11institutetext: Department of Astrophysics/IMAPP, Radboud University, PO Box
9010, 6500 GL, The Netherlands
11email<EMAIL_ADDRESS>22institutetext: National Astronomical
Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588,
Japan33institutetext: Department of Physics and Astronomy, The University of
Calgary, 2500 University Drive NW, Calgary, Alberta, T2N1N4,
Canada44institutetext: School of Physics and Astronomy, Yunnan University,
Kunming 650500, China55institutetext: National Astronomical Observatories,
Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing
100101, China 66institutetext: School of Astronomy, University of Chinese
Academy of Sciences, Beijing 100049, China77institutetext: INAF Istituto di
Radioastronomia, via Gobetti 101, 40129 Bologna, Italy88institutetext: INAF –
Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze,
Italy99institutetext: Department of Physics, Stanford University, Stanford, CA
94305, USA1010institutetext: Kavli Institute for Particle Astrophysics &
Cosmology, P.O. Box 2450, Stanford University, Stanford, CA 94305,
USA1111institutetext: School of Natural Sciences, Private Bag 37, University
of Tasmania, Hobart, TAS, Australia1212institutetext: Dunlap Institute for
Astronomy & Astrophysics, University of Toronto, 50 St. George Street,
Toronto, ON M5S 3H4, Canada1313institutetext: David A. Dunlap Department of
Astronomy & Astrophysics, University of Toronto, 50 St. George Street,
Toronto, ON M5S 3H4, Canada1414institutetext: Division of Physical and
Biological Sciences, University of California Santa Cruz, 1156 High Street,
Santa Cruz, CA 95064, USA1515institutetext: Dominion Radio Astrophysical
Observatory, Herzberg Astronomy and Astrophysics Research Centre, National
Research Council Canada, PO Box 248, Penticton, B.C. Canada, V2A
6J91616institutetext: Department of Computer Science, Math, Physics, &
Statistics, The University of British Columbia, Okanagan Campus, Kelowna, BC
V1V 1V7, Canada1717institutetext: Research School of Astronomy and
Astrophysics, Australian National University, Canberra, ACT 2611,
Australia1818institutetext: Banting and KIPAC Fellowships: Kavli Institute for
Particle Astrophysics & Cosmology (KIPAC), Stanford University, Stanford, CA
94305, USA1919institutetext: Skaha Remote Sensing Ltd., 3165 Juniper Drive,
Naramata, British Columbia V0H 1N0, Canada

# Faraday moments of the Southern Twenty-centimeter All-sky Polarization
Survey (STAPS)

N. Raycheva 11 M. Haverkorn 11 S. Ideguchi 22 J. M. Stil 33 X. Sun 44 J. L.
Han 5566 E. Carretti 77 X. Y. Gao 55 A. Bracco 88 S. E. Clark 991010 J. M.
Dickey 1111 B. M. Gaensler 121213131414 A. Hill 15151616 T. Landecker 1515 A.
Ordog 15151616 A. Seta 1717 M. Tahani 1818 M. Wolleben 1919

(Received 2023; accepted 2024)

###### Abstract

Context. Faraday tomography of broadband radio polarization surveys enables us
to study magnetic fields and their interaction with the interstellar medium
(ISM). Such surveys include the Global Magneto-Ionic Medium Survey (GMIMS),
which covers the northern and southern hemispheres at $\sim$ 300–1800 MHz.

Aims. In this work, we used the GMIMS High Band South (1328–1768 MHz), also
named the Southern Twenty-centimeter All-sky Polarization Survey (STAPS),
which observes the southern sky at a resolution of 18$\arcmin$.

Methods. To extract the key parameters of the magnetized ISM from STAPS, we
computed the Faraday moments of the tomographic data cubes. These moments
include the total polarized intensity, the mean Faraday depth weighted by the
polarized intensity, the weighted dispersion of the Faraday spectrum, and its
skewness. We compared the Faraday moments to those calculated over the same
frequency range in the northern sky (using the Dominion Radio Astrophysical
Observatory, DRAO), in a strip of $360\degr\times 30\degr$ that overlaps with
STAPS coverage.

Results. We find that the total polarized intensity is generally dominated by
diffuse emission that decreases at longitudes of $l\leq 300\degr$. The Faraday
moments reveal a variety of polarization structures. Low-intensity regions at
high latitudes usually have a single Faraday depth component. Due to its
insufficiently large frequency coverage, STAPS cannot detect Faraday thick
structures. Comparing the Faraday depths from STAPS to total rotation measures
from extragalactic sources suggests that STAPS frequencies are high enough
that the intervening ISM causes depolarization to background emission at
intermediate and high Galactic latitudes. Where they overlap, the STAPS and
DRAO surveys exhibit broad correspondence but differ in polarized intensity by
a factor of $\sim$1.8.

###### Key Words.:

ISM: magnetic fields – polarization – techniques: polarimetric – turbulence

##  1 Introduction

Knowledge of Galactic magnetism is crucial to interpreting many astrophysical
processes observed in the Galaxy, such as star formation (e.g., Li et al.
2009; Wurster & Li 2018), cosmic ray propagation (e.g., Minter & Spangler
1996; Giacalone 2017), and cloud formation (e.g., Tahani et al. 2022). The
interpretation of diffuse polarized synchrotron emission in the Galaxy is
aided by broad wavelength ranges, allowing us to use the Faraday tomography
technique (Burn, 1966; Brentjens & de Bruyn, 2005; Brentjens, 2011), which can
disentangle the polarized synchrotron emission through the frequency-dependent
Faraday rotation it experiences along the line of sight (LoS), caused by
magnetic fields.

The change in orientation of polarization angle ($\Delta\chi$) at a certain
wavelength ($\lambda$) through a magneto-ionic medium in an LoS between a
source and an observer due to Faraday rotation is an efficient tool with which
to study the properties of the Galactic magnetic fields at radio frequencies
(e.g., Burn 1966; Brentjens & de Bruyn 2005). The observed polarization angle
is defined with Stokes parameters $Q$ and $U$ as (IAU convention)

$\chi=\frac{1}{2}\,\rm{arctan}\left(\frac{\rm U}{\rm Q}\right){\rm(rad)}.$ (1)

If a signal is Faraday-rotated, its change in polarization angle is

$\Delta\chi=0.812\lambda^{2}\int_{d}^{0}n_{e}(r)B_{\parallel}(r)dr=\lambda^{2}RM,$
(2)

where $n_{e}$ is the free thermal electron density (cm-3), $B_{\parallel}$ is
the magnetic field component parallel to the LoS ($\mu$G), and the bounds of
the integration cover the entire LoS (pc) from the source at a distance, $d$
(pc), to the observer. Here, $r$ is a location along the path length and $dr$
is an incremental displacement between a source and an observer along the LoS.
If the polarization angle varies linearly with $\lambda^{2}$, the rotation
measure (RM) can be calculated as

$RM\equiv\frac{d\chi(\lambda^{2})}{d\lambda^{2}}\leavevmode\nobreak\
\leavevmode\nobreak\ \rm(rad\leavevmode\nobreak\ m^{-2}),$ (3)

where the sign of RM provides the direction of $B_{\parallel}$; that is, a
negative [positive] sign of RM signifies a $B_{\parallel}$ pointing away from
[toward] the observer.

If linearly polarized emission and Faraday rotation are mixed along an LoS,
Faraday rotation is described not by a single number, RM, but as a distance-
dependent Faraday depth ($\phi$),

$\phi(d)\ =\ 0.812\intop_{d}^{0}n_{e}(r)B_{\parallel}(r)dr\leavevmode\nobreak\
\leavevmode\nobreak\ \rm(rad\leavevmode\nobreak\ m^{-2}).$ (4)

The difference between this equation and Eq. 2 is that the distance, $d$, to
the emitting volume is variable, resulting in a spectrum of Faraday depths
rather than a single RM value.

The RM-synthesis algorithm is a technique utilized to analyze and extract
information about the observed polarized intensity (PI),
$(P(\phi)=\sqrt{Q(\phi)^{2}+U(\phi)^{2}})$, by reconstructing the distribution
of polarized emission as a function of $\phi$ (Burn, 1966; Brentjens & de
Bruyn, 2005). The output Faraday spectrum is deconvolved to recover the
underlying polarized emission profile and separate the contributions from
different Faraday depths. The complex mixture of Faraday rotations can be
disentangled using this process. $RM$-synthesis computes a unique sampling
function representing the finite frequency coverage, called the RM spread
function (R($\phi$), RMSF), for each image pixel in a data cube, which
characterizes the resolution of the observation in $\phi$. The RMSF is
convolved with the true Faraday spectrum. It can be deconvolved, using
algorithms like RM-CLEAN for partial correction of distortion from the dirty
beam (see Heald, 2009; Kumazaki et al., 2014, for details). We used RM
synthesis to perform Faraday tomography, which allows us to examine the
complex distributions of magnetized and ionized gas in the Faraday spectrum
along each LoS.

As was described by Brentjens & de Bruyn (2005) in equations 61–63,
$RM$-synthesis has limitations. The full width at half maximum of the main
lobe of R($\phi$), the Faraday resolution, is (Dickey et al., 2019)

$\delta\phi=\frac{3.79}{\Delta\lambda^{2}},$ (5)

where $\Delta(\lambda^{2})$ is the squared wavelength range of the observing
bandwidth. The maximum Faraday depth that can be detected is given by

$\phi_{max}\ =\ \frac{1.9}{\delta\lambda^{2}},$ (6)

where $\delta\lambda^{2}$ is the squared wavelength range of each channel.
Additionally, the broadest feature that can be detected is given by the
squared minimum wavelength ($\lambda_{min}^{2}$) as

$\phi_{max-scale}\ \simeq\ \frac{\pi}{\lambda_{min}^{2}}.$ (7)

We can see two observational features within Faraday spectra: Faraday
simplicity and Faraday complexity (Brentjens & de Bruyn, 2005). A Faraday
simple spectrum contains one component, indicating a synchrotron emitting
source with Faraday rotation (by a ‘Faraday screen’) between the source and
observer (Brentjens & de Bruyn, 2005). A Faraday complex spectrum occurs when
polarized emission and Faraday rotation are mixed along the LoS (or within the
observation beam). In terms of observations, a feature is Faraday thick if it
is broad compared to the Faraday resolution, and the ability to detect a
Faraday thick structure depends on the observing wavelength, which is
specifically limited by Eq. (7). These sources extend in the Faraday spectrum
over a width, $\Delta\phi$, which occurs when $\lambda^{2}\Delta\phi\gg 1$. As
is seen in the wavelength dependence of the thickness of a Faraday spectrum,
the higher the frequency, the broader the minimum observable width of the
Faraday spectrum.

