, causing the
inconsistency in the prediction for the future experiments.

Another method of evaluation of the particle fluence is based on the
reconstruction of particle tracks by standard algorithms. This method
considers only charged particles and strongly depends on the detector
acceptance and experiments’ software. Therefore a comparison of such
simulation between experiments is not reliable.

##  2 Physics models for hadronic interaction

Proton is not a fundamental particle but is comprised of quarks and gluons
(called partons) interacting via the strong and Coulomb force. The type of the
interaction and the composition of the final state depend on the proton
energy. At the lowest energies ($\sqrt{s}$ below 1.5 GeV) protons are
considered as charged point-like objects that scatter elastically without any
loss of energy. As the energy increases, the inelastic processes prevail,
driven by the interaction at the partonic level. With further rise of
available energy, the resolution of the projectile parton reveal finer and
finer structure of the proton as in Deep Inelastic Scattering (DIS). At the
LHC energies the total proton-proton cross-section is dominated by the
inelastic proton-proton scattering and the contribution of elastic part is
about 25%.

Kinematics of $pp$ collision is described by four-momentum transfer $Q^{2}$,
that is regarded as a scale at which proton is probed by the parton, and
Mandelstam variable $t$ that describes the momentum transfer between partons.
In the scattering experiments $Q^{2}=-t$ and, due to the uncertainty
principle, the resolution power of the parton is proportional to $1/\sqrt{t}$.
With the increase of centre-of-mass energy $\sqrt{s}$ and the four-momentum
transfer $Q^{2}$, the finer structure of protons is resolved and more gluons
and pairs of sea quarks take part in the scattering. Each of these partons
carry fraction of protons’ momentum, denoted as $x$. At high energy, gluon
density grows much faster than the quark density therefore LHC is essentially
a gluon-gluon collider and provides access to the kinematic region never
reached before: high $Q^{2}$ and low $x$. Due to the high energy and high
density of partons, multiparton scattering at LHC occur very often.

The interaction between protons involve wide spectrum of processes highly
dependent on the fraction of the proton’s momenta carried by the partons $x$,
and the momentum transfer $t$ between partons. Process is usually regarded as
"hard" if is characterised by a large momentum component that is transverse to
the beam $p_{T}$, see Fig. 1. This is equivalent to the parton interactions
with a large $t$, since $t\simeq p_{T}^{2}$. However, the $pp$ collisions,
even at high energy, are dominated by soft processes.

Figure 1: a) Schema of typical $pp$ collision at LHC energy. When scattered
parton has large transverse momentum $p_{T}$ hadron is produced at large polar
angle (small pseudorapidity). Such an interaction is called "hard". The
majority of interactions are between partons that carry small fraction of
proton momentum and give rise to the flux of hadrons (mainly pions) produced
at lower angles (high pseudorapidity). The Higgs boson, or any other
interesting massive particle $M$, is rather produced in hard interactions,
whereas particles originating from soft interactions are the main source of
radiation damage in detector closest to the interaction point. b) The range of
pseudorapidity for different components of the total $pp$ cross-section with
the LHC detectors coverage.

Detectors at the LHC experiments are situated around proton collision points,
along the beam line (usually considered as the "$z$" axis in the Cartesian
coordinate system). The acceptance of the ATLAS and CMS tracking detectors
covers the pseudorapidity region222Pseudorapidity is defined as:
$\eta=-ln(tg~{}\theta/2)$ and reflects the distribution of the particles polar
angle with respect to the beam axis. $|\eta|<2.5$ (central part), whereas LHCb
is capable to track particles at lower angles: $2<\eta<5$ (forward). Hard
processes are therefore reconstructed in the central part of the detectors,
whereas most of the soft interaction result in the final states that are
produced in the forward direction. The schema of a LHC $pp$ event and the
pseudorapidity intervals in the LHC experiments are shown in Fig. 1.

###  2.1 Hard and soft regime in proton-proton cross-section

The interaction among hadrons consist of the mix of processes that are driven
by a few interactions at the partonic level. The hard (or "semi-hard")
interactions involve parton-parton interactions with large momentum transfer
that is sufficient to resolve the structure of a hadron. Hard interactions are
driven by processes with high momentum transfer $p_{T}>$2 GeV/c and can be
described by perturbative QCD (pQCD). Hard scattering results in jets in the
final state. Soft interactions are characterized by an energy scale of an
order of the hadron size (1 fm), what correspond to the momentum transfer
$|t|\propto 1/R^{2}\approx$ 1 (GeV/c)2. At such large distances perturbative
QCD cannot be applied. In such events particles in the final state have large
longitudinal, but small transverse momentum.

Each hard interaction between partons from two protons is usually accompanied
by the soft interaction between the beam remnants (so-called Underlying Event,
UE). Radiation of gluons in the initial or final state also give rise to the
particles with low $p_{T}$. Therefore one can assume that $pp$ collisions are
dominated by soft processes and the description of the proton interactions
involve both perturbative and non-perturbative components, the latter being
the subject of modelling and introduces considerable uncertainties. Modelling
of the hadronic collisions includes the description of the initial state
partons density, process of interaction (creation of parton showers) and
hadronisation (formation of the stable hadrons).

It is hardly possible to establish one, exclusive and exhaustive theory for
the description of proton-proton interaction at both hard and soft regions.
Proton is not only composite but also extended object, which transverse
dimension is Lorentz contracted when it is accelerated to the LHC energies.
Thus, accelerated protons look like flat disks traveling in a direction
perpendicular to the disk surfaces and, due to the time dilatation, quarks and
gluons in the same hadron do not interact among themselves (partons are
"frozen" as in Color Glass Condensate model CGC ). Therefore, there is neither
no interference between the initial state interaction, nor final state
interaction in the same hadron. Thus, we can assume that the interaction
occurs only between partons from different protons and happens in a very short
time during collision. Hence it can be considered as probing of the internal
structure of the proton bound state with the resolution depending on the
momentum transfer between scattered partons. The probability of a hard
scattering with a large momentum transfer depends on the chance of finding two
partons, one from each proton, within the short distance. At the LHC energies
it is quite high and, occasionally, even more than one hard scattering may
occur in addition to soft interactions. This phenomenon is referred as double
parton-scattering (DPS) and is considered in the current physics models for
the LHC particle production.

Thanks to the lack of the interference in the scattering of partons from
different hadrons, the description of the inelastic proton-proton collision is
based on the concept of factorization of the cross-section fact_collins . It
is assumed that the cross-section can be computed as a convolution product of
perturbative hard scattering cross-section of point-like partons and universal
factors called parton density functions (PDF) that are assumed to be long-
distance non-perturbative quantities and are conveniently written in terms of
kinematic variables $x$ and $Q^{2}$:

$\sigma=\sum_{i,j,k}\int dx_{1}dx_{2}~{}dt~{}\sigma^{k}_{ij}\times
f_{i}^{1}(x_{1},Q^{2})f_{j}^{2}(x_{2},Q^{2}),$ (1)

where $\sigma^{k}_{ij}$ is the QCD hard-scattering cross-section for the $k$th
process between parton $i$ and $j$, with a momentum transfer $t$. Functions
$f_{i}^{1,2}(x_{1,2},Q^{2})$ describe the probability for finding a parton $i$
with a fraction $x$ of the proton momentum ($0<x\leq 1$) when the proton is
probed at the scale of $Q^{2}$. The main problem in the total hadron cross-
section is that it can be neither calculated not measured for the full range
of $x$ and $Q^{2}$. One need to know the distribution of partons in a hadron
and partonic cross-section, both for very small and maximal momentum
transfers.

