YITP-22-159

11institutetext: Yukawa Institute for Theoretical Physics, Kyoto University,
Kyoto 606-8502, Japan 22institutetext: Department of Physics, Kyoto
University, Kitashirakawa-oiwake, Sakyo, Kyoto 606-8502, Japan

# Study of Hadron Masses with Faddeev-Popov Eigenmode Projection in the
Coulomb Gauge

Hiroki Ohata 11<EMAIL_ADDRESS>Hideo Suganuma 22
<EMAIL_ADDRESS>
###### Abstract

Using SU(3) lattice QCD, we investigate role of spatial gluons for hadron
masses in the Coulomb gauge, considering the relation between QCD and the
quark model. From the Coulomb-gauge configurations at the quenched level on a
$16^{3}\times 32$ lattice at $\beta$ = 6.0, we consider the $\vec{A}=0$
projection, where all the spatial gluon fields are set to zero. In this
projection, the inter-quark potential is unchanged. We investigate light
hadron masses and find that nucleon and delta baryon masses are almost
degenerate. This result suggests that the N-$\Delta$ mass difference arises
from the color-magnetic interactions, which is consistent with the quark model
picture. Next, as a generalization of this projection, we expand spatial gluon
fields in terms of Faddeev-Popov eigenmodes and leave only some partial
components. We find that the ${\rm N}-\Delta$ and $0^{++}-2^{++}$ glueball
mass splittings are almost reproduced only with 1 % low-lying components. This
suggests that low-lying color-magnetic interaction leads to the hadron mass
splitting.

##  1 Introduction

In spite of the great success of the quark model DeRujula:1975qlm , its
connection with quantum chromodynamics (QCD) is not yet clear, and to clarify
the relation between these theories is one of the most important problems in
hadron physics. A significant difference between them is the main dynamical
degrees of freedom: while QCD is formulated with current quarks and gluons,
the quark model is described only with the massive constituent quarks. The
other significant difference would be the symmetries they possess. QCD has
local color $\mathrm{SU}(3)$ symmetry, whereas the quark model only has global
color $\mathrm{SU}(3)$ symmetry reflecting the absence of dynamical gluons.
This suggests that the quark model is a low-energy effective theory of QCD in
some gauge.

Then, what gauge corresponds to the quark model? Here, we need not respect
full Lorentz symmetry, since the nonrelativistic quark model has only the
spatial-rotation symmetry. We consider that gauge to be the Coulomb gauge.
There are several reasons for this choice. First, the Coulomb gauge leaves
global color $\mathrm{SU}(3)$ symmetry at each time slice, which might
correspond to the global color symmetry in the quark model. Second, the
Coulomb gauge is globally defined by minimizing spatial gauge-field
fluctuations, and therefore one expects only a small fluctuation of the
spatial gluon field, which might explain the absence of dynamical gluons in
the quark model. Third, in the Coulomb gauge, the temporal gauge field $A_{0}$
is no more dynamical but appears as a potential for quarks, which might give
the static potential in the quark model. Several researchers have already
studied the relation between the quark model and Coulomb gauge QCD
Szczepaniak:2001rg .

We investigate the relation between the quark model and Coulomb gauge QCD
using gauge configurations generated in SU(3) lattice QCD. The organization of
this paper is as follows. In Sec. 2, we briefly review QCD in the Coulomb
gauge. In Sec. 3, we propose $\vec{A}=0$ projection, i.e., removal of spatial
gluons from lattice gauge configurations in the Coulomb gauge, and investigate
the static inter-quark potential and hadron masses. In Sec. 4, we propose a
generalization of the $\vec{A}=0$ projection in the Coulomb gauge utilizing
the Faddeev-Popov eigenmodes. In Sec. 5, we apply this projection to light
hadron and glueball masses. Section 6 is devoted to Summary and Conclusion.
This paper is based on Ref. Ohata:2022iqm .

##  2 Coulomb gauge QCD

We briefly review Coulomb gauge QCD, which is locally defined as
$\partial_{i}A_{s,i}^{a}=0$ for the gluon field
$A_{s,\mu}=A_{s,\mu}^{a}T^{a}\in{\rm su}(N_{c})$. In the Coulomb gauge, the
spatial gluon fields $A_{i}^{a}$ are canonical variables and behave as
dynamical variables, whereas the temporal gluon field $A_{0}^{a}$ just behaves
as a potential field. The global definition of the Coulomb gauge is to
minimize

$R_{\mathrm{C}}[A]\coloneqq\int dt\leavevmode\nobreak\
d^{3}s\sum_{i,a}\leavevmode\nobreak\ [A_{s,i}^{a}(t)]^{2}$ (1)

under the gauge transformation, which satisfies the local Coulomb gauge
condition. In fact, the gauge fluctuation of the spatial gluon field is
strongly suppressed in the Coulomb gauge.

Thus, in the Coulomb gauge, spatial gluons are forced to be minimized and the
temporal gluon just provides a potential. This situation might be preferable
to match QCD and the quark model, since the quark model only has quark degrees
of freedom and an inter-quark potential.

In the path-integral formalism, the generating functional of the Yang-Mills
(YM) theory in the Coulomb gauge is expressed by

$Z=\int DA\,e^{iS[A]}\leavevmode\nobreak\
\delta(\partial_{i}A_{i}^{a})\leavevmode\nobreak\ {\rm
Det}(D_{i}\partial_{i})$ (2)

with the YM action $S[A]$. There appears the Faddeev-Popov (FP) operator,

$M^{ab}\coloneqq
D_{i}^{ab}\partial_{i}=\partial_{i}^{2}\delta^{ab}+gf^{abc}A^{c}_{i}\partial_{i},$
(3)

as a weight of the gauge orbital in the Coulomb gauge. This FP operator is one
of the key operators in the Coulomb gauge, and its inverse gives the
propagator of the FP ghost field.

In lattice QCD, the gauge degrees of freedom is described by the link variable
$U_{s,\mu}\coloneqq e^{iagA_{s,\mu}}\in{\rm SU}(N_{c})$, where $g$ is the
gauge coupling and $a$ the lattice spacing, and the global definition of the
Coulomb gauge is to maximize

$R_{\mathrm{C}}[U]\coloneqq\sum_{t,s,i}{\rm Re}\leavevmode\nobreak\ {\rm
Tr}U_{s,i}(t)$ (4)

under the gauge transformation. This global definition leads to the local
gauge condition $\partial^{B}_{i}{\cal A}_{s,i}=0$, where $\partial^{B}$
denotes the backward derivative and

${\cal
A}_{s,i}\coloneqq\frac{1}{2iag}\\{U_{s,i}-U_{s,i}^{\dagger}\\}\bigg{|}_{\rm
traceless}\in{\rm su}(N_{c})$ (5)

goes to the original gluon field $A_{i}(s)$ in the continuum limit.

In lattice QCD in the Coulomb gauge, the generating functional of the gauge
sector is written by

$Z=\int DU\,e^{-S[U]}\leavevmode\nobreak\ \delta(\partial_{i}{\cal
A}_{i}^{a})\leavevmode\nobreak\ {\rm Det}(M),$ (6)

where the FP operator in the Coulomb gauge takes the form Iritani:2012bc  of

$M_{x,y}^{a,b}=\sum_{i}A_{x,i}^{a,b}\delta_{x,y}-B_{x,i}^{a,b}\delta_{x+\hat{i},y}-C_{x,i}^{a,b}\delta_{x-\hat{i},y},$
(7)

with

$\displaystyle\\!\\!\\!\\!A_{x,i}^{a,b}={\rm Re}{\rm
Tr}[\\{T^{a},T^{b}\\}(U_{x,i}+U_{x-\hat{i},i})],\leavevmode\nobreak\
B_{x,i}^{a,b}=2{\rm Re}{\rm Tr}[T^{b}T^{a}U_{x,i}],\leavevmode\nobreak\
C_{x,i}^{a,b}=2{\rm Re}{\rm Tr}[T^{a}T^{b}U_{x-\hat{i},i}],\\!$ (8)

at each time slice, which leads to Eq. (3) in the continuum limit. Note that
$M$ is a real symmetric matrix, i.e., $M^{a,b}_{x,y}=M^{b,a}_{y,x}\in{\bf R}$,
and hence its eigenvalues $\lambda_{n}$ are real.

In the Coulomb gauge, this FP operator is a key quantity to control the gauge-
orbital weight, and has been studied in the context of the Gribov horizon
Gribov:1977wm ; Zwanziger:1995cv , the instantaneous color-Coulomb potential
Zwanziger:1995cv ; Zwanziger:1998ez ; Szczepaniak:2001rg ; Zwanziger:2002sh ,
and a confinement scenario of the gluon-chain picture Greensite:2001nx ;
tHooft:2002pmx .

##  3 Lattice QCD simulation and $\vec{A}=0$ projection

We perform SU(3) quenched lattice QCD calculation at $\beta=6.0$ on a
$L_{s}^{3}\times L_{t}=16^{3}\times 32$ lattice. We impose periodic boundary
conditions for link variables in all directions. We generate 500 gauge
configurations, which are picked up every $500$ sweeps after a thermalization
of $5000$ sweeps. For these gauge configurations, we perform Coulomb gauge
fixing by maximizing Eq. (4) under the SU(3) gauge transformation. Thus, we
obtain representatives of the QCD vacuum in the Coulomb gauge.

The static inter-quark potential is one of the most fundamental constituents
in the quark model. In QCD, it can be calculated from correlator of the
Polyakov loops. Since the Polyakov loop is defined only by temporal gauge
fields, the static quark-antiquark potential is independent of spatial gauge
fields in the gauge configurations. To see this, we define $\vec{A}=0$
projection as a simple removal of spatial gluons in the generated gauge
configurations in the Coulomb gauge, and apply it to the static quark-
antiquark potential. Figure 1 shows the static quark-antiquark potentials
calculated from Wilson loops, evaluated from the original gauge configurations
and $\vec{A}=0$ projected ones in the Coulomb gauge.

Figure 1:  The static quark-antiquark potentials evaluated from the original
configurations (black) and $\vec{A}=0$ projected ones in the Coulomb gauge
(red). The black and red curves are the fit results with the Cornell potential
$V(r)=-A/r+\sigma r+\mathrm{const}$.

They agree very well in the whole distance region as expected. In the quark
model, the inter-quark potential is introduced by hand. In lattice QCD in the
Coulomb gauge, we can keep the original static inter-quark potential by just
leaving the temporal gauge fields unchanged.

Using the $\vec{A}=0$ projection in the Coulomb gauge, we next investigate
hadron masses to see the role of spatial gluons, which gives some hadron-mass
splitting via the color-magnetic interactions in the quark model. Since hadron
masses are not purely static quantities, their values under the $\vec{A}=0$
projection are non-trivial. We use the clover fermion ($\mathcal{O}(a)$
improved Wilson fermion) with a non-perturbatively determined clover
coefficient $c_{\mathrm{SW}}=1.769$ Luscher:1996ug . We consider the light
hadrons of the pion, rho meson, nucleon, and delta baryon. We extract pion
mass $m_{\pi}$ and rho meson mass $m_{\rho}$ from correlators defined as

$\displaystyle
C(t)=\sum_{\vec{x}}M^{\mathrm{ud}}(\vec{x},t)M^{\mathrm{ud}}(\vec{0},0)^{\dagger},\quad
M^{\mathrm{ud}}=\begin{cases}\overline{u}\gamma_{5}d&\text{for pion
($\pi^{-}$)},\\\ \overline{u}\gamma_{i}d&\text{for rho meson
($\rho^{-}$)}.\end{cases}$ (9)

For nucleon mass $m_{\mathrm{N}}$ and delta baryon mass $m_{\Delta}$, we use
correlators

$\displaystyle\\!\\!C(t)=\Gamma_{\alpha\beta}\sum_{\vec{x}}B^{\mathrm{udu}}_{\alpha}(\vec{x},t)\overline{B}_{\beta}^{\mathrm{udu}}(\vec{0},0),\quad
B^{\mathrm{udu}}=\begin{cases}\epsilon^{abc}((u^{\mathrm{T}})^{a}C\gamma_{5}d^{b})u^{c}\\\
\qquad\qquad\qquad\text{for nucleon (proton)},\\\
\epsilon^{abc}\\{2((u^{\mathrm{T}})^{a}C\gamma_{i}d^{b})u^{c}+((u^{\mathrm{T}})^{a}C\gamma_{i}u^{b})d^{c}\\}\\!\\!\\!\\!\\\
\qquad\qquad\qquad\text{for delta baryon ($\Delta^{+}$)}.\end{cases}$ (10)

Here $C=\gamma_{0}\gamma_{2}$ is the charge conjugation matrix, and we set
$\Gamma=(1\pm\gamma_{0})$ to extract components proportional to these parity
projection operators.

