# Improved simulation of beam backgrounds and collimation at SuperKEKB

A. Natochii<EMAIL_ADDRESS>University of Hawaii, Honolulu, Hawaii 96822,
USA  S. E. Vahsen  University of Hawaii, Honolulu, Hawaii 96822, USA  H.
Nakayama  High Energy Accelerator Research Organization (KEK), Tsukuba
305-0801, Japan  The Graduate University for Advanced Studies (SOKENDAI),
Hayama 240-0193, Japan  T. Ishibashi  S. Terui  High Energy Accelerator
Research Organization (KEK), Tsukuba 305-0801, Japan

###### Abstract

Mitigation of beam backgrounds via collimators is critical for the success of
the Belle II experiment at the SuperKEKB electron-positron collider. We report
on an improved simulation methodology, which includes a refined physical
description of the collimators and beam pipe, our first implementation of
collimator tip scattering, and in which the existing beam particle tracking
software has been embedded into a new sequential tracking framework. These
improvements resolve longstanding discrepancies between measured and predicted
Belle II background levels, and significantly reduce the computing time
required to optimize the collimation system in simulation. Finally, we report
on collimator aperture scans, which confirm the accuracy of the simulation and
suggest a new method for aligning the collimators.

Particle Tracking, Collimation System, Accelerator Background.

††preprint: PRAB

##  I Introduction

The KEKB accelerator complex provided a world-record instantaneous luminosity
of
$2.11\text{\times}{10}^{34}\text{\,}\mathrm{c}\mathrm{m}^{-2}\,\mathrm{s}^{-1}$
to the Belle experiment, which operated from 1999 through 2010 at the High
Energy Accelerator Research Organization (KEK) in Japan. The upgraded Belle II
experiment served by the SuperKEKB electron-positron collider [1] seeks to
achieve an unprecedented instantaneous luminosity of
$8.0\text{\times}{10}^{35}\text{\,}\mathrm{c}\mathrm{m}^{-2}\,\mathrm{s}^{-1}$
and to collect $50\text{\,}\mathrm{a}\mathrm{b}^{-1}$ of data in 10 years of
stable operation. Recently, SuperKEKB achieved a new world record luminosity
of
$2.4\text{\times}{10}^{34}\text{\,}\mathrm{c}\mathrm{m}^{-2}\,\mathrm{s}^{-1}$
[2]. In order to increase the luminosity by a factor of 40 compared to KEKB,
the SuperKEKB design involves new beam optics that utilize a nano-beam scheme
[3] and higher beam currents of $2.6\text{\,}\mathrm{A}$ and
$3.6\text{\,}\mathrm{A}$ for the electron and positron beam, respectively.
These changes will also significantly increase the backgrounds from the
machine. In particular, large beam losses near the interaction region where
Belle II is located, can adversely affect operational stability, quality of
data, and detector longevity. The main goal of the collimation system is to
protect the Belle II detector and delicate machine components such as
superconducting magnets, while maintaining practical beam lifetimes, beam
impedance, and injection performance.

In this article, we describe beam backgrounds caused by circulating beam
particles interacting with their surroundings, such as the beam pipes,
residual gas molecules, charges in the same bunch, and crossing beams. These
interactions all involve elastic or inelastic scattering, which causes beam
particles to deviate from their nominal trajectories. Some fraction of these
particles end up being lost fully from the beam and hit the beam pipe, which
produces showers of secondary background particles. Simulating machine-induced
backgrounds is challenging and requires a good understanding of all processes
causing beam losses. During the early commissioning stages of Belle II and
SuperKEKB, simulated and measured background rates differed by factors ranging
from $\rm 10^{-2}-10^{3}$ [4, 5]. For collisions of the beam with residual gas
molecules, discrepancies between simulation and measurement were expected, as
details such as the pressure distribution and measured gas composition in the
beam pipe had not been simulated. Subsequent work has steadily improved the
understanding of the beam-gas component and will be reported in detail
separately. For Touschek (intrabeam) scattering, however, the observed
discrepancy was not expected, and hard to explain. At that time, simulated
collimators would stop any incident particle hitting a collimator. One
hypothesis to explain the Touschek discrepancy was, therefore, collimator
leakage, where surviving particles scattering off of the collimator jaw (a
process known as tip scattering) reach the interaction region. In this work,
we finally resolve this Touschek discrepancy and show that its origin is
indeed collimator leakage, however not via tip scattering as originally
expected. We demonstrate instead that an improved simulation of the shape of
each collimator leads to considerable changes in predicted background rates.

This article is structured as follows. In Section II, we begin with an
overview of the main SuperKEKB background processes, their measurement, and
their simulation. We focus on measurements with a background monitoring system
based on diamond detectors. For that reason, those detectors are described in
detail in Section II.2. Next, in Section III, we review the collimation
system. Given that the improved simulation of this system had a particularly
large impact and resolved the Touschek data/MC discrepancy for the electron
beam, Section IV documents the detailed changes made to the simulation
procedure, including the exact model for each individual collimator, which
turned out to be a critical ingredient. Finally, the simulation is validated
with collimator scans described in Section V, using diamond detectors to
measure dose rates. The major conclusions and a summary of the research are
provided in Section VI. Details on improved beam-gas modelling, and more
extensive validation measurements by all Belle II sub-detectors, will be
published separately in a forthcoming article.

##  II Beam background

Here we give a brief overview of the major sources of beam-induced backgrounds
at the SuperKEKB collider, their measurements using a radiation monitoring
system, and the background simulation procedure.

###  II.1 Background sources

SuperKEKB is a high energy circular collider that consists of two rings, a
$7\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ high energy electron ring (HER) and
$4\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ low energy positron ring (LER). A
comprehensive overview of the machine design is given in Ref. [1]. Figure 1
depicts the SuperKEKB accelerator and related facilities, including the
interaction region (IR) where the Belle II detector is located. Belle II
extends approximately $\pm 4\text{\,}\mathrm{m}$ from the interaction point
(IP), where the two beams collide.

Figure 1: Schematic drawing of the SuperKEKB collider complex, with Belle II
shown in the interaction region.

Compared to its predecessor, SuperKEKB is designed to operate with double the
beam currents and twenty times smaller vertical beam sizes at the IP, both of
which imply a significant increase of beam-induced backgrounds in the
interaction region. The dominant expected background sources are 1) collision
processes such as Radiative Bhabba scattering and two-photon processes,
collectively referred to as luminosity backgrounds, 2) single-beam processes
such as beam-gas scattering, including Bremsstrahlung and Coulomb scattering
of beam particles with residual gas molecules, and Touschek scattering, which
denotes Coulomb scattering between particles in the same bunch, 3) synchrotron
radiation and 4) injection backgrounds induced by injected charges with large
amplitudes of oscillation due to injection kicker errors [4].

We focus here primarily on Touschek and beam-gas scattering, as these two
processes lead to off-orbit beam particles that form a beam halo, which then
interacts with the machine aperture and leads to beam losses.

A set of countermeasures are used to mitigate beam-induced backgrounds. To
suppress off-orbit particles, sets of 20 and 10 collimators with movable jaws
are installed around the HER and LER, respectively. Vacuum scrubbing reduces
the residual gas pressure in the beam pipe, thus suppressing beam-gas
scattering. Heavy metal shields outside the IR beam pipe protect Belle II
against electromagnetic showers. A thin layer of gold on the inner surface of
the beam pipe suppresses synchrotron radiation. More information about beam
background mitigation at SuperKEKB can be found in Ref. [6].