Visualization and interpretation of data cubes produced by Faraday Tomography
can be enhanced by moment mapping (Dickey et al., 2019). This involves the
integration of various quantities (weighted by the PI) along the Faraday depth
axis, which produces a scalar quantity per spatial pixel known as a
statistical Faraday moment. The analysis of statistical moment maps simplifies
the understanding of some main characteristics of the Faraday cubes and, by
extension, interstellar magnetic fields.

An observational project in the last two decades, known as the Global Magneto-
Ionic Medium Survey (GMIMS), has given us a better understanding of the role
of magnetic fields in the interstellar medium (ISM). It measures diffuse
polarized emission from approximately 300 MHz to 1800 MHz across the entire
sky, probing both the southern (Wolleben et al., 2009, 2019) and northern
(Wolleben et al., 2021) hemispheres. Many studies have already drawn attention
to the technique of moment mapping using radio polarization surveys. Hill
(2018) modeled a dense ionized ISM component embedded in warm ionized diffuse
gas and calculated Faraday spectra and moments at various GMIMS bands. He
found that depolarization at larger distances was lower than expected, and
observed that Faraday depths could even be randomized. Thomson et al. (2018)
studied the $B_{\parallel}$ of the Galactic supershell GSH 006$-$15$+$7 using
the S-band Polarization All Sky Survey (S-PASS) and generated maps of the
first moment for the parameters of the Faraday rotation model and Faraday
thickness. Wolleben et al. (2019) mapped the zeroth and first moments of the
southern sky at 300$-$480 MHz with the 64m Parkes telescope Murriyang as part
of the GMIMS Low-Band survey. Dickey et al. (2019) presented a detailed
comparison between the moments of this survey and the GMIMS-North High-band
survey, taken with the 26m Dominion Radio Astrophysical Observatory (DRAO)
telescope at 1270$-$1750 MHz. Thomson et al. (2021) used the same data as
Dickey et al. (2019) to study the Faraday moments of the G150$-$50 region in
detail. Inspecting its magnetic field properties, they found that this region
is related to Radio Loop II. For a statistical examination in multiphase ISM,
Bracco et al. (2022) computed moment maps on the Faraday cubes of mock
observations of colliding supershells generated by stellar feedback with
frequency ranges of 115–170 MHz adapted for LOFAR observations (see references
therein). They found that radio sky observations at low frequencies require
multiphase modeling approaches to magnetohydrodynamic processes. Using moment
maps from the LOFAR Two-meter Sky Survey, Erceg et al. (2022) found a
correlation with the extragalactic RM of Hutschenreuter et al. (2022) and with
the DRAO GMIMS at 1.4 GHz (Dickey et al., 2019) in the Loop III region.
Šnidarić et al. (2023) also used the moments of the LOFAR data to inspect the
ELAIS-N1 field and found a connection to the polarized emission from Loop III.

This paper studies Faraday depth computed from GMIMS High Band South (GHBS),
also known as the Southern Twenty-centimetre All-Sky Survey (STAPS). The
survey covers 1328$-$1768 MHz. For data verification, we compare it with
moment maps from GMIMS High Band North (GHBN, Wolleben et al. 2021) in the sky
common to both surveys. Here, we only focused on the survey data in Faraday
depth space, and Sun et al. (in prep.) present the general survey details and
overview.

This study is organized as follows. We present the details of the data used in
Sect. 2. The techniques to calculate the moment maps from the Faraday cubes of
STAPS are explained in Sect. 3. We present the Faraday moments of STAPS in
Sect. 4. We give our data verification results in Sect. 5. We discuss our
findings in Sect. 6. Finally, Sect. 7 presents the summary and conclusions.

Table 1: Survey properties of STAPS and GHBN. | STAPS | GHBN
---|---|---
Declination range | $\delta<0\degr$ | $-30\degr<\delta<+87\degr$
Angular resolution | 18$\arcmin$ | 40$\arcmin$
Frequency range | 1328 $-$ 1768 MHz | 1280 $-$ 1750 MHz
Frequency resolution | 1 MHz | 1 MHz
$\lambda^{2}$ range | 0.029 m2 $-$ 0.051 m2 | 0.029 m2 $-$ 0.055 m2
$\Delta\lambda^{2}$ | 0.022 m2 | 0.026 m2
$\delta\lambda^{2}$ | (3.2 $-$ 7.7) $\times$ 10-5 m2 | (3.3 $-$ 8.6) $\times$ 10-5 m2
RM resolution ($\delta\phi$) | 170 rad m-2 | 148 rad m-2
RM range ($\phi_{max}$) | (2.2 $-$ 5.3) $\times$ 104 rad m-2 | (2.1 $-$ 5.2) $\times$ 104 rad m-2
RM feature width ($\phi_{max-scale}$) | 1.1 $\times$ 102 rad m-2 | 1.1 $\times$ 102 rad m-2
Cleaned $\phi$ spectral range (for the entire survey) |  $\pm$750 rad m-2 | $-$
Cleaned $\phi$ spectral range (for the overlap region) |  $\pm$500 rad m-2 |  $\pm$500 rad m-2
$\phi$ sampling (for the entire survey) | 5 rad m-2 | $-$
$\phi$ sampling (for the overlap region) | 1 rad m-2 | 1 rad m-2

##  2 Data

###  2.1 Southern Twenty-centimetre All-Sky Survey

The data presented here are part of the GMIMS project (Wolleben et al., 2009).
This survey was started in 2008 as an independent project named STAPS and
observed mostly piggybacking on its sister survey, S-PASS (Carretti et al.,
2019), which covered the frequency range of 2176$-$2400 MHz. Later, STAPS was
incorporated into GMIMS as the GMIMS High Band South survey.

The STAPS observations were made with the 64m Parkes telescope Murriyang using
an H-OH receiver. The S-PASS Galileo receiver was placed in prime focus, and
the H-OH receiver was located off-axis next to the Galileo receiver, displaced
630 mm from the prime focus horizontally, and 7.6 mm vertically to keep the
feed in focus. In this way, STAPS was able to piggyback on the S-PASS survey
without requiring additional observing time, with some penalty in the form of
beam distortions (see below). The complete survey and data products will be
presented by Sun et al. (in prep).

The Southern Twenty-centimetre All-Sky Survey covers the frequency range of
1328$-$1768 MHz with a frequency resolution of 1 MHz, and the whole sky south
of a declination of $0{\degr}$. The data reduction and map-making process were
analogous to the data processing and mapping of the S-PASS survey (Carretti et
al., 2019): the scanning was done using fast azimuth scans at a constant
elevation of $EL=33{\degr}$, the elevation of the South Celestial Pole as seen
from Parkes. Scanning at this elevation allows data to be gathered as far
south in declination as $-90\degr$. The northern limit was set by choosing the
azimuth coverage of the scans. The advantage of scanning at a constant
elevation is that ground emission is constant. The scans were as long as
possible ($\sim$120${\degr}$) to reduce the chances of missing large-scale
structures and to perform a polarization calibration using the parallactic
angle modulation. The complete southern sky was scanned both toward the east
and the west, after which the east and west scans were basket-woven into a sky
map. This technique was developed for S-PASS (see Carretti et al. 2019 for
details).

The offset position of the feed led to a coma lobe, and a concomitant
broadening of the main beam, together with a decrease in the forward gain of
the telescope. The peak strength of the coma lobe is about 6% to 9% of the
total intensity from the highest to the lowest frequency. As the angular
location of the coma lobe varies per observing beam, its effect on the diffuse
polarized emission is thought to be between 2% and 4% off-axis, increasing
with frequency (Sun et al., in prep). The STAPS total-intensity scale is tied
to the GMIMS HBN total-intensity scale, which is absolutely calibrated. In
STAPS, the PI scale is tied to the total-intensity scale, as is detailed in
the survey paper (Sun et al., in prep). Here, the polarization data were
convolved to a common resolution of 18$\arcmin$, assuming a Gaussian observing
beam. The maps in Jy beam-1 were converted into K using the conversion factor
of $0.537\times(\nu_{0}/\nu)^{2}$ K/Jy, where $\nu$ is any frequency in the
coverage, and $\nu_{0}$ is 1400 MHz. This conversion ignores the partial
redistribution of intensity outside the main beam (predominantly into the coma
lobe), which overestimates the intensity scale in K. Observational details
that are relevant to this paper and parameters of the Faraday cubes are given
in Table 1.

###  2.2 The Global Magneto-Ionic Medium Survey High Band North

The GHBN survey (Wolleben et al., 2021) in the northern sky covers
approximately the same frequency range as STAPS, with a resolution of
40$\arcmin$. As GHBN extends down to declinations of $-30{\degr}$, there is an
overlapping sky region of $\sim$30${\degr}$ in declination, which is observed
by both the northern and the southern surveys. The Faraday cubes and Faraday
moments have been analyzed by Dickey et al. (2019), Ordog (2020) and Dickey et
al. (2022). In this paper, we compare the Faraday cubes and moments of the
GHBN survey with those of the GHBS. (see Table 1 for the observational
parameters of the GHBN survey).

##  3 Producing Faraday spectra

We now describe the computations of RM synthesis, noise map, and Faraday
moments, including any preprocessing steps undertaken for analysis.