Figure 2 (a) shows light hadron mass measurement from the original gauge
configuration. For each hoppoing parameter $\kappa$ and sector, hadron mass is
obtained from single cosh / exponential fit to the correlation function. From
linear chiral extrapolations of $m_{\pi}^{2}$ and $m_{\rho}$, the physical
hopping parameter and the lattice spacing are determined so that
$m_{\pi}=0.14{\rm GeV}$ and $m_{\rho}=0.77{\rm GeV}$. The lattice spacing is
found to be $a=0.1051(15){\rm fm}$. Nucleon and delta baryon masses are found
to be $m_{\mathrm{N}}=1.050(24){\rm GeV}$ and $m_{\Delta}=1.334(31){\rm GeV}$,
which are consistent with a quenched lattice QCD simulation with a similar
lattice setup JLQCD:2002zto .

Figure 2:  Hadron mass measurement (a) from the original gauge configuration;
(b) under the $\vec{A}$ = 0 projection in the Coulomb gauge; (c) under the
eigenmode projection with 5 % low-lying FP eigenmodes; (d) under the eigenmode
projection with 95 % high-lying FP eigenmodes.

Figure 2 (b) shows light hadron mass measurement under the $\vec{A}=0$
projection. Here we assume that the lattice spacing is the same as the one
obtained from the original gauge configuration because the static quark-
antiquark potential is unchanged. The physical hopping parameter is determined
so that $m_{\pi}=0.14{\rm GeV}$. Rho meson, nucleon, and delta baryon masses
are found to be $m_{\rho}=0.409(17){\rm GeV},m_{\mathrm{N}}=0.607(44){\rm
GeV}$, and $m_{\Delta}=0.658(49){\rm GeV}$, respectively under the $\vec{A}=0$
projection. The overall decrease of hadron masses suggests a constituent quark
with smaller mass of $m_{\mathrm{Q}}\simeq 0.2{\rm GeV}$.

Interestingly, we find that nucleon and delta baryon masses are almost
degenerate under the $\vec{A}=0$ projection. Here, we consider the physical
meaning of the $\vec{A}=0$ projection. Under the $\vec{A}=0$ projection, while
color-electric fields can remain, color-magnetic fields
$H_{i}^{a}=\epsilon_{ijk}(\partial_{j}A_{k}^{a}-\partial_{k}A_{j}^{a}-gf^{abc}A_{j}^{b}A_{k}^{c})$
are inevitably zero, and therefore color-magnetic interactions vanish. Since
only the color-magnetic interactions are spin-dependent for static systems in
QCD, mass splitting between different spin states are universally expected to
vanish or to decrease under the $\vec{A}=0$ projection. This correspondence
between color-magnetic interactions and spin-dependent hadron mass splitting
seems consistent with the standard quark-model picture, since the
$\mathrm{N}-\Delta$ mass splitting is generated by the color-magnetic
interactions in the quark model.

##  4 A generalization of the $\vec{A}=0$ projection in the Coulomb gauge

To investigate the role of spatial gluons in the Coulomb gauge quantitatively,
we generalize the $\vec{A}=0$ projection so as to eliminate the spatial gluons
contribution continuously. To this end, we utilize the Faddeev-Popov (FP)
operator $M$ and consider expansion of the spatial gluon fields in terms of
its eigenmodes. Roughly, the FP operator defined by Eq. (3) resembles the
Laplacian in the Coulomb gauge, where spatial gluon amplitude is fairly
suppressed, and low-lying FP modes correspond to low-energy components. In
lattice QCD, the FP operator $M$ is expressed as a $8L_{s}^{3}\times
8L_{s}^{3}$ matrix $M_{IJ}=M_{x,y}^{a,b}$ in Eq. (7), with the index $I=(x,a)$
and $J=(y,b)$.

Since the FP operator is a real symmetric matrix, its eigenmodes $\psi_{n}$
satisfying

$M\psi_{n}=\lambda_{n}\psi_{n}\quad(n=1,2,\dots,8L_{s}^{3})$ (11)

can be taken real and normalized eigenvectors $\psi_{n}(t)$ at each time $t$
form a complete set

$\sum_{n}\psi_{n,s}^{a}(t)\leavevmode\nobreak\
\psi_{n,s^{\prime}}^{a^{\prime}}(t)=\delta_{s,s^{\prime}}\leavevmode\nobreak\
\delta^{a,a^{\prime}}.$ (12)

With these eigenvectors, spatial gluon fields at each time $t$ can be uniquely
expanded as

$\displaystyle
A_{s,i}^{a}(t)=\sum_{n=1}^{8L_{s}^{3}}c_{n,i}(t)\leavevmode\nobreak\
\psi_{n,s}^{a}(t),\qquad
c_{n,i}(t)\coloneqq\sum_{s,a}A_{s,i}^{a}(t)\leavevmode\nobreak\
\psi_{n,s}^{a}(t).$ (13)

Among the $8L_{s}^{3}$ eigenmodes, we consider a restriction of these modes,
and define FP eigenmode projection for spatial gluon fields $A_{s,i}(t)$ as a
replacement by

$\displaystyle A_{s,i}^{a,\mathrm{projected}}(t)=\sum_{n\in
S}c_{n,i}(t)\leavevmode\nobreak\ \psi_{n,s}^{a}(t)$ (14)

in the Coulomb gauge. Here, $S$ denotes a subset of the whole set
$\\{1,\dots,8L_{s}^{3}\\}$. In this projection, the temporal gluon $A_{4}$ is
unchanged. The eigenmode projection with whole set
$S=\\{1,\dots,8L_{s}^{3}\\}$ gives the original gauge configuration, whereas
the projection with no eigenmode ($S=\phi$) becomes the $\vec{A}=0$
projection. Then, we can connect the $\vec{A}=0$ projected configuration with
the original one by smoothly changing the subset $S$.

In this study, we only consider eigenmode projections with $N_{\mathrm{low}}$
low-lying or $N_{\mathrm{high}}$ high-lying eigenmodes. Specifically, we set

$\displaystyle S_{\mathrm{low}}=\\{1,\dots,N_{\mathrm{low}}\\},\qquad
S_{\mathrm{high}}=\\{8L_{s}^{3}-N_{\mathrm{high}}+1,\dots,8L_{s}^{3}\\}$ (15)

for eigenmodes which are ordered as
$\lambda_{1}\leq\lambda_{2}\leq\dots\leq\lambda_{8L_{s}^{3}}$. For each gauge
configuration and time $t$, we calculate 1638 low-lying and high-lying
eigenmodes for the FP operator by using the Krylov-Schur solver implemented in
SLEPc Hernandez:2005:SSF . It is found that the 1638-th smallest eigenvalue
corresponds to the low-lying eigenmode of about $(2{\rm GeV})^{2}$.

Note that, unlike the Fourier expansion Yamamoto:2008am ; Yamamoto:2008ze ,
the current expansion has no practical eigenvalue degeneracy except for the
trivial eight zero-modes, and one can smoothly connect the $\vec{A}=0$
projected gauge configuration with the original one.

##  5 hadron and glueball masses under the eigenmode projection

We calculate light hadron masses under the eigenmode projection. Figure 2 (c)
and (d) show light hadron masses under the eigenmodes projection with 5% low-
lying and with 95% high-lying eigenmodes, respectively. Figure 3 shows hadron
masses under the eigenmode projection plotted against the number of low-lying
or high-lying FP eigenmodes. Some of these results are listed in Table 1.

Table 1:  Hadron masses under the eigenmode projection with low-lying FP eigenmodes.  num. of low-lying FP modes |  $m_{\rho}$ [GeV] |  $m_{\mathrm{N}}$ [GeV] |  $m_{\Delta}$ [GeV]
---|---|---|---
0 (0 %) | 0.409(17) | 0.607(44) | 0.658(49)
8 (0.02 %) | 0.569(23) | 0.788(86) | 0.887(68)
33 (0.10 %) | 0.675(25) | 0.984(68) | 1.122(77)
164 (0.50 %) | 0.772(20) | 1.045(35) | 1.306(50)
328 (1.00 %) | 0.785(21) | 1.051(36) | 1.372(54)
$8L_{s}^{3}$ (100 %) | 0.77 | 1.050(24) | 1.334(31)
Figure 3:  Hadron masses under the eigenmode projection in GeV unit plotted
against the number of low-lying or high-lying FP eigenmodes in the projection.
Pion mass $m_{\pi}=0.14{\rm GeV}$ is used to set the physical hopping
parameter for each measurement. The lattice spacing $a=0.1051(15){\rm fm}$ is
used to set the scale. The solid and dashed lines are connecting adjacent data
points for visibility.

We find monotonically increasing behaviors in Fig. 3. The original light
hadron masses are approximately reproduced with only one percent of low-lying
eigenmodes. Interestingly, Figs. 2 (c) and (d) are respectively very similar
with Figs. 2 (a) and (b) in Sec. 3, except for the values of hopping parameter
$\kappa$. These results indicate an important role of low-lying eigenmodes on
light hadron masses. From Fig. 3 and Table 1, we also find that the
$\mathrm{N}-\Delta$ mass splitting becomes evident when projected with more
than 0.1 % low-lying eigenmodes.

Finally, to go beyond the quark model, we investigate $0^{++}$ and $2^{++}$
glueball masses in the FP eigenmode projection in the Coulomb gauge. The mass
splitting of $0^{++}$ and $2^{++}$ glueballs are also considered to be due to
the color-magnetic interaction.

These glueball masses can be extracted from correlators

$\displaystyle
C(t)=\phi(t)\phi(0),\qquad\phi(t)=\begin{cases}\sum_{\vec{x}}{\rm
Re}\leavevmode\nobreak\ {\rm tr}(P_{12}+P_{23}+P_{31})(\vec{x},t)&\text{for
$0^{++}$},\\\ \sum_{\vec{x}}{\rm Re}\leavevmode\nobreak\ {\rm
tr}(P_{12}-P_{13})(\vec{x},t)&\text{for $2^{++}$},\end{cases}$ (16)

where $P_{ij}$ is the plaquette in $i-j$ plane and the vacuum contribution
must be subtracted from the correlator for $0^{++}$ glueball mass
Ishikawa:1982tb . For the glueball calculation, although 500 gauge
configurations might not be enough statistically Teper:1987wt , we calculate
approximate $0^{++}$ and $2^{++}$ glueball masses under the eigenmode
projection with low-lying eigenmodes, from single exponential fits to the
correlators (16). Those results are summarized in Fig. 4.

Figure 4:  Glueball masses under the eigenmode projection in GeV unit plotted
against the number of low-lying FP eigenmodes in the projection.

We also show glueball mass mesurement under the eigenmode projection with 0.1
% low-lying eigenmodes in Fig. 5. In the case of projection with high-lying
eigenmodes, the correlation functions are too noisy that we can not extract
the masses at all.

Figure 5:  Glueball mass measurement under the eigenmode projection with 0.1 %
low-lying FP eigenmodes. Correlation functions are normalized such that
$C(0)=1$.

