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dark matter — galaxies: clusters: general — gravitational lensing: weak —
large-scale structure of universe

# Hundreds of weak lensing shear-selected clusters from the Hyper Suprime-Cam
Subaru Strategic Program S19A data

Masamune Oguri11affiliation: Research Center for the Early Universe,
University of Tokyo, Tokyo, 113-0033, Japan 22affiliation: Department of
Physics, University of Tokyo, Tokyo 113-0033, Japan 33affiliation: Kavli
Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI),
University of Tokyo, Chiba 277-8582, Japan Satoshi Miyazaki44affiliation:
National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan  55affiliation: SOKENDAI (The Graduate University for Advanced
Studies), Mitaka, Tokyo, 181-8588, Japan Xiangchong Li22affiliationmark:
33affiliationmark:  Wentao Luo33affiliationmark:  Ikuyuki
Mitsuishi66affiliation: Division of Physics and Astrophysical Science,
Graduate School of Science, Nagoya University, Nagoya 464-8602, Japan Hironao
Miyatake77affiliation: Institute for Advanced Research, Nagoya University,
Nagoya 464-8601, Japan 66affiliationmark:  33affiliationmark:  Surhud
More88affiliation: The Inter-University Center for Astronomy and Astrophysics
(IUCAA), Post Bag 4, Ganeshkhind, Pune 411007, India 33affiliationmark:
Atsushi J. Nishizawa77affiliationmark:  Nobuhiro Okabe99affiliation:
Department of Physical Science, Hiroshima University, 1-3-1 Kagamiyama,
Higashi-Hiroshima, Hiroshima 739-8526, Japan 1010affiliation: Hiroshima
Astrophysical Science Center, Hiroshima University, 1-3-1 Kagamiyama, Higashi-
Hiroshima, Hiroshima 739-8526, Japan 1111affiliation: Core Research for
Energetic Universe, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima,
Hiroshima 739-8526, Japan Naomi Ota1212affiliation: Department of Physics,
Nara Women’s University, Kitauoyanishi-machi, Nara, Nara 630-8506, Japan
Andrés A. Plazas Malagón1313affiliation: Department of Astrophysical Sciences,
Princeton University, Peyton Hall, Princeton, NJ 08544, USA Yousuke
Utsumi1414affiliation: Kavli Institute for Particle Astrophysics and
Cosmology, SLAC National Accelerator Laboratory, Stanford University, Menlo
Park, CA 94025, USA<EMAIL_ADDRESS>
###### Abstract

We use the Hyper Suprime-Cam Subaru Strategic Program S19A shape catalog to
construct weak lensing shear-selected cluster samples. From aperture mass maps
covering $\sim 510$ deg2 created using a truncated Gaussian filter, we
construct a catalog of 187 shear-selected clusters that correspond to mass map
peaks with the signal-to-noise ratio larger than 4.7. Most of the shear-
selected clusters have counterparts in optically-selected clusters, from which
we estimate the purity of the catalog to be higher than 95%. The sample can be
expanded to 418 shear-selected clusters with the same signal-to-noise ratio
cut by optimizing the shape of the filter function and by combining weak
lensing mass maps created with several different background galaxy selections.
We argue that dilution and obscuration effects of cluster member galaxies can
be mitigated by using background source galaxy samples and adopting the filter
function with its inner boundary larger than about $2^{\prime}$. The large
samples of shear-selected clusters that are selected without relying on any
baryonic tracer are useful for detailed studies of cluster astrophysics and
cosmology.

##  1 Introduction

Clusters of galaxies are the most massive gravitationally bound objects in the
Universe and have proven to be a key class of objects for establishing the
standard cosmological model that consists of dark matter and dark energy (see
e.g., Allen et al., 2011; Kravtsov & Borgani, 2012, for reviews). We can study
the internal structure and statistical property of clusters of galaxies with
multi-wavelength datasets, including optical, X-ray, and radio. For instance,
a massive cluster of galaxies can be securely identified from an overdensity
of cluster member galaxies with similar colors (e.g., Gladders & Yee, 2000).
X-ray from hot gas in clusters of galaxies provides an important means of
finding and studying clusters of galaxies (e.g., Ebeling et al., 2001). Large
samples of clusters are being constructed via the Sunyaev-Zel’dovich effect on
cosmic microwave background fluctuations (e.g., Planck Collaboration et al.,
2016; Hilton et al., 2021).

The abundance and internal structure of clusters of galaxies are mainly
determined by the dynamics of dark matter, which makes it critically important
to study the distribution of dark matter in clusters in great detail. Weak
gravitational lensing directly probes the dark matter distribution in clusters
of galaxies and hence plays a key role in characterizing clusters (see e.g.,
Umetsu, 2020, for a review). Among others, weak gravitational lensing plays an
essential role for the use of the cluster population as a probe of
cosmological parameters, because cluster observables must be linked to cluster
masses in order to compare the observed abundance of clusters with theoretical
predictions (see e.g., Pratt et al., 2019, for a review). Indeed, attempts to
use clusters as an accurate cosmological probe have often been hampered by the
uncertainty of mass calibrations of clusters for which complicated selection
biases in cluster surveys must be taken into account.

A new approach that has been explored less extensively is finding clusters
directly from weak gravitational lensing shear data by identifying peaks in
weak lensing mass maps (e.g., Schneider, 1996; White et al., 2002; Hamana et
al., 2004; Hennawi & Spergel, 2005; Maturi et al., 2005, 2010; Fan et al.,
2010; Marian et al., 2012; Lin et al., 2016). The abundance of the peaks
contains information on the abundance of massive dark matter halos and hence
on cosmological parameters (e.g., Jain & Van Waerbeke, 2000; Dietrich &
Hartlap, 2010; Maturi et al., 2010; Shan et al., 2014; Liu et al., 2015a, b;
Hamana et al., 2015; Kacprzak et al., 2016; Shan et al., 2018). In addition,
the relatively simple and clean selection function of weak lensing shear-
selected clusters (e.g., Hamana et al., 2012; Chen et al., 2020) enables to
use them for better understanding of cluster astrophysics. For instance, the
X-ray analysis of shear-selected clusters suggests that they tend to be X-ray
underluminous compared with clusters found in other techniques (Giles et al.,
2015; Miyazaki et al., 2018a).

However a challenge lies in the requirement of wide and deep imaging for
finding a significant number of weak lensing shear-selected clusters. First
attempts identified only a handful of such clusters, if restricted to those
with a sufficiently high signal-to-noise ratio (see e.g., Appendix A) of
$\gtrsim 5$ (Wittman et al., 2001; Miyazaki et al., 2002; Hetterscheidt et
al., 2005; Wittman et al., 2006; Miyazaki et al., 2007; Schirmer et al., 2007;
Gavazzi & Soucail, 2007; Utsumi et al., 2014). On the other hand, more
systematic search of weak lensing shear-selected clusters is made possible
thanks to recent progress of wide-field imaging surveys. For instance, Shan et
al. (2012) constructed a sample of 51 shear-selected clusters with signal-to-
noise ratios larger than 4.5 from the Canada-France-Hawaii Telescope Legacy
Survey (Heymans et al., 2012).

Hyper Suprime-Cam (HSC; Miyazaki et al., 2018b), which is a wide-field optical
imager mounted on the Subaru 8.2-meter telescope, offers a unique opportunity
for constructing a large sample of weak lensing shear-selected clusters
(Miyazaki et al., 2015). In particular, the HSC Subaru Strategic Program (HSC-
SSP; Aihara et al., 2018b, a, 2019), which is a deep multi-band imaging survey
of 1400 deg2 of the sky, is an ideal survey for this purpose. Miyazaki et al.
(2018a) presented a sample of 65 weak lensing shear-selected clusters with
signal-to-noise ratios larger than 4.7 from the HSC-SSP first year shape
catalog covering $\sim 160$ deg2 (Mandelbaum et al., 2018a). Using the same
HSC S16A data, Hamana et al. (2020) constructed a sample of 124 shear-selected
clusters with signal-to-noise ratios larger than 5 by mitigating the dilution
effect of foreground and cluster member galaxies.

In this paper, we present updated catalogs of weak lensing shear-selected
clusters from the latest HSC-SSP S19A shape catalog covering $\sim 433$ deg2
(Li et al., in prep.). We construct catalogs using three different approaches
adopting different shapes of filters. We also assign redshifts of individual
clusters by cross matching the shear-selected clusters with optically-selected
clusters.

This paper is organized as follows. The data used for our analysis are
summarized in section 2. Our method to construct mass maps is detailed in
section 3. We present our main results in section 4, and conclude in section
5. Throughout the paper we assume the matter density $\Omega_{\rm m}=0.3$, the
cosmological constant $\Omega_{\Lambda}=0.7$, the baryon density $\Omega_{\rm
b}=0.05$, the dimensionless Hubble constant $h=0.7$, the spectral index
$n_{\rm s}=0.96$, and the normalization of the matter power spectrum
$\sigma_{8}=0.81$.

##  2 Data

###  2.1 Weak lensing shape catalog

The HSC-SSP S19A shape catalog (Li et al., in prep.) is constructed in a
manner similar to the HSC-SSP S16A shape catalog presented in Mandelbaum et
al. (2018a) that takes a moment-based approach of Hirata & Seljak (2003). The
multiplicative and additive biases are derived with realistic image
simulations (Mandelbaum et al., 2018b). The detailed systematics tests
presented in Li et al., in prep. indicate that the S19A shape catalog is
sufficiently accurate and is ready for various cosmological and astrophysical
analyses. The catalog contains $\sim 36$ million galaxies.

Photometric redshifts of individual galaxies are also measured with various
methods using the HSC $grizy$-band photometry (see Tanaka et al., 2018, for
photometric redshifts of galaxies in the S16A data). In this paper we may use
subsamples of the S19A shape catalog that are defined based on these
photometric redshifts. Specifically, we define subsamples using the so-called
P-cut method (see Oguri, 2014; Medezinski et al., 2018), for which galaxies
satisfying

$\int_{z_{\rm min}}^{z_{\rm max}}P(z)dz>P_{\rm th},$ (1)

where $P(z)$ the probability distribution function (PDF) of the photometric
redshift of each galaxy, are included. This P-cut allows us to securely and
flexibly select galaxies behind clusters of interest by choosing the
parameters $z_{\rm min}$, $z_{\rm max}$, and $P_{\rm th}$ appropriately (see
Medezinski et al., 2018), and hence mitigate the dilution effect by cluster
member galaxies (see also Hamana et al., 2020). The performance of the P-cut
method is shown to be comparable to the selection of background galaxies in
color-color space (Medezinski et al., 2018).

Throughout the paper we adopt the dNNz photometric redshift measurement
(Nishizawa et al., in prep.) for defining background galaxy samples with the
P-cut method. The dNNz is based on the multiple-layer perceptron that consists
of six hidden layers where each layer contains 100 nodes. Input attributes are
cmodel magnitude, size, and point spread function matched aperture magnitude
in five broad bands, leading to 15 attributes in total, for each galaxy. The
outputs are probabilities of the galaxy lying at the redshift bin spanning
from $z=0$ to $7$ divided into 100 bins. The code is trained to minimize the
total difference between the output probabilities and input delta-function
like probability summed over all training sample and redshift ranges. From the
analysis on the test sample that has not been used for training, it is found
that redshifts are accurate at the bias of $10^{-4}$, the scatter of $3\%$,
and the outlier fraction of less than $10\%$ . The dNNz can avoid the
overfitting even when the part of the data is missing (e.g., due to the lack
of images in some broadband filters), by introducing a dropout layer right
after the input layer with missing rate being 0.2. When applying the dNNz on
the real data, the dNNz with the dropout layer is applied only when the part
of the data is missing, otherwise the dNNz without the dropout layer is
employed for the complete data to maximize the performance of the photometric
redshift measurement.

###  2.2 Optically-selected cluster catalogs

We use several optically-selected cluster catalogs to assign redshifts for
individual peaks in weak lensing mass maps. First, we adopt a cluster catalog
(S20A version 1111While the S20A version 2 CAMIRA cluster catalog for which
photometry corrections to mitigate various photometry issues (Aihara et al.,
2019) are included is currently available, in this paper we use the version 1
catalog that was the latest version when the shear-selected cluster catalogs
were constructed.) constructed with the photometric data from the HSC-SSP S20A
internal data release using the CAMIRA algorithm (Oguri, 2014). In Oguri et
al. (2018a), the CAMIRA algorithm has been applied to the HSC-SSP S16A data
covering $\sim 230$ deg2 to construct a catalog of 1921 clusters at redshift
$0.1<z<1.1$ and richness $N$ greater than 15 (see Oguri, 2014, for the
definition of richness $N$ in the CAMIRA algorithm). In this paper, we adopt
an updated catalog of 8910 clusters at redshift $0.1<z<1.38$ and richness
greater than 15 from the HSC-SSP S20A data covering $\sim 830$ deg2.

