Tri-Scalar CFT and Holographic Bi-Fishchain Model

Wung-Hong Huang*

Department of Physics, National Cheng Kung University,

No.1, University Road, Tainan 701, Taiwan

Abstract

Bi-scalar CFT from $\gamma$ deformed $\cal N$=4 SYM describes the fishnet
theory which is integrable in the planar limit. The holographic dual of the
planar model is the fishchain model. The derivation of the weak-strong duality
from the first principle was presented in a recent paper (”The Holographic
Fishchain” arXiv:1903.10508). In this note we extend the investigation to the
tri-scalar CFT which raises from the large twist limit of ABJM theory. We show
that it becomes tri-scalar fishnet theory in planar limit and the dual theory
is the holographic bi-fishchain model.

* Retired Professor of NCKU, Taiwan.

* E-mail<EMAIL_ADDRESS>
##  1 Introduction

The AdS/CFT correspondence relates bulk AdS physics to a boundary conformal
field theory. The archetypal example of the gauge-string correspondence is
that of N = 4 super Yang-Mills theory, which regarded as the boundary CFT,
describes the type IIB superstrings in the $AdS_{5}\times S^{5}$ bulk [1, 2,
3]. The correspondence has been applied to investigate several problems in
large $N_{c}$ QCD such as the Wilson loop [4], the meson spectra [5], baryon
dynamics [6] and so on.

Gauge-string correspondence, despite go through many successful checks and has
been applied to study, again successfully, both strongly interacting QFTs and
quantum gravity, still lacks a satisfactory microscopic derivation. One of the
basic question is how to reformulate CFT correlation functions and then, from
which the emergence of gravitational dynamics becomes manifest. In other
words, given a CFT and assuming a limit where the dual geometry is well
defined, how to find an algorithm to determine this dual background directly
from the correlators? In a recent paper [7] Gromov and Sever showed a possible
way to achieve the goal! The model theory they considered is a specially
deformed SYM theory.

It is known that the $\cal N$= 4 SYM theory after $\gamma$ deformed in double
scaling limit is conformal and integrable in the planar limit [8]. In this
limit, the gauge fields decouple and the theory contains three complex scalars
and fermions interacting through quartic and “chiral” Yukawa couplings. After
turning off some couplings the fermions and scalar decouple and left the
following bi-scalar Lagrangian [8]:

$\displaystyle{\cal
L}_{\phi}={N_{c}}\text{Tr}\,\Big{(}\partial^{\mu}\phi^{\dagger}_{1}\partial_{\mu}\phi_{1}+\partial^{\mu}\phi^{\dagger}_{2}\partial_{\mu}\phi_{2}+4(2\pi)^{2}\xi^{2}\phi^{\dagger}_{1}\phi^{\dagger}_{2}\phi_{1}\phi_{2}\Big{)}~{}~{}$
(1.1)

where where $\phi_{i}^{a}=\phi^{a}_{i}T^{a}$ are complex scalar fields and
$T^{a}$ are the generators of the SU($N_{c}$) gauge group. The 4-dimensional
QFT is conformal in the large $N_{c}$ limit and integrable at any coupling in
the ’t Hooft limit, even in the absence of supersymmetry and gauge symmetry
[8].

Since $\phi_{i}^{a}$ are complex matrix fields the interaction term in the
Lagrangian ${\cal L}_{\phi}$ is not real and the theory described by (1.1) is
not unitary. Non-unitary CFTs appear in a condensed matter context, for
example in [9] and references therein. The property of non-unitary CFTs is not
clear as the bootstrap methods are not applicable for the non-unitary CFTs.

In the large $N_{c}$ limit the bi-scalar Lagrangian becomes planar model which
is called as fishnet theory. Authors of [7] presented the first-principle
derivation of a weak-strong duality between the fishnet theory in four
dimensions and a discretized string-like model living in five dimensions,
which is called as fishchain mode.

It is known that the analogous fishnet diagrams also exist in 2,3 and 6
dimensions graphs considered by Zamolodchikov [10] and it is interesting to
consider the ABJM model which is a 3-dimensional theory [11]. In this note we
extend the investigation of [7] to the tri-scalar CFT which raises from the
large twist limit of ABJM theory. We will show, in step by step, that the dual
theory is the holographic bi-fishchain model111Authors of [12] expected that
their techniques used in [7] and [12] can be generalized rather
straightforwardly to this class of graphs. Our note explicitly confirm their
expectation. The derivation of each equation in this note is written in
detailed and is easy to read for the readers..

In section 2 we describe the anomalous dimension of correlation function,
Feynman diagram and the wheel-type Feynman graph. We plot several diagrams to
illuminate how to use the graph-building element to build the Feynman graph
and associated fishnet diagram. We first discuss the bi-scalar theory and then
extend it to the tri-scalar theory. In section 3 we use the diagram to find
the associated Lagrangian, which shows the strong-weak duality after using the
gauge symmetry in the system. In section 4, from the property that the
D-dimensionally conformal invariant quantity is a D+2 scalar on SO(D+1,1) we
uplifte the action into a 5D embedding space, which leads to the holographic
fishchain model. These then explicitly derive holographic bi-fishchain model
from tri-scalar CFT. A short conclusion is made in the last section.

##  2 Anomalous Dimension and Graph-building Method

A family of scalar single-trace operators to be studied is [13]

$\displaystyle{\cal
O}_{J,n,\ell}=P_{2\ell}\text{tr}\Big{[}\phi^{J}_{1}\phi^{n}_{2}(\phi^{\dagger}_{2})^{n}\Big{]}+....$
(2.1)

where $P_{2\ell}$ denotes $2\ell$ derivatives acting on scalar fields inside
the trace with all Lorentz indices contracted. The dots stand for similar
operators with the scalar fields $\phi_{1},~{}\phi_{2}$ and
$\phi^{\dagger}_{2}$ exchanged inside the trace. The operators have scaling
dimension $\Delta=J+2(n+\ell)$ for zero coupling $\xi=0$. We consider the
simplest case of $n=\ell=0$ and the operator takes the form

$\displaystyle{\cal O}_{J}=\text{tr}[\phi^{J}_{1}]$ (2.2)

In $\cal N$= 4 SYM the similar operator is protected from quantum corrections
dues to the suppersymmetry therein and is known as the BMN vacuum operator. In
the $\gamma$-deformed $\cal N$= 4 SYM, the operator is not protected and its
scaling dimension will depend on the coupling strength $\xi$.

The coupling dependent corrections to the scaling dimension $\Delta$ is
defined by

$\displaystyle D(x)=\langle{\cal O}_{J}(x)\bar{\cal
O}_{J}(0)\rangle={d(\xi)\over(x^{2})^{J+\gamma_{J}(\xi)}}~{}~{}~{}~{}$ (2.3)

where the normalization constant $d(\xi)$ and the anomalous dimension
$\gamma_{J}(\xi)$ depend on the coupling constant $\xi$. In the large $N_{c}$
limit both only come from the wheel-type Feynman graphs. Each wheel contains J
interaction vertices and the contribution to the scaling dimension of the
wheel graph with M frames scales as $(\xi^{2})^{JM}$, by the Feymnan rule read
from the Lagrangian of bi-scalar theory (1.1). For example Fig. 1 is the J=M=3
wheel-type Feynman graph in which three $\phi^{\dagger}_{1}(0)$ locate
together at origin while three $\phi_{1}(x_{1})$ locate separately at radius
r=$x_{1}$. There are $3\times 3$ vertices and is order of $(\xi^{2})^{9}$
wheel-type Feynman graph which is used to calculate $\langle{\cal
O}_{3}(x)\bar{\cal O}_{3}(0)\rangle$ in (2.3).

Figure 1: J=M=3 wheel-type Feynman graph. There are $3\times 3=9$ vertices.

The Feynman rules associated to the bi-scalar Lagrangian (1.1) are :

Propagator $\displaystyle:$
$\displaystyle\langle\phi_{1}(x_{i})\phi^{\dagger}_{1}(y_{i})\rangle={1\over(2\pi)^{2}(x_{i}-y_{i})^{2}};~{}\langle\phi_{2}(y_{i})\phi^{\dagger}_{2}(y_{i+1})\rangle={1\over(2\pi)^{2}(y_{i}-y_{i+1})^{2}}~{}~{}~{}$
(2.4) Vertex $\displaystyle:$
$\displaystyle(4\pi)^{2}\xi^{2}\phi^{\dagger}_{1}\phi^{\dagger}_{2}\phi_{1}\phi_{2}$
(2.5)

Using above Feynman rules we can resume the wheel-type Feynman graphs from the
“graph-building” operator $\widehat{B}$ by defining its integral kernel

$\displaystyle\widehat{B}(\\{\vec{y}_{i}\\}_{i=1}^{J},\\{\vec{x}_{j}\\}_{j=1}^{J})=\prod_{i=1}^{J}\frac{(4\pi)^{2}\xi^{2}}{(2\pi)^{2}(\vec{y}_{i}-\vec{y}_{i+1})^{2}(2\pi)^{2}(\vec{x}_{i}-\vec{y}_{i})^{2}}=\prod_{i=1}^{J}\frac{\xi^{2}/\pi^{2}}{(\vec{y}_{i}-\vec{y}_{i+1})^{2}(\vec{x}_{i}-\vec{y}_{i})^{2}}~{}~{}~{}$
(2.6)

which is simply a product of several “graph-building element”. For example

$\displaystyle\widehat{B}(y_{1},y_{2},y_{3},x_{1},x_{2},x_{3})=\prod_{i=1}^{3}\frac{\xi^{2}/\pi^{2}}{(\vec{y}_{i}-\vec{y}_{i+1})^{2}(\vec{x}_{i}-\vec{y}_{i})^{2}}$
(2.7)

is shown in figure 2.

Figure 2: Graph-building method. Use “graph-building element” to build Feynman
graph.

Note that we use periodic boundary condition $y_{1}=y_{J+1}$. Each horizontal
link is a free scalar propagator $(2\pi)^{-2}(y_{j}-y_{j+1})^{-2}$ while each
vertical line has similar propagator connecting points $x_{i}$ and $y_{i}$, as
described in (2.4). Applying this operator once, we add one wheel to the graph
on above, thus the sum of all wheels inside the graph forms a geometric series
:

$\displaystyle\text{All wheels}=\frac{1}{1-\widehat{B}}~{}~{}~{}$ (2.8)

For example, by the definition (2.3)

$\displaystyle\frac{1}{1-\widehat{B}_{3}}=\langle{\cal O}_{3}(x)\bar{\cal
O}_{3}(0)\rangle$ $\displaystyle=$
$\displaystyle\langle\text{tr}[\phi_{1}(x_{1})\phi_{1}(x_{2})\phi_{1}(x_{3})]\text{tr}[(\phi^{\dagger}_{1}(0))^{3}]\rangle={d(\xi)\over(x^{2})^{3+\gamma_{3}(\xi)}}$
(2.9)

which is used to calculate the correlation function of J=M=3 wheel-type
Feynman graph in Figure 1. The zeros of the denominator in (2.8) can be
identified as the anomalous dimensions $\gamma(\xi)$ of the local operators
[15], which plays a special role in CFT.

