# Study of time and energy resolution of an ultra-compact sampling
calorimeter (RADiCAL) module at EM shower maximum
over the energy range 25 GeV $\leq$ E $\leq$ 150 GeV

Carlos Perez-Lara  James Wetzel  Ugur Akgun  Thomas Anderson  Thomas Barbera
Dylan Blend  Kerem Cankocak  Salim Cerci  Nehal Chigurupati  Bradley Cox  Paul
Debbins  Max Dubnowski  Buse Duran  Gizem Gul Dincer  Selbi Hatipoglu  Ilknur
Hos  Bora Isildak  Colin Jessop  Ohannes Kamer Koseyan  Ayben Karasu Uysal
Reyhan Kurt  Berkan Kaynak  Alexander Ledovskoy  Alexi Mestvirishvili  Yasar
Onel  Suat Ozkorucuklu  Aldo Penzo  Onur Potok  Daniel Ruggiero  Randal Ruchti
Deniz Sunar Cerci  Ali Tosun  Mark Vigneault  Yuyi Wan  Mitchell Wayne  Taylan
Yetkin  Liyuan Zhang  Renyuan Zhu  Caglar Zorbilmez

###### Abstract

The RADiCAL Collaboration is conducting R&D on high performance
electromagnetic (EM) calorimetry to address the challenges expected in future
collider experiments under conditions of high luminosity and/or high
irradiation (FCC-ee, FCC-hh and fixed target and forward physics
environments). Under development is a sampling calorimeter approach, known as
RADiCAL modules, based on scintillation and wavelength-shifting (WLS)
technologies and photosensor, including SiPM and SiPM-like technology. The
modules discussed herein consist of alternating layers of very dense (W)
absorber and scintillating crystal (LYSO:Ce) plates, assembled to a depth of
25 $X_{0}$. The scintillation signals produced by the EM showers in the region
of EM shower maximum (shower max) are transmitted to SiPM located at the
upstream and downstream ends of the modules via quartz capillaries which
penetrate the full length of the module. The capillaries contain DSB1 organic
plastic WLS filaments positioned within the region of shower max, where the
shower energy deposition is greatest, and fused with quartz rod elsewhere. The
wavelength shifted light from this spatially-localized shower max region is
then propagated to the photosensors. This paper presents the results of an
initial measurement of the time resolution of a RADiCAL module over the energy
range 25 GeV $\leq$ E $\leq$ 150 GeV using the H2 electron beam at CERN. The
data indicate an energy dependence of the time resolution that follows the
functional form: $\sigma_{t}=a/\sqrt{E}\oplus b$, where a = 256 $\sqrt{GeV}$
ps and b = 17.5 ps. The time resolution measured at the highest electron beam
energy for which data was currently recorded (150 GeV) was found to be
$\sigma_{t}$ = 27 ps.

###### keywords:

calorimetry , detector , fast-timing , FCC , ILC , 10 TeV pCM

††journal: NIM A

[label1]organization=Adiyaman University, city=Adiyaman, country=Turkiye
[label2]organization=California Institute of Technology, city=Pasadena,
country=CA USA [label3]organization=Coe College, city=Cedar Rapids, state=IA,
country=USA [label4]organization=Hofstra University, city=Hempstead, state=NY,
country=USA [label5]organization=University of Iowa, city=Iowa City, state=IA,
country=USA [label6]organization=Istanbul University, city=Istanbul,
country=Turkiye [label7]organization=Istanbul University - Cerrahpasa,
city=Istanbul, country=Turkiye [label8]organization=Istanbul Technical
University, city=Istanbul, country=Turkiye [label9]organization=University of
Notre Dame, city=Notre Dame, state=IN, country=USA
[label10]organization=University of Virginia, city=Charlottesville, state=VA,
country=USA [label11]organization=Yildiz Technical University, city=Istanbul,
country=Turkiye

##  1 Introduction

The R&D objectives of the RADiCAL collaboration are focused on the development
of precision EM calorimetry, based primarily on optical techniques, for future
colliding beam experiments at facilities such as FCC-ee, FCC-hh [1, 2, 3, 4,
5], as well as fixed target and forward direction/small angle experiments. The
effort is directed toward addressing the Priority Research Directions (PRD)
for calorimetry listed in the US Department of Energy Basic Research Needs
(BRN) workshop for HEP Instrumentation [6], as well as active participation in
collaborative efforts currently in progress through ECFA DRD6 [7] and CPAD
CRD9 programs [8, 9, 10, 11].

Through the development of fast, radiation-hard scintillators, wavelength
shifters, and optical transmission elements, performance goals include
producing modular structures capable of achieving: an EM energy resolution of
$\sigma_{E}/E=10\%/\sqrt{E}~{}\oplus~{}0.3/E~{}\oplus~{}0.7\%$ [5]; a timing
resolution of $\sigma_{t}~{}\leq~{}20$ ps at high energy [12]; and EM shower
centroid position localization to within a few mm [12, 13]. Measurements of
the radiation tolerance of various fluorescent and photosensing elements [14]
along with initial measurements of the energy resolution and timing resolution
have been presented elsewhere [14, 15, 16] and remain under active development
and further study. This paper presents first results from a beam test of the
RADiCAL module concept for timing resolution over an extensive range of
electron beam energies:
$25~{}\text{GeV}~{}\leq~{}\text{E}~{}\leq~{}150~{}\text{GeV}$ [17, 18].

##  2 RADiCAL module under test

For these studies, the RADiCAL prototype, Fig. 1, is a Shashlik/Kebap-style
module of overall dimensions 14 x 14 x 135 mm3 that consists of a layered
structure of 29 LYSO:Ce crystal plates of 1.5 mm thickness interleaved with 28
W plates of 2.5 mm thickness and assembled to a depth of $\approx 25~{}X_{0}$
and corresponding to an absorption length of $\approx 0.9~{}\lambda$. Thin
Tyvek sheets were inserted between successive layers to act as reflective
spacers to avoid optical absorption between LYSO:Ce and W surfaces. Overall,
the structure has a radiation length $X_{0}$ = 5.4 mm and a Moliére radius
$R_{M}$ = 13.7 mm. Holes were drilled through five locations in all of the
tiles as shown schematically in Fig. 2, to facilitate the penetration of
quartz capillaries through the layered structure, each capillary containing
wavelength shifter (WLS) filaments positioned strategically to facilitate
energy or timing measurement. SiPM photosensors were mounted on front-end
electronic cards located at both upstream and downstream ends of the module to
detect the optical signals from the capillaries.

Figure 1: A schematic of a RADiCAL module for ultra-compact EM calorimetry
based upon interleaved layers of W and LYSO:Ce crystal in a Shashlik/Kebap-
like structure and read out via specialized WLS quartz capillary elements
which penetrate through the module to SiPM photosensors, positioned at both
ends of the module. In this schematic, the beam enters the module from the
upper left. The cross-sectional dimensions of the module are set by the
Molière radius of the structure.  Figure 2: Cross section of the RADiCAL
module tiles, indicating the five locations (holes) for capillary waveguide
insertion. Tungsten plates and LYSO:Ce scintillation plates had similar cross-
sectional dimensions. The LYSO:Ce plates, as shown, had a thickness of 1.5 mm.
The tungsten plates had a thickness of 2.5 mm. The central hole location,
available for calibration or additional measurement, was not used in these
tests reported here.

To date, several types of WLS capillaries have been fabricated for the ongoing
testing, as shown in Fig. 3. They include: E-Type capillaries for energy
measurement, where the capillary cores are filled either with a liquid
wavelength shifter or with a solid WLS filament that runs the full length of
the module; and T-type capillaries for timing measurement, in which solid WLS
filaments of short length are inserted into the capillary cores to a position
corresponding to the region of EM shower max. The remaining core volume of
such T-type capillaries are filled with quartz rods which are inserted and
fused to the quartz walls of the capillaries, creating a solid quartz
waveguides upstream and downstream of the WLS filaments.

Using a custom-built capillary-tube tester, the optical responses of both
E-Type and T-Type capillaries containing DSB1 wavelength shifters were
assessed in laboratory bench tests [19]. DSB1 absorbs strongly at
$\lambda=425~{}nm$, emission peak at $\lambda=495~{}nm$ and fast fluorescence
decay $\tau=3.5~{}ns$. Fig. 4 shows the response of such capillaries when the
wavelength shifter is excited by an LED at $\lambda=425~{}nm$. Fig. 4
indicates that the response of an E-type capillary is very uniform over the
full capillary length, desirable for longitudinal uniformity for energy
measurement. Also shown is the highly localized response of the T-type,
pronounced in the shower max region and negligible elsewhere, desirable for
precision timing. It should be noted that the T-type capillaries also provide
a localized measurement of the shower energy at shower max, important for
shower position determination at that location.

Figure 3: Response of WLS Capillaries containg DSB1 waveshifter to UV light at
365 nm. Upper and lower are E-Type capillaries. Middle is a T-Type capillary.
Figure 4: Example of the optical response to LED light of 425 nm as a function
of position for capillaries of E-type (blue) and T-type (orange) with the
photodetector placed at the left (0 mm location).

Fig. 5 and Fig. 6 show schematics of a RADiCAL module instrumented with E-type
and T-type capillaries appropriate for the testing of energy or timing
resolution. Ultimately, an experimental application would utilize both
capillary types positioned strategically within the module. This evolution is
discussed in Section 7.

Figure 5: RADiCAL module with E-type capillaries where the WLS runs the full
length of the modules. Figure 6: RADiCAL module with T-type capillaries where
the WLS is located in the region of shower max.

For the tests reported in this paper, the RADiCAL module was instrumented with
four T-type capillaries as indicated in Fig. 6. The capillaries were of 183 mm
length, having an outer diameter of 1150 $\mu$m and an inner (core) diameter
of 950 $\mu$m. Within each core, a DSB1 WLS filament of 900 $\mu$m diameter
and 15 mm length was positioned in the region of shower max for EM showers of
25 GeV $\leq$ E $\leq$ 150 GeV. The remainder of each core was filled and
fused with quartz rods as described above. The capillaries then transmitted
the wave-shifted light to Hamamatsu HDR2 SiPM photosensors which were mounted
on readout cards postioned at both the upstream and downstream ends of the
module. The central hole in the module (as shown in Fig. 2 was unused in these
studies, but is reserved for future use for timing measurement, energy
measurement or calibration. A GEANT4 study (Fig. 7) shows the fraction of EM
shower energy deposited in successive LYSO:Ce tiles in the module, indicating
that the location of shower max varies slowly with energy, allowing useful
placement of wavelength shifting filaments for timing.

Figure 7: GEANT4 simulation of the fraction of EM shower energy in a given
LYSO:Ce tile within a RADiCAL module schematically shown in Figs. 1 and 6. The
Beam is incident from the left, impacting Layer 0 first. At the bottom of the
figure is a schematic of the capillary with the location of the WLS filament
timing element marked. This location within the capillary was optimized for
earlier studies at the Fermilab FTBF, where the electron energy was in the
range $20~{}\leq~{}E~{}\leq~{}30~{}GeV$ and where shower max occurred in
layers 8-10. This position is adequate (although not optimized) for higher
energy showers which occur deeper in the module in layers 11-13, and will be
corrected in future work, by repositioning and lengthening slightly the WLS
filament location.