Since the HSC-SSP survey region is chosen to overlap with the Sloan Digital
Sky Survey (SDSS; York et al., 2000), we also use cluster catalogs from the
SDSS for assigning cluster redshifts. Specifically, we use the redMaPPer
cluster catalog (Rykoff et al., 2014) that contains clusters at $0.08<z<0.6$
as well as the WHL15 cluster catalog (Wen et al., 2012; Wen & Han, 2015) that
contains clusters at $0.05<z<0.79$. Both of these cluster catalogs were
constructed based on the photometric galaxy catalog covering $\sim 14,000$
deg2 from SDSS Data Release 8 (DR8; Aihara et al., 2011). In addition to these
purely optically-selected clusters, we also adopt the CODEX cluster catalog
(Finoguenov et al., 2020) that contains clusters at $0.05<z<0.69$ from ROSAT
all-sky survey (RASS; Voges et al., 1999) with optical confirmations using the
redMaPPer algorithm applied to the SDSS DR8 data. Throughout the paper we use
X-ray centroids as centers of the CODEX clusters.

All the cluster redshifts we adopt throughout this paper are photometric
redshifts of clusters derived from the HSC $grizy$-band photometry for CAMIRA
or from the SDSS $ugriz$-band photometry for redMaPPer, WHL15, and CODEX. The
typical accuracy of these cluster photometric redshifts is
$\sigma_{z}/(1+z)\sim 0.01$, which is sufficiently accurate for our current
purpose.

##  3 Mass maps

###  3.1 Introduction

Since both convergence $\kappa$ and shear $\gamma$ are derived by the second
derivatives of the lens potential, we can derive a convergence (mass) map from
the shear map by a convolution of the shear map with a kernel (Kaiser &
Squires, 1993). More generally, in the flat sky coordinate
$\boldsymbol{\theta}$, we can construct a map of the aperture mass $M_{\rm
ap}(\boldsymbol{\theta})$ (Schneider, 1996), which is a convergence convolved
with a spatial filter $U$

$M_{\rm ap}(\boldsymbol{\theta})=\int
d\boldsymbol{\theta}^{\prime}\kappa(\boldsymbol{\theta}^{\prime})U(|\boldsymbol{\theta}-\boldsymbol{\theta}^{\prime}|).$
(2)

Provided that the spatial filter is compensated

$\int d\theta\,\theta\,U(\theta)=0,$ (3)

it is equivalent to the convolution of the tangential shear with a kernel $Q$

$M_{\rm ap}(\boldsymbol{\theta})=\int
d\boldsymbol{\theta}^{\prime}\gamma_{+}(\boldsymbol{\theta^{\prime}};\boldsymbol{\theta})Q(|\boldsymbol{\theta}-\boldsymbol{\theta^{\prime}}|),$
(4)

where $\gamma_{+}(\boldsymbol{\theta^{\prime}};\boldsymbol{\theta})$ is the
tangential shear at $\boldsymbol{\theta^{\prime}}$ defined with respect to
$\boldsymbol{\theta}$ and $Q$ is related with $U$ as

$Q(\theta)=\frac{2}{\theta^{2}}\int_{0}^{\theta}d\theta^{\prime}\theta^{\prime}U(\theta^{\prime})-U(\theta).$
(5)

In this paper, we consider two types of the spatial filters. One is a
truncated Gaussian filter, which resembles the one adopted in Miyazaki et al.
(2018a) to construct a shear-selected cluster sample from the HSC S16A data
(see also Hamana et al., 2020). The other is a filter introduced by Schneider
(1996), which we call a truncated isothermal filter throughout the paper and
is designed to optimize the detection of halos from mass maps. In the next
subsections, we describe these filters in more detail.

###  3.2 Truncated Gaussian filter

We use the following kernel function to define the truncated Gaussian filter

$Q(\theta)=\frac{1}{\pi\theta^{2}}\left[1-\left(1+\frac{\theta^{2}}{\theta_{0}^{2}}\right)e^{-\theta^{2}/\theta_{0}^{2}}\right]e^{-\theta^{4}/\theta_{\rm
out}^{4}},$ (6)

which is smoothly truncated at $\theta=\theta_{\rm out}$. Thus the filter is
slightly different from the one adopted in Miyazaki et al. (2018a) and Hamana
et al. (2020) for which the kernel function is sharply truncated at
$\theta=\theta_{\rm out}$. We adopt this smoothly truncated form for the
numerical stability of our approach to derive mass maps using the Fast Fourier
Transform (see subsection 3.4). The corresponding spatial filter $U(\theta)$
can be derived as

$U(\theta)=-Q(\theta)-\int_{0}^{\theta}d\theta^{\prime}\frac{2}{\theta^{\prime}}Q(\theta^{\prime}).$
(7)

Throughout the paper we adopt $\theta_{0}=1\farcm 5$ and $\theta_{\rm
out}=13^{\prime}$ so that resulting signal-to-noise ratios roughly matches
those in Miyazaki et al. (2018a) and in Chen et al. (2020). Following Miyazaki
et al. (2018a), we also do not apply any cut in the source galaxy sample.
Hereafter this set-up is referred to as TG15.

###  3.3 Truncated isothermal filter

Schneider (1996) introduced the following form of the spatial filter for the
aperture mass

$U(\theta)=\left\\{\begin{array}[]{ll}1&(\theta\leq\nu_{1}\theta_{R})\\\
\frac{1}{1-c}\left(\frac{\nu_{1}\theta_{R}}{\sqrt{(\theta-\nu_{1}\theta_{R})^{2}+(\nu_{1}\theta_{R})^{2}}}-c\right)&(\nu_{1}\theta_{R}\leq\theta\leq\nu_{2}\theta_{R})\\\
\frac{b}{\theta_{R}^{3}}(\theta_{R}-\theta)^{2}(\theta-\alpha\theta_{R})&(\nu_{2}\theta_{R}\leq\theta\leq\theta_{R})\\\
0&(\theta_{R}\leq\theta)\end{array}\right.$ (8)

where $c$, $b$, $\alpha$ are determined from the condition that $U(\theta)$
and its first derivative are continuous at $\theta=\nu_{2}\theta_{R}$ as well
as the compensation condition given by equation (3). This means that the shape
of $U(\theta)$ is specified by three parameters, $\nu_{1}$, $\nu_{2}$, and
$\theta_{R}$. The corresponding kernel function is derived using equation (5).

This filter has several desirable properties. First, we can tweak the shape of
the filter quite flexibly by adjusting the three parameters $\nu_{1}$,
$\nu_{2}$, and $\theta_{R}$. Second, it has $Q(\theta)=0$ at
$\theta<\nu_{1}\theta_{R}$ and hence allows us to efficiently remove the
contribution from the innermost part of halos, which is a source of various
systematic effects such as the dilution effect by cluster member galaxies, the
effect of reduced shear, and the magnification bias, to the signal. Third, the
filter is confined within a finite radius (i.e., $U(\theta)=Q(\theta)=0$ at
$\theta>\theta_{R}$) and hence mitigate the impact of e.g., the boundary of
the survey region to the map.

Table 1: Summary of set-ups to construct shear-selected cluster samples.  Name | Filter | Parameter values | Source galaxy selection | Num. of galaxies
---|---|---|---|---
TG15 | Truncated Gaussian |  $\theta_{0}=1\farcm 5$, $\theta_{\rm out}=13^{\prime}$ | No cut | $35804886$
TI05 | Truncated isothermal |  $\nu_{1}=0.021$, $\nu_{2}=0.36$, $\theta_{R}=23\farcm 8$ |  $z_{\rm min}=0.2$, $z_{\rm max}=7$, $P_{\rm th}=0.95$ | 32750421
| Truncated isothermal |  $\nu_{1}=0.025$, $\nu_{2}=0.36$, $\theta_{R}=20\farcm 0$ |  $z_{\rm min}=0.3$, $z_{\rm max}=7$, $P_{\rm th}=0.95$ | 26024748
| Truncated isothermal |  $\nu_{1}=0.027$, $\nu_{2}=0.36$, $\theta_{R}=18\farcm 5$ |  $z_{\rm min}=0.5$, $z_{\rm max}=7$, $P_{\rm th}=0.95$ | 20664165
| Truncated isothermal |  $\nu_{1}=0.027$, $\nu_{2}=0.36$, $\theta_{R}=18\farcm 5$ |  $z_{\rm min}=0.7$, $z_{\rm max}=7$, $P_{\rm th}=0.95$ | 15564223
TI20 | Truncated isothermal |  $\nu_{1}=0.095$, $\nu_{2}=0.36$, $\theta_{R}=21\farcm 1$ |  $z_{\rm min}=0.2$, $z_{\rm max}=7$, $P_{\rm th}=0.95$ | 32750421
| Truncated isothermal |  $\nu_{1}=0.110$, $\nu_{2}=0.36$, $\theta_{R}=18\farcm 2$ |  $z_{\rm min}=0.3$, $z_{\rm max}=7$, $P_{\rm th}=0.95$ | 26024748
| Truncated isothermal |  $\nu_{1}=0.121$, $\nu_{2}=0.36$, $\theta_{R}=16\farcm 6$ |  $z_{\rm min}=0.5$, $z_{\rm max}=7$, $P_{\rm th}=0.95$ | 20664165
| Truncated isothermal |  $\nu_{1}=0.121$, $\nu_{2}=0.36$, $\theta_{R}=16\farcm 6$ |  $z_{\rm min}=0.7$, $z_{\rm max}=7$, $P_{\rm th}=0.95$ | 15564223
Figure 1:  The kernel function $Q(\theta)$, which is used to derive the
aperture mass map (see equation 4), is shown. Solid, dashed, and dash-dotted
lines are $Q(\theta)$ for the TG15, TI05, and TI20 set-ups summarized in Table
1. Note that we adopt slightly different shapes of $Q(\theta)$ for different
source galaxy selections. We show $Q(\theta)$ normalized by its maximum value.

To explore the impact of the different inner boundary of the filter,
$\nu_{1}\theta_{R}$, in this paper we consider two different values of the
inner boundary, $\nu_{1}\theta_{R}=0\farcm 5$, and $2^{\prime}$, which are
referred as TI05 and TI20, respectively. For both TI05 and TI20, we construct
mass maps with four different source galaxy subsamples defined using
photometric redshifts of source galaxies (see subsection 2.1), in order to
enhance the detection efficiency particularly at high redshifts (see also
Hamana et al., 2020). For each inner boundary of the filter and the source
galaxy sample, we carefully choose parameter values of the filter to maximize
the expected signal-to-noise ratio and to mitigate the impact of density
fluctuations along the line-of-sight on cluster finding. The specific
procedure of the optimization of the parameters is detailed in Appendix A.
Table 1 summarizes set-ups to construct shear-selected cluster samples. The
shapes of the kernel function $Q(\theta)$ used in this paper are also
presented in Figure 1.

###  3.4 Practical procedure

Following Oguri et al. (2018b), we adopt the Fast Fourier Transform (FFT) to
derive aperture mass maps from the shape catalog. For a given source galaxy
sample, we first create a shear map in a two-dimensional rectangular grid by a
simple tangent plane projection. Throughout the paper, a pixel scale of the
grid ($\boldsymbol{\theta}$-coordinate) of $\Delta\theta=0\farcm 25$ is
adopted. We use a discrete version of equation (4) with weights to derive an
aperture mass map i.e.,

$M_{\rm
ap}(\boldsymbol{\theta}_{i})=\frac{\left(\Delta\theta\right)^{2}}{W(\boldsymbol{\theta}_{i})}\sum_{j}w_{j}\left\\{\gamma_{+}(\boldsymbol{\theta}_{j};\boldsymbol{\theta}_{i})-c_{+,j}\right\\}Q(|\boldsymbol{\theta}_{i}-\boldsymbol{\theta}_{j}|),$
(9)
$W(\boldsymbol{\theta}_{i})=\frac{\sum_{j}w_{j}(1+m_{j})Q(|\boldsymbol{\theta}_{i}-\boldsymbol{\theta}_{j}|)}{\sum_{j}Q(|\boldsymbol{\theta}_{i}-\boldsymbol{\theta}_{j}|)},$
(10)

where $i$ and $j$ label each grid, $w_{i}$ is the weight of $i$-th grid that
is computed from the sum of weights of sources galaxies in the grid, and
$c_{+,i}$ and $m_{i}$ denote the additive and multiplicative biases in the
$i$-th grid, respectively. We evaluate these summations by FFT with an
appropriate zero padding beyond the boundary. We also note that, due to the
tangent plane projection, the principal axes of the
$\boldsymbol{\theta}$-coordinate are not necessarily aligned with the North
and West directions, even for the HSC-SSP S19A patches (see below) whose areas
are relatively small. Since the shear in the HSC-SSP shape catalog is defined
with respect to the equatorial coordinate system, we rotate the shear so that
the shear is converted to the one defined with respect to the
$\boldsymbol{\theta}$-coordinate. Furthermore, since the operation to create
an aperture mass map (equation 9) is confined within a relatively small sky
area thanks to the compact size of the convolution kernel $Q(\theta)$, the
flat-sky approximation is expect to be locally accurate in deriving each pixel
value of the aperture mass map.