An important observation is that the zeros of the denominator can be found by
solving the eigenvalue equation

$\displaystyle(-1+\widehat{B})\Psi=0~{}~{}~{}$ (2.10)

To proceed we can use the 4 dimmentional Green function

$\displaystyle\Box_{\vec{x}}\,{1\over
4\pi^{2}(\vec{x}-\vec{y})^{2}}=-\delta^{4}(\vec{x}-\vec{y})~{}~{}$ (2.11)

and definition of $\widehat{B}$ in (2.6) to construct the bi-scalar
Hamiltonian ${\cal H}_{\text{bi}}$ form below relation

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\prod_{i}\Box_{i}\Big{(}(-1+\widehat{B})\Psi\Big{)}=-\Big{(}\prod_{i}\Box_{i}\Big{)}\Psi+\Big{(}\prod_{i}\Box_{i}\,\widehat{B}\Big{)}\Psi$
(2.12) $\displaystyle=$ $\displaystyle{\cal
H}_{\text{bi}}\circ\Psi(\\{x_{i}\\}),~{}~{}~{}~{}{\cal
H}_{\text{bi}}=\prod_{i=1}^{J}\vec{p}_{i}^{2}-\prod_{i=1}^{J}{4\xi^{2}\over(\vec{x}_{i}-\vec{x}_{i+1})^{2}}~{}~{}~{}~{}$

To obtain above result we use the relation
$\vec{p}_{i}=-i\vec{\partial}_{x_{i}}$ and property of delta function in Green
function relation (2.11).

In this note we will extend the method to the large twist limit of ABJM
theory. The associated Lagrangian of tri-scalar CFT is [14].

$\displaystyle{\cal
L}_{\phi}={N_{c}}\text{Tr}\,\Big{(}\partial^{\mu}\phi^{\dagger}_{1}\partial_{\mu}\phi_{1}+\partial^{\mu}\phi^{\dagger}_{2}\partial_{\mu}\phi_{2}+\partial^{\mu}\phi^{\dagger}_{3}\partial_{\mu}\phi_{3}+4(2\pi)^{3}\xi^{2}\,\phi^{1}\phi^{\dagger}_{3}\phi^{2}\phi^{\dagger}_{1}\phi^{3}\phi^{\dagger}_{2}\Big{)}~{}~{}~{}$
(2.13)

The Lagrangian describes a pure 3D tri-scalar interacting theory and is
integrable in the planar limit. We will first follow the previous method to
find how the ${\cal H}_{\text{bi}}$ in (2.12) will be modified.

It is easy to see that the “graph-building” operator $\widehat{B}$ in (2.6)
now becomes

$\displaystyle\widehat{B}(\\{\vec{y}_{i}\\}_{i=1}^{J},\\{\vec{x}_{j}\\}_{j=1}^{J})=\prod_{i=1}^{J}\frac{4(2\pi)^{3}\xi^{2}}{(2\pi)^{2}(\vec{y}_{i}-\vec{y}_{i+1})^{2}(2\pi)^{2}(\vec{x}_{i}-\vec{y}_{i})^{2}(2\pi)^{2}(\vec{y}_{i}-\vec{z}_{i+1})^{2}}~{}~{}~{}$
(2.14)

which is simply a product of several “graph-building element”, as can be seen
in Figure 3 for bi-scalar CFT and Figure 4 for tri-scalar CFT.

Figure 3: Use “graph-building element” to build Feynman graph of bi-scalar
CFT.

Figure 4: Use “graph-building element” to build Feynman graph of tri-scalar
CFT.

Note that using the Feynman graph in figure 4 we can construct the wheel-type
Feynman graph for bi-scalar CFT and tri-scalar CFT, as shown in figure 5. By
gluing the arrow of each color and putting green stars on the central of a
wheel we can see that the left-hand side becomes wheel-type Feynman graph for
bi-scalar CFT, as the Figure 1, and the right-hand side becomes wheel-type
Feynman graph for tri-scalar CFT 222The paper [14] had also plotted the wheel-
type Feynman graph for tri-scalar CFT (figure 11), which using the curved
line. Our graph in figure 5 is plotted in the strength line and is easy to
imagine for some readers.

Figure 5: Use “ Feynman graph” to build “wheel-type Feynman graph“. Wheel-type
Feynman graph is obtained by gluing the arrow of each color and putting green
stars on central of a wheel.

To proceed we see that the relation
$\vec{z}_{1}=\vec{x}_{1}+\vec{y}_{2}-\vec{y}_{1}$ can simplify the propagator
relation

$\displaystyle{1\over(2\pi)^{2}(z_{1}-y_{1})^{2}}$ $\displaystyle=$
$\displaystyle{1\over(2\pi)^{2}(x_{1}+y_{2}-2y_{1})^{2}}={1\over(2\pi)^{2}(x_{1}+x_{2}-2x_{1})^{2}}~{}$
(2.15) $\displaystyle=$ $\displaystyle{1\over(2\pi)^{2}(x_{1}-x_{2})^{2}}$
(2.16)

where the last relation in (2.15) is the result of delta function property in
Green function relation (2.11). Therefore the eigenvalue equation associated
to bi-scalar CFT in (2.12) now becomes

$\displaystyle 0$ $\displaystyle=$ $\displaystyle{\cal
H}_{\text{tri}}\circ\Psi(\\{x_{i}\\}),~{}~{}~{}{\cal
H}_{\text{tri}}=\left(\prod_{i=1}^{J}\vec{p}_{i}^{2}-\prod_{i=1}^{J}{4\xi^{2}\over(\vec{x}_{i}-\vec{x}_{i+1})^{4}}\right)~{}~{}$
(2.17)

in tri-scalar CFT. We see that just changes the factor
$(\vec{x}_{i}-\vec{x}_{i+1})^{-2}$ in ${\cal H}_{\text{bi}}$, i.e. eq.(2.12),
to $(\vec{x}_{i}-\vec{x}_{i+1})^{-4}$ we then obtain ${\cal H}_{\text{tri}}$ ,
i.e. eq.(2.17).

In the following sections we will first finds the associated Lagrangian of
tri-scalar CFT from ${\cal H}_{\text{tri}}$ and proves that the corresponding
dual Lagrangian describes a bi-fishchain model, which is plotted in figure 6.
Below derivations totally follow the method in the original paper [7] while
present them in step by step in mathematical relations and is more easy to
read.

##  3 Fishnet Lagrangian and Strong-Weak Duality

Through the following standard calculations one can find the associated
Lagrangian. First, using $H_{\text{tri}}$ in (2.17) we have a relation

$\displaystyle\dot{\vec{x}}_{k}$ $\displaystyle\equiv$ $\displaystyle{\partial
H_{\text{tri}}\over\partial{\vec{p}_{k}}}={2\prod_{i=1}^{J}\vec{p}_{i}^{2}\over\vec{p}_{k}}$
(3.1)

which leads to

$\displaystyle\prod_{i=1}^{J}\,\dot{\vec{x}}_{i}=2^{J}\,\Big{(}\prod_{i=1}^{J}\vec{p}_{i}^{2}\Big{)}^{J-{1\over
2}}~{};~{}~{}~{}\prod_{i=1}^{J}\vec{p}_{i}^{2}={1\over 2^{2J\over
2J-1}}\,\Big{(}\prod_{i=1}^{J}\,\dot{\vec{x}}_{i}^{2}\Big{)}^{1\over 2J-1}$
(3.2)
$\displaystyle\sum_{k=1}^{J}\Big{(}\vec{p}_{k}\dot{\vec{x}}_{k}\Big{)}=\sum_{k=1}^{J}\Big{(}2\prod_{i=1}^{J}\vec{p}_{i}^{2}\Big{)}=2J\prod_{i=1}^{J}\vec{p}_{i}^{2}$
(3.3)

The Lagrangian then becomes

$\displaystyle{L}(\gamma)=\sum_{k}\vec{p}_{k}\dot{\vec{x}}_{k}-{\cal
H}_{\text{tri}}=\frac{2J-1}{2^{\frac{2J}{2J-1}}}\left(\frac{1}{\gamma}\prod_{i=1}^{J}\vec{\dot{x}}_{i}^{2}\right)^{\frac{1}{2J-1}}+{\gamma}\prod_{i=1}^{J}\frac{4\xi^{2}}{(\vec{x}_{i}-\vec{x}_{i+1})^{4}}~{}~{}~{}$
(3.4)

where we add a gauge parameter $\gamma$. If we consider the following gauge
transformations (time reparameterization)

$\displaystyle t$ $\displaystyle\rightarrow$ $\displaystyle
f(t);~{}~{}~{}\gamma\rightarrow f^{\prime}(t)~{}~{}$ (3.5)

which imply

$\displaystyle dt\rightarrow f^{\prime}dt;~{}~{}~{}~{}\dot{x}^{2}={dx^{2}\over
dt^{2}}\rightarrow{dx^{2}\over df^{2}}{1\over f^{\prime 2}}{\dot{x}}^{2}$
(3.6)

the Lagrangian and action then transform as

$\displaystyle L(\gamma)$ $\displaystyle\equiv$
$\displaystyle\left(\frac{2J-1}{2^{\frac{2J}{2J-1}}}\left(\frac{1}{\gamma}\prod_{i=1}^{J}\vec{\dot{x}}_{i}^{2}\right)^{\frac{1}{2J-1}}+{\gamma}\prod_{i=1}^{J}\frac{4\xi^{2}}{(\vec{x}_{i}-\vec{x}_{i+1})^{4}}\right)\to{1\over
f^{\prime}}\,L(\gamma)$ (3.7) $\displaystyle S$ $\displaystyle\equiv$
$\displaystyle\int dt\,L(\gamma)\rightarrow\int dt\,f^{\prime}\,\Big{(}{1\over
f^{\prime}}\,L(\gamma)\Big{)}=\int dt\,L(\gamma)\equiv S$ (3.8)

We see that the action is invariant under the gauge transformations (3.5).

Now, we use the gauge freedom of $\gamma$ to choose the its extremum by
$\partial_{\gamma_{max}}L(\gamma_{max})=0$. The action and Lagrangian then
become

$\displaystyle S(\gamma_{max})$ $\displaystyle=$ $\displaystyle\xi\,\int
dt\,L(\gamma_{max})~{}~{}$ (3.9) $\displaystyle L(\gamma_{max})$
$\displaystyle=$ $\displaystyle
2J\,\left(\prod_{i=1}^{J}{\vec{\dot{x}}_{i}^{2}\over(\vec{x}_{i}-\vec{x}_{i+1})^{4}}\right)^{1\over
2J}~{}~{}$ (3.10)

Two interesting properties are shown in above action and Lagrangian:

1\. In path integration the action factor $e^{-{S\over\hbar}}$ tells us that
the overall factor of coupling $\xi$ in action (3.9) plays the role of
${1\over\hbar}$. Since that small $\hbar$ is classical the large $\xi$ is
classical too. Therefore, the strong coupling of Lagrangian (3.4) can be
studied by classical model of (3.10) and strong-weak duality is shown up in
here.

2\. The action S in (3.9) is also invariant under global conformal
transformations, which maybe relates to the property a CFT dual. We will,
next, use the conformal symmetry to uplifte the action into a 5D embedding
space, which leads to the holographic fishchain model.

##  4 Holographic Bi-Fishchain Mode

It is known that conformal algebra on D dimension is same as the algebra of
SO(D+1,1) and D-dim conformal invariant quantity is a D+2 scalar on SO(D+1,1).
Using this property we can express the Lagranginal $L_{\gamma_{max}}$ of
(3.10) in higher dimension, which is called as Fishchain.