###  II.2 Radiation monitors

To ensure safe operation of the Belle II detector, a dedicated background
monitoring and beam abort system was installed at SuperKEKB. Its goal is to
monitor the radiation dose rates around the IR beam pipe. The system consists
of 28 diamond detectors mounted around the outside of the beam pipe in the
interaction region, as shown in Figure 2. These detectors are grouped and
named based on their location, as follows: four backward
($-56.8\text{\,}\mathrm{c}\mathrm{m}$) QCS detectors (QCS BW); six backward
($-27.4\text{\,}\mathrm{c}\mathrm{m}$) SVD detectors (SVD BW); four backward
($-9.8\text{\,}\mathrm{c}\mathrm{m}$) and four forward
($+13.6\text{\,}\mathrm{c}\mathrm{m}$) Beam Pipe (BP) detectors; six forward
($+29.2\text{\,}\mathrm{c}\mathrm{m}$) SVD detectors (SVD FW); four forward
($+56.8\text{\,}\mathrm{c}\mathrm{m}$) QCS diamond detectors (QCS FW), where
the numbers in parentheses are distances from the IP along the beam orbit.

Figure 2: Location of diamond detectors in the interaction region. Numbers in
rectangles indicate each detector’s azimuth angle. Blue and green rectangles
indicate diamond detectors dedicated to dose rate monitoring at
$10\text{\,}\mathrm{H}\mathrm{z}$ and reserved for the beam abort function,
respectively.

The diamond sensor packaged into a detector unit is an artificial single-
crystal produced by chemical vapor deposition (sCVD) [7]. Each sensor has a
volume of $4.5\text{\times}4.5\text{\times}0.5\text{\,}{\mathrm{mm}}^{3}$ and
a mass of $35.6\text{\,}\mathrm{m}\mathrm{g}$. For all 28 detectors dose rate
data are read out at $10\text{\,}\mathrm{H}\mathrm{z}$ for monitoring; buffer
memories with data sampled at $400\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$ are
read out after beam aborts for post-abort analysis. The four upper BP diamond
detectors (highlighted in green in Fig. 2) are enabled to generate beam abort
request signals, when sums of $400\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$
data in predefined moving time windows exceed programmed thresholds; they use
a measurement range compatible with signals from large beam losses, with
reduced sensitivity. The remaining 24 diamond detectors use a measurement
range providing the highest sensitivity of about
$10\text{\,}\upmu\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{/}\mathrm{s}$111$1\text{\,}\mathrm{r}\mathrm{a}\mathrm{d}$
= ${10}^{-2}\text{\,}\mathrm{G}\mathrm{y}$ is used as the dose unit in the
text. in dose rates read out at $10\text{\,}\mathrm{H}\mathrm{z}$, and are
dedicated to monitoring. More information can be found in Ref. [8]. In the
present article, monitoring diamond detectors are used to validate the
improved beam background simulation as reported in Section V.

##  III Collimation system

Figure 3: Location of the SuperKEKB collimators as on June 2020. Vertical and
horizontal collimator names contain the letters V and H, respectively. The
storage ring is divided into twelve sections, which are referred to as D01
through D12.

At a high luminosity machine such as SuperKEKB, beam collimators are critical
components necessary for reducing backgrounds in the interaction region to
acceptable levels. They help to avoid superconducting magnet quenches and
protect sensitive Belle II electronics and sensors from stray beam particles
and resulting background showers. Figure 3 illustrates the current positions
of SuperKEKB collimators. Each ring is divided into twelve sections, labeled
D01 through D12. Sections D09 and D12 contain collimators inherited from KEKB,
which have a jaw only on one side of the beam (KEKB-type, Fig. 5a, 5b) while
all other collimators are new (SuperKEKB-type, Fig. 6a, 6b) and have two jaws.

###  III.1 Description of collimators

Before we proceed, let us first introduce the terminology used to describe the
collimators, see Fig. 4: the collimator chamber is an element of the
accelerator vacuum system which hosts movable jaws; the movable jaw is a
tapered copper or aluminium block that absorbs stray beam particles far from
the beam core; the collimator head is usually a high atomic number (high-Z)
material block used to collimate the particle beam, it is the part of the
movable jaw closest to the beam core; the tip of the collimator jaw is the
shortest surface of the head on the beam side, the collimator aperture is
defined as the distance from the beam core to this surface of the head.

Figure 4: A typical collimator structure: chamber, movable jaw, head, tip. Table 1: Properties of the SuperKEKB collimators. The first column indicates the collimator type. Names of collimators with identical orientations and materials are grouped in the second column. The third column describes the orientation of each such group of collimators, while the fourth and fifth columns list the collimator jaw body and head materials. The radiation lengths (RLs) of these materials are [9]: RL(Cu) = $14.4\text{\,}\mathrm{m}\mathrm{m}$, RL(Al) = $89.0\text{\,}\mathrm{m}\mathrm{m}$, RL(W) = $3.5\text{\,}\mathrm{m}\mathrm{m}$, RL(Ta) = $4.1\text{\,}\mathrm{m}\mathrm{m}$, RL(Ti) = $35.6\text{\,}\mathrm{m}\mathrm{m}$. The sixth column shows the length of the collimator tip in both millimetres and number of RLs. Collimator | Orientation | Body | Head | Tip length
---|---|---|---|---
Type | Section/name | material | material | [mm]/[RL]
SuperKEKB | D01-D06 | Hor. | Cu | W | 10.0 / 2.9
D01-D02 | Vert. | Cu | W | 10.0 / 2.9
D06V1 | Vert. | Cu | Ta | 5.0 / 1.2
D06V2 | Vert. | Cu | Ta | 10.0 / 2.4
KEKB | D09-D12 | Hor. | Cu | Ti | 80.0 / 2.2
D09-D12 | Vert. | Al | Ti | 40.0 / 1.1

Here it is worth mentioning the main differences between collimator types used
in the collider, which are summarized in Table 1. All SuperKEKB-type
collimators (vertical and horizontal), with the exception of D06V1 and D06V2,
have a tungsten (W) head with a $10\text{\,}\mathrm{m}\mathrm{m}$ long tip.
The vertical collimator D06V1, which was installed during the 2019-2020 winter
shutdown, has a tantalum (Ta) head with a shorter,
$5\text{\,}\mathrm{m}\mathrm{m}$, tip, while the vertical collimator D06V2 has
a tantalum head with a $10\text{\,}\mathrm{m}\mathrm{m}$ tip. The jaw of
SuperKEKB-type collimators (i.e., disregarding the head made of high-Z
material) is made of copper (Cu). While SuperKEKB-type collimators have heads
that are $12\text{\,}\mathrm{m}\mathrm{m}$ wide transverse to the beam
direction, KEKB-type collimators have race-track-shaped cross-sections and
much wider heads. The jaws of KEKB-type horizontal and vertical collimators
are made of Cu and aluminum (Al), respectively. Their heads are made of
titanium (Ti) with a $80\text{\,}\mathrm{m}\mathrm{m}$ tip length for
horizontal and $40\text{\,}\mathrm{m}\mathrm{m}$ tip length for vertical
collimators. Each movable jaw is equipped with a linear variable differential
transformer (LVDT) with a position accuracy better than
$30\text{\,}\upmu\mathrm{m}$ and $80\text{\,}\upmu\mathrm{m}$ while the
repeatability is about $50\text{\,}\upmu\mathrm{m}$ and
$100\text{\,}\upmu\mathrm{m}$ for SuperKEKB-type and KEKB-type collimators,
respectively. The alignment precision of the collimator chamber with respect
to the nearest quadrupole magnet, which defines the transverse beam position,
varies from $200\text{\,}\upmu\mathrm{m}$ up to $500\text{\,}\upmu\mathrm{m}$
for vertical SuperKEKB-type collimators. For horizontal SuperKEKB-type and all
KEKB-type collimators the precision varies from $500\text{\,}\upmu\mathrm{m}$
up to $1000\text{\,}\upmu\mathrm{m}$. This uncertainty may cause an unintended
offset between the reference of the collimator head position and the beam
core.