Since the HSC-SSP S19A shape catalog consists of six disjoint patches (XMM,
VVDS, WIDE12H, GAMA09H, GAMA15H, and HECTOMAP), we create mass maps for each
of these patches to search for peaks.

We define signal-to-noise ratios of peaks using local estimates of the shape
noise from the “sigma map” (Oguri et al., 2018b). We derive the sigma map by
randomly rotating the orientations of source galaxies before constructing the
mass map, and repeating this procedure 500 times. The sigma map is given by
the square root of the variance of the randomized mass maps. The signal-to-
noise ratio $\nu$ is defined by the ratio of the peak value of the mass map to
the noise value at the peak position from the sigma map. We note that the
sigma map derived by this procedure includes only the shape noise and hence
does not include cosmic shear from the large-scale structure (see also
Appendix A).

We mask the boundary of the survey region as follows. We first derive the
smoothed number density map by first deriving the pixelized number density
maps and then smoothing it by a Gaussian kernel with a standard deviation of
$8^{\prime}$. The average number density is derived from the smoothed map with
$3\sigma$ clipping. We mask pixels that have values less than 0.5 times the
average number density. We also mask pixels with values of the sigma map more
than 1.5 times higher than the average value.

From the map of the signal-to-noise ratio $\nu$, we select peaks with $\nu\geq
4.7$, which is the threshold adopted also in Miyazaki et al. (2018a). To avoid
double counting of clusters, we discard any peaks that have other peaks with
higher $\nu$ within $4^{\prime}$ from the peaks. For mass maps with the
truncated isothermal filter (TI05 and TI20), we create mass maps with four
different values of $z_{\rm min}$ as shown in Table 1. We create a list of
peaks for each value of $z_{\rm min}$, and combine four lists of peaks with
matching radius of $4^{\prime}$. For a peak that is detected in multiple mass
maps from different source galaxy selections, we adopt the highest value of
$\nu$ among the mass maps as the signal-to-noise ratio of that peak.

Figure 2: The spatial distribution of identified shear-selected clusters i.e.,
peaks in mass maps, in six disjoint patches. Middle green, small red, large
cyan circles show clusters by TG15, TI05, and TI20 set-ups (see Table 1),
respectively. They are overlaid in mass maps created by the Gaussian filter
without the truncation ($\theta_{\rm out}\rightarrow\infty$ in equation 6)
with $\theta_{0}=2^{\prime}$.  Table 2: Summary of shear-selected cluster
samples and matching with optically-selected cluster catalogs. The last column
shows the number of shear-selected clusters that are matched with any of
optically-selected cluster catalogs and thus with redshift assignments.

Name | Num. of clusters | CAMIRA match∗ | redMaPPer match∗ | WHL15 match∗ | CODEX match∗ | Any match
---|---|---|---|---|---|---
TG15 | 187 | 163 (21) | 135 (7) | 177 (50) | 59 (6) | 182
TI05 | 418 | 353 (51) | 271 (11) | 364 (104) | 124 (6) | 392
TI20 | 200 | 173 (33) | 138 (10) | 184 (69) | 79 (3) | 193

∗ The number in parentheses indicates shear-selected clusters that are matched
with multiple optically-selected clusters.

##  4 Results

###  4.1 Shear-selected cluster catalogs and redshift assignments

We construct shear-selected cluster catalogs from peaks with $\nu\geq 4.7$ in
mass maps covering $\sim 510$ deg2 following the procedure described in
subsection 3.4. The catalogs are constructed for the three set-ups (TG15,
TI05, and TI20) summarized in Table 1. The catalogs contain 187, 418, and 200
clusters for TG15, TI05, and TI20, respectively. Compared with the TG15 set-up
adopted in Miyazaki et al. (2018a), we find roughly twice the number of shear-
selected clusters for TI05, because the shape of the kernel function for TI05
follows the expected tangential shear profile more closely than for TG15 and
therefore is more optimal. The small number of shear-selected clusters for
TI20, on the other hand, is due to the removal of the large central region
($<2^{\prime}$) where a significant tangential shear signal is observed for
many clusters. Figure 2 shows the spatial distribution of these clusters from
peaks in mass maps. We note that, even though the numbers of shear-selected
clusters are similar between TG15 and TI20, roughly half of the clusters are
detected both in TG15 and TI20, partly because most of the clusters have
signal-to-noise ratios near the threshold (see below).

To check whether they are indeed associated with concentrations of red
galaxies and to assign redshifts to these clusters, we cross match the shear-
selected cluster catalogs with four optically-selected cluster catalogs, HSC
CAMIRA, SDSS redMaPPer, SDSS WHL15, and SDSS CODEX (see subsection 2.2 for
concise descriptions of these catalogs). We regard any optically-selected
clusters that are located within the physical transverse distance of
1$h^{-1}$Mpc computed at the cluster redshifts from each shear-selected
cluster as matched clusters. It is possible that multiple optically-selected
clusters are matched with a single shear-selected cluster, and in that case we
regard an optically-selected cluster that is located closest to the shear-
selected cluster, whose location refers to a peak position in a mass map, in
terms of the physical transverse distance as a primary match and assign the
photometric redshift of the primarily matched cluster to the shear-selected
cluster.

The result of matching is summarized in Table 2. For TG15 and TI20, $\sim 97$%
of shear-selected clusters have counterparts in optically-selected cluster
catalogs and hence have redshift assignments. The fraction slightly decreases
to $\sim 94$% for TI05. For comparison, we generate a random catalog of shear-
selected clusters by randomly drawing points in unmasked regions of mass maps
with the number density of 5 deg-2, and apply the same method for matching
shear-selected clusters with the optically-selected clusters to find that
$\sim 33\%$ of the random shear-selected clusters are matched with any of
optically-selected clusters. Since this fraction represents the chance
probability of matching with the optically-selected clusters, we can argue
that the true fraction of unmatched shear-selected clusters may be as high as
$3\times 3/2\sim 5$% for TG15 and TI20, and $6\times 3/2\sim 9$% for TI05.
Taking also the incompleteness of our optically-selected cluster samples used
for matching (e.g., no cluster at $z<0.05$) into consideration, we can argue
that the purity of our shear-selected cluster catalogs is higher than 95% for
TG15 and TI20 and more than 91% for TI05. The catalogs of the shear-selected
clusters for TG15, TI05, and TI20 set-ups including the results of cross
matching are shown in Supplementary Tables 1, 2, and 3.

Figure 3: Distributions of the signal-to-noise ratio $\nu$. Solid, dashed, and
dash-dotted lines show distributions for the TG15, TI05, TI20 set-ups
summarized in Table 1, respectively. Thin lines indicates distributions of the
signal-to-noise ratio $\nu$ from B-mode mass map peaks.  Figure 4: Similar to
Figure 3, but distributions of cluster redshift $z_{\rm cl}$ are shown.
Figure 5: Similar to Figure 3, but distributions of the physical transverse
distance $d$ between the shear-selected cluster and the primarily matched
optically-selected cluster are shown. The distribution of $d$ for the case
using the random catalog is also shown by the dotted line for reference.

We show distributions of the signal-to-noise ratio $\nu$ and the redshift
$z_{\rm cl}$ of shear-selected clusters for all the three set-ups in Figures 3
and 4, respectively. Number counts rapidly decrease with increasing $\nu$ and
the redshift distributions are peaked at $z_{\rm cl}\sim 0.2-0.3$, both of
which are consistent with theoretical predictions (see e.g., Miyazaki et al.,
2018a).

While weak lensing mass maps are constructed from E-mode shear, B-mode mass
maps generated from B-mode shear provide an important means of checking the
validity of the analysis (e.g., Utsumi et al., 2014). As a sanity check, we
select mass map peaks from B-mode mass maps adopting the same signal-to-noise
ratio threshold of $\nu\geq 4.7$. The distribution of B-mode mass map peaks
are also shown in Figures 3. In total there are 6, 17, and 3 B-mode mass map
peaks with $\nu\geq 4.7$ for TG15, TI05, and TI20, respectively. We find that
the numbers of B-mode mass map peaks are sufficiently small, $\lesssim 4$%,
compared with those of E-mode mass map peaks, which support the high purity of
our shear-selected cluster catalogs as estimated above.

In Figure 5, we show distributions of the physical transverse distance between
the shear-selected cluster and the primarily matched optically-selected
cluster. We find that in most case the distance is small, $\lesssim
0.3h^{-1}$Mpc. The mean distance is higher for TI20 than in the other set-ups,
which can be understood by the conservative choice of $Q(\theta)$ to remove
the small-scale information (see Figure 1), which naturally leads to the
degraded angular resolution of the resulting mass maps. For comparison, we
also plot the distribution in the case of using the random catalog mentioned
above. Since the spatial distribution of the random catalog is not correlated
with any of optically selected clusters, the resulting distribution indicates
that expected for matching by chance. We find that the distribution for the
random catalog differs considerably from those for shear-selected clusters,
supporting the high purity of our shear-selected cluster catalogs.

Figure 6: Examples of tangential shear profiles and fitting them with an NFW
profile. We show examples for high and low signal-to-noise ratio $\nu$ from
the TI05 catalog. Symbols with errors show observed tangential shear profiles,
and lines indicate best-fit NFW profiles. Errors include only the shape noise.

###  4.2 Weak lensing mass measurements

Following Miyazaki et al. (2018a), we derive weak lensing masses of all the
shear-selected clusters with redshift assignments by fitting their
differential surface density profiles. We use the P-cut method (Oguri, 2014;
Medezinski et al., 2018) to securely select background galaxies for each
shear-selected cluster, adopting $z_{\rm min}=z_{\rm cl}+0.2$ and $P_{\rm
th}=0.98$ in equation (1). We derive the differential surface density profile
$\Delta\Sigma(R)$ by fully taking account of the PDF of the photometric
redshift of each galaxy (see e.g., Medezinski et al., 2018, for a specific
procedure), again adopting the dnnz photometric redshift measurements. For
TG15, we derive differential surface density profiles in the range
$R=[0.3,\,7]h^{-1}$Mpc with a spacing of $\Delta\log R=0.09$. The outer radius
is chosen to be same as that used in Miyazaki et al. (2018a) and Chen et al.
(2020) so that the mass bias derived in Chen et al. (2020) can be applied. We
adopt a slightly larger inner boundary of $R=0.3h^{-1}$Mpc to mitigate the
dilution effect by cluster member galaxies (Medezinski et al., 2018). For TI05
and TI20, we adopt a more conservative radius range of $R=[0.3,\,3]h^{-1}$Mpc.
In all cases, we consider the shape noise and ignore the cosmic shear error,
again to follow the set-up assumed in Chen et al. (2020).

The differential surface density profiles are fitted with an NFW profile
(Navarro et al., 1997), which describes differential surface density profiles
of clusters in numerical simulations reasonably well out to $\sim 10$ times
the virial radius (Oguri & Hamana, 2011). We parameterize the NFW profile by
$M_{\rm 500c}$ and $c_{\rm 500c}$, which describe the mass and concentration
parameter for the critical overdensity of $500$. We restrict the range of the
concentration parameter to $0.5<c_{\rm 500c}<10$ and derive both $M_{\rm
500c}$ and $c_{\rm 500c}$ from fitting to the observed differential surface
density profile of each cluster. We show some examples of our tangential shear
profile fitting in Figure 6.

Figure 7: Weak lensing masses and redshifts of shear-selected clusters in the
TG15 (left), TI05 (middle), and TI20 (right) catalogs. Red filled circles with
$1\sigma$ errorbars show $M_{\rm 500c}$ derived from differential surface
density profile fitting. Blue open circles indicate clusters that are matched
with the CODEX catalog.

The derived weak lensing masses for the TG15, TI05, and TI20 catalogs as a
function of the cluster redshift are shown in Figure 7. The fitting results
are also given in Supplementary Tables 1, 2, and 3. The increasing trend of
weak lensing masses with increasing cluster redshift is theoretically expected
for shear-selected clusters (see e.g., Miyazaki et al., 2018a). The massive
($M_{\rm 500c}\sim 10^{15}h^{-1}M_{\odot}$) cluster at $z_{\rm cl}\sim 0.18$
is the well-known cluster Abell 1689 that has one of the largest Einstein
radii known (e.g., Oguri & Blandford, 2009). Our weak lensing mass estimation
of Abell 1689 is consistent with more careful lensing mass estimates in the
literature (e.g., Umetsu & Broadhurst, 2008; Umetsu et al., 2015).