Let us denote $X^{A}$ as D+2 dim coordinate

$\displaystyle X^{A}=(X^{1},.....X^{D},X^{D+1},X^{D+2})$ (4.1)

In light cone cordinate

$\displaystyle X^{\pm}=X^{5}\pm
X^{6},~{}~{}~{}~{}ds^{2}=-dX^{+}dX^{-}+\sum_{\mu}(dX^{\mu})^{2}$ (4.2)

To project it to 4 dimension we can choose it on null cone, since null is
invariant. I.e. we choose

$\displaystyle X^{A}$ $\displaystyle=$
$\displaystyle(X^{+},X^{-},x^{\mu})=(1,x^{2},x^{\mu})~{}~{}$ (4.3)
$\displaystyle\to X^{2}$ $\displaystyle=$ $\displaystyle X^{A}X_{A}=-1\cdot
x^{2}+x_{\mu}x^{\mu}=0~{}~{}~{}$ (4.4)

Above choice leads to

$\displaystyle\dot{X}^{A}$ $\displaystyle=$
$\displaystyle(0,2\dot{x}^{\mu}x_{\mu},\dot{x}^{\mu})~{}~{}\to~{}~{}\dot{X}^{A}\dot{X}_{A}=-0\times
2\dot{x}^{\mu}x_{\mu}+\dot{x}^{\mu}\dot{x}_{\mu}=\dot{x}^{\mu}\dot{x}_{\mu}$
(4.5)

and Lagranginal $L(\gamma_{max})$ of (3.10) can be expressed in 5D coordinate
$X^{A}$ :

$\displaystyle
L_{\text{5D}}=2J\left(\prod_{i=1}^{J}{\vec{\dot{X}}_{i}^{2}\over(-2X_{i}\cdot
X_{i+1})^{2}}\right)^{1\over 2J}~{}~{}$ (4.6)

Having above 5D Lagrangian the next work work is to find a conventional
Lagrangian, with both of kinetic term and interaction term, which describes
above Lagrangian.

With the guiding in [7, 12], which investigate the bi-scalar CFT case, the
associated Lagrangian in tri-scalar CFT case is

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle-\left(\sum_{i}\Big{[}{\dot{X}_{i}^{2}\over
2^{2}\,\alpha_{i}}+\eta_{i}X_{i}^{2}\Big{]}\right)-J\Big{(}\prod_{k}\alpha_{k}\Big{)}^{1\over
J}\prod_{k}\Big{(}-X_{k}\cdot X_{k+1}\Big{)}^{-{2}\over J}~{}~{}~{}$ (4.7)

The $\eta_{i}$ is a Lagrangian multiple used to constrain the theory on null
plane $X_{i}^{2}=0$. The Lagrangian has several symmetry : 1. Conformal
symmetry. 2. Time reparameterization of $t\rightarrow
f(t),\alpha_{i}\rightarrow\alpha_{i}/f(t),\eta_{i}\rightarrow\eta_{i}/f(t)$.
3. Translation symmetry $X_{i}\to X_{i+1}$.

To proceed we first extremeize (4.7) with respective to parametre $\alpha_{i}$
we find

$\displaystyle 0$ $\displaystyle=$ $\displaystyle\partial_{\alpha_{i}}{\cal
L}$ $\displaystyle\to~{}~{}{\dot{X}_{i}^{2}\over{2^{2}}\,\alpha_{i}}$
$\displaystyle=$ $\displaystyle\Big{(}\prod_{k}\alpha_{k}\Big{)}^{1\over
J}\prod_{k}\Big{(}-X_{k}\cdot X_{k+1}\Big{)}^{-{{2}\over J}}$ (4.8)

Note that the right-hand side in the last equation is independent of index
$"i"$ . Above relation leads to

$\displaystyle\prod_{i}{\dot{X}_{i}^{2}}$ $\displaystyle=$
$\displaystyle\,2^{2J}\Big{(}\prod_{k}\alpha_{k}\Big{)}^{2}\prod_{k}\Big{(}-X_{k}\cdot
X_{k+1}\Big{)}^{-2}$ (4.9)

Using above last two relations and choosing the gauge of $\eta_{i}=1$ we find
that (4.7) becomes

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle
2{J}\,\Big{(}\prod_{k}\alpha_{k}\Big{)}^{1\over J}\prod_{k}\Big{(}-X_{k}\cdot
X_{k+1}\Big{)}^{-{2\over J}}~{}~{}~{}$ $\displaystyle=$ $\displaystyle
2J\,\left(\Big{(}\prod_{i}{\dot{X}_{i}^{2}}\Big{)}\,\,2^{-2J}\,\prod_{k}\Big{(}-X_{k}\cdot
X_{k+1}\Big{)}^{2}\right)^{1\over 2J}\,\prod_{k}\Big{(}(-X_{k}\cdot
X_{k+1})^{2}\Big{)}^{-{1\over J}}$ $\displaystyle=$
$\displaystyle{2J\,\prod_{k}\left({\dot{X}_{k}^{2}}\over(-2X_{k}\cdot
X_{k+1})^{2}\right)^{1\over 2J}}$ (4.11)

and ${\cal L}$ goes back to $L_{\text{5D}}$. In the gauge $\alpha_{i}=1$
equation (4) can be written as

$\displaystyle S$ $\displaystyle=$ $\displaystyle\xi\,J\int dt\,{\cal
L},~{}~{}~{}~{}{\cal L}=4\prod_{k}\Big{(}-X_{k}\cdot X_{k+1}\Big{)}^{-{2\over
J}}~{}~{}$ (4.12)

Therefore, the dual tri-scalar CFT is the the holographic bi-fishchain model
on the lightcone of 5D spacetime and it’s Lagrangian is (4.12). Note that the
Lagrangian of the holographic fishchain mode related to bi-scalar CFT derives
in [7] is ${\cal L}=2\prod_{k}\Big{(}-X_{k}\cdot X_{k+1}\Big{)}^{-{1\over
J}}$.

We plot the model diagrams in figure 6 in which the black circle represents
the lattice point $X_{k}$. The strength of link between two lattice point
$X_{k}$ and $X_{k+1}$ is $\Big{(}-X_{k}\cdot X_{k+1}\Big{)}^{-{1\over J}}$.
The holographic fishchain (upper diagram) has only one link, while holographic
bi-fishchain (down diagram) has two links, between two nearest lattice points.
The model chain has $J$ sites and is closed.

Figure 6: Fishchain and bi-fishchain Models. Fishchain has one link. Bi-
fishchain has two links. Each chain has J sites and is closed.

##  5 Conclusion

We derive the dual model of 3D tri-scalar CFT which raises from the large
twist limit of ABJM theory. It is a weak to strong coupling duality between
the single trace operators of the tri-fishnet theory and a quantum-mechanical
system of particles which forms a bi-fishchain in light-cone limit of five
dimensions. The gauge symmetry in the system plays important role to have the
strong-weak duality between them.

Note that our derivations are totally following the original paper [7] and
thus the derivations of holographic single fishchain and bi-fishchain can be
written in a unified form. One can begin with the eigenvalue equation
associated to bi-scalar or tri-scalar CFT which is written as

$\displaystyle 0$ $\displaystyle=$ $\displaystyle{\cal
H}_{\alpha}\circ\Psi(\\{x_{i}\\}),~{}~{}~{}{\cal
H}_{\alpha}=\left(\prod_{i=1}^{J}\vec{p}_{i}^{2}-\prod_{i=1}^{J}{4\xi^{2}\over(\vec{x}_{i}-\vec{x}_{i+1})^{2\alpha}}\right)~{}~{}$
(5.1)

where $\alpha=1$ describes bi-scalar CFT in (2.12) while $\alpha=2$ describes
tri-scalar CFT in (2.17). In this way, we can perform the similar derivations
and find that the Lagrangian of the chain becomes

$\displaystyle{\cal L}=2\alpha\prod_{k}\Big{(}-X_{k}\cdot
X_{k+1}\Big{)}^{-{\alpha\over J}}$ (5.2)

in which $\alpha=1$ describes fishchain in Gromov and Sever paper [7] while
$\alpha=2$ describes bi-fishchain in (4.12).

Finally, we can furthermore follow the method in [12] to quantize the bi-
fishchain Model. The result is like as that in fishchain model : While the
classical model was formulated on the lightcone of five dimensional flat
spacetime, described in (4.4), after the quantization the quantum bi-fishchain
model will live on $AdS_{4}$ with radius $\sim{1\over\xi}$, where $\xi$ is the
‘t Hooft coupling defined in tri-scalar Lagrangian (2.13).

References

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# Magellanic System Stars Identified in SMACS J0723.3-7327 James Webb Space
Telescope Early Release Observations Images

Jake Summers School of Earth and Space Exploration, Arizona State University,
Tempe, AZ 85287-1404, USA  Rogier A. Windhorst School of Earth and Space
Exploration, Arizona State University, Tempe, AZ 85287-1404, USA  Seth H.
Cohen School of Earth and Space Exploration, Arizona State University, Tempe,
AZ 85287-1404, USA  Rolf A. Jansen School of Earth and Space Exploration,
Arizona State University, Tempe, AZ 85287-1404, USA  Timothy Carleton School
of Earth and Space Exploration, Arizona State University, Tempe, AZ
85287-1404, USA  Patrick S. Kamieneski School of Earth and Space Exploration,
Arizona State University, Tempe, AZ 85287-1404, USA  Benne W. Holwerda
Department of Physics and Astronomy, University of Louisville, Louisville KY
40292, USA  Christopher J. Conselice Jodrell Bank Centre for Astrophysics,
Alan Turing Building, University of Manchester, Oxford Road, Manchester M13
9PL, UK  Nathan J. Adams Jodrell Bank Centre for Astrophysics, Alan Turing
Building, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Brenda Frye Steward Observatory, University of Arizona, 933 N Cherry Ave,
Tucson, AZ, 85721-0009, USA  Jose M. Diego Instituto de Física de Cantabria
(CSIC-UC). Avenida. Los Castros s/n. 39005 Santander, Spain  Christopher N. A.
Willmer Steward Observatory, University of Arizona, 933 N Cherry Ave, Tucson,
AZ, 85721-0009, USA  Rafael Ortiz III School of Earth and Space Exploration,
Arizona State University, Tempe, AZ 85287-1404, USA  Cheng Cheng Chinese
Academy of Sciences, National Astronomical Observatories, CAS, Beijing 100101,
China  Alex Pigarelli School of Earth and Space Exploration, Arizona State
University, Tempe, AZ 85287-1404, USA  Aaron Robotham International Centre for
Radio Astronomy Research (ICRAR) and the International Space Centre (ISC), The
University of Western Australia, M468, 35 Stirling Highway, Crawley, WA 6009,
Australia  Jordan C. J. D’Silva International Centre for Radio Astronomy
Research (ICRAR) and the International Space Centre (ISC), The University of
Western Australia, M468, 35 Stirling Highway, Crawley, WA 6009, Australia  ARC
Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D),
Australia  Scott Tompkins School of Earth and Space Exploration, Arizona State
University, Tempe, AZ 85287-1404, USA  Simon P. Driver International Centre
for Radio Astronomy Research (ICRAR) and the International Space Centre (ISC),
The University of Western Australia, M468, 35 Stirling Highway, Crawley, WA
6009, Australia  Haojing Yan Department of Physics and Astronomy, University
of Missouri, Columbia, MO 65211, USA  Dan Coe AURA for the European Space
Agency (ESA), Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21210, USA  Norman Grogin Space Telescope Science Institute,
3700 San Martin Drive, Baltimore, MD 21210, USA  Anton Koekemoer Space
Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21210, USA
Madeline A. Marshall National Research Council of Canada, Herzberg Astronomy &
Astrophysics Research Centre, 5071 West Saanich Road, Victoria, BC V9E 2E7,
Canada  ARC Centre of Excellence for All Sky Astrophysics in 3 Dimensions
(ASTRO 3D), Australia  Nor Pirzkal Space Telescope Science Institute, 3700 San
Martin Drive, Baltimore, MD 21210, USA  Russell E. Ryan, Jr. Space Telescope
Science Institute, 3700 San Martin Drive, Baltimore, MD 21210, USA
<EMAIL_ADDRESS>
###### Abstract