Therefore, the realisation of the factorization idea in generators depends on
the physics model and requires fixing the momentum scale for hard and soft
interaction. MC event generators are used to model the final state particle
systems produced in the collision and to obtain predictions for the parameters
that cannot be determined experimentally, especially in low-pT region.
Phenomenological models for soft hadron interaction used in general-purpose
event generator like PYTHIA, SHERPA of HERWIG takes as a start the
perturbative parton-parton interaction with the models of parton showers,
hadronisation and multiparton (MPI) interaction. This approach enables tuning
of the parameters for the better agreement with data and, thanks to the well-
known cross-section in the hard regime, provides the predictive power for the
LHC physics. It however does not describe well the very soft region.

Another approach is undertaken by the PHOJET, EPOS or DPMJET generators. These
tools use Regge theory, when the interaction between partons comes from the
color chains which hadronize and produce particles. MPIs are introduced by the
exchange of additional objects, like hard pomeron. Higher multiplicities in
Run II LHC data enforced a general revision of this approach generators .

###  2.2 Proton-proton cross-section

The total $pp$ cross-section $\sigma_{tot}$ includes elastic $\sigma_{el}$ and
inelastic $\sigma_{in}$ processes. The latter contains several components:
diffractive $\sigma_{diff}$ (single, double diffractive and central
production) and non-diffractive $\sigma_{ND}$ processes:

$\sigma_{tot}=\sigma_{el}+\sigma_{in}=\sigma_{el}+\sigma_{diff}+\sigma_{ND}$
(2)

In elastic collision protons do not change the initial energy and move almost
along the beam line after the interaction, see Fig 1. Elastic interactions are
driven by the long-ranged Coulomb interaction and, much stronger, short-ranged
strong interaction. The former is well-known and predictable within the QED,
the latter is hard to describe since the perturbative QCD is not applicable in
the low-momentum-transfer region, where most of the elastic hadron interaction
occur.

Inelastic processes are driven by parton-parton interaction described in Eq. 1
and contain both soft (like diffractive) and hard components. $\sigma_{tot}$
is not calculable in the framework of the perturbative QCD. Hence the Regge
theory, which considers exchange of set particles that belong to the Regge
trajectory as a cause of interaction, is used for the prediction of the
$\sigma_{tot}$ high energy behaviour.

The soft contribution is parametrised by the soft pomeron exchange which leads
to the peripherial production of partons and, eventually, hadrons. In hard
interactions, mainly among valence quarks, all intermediate partons are
characterised by large virtualities and all scattering amplitudes can be
calculated using the pQCD. Sea quarks and gluons share small fraction of
proton’s energy and interact via non-perturbative small momentum transfer
reactions that appear as long cascades of partons. Created sea quark-antiquark
pairs are connected by soft pomeron and undergo both soft and hard
interactions gribov_regge . In pQCD pomeron is regarded as series of gluon
ladders.

In single diffractive (SD) processes one of the protons survives the collision
while the other breaks up and produces some particles in a limited
pseudorapidity region. The intact proton moves at very small angle with
respect to proton beam line. The SD events are characterised by the large
rapidity gap between the outgoing proton and the produced system. In double
diffractive (DD) events both protons dissociate into system of particles which
are separated by the rapidity gaps. As a special case of diffractive reaction
central production (CD) may be considered. In these processes both protons
survive the collision losing only small amount of energy. A few particles
(most often two) are produced centrally, with a rapidity gap from each of the
proton. CD process are much less frequent from the SD and DD but can provide
valuable information of the proton’s structure and dynamics. In non-
diffractive events a large number of particles are produced but, opposite to
the diffractive events, no gap in rapidity occurs.

The relative contribution of different terms is established based on physical
models with experimental inputs. In general, it can be evaluated, that at the
LHC energies the elastic cross-section amounts to about 25% of the total
cross-section, non-diffractive part is about 60% and diffractive interactions
about 15%. These values vary depending on the applied model and $\sqrt{s}$.

The properties of various final states that contribute to the total proton-
proton cross-section differ significantly, as indicated in Fig 1. There are
two protons with the energy that is equal the beam energy outgoing almost
parallel to the beam in elastic interaction, in the SD events a several
particles (mostly pions) are produced except one proton at very small polar
angle, DD processes provides multiplicities of the order few dozens, whereas
most of particles are produced in the detector acceptance through the
inelastic ND processes. The particles’ energy spectrum and the distribution of
the event multiplicities in different type of proton-proton interactions at
$\sqrt{s}$=14 TeV are shown in Fig. 2.

Figure 2: Left: Energy spectrum, right: distribution of particle multiplicity
for events generated in non-diffractive (ND), single diffractive (SD) and
double-diffractive (DD) processes in 0.5$\times$106 proton-proton
interactions.

The various components of the total $pp$ cross-section have also a different
pseudorapidity distributions, see Fig. 3. These plots show particles produced
in half milion proton-proton collision at $\sqrt{s}$=14 TeV, generated by
Pythia 8.2 generator.

Figure 3: Distribution of the pseudorapidity of the particles participating
in: single diffraction (left) non-diffractive inelastic interactions (right).
More than 70% of produced hadrons are pions, 40% of particles are produced in
the central region and 25% (35% of total number of protons and 20% of pions)
at $|\eta|>5$ (along the beam line).

It can be noted that most of the particles reconstructed from $pp$ collision
at the LHC energies originate from the inelastic ND processes or are the
products of protons dissociation in diffractive events. Since the central
collisions are more likely to contain high-pT partons, most of the particles
with higher transverse momentum are produced in the central region of the
detectors.

###  2.3 Pythia 8.2

Pythia is a Monte Carlo (MC) event generator of hadron-hadron collision based
on perturbative QCD which describe properly parton scattering above some
$p_{T}$ value ($p_{T}\gtrsim 2$ GeV/c), extended to the low-$p_{T}$ region
with a models for the soft part. It generates a wide spectrum of processes,
starting from both soft- and hard-QCD processes to Higgs production and exotic
particle production and covers all components of the total hadronic cross-
section. This generator keeps on evolving and the currently used version
Pythia 8.2 includes tuning originating from the latest LHC results from Run II
data taking period Pythia8 .

The total proton-proton cross-section is described in frame of Donnachie-
Landshoff parametrisation as Donn_Lann :

$\sigma_{tot}^{pp}(s)=As^{P}+Bs^{-R}$ (3)

All topologies represented by relation 2 are included with the amplitudes
within Regge theory with one pomeron ($P$) and one Reggeon ($R$) term. The
parameters $A,B,P$ and $R$ obtained from the Regge fits to data and give:
$A=21.70$, $P=0.0808$, $B=56.08$, $R=0.4525$ Pythia8 . The prediction of the
the $\sigma_{tot}$ behaviour as the $\sqrt{s}$ increases are revised as soon
as new experimental results are available. One of the major correction had to
be applied after the observation from the first runs of LHC at higher
$\sqrt{s}$, that the parton-parton cross section became larger than the total
proton-proton cross-section. It indicated that more than one interaction among
the partons occurred in one $pp$ collision. Multiple Parton Interaction (MPI)
phenomenon was expected and therefore, based on experimental evidences,
generators had to be corrected for the better agreement.

The elastic cross-section is related to the total cross-section via the
optical theorem Coll_Regge  and is parametrised as an exponentially decreasing
function of the Mandelstam invariant $t$. The inelastic component is obtained
by the subtraction of the elastic cross-section from the total cross-section:
$\sigma_{in}^{pp}(s)=\sigma_{tot}^{pp}(s)-\sigma_{el}^{pp}(s)$. The split of
the inelastic cross-section into SD, DD and CD components is based on the
specific Pythia parametrisations which are available for possible tuning.
Eventually, the non-diffractive part of hadronic interaction (often called in
experiment minimum bias event) is evaluated by integrating the diffractive
components and subtracting them from $\sigma_{in}^{pp}$.