Figure 8: Weak lensing masses of shear-selected clusters in the TI05 catalog
are compared with X-ray luminosities of CODEX clusters that are matched with
the shear-selected clusters, with the correction of the dimensionless Hubble
parameter $E(z)$. The X-ray luminosities are measured in the rest-frame
$0.1-2.4$ keV (Finoguenov et al., 2020).

In Figure 7, we indicate clusters that are matched with the CODEX catalog.
Only less than half of shear-selected clusters are matched with CODEX clusters
(see also Table 1), which is partly due to the shallow RASS X-ray data that is
used to construct the CODEX catalog (Finoguenov et al., 2020). As a sanity
check, we compare our weak lensing masses with X-ray luminosities provided by
the CODEX catalog. Figure 8 shows the comparison of weak lensing masses with
X-ray luminosities for the TI05 catalog. As expected, we find a good
correlation between the masses and the X-ray luminosities. We note that
deriving the underlying scaling relation requires the correction of selection
biases of both the weak lensing selections in HSC-SSP and the X-ray selection
in RASS. We plan to study X-ray properties of these shear-selected clusters in
more detail using the eROSITA Final Equatorial Depth Survey data that are much
deeper than the RASS X-ray data, which will be reported in a separate paper
(Ramos-Ceja et al., in prep.).

Figure 9:  Stacked number density profiles of source galaxies around CODEX
clusters. Here we show number density profiles relative to those from random
source catalogs with their total numbers matched to total numbers of source
galaxy samples such that the normalized number density profile should be unity
if there is no effect of cluster member galaxies. Different lines show results
for different source galaxy samples summarized in Table 1. From left to right
panels, stacking is conduced around low redshift clusters at $0.05<z_{\rm
cl}<0.2$ with richness $\lambda>25$, intermediate redshift clusters at
$0.2<z_{\rm cl}<0.3$ with $\lambda>30$, and high redshift clusters at
$0.3<z_{\rm cl}<0.4$ with $\lambda>40$. Results for background source galaxy
samples satisfying $z_{\rm min}\geq z_{\rm cl}+0.1$ for all the clusters are
highlighted by thick lines. The narrow and wide shaded regions indicate the
non-zero ranges of the kernel function $Q(\theta)$ for TI20 and TI05,
respectively (see also Figure 1).

###  4.3 Effects of cluster member galaxies

Cluster member galaxies affect weak lensing measurements mainly in two ways.
One is the enhancement of the number density of source galaxies due to the
contribution of cluster member galaxies, which dilutes weak lensing signals.
The other is the diminishment of the number density of source galaxies due to
the obscuration of small galaxies by cluster member galaxies. The former
effect can be mitigated by using only source galaxies located behind clusters
for weak lensing measurements. Since these effects are more pronounced near
centers of clusters, choosing the kernel function $Q(\theta)$ that has a
smaller contribution from small $\theta$ can also mitigate the effects of
cluster member galaxies. Here we check number density profiles of our source
galaxy samples around massive clusters to check how our weak lensing mass maps
are affected by cluster member galaxies.

To reduce the statistical noise, we derive stacked number density profiles of
source galaxies for samples of massive clusters. Purely optically-selected
clusters are not ideal for this purpose, because some of optically-selected
clusters exhibit large off-centering up to $\sim 1$ Mpc, which needs to be
taken into account when interpreting observed stacked number density profiles.
Thus we adopt the CODEX cluster catalog and stack number density profiles
around X-ray centroids of CODEX clusters, because X-ray emission peaks are
expected to be close to halo centers (e.g., Zhang et al., 2019).

Figure 9 shows stacked number density profiles for three CODEX cluster samples
with different cluster redshifts. We also apply the richness cut for each
cluster sample in order to select clusters with many cluster member galaxies,
where the richness $\lambda$ for CODEX clusters is measured using the
redMaPPer algorithm (Rykoff et al., 2014). The richness threshold is
determined so that a sufficient ($\gtrsim 50$) number of clusters are included
in each of the cluster subsamples. We show stacked number density profiles
normalized by those computed with random source galaxy catalogs in order to
highlight effects of cluster member galaxies. We find a clear signature of the
enhancement of number density profiles without any background galaxy
selection. On the other hand, when source galaxies behind clusters are
selected, we see decrements toward cluster centers due to the obscuration by
cluster member galaxies (or magnification effects, see e.g., Chiu et al.,
2020). We find that decrements are negligibly small at $\theta>2^{\prime}$,
suggesting that the TI20 catalog is little affected by cluster member galaxy
obscurations. In contrast, the TG15 and TI05 catalogs are more or less
affected by cluster member galaxies in the sense that the observed signal-to-
noise ratios may be affected by the dilution or obscuration effect due to
cluster member galaxies. Since cluster member galaxies do not contribute to
the signal, the dilution effect enhances the noise and reduce the signal-to-
noise ratio. The obscuration affects both the signal and noise in a
complicated manner. These effects may be needed to be taken into account when
deriving an accurate selection function of shear-selected clusters in the TG15
and TI05 catalogs.

##  5 Conclusion

We have constructed shear-selected cluster catalogs by selecting peaks in weak
lensing aperture mass maps covering $\sim 510$ deg2 reconstructed by the HSC-
SSP S19A shape catalog. Aperture mass maps are constructed using the truncated
Gaussian filter (TG15) as well as the truncated isothermal filter with the
inner boundary of $0\farcm 5$ (TI05) and $2^{\prime}$ (TI20). For TI05 and
TI20, we employ multiple source galaxy subsamples for which galaxies below
redshift $z_{\rm min}$ are removed to improve the efficiency. With the signal-
to-noise ratio threshold of 4.7, our shear-selected cluster catalogs contain
187, 418, and 200 clusters for the TG15, TI05, and TI20 set-ups, respectively.
Cross matching with optically-selected cluster catalogs suggests that the
purity of the catalogs is high, more than 95% for TG15 and TI20 and more than
91% for TI05.

These catalogs represent by far the largest catalogs of shear-selected
clusters to date with such high signal-to-noise threshold, and will be useful
for detailed studies of cluster astrophysics and cosmology. In this paper, we
have demonstrated how the shape of the kernel function for constructing the
aperture mass map can be optimized adopting a flexible functional form of the
filter function proposed by Schneider (1996). In particular, we have found
that it is possible to choose the filter function such that it is almost free
from effects of cluster member galaxies yet can select a sufficiently large
number of clusters. Such a clean shear-selected cluster sample will be useful
for obtaining accurate and robust constraints on cosmological parameters from
the cluster abundance, in contrast to optically-selected clusters for which
constraining power appears to be limited by various systematic effects (Abbott
et al., 2020). We will explore cosmological constraints with shear-selected
clusters in a forthcoming paper.

We thank T. Hamana and K. Umetsu for useful discussions and comments. This
work was supported in part by the World Premier International Research Center
Initiative (WPI Initiative), MEXT, Japan, and JSPS KAKENHI Grant Nos.
JP18K03693, JP20H00181, JP20H05856. This work was supported in part by Japan
Science and Technology Agency (JST) CREST JPMHCR1414, and by JST AIP
Acceleration Research Grant No. JP20317829, Japan.

The Hyper Suprime-Cam (HSC) collaboration includes the astronomical
communities of Japan and Taiwan, and Princeton University. The HSC
instrumentation and software were developed by the National Astronomical
Observatory of Japan (NAOJ), the Kavli Institute for the Physics and
Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High
Energy Accelerator Research Organization (KEK), the Academia Sinica Institute
for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University.
Funding was contributed by the FIRST program from the Japanese Cabinet Office,
the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the
Japan Society for the Promotion of Science (JSPS), Japan Science and
Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK,
ASIAA, and Princeton University.

This paper makes use of software developed for the Large Synoptic Survey
Telescope. We thank the LSST Project for making their code available as free
software at http://dm.lsst.org

This paper is based on data collected at the Subaru Telescope and retrieved
from the HSC data archive system, which is operated by Subaru Telescope and
Astronomy Data Center (ADC) at NAOJ. Data analysis was in part carried out
with the cooperation of Center for Computational Astrophysics (CfCA), NAOJ.

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.

## References

  * Abbott et al. (2020) Abbott, T. M. C., Aguena, M., Alarcon, A., et al. 2020, Phys. Rev. D, 102, 023509
  * Aihara et al. (2011) Aihara, H., Allende Prieto, C., An, D., et al. 2011, ApJS, 193, 29
  * Aihara et al. (2018a) Aihara, H., Armstrong, R., Bickerton, S., et al. 2018a, PASJ, 70, S8
  * Aihara et al. (2018b) Aihara, H., Arimoto, N., Armstrong, R., et al. 2018b, PASJ, 70, S4
  * Aihara et al. (2019) Aihara, H., AlSayyad, Y., Ando, M., et al. 2019, PASJ, 71, 114
  * Allen et al. (2011) Allen, S. W., Evrard, A. E., & Mantz, A. B. 2011, ARA&A, 49, 409
  * Bartelmann (1996) Bartelmann, M. 1996, A&A, 313, 697
  * Chen et al. (2020) Chen, K.-F., Oguri, M., Lin, Y.-T., & Miyazaki, S. 2020, ApJ, 891, 139
  * Chiu et al. (2020) Chiu, I. N., Umetsu, K., Murata, R., Medezinski, E., & Oguri, M. 2020, MNRAS, 495, 428
  * Diemer & Joyce (2019) Diemer, B., & Joyce, M. 2019, ApJ, 871, 168
  * Diemer & Kravtsov (2015) Diemer, B., & Kravtsov, A. V. 2015, ApJ, 799, 108
  * Dietrich & Hartlap (2010) Dietrich, J. P., & Hartlap, J. 2010, MNRAS, 402, 1049
  * Ebeling et al. (2001) Ebeling, H., Edge, A. C., & Henry, J. P. 2001, ApJ, 553, 668
  * Fan et al. (2010) Fan, Z., Shan, H., & Liu, J. 2010, ApJ, 719, 1408
  * Finoguenov et al. (2020) Finoguenov, A., Rykoff, E., Clerc, N., et al. 2020, A&A, 638, A114
  * Gavazzi & Soucail (2007) Gavazzi, R., & Soucail, G. 2007, A&A, 462, 459
  * Giles et al. (2015) Giles, P. A., Maughan, B. J., Hamana, T., et al. 2015, MNRAS, 447, 3044
  * Gladders & Yee (2000) Gladders, M. D., & Yee, H. K. C. 2000, AJ, 120, 2148
  * Hamana et al. (2012) Hamana, T., Oguri, M., Shirasaki, M., & Sato, M. 2012, MNRAS, 425, 2287
  * Hamana et al. (2015) Hamana, T., Sakurai, J., Koike, M., & Miller, L. 2015, PASJ, 67, 34
  * Hamana et al. (2020) Hamana, T., Shirasaki, M., & Lin, Y.-T. 2020, PASJ, 72, 78
  * Hamana et al. (2004) Hamana, T., Takada, M., & Yoshida, N. 2004, MNRAS, 350, 893
  * Hennawi & Spergel (2005) Hennawi, J. F., & Spergel, D. N. 2005, ApJ, 624, 59
  * Hetterscheidt et al. (2005) Hetterscheidt, M., Erben, T., Schneider, P., et al. 2005, A&A, 442, 43
  * Heymans et al. (2012) Heymans, C., Van Waerbeke, L., Miller, L., et al. 2012, MNRAS, 427, 146
  * Hilton et al. (2021) Hilton, M., Sifón, C., Naess, S., et al. 2021, ApJS, 253, 3
  * Hirata & Seljak (2003) Hirata, C., & Seljak, U. 2003, MNRAS, 343, 459
  * Jain & Van Waerbeke (2000) Jain, B., & Van Waerbeke, L. 2000, ApJ, 530, L1
  * Kacprzak et al. (2016) Kacprzak, T., Kirk, D., Friedrich, O., et al. 2016, MNRAS, 463, 3653
  * Kaiser & Squires (1993) Kaiser, N., & Squires, G. 1993, ApJ, 404, 441
  * Kravtsov & Borgani (2012) Kravtsov, A. V., & Borgani, S. 2012, ARA&A, 50, 353
  * Lin et al. (2016) Lin, C.-A., Kilbinger, M., & Pires, S. 2016, A&A, 593, A88
  * Liu et al. (2015a) Liu, J., Petri, A., Haiman, Z., et al. 2015a, Phys. Rev. D, 91, 063507
  * Liu et al. (2015b) Liu, X., Pan, C., Li, R., et al. 2015b, MNRAS, 450, 2888
  * Mandelbaum et al. (2018a) Mandelbaum, R., Miyatake, H., Hamana, T., et al. 2018a, PASJ, 70, S25
  * Mandelbaum et al. (2018b) Mandelbaum, R., Lanusse, F., Leauthaud, A., et al. 2018b, MNRAS, 481, 3170
  * Marian et al. (2012) Marian, L., Smith, R. E., Hilbert, S., & Schneider, P. 2012, MNRAS, 423, 1711
  * Maturi et al. (2010) Maturi, M., Angrick, C., Pace, F., & Bartelmann, M. 2010, A&A, 519, A23
  * Maturi et al. (2005) Maturi, M., Meneghetti, M., Bartelmann, M., Dolag, K., & Moscardini, L. 2005, A&A, 442, 851
  * Medezinski et al. (2018) Medezinski, E., Oguri, M., Nishizawa, A. J., et al. 2018, PASJ, 70, 30
  * Miyazaki et al. (2007) Miyazaki, S., Hamana, T., Ellis, R. S., et al. 2007, ApJ, 669, 714
  * Miyazaki et al. (2002) Miyazaki, S., Hamana, T., Shimasaku, K., et al. 2002, ApJ, 580, L97
  * Miyazaki et al. (2015) Miyazaki, S., Oguri, M., Hamana, T., et al. 2015, ApJ, 807, 22
  * Miyazaki et al. (2018a) —. 2018a, PASJ, 70, S27
  * Miyazaki et al. (2018b) Miyazaki, S., Komiyama, Y., Kawanomoto, S., et al. 2018b, PASJ, 70, S1
  * Navarro et al. (1997) Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, ApJ, 490, 493
  * Oguri (2014) Oguri, M. 2014, MNRAS, 444, 147
  * Oguri & Blandford (2009) Oguri, M., & Blandford, R. D. 2009, MNRAS, 392, 930
  * Oguri & Hamana (2011) Oguri, M., & Hamana, T. 2011, MNRAS, 414, 1851
  * Oguri et al. (2018a) Oguri, M., Lin, Y.-T., Lin, S.-C., et al. 2018a, PASJ, 70, S20
  * Oguri et al. (2018b) Oguri, M., Miyazaki, S., Hikage, C., et al. 2018b, PASJ, 70, S26
  * Planck Collaboration et al. (2016) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, A&A, 594, A27
  * Pratt et al. (2019) Pratt, G. W., Arnaud, M., Biviano, A., et al. 2019, Space Sci. Rev., 215, 25
  * Rykoff et al. (2014) Rykoff, E. S., Rozo, E., Busha, M. T., et al. 2014, ApJ, 785, 104
  * Schirmer et al. (2007) Schirmer, M., Erben, T., Hetterscheidt, M., & Schneider, P. 2007, A&A, 462, 875
  * Schneider (1996) Schneider, P. 1996, MNRAS, 283, 837
  * Shan et al. (2012) Shan, H., Kneib, J.-P., Tao, C., et al. 2012, ApJ, 748, 56
  * Shan et al. (2018) Shan, H., Liu, X., Hildebrandt, H., et al. 2018, MNRAS, 474, 1116
  * Shan et al. (2014) Shan, H. Y., Kneib, J.-P., Comparat, J., et al. 2014, MNRAS, 442, 2534
  * Takahashi et al. (2012) Takahashi, R., Sato, M., Nishimichi, T., Taruya, A., & Oguri, M. 2012, ApJ, 761, 152
  * Tanaka et al. (2018) Tanaka, M., Coupon, J., Hsieh, B.-C., et al. 2018, PASJ, 70, S9
  * Umetsu (2020) Umetsu, K. 2020, A&A Rev., 28, 7
  * Umetsu & Broadhurst (2008) Umetsu, K., & Broadhurst, T. 2008, ApJ, 684, 177
  * Umetsu et al. (2015) Umetsu, K., Sereno, M., Medezinski, E., et al. 2015, ApJ, 806, 207
  * Utsumi et al. (2014) Utsumi, Y., Miyazaki, S., Geller, M. J., et al. 2014, ApJ, 786, 93
  * Voges et al. (1999) Voges, W., Aschenbach, B., Boller, T., et al. 1999, A&A, 349, 389
  * Wen & Han (2015) Wen, Z. L., & Han, J. L. 2015, ApJ, 807, 178
  * Wen et al. (2012) Wen, Z. L., Han, J. L., & Liu, F. S. 2012, ApJS, 199, 34
  * White et al. (2002) White, M., van Waerbeke, L., & Mackey, J. 2002, ApJ, 575, 640
  * Wittman et al. (2006) Wittman, D., Dell’Antonio, I. P., Hughes, J. P., et al. 2006, ApJ, 643, 128
  * Wittman et al. (2001) Wittman, D., Tyson, J. A., Margoniner, V. E., Cohen, J. G., & Dell’Antonio, I. P. 2001, ApJ, 557, L89
  * York et al. (2000) York, D. G., Adelman, J., Anderson, John E., J., et al. 2000, AJ, 120, 1579
  * Zhang et al. (2019) Zhang, Y., Jeltema, T., Hollowood, D. L., et al. 2019, MNRAS, 487, 2578