###  3.1 Rotation measure synthesis

We used the RM-Tools package111https://github.com/CIRADA-Tools/RM-Tools,
v1.1.1 from the Canadian Initiative for Radio Astronomy Data Analysis (CIRADA)
(Purcell et al., 2020) to perform RM synthesis (Brentjens & de Bruyn, 2005)
and RM-CLEAN (Heald, 2009) to calculate the Faraday depth spectra using Stokes
$Q$ and $U$ data cubes. The resulting Faraday depth cubes cover the Faraday
depth range of $\pm$1000 rad m-2 in steps of 5 rad m-2.

###  3.2 Estimation of the RM-CLEAN threshold

As the reliability of detecting the Faraday complexity of the spectrum depends
on the choice of RM-CLEAN threshold (Anderson et al., 2015), this threshold
needs to be carefully estimated. We chose an RM-CLEAN threshold based on the
noise per pixel calculation, as is explained below.

The RM-CLEAN threshold (intensity threshold) has been chosen in various ways
in earlier GMIMS surveys. Wolleben et al. (2019) used 0.06 K as their RM-CLEAN
threshold, which is the survey rms. Ordog (2020) used the Faraday spectra
themselves to estimate the RM-CLEAN threshold. She calculated noise ($\sigma$)
as the mean polarized emission at a Faraday depth of $|\phi|\in[500,1000]$ rad
m-2, where no real signals are expected, to arrive at a $5\sigma$ threshold
level of 0.03 K. As the emission varies greatly across the map, we decided to
evaluate a pixel-by-pixel RM-CLEAN threshold instead.

Assuming that the noise in Stokes $Q$ and $U$ is Gaussian, the noise in the PI
distribution as a function of Faraday depth, for each pixel, should follow a
Rayleigh distribution (Hales, 2012; Van Eck et al., 2018), which is a special
case of the Rice distribution (Rice, 1945):

$R(x;\sigma)=\frac{x}{\sigma^{2}}e^{-x^{2}/2\sigma^{2}},$ (8)

where $0\leq x<\infty$, and $\sigma>0$ defines the noise limit. Therefore, for
every pixel we fitted the distribution of PI over the Faraday depth with a
Rayleigh distribution, using only Faraday depths where we expected no signal
and only noise. We fitted Rayleigh distributions to each pixel in the range of
$|\phi|\in[500,1000]$ rad m-2 and also in the range of $|\phi|\in[750,1000]$
rad m-2 and mapped the corresponding $\sigma$ values. The map calculated in
the range of $|\phi|\in[500,1000]$ rad m-2 gave clearly higher $\sigma$ levels
than the one calculated in $|\phi|\in[750,1000]$ rad m-2. We also performed a
Kolmogorov-Smirnov goodness-of-fit test per pixel on each map. The results
show a better fit between the data and the Rayleigh distribution in the range
of $|\phi|\in[750,1000]$ rad m-2 than those in the range of
$|\phi|\in[500,1000]$ rad m-2. These indicate that the latter map contains
more signal (likely in side lobes that were not cleaned perfectly) than the
former. As a result, we included the range of $|\phi|\in[750,1000]$ rad m-2 in
our estimation of $\sigma$.

The resulting map of $\sigma$ is shown in Fig. 1. It shows the 1$\sigma$
intensity threshold for each pixel. As is seen in this map, it contains
features of the PI map (Sun et al., in prep), which means that in addition to
instrumental noise, there are low-level side lobes of signals at lower
absolute Faraday depths. We used this map to estimate the RM-CLEAN threshold.
Since the level of side lobes in the range of $[750,1000]$ rad m-2 is up to
10% of the peak intensity ($P_{peak}$, the Faraday depth in steps of 5 rad m-2
that has the highest $P$ in the spectrum) in Fig. 3, the noise is seen to be
correlated with the peak intensity. We did not use a Ricean polarization bias
correction to reduce the noise contributions to a polarized brightness. This
is because we applied a 6$\sigma$ intensity threshold for moment map
computations, at which debiasing is negligible (Wardle & Kronberg, 1974;
George, Stil, & Keller, 2012).

Figure 1: $\sigma$ map of STAPS showing the estimated noise level for each
pixel in Galactic coordinates, shown in the Mollweide projection with the
Galactic center at the center, and calculated in the range of
$|\phi|\in[750,1000]$ rad m-2 using the technique described in Sect. 3.2. The
color bar is saturated to 0.01 K and the maximum is 0.2 K. Figure 2: Faraday
spectra of four sample pixels in STAPS. The orange line represents the
3$\sigma$ RM-CLEAN cutoff limit. The purple line shows the 6$\sigma$ intensity
limit applied during Faraday moment calculations. The gray line represents the
dirty Faraday depth spectrum. The black line presents the RM-CLEANed Faraday
depth spectrum, and the RM-CLEAN model components (CC) are shown in green. The
Galactic coordinates of the pixels from which the spectra were taken are shown
in the top left of the panels. The legends in the top right show the Faraday
moment results for the related pixels. We note the different scales on the $y$
axis in each plot.

###  3.3 RM-CLEAN

In the RM-CLEAN process, we adopted a loop gain of 0.1, 1000 iterations, and a
3$\sigma$ RM-CLEAN threshold per pixel, using the $\sigma$ map. Fig. 2
displays the dirty and clean Faraday spectra of four pixels chosen from
representative regions, selected to show different spectra characteristics in
the survey. The first spectrum in the upper panel consists of a single peak
with one RM-CLEAN model component. The second panel’s complex spectrum
contains two low-intensity, unresolved peaks. The third panel has a resolved
spectrum, broader than the RMSF, with a tail toward negative $\phi$ and with
many RM-CLEAN model components. The fourth panel displays a double-peaked
spectrum in the direction of the H ii region Sh 2$-$27 (Sharpless, 1959) at
$(l,b)\sim(5\aas@@fstack{\circ}7,22\degr)$. The figure shows that side lobes
are significantly suppressed by the RM-CLEAN process. The maximum width of
Faraday depth features, $\phi_{max-scale}$, in STAPS is smaller than the
Faraday depth resolution, $\delta\phi$, as is the case with GHBN, as Wolleben
et al. (2010) noted. Therefore, Faraday thick structures will not be
detectable. A Faraday thick structure with sharp edges may show up as two
seemingly unrelated features at the edges, such as the fourth panel of Fig. 2.
However, these features could also be two unrelated and unresolved emission
features. Depending on the shape of the feature’s edge, a Faraday-thick
structure can also look like a slightly resolved component, as in the third
panel.

###  3.4 Faraday moments

We computed four moments of the RM-CLEANed image cubes by collapsing the
Faraday depth cube into a Faraday moment image (Dickey et al. 2019) using a
cutoff, which is discussed in Sect. 3.5: the total polarized brightness
integrated over the Faraday spectrum (the zeroth moment, $M_{0}$); the mean
Faraday depth weighted by the PI (the first moment, $M_{1}$); the standard
deviation or width of the Faraday spectra weighted by the PI (the second
moment, $M_{2}$); and the skewness (shape asymmetry) of Faraday spectra
weighted by the PI (the third moment, $M_{3}$). For the discrete distribution
of $P(\phi)$, these maps are respectively defined as (Dickey et al., 2019)

$M_{0}\ =\ \Delta\phi\sum_{i=1}^{n}P_{i}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \rm(K\leavevmode\nobreak\
rad\leavevmode\nobreak\ m^{-2}\leavevmode\nobreak\ RMSF^{-1}),$ (9)

$M_{1}\ =\
\frac{\Delta\phi\sum_{i=1}^{n}P_{i}\phi_{i}}{M_{0}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \rm(rad\leavevmode\nobreak\
m^{-2}),$ (10)

$M_{2}\ =\
\frac{\Delta\phi\sum_{i=1}^{n}P_{i}\left(\phi_{i}-M_{1}\right)^{2}}{M_{0}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \rm(rad\leavevmode\nobreak\
m^{-2})^{2},$ (11)

and

$M_{3}\ =\
\frac{\Delta\phi\sum_{i=1}^{n}P_{i}\left(\phi_{i}-M_{1}\right)^{3}}{M_{0}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \rm(rad\leavevmode\nobreak\
m^{-2})^{3},$ (12)

where $n$ is the number of $\phi$ values (the length of the $z$ axis in the
image cube), $\phi_{i}$ is $\phi$ of the $i$th Faraday depth component, and
$\Delta\phi$ is the step size in the $\phi$ interval
$(|\phi_{i}-\phi_{i+1}|)$. $M_{2}$ shows the spread of the PI from $M_{1}$,
and, following Dickey et al. (2019), we also use its square root-normalized
version ($m_{2}=\sqrt{M_{2}}$). Here, $M_{3}$ demonstrates a Faraday
spectrum’s asymmetry about its mean, where $M_{3}=0$ for a symmetrical
$|F|(\phi_{i})$ distribution. Again, as in $m_{2}$, we used its root-
normalized version ($m_{3}=\sqrt[3]{M_{3}}$). $m_{3}>0$ indicates positive
[right] skewness, while $m_{3}<0$ indicates negative [left] skewness.
Throughout this paper, all plots and explanations involving second and third
moments use $m_{2}$ and $m_{3}$, respectively, which have units of rad m-2.

For a spectrum with a single Faraday-thin peak, $M_{1}$ is $\phi$, at which
this peak polarization happens ($M_{1}=\phi_{peak}$, Fig. 4), $m_{2}$ is the
width of the peak, and $m_{3}$ is zero. At STAPS frequencies, this is a
possible scenario for high Galactic latitudes. However, $M_{1}\neq\phi_{peak}$
and typically $m_{2}$ have high values for more complex spectra with multiple
emission components.

###  3.5 Faraday depth cutoff in moment maps

A cutoff in the Faraday depth range needs to be applied, to exclude Faraday
depths dominated by noise as much as possible in the Faraday moment
calculations. This cutoff in the Faraday depth range is commonly defined as
the Faraday depth range within which a pixel has an intensity above a certain
level, with additional criteria to consider Faraday complexity.