Recent results from TOTEM, ALPHA and LHCb updated information on
$\sigma_{tot}^{pp}(s)$ and $\sigma_{el}^{pp}(s)$ at LHC energies $\sqrt{s}$
=7-8 TeV and 13 TeV MPI . It resulted in more reliable parametrisation of the
Eq. 3 and revision of the previous version of the generators. The models for
diffractive interactions were populated with hard-pomeron exchange with the
additional possibility of MPIs and initial and final state radiation as well
MPI .

The limitation of the Eq. 1 comes from the substantial contribution of the
soft, non-perturbative component of the $pp$ interaction. The cross-section
above arbitrary chosen $p_{Tmin}$ provides the divergence as $p_{T}\rightarrow
0$ ($p_{T}$ is the transverse momentum of the outgoing parton in the parton-
parton center-of-mass frame):

$\sigma(p_{Tmin})=\int^{s/4}_{p_{Tmin}}dp_{T}^{2}\frac{d\sigma}{dp_{T}^{2}}.$
(4)

The scale $Q_{min}^{2}=p_{Tmin}^{2}$ can be regarded as a point that connects
the soft and hard components of the generator. The choice of $p_{Tmin}$ is of
a crutial meaning - the lower it is chosen the higher average number of MPIs
and particles in the final state. The perturbative parton-parton scattering
cross-section given by Eq. 1 and 4, dominated by the $t$-channel gluon
exchange, is divergent at low-momentum transfers as $1/t^{2}\sim 1/p_{T}^{4}$,
The divergence in Eq. 4 may be regularised by the introduction of the
threshold parameter $p_{T0}$ as: $1/p_{T}^{4}\rightarrow
1/(p_{T}^{2}+p_{T0}^{2})^{2}$. In such a way the perturbative limit is reached
as $p_{T}\rightarrow 0$ with a smooth fall at a scale $p_{T0}$ MPI . This
approach is supported by the fact that at low $Q^{2}$ (low $p_{T}$) the
resolution of the probing projectile is poor and the partons in proton are
screened by one another. Therefore as $p_{T}\rightarrow 0$ the cross-section
takes small but non-zero value that depends on the matter distribution inside
the hadron minbias_pred .

A dependency on the proton-proton centre-of-mass energy should be introduced
in addition since at higher energies partons are probed at smaller $x$, where
the parton density increases and the distance of colour screening decreases:
$p_{T0}=p_{T0}^{ref}(\sqrt{s}/\sqrt{s}_{0})^{\epsilon}$. The parameter
$\sqrt{s}_{0}$ is given at a reference energy and $p_{T0}^{ref}$ is $p_{T0}$
at $\sqrt{s}_{0}$. It means that at given $\sqrt{s}$ the number of MPIs
depends on the $p_{T0}$, smaller values of $p_{T0}$ result in more MPIs
because of the increase of the MPI cross-section and higher value of the
average particle multiplicity. In that way the parameter $p_{T0}$ is scaled
with the respect to reference value $p_{T0}^{ref}$ that is tuned to the
experimental results at given $s_{0}$. The introduction of the $p_{T0}$ in
Pythia generator allowed to include the soft scattering regions into hard
scattering regime with the price of a special parameter tuning.

The strong couple constant $\alpha_{s}$ become also divergent at low $p_{T}$
and had to be regulated by the cut-off parameter $p_{T0}$ as well. It resulted
in change of the scale of $\alpha_{s}$ since it had to be determined in the
scale of $\alpha_{s}(p_{T}^{2}+p_{T0}^{2})$ instead of
$\alpha_{s}(p_{T}^{2})$.

Quite a lot parton distribution functions (PDFs) are recorded in the LHAPDF
libraries lhapdf  and available in Pythia 8.2. Initially PDFs included a
leading-order contribution only but the recent event tunes provided by ATLAS
and CMS included next-to-leading-order (NLO). The parameters were tuned to the
latest LHC results and improved the charged multiplicity distribution from
both, low- and high- side, especially for the central rapidity values.

To summarize: the production of particles in proton-proton collisions
simulated by Pythia 8.2 generator depends on a number of parameters. The most
significant are: $p_{T0}^{ref}$, $\epsilon$, $\alpha_{s}$. The parameter
$p_{T0}$ separates the perturbative from the non-perturbative regions and
depends on the centre-of-mass energy $\sqrt{s}$. Therefore, the reference
value $p_{T0}^{ref}$ at chosen reference value $\sqrt{s_{0}}$ is set with a
power-like dependency on $\sqrt{s}$ with parameter $\epsilon$ .

The comparison of the hadron (pions, protons, kaons, neutrons) multiplicity
generated with different settings of $p_{T0}$ and $\epsilon$ is shown in Fig.
4. The values of parameters were chosen according to the tuning provided by
the LHC experiments after Run I MPI . The value $\alpha_{s}$ was set to
$\alpha_{s}$=0.130 at $\sqrt{s}_{0}$=7 TeV according to the experimental
tuning.

All distributions were generated at the proton-proton centre-of-mass-energy
$\sqrt{s}$=14 TeV. Three, important for the experiments, regions in
pseudorapidity were defined: $|\eta|$>5 - particles are produced very close to
the beam line, beyond the acceptance of any experiment, 2.5<$|\eta|$<5 - in
the forward direction, within the LHCb acceptance and $|\eta|$<2.5 - the
central detectors.

Figure 4: Distribution of hadron multiplicities for different settings of
Pythia 8.2 parameters $p_{T0}^{ref}$ and $\epsilon$. Multiplicity is
calculated for: full polar angle, $|\eta|$>5, 2.5<$|\eta|$<5, and
$|\eta|$<2.5.

Smaller values of $p_{T0}$ result in higher number of particles produced and
higher hadron multiplicities. The power $\epsilon$ has less influence on the
multiplicity distributions, albeit higher value increases the $p_{T0}$ cut-off
thus reduces the number of particles from collision.

In one proton-proton collision at $\sqrt{s}$=14 TeV about 100 hadrons (charged
pions, kaons, protons and neutrons) are produced. The mean multiplicity varies
between 81-105 for the $p_{T0}^{ref}$=2.0-2.5 GeV/c. The difference in
settings of the $\epsilon$ parameter from the range $\epsilon$=0.2-0.3 (for
the same $p_{T0}^{ref}$) are at the level of 5%. The distributions of hadron
multiplicities generated with different settings are depicted in Fig. 4. This
analysis showed, that one set that result in much higher multiplicities (above
110): $p_{T0}^{ref}$= 1.9 GeV and $\epsilon$=0.2. This is displayed for better
understanding on the tendency: if the cut-off of hard scale was set as a low
value, the number of produced particles turned out to increase substantially.
The fist estimation based on predictions before the LHC era assumed the
$p_{T0}^{ref}$ to be slightly above 1.2 and provided more than 150 hadrons in
one proton-proton collision at $\sqrt{s}$=14 TeV. The results from LHC Run I
and first part of Run II shifted the parameters towards higher values and
events were simulated with lower multiplicities.

###  2.4 DPMJET 3

The concept of the DPMJET event generator is based on two-component Dual
Parton Model (DPM) which consist of soft hadronic processes described by the
exchange of pomerons within the Regge theory Coll_Regge , and hard processes
described by perturbative parton scattering DPM . Description of the hadronic
final states is provided by both non-perturbative and perturbative QCD
included in one common framework and separated by the parameter
$p_{Tcuttoff}$. DPMJET inherited the model for the proton-proton interaction
from the PHOJET generator DPMJET3 .