##  Appendix A Optimization of the truncated isothermal filter

We use signal-to-noise ratios of mass map peaks computed assuming the NFW
profile (Navarro et al., 1997) to optimize parameters of the truncated
isothermal filter presented in subsection 3.3. For each set of parameters, we
compute the aperture mass $M_{\rm ap,NFW}$ at the center of a halo using
equation (2) with the convergence assuming an NFW profile (e.g., Bartelmann,
1996). For each subsample of source galaxies given in Table 1, we fully take
account of the source redshift distribution by adopting the stacked PDF of the
photometric redshift PDFs of all the source galaxies. We also adopt the
fitting form of the mass-concentration relation presented by Diemer & Kravtsov
(2015) and Diemer & Joyce (2019). The signal-to-noise ratio is then computed
as

$\nu_{\rm NFW}=\frac{M_{\rm ap,NFW}}{\sigma_{\rm shape}},$ (11)

where $\sigma_{\rm shape}$ describes the shape noise of the filtered
convergence field that is computed as

$\sigma_{\rm shape}=\sigma_{e}\sqrt{\frac{\pi\int
d\theta\,\theta\,Q^{2}(\theta)}{n_{\rm gal}}},$ (12)

with $\sigma_{e}$ and $n_{\rm gal}$ being the root-mean-square of the
ellipticity and the number density of source galaxies, respectively. In this
calculation we simply assume $\sigma_{e}=0.4$ and $n_{\rm gal}=22$ arcmin-2
before any source galaxy selection so that the resulting noise of the mass map
roughly coincides with that from the real data. For each source galaxy
selection we reduce $n_{\rm gal}$ according to the weighted sum of the number
of galaxies after the source galaxy selection.

While it is customary to define a sample of shear-selected clusters by
applying a threshold to the signal-to-noise ratio that is defined in a manner
similar to equation (11) where only the shape noise is considered, it is known
that the accumulated density fluctuations along the line-of-sight (i.e.,
cosmic shear) also contribute to the noise.

$\sigma_{\rm LSS}=\sqrt{\int\frac{\ell
d\ell}{2\pi}|\hat{U}(\ell)|^{2}C_{\ell}},$ (13)

where $\hat{U}(\ell)$ is the Fourier counterpart of the filter $U(\theta)$ and
the cosmic shear power spectrum $C_{\ell}$ is related to the nonlinear matter
power spectrum $P_{\rm m}(k;z)$ as

$C_{\ell}=\int d\chi\frac{\left[W^{\kappa}(\chi)\right]^{2}}{\chi^{2}}P_{\rm
m}(k=\ell/\chi;z),$ (14) $W^{\kappa}(\chi)=\int_{z}^{\infty}dz_{\rm
s}\frac{4\pi G}{c^{2}}\frac{(\chi_{\rm s}-\chi)\chi}{\chi_{\rm
s}(1+z)^{2}}\bar{\rho}_{\rm m}p(z_{\rm s}),$ (15)

where $\chi$ and $\chi_{\rm s}$ are the comoving radial distances
corresponding to redshift $z$ and $z_{\rm s}$, respectively, and $p(z_{\rm
s})$ denotes the redshift distribution of source galaxies. We use the revised
halofit model of Takahashi et al. (2012) to compute $P_{\rm m}(k;z)$. The
signal-to-noise ratio including the large-scale structure noise is simply
calculated as

$\nu_{\rm NFW,wLSS}=\frac{M_{\rm ap,NFW}}{\sqrt{\sigma_{\rm
shape}^{2}+\sigma^{2}_{\rm LSS}}}.$ (16)

We optimize parameters of the truncated isothermal filter as follows. For each
source galaxy selection listed in Table 1, we consider NFW halos located at
$z=z_{\rm min}-0.1$ with varying halo mass as representative halos detected in
mass maps with the source galaxy selection characterized by $z_{\rm min}$. For
each mass of the NFW halo, we vary $\nu_{1}$, $\nu_{2}$, and $\theta_{R}$ to
search for the optimal set of parameters that maximize $\nu_{\rm NFW,wLSS}$
given by equation (16). Since the combination of $\nu_{1}\theta_{R}$
determines the inner boundary of the filter (see subsection 3.3), in this
paper we consider two cases, $\nu_{1}\theta_{R}=0\farcm 5$ and $2^{\prime}$.
The former is chosen such that the tangential shear at $\theta\sim
1^{\prime}-1\farcm 5$, where the contribution to the signal is large, is
included. The latter removes the significant fraction of the inner part of the
profile from the calculation, and hence is much less affected by various
systematic effects as discussed in subsection 3.3 (see also subsection 4.3).
First we vary all the three parameters with the constraint on
$\nu_{1}\theta_{R}$ to find that $\nu_{\rm NFW,wLSS}$ is generally maximized
for $\nu_{2}\sim 0.3-0.4$. We thus fix $\nu_{2}=0.36$ throughout the paper and
derive the optimal choice of $\nu_{1}$ and $\theta_{R}$, as well as $\nu_{\rm
NFW}$ and $\nu_{\rm NFW,wLSS}$, as a function of the halo mass. We adopt
values of $\nu_{1}$ and $\theta_{R}$ for the halo mass that yields $\nu_{\rm
NFW}\sim 5$, roughly corresponding to threshold of constructing shear-selected
clusters used in the literature. Parameters determined by this procedure for
each $\nu_{1}\theta_{R}$ and the source galaxy selection are presented in
Table 1.

##  Appendix B Supporting Information

Supplementary Tables 1, 2, and 3 are available online.


# Cancellation of the sigma mode in the thermal pion gas by quark Pauli
blocking

David Blaschke<EMAIL_ADDRESS>Institute of Theoretical Physics,
University of Wroclaw, Max Born Pl. 9, 50-204 Wroclaw, Poland  Alexandra
Friesen<EMAIL_ADDRESS>Joint Institute for Nuclear Research, 141980,
Dubna, Russia  Dubna State University, 141982, Dubna, Russia  Yuriy Kalinovsky
<EMAIL_ADDRESS>Joint Institute for Nuclear Research, 141980, Dubna, Russia
Dubna State University, 141982, Dubna, Russia

###### Abstract

We calculate the pressure of the interacting pion gas using the Beth-Uhlenbeck
approach to the relativistic virial expansion with Breit-Wigner phase shifts
for the $\sigma$\- and $\varrho$\- meson resonances. The repulsive phase shift
$\delta^{2}_{0}$ is taken from quark interchange model of Barnes and Swanson
[Phys. Rev. D 46 (1992) 131] in very good agreement with experimental data. In
this work we show that the cancellation of the attractive (I = 0) and
repulsive (I = 2) isospin channel contributions to the scalar $\pi-\pi$
interaction in the low-energy region that is known for the vacuum phase
shifts, takes place also at finite temperature. This happens despite the
strong medium dependence of these phase shifts that enters our model by the
temperature dependence of the $\sigma$\- meson and constituent quark masses
because for these masses the relation $M_{\sigma}(T)\approx 2m(T)$ holds and
the scattering length approximation is valid as long as the strong decay
channel $\sigma\to\pi\pi$ is open. Exploiting the Nambu–Jona-Lasinio model for
describing the dynamical breaking of chiral symmetry in the vacuum and its
restoration at finite temperature, we justify with our approach that the
$\sigma$-meson should be absent from the hadron resonance gas description at
low temperatures because the above cancellation holds. However, since this
cancellation breaks down in the vicinity of the hadronization transition,
where due to chiral symmetry restoration the decay channel $\sigma\to\pi\pi$
closes and the $\sigma$\- meson becomes a good resonance, the latter should be
included into the statistical model description of chemical freeze-out in
heavy-ion collisions.

##  I Introduction

The thermodynamics of the hadron resonance gas (HRG) is of crucial importance
for the interpretation of the results of the ab-initio evaluation of the QCD
partition function by simulations of the lattice gauge theory as well as for
the explanation of yields of hadrons produced in ultrarelativistic heavy-ion
collisions at the chemical freeze-out. This HRG model makes the simplifying
assumption that instead of accounting for the interactions among hadrons one
may just evaluate the statistical sum over all known resonances in the
spectrum of hadrons as eigenstates of the QCD Hamiltonian. While such a
procedure would account for the attractive interaction leading to the
formation of resonances in the spectrum, one may additionally account for
repulsion by invoking an excluded volume of the hadrons treated, e.g., as hard
spheres Venugopalan and Prakash (1992).