As Dickey et al. (2019) mentioned, at high $\pm\phi$, noise or spurious
features in the $P$ distribution strongly affect $M_{1}$ and $m_{2}$, which
becomes more drastic in higher moments. Thus, they restricted the range of
$\phi$ channels according to some Faraday moment map calculation criteria.
They used the following Faraday depth threshold methods for GHBN and GMIMS Low
Band South (GLBS) Faraday spectra. It was calculated that each pixel should be
within the intensity threshold of 5$\sigma$ (0.04 K), or 15% of the Faraday
spectrum peak, whichever is the largest, and only Faraday depth channels
meeting these criteria were included. However, as they found second peaks in
GLBS, they added another criterion to thresholding by fitting a Gaussian to
the highest peak in the concerned spectrum and subtracting it from the entire
data of that spectrum to find the second peak and again only included the
Faraday depths that meet this criterion. Studying GHBN, Ordog (2020) and
Wolleben et al. (2021) based their criteria for thresholding on Dickey et al.
(2019), although they did not apply any Gaussian fitting to locate multiple
peaks. Additionally, they ensured that the artificial features found in
Faraday spectra as peaks above their 5$\sigma$ RM-CLEAN threshold (0.03 K)
would be eliminated by adding a 33% margin to the RM-CLEAN threshold (0.04 K).

In some Faraday spectra of STAPS, there are multiple peaks (more than two)
well above the 3$\sigma$ level per pixel after the RM-CLEAN process.
Consequently, in our situation, the methods described above could cause an
underestimation of the number of Faraday depth peaks, leading to the missing
complexity of spectra. Hence, we did not directly utilize the exact
thresholding done by Dickey et al. (2019) and Ordog (2020), but adapted them
to our analysis: we used a 6$\sigma$ threshold to each pixel’s intensity to
minimize any possible noise peaks that can occur in the case of a polarized
peak that is surrounded by high noise levels. Eventually, to compute the
Faraday moments, we only incorporated the Faraday depths in the intensity
peak(s) above the 6$\sigma$ threshold and masked the Faraday spectra below
this threshold. During the moment map computations, we included Faraday depth
channels of $\pm$500 rad m-2 for the overlap region, as GHBN is only reliable
in this Faraday depth span (as is explained in Sect. 5.1), and $\pm$750 rad
m-2 for STAPS, high-resolution data.

##  4 Faraday moment maps of the Southern Twenty-centimetre All-Sky Survey

The Faraday moment maps of STAPS are presented in Fig. 5. In the following
paragraphs, we describe each moment map.

Figure 3: $P_{peak}$ map of STAPS shown in Mollweide projection. The white
areas are either outside the survey coverage, or masked pixels where S/N$<$6\.
The color bar is saturated for presentation clarity. Figure 4: Map of the
STAPS peak Faraday depth, $\phi_{peak}$, as fitted from the Faraday spectrum.
The figure parameters are the same as in Fig. 3. Figure 5: Faraday moment maps
of STAPS shown in the Mollweide projection. (a) $M_{0}$; (b) $M_{1}$; (c)
$m_{2\leavevmode\nobreak\ }$($=\sqrt{M_{2}}$); (d) $m_{3}$
($=\sqrt[3]{M_{3}}$). The color bars are saturated to reveal the finer
details. All the figure parameters are the same as in Fig. 3. The figure
continues on the next page.

Continued.

In the $M_{0}$ map (Fig. 5a), there is a sharp decrease in high and/or low
total polarized brightness regions: from high $M_{0}$ values (left-hand side)
to low $M_{0}$ values at $l<315\degr$ (right-hand side). The high PI in the
inner Galactic disk is due to instrumental polarization. Even though
instrumental polarization is low (see Sect. 2), it will be significant at the
location of the very high-intensity synchrotron sources in the inner Galactic
disk. Moreover, we can see the Galactic center spur (Sofue, Reich, & Reich,
1989), and the northern and southern ridges identified in S-PASS (Carretti et
al., 2013). Other than these, the high $M_{0}$ (30 K rad m-2) around the
supernova remnant candidate G353$-$34 (Testori, Reich, & Reich, 2008) at
$(l,b)\sim(353\degr,-34\degr)$ is quite prominent.

The $M_{1}$ map is shown in Fig. 5(b). In the case of a unimodal unresolved
Faraday spectrum, $M_{1}$ equals $\phi_{peak}$. However, the Faraday rotation
sky observed with STAPS is often more complex than a single peak, except at
high Galactic latitudes. In this map, the sky separates into positive and
negative patches of RM over large areas, but individual structures also stand
out with their $M_{1}$ values (e.g., Sh 2$-$27, G353$-$34).

The $m_{2}$ map is shown in Fig. 5(c). It shows the width of the Faraday
spectrum peaks. High values indicate a wide distribution of polarized
brightness and the possibility that multiple peaks may be present in Faraday
spectra; that is, Faraday complexity (Alger et al., 2021). $m_{2}$ is the
smallest in an unresolved spectrum. The FWHM of the RMSF of 170 rad m-2
corresponds to $m_{2}$ for a Gaussian unresolved spectrum of
$\sigma=\mbox{FWHM}/2\sqrt{2\ln 2}\approx 72$ rad m-2. We note that for some
LoSs (shown on a gray color scale) the corresponding spectra are slightly
narrower than the RMSF due to inaccuracies in the fitting.

Many diffuse structures stand out due to their high $m_{2}$ values compared to
their surroundings in the $m_{2}$ map. This indicates spectral complexity,
likely multiple Faraday depth components (see Sect. 6.1), such as toward the
supernova remnant candidate G353$-$34.

The $m_{3}$ map in Fig. 5(d) shows the skewness of the spectrum. Two spectra
that have the same $M_{1}$ and $m_{2}$ can still have different $m_{3}$,
reflecting variations in their tail behaviour and overall shape. We also see
certain regions separated by their skewness, such as Sh 2$-$27 with positive
[right] skewness, while the higher-latitude sky contains large patches of
either positive or negative skewness. The interpretation of these maps is
presented in Sect. 6.

##  5 Data verification

Since the GHBN survey has a similar frequency coverage and partially
overlapping declination range, we can compare the two surveys in the overlap
region. Sun et al. (in prep) compare the surveys for the polarized emission in
frequency space, and we compare the surveys in Faraday space in this section.

Since the two surveys cover different hemispheres, the overlap region is
necessarily at the edge of both surveys. In particular, for the GHBN survey,
this means a deterioration of data quality that we will have to take into
account (Wolleben et al., 2021).

###  5.1 Extracting the overlap region between STAPS and GHBN

STAPS and GHBN overlap in the region $\delta=[-30{\degr},0{\degr}]$, which we
refer to as “the overlap region” hereafter.

The frequency coverage, the pixel size, the angular resolution, and the units
must be the same in the overlap region to compare the two surveys.
Accordingly, we applied the following steps:

  1. i)

The Stokes $Q$ and $U$ maps of STAPS were smoothed to the GHBN resolution,
ensuring flux conservation, and re-gridded onto the GHBN pixel grid.

  2. ii)

The overlap region was extracted from both surveys and projected onto
equatorial coordinates.

  3. iii)

STAPS and GHBN cover similar frequencies with slight differences (see Table
1), but use the same frequency channel width. Thus, we only used the frequency
range common to the two surveys.

  4. iv)

We repeated the RM-synthesis procedure and the cleaning described in Sect. 3.3
to produce new cleaned Faraday spectra and moment maps of the overlap region
using the smoothed $Q$ and $U$ STAPS maps.

  5. v)

Faraday depths in GHBN beyond $\pm$500 rad m-2 are dominated by spurious
features, especially in the southern limit of the survey (in the area of the
overlap region in Fig. 6 of Wolleben et al. 2021). Thus, we did not include
these Faraday depth ranges in our analysis of the overlap region.

Fig. 6 shows the results of steps 1–3 of the procedure for an example pixel
that is typical of high Galactic latitudes, where both the frequency coverages
are equalized and STAPS data are smoothed to the GHBN resolution. Fluctuations
in STAPS data are significantly lower than in GHBN due to the higher noise in
GHBN than in STAPS.

We performed RM synthesis and RM-CLEAN to compute the Faraday depth cubes of
the overlap region. We adopted pixel-wise 6$\sigma$ cutoff parameters from the
$\sigma$ map calculated for each survey to determine the Faraday depth cutoffs
needed to calculate the moment maps, as is explained in Section 3.4. The STAPS
and GHBN maps are displayed in Fig.7. Unlike GHBN, the Stokes $I$ leakage into
the Stokes $Q$ and $U$ are not subtracted in the current STAPS data. Since
this effect is thought to be most significant in the Galactic plane, we masked
the Galactic plane in the comparison. The area in GHBN most affected by ground
emission, at declinations below $-25\degr$ (Wolleben et al., 2021), is also
masked.

Figure 6: Representation of the frequency coverage of STAPS and GHBN at the
Galactic location of $l=310\aas@@fstack{\circ}2,b=52\aas@@fstack{\circ}5$. The
top panel plots (a) present the original Stokes $Q$ and $U$ of STAPS and GHBN,
and the bottom panel plots (b) present the data modified after the procedure
explained in Sect. 5. The area highlighted in yellow represents the STAPS
frequency coverage. The hatched gray areas indicate the frequency ranges for
which GHBN lacks adequate data in the overlap region for proper comparison.
These pixels were discarded in the original data due to the localized RFI, as
is described by Ordog (2020).

###  5.2 Comparison between STAPS and GHBN

The Faraday moment maps of the overlap region of both surveys are presented in
Fig. 7. The moment maps in Figs. 7(a) and 7(b) show that STAPS and GHBN show
similarities in many places, such as, in the $M_{0}$ maps, the bright regions
centered at
$(\alpha,\delta)\sim(15^{h},-15\degr),(22^{h},-6\degr),(6^{h},-4\degr)$.
However, there are significant differences between the $M_{0,STAPS}$ and
$M_{0,GHBN}$ maps, whereas both surveys agree on many features that are
considered real Galactic signals. However, spurious signals and low signal-to-
noise ratios (S/Ns) in either or both maps can easily cause significant
deviations in the Faraday moment maps. Below, we discuss the agreements
between the maps and the differences, with their likely causes.