We identify 71 distant stars in JWST/NIRCam ERO images of the field of galaxy
cluster SMACS J0723.3-7327 (SMACS 0723). Given the relatively small
($\sim$$10^{\circ}$) angular separation between SMACS 0723 and the Large
Magellanic Cloud, it is likely that these stars are associated with the LMC
outskirts or Leading Arm. This is further bolstered by a spectral energy
distribution analysis, which suggests an excess of stars at a physical
distance of $40-100$ kpc, consistent with being associated with or located
behind the Magellanic system. In particular, we find that the overall surface
density of stars brighter than 27.0 mag in the field of SMACS 0723 is
$\sim$2.3 times that of stars in a blank field with similar galactic latitude
(the North Ecliptic Pole Time Domain Field), and that the density of stars in
the SMACS 0723 field with SED-derived distances consistent with the Magellanic
system is $\sim$6.1 times larger than that of the blank field. The candidate
stars at these distances are consistent with a stellar population at the same
distance modulus with [Fe/H] $=-1.0$ and an age of $\sim$$5.0$ Gyr. On the
assumption that all of the 71 stars are associated with the LMC, then the
stellar density of the LMC at the location of the SMACS 0723 field is
$\sim$$740$ stars kpc-3, which helps trace the density of stars in the LMC
outskirts.

Galactic archaeology(2178) — Stellar spectral types(2051) — James Webb Space
Telescope(2291)

††software: Astropy (Astropy Collaboration et al., 2013, 2018, 2022); EAZY
(Brammer et al., 2008); JWST Calibration Pipeline (Bushouse et al., 2022);
Photutils (Bradley et al., 2022); ProFound (Robotham et al., 2018); Pypher
(Boucaud et al., 2016); Pysynphot (STScI Development Team, 2013);
SourceExtractor (Bertin, E. & Arnouts, S., 1996); WebbPSF (Perrin et al.,
2012)††facilities: Hubble and James Webb Space Telescope Mikulski Archive
https://archive.stsci.edu. All of the JWST data used in this paper can be
found on MAST: http://dx.doi.org/10.17909/b7hf-he53 (catalog 10.17909/b7hf-
he53). The RELICS HST data used in this paper can be found here:
http://dx.doi.org/10.17909/T9SP45 (catalog 10.17909/T9SP45).

##  1 Introduction

The Magellanic Clouds are the most massive satellite galaxies of the Milky Way
(McConnachie, 2012), so they can provide a uniquely detailed look at how
massive satellite galaxies and Milky Way-like host galaxies interact. Also,
they are the only nearby satellites of the Milky Way that are not devoid of
gas (Putman et al., 2021). Those two seem connected—most satellite galaxies of
this mass are star forming (Wheeler et al., 2014), but how tidal and ram-
pressure stripping processes affect galaxies like the LMC and SMC in orbit
around the Milky Way is not fully understood. A more detailed view of the
structure and dynamics of the Magellanic Clouds can help improve our
understanding of the interaction between the Clouds and the Milky Way, and in
turn these environmental processes. The detailed orbital history of the
Magellanic Clouds can provide clues about the properties (mass and
concentration) of the Milky Way (e.g., Santos-Santos et al. 2021).

The Magellanic system (See D'Onghia & Fox 2016 for a review) consists of the
Large Magellanic Cloud (LMC), the Small Magellanic Cloud (SMC), the Bridge
(between the LMC and SMC; Hindman et al. 1963), the Magellanic Stream (behind
the LMC and SMC; Mathewson et al. 1974), and the Leading Arm (ahead of the LMC
and SMC; Putman et al. 1998). The Leading Arm was first identified by Putman
et al. (1998) in the H I Parkes All-Sky Survey as a thin region leading the
LMC-SMC system at higher velocities. Lu et al. (1998) further connected the
Leading Arm to the LMC and SMC by finding similarities to LMC and SMC S/Fe
abundance ratios in a high-velocity cloud in the Leading Arm.

As a key rung in the distance ladder (Riess et al., 2022), the stellar
populations of the LMC and SMC have been studied in detail. The distance
modulus to the LMC has been measured to be $18.48\pm 0.05$ (based on a
combination of Cepheids, Luminous Red Variables, RR Lyrae Stars, Red Clump
Stars, and Eclipsing Binaries), which corresponds to a distance between $48.5\
\text{kpc}$ and $50.8\ \text{kpc}$ (Walker, 2011; Pietrzyński et al., 2019;
Riess et al., 2019). The SMC is around $10\ \text{kpc}$ farther away, measured
with similar techniques (Graczyk et al., 2013). The velocity distribution of H
I clouds in the Leading Arm is consistent with a distance between $40$ and
$70$ kpc (Venzmer et al., 2012). Some simulations of the Magellanic Stream
predict that it is $\sim$$20\ \text{kpc}$ from the Sun at its closest point,
with a stellar component that is even closer (Lucchini et al., 2021).

The Leading Arm spans around $70^{\circ}$ in the sky, has an inclination
against the sky of $\sim$$13.6^{\circ}$ (almost face-on), and spans around 52
kpc (Brüns et al., 2005; Venzmer et al., 2012). Most observations of the
Leading Arm are of H I gas, of which Venzmer et al. (2012) estimated a lower
limit H I mass of $3.8\times 10^{7}\ M_{\odot}$, compared to $4.4\times
10^{8}\ M_{\odot}$ for the LMC and $4.0\times 10^{8}\ M_{\odot}$ for the SMC
(Brüns et al., 2005). However, the overall properties of the Leading Arm are
not as well studied as the Clouds themselves.

Figure 1:

Figure 2: Map of H I in the LMC-SMC system, in galactic coordinates, showing
the location of the field of SMACS 0723. The SMACS 0723 field is near the
Leading Arm and is part of the LMC outskirts, only $\sim$$10\arcdeg$ from the
center of the LMC. This image is reproduced from Fig. 3 of Mathewson (2012),
with explicit permission from the editors of JAHH, with additional annotations
showing the location of the Bridge, Leading Arm, Magellanic Stream, and SMACS
0723 field. The outermost contours represent $10^{19}$ atoms cm-2.

The formation of the Leading Arm must involve tidal forces, although
additional forces such as ram pressure stripping may have been involved in its
formation (Nidever et al., 2008). At the same time, the Magellanic Stream
could have formed due to ram pressure stripping (Meurer et al., 1985; Moore &
Davis, 1994) or by blowout from supernovae (Nidever et al., 2008). With ram
pressure forces and blowout, a stellar component is not expected, while tidal
models do predict a stellar component for the Leading Arm. Initial searches
did not detect a stellar component of the Magellanic Stream (Putman et al.,
2003; Nidever et al., 2008). However, Casetti-Dinescu et al. (2014) observed 6
young stars in the Leading Arm, between $10$ and $40$ kpc from the Sun, as
well as one low radial velocity O6V star at $\sim$$40$ kpc, providing evidence
for recent star formation within the Leading Arm. Thus, the properties of the
Leading Arm can be inferred from the stellar populations that are observed
within it.

In addition to the Magellanic Stream and Leading Arm, the Magellanic outskirts
are of great interest (e.g., Saha et al., 2010; Mackey et al., 2016; Nidever
et al., 2017; Choi et al., 2018a, b; Mackey et al., 2018; Nidever et al.,
2019; Cullinane et al., 2020; Drlica-Wagner et al., 2021; Petersen et al.,
2022). Studying the outskirts of the LMC can reveal the extent of Magellanic
stellar populations in the sky as well as the star formation history of these
regions (Nidever et al., 2017). Furthermore, surveys around the LMC can help
identify dwarf galaxies bound to the LMC/SMC (Drlica-Wagner et al., 2016),
stellar streams neighboring the Clouds (Drlica-Wagner et al., 2021), and faint
star clusters at large physical distances (Nidever et al., 2017). Kinematic
information, combined with chemical information, from the periphery of the
Clouds can constrain the evolutionary history of the LMC-SMC-MW system (Muñoz
et al., 2023). Cullinane et al. (2022) used data from fields between
$\sim$$8^{\circ}$ and $\sim$$11^{\circ}$ from the center of the LMC to
constrain the kinematics of the LMC-SMC-MW system. They also observed a mean
metallicity of [Fe/H] $\approx-1$ across the 17 LMC fields analyzed. Petersen
et al. (2022) showed that the LMC halo extends out to 30$\arcdeg$ from the LMC
center.

The James Webb Space Telescope (JWST) provides a unique opportunity to study
the stellar component of the LMC outskirts and Magellanic Leading Arm. The
high resolution of the JWST, combined with its ability to search for faint
stars out to $m\sim 27$ mag (with S/N $\approx 35$ for a 3000 s exposure in
F200W; e.g., Fig. 6–8 of Windhorst et al., 2023), allows us to search for
late-type stellar populations within the Magellanic system.

In this paper, we use JWST observations of the SMACS J0723.3-7327 galaxy
cluster to search for stars that belong to the Magellanic system. Such stars
likely belong to the Magellanic outskirts, given that the field of SMACS
J0723.3-7327 is located between two of the 17 fields observed by Cullinane et
al. (2022) (see Figure 2 for a diagram of its location relative to the
Magellanic system). In Section 2 we describe the observations, the data
reduction process, and the source detection methods. In Section 3 we describe
the SED-fitting methods used. In Section 4 we describe the results from SED-
fitting and the associated uncertainties. In Section 5 we provide, in Table 1,
a list of 71 candidate stars that are within the expected distance range of
the Magellanic system and we compare these candidate stars to theoretical
isochrones to estimate metallicities and ages. We conclude with a summary and
prospects for future studies in Section 6. We express all magnitudes in the AB
system (Oke & Gunn, 1983) unless otherwise noted.