The scattering of protons in DPM is described by the topologically complicated
diagrams with multiple exchanges of pomerons in the $t$-channel. The dominant
contribution includes elementary soft interaction between protons which each
are split into coloured systems of valence quark and diquark. Two chains are
stretched between quark and diquark from different protons forming color
neutral systems that eventually move apart and hadronize. As energy of protons
increases, the diagrams with topological expansion of the chains between sea
partons give rise to multiparton scattering. In hadron-hadron interactions
several pomerons can be exchanged in parallel, what stands for MPIs. This
concept comprises exchange of a single, double, triple or even loop of
pomerons, hard pomerons and soft pomerons. Any of experimentally known
process, like single or double diffraction, can be attributed to one of the
types of exchanged pomerons, or combination of them.

Differential elastic proton-proton cross-section is evaluated as a sum of
scattering amplitudes expanded as a series of partial waves with different
angular momentum. The total cross section is related to the elastic cross-
section via optical theorem, therefore the inelastic cross-section
$\sigma_{in}$ in DPMJET 3 generator is an internal element of the model and
includes diffractive processes without any external tuning. The model can be
changed as soon as one can obtain the appropriate measurements of
$\sigma_{tot}$ or $\sigma_{el}$.

The DPMJET 3 is the latest version embedded in the FLUKA package FLUKA ;
FLUKA_rev . The current revision of the program, bundled with FLUKA, was
revised and updated to face up with the comparison with LHC data at
$\sqrt{s}$=7 and 13 TeV and differs significantly from the standalone, not
tuned, version of the program DPMJET3_rev . The revised model includes one
soft and hard pomerons together with a "effective" reggeon. Therefore, the
same model is used for soft and hard components, with fixed, but tuned to
data, parameters. This differentiates DPMJET 3 from Pythia, where a vast of
parameters can be changed according to the experiments’ preferences and
different purposes.

##  3 Comparison of generators for particle production

Both generators, DPMJET 3 and Pythia, use factorisation theorem, which allows
hadronic cross-sections to be expressed in terms of parton-level cross-
sections convoluted with hadron partonic densities. DPMJET 3 originally
contained a description of the soft interaction only, and the hard QCD
processes were included at the later stage. Pythia, on the contrary, was
intentionally dedicated to model the high energy collisions with high
transverse momentum transfers, and the size of the soft component was tuned to
follow the experimental results.

Both generators have the same model of hadronisation and use PDF
parametrisations from the same LHAPDF library lhapdf .

Although physics models employed in the Pythia 8.2 and DPMJET 3 generators
differ in the approach to the soft and hard components of the total cross-
section, both are tuned to the latest experimental data and should not provide
substantially different results. Regardless of the total $pp$ cross-section is
the same, the contribution of diffractive events may be different, leading to
different angular distributions.

Radiation damage in the bulk of silicon sensors depends on the number of
particles produced in the collision, the angle with respect of the proton
beamline and the energy spectrum of particles traversing the detector.
Therefore, in this study, the comparison of event multiplicities, particles
transverse momenta, angular distribution and energy spectra are provided.

###  3.1 Multiplicity distribution and energy spectra

Depending on the version of the program and parameters used, the number of
particles created in one proton-proton collision may differ significantly.
Protons, neutrons, pions and kaons, which are regarded as hadrons in this
document, are particles which are of the crucial meaning for the radiation
damage in bulk of silicon.

The average multiplicity of hadrons produced in both generators is shown in
Table 1. This table also contains the composition of hadron flux given with
respect to the number of hadrons produced in one interaction. Elastic proton-
proton collisions are excluded from this comparison.

| $N_{h}/pp$ |  $N_{\pi}/N_{h}$ [%] |  $N_{p}/N_{h}$ [%] |  $N_{n}/N_{h}$ [%] |  $N_{K}/N_{h}$ [%]
---|---|---|---|---|---
Pythia 8.2 | 81-105 | 73.6 | 8.8 | 8.2 | 9.4
DPMJET 3 | 78 | 82.7 | 6.3 | 4.5 | 6.5
Table 1: The average number of hadrons $N_{h}/pp$ produced in one proton-
proton collision and fraction of pions, protons, neutrons and kaon calculated
with respect to total number of hadrons. In case of Pythia 8.2 values are
calculated for $p_{T0}^{ref}$ from the interval 2.0-2.5 GeV/c. For DPMJET 3
the default FLUKA 2011.2x.6 settings are considered FLUKA ; FLUKA_rev .

The average number of hadrons produced in one proton-proton collision is
slightly higher (5% for $p_{T0}^{ref}$=2.3 GeV/c) in the case of Pythia 8.2
with respect to DPMJET 3. This effect reflects the origin of generators - the
family of Pythia generators were created to describe the high-multiplicity
high-pT events with the extension to the soft processes. Probabilities of
protons and neutrons production are similar due to isospin invariance; kaons
are more often produced because of their lower mass. DPMJET 3 produces a
higher number of pions than Pythia 8.2, which is understandable in soft
processes, whereas the production of heavier baryons requires more momentum
transfer and occurs less often.

Distribution of the multiplicity for one of Pythia settings
($p_{T0}^{ref}$=2.3 GeV/c, $\epsilon$=0.2) and DPMJET 3 is depicted in Fig. 5.
It shows that the choice of parameters affects the number of produced
particles considerably. Multiplicity is also given in three experimentally
important regions of pseudorapidity: particles produced in $|\eta|$>5 that are
too close to the beamline to be detected, the region $2.5<|\eta|<5$ is
considered as the forward production, whereas $|\eta|<2.5$ represents a
typical central detector, see Fig. 5.

Figure 5: Distribution of multiplicity, i.e. number of hadrons (pions,
protons, neutrons, kaons) produced in one proton-proton collision in Pythia
8.2 with $p_{T0}^{ref}$=2.3 GeV/c and DPMJET 3. Multiplicity is calculated
for: full polar angle, $|\eta|$>5, 2.5<$|\eta|$<5, and $|\eta|$<2.5.

DPMJET 3 generator generates more low-multiplicity events than Pythia 8.2 and
very few events with more than 400 hadrons per interaction. The majority of
particles are produced in the central region because partons in proton
collisions may either propagate as beam remnants moving in the forward
direction or populate the central regions with a high number of low-pT
particles.

The energy spectra plotted separately for each type of particles, are shown in
Fig. 6. Although both generators produce a similar number of pions with
comparable spectra shapes, apparently the number of protons with the highest
energies (from diffractive events) produced by DPMJET 3 exceeds the Pythia 8.2
case. Observed differences in the energy spectrum are small and caused only by
the differences in the total number of particles. This effect is
understandable since parameters of the models in both generators are tuned to
the LHC data and samples correspond to the same centre-of-mass energy
$\sqrt{s}$=14 TeV.

Figure 6: Energy spectrum of pions, protons, neutrons and kaone produced in
proton-proton collisions at $\sqrt{s}$=14 TeV by Pythia 8.2 and DPMJET 3
generators. Distributions show spectrum of particles produced in one $pp$
collision (logarithmic scale).

Distribution of pseudorapidity for all particles with a subset of particles
with the energy above 100 GeV333Tracks reconstructed with good quality usually
have momentum below 100 GeV/c. Therefore particles with energy above 100 GeV
may not be reconstructed by standard tracking algorithms but still contribute
to the fluence and interact with the tracking detectors.  is shown in Fig. 7.
The majority of particles are produced in the central and forward region, but
protons carry the highest energies in the beam pipe region. They originate
from the diffractive low-multiplicity events and very rarely can be detected
in LHC silicon trackers.

Mean energy of particles is about 250 MeV and most of the energy is carried by
particles produced in the central values of pseudorapidity, see Fig. 7 and 8.