---
(a)
(a)

---
(b)
(b)

Figure 5: KEKB-type collimators have a single jaw, on only one side of the
beam. All dimensions are in mm. (a) Conceptual drawing and photo of the
collimator assembly [10]. (b) Schematic drawing of the horizontal collimator
chamber.

---
(a)
(a)

---
(b)
(b)

Figure 6: SuperKEKB-type collimators have two jaws, one on each side of the
beam. All dimensions are in mm. (a) Conceptual drawings of the collimator
assembly [11]. (b) Schematic drawing of the collimator jaw.

###  III.2 Collimation system optimization

Careful adjustment of collimators is necessary to reduce experimental
backgrounds to acceptable levels for stable operation of Belle II. SuperKEKB
thus follows a collimation system optimization procedure where an operator
manipulates collimator jaw positions to fulfil a set of requirements. The aim
is to find suitable positions for each collimator jaw that both reduce
backgrounds and maintain adequate beam lifetimes. Since SuperKEKB maintains
constant beam currents by performing continuous top-up injections, it is also
necessary to achieve a high injection efficiency, which is defined as the
ratio of stored to injected charge. A lower injection efficiency requires a
higher time-averaged (net) injection rate, which in turn leads to a reduction
in Belle II data taking efficiency, as Belle II uses a Level-1 trigger veto
synchronized with injections. Aggressively closing the collimators reduces the
machine’s physical aperture, which is determined by the beam pipe’s cross-
section, including collimators. This results in lower injection efficiency, as
well as higher local losses at collimators, leading to undesirable activation,
heating, and potential collimator damage. On the other hand, a wide open
configuration leads to an increase of injection and storage backgrounds.
Therefore, the collimation system optimization requires a compromise between
(i) maintaining good injection performance and beam lifetime, both of which
favor wider collimators, and (ii) reducing backgrounds in Belle II, which
favors narrower collimators. Relying on a well-known algorithm based on phase
advance analysis, which is also known as betatron collimation [12], operators
aim to find the most optimal aperture of collimators. A half-integer phase-
advance between the IR and a given collimator (see Fig. 3 in Ref. [11])
provides an effective reduction of IR losses and hence of Belle II
backgrounds. This optimization procedure is performed at low beam currents to
avoid accidental collimator head damage, and requires an accurate tuning of
the radial position of each jaw, one by one, by monitoring interaction region
losses and injection efficiency.

##  IV Particle tracking simulation and recent improvements

The beam background simulation starts with a multi-turn particle tracking
framework named Strategic Accelerator Design (SAD) [13]. SAD is initialized
with a machine optics file, which describes the collider configuration and all
collimator apertures during a dedicated measurement. The program tracks
scattered particles through the machine lattice. Particles lost in the
interaction region are exported from SAD to Geant4 [14, 15, 16] for simulation
of the detector response. Geant4 scripts realistically describe the geometry
of the detector and produce digitized observables (e.g., dose rate) to be
compared against the experimental data.

Here we focus on two single-beam processes, beam-gas scattering and Touschek
scattering, and their interaction with collimators. These two processes are
the dominant sources of beam backgrounds at low luminosities. Luminosity
backgrounds, which we expect to dominate at higher luminosities, are not
affected by the collimators.

In the Belle II collaboration, the ratio of measured to MC simulated rates for
a given background component is colloquially referred to as the ”data/MC”
ratio. Such ratios are obtained from dedicated single-beam and luminosity
background studies conducted at SuperKEKB. A description of such measurements
and of the data/MC calculation procedure can be found in Ref. [4]. The
ultimate goal of the simulation is to reach data/MC $\approx$ 1, which would
demonstrate that our understanding of the machine is sufficient to predict the
background behaviour at different beam currents and optics. In the present
paper, we focus on the MC side, and explain in detail how the simulation has
been improved.

###  IV.1 Existing simulation infrastructure

The production of an MC sample for a given machine configuration, including
beam optics, collimation system settings, and beam currents, involves two main
steps. A multi-turn beam simulation tracks scattered beam particles through
the lattice of the accelerator. This involves the software package SAD, which
is described below. Beam particles lost near the IR are tracked in Geant4,
which predicts secondary showers due to interaction with material, and the
subsequent response of Belle II sub-detectors and diamond beam background
monitors. The present article is concerned mainly with the first step, which
we will now describe in more detail.

SAD is a simulation framework for accelerator design and particle tracking,
which has been developed at KEK since 1986. Single-beam particle tracking
starts by defining about 500 equidistant scattering positions along the beam
axis. The coordinate advancing along the nominal beam particle trajectory is
named S. At each scattering position, the scattering of electrons or positrons
is simulated. A vertical offset with respect to the beam center is applied to
each sextupole magnet position. These offsets generate machine errors referred
to as X-Y coupling. The generated offsets are normally distributed with a
standard deviation of $50\text{\,}\upmu\mathrm{m}$ around the beam center. We
assume particle bunches are 3D Gaussian distributions with a volume of
$\sigma_{\rm X}\times\sigma_{\rm Y}\times\sigma_{\rm S}$, where $\sigma_{\rm
X/Y}=\sqrt{\beta_{\rm X/Y}\times\varepsilon_{\rm X/Y}}$ is the standard
deviation of the normal distribution in the XY-plane, while $\sigma_{\rm S}$
is the standard deviation along the beam axis and hence is the length of the
bunch, $\beta$ is the beta function at the scattering position, and
$\varepsilon$ is the emittance of the beam. Randomly distributed particles are
generated at each scattering position with a modified momentum and a
statistical weight factor, both calculated based on beam-gas and Touschek
scattering theories [1].

We recently improved the IR beam pipe geometry in SAD, so that it agrees
better with the experimental reality and the shape used in the Belle II Geant4
model, see Fig. 7. When running SAD, a set of input parameters is specified.
These include $Z_{\rm eff}$, the effective atomic number of the residual gas
in the beam pipe. We assume $Z_{\rm eff}=7$, mostly due to CO molecules [17].
The number of simulated machine turns is usually set to 1000. We also enable
synchrotron radiation and acceleration by radiofrequency cavities in the SAD
simulation. Beam particles that cross the inner beam pipe surface or hit a
collimator aperture are denoted as lost beam particles, and these are not
tracked further by SAD. Instead, the simulation passes lost particles into
Geant4, which simulates their interaction with surrounding materials and
detectors.

---
(a) HER
(a)

---
(b) LER
(b)

Figure 7: Physical aperture of the HER (a) and LER (b) beam pipes in the
interaction region, as implemented in SAD. Colored lines represent the beam
envelope, which is limited by collimators. Arrows indicate the beam direction.
The beam optics simulated are those of June 2020: a beta function at the IP of
$\beta^{*}_{\rm H/V}=$60\text{/}0.8\text{\,}\mathrm{m}\mathrm{m}$$ and crab
sextupoles turned on for a so-called Crab Waist collision scheme [18].

Historically at KEK, the particle tracking simulation was organised so that
all lost particles were collected after a given number of machine turns in
SAD. Only elliptical shapes, rectangular shapes, and a mix of both shapes
could be used to describe collimators. By default, collimators were
implemented as infinitely thin planes that absorb all particles outside the
elliptical aperture defined by the half-axes $d_{\rm 1}$ and $d_{\rm 2}$, as
shown in Fig. 8.

---
(a) Horizontal
(a)

---
(b) Vertical
(b)

Figure 8: Default horizontal (a) and vertical (b) collimator models in SAD.
The ellipse half-axes $d_{\rm 1}$ and $d_{\rm 2}$ represent the distances from
the beam axis to the left/top and right/bottom tip of the collimator,
respectively. In each case, the second half-axis (vertical for horizontal
collimator and horizontal for vertical collimator) is set equal to the beam
pipe radius $45\text{\,}\mathrm{m}\mathrm{m}$. Particles hitting the grey area
are classified as lost.