Figure 7: Moment maps of the overlap region. From top to bottom: (a)
$M_{0,STAPS}$; (b) $M_{0,GHBN}$; (c) $M_{1,STAPS}$; (d) $M_{1,GHBN}$; (e)
$m_{2,STAPS}$; (f) $m_{2,GHBN}$; (g) $m_{3,STAPS}$; (h) $m_{3,GHBN}$. The
white areas show either the masked values or the value of zero. The color bars
are saturated to present the finer details of structures, and the $M_{0}$
color maps specifically are saturated at 50$\%$ of the maximum. Figure 8: Maps
of $P_{peak}$ of the STAPS (top) and GHBN (bottom) overlapping region. Each
has the same masked values for comparison. All the figure parameters are the
same as in Fig. 7. Figure 9: Scatter plot of $P_{peak}$ of STAPS and GHBN over
the region where the surveys overlap. The green line corresponds to the linear
fit of the data. The color bar corresponds to the number of data points in
each hexbin. $R$ is the coefficient of correlation.

  * i)

Artifacts in the GHBN maps exist that are not present in STAPS, as Wolleben et
al. (2021) have already pointed out. Most notable are the zigzag patterns of
the basket-weaving scanning, predominantly visible at right ascensions between
about 3h and 20h.

  * ii)

The level of PI in the GHBN survey is almost twice that of STAPS. This is seen
clearly in Figs. 9 and 10, which show, on average, a factor of 1.8 difference
between the maximum PI in the Faraday spectra of the two surveys. There is
evidence that the intensity scale of GHBN is generally correct. Wolleben et
al. (2021) show that the total-intensity scale over the entire GHBN survey
area is within 3% of the highly reliable scale of the Stockert survey (Reich,
1982; Reich & Reich, 1986) when main-beam brightness temperatures are
compared. Additionally, Wolleben et al. (2021) compared polarized intensities
over the brightest features, the Fan Region and the North Polar Spur, with
polarized intensities from the Dwingeloo 1411 MHz data (Brouw & Spoelstra,
1976) and found an agreement within 10% over both regions. The problem with
GHBN polarized intensities appears to be confined to declinations below 0
degrees, getting progressively worse toward the southern limit of the GHBN
survey, as is indicated by comparisons of polarized intensities between GHBN
and the data of Wolleben et al. (2006) at 1410 MHz. On the other hand, the
STAPS data were compared to polarization data from the Villa Elisa 1420 MHz
survey (Testori, Reich, & Reich, 2008) and found to be consistent for most of
the sky. However, at high declinations, specifically at the edge of the
survey’s declination range, the STAPS Stokes $Q$ and $U$ signal decreased by a
factor of $\sim$1.6 compared to the Villa Elisa 1420 MHz survey, indicating
that the STAPS intensity may be underestimated at the edges of the survey.
These two effects, together with increased artifacts at the GHBN’s spatial
edges and higher side lobes in the GHBN, may explain the differences in
intensity between the two surveys in the overlap region. Despite this
disagreement between the two surveys, we can continue to compare the higher
moments, which are independent of amplitude, depending only on the change of
polarization angle as a function of frequency.

Figure 10: Representative example spectrum of the overlap region of each
survey. The solid black line shows the CLEAN Faraday spectrum of the GHBN
overlap region, and the solid red line shows that of STAPS.

The GHBN $M_{1}$ map is visibly affected by zigzag striping due to the
scanning technique in low-intensity regions, particularly between right
ascensions 3h and 20h, where GHBN has more negative $M_{1}$ values than STAPS
in that direction. A similar situation occurs between 4h and 12h, which is a
low-intensity region. Furthermore, around the right ascensions of 7h to 8h,
where the RA strip crosses the outer Galactic plane, differences between the
surveys can be attributed to instrumental polarization.

The $m_{2}$ maps in Figs. 7(e) and 7(f), and the $m_{3}$ maps in Figs. 7(g)
and 7(h), are also affected by the issues explained above. In general, STAPS
$m_{2}$ is higher than GHBN $m_{2}$. This might be due to some real peaks in
the STAPS data that may be missed in the GHBN, as it has a lower S/N than
STAPS. The above-mentioned issues can cause a reversal in the skewness, as is
seen in local anti-correlations of $m_{3}$ in STAPS and GBHN.

Fig. 8 shows the maps of peak PI for STAPS (top) and GHBN (bottom). As can be
seen in the $M_{0}$ maps, an intensity difference between the two of a factor
of $\sim$1.8 is also apparent in the scatter plot (Fig. 9). This situation is
also demonstrated in the representative Faraday spectrum in Fig. 10, where the
GHBN intensity is higher than STAPS. This intensity difference increases from
a factor of 1.7 at $0^{\circ}<\delta<-10^{\circ}$ to a factor of two at
$-20^{\circ}<\delta<-25^{\circ}$, supporting our conclusion that this is an
edge effect.

In summary, this comparison of $PI$ and Faraday moment maps in the STAPS and
GHBN surveys shows broad correspondences between the two surveys. However, in
many locations, these are overshadowed by artifacts, mostly in GHBN, possibly
due to incomplete removal of ground radiation (as is explained in Wolleben et
al. 2021). Also, there is a difference in PI of about a factor of 1.8 between
the two surveys. The cause of this difference is unknown, but likely related
to inaccuracies because the overlap region is at the edge of the sky coverage
of either survey.

##  6 Discussion

The maps in Figs. 3 and 4 and a comparison of the moment maps in Fig. 5
illustrate properties of the magneto-ionic medium. Care has to be taken, as
low-S/N regions may result in a signal in the Faraday moment maps that does
not reflect structures in the ISM. For example, in the southern sky at
$l\lesssim 300{\degr}$, $M_{0}$ mainly shows a very low PI, except for a few
regions, including the Vela supernova remnant.

Comparing the first and second moment maps of Faraday depth cubes in the GMIMS
Low Band South, (Wolleben et al., 2019) shows no correspondence with the STAPS
moment maps. As the GMIMS Low Band South has frequency coverage of 300 to 480
MHz, emission from distances greater than a few hundred parsecs is heavily
depolarized, and is sensitive only to much lower Faraday depths of $\pm$10 rad
m-2. It therefore probes only the very nearby gas (Dickey et al., 2019). Due
to the lower STAPS Faraday depth resolution, STAPS is barely sensitive to the
Faraday depths detected in GMIMS Low Band South. Hence, STAPS could not
distinguish the multiple closely spaced peaks in some regions in GMIMS Low
Band South. For the specific case of the H ii region Sh 2$-$27, the $M_{1}$
map indicates a median RM of around $-$155 rad m-2, which is consistent with
the value of RM found from S-PASS polarimetry in Raycheva et al. (2022) and
Wolleben et al. (2021) with GHBN data.

###  6.1 Complexity in Faraday spectra

The $m_{2}$ map is an indicator of complexity in the Faraday spectra.
Complexity could be caused by Faraday thick components (broad peaks), multiple
Faraday thin components, or a combination. Multiple components could arise
from structures along the LoS, but also within the telescope beam. Due to the
frequency coverage of STAPS, the maximum detectable Faraday thickness is
comparable to the resolution of Faraday depth (see Table 1), and the
complexity of spectra will consist of multiple, potentially overlapping,
unresolved peaks. In addition, distinct objects such as Sh 2$-$27 or G353$-$34
stand out, and have multiple Faraday depth peaks representing the object and
the background.

Fig. 11 shows the absolute difference between $\phi_{peak}$ and $M_{1}$ (the
position of the mean relative to the peak), which is close to zero for a
spectrum with a single unresolved peak or a symmetric complex spectrum, but
nonzero for asymmetric complex spectra. In this figure, some parts have high
$\phi_{{peak}}-M_{1}$ values and most of these are observed at low intensities
(see Fig. 5a). As is seen in Fig. 12, these pixels predominantly represent
regions with a low S/N. This observation indicates that in the regions where
the S/N is high, the $\phi_{{peak}}-M_{1}$ value tends to be low, indicating
the presence of single peaks, except for certain discrete areas (e.g., the
Galactic plane, Sh 2$-$27, and G353$-$34).

Figure 11: Results of $(\phi_{peak}-M_{1})$. The parameters are the same as in
Fig. 5. Figure 12: Comparison between $|\phi_{peak}-M_{1}|$ and $M_{0}$. The
inner panel shows the values zoomed in between $(\phi_{peak}-M_{1})<100$ rad
m-2 and $M_{0}<100$ mK. The Galactic plane $(-6\degr<b<5\degr)$ and discrete
objects such as Sh 2$-$27 and G353$-$34 are excluded. The color bar
corresponds to the number of data points in each hexbin.

Bell, Junklewitz, & Enßlin (2011) introduced and explained “Faraday caustics”
as an observational signature in Faraday spectra of LoS magnetic field
reversals. These caustics would be visible as Faraday depth peaks with an
asymmetric tail. Although we see spectra with asymmetric tails, this asymmetry
might also be due to a second overlapping Faraday depth structure. According
to Bell, Junklewitz, & Enßlin (2011), the highest to lowest frequency ratio
must be at least 1.5 to resolve this structure (Van Eck et al., 2019). Since
STAPS’s ratio is 1.33, we are unlikely to resolve Faraday caustics. Even if we
resolved Faraday caustics, these are at this resolution impossible to
distinguish from blended double peaks without additional information such as
high-resolution multifrequency data.

The $m_{3}$ map in Fig.5d can give unique information about the spectrum’s
shape; namely, in which direction it is skewed. In low-S/N regions, the
$m_{3}$ map generally shows high skewness, which represents only noise. As
$m_{2}$ and $m_{3}$ include increasing powers of intensity differences, the
noise will dominate the low-S/N pixels in $m_{2}$ and especially $m_{3}$.
Hence, according to our estimation, it is suggested that the $m_{3}$ map be
treated with caution for signals with an S/N lower than 10. It is noticeable
that most of the sky away from the plane ($|b|>30\degr$) shows a negative
skewness (the tail to the left, represented in the shades of blue color),
which implies an additional low-level component toward negative Faraday depths
over most of the high-(negative-)latitude southern sky. Clearly, it shows the
additional Faraday depth component in discrete objects.