Figure 7: Left: distribution of the pseudorapidity of particles produced in
Pythia 8.2 and DPMJET 3 generators. Particles with energy greater than 100 GeV
have pseudorapidity $|\eta|$>4, therefore are beyond the acceptance of the LHC
tracking detectors. Distributions are scaled to one proton-proton collision.
Right: dependency of the kinetic energy on pseudorapidity (DPMJET 3). Most of
the energy is carried by particles with energy below 3 GeV in the central
region of the pseudorapidity.

Figure 8: Low-energy spectrum of hadrons produced in one proton-proton
collisions at $\sqrt{s}$=14 TeV by Pythia 8.2 and DPMJET 3 generators (left).
Dependency of the kinetic energy on pseudorapidity (right).

###  3.2 Transverse momenta distribution

Distribution of the particles’ transverse momenta $p_{T}$ shows agreement
between both generators with the small excess of DPMJET 3 events with higher
$p_{T}$ which originate from particles with high energy, see the left pane of
Fig. 9. More distinctive difference is visible in the distribution of
transverse momenta $p_{T}$ as a function of pseudorapidity $\eta$ (right plot
in Fig. 9). DPMJET 3 generates relatively more low-$p_{T}$ particles at higher
pseudorapidity, i.e. in the very forward region. In consequence, one observes
a drop in production of such particles in the central region where the
majority of Pythia 8.2 particles are generated.

The correlation of the mean momentum of the produced particles with the
corresponding event multiplicity $N_{h}$ is shown in Fig. 10, where the
comparison of mean $p_{T}$ of events generated by DPMJET 3 and Pythia 8.2 is
presented. It is noticeable, especially on the projection of these
distributions in Fig. 11, that Pythia 8.2 events contain more events with
higher $\langle p_{T}\rangle$.

Figure 9: Left: distribution of the transverse momentum pT of hadrons produced
in one proton-proton collisions at $\sqrt{s}$=14 TeV by Pythia 8.2 and DPMJET3
generators. Right: dependency of the particles’ transverse momentum pT on
pseudorapidity $\eta$.

Figure 10: Event multiplicity as a function of mean transverse momentum
$\langle p_{T}\rangle$ of hadrons in the event: DPMJET 3 (left), Pythia 8.2
(right).

Figure 11: Left: multiplicity of the DPMJET 3 and Pythia 8.2 events as a
function of the mean event transverse momentum $\langle p_{T}\rangle$. Right:
comparison of $\langle p_{T}\rangle$ distribution of events produced in one
proton-proton collision between DPMJET 3 and PYTHIA 8.2.

###  3.3 Minimum bias events

Minimum-bias data comprise of the events selected with the minimal
experimental impact, i.e., without applying any triggers. This class of events
is regarded as an experimental realisation of non-diffractive inelastic
interactions and is usually used for data-MC comparison. The straightforward
experimental selection criteria for each particle are used for comparison: at
least one the particle in the event must be within detector acceptance
$|\eta|$<5, the kinetic energy must be lower than 100 GeV and the transverse
momentum $p_{T}$ greater than 250 MeV/c.

The spectrum of kinetic energy and pseudorapidity distribution, depicted in
Fig. 12, indicate that minimum bias cuts have more severe effects in case of
the particles produced by DPMJET 3 generator. In the case of Pythia 8.2,
almost 90% of particles passed these cuts, whereas, in the case of DPMJET 3,
this is only 70%. After applying the cuts, the maximum of $\eta$ distribution
in case of events generated by Pythia 8.2 is higher by approximately 20% with
respect to DPMJET 3, whereas the difference for all generated particles,
before the minimum bias selection criteria, is only 4% (compare Fig. 7).

Figure 12: Spectrum of particles’ kinetic energy (left) and distribution of
their pseudorapidity (right) in one proton-proton interaction for minimum bias
events. Events are produced by Pythia 8.2, and DPMJET 3 generators.

A more significant rejection rate of particles produced by DPMJET 3 reflects
the softer structure of $\langle p_{T}\rangle$ distribution. The most
substantial effect pertains for pions and protons from diffractive events.

The distribution of $\langle p_{T}\rangle$ and the dependency of the minimum-
bias events multiplicity $N_{h}$ on the $\langle p_{T}\rangle$, defined with
above criteria, are depicted in Fig. 13. The comparison illustrates that the
Pythia 8.2 generates minimum bias events that are characterised by the harder
structure.

Figure 13: Transverse momentum distribution (left) and hadron multiplicity as
a function of $\langle p_{T}\rangle$ (right) of particles from minimum bias
events generated by DPMJET 3 and Pythia 8.2.

The comparison of selected features of particles produced by the respective
generators such as energy, momentum, transverse momentum and pseudorapidity
has been done. The most interesting result shows about 10% higher event
multiplicity prediction of Pythia 8.2. There are no significant differences in
the energy spectrum, but events generated by Pythia 8.2 have higher mean
transverse momentum and populates more the central region of pseudorapidity.
Very generic experimental criteria imposed on generated events show their
influence on pseudorapidity and transverse momentum distribution - in both
generators particles produced at the very forward region are not accepted by
the minimum bias cuts, but, also, the number of DPMJET 3 particles is further
reduced, mainly from the central region. This result shows the potential
impact on the calculation of fluence because the minimum bias criteria applied
at the generator level might reduce the number of reconstructable particles in
case of DPMJET 3.

##  4 Fluence simulation in the LHC environment

The impact of different physics models used for the generation of LHC events
may influence the distributions of particle fluence predicted for the
detectors, especially at small distances from the interaction point (IP). The
impact of secondary particles created in nuclear reactions with the material
present within the detector acceptance or stray radiation becomes more
important in more distant places of the detector,

For this study, a simulation tool FLUKA is used FLUKA ; FLUKA_rev . This is a
diverse software platform that includes event generator, a package for
geometry description and finally, the particle transport code to simulate
interaction with matter. DPMJET 3 generator is embedded in FLUKA and is used
as a default generator for the proton-proton collisions. It can be replaced
with other generators, like Pythia 8.2, by modifying a specialised interface
routine.

The FLUKA package provides a set of tools for the visualisation and
calculation of the variety of dosimetry parameters, like neutron equivalent
and particle fluence, total ionising dose, deposited energy and more. Fluence
is defined as the number of particles incident on a sphere of given cross-
sectional area perpendicular to the direction of each particle. Fluence can
also be expressed equivalently, as the lengths of tracks of the particles’
trajectories in a unit of volume icru . Thus, in general, the fluence depends
on particle multiplicity in an event, energy and polar angle of their tracks
with respect to the beamline ($z$-axis).

The damage occurring to the device is usually normalised to the fluence of
neutrons with the kinetic energy of 1 MeV, which would result in the
deposition of the same nonionising energy causing equivalent damage to the
material. The 1 MeV equivalent neutron fluence $\phi_{eq}$ is calculated
internally in the FLUKA program and considers the energy spectrum of the
damaging particles and the standard tables with the damage functions niel .

###  4.1 Geometry of the typical LHC experiment

The physics program of any LHC experiment requires accurate information on the
position of production and decay vertices and very efficient track
reconstruction. For better precision, the first measured point on track should
be as close as possible to the point of protons collision (IP). This is
necessary for the life-time measurements and to resolve multiple primary
vertices that are created in collision with more than one proton-proton
interaction (so-called pile-up events). Therefore, highly granulated silicon
detectors are usually situated in the very close proximity to the IP.

A typical LHC silicon tracker comprises of the cylindrical barrels with
silicon sensors placed parallel to the beamline (along the $z$ axis), see Fig.
14. For the analysis presented in this paper we created a hypothetical
detector with five such layers including 300-$\mu$m silicon sensors. The radii
of the barrels are: 6 cm, 8 cm, 10 cm, 12 cm and 14 cm. The collisions of
protons at the $\sqrt{s}$=14 TeV are provided by the DPMJET 3 and Pythia 8.2
generators, the particle transport and track reconstruction is performed
within FLUKA package.