To predict the losses in the IR with reasonable statistics, we typically
generate 100 particles per scattering position per ring, for every single-beam
scattering process (i.e., Coulomb, Bremsstrahlung, and Touschek). The KEK
Central Computer system [19, 20] enables us to run more than 1000 simulations
in parallel, thereby improving statistics and reducing computation time. Each
final tracking simulation sample contains about $10^{6}-10^{8}$ generated
particles per ring.

SAD stores beam particle coordinates at the end of lattice elements that have
an aperture narrower than the transverse particle position. Such particles are
considered lost, traced back to the inner beam pipe surface, and passed to
Geant4, which is used to simulate interactions with the beam pipe walls.

###  IV.2 New sequential tracking algorithm

To augment the functionality of the particle tracking in SAD, we developed a
new sequential tracking algorithm, which reports on stray beam particles turn-
by-turn. The idea of this method is as follows: after beginning particle
tracking, SAD stops at each collimator and stores the 6D coordinates of each
beam particle. Afterwards, the simulation continues tracking until the next
collimator. This process continues, with the algorithm sequentially stopping
at collimators until 1000 machine turns have been reached, upon which the
routine is terminated. At the end of the code execution, there is now not only
information about lost particles, but also a history of the beam dynamics
across all simulated machine revolutions. This straightforward algorithm gives
many benefits and allows for the study of beam evolution. Below, we discuss
some of the essential features resulting from the sequential tracking
algorithm.

###  IV.3 Corrected collimator profiles

Collimators used at KEKB all had heads with race-track shaped cross sections,
see Fig. 5. A fairly accurate elliptical approximation to this was therefore
implemented as the default collimator shape in SAD, see Fig. 8. At SuperKEKB,
however, only collimators D09 and D12 are of the original KEKB-type, while all
the others are of the new SuperKEKB-type. These newer collimators have two
opposite jaws with a rectangular head shape of
$12\text{\,}\mathrm{m}\mathrm{m}$ width transverse to the beam axis direction
(Fig. 6). To improve the simulation accuracy, we implemented a more realistic
SuperKEKB-type collimator shape in the SAD particle tracking scripts. This was
enabled by the sequential tracking technique introduced above.

At the tracking stage, the simulation program uses information about the
particle distribution at each collimator to define which part of the beam
interacts with a given collimator. Particles outside the edge of the jaw are
considered as not absorbed, and the routine continues to track them around the
ring.

Figure 9: Distribution of collimated beam particles at the LER D06H1
horizontal collimator. The red, dashed ellipse shows the original SAD
collimator. The two black, dot-dashed rectangles show the newly implemented,
more realistic collimator edge. The relative particle flux is shown in color.
The bin size is
$$0.2\text{\,}\mathrm{m}\mathrm{m}$\times$0.2\text{\,}\mathrm{m}\mathrm{m}$$.

Figure 9 illustrates a particle distribution simulated with default
(elliptical) and realistic (rectangular) profiles of the collimator. Particles
that would previously hit the elliptical collimator and be classified as lost
from the beam, can now pass the more realistic collimator and remain in the
beam to be tracked by the simulation. Although the majority of the collimated
beam is located at the tip of the collimator, particles far from the beam axis
and outside the real edge of the collimator jaw can contribute to IR losses.

###  IV.4 New collimator tip scattering simulation

To implement beam particle scattering off of collimators jaws, referred to as
collimator tip scattering, we carried out a dedicated Geant4 simulation and
parameterized the results. Targets made of tungsten, tantalum, and copper were
bombarded by $47\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$
positron and electron beams, respectively. The simulation results were then
transformed into a matrix form and used in SAD with the new, more realistic
collimator profiles described above.

Due to the tapered collimator structure, particles intercepted by the jaw at
different radial distances from the beam center traverse different path
lengths, $L_{\rm Z}$, inside the head along the direction of the beam.
Therefore, the survival probability as a function of $L_{\rm Z}$ was included
in SAD as well (Figure 10a). The induced angular kick and momentum change in
the case of scattering are shown in Figure 10b. When a simulated beam particle
collides with a collimator jaw, we apply a change to the particle’s momentum
($\Delta p_{\rm X}/p_{\rm 0}$, $\Delta p_{\rm Y}/p_{\rm 0}$ and $\Delta
p/p_{\rm 0}$), which depends on the path length inside the jaw. The momentum
change is drawn from a parameterized probability density function which is
different for each slice of $L_{\rm Z}$, and has a maximum equal to one (Fig.
10b). Afterwards, the particle is tracked again through the machine lattice
until it is lost or the simulation reaches the maximum number of turns. Due to
the Mathematica-like nature of SAD, which facilitates matrix calculations, our
enhanced algorithm does not significantly increase the computing time for
typical simulation statistics.

---
(a) Survival probability
(a)

---
(b) Momentum change
(b)

Figure 10: Geant4 simulation results for
$4\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$ positrons
interacting with tungsten. (a) Particle survival probability versus path
length inside the jaw, $L_{\rm Z}$. Statistical error $<1\%$. (b) Momentum
change, $\Delta P$, versus scattering angle, $\theta_{\rm Scat.}$, and path
length inside the jaw. Each slice of $L_{\rm Z}$ is normalized so that its
maximum value in the $\Delta P$-$\theta_{\rm Scat.}$ plane is unity.
Therefore, the color and the size of each box (bin) represent the relative
probability for a scattered particle with a given given $L_{\rm Z}$ to obtain
a particular $\Delta P$ and $\theta_{\rm Scat.}$. Bin size is
$$1\text{\,}\mathrm{m}\mathrm{m}$\times$40\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$\times$100\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{/}\mathrm{c}$$.

The realistic collimator shape and tip scattering were implemented only for
SuperKEKB-type collimators, while KEKB-type collimators are modelled as ideal
absorbers of particles outside the default elliptical aperture.

##  V Improved predictions and experimental validation

In this section, we review the key results of the improved particle tracking
and collimator simulations. The implementation of more realistic collimator
shapes solves a longstanding problem with the predicted background rates. In
addition, we introduce a new method for the collimation system optimization,
based on the new sequential tracking algorithm, which significantly reduces
computing time and helps operators set collimator positions for efficient
background reduction. Finally, we experimentally validate the improved
collimator simulation by comparing simulated backgrounds against measured IR
dose rates during a collimator aperture scan. We find good agreement between
experimental data and Monte-Carlo simulation results.

###  V.1 Resolving a longstanding problem

The improved particle tracking and collimator shape implementation
significantly improve the agreement between experimental data and Monte-Carlo
simulation. Whereas previously discrepancies as large as two or three orders
of magnitude were observed [4], these have now been reduced to a factor of ten
or less.

Table 2: Comparison of the predicted SuperKEKB beam particle IR loss rates ($\mathcal{R}$) and beam lifetime ($\tau$) from SAD simulations with elliptical (ellipt.) and realistic (real.) collimator profiles. Simulated collimator apertures and optics were those of May 2020: LER $\beta^{*}_{\rm H/V}=$80\text{/}1\text{\,}\mathrm{m}\mathrm{m}$$, HER $\beta^{*}_{\rm H/V}=$60\text{/}1\text{\,}\mathrm{m}\mathrm{m}$$ with the Crab Waist scheme. Background | Parameters | HER | HER | HER ratio | LER | LER | LER ratio
---|---|---|---|---|---|---|---
source | ellipt. | real. | real./ellipt. | ellipt. | real. | real./ellipt.
Coulomb |  $\mathcal{R}$ [MHz] | $0.23\pm 0.01$ | $9.38\pm 0.30$ | $40.78\pm 2.20$ | $55.25\pm 0.99$ | $57.48\pm 1.03$ | $1.04\pm 0.02$
$\tau$ [min] | $261.00\pm 20.26$ | $247.81\pm 18.67$ | $0.95\pm 0.10$ | $28.61\pm 1.71$ | $28.34\pm 1.67$ | $0.99\pm 0.08$
Bremsstrahlung |  $\mathcal{R}$ [MHz] | $0.52\pm 0.01$ | $0.55\pm 0.01$ | $1.06\pm 0.03$ | $4.74\pm 0.06$ | $4.66\pm 0.06$ | $0.98\pm 0.02$
$\tau$ [min] | $7513.63\pm 86.29$ | $7532.44\pm 92.27$ | $1.00\pm 0.02$ | $1524.39\pm 8.93$ | $1523.32\pm 8.60$ | $1.00\pm 0.01$
Touschek |  $\mathcal{R}$ [MHz] | $21.46\pm 0.74$ | $48.06\pm 8.93$ | $2.24\pm 0.42$ | $26.80\pm 2.33$ | $33.93\pm 2.77$ | $1.27\pm 0.15$
$\tau$ [min] | $59.48\pm 0.46$ | $58.65\pm 0.46$ | $0.99\pm 0.01$ | $6.75\pm 0.04$ | $6.75\pm 0.04$ | $1.00\pm 0.01$