From Fig. 4, it seems that $0^{++}$ and $2^{++}$ glueball masses are
approximately degenerate near the $\vec{A}=0$ projection. This can be also
seen from Fig. 5, where correlation functions are very alike. They
approximately converge to the original values with just one percent of low-
lying eigenmodes. We also find that the $0^{++}-2^{++}$ glueball mass
splitting becomes evident when projected with more than 0.1 % low-lying
eigenmodes. Thus, we obtain similar results also in the case of glueball
masses.

##  6 Summary and Conclusion

In this paper, aiming at the relation between QCD and the quark model, we have
investigated the role of spatial gluons for hadron masses in the Coulomb
gauge. In the $\vec{A}=0$ projection, where all the spatial gluon fields are
set to zero, we have found that nucleon and delta baryon masses are almost
degenerate. This suggests that the N-$\Delta$ mass difference arises from the
color-magnetic interactions, which is consistent with the quark model picture.
Next, as a generalization of this projection, we expanded spatial gluon fields
in terms of Faddeev-Popov eigenmodes and left only some partial components. We
have found that the original ${\rm N}-\Delta$ and $0^{++}-2^{++}$ glueball
mass splittings are almost reproduced only with 1 % low-lying components. This
suggests that low-lying color-magnetic interaction leads to the hadron mass
splitting.

Our study suggests a possibility that the quark model can be derived from
Coulomb gauge QCD by a reduction of high-lying spatial gluons above $1.3{\rm
GeV}$. Also, a QCD-based constituent gluon model might be formulated from
Coulomb gauge QCD with the reduction of high-lying spatial gluons, and its
combination with the quark model might describe wide category of hadrons
including glueballs and hybrids.

H.O. is supported by a Grant-in-Aid for JSPS Fellows (Grant No.21J20089). H.S.
is supported by a Grants-in-Aid for Scientific Research [19K03869] from JSPS.
This work was in part based on Bridge++ code Ueda:2014rya . We used SLEPc
Hernandez:2005:SSF  to solve eigenvalue problems. The numerical simulations
was carried out on Yukawa-21 at YITP, Kyoto University.

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# An ALMA 1.3 millimeter Search for Debris Disks around Solar-type Stars in
the Pleiades

Devin Sullivan  Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge, MA 02138, USA  David Wilner Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge, MA 02138, USA  Luca Matrà School of Physics, Trinity College Dublin, The University of Dublin, College Green, Dublin 2, Ireland  Centre for Astronomy, School of Physics, National University of Ireland Galway, University Road, Galway, Ireland H91 TK33  Mark C. Wyatt Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK  Sean M. Andrews Center for Astrophysics | Harvard & Smithsonian, 60 Garden St., Cambridge, MA 02138, USA  Meredith A. MacGregor Department of Astrophysical and Planetary Sciences, University of Colorado, 2000 Colorado Avenue, Boulder, CO 80309, USA  Brenda Matthews Herzberg Astronomy & Astrophysics Research Centre, National Research Council of Canada, 5071 West Saanich Road, Victoria, BC V9E 2E7, Canada

(Accepted July 11, 2022)

###### Abstract

Millimeter emission from debris disks around stars of different ages provides
constraints on the collisional evolution of planetesimals. We present ALMA 1.3
millimeter observations of a sample of 76 Solar-type stars in the $\sim 115$
Myr-old Pleiades star cluster. These ALMA observations complement previous
infrared observations of this sample by providing sensitivity to emission from
circumstellar dust at lower temperatures, corresponding to debris at radii
comparable to the Kuiper Belt and beyond. The observations obtain a beam size
of $1\farcs 5$ (200 au) and a median rms noise of $54$ $\mu$Jy beam-1, which
corresponds to a fractional luminosity $L_{dust}/L_{star}\sim 10^{-4}$ for 40
K dust for a typical star in the sample. The ALMA images show no significant
detections of the targeted stars. We interpret these limits in the context of
a steady-state collisional cascade model for debris disk evolution that
provides a good description of observations of the field population near the
Sun but is not well-calibrated on younger populations. The ALMA non-detections
of the Pleiades systems are compatible with the disk flux predictions of this
model. We find no high fractional luminosity outliers from these ALMA data
that could be associated with enhanced collisions resulting from activity not
accounted for by steady-state evolution. However, we note that two systems
(HII 1132 and HD 22680) show 24 $\mu$m excess much higher than the predictions
of this model, perhaps due to unusually high dust production from dynamical
events involving planets.

††software: CASA (McMullin et al. 2007), UVMultiFit (Marti-Vidal et al. 2014),
astropy (The Astropy Collaboration 2013, 2018)

##  1 Introduction

The collisions between planetesimals orbiting main-sequence stars produce dust
heated by stellar radiation that is manifested as debris disk emission. Debris
disk detection statistics together with trends of dust excess with stellar age
probe the evolution of planetesimal belts, which may be influenced by planets
(see reviews by Wyatt (2008); Matthews et al. (2014); Hughes et al. (2018)).
Studies of 24 and 70 $\mu$m dust emission around stars with a range of ages
and spectral types show that debris disk detection frequency and fractional
luminosity ($L_{dust}/L_{star}$) rise over perhaps 10’s of Myr and then
decline slowly over 100’s of Myr (Rieke et al., 2005; Su et al., 2006;
Carpenter et al., 2009). These data can be explained in a theoretical
framework that invokes an early phase of runaway growth and merging of
planetesimals (Kenyon & Bromley, 2008) followed by a steady state destructive
collisional cascade (Dominik & Decin, 2003). In this picture, planetesimals
collide and fragment into smaller objects that in turn collide and fragment,
ending in circumstellar dust small enough to be removed by stellar winds and
radiation. Because collisional depletion is faster for smaller disks and more
massive disks, the wide dispersion of fractional luminosity and dust
temperature observed at a given age may be accounted for by variations in the
initial planetesimal belt masses and radii (Wyatt et al., 2007a; Löhne et al.,
2008). This standard collisional evolution model for debris disks predicts a
maximum fractional luminosity of $10^{-3}$ at early times, in line with the
most extreme systems known (Wyatt et al., 2007b).

Observational constraints on evolutionary models are conspicuously sparse for
debris disks around Solar-type stars at large radii, beyond 10 au, for ages
near 100 Myr, i.e. older than nearby moving groups but younger than the nearby
field. This is a critical epoch when young planetary systems may be
dynamically active. Such systems are especially relevant to provide context
for the evolution of our Kuiper Belt, which apparently was destabilized by the
outward migration of Neptune and underwent a dramatic dynamical depletion,
from an initial mass of more than 10 M⊕ at radii 30 to 50 au, to less than 0.1
M⊕ today; modeling suggests that the corresponding fractional luminosity
decreased from $10^{-3}$ to $10^{-7}$ (Booth et al., 2009). This is also the
epoch of the formation of the Earth’s moon (e.g. see review by Asphaug, 2014).
Sensitive surveys of the (older) field population of Solar-type stars (FGK
type) within 20 pc show the occasional system (5 of 275) with debris at large
radii and high fractional luminosity ($>5\times 10^{-5}$) whose presence is
not predicted by standard steady-state collisional evolution models (Sibthorpe
et al., 2018). A plausible hypothesis for such large excess is an ongoing
system-wide dynamical event involving planets, akin to the one that shaped the
Kuiper Belt. Another possibility is that these systems follow the “late
bloomer” path revealed by coagulation models, where protoplanetary disk rings
with nearly completely efficient planetesimal formation slowly rise to these
high fractional luminosities over $\sim 1$ Gyr (Najita et al., 2022).

The Pleiades cluster has long provided a benchmark for debris disk evolution
studies because of its combination of (1) age: $115\pm 10$ Myr (Stauffer et
al., 1998), the epoch of maximum dust production rate in “self-stirred”
evolutionary models (Kenyon & Bromley, 2008), (2) proximity: $133.5\pm 1.2$ pc
(Melis et al., 2014; Gaia Collaboration et al., 2018), for maximum
sensitivity, and (3) a sufficiently large sample: $\mathcal{O}(100)$ Solar-
type stars, to obtain meaningful statistics to compare with evolutionary
models. No other star cluster satisfies these criteria. Sensitive searches for
warm ($>100$ K) circumstellar dust in the Pleiades were performed in a series
of Spitzer 24 $\mu$m surveys that include members of spectral type F5 to K1
(Stauffer et al., 2005; Gorlova et al., 2006; Sierchio et al., 2010). For
systems observed at 24 $\mu$m, 23/71 (32%) show excesses greater than 10%
above the photospheres, attributed to dusty debris at blackbody radii from
$\sim 1$ to 10 au. By comparison, only 3/164 (2%) of the Solar-type stars in
the 750 Myr old Praesepe cluster (mean distance 187.3 pc, Lodieu et al., 2019)
show 24 $\mu$m excesses, as dust production declines with time due to
collisional depletion in the inner regions (e.g. Gáspár et al., 2009). All
models that explain debris disk evolution predict that the Pleiades should
harbor a population of colder ($<100$ K) debris belts at larger radii ($>10$
au).

There are few existing constraints on cold dust belts in the Pleiades.
Emission at 24 $\mu$m is on the Wien side of the spectral peak for dust at
radii of 10’s of au and therefore exponentially depressed. A subset of 20
Pleiades systems was observed by Spitzer at 70 $\mu$m as a part of the FEPS
program, which targeted 328 Solar-type stars spanning ages of 3 Myr to 3 Gyr
to study debris disk evolution (Carpenter et al., 2009). No excess emission
was detected from these 20 Pleiades targets at 70 $\mu$m, but with only
marginal sensitivity to the outer belts of interest (Stauffer et al., 2005).
Millimeter wavelengths provide direct access to cold dust, but millimeter
observations of the Pleiades have yet to reach relevant levels of sensitivity
to $L_{dust}/L_{star}$ (Zuckerman & Becklin, 1993; Greaves et al., 2009;
Roccatagliata et al., 2009).

The Survey of Nearby Stars (SONS) with the JCMT provided 0.85 (and 0.45)
millimeter follow-up of debris disks near the Sun known from far-infrared
surveys, resulting in 31 detections of FGK stars (Holland et al., 2017).
Nearly all systems with ages $<50$ Myr were detected, with lower detection
rates at older ages, in line with expectations for collisional evolution. Only
6 systems with ages in the range 50 to 200 Myr were detected. Now, the
unprecendented sensitivity of ALMA allows for unbiased surveys of millimeter
emission from circumstellar to constrain debris disk evolutionary models. For
example, Lovell et al. (2021) used ALMA to detect 4/30 Class III systems in
the Lupus clouds and concluded that the debris belts found around older stars
are already present at an age of $\sim 2$ Myr.

In this paper, we report the results of an ALMA survey of Solar-type stars in
the Pleiades cluster at 1.3 millimeters wavelength. These observations improve
on previous millimeter observations by about an order of magnitude in
sensitivity and for a much larger sample, and they complement previous
infrared observations of the same stars. In §2, we describe the details of the
ALMA observations. In §3, we present the results for the sample. In §4, we
discuss these results in the framework of steady-state collisional evolution
developed by Wyatt & Dent (2002) and previously applied to Solar-type stars by
Kains et al. (2011) and Sibthorpe et al. (2018). Finally, §5 summarizes the
conclusions from this study.