Figure 14: Two-dimensional distribution of pion (left) and proton (right)
fluence. Detector consists of five 300 $\mu$m thick silicon cylindrical
barrels of radius 6 cm, 8 cm. 10 cm, 12 cm and 14 cm. Simulation is based on a
sample with 105 events from $pp$ interactions at $\sqrt{s}$=14 TeV generated
by DPMJET 3 and reconstructed with the FLUKA program. IP is at the $(x,y,z)$ =
$\\{0,0,0\\}$ cm.

Two-dimensional distribution of pion and proton fluence obtained for this
geometry is depicted in Fig. 14. Dependency of pions and protons fluence on
the distance from IP, for the increasing radius of the detector, is shown in
Fig. 15. Fluence is calculated for the amount of data that correspond to 1
fb-1 of luminosity assuming the same value for the inelastic cross-section for
both generators, $\sigma_{pp}$= 80 mb.

Figure 15: Pion (left) and proton (right) fluence as a function of the
distance from the IP. Events are simulated by DPMJET 3 generator and tracks
are reconstructed in FLUKA. Plots show the fluence at five radii in five
layers of cylindrical barrels (geometry from Fig. 14). Values are scaled to 1
fb-1 ($\sigma_{pp}$= 80 mb).

It is evident that in any detector geometry, the highest fluence is expected
at small radial distances from the IP because of the highest density of
tracks. However, the $z$-dependency has more complex behaviour. It is observed
that the particle fluence has a minimum at $z$ = 0 cm since there are hardly
any particles produced at large angles to the proton beam direction. This is
especially visible in case of protons, which are more often produced at lower
angles, whereas pions are more abundant at higher angles. At the high-$z$ tail
the distribution of fluence saturates, see Fig. 15. One can understand it as a
purely geometrical effect: when particle traverses consecutive layers at
different radii, the track lengths measured in equal detector thickness is the
same. The volume element in which tracks’ length is measured, increases with
the radius, so the tracks’ length density decreases for higher radius. But
when fluence is evaluated further from the IP in $z$-direction, this effect is
compensated by the higher density of particles produced at lower polar angles.
Therefore, the particle fluence, calculated as the track-length density, is
approximately constant at detectors further from the IP than about $z$>40 cm.

The fluence of pions, protons, neutrons and kaons is usually expressed in the
standard neutron equivalence units ($\phi_{eq}$), considering the number of
particles, their energy and polar angle. Particle fluence distributions, for
respective particle types, are weighted with the energy-dependent displacement
damaged functions $D(E)$ and combined niel . The damage function accounts for
both the cross-section for displacing silicon atoms and the energy released in
creating displacements. Most of the particles produced in the collision have
energy below 3 GeV. Damage functions are experimentally determined curves that
weight the damage caused by a different type of particles with respect to a
neutron of 1 MeV kinetic energy niel . In the case of particles with kinetic
energies above 1 GeV, this function is constant but for lower energies is
higher and depends strongly on energy and varies significantly between
different types of particles. Therefore, low energy particles contribute the
most significant to the neutron equivalence fluence.

The $z$-dependency of $\phi_{eq}$ for layers of detectors with different radii
is presented in Fig. 16. Particles traversing the detector in the central
region have lower energies than those at very low polar angles (compare Fig. 7
and 16). Therefore, particles produced in at the highest angles, although they
constitute less than 25% of the total amount of particles, contribute the most
significant to $\phi_{eq}$. The neutron equivalence fluence peaks at the $z$=0
cm (contrary to the particle fluence) and saturates slower at higher distances
than distributions of particles fluence. The decrease of $\phi_{eq}$ along the
$z$ direction is due to particles with higher energy which are more often
produced at very low polar angles and contribute to $\phi_{eq}$ with the lower
value of the damage function $D(E)$.

Figure 16: Dependency of the neutron equivalence fluence simulated with DPMJET
3 on radius and distance from IP (left). The dependence of the kinetic energy
on the polar angle (right). Values are scaled to 1 fb-1 ($\sigma_{pp}$= 80
mb).

###  4.2 Comparison of particle fluence

Dependency of the neutron equivalent fluence $\phi_{eq}$ on the radius and the
distance from the IP is shown in Fig. 17 (left). At the radius $R$= 6 cm
fluence reaches the value $\phi_{eq}$=1.7$\times$1012 neq/cm2 for 1 fb-1 of
data, whereas the detectors at $z$=100 cm are 30% less irradiated. This ratio
drops to 16% in case of stations at $R$=12 cm and decreases as radius
increases. Fluence, averaged over the radius between 0.5 cm and 14 cm, is
depicted in the right pane of Fig. 17. The lower radius is taken for the
calculation, the greater the $z$-dependency in fluence is visible.

Figure 17: Comparison of DPMJET 3 and Pythia 8.2 simulation of neutron
equivalence fluence as a function of $z$ (distance from IP along the
beamline). Fluence is calculated at radius 6 cm (solid line) and 12 cm (dashed
line) (left). Distribution of fluence averaged over radii 0.5 cm to 14 cm
(right). Values are scaled to 1 fb-1 ($\sigma_{pp}$= 80 mb).

Comparing the DPMJET 3 and Pythia 8.2 simulations one can see agreement in
detectors very close to the IP but in the more distant places, small excess
(8% at $R$=6 cm) of Pythia 8.2 fluence is noticeable. This can be explained
while comparing the number and energy of pions produced in each generator.
Although Pythia 8.2 generates up to 10% (depending on the tuning) more
particles than DPMJET 3 in one $pp$ collision, for the latter the fraction of
generated pions with respect to all hadrons is higher. Therefore, the number
of pions expected for the amount of data that corresponds to 1 fb-1 of
integrated luminosity in case of Pythia 8.2 is almost 10% lower than in DPMJET
3. In the low-$z$ regions (high polar angle $\theta$) the highest contribution
to $\phi_{eq}$ comes from pions; therefore DPMJET 3 shows small excess of
$\phi_{eq}$ at $z$=0 cm in comparison with Pythia 8.2. In the case of protons,
which populates small polar angles, Pythia 8.2 generates about 40% more
particles than DPMJET 3, which give higher fluence at a higher $z$.

The comparison of particle fluence for all types of hadrons simulated with the
two generators is shown in Fig. 18. It is visible that fluence of protons,
neutrons and kaons diverges significantly among generators, but in case of
pions, the difference is much smaller. The contribution of pions in the
neutron equivalence fluence is the most substantial, therefore smaller
difference between generators is caused mainly by the smaller difference in
the pion component. In the more distant places, protons and neutrons
contribute in larger amount and differences are more significant.

Figure 18: Fluence of protons, neutrons, pions and kaons simulated in FLUKA
with DPMJET 3 and Pythia 8.2 generators as a function of the $z$-distance from
IP. Fluence is averaged over radius from 0.5 cm to 14 cm. Values are scaled to
1 fb-1 ($\sigma_{pp}$= 80 mb).

Comparison of neutron equivalence fluence simulated by DPMJET 3 and Pythia 8.2
in various radius $R$ and $z$-positions is summarised in Table 2.

$\phi_{eq}$ [1012/cm2] |  $R$ = 0.5 cm |  $R$ = 6 cm |  $R$ = 12 cm
---|---|---|---
D | P | D | P | D | P
z = 0 cm | 216 | 214 | 1.71 | 1.66 | 0.45 | 0.44
z = 100 cm | 80 | 86 | 1.22 | 1.32 | 0.36 | 0.37
Table 2: Neutron equivalent fluence $\phi_{eq}$ calculated in DPMJET 3 (D) and
Pythia 8.2 (P) simulations at different radii ($R$) and distance from IP in
$z$-direction. Fluence is calculated for 1 fb-1 of data ($\sigma_{pp}$= 80
mb).