Table 2 lists the rates of beam particles lost at the IR, defined as
$|S|<$4\text{\,}\mathrm{m}$$, for two different profiles of the simulated
collimators in SAD. We see that switching from the default (elliptical) to the
improved (realistic) profile, Touschek and beam-gas IR loss rates increase by
factors up to 40, while the beam lifetimes are unchanged. Note that measured
IR backgrounds are proportional to the IR loss rate, but also depend on the
detailed IR loss distribution. The increased IR loss rate is mainly due to the
beam particles, which were previously stopped in simulation by the idealized,
elliptical collimators, but which in reality are able to pass outside the
rectangular collimator jaw. This component is identified with the ”keep
tracking” label in Figure 9.

Although only four out of twenty collimators in the HER are SuperKEKB-type,
placed in the D01 section, we find that the more realistic collimator shapes
result in significantly larger HER beam losses; both the Coulomb and Touschek
components increase. We explain the more substantial background increase in
the HER compared to the LER as a net contribution from collimators narrower
than the interaction region aperture. All horizontal collimators in both rings
and three vertical collimators in the LER are narrower than the IR. Thus stray
particles in the vertical plane not absorbed by the most upstream vertical LER
collimator can be stopped by the other two downstream collimators before
reaching the IR. In contrast, in the HER, only one vertical collimator, D01V1,
is narrower than the IR aperture, while other HER vertical collimators in D09
and D12 sections are wider than the IR. Although Coulomb losses are
distributed in both planes, Touschek losses mostly occur in the horizontal
plane due to the non-zero horizontal dispersion and momentum deviation of
scattered particles. However, after hundreds of machine turns, Touschek
scattered particles that are not absorbed by horizontal collimators propagate
from the horizontal to the vertical plane due to the coupling between
horizontal and vertical betatron motions. Therefore, the change of the D01V1
collimator shape affects not only Coulomb but also Touschek beam losses. Also,
D01V1 has a larger horizontal beta function (Tab. 3) than LER vertical
collimators. Therefore, the modification of its shape in simulation enhances
the rate of particles passing outside the rectangular collimator jaw in the
horizontal plane, as there the beam is wider than the jaw’s transverse width.
The combination of all effects described above leads to a higher increase in
HER backgrounds compared to the LER case.

Table 3: SuperKEKB vertical collimator settings in May 2020, when the Crab Waist optics were used. The third and forth columns list the vertical (V) and horizontal (H) beta functions at the collimators, $\beta$. The fifth column shows the collimator aperture in units of the betatron beam size, $\sigma_{\rm\beta}=\sqrt{\beta\times\varepsilon}$, where $\varepsilon$ is the beam emittance. The smallest vertical apertures in the IR are $106\sigma_{\beta\rm V}$ and $55\sigma_{\rm\beta V}$ for LER and HER, respectively. Ring | Collimator |  $\beta_{\rm V}$ [m] |  $\beta_{\rm H}$ [m] |  d [$\sigma_{\rm\beta V}$]
---|---|---|---|---
LER | D06V1 | 67.35 | 14.63 | 58
D06V2 | 20.57 | 9.96 | 82
D02V1 | 13.95 | 10.89 | 75
HER | D01V1 | 46.46 | 40.13 | 41

The improved collimator shapes in simulation thus resolved the longstanding
deficit in predicted backgrounds described in Section I, which became apparent
with the first SuperKEKB background measurements in 2016 [4]. The most recent
detailed results, including comprehensive data/MC calculations for all Belle
II sub-detectors for the 2019-2020 data-taking period will be reported in a
future publication.

###  V.2 Collimation system optimization

The collimation system optimization described in Section III.2 involves the
manual adjustment of each collimator aperture, making the procedure very time
consuming. Due to many degrees of freedom present in the collimation system
(10 and 20 collimators for LER and HER, respectively), performing such an
optimization requires several hours of SuperKEKB operation without Belle II
data taking. Furthermore, non-optimal collimation system settings can occur
due to local minima, and the phase advance collimation is only effective
against scattering upstream of the collimator. Therefore, up to now, a
dedicated Monte-Carlo code was used at KEK to run a separate simulation in SAD
for each set of collimator potential apertures. Unfortunately, finding the
optimal collimator configuration with this code takes several days. Our new
sequential tracking framework, described above, now greatly reduces the
computational time required for collimator optimization in simulation. The new
procedure collects particle distributions at each collimator with fully open
collimators, running the SAD simulation only once. Afterwards, an off-line
linear scan of collimators is performed using C/C++ and CERN-ROOT based
scripts. The goal is to maximize the beam lifetime while maintaining the
lowest possible IR losses. This new method allows us to find an optimal
position for all jaws using just a few hours of CPU time. This new procedure
was not possible with the original particle tracking implementation, and
directly enabled by the sequential tracking framework.

Figure 11: Example of SAD simulation results for the optimization of an LER
horizontal collimator, showing beam lifetime (top) and IR loss rate (bottom)
versus collimator aperture. The blue, solid line with rectangular markers
shows Touschek scattering; the red, dashed line with circles shows Coulomb
scattering; the green, dot-dashed line with triangles shows Bremsstrahlung
scattering. The vertical black, solid line indicates the optimal aperture of
the collimator.

Figure 11 demonstrates the off-line collimator scan procedure in simulation
for an LER collimator. As the collimators are gradually closed, going from
right to left in the figure, the collimator aperture at which the beam
lifetime starts to decrease significantly is the optimal one. In this optimal
configuration, the collimator mainly absorbs particles that potentially hit
the IR aperture. Since these particles are going to be lost anyway, the beam
lifetime is constant even if IR losses decrease for the aperture scan
$d=$10\text{\,}\mathrm{m}\mathrm{m}$\rightarrow$7\text{\,}\mathrm{m}\mathrm{m}$$
in Figure 11. For a narrower aperture, the collimator stops particles that are
not lost even after 1000 turns. Therefore, for the scan
$d=$7\text{\,}\mathrm{m}\mathrm{m}$\rightarrow$2\text{\,}\mathrm{m}\mathrm{m}$$,
we see beam lifetime degradation while the IR losses stay almost flat.
Applying the same logic to all collimators, and iterating as needed, the
optimal combined collimator configuration for each ring can be found.