##  2 Observations

###  2.1 ALMA Sample Selection

We based the ALMA survey targets on the sample of 76 Solar-type Pleiades stars
with reliable 24 $\mu$m photometry compiled by Sierchio et al. (2010).111We
note two modifications of the ALMA sample from the compilation of Sierchio et
al. (2010): AK 1B 146 (HD 23170, HII 177) was not included as it is not a
Pleiades member (Torres et al., 2021), and HII 1776 was included as it was in
the FEPS sample observed at 70 $\mu$m by Stauffer et al. (2005). The sample
was derived from the work of Stauffer et al. (2005) that determined each star
to be a likely Pleiades cluster member based on radial velocities, proper
motions, chromospheric and coronal activity indicators, and lithium abundance.
These stars have colors $1.5<V-K_{S}<2.15$, corresponding to spectral types F5
to K1. None of the stars in this sample shows evidence for hot ($>300$ K)
dust, from stringent limits of excess emission at wavelengths $\lesssim 8$
$\mu$m. This is consistent with the expected clearing of inner disk regions
($<1$ au) by collisions over $\sim 100$ Myr (and perhaps also by planets).
Table 1 lists the target stars and their coordinates, together with the
stellar effective temperatures and luminosities from the analysis provided in
the Gaia DR2 catalog (Gaia Collaboration et al., 2018) where available Table 1
also lists estimates of the stellar mass based on the luminosity-mass relation
of Demircan & Kahraman (1991) for zero-age main sequence stars in this mass
range. The last column lists the 24 $\mu$m excess ratio ($F_{obs}/F_{star}$)
determined by Sierchio et al. (2010), with significant detections of
circumstellar dust emission indicated in bold (excess ratio $>1.1$, i.e.
exceeding $3\sigma$). The 20 systems observed at 70 $\mu$m by Stauffer et al.
(2005) are at the end of Table 1 (demarcated by a line).

Table 1: ALMA Pleiades Targets

Name | Alternate | RA | Dec | DistanceaaPositions and flux densities obtained from Gaussian model fits to the visibilities and corrected for primary beam attenuation. The positional uncertainties are typically $<0\farcs 2$ ($1\sigma$) in each coordinate. |  $T_{eff}$bbStellar luminosities and effective temperatures from the Gaia DR2 catalog (Gaia Collaboration et al., 2018).  |  $L_{star}$bbStellar luminosities and effective temperatures from the Gaia DR2 catalog (Gaia Collaboration et al., 2018).  | $M_{\star}$ | 24 $\mu$m
---|---|---|---|---|---|---|---|---
| Name | (J2000) | (J2000) | (pc) | (K) | ($L_{\odot}$) | ($M_{\odot}$) | excess ratiocc24 $\mu$m excess ratio reported by Sierchio et al. (2010) except for HII 1776 that is reported by Carpenter et al. (2009); significant detections are indicated in bold.
HIP 17317 | AK IA 36 | 3:42:24.05 | 22:25:15.2 | $137.4_{-1.0}^{+1.0}$ | $5626_{-113}^{+101}$ | $1.19_{-0.01}^{+0.01}$ | 1.04 | 1.03
BD+21 508 | AK IA 56 | 3:43:31.15 | 22:09:29.2 | $135.0_{-1.1}^{+1.2}$ | $5468_{-123}^{+379}$ | $1.14_{-0.01}^{+0.01}$ | 1.03 | 1.13
HD 23312 | AK IA 76 | 3:44:58.94 | 22:01:56.0 | $141.1_{-1.2}^{+1.2}$ | $6253_{-147}^{+134}$ | $2.66_{-0.03}^{+0.03}$ | 1.32 | 1.26
HD 24463 | AK IA 317 | 3:54:21.62 | 24:04:31.6 | $135.3_{-1.0}^{+1.0}$ | $6136_{-190}^{+201}$ | $1.97_{-0.02}^{+0.02}$ | 1.21 | 1.01
HD 22627 | AK IB 7 | 3:39:09.15 | 24:22:03.3 | $143.2_{-1.0}^{+1.0}$ | $5924_{-92}^{+81}$ | $2.01_{-0.02}^{+0.02}$ | 1.22 | 1.12
HIP 17044 | AK IB 8 | 3:39:13.51 | 24:27:58.6 | $141.7_{-0.9}^{+0.9}$ | $5733_{-761}^{+181}$ | $1.24_{-0.01}^{+0.01}$ | 1.06 | 1.04
HD 23598 | AK IB 365 | 3:47:20.91 | 25:31:32.1 | $136.2_{-0.9}^{+1.0}$ | $5892_{-266}^{+113}$ | $1.84_{-0.02}^{+0.02}$ | 1.18 | 1.01
HD 24086 | AK IB 590 | 3:51:06.36 | 25:35:40.0 | $139.0_{-1.3}^{+1.3}$ | $6430_{-149}^{+270}$ | $3.55_{-0.04}^{+0.04}$ | 1.44 | 0.99
HD 23975 | AK IB 560 | 3:50:17.70 | 25:22:45.5 | $137.3_{-1.2}^{+1.3}$ | $6155_{-212}^{+283}$ | $2.17_{-0.03}^{+0.03}$ | 1.24 | 1.34
HD 22444 | AK II 34 | 3:37:24.07 | 22:21:02.6 | $132.3_{-1.5}^{+1.6}$ | $6135_{-101}^{+102}$ | $3.04_{-0.04}^{+0.04}$ | 1.37 | 1.06
TYC 1798-465-1 | AK II 359 | 3:37:34.93 | 24:14:10.8 | $155.4_{-1.2}^{+1.2}$ | $5713_{-252}^{+156}$ | $1.21_{-0.01}^{+0.01}$ | 1.05 | 0.99
HIP 16979 | AK II 383 | 3:38:22.61 | 22:29:58.0 | $137.4_{-0.9}^{+0.9}$ | $5880_{-129}^{+118}$ | $1.50_{-0.01}^{+0.01}$ | 1.12 | 1.12
HD 22680 | AK II 437 | 3:39:41.22 | 23:17:26.3 | $140.5_{-1.1}^{+1.1}$ | $6057_{-217}^{+181}$ | $1.75_{-0.02}^{+0.02}$ | 1.17 | 3.79
HIP 16639 | AK III 288 | 3:34:07.34 | 24:20:39.1 | $137.2_{-0.9}^{+0.9}$ | $6086_{-171}^{+160}$ | $2.36_{-0.02}^{+0.02}$ | 1.27 | 0.99
HII 25 | HD 23061 | 3:42:55.15 | 24:29:34.2 | $138.4_{-1.0}^{+1.0}$ | $6197_{-137}^{+464}$ | $2.54_{-0.02}^{+0.02}$ | 1.30 | 1.12
HII 102 | TYC 1799-118-1 | 3:43:24.56 | 23:13:32.5 | $138.8_{-0.7}^{+0.7}$ | $5338_{-60}^{+529}$ | $1.09_{-0.01}^{+0.01}$ | 1.02 | 1.04
HII 1132 | HD 23514 | 3:46:38.43 | 22:55:10.4 | $138.2_{-1.0}^{+1.0}$ | $6269_{-196}^{+392}$ | $2.68_{-0.03}^{+0.03}$ | 1.32 | 17.44
HII 1139 | HD 23513 | 3:46:40.02 | 23:06:36.3 | $137.2_{-1.1}^{+1.1}$ | $6365_{-150}^{+179}$ | $2.67_{-0.03}^{+0.03}$ | 1.32 | 1.09
HII 1766 | HD 23732 | 3:48:16.91 | 25:12:53.6 | $137.8_{-1.4}^{+1.5}$ | $6614_{-336}^{+430}$ | $3.19_{-0.04}^{+0.04}$ | 1.39 | 1.60
HII 2172 | HD 282965 | 3:49:11.77 | 24:38:10.9 | $139.1_{-1.1}^{+1.1}$ | $5809_{-190}^{+50}$ | $1.10_{-0.01}^{+0.01}$ | 1.02 | 1.10
HII 3031 | HD 24132 | 3:51:27.25 | 24:31:06.2 | $135.9_{-1.2}^{+1.2}$ | $6753_{-201}^{+239}$ | $4.25_{-0.05}^{+0.05}$ | 1.51 | 1.01
TYC 1802-95-1 | PELS 7 | 3:34:47.23 | 26:05:40.3 | $177.2_{-1.7}^{+1.8}$ | $5596_{-105}^{+163}$ | $1.79_{-0.02}^{+0.02}$ | 1.18 | 0.99
HIP 17020 | PELS 20 | 3:38:56.90 | 24:34:10.4 | $139.1_{-1.3}^{+1.3}$ | $5723_{-170}^{+127}$ | $1.02_{-0.01}^{+0.01}$ | 1.00 | 1.19
HIP 17245 | PELS 23 | 3:41:36.19 | 25:37:08.6 | $138.5_{-1.0}^{+1.0}$ | $5800_{-175}^{+124}$ | $1.54_{-0.01}^{+0.01}$ | 1.12 | 1.06
HIP 17125 | PELS 25 | 3:40:03.11 | 27:44:24.9 | $138.8_{-2.5}^{+2.6}$ | $6220_{-133}^{+95}$ | $2.26_{-0.05}^{+0.05}$ | 1.26 | 1.02
BD+21 516 | PELS 40 | 3:45:09.91 | 21:42:15.5 | $127.3_{-0.8}^{+0.8}$ | $5837_{-156}^{+267}$ | $1.41_{-0.01}^{+0.01}$ | 1.10 | 1.03
HIP 18544 | PELS 86 | 3:58:01.72 | 20:40:35.5 | $125.1_{-1.1}^{+1.1}$ | $6185_{-102}^{+127}$ | $2.24_{-0.03}^{+0.03}$ | 1.26 | 1.10
BD+23 455 | PELS 121 | 3:27:42.07 | 23:48:12.4 | $195.1_{-1.7}^{+1.7}$ | $5676_{-111}^{+155}$ | $2.58_{-0.03}^{+0.03}$ | 1.31 | 0.97
HIP 16753 | PELS 124 | 3:35:31.73 | 22:49:24.0 | $131.5_{-0.8}^{+0.8}$ | $6094_{-135}^{+196}$ | $1.64_{-0.01}^{+0.01}$ | 1.15 | 1.06
BD+26 592 | PELS 128 | 3:39:53.74 | 26:43:00.3 | $221.7_{-2.6}^{+2.7}$ | $5633_{-101}^{+69}$ | $3.40_{-0.06}^{+0.06}$ | 1.42 | 1.05
TYC 1256-516-1 | PELS 135 | 3:45:01.68 | 19:33:32.8 | $137.4_{-1.0}^{+1.0}$ | $6190_{-167}^{+257}$ | $2.67_{-0.03}^{+0.03}$ | 1.32 | 1.10
HIP 18091 | PELS 146 | 3:52:00.83 | 19:35:47.8 | $143.1_{-0.8}^{+0.8}$ | $5593_{-205}^{+234}$ | $1.16_{-0.01}^{+0.01}$ | 1.04 | 1.43
HD 23935 | PELS 150 | 3:49:52.94 | 25:38:49.9 | $138.5_{-0.8}^{+0.8}$ | $6008_{-151}^{+163}$ | $2.41_{-0.02}^{+0.02}$ | 1.28 | 1.26
BD+22 617C | PELS 173 | 4:00:53.14 | 23:11:37.5 | $184.3_{-2.2}^{+2.2}$ | $6457_{-112}^{+205}$ | $3.95_{-0.06}^{+0.06}$ | 1.48 | 1.18
HIP 18955 | PELS 174 | 4:03:44.20 | 22:56:38.5 | $137.0_{-1.1}^{+1.1}$ | $5767_{-266}^{+144}$ | $2.20_{-0.02}^{+0.02}$ | 1.25 | 1.06
HIP 17316 | TrS 42 | 3:42:24.03 | 21:28:23.6 | $129.1_{-0.8}^{+0.8}$ | $5954_{-148}^{+274}$ | $1.62_{-0.01}^{+0.01}$ | 1.14 | 1.10
HD 24302 | Tr 60 | 3:52:53.50 | 24:42:55.7 | $140.5_{-1.2}^{+1.2}$ | $6376_{-108}^{+181}$ | $2.68_{-0.03}^{+0.03}$ | 1.32 | 1.01
HII 293 | V* V1169 Tau | 3:44:13.95 | 24:46:44.9 | $136.1_{-1.0}^{+1.0}$ | $5503_{-98}^{+145}$ | $0.82_{-0.01}^{+0.01}$ | 0.94 | 0.98
HII 405 | HD 23269 | 3:44:40.78 | 24:49:05.8 | $134.5_{-1.0}^{+1.0}$ | $6072_{-86}^{+68}$ | $1.74_{-0.02}^{+0.02}$ | 1.17 | 0.98
HII 489 | TYC 1803-808-1 | 3:44:56.40 | 24:25:57.4 | $134.2_{-1.2}^{+1.2}$ | $5693_{-189}^{+464}$ | $1.06_{-0.01}^{+0.01}$ | 1.01 | 1.20
HII 571 | TYC 1803-1156-1 | 3:45:15.39 | 25:17:21.2 | $134.4_{-0.9}^{+0.9}$ | $5167_{-127}^{+230}$ | $0.57_{-0.01}^{+0.01}$ | 0.84 | 1.05
HII 727 | V* V855 Tau | 3:45:40.20 | 24:37:37.3 | $131.6_{-1.1}^{+1.1}$ | $5927_{-60}^{+156}$ | $1.89_{-0.02}^{+0.02}$ | 1.19 | 1.00
HII 739 | V* V969 Tau | 3:45:42.15 | 24:54:20.7 | $184.8_{-20.9}^{+27.0}$ | $5820_{-170}^{+480}$ | $4.40_{-0.58}^{+0.58}$ | 1.53 | 1.00
HII 923 | BD+22 548 | 3:46:10.07 | 23:20:23.0 | $122.5_{-0.9}^{+0.9}$ | $5775_{-108}^{+67}$ | $1.13_{-0.01}^{+0.01}$ | 1.03 | 1.06
HII 996 | V* V1045 Tau | 3:46:22.69 | 24:34:11.6 | $138.7_{-0.9}^{+0.9}$ | $5775_{-223}^{+317}$ | $1.10_{-0.01}^{+0.01}$ | 1.02 | 1.21
HII 1117 | HD 282975 | 3:46:37.73 | 23:47:15.0 | $134.6_{-0.9}^{+0.9}$ | $5535_{-116}^{+45}$ | $1.34_{-0.01}^{+0.01}$ | 1.08 | 1.04
HII 1309 | HD 23584 | 3:47:10.09 | 24:16:35.1 | $136.1_{-1.1}^{+1.2}$ | $6165_{-71}^{+175}$ | $2.45_{-0.03}^{+0.03}$ | 1.29 | 0.97
HII 1338 | HD 23608 | 3:47:16.59 | 24:07:41.5 | $95.1_{-8.4}^{+10.2}$ | $6355_{-105}^{+141}$ | $2.34_{-0.24}^{+0.24}$ | 1.27 | 0.96
HII 1514 | HD 282967 | 3:47:40.45 | 24:21:52.5 | $136.2_{-1.2}^{+1.2}$ | $5629_{-136}^{+75}$ | $0.97_{-0.01}^{+0.01}$ | 0.98 | 1.07
HII 1613 | HD 282973 | 3:47:52.51 | 23:56:28.4 | $134.3_{-1.0}^{+1.0}$ | $6003_{-95}^{+121}$ | $1.68_{-0.02}^{+0.02}$ | 1.15 | 1.01
HII 1726 | HD 23713 | 3:48:07.12 | 24:08:30.9 |  $\cdots$ | $6444_{-446}^{+455}$ |  $\cdots$ |  $\cdots$ | 1.04
HII 1797 | BD+23 551 | 3:48:16.91 | 23:38:12.5 | $140.4_{-0.9}^{+0.9}$ | $5886_{-64}^{+84}$ | $1.49_{-0.01}^{+0.01}$ | 1.11 | 1.52
HII 1856 | HD 282971 | 3:48:26.20 | 24:02:53.5 | $135.0_{-1.1}^{+1.1}$ | $5956_{-103}^{+93}$ | $1.49_{-0.02}^{+0.02}$ | 1.11 | 0.94
HII 1912 | HD 23778 | 3:48:34.81 | 24:10:51.2 | $221.4_{-41.8}^{+66.8}$ | $6256_{-173}^{+365}$ | $8.59_{-1.76}^{+1.76}$ | 1.86 | 0.97
HII 1924 | Cl Melotte 22 1924 | 3:48:34.54 | 23:26:04.4 | $136.8_{-0.9}^{+0.9}$ | $5741_{-93}^{+177}$ | $1.16_{-0.01}^{+0.01}$ | 1.04 | 1.04
HII 2027 | Cl Melotte 22 2027 | 3:48:48.95 | 24:16:02.8 | $143.0_{-1.0}^{+1.0}$ | $5066_{-82}^{+363}$ | $0.88_{-0.01}^{+0.01}$ | 0.96 | 0.97
HII 120 | TYC 1799-102-1 | 3:43:31.98 | 23:40:25.7 | $135.7_{-1.0}^{+1.1}$ | $5524_{-94}^{+688}$ | $0.78_{-0.01}^{+0.01}$ | 0.92 | 1.04
HII 152 | V* V963 Tau | 3:43:37.77 | 23:32:08.7 | $136.0_{-1.0}^{+1.0}$ | $5502_{-229}^{+161}$ | $0.83_{-0.01}^{+0.01}$ | 0.94 | 1.12
HII 173 | Cl Melotte 22 173 | 3:43:48.44 | 25:11:23.6 | $130.4_{-1.6}^{+1.6}$ | $5206_{-159}^{+305}$ | $0.76_{-0.01}^{+0.01}$ | 0.91 | 1.00
HII 174 | V* V1271 Tau | 3:43:48.38 | 25:00:14.8 | $136.3_{-1.0}^{+1.0}$ | $4991_{-32}^{+60}$ | $0.41_{-0.01}^{+0.01}$ | 0.76 | 0.99
HII 250 | Cl Melotte 22 250 | 3:44:04.26 | 24:59:22.5 | $129.8_{-1.1}^{+1.1}$ | $5518_{-135}^{+166}$ | $0.76_{-0.01}^{+0.01}$ | 0.91 | 1.12
HII 314 | V* V1038 Tau | 3:44:20.19 | 24:47:45.4 | $134.2_{-0.9}^{+0.9}$ | $5431_{-107}^{+87}$ | $0.90_{-0.01}^{+0.01}$ | 0.96 | 1.02
HII 514 | Cl Melotte 22 514 | 3:45:04.03 | 25:15:27.4 | $135.3_{-0.8}^{+0.9}$ | $5516_{-71}^{+398}$ | $0.85_{-0.01}^{+0.01}$ | 0.95 | 1.19
HII 1015 | HD 282952 | 3:46:27.37 | 25:08:07.1 | $135.0_{-1.0}^{+1.0}$ | $5626_{-117}^{+101}$ | $0.95_{-0.01}^{+0.01}$ | 0.98 | 0.98
HII 1101 | HD 282954 | 3:46:38.80 | 24:57:33.8 | $136.3_{-1.0}^{+1.0}$ | $5900_{-271}^{+206}$ | $1.26_{-0.01}^{+0.01}$ | 1.06 | 1.48
HII 1182 | BD+22 552 | 3:46:47.08 | 22:54:51.5 | $138.5_{-1.0}^{+1.0}$ | $5590_{-225}^{+264}$ | $1.06_{-0.01}^{+0.01}$ | 1.01 | 1.03
HII 1200 | BD+22 553 | 3:46:50.56 | 23:14:20.3 | $153.2_{-1.5}^{+1.6}$ | $5946_{-79}^{+94}$ | $2.14_{-0.03}^{+0.03}$ | 1.24 | 1.15
HII 1776 | HD 282958 | 3:48:17.72 | 25:02:51.4 | $133.5_{-1.0}^{+1.0}$ | $5499_{-107}^{+101}$ | $0.69_{-0.01}^{+0.01}$ | 0.89 | 1.02
HII 2147 | V* V1282 Tau | 3:49:06.14 | 23:46:51.8 | $138.2_{-1.0}^{+1.0}$ | $5305_{-199}^{+738}$ | $1.02_{-0.01}^{+0.01}$ | 1.00 | 1.00
HII 2278 | HD 282960 | 3:49:25.75 | 24:56:14.5 |  $\cdots$ | $5080_{-81}^{+324}$ |  $\cdots$ |  $\cdots$ | 0.98
HII 2506 | BD+22 574 | 3:49:56.52 | 23:13:06.2 | $146.5_{-1.1}^{+1.1}$ | $5849_{-92}^{+104}$ | $1.42_{-0.01}^{+0.01}$ | 1.10 | 1.00
HII 2644 | Cl Melotte 22 2644 | 3:50:20.90 | 24:28:00.3 | $137.4_{-1.0}^{+1.1}$ | $5469_{-62}^{+166}$ | $0.63_{-0.01}^{+0.01}$ | 0.87 | 1.01
HII 2786 | V* V1175 Tau | 3:50:40.11 | 23:55:58.1 | $139.1_{-1.2}^{+1.2}$ | $5852_{-103}^{+199}$ | $1.25_{-0.01}^{+0.01}$ | 1.06 | 0.99
HII 2881 | V* V1176 Tau | 3:50:54.32 | 23:50:05.6 | $147.1_{-13.5}^{+16.5}$ | $4917_{-57}^{+122}$ | $0.63_{-0.07}^{+0.07}$ | 0.87 | 1.00
HII 3097 | Cl Melotte 22 3097 | 3:51:40.48 | 24:58:58.6 | $134.6_{-1.3}^{+1.3}$ | $5441_{-143}^{+184}$ | $0.74_{-0.01}^{+0.01}$ | 0.91 | 1.01
HII 3179 | HD 24194 | 3:51:56.89 | 23:54:06.1 | $133.1_{-1.1}^{+1.1}$ | $5845_{-172}^{+144}$ | $1.42_{-0.02}^{+0.02}$ | 1.10 | 1.00