The statistical uncertainties in case of Monte Carlo samples are negligible.
The most significant contribution to systematic uncertainty comes from the
damage functions parametrisation used for the neutron equivalence calculation.
According to the latest recommendations it may reach as much as 30% rd50 . In
presented comparison this effect is negligible since the same (embedded in
FLUKA) damage functions were applied for both generators.

##  5 Summary

The current and future High Energy experiments require precise reconstruction
of tracks, vertices of production and decay in events that comprise hundreds
of particles. Silicon trackers are usually situated in the densest radiation
environment and are under the influence of the mixed radiation field. It has
been proved that the currently used silicon structures should operate without
compromising the quality of the physics data up to $\phi_{eq}$=1016 neq/cm2
rd50 . The fluence in the proximity to the IP, predicted for Run III at LHC,
may reach this limit and will be well above it in the foreseen future
experiments like HL-LHC HL  and FCC (Future Circular Collider) FCC ; FCC2 .
Fluence can hardly be measured and must be rather simulated instead.
Therefore, it is vital to perform a detailed comparison of available tools for
the generation of proton-proton events at LHC energies and transport codes
used for the simulation of track reconstruction.

Extensive studies of the most commonly used generators: Pythia 8.2 and DPMJET
3 showed that the former produces about 10% more hadrons in each proton-proton
collision. This difference originates from different physics model used to
describe proton-proton interactions: Pythia generators use pQCD for the
parton-parton interactions, DPMJET 3 model originates from soft hadronic
interactions populated with hard pomeron exchange and multiple parton
scattering for high-$p_{T}$ processes. Pythia models are tuned to the LHC
data, which is predominantly obtained from the central detectors minbias_pred
. DPMJET 3 was revised after the LHC Run I to correct the problem of deficit
of the high-multiplicity events FLUKA_rev . There are no significant
differences in the distributions of pseudorapidity, energy and transverse
momentum in the central regions, but in DPMJET 3 more high energy particles in
the very forward direction are generated.

The fluence of particles in the inner part of detectors consists almost
entirely of the primary particles produced in proton-proton interaction. Since
pions constitute about 80% of the total number of hadrons, the distribution of
$\phi_{eq}$ is the most sensitive to the discrepancy in the production of
pions. DPMJET 3 generates less particles per event but with a higher fraction
of pions than Pythia 8.2, hence both generators show a reasonable agreement in
the distribution of $\phi_{eq}$ for the regions very close to the IP. The
influence of protons and neutron grows up for regions at high-$z$ tails.
Therefore the difference in fluence simulated by Pythia 8.2 in a given
$z$-position amounts to 7% over the DPMJET 3. The $z$-dependency of the
$\phi_{eq}$ becomes almost uniform for the more distant radial regions of the
detector.

This analysis also showed the significant impact of Pythia generator
parameters settings on the multiplicity distributions and the effect of
minimum bias selection criteria on the number of particles that are further
transported through the detector material to perform track recosntruction.
Since DPMJET 3 generates more particles in the forward direction, a larger
fraction of them is rejected by the selection cuts. This observation is
important if $\phi_{eq}$ is simulated by methods that depend on the tracks
reconstructed by standard algorithms since only tracks reconstructed with good
quality and within the detector acceptance are further processed. It should
also be considered while comparing the simulation among LHC experiments due to
differences in the definition of minimum bias events.

###### Acknowledgements.

This work was supported by the National Research Centre, Poland (NCN), grants
No. UMO-2016/21/B/ST2/01083.

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# Predicting Generalization in Deep Learning via
Local Measures of Distortion

Abhejit Rajagopal
University of California, San Francisco
<EMAIL_ADDRESS>
Vamshi C. Madala
Universtiy of California, Santa Barbara
<EMAIL_ADDRESS>
Shivkumar Chandrasekaran
University of California, Santa Barbara
<EMAIL_ADDRESS>
Peder E. Z. Larson
Universtiy of California, San Francisco
<EMAIL_ADDRESS>

###### Abstract

We study generalization in deep learning by appealing to complexity measures
originally developed in approximation and information theory. While these
concepts are challenged by the high-dimensional and data-defined nature of
deep learning, we show that simple vector quantization approaches such as PCA,
GMMs, and SVMs capture their spirit when applied layer-wise to deep extracted
features giving rise to relatively inexpensive complexity measures that
correlate well with generalization performance. We discuss our results in 2020
NeurIPS PGDL challenge.

##  1 Background

In approximation theory, generalization error is measured by the maximum error
between an approximation $f^{*}(x)$ and the true target function $f(x)$ in a
domain $\mathcal{D}$, as $\min_{x\in\mathcal{D}}||f(x)-f^{*}(x)||_{\infty}$.
For classification problems, this definition is often relaxed to estimate how
often the target function is matched over the whole domain, e.g. using a
normalized 0-norm to measure train/test classification “accuracy”, or a more-
continuous $p$-norm to gauge average train/test error. In any case, measuring
the true generalization performance requires knowledge of the target function
over the whole domain, which presents an issue for deep learning problems with
finite high-dimensional data.

Based on recent works [1, 2], however, the 2020 NeurIPS PGDL Challenge seeks
predictors of deep neural network (DNN) generalization performance that are
based solely on training data, DNN parameters, and training hyperparameters
(e.g. learning rate, optimizer, training loss value, etc). Here, the true
generalization is measured as the accuracy of the model on a large withheld
set of test data. For each architecture (e.g. fully-convolutional or CNN +
fully-connected) and task (e.g. image classification on CIFAR10 or SVNH), the
challenge provides several trained networks with comparable training
performance ($\geq 90\%$) but significantly different testing performance. The
task is then to develop a method for predicting the generalization error for a
withheld set of networks trained for the same or new tasks with the same or
different hyperparameters.

Our approach to this challenge is based on recent work in approximation theory
[3, 4, 5], which suggests that deep networks constructed using the true
compositional structure of the target function can not only escape the curse
of dimensionality but also converge to the target with provably good
generalization error by penalizing the roughness of the interpolant, even when
only given point-data. For approximation from point data, there is an inherent
trade-off between (a) fitting all the training points, (b) the order or
complexity of the approximation, and (c) the rate of convergence. Thus, for a
given parameterization (DNN with weights), we seek to predict its
generalization performance using measures of the model’s regularity or
smoothness. While a comprehensive approach would count oscillations at every
node (e.g. neuron) in the feedforward directed acyclic graph $\mathcal{G}$
shared by the target function and its neural network approximation, due to the
large nodal degree of typical DNNs and for the purpose of this challenge, we
seek a less expensive measure of the network’s smoothness that can operate
within the challenge’s computational parameters ($\leq 5$min/model, $\leq
20$GB RAM).

##  2 Measures of Distortion

Vector quantization (VQ) is intimately connected to both statistical learning
theory and pattern recognition. In communications, VQ is used at the receiver
to determine which (possibly high-dimensional) codewords were transmitted. If
we view image classification through the same lens, we see that deep networks
are simply an elaborate decoder for the classification labels (e.g. one-hot
encoded or multi-level). For deterministic feedforward networks that are
trained well, features extracted in the intermediate layers are drop-in
replacements for the inputs. That is, the role of the deep network is to
degeneratively transform seemingly disparate, high dimensional image vectors
to clustered representations that are ultimately quantized to the
classification labels.