##  2 Observations

###  2.1 ERO and PEARLS Observations

The primary JWST data used in this study are NIRCam images of the SMACS
J0723.3-7327 (hereafter SMACS 0723) field from the JWST Early Release
Observations (ERO 2736, P.I. K. Pontoppidan). For comparison, we also use
NIRCam images of three fields from the Prime Extragalactic Areas for
Reionization and Lensing Science (hereafter PEARLS) survey (GTO 1176, P.I. R.
Windhorst; GTO 2738, P.I. R. Windhorst & H. Hammel): the North Ecliptic Pole
Time Domain Field (hereafter NEP TDF), the IRAC Dark Field (hereafter IDF),
and the MACS J0416.1-2403 (hereafter MACS 0416) galaxy cluster. We refer the
reader to Pontoppidan et al. (2022) for more details on the SMACS 0723
observations and Windhorst et al. (2023) for details about the PEARLS program.
The PEARLS images used cover the first two spokes of the NEP TDF, the first
two epochs of the IDF, and all three epochs of MACS 0416. The NEP TDF and IDF
are medium-deep blank fields in the North Ecliptic Pole region and MACS 0416
is a galaxy cluster at $z=0.4$ (Caminha et al., 2017). SMACS 0723 is a galaxy
cluster at $z=0.39$ (e.g., Repp & Ebeling, 2018; Coe et al., 2019; Pascale et
al., 2022; Mahler et al., 2023). SMACS 0723 and the North Ecliptic Pole have
roughly similar angular separations from the Galactic bulge of $75\arcdeg$ and
$95.8\arcdeg$, respectively.111The galactic coordinates for SMACS 0723 and the
NEP TDF are $l=285.0\arcdeg$, $b=-23.76\arcdeg$ and $l=95.8\arcdeg$,
$b=33.6\arcdeg$, respectively.

The SMACS 0723 field was observed on 2022 June 7 in the F090W, F150W, F200W,
F277W, F356W, and F444W NIRCam filters, with an exposure time of 7537 s in
each band. These observations reach a $5\sigma$ AB magnitude limit of
$\sim$$29.5$ mag for point sources with a $0\farcs 08$ aperture radius
according to the JWST Exposure Time Calculator (ETC). Module B of NIRCam is
centered on the galaxy cluster, while module A is the noncluster module. The
coordinates of SMACS 0723 are RA=$07^{h}\ 23^{m}\ 19.5^{s}$, DEC=$-73\arcdeg\
27\arcmin\ 15\farcs 6$.

Spoke 1 of the NEP TDF was observed with JWST/NIRCam on 2022 August 26 and
2022 September 14 with an observation time of 2920 s in the F090W, F115W,
F410M, and F444W bands, and an observation time of 3350 s in the F150W, F200W,
F277W, and F356W bands for each pointing; spoke 2 was observed on 2022
November 26-27 with the same exposure parameters. The NEP TDF has coordinates
of RA=$17^{h}\ 22^{m}\ 47.9^{s}$, DEC=$+65\arcdeg\ 49\arcmin\ 21\farcs 5$
(Windhorst et al., 2023). Epoch 1 of the IDF was observed by NIRCam on 2022
July 8 with an observation time of 3157 s in the F150W, F200W, F356W, and
F444W filters; epoch 2 was observed on 2023 January 6 with the same exposure
parameters. The coordinates of the IDF are RA=$17^{h}\ 40^{m}\ 8.5^{s}$,
DEC=$+68\arcdeg\ 58\arcmin\ 27\farcs 0$ (Windhorst et al., 2023). MACS 0416
was observed by NIRCam for three epochs with, for each epoch, an observation
time of 3779 s in F090W, F115W, F410M, and F444W, and an observation time of
2920 s in F150W, F200W, F277W, and F356W. Epoch 1 was taken on 7 October 2022,
epoch 2 on 29 December 2022, and epoch 3 on 10 Februrary 2023. The coordinates
of MACS 0416 are RA=$4^{h}\ 16^{m}\ 8.9^{s}$, DEC=$-24\arcdeg\ 4\arcmin\
28\farcs 7$ (Windhorst et al., 2023). The NEP TDF, the IDF, and MACS 0416
reach approximately 29.0 mag at $5\sigma$ for point sources in F200W with an
aperture radius of $0\farcs 08$ (Windhorst et al., 2023).

In addition to JWST NIRCam data of the SMACS 0723 cluster, the MACS 0416
cluster, the NEP TDF, and the IDF, we also use public Hubble Space Telescope
(HST) data of SMACS 0723 from the “Reionization Lensing Cluster Survey”
(RELICS) program (GO 14096, P.I. D. Coe). RELICS provides HST images and
catalogs, of which we use the combined ACS and WFC3/IR catalog (Cerny et al.,
2018; Salmon et al., 2020), with magnitudes in the F435W, F606W, F814W, F105W,
F125W, F140W, and F160W filters. These observations have $5\sigma$ point
source AB magnitude limits between 26.0 mag and 27.6 mag (Coe et al., 2019),
and provide useful information at optical wavelengths for objects detected in
the JWST data. For stars, these shorter wavelengths are critical for
distinguishing between the hottest spectral types. These data are used in
Section 3.3 as consistency checks, but are not used in the final analysis. We
matched the positions of stars and galaxies in the NIRCam images to those of
the RELICS images. Given the one sigma astrometric residuals of $\sim$$148$
milliarcseconds between the two data sets, we found only one star to have
exhibited possible proper motion of more than 3$\sigma$ and therefore
concluded that proper motion is not significant between the HST and JWST
observations.

###  2.2 Data Reduction

All NIRCam uncal data were retrieved from the Mikulski Archive for Space
Telescopes (MAST). For all stages of the JWST calibration pipeline, data from
SMACS 0723, the first epoch of the NEP TDF, the IDF, and MACS 0416, were
reduced using version 1.7.2 of the pipeline and pmap_0995. Version 1.8.4 and
pmap_1017 were used for epoch 2 of the NEP TDF/IDF. Finally, version 1.8.4 and
pmap_1027 were used to reduce epoch 2 and pmap_1041 was used for epoch 3 of
MACS 0416. The difference between the various pipeline versions was minimal as
no major NIRCam reference file updates were released between pmap_0995 and
1041. The data was then processed through Stage 1 (detector1pipeline) of the
JWST calibration pipeline, with a snowball flagging step performed before the
jump detection step. Snowballs are caused by intense cosmic rays and are dealt
with by expanding the data quality flags around such cosmic rays. C. Willott’s
dosnowballflags.py algorithm is used to subtract
snowballs.222https://github.com/chriswillott/jwst Next, wisp templates, which
were constructed by CNAW from all publicly available NIRCam SW images, were
subtracted from the NRCA3, NRCA4, NRCB3, and NRCB4 detectors for the F150W and
F200W filter exposures (see Robotham et al. 2023 for an alternative method of
removing wisps). Wisps are also present in F090W and F115W in the NEP TDF, but
there is presently not enough public data that contains these wisps to make a
high-S/N template. Following the wisp subtraction, the rate files were reduced
with the default Stage 2 pipeline (calwebbimage2). After this, a correction
for 1/f noise was applied using the ProFound package (Robotham et al., 2018),
which also performed sky subtraction. Both an overall sky level and a sky
gradient were fit with ProFound and subtracted from each image (see Windhorst
et al. 2023 for more details). Finally, the data were astrometrically aligned
and drizzled to mosaics with a pixel size of $0\farcs 03$ using the JWST Stage
3 pipeline (calwebbimage3). SMACS 0723 data was relatively aligned, whereas
NEP TDF, IDF, and MACS 0416 data were aligned to Gaia DR3.

###  2.3 Photometry

Figure 3:

Figure 4: (left) SourceExtractor MAGERR_AUTO vs MAG_AUTO for the F150W NIRCam
filter, with a horizontal dashed line establishing the uncertainty needed for
a source to have a signal-to-noise ratio of 5. (middle) MAG_AUTO vs
FWHM_IMAGE, with stars plotted in red based on the FWHM and magnitude
separation criteria. The green dashed line shows the point source and surface
brightness sensitivity limit for the corresponding image (for details, see
Windhorst et al. 2023). (right) Star (red) and galaxy (blue) counts per square
degree vs MAG_AUTO. Best-fit lines of the star counts are plotted for SMACS
0723, the NEP TDF, and the IDF for comparison. The slope of the best-fit line
for SMACS 0723 stars is greater than that of the other fields—there is an
excess of faint stars.

We use SourceExtractor (Bertin, E. & Arnouts, S., 1996) for source detection
in the NIRCam images. First, a set of PSF-matched (to F444W) images is
created. These are made by using pypher (Boucaud et al., 2016) to calculate
the necessary convolution kernel to match each filter’s PSF to that of the
F444W PSF, where the PSFs are simulated using WebbPSF (Perrin et al., 2012).
This convolution results in all PSFs having approximately the same FWHM and
shape. SourceExtractor is then run on the images in dual-image mode with F444W
as the detection filter and with DETECT_MINAREA=9, DETECT_THRESH=1.5, and a
5$\times$5 pixel gaussian convolution kernel with a FWHM of 3 pixels. F444W is
among the deepest of the images, and using it results in the greatest number
of stars being found, so it is used as the detection image. The MAG_AUTO of
each source is taken to be its observed magnitude.

To select stars from the object catalogs, we use similar methods to Windhorst
et al. (2011). We require MAG_AUTO $<$ 27 and that FWHM_IMAGE lies within the
stellar locus of the middle plot in Figure 4. i.e., the FWHM fell between
$0\farcs 14$ and $0\farcs 18$ in all filters (widening for stars brighter than
20 mag). The magnitude requirement is used to avoid classifying small, faint
galaxies as stars. An object is classified as a star if it fits these criteria
in at least three NIRCam filters.

##  3 SED Fitting

Near-IR spectral energy distributions are fit to the stars to estimate their
distances and spectral types. To conduct the SED fitting, we use the framework
of EAZY (Brammer et al., 2008) given its flexibility at conducting this
analysis, though we emphasize that we use separate stellar SED templates
rather than galaxy SED templates. To do this, the maximum redshift is set to
$z=0$ and the minimum filters used in the fitting is set to 4. Before fitting
SEDs, the catalogs (other than the IDF, which is only used for star counts)
are corrected for galactic extinction ($A_{V}=0.587$ mag for SMACS 0723,
$A_{V}=0.082$ mag for the NEP TDF, and $A_{V}=0.101$ mag for MACS 0416). We
apply the full value of $A_{V}$ to better fit distant stars which have more
galactic dust in front of them. The $A_{V}$ values are likely quite different
between fields due to LMC dust in the field of SMACS 0723, although the
correction only causes an adjustment of $\sim$$0.05$ mag in F200W for SMACS
0723. The SMACS 0723 RELICS catalog has the same $A_{V}$ correction by
default. We use fluxes for the EAZY SED fitting.

###  3.1 Template Construction

Instead of the default galaxy SED templates, we use stellar SED templates from
the SpeX IRTF Spectral Library from Rayner et al. (2009) and Cushing et al.
(2005). The SpeX library covers wavelengths of $0.8-5.0\ \micron$, with
templates for stars with MW abundances, cooler in temperature than type
A.333Tempaltes for F2V, F3V, F4V, F5V, F7V, F8V, F9.5V, G0V, G1V, G2V, G3Va,
G5V, G6.5V, G8V, K0V, K1V, K2V, K3V, K5V, K7V, M0V, M0.5V, M1V, M1.5V, M2.5V,
M3V, M3.5V, M4V, M5V, M6V, M6.5V, M7V, M8V, M9V, M9.5V, L1, L3, L3.5, and L5
stars are used from
http://irtfweb.ifa.hawaii.edu/~spex/IRTF_Spectral_Library/. We only fit main
sequence stars because they account for most stars in the LMC field. If we
assume there are $25,000$ AGB stars per 16 deg2, as in Wiśniewski et al.
(2011), then there could be around 3 AGB stars in the NIRCam SMACS 0723 data.
This is insignificant compared to the number of total stars observed, and
since we are focusing on the overall trend in star distances, these stars can
safely be ignored.