aaPositions and flux densities obtained from Gaussian model fits to the
visibilities and corrected for primary beam attenuation. The positional
uncertainties are typically $<0\farcs 2$ ($1\sigma$) in each coordinate.

Distances from Gaia DR2 catalog parallaxes (Gaia Collaboration et al., 2018)
using the method of Bailer-Jones et al. (2018).

###  2.2 ALMA Parameters and Observations

The ALMA Band 6 (1.3 mm) observations were performed in 7 execution blocks in
the period from 2019 December 12 to 2019 December 18 with 42 to 44 available
antennas (program 2019.1.00251.S). The array was in configuration C43-2, which
provided a beam size of $\sim 1\farcs 5$ (200 au) and $12^{\prime\prime}$
(1600 au) maximum recoverable scale. Each star was observed for 4.2 minutes.
The correlator setup included three spectral windows centered at 232.5, 247.0,
and 245.0 GHz with 2.0 GHz bandwidth and 128 channels (TDM mode), and one
spectral window centered at 230.538 GHz with 1.875 GHz bandwidth and 3840
channels (FDM mode). The field of view was set by the $27^{\prime\prime}$
(FWHM) primary beam of the ALMA antennas at 1.3 mm. The datasets were
processed by the ALMA standard calibration pipeline using the Common Astronomy
Software Applications ($\mathtt{CASA}$) software package (version 5.6.1)
(McMullin et al., 2007). Table 2 presents a log of the ALMA observations. In
brief, the time dependence of phase and amplitude were calibrated with
frequent observations of the nearby quasar J0357+2319, the passband response
was calibrated with observations of the strong source J04323-0120, and the
absolute flux scale was referenced to ALMA observations of Solar System
objects with an estimated accuracy of 10%.