A simple model system for the study of the interplay between attraction and
repulsion in hot hadronic matter is the interacting pion gas at finite
temperature. For its description one can use the Beth-Uhlenbeck approach with
the well measured phase shifts of the pion-pion interactions in free space
that allow to evaluate the second virial coefficient of the partition function
Welke et al. (1990). It has soon be realized Venugopalan and Prakash (1992)
that the contributions from phase shifts in the attractive isospin-zero
$\sigma$-meson channel ($\delta_{0}^{0}$) and in the repulsive isospin-2
channel ($\delta_{0}^{2}$) largely compensate each other in the partition
function. This cancellation has also been obtained within the Nambu-Jona-
Lasinio (NJL) model description of pion-pion scattering lengths at finite
temperature Quack et al. (1995) where it could be traced to the manifest
chiral symmetry in this field-theoretic quark model of the interacting pion
gas. This model reproduces the Weinberg relations for the scattering lengths
which are the basis for the cancellation. On the quark level of description,
the repulsion in the $\pi\pi$ scattering is due to the quark exchange
interaction between pions, represented on the quark one-loop level by the so-
called box diagrams. It can be shown that this quark exchange interaction,
also denoted as quark Pauli blocking, leads to a repulsive phase shift well in
accordance with the experimental data Barnes and Swanson (1992).

The fact of the cancellation of the scalar isoscalar $\sigma$-meson (f0(500))
against the repulsion in the scalar isotensor channel has recently been
rediscovered in Broniowski et al. (2015) when it was discussed whether the
$\sigma$ meson should be included to the HRG thermodynamics. The authors
concluded that one should not incorporate f0(500) in standard HRG models for
studies of isospin averaged quantities Broniowski et al. (2015); Giacosa et
al. (2016). They also suggest that a similar cancellation should take place in
the corresponding channels for kaon interactions with the $\kappa$ as the
chiral partner state of the kaon, see also Friman et al. (2015) for a
formulation within the S-matrix approach Dashen et al. (1969).

On the other hand, in Andronic et al. (2009) it was pointed out that the
inclusion of the $\sigma$-meson to the HRG gives a significant improvement of
the statistical model description of the ”horn” structure in the beam energy
dependence of the kaon to pion ratio $K^{+}/\pi^{+}$. The question arises
whether both statements can be true simultaneously. Namely, that on the one
hand in the Beth-Uhlenbeck (or S-matrix) formulation of the thermodynamics of
the interacting HRG a cancellation of the attractive $\delta_{0}^{0}$ channel
($\sigma$-meson) against the repulsive $\delta_{0}^{2}$ channel (Pauli
blocking) occurs, and that on the other hand a sigma meson has to be
considered as an important degree of freedom in the statistical model of
particle production where a sudden (chemical) freeze-out of particle species
occurs in the vicinity of the QCD chiral restoration/quark deconfinement
temperature.

We want to answer this question affirmative in the following way. When
formulating the hadron resonance gas in the S-matrix formalism one has to take
into account the medium effects on the scattering phase shifts, i.e. to apply
the generalized Beth-Uhlenbeck approach. This should then in particular take
into account the effects of the chiral symmetry restoration, namely that the
sigma meson becomes degenerate with the pion (parity doubling) above the
chiral restoration transition that entails dropping quark masses. Before the
sigma meson mass becomes degenerate with that of the pion it has to cross the
two-pion mass threshold where the strong two-pion decay channel of the sigma
meson closes and it becomes a sharp resonance Volkov et al. (1998a). For a
resonance at the threshold the scattering length approximation breaks down,
while a Breit-Wigner ansatz for the phase shifts will be appropriate. The
dropping quark masses entail a Mott effect for both, the pion and the sigma.
This character change of the mesons from bound states to resonances in the
continuum leads to the ceasing of the quark Pauli blocking effect and thus the
resulting phase shift should turn to zero.

So, the cancellation of contributions from the S-wave resonances in the
attractive (I = 0) and the repulsive (I = 2) channels to the thermodynamics of
the pion gas discussed on the basis of the Beth-Uhlenbeck approach to the
second virial coefficient which uses the known free-space phase shifts
$\delta_{l}^{I}(\sqrt{s_{\pi\pi}})$ in the dominant channels
$(l,I)=(0,0),(1,1),(0,2)$ can be applicable also at finite temperature. This
finite-temperature cancellation can be seen in the framework of the NJL model
Quack et al. (1995), where the pion-pion scattering lengths were obtained at
finite temperature.

The goal of this work is to show how the $\sigma$-meson cancellation works at
finite temperature when one uses the NJL model results for the temperature
dependence of the $\sigma$-meson mass and width parameter in the Breit-Wigner
ansatz for $\delta_{0}^{0}$ phase shift on the one hand and the temperature
dependent quark mass in the non-relativistic quark exchange model for
$\delta_{0}^{2}$ channel on the other.

The paper is organised as follows. We start from the Breit-Wigner phase shifts
for the $\delta_{1}^{1}$ ($\rho$-meson) and $\delta_{0}^{0}$ ($\sigma$-meson)
channels Welke et al. (1990) and use the result of the nonrelativistic
potential model calculation of the repulsive $\delta_{0}^{2}$ channel from the
quark-exchange Born diagrams Barnes and Swanson (1992) that takes into account
the quark substructure of the pion. In Sec. III the temperature dependence of
the mass spectra and phase shifts is calculated in the framework of the NJL
model. The results are presented and discussed in Sec. IV.

##  II The interacting pion gas

###  II.1 Beth-Uhlenbeck equation and cancellation of the $\sigma$ meson

The virial expansion of the grand canonical partition function of the system
with the known S-matrix for two-particle scattering can be written as a virial
expansion Dashen et al. (1969); Venugopalan and Prakash (1992)

${\rm ln}Z={\rm
ln}Z_{0}+\sum_{i_{1},i_{2}}z_{1}^{i_{1}}z_{2}^{i_{2}}b(i_{1},i_{2}),$ (1)

where $z_{j}=\exp(\beta\mu_{j})$ for $j=1,2$ and $b(i_{1},i_{2})$ is the
second virial coefficient defined by the S-matrix with labels $i_{1},i_{2}$
referring to a channel of the S-matrix with $i_{1}+i_{2}$ particles in initial
state.

$b(i_{1},i_{2})=\frac{V}{4\pi i}\int\frac{d^{3}P}{(2\pi)^{3}}\int
d\epsilon\exp(-\beta\sqrt{P^{2}+\epsilon^{2}}){\rm
Tr}_{i_{1},i_{2}}\left[AS^{-1}\frac{\overleftrightarrow{\partial}}{\partial\epsilon}S\right].$
(2)

Here $\beta=T^{-1}$ is the inverse temperature, V is the volume, P is the
centre-of-mass (total) momentum and $\epsilon$ the energy of the two-particle
system. The symbol $A$ denotes the symmetrization/antisymmetrization operator
for a system of bosons/fermions. The trace is taken over all combinations of
particle number. For a one-component system under the assumption that hadrons
interact mainly via elastic collisions, the second virial coefficients of Eq.
(2) can be simplified by chosing the representation of the S-matrix in terms
of the two-particle phase shifts. The lowest virial coefficient $b_{2}$
corresponds to the case $i_{1}=i_{2}=1$. Then Eq. (2) can be written as

$b_{2}=\frac{1}{2\pi^{3}\beta}\int_{M}^{\infty}d\epsilon\epsilon^{2}K_{2}(\beta\epsilon)\sum_{l,I}{{}^{\prime}}g_{l,I}\frac{\partial\delta_{l}^{I}(\epsilon)}{\partial\epsilon},$
(3)

with the modified Bessel function $K_{2}$ and the degeneracy factor
$g_{l}^{I}=(2l+1)(2I+1)$. $M$ is the invariant mass of the interacting pair at
threshold. For given $l$ the sum over $I$ is restricted to values consistent
with statistics. In the limit that only the second virial coefficient is
considered, the interaction pressure (as well as all the other thermodynamic
variables) can be obtained from Eq. (1) in the form of a Beth-Uhlenbeck
equation,

$P_{int}=Tz^{2}b_{2}=\frac{z^{2}}{2\pi^{3}\beta^{2}}\int_{M}^{\infty}d\epsilon\epsilon^{2}K_{2}(\beta\epsilon)\sum_{l,I}{{}^{\prime}}g_{l,I}\frac{\partial\delta_{l}^{I}(\epsilon)}{\partial\epsilon}.$
(4)

For the one component pion gas, the total center of mass energy is chosen as
$\epsilon=2(q^{2}+M_{\pi}^{2})^{1/2}$ and the threshold mass is $M=2M_{\pi}$
in Eqs. (3) and (4). For the case $\delta_{l}^{I}\rightarrow 0$ at low
energies $\epsilon\rightarrow M$ the equation for the second virial
coefficient $b_{2}$ can be simplified using integration by parts to

$\displaystyle b_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{3}}\int_{2M_{\pi}}^{\infty}d\epsilon\epsilon^{2}K_{1}(\beta\epsilon)\sum_{l,I}{{}^{\prime}}g_{l,I}\delta_{l}^{I}(\epsilon).$
(5)

The corresponding pressure contribution from the two-particle interactions is

$\displaystyle P_{int}$ $\displaystyle=$ $\displaystyle
P^{00}+P^{02}+P^{11}~{},$ (6)

where the partial pressure contributions at vanishing chemical potential
($z=1$) are defined by the phase shift,

$\displaystyle P^{lI}$ $\displaystyle=$
$\displaystyle\frac{g_{l,I}}{2\pi^{3}\beta}\int_{2M_{\pi}}^{\infty}d\epsilon\epsilon^{2}K_{1}(\beta\epsilon)\delta_{l}^{I}(\epsilon).$
(7)

The second virial coefficient $b_{2}$ and therefore the thermodynamic
functions are . The phase shift in free space (vacuum) can be obtained from
experiments and be compared with theoretical models for it. The latter could
then also be used to predict its in-medium modifications. The low-energy
$\pi\pi$ interaction includes the
$\delta_{0}^{0},\delta_{1}^{1},\delta_{0}^{2}$ phase shifts. The repulsive
isospin-2, S-wave phase shift can be well described in the scattering length
approximation by $\delta_{0}^{2}=-0.12q/M_{\pi}$. The phase shift
$\delta_{1}^{1}$ contains the $\rho$-meson resonance and $\delta_{0}^{0}$
contains the $\sigma$-meson resonance. These resonant phase shifts can be
chosen in the simple Breit-Wigner form Welke et al. (1990),

$\displaystyle\delta_{0}^{0}(\epsilon)$ $\displaystyle=$
$\displaystyle\frac{\pi}{2}+\arctan\left(\frac{\epsilon-
M_{\sigma}}{\Gamma_{\sigma}/2}\right),$ (8)
$\displaystyle\delta_{1}^{1}(\epsilon)$ $\displaystyle=$
$\displaystyle\frac{\pi}{2}+\arctan\left(\frac{\epsilon-
M_{\rho}}{\Gamma_{\rho}/2}\right).$ (9)

For the case $M_{\pi}=0.138$ GeV, $M_{\sigma}=5.8~{}M_{\pi}$,
$M_{\rho}=5.53~{}M_{\pi}$ and

$\displaystyle\Gamma_{\sigma}$ $\displaystyle=$
$\displaystyle\beta_{\sigma}q,$ (10) $\displaystyle\Gamma_{\rho}$
$\displaystyle=$
$\displaystyle\beta_{\rho}q\left(\frac{q/M_{\pi}}{1+(q/M_{\rho})^{2}}\right)^{2},$
(11)

with $\beta_{\sigma}=2.06$ and $\beta_{\rho}=0.095$. In this approximation,
the well-known picture for the phase shifts
$\delta_{0}^{0},\delta_{1}^{1},\delta_{0}^{2}$ is obtained, see Fig. 1.

In the work Barnes and Swanson (1992) the phase shift for $I=2$ low-energy
$\pi\pi$ scattering was obtained within a diagrammatic approach to the quark
exchange interaction between $q\bar{q}$ mesons in the framework of a
nonrelativistic potential model for their quark substructure, see also
Blaschke and Röpke (1993). The phase shift, obtained by the so-called quark
Born diagrams has the form

$\sin\delta_{0}^{2}=-\left\\{\frac{\alpha_{s}}{9\lambda
m^{2}}{\frac{\sqrt{q^{2}+M_{\pi}^{2}}}{q}}\left(1-e^{-2\lambda
q^{2}}+\frac{16\lambda q^{2}}{3\sqrt{3}}e^{-4\lambda
q^{2}/3}\right)\right\\}.$ (12)

The parameter $\alpha_{s}$ is fixed to the value $\alpha_{s}/m^{2}=4.48$ GeV-2
so that $\alpha_{s}=0.71$ for $m=0.40$ GeV. The parameter $\lambda$ is related
to the Gaussian wave function of the nucleon in the usual simple-harmonic-
oscillator (SHO) model with the quark-model parameter
$\beta_{SHO}=1/(2\sqrt{\lambda})=0.337$ Barnes and Swanson (1992). The
resulting phase shift $\delta_{0}^{2}$ is shown in Fig. 1 as a blue dashed
line.