Although the final quantization rule can be quite complicated due to the high
nodal degree and nonlinear activation of many DNNs, [5] provides the insight
that for valid compositional architectures (decoders), minimizing the Sobolev
norm of the nodal functions yields provably good performance. This implies
that for two networks with the same training performance, the network with
smoother nodes/layers will tend to have better generalization performance as
the number of samples $N\rightarrow\infty$. We postulate that a
computationally efficient alternative to computing the Sobolev norm at each
node is to instead estimate the complexity of the required VQ rule at that
layer. For example, if we view the feature distribution of training data at an
intermediate layer $\ell$ as a symbol constellation, a complicated VQ rule
would correspond to a highly oscillatory (rough) function at the next layer
$\ell+1$.

###  2.1 Symbol Distortion and Euclidean Metrics

We can use $p$-norms to measure the expected complexity or distortion of VQ at
an intermediate layer, starting with the input layer (prior to any network
operation). This is commonly used in VQ to measure the mismatch between the
source symbols and reconstructed symbols [6], and also has connections to the
mesh norm $\min_{x,y\in T,x\neq y}\|x-y\|$ that is central to approximation
theory. As seen in Figure 1, the mean minimum intra-class distance of training
data reduces deeper in the network.

(a) Figure 1: $L_{2}$ distortion matrices as a function of layer (PGDL Model
668). The diagonal and off-diagonal elements represent the mean minimum intra-
and inter-class $2$-norm distance, respectively.

###  2.2 Label Distortion

Due to the high nodal degree of current DNNs, $p$-norms may systematically
over estimate the complexity of the required VQ. A simpler metric closer to
the classification task is the accuracy of a nearest-neighbor-type classifier.
Good networks will have diagonally dominant confusion matrices, corresponding
to simple classification boundaries even if the distance between samples is
large. However, such classifiers also depend on $p$ norms. To better capture
distances between the high dimensional feature vectors, we can use
dimensionality reduction techniques, e.g. kernel PCA.

####  2.2.1 Kernel Principal Component Analysis (kPCA) and Gaussian Mixture
Models (GMMs)

Unfortunately, computing the accuracy of a nearest-neighbor classifier using
transformed coordinates of the training data’s features does not correlate
well with generalization (test) performance of models with good training
accuracy, although it can help identify networks with unwanted degeneracy.
Instead, we compute the principal components and evaluate the “validation”
accuracy of the resulting nearest neighbor classifier using different
mutually-exclusive subsets of the training data. Since each kernel PCA model
is trained on only a subset of the training data’s features (e.g. at layer
$\ell$), the resulting confusion matrix measures how disparate the features of
training data are, or how generalizable the features are across different
subsets.

To enable faster computation during the competition, we additionally employ
Gaussian mixture models (GMMs) that are initialized and trained on each class
independently with 3-5 components per class. In addition to enabling a fast
classification computation (e.g. closest mixture centroid), GMMs can be used
to measure the confidence in each prediction with respect to the distribution
of the chosen training subset. That is, for each class $i$, we compute the
distortion $d_{i}=\mathbb{E}_{x_{j}}\\{\max_{k\in
GMM_{j}}p(\Theta_{k}|x_{i})\\}\;\forall\;j$ where $j$ represents each class in
the dataset; $k$ corresponds to each of the GMM components of class $j$
represented by $GMM_{j}$; and the expectation is approximated by calculating
the mean of inner maximum values for all data points in class $j$.

###  2.3 Support Vector Machines (SVMs)

To provide additional supervision into the classifier, we can employ support
vector machines (SVMs). Unlike a nearest-neighbor or GMM-based classifiers,
kernel SVMs can easily achieve arbitrary training accuracy, approaching the
performance of neural network layers at the expense of more parameters. That
is, in addition to computing confusion matrices and confidence measures based
on the distance to the boundary (e.g. margin error), the complexity of the SVM
can be directly measured using the number of support vectors required for
$\epsilon$ training error. The more complicated the decision function, the
more support vectors that are required, and higher the expected Sobolev norm.

##  3 Results

For the PGDL competition, we evaluated how well the aforementioned measures of
distortion correlate with test performance. With the exception of the direct
SVM complexity metric (number of support vectors), for each model we computed
the layer-wise symbol and label distortion using intermediate features
($L_{2}$ distance), their projected (kPCA) coordinates relative to GMM
centroids, and kernel (radial basis function) SVMs. The normalized trace (mean
of the diagonal) of these matrices was used as the final complexity measure
for these cases. The correlation with the test accuracy for each Task-1 and
Task-2 DNN model in the competition’s public dataset is shown in Figures
2(a)-2(b), while a comparison of correlation values is displayed in Table 1.
As can be seen, the number of support vectors (#SVs) provides the best
correlation for both Task-1 and Task-2 models.

(a) Task-1 models.

(b) Task-2 models.

Figure 2: Distortion-test accuracy correlations for different distortion measures. Table 1: $R^{2}$ values of distortion-test accuracy correlations Distortion measure | Task-1 | Task-2
---|---|---
Trainset | Testset | Trainset | Testset
Intermediate features | 0.53 | 0.61 | 0.56 | 0.9
GMM | 0.31 | 0.39 | 0.10 | 0.43
SVM | 0.50 | 0.71 | 0.44 | 0.93
#SVs | 0.59 | 0.93 | 0.81 | 0.98

To further demonstrate the properties of our complexity measures, we analyse
the distortion matrices for four representative models from the Task 1 public
dataset, indicated by stars in the correlation plots of Figure 2(a). Despite
having similar training performance, these models differ in their
generalization accuracy on the test set, as: model 668: 86.3%; model 542:
83.3%; model 152: 71.7% and model 90: 66.8%. The corresponding distortion
matrices for these models are shown in Figure 3. While the $L_{2}$ distortion
measure (Fig. 3(a)) is able to distinguish a well generalized model from a
poorly generalized one (by comparing the relative normalized trace), it is
harder to draw a conclusion by just looking at a single model’s distortion
matrix. This may be attributed to the limitation of the $2$-norms in computing
representative distances between high dimensional features, as previously
mentioned. The SVM distortion measure overcomes this problem, as can be seen
from Figure 3(c). SVM distortions are able to quantify the distances between
the points within a class and also between the point clusters of different
classes while maintaining a better correlation with model’s generalization.

(a) $L_{2}$ Distortion.

(b) GMM label distortion.

(c) SVM label distortion.

Figure 3: Comparison of distortion measures.

The kPCA + GMM distortion measure (Figure 3(b)) tends to approach this but
fails to quantify the distortion for certain models. This might be due to the
constraint imposed by the radial basis function (RBF) kernel used in kPCA, and
a forced reduction in the dimensionality (from 100s to just 3). This can be
seen from Figure 4, which depicts the feature vectors for the above models
after applying kPCA and fitting GMMs. The ellipses of same color but with
different gradients represent the clusters within a class. These plots provide
a visual validation to our claim that better generalized models have smoother
intermediate feature representations and thus better separated class clusters.

(a) Figure 4: Label wise GMM clusters for different models.

##  4 Conclusion

We showed that simple measures of regularity can be used to predict
generalization, laying the foundation for exploring generalization in deep
learning from the point of view of approximation theory and computational
harmonic analysis. We developed simple metrics to measure the smoothness of
DNN layers by exploiting the similarity between intermediate feature
representations extracted by the models. We evaluated these metrics against
VGG-like CNN models and Network-in-network models and demonstrated that these
metrics correlate well with the models’ generalization, independent of the
training strategy. Currently our proposed metrics do not take into account the
notion of adversarial performance or architectural hyperparameters, and so in
our future work we aim to incorporate these notions to develop more accurate
complexity measures that better match theoretical results.

## Acknowledgments and Disclosure of Funding

This research was supported by AFRL grant #FA8650-18-C-1137, NIH/NCI grant
#R01CA229354, and NIH/NIBIB grant #1F32EB030411-01.

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