# Exposing a fibre-based dual-readout calorimeter to a positron beam

N. Ampilogov  S. Cometti  J. Agarwala  V. Chmill  R. Ferrari  G. Gaudio  P.
Giacomelli  A. Giaz  A. Karadzhinova-Ferrer  A. Loeschcke-Centeno  A. Negri
L. Pezzotti  G. Polesello  E. Proserpio  A. Ribon  R. Santoro  I. Vivarelli

###### Abstract

A prototype of a dual-readout calorimeter using brass capillary tubes
surrounding scintillating and clear plastic optical fibres was tested using
beams of particles with energies between 10 and 100 GeV produced by the CERN
SPS. The scope of the test was to characterise the performance of the tube-
based detector response to positrons in terms of linearity, energy resolution,
and lateral granularity. After calibrating the detector and processing the
output signal to correct for the energy dependency on the particle impact
point, the linearity of the measurement was found to be better than 1%. The
positron response was compared to that predicted by a Geant4-based simulation,
finding good agreement both in terms of energy resolution and shower profile.
The detector resolution was estimated to be well described by a stochastic
term of 14.5% with a negligible constant term.

##  1 Introduction

One of the requirements of the desired physics exploitation programme of
future $e^{+}e^{-}$ colliders (such as, for instance, FCC-ee and CEPC [1, 2])
is a high-precision measurement of the jet four-momenta. Dual-readout
calorimetry relies on a dual sampling of the calorimeter signal, with two
sensitive media having a different $h/e$ factor. The combined information from
the two signals allows for the effective correction of fluctuations of the
hadronic shower’s electromagnetic fraction, therefore dramatically improving
the energy measurement while recovering the linearity of the response to
hadrons. Dual-readout calorimetry [3] is a well-established technique: a
20-year-long experimental programme established the proof-of-principle [4, 5,
6], converging on the solution of employing two sets of optical fibres
embedded in the absorber, running parallel to the impinging direction of the
particles. Scintillating fibres are utilised to sample all charged components
of the shower, while plastic un-doped fibres collect Cherenkov light emitted
mainly by electrons and positrons, and are therefore mostly sensitive to the
electromagnetic component of the hadronic shower. Recent developments of the
dual-readout technique saw the test of Silicon PhotoMultipliers (SiPM) as
light detectors capable to push the granularity of the calorimeter at the
level of independently reading out each individual fibre [7]. A simulation of
a full 4$\pi$ dual-readout calorimeter, using a mechanics and geometry
different from those tested in this paper is documented in Refs. [8, 9], where
a hadronic energy resolution of $\sigma/E\sim$30$\%$/$\sqrt{E}$ for single
hadron and $\sigma/E\sim$38$\%$/$\sqrt{E}$ for jets was demonstrated, while
maintaining good electron and photon energy resolution (of the order of
$\sigma/E\sim 15\%/\sqrt{E}$). The baseline IDEA detector concept for future
$e^{+}e^{-}$ colliders [10] utilises a dual-readout fibre-based calorimeter
for the energy and position measurement of electrons, photons and hadrons. An
option utilising a crystal-based dual-readout electromagnetic section in front
of the fibre-based calorimeter was also studied, and its performance is
described in Refs. [11, 12].

This paper focuses on the test of a new prototype employing a new scalable
concept for the mechanical construction: individual brass cylindrical
capillary tubes surrounding the optical fibres are glued together to form
calorimeter modules. These capillary tubes offer a flexible solution for
large-scale construction, at an affordable cost. A dual-readout calorimeter
using brass capillary tubes as absorber and two types of optical fibres as
active medium was built [13] and tested using beams of particles during two
different test-beam campaigns. The size of the prototype was such that
electromagnetic showers can be efficiently contained. The objective of the
test beams was to assess the calorimeter prototype performance in terms of
linearity and resolution of its response to electrons. The excellent lateral
granularity offered by the capillary tube solution is also exploited to
perform a measurement of the electron shower shape. Section 2 describes the
details of the experimental setup, including the calorimeter layout and its
readout, and the setup of the auxiliary detectors used in particular for
boosting the capability to select electrons among the interacting beam
particles. Section 3 describes the calibration procedure followed to first
equalise the response of the calorimeter modules, then calibrate the whole
calorimeter to the electromagnetic scale. The calorimeter response to
positrons is discussed in Section 4, and finally conclusions are drawn in
Section 5.

##  2 Experimental setup and detector simulation

The prototype tested on beam in 2021 is shown in Figure 1. Its size is about
$100\times 10\times 10\ \mathrm{cm}^{3}$. It consists of nine identical
modules, arranged as detailed in Figure 1(a). Each module is made of 320 brass
(63% Cu, 37% Zn) capillaries (with external diameter of 2 mm and internal
diameter of 1.1 mm) equipped, alternatively, with scintillating (BC-10 from
Saint Gobain [14]) and clear undoped (SK-40 [15] from Mitsubishi) optical
fibres. The Molière radius of such prototype is calculated to be 23.8 mm while
the effective radiation length is 22.7 mm. The layout of the scintillating and
clear fibres is shown in Figure 1(c).

The external modules $\mathrm{M}_{1}-\mathrm{M}_{8}$ are instrumented with
R-8900 PMTs [16]. The scintillating and clear fibres are separated and bundled
in two groups on the back side of each module to match the PMTs’ window. A
yellow filter (Kodak, Wratten nr 3, with nominal transmission of about 7% at
425 nm and 90% at 550 nm, visible in Figure 2(a)) is placed between the
scintillating fibres and the detector to cut off the short wavelength
component of the scintillating signal. The presence of the yellow filters
helps reducing the calorimeter response dependence on the shower depth and
starting point by selecting wavelength with a longer fibre attenuation length.
The PMTs are read out with V792AC QDC modules produced by CAEN S.p.A.

Figure 1: (a) Front and (b) side view of the prototype before the connection
to the light sensors (PMTs and SiPMs). The fibres from the external modules
are bundled to match the PMTs’ window while the longer fibres from the central
module are connected to a patch panel to be interfaced with SiPMs. (c) Sketch
of the front face of the calorimeter detailing the relative positions of the
cherenkov and scintillating fibres.

For the central module $\mathrm{M}_{0}$, each individual fibre is read out by
a Hamamatsu (S14160-1315 PS [17]) Silicon Photomultiplier (SiPM) with a
sensitive area of 1.3 $\times$ 1.3 mm2. Since almost 10$\%$ of the entire
energy is released within one millimeter from the core of the shower (1–2
fibres) [7, 8], SiPMs with a wide dynamic range (i.e., 7284 cells, 15 $\mu$m
pitch) were selected. SiPMs with compact packaging were not available at the
time of the construction, therefore it was decided to fan out the fibres on
the back sides of the calorimeter to match the front-end boards housing 64
SiPMs. The SiPMs on the front-end board are separated in two groups (32 SiPMs
each) insulated with a light tight frame (Figure 2(a)) to avoid optical
crosstalk between the Cherenkov and scintillation signals. As for the modules
$\mathrm{M}_{1}-\mathrm{M}_{8}$, yellow filters are placed between the
scintillating fibres and the SiPMs. In addition, for $\mathrm{M}_{0}$ a
transparent paper is used between the clear fibres and the SiPMs for
mechanical reason and to avoid any air gap between the fibres and the light
sensors.

The SiPM readout is based on the FERS system produced by CAEN S.p.A. [18] to
fully exploit the Citiroc-1A [19] performance: i.e., wide dynamic range,
linearity and multi-photon quality even with SiPMs with small pitch size and
small gain (1–3 $\times$ 105 at nominal settings). The SiPM readout setup is
shown in Figure 2 (b). Each readout board (A5202) is equipped with two
Citiroc-1A to operate 64 SiPMs. The signal produced by each SiPM feeds two
charge amplifiers with tunable gains. The range accessible by one of the two
amplifiers (namely the High Gain, HG in the following) is almost 10 times
higher than the other (the Low Gain, LG). This feature allows two spectra for
each SiPM to be stored on disk. The spectrum acquired via the HG chain is
useful to analyse the multi-photon signal and to extract the ADC-to-
photoelectrons (p.e.) conversion constant. The LG spectrum extends the overall
dynamic range. The settings for the two charge amplifiers were chosen to
guarantee good quality HG spectra and a wide dynamic range, while maintaining
an overlap between the signals acquired with the two different gains to be
used for their intercalibration. The settings chosen allow to read signals
from 1 to almost 4000 p.e. which is about 55% of the SiPM occupancy
considering the microcells available in the sensitive area.