##  3 Expectations from simulation

A GEANT4 simulation [12] was used to assess the potential for precision timing
resolution when measured from the shower max region. For this study, the
simulation assumed only one SiPM readout at the downstream end of a T-type
capillary inserted through the center of the module. The simulation indicated
excellent timing resolution could be achieved dependent upon detected light
level, Fig 8. Also noteworthy was that the shower max timing signal is derived
from a region of very small transverse size r $\leq$ 5 mm, Fig. 9, with a
shower radius at that location, which is significantly smaller than the
Molière Radius and more closely approximating the radiation length of the
module. This compact region contains one to two orders of magnitude more
charged shower particles than would be created by the passage of a minimum
ionizing particle (MIP) in that location, creating a very large and localized
optical pulse. The timing resolution is dominated by the measured rise time of
this signal. Fig. 8 indicates that, in simulation, a timing resolution of
significantly less than 50 ps could be achievable with RADiCAL modules
depending upon the detected light level. By reading out the light from both
upstream and downstream ends of timing capillaries, and by using several
timing capillaries per module, the aim has been to reach or exceed this
expectation for the timing resolution of a RADiCAL module.

Figure 8: Timing resolution vs detected light yield in photoelectrons per MeV,
simulated for a 50 GeV electron shower. Only the downstream readout was used
in this GEANT4 simulation study. Figure 9: Location of the energy in a RADiCAL
module at shower max for 50 GeV electrons in GEANT4 simulation. The boundary
of the module itself is a square of area 14 x 14 mm2 located at the center of
the figure.

The shower max method also provides an additional pattern recognition
constraint. In the modular array, the module that is struck by an incoming
electron (or gamma) provides the predominant timing signal for the EM shower.
In adjacent modules, the timing signals are effectively absent, since there is
very little energy deposited in them in that region, given the very compact
transverse size of the shower at the shower max location, Fig. 9. This is a
potentially powerful locational tool to identify which module is struck by an
incoming EM particle in the higher $\eta$ (more forward) regions of collider
detectors or fixed target experiments where separation and reconstruction of
overlapping showers would be potentially challenging.

These simulations suggested that the shower max timing technique, combined
with fast, local energy measurement, would be a potentially powerful technique
for measuring both precision timing and shower location for EM particles
(electrons, positrons and gammas).

##  4 Module assembly and beam test arrangement

The RADiCAL module shown schematically in Fig. 1 and under UV illumination in
Fig. 10a was assembled in a Delrin structure. WLS capillary tubes were then
inserted and front end electronics cards supporting the SiPM were attached to
the upstream and downstream ends of the structure as shown in Fig. 10b. The
module was then mounted on a 3D-printed base plate which held the RADiCAL
module and, just upstream, a micro channel plate (MCP) phototube which
provided high resolution reference timing, Fig. 10c. Once bias voltage cables
and readout cables to CAEN digitizers were installed, the RADiCAL module was
covered with a 3D-printed light-tight cover that fit securely into grooves in
the base plate. The assembly as then taken to the North Area/Prèvessin Site at
the CERN SPS, where it was installed on the CMS/HF Motion Table in the H2 beam
line (see Fig. 11a and Fig. 11b).

The final experimental arrangement of detectors in the beam line is shown in
Fig. 11c. Specifically, scintillation counters of dimension 1 x 1 cm2 and 2 x
2 cm2 (labeled A1 and A2 respectively in the figure) provided triggering
options. The A2 counter served as the primary trigger for the data taking,
since its area fully covered the entire active area of the RADiCAL module. The
MCP tube (Hamamatsu R3809U-50) with a timing resolution of 10 ps
$<\sigma_{t}<20$ ps depending on detection region within the tube, and labeled
B in the figure, was used as an independent precision timing reference. The
RADiCAL module, labeled C, was next in the sequence and was followed by a 2 x
2 array of Pb glass counters, labeled D, forming a detector of 4 x 4 x 40 cm3
which served as a backing calorimeter to measure any shower energy leakage
from the RADiCAL module and for identification of any potential hadron (MIP)
background in the beam. Not shown in Fig. 11c was a beam chamber provided by
the CERN SPS Coordination Team, which provided sub-millimeter horizontal and
vertical (x, y) position information of the incoming beam at the location of
the RADiCAL module.

(a)

(b)

(c)

Figure 10: The RADiCAL modular structure exposed to UV light at 365 nm,
showing fluorescence of LuAG:Ce tiles at 425nm and DSB1 WLS emission at 495nm
at the ends of the capillaries, (a). Assembly of a RADiCAL module. The top
shows the tiles being stacked; the bottom shows the Delrin assembly, (b). The
3D printed mounting structure that supported the MCP tube, in the cylindrical
enclosure on the left, and the RADiCAL module on the right with front end
readout cards installed, (c).

(a)

(b)

(c)

Figure 11: Installation of the RADiCAL module on the CMS/HF Lift Table in the
H2 beam line at the North Area Site at the CERN SPS. Beam direction is from
lower right to upper left, (a). Overhead closeup view of the relative
placement of the MCP tube (lower left), the 2 x 2 cm2 trigger counter, the
RADiCAL module underneath the 3D-printed enclosure and the upstream end of the
Pb glass array concealed beneath black cloth (upper right). The beam enters
this picture from the lower left corner and exits upper right., (b). The 3D
printed mounting structure that supported the MCP tube, in the cylindrical
enclosure on the left, and the RADiCAL module on the right with front end
readout cards installed, (c).

The custom-built front-end electronics cards each supported four Hamamatsu
HDR2 SiPM and supported on-board low-gain and high-gain amplification of the
SiPM signals. The low gain amplification was for local energy measurement in
the shower max region; the high gain signals were differentially amplified to
provide fast rise-time pulses for timing measurement and were modeled after a
CERN design [20]. All signals were digitized with CAEN DT5742 digitizers. Fig.
12 shows characteristic pulse shapes of these signals recorded in an earlier
data run at Fermilab [16] and which were similar for this data. The local
shower energy (at Shower max) could be determined by the pulse height or pulse
area of the low gain signals, Fig. 12 Left; the shower timing for each SiPM
channel was determined by setting a fixed threshold on the leading edge of the
pulse rise of the high gain signals, Fig. 12 Right. For the data reported
here, the high gain signals were driven further into saturation, to maximize
the signal rise of the leading edge of the pulses for improved timing
response.

Figure 12: Average normalized waveforms for signals from the SiPM amplified at
low gain (left) and high gain (right). The low gain signals were used for
localized energy measurement from the shower max region. The high gain signals
were used for fast timing using the fixed threshold on the rise of the leading
edge of the pulse.

##  5 Module performance

The CERN H2 electron beam was directed to the detector apparatus shown in Fig.
11c, in successive 25 GeV steps in beam energy, covering the range 25 GeV
$\leq E_{beam}\leq$ 150 GeV. At least ${10^{6}}$ triggers were recorded at
each step for data analysis.

###  5.1 Measurement of EM shower energy at shower max

As shown in Fig. 7 above, simulation studies indicated that a portion of the
total EM shower energy was sampled by $\approx$ 3 LuAG:Ce tiles at shower max,
and the WLS filaments positioned in the T-type capillaries at that location
provided a wave shifted optical signal from that location which was then
transmitted via the capillaries to the SiPM readout. The SiPM signals were
then amplified at low-gain for position-based energy measurement in the shower
max region and at high-gain, with differential amplification for fast timing
measurement.

Fig. 13 shows the response from each of the capillaries in the module as a
function of electron beam (x, y) position as determined from the beam line
wire chamber and for two beam energies, 25 GeV and 125 GeV. The responses
displayed are based on the amplitudes of the individual low-gain signals from
each of the four T-type capillaries and their SiPM readout at both upstream
and downstream ends of the module. The locations of the four capillaries and
the (uninstrumented) central hole are clearly identifiable, since the WLS
material is not scintillating, but rather wave shifting. It is also clear that
the closer the electron beam is to a given capillary, the stronger the WLS
fiber signal is seen. Also seen in the images at 25 GeV is a faint ring, which
is suggestive of an internal seal within the upstream MCP tube, causing some
showers to be initiated upstream of the RADiCAL module, reducing the detected
energy at shower max. The presence of the ring indicates that the RADiCAL
module and the MCP are very well aligned in the beam.

Figure 13: For $E_{beam}$ = 25 GeV: Beam particle location horizontally and
vertically (x, y) at the RADiCAL module provided by the upstream beam chamber,
and with the data triggered by the A2 counter. Color density is given by the
pulse height in the low-gain amplifiers for each SiPM. Left four plots are for
the upstream SiPM and the right four plots are for the downstream SiPM. The
capillaries and central (unused hole) are clearly seen. The faint ring is
likely due to an internal seal within the MCP. The MCP was located just
upstream of the RADiCAL module in the beamline.

####  5.1.1 Energy measurement in the module for $E_{beam}$ = 25 GeV

Fig. 14 shows the distributions of the amplitudes of the low-gain signals for
each SiPM channel for a beam energy of 25 GeV. In the analysis, a cut was
applied to remove events for which there was a significant response in the
downstream Pb glass array. This was to exclude triggered events that either
missed the RADiCAL module or were associated with non-EM particles such as
hadrons (primarily pions) in the beam. The peaks of these distributions were
then scaled to a common value, to adjust for any variation. Then the low gain
amplitudes from the downstream SiPM were summed (Fig. 15 Left), and separately
the the low gain amplitudes from the upstream SiPM were summed (Fig. 15
Center). There were no obvious differences in these distributions, and hence
downstream and upstream signals were combined (summed) together (Fig. 15
Right). Hence the measured energy in the RADiCAL module $E_{meas}$ was given
by the summation of the amplitudes of all eight low-gain SiPM signals. This
resultant distribution has a significant peak near $E_{meas}$ $\approx{900}$
mV, and a tail that extends down to lower values of measured energy. A
Gaussian fit to the peak of the distribution is shown in Fig. 15 Right. The
central value of the fit is assigned to the energy of the beam, $E_{beam}$ =
25 GeV, and the standard deviation of the fit is associated with the measured
energy resolution in the module at shower max for this beam energy,
$\sigma_{meas}$. Several factors contribute to the shape of the observed
distribution in Fig. 15 Right. These include: the particle statistics of the
EM showers; shower leakage; photo efficiency and optical coupling of the
scintillator and wavelength shifter, and optical transport via the capillaries
to the SiPM. These were considered in the simulation discussion above in
Section 3 and Fig. 8. Beam momentum resolution, while present, was but a small
contributor and understood to be $\leq$ 2%, dependent upon collimation and
transport settings in the H2 beam line.