Table 2: ALMA Observations Log  Date | Antennas | Baselines | pwv | Passband | Phase
---|---|---|---|---|---
|  | (m) | (mm) |  |
2019 December 12 | 43 | 15.0 - 313.7 | 1.0 | J0423-0120 | J0357+2319
2019 December 13 | 43 | 15.0 - 313.7 | 1.2 | J0423-0120 | J0357+2319
2019 December 14 | 44 | 15.0 - 313.7 | 2.8 | J0423-0120 | J0357+2319
2019 December 15 | 42 | 15.1 - 313.7 | 0.8 | J0423-0120 | J0357+2319
2019 December 16 | 43 | 15.1 - 313.7 | 0.8 | J0423-0120 | J0357+2319
2019 December 17 | 43 | 15.1 - 312.7 | 1.1 | J0423-0120 | J0357+2319
2019 December 18 | 42 | 15.1 - 312.7 | 1.0 | J0423-0120 | J0357+2319

###  2.3 ALMA Sensitivity in Context

A useful way to contextualize the sensitivity of these ALMA observations is to
compare detection limits from surveys at different wavelengths in terms of
fractional luminosity, which can be expressed for blackbody emission simply as
a function of dust temperature, stellar effective temperature, and dust-to-
star flux ratio (e.g. Beichman et al., 2006; Greaves et al., 2009). The ring-
like appearance of most debris disks supports the assumption of a single value
for the dust temperature as a reasonable approximation, even as some debris
disks sometimes do span a broad range of radii (e.g. Kalas et al., 2006). The
fractional luminosity provides a proxy for the total cross-sectional area of
dust. Figure 1 shows curves for the minimum detectable values of
$L_{dust}/L_{star}$ for a typical Solar-type star in the Pleiades sample
($L_{star}=1.8~{}L_{\odot},T_{eff}=5825$ K) as a function of $T_{dust}$ for
(1) the Pleiades sample observed at 24 $\mu$m by Spitzer with flux density
more than 10% in excess of the extrapolated photospheric flux density (32%
detected, see Sierchio et al. (2010)), (2) the subsample of 20 systems
observed at 70 $\mu$m by Spitzer with $3\times rms$ = 21 mJy (no detections,
Stauffer et al. (2005)), (3) the small group of 5 systems observed from the 70
$\mu$m subsample at 1.2 mm with the IRAM 30 meter telescope with $3\times rms$
= 2.1 mJy, which are the most sensitive millimeter observations to date (no
detections, Roccatagliata et al. (2009)), and (4) the 1.3 mm ALMA observations
for $3\times rms$ = 160 $\mu$Jy beam-1, representative of the Pleiades sample
(see Table 3, where the rms value for each target was determined by the noise
statistics of the entire image), as well as an additional fall off as the
largest belts become spatially resolved (see §3.1). The millimeter curves also
account for steeper than blackbody spectra at these long wavelengths (see
§4.2.2 for details). The ALMA detection limit expressed in fractional
luminosity is $L_{dust}/L_{star}=2\times 10^{-4}$ for $T_{dust}=50$ K. This
limit is comparable to the Spitzer 24 $\mu$m threshold for similar values of
$T_{dust}$, and substantially better for dust belts at lower temperatures.

Figure 1: Detection limits for various debris disk surveys of Solar-type stars
in the Pleiades, in terms of fractional luminosity ($L_{dust}/L_{star}$). The
colored curves show the typical limits from the Spitzer 24 $\mu$m surveys of
the Pleiades sample targeted for ALMA (green, Sierchio et al., 2010), the
Spitzer 70 $\mu$m observations of a subsample of 20 systems (red, Stauffer et
al., 2005), and IRAM 30 meter 1.2 mm observations of 5 of those systems (blue,
Roccatagliata et al., 2009). The purple line shows the ALMA detection limit,
which improves on the millimeter sensitivity by about an order of magnitude
(for more than an order of magnitude more systems).

###  2.4 Background Source Confusion

The Pleiades cluster shows a reflection nebula from a nearby dust cloud, and
the presence of this dusty cloud has the potential to confuse searches for
excess emission from stars in the cluster. For ALMA, the interferometer
response strongly spatially filters extended emission, so confusion from this
cloud is unlikely to be problematic. Moreover, with a $1\farcs 5$ beam,
extragalactic millimeter source counts show that chance alignments with
background contaminants are insignificant (e.g. Carniani et al., 2015;
González-López et al., 2020). The larger Spitzer beams at 24 $\mu$m
($6^{\prime\prime}$) and 70 $\mu$m ($20^{\prime\prime}$) clearly encompass any
circumstellar dust emission around Pleiades members but also raise the spectre
of confusion with background sources. Stauffer et al. (2005) suggest that up
to 2 out of 5 excess detections at 24 $\mu$m in their sample of 20 stars could
be due to confusion with background Active Galactic Nuclei or Galactic cirrus.

##  3 Results

We used the tclean task in CASA to generate continuum images for each star in
the sample. We adopted natural weighting to obtain the best point source
sensitivity, iterated to a threshold of $2\times$ the rms noise level, and
applied a primary beam correction. Table 3 lists the beam size and rms noise
for the image for each star, which ranges from 48 to 73 $\mu$Jy beam-1. The
median rms noise was 54 $\mu$Jy beam-1. Figures 2 and 3 shows galleries of
ALMA 1.3 mm images centered on each of the target stars in the full Pleiades
sample.

Table 3: ALMA 1.3 millimeter Imaging

Name | Beam FWHM | P.A. | rms noise
---|---|---|---
| (arcsec) | (deg) | ($\mu$Jy beam-1)
HIP 17317 | $1.56\times 1.42$ | $0.8$ | 48
BD+21 508 | $1.56\times 1.42$ | $1.4$ | 50
HD 23312 | $1.55\times 1.42$ | $2.5$ | 50
HD 24463* | $1.62\times 1.41$ | $10.6$ | 58
HD 22627 | $1.60\times 1.41$ | $-6.6$ | 51
HIP 17044 | $1.60\times 1.41$ | $-7.7$ | 51
HD 23598 | $1.63\times 1.40$ | $-0.8$ | 53
HD 24086 | $1.63\times 1.40$ | $0.3$ | 51
HD 23975 | $1.62\times 1.40$ | $-0.2$ | 51
HD 22444 | $1.56\times 1.42$ | $-12.7$ | 51
TYC 1798-465-1 | $1.64\times 1.38$ | $-26.6$ | 57
HIP 16979 | $1.56\times 1.42$ | $-14.4$ | 49
HD 22680 | $1.58\times 1.42$ | $-12.5$ | 52
HIP 16639 | $1.62\times 1.40$ | $-18.9$ | 55
HII 25 | $1.61\times 1.40$ | $-12.3$ | 54
HII 102 | $1.58\times 1.42$ | $-11.1$ | 52
HII 1132 | $1.57\times 1.43$ | $-9.1$ | 50
HII 1139* | $1.58\times 1.42$ | $-9.1$ | 55
HII 1766 | $1.63\times 1.39$ | $-8.2$ | 54
HII 2172 | $1.61\times 1.40$ | $-9.0$ | 52
HII 3031 | $1.60\times 1.41$ | $-6.5$ | 52
TYC 1802-95-1* | $1.65\times 1.38$ | $-17.2$ | 59
HIP 17020 | $1.64\times 1.39$ | $-19.5$ | 50
HIP 17245 | $1.66\times 1.39$ | $-16.7$ | 54
HIP 17125 | $1.71\times 1.37$ | $-12.5$ | 55
BD+21 516 | $1.56\times 1.42$ | $-18.8$ | 49
HIP 18544 | $1.52\times 1.43$ | $-3.1$ | 50
BD+23 455 | $1.64\times 1.37$ | $-27.2$ | 54
HIP 16753 | $1.61\times 1.40$ | $-25.6$ | 52
BD+26 592* | $1.68\times 1.38$ | $-15.7$ | 61
TYC 1256-516-1 | $1.52\times 1.41$ | $-30.1$ | 51
HIP 18091* | $1.51\times 1.42$ | $-27.3$ | 73
HD 23935 | $1.65\times 1.39$ | $-14.9$ | 52
BD+22 617C | $1.58\times 1.43$ | $-7.6$ | 50
HIP 18955 | $1.57\times 1.43$ | $-5.1$ | 53
HIP 17316 | $1.57\times 1.40$ | $-27.6$ | 52
HD 24302 | $1.63\times 1.40$ | $-17.0$ | 49
HII 293 | $1.64\times 1.38$ | $-22.2$ | 58
HII 405 | $1.63\times 1.38$ | $-21.1$ | 53
HII 489 | $1.63\times 1.38$ | $-22.2$ | 54
HII 571 | $1.64\times 1.38$ | $-20.5$ | 57
HII 727 | $1.63\times 1.38$ | $-22.6$ | 56
HII 739 | $1.64\times 1.38$ | $-21.9$ | 54
HII 923 | $1.61\times 1.41$ | $-24.9$ | 53
HII 996 | $1.66\times 1.35$ | $-29.8$ | 60
HII 1117 | $1.62\times 1.38$ | $-25.0$ | 52
HII 1309 | $1.63\times 1.38$ | $-23.8$ | 55
HII 1338 | $1.63\times 1.38$ | $-24.7$ | 54
HII 1514 | $1.63\times 1.38$ | $-24.2$ | 55
HII 1613 | $1.62\times 1.38$ | $-25.4$ | 55
HII 1726 | $1.63\times 1.37$ | $-25.0$ | 56
HII 1797 | $1.62\times 1.38$ | $-26.6$ | 52
HII 1856 | $1.63\times 1.38$ | $-25.8$ | 55
HII 1912 | $1.63\times 1.37$ | $-25.6$ | 52
HII 1924 | $1.62\times 1.39$ | $-28.1$ | 53
HII 2027 | $1.63\times 1.37$ | $-26.0$ | 54
HII 120 | $1.63\times 1.35$ | $-31.4$ | 56
HII 152 | $1.63\times 1.36$ | $-31.9$ | 53
HII 173 | $1.66\times 1.36$ | $-25.8$ | 58
HII 174 | $1.66\times 1.35$ | $-27.9$ | 56
HII 250 | $1.65\times 1.35$ | $-28.3$ | 55
HII 314 | $1.65\times 1.35$ | $-28.4$ | 55
HII 514 | $1.67\times 1.35$ | $-28.4$ | 57
HII 1015 | $1.66\times 1.35$ | $-27.9$ | 53
HII 1101 | $1.65\times 1.35$ | $-28.2$ | 57
HII 1182 | $1.64\times 1.39$ | $-33.0$ | 56
HII 1200 | $1.64\times 1.35$ | $-35.0$ | 56
HII 1776 | $1.66\times 1.35$ | $-29.0$ | 55
HII 2147 | $1.64\times 1.35$ | $-32.3$ | 56
HII 2278 | $1.67\times 1.35$ | $-29.7$ | 55
HII 2506 | $1.64\times 1.36$ | $-33.6$ | 55
HII 2644 | $1.66\times 1.35$ | $-29.9$ | 52
HII 2786 | $1.65\times 1.34$ | $-31.9$ | 57
HII 2881 | $1.65\times 1.34$ | $-32.2$ | 53
HII 3097 | $1.67\times 1.35$ | $-29.9$ | 54
HII 3179 | $1.65\times 1.34$ | $-32.2$ | 54

Note. — ∗ Serendipitous emission source in the field of view

Figure 2:  ALMA 1.3 mm images of the Pleiades FGK star sample, with typical at
the origin of its panel. The synthesized beam is indicated by the ellipse in
the lower left corner. No circumstellar dust emission is detected from any of
the target stars.  Figure 3:  ALMA 1.3 mm images of the Pleiades FGK star
sample (continued).

###  3.1 Pleiades 1.3 mm Dust Continuum Emission Limits

We visually inspected the images and did not identify any significant sources
of emission at the target star locations. We consider the ALMA detection limit
to be $3\times$ the rms noise level, an appropriate metric given that we have
accurate a priori knowledge of the stellar positions. As an additional check,
we fitted a point source model to the visibilities for each target star using
the UVMultiFit library (Martí-Vidal et al., 2014), restricting the fit source
location to within $0\farcs 2$ offsets of the stellar position; this procedure
also did not reveal any significant sources of emission.