The prototype was qualified on beam at the Deutches Elektronen-SYnchrotron
(DESY) [20] using electron beams with energies $1\ \mathrm{GeV}\leq
E_{\mathrm{beam}}\leq 6\ \mathrm{GeV}$, and at CERN, using positron beams of
the SPS H8 beam line with energies $10\ \mathrm{GeV}\leq E_{\mathrm{beam}}\leq
100\ \mathrm{GeV}$. The results of this paper use only data collected at CERN.

Figure 2: (a) Back side of $\mathrm{M}_{0}$ and a front-end board with 64
SiPMs before the installation. (b) The full system connected to the five A5202
boards utilised to read out the 320 SiPMs.

###  2.1 Beam setup

A set of auxiliary detectors present on the beam line were used to improve
particle identification and support the data taking.

  * •

Upstream the beam, two Cherenkov threshold counters [21] were available. They
were filled with $\mathrm{He}$. The pressure of the gas in the counters was
chosen depending on the beam energy to optimise the separation between
electrons and pions.

  * •

Two scintillation counters ($\mathrm{T_{1}}$ and $\mathrm{T_{2}}$ in the
following), each 2.5 mm thick, with an area of overlap of about $4\times 4\
\mathrm{cm^{2}}$ were used in coincidence
($\mathrm{T_{1}}\land\mathrm{T_{2}}$). A third scintillator counter
($\mathrm{T_{3}}$), placed downstream the beam, had a 10 mm radius hole. Its
purpose was to veto particles travelling far from the beam axis. The particle
trigger signal was defined to be
$\left(\mathrm{T_{1}}\land\mathrm{T_{2}}\right)\land\bar{T}_{3}$ (physics
trigger in the following).

  * •

A pair of Delay Wire Chambers (DWC) was placed downstream the beam with
respect to the trigger scintillators to be able to determine the location of
the impact point of the beam particles at the calorimeter surface with a
precision of a few mm, depending on the beam energy.

  * •

A preshower detector (PS in the following), consisting of 5 mm of lead and a
scintillator slab read out with a photomultiplier was located at 285 cm from
the face of the calorimeter. Its purpose was to help with the identification
of high-energy positrons, which would be the only particles in the beam with a
significant radiation probability in the lead. The positioning of the
calorimeter at such a distance from the PS was forced due to access
limitations to the available test-beam area.

The signals from the auxiliary detectors and the PMTs reading modules
$\mathrm{M}_{1}-\mathrm{M}_{8}$ were collected using a single data acquisition
system. The system was set up so that every ten physics triggers, a “pedestal”
trigger was produced. All trigger signals (physics and pedestals) were also
sent to a second independent data acquisition, reading out the signals from
the SiPMs of module $\mathrm{M}_{0}$. The synchronisation of the two data
acquisition systems was done offline, by making use of the pedestal events.

In the following, a right-handed orthogonal system of coordinates with the $z$
axis running along the beam line, and with the $y$ axis pointing vertical
upwards is used. The origin of the coordinate system is taken to be on the
front face of the calorimeter, at the geometrical centre of $\mathrm{M}_{0}$.
The calorimeter prototype was placed on beam so that its longest side was
forming an angle of about $1^{\circ}$ with the $z$ axis in the $x-z$ plane.
This is to avoid channeling effects (particles entering and travelling long
distances in an optical fibre, causing abnormally large scintillation
signals). No angle was introduced between the calorimeter long axis and the
$z$ axis in the $y-z$ plane.

###  2.2 Test beam simulation

The full test beam setup (including the calorimeter prototype and the
auxiliary detectors) was reproduced in a software package based on Geant4
`10.7.p01` [22]. The software package can simulate the detector response in a
“fast” or “full” mode. For the full mode, the photon propagation within the
optical fibres is completely delegated to Geant4. In the fast mode, such
propagation is parametrised. In either mode, the scintillation and Cherenkov
light yields have been tuned to correctly reproduce the average amount of
light. To take into account the Poissonian fluctuations of light emission, the
simulated light emission at every hit was smeared accordingly. The simulation
results in the following sections have been obtained using the fast simulation
mode, using the `FTFP_BERT` physics list. The response of the scintillating
fibres takes into account the Birks’ effects on light emission. The output of
the simulation is the number of photoelectrons expected at the end of each
optical fibre. No emulation of the SiPMs is included - therefore the
simulation does not include, e.g., the noise introduced by the readout
electronics.

##  3 Calorimeter response equalisation and calibration

The calibration of the prototype was performed in three steps. First, the gain
of all SiPMs in $\mathrm{M}_{0}$ was equalised, and the conversion factor
between ADC counts and p.e. derived, by making use of the SiPM multi-photon
spectrum. Then, the response of all modules $\mathrm{M}_{0}$-$\mathrm{M}_{8}$
was equalised by making use of a positron beam with an energy of 20 GeV.
Finally, the overall calorimeter energy scale was set by looking at the
response of the whole calorimeter prototype to beams of positrons.

###  3.1 SiPM equalisation using the multi-photon spectrum

A first equalisation of the SiPMs was performed in the laboratory before the
beam tests, by making use of an ultra-fast LED emitting at 420 nm. The SiPM
response was equalised by applying the same overvoltage and by tuning the
amplifier settings. The procedure allowed to operate all SiPMs with about the
same photon detection efficiency (PDE) and with signals equalised in
amplitude. The SiPMs were operated with a voltage set at $+7\ \mathrm{V}$ over
the breakdown voltage. Even if it is not a typical setting for a SiPM, it
guarantees a stable PDE (against small temperature variations) and a
multiplication factor in the avalanche region of the order of $0.5\cdot
10^{6}$ per each detected photon. The spurious effects (i.e. dark count rate
and crosstalk) have limited impact on the measurements.

Figure 3: (a) High gain (HG) and (b) low gain (LG) spectra obtained for the
same SiPM with 6 GeV electrons at the DESY test beam.

Calibration in p.e. of the LG response is required in order to sum signals
from different SiPMs and to correct for non-linearity due to the limited
number of cells available in each detector, if needed. Figure 3 shows the
typical spectra measured in the two gains by one SiPM connected to a
scintillating fibre in response to 6 GeV electrons at the DESY test beam. The
pedestal, the multi-photon spectrum and the ADC saturation are clearly visible
for the HG plot. The saturation is not affecting the measurement since the
information is still available in the LG spectrum. The strategy used to
calibrate both spectra in p.e. was the following:

  * •

The pedestal and the multi-photon were fitted with Gaussian functions. The
results were used to convert the ADC channels in p.e. by using the mean value
of the pedestal and the average peak-to-peak distance obtained by fitting
three consecutive peaks (Figure 4(a)).

  * •

The HG values, converted in p.e., were plotted against those of the ADC counts
of the LG channel (Figure 4(b)). The points in the plot exceeding 125 p.e.
were not considered in the fit and the slope was used to extract the ADC-
to-p.e. conversion for the LG even if the multi-photon structure was not
accessible in this regime. The typical conversion factor was about 1 p.e./ADC
count with uncertainties of the order of 0.1%.

Figure 4: (a) Zooming of the region below 250 ADC counts of the HG spectrum in
Figure 3(a) showing the multi-photon fit. (b) Scatter plot of HG (in p.e.)
signal against the LG signal for the same SiPM.

The procedure was performed for all SiPMs and the parameters were extracted on
a run-by-run basis, without having the need to collect dedicated data for
calibration. The stability of the calibration performed with the multi-photon
spectrum was monitored over several runs. As an example, Figure 5 shows the
pedestal and the peak-to-peak distance for one SiPM in each readout board,
using the HG data collected during a series of consecutive runs. Variations
below 2$\%$ were measured for all SiPMs. It was therefore decided to use a
single calibration constant for each SiPM for all the data taking period.