Figure 14: For $E_{beam}$ = 25 GeV: Distribution of the pulse heights from the
low gain signals from each SiPM channel. The four downstream SiPM channels at
the top. The four upstream SiPM channels at the bottom. The SiPM channel
labeled UPSE in the lower right had a lower gain than the others. The Gaussian
fits to the peak values of all the channels were subsequently scaled to a
common value for local energy and position measurement. Figure 15: For
$E_{\text{beam}}$ = 25 GeV: Sum of the calibrated low-gain signals: Left: for
the downstream SiPM channels; Middle: for the upstream channels. Right: the
sum of all channels: the peak of the Gaussian fit to this distribution
$E_{\text{meas}}$ (in mV) is set to the beam energy $E_{\text{beam}}$ (in GeV)
and the width of the Gaussian fit is the measured energy resolution
$\sigma_{\mathrm{E_{\text{meas}}}}$.

####  5.1.2 Energy measurement in the module for
$25~{}GeV~{}\leq~{}E_{beam}~{}\leq~{}150~{}GeV$

The procedure described in Section 5.1.1 was then applied to the data taken in
each successive step in beam energy. The summed amplitudes of all eight low-
gain SiPM signals for each triggered event were identified as Emeas in mV
units for each beam energy step Ebeam in GeV units from 25 GeV up to 150 GeV.
These distributions are shown in Fig. 16. Notable features in each
distribution are a strong peak region which narrows with increasing beam
energy, and tails in each distribution to lower measured energy. Gaussian fits
were made to each peak region, with the central values of the Gaussians (in
mV) associated with the values of the delivered beam energy (in GeV). The
standard deviation of each Gaussian was identified as the corresponding energy
resolution $\sigma_{meas}$ for that measured energy.

####  5.1.3 Overall behavior of energy measurement at shower max

For each step in beam energy $E_{beam}$ the central value of the Gaussian fit
to each measured energy peak $E_{meas}$ (in mV) was plotted versus the nominal
beam energy $E_{beam}$ (in GeV) as shown in Fig. 17 Left. Linear behavior was
observed, with a fitted slope of: $E_{meas}/E_{beam}=29.4$ mV/GeV. Then in
Fig. 17 Right, for each step in beam energy $E_{beam}$, each measured
resolution $\sigma_{E_{meas}}$ (in mV) divided by the central value of the
corresponding Gaussian peak $E_{meas}$ (in mV) was plotted as a function of
the delivered beam energy. The fit to the data follows the functional form:

$\frac{\sigma_{E_{meas}}}{E_{meas}}=\frac{52.04\%}{\sqrt{E_{beam}}}\oplus\frac{31.62\%}{E_{beam}}\oplus
9.31\%$ (1)

Note that this does not represent the energy resolution that would result from
a high-resolution measurement of the energy of an EM shower in which all 29
LYSO:Ce layers in the full module would contribute to the measurement, as well
as energy from surrounding modules in a 3 x 3 or larger array. Rather, the
measurement here corresponds to the energy and energy resolution derived from
the restricted region of shower max, corresponding to sampling over $\approx$
3 LYSO:Ce tiles within a single module. The key value of this energy
measurement is for its importance for the determination of EM shower position
localization and for the correlation of the timing resolution with the
detected energy at shower max. The former topic, spatial localization, will be
discussed in a future paper. The latter topic, correlation of timing
resolution with detected energy, is the subject of this paper and elaborated
below.

Figure 16: Distributions of summed energy $E_{meas}$ (in mV) from all eight
SiPM channels in the RADiCAL module for the six beam energies $E_{beam}$
utilized in this study. The central value of the Gaussian fits (red lines) for
each energy (in mV) were then associated with the given nominal energy of the
beam (in GeV).

Figure 17: Left: Plot of the summed energy from all SiPM channels $E_{meas}$
(in mV) vs the nominal beam energy $E_{beam}$ (in GeV), showing a linear
slope. Right: Plot of the measured ratio $\sigma_{E_{meas}}/E_{meas}$ from the
shower max region as a function of the nominal beam energy.

###  5.2 Measurement of shower timing.

To determine the signal timing for each SiPM channel, fixed thresholds were
set for the high-gain signals for the upstream and downstream SiPM. When the
leading edge of the high-gain pulses from a given channel exceeded the
threshold, the timing for that channel was determined and compared with the
reference timing provided by the MCP tube. This allowed for confirmation of
the synchronization of arrival times the timing signals, and allowed for
averaging of the timing signals.

####  5.2.1 General Characteristics of Timing Measurement at $E_{beam}$ = 25
GeV

In Fig. 18 are shown the timing differences between each of the eight SiPM
channels compared with the MCP timing: $\Delta
t(i)~{}=~{}(t_{SiPM~{}(i)}~{}-~{}t_{MCP})$ for (i=1,8). These timing
differences are then averaged separately over the four downstream SiPM
channels (i = 1,4): $\Delta t_{DW}=\Sigma~{}\Delta t(i)/4$, and over the four
upstream SiPM channels (j = 5,8): $\Delta t_{UP}=\Sigma~{}\Delta t(j)/4$.
These distributions are shown in Fig. 19 in the Upper Left and Upper Right
hand plots. Fig. 19 Lower Left shows the average difference event by event for
these downstream and upstream timing averages: $(\Delta t_{DW}~{}-~{}\Delta
t_{UP})/2$, and Fig. 19 Lower Right shows the overall average of both upstream
and downstream timings: $(\Delta t_{DN}~{}+~{}\Delta t_{UP})/2$. The timing
resolution of the RADiCAL module can be determined from Gaussian fits to
either of these distributions. Notable in Fig. 19 Lower Right is a small
amplitude satellite bump whose centroid is shifted by $\approx~{}0.2~{}ns$
relative to the main peak. This small peak corresponds to triggered events for
which the timing algorithm, which uses a fixed threshold, is shifted due to
fluctuation by one sample segment in the CAEN digitzer that is used to
digitize the signal timing. This condition affected roughly 15% of the timing
signals of all SiPM channels, causing the shift of $\approx~{}0.2~{}ns$
relative to the MCP reference time. The effect appears only the average of the
sum of all channels relative to the MCP. It does not appear in Fig. 19 Lower
Left, as this is the timing difference between downstream and upstream SiPM
channels, which is independent of the MCP timing. In any case, the timing
resolution is determined by Gaussian fit to the main peaks, and hence the
calculation of the timing resolution calculation in the Lower Right hand plot
is generally unaffected by the presence of this satellite peak. The main peak
in Fig. 19 Lower Right is more broad, but this is due primarily to the
presence and contribution of the MCP resolution to this distribution, rather
than from a contribution from the satellite peak.

The timing resolution, in a generalized way over the energy range 25 GeV$\leq$
Ebeam $\leq$150 GeV, could then be assessed by taking the average timing of
all eight timing differences with respect to the MCP timing: $(\Delta
t_{DW}~{}+~{}\Delta t_{UP})/2$; or by taking the average of the difference in
the time average of the four downstream high gain signals and the time average
of the four upstream high gain signals: $(\Delta t_{DW}~{}-~{}\Delta
t_{UP})/2$. The second method, through subtraction, gives a result which is
independent of the MCP timing. Fig. 20 displays the results of the second
method for each step in beam energy. The Gaussian fits to the peaks indicate
steady improvement in the time resolution as a function of beam energy.

Figure 18: Ebeam = 25 GeV. The timing distributions for each SiPM relative to
the MCP timing: $\Delta t$ (i) = (tSiPM (i) \- tMCP). Left: for the four
downstream SiPM. Right for the four upstream SiPM. Figure 19: Ebeam = 25 GeV.
Upper Left: The average of the timing distributions for the four downstream
SiPM. Upper Right: The average of the timing distributions for the four
upstream SiPM. Lower Left: The average of the difference of downstream and
upstream timing distributions. Lower Right: The average of downstream and
upstream timing distributions. Note in this plot the appearance a small
satellite peak, which has an area roughly 15% of the larger peak. Given its
small size, its presence has little effect on the timing resolution determined
from this distribution (See text) Figure 20: Timing differences between the
average times of the four downstream SiPM and the four upstream SiPM in
successive steps in beam energy over the range: 25 GeV $\leq$ E $\leq$ 150
GeV. The Gaussian fits to these distributions provide a preliminary measure of
the timing resolution of the module. The peaks narrow with energy, indicating
improvement in timing resolution.

###  5.3 Refined timing analysis

What the above overview neglects to consider is that the resolution of the
timing signals depends on the detected/measured energy ($E_{meas}$) in the
RADiCAL module event by event for a given beam energy ($E_{beam}$), rather
than simply given by a unique and fixed energy value for all events. This
consideration is based upon earlier study of the timing resolution of the
RADiCAL module by our group at the Fermilab Test Beam Facility for which a
negative beam of $E_{beam}$ = 28 GeV was available [9], and where a strong
dependence was observed for the timing resolution as a function of the
detected energy in the RADiCAL module (in mV).

This issue is illustrated by examination of the measured energy signals shown
in Fig. 16, where each of the distributions exhibits a low energy tail. For
example, in the case of $E_{beam}$ = 25 GeV, the detected energy at the peak
of the distribution is $E_{meas}~{}\approx$ 900 mV, corresponding to the
delivered beam energy value, whereas the low energy tail extends down to
$E_{meas}~{}\approx$ 200 mV, which corresponds to an observed/detected energy
of $E_{meas}$ = 6.7 GeV, based upon the slope of the fit in Fig.17 Left,
relating measured energy $E_{meas}$ (in mV) to Energy (in GeV). To account for
such variation, the timing resolution for a given delivered beam energy
$E_{beam}$ was therefore studied in bins of measured energy $E_{meas}$ across
the peak regions of the energy distributions shown in Fig. 16.

For each step of electron beam energy $E_{beam}$, the corresponding measured
energy distribution $E_{meas}$ in the RADiCAL module shown in Fig. 16 was
divided into nine bins around the peak of the distribution, and the technique
described in Section 5.2.1 to determine the timing resolution was applied bin-
by-bin. These studies are shown in Figs. 21, 22, 23, 24, 25 and 26. Displayed
in each of the figures are the average of the timing difference between
downstream and upstream averaged signals, $(\Delta t_{DW}~{}-~{}\Delta
t_{UP})/2$, as well as the average of the sum of downstream and upstream
averaged signals, $(\Delta t_{DW}~{}+~{}\Delta t_{UP})/2$. As noted in Section
5.2, the former distribution is a direct measure of the RADiCAL timing
resolution, whereas the latter distribution includes a non-negligible
contribution due to the resolution of the MCP. The standard deviations of the
Gaussian fits to these distributions were then plotted bin by bin as a
function of measured energy for each of nine bins.

Figure 21: For Ebeam = 25 GeV. The measured energy Emeas (in mV) has been
subdivided into 9 bins, bin 1 at the left, bin 9 at the right in the lower
figure. Top: the distributions of $(\Delta t_{DW}-\Delta t_{UP})/2$ for each
bin of observed energy Eobs in mV as indicated by the horizontal error bars in
the lower plot. Middle: the distributions of $(\Delta t_{DW}+\Delta t_{UP})/2$
for each bin of observed energy. Lower: the measured time resolution
$\sigma_{t}$ in ps for each bin of observed energy obtained from the standard
deviation of the Gaussian fit to the corresponding timing peaks above in the
figure. Blue values correspond to $(\Delta t_{DW}-\Delta t_{UP})/2$ and orange
values correspond to $(\Delta t_{DW}-\Delta t_{UP})/2$.