Although minimum IR losses and a high, acceptable lifetime can be achieved in
simulation by this procedure, the collimators also have to satisfy specific
requirements to avoid what is known as transverse mode coupling instability
(TMCI) [21]. TMCI is a result of the wake-field effect from bunches of charge
travelling through the collimator aperture leading to the onset of the bunch
current head-tail instability, which may enlarge the beam size. According to
Ref. [22], the bunch current must be limited, depending on the collimator
aperture, in order to avoid TMCI:

$I_{\rm thresh.}=\frac{8f_{\rm
s}E/e}{\sum\limits_{j}\beta_{j}k_{j}(\sigma_{\rm S},d)},$ (1)

where $I_{\rm thresh.}$ is the upper limit on the bunch current, $f_{\rm
s}=$2.13\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ or $f_{\rm
s}=$2.80\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$$ is the synchrotron frequency
for LER or HER, respectively, $E$ is the beam energy, $e$ is the unit charge,
$\beta_{j}$ and $k_{j}$ are the beta function and kick factor of the $j$-th
collimator, respectively. The analytical formula for the kick factor as a
function of the longitudinal beam size, $\sigma_{\rm S}$, and the collimator
aperture, $d$, is given in Ref. [22]. Here, however, we account for the
geometry of each collimator by using GdfidL [23] simulation results for the
$k_{j}$ values, see Fig. 12.

---
(a) Horizontal collimators
(a)

---
(b) Vertical collimators
(b)

Figure 12: GdfidL simulation results for the horizontal (a) and vertical (b)
collimator kick factor versus collimator aperture. The assumed bunch length is
$6\text{\,}\mathrm{m}\mathrm{m}$. The collimator cross-section is specified in
brackets [11].

---
(a) HER
(a)

---
(b) LER
(b)

Figure 13: SuperKEKB HER (a) and LER (b) collimator apertures and their
constraints. Black, solid (green, triple-dot-dashed) and magenta, double-dot-
dashed (gray, dotted) lines show the minimum allowed SuperKEKB-type and KEKB-
type vertical (horizontal) collimator apertures at different beta function
values, respectively, to avoid TMCI from a single collimator. The red, dashed
and blue, dot-dashed lines show the maximum collimator aperture for horizontal
and vertical collimators, respectively, beyond which we expect increased
losses due to the IR aperture. Filled, red circles and filled, blue squares
are optimized apertures of the horizontal and vertical collimators,
respectively, based on simulation only; magenta, open circles and black, open
squares are the experimental settings of the horizontal and vertical
collimators used in June 2020.

Equation 1 defines the highest acceptable bunch current ($I_{\rm b}$) and the
narrowest possible collimator aperture before TMCI are expected. On the other
hand, the aperture of the IR beam pipe dictates the maximum aperture of the
collimator. Figure 13 shows optimized apertures of LER and HER collimators,
together with these two aperture constraints. To avoid TMCI and to protect the
IR, horizontal (red circles) and vertical (blue squares) collimators have to
be inside red, hatched and blue, filled areas, respectively. For a particular
set of machine parameters, which are traditionally used in Belle II for the
beam background simulation, $I_{\rm
LER/HER}=$1.2\text{/}1.0\text{\,}\mathrm{A}$$ and $N_{\rm bunch}=1576$, the
simulation-optimized collimator settings shown satisfy the bunch current
requirements for both rings: (i) LER $I_{\rm
b}=$0.76\text{\,}\mathrm{m}\mathrm{A}$<I^{opt.}_{\rm
thresh.}=$1.15\text{\,}\mathrm{m}\mathrm{A}$$; (ii) HER $I_{\rm
b}=$0.63\text{\,}\mathrm{m}\mathrm{A}$<I^{opt.}_{\rm
thresh.}=$1.19\text{\,}\mathrm{m}\mathrm{A}$$. For SuperKEKB operation in June
2020, however, the experimental collimator settings used during operation are
outside the allowed areas (Fig. 13, magenta opened circles and black opened
squares), which implies they are ineffective for stray particle collimation.
However, these settings permit the increase of the stored bunch current if the
background level is acceptable. The maximum bunch current satisfying TMCI
constraints would be $I^{\rm exp.}_{\rm
thresh.}=$1.52\text{\,}\mathrm{m}\mathrm{A}$$ in the LER and $I^{\rm
exp.}_{\rm thresh.}=$1.73\text{\,}\mathrm{m}\mathrm{A}$$ in the HER.

The optimized collimator configuration obtained from simulation can be
considered an idealized target that would provide the lowest feasible IR
backgrounds. However, the quality of the injected beam is one of the stumbling
blocks, as it imposes additional restrictions on the range of jaw movement
during the manipulation of the collimators. Understanding this limitation from
first principles would be highly beneficial for realistic background planning,
and will be pursued in the future.

Table 4 compares the simulated IR loss rates for an experimental configuration
of collimators (i.e., found by an operator) and an optimized collimator
configuration obtained from simulation. We find that starting from the
experimental configuration, an additional squeezing of collimators should
reduce IR losses further. However, this is just a reference for operators
since injection errors were not simulated. Therefore, even if the storage
background can be significantly reduced by closing a collimator, unstable
injection limits the aperture of collimators.

Table 4: Comparison of the SAD simulated IR losses ($\mathcal{R}$) and beam lifetime ($\tau$) for the experimental (exp.) and optimized (opt.) collimator settings. Beam optics were those of June 2020: $\beta^{*}_{\rm H/V}=$60\text{/}0.8\text{\,}\mathrm{m}\mathrm{m}$$ with the Crab Waist scheme. Background | Parameters | HER | HER | LER | LER
---|---|---|---|---|---
source | exp. | opt. | exp. | opt.
Coulomb |  $\mathcal{R}$ [MHz] | $13.18\pm 0.63$ | $3.71\pm 0.23$ | $68.93\pm 2.58$ | $13.66\pm 0.74$
$\tau$ [min] | $281.54\pm 31.15$ | $263.58\pm 28.01$ | $38.33\pm 3.08$ | $33.24\pm 2.72$
Bremsstrahlung |  $\mathcal{R}$ [MHz] | $0.43\pm 0.01$ | $0.41\pm 0.01$ | $3.19\pm 0.08$ | $2.53\pm 0.05$
$\tau$ [min] | $7921.77\pm 122.92$ | $7819.97\pm 122.22$ | $2326.91\pm 21.33$ | $2191.14\pm 24.68$
Touschek |  $\mathcal{R}$ [MHz] | $32.36\pm 9.46$ | $0.57\pm 0.74$ | $67.42\pm 3.81$ | $0.93\pm 0.07$
$\tau$ [min] | $45.62\pm 0.79$ | $41.34\pm 0.51$ | $13.82\pm 0.09$ | $8.59\pm 0.06$

###  V.3 Experimental validation

In order to validate the improvements to the particle tracking software
described above, we performed two consecutive SuperKEKB collimator aperture
scans on June 27th, 2020. Two specific LER collimators were chosen for this
study. D06V1 was chosen because it was the most recently installed vertical
collimator at the time, intended to suppress the dominant IR background
component from LER beam-gas scattering. D02H4 was chosen because it is the
closest horizontal collimator to the IR, allowing us to check for IR
contributions from tip scattering, and to validate the simulation of this
background component. We scanned the aperture of each collimator in steps,
measuring IR losses using diamonds detectors. The high voltage was kept off in
all Belle II sub-detectors to avoid damage to sensitive electronics during
collimator manipulation. Table 5 summarizes the machine settings, where the
narrowest IR element is QC1 at $S\approx$1\text{\,}\mathrm{m}$$ from the IP.
QC1 is a superconducting quadrupole magnet of the final focusing system that
has an aperture of 34$\sigma_{\rm\beta H}$ and 92$\sigma_{\rm\beta V}$ in the
horizontal and vertical planes, respectively, where
$\sigma_{\rm\beta}=\sqrt{\beta\times\varepsilon}$ stands for the betatron beam
size.