Figure 5: (a) Pedestal and (b) peak-to-peak distance measured for one SiPM per
each readout board during a series of consecutive runs.

The channel signal used in the following for the data analysis is the SiPM
signal in p.e. from the HG, unless the HG is found to be saturated, in which
case the LG signal in p.e. is used.

###  3.2 Calorimeter response equalisation

The second step in the calibration was the equalisation of the modules
$\mathrm{M}_{0}-\mathrm{M}_{8}$. The equalisation was performed by means of
nine different runs, each collected by translating the calorimeter prototype
so that a beam with energy $E_{\mathrm{beam}}=20\ \mathrm{GeV}$ was hitting in
the centre of a different module. The positron component of the beam was
selected by making use of the auxiliary Cherenkov counters. Selection are
applied to both counters to reject particles leaving a pedestal signal in each
of them. The beam composition could be studied by making use of the counters
and of the PS auxiliary detector: it was estimated that the Cherenkov
selection yields a pure positron beam with negligible pion contamination for
$E_{\mathrm{beam}}=20\ \mathrm{GeV}$.

The equalisation procedure assumed an equal tower response to positrons, in
runs where the beam is hitting the module centre. The individual module
containment was estimated (from the simulation) to be
$\epsilon_{\mathrm{module}}=72\%$. We define the average scintillation
(Cherenkov) response in ADC counts of a module $i$ ($i=1,8$) to a positron
beam of energy $E_{\mathrm{beam}}$ hitting its centre as
$A^{S}_{\mathrm{M}_{i}}\left(E_{\mathrm{beam}}\right)$
($A^{C}_{\mathrm{M}_{i}}\left(E_{\mathrm{beam}}\right)$). The response of the
module in p.e. $P^{S,C}_{\mathrm{M}_{i}}\left(E_{\mathrm{beam}}\right)$ is
obtained by multiplying $A^{S,C}_{\mathrm{M}_{i}}$ by a constant $a_{i}^{S,C}$
independent of the beam energy. The constants $a_{i}^{S,C}$ are determined by
imposing that

$\displaystyle P^{S,C}_{\mathrm{M}_{i}}\left(20\
\mathrm{GeV}\right)=a_{i}^{S,C}\times A^{S,C}_{\mathrm{M}_{i}}\ \left(20\
\mathrm{GeV}\right)=P^{S,C}_{\mathrm{M}_{0}}\left(20\ \mathrm{GeV}\right),$

that is, by imposing that the module response is the same as that of module
$\mathrm{M}_{0}$ in identical conditions.

Figure 6 shows the scintillation signal in each tower of the calorimeter after
equalisation, when the 20-GeV positron beam is hitting in $\mathrm{M}_{0}$.
The left-right asymmetry in the energy deposit is due to the angle between the
beam line and the prototype main axis.

Figure 6: Energy deposited in each module during a run where the 20-GeV
positron beam was aimed at the centre of $\mathrm{M}_{0}$ for the
scintillation channel. The scale on the $x$-axis corresponds to p.e., while
for the modules $\mathrm{M}_{1}-\mathrm{M}_{8}$ the response is equalised to
that of $\mathrm{M}_{0}$ following the procedure described in the text.

After the equalisation was performed, all modules were expected to respond the
same to a signal of a positron beam, and the signal was referred to the
numbers of p.e. seen by $\mathrm{M}_{0}$ when hit by a positron beam of the
same energy. It is interesting to evaluate how many p.e. $\mathrm{M}_{0}$ sees
per GeV of input energy. The values of the p.e. to GeV conversion factor for
$\mathrm{M}_{0}$, $F^{S(C)}_{\mathrm{PheToGeV}}$ are extracted as

$\displaystyle
F^{S,C}_{\mathrm{PheToGeV}}=\frac{P^{S,C}_{\mathrm{M}_{0}}\left(20\
\mathrm{GeV}\right)}{\epsilon_{\mathrm{module}}\times 20\ \mathrm{GeV}}$

The measured values are $F^{S}_{\mathrm{PheToGeV}}=277$,
$F^{C}_{\mathrm{PheToGeV}}=48$. These values are found to be compatible with
the ones obtained in Ref. [23], which were obtained as an average over many
runs from both test beam periods.

###  3.3 Calorimeter calibration

The final step for the prototype calibration was to rescale the response of
each module by a single constant ($\delta_{S}$ for the scintillation signal
and $\delta_{C}$ for the Cherenkov signal), so that the total energy measured
in the calorimeter corresponded to the beam energy separately for the
scintillation and Cherenkov signal111The simulation predicts a shower
containment of the full prototype of $\epsilon=94\%$ at $E_{\mathrm{beam}}=20\
\mathrm{GeV}$. This number was found to be nearly independent with
$E_{\mathrm{beam}}$. This lateral leakage was therefore de-facto reabsorbed in
the determination of $\delta_{S,C}$.. The constants $\delta_{S}$ and
$\delta_{C}$ are determined as

$\displaystyle\delta_{S,C}=\frac{20\
\mathrm{GeV}}{\langle\sum_{i=0}^{8}P^{S,C}_{\mathrm{M}_{i}}\rangle},$

where the average is computed over all selected positrons from a 20-GeV run
with the beam pointing to the geometric centre of $\mathrm{M}_{0}$.

The individual module energy (in GeV) can therefore be defined as

$\displaystyle E^{S,C}_{i}=\delta_{S,C}\times P^{S,C}_{\mathrm{M}_{i}},$

##  4 Results

After the calibration procedure was applied, the data were analysed to measure
the performance of the prototype for positrons. These were characterised in
terms of the linearity of the response as a function of energy and its
resolution. Moreover, exploiting the high granularity provided by the SiPM
readout, the shape of the electromagnetic showers in the calorimeter was
studied in detail, and compared to the Geant4 simulation.

###  4.1 Positron selection

Despite a beam configuration optimised to yield positronss, the beam of the H8
beam line at CERN contained in general positrons, muons and hadrons, in
proportions which are strongly energy dependent. The definition of a procedure
to select positrons was necessary. The first step in data selection was the
definition of a fiducial area for the impact of the beam on the calorimeter
ensuring good containment of the beam in $\mathrm{M}_{0}$.

Due to a malfunctioning, the DWCs were determined to have a space resolution
of about 2 mm, not adequate for studies of the dependence of the calorimeter
response on the impact point and angle of the beam. The chambers were thus
only used to clean the beam spot by requiring that the radius of the beam in
both chamber was smaller than 15 mm, and that for both coordinates, the
difference of the coordinates measured in the two chambers was less than $3\
\mathrm{mm}$.

Studies which require a precise knowledge of the position of the shower inside
the calorimeter are based on the barycentre of the shower
($X_{\mathrm{CALO}}$,$Y_{\mathrm{CALO}}$). This variable is calculated from
the formula:

$X_{\mathrm{CALO}}=\frac{\sum_{i}x_{i}E_{i}}{\sum_{i}E_{i}};\hskip
28.45274ptY_{\mathrm{CALO}}=\frac{\sum_{i}y_{i}E_{i}}{\sum_{i}E_{i}}$ (4.1)

where the sum runs on all of the 320 fibres in $\mathrm{M}_{0}$, $E_{i}$ is
the energy deposited in each fibre, with coordinates ($x_{i}$, $y_{i}$). In
order to guarantee good containment in the central module, a selection was
applied event-by-event to the position of the barycentre.

For the selection of a pure positron beam, two Cherenkov counters and a PS
detector were available, as described in Section 2. A fraction of the
positrons starts showering in the PS, yielding a signal proportional to the
number of secondary positrons produced in the shower. The requirement of a
lower limit on the observed signal in the PS allows the separation of
positrons from hadrons which approximately behave as a Minimum Ionising
Particle (MIP). However, the secondary positrons and photons emerge from the
PS at an angle with respect to the primary positron. Given the significant
distance of the PS from the calorimeter face, the fraction of the beam energy
collected in the calorimeter, and the distribution of energy among the central
cell and the surrounding ones depend on the amount of signal deposited in the
PS. This is shown in Figure 7, where the average energy in the calorimeter is
displayed as a function of the energy deposit in the PS.