Figure 22: For Ebeam = 50 GeV. The measured energy Emeas (in mV) has been
subdivided into 9 bins, bin 1 at the left, bin 9 at the right in the lower
figure. Top: the distributions of $(\Delta t_{DW}-\Delta t_{UP})/2$ for each
bin of observed energy Eobs in mV as indicated by the horizontal error bars in
the lower plot. Middle: the distributions of $(\Delta t_{DW}+\Delta t_{UP})/2$
for each bin of observed energy. Lower: the measured time resolution
$\sigma_{t}$ in ps for each bin of observed energy obtained from the standard
deviation of the Gaussian fit to the corresponding timing peaks above in the
figure. Blue values correspond to $(\Delta t_{DW}-\Delta t_{UP})/2$ and orange
values correspond to $(\Delta t_{DW}-\Delta t_{UP})/2$.

Figure 23: For Ebeam = 75 GeV. The measured energy Emeas (in mV) has been
subdivided into 9 bins, bin 1 at the left, bin 9 at the right in the lower
figure. Top: the distributions of $(\Delta t_{DW}-\Delta t_{UP})/2$ for each
bin of observed energy Eobs in mV as indicated by the horizontal error bars in
the lower plot. Middle: the distributions of $(\Delta t_{DW}+\Delta t_{UP})/2$
for each bin of observed energy. Lower: the measured time resolution
$\sigma_{t}$ in ps for each bin of observed energy obtained from the standard
deviation of the Gaussian fit to the corresponding timing peaks above in the
figure. Blue values correspond to $(\Delta t_{DW}-\Delta t_{UP})/2$ and orange
values correspond to $(\Delta t_{DW}-\Delta t_{UP})/2$.

Figure 24: For Ebeam = 100 GeV. The measured energy Emeas (in mV) has been
subdivided into 9 bins, bin 1 at the left, bin 9 at the right in the lower
figure. Top: the distributions of $(\Delta t_{DW}-\Delta t_{UP})/2$ for each
bin of observed energy Eobs in mV as indicated by the horizontal error bars in
the lower plot. Middle: the distributions of $(\Delta t_{DW}+\Delta t_{UP})/2$
for each bin of observed energy. Lower: the measured time resolution
$\sigma_{t}$ in ps for each bin of observed energy obtained from the standard
deviation of the Gaussian fit to the corresponding timing peaks above in the
figure. Blue values correspond to $(\Delta t_{DW}-\Delta t_{UP})/2$ and orange
values correspond to $(\Delta t_{DW}-\Delta t_{UP})/2$.

Figure 25: For Ebeam = 125 GeV. The measured energy Emeas (in mV) has been
subdivided into 9 bins, bin 1 at the left, bin 9 at the right in the lower
figure. Top: the distributions of $(\Delta t_{DW}-\Delta t_{UP})/2$ for each
bin of observed energy Eobs in mV as indicated by the horizontal error bars in
the lower plot. Middle: the distributions of $(\Delta t_{DW}+\Delta t_{UP})/2$
for each bin of observed energy. Lower: the measured time resolution
$\sigma_{t}$ in ps for each bin of observed energy obtained from the standard
deviation of the Gaussian fit to the corresponding timing peaks above in the
figure. Blue values correspond to $(\Delta t_{DW}-\Delta t_{UP})/2$ and orange
values correspond to $(\Delta t_{DW}-\Delta t_{UP})/2$.

Figure 26: For Ebeam = 150 GeV. The measured energy Emeas (in mV) has been
subdivided into 9 bins, bin 1 at the left, bin 9 at the right in the lower
figure. Top: the distributions of $(\Delta t_{DW}-\Delta t_{UP})/2$ for each
bin of observed energy Eobs in mV as indicated by the horizontal error bars in
the lower plot. Middle: the distributions of $(\Delta t_{DW}+\Delta t_{UP})/2$
for each bin of observed energy. Lower: the measured time resolution
$\sigma_{t}$ in ps for each bin of observed energy obtained from the standard
deviation of the Gaussian fit to the corresponding timing peaks above in the
figure. Blue values correspond to $(\Delta t_{DW}-\Delta t_{UP})/2$ and orange
values correspond to $(\Delta t_{DW}-\Delta t_{UP})/2$.

##  6 Analysis summary and key results

Given the extensive data sets recorded in the six energy steps of beam energy
over the range 25 GeV $\leq$ E $\leq$ 150 GeV, the timing resolution of the
RADiCAL module can be measured in a variety of ways. These methods are
displayed graphically in Fig. 27 Left and are described here:

  1. 1.

Downstream in Fig. 27 Left. The module timing is determined from the average
of the timing from each of four downstream SiPMs only, through the calculation
of the quantity: $\Delta t_{DW}=\Sigma(t_{SiPM(i)}~{}-~{}t_{MCP})/4$ and
fitting the peak of this distribution for which an example is shown in Fig. 19
Upper Left for $E_{beam}$ = 25 GeV. For these calculations, the full spectrum
of measured energy values corresponding to a given beam energy were used as
shown in Figure 16.

  2. 2.

Upstream in Fig. 27 Left. The module timing is determined from the average of
the timing from each of four upstream SiPMs only, through the calculation of
the quantity: $\Delta t_{UP}=\Sigma(t_{SiPM(i)}~{}-~{}t_{MCP})/4$ and fitting
the peak of this distribution for which an example is shown in Fig. 19 Upper
Right for $E_{beam}$ = 25 GeV. For these calculations, the full spectrum of
measured energy values corresponding to a given beam energy were used as shown
in Fig. 16.

  3. 3.

(DW-UP)/2 in Fig. 27 Left. The module timing is determined from the difference
of the average timing of the downstream SiPMs relative to the MCP and the
average timing of the upstream SiPMs relative to the MCP. The fits to these
values are shown in Fig. 20. Again, for these calculations, the full spectrum
of of observed energy values corresponding to a given beam energy value were
used as shown in Fig. 16.

  4. 4.

BestMinus in Fig. 27 Left. Here, as for Method 3 above, the module timing is
determined from the difference of the average timing of the downstream SiPM
relative to the MCP and the average timing of the upstream SiPM timing
relative to the MCP, but using the detailed energy analysis shown in Figs. 21,
22, 23, 24, 25 and 26. This analysis selects the measured time differences in
bins 6-8 in each figure and computes the average of these time differences
$(\Delta t_{DW}~{}-~{}\Delta t_{UP})/2$ to determine the timing resolution.
This approach assures that the timing resolution is determined when the
observed energy in the RADiCAL module corresponds very closely to the actual
delivered beam energy.

  5. 5.

BestPlus in Fig. 27 Left. Here the module timing is determined from the sum of
the average of the downstream SiPM timing relative to the MCP and the upstream
SiPM timing relative to the MCP, using the detailed energy analysis shown in
Figs. 21, 22, 23, 24, 25 and 26. This analysis selects the measured time
averages in bins 6-8 in each figure and computes the overall average timing
$(\Delta t_{DW}~{}+~{}\Delta t_{UP})/2$ to determine the timing resolution.
This approach assures that the timing resolution is determined when the
observed energy in the RADiCAL module corresponds closely to the actual
delivered beam energy. However, this method includes an unsubtracted
resolution contribution due to the MCP tube.

  6. 6.

BestPlus-MCP in Fig. 27 Left. This analysis is identical to that of method 5
above, but subtracts a value of $\sigma_{t,MCP}~{}\approx$ 10 ps to correct
for the presence of the MCP in the $(\Delta t_{DW}~{}+~{}\Delta t_{UP})/2$
approach to determination of the timing resolution of the module.

Of these six approaches, method 4 (BestMinus in Fig. 27 Left) provides the
“best estimator” of the timing resolution of the RADiCAL module in which the
measured energy in the module $E_{meas}$ is effectively the same as the
nominal beam energy $E_{beam}$ delivered to the module. Hence, we identify
this methodology as the best estimator of the timing resolution of the RADiCAL
module in its current form, and the results of method 4 are plotted in Fig. 27
Right.

The fit to the data points in Fig. 27 Right indicates that the functional
dependence of the timing resolution with respect to the energy measured in the
RADiCAL module follows the form:

$\sigma_{t}=\frac{a}{\sqrt{E}}~{}\oplus~{}b$ (2)

where: a = $256\sqrt{GeV}$ ps, b = 17.5 ps.

Additionally, the time resolution measured at the highest electron beam energy
for which data was currently recorded (150 GeV) was found to be
$\sigma_{t}=27$ ps. The data also indicates that in the limit of high energy,
the timing resolution is $\sigma_{t}=17.5$ ps under the current measurement
conditions.

Figure 27: Left: Various methodologies for determining the timing resolution
of the RADiCAL module (See Section 6), with the legend identifying each
method. Right: Best Estimator of the timing Resolution for the RADiCAL module,
which is the BestMinus method, is displayed as a function of detected wnergy
in the module for all six beam energies 25 GeV $\leq$ E $\leq$ 150 GeV.

Figure 28: Potential capillary positioning in an enhanced RADiCAL module
design that allows for simultaneous high-resolution timing and energy
measurements. The lighter shaded circles represent T-type capillaries, while
the darker shaded circles represent E-type capillaries. The center circle
houses a capillary that could either of T-type or E-Type, or alternatively
could be used for calibration.

##  7 Ongoing research and development

The ongoing program of RADiCAL R&D includes data analysis, instrumentation
development, and simulation and beam testing activities.

  1. 1.

Data Analysis: EM Shower Localization. This is a systematic study of spatial
localization of EM showers over the energy range 25 GeV $\leq$ E $\leq$ 150
GeV. This analysis correlates energy information from the capillaries in the
RADiCAL module with the spatial information of the incoming beam provided by a
wire chamber positioned $\approx~{}{3~{}m}$ upstream of the RADiCAL module in
the H2 beam line at CERN.

  2. 2.

Data Analysis: Comparison of DSB1 and LuAG:Ce Wavelength Shifters. This is a
comparative study of timing resolution of the RADiCAL module in which T-type
capillaries containing filaments of the ceramic wavelength shifter LuAG:Ce are
compared with the data presented here in which DSB1 organic fiber filaments
were utilized. This comparison will be in 25 GeV steps in electron beam energy
over the range 25 GeV $\leq$ E $\leq$ 150 GeV.

  3. 3.