###  6.2 Galactic center spur and S-PASS northern and southern ridges

The large loops and spurs identified with S-PASS in Carretti et al. (2013)
(such as the northern and southern ridges (extending from the center with high
PI) are discernible in the STAPS $M_{0}$ map as well. The northern and
southern ridges coincide with the high-intensity regions in $M_{0}$ centered
on $(l,b)\sim(354\degr,35\degr)$ and $(l,b)\sim(355\degr,-30\degr)$. The
Galactic center spur is visible as the arch in $M_{0}$ running from about
$(0,10)\degr$ to $(350,25)\degr$.

However, there is more depolarization in the band below $\sim$30$\degr$ around
the Galactic plane. This was first identified by Wolleben et al. (2006) in the
northern sky in the same wavelength range, partially depolarizing these
ridges. If the ridges were formed by outflows of star formation from the
Galactic center, the visible parts (at $|b|\gtrsim 30\degr$) would be many
kiloparsecs above the Galactic plane. As the scale height of thermal free
electrons is estimated to be between 1000 pc (Reynolds, 1991) and 1830 pc
(Gaensler et al., 2008), this suggests that at high distances above the plane,
the ridges would exhibit negligible electron density and negligible Faraday
rotation. In fact, the $M_{1}$ map shows that the Faraday depth structure of
these ridges does not stand out from the background, and $m_{2}$ shows
unresolved spectra in these directions, indicating that the ridges do not give
an additional Faraday depth component besides the background. In the data of
Carretti et al. (2019), the ridges are also not visible in the RM map between
2.3 GHz and 22 GHz and 30 GHz, which is confirmed by our conclusion that RM is
negligible in these features.

###  6.3 Comparison to point source rotation measures

From surveys of polarized extragalactic background sources, Oppermann et al.
(2015) derived the Faraday sky map of the Galaxy by removing the intrinsic
Faraday rotation component of the background sources, which has recently been
improved by Hutschenreuter et al. (2022),222https://wwwmpa.mpa-
garching.mpg.de/~ensslin/research/data/faraday2020.html hereafter referred to
as $RM_{F.sky}$.

We smoothed STAPS $M_{1}$ and $RM_{F.sky}$ to a common resolution of $\sim$134
arcmin, chosen as the average difference between data points in the
S-PASS/ATCA survey (Schnitzeler et al., 2019), which dominates the southern
sky except in the Galactic plane and Magellanic Clouds where the source
density is much higher. Fig. 13 shows these two maps. We chose to apply this
smoothing to the Faraday moments and not to Stokes $Q$ and $U$ to make it as
similar as possible to the process applied to obtain the Faraday sky, whereby
RM values of point sources were interpolated over the sky.

Figure 13: Comparison between $RM_{F.sky}$ and STAPS $M_{1}$ RMs. The top
panel: The STAPS $M_{1}$ map smoothed onto the Faraday sky map of
Hutschenreuter et al. (2022) at the bottom panel. All the other figure
parameters are the same as in Fig. 5).

We present a comprehensive comparison of the two datasets (STAPS and the
Faraday sky of Hutschenreuter et al. 2022) using a scatter plot, as is shown
in Fig. 14. In both datasets, we masked the following: i) the PI data points
with low S/N ($<$6); ii) data points in the Galactic plane with extreme
$RM_{F.sky}$ values and the region of G353$-$34 covering a diameter of $\sim
10\degr$; iii) the Sh 2$-$27 H ii region as there is a very good correlation
between $RM_{F.sky}$ and STAPS $M_{1}$ (where the Pearson correlation
coefficient, $R$, is 0.74), as is shown in Fig. 13. The remaining data points
were then compared with the Faraday sky model.

The Faraday sky map exhibits higher $|RM|$ values than STAPS $|M_{1}|$, which
is particularly evident within the Galactic plane. This discrepancy is
attributed to the Faraday sky probing the complete LoS through the Galaxy,
while the STAPS signal is depolarized at large distances, possibly due to
small angular-scale structures in RM. Nevertheless, across most of the high-
latitude sky, the features in both maps generally align well. The RMs from
extragalactic sources are expected to be twice that of Galactic diffuse
emission if the medium behaves as a Burn slab (Burn, 1966). However, STAPS
$M_{1}$ Faraday depths and the Faraday sky disagree with the expected Burn
slab behaviour. Specifically, at intermediate and upper latitudes, the
comparison between STAPS $M_{1}$ and the Faraday sky has a slope of 0.9 and a
correlation coefficient of 0.36. This shows that the Burn slab is not a good
approximation for the Galaxy, as is pointed out by Dickey et al. (2022).
However, it is consistent with a model of several emitting and rotating
layers, which can produce slopes smaller than 1 (Erceg et al., 2022).
Additionally, deviations from the linear relation at high $RM_{F.sky}$ are
likely attributed to structures at low latitudes, such as the G353$-$34
region, where the LoS differs between STAPS $M_{1}$ and $RM_{F.sky}$.

Figure 14: Comparison between $M_{1}$ of STAPS ($\phi_{M_{1}}$) and the
Faraday sky map of Hutschenreuter et al. (2022). The green line corresponds to
the linear fit to the entire data. For display purposes, 10$\times$10 pixels
are binned into one larger pixel of 30$\arcmin$. The color bar corresponds to
the number of data points in each hexbin. $R$ is the coefficient of
correlation.

##  7 Summary and conclusion

We have presented the Faraday moments observed at 1328$-$1768 MHz from the
spectro-polarimetric STAPS survey, also called the GMIMS High-Band South
survey, to investigate magnetic structures in the Galaxy. Faraday moments
provide a great understanding of radio polarization in the ISM by summarizing
the Faraday spectra in the total polarized brightness ($M_{0}$), the average
Faraday depth weighted by the PI ($M_{1}$), the weighted width of the Faraday
spectrum for each pixel ($m_{2}$), and its skewness ($m_{3}$). We have used
the survey data from STAPS and compared them to relevant previous studies. A
comparison of the moment maps with the GHBN survey at the same frequency range
in the northern sky shows broad agreement, but also points to significant
ground emission in GHBN (Wolleben et al., 2021) and edge effects in both
surveys.

The conclusions reached in this work are:

  1. 1.

The weighted PI $M_{0}$ shows a generally low polarization at longitudes of
$l\lesssim 300\degr$ in the fourth quadrant, but does show polarization
imprints from structures such as the Vela supernova remnant. The S-PASS ridges
are visible as high-polarization structures, but only at absolute latitudes
higher than the depolarization strip at $-30^{\circ}<b<30^{\circ}$.

  2. 2.

Large parts of the southern sky can be described by single Faraday components,
with apparent exceptions being mostly specific structures (supernova remnants,
H ii regions). As a result of the limited frequency range of the survey,
Faraday thick structures cannot be detected, or can only be detected at their
edges. Therefore, we cannot distinguish between two separate Faraday-thin
components in a Faraday spectrum and an edge-detected Faraday-thick structure,
and cannot reliably detect Faraday caustics.

  3. 3.

The ratio between the $RM_{F.sky}$ presented by Hutschenreuter et al. (2022)
and STAPS $M_{1}$ RMs is 0.9 so that the first moment value is almost
equivalent to the total RM value. This indicates a deviation from the expected
ideal Burn slab model, but is consistent with a model of multiple emitting
elements in a Faraday rotating medium, as is shown by Erceg et al. (2022).

The Southern Twenty-centimeter All-sky Polarization Survey offers a remarkable
opportunity to delve into the intricate characteristics of the Galactic
magneto-ionic medium and diffuse sources at frequencies between 1328 and 1768
MHz, with a resolution of 18$\arcmin$. Further research should include
exploring individual diffuse sources that could be revealed by STAPS, but also
in combination with and comparison to the other GMIMS surveys.

###### Acknowledgements.

This work is part of the joint NWO-CAS research programme in the field of
radio astronomy with project number 629.001.022, which is (partly) financed by
the Dutch Research Council (NWO). We thank Karel D. Temmink for the help with
coding/parallelization in the astronomy department’s coma computing cluster.
MH acknowledges funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (grant
agreement No 772663). JMS acknowledges the support of the Natural Sciences and
Engineering Research Council of Canada (NSERC), 2019-04848. XHS, JLH and XYG
are supported by the International Partnership Program of Chinese Academy of
Sciences, Grant No. 114A11KYSB20170044. The Dunlap Institute is funded through
an endowment established by the David Dunlap family and the University of
Toronto. AO is partly supported by the Dunlap Institute. BMG acknowledges the
support of the Natural Sciences and Engineering Research Council of Canada
(NSERC) through grant RGPIN-2022-03163, and of the Canada Research Chairs
program. AB acknowledges financial support from the INAF initiative ”IAF
Astronomy Fellowships in Italy” (grant name MEGASKAT). MT is supported by the
Banting Fellowship (Natural Sciences and Engineering Research Council Canada)
hosted at Stanford University and the Kavli Institute for Particle
Astrophysics and Cosmology (KIPAC) Fellowship. The authors acknowledge
Interstellar Institute’s program ”With Two Eyes” and the Paris-Saclay
University’s Institute Pascal for hosting discussions that nourished the
development of the ideas behind this work. This work made use of
Astropy333http://www.astropy.org a community-developed core Python package for
Astronomy (Astropy Collaboration et al., 2018) and Python modules of
Matplotlib444http://www.matplotlib.org/ (Hunter, 2007),
NumPy555https://numpy.org/ (Oliphant, 2006; van der Walt et al., 2011), and
SciPy666https://www.scipy.org/index.html (Virtanen et al., 2020). This
research made use of the software package
Montage777http://montage.ipac.caltech.edu/. Parts of this work’s results use
the color maps in the CMasher package (van der Velden, 2020). The Parkes 64m
radio-telescope (Murriyang) is part of the Australia Telescope National
Facility 888https://ror.org/05qajvd42 which is funded by the Australian
Government for operation as a National Facility managed by CSIRO. We
acknowledge the Wiradjuri people as the Traditional Owners of the Observatory
site.