Figure 1: Left panel: $\pi\pi$-scattering phase shifts as functions of the
total center-of-mass-momentum. The blue dashed line corresponds to the phase
shift $\delta_{0}^{2}$ calculated using Eq. (12). Right panel: Interaction
contribution to the pressure according to the Beth-Uhlenbeck formula with the
medium-independent phase shifts of the left panel (without the contribution
from the $\rho$-meson channel). The almost perfect cancellation of the
$\sigma$-meson (green dashed line) by the quark Pauli blocking (red dashed-
dotted line) is demonstrated by the blue solid line.

Inserting the phase shifts to the Beth-Uhlenbeck formula (7), one observes an
almost perfect cancellation of the sigma meson by the quark Pauli blocking,
see the right panel of Fig. 1. This result has been obtained under the
assumption that the phase shifts are independent of the temperature which is
no longer valid when approaching the pseudocritical temperature $T_{c}$, where
chiral symmetry gets restored and the pion undergoes a Mott dissociation. This
pion dissociation effect on the Pauli blocking phase shift $\delta_{0}^{2}$
can be accounted for by multiplying it with a factor

$c(T)=\begin{cases}|2m(T)-M_{\pi}(T)|/|2m(0)-M_{\pi}(0)|,&\text{if $T<T_{{\rm
Mott},\pi}$}.\\\ 0,&\text{otherwise}.\end{cases}$ (13)

that quantifies the reduction of the binding energy when approaching the Mott
temperature. The relation to the fourth power of the pion-quark-antiquark
coupling $g_{\pi qq}(T)=|2m(T)-M_{\pi}(T)|^{1/4}$ is plausible because the
Pauli blocking effect is encoded in the quark box diagram for the four pion
vertex, see Eq. (3.4) of Ref. Hüfner et al. (1996).

###  II.2 Analytic justification for the cancellation

The fact of the $\sigma$ cancellation at low energies and the robustness of
this result at finite temperatures despite the in-medium modification of the
phase shifts can be understood also analytically in the low-momentum expansion
of the phase shifts of Eqs. (8) and (12) that introduces the scattering
lengths $a^{I}_{l}$,

$\frac{d\delta^{i}_{l}}{dq}\bigg{|}_{q=0}\approx a^{I}_{l}.$ (14)

The $\sigma$ meson scattering length is then

$a^{0}_{0}=\frac{\beta_{\sigma}}{2(M_{\sigma}-2M_{\pi})},~{}~{}\beta_{\sigma}=2.06,$
(15)

which becomes singular in the vicinity of the chiral transition, when
$M_{\sigma}\to 2M_{\pi}$ and the dominant $\sigma\to 2\pi$ decay channel
closes so that the $\sigma$ meson becomes a good resonance at the threshold.
This is the limitation for the present considerations because the scattering
length approximation breaks down in this case.

The scattering length for the phase shift of the quark exchange process (quark
Pauli blocking) from the Eq.(12) gives

$a^{2}_{0}=-\frac{2}{9}\left(1+\frac{8}{3\sqrt{3}}\right)\alpha_{s}\frac{M_{\pi}}{m^{2}},$
(16)

so that we obtain for the relevant ratio of scattering lengths

$\displaystyle\frac{a^{0}_{0}}{5a^{2}_{0}}$ $\displaystyle=$
$\displaystyle\frac{1.03}{-2M_{\pi}+M_{\sigma}}\frac{1}{5}\left(-\frac{0.565\,\alpha_{s}M_{\pi}}{m^{2}}\right)^{-1}=-\frac{1.03}{-2M_{\pi}^{2}+M_{\sigma}M_{\pi}}\frac{m^{2}}{2.82\alpha_{s}}$
(17) $\displaystyle=$
$\displaystyle\frac{1.03}{2(M_{\pi}-M_{\sigma}/4)^{2}-M_{\sigma}^{2}/8}\frac{m^{2}}{2.82\alpha_{s}}\approx-\frac{4m^{2}}{1.41\,\alpha_{s}M_{\sigma}^{2}},$

where in the last step the approximation $(m_{\pi}-m_{\sigma}/4)^{2}\approx 0$
has been used.

The wanted result of the cancellation by quark Pauli blocking of the $\sigma$
meson contribution to the thermodynamics of the pion gas is obtained when
$\alpha_{s}\sim 0.7$. Then, because $M_{\sigma}\approx 2m$, the result is
$a^{0}_{0}/5a^{2}_{0}=-0.99$, which means the total cancellation of
contributions from $\sigma-$channel by the contribution from the repulsive
Pauli-blocking channel. For the parameter set given by Welke Welke et al.
(1990) $\delta^{0}_{0}/5\delta^{2}_{0}=-1.18$ and for the Weinberg results
$a_{0}=0.158$ and $a_{2}=-0.045$ follows $a^{0}_{0}/5a^{2}_{0}=-0.708$
Weinberg (1966).

We observe that the result of the cancellation of the $\sigma$ meson as a
resonance in the interacting pion gas by quark Pauli blocking depends only on
the relation $M_{\sigma}=2m$ between the masses of the $\sigma$ and the quark
as well as the approximate validity of $(M_{\pi}-M_{\sigma}/4)^{2}\approx 0$.
Therefore, one can expect that it should hold also when going beyond the
standard Beth-Uhlenbeck approach and taking into account an explicit
temperature dependence of the phase shifts, as long as the above relations
between quark and meson masses remain intact. In the following, we consider
the Nambu – Jona-Lasinio model for describing the temperature dependence of
quark and meson masses as well as meson decay constants, so that the limits of
the cancellation effect can be estimated.

##  III Mesons at finite temperature in the Nambu – Jona-Lasinio model

In Eqs. (5) and (7) the dependence on the temperature appears due to the
parameter $\beta=1/T$ only and it is assumed there that other quantities (like
masses and $\delta_{l}^{I}$) are temperature independent. Since the
constituent quark mass in (12) is generated by dynamical chiral symmetry
breaking in the vacuum, one expects its approximate restoration at finite
temperatures (and densities). This effect of chiral symmetry restoration
($\chi$SR) in a hot and dense medium can be modeled with the Nambu-Jona -
Lasinio (NJL) model Klevansky (1992).

We employ here the NJL model with two flavours of quarks defined by the
Lagrangian

$\mathcal{L}_{\rm
NJL}=\bar{q}\left(i\gamma_{\mu}\partial^{\mu}-\hat{m}_{0}\right)q+G_{s}\left[\left(\bar{q}q\right)^{2}+\left(\bar{q}i\gamma_{5}\vec{\tau}q\right)^{2}\right]-G_{\rm
v}\left[(\bar{q}\gamma^{\mu}\tau^{a}q)^{2}+(\bar{q}\gamma^{\mu}\gamma^{5}\tau^{a}q)^{2}\right],$
(18)

with chirally symmetric four-quark interactions in the scalar, pseudo-scalar,
vector and axial-vector channels. $G_{s}$ and $G_{\rm v}$ are the scalar and
vector coupling constants, $\bar{q}$ and $q$ \- the quark spinor fields,
$\hat{m}_{0}$ is the diagonal matrix of the current quark mass,
$\hat{m}_{0}={\rm diag}(m_{u}^{0},m_{d}^{0})$ with
$m_{u}^{0}=m_{d}^{0}=m_{0}$, and $\overrightarrow{\tau}$ are the SU(2) Pauli
matrices in flavor space with the components $\tau^{a}(a=1,2,3)$.

In the mean-field approximation the constituent quark mass is obtained by
solving the gap equation at the mean field level

$\displaystyle
m=m_{0}+8G_{s}N_{c}N_{f}\int_{\Lambda}\dfrac{d^{3}p}{(2\pi)^{3}}\dfrac{m}{E_{p}}\left[1-f(E^{+})-f(E^{-})\right]~{},$
(19)

where the dependence on temperature and chemical potential is modeled in the
Fermi-functions $f(E^{\pm})=(1+e^{\beta E_{p}^{\pm}})^{-1}$ and the quark
(antiquark) energy dispersion relation $E_{p}^{\pm}=E_{p}\pm\mu$.

Mesons are considered as quark-antiquark bound states and their properties are
described by the Bethe-Salpeter equation in the pole approximation

$1-2G_{s}\ \Pi_{M}(k^{2})|_{k^{2}=M_{M}^{2}}=0~{}~{},~{}M=\pi,\sigma,$ (20)

with the polarization operator $\Pi_{M}(k^{2})$ determining the meson
properties being defined as

$\Pi_{M}(k^{2})=i\int\frac{d^{4}p}{(2\pi)^{4}}\
\mbox{Tr}\,\left[\Gamma_{M}S(p+k)\Gamma_{M}S(p)\right],$ (21)

where the vertex factor $\Gamma_{M}$ depends on the meson species $M=\pi$,
$\sigma$, $\rho$ and $a_{1}$. For the pseudo-scalar $\pi$ meson
$\Gamma_{\pi}=i\gamma_{5}\tau^{a}$ and for the scalar $\sigma$ meson
$\Gamma_{\sigma}={\bf 1}\tau^{a}$; $S(q)$ is the quark propagator and the
trace is being taken over color, flavor and spinor indices. For mesons at rest
(${\mathbf{P}}=0$) in the medium, these conditions correspond to the
equations:

$\displaystyle
1+8G_{s}N_{c}N_{f}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{E_{p}}\frac{E_{p}^{2}}{M_{\pi}^{2}-4E_{p}^{2}}\left(1-f(E^{+})-f(E^{-})\right)$
$\displaystyle=$ $\displaystyle 0,$ (22) $\displaystyle
1+8G_{s}N_{c}N_{f}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{E_{p}}\frac{E_{p}^{2}-m^{2}}{M_{\sigma}^{2}-4E_{p}^{2}}\left(1-f(E^{+})-f(E^{-})\right)$
$\displaystyle=$ $\displaystyle 0~{}.$ (23)

Both pion-quark $g_{\pi qq}(T,\mu)$ and sigma-quark $g_{\sigma qq}(T,\mu)$
coupling strengths can be defined from $\Pi_{M}$ by the residuum of the mass
pole approximation (20)

$\displaystyle g_{Mqq}^{-2}(T,\mu)=\frac{\partial\Pi_{M}(k^{2})}{\partial
k^{2}}|_{{k^{2}=M_{M}^{2}}}~{}~{},~{}M=\pi,\sigma.$ (24)

Generally, the pole mass equation (20) can be extended to the vector and
axial-vector case and the set of equations Eqs. (19), (22), (23) together with
an equation for the vector meson is solved self-consistently. To simplify the
calculations for the $\rho$ meson, the relations suggested in the works Volkov
et al. (1998b); Volkov and Radzhabov (2006); Ebert et al. (1993) are used. The
mass of the $\rho$-meson and its width can be calculated as

$\displaystyle M_{\rho}^{2}$ $\displaystyle=$ $\displaystyle\frac{g_{\rho
qq}^{2}}{4G_{\rm v}},$ (25) $\displaystyle g_{\rho qq}$ $\displaystyle=$
$\displaystyle\sqrt{6}g_{\sigma qq},$ (26) $\displaystyle\Gamma_{\rho\pi\pi}$
$\displaystyle=$ $\displaystyle\frac{g_{\rho\pi\pi}^{2}}{48\pi
M_{\rho}^{2}}\sqrt{\left(M_{\rho}^{2}-4M_{\pi}^{2}\right)^{3}}.$ (27)

The decay width $\Gamma_{\sigma\pi\pi}$ within the NJL model is defined by the
triangle Feynman diagram treating the sigma meson as quark-antiquark system
Zhuang and Yang (2001); Friesen et al. (2012):

$\displaystyle\Gamma_{\sigma\pi\pi}=\frac{3}{2}\ \frac{(2g_{\sigma
qq}g^{2}_{\pi qq}A_{\sigma\pi\pi}(T,\mu))^{2}}{16\pi\
M_{\sigma}}\sqrt{1-\frac{4M_{\pi}^{2}}{M_{\sigma}^{2}}}~{},$ (28)

where the factor 3/2 takes into account the isospin conservation and
$g_{\sigma qq}$ and $g_{\pi qq}$ are coupling constants defined from Eq. (24).
The amplitude of the triangle vertex $A_{\sigma\rightarrow\pi\pi}$ is

$\displaystyle A_{\sigma\pi\pi}=\int\frac{d^{4}q}{(2\pi)^{4}}\ Tr\\{S(q)\
\Gamma_{\pi}\ S(q+P)\ \Gamma_{\pi}\ S(q)\\}.$ (29)

The kinematic factor $\sqrt{1-{4M_{\pi}^{2}}/{M_{\sigma}^{2}}}$ in (28) leads
to the constraint $M_{\sigma}>2M_{\pi}$, if this condition is broken, the
decay $\sigma\rightarrow\pi\pi$ is forbidden and the $\sigma$ meson becomes a
good bound state of the $\pi\pi$ interaction with only a negligible
electromagnetic decay width from the $\sigma\to\gamma\gamma$ process Volkov et
al. (1998a).