We can translate the ALMA upper limits into estimates of dust mass by making
standard assumptions, i.e.
$M_{dust}=F_{\nu}d^{2}/(\kappa_{\nu}B_{\nu}(T_{dust}))\approx
0.012~{}(F_{1.3~{}mm}/54~{}\mu
Jy)(d/133.5~{}pc)^{2}(T_{dust}/40~{}K)~{}M_{\oplus}$ where $F_{1.3~{}mm}$ is
the 1.3 mm flux density, $\kappa_{1.3~{}mm}=2.0$ cm2 g-1 is the dust mass
opacity (Beckwith et al., 1990), and $B_{\nu}(T_{dust})$ is the Planck
function taken to be in the Rayleigh-Jeans regime. We have adopted the dust
mass opacity for circumstellar disks advocated by Beckwith et al. (1990) in
order to facilitate comparisons with other studies that make this common
assumption. With this assumption, taking $3\times$ the median rms noise level
of the sample and $M_{dust}=40$ K leads to a dust mass limit of 0.036 M⊕
(about $3\times$ the mass of the Moon). We also made a stacked image from the
76 continuum images, to obtain a deeper constraint on the average flux density
of the undetected sources. This stacked image shows no significant emission at
the center and has an rms noise level of 6.1 $\mu$Jy beam-1. Adopting the same
assumptions leads to a dust mass limit of 0.004 M⊕ for the mean level of dust
around stars in this sample.

The typical FWHM beam size of $1\farcs 5$ (200 au) comfortably encompasses
dust belts with radii up to 100 au. For larger dust belts, the emission
spreads across multiple beams, and the sensitivity for detection is generally
lower, depending in detail on the radius, width and viewing geometry. To
quantify the dust belt detection threshold as a function of size in a simple
way, we made a series of natural weight images with Gaussian tapers that
increased in $0\farcs 5$ steps from 0 to $12\farcs 5$. This process produces a
sequence of larger beams to cover larger dust belts (at any viewing geometry)
at the cost of poorer point source sensitivity. This sensitivity as a function
of size scale has a similar shape for all of the targets because the $u,v$
plane coverage of the ALMA observation is very similar. The sensitivity
decreases by a factor of 2 at a beam size of $4\farcs 4$ and a factor of 3 at
$6\farcs 9$. While there is some dispersion in the noise levels at the largest
tapers among the different targets, the increase in rms as a function of
$\theta$, the beam size FWHM (arcsec), is described well by the expression
$1.000+0.32(\theta-1.5)+0.010(\theta-1.5)^{2}$ for beam sizes in the range of
interest. We will adopt this dependence of sensitivity with size scale in
comparing the observations to population synthesis models in §4.2.

###  3.2 Serendipitous Source Detections

For five targets, we identified significant ($>5\sigma$) sources of emission
within the field of view but not associated with the centrally located star
(HD 24463, HII 1139, TYC 1802-95-1, BD+26 592, and HIP 18091). For these
serendipitous sources, we used $\mathtt{UVMultiFit}$ to fit a Gaussian model
to the detected source(s), simultaneous with a central point source. Table 4
lists fitted offsets (east and north) from the field center, absolute
coordinates, and flux densities. Figure 4 shows the images that contain these
sources. The source in the field of TYC 1802-95-1 is clearly resolved. We
checked the 2MASS infrared K band images of all of these sources for near-
infrared counterparts and did not find any, although blending made the
relevant region difficult to assess in some cases. The lack of any obvious
near-infrared counterparts suggests the emission is not associated with stars
located in the Pleiades cluster. Most likely they are dusty galaxies in the
background. According to the 1.2 mm source counts of González-López et al.
(2020), approximately 5 sources exceeding $\gtrsim 500~{}\mu$Jy are expected
across the sample ($\sim 5\sigma$ at the primary beam half power width),
consistent with these detections.

Table 4: Serendipitous Source DetectionsaaPositions and flux densities obtained from Gaussian model fits to the visibilities and corrected for primary beam attenuation. The positional uncertainties are typically $<0\farcs 2$ ($1\sigma$) in each coordinate. Target Field | offset | RA | Dec | Flux Density
---|---|---|---|---
| (E′′, N′′) | (J2000) | (J2000) | (mJy)
HD 24463 | -1.1, -5.6 | 3:54:21.52 | 24:04:26.0 | 1.48$\pm$0.22
HII 1139 | -8.6, -13.1 | 3:46:39.43 | 23:06:23.2 | 1.84$\pm$0.31
HII 1139 | 11.1, -4.4 | 3:46:40.74 | 23:06:31.9 | 1.38$\pm$0.22
TYC 1802-95-1 | -2.2, -2.3 | 3:34:47.05 | 26:05:38.0 | 1.25$\pm$0.15
BD+26 592 | 3.0, -4.1 | 3:39:53.90 | 26:42:56.2 | 0.95$\pm$0.11
HIP 18091 | 11.7, -0.6 | 3:52:01.57 | 19:35:47.2 | 2.58$\pm$0.16
Figure 4: ALMA 1.3 mm continuum images for Pleiades fields where significant
($>5\sigma$) emission sources were detected that are not associated with the
target stars located at the field center. Table 4 lists the locations and flux
densities of these sources. The beam sizes are indicated in the lower left
corner of each panel.

##  4 Discussion

The ALMA 1.3 mm survey of a sample of FGK type stars in the 115 Myr-old
Pleiades cluster did not result in any detections of circumstellar dust
emission. These data provide new limits on dust belt fractional luminosities
and masses in a regime of low temperatures and large radii that significantly
improve on previous observations of Solar-type stars of Pleiades age. In
particular, the dust mass limit derived from stacking the non-detections is
$<0.004$ M⊕, about two orders of magnitude lower than from previous millimeter
observations of such stars in the Pleiades (for the same dust opacities and
temperatures; Greaves et al. 2009; Roccatagliata et al. 2009). While a handful
of Solar analogs believed to be at an age similar to the Pleiades are known to
harbor massive ($\gtrsim 0.10$ M⊕) dust belts at Kuiper Belt scales and beyond
(Holland et al., 2017), such systems are evidently rare and not found in this
Pleiades sample. For example, the disk around the nearby $\sim 80-200$ Myr-old
G2 star HD 107146 would be 500 $\mu$Jy at 1.3 mm at Pleiades distance
(Williams et al., 2004; Ricci et al., 2015) and readily detected in this
survey. We examine these Pleiades survey results in the context of debris disk
evolution models based on collisional evolution.

###  4.1 Limits on Fractional Luminosity

As Figure 1 shows, these ALMA observations provide new sensitivity to dust
belts with $T\lesssim 40$ K and fractional luminosity $>10^{-4}$. For warm
disks with blackbody radii $\sim 5$ au, the fractional luminosity limit
implied by the observations is $f\sim 5\times 10^{-3}$. For colder disks with
blackbody radii $\sim 100$ au, the fractional luminosity limit is $f\sim
5\times 10^{-5}$. None of the Pleiades stars with significant 24 $\mu$m excess
emission are detected at 1.3 mm. Given that these systems are also not
detected by Spitzer at the shorter wavelength of 8 $\mu$m, the circumstellar
dust must be at intermediate temperatures, and therefore concentrated at
intermediate radii (from a few au to a few 10’s of au). The previous non-
detections of Pleiades targets at 70 $\mu$m and 1.2 mm are all consistent with
the ALMA results (e.g., see Figure. 1). The lack of 1.3 mm dust continuum
emission detections implies an absence of massive and cold debris belts at
large radial distances around the target stars.

###  4.2 Collisional Evolution Models

Multi-wavelength observations of stars of different ages and spectral types
provide constraints on debris disk incidence and fractional luminosity that
have led to the development of various models to describe debris disk
evolution based on collisions and “stirring” mechanisms that dynamically
excite planetesimals leftover from the planet formation process (e.g., Wyatt &
Dent, 2002; Dominik & Decin, 2003; Kenyon & Bromley, 2004; Wyatt et al.,
2007a).

Najita et al. (2022) used new coagulation models to assess the possibility
that the observed population of cold debris disks around Sun-like stars evolve
from the $\sim 25$% of T Tauri systems that harbor large disks with ringed
substructures. They show that the resulting “bright stalwart” debris disks by
themselves can account for the available constraints on cold debris disk
incidence and luminosities. In particular, models that start with rings at
radii of 45 or 75 au with at least a few Earth masses in solids, using
intermediate planetesimal formation efficiencies, readily produce cold belts
with $L_{disk}/L_{star}\sim 10^{-3}$ at 10-100 Myr that subsequently fade. In
this scenario, one would expect the ALMA 1.3 mm survey of the Pleiades to
reveal many such belts. Since none were detected, the descendents of the large
and ringed protoplanetary disks, if present, must be at least an order of
magnitude less luminous at Pleiades age. These limits suggest different
initial conditions or evolutionary pathways.

The analytical evolutionary model of Wyatt et al. (2007a), building on Wyatt &
Dent (2002) and Dominik & Decin (2003), provides a practical framework for
comparison with observations. In brief, this model assumes steady-state
evolution in a catastrophic collisional cascade. Every star is assumed to be
born with a planetesimal belt with radius $r$, narrow width $dr$, mass
$M_{disk}$, and a size distribution of colliding objects $n(D)\propto
D^{2-3q}$, where $q=11/6$ (i.e. collisional equilibrium, Dohnanyi, 1969;
Tanaka et al., 1996). In this analytical model, the belt properties are an
initial condition; no physical explanation is provided for the origin of the
belts. The production of dust is a top-down process that requires
fragmentation of larger belt objects, and the evolution on long timescales is
therefore determined by the collisional lifetime of the largest planetesimals,
$t_{c}$ (see equation 7 of Wyatt et al. (2007a)). Then the time evolution of
the disk mass is determined by the differential equation
$dM_{tot}/dt=-M_{tot}/t_{c}$, which implies
$M_{tot}(t)=M_{tot}(0)/[1+t/t_{c}(0)]$, i.e. the debris disk mass (and
fractional luminosity) decrease as $t^{-1}$. This is an approximation, as
catastrophic collisions gradually reduce the size of the largest planetesimals
over time, which results in a shorter collisional timescale and a faster
decrease of disk mass (Kenyon & Bromley, 2016). Nonetheless, this basic model
has been successful in reproducing many observations of debris disks,
including the properties of debris disks around A type stars of various ages
at 24 $\mu$m and 70 $\mu$m (Wyatt et al., 2007b), the incidence and excess
emission of nearby FGK type stars at 24, 70, 100 and 160 $\mu$m (Kains et al.,
2011; Sibthorpe et al., 2018), and the (low) detection rate of debris disks
around M stars (Morey & Lestrade, 2014). Additional intuition about the time
evolution behavior may be obtained by noting that $t_{c}\propto
r^{3.5}/M_{tot}$ in this model, or $t_{c}\propto 1/(M_{tot}T^{7})$ using
blackbody radiative equilibrium to relate dust temperature to belt radius,
i.e. $T\propto L_{star}^{1/4}r^{-1/2}$. The extremely steep dependence of
timescale on dust temperature implies that warm (small radius) belts evolve
much more rapidly than cold (large radius) belts.

####  4.2.1 Steady-State Evolution Parameters

We applied the collisional evolution model as described by Wyatt et al.
(2007b) to the population of FGK stars in Pleiades observed by ALMA. The
initial disk parameters are the disk mass $M_{tot}(0)$, the disk radius $r$
and width $dr$, the material density $\rho$, maximum planetesimal diameter
$D_{c}$, minimum dust grain diameter, $D_{bl}$ and power-law index of the
planetesimal size distribution, $q$. The time evolution is determined by the
dispersal threshold $Q_{D}^{*}$, eccentricity $e$, and inclination $i$. We fix
$dr=r/2$, $\rho=\textrm{2700 kg m}^{-3}$, $q=11/6$, and $e/i=1$, following the
analysis of nearby FGK stars by Kains et al. (2011). The disk masses are drawn
from a log-normal distribution with mean Mmid and and width 1.14 dex (as
determined by Andrews & Williams 2005), and the disk radii are drawn from a
power law distribution $N(r)\propto r^{\gamma}$ from 1 to 1000 au. The five
free parameters in the model are Mmid, $\gamma$, $Q_{D}^{*}$, $e$ and $D_{c}$,
and we set these to the values determined by Sibthorpe et al. (2018) from
fitting to Herschel observations of debris disks around FGK stars near the
Sun, i.e. M${}_{mid}=2.1$ M⊕, $\gamma=-1.7$, $Q_{D}^{*}=500$ J kg-1, $e=0.05$,
and $D_{c}=450$ km. With these assumptions, we can test if the same
evolutionary model is consistent with observations of the Pleiades.