Figure 7: Average value of the measured energy in the calorimeter for the
scintillation signal as a function of the size of the signal in the PS
detector.

There is a nearly linear anti-correlation between the calorimeter and the PS
signal: the calorimeter signal corresponding to, for example, 14 MIPs in the
PS is about 1 GeV lower than that corresponding to 1 MIP for a positron beam
of $20\ \mathrm{GeV}$. By making use of the simulation, it was determined that
this effect is dominated by the additional lateral leakage introduced by the
opening of shower started in the PS. While this effect can be corrected in
principle for what concerns the assessment of the linearity of the response,
it is significantly more complex to account for it correctly when assessing
the resolution of the energy measurement.

One particular run was taken with a 20-GeV positron beam and where the PS
detector was out of the beam line. After the requirement on the Cherenkov
counter is applied, about $20000$ events are available for analysis, with
negligible hadron contamination. Of these, about $4300$ are in the beam
fiducial window defined above. This run (referred to as “reference run” in the
following) was used to understand key features of the calorimeter response.

###  4.2 Positron energy measurement

The selection of the positron component of the beam for the reference run was
performed by requiring that the signal in each of the Cherenkov threshold
counters to be larger than five times the RMS of the corresponding pedestal.

The distribution of the calibrated SiPM signals for the reference run is shown
in Figure 8 for the scintillation and Cherenkov signals.

Figure 8: Distribution of the sums of the calibrated SiPM signals for the (a)
scintillating and (b) Cherenkov signals for the reference run.

These distributions are far from being Gaussian. In order to understand this
feature, the dependence of the energy measurement on the position of the
barycentre of the shower was studied, and is shown in Figure 9.

Figure 9: Average of the summed SiPM signals as a function of (a)
$Y_{\mathrm{CALO}}$ and (b) $X_{\mathrm{CALO}}$ for the reference run. The red
points refer to the scintillation signal, while the blue ones refer to the
Cherenkov.

A periodical modulation is observed in the $Y_{\mathrm{CALO}}$ coordinate,
with a period corresponding to the spacing between different types of fibres
(Figure 1(c)) and an amplitude which is approximately $\pm 10\%$ and $\pm 5\%$
of the signal for scintillator and Cherenkov light, respectively. The phases
of scintillator and Cherenkov signals are opposite. No such modulation is
observed in the $x$ coordinate. The modulation in $Y_{\mathrm{CALO}}$ is due
to the geometrical structure of the calorimeter with alternate rows of
scintillating and Cherenkov fibres. Because of the absence of tilt between the
fibres and the beam line in the $y$-direction, the scintillation signal is
higher for positrons depositing most of the energy in a scintillation row. By
summing the Cherenkov and scintillation signals which have opposite phases the
effect is mitigated, but it does not disappear, as the observed oscillation
amplitude is smaller for the Cherenkov signal. This difference is ascribed to
the fact that Cherenkov light has a well defined geometrical relationship with
the direction of the particle producing it, whereas scintillation light is
produced nearly isotropic.

This result is correctly reproduced by the simulation, provided the correct
relative orientation of the beam and calorimeter are reproduced. It was
observed with the simulation that the amplitude and phase of the modulation
have a strong dependence on the impact angles of the beam on the calorimeter.
A simulation scan aimed at optimising the match of the Monte Carlo predictions
to the data for the distribution in the left plot of Figure 9 yielded an
optimal angle of 1.5∘ in $x$ and -0.4∘ in $y$, compatible with the setup
described in Section 2 within uncertainties.

The amplitude of the energy measurement modulation was studied as a function
of the impact angle of the beam onto the calorimeter. Figure 10 shows the
simulated scintillation energy measurement as a function of the
$Y_{\mathrm{CALO}}$ for three different beam angles in the $(y,z)$ plane. An
angle of $2.5^{\circ}$ is sufficient to completely cancel the modulation.

Figure 10: Average of the energy in $\mathrm{M}_{0}$ as a function of
$Y_{\mathrm{CALO}}$ for a simulated 20-GeV positron beam. The distribution is
shown for three different configurations of the beam angle into the
calorimeter. The number in the legend correspond to the angle in the $x-z$
(indicated by $x$) and $y-z$ (indicated by $y$) planes. The same fiducial
selection is applied as for data.

A correction procedure for the dependence of the calorimeter energy
measurement on the particle impact point was developed by making use of the
reference run. A useful variable to this aim is $R_{\mathrm{max}}$, defined as
the ratio of the energy deposited in $\mathrm{M}_{0}$ in the row of
scintillating fibres with the highest signal to the total scintillation signal
in $\mathrm{M}_{0}$.

Figure 11: (a) Distribution of $R_{\mathrm{max}}$ for positrons in the
reference run. (b) Distribution of the total scintillation energy signal for
events with $R_{\mathrm{max}}<0.4$ (red) and $R_{\mathrm{max}}>0.4$ (blue).

The distribution of $R_{\mathrm{max}}$ for the reference run is shown in
Figure 11(a), while Figure 11(b) shows that the events with high value of
$R_{\mathrm{max}}$ correspond to the configurations with high energy deposits
in the scintillation channel. This can be expected, as if a positron running
approximately parallel to the fibre direction in the vertical plane initiates
a shower in a row corresponding to scintillating fibres, that will correspond
to a high scintillation response overall and high values of
$R_{\mathrm{max}}$. On the other hand, if the positron initiates the shower in
a row of Cherenkov fibres, the bulk of the energy deposition in the
scintillation signal will be shared between two rows, yielding a lower value
for $R_{\mathrm{max}}$.

Figure 12: Average energy measured in the central cell normalised to the
expected as a function of $R_{\mathrm{max}}$ for the (a) scintillation and (b)
Cherenkov signal.

The dependence on $R_{\mathrm{max}}$ of the energy measured in
$\mathrm{M}_{0}$ is shown in Figure 12 for the scintillation and the Cherenkov
signals. Such dependence is parametrised with a second-degree polynomial, and
the resulting correction factor is applied to the measured energy. The result
of the procedure is shown in Figure 13 left, which shows that the position
dependence of the measurement is almost completely smoothed out. The
distribution for the measured energies in $\mathrm{M}_{0}$ after this
correction is shown in the right side of Figure 13, for both the scintillation
and Cherenkov signals. The comparison with Figure 8 shows that the correction
recovers the Gaussian distribution for both the scintillation and Cherenkov
signals.

Figure 13: (a) Average of the summed SiPM signals in $\mathrm{M}_{0}$ as a
function of $Y_{\mathrm{CALO}}$ for the reference run after the correction
procedure described in the text. The scintillation channels are in red, the
Cherenkov ones in blue. (b) Distribution of the sums of the calibrated SiPM
signals for scintillating (red) and Cherenkov (blue) signals in
$\mathrm{M}_{0}$ after corrections for the reference run.

For the following results, the procedure just described is applied to the data
and, where relevant, to the simulation.

###  4.3 Determination of the calorimeter response to positrons

Pure samples of positrons for all available beam energies between 10 and 100
GeV can be selected by applying the fiducial selection described in Section
4.1. For beam energies up to $E_{\mathrm{beam}}=30\ \mathrm{GeV}$, the
positron component of the beam was selected by requiring the signal in each of
the Cherenkov threshold counters to be larger than five times the RMS of the
corresponding pedestal. For energies $E_{\mathrm{beam}}>30\ \mathrm{GeV}$, the
positron selection was done by requiring a signal in the PS larger or equal to
three MIPs. The PS requirement induces a large event-by-event fluctuations in
the amount of energy deposited in the calorimeter due to the fluctuations in
the angular opening of the shower initiated in the preshower detector. The
dependence of the average energy deposit on the number of MIPs deposited in
the preshower shown in Figure 7 is corrected for by adding to the total
calorimeter energy a linear function of the PS signal, determined with a
linear fit. This correction is indicated in the following as
$\delta_{\mathrm{PS}}$. The correction is about $80\ \mathrm{MeV/MIP}$. The
variable $E_{\mathrm{meas}}$ is therefore defined as

$\displaystyle
E_{\mathrm{meas}}=\delta_{\mathrm{PS}}+\frac{\sum_{i}E^{S}_{i}+\sum_{i}E^{C}_{i}}{2}$

The linearity of the calorimeter response is shown in Fig 14. The linearity of
the response was better than 1% over the full explored range.