Instrumentation Development: Enhanced Modular Design. Preparations are
underway to improve the timing resolution and energy resolution of RADiCAL
modules, through revised modular design, improved front-end electronics and
Data Acquisition. Enhanced timing resolution can be realized by increasing the
thickness of the LYSO:Ce crystal layers in the region of shower max by a
factor of 2 to improve light collection. Energy resolution can be enhanced by
increasing the thickness of the LYSO:Ce crystal layers throughout the module
to increase the sampling fraction for EM showers. This will reduce the
stochastic term in the energy resolution to meet the FCC-hh requirements for
EM energy resolution described in Sect. 1 above [5]. The goals are to further
improve (reduce) the timing resolution $\sigma_{t}$ of a single RADiCAL module
by a factor of $\approx 1/\sqrt{2}$ compared to its current value. To improve
simultaneously the stochastic term in the overall energy resolution, by a
similar factor, would require a 3 x 3 array of RADiCAL modules of updated
design, as the full shower development is set by the Moliére radius $R_{M}$
and a modular array is needed for transverse containment of the full EM shower
[13]. This is described next in this section.

  4. 4.

Simulation and Beam Test: Enhanced Modular Design. A new modular structure
with the potential to meet simultaneously the RADiCAL goals for both timing
and energy resolution is currently under investigation through Geant4
simulation and is shown schematically in Fig. 28. In this module there are
five T-type capillaries and four E-Type capillaries, each with SiPM readout at
both ends and with high and low gain signals digitized. As the figure
indicates, the capillaries can be deployed in various configurations to
benefit timing, energy and positional analysis. For this structure, the E-Type
capillaries will be filled with solid WLS filaments of LuAG:Ce ceramics
running the full length of the module. LuAG:Ce is favored for its solid
ceramic nature, ability to be drawn into filaments and excellent radiation
hardness [14, 21]. The T-Type capillaries will be filled with either DSB1 or
LuAG:Ce WLS filaments for timing resolution comparison. The cross section of
the new module (18 x 18 mm2) will be larger than that of the current module
(14 x 14 mm2) due to the increase in the radiation length and hence Moliére
radius of the new design. This is a result of the thickness changes in the
LYSO:Ce layers of the new structure. The intent is to conduct a comparative
test of both the current and new modular structures for timing, energy and
position localization in high energy electron beam over the energy range 25
GeV $\leq$ E $\leq$ 200 GeV. Once qualified in single module tests, a 3 x 3
array of such modules will be constructed and tested.

##  8 Conclusions

The results presented in Section 6, and Fig. 27 indicate that the RADiCAL
modular calorimetry technique has the potential to meet and exceed the timing
requirements for FCC-ee and FCC-hh applications, and additionally would be a
potentially useful EM calorimetry approach for forward physics applications
and fixed-target style experiments.

Additionally, the impact of the RADiCAL research and development program,
while directed toward future particle physics experimental applications, is
not experiment specific, is potentially broad and significant, and can be
expected to inform further developments in high energy physics instrumentation
through ECFA DRD6 and CPAD9 programs. The technological innovations are
versatile and can be applied in particle physics, nuclear physics, materials
science, and medical physics.

##  9 CRediT authorship contribution statement

Carlos Perez Lara: Conceptualization, Methodology, Software, Validation,
Formal Analysis, Investigation, Supervision, Data Curation, Writing – Original
Draft, Writing – Review and Editing. James Wetzel: Conceptualization,
Methodology, Software, Validation, Formal Analysis, Investigation,
Supervision, Data Curation, Writing – Original Draft, Writing – Review and
Editing. Ugur Akgun: Resources. Thomas Anderson: Investigation, Resources.
Thomas Barbera: Investigation, Formal Analysis. Dylan Blend: Investigation.
Kerem Cankocak: Investigation, Supervision. Salim Cerci: Investigation,
Writing – Review and Editing. Nihal Chigurupati: Investigation, Writing –
Review and Editing. Bradley Cox: Conceptualization, Methodology, Validation,
Investigation, Resources, Writing – Review and Editing, Supervision, Funding
Acquisition. Paul Debbins: Investigation, Resources. Max Dubnowski:
Investigation. Buse Duran: Investigation. Gizem Gul Dincer: Investigation.
Selbi Hatipoglu: Investigation. Ilknur Hos: Investigation, Writing – Review
and Editing. Bora Isildak: Investigation, Writing – Review and Editing. Colin
Jessop: Funding Acquisition, Data Curation, Writing – Review and Editing.
Ohannes Kamer Koseyan: Investigation. Ayben Karasu Uysal: Writing – Review and
Editing. Reyhan Kurt: Writing – Review and Editing. Berkan Kaynak: Validation,
Investigation, Resources, Writing – Review and Editing. Alexander Ledovskoy:
Conceptualization, Methodology, Software, Validation, Formal Analysis,
Investigation, Writing – Review and Editing. Alexi Mestvirishvili: Validation,
Investigation, Supervision, Resources. Yasar Onel: Conceptualization,
Validation, Investigation, Resources, Supervision, Funding Acquisition. Suat
Ozkorucuklu: Investigation, Resources, Writing – Review and Editing,
Supervision. Aldo Penzo: Investigation, Resources, Writing – Review and
Editing. Onur Potok: Validation, Investigation, Resources, Writing – Review
and Editing. Daniel Ruggiero: Validation, Investigation, Resources. Randal
Ruchti: Conceptualization, Methodology, Validation, Investigation, Resources,
Writing – Original Draft, Writing – Review and Editing, Visualization,
Supervision, Project Administration, Funding Acquisition. Deniz Sunar Cerci:
Investigation, Writing – Review and Editing. Ali Tosun: Investigation. Mark
Vigneault: Investigation, Resources. Yuyi Wan: Investigation. Mitchell Wayne:
Writing – Review and Editing, Project Administration, Funding Acquisition.
Taylan Yetkin: Investigation, Writing – Review and Editing. Liyuan Zhang:
Investigation, Resources, Writing – Review and Editing. Renyuan Zhu:
Conceptualization, Methodology, Validation, Formal Analysis, Investigation,
Resources, Writing – Review and Editing, Supervision, Funding Acquisition.
Caglar Zorbilmez: Investigation, Writing – Review and Editing.

##  10 Acknowledgements

We thank the CERN SPS Coordination Team for their expert assistance in
providing excellent beam conditions and logistics to make the experimental
tests described above very successful. And We thank colleagues from the
COMPASS experiment for the loan of Pb glass detectors which were used for
backing calorimetry.

This is work has been supported in part by: US DOE grant DE-SC0017810, US NSF
grant NSF-PHY-1914059, the University of Notre Dame through its Resilience and
Recovery Grant Program, and QuarkNet for high school teacher and student
participation.

## References

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URL https://doi.org/10.1140/epjc/s10052-019-6904-3

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URL https://doi.org/10.1140/epjst/e2019-900045-4

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URL https://www.osti.gov/biblio/1659761

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URL https://committees.web.cern.ch/drdc

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URL https://indico.fnal.gov/event/58909/

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URL https://indico.slac.stanford.edu/event/8288/

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URL https://indico.slac.stanford.edu/event/8288/sessions/589/#20231109

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URL https://notredame.box.com/s/vjpbvpxn2sff72y6y4wvue3nnhcucgtx

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URL https://notredame.box.com/s/p9afzt9hxp2yg22vmoycfir8fnnt7tzk

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URL https://www.mdpi.com/2410-390X/6/3/27

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URL https://meetings.aps.org/Meeting/MAS23/Session/G01.3

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URL https://indico.slac.stanford.edu/event/8288/sessions/589/#20231107

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URL https://eljentechnology.com

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URL https://dx.doi.org/10.1088/1361-6560/aadbcd

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# Photometric Metallicity and Distance for the Two RR Lyrae in Segue II and
Ursa Major II Dwarf Galaxies Based on Multi-Band Light-Curves

Chow-Choong Ngeow Graduate Institute of Astronomy, National Central
University, 300 Jhongda Road, 32001 Jhongli, Taiwan  Anupam Bhardwaj INAF-
Osservatorio astronomico di Capodimonte, Via Moiariello 16, 80131 Napoli,
Italy

###### Abstract

Multi-band light curves of two RR Lyrae variables in Segue II and Ursa Major
II ultra-faint dwarf (UDF) galaxies were collected from near simultaneous
observations using the Lulin One-meter Telescope in $Vgri$ bands. Together
with Gaia $G$-band light curves, we determined photometric metallicities using
empirical relations involving pulsation period and Fourier parameter as
dependent parameters. We demonstrated that the RR Lyrae photometric
metallicity can be determined accurately when these empirical relations were
employed at multiple wavelengths, which can potentially improve the distance
determination based on RR Lyrae stars. The photometric metallicities based on
our approach were found to be $-2.27\pm 0.13$ dex and $-1.87\pm 0.16$ dex for
the RR Lyrae in Segue II and Ursa Major II UFD, respectively, with
corresponding distance moduli of $17.69\pm 0.15$ mag and $17.58\pm 0.15$ mag,
in agreement with previous literature determinations. This approach of
photometric metallicity of RR Lyrae based on multi-band optical light curves
will be particularly relevant for distance measurements in the era of the Vera
C Rubin’s Legacy Survey of Space and Time.

††software: astropy (Astropy Collaboration et al., 2013, 2018, 2022), PSFEx
(Bertin, 2011) , SCAMP (Bertin, 2006), Source-Extractor (Bertin & Arnouts,
1996)

##  1 Introduction

The old-population pulsating stars RR Lyrae are well-known distance indicators
because they exhibit an absolute $V$-band magnitude-metallicity relation and
well-defined period-luminosity(-metallicity) relations in other filters (for a
review, see Bhardwaj, 2020). In most cases, the metallicity of RR Lyrae,
parameterized as $\mathrm{[Fe/H]}$, needs to be known a priory to apply these
relations. The best $\mathrm{[Fe/H]}$ measurements are based on (high-
resolution) spectroscopic observations. However, such observations could be
time-consuming and are only limited to relatively bright RR Lyrae in the Milky
Way. Alternatively, $\mathrm{[Fe/H]}$ can be obtained using the light-curve
information, as shown in the seminar paper by Jurcsik & Kovacs (1996). The
$\mathrm{[Fe/H]}$ measured using this approach, known as photometric
metallicity, is expected to be widely applied to the distant RR Lyrae
discovered from the multi-band time-domain sky surveys, especially the Vera C.
Rubin Observatory’s Legacy Survey of Space and Time (LSST, Ivezić et al.,
2019), because only few spectrographs on large-aperture telescopes, or perhaps
none, can be used to collect high-resolution spectra of very distant RR Lyrae
stars.

In the past, RR Lyrae photometric metallicities were typically obtained using
light curve structure information in a single filter. On the other hand,
empirical relations that use the light-curve information to estimate
$\mathrm{[Fe/H]}$ have been derived in several filters, ranging from optical
$V$-band to infrared WISE filters (some examples can be found in Section 2).
Higher precision photometric metallicity can be achieved by averaging out the
$\mathrm{[Fe/H]}$ obtained from several such relations in the same or
different filters mitigating possible systematic effects. In this work, we
demonstrate this is indeed the case based on the multi-band observations of
two ab-type (i.e. fundamental-mode) RR Lyrae discovered in the ultra-faint
dwarf galaxies SEGUE II (Boettcher et al., 2013) and Ursa Major II (hereafter
UMaII, Dall’Ora et al., 2012). These near-simultaneous time-series
observations were carried out using the Lulin One-meter Telescope (LOT),111We
note that Vivas et al. (2020) discovered three additional RR Lyrae associated
with UMaII. However, these three RR Lyrae were located outside the field-of-
view (FOV) for LOT. In contrast, no additional RR Lyrae were found for SEGUE
II (Vivas et al., 2020). a general-purpose Cassegrain reflector located at the
Lulin Observatory in central Taiwan. The collected data allow us to obtain the
photometric metallicities for these two RR Lyrae based on the homogeneous
light curve data at multiple wavelengths.