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11institutetext: IMAPP, Radboud University, NL-6525 AJ Nijmegen, The
Netherlands 22institutetext: Wigner RCP, Konkoly-Thege 29-33, H-1121 Budapest
XII, Hungary 33institutetext: Eszterházy University, Mátrai út 36, H-3200
Gyöngyös, Hungary 44institutetext: Eötvös University, Pázmány Péter Sétány
1/A, H-1111 Budapest, Hungary

# Interpreting BEC in e+e- annihilation 111Talk given at XLVIII International
Symposium on Multiparticle Dynamics, Singapore, 3–7 Sep. 2018

W.J. Metzger 11<EMAIL_ADDRESS>T. Csörgő 2233 T. Novák 33 S. Lökös
3344

###### Abstract

The usual interpretation of Bose-Einstein correlations (BEC) of identical
boson pairs relates the width of the peak in the correlation function at small
relative four-momentum to the spatial extent of the source of the bosons.
However, in the $\tau$-model, which successfully describes BEC in hadronic
$\mathrm{Z}$ decay, the width of the peak is related to the temporal extent of
boson emission. Some new checks on the validity of both the $\tau$-model and
the usual descriptions are presented.

##  1 Introduction

First a brief review of ‘classic’ parametrizations of Bose-Einstein
Correlations (BEC) is given and contrasted with the parametrization of the
$\tau$-model Tamas;Zimanji:1990 ; ourTauModel . which has been found
L3_levy:2011  to describe well Bose-Einstein correlations in hadronic
$\mathrm{Z}$ decay.

The data used in this paper are taken from Ref. (L3_levy:2011, ). They
comprise both two-jet and three-jet events, as determined using the Durham jet
algorithm durham ; durham2 ; durham3  with resolution parameter
$y_{\mathrm{cut}}$=0.006, from e+e- annihilation at the $\mathrm{Z}$-pole.

###  1.1 ‘Classic’ Parametrizations

The Bose-Einstein correlation function, $R_{2}$, is measured by
$R_{2}(Q)=\rho(Q)/\rho_{0}(Q)$, where $\rho(Q)$ is the density of identical
boson pairs with invariant four-momentum difference
$Q=\sqrt{-(p_{1}-p_{2})^{2}}$ and $\rho_{0}(Q)$ is the similar density in an
artificially constructed reference sample, which should differ from the data
only in that it does not contain the effects of Bose symmetrization of
identical bosons. It is often parametrized as

$R_{2}=\gamma\left[1+\lambda G\right](1+\epsilon Q)\;,$ (1)

with

$G=\exp\left(-\left(rQ\right)^{2}\right)\;.$ (2)

The corresponding distribution of boson emission points in space-time is a
spherically symmetric Gaussian with standard deviation $r$.

The factor $(1+\epsilon Q)$ is included to account for non-BEC which are not
removed by $\rho_{0}$, i.e., to make up for slight inadequacies in $\rho_{0}$,
and $\gamma$ is a normalization parameter. The parameter $\lambda$ is
introduced to account for effects reducing the amount of BEC, e.g., some of
the bosons coming from resonance decays, or the identical bosons being
partially coherent.

However, this description was found L3_levy:2011  not to describe the
$\mathrm{Z}$-decay data, a better description being provided by the Edgeworth
expansion about the Gaussian L3_3D:1999 :

$G=\exp\left(-\left(rQ\right)^{2}\right)\cdot\left[1+\frac{\kappa}{3!}H_{3}(rQ)\right]\;,$
(3)

where $H_{3}$ is the third-order Hermite polynomial.

A different way to depart from the Gaussian is the generalization to a
symmetric Lévy stable distribution. Then

$G=\exp\left(-\left(rQ\right)^{\alpha}\right)\>,$ (4)

where $0<\alpha\leq 2$ is the so-called index of stability, which was
introduced to BEC in Ref. Tamas:Levy2004 .

A fit of the Edgeworth parametrization to the two-jet data of Ref.
(L3_levy:2011, ) finds $\kappa=0.71\pm 0.06$, while a fit of the symmetric
Lévy parametrization yields $\alpha=1.34\pm 0.04$. Both values are far from
the corresponding Gaussian values of $\kappa=0$ and $\alpha=2$, respectively.
Although the $\chi^{2}$ of these fits are a great improvement over that of the
Gaussian fit, they are still very large. The corresponding confidence levels
are $10^{-15}$ for the Gaussian and $10^{-5}$ and $10^{-8}$ for the Edgeworth
and Lévy fits, respectively. The symmetric Lévy fit is shown in Fig. 1a.

Figure 1: (a) Symmetric Lévy fit for two-jet events; (b) Simplified
$\tau$-model fit for two-jet events.

The reason for the failure of the ‘classic’ parametrizations is readily
apparent from Fig. 1a. There is a region of anti-correlation ($R_{2}<1$)
extending from about $Q=0.5$ to
$1.5\,\hbox{$\mathrm{G}$\hbox{$\mathrm{e\kern-1.00006ptV}$}}$. The ‘classic’
parametrizations, which are of the form $1+P$, where $P$ is a positive-
definite quantity, are unable to accomadate the anti-correlation. This was not
realized for a long time because experiments only plotted the correlation
function up to $Q=2\;\hbox{$\mathrm{G}$\hbox{$\mathrm{e\kern-1.00006ptV}$}}$
or less. In Ref. (L3_levy:2011, ) $Q$ was plotted to 4
$\mathrm{G}$$\mathrm{e\kern-1.00006ptV}$, and the anti-correlation became
apparent. This anti-correlation, which one might term Bose-Einstein Anti-
Correlations (BEAC), as well as the Bose-Einstein correlations (BEC) are both
well described by the $\tau$-model.

###  1.2 The $\tau$-model

The $\tau$-model Tamas;Zimanji:1990 ; ourTauModel  rests on several
assumptions:

  * •

The average production point is proportional to the momentum of the emitted
boson. Dimensionally the momentum must be multiplied by a time divided by a
mass to yield a spatial dimension. The description of a two-jet event is
invariant to Lorentz boosts along the direction of the colour field. The
relevant boost-invariant quantities are the “longitudinal” proper time,
$\tau=\sqrt{\overline{t}^{2}-\overline{r}^{2}_{z}}$ and the “transverse” mass,
$m_{\mathrm{t}}=\sqrt{E^{2}-p_{z}^{2}}$, resulting in

$\overline{x}^{\mu}(p^{\mu})=a\tau\ p^{\mu}\;,\qquad a=1/m_{\mathrm{t}}\;.$
(5)

  * •

The spatial distribution of production points about their mean is very narrow,
although the distribution of proper time may be broad.

  * •

The distribution of $\tau$ is a one-sided Lévy distribution, one-sided because
no particles are emitted before the e+e- collision.

Then $R_{2}$ turns out to depend on three variables, $Q$ and the transverse
mass of each of the particles making up the pair:

$\displaystyle R_{2}(Q,a_{1},a_{2})$ $\displaystyle=$
$\displaystyle\gamma\left\\{1+\lambda\cos\left[\frac{\tau_{0}Q^{2}(a_{1}+a_{2})}{2}+\tan\left(\frac{\alpha\pi}{2}\right)\left(\frac{\Delta\tau{Q^{2}}}{2}\right)^{\\!\alpha}\frac{a_{1}^{\alpha}+a_{2}^{\alpha}}{2}\right]\right.$
(6)
$\displaystyle\hfill\left.\cdot\exp\left[-\left(\frac{\Delta\tau{Q^{2}}}{2}\right)^{\\!\alpha}\frac{a_{1}^{\alpha}+a_{2}^{\alpha}}{2}\right]\right\\}\cdot\left(1+\epsilon
Q\right)\;.$

Fits in three dimensions are problematic with the available statistics. Hence
we simplify this expression by introducing an effective radius, $R$, defined
by

$R=\left(\frac{\Delta\tau}{2}\right)^{\\!\alpha}\frac{a_{1}^{\alpha}+a_{2}^{\alpha}}{2}\;.$
(7)

Further, we assume that particle production begins immediately, i.e.,
$\tau_{0}=0$. Then

$\displaystyle R_{2}(Q)$
$\displaystyle=\gamma\left[1+\lambda\cos\left(\left(R_{\mathrm{a}}Q\right)^{2\alpha}\right)\exp\left(-\left(RQ\right)^{2\alpha}\right)\right](1+\epsilon
Q)\;,$ (8a) $\displaystyle R_{\mathrm{a}}^{2\alpha}$
$\displaystyle=\tan\left(\frac{\alpha\pi}{2}\right)R^{2\alpha}\;.$ (8b)

The fit of the simplified $\tau$-model to the two-jet data is shown in Fig.
1b. Unlike the fits of the classic parametrizations, the $\chi^{2}$ is
acceptable, and the residuals lack structure.

Note that the difference between the parametrizations of Eqs. (4) and (8) is
the presence of the cosine term, which provides the description of the BEAC
dip. The parameter $R$ describes the BEC peak, and $R_{\mathrm{a}}$ describes
the anti-correlation region. While one might have had the insight to add, ad
hoc, a $\cos$ term to Eq. (4), it is the $\tau$-model which provides a
physical reason for it and which predicts a relationship, Eq. (8b), between
$R$ and $R_{\mathrm{a}}$, i.e., between the correlation and the anti-
correlation.

##  2 Expansions

Recall that the Edgeworth expansion of the Gaussian parametrization provided
evidence (in addition to the poor $\chi^{2}$) that the Gaussian was
inadequate. In this section we look at expansions of the Symmetric Lévy and
the $\tau$-model parametrizations.

###  2.1 Symmetric Lévy

The symmetric Lévy distribution can be expanded in terms of Lévy polynomials
DeKock:WPCF2011 ; Csorgo:Odderon , $l_{i}$, which are orthonormal. The
resulting expression for $R_{2}$ is

$R_{2}(Q)=\gamma\left[1+\lambda\exp\left(-\left|rQ\right|^{\alpha}\right)(1+\sum
c_{i}l_{i})\right]\cdot\left(1+\epsilon Q\right)\;.$ (9)

Fits to the two-jet data are shown in Fig. 2a for orders 0 through 3 of the
Lévy polynomials. The order-0 fit (also shown in Fig. 1a) has a very poor
$\chi^{2}$, but the order-1 fit has a good $\chi^{2}$. Higher orders show only
marginal further improvement.