Table 5: SuperKEKB settings for the LER collimator studies conducted on June 27, 2020 with the Crab Waist scheme. Indexes $H$ and $V$ indicate horizontal and vertical planes, respectively. $\beta$ is the beta function; $\alpha=-0.5\times d\beta/dS$ is the correlation function; $D$ is the dispersion function; $D^{\prime}$ is the chromatic derivative of $D$; $\Delta\mu$ is the phase advance; $\varepsilon$ is an emittance; $I$ is the beam current; $N_{\rm b}$ is the number of bunches; Inj. rep. is the injection repetition rate; $Q$ is the betatron tune; $\sigma_{\rm S}$ is the bunch length; P is the particle momentum. Element |  $S$ [m] |  $\beta_{\rm H/V}$ [m] | $\alpha_{\rm H/V}$ |  $D_{\rm H/V}$ [m] | $D_{\rm H/V}^{\prime}$ |  $\Delta\mu_{\rm H/V}$ [$1/2\pi$]
---|---|---|---|---|---|---
QC1 | 0 | 28.56/978.06 | -113.30/-20.80 | -2.60e-5/-2.70e-4 | 1.34e-4/6.10e-4 | 0/0
D02H4 | 15 | 26.40/16.50 | -0.54/-11.62 | -4.43e-1/-1.72e-3 | 8.41e-2/-2.50e-4 | 0.04/0.98
D06V1 | 1147 | 14.64/67.35 | -3.42/20.26 | 5.16e-1/-7.20e-9 | -7.28e-2/-1.10e-9 | 0.81/0.98
$\varepsilon_{\rm H/V}$ $[\rm pm\cdot rad]$ |  $I$ [mA] | $N_{\rm b}$ | Inj. rep. [Hz] | $Q_{\rm H/V}$ |  $\sigma_{\rm S}$ [mm] |  P [GeV/c]
3300/22 | 200 | 978 | 12.5 | 45.525/43.581 | 5.0 | 4

####  V.3.1 Vertical collimator scan

D06V1 was installed in January 2020 with a tungsten head and was upgraded with
a low atomic number (low-Z) material (graphite) head in September 2020. Since
this upgrade occurred after the June 27th 2020 study described above, it is
possible to compare both measured and simulated outcomes for a given
collimator before and after this upgrade, making D06V1 an ideal candidate for
particle tracking code verification.

Figure 14 shows the measured IR background versus D06V1 aperture in units of
$\sigma_{\rm\beta V}$, where we assume that the aperture is symmetric, i.e.
$|d_{1}|=|d_{2}|$. Two specific sets of diamond detectors, QCS-FW and BP-
FW/BW, were chosen for the background monitoring since they are the most
sensitive to the LER background in the interaction region. The name of each
diamond detector indicates its location: QCS is the superconducting magnet of
the final focusing system; FW (BW) denotes the forward (backward) side of the
Belle II detector; 135, 215 or 225 are azimuth angle positions in degrees, see
Fig. 2. To compare simulation and experimental data accurately, we normalize
the two distributions. First, we define the average value in the valley
region, $d<52\sigma_{\rm\beta V}$, as a pedestal, which we subtract from all
data points. This ensures both pedestals are equal to zero. Then each data
point is divided by the average value of the plateaus at $91\sigma_{\rm\beta
V}<d<104\sigma_{\rm\beta V}$, which makes the two plateau equal to one.

---
(a) QCS-FW detectors (QCS-FW-135 & QCS-FW-225)
(a)

---
(b) BP-FW/BW detectors (BP-BW-215, BP-BW-325 & BP-FW-215)
(b)

Figure 14: Relative total dose rate for two groups of diamond detectors in the
interaction region versus D06V1 collimator aperture, $d=(|d_{1}|+|d_{2}|)/2$,
in units of $\sigma_{\rm\beta V}$. Top: the red, dashed line shows simulation
results; the blue, solid line is experimental data; the grey, vertical band is
the aperture of the QC1 magnet and D06V2 collimator. Bottom: Residual dose
rate (experimental minus simulation).

With a sufficiently small aperture, scattered particles that could have hit
the IR are collimated (i.e., absorbed or scattered), and hence the background
remains roughly constant. At about $60\sigma_{\rm\beta V}$, IR backgrounds
start to increase as D06V1 intercepts a smaller fraction of the stray beam
particles. The resulting observed dose rate versus aperture is close to
S-shaped because of the Gaussian-like beam profile. At this point, the
collimator still represents the primary obstacle on the beamline since it has
the narrowest aperture. A wider opening increases the number of particles that
hit the IR vacuum pipe. At about $85\sigma_{\rm\beta V}$, background losses
begin to saturate because particles that were previously collimated by D06V1
are now redistributed and lost at other apertures after several turns in the
machine. Around $90\sigma_{\rm\beta V}$, however, there is a noticeable $\sim
10\%$ increase in backgrounds, which does not follow the S-shaped trend. This
is caused both by QC1, which has an aperture of $92\sigma_{\rm\beta V}$, and
by the D06V2 collimator with an aperture of $91-94\sigma_{\rm\beta V}$
($\Delta\mu_{\rm V}=0.34~{}[1/2\pi]$). D06V2 is $20\text{\,}\mathrm{m}$
downstream of D06V1 (Fig. 3). The range $91-94\sigma_{\rm\beta V}$, which
includes the QC1 and D06V2 apertures, is highlighted in grey in Figure 14.

The discrepancy between experimental and simulated data in Fig. 14 may be due
to the limited vertical SuperKEKB-type collimator alignment accuracy of $\pm
0.5\text{\,}\mathrm{m}\mathrm{m}$. Simulation shows that a vertical offset
($\Delta d$, Fig. 15a) between the coordinate reference of D06V1 and the beam
center (defined by the beam position monitors of the nearest quadrupole
magnets) can cause exactly such differences, specifically in the form of a
modified slope of the measured S-shape. To find the true vertical offset, a
set of MC simulations was performed, where we apply different values of
$\Delta d$ to the collimator aperture in SAD. We define the goodness of the
simulation for each of the two sets of diamond detectors as:

$\chi^{2}/N=\frac{1}{N}\sum_{i=1}^{N}\left(\frac{Res.[i]}{\Delta
Res.[i]}\right)^{2},$ (2)

where $Res.[i]$ and $\Delta Res.[i]$ are the dose rate residual and its
statistical uncertainty at a given aperture $d[i]$, while $N$ is the number of
data points where $Res.$ was calculated (Fig. 14, bottom). Fig. 15b shows the
goodness of the simulation versus vertical offset.

---
(a) Collimator misalignment
(a)

---
(b) Simulation goodness
(b)

Figure 15: Vertical misalignment of the D06V1 collimator and its effect. (a)
Illustration of the vertical offset ($\Delta d$) between the position
reference of the collimator and the beam core. (b) The goodness of the
simulation versus vertical offset. Top: reduced chi-square for two sets of
diamond detectors; bottom: sum of those two reduced chi-squares versus
vertical offset.

---
(a) QCS-FW diamond detectors
(a)

---
(b) BP-FW/BW diamond detectors
(b)

Figure 16: Relative total dose rate in the interaction region versus D06V1
collimator aperture with a vertical offset $\rm\Delta
d=$0.15\text{\,}\mathrm{m}\mathrm{m}$$.

The simulated collimator offset where $\sum\chi^{2}/N$ reaches a minimum
suggests the optimal setting for the most accurate simulation of the
experiment. Figure 15b shows that the optimal vertical offset is $\Delta
d_{\rm opt.}=$150\pm 25\text{\,}\upmu\mathrm{m}$$, which means that the top
jaw is closer to the beam core than the bottom jaw. Figure 16 shows the
simulated collimator scan when this offset value is used to simulate the D06V1
collimator, i.e. so that $|d_{1}|+2\Delta d_{\rm opt.}=|d_{2}|$ in the
simulation.

Figure 17: LER beam lifetime versus D06V1 collimator aperture for a vertical
collimator misalignment of $\rm\Delta d=$0.15\text{\,}\mathrm{m}\mathrm{m}$$.