Section 2 presents multi-band light curves collected from LOT, together with
archival Gaia light curves, for these two RR Lyrae. Based on these multi-band
light curves, we obtained their photometric metallicity in Section 3. A by-
product of our work is the derivation of distance moduli to these dwarf
galaxies, which will be discussed in Section 4. The conclusions of our work is
presented in Section 5. Throughout this work, these two RR Lyrae will be
referred as SEGUE II-V1 and UMaII-V1, respectively.

##  2 Multi-Band Light Curves

Figure 1: The $grVI$-band light curves based on LOT observations and the
$G$-band light curves retrieved from Gaia DR3 for SEGUE II-V1 (left panel) and
UMaII-V1 (right panel). For a better visualization, these light curves, except
the $g$-band, were offsetted. The black curves represent the best-fit light
curves based on the Fourier-expansion.

Time-series observations of SEGUE II-V1 and UMaII-V1 were carried out with LOT
in 12 and 18 nights, respectively, between 2019 November 14 and 2020 April 15.
The Princeton Instruments SOPHIA 2048B back-illuminated CCD was mounted on LOT
during these observations, providing a FOV of $13.2\arcmin\times 13.2\arcmin$
(with a pixel scale of $0.386\arcsec$/pixel) on the CCD images. A sequence of
$Vgri$ exposures, with identical exposure time of 360 second in all filters,
were taken multiple times for each RR Lyrae in a given night when they were
visible from the Lulin Observatory. All of the collected images were reduced
in a standard manner (including bias- and dark-subtraction, followed by flat-
fielding). Astrometric calibration on the reduced images were done using the
SCAMP package (Bertin, 2006). The PSF (point-spread function) photometry of RR
Lyrae in the reduced images were measured using the combined PSFEx (Bertin,
2011) and Source-Extractor (Bertin & Arnouts, 1996) package, at which the
details on the photometric calibration were presented in the Appendix A. Table
1 and 2 provide multi-band light curves collected from the LOT observations
for SEGUE II-V1 and UMaII-V1, respectively.

Table 1: Light Curves for SEGUE II-V1.

$JD$ | mag | mag_err | Filter
---|---|---|---
2458831.0928240740 | 19.150 | 0.042 | g
2458839.9920949074 | 18.961 | 0.014 | g
2458802.1246759258 | 18.824 | 0.025 | g
2458831.1647916664 | 19.078 | 0.051 | g
2458830.0386458333 | 18.798 | 0.026 | g
$\cdots$ | $\cdots$ | $\cdots$ | $\cdots$

Note. — This table is available in its entirety in machine-readable form, a
portion is shown here for guidance on its content.

Table 2: Light Curves for UMaII-V1.

$JD$ | mag | mag_err | Filter
---|---|---|---
2458911.1549074072 | 18.298 | 0.017 | g
2458909.9914583331 | 18.119 | 0.029 | g
2458931.0296990741 | 18.653 | 0.024 | g
2458850.3490856481 | 18.959 | 0.156 | g
2458885.1327662035 | 18.153 | 0.019 | g
$\cdots$ | $\cdots$ | $\cdots$ | $\cdots$

Note. — This table is available in its entirety in machine-readable form, a
portion is shown here for guidance on its content.

In addition to the new LOT observations, we have also downloaded $G$-band
light curves released from the Gaia Data Release 3 (DR3, Gaia Collaboration et
al., 2016, 2023) for these two RR Lyrae, with Gaia DR3 source ID for SEGUE
II-V1 and UMaII-V1 as 87207528534363264 and 1043841876592990208, respectively.
We have also adopted their periods as derived in Gaia DR3 (Clementini et al.,
2023; Eyer et al., 2023, from the I/358/vrrlyr Table available in the
SIMBAD/VizieR database), because the adopted periods are adequate to fold the
light curves and there is no need to re-derive their periods. The adopted
periods are $P(\mathrm{SEGUE\ II}$-$\mathrm{V1})=0.7494938\pm 0.0000164$ days
and $P(\mathrm{UMaII}$-$\mathrm{V1})=0.5651226\pm 0.0000037$ days.

Figure 1 presents multi-band phase folded light curves with the adopted
periods. These light curves were then fitted with a low-order Fourier-
expansion (see Ngeow, 2022, and reference therein). In brief, we fitted a
Fourier series of the following form:

$\displaystyle m(\Phi)$ $\displaystyle=$ $\displaystyle
m_{0}+\sum^{n}_{i=1}A_{i}\sin(2\pi i\Phi+\phi_{i}),$

to the phased light curves, where $\Phi$ is pulsational phase from 0 to 1, and
Fourier order $n$ varied from 4 to 9. The best order of fit was determined
based on the method as described in Ngeow et al. (2023). From the best-fitted
Fourier expansions, we determined the $\phi_{31}=\phi_{3}-3\phi_{1}$ Fourier
parameters which are needed to estimate the photometric metallicity for RR
Lyrae (see next Section). The determined $\phi_{31}$ from the multi-band light
curves are provided in Table 3.

##  3 The Metallicity of RR Lyrae

###  3.1 The Adopted Relations

It is well-known that $\mathrm{[Fe/H]}$ of a RR Lyrae can be obtained from its
pulsation period $P$ and its light curve parameter, primarily the Fourier
parameter $\phi_{31}$ (Jurcsik & Kovacs, 1996). We collected the $grGVI$-band
$[\mathrm{Fe/H}]$-$\phi_{31}$-$P$ relations that are available recently in the
literature. These include the $grV$-band relations taken from Ngeow (2022,
hereafter N22):

$\displaystyle[\mathrm{Fe/H}]^{g}_{N22}$ $\displaystyle=$
$\displaystyle-1.01[\pm 0.06]-7.43[\pm 0.80](P-0.58)$ $\displaystyle+1.69[\pm
0.16](\phi_{31}-5.25),\ \ \sigma=0.24,$
$\displaystyle[\mathrm{Fe/H}]^{r}_{N22}$ $\displaystyle=$
$\displaystyle-1.54[\pm 0.07]-8.28[\pm 1.04](P-0.58)$ $\displaystyle+1.30[\pm
0.15](\phi_{31}-5.25),\ \ \sigma=0.30,$
$\displaystyle[\mathrm{Fe/H}]^{V}_{N22}$ $\displaystyle=$
$\displaystyle-1.21[\pm 0.05]-7.67[\pm 0.78](P-0.58)$ $\displaystyle+1.50[\pm
0.14](\phi_{31}-5.25),\ \ \sigma=0.24.$

These relations were derived based on a sample of 30 RR Lyrae located in the
Kepler Field, and their spectroscopic $[\mathrm{Fe/H}]$ were derived from
high-resolution spectra (Nemec et al., 2013). The metallicity scale for N22,
$[\mathrm{Fe/H}]_{N22}$, is in the Carretta et al. (2009) scale. We have also
adopted $V$-band $[\mathrm{Fe/H}]$-$\phi_{31}$-$P$ relation from Mullen et al.
(2021, hereafter M21), derived from 1980 field RR Lyrae, as:

$\displaystyle[\mathrm{Fe/H}]^{V}_{M21}$ $\displaystyle=$
$\displaystyle-1.22[\pm 0.01]-7.60[\pm 0.24](P-0.58)$ $\displaystyle+1.42[\pm
0.05](\phi_{31}-5.25),\ \ \sigma=0.41.$

The M21 metallicity is in Crestani et al. (2021) scale, therefore we applied
an offset of $-0.08$ dex to convert $[\mathrm{Fe/H}]^{V}_{M21}$ to the
Carretta et al. (2009) scale as determined in Mullen et al. (2021). We did not
include $V$-band relation from Jurcsik & Kovacs (1996) because the qualities
for some of the light curves used in their work (as commented in Mullen et
al., 2021; Zong et al., 2023) are not as good as those studied either in M21
or N22.

In the Gaia $G$-band, Iorio & Belokurov (2021, hereafter I21) derived the
relation using a set of 86 field RR Lyrae, as given below:

$\displaystyle[\mathrm{Fe/H}]^{G}_{I21}$ $\displaystyle=$
$\displaystyle-1.68[\pm 0.05]-5.08[\pm 0.5](P-0.60)$ $\displaystyle+0.68[\pm
0.11](\phi_{31}-2.00-\pi),\ \ \sigma=0.31.$

Mullen et al. (2021) pointed out that a $\pi$ is needed to be subtracted from
$\phi_{31}$ when using their relation, hence we included it in the above
relation. The metallicity scale for the I21 relation is in Zinn & West (1984)
scale, therefore we converted the Zinn & West (1984, ZW84) scale to the
Carretta et al. (2009, C09) scale using the relation given in Carretta et al.
(2009): $[\mathrm{Fe/H}]_{C09}=1.105[\mathrm{Fe/H}]_{ZW84}+0.160$.
Independently, Jurcsik & Hajdu (2023, hereafter J23) derived the $G$-band
relation using RR Lyrae-hosted globular clusters. Since it is well-known that
globular clusters can be divided into Oosterhoff-I (OoI, with higher
metallicity) and Oosterhoff-II (OoII, with lower metallicity) types, Jurcsik &
Hajdu (2023) derived three sets of $G$-band relation: one for each Oosterhoff
types and a relation for the combined sample. Since both SEGUE II-V1 and
UMaII-V1 are OoII RR Lyrae (Boettcher et al., 2013; Dall’Ora et al., 2012), we
adopted the OoII relation:

$\displaystyle[\mathrm{Fe/H}]^{G}_{J23}$ $\displaystyle=$
$\displaystyle-4.324[\pm 0.157]-7.212[\pm 0.248]P$ $\displaystyle+1.353[\pm
0.048]\phi_{31},\ \ \sigma=0.17.$

As Jurcsik & Hajdu (2023) adopted a slightly different Solar compositions, and
offset of $-0.04$ dex was added to $[\mathrm{Fe/H}]^{G}_{J23}$ to bring the
metallicity to the Carretta et al. (2009) scale. Besides these two $G$-band
relations, another relation is available from Li et al. (2023, hereafter L23),
which includes an additional $R_{21}=A_{2}/A_{1}$ Fourier parameter:

$\displaystyle[\mathrm{Fe/H}]^{G}_{L23}$ $\displaystyle=$
$\displaystyle-1.888[\pm 0.002]-5.772[\pm 0.026](P-0.6)$
$\displaystyle+1.090[\pm 0.005](\phi_{31}-2.0-\pi)$ $\displaystyle+1.065[\pm
0.030](R_{21}-0.45),\ \ \sigma=0.24.$

The $R_{21}$ values from the $G$-band light curve are $0.495\pm 0.022$ and
$0.423\pm 0.006$ for SEGUE II-V1 and UMaII-V1, respectively. Similar to the
I21 relation, a $\pi$ is needed to be subtracted from the $\phi_{31}$ value.
The above relation was derived from a sample of $\sim 2000$ RR Lyrae with
spectroscopic $[\mathrm{Fe/H}]$ measured from Liu et al. (2020). According to
Liu et al. (2020), the difference between their $[\mathrm{Fe/H}]$ and
published $[\mathrm{Fe/H}]$, in the Carretta et al. (2009) scale, is
negligible. Hence, we assume $[\mathrm{Fe/H}]^{G}_{L23}$ is in the Carretta et
al. (2009) scale.