The relevant stellar parameters in the models are luminosity, $L_{star}$,
mass, $M_{star}$, and effective temperature, $T_{eff}$. We adopted values for
$T_{eff}$, $L_{star}$ and $M_{star}$ from Table 1. (Highly accurate values for
the stellar masses are not critical as the range across the sample is modest
and model dependencies are weak.)

We omitted Pels 173 from our analysis for the same reasons it was excluded by
Sierchio et al. (2010): the reddening to this star suggests it lies behind the
Pleiades cluster and is not a member. We also omitted HII 1726 and HII 2278
because the Gaia DR2 catalog does not provide a set of stellar parameters.

####  4.2.2 Modified Blackbody Emission

To calculate the spectrum of each dust belt, we assumed modified blackbody
emission and obtain the flux from $F_{\nu}=(2.95\times
10^{10})B_{\nu}(\lambda,T)fr^{2}d^{-2}X_{l}^{-1}$ where $F_{\nu}$ is the
disk’s flux in units of Janskys, $f$ is the fractional luminosity, $r$ is the
belt radius in au, $d$ is distance to the star in parsecs, and $X_{l}$ is an
additional factor included to account for the steeper fall-off due to
departures from blackbody emission at long wavelengths, $X_{l}=(\lambda\
/\lambda_{0})^{\beta}$, where $\lambda_{0}$ is a turnover wavelength (adopted
to be 0.21 mm) and $\beta$ is power-law index that reflects the dust
emissivity citep[see Section 2.3 of ][]Wyatt2008. A value of $\beta=1$ is
often assumed for debris disks (e.g., Holland et al., 2017). However, detailed
observations of the millimeter spectral slopes of nearby debris disks suggest
that $\beta=0.5$ provides a more appropriate empirical description (MacGregor
et al., 2016). An additional effect is that the blackbody radius determined
from radiative equilibrium tends to underestimate the true dust belt radius
since small grains do not emit efficiently at long wavelengths (e.g., Booth et
al., 2013). This dependence acts to increase ALMA sensitivity to dust belts of
larger radii.

####  4.2.3 Pleiades Sample Population Synthesis

We took random draws from the distributions of disk masses and radii to
generate a population of belts around each of the 73 stars considered in the
sample. These disks were then evolved to the Pleiades age of 115 Myr following
collisional evolution, to produce a synthetic dust belt associated with each
star. Then fluxes in the relevant bands were calculated for each one. This
process was repeated 10,000 times to build up meaningful statistics.

Figure 5 illustrates the evolution of belt masses and fractional luminosities
as a function of belt blackbody temperature for a single population of 73
stars at several ages. The initial disk masses arise from sampling the log-
normal distribution. By 20 Myr, the warmest belts already have significantly
depleted mass, due to the steep dependence on dust temperature of the
collisional lifetime of the largest planetesimals. By contrast, the coldest
belts ($T\lesssim 75K)$ retain most of their mass at 20 Myr, at 115 Myr, and
even on Gyr timescales. For a belt around a 1.0 M⊙ star at 5 au radius,
$t_{c}=5.8/(M_{tot}/M_{\oplus})$ Myr, while for a belt at 50 au radius,
$t_{c}=8.8\times 10^{4}/(M_{tot}/M_{\oplus})$ Myr and the initial mass is a
dominant factor. This behavior is also reflected in the fractional
luminosities. At late times, the fractional luminosity is strongly dependent
on blackbody temperature (or radius) as the belts are collisionally depleted.
The coldest belts are governed by very long collisional lifetimes.

Figure 5: (top) Belt masses plotted against blackbody temperature for an
example population of simulated debris disks at four representative ages. The
leftmost panel shows the initial masses, and subsequent panels show the belt
masses after collisional evolution at 20 Myr, 115 Myr, and 4.6 Gyr. By 20 Myr,
the warmest belts are already significantly depleted, while the coldest belts
experience negligible mass depletion. (bottom) The fractional luminosities
plotted against disk blackbody temperature for the same example population.

Figure 6 (left panel) shows the frequency of 1.3 mm detections of dust belts
for the sample of 73 Pleiades stars, given the sensitivity achieved in the
ALMA survey. From the 10,000 realizations of this Pleiades sample, there are
9670 with zero detections (96.7%), 325 with one detection (3.25%), and 5 with
2 detections (0.05%). Thus the ALMA nondetection of any significant disk
emission from the Pleiades sample is in agreement with the most likely outcome
of these models. Much deeper integrations would be needed to detect a
significant number of dust belts in this model population. Only about 5% of
the targets have 1.3 mm flux densities that exceed 20 $\mu$Jy, a level that
would require almost an order of magnitude better sensitivity for detection
(corresponding to $\sim 100$ times longer integration times, or $\sim 7$ hours
on source with ALMA). This aspect of the modeled population is consistent with
the flux density limit obtained from stacking the individual ALMA images of
Pleiades stars.

The assumed long wavelength slope of the millimeter emission, encapsulated in
the parameter $\beta$, has a modest impact on the 1.3 mm emission and the
resulting detection statistics. In particular, a shallower unmodified
blackbody spectrum ($\beta=0$) extended (unrealistically) to the millimeter
regime would result in at least one ALMA detection in 34% of the simulated
populations. For the steeper $\beta=1$, at least one ALMA detection from the
sample would occur less than 1% of the time.

Figure 6:  The hatched histograms show population synthesis models predictions
for the detection frequency of debris disks in the Pleiades Solar-type star
sample (left) by ALMA at 1.3 mm at the achieved sensitivity, and (right) by
Spitzer at $24~{}\mu$m, following the detection criterion of Sierchio et al.
(2010). The grey shaded regions in each panel indicate the observed number
(with associated uncertainties).

The population synthesis also provides predictions for 24 $\mu$m emission to
compare with the Spitzer observations of the Pleiades sample. The 24 $\mu$m
observations are much more sensitive than the 1.3 mm survey to the bulk of the
dust belts. Sierchio et al. (2010) found significant excess emission from
23/71 stars, for a detection rate of 32$\pm$6.8% of the sample (where an
excess ratio $>1.1$ was considered significant, i.e. exceeding $3\sigma$).
Figure 6 (right panel) shows the distribution of 24 $\mu$m detections for the
10,000 simulated populations. The mean of this distribution is 15.1, with
standard deviation of 3.4, or 21$\pm$4.7% of the sample (taking into account
that two targets were dropped because they lack Gaia catalog stellar
properties). Given the dispersion in the model prediction and the
observational uncertainties, the population synthesis model predictions are
consistent with the observations. The trend of the data to the high side of
the model predictions may indicate a contribution from background confusion.
As noted in §2.4, Stauffer et al. (2005) cautioned that 10% of their sample
(2/20) may be contaminated by background 24 $\mu$m emission. If this same
fraction were to hold for the full sample of Sierchio et al. (2010), then the
number of stars with significant 24 $\mu$m dust belt detections would be
lower, conceivably 16/71, close to the population synthesis prediction.
However, the difference between the nominal number of 24 $\mu$m detections and
the model prediction is not statistically significant and should not be
overinterpreted.

At 24 $\mu$m, the photospheric excess ratios predicted by the population
synthesis models are always less than $2$, even in the most extreme systems.
From this perspective, the HII 1132 and HD 22680 systems are notable, with
excess ratios of 17.44 and 3.79 respectively reported by Sierchio et al.
(2010). These values are much higher than any of the other dust belts in the
sample and not expected in the passive, steady-state collisional evolution
scenario at Pleiades age. The corresponding fractional luminosity for HII
1132, the largest outlier, is $1.9\times 10^{-3}$ (from Table 2 of Sierchio et
al., 2010). If the radius of this belt is 5 au, then it falls below the ALMA
detection threshold at the sensitivity of our 1.3 mm survey (although it would
have been readily detected at 70$~{}\mu$m were it included in the FEPS sample
for observation at that wavelength). These systems are candidates of interest
for deeper observations at longer wavelengths, to characterize their excess
emission and to understand their physical nature. One intriguing possibility
is that dynamical disturbances in the associated planetary systems have
resulted in an epoch of unusually high dust production.

Given the absence of any ALMA detections, consistency of the observations with
the population synthesis prediction is not especially constraining to the
collisional evolution model. An implicit assumption that remains to be
explored is whether or not progenitors of debris disks in star clusters like
the Pleiades could be different than those for the field population near the
Sun. The recent recognition of a high incidence of bright debris disks in
F-type stars in the nearby 23 Myr-old $\beta$ Pictoris moving group compared
with older moving groups and the field suggests a faster evolution than
steady-state collisional evolution in the first $\sim 100$ Myr, perhaps due to
different initial disk properties or perhaps other factors, such as the
influence of planets (Pawellek et al., 2021). Since most field stars are
thought to form in groups with $<100$ members like the nearby moving groups
(Adams & Myers, 2001), it seems unlikely that initial conditions are
responsible. By contrast, about 10% of stars form in dense open clusters like
the Pleiades, where it is more plausible that birth environment could be
important. Studies of protoplanetary disk properties in rich cluster
environments, in particular the Orion Nebula Cluster, hint at different disk
bulk properties than in more distributed star-forming regions like Taurus, in
particular smaller sizes and lower masses in close proximity to the harsh
radiation fields of O-type stars (e.g., Eisner et al., 2018). If birth
environment is not responsible, then the consistency of the Pleiades results
with the population synthesis models suggests that whatever process results in
overbright disks at 23 Myr has ended by $\sim$100 Myr.

##  5 Conclusions

The 115 Myr-old Pleiades cluster provides an important testbed for models of
debris disk evolution. Of particular interest are Solar-type stars surrounded
by possible analogs of the early Kuiper Belt. We used ALMA to survey 76 FGK
type stars in the Pleiades at 1.3 mm to improve on previous observations at
this long wavelength in both sensitivity and number to probe the presence of
massive cold dust belts. These observations obtained a typical beam size $\sim
1\farcs 5$ (200 au) and a median rms noise 54$~{}\mu$Jy, and they showed no
significant detections of circumstellar dust. Several sources detected within
the ALMA fields of view of the Pleiades stars are consistent with the expected
background of dusty galaxies. For a dust belt at 40 K, the $3\sigma$ limit on
1.3 mm flux density corresponds to a fractional luminosity,
$L_{dust}/L_{star}<10^{-4}$. These sensitive 1.3 mm observations provide a
testbed for debris disk evolutionary models for a population at an age where
there are few existing constraints. A standard, passive collisional cascade
model for steady state evolution is statistically consistent with the ALMA 1.3
mm flux density limits, as well as previous detection statistics at 24 $\mu$m
and shorter infrared wavelengths. The millimeter observations reveal no
outliers from the model expectations, in particular no high fractional
luminosity cold belts similar to those found around a few Solar-type stars in
the local field population. Two of the Pleiades systems shows extreme
$24~{}\mu$m excess emission that suggest significant departures from steady
state evolution, perhaps resulting from dynamical disturbances involving
planets.

This paper makes use of the following ALMA data: ADS/JAO.ALMA#2019.1.00251.S.
ALMA is a partnership of ESO (representing its member states), NSF (USA) and
NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio
Astronomy Observatory is a facility of the National Science Foundation
operated under cooperative agreement by Associated Universities, Inc. D.S.
thanks the Smithsonian Astrophysical Observatory for summer research support
to carry out parts of this project. L.M. acknowledges funding from the
European Union’s Horizon 2020 research and innovation programme under the
Marie Sklodowska-Curie grant agreement No. 101031685. MAM acknowledges the
National Aeronautics and Space Administration under award number
19-ICAR19_2-0041. We thank Scott Kenyon and Qizhou Zhang for many valuable
comments on an early version of the manuscript. We also thank Joan Najita for
stimulating discussion.

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