We then use pysynphot (STScI Development Team, 2013) to compute the apparent
magnitude in every NIRCam filter for each stellar SED template. The Gaia
parallax of each SpeX star is then used to determine its absolute magnitude in
each NIRCam filter (Gaia Collaboration et al., 2016, 2022; Babusiaux et al.,
2022). Since spectral type for main sequence stars roughly correlates with
their absolute magnitudes, the absolute magnitude of each SpeX star can be
estimated given its parallax-based distance and its apparent magnitude.

###  3.2 Bayesian Probabilities

When fitting stellar SEDs with EAZY, the code is ran once for each individual
template, the template is fit to every star and the spectral type, $\chi^{2}$,
and observed/template SEDs are recorded. All stars with best-fit
$\chi^{2}_{\text{red}}>2$ are removed from the final catalog. This cut removes
21 objects from the SMACS 0723 catalog. Most high-$\chi^{2}$ objects are
likely either not main-sequence stars or have questionable photometry.

To provide an estimate of the distance moduli, $\mu$, of the stars in SMACS
0723, we measure the posterior probability distributions of the distance
modulus for each star. To do this, we use the output from EAZY, which provides
the fitting parameters for every star for each stellar SED template. Bayes’
Theorem is then used:

$P(\text{Template}|\text{Fluxes})=\frac{P(\text{Fluxes}|\text{Template})P(\text{Template})}{\sum_{i}P(\text{Fluxes}|\text{Template}_{i})},$
(1)

where each term is a probability. The measured fluxes in each NIRCam filter
for a given star are the “Fluxes” and the spectra for a given spectral type
(SpeX star) is the “Template.” Equation 1 solves for
$P(\text{Template}|\text{Fluxes})$ (the posterior probability), which is the
probability that a star fits a particular SpeX SED template, assuming that the
fluxes are accurate. $P(\text{Fluxes}|\text{Template})$ is the probability
that the fluxes are correct, assuming that the star fits a particular SpeX SED
template (which corresponds to a spectral type), and $P(\text{Template})$ is
the probability of a random star fitting the SpeX SED template (the prior). We
assume a constant prior probability model (changing this to reflect the MW IMF
has almost no effect on the results) on absolute magnitude, turning Equation 1
into the following:

$P(\text{Template}|\text{Fluxes})=\frac{P(\text{Fluxes}|\text{Template})}{\sum_{i}P(\text{Fluxes}|\text{Template}_{i})}.$
(2)

We calculate $P(\text{Fluxes}|\text{Template})$ as the product of gaussians
(one per filter) centered at each predicted flux, with a standard deviation
equal to EAZY err_full (includes photometric uncertainty and template
uncertainty), evaluated at the measured flux. The posterior probability for
each template fitting a given star is equal to the probability that the
template fits the star divided by the sum of the probabilities over all of the
templates. This posterior probability distribution (as a function of spectral
type, which each have a corresponding absolute magnitude) is then calculated
for all stars in all filters and is converted from spectral types to absolute
magnitudes. The probability distributions are then converted to distance
moduli by subtracting the distribution from each filter’s apparent magnitude.
Therefore each distribution gives the probability that a given star is fit
with a given distance modulus. These probability distributions are then
linearly interpolated and all 6-8 of them (6-8 NIRCam filters) are summed
together for each star. These probability density functions are normalized to
an area of one, and are used to calculate the lower bound, median, and upper
bound of the distance modulus for each star. This is done by measuring the
16th, 50th, and 84th percentile in the cumulative probability distribution,
respectively. Thus the reported uncertainties correspond to $1\sigma$
uncertainty. Measuring the probability densities of each star allows for more
complete assessment of uncertainties, reported in Table 1.

###  3.3 HST Checks

To confirm the SED fits from the SpeX library of stellar SEDs, we use stellar
SED templates from the optical library from Pickles (1998). For stars in the
field of SMACS 0723 that have HST data available (there are 90 of them), the
optical HST data (F435W, F606W, F814W, F105W, F125W, F140W, and F160W) and
short wavelength NIRCam data (F090W, F150W, and F200W) are fit with EAZY by
the Pickles (1998) templates (the templates cover $0.12-2.50\ \micron$ in
wavelength) for a variety of main sequence stars.444Available templates
include those of O5V, O9V, B0V, B1V, B3V, B6V, B8V, A0V, A2V, A3V, A5V, F0V,
F2V, F5V, F8V, G0V, G2V, G5V, G8V, K0V, K2V, K5V, K7V, M0V, M4V, and M5V stars
from https://archive.stsci.edu/hlsps/reference-atlases/cdbs/grid/pickles/. The
Pearson correlation coefficient ($r$) was calculated between the shared
templates for the two spectral libraries over their shared wavelengths, and
all templates except for the M5V ($r\approx 0.76$) template agree to $r\gtrsim
0.97$. However, the M5V template is the last template available from the
Pickles (1998) library, so this is to be expected. Since the Pickles (1998)
templates are normalized and combine multiple stars for the creation of each
template, we assume each template has the same absolute magnitude as the SpeX
star with the same spectral type. The absolute magnitudes fit by the two
libraries have the greatest agreement at $M_{\text{F090W}}\lesssim 7$ mag,
where the standard deviation of their difference is $\sim$$0.35$ mag. This
limited range of agreement is to be expected because the SpeX library offers a
greater variety of late-type stellar SED templates compared to the Pickles
(1998) library. Since the SpeX library leads to similar results as the Pickles
(1998) library, covers more near-IR wavelengths, and has a wider variety of
M-star templates, it is reasonable to use the SpeX library with all of the
SMACS 0723, NEP TDF, and MACS 0416 photometry.

##  4 Magellanic Star Candidate Identification

###  4.1 The Distance Modulus

Figure 5:

Figure 6: Distribution of distance moduli, $\mu$, (calculated from the 50th
percentile in the distance modulus PDF) for stars in the SMACS 0723, the NEP
TDF, and the MACS 0416 fields. (top) Binned distance moduli for stars in each
field. (middle) Ratio of stars found at each distance in SMACS 0723 compared
to the NEP TDF and MACS 0416, normalized to have a sum of one. (bottom)
Difference in probability density between SMACS 0723 and the NEP TDF/MACS 0416
fields, where the data from each field is normalized to have a sum of one
(resulting in a difference of zero between fields). The vertical black dashed
lines at a distance modulus of 18 and 20 mag show the ranges of candidates
taken, the vertical red dashed lines shows the distance to the LMC, and the
vertical blue dashed lines show the distance to the SMC.

We consider two ways of estimating the distribution of distance moduli of
stars in the SMACS 0723 field. First, we consider the distribution of distance
estimates of each star (using the 50th percentile of the distance modulus
PDF). Alternatively, all of the probability distributions of each star can be
added up to form one probability distribution for each field as a whole.
Figure 6 shows the distribution of 50th percentile distance moduli for each
field. The ratio of the distributions, shown in the middle plot, shows a clear
surplus of stars in the field of SMACS 0723 at $18\lesssim\mu\lesssim 20$ mag,
which corresponds to a distance between $\sim$$39.8\ \text{kpc}$ and
$\sim$$100.0\ \text{kpc}$. We refer to the stars in this range of distances as
stellar candidates, however the properties of each individual star are more
uncertain compared to the population as a whole. Additionally, some candidate
stars could be part of the MW Halo since the JWST photometry does not have the
spectral resolution to detect precise metallicity differences between
individual stars. The integrated surface density between these distance moduli
is $\sim$$50,500$ stars deg-2 in the SMACS 0723 field, compared to
$\sim$$8200$ stars deg-2 in the NEP TDF. This means that the field of SMACS
0723 has $\sim$$6.1$ times as many stars per sq. degree as the NEP TDF between
39.8 and 100 kpc, compared to an overall ratio of $\sim$$2.3$ and a ratio at
all distances other than $18\leq\mu\leq 20$ mag of $\sim$$1.9$. Overall, the
probability distribution for the SMACS 0723 field has $\sim$$7600$ more stars
deg-2 mag-1 than the NEP TDF. However, between 39.8 and 100 kpc the difference
is $\sim$$21,100$ stars deg-2 mag-1. Figure 6 shows the difference in 50th
percentile star counts between the SMACS 0723 field and the two other fields
in the bottom plot. The field of SMACS 0723 has a clear excess between 39.8
and 100 kpc, whereas the NEP TDF and MACS 0416 field have a minor excess
relative to the SMACS 0723 field at closer distances. Hence, there is a
significant surplus of stars ($15.3\sigma$ compared to the NEP TDF,
$7.2\sigma$ compared to the field of MACS 0416) in the field of SMACS 0723 at
this distance. The range of distances found in this paper is different
compared to Casetti-Dinescu et al. (2014), which observed early-type stars at
a distance of up to $\sim$$40\ \text{kpc}$. The stars observed in this paper
are farther than $40$ kpc, likely because the field of SMACS 0723 is closer in
the sky to the LMC than those in Casetti-Dinescu et al. (2014). The star
candidates are expected to be closer to $\sim$$50\ \text{kpc}$ because they
are only $\sim$$10^{\circ}$ from the LMC and the field of SMACS 0723 is near
the line-of-nodes for the LMC’s inclination (Cullinane et al., 2022), unlike
the stars in Casetti-Dinescu et al. (2014), which were further stars in the
Leading Arm.

###  4.2 Contamination

Figure 7:

Figure 8: Histogram of X-positions of candidate stars in the SMACS 0723
mosaics. Error bars show $\sqrt{N}$ uncertainties. The noncluster module is on
the left of the vertical black dashed line and the cluster module is on the
right, with the position of the brightest cluster galaxy shown as a vertical
red dashed line. No significant excess of candidate LMC stars around the BCG
is detected.

Since SMACS 0723 is a galaxy cluster, an obvious source of contamination would
be dwarf galaxies or globular clusters within the galaxy cluster. We would
expect foreground LMC stars to be distributed randomly throughout the SMACS
0723 images, and not concentrated in the NIRCam module that contains the SMACS
0723 cluster center. As a test we counted the numbers of stellar candidates in
the NIRCam module centered on the cluster (32 stellar candidates) and offset
from the SMACS 0723 cluster (39 stellar candidates), obtaining numbers which
are less concentrated towards the cluster center. Figure 8 shows the number of
candidates as a function of X-coordinate. Since the mosaics are not rotated to
be north-up, the X-coordinate is equivalent to the lengthwise distance along
the NIRCam field of view. As expected, there is a dip in star counts close to
the module gap. There is an excess of candidate stars near the BCG, but the
noncluster module has a similar spike in stars elsewhere, so we conclude that
contamination in this catalog is unlikely. The centroid of LMC candidates for
each NIRCam module is $\lesssim$$250$ pixels from the center of each module.
Thus the candidates are not skewed a certain direction and are randomly
distributed as expected.