Finally, the $I$-band $[\mathrm{Fe/H}]$-$\phi_{31}$-$P$ relations were
published in Smolec (2005, hereafter S05) and Dékány et al. (2021, hereafter
D21). S05 provides a two-parameter and a three-parameter relation. However,
Hajdu et al. (2018) demonstrated that the two-parameter relation suffers from
a systematic bias, therefore we adopted the three-parameter relation (which
also has a smaller dispersion):

$\displaystyle[\mathrm{Fe/H}]^{I}_{S05}$ $\displaystyle=$
$\displaystyle-6.125[\pm 0.832]-4.795[\pm 0.285]P+1.181$ $\displaystyle[\pm
0.113]\phi_{31}+7.876[\pm 1.776]A_{2},\ \sigma=0.14.$

The metallicity scale for S05 relation is in the Jurcsik (1995, J95) scale,
and we converted it to the Carretta et al. (2009) scale using
$[\mathrm{Fe/H}]_{C09}=1.001[\mathrm{Fe/H}]_{J95}-0.112$ (Kapakos et al.,
2011). Dékány et al. (2021) performed a feature selection and found that the
optimal relation could also be obtained with three parameters, and
subsequently applied a Bayesian linear regression on a sample of $\sim 80$ RR
Lyrae with good quality $I$-band light curves to derive the following
relation, which is similar to the S05 relation:

$\displaystyle[\mathrm{Fe/H}]^{I}_{D21}$ $\displaystyle=$
$\displaystyle-5.819[\pm 0.149]-6.350[\pm 0.067]P+1.248$ $\displaystyle[\pm
0.020]\phi_{31}+5.785[\pm 0.320]A_{2},\ \sigma=0.20.$

According to Dékány et al. (2021), the D21 metallicity scale is in, or has
been shifted to, the Crestani et al. (2021) scale. Hence, we applied the same
offset of $-0.08$ dex as in the case of M21 relation to bring the D21
metallicity scale to the Carretta et al. (2009) scale. For SEGUE II-V1 and
UMaII-V1, the $A_{2}$ parameters in the $I$-band are $0.069\pm 0.003$ and
$0.114\pm 0.003$, respectively.

Table 3: The determined $\phi_{31}$ Fourier parameters and photometric metallicities for SEGUE II-V1 and UMaII-V1. Filter / | SEGUE II-V1 | UMaII-V1
---|---|---
Relation | $\phi_{31}$ | $[\mathrm{Fe/H}]$ | $\phi_{31}$ | $[\mathrm{Fe/H}]$
$g$/N22 | $5.170\pm 0.057$ | $-2.40\pm 0.30$ | $4.703\pm 0.024$ | $-1.82\pm 0.27$
$V$/N22 | $5.491\pm 0.060$ | $-2.15\pm 0.29$ | $4.811\pm 0.028$ | $-1.75\pm 0.26$
$V$/M21 | $\cdots$ | $-2.25\pm 0.42$ | $\cdots$ | $-1.81\pm 0.41$
$G$/I21 | $5.615\pm 0.091$ | $-2.18\pm 0.33$ | $4.908\pm 0.030$ | $-1.68\pm 0.32$
$G$/J23 | $\cdots$ | $-2.17\pm 0.42$ | $\cdots$ | $-1.80\pm 0.36$
$G$/L23 | $\cdots$ | $-2.19\pm 0.26$ | $\cdots$ | $-1.97\pm 0.24$
$r$/N22 | $5.678\pm 0.056$ | $-2.39\pm 0.37$ | $4.949\pm 0.027$ | $-1.81\pm 0.31$
$I$/S05 | $6.212\pm 0.114$ | $-1.95\pm 1.13$ | $5.304\pm 0.056$ | $-1.79\pm 1.07$
$I$/D21 | $\cdots$ | $-2.51\pm 0.32$ | $\cdots$ | $-2.21\pm 0.29$

###  3.2 The Derived Metallicity

Figure 2: The derived photometric metallicities in various filters, as
summarized in Table 3, for SEGUE II-V1 (left panel) and UMaII-V1 (right
panel). The orange horizontal lines represent the weighted means based on the
nine adopted relations, and the orange shaded regions are the $\pm 1\sigma$
dispersion of the weighted means. Horizontal lines in green and magenta colors
are the published values, together with their $\pm 1\sigma$ dispersion shown
as dashed lines, based on spectroscopic observations of a number of bright
member stars in these two dwarf galaxies. For a better visualization, we only
plotted the values taken from two publications for these two RR Lyrae.

The derived multi-band photometric metallicities, using the nine relations
listed in the previous subsection, are listed in Table 3 and graphically
presented in Figure 2. As mentioned, all of these photometric metallicities
are in the Carretta et al. (2009) scale. For both RR Lyrae, all of the derived
photometric metallicities are mutually consistent with each others from the
nine relations adopted in the previous subsection. The errors on the derived
metallicities are the quadrature sum of the propagated errors of each
parameters (except we omitted the errors on period as they are negligible) and
the dispersion $\sigma$ for each relations. The large errors on the
$[\mathrm{Fe/H}]^{I}_{S05}$ values are mainly due to the large error on the
constant term, $\pm 0.832$, in the S05 relation. Errors from other relations
are around $\sim 0.3$ to $\sim 0.4$ dex.

We assume that the adopted nine relations in the previous subsection are
equally well calibrated, although each relation may have their own systematic
uncertainties (Jurcsik & Hajdu, 2023). The uncertainties on the coefficients
of these relations were propagated to the derived photometric metallicities.
Therefore, we took the weighted means for the photometric metallicities listed
in Table 3 giving smaller weights to more uncertain measurements. When taking
a weighted mean, we obtained $[\mathrm{Fe/H}]=-2.27\pm 0.11$ dex and $-1.87\pm
0.10$ dex for SEGUE II-V1 and UMaII-V1, respectively. Dispersions around these
mean values were $0.13$ dex and $0.16$ dex, respectively, and will be adopted
as the errors on these weighted means, which are $\sim 2$ to $\sim 3$ times
smaller that those listed in Table 3 (except for the S05 relation). We have
also tried to eliminate certain data points, such as excluding those based on
S05 relation or with the largest deviations, the resulted weight means do not
differ by more than $\pm 0.03$ dex from our preferred values. Small statistics
of two RR Lyrae precludes us from quantifing the optimal number of relations
or the preferred relation for accurate photometric metallicity determinations.

In case of SEGUE II-V1, our derived photometric metallicity is in good
agreement with the values determined in Belokurov et al. (2009) and Kirby et
al. (2013), as shown in the left panel of Figure 2, and almost identical to
the value recalculated by Boettcher et al. (2013, $-2.26\pm 0.14$ dex). For
UMaII-V1, our photometric metallicity is close to the value given by Simon &
Geha (2007), but more metal rich than those determined in Kirby et al. (2013)
or Willman & Strader (2012, $-2.36$ dex with $\sigma=0.56$). However, dwarf
galaxies are known to exhibit a wide spread in metallicity (for examples, see
Simon & Geha, 2007; Willman & Strader, 2012; Kirby et al., 2013, and reference
therein). The published metallicities are mostly measured from the
spectroscopic observations of red giant branch stars and/or blue horizontal
branch stars, and hence they may not necessarily represent the photometric
metallicities for RR Lyrae.

##  4 The Distances of the Two RR Lyrae

Table 4: Mean Magnitudes for SEGUE II-V1 and UMaII-V1.

RR Lyrae | $\langle V\rangle$ | $\langle I\rangle$ | $\langle g\rangle$ | $\langle r\rangle$
---|---|---|---|---
SEGUE II-V1 | 18.74 | 17.91 | 18.93 | 18.53
UMaII-V1 | 18.48 | 17.86 | 18.59 | 18.36

Note. — An error of $0.02$ mag is adopted on these mean magnitudes.

Since the Wesenheit magnitude, denoted as $W$, is reddening-free by
construction (see the Appendix in Madore & Freedman, 1991, 2023), we adopted
the recent period-Wesenheit-metallicity (PWZ) relations available in the
literature to derive the distances to these two RR Lyrae with our photometric
metallicities determined in the previous Section. For our $VIgr$-band light
curves, the number of data points per light curves is more than 100, hence
according to Ngeow et al. (2022) the errors on the mean magnitudes derived
from these light curves are negligible. Nevertheless, we adopted a
conservative error of $0.02$ mag on the intensity-averaged mean magnitudes
derived from the fitted Fourier-expansion (see Figure 1) on our multi-band
light curves. The $VIgr$-band mean magnitudes are listed in Table 4.

Neeley et al. (2019) calibrated multi-band PWZ relations based on the 55 Milky
Way RR Lyrae together with their Gaia DR2 parallaxes. According to Neeley et
al. (2019), metallicities for these 55 RR Lyrae are assumed to be in the Zinn
& West (1984) scale. Therefore, we converted our derived metallicities to the
Zinn & West (1984) scale by inverting the
$[\mathrm{Fe/H}]_{C09}$-$[\mathrm{Fe/H}]_{ZW}$ conversion. After applying the
$VI$-band PWZ relation from Neeley et al. (2019), we determined the $VI$-band
based distance moduli ($\mu_{VI}$) to SEGUE II-V1 and UMaII-V1 as
$\mu_{VI}\mathrm{(SEGUE\ II}$-$\mathrm{V1)}=17.67\pm 0.20$ mag and
$\mu_{VI}\mathrm{(UMaII}$-$\mathrm{V1)}=17.56\pm 0.19$ mag.

As an independent check, we have also obtained the distance moduli for the two
RR Lyrae using the $gr$-band PWZ relation as derived in Ngeow et al. (2022,
using the RR Lyrae in globular clusters), where the metallicity scale is in
the Dias et al. (2016, D16) scale. The conversion of such metallicity scale to
the Carretta et al. (2009) scale is given as
$[\mathrm{Fe/H}]_{C09}=0.99[\mathrm{Fe/H}]_{D16}-0.05$ (Dias et al., 2016).
After we inverted our derived metallicity to the Dias et al. (2016) scale, we
obtained $gr$-band based distance moduli ($\mu_{gr}$) as
$\mu_{gr}\mathrm{(SEGUE\ II}$-$\mathrm{V1)}=17.71\pm 0.24$ mag and
$\mu_{gr}\mathrm{(UMaII}$-$\mathrm{V1)}=17.61\pm 0.24$ mag. The derived
distance moduli based on the $VI$-band and the $gr$-band PWZ relations are
summarized in Table 5.