In order to describe the mass spectra in the NJL model, a set of parameters is
required. Since we are interested in a phenomenology for the case that the
constituent quark mass in the vacuum is $m=400$ MeV, we choose the
corresponding parameterization from table V of Ref. Grigorian (2007) with the
cutoff parameter $\Lambda=0.5879$ GeV, the current quark mass $m_{0}=5.582$
MeV and the scalar-pseudoscalar coupling constant $G_{s}\Lambda^{2}=2.442$.
The vector meson coupling constant $G_{\rm v}\Lambda^{2}=2.4$ is found from
fitting the $\rho$ meson mass in vacuum $M_{\rho}=780$ MeV.

The double quark mass, masses of scalar, pseudoscalar and vector mesons as
functions of the temperature at zero chemical potential are presented in Fig.
2 (left panel). The temperature is normalized to the pseudocritical
temperature $T_{c}=0.225$ GeV of the chiral crossover transition at zero
chemical potential. It is clearly seen, that the masses of the $\pi$\- and
$\rho$-meson stay almost constant when $T<T_{c}$. Then the pion mass increases
and exceeds the mass of two constituent quarks at the Mott temperature
$T^{\pi}_{\rm Mott}=0.244$ GeV where the dissociation of the pion occurs (also
called ”soft deconfinement” Hüfner et al. (1996)) that is analogous to the
Mott effect in solid-state and plasma physics Zablocki et al. (2010).

Figure 2: Left panel: the temperature dependence $M_{i}(T)$ of mass spectra
for mesons ($i=\pi,\sigma,\rho$) and doubled quark mass $m(T)$). Right panel:
scaled temperature dependence of total decay widths
$\Gamma_{i}(T)/\Gamma_{i}(0)$ for mesons ($i=\sigma,\rho$) calculated within
NJL model.

The temperature dependencies for $\Gamma_{\sigma\pi\pi}$ and
$\Gamma_{\rho\pi\pi}$, calculated in the framework of the NJL model are shown
in the Fig. 2 (right panel) as black and blue solid lines, accordingly. It can
be seen that the $\sigma\rightarrow\pi\pi$ decay is forbidden after
$T^{\sigma}_{\rm diss}=0.228$ GeV, where $\Gamma_{\sigma\pi\pi}$ drops to
zero. A center of mass momentum dependence for $\Gamma_{\rho}$ and
$\Gamma_{\sigma}$, are introduced by using of Eqs. (10) and (11).

The phase shift $\delta_{0}^{0}$ calculated using Eq. (8) for the temperatures
$T=0$, 0.15, 0.185 GeV is shown in left panel of Fig. 3. The blue line
corresponds to the Eq. (8) without T-dependence and the results almost
coincide due to the choice of the NJL parameters giving the mass
$M_{\sigma}\sim 0.8$ GeV. The phase shift $\delta_{0}^{2}$ calculated by Eq.
(12) for the temperatures $T=0$, 0.15, 0.185 GeV is shown in the right panel
of Fig. 3. The solid blue line reproduces the scattering length approximation
$\delta_{0}^{2}=-0.12q/m_{\pi}$. The experimental data are taken from Belkov
et al. (1979); Grayer et al. (1974); Hyams et al. (1973); Srinivasan et al.
(1975) for $\delta_{0}^{0}$ and Hoogland et al. (1977) for $\delta_{0}^{2}$.

Figure 3: The phase shifts $\delta_{0}^{0}$ (left panel) and $\delta_{0}^{2}$
(right panel) for different temperatures $T=0$, 0.15, 0.185 GeV. The solid
blue line in the right panel corresponds to the scattering length
approximation $\delta_{0}^{2}=-0.12q/M_{\pi}$. Experimental data are taken
from Belkov et al. (1979); Grayer et al. (1974); Hyams et al. (1973);
Srinivasan et al. (1975) for $\delta_{0}^{0}$ and from Hoogland et al. (1977)
for $\delta_{0}^{2}$.

The partial contributions to the total pressure for mesons were calculated
according to Eq. (7). The repulsive isospin-2 channel phase shift was
calculated according to Eq. (12). The partial contributions to the total
pressure are presented in the left panel of the Fig. 4 as functions of the
temperature. The contributions from $\delta^{0}_{0}$ and $5\delta^{2}_{0}$
compensate each other and the main virial correction to the pressure is given
by the $\rho$ meson. The pressure of the ideal pion gas (red dashed line) and
the interacting pion gas (black solid line) are presented in the right panel
of Fig. 4.

Figure 4: Left panel: Contributions to the pressure of the pion gas at second
order of the virial expansion ($\pi\pi$ scattering) using the Beth-Uhlenbeck
equation with medium dependent phase shifts. The quark Pauli blocking term
$P^{02}$ (blue solid line) is modified to take into account that it vanishes
when the pion bound state is dissociated. Right panel: Pressure of the ideal
pion gas (red dashed line), compared to the total pressure with all three
interaction channels (black solid line) and with just the $\rho$-meson channel
(blue dotted line). The $\rho$-meson contribution is also shown separately by
the blue dotted line.

##  IV Results and conclusion

In this paper we discussed that the contribution to the pressure from the
S-wave channel is small due to an approximate cancellation of the attractive
(I = 0) sigma resonance contribution against the repulsive (I = 2)
contribution that is explained by quark Pauli blocking. Using the simple
Breit-Wigner approximation for the meson phase shifts Welke et al. (1990) and
the nonrelativistic potential model result Barnes and Swanson (1992) for the
phase shift in the repulsive channel (I=2) together with the temperature
dependence of the mass spectra from the NJL model, we show that this
cancellation appears not only in low-energy and low-temperature region as it
was discussed before Venugopalan and Prakash (1992); Broniowski et al. (2015),
but also takes place at finite temperature as long as $T\lesssim 0.75T_{c}$.

In Table 1 the relation $a^{0}_{0}/(5a^{0}_{2})$ obtained in the frame of NJL
model for different temperatures is shown Quack et al. (1995).

Table 1: The relation $a^{0}_{0}/(5a^{0}_{2})$ in the frame of NJL model. | T=0 | T = 0.14 | T = 0.16 | T = 0.19 | T = 0.2
---|---|---|---|---|---
$a^{0}_{0}$ | 0.148 | 0.155 | 0.164 | 0.202 | 0.237
$a^{0}_{2}$ | -0.036 | -0.037 | -0.037 | -0.041 | -0.043
$a^{0}_{0}/(5a^{0}_{2})$ | -0.826 | -0.85 | -0.88 | -0.996 | -1.11

Results obtained in the Table 1 and results are shown in the Fig.4 show that
the cancellation works also at finite temperature. As can be seen from the
Fig. 4, at high temperatures near the phase transition the difference
$P^{00}+5P^{20}$ becomes finite and $\sigma$-meson channel should be taken
into account. Similar picture appears in the NJL model, where the scattering
length approximation is used Quack et al. (1995). It was discussed in
Introduction, that singularity in scattering lengths appears near the $T_{c}$
due to the cancellation of the input from the $\sigma$-exchange diagram to the
total amplitude. Near the critical temperature there appears the interplay
between the pion Pauli blocking process and the creation of $\sigma$ meson as
a bounding state just before turning back into resonance state Volkov et al.
(1998a).

That leads to the fact, that deep inside the HRG phase for $T\ll T_{c}$ a
complete cancellation of the $\sigma$ meson against the repulsive channel
appears so that the $\sigma$ meson should not be included to the HRG model
Broniowski et al. (2015). In the hadronization region for $T\approx T_{c}$,
however, it is important to take into account the $\sigma$ meson as the degree
of freedom Andronic et al. (2009), since there it is a good resonance and the
quark Pauli blocking ceases because of the Mott dissociation of the pion.

###### Acknowledgements.

D.B. acknowledges discussions with Anton Andronic and Wojciech Broniowski at
Quark Matter 2022 in Cracow.

## References

  * Venugopalan and Prakash (1992) R. Venugopalan and M. Prakash, Nucl. Phys. A 546, 718 (1992).
  * Welke et al. (1990) G. M. Welke, R. Venugopalan, and M. Prakash, Phys. Lett. B 245, 137 (1990).
  * Quack et al. (1995) E. Quack, P. Zhuang, Y. Kalinovsky, S. P. Klevansky, and J. Hufner, Phys. Lett. B 348, 1 (1995), eprint hep-ph/9410243.
  * Barnes and Swanson (1992) T. Barnes and E. S. Swanson, Phys. Rev. D 46, 131 (1992).
  * Broniowski et al. (2015) W. Broniowski, F. Giacosa, and V. Begun, Phys. Rev. C 92, 034905 (2015), eprint 1506.01260.
  * Giacosa et al. (2016) F. Giacosa, V. Begun, and W. Broniowski, Acta Phys. Polon. Supp. 9, 213 (2016), eprint 1603.07687.
  * Friman et al. (2015) B. Friman, P. M. Lo, M. Marczenko, K. Redlich, and C. Sasaki, Phys. Rev. D 92, 074003 (2015), eprint 1507.04183.
  * Dashen et al. (1969) R. Dashen, S.-K. Ma, and H. J. Bernstein, Phys. Rev. 187, 345 (1969).
  * Andronic et al. (2009) A. Andronic, P. Braun-Munzinger, and J. Stachel, Phys. Lett. B 673, 142 (2009), [Erratum: Phys.Lett.B 678, 516 (2009)], eprint 0812.1186.
  * Volkov et al. (1998a) M. K. Volkov, E. A. Kuraev, D. Blaschke, G. Ropke, and S. M. Schmidt, Phys. Lett. B 424, 235 (1998a), eprint hep-ph/9706350.
  * Blaschke and Röpke (1993) D. Blaschke and G. Röpke, Phys. Lett. B 299, 332 (1993).
  * Hüfner et al. (1996) J. Hüfner, S. P. Klevansky, and P. Rehberg, Nucl. Phys. A 606, 260 (1996).
  * Weinberg (1966) S. Weinberg, Phys. Rev. Lett. 17, 616 (1966).
  * Klevansky (1992) S. P. Klevansky, Rev. Mod. Phys. 64, 649 (1992), URL https://link.aps.org/doi/10.1103/RevModPhys.64.649.
  * Volkov et al. (1998b) M. K. Volkov, D. Ebert, and M. Nagy, Int. J. Mod. Phys. A 13, 5443 (1998b), eprint hep-ph/9705334.
  * Volkov and Radzhabov (2006) M. K. Volkov and A. E. Radzhabov, Phys. Usp. 49, 551 (2006), eprint hep-ph/0508263.
  * Ebert et al. (1993) D. Ebert, Y. L. Kalinovsky, L. Munchow, and M. K. Volkov, Int. J. Mod. Phys. A 8, 1295 (1993).
  * Zhuang and Yang (2001) P. Zhuang and Z. Yang, Chin. Phys. Lett. 18, 344 (2001), eprint nucl-th/0008041.
  * Friesen et al. (2012) A. V. Friesen, Y. L. Kalinovsky, and V. D. Toneev, Phys. Part. Nucl. Lett. 9, 1 (2012), eprint 1104.2698.
  * Grigorian (2007) H. Grigorian, Phys. Part. Nucl. Lett. 4, 223 (2007), eprint hep-ph/0602238.
  * Zablocki et al. (2010) D. Zablocki, D. Blaschke, and G. Röpke, in _Metal-to-Nonmetal Transitions. Springer Series in Materials Science, vol 132_ , edited by R. Redmer, F. Hensel, and B. Holst (Springer, Berlin, Heidelberg, 2010), pp. 161–182.
  * Belkov et al. (1979) A. A. Belkov, S. A. Bunyatov, K. N. Mukhin, O. O. Patarakin, V. M. Sidorov, M. M. Sulkovskaya, A. F. Sustavov, and V. A. Yarba, JETP Lett. 29, 597 (1979).
  * Grayer et al. (1974) G. Grayer et al., Nucl. Phys. B 75, 189 (1974).
  * Hyams et al. (1973) B. Hyams et al., Nucl. Phys. B 64, 134 (1973).
  * Srinivasan et al. (1975) V. Srinivasan et al., Phys. Rev. D 12, 681 (1975).
  * Hoogland et al. (1977) W. Hoogland et al., Nucl. Phys. B 126, 109 (1977).