Figure 14: Dependency of the calorimeter response on the beam energy.

###  4.4 Resolution measurement

The impact of the PS selection on the resolution of the energy measurement is
very strong, and it leads to calorimeter performance which are significantly
worse than those that one would have obtained in absence of PS-induced lateral
leakage. For this reason, a different strategy was adopted: for the
determination of the energy resolution, only datasets corresponding to beam
energies where the positron selection can reliably be done only by using the
Cherenkov counters were used. This limits the available energies to 10, 20 and
30 GeV. On top of the Cherenkov counter selection, an additional requirement
to accept only events leaving a signal equivalent to one MIP in the PS was
imposed. The obtained resolution was compared to that of the simulation.
Finally, the simulation is used to extrapolate at higher energies.

The energy measurement of the calorimeter distribution of the energy
measurement response obtained for a 20-GeV run with the PS is shown in Figure
15 for data and simulation.

Figure 15: Distribution of the energy measured after all corrections for a
20-GeV positron beam for (a) data and (b) the simulation.

The calorimeter resolution is evaluated as the ratio of $\sigma/E$ of the RMS
width to the mean value of a Gaussian fit to the energy measurement
distribution for data and simulation. The results are summarised in Figure 16.
For the three available experimental points the measured resolution is
compatible with the simulated one. For comparison, the resolution of the same
module in a setup with no preshower in the beam is also calculated, with an
angle of impact of 2.5∘ in both the horizontal and vertical direction.

Figure 16: Measured resolution of the calorimeter prototype for positrons as a
function of the reciprocal of the square root of the beam energy. The black
points represent the data, the blue ones correspond to the simulation
reproducing the test beam geometry including the PS, the red ones correspond
to a simulation with the beam impinging on the calorimeter with an angle of
2.5∘ both in the horizontal and in the vertical direction and no PS on the
beam line.

The resolution for the simulation reproducing the layout of the test beam has
a stochastic term of 16.4% and a constant term of 0.1%. The three available
data points yield resolutions compatible with the simulation. Because of the
small constant term, fits with a linear or quadrature sum of the stochastic
and constant term yield similar results.

The simulation with a 2.5∘ angle yields a stochastic term of 14.5% which we
consider representative of the electromagnetic resolution of the tested
geometry in a real experiment.

###  4.5 Positron Shower Profile

Thanks to the single fibre readout of the central module and the small
distance of 2 mm between the individual fibres, a high-granularity measurement
of the lateral development of the electromagnetic shower could be performed
and compared to the predictions of the Geant4 simulation. The measurement was
performed independently for the scintillation and Cherenkov signal. Due to the
high statistics needed to extract shower profiles, only the reference run,
i.e. the run steering $20$ GeV positrons without the preshower, was used for
this measurement. Similarly to the event selection described before, positrons
are selected by the usage of the Cherenkov counter, moreover only events with
particles hitting the center of the calorimeter front face within a radius of
$5$ mm are kept for analysis. About $4\times 10^{3}$ events from the reference
run are selected for analysis and are compared to $50\times 10^{3}$ simulated
events222For this simulation the patch03 of the Geant4 10.7 release was used..
Only the SiPMs’ signals were considered, therefore the profiles in the
following refer to the shower’s core with an energy containment of $72\%$.
Since one run only was used, SiPMs’ signals were calibrated using dedicated
calibration constants obtained from events of the reference run only.

The radial distance ($r_{i}$) between each fiber $i$ and the shower axis was
computed for each individual fibre as

$\displaystyle
r_{i}=\sqrt{\left(x_{i}-X_{\mathrm{CALO}}\right)^{2}+\left(y_{i}-Y_{\mathrm{CALO}}\right)^{2}}$

With such measurement, the average signal in a fibre is determined as a
function of $r$, and this represents the lateral shower profile. The result is
summarised in Figure 17 which represents a millimeter sampling of a $20$ GeV
positron-induced shower. It is remarkable how the average signal carried by a
fiber drops by two orders of magnitude over a radial distance of only $25$ mm.
For the simulated results, an error of $\pm 0.4^{\circ}$ in the orientation of
the calorimeter with respect to beam axis is considered: the horizontal angle
was assumed to be within $1.0^{\circ}$ and $1.8^{\circ}$ while the vertical
angle was not changed with respect to previous simulations. This assumption is
compatible with the experimental setup described Section 4.1 and leads to the
colored error bands. We found a good MC-to-data agreement over the entire
distance range.

Figure 17: Lateral shower profile of positrons with an energy of 20 GeV. The
black circles (triangles) refer to the scintillation (Cherenkov) signal in
$\mathrm{M}_{0}$. While the red (blue) band corresponds to the simulation
predictions for the scintillation (Cherenkov) signal respectively.

The shower profile as seen by the Cherenkov signal is wider than that measured
with scintillation. This observation confirms that of Ref. [7]. This is
interpreted as a consequence of the fact that the early components of the
electromagnetic shower are collimated with the incoming positron direction,
therefore the Cherenkov light that they emit does not contribute, since it
falls outside the numerical aperture of the fibre.
In Figure 18, derived from the same experimental data, the signals from
individual fibers located in the same $r$-bin are summed, and the average
signals are plotted as a function of $r$. This is a measurement of the
positron-induced shower radial profile. Figure 18 also illustrates the
fraction of SiPMs’ signal deposited within a certain distance from the shower
axis, we call it the electromagnetic shower cumulative profile.

Figure 18: Radial and cumulative shower profiles (left and right respectively)
of positrons with an energy of 20 GeV. The red circles (blue triangles) refer
to the scintillation (Cherenkov) signal in $\mathrm{M}_{0}$.

##  5 Conclusions

A dual-readout sampling calorimeter prototype using brass capillary tubes as
absorber and optical fibers as active medium was tested using beams of
particles at DESY and at the H8 beam line at CERN. The dual readout was
realised by making use of two different types of fibers: doped scintillating
Saint-Gobain BF-10 fibers, and clear “Cherenkov” Mitsubishi SK40 fibres. The
prototype (with a total size of about $10\times 10\times 100\
\mathrm{cm^{3}}$) was composed by nine modules. For the central module, the
individual fibres were read out by means of Hamamatsu S14160-1315 PS SiPMs,
while for the surrounding eight modules the two sets of fibres were bundled
together and read out by Hamamatsu R-8900 PMTs.

After having calibrated the detector by making use of the SiPM multi-photon
spectrum and of beams of positrons, a procedure was developed to correct the
detector response for the dependency of the energy measurement on the particle
impact point on the calorimeter front face. Then, the detector response was
studied using beams of positrons with energies between 10 and 100 GeV: the
linearity of the energy measurement was found to be within 1%, the measured
response was found to be in agreement with that predicted by a dedicated
Geant4 simulation. The energy resolution was found to have a significant
dependence on the impact angle of the beam on the calorimeter front face for
angles between $\theta=0^{\circ}$ and $\theta=2.5\%$. The energy resolution on
data was evaluated by making use of beams of positrons with
$E_{\mathrm{beam}}=10,20\ \mathrm{and}\ 30\ \mathrm{GeV}$, and determined to
be

$\displaystyle\frac{\sigma}{E}=\frac{\left(17.5\pm
2.2\right)\%}{\sqrt{E}}+\left(0.1\pm 0.5\right)\%,$

in agreement with that predicted by the simulation for the same beam impact
angle. The extrapolation performed with the simulation up to impact angles
$\theta=2.5^{\circ}$ showed that the optimal resolution of the prototype is
expected to be

$\displaystyle\frac{\sigma}{E}=\frac{14.5\%}{\sqrt{E}}+0.1\%.$

Finally, a precise measurement of the electromagnetic shower profile of 20-GeV
positrons was compared with the predictions of the simulation, finding good
agreement. The shower development as seen by the scintillation signal is
narrower than that seen by the Cherenkov one, as a consequence of the loss of
Cherenkov light from particles in the early phases of the shower development.