The measured and simulated LER beam lifetime, which reflects the integral of
the beam particle loss rate around the entire ring, is shown in Figure 17. The
simulation results were scaled to the experimental data by an empirical scale
factor of 0.64 for better visualisation. The simulated trend versus D06V1
collimator aperture reproduces the experimental trend quite well, which
suggests that the machine and the interaction of the beam with this collimator
are modeled correctly. The need for the scale factor is not unexpected, as the
SAD model of the ring does not include all detailed machine imperfections,
such as misalignment of machine lattice components and noise of the current in
the magnets. Our assumption that the effective atomic number of the residual
gas in the beam pipe is uniform and equal to $Z_{\rm eff.}=7$ also potentially
affects the absolute scale of the simulated beam losses.

####  V.3.2 Tip scattering

Since a high-energy electron or positron can penetrate a high-Z collimator
head (Fig. 10a) and scatter without getting absorbed, the tracking code has to
be able to follow tip scattered beam particles. Figure 10b predicts that
tungsten will impart a significant momentum and angular change to tip
scattered beam particles, which makes it difficult for such particles to
remain in the ring for a long time/distance. Therefore, the collimator closest
to the IR is expected to be the primary source of IR backgrounds specifically
due to tip scattering. We therefore performed an aperture scan of the LER
collimator D02H4 to look for potential contributions from tip scattered
particles to IR losses. As mentioned above, the horizontal collimator has an
alignment uncertainty of up to $1\text{\,}\mathrm{m}\mathrm{m}$, hence we
first repeat the simulation goodness analysis, i.e. we utilize
$\sum\chi^{2}/N=f(d)$ to determine the most probable horizontal misalignment
between D02H4 and the center of the beam core. We obtain a horizontal offset
of $\rm\Delta d=$-0.4\text{\,}\mathrm{m}\mathrm{m}$$ for the estimated
horizontal misalignment between the D02H4 collimator chamber and the nearest
Q-magnet in the LER. This means that the outer collimator jaw ($d_{2}$) is
closer to the beam than the inner jaw ($d_{1}$) by $\rm 2|\Delta
d|=$0.8\text{\,}\mathrm{m}\mathrm{m}$$.

---
(a) QCS-FW diamond detectors (QCS-FW-135 & QCS-FW-225)
(a)

---
(b) BP-FW/BW diamond detectors (BP-BW-215, BP-BW-325 & BP-FW-215)
(b)

Figure 18: Simulated and observed total dose rate for two sets of diamond
detectors in the interaction region versus D02H4 collimator aperture, for a
simulated horizontal offset $\rm\Delta d=$-0.4\text{\,}\mathrm{m}\mathrm{m}$$.

Figure 18 compares the simulated and measured dose rates versus collimator
aperture, where the simulation now includes the alignment offset obtained
before. The measured and simulated results were normalized to have equal
relative does rates in the plateau ($29\sigma_{\rm\beta
H}<d<30\sigma_{\rm\beta H}$) and valley ($26\sigma_{\rm\beta
H}<d<27\sigma_{\rm\beta H}$) regions. These results show that as the
collimator is gradually closed, the forward QCS diamond detectors measure a
decrease in the IR beam background. The beam pipe detectors, also initially
observe a decrease in backgrounds. After an aperture smaller than
$27\sigma_{\beta H}$ is reached, however, the beam pipe detector dose rate
increases again, both in experiment and in simulation. The relative dose rate
at about $25-26\sigma_{\beta H}$ is noticeable higher than the expected
gradually decreasing trend of IR losses. We explain this behaviour of the
background rate versus collimator aperture as due to a significant background
contribution from tip scattered particles. At the narrowest measured
apertures, a broader fraction of the beam halo interacts with the collimator
jaws. As a consequence, a higher flux of non-absorbed particles escapes the
collimator, increasing the probability that they reach Belle II. According to
the SAD simulation, most resulting beam losses are located immediately
downstream of the QCS-FW detectors. Therefore, these diamond detectors are not
very sensitive to this relatively small effect, while the BP-FW/BW diamond
detectors do detect this phenomenon.

Figure 19: Example results from a collimator optimization that utilizes the
newly implemented tip scattering simulation. The solid, black line shows
expected total single-beam IR loss rate versus D06V1 collimator head length,
for a head made of graphite. For comparison, the grey horizontal line shows
results for a $5\text{\,}\mathrm{m}\mathrm{m}$ tantalum head.

Our addition of tip scattering to the background simulation has already proven
useful when deciding on future collimator designs and configurations. One
example is the September 2020 replacement of the D06V1 tantalum collimator
head with graphite. The initial design length of the graphite collimator tip
was $60\text{\,}\mathrm{m}\mathrm{m}$. We performed a set of simulations with
different lengths ($L_{\rm tip}$) to see if this dimension could be reduced,
to minimize the impedance of the collimator [24]. Figure 19 shows predicted
beam loss rates versus $L_{\rm tip}$. It can be seen that for $L_{\rm
tip}>$50\text{\,}\mathrm{m}\mathrm{m}$$ the effect of the graphite head is
almost the same as for a tantalum head of $5\text{\,}\mathrm{m}\mathrm{m}$
length (grey horizontal line). This outcome justifies the choice of
$60\text{\,}\mathrm{m}\mathrm{m}$ for graphite. Unfortunately, for $L_{\rm
tip}<$50\text{\,}\mathrm{m}\mathrm{m}$$ IR losses start to increase due to tip
scattered particles. The induced momentum and angle changes are insufficient
to kick scattered particles out of the machine’s aperture before they reach
the interaction region. A shorter head leads to higher Coulomb scattering
losses, as the vertical collimator does not significantly affect the Touschek
halo, which is distributed mainly in the horizontal plane. As a result, the
D06V1 graphite head length cannot be shorter than about
$50\text{\,}\mathrm{m}\mathrm{m}$ if we wish to keep the same level of LER
backgrounds as with a $5\text{\,}\mathrm{m}\mathrm{m}$ tantalum head.

##  VI Summary and conclusions

As SuperKEKB improves further on its world record luminosity, continued
simulation and mitigation of beam-induced backgrounds will be essential.
Collimators play a crucial role in Belle II background mitigation and in
cleaning up the halo of the circulating beams. To find an optimal collimator
configuration and to predict beam particle losses in the machine, an extended
and improved Monte-Carlo simulation framework was developed. This version
includes more realistic shapes of all collimator jaws, a new sequential
tracking procedure that enables off-line collimator scans and optimization,
and a model for the beam particle scattering in the collimator material.
Several other crucial improvements were recently implemented, for example
position-dependent simulation of measured vacuum pressure in each ring, but
these will be reported separately. The new simulation framework enables
optimization of collimator settings in a few hours, while the old framework
required several days of CPU time.

The described implementations and improvements of the accelerator particle
tracking result in greatly improved agreement with experimental data.
Background simulation validation with Belle II sub-detectors will be reported
separately in the future. Here, we have focused on validating the effect of
collimators in the simulation, as this is where we found the largest
discrepancies with experiment. Simulation of LER collimator aperture scans
reproduce measurements well, confirming the newly implemented collimator
shapes and tip scattering models. These scans are sensitive to misalignments
of collimator chambers with respect to the beam center. The presented method
of utilizing the matching to simulation may thus be used for improved
collimator alignment in the future. The findings also emphasize the importance
of more precise positioning of accelerator components.

Finally, the enhanced simulation framework also allows us to predict the
effect of new machine elements and to study possible improvements for further
background reduction in the future.

###### Acknowledgements.

We enthusiastically thank the SuperKEKB commissioning group for operation of
the accelerator; the KEK computing group for on-site support; the Belle II
diamond detector group for assistance with dose rate calibrations; J. Schueler
and T. E. Browder (University of Hawaii) and L. Lanceri (INFN, Sezione di
Trieste) for their feedback on the manuscript. This work was supported by the
U.S. Department of Energy (DOE) via Award Number DE-SC0007852 and via U.S.
Belle II Operations administered by Brookhaven National Laboratory (DE-
SC0012704).

.

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