In addition to the spatial distribution of candidate stars in the field of
SMACS 0723, the inclusion of MACS 0416 (which has approximately the same
redshift at $z=0.4$, as SMACS 0723 at $z=0.38$) data further corroborates that
there is little to no contamination. In particular, Figure 6 shows that there
is no excess of stars in the field of MACS 0416 at $18\leq\mu\leq 20$ mag.
Thus the excess in point sources at $18\leq\mu\leq 20$ mag is not caused by
contamination from the SMACS 0723 galaxy cluster.

###  4.3 Uncertainty in $\mu$

There are various contributions in the uncertainty budget of the calculated
distance moduli, including photometric uncertainties, SpeX calibration
uncertainties, and SED fit uncertainties, which we account for. Unaccounted
for uncertainties include distance-dependent dust extinction as well as
metallicity differences between the stars observed and the EAZY templates
used. Photometric uncertainties in the NIRCam data and the SpeX calibration
only contribute $\lesssim$$3\%$ uncertainties to the distance moduli. This was
verified using Monte Carlo methods: the observed magnitudes were resampled
based on their SourceExtractor uncertainties 1000 times for each star, the
1000 probability distributions for each star were averaged together, and the
50th percentile in the PDF of distance moduli was recalculated to be
$\lesssim$$3\%$ of the distance modulus calculated with only the observed
magnitudes. The calibration for the IRTF spectral library is within a few
percent (Rayner et al., 2009). For a conservative estimate on uncertainty in
the SED templates, the 2MASS photometry typically has uncertainties of
$\lesssim$$0.25$ mag, so we add this number in quadrature to the final upper
and lower uncertainties. The uncertainty in parallax for each of the SpeX
stars used is $\lesssim$$0.01$ mag, and is therefore negligible. Finally, the
SED fit uncertainties are described in section 3.2 and generally have the
greatest contribution to the overall uncertainty in $\mu$. Thus the 2MASS
photometric uncertainty and the SED fit uncertainties are used to calculate
the final uncertainties in $\mu$, reported in Table 1.

###  4.4 Gaia Stars

A Gaia DR3 (Gaia Collaboration et al., 2016, 2022; Babusiaux et al., 2022)
query555The Gaia DR3 catalog can be queried at
https://gea.esac.esa.int/archive/. was performed in each of the fields studied
in this paper to check if any of the detected stars had measured parallaxes. A
handful of the JWST stars from each field have parallaxes and proper motions,
but the uncertainties are large enough to render them unhelpful. As a separate
check, a Gaia DR3 query with a larger area was undertaken around the field of
SMACS 0723 and the NEP TDF to check for an excess of stars at the distance of
the LMC. However, due to the shallower depth of Gaia, no significant excess of
low-parallax stars nor stars with small proper motion characteristic of the
LMC was found. Thus Gaia data alone is not enough to detect an excess of stars
around the field of SMACS 0723 at the distance of the LMC.

##  5 Magellanic Star Candidate Properties

Figure 9:  Figure 10:

Figure 11:

Figure 12: (top) Color-magnitude diagrams including the 71 candidate stars
(with colored shapes corresponding to different spectral types) and the rest
of the detected stars (black dots). Plotted on them are shaded isochrones of
various ages and metallicities for distance moduli between 18 and 20 mag.
(bottom) Color-color diagram with the 71 candidate stars as well as an
isochrone of $t=10.0$ Gyr and [Fe/H] $=-1.0$. A few candidates typed as G
stars must be younger than this age.

###  5.1 Age and Metallicity Estimates

We now set out to estimate the approximate ages and metallicities of the LMC
candidate stars. When selecting star candidates that are part of the
Magellanic system, we required that $18\leq\mu\leq 20$ mag and
$\chi^{2}_{\text{red}}<1$. This range of distances allows for the detection of
nearby stars in the Leading Arm (possibly around $40$ kpc) as well as farther
stars in the Magellanic outskirts closer to $50$ kpc. The smaller
$\chi^{2}_{\text{red}}$ value selects only the candidates that fit their
stellar template the best. These LMC stellar candidates are presented in Table
1.

Figure 12 shows all of the stars on color-magnitude diagrams, with isochrones
corresponding to stellar populations between $\mu=18$ mag and $\mu=20$ mag.
The isochrones are created using MESA Isochrones & Stellar Tracks data
(Dotter, 2016; Choi et al., 2016; Paxton et al., 2011, 2013, 2015). The
isochrones assume [Fe/H] $=-1.0$ and $v/v_{crit}=0.4$ rotation rates. We also
generate isochrones of [Fe/H] $=0.0$ to represent MW stars. Most of the
candidate stars (colored shapes) fall in the range of the isochrones,
reflecting that the candidates form their own stellar population. The other
detected stars (black dots) that fall around the isochrones generally have a
distance modulus close to that of the LMC, but outside the range of distance
moduli used to select candidates.

The characteristics of the isochrones reflect the metallicity of this stellar
component in the LMC. The implied metallicities seem to be consistent with
[Fe/H] $=-1.0$. This metallicity is consistent with that of stars observed in
the LMC outskirts (Cullinane et al., 2022). Many candidates have colors and
magnitudes that are not consistent with having [Fe/H] $=0.0$, so are not
likely to be MW stars. Some other candidates presented could be MW stars, but
that information could only be distinguished via precise metallicity or proper
motion measurements. The stars that are almost considered candidates, with
$\mu\lesssim 18$ mag or $\mu\gtrsim 20$ mag, could also be LMC stars, but we
cannot be certain of this since the difference between the SMACS 0723 field
and the NEP TDF at that physical distance is not significant.

Figure 12 also shows a color-color diagram with the candidate stars as colored
shapes and the other detected stars as black dots. In general, the candidates
are closer to the lower-left (hotter) than the bulk of observed stars, Many
non-candidate stars are closer to the upper right (type M). Additionally, the
isochrone of $10$ Gyr (with [Fe/H] $=-1.0$) shows that at least a few
candidates are inconsistent with the ages of MW halo stars.

Of all 204 stars identified in the field of SMACS 0723, only two are fit as
type F stars. These two stars have relatively blue colors, and can be seen as
black dots in the the bottom left corner of the color-color diagram in Figure
12. These stars are the bluest two stars in the color-magnitude diagrams in
Figure 12. Both stars have HST data available. The first of the two type F
stars was fit as F9.5V with $\mu=20.2^{+0.8}_{-0.3}$ mag using the SpeX
templates and G6.5V with $\mu=19.9^{+0.3}_{-0.3}$ mag using the Pickles (1998)
templates. The second star was fit as type F5V with both libraries, and had
$\mu=18.0^{+0.4}_{-0.7}$ mag using the SpeX templates and
$\mu=18.5^{+0.3}_{-0.6}$ mag using the Pickles (1998) templates. Thus both
stars are slightly outside of the criteria used for selecting Magellanic
system candidates. However, it is still possible that these stars are part of
the LMC outskirts. If this is the case, then there could be recent star
formation within the LMC outskirts from $<5.0$ Gyr ago.

###  5.2 Absolute Magnitudes and Masses

Figure 13:

Figure 14: HR Diagram of all stars observed in the field of SMACS 0723.
Distance modulus ($\mu$) is indicated by the color-scheme, and isochrones are
plotted to indicate the expected positions of an old MW stellar population and
a younger LMC population.

Figure 15:

Figure 16: Histogram of absolute magnitudes (in the 2MASS Ks bandpass) for
both the LMC candidate stars (in red) and the remaining MW stars (in blue).
The MW luminosity function measured in Just et al. (2015) is plotted in
purple. Dashed lines indicate the incompleteness limit of our sample (due to
the magnitude cutoff of $m<27$ mag and the distance cutoffs for each
population).

We now use the measured absolute magnitudes of the stars in the field of SMACS
0723 to evaluate the properties of each stellar population (MW and LMC) as
well as estimate stellar masses. Figure 14 shows the HR diagram for the stars
observed in this study. Bluer stars tend to be fit by either [Fe/H] $=-1.0$ or
[Fe/H] $=0$. However, redder stars are better fit by the [Fe/H] $=-1.0$ model.

We now use the measured absolute magnitudes to compare our observed stellar
luminosity function (LF) to that in Figure 6 of Just et al. (2015). Figure 16
shows the luminosity function for the set of MW stars, the set of LMC stars,
and the Just et al. (2015) results. The data is normalized to have a sum of 1
for stars brighter than the completeness limits. Just et al. (2015) plots
nearly all stars in the $20$ pc sphere around the Sun, whereas the field of
SMACS 0723 is a single pointing outside the MW disk. The LF observed for the
LMC candidates is $\sim$$2$ mag brighter than that of the closer MW stars
($\mu<18$ mag). Additionally, the LF observed for the MW stars matches up
better with the Just et al. (2015) LF, although it is incomplete for
$M_{K_{s}}\gtrsim 9$ mag.

Figure 17:

Figure 18: Histogram of masses observed in each stellar population (MW, blue:
$\mu<18$ mag; LMC, red: $18<\mu<20$ mag). The MW initial mass function (IMF)
is plotted in purple using the Salpeter (1955) IMF for $\log(M/M_{\odot})\geq
0$ and the Chabrier (2003) IMF for $\log(M/M_{\odot})\leq 0$. The IMF observed
in Sirianni et al. (2000) for LMC star clusters is plotted in green for
$0.13\leq\log(M/M_{\odot})\leq 0.81$. Graphs are normalized to a sum of 1 for
masses greater than the completeness limits. Any mass bins with counts less
than $10^{-2}$ have zero stars in them and are not plotted.

Using the absolute magnitudes derived for each star, we then estimate stellar
masses. To do this, we measure the Ks absolute magnitude for each SpeX
template using pysynphot (STScI Development Team, 2013). This allows us to use
the mass relations in Henry & McCarthy (1993). The results of this analysis
are plotted in Figure 18. We compare the observed IMFs to the Salpeter (1955)
and Chabrier (2003) MW IMF, plotted in purple, as well as the Sirianni et al.
(2000) LMC IMF, plotted in green. The results are not complete for lower mass
stars, but the MW star masses ($\mu<18$ mag) show agreement for
$-0.6\leq\log(M/M_{\odot})\leq 0.2$. The LMC stellar masses ($18<\mu<20$ mag),
on the other hand, are more biased towards higher mass stars and do not fit
the MW IMF well. The LMC IMF is not complete for low masses and so does not
overlap with our data.

###  5.3 Stellar Density

Taking the areal coverage of SMACS 0723 to be $\sim$$0.002$ sq. deg., our 71
candidates would lead to a stellar density of $\sim$$34,000\ \text{deg}^{-2}$.
This corresponds to a spatial density at a distance of 50 kpc (between 40 and
100 kpc from the Sun) of 740 stars kpc-3. This spatial density will further
help models constrain the orbital history of the Magellanic system around the
Milky Way. In particular, measuring the stellar density profile of the LMC can
help reveal faint features of the LMC other than the primary elliptical
profile (Massana et al., 2020). With shallower data, the surface brightness
profile of the LMC was not detected beyond galactocentric radii of
$\sim$$8\arcdeg$ (Massana et al., 2020); deeper data at farther radii, such as
with the field of SMACS 0723, can reveal a more complete LMC profile. Compared
to Cullinane et al. (2022), we observe many more faint stars, since the
angular density of Gaia stars around this area of the LMC outskirts (around
the field of SMACS 0723 and two Cullinane et al. 2022 fields) is only
$\sim$$45$ stars/deg-2. The ages observed are inconsistent with $t=10$ Gyr
because there are a few type G stars (plotted as stars and triangles in Figure
12) that are likely relatively young. This suggests that the group of stars
observed has a similar dynamical age as the body of the LMC.