The electromagnetic performance of the dual-readout sampling calorimeter
prototype is overall found to be satisfactory and in line with the
expectations, making this mechanical and readout solution a promising one for
future developments.

## References

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[e]Niccolò Forzano

# Lattice studies of Sp(2N) gauge theories using GRID

Ed Bennett  Peter Boyle  Luigi Del Debbio  Deog Ki Hong  Jong-Wan Lee  Julian
Lenz  C.-J. David Lin  Biagio Lucini  Alessandro Lupo  Maurizio Piai  Davide
Vadacchino

###### Abstract

Four-dimensional gauge theories based on symplectic Lie groups provide elegant
realisations of the microscopic origin of several new physics models.
Numerical studies pursued on the lattice provide quantitative information
necessary for phenomenological applications. To this purpose, we implemented
$Sp(2N)$ gauge theories using Monte Carlo techniques within Grid, a performant
framework designed for the numerical study of quantum field theories on the
lattice. We show the first results obtained using this library, focusing on
the case-study provided by the $Sp(4)$ theory coupled to $N_{as}=4$ Wilson-
Dirac fermions transforming in the 2-index antisymmetric representation. In
particular, we discuss preliminary tests of the algorithm and we test some of
its main functionalities.

##  1 Introduction

Four-dimensional symplectic gauge theories stand out in the literature for
their relevance in the context of new physics models. For this reason, a first
quantitative study of the strongly coupled dynamics based on $Sp(2N)~{}$gauge
theories was obtained using lattice field theory methods [1, 2, 3, 4, 5, 6].
The $Sp(4)~{}$theory with $N_{f}=2$, and $N_{as}=3$ is particularly
interesting [4]: it gives rise, at low energies, to the effective field theory
entering the minimal Composite Higgs model [7, 8] (see Refs. [9, 10] and
references therein), and also realises top (partial) compositeness [11]. In
this work, we make preliminary, somewhat technical, progress to lay the
foundation for future large-scale studies for $Sp(2N)~{}$theories using Grid
[12, 13]. To this end, we wrote and tested new code [14] that supports the
study of $Sp(2N)~{}$theories with matter fields in multiple representations.
This report is structured as follows. In Sect. 2, we define the
$Sp(2N)~{}$gauge theories of interest, both in the continuum and on the
lattice. Sect. 3 discusses all the tests we performed on the algorithm, and we
test some of its main functionalities. Finally, we draw our conclusions in
Sect. 4.

##  2 Symplectic gauge theories and lattice setup

In the continuum, we consider $Sp(2N)~{}$field theories ($N>1$), having as
Lagrangian densities

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\hbox{Tr}\
G_{\mu\nu}G^{\mu\nu}\,+\,\frac{1}{2}\sum_{i}^{N_{\rm
f}}\left(i\overline{Q^{i}}_{a}\gamma^{\mu}\left(D_{\mu}Q^{i}\right)^{a}\,-\,i\overline{D_{\mu}Q^{i}}_{a}\gamma^{\mu}Q^{i\,a}\right)\,-\,m^{\rm
f}\sum_{i}^{N_{\rm f}}\overline{Q^{i}}_{a}Q^{i\,a}+$ (1)
$\displaystyle\,+\,\frac{1}{2}\sum_{k}^{N_{\rm
as}}\left(i\overline{\Psi^{k}}_{ab}\gamma^{\mu}\left(D_{\mu}\Psi^{k}\right)^{ab}\,-\,i\overline{D_{\mu}\Psi^{k}}_{ab}\gamma^{\mu}\Psi^{k\,ab}\right)\,-\,m^{\rm
as}\sum_{k}^{N_{\rm as}}\overline{\Psi^{k}}_{ab}\Psi^{k\,ab}\,,$

where the mass-degenerated Dirac fermions $Q^{ia}$, with $i=1,\cdots,N_{\rm
f}$, and $a=1,\cdots,2N$, transform in the fundamental representation, whereas
$\Psi_{k\,ab}$, with $k=1,\cdots,N_{\rm as}$, transform in the $2$-index
antisymmetric representation. The covariant derivatives are defined through
the transformation properties under the action of an element $U$ of the
$Sp(2N)~{}$gauge group—$Q\to UQ,\,\Psi\to U\Psi U^{T}$.
We consider the system discretised on a lattice of size
$\tilde{V}/a^{4}=T\times L^{3}$, $a$ being the lattice spacing. The action on
the lattice is the sum of two terms, $S\equiv S_{g}+S_{f}$, where $S_{g}$ is
the gauge action
$S_{g}\equiv\beta\sum_{x}\sum_{\mu<\nu}\left(1-\frac{1}{2N}{\rm
Re}\,{\hbox{Tr}}\,{\cal P}_{\mu\nu}(x)\right)$ where ${\cal
P}_{\mu\nu}(x)\equiv
U_{\mu}(x)U_{\nu}(x+\hat{\mu})U_{\mu}^{\dagger}(x+\hat{\nu})U_{\nu}^{\dagger}(x)$
is the _elementary plaquette_ , $U_{\mu}(x)\in Sp(2N)$ is the _link variable_
, and $\beta\equiv 4N/g_{0}^{2}$ is the _inverse bare gauge coupling_. The
fermion action $S_{f}$ is $S_{f}\equiv a^{4}\sum_{j=1}^{N_{\rm
f}}\sum_{x}\overline{Q}^{j}(x)D^{\mathrm{(f)}}_{m}Q^{j}(x)+a^{4}\sum_{j=1}^{N_{\rm
as}}\sum_{x}\overline{\Psi}^{j}(x)D^{\mathrm{(as)}}_{m}\Psi^{j}(x)$, where the
covariant derivatives $D^{(f)}_{m}$ and $D^{(as)}_{m}$ are built using the
links in the fundamental and 2-index antisymmetric representations,
respectively. We shall indicate with $m_{0}^{\mathrm{f}}$ and
$m_{0}^{\mathrm{as}}$ are the bare masses of the fermions in the fundamental
and $2$-index antisymmetric representation,

Figure 1: Test of Creutz equality [15] (left panel), $\langle\exp(-\Delta
H)\rangle=1$ dependence of $\langle\exp(-\Delta H)\rangle$ on the time–step,
$\Delta\tau$, in the MD integration, for $N=2$, $N_{\rm f}=0$, and $N_{\rm
as}=4$. Test of independence of the plaquette (right panel) on the time–step
$\Delta\tau$. The relevant parameters are the trajectory length $\tau=1$,
number of steps $n_{\rm steps}=14,16,18,22,26$ ($\Delta\tau=\tau/n_{\rm
steps}$), for an ensemble with lattice volume $\tilde{V}/a^{4}=8^{4}$,
$\beta=6.8$, and $am_{0}^{\rm as}=-0.6$. The horizontal line in the right
panel represents to the plaquette value obtained averaging over trajectories
having different a number of step values, $n_{\rm steps}$. See, for
comparison, Ref. [16]. (Figure taken from Ref. [17]).

Figure 2: Left panel: Dependence of $\langle\Delta H\rangle$ on the time-step,
$\Delta\tau$, used for the MD integration. Right panel: Test of the relation
between acceptance probability and $\Delta H$ [18]. The theory considered and
the relevant parameters of this study are the same as Fig. 1. These tests
follow closely Ref. [16]. (Figure taken from Ref. [17]).