(a) (b)

Figure 2: (a) Symmetric Lévy fit for two-jet events up to orders 0 through 3
Lévy polynomials; (b) First-order Lévy polynomial fit (black) for two-jet
events compared to the $\tau$-model fit (red).

The first-order symmetric Lévy polynomial fit is shown together with the
simplified $\tau$-model fit in Fig. 2b. The $\chi^{2}$ of the Lévy polynomial
fit is slightly better than the $\tau$-model fit, but the fit curves are
nearly identical, what difference there is being mainly for
$Q>1.5\,\hbox{$\mathrm{G}$\hbox{$\mathrm{e\kern-1.00006ptV}$}}$.

Comparing Eqs. (8) and (9), we see that in the symmetric Lévy parametrization
the cosine of the $\tau$-model parametrization is replaced by the Lévy
polynomial expansion. Also, in the exponential $2\alpha$ becomes simply
$\alpha$. Fig. 3 compares the cosine and the Lévy polynomials. We see a rather
similar behaviour: Both decrease more or less linearly with $Q$, which
explains why both fit the data approximately equally well.

Figure 3: Comparison of the cosine term of the $\tau$-model parametrization
with the Lévy polynomial term of the symmetric Lévy parametrization.

###  2.2 $\tau$-model (asymmetric Lévy)

Lacking an orthogonal polynomial expansion for the asymmetric Lévy
distribution $H(\tau)$ of the $\tau$-model, we use, motivated by the results
of Ref. (DeKock:WPCF2011, ), a derivative expansion:

$\displaystyle R_{2}(Q)$ $\displaystyle=$
$\displaystyle{\gamma}\left[1+\lambda\left\\{\cos\left(\left(R_{\mathrm{a}}Q\right)^{2\alpha}\right)\exp\left(-\left(RQ\right)^{2\alpha}\right)\right.\right.\qquad\qquad\qquad$
(10) $\displaystyle\left.\left.+\sum
c_{n}\frac{{\textrm{d}}^{n}\phantom{Q}}{{\textrm{d}}Q^{n}}\cos\left(\left(R_{\mathrm{a}}Q\right)^{2\alpha}\right)\exp\left(-\left(RQ\right)^{2\alpha}\right)\right\\}\right]\cdot\left(1+\epsilon
Q\right)\;.$

We also consider letting $R_{\mathrm{a}}$ be a free parameter rather than as
defined in Eq. (8b). The results of these fits are shown in Table 1 and Fig.
4a. The $\chi^{2}$ of the order-1 fit is, of course, smaller than that of the
order-0 fit, as is the $R_{\mathrm{a}}$-free fit, and the confidence levels
are somewhat better. To test the significance of the improvement in
$\chi^{2}$, we make use of the $\chi^{2}$-difference. For the order-0 and
order-1 fits this is $\chi^{2}_{\text{diff}}=94.7-90.9=3.8$, and the
difference in the number of degrees of freedom is 1. The confidence level for
a $\chi^{2}$ of 3.8 with 1 degree of freedom is 5.1%. A small value of this
confidence level, say less than 5%, would be grounds for rejecting the order-0
parametrization. (This corresponds to the 95% commonly used in making
decisions.) Thus we conclude that for two-jet events the order-0 fit is
adequate.

Table 1: Fit results of $\tau$-model parametrizations for two-jet events.

$\begin{array}[]{lccc}\hline\cr&\text{order 0}&\text{order 1}&\text{order 0,
$R_{\mathrm{a}}$\ free}\\\ \hline\cr\alpha&0.44\pm 0.01&0.43\pm 0.01&0.41\pm
0.02\\\ R\ \text{(fm)}&0.78\pm 0.04&0.84\pm 0.05&0.79\pm 0.04\\\
R_{\mathrm{a}}\ \text{(fm)}&-&-&0.69\pm 0.04\\\ \lambda&0.61\pm 0.03&0.67\pm
0.05&0.63\pm 0.03\\\ \gamma&0.979\pm 0.002&0.979\pm 0.002&0.988\pm 0.005\\\
\epsilon&0.005\pm 0.001&0.005\pm 0.001&0.001\pm 0.002\\\ c_{1}&-&0.0008\pm
0.0005&-\\\ \hline\cr\chi^{2}/{\text{DoF}}&94.7/95&90.9/94&91.0/94\\\
\text{CL}&49\%&57\%&57\%\\\ \hline\cr\end{array}$

The order-0, $R_{\mathrm{a}}$-free fit also provides an adequate description,
having a $\chi^{2}$ nearly identical to that of the order-1,
$R_{\mathrm{a}}$-constrained fit. Note that the physical parameters ($\alpha$,
$R$, $\lambda$) differ at most by about 1 standard deviation in going from
order-0 to order-1 or to $R_{\mathrm{a}}$ free. Thus conclusions based on
these values, such as the reconstructions of the space-time picture in Ref.
(L3_levy:2011, ), remain valid.

(a) (b)

Figure 4: (a) Fits of the $\tau$-model (black), the order-1 expansion of the
$\tau$-model (red), and the $\tau$-model with $R_{\mathrm{a}}$ free (green),
for two-jet events; (b) Fits of the $\tau$-model (black), the order-1
expansion of the $\tau$-model (red), the $\tau$-model with $R_{\mathrm{a}}$
free (green), and the order-1 expansion of the $\tau$-model with
$R_{\mathrm{a}}$ free (blue), for three-jet events.

##  3 Conclusions for two-jet events

We have used expansions about the hypothesized form to test whether it
provides an adequate description of the data or is only a (poor)
approximation. In the latter case the shape of the expansion terms provide an
indication of how to modify the original parametrization. For the symmetric
Lévy parametrization this showed that an approximately linearly decreasing
function of $Q$ is necessary, which in fact is what is provided by the
$\tau$-model.

An expansion in the case of the $\tau$-model is found not to be significant,
i.e., the one-sided Lévy distribution of the $\tau$-model is adequate.

##  4 Three-jet events

For two-jet events hadronization occurs basically in 1+1 dimensions, which
lead to the dependence of $R_{2}$ on $\tau$, the longitudinal proper time and
$m_{\mathrm{t}}$, the transverse mass. For three-jet events, the
$\mathrm{q}$$\bar{\mathrm{q}}$$\mathrm{g}$ system no longer forms a linear
system (in the overall centre of mass), but a planar one. There is no event
axis by which the transverse mass and longitudinal proper time are defined.
Therefore we might expect the $\tau$-model, as formulated for a two-jet
system, to work less well.

The results of fits of the $\tau$-model and its first-order expansion, without
and with $R_{\mathrm{a}}$ a free parameter, are shown in Fig. 4b and Table 2.
Applying the $\chi^{2}$-difference test to the order-0 and order-1 fits yields
a confidence level of 37% for the case that $R_{\mathrm{a}}$ is constrained
and 58% when $R_{\mathrm{a}}$ is free.

However, the $\chi^{2}$-difference between the order-0 $R_{\mathrm{a}}$
constrained and free cases yields a confidence level of $6\cdot 10^{-8}$. Thus
regarding $R_{\mathrm{a}}$ as a free parameter does give significant
improvement.

But it must be pointed out that the parameters $R_{\mathrm{a}}$ and $\tau_{0}$
are expected to be highly correlated. While for two-jet events it was found in
Ref. (L3_levy:2011, ) that $\tau_{0}$ is consistent with zero, such studies
have not yet been performed for three-jet events. Further investigation is
ongoing. Also, note that the value of $\alpha$ is significantly less for the
fits with $R_{\mathrm{a}}$ free than for those with $R_{\mathrm{a}}$
constrained. This was not the case for two-jet events. This too requires
additional investigation.

Table 2: Fit results of $\tau$-model parametrizations for three-jet events.

$\begin{array}[]{lcccc}\hline\cr&\text{order 0}&\text{order 1}&\text{order 0,
$R_{\mathrm{a}}$\ free}&\text{order 1, $R_{\mathrm{a}}$\ free}\\\
\hline\cr\alpha&0.42\pm 0.01&0.42\pm 0.01&0.35\pm 0.01&0.35\pm 0.01\\\ R\
\text{(fm)}&0.98\pm 0.04&0.96\pm 0.05&1.06\pm 0.05&1.00\pm 0.11\\\
R_{\mathrm{a}}\ \text{(fm)}&-&-&0.87\pm 0.04&0.96\pm 0.30\\\ \lambda&0.84\pm
0.04&0.81\pm 0.05&0.92\pm 0.05&0.65\pm 0.49\\\ \gamma&0.977\pm 0.001&0.977\pm
0.001&0.997\pm 0.005&0.994\pm 0.007\\\ \epsilon&0.008\pm 0.001&0.008\pm
0.001&0.0003\pm 0.0017&0.001\pm 0.002\\\ c_{1}&-&-0.0005\pm 0.0005&-&-0.04\pm
0.08\\\ \hline\cr\chi^{2}/{\text{DoF}}&113.2/95&112.4/94&83.7/94&83.4/93\\\
\text{CL}&10\%&9\%&77\%&75\%\\\ \hline\cr\end{array}$

##  5 Conclusions for three-jet events

As for the two-jet case, expansion of the $\tau$-model expression does not
lead to significant improvement in the fits. This validates the use of an
asymmetric Lévy distribution for the longitudinal proper time.

However, significant improvement of the fit is obtained by letting
$R_{\mathrm{a}}$ be a free parameter. i.e., by lessening the connection of the
simplified $\tau$-model between the BEC peak and the antisymmetric dip.
Whether letting $\tau_{0}$ also be a free parameter would also give
significant improvement is the subject of ongoing investigation, as is the
question whether $\alpha$ decreases as the number of jets increases.

S. Lökös greatfully acknowledges the support from the ERASMUS project of the
EU and thanks W. Metzger for his kind hospitality at the Radboud University
Nijmegen, The Netherlands. This research was partially supported by the NKTIH
FK 123842 and FK 123 959 grants (Hungary), by the Circles of Knowledge Club
(Hungary) and by the EFOP 3.6.1-16-2016-00001 project (Hungary).

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