Table 5: Derived Distance Moduli for SEGUE II-V1 and UMaII-V1. $\mu$ (mag.) | SEGUE II-V1 | UMaII-V1
---|---|---
$\mu_{VI}$ | $17.67\pm 0.20$ | $17.56\pm 0.19$
$\mu_{gr}$ | $17.71\pm 0.24$ | $17.61\pm 0.24$
$\langle\mu\rangle$ | $\mathbf{17.69\pm 0.15}$ | $\mathbf{17.58\pm 0.15}$

It is worth to mention that the impact on the determined distance moduli is
negligible ($\pm 0.01$ mag at most) if we did not convert the metallicity to
the Zinn & West (1984) or Dias et al. (2016) scale. This is mainly due to the
small metallicity terms (0.13 and 0.05 dex/mag, respectively) in the adopted
PWZ relations. Since both of the $\mu_{VI}$ and $\mu_{gr}$ were determined
from independent PWZ relations and in different filters, the averages of
$\mu_{VI}$ and $\mu_{gr}$ were adopted as the final distance moduli for these
two RR Lyrae. For SEGUE II-V1, we obtained $\langle\mu\rangle=17.69\pm 0.15$
mag, which is almost identical to the value found in Belokurov et al. (2009,
$17.7\pm 0.1$ mag) and Boettcher et al. (2013, $D=34.4\pm 2.6$ kpc or
$\mu=17.68^{+0.16}_{-0.17}$ mag) measured from the blue horizontal-branch
stars. Similarly, we obtained $\langle\mu\rangle=17.58\pm 0.15$ mag for
UMaII-V1, and it is in good agreement with the estimated value given by Zucker
et al. (2006, $\sim 17.5\pm 0.3$ mag), based on the isochrone fitting, and is
almost the same as the one determined in Vivas et al. (2020, $17.60\pm 0.20$
mag) when including additional RR Lyrae for UMaII. We also found statistically
consistent distance measurements to these two dwarf galaxies if independent
theoretical PWZ relations from Marconi et al. (2015) and Marconi et al. (2022)
were adopted instead of the empirical relations.

##  5 Conclusions

In this work, we obtained homogeneous $grVI$-band light curves using LOT for
two RR Lyrae in SEGUE II and UMaII ultra-faint dwarf galaxies. Together with
their Gaia $G$-band light curves, we derived the photometric metallicities for
these two RR Lyrae using nine $[\mathrm{Fe/H}]$-$\phi_{31}$-$P$ relations
(some of them include an additional Fourier parameter) recently published in
the literature. Hence, the weighted means of the photometric metallicities for
SEGUE II-V1 and UMaII-V1 were found to be $[\mathrm{Fe/H}]=-2.27\pm 0.13$ dex
and $-1.87\pm 0.16$ dex, respectively. Using these photometric metallicities,
together with empirical PWZ relations, we determined the distance moduli to
these two RR Lyrae as $\mu\mathrm{(SEGUE\ II}$-$\mathrm{V1)}=17.69\pm 0.15$
mag and $\mu\mathrm{(UMaII}$-$\mathrm{V1)}=17.58\pm 0.15$ mag. These distance
and photometric metallicity measurements are in good agreement with previous
determinations in the literature.

As mentioned in the Introduction, the motivation of this work is to
demonstrate that a higher precision on photometric metallicity for RR Lyrae
can be acheived when averaging the values derived from multiple metallicity-
Fourier parameter(s)-period relations. This is indeed the case for the two RR
Lyrae, SEGUE II-V1 and UMaII-V1, investigated in this work. Our proposed
approach will be particularly important for various ongoing and upcoming
multi-band time-domain sky surveys providing well-sampled RR Lyrae light
curves in the Milky Way. This is especially true for LSST which will provide
unprecedentedly large samples of distant RR Lyrae beyond our Galaxy and
particularly in the LSST-discovered new dwarf galaxies. The spectroscopic
follow-up observations of these variables will not be feasible either because
they are too far/faint or due to lack of rather competitive observational
time. In this scenario, photometric metallities for RR Lyrae can be determined
using the proposed approach, since LSST will provide multi-filter light curves
of RR Lyrae variables. However, it is critically important to adopt a
homogeneous metallicity scale or homogenize metallicity measurements based on
different empirical relation for better constrain on the photometric
metallicities, as shown in detail in the manuscript. These photometric
metallicities can then be used to derive the RR Lyrae-based distances, as well
as to compare and/or constraint the inferred metallicity based on isochrone
fitting to the color-magnitude diagrams of the newly-discovered dwarf
galaxies. The large number of expected RR Lyrae in dwarf galaxies with LSST
will also allow to derive photometric metallicity maps and gain insight into
chemical and morphological structure of these stellar systems.

Figure 3: The $W1$-band light curves for SEGUE II-V1 and UMaII-V1 extracted
from the ALLWISE (WISE Team, 2019) database, which is composed of data
collected from the WISE (Wright et al., 2010) and the NEOWISE (Mainzer et al.,
2011) mission. The red triangles indicate data-points with “null” entries in
the error bar. Both light curves were folded with periods adopted from Gaia
DR3 (see Section 2).

Finally, it is worth to remind that the $gr$-band and the $VI$-band
metallicity-Fourier parameter(s)-period relations used in Section 3 are in the
Pan-STARRS1 and the Johnson-Kron-Cousins photometric systems, respectively.
Therefore, the $gri$-band light curves based on the LSST observations need be
photometrically transformed to the $grVI$-band, as well as the Gaia $G$-band,
by properly taking the color-terms into account (as discussed in Ngeow, 2022).
We expect such photometric transformations would be available after the first-
light of LSST, and hence it will be straight forward to apply our proposed
approach to the LSST light curve data for RR Lyrae.

We thank the useful discussions and comments from an anonymous referee to
improve the manuscript. CCN is thankful for funding from the National Science
and Technology Council (NSTC, Taiwan) under the contract
109-2112-M-008-014-MY3. AB acknowledges funding from the European Union’s
Horizon 2020 research and innovation program under the Marie Skłodowska-Curie
grant agreement No. 886298. This publication has made use of data collected at
Lulin Observatory, partly supported by NSTC grant 109-2112-M-008-001. This
research has made use of the SIMBAD database and the VizieR catalogue access
tool, operated at CDS, Strasbourg, France. This research made use of
Astropy,222http://www.astropy.org a community-developed core Python package
for Astronomy (Astropy Collaboration et al., 2013, 2018, 2022). This work has
made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. The Pan-STARRS1 Surveys
(PS1) and the PS1 public science archive have been made possible through
contributions by the Institute for Astronomy, the University of Hawaii, the
Pan-STARRS Project Office, the Max-Planck Society and its participating
institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max
Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins
University, Durham University, the University of Edinburgh, the Queen’s
University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las
Cumbres Observatory Global Telescope Network Incorporated, the National
Central University of Taiwan, the Space Telescope Science Institute, the
National Aeronautics and Space Administration under Grant No. NNX08AR22G
issued through the Planetary Science Division of the NASA Science Mission
Directorate, the National Science Foundation Grant No. AST-1238877, the
University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos
National Laboratory, and the Gordon and Betty Moore Foundation. This
publication makes use of data products from the Wide-field Infrared Survey
Explorer, which is a joint project of the University of California, Los
Angeles, and the Jet Propulsion Laboratory/California Institute of Technology,
and NEOWISE, which is a project of the Jet Propulsion Laboratory/California
Institute of Technology. WISE and NEOWISE are funded by the National
Aeronautics and Space Administration. LO:1m, Gaia, PS1

##  Appendix A Details on Photometric Calibration

Same as in Ngeow (2022), photometric calibrations on the LOT reduced images
were done using the Pan-STARRS1 (PS1) Data Release 1 photometric data
(Chambers et al., 2016; Flewelling et al., 2020) as reference catalogs. The
selection criteria of the PS1 reference stars are identical to Ngeow (2022,
whenever applicable), and will not be repeated here. In the $gr$-band, the
photometric calibration was performed via the following equations:

$\displaystyle g^{PS1}-g^{\mathrm{instr}}$ $\displaystyle=$ $\displaystyle
ZP_{g}+C_{g}(g^{PS1}-r^{PS1}),$ (A1) $\displaystyle
r^{PS1}-r^{\mathrm{instr}}$ $\displaystyle=$ $\displaystyle
ZP_{r}+C_{r}(g^{PS1}-r^{PS1}).$ (A2)

These equations also allow the $(g^{PS1}-r^{PS1})$ colors to be measured via

$\displaystyle(g^{PS1}-r^{PS1})$ $\displaystyle=$
$\displaystyle\frac{ZP_{g}-ZP_{r}+(g^{\mathrm{instr}}-r^{\mathrm{instr}})}{1-C_{g}+C_{r}}.$
(A3)

For the $V$-band, we first transformed the $gr$-band photometry in the PS1
reference stars catalog to the $V$-band using the transformation equation
provided by Tonry et al. (2012), followed by fitting the reference stars with
the following equation:

$\displaystyle V-v^{\mathrm{instr}}$ $\displaystyle=$ $\displaystyle
ZP_{V}+C_{V}(g^{PS1}-r^{PS1}).$ (A4)

Calibration for the $i$-band data required a further treatment, this is
because we would like to calibrate the $i$-band photometry to the Johnson-
Cousin $I$-band magnitudes. Combining
$i^{PS1}-i^{\mathrm{instr}}=ZP_{i}+C_{i}(g^{PS1}-r^{PS1})$ with the Tonry et
al. (2012) transformation, the $i$-band data can be calibrated using the
following equation:

$\displaystyle I-i^{\mathrm{instr}}$ $\displaystyle=$
$\displaystyle(ZP_{i}-0.367)+(C_{i}-0.149)\times(g^{PS1}-r^{PS1}).$ (A5)

In equations A1 to A5, $\\{g,r,v,i\\}^{\mathrm{instr}}$ are instrumental
magnitudes measured with $MAG\\_PSF$ implemented in the Source-Extractor
package (together with the PSF models derived from the PSFEx package, Bertin,
2011), while $g^{PS1}$ and $r^{PS1}$ are magnitudes taken from the PS1
reference catalog. After solving the coefficients $ZP$ and $C$ in these
equations, we then apply them to the RR Lyrae by first obtaining the
$(g^{PS1}-r^{PS1})$ colors using equation A3, and subsequently calibrate the
$grVI$-band photometry from the same set of $Vgri$ images.

##  Appendix B The Light-Curves in the WISE-Filter

Besides the $V$-band relation, Mullen et al. (2021) have also derived a
similar relation in the infrared WISE filter. However, the $W1$-band light
curves for both RR Lyrae, extracted from the ALLWISE database, do not show a
RR Lyrae-like light curve in the $W1$-band, rather the extracted light curves
are quite scatter with large error bars (see Figure 3). As a result, we
omitted the $W1$-band $[\mathrm{Fe/H}]$-$\phi_{31}$-$P$ relation in this work.

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