##  3 Tests of the algorithm

Focussing primarily on the $Sp(4)~{}$theory with $N_{\rm f}=0$, and $N_{\rm
as}=4$ Dirac fermions, we perform preliminary algorithm tests, to check the
correct implementation of the new code. First of all, we check the sanity of
the integrators we use for the molecular dynamics (MD) evolution. The
numerical results are presented in Figs. 1 and 2. Their correspondent ensemble
is obtained evolving the system for $3400$ trajectories and has Madras-Sokal
[19] integrated auto-correlation time $\tau_{c}=6.1(2)$. The first test
verifies whether the Creutz equality [15] is satisfied. This can be done by
measuring the value of the Hamiltonian, $\Delta H$, before and after each
trajectory in the HMC evolution to find $\left\langle\exp\left(-\Delta
H\right)\right\rangle=1$. This is supported by our numerical results (left
panel of Fig. 1). As a second test, we verify that quantities computed through
hybrid Monte-Carlo (HMC and RHMC) updates do not depend on the MD time-size
step, $\Delta\tau$, as our updates are obtained through exact algorithms. To
do so, we use the elementary plaquette and verify (right panel of Fig. 1) the
independence of such quantity on $\Delta\tau$. As a third test, we verify the
relation between $\langle\Delta H\rangle$ and $\Delta\tau$: for a second-order
integrator it is supposed to scale as $\langle\Delta
H\rangle\propto(\Delta\tau)^{4}$. In left panel of Fig. 2, we show the lattice
result, together with a best-fit to the curve $\log\langle\Delta
H\rangle={\cal K}_{1}\,\log(\Delta\tau)+{\cal K}_{2}$, with ${\cal
K}_{1}=3.6(4)$. The fit value of the reduced is $\chi^{2}/N_{\rm d.o.f.}=0.6$,
and ${\cal K}_{1}$ is compatible with $4$. As a last check, we show in right
panel of Fig. 2 results confirming the predicted relation for the acceptance
probability, $P_{\rm acc}={\rm
erfc}\left(\frac{1}{2}\sqrt{\left\langle\Delta\frac{}{}H\right\rangle}\right)$
[18].

We monitored the contribution to the MD of the fields, and how it changes with
bare fermion masses. We show in Fig. 3 the force split in gauge and fermion
contributions, $F(x,\mu)=F_{g}(x,\mu)+F_{f}(x,\mu)$. The latter includes all
four fermions. As shown in Fig. 3, for large and positive values of
$am_{0}^{\rm as}$ the dynamics is led by the gauge degrees of freedom, as one
would expect from a system approaching the quenched regime. Conversely,
decreasing the mass, the fermion contribution increases and for negative
values of the Wilson bare mass corresponding to small PCAC masses, the fermion
contribution dominates.

Figure 3: Field contribution to the MD force for the theory with $N=2$,
$N_{\rm f}=0$, and $N_{\rm as}=4$, on isotropic lattice with
$\tilde{V}=(8a)^{4}$, and lattice coupling $\beta=6.8$. The two blocks are
respectively indicating the gauge (light shading, left) and the fermion (dark
shading, right) contribution. Fermion contributions are summed over flavor.
The six panels correspond to different choices of bare mass: $am_{0}^{\rm
as}=-0.9,\,-0.1,\,+0.6,\,+1.8,\,+15,\,+50$ (left to right, top to bottom). The
results are normalised so that the gauge contribution is held constant. The
computation of the force contributions is made in Ref. [16]. (Figure taken
from Ref. [17]).

As a last test, we verify that our implementation of the Wilson-Dirac
operators is correct. We consider the $Sp(4)~{}$theory with quenched fermions
in either the fundamental or 2-index antisymmetric representation. We compute
the spectrum of eigenvalues of the hermitian Wilson-Dirac operator
$Q_{m}=\gamma_{5}D_{m}$. Following the procedure discussed in Ref. [20], we
compute the distribution of the unfolded density of spacing, $P(s)$, and
compare our results to the predictions of chiral Random Matrix Theory [21].
The unfolded density of spacing is

$P(s)=N_{\tilde{\beta}}s^{\tilde{\beta}}\exp\left(-c_{\tilde{\beta}}s^{2}\right)\,,\quad\text{where}\quad
N_{\tilde{\beta}}=2\dfrac{\Gamma^{\tilde{\beta}+1}\left(\frac{\tilde{\beta}}{2}+1\right)}{\Gamma^{\tilde{\beta}+2}\left(\frac{\tilde{\beta}+1}{2}\right)},\,c_{\tilde{\beta}}=\dfrac{\Gamma^{2}\left(\frac{\tilde{\beta}}{2}+1\right)}{\Gamma^{2}\left(\frac{\tilde{\beta}+1}{2}\right)}\,,$
(2)

where $\tilde{\beta}$ is the Dyson index. As the spectrum is linked to the
chiral symmetry-breaking pattern, the distribution $P(s)$ discriminates
between the symmetry-breaking patterns associated to different representations
of groups. Due to this property, the Dyson index takes different values:
$\tilde{\beta}=4$ corresponds to $SU(2N_{f})\to SO(2N_{f})$, $\tilde{\beta}=2$
to $SU(N_{f})\times SU(N_{f})\to SU(N_{f})$, and $\tilde{\beta}=1$ to
$SU(2N_{f})\to Sp(2N_{f})$. To compare our results on the lattice with Eq.
(2), we compute the eigenvalues of $Q_{m}$ for $N_{\rm conf}$ configurations.
Then, following the procedure described in Ref. [20], we find the discretised
unfolded density of spacings, $P(s)$. In Fig. 4, we show our numerical
results: one finds a distribution that is compatible with the expected
symmetry-breaking patterns. The observed agreement with the predicted model
gives us strong indication that we correctly implemented the Wilson-Dirac
operators. 111The correctness of the Wilson–Dirac operator could also be
checked via consistency with the Feynman rules, both by using the free
propagator obtained from Fourier transforms, and by comparing the results of
gauge transformation. This has not been done in this work.

Figure 4: Distribution of the unfolded density of spacing between subsequent
eigenvalues of the hermitian Dirac-Wilson operator $Q_{m}=\gamma_{5}D_{m}$,
obtained as in Ref. [20], and comparison with predictions from chRMT [21],
computed in the quenched approximation, with ensembles having $\beta=8.0$,
$am_{0}=-0.2$, and lattice volume $\tilde{V}=(4a)^{4}$, in the $Sp(4)$ theory.
The left panel shows the case of fermions transforming in the fundamental
representation, and the right is for fermions in the 2-index antisymmetric
one. The numbers of configurations are $N_{\rm{conf,f}}=88$ and
$N_{\rm{conf,as}}=47$, while the number of eigenvalues in each configuration
used is $3696$ for fundamental fermions and $5120$ for antisymmetric fermions.
(Figure taken from Ref. [17]).

We performed a lattice parameter space scan, to identify possible phase
transitions happening while varying lattice parameters, by studying the
average plaquette, $\langle P\rangle$, and its hysteresis. Performing such a
study we know where there is no bulk phase transition, and one can safely
perform lattice numerical calculations. The left panel of Fig. 5 displays the
average plaquette, $\langle P\rangle$, in ensembles generated using a cold
start. For this theory, the average plaquette is a smooth function everywhere,
except for precise values of $\beta_{*}$ and $am_{0}^{{\rm as}\,\ast}$, where
it shows an abrupt change—this gives us a strong indication of a first-order,
bulk phase transition.

Figure 5: Left panel: parameter scan of the $Sp(4)$ theory with $N_{\rm f}=0$,
$N_{\rm as}=4$ fermions, with ensembles generated from a cold start, using the
HMC. We show the value of the average plaquette, $\langle P\rangle$, as a
function of the bare mass, for a few representative values of the coupling
$\beta$ as shown in the legend. The lattice size is $\tilde{V}=(8a)^{4}$, and
each point is obtained by varying the lattice coupling $\beta=5.6\hbox{ to
}7.0$. Right panel: Hysteresis between hot (red) and cold (other colors)
starts for the $Sp(4)$ theory. The lattice coupling is $\beta=6.4\hbox{ to
}5.6$ (left to right, and top to bottom). (Figure taken from Ref. [17]).

As a further verification, we re-generated the same ensembles with hot starts
and repeated the measurements using the same lattice parameters. The right
panel of figure 5 shows the comparison of the average plaquette values,
$\langle P\rangle$, between hot and cold starts, using the same bare lattice
parameters and hysteresis is clearly visible, for $\beta<\beta_{*}\simeq 6.4$.

We adopt a scale setting procedure that makes use of the Wilson flow. [22].
One introduces the fifth-dimension, flow time, $t$, and solves the defining
diffusion differential equation
$\dfrac{\mathrm{d}B_{\mu}(x,\,t)}{\mathrm{d}t}=D_{\nu}G_{\nu\mu}(x,\,t)$, with
the boundary conditions, $B_{\mu}(x,\,0)=U_{\mu}(x)$. Then, one defines the
quantities ${\cal
E}(t)\equiv\frac{t^{2}}{2}\left\langle\mathrm{Tr}\,\left[G_{\mu\nu}(t)G_{\mu\nu}(t)\right]\right\rangle\,,{\cal
W}(t)\equiv t\frac{d}{dt}\mathcal{E}(t)$ and introduces a prescription
$\left.\mathcal{E}(t)\right|_{t=t_{0}}=\mathcal{E}_{0}$ that sets the scale
$t_{0}$, or alteratively $\left.{\cal
W}(t)\right|_{t=w_{0}^{2}}=\mathcal{W}_{0}$ that sets the scale $w_{0}^{2}$.
Both $\mathcal{E}_{0}$ and $\mathcal{W}_{0}$ are chosen conventionally.

Figure 6: Wilson Flow [22] energy density ${\cal E}(t)$ (left panel) and
${\cal W}(t)$ (right), computed as in Refs. [1, 23], from the standard (pl)
and the clover-leaf (cl) plaquette, for the $Sp(4)$ theory with $N_{\rm as}=4$
fermions transforming in the 2-index antisymmetric representation. The lattice
size is $\tilde{V}=(12a)^{4}$, $\beta=6.8$ and $6.9$, $am_{0}^{\rm as}=-0.8$.
The time step is $0.01$, $t_{max}=4.5$ to reduce finite-size effects. We
choose ${\cal W}_{0}=\frac{1}{2}C_{2}(F)$ for the scale setting. The
corresponding values of $w_{0}$ from the plaquette and the clover-leaf are
$w_{0,pl.}=1.485(3)$ and $w_{0,cl.}=1.495(2)$ for $\beta=6.8$ and
$w_{0,pl.}=2.005(2)$ and $w_{0,cl.}=2.026(2)$ for $\beta=6.9$. We have set
$a=1$, for notational convenience. (Figure taken from Ref. [17]).

In Fig. 6 we show ${\cal E}(t)$ and ${\cal W}(t)$ as functions of the flow
time, $t$. On the lattice, the calculation of ${\cal E}(t)$ and ${\cal W}(t)$
depends on a definition of $G_{\mu\nu}$, and we display explicitly two
choices: the elementary, $\mathcal{P}_{\mu\nu}(x)$, and the clover-leaf
plaquette, $C_{\mu\nu}(x)$. The plots show results agreeing with previous
findings in the literature, according to which at early flow times ${\cal
E}(t)$ and ${\cal W}(t)$ strongly differ due to UV fluctuations. Then, the
cut-off effects are smoothened and the two curves become closer to each other.
Moreover, we notice that the function ${\cal W}(t)$ displays a milder
dependence. For this reason, we set the scale $w_{0}$ using ${\cal W}(t)$, by
conventionally setting ${\cal W}_{0}=\frac{1}{2}C_{2}(F)$. Having set the
scale, one can define the topological charge. For gauge configurations
generated by Monte Carlo simulation, this observable is dominated by UV
fluctuations, hence it will be regulated defining it through $B_{\mu}(x,t)$,
obtaining
$Q_{L}(t)\equiv\frac{1}{32\pi^{2}}\varepsilon^{\mu\nu\rho\sigma}\sum_{x}\hbox{Tr}\,\left[{\cal
C}_{\mu\nu}(x,t){\cal C}_{\rho\sigma}(x,t)\right]$.

Figure 7: Evolution with the ensemble trajectories of the topological charge
computed at flow time $t=w_{0}^{2}$, as shown in Ref. [23], for the $Sp(4)$
theory with $N_{\rm as}=4$ fermions transforming in the 2-index antisymmetric
representation. The lattice parameters and size are the same as Fig. 6. The
histograms of the measurements (right panels) are compatible with a normal
distribution centered at zero, with reduced chi-square $\chi^{2}/N_{\rm
d.o.f}=\tilde{\chi}^{2}=1.1$ for both panels. The integrated autocorrelation
time computed using the Madras-Sokal windowing algorithm is $\tau_{Q}=31(3)$
(left panel) and $\tau_{Q}=238(12)$ (right panel). (Figure taken from Ref.
[17]).

In Fig. 7 we display the value of $Q_{L}(t=w_{0}^{2})$, for the same ensembles
of the Wilson flow study. There is no evidence of topological freezing, as the
Madras-Sokal integrated autocorrelation time [19], $\tau_{Q}$, of the
topological charge turns out to be many orders of magnitude smaller than the
number of trajectories.

##  4 Summary and outlook

Symplectic gauge theories have a variety of phenomenological applications in
many contexts such as Composite Higgs Models and top partial compositeness [7,
8], strongly interacting dark matter models [24, 25]. We developed and tested
new software, embedded into the Grid environment to take full advantage of its
flexibility. We reported the positive results of our tests of the algorithms,
particularly focusing on the $Sp(4)$ theory coupled to $N_{as}=4$ (Dirac)
fermions transforming in the antisymmetric representation. This work and the
software we developed for it set the stage needed to explore and quantify
future large-scale studies—among them, the study of the conformal window
extent in strongly coupled gauge theories with matter field content. Moreover,
the tools we developed will be useful in the context of the recent literature
discussing the spectroscopy of $Sp(2N)~{}$theories with various
representations (see, e.g. Refs.[1, 2, 4]), and can be further extended by
applying new techniques based on the spectral densities [26].

## Acknowledgments

The work of EB, JL and BL has been funded by the ExaTEPP project EP/X017168/1.
The work of EB and JL has also been supported by the UKRI Science and
Technology Facilities Council (STFC) Research Software Engineering Fellowship
EP/V052489/1. The work of NF has been supported by the STFC Consolidated Grant
No. ST/X508834/1. The work of PB was supported in part by US DOE Contract
DESC0012704(BNL), and in part by the Scientific Discovery through Advanced
Computing (SciDAC) program LAB 22-2580. The work of DKH was supported by Basic
Science Research Program through the National Research Foundation of Korea
(NRF) funded by the Ministry of Education (NRF-2017R1D1A1B06033701). The work
of LDD and AL was supported by the ExaTEPP project EP/X01696X/1. The work of
JWL was supported in part by the National Research Foundation of Korea (NRF)
grant funded by the Korea government(MSIT) (NRF-2018R1C1B3001379) and by IBS
under the project code, IBS-R018-D1. The work of DKH and JWL was further
supported by the National Research Foundation of Korea (NRF) grant funded by
the Korea government (MSIT) (2021R1A4A5031460). The work of CJDL is supported
by the Taiwanese NSTC grant 109-2112-M-009-006-MY3. DV is supported by a STFC
new applicant scheme grant. The work of BL and MP has been supported in part
by the STFC Consolidated Grants No. ST/P00055X/1, ST/T000813/1, and
ST/X000648/1. BL, MP, AL and LDD received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
program under Grant Agreement No. 813942. The work of BL is further supported
in part by the EPSRC ExCALIBUR programme ExaTEPP (project EP/X017168/1), by
the Royal Society Wolfson Research Merit Award WM170010 and by the Leverhulme
Trust Research Fellowship No. RF-2020-4619. LDD is supported by the UK Science
and Technology Facility Council (STFC) grant ST/P000630/1. Numerical
simulations have been performed on the Swansea SUNBIRD cluster (part of the
Supercomputing Wales project) and AccelerateAI A100 GPU system, and on the
DiRAC Extreme Scaling service at the University of Edinburgh. Supercomputing
Wales and AccelerateAI are part funded by the European Regional Development
Fund (ERDF) via Welsh Government. The DiRAC Extreme Scaling service is
operated by the Edinburgh Parallel Computing Centre on behalf of the STFC
DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BEIS
capital funding via STFC capital grant ST/R00238X/1 and STFC DiRAC Operations
grant ST/R001006/1. DiRAC is part of the National e-Infrastructure.
Research Data Access Statement and Open Access Statement—The data shown in
this manuscript can be downloaded from the data release of Ref. [17]. For the
purpose of open access, the authors have applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising.

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