The expected number of main sequence turnoff (MSTO) stars at an angular
distance of $10\arcdeg$ from the LMC center is around $600$ stars/deg2
(Nidever et al., 2019). Since our data goes deeper, it is difficult to
directly compare our results with other studies. However, if we take the
number of G stars (since the MSTO in Nidever et al. 2019 go to $\sim$$9$ Gyr)
observed in the field of SMACS 0723 at near LMC distance, we find $\sim 5300$
stars/deg2. If instead we consider stars with $2.5$ mag $\leq
M_{\text{F090W}}\leq$ $4.5$ mag, we find a stellar density of $\sim 1900$
stars/deg2. These stellar densities are somewhat different due to using
different methods to select stars. This could mean that there are more LMC
stars in the direction of SMACS 0723, or that the comparison is inexact.

##  6 Conclusion

We observe a total of 71 candidate stars that are between 39.8 and 100 kpc
from the Sun, according to their SED-derived distance moduli. These stars are
in a field that is $\sim$$10^{\circ}$ or $\sim$$9$ kpc from the LMC. We
present information about these candidates in Table 1. The candidate stars
could be in the outskirts of the LMC or part of the MW halo. They could also
be part of the Leading Arm, but the distances are too uncertain to separate
the Leading Arm from the LMC outskirts. The stars lie around isochrones of
this distance with [Fe/H] $=-1.0$, which is the expected metallicity for stars
in the LMC outskirts (Cullinane et al., 2022). While our identification of a
stellar overdensity is highly significant, the association of any individual
object to the Magellanic system, as well as the exact distances/spectral types
of individual objects are more uncertain. Spectroscopy of these faint and
crowded objects requires JWST to further constrain their stellar types,
distances, and dynamics, allowing for a more complete picture of this
interesting system.

Acknowledgements: We thank Ian Smail for helpful suggestions regarding the
source detection methods. We also thank Steven Willner and the referee for
helpful suggestions that improved the submitted manuscript. JS acknowledges
support from an undergraduate Arizona NASA Space Grant, Cooperative Agreement
80NSSC20M0041. RAW, SHC, and RAJ acknowledge support from NASA JWST
Interdisciplinary Scientist grants NAG5-12460, NNX14AN10G and 80NSSC18K0200
from GSFC. CJC and NJA acknowledge support from the European Research Council
(ERC) Advanced Investigator Grant EPOCHS (788113). MAM acknowledges the
support of a National Research Council of Canada Plaskett Fellowship, and the
Australian Research Council Centre of Excellence for All Sky Astrophysics in 3
Dimensions (ASTRO 3D), through project number CE17010001. CNAW acknowledges
funding from the JWST/NIRCam contract NASS-0215 to the University of Arizona.
We also acknowledge the indigenous peoples of Arizona, including the Akimel
O’odham (Pima) and Pee Posh (Maricopa) Indian Communities, whose care and
keeping of the land has enabled us to be at ASU’s Tempe campus in the Salt
River Valley, where much of our work was conducted. This work is based on
observations made with the NASA/ESA/CSA James Webb Space Telescope. The data
were obtained from the Mikulski Archive for Space Telescopes (MAST) at the
Space Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127
for JWST. These observations are associated with JWST programs 1176, 2736, and
2738. This work is based on observations taken by the RELICS Treasury Program
(GO 14096) with the NASA/ESA HST, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555.
This work has made use of data from the European Space Agency (ESA) mission
Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing
and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. We also acknowledge the JWST
ERO team responsible for providing these
products.666https://www.stsci.edu/jwst/science-execution/approved-
programs/webb-first-image-observations

Table 1: Candidate LMC stars identified with star-galaxy separation and fit with IRTF spectral templates. | ———————— PHOTOMETRYaaReported photometry was measured as discussed in Section 2.3 on PSF-matched NIRCam mosaics. Apparent magnitude uncertainties are not displayed because they are very small ($\lesssim$$0.01$ mag). All reported magnitudes use the AB system. ———————– | ——– PROPERTIES ——–
---|---|---
id | RA | Dec | F090W | F150W | F200W | F277W | F356W | F444W | Type | $\chi^{2}_{\text{red}}$ |  $\mu$bbFrom the SED fits using templates from the IRTF Spectral Library (Cushing et al., 2005; Rayner et al., 2009). Distance moduli are calculated individually for each filter and averaged, where the lower uncertainty is the 16th percentile, the reported value is the 50th percentile, and the upper uncertainty is the 84th percentile of the cumulative probability distribution for a given star. The uncertainties are added in quadrature with an additional uncertainty of $0.25$ mag due to the SpeX flux calibration uncertainties. Only stars with $18\leq\mu\leq 20$ mag are taken as candidates.
| (deg) | (deg) | (mag) | (mag) | (mag) | (mag) | (mag) | (mag) |  | (mag)
1 | 110.86216700 | -73.47061220 | 24.77 | 24.33 | 24.53 | 25.24 | 25.49 | 25.82 | K7V | 0.66 | $18.1^{+0.3}_{-0.3}$
2 | 110.85557140 | -73.47091030 | 24.95 | 24.69 | 24.97 | 25.58 | 26.05 | 26.56 | K3V | 0.66 | $19.0^{+0.3}_{-0.3}$
3 | 110.71958260 | -73.49692810 | 25.05 | 24.82 | 25.03 | 25.67 | 26.16 | 26.50 | K3V | 0.48 | $19.1^{+0.3}_{-0.3}$
4 | 110.91536500 | -73.45760250 | 26.30 | 25.84 | 26.04 | 26.78 | 26.89 | 27.34 | K7V | 0.89 | $19.4^{+0.4}_{-1.2}$
5 | 110.67205510 | -73.50673290 | 26.63 | 26.24 | 26.26 | 26.70 | 27.31 | 27.31 | M1V | 0.81 | $18.7^{+0.5}_{-1.0}$
6 | 110.70759900 | -73.49949690 | 26.50 | 25.99 | 26.16 | 26.69 | 27.21 | 27.55 | M0V | 0.27 | $19.2^{+0.5}_{-0.7}$
7 | 110.67496690 | -73.50461890 | 25.96 | 25.68 | 25.89 | 26.54 | 26.99 | 27.46 | K3V | 0.19 | $19.9^{+0.3}_{-0.3}$
8 | 110.66512020 | -73.50600780 | 26.55 | 26.06 | 26.24 | 26.79 | 27.15 | 27.29 | M3V | 0.38 | $18.3^{+0.7}_{-0.7}$
9 | 110.70289650 | -73.49699630 | 25.90 | 25.62 | 25.84 | 26.37 | 26.86 | 27.11 | K3V | 0.33 | $19.8^{+0.3}_{-0.4}$
10 | 110.68002770 | -73.50094110 | 24.80 | 24.62 | 24.87 | 25.52 | 26.08 | 26.39 | K2V | 0.52 | $19.3^{+0.3}_{-0.3}$
11 | 110.89040420 | -73.45783110 | 26.02 | 25.52 | 25.64 | 26.25 | 26.63 | 27.08 | M0.5V | 0.30 | $18.6^{+0.4}_{-0.7}$
12 | 110.84920430 | -73.46677800 | 23.09 | 23.07 | 23.39 | 24.00 | 24.48 | 24.91 | G3Va | 0.18 | $18.1^{+1.5}_{-0.4}$
13 | 110.86553090 | -73.46272980 | 25.87 | 25.63 | 25.89 | 26.46 | 26.82 | 27.26 | K3V | 0.36 | $19.9^{+0.3}_{-0.3}$
14 | 110.72000470 | -73.49268410 | 25.59 | 25.15 | 25.32 | 25.84 | 26.24 | 26.62 | M0V | 0.24 | $18.2^{+0.6}_{-0.6}$
15 | 110.66805300 | -73.50092820 | 22.86 | 22.99 | 23.36 | 24.04 | 24.57 | 24.94 | G6.5V | 0.30 | $18.8^{+0.3}_{-0.5}$
16 | 110.72784760 | -73.48935170 | 26.17 | 25.77 | 25.96 | 26.51 | 26.95 | 27.29 | K7V | 0.06 | $19.5^{+0.3}_{-0.3}$
17 | 110.76571260 | -73.48077720 | 25.91 | 25.47 | 25.67 | 26.11 | 26.55 | 26.84 | M1V | 0.35 | $18.2^{+0.7}_{-0.9}$
18 | 110.87489520 | -73.45841050 | 25.19 | 25.08 | 25.36 | 25.99 | 26.40 | 26.88 | K2V | 0.03 | $19.8^{+0.3}_{-0.3}$
19 | 110.73000210 | -73.48620160 | 25.43 | 25.02 | 25.28 | 25.83 | 26.36 | 26.71 | K7V | 0.48 | $19.0^{+0.4}_{-0.3}$
20 | 110.71625750 | -73.48854100 | 26.05 | 25.60 | 25.79 | 26.36 | 26.69 | 26.99 | M0V | 0.47 | $18.6^{+0.7}_{-1.0}$
21 | 110.72057470 | -73.48737060 | 26.53 | 26.16 | 26.36 | 26.91 | 27.34 | 27.80 | K7V | 0.10 | $19.9^{+0.4}_{-0.3}$
22 | 110.80594110 | -73.46958100 | 25.72 | 25.34 | 25.58 | 26.18 | 26.53 | 26.90 | K7V | 0.43 | $19.2^{+0.5}_{-0.3}$
23 | 110.68128930 | -73.49252790 | 25.04 | 24.91 | 25.19 | 25.76 | 26.25 | 26.52 | K2V | 0.38 | $19.5^{+0.3}_{-0.3}$
24 | 110.80106310 | -73.46821900 | 24.89 | 24.63 | 24.82 | 25.40 | 25.87 | 26.26 | K3V | 0.09 | $18.8^{+0.3}_{-0.3}$
25 | 110.75536650 | -73.47725180 | 25.74 | 25.40 | 25.60 | 26.06 | 26.45 | 26.75 | K7V | 0.69 | $19.0^{+0.4}_{-0.8}$
26 | 110.68680600 | -73.49055430 | 26.78 | 26.26 | 26.46 | 26.97 | 27.35 | 28.14 | M0V | 0.82 | $19.8^{+0.4}_{-0.4}$
27 | 110.64580220 | -73.49817030 | 24.37 | 24.30 | 24.52 | 25.16 | 25.70 | 25.98 | K2V | 0.25 | $19.0^{+0.3}_{-0.3}$
28 | 110.82877260 | -73.46026270 | 22.95 | 23.07 | 23.44 | 24.08 | 24.55 | 24.97 | G1V | 0.17 | $18.9^{+0.3}_{-0.5}$
29 | 110.72096780 | -73.48176010 | 25.78 | 25.40 | 25.60 | 26.10 | 26.52 | 26.87 | K7V | 0.21 | $19.1^{+0.3}_{-0.3}$
30 | 110.67001300 | -73.49191860 | 26.37 | 25.94 | 26.09 | 26.62 | 27.05 | 27.44 | M0V | 0.12 | $19.3^{+0.4}_{-0.8}$
31 | 110.64265760 | -73.49445510 | 23.86 | 23.67 | 24.01 | 24.65 | 25.20 | 25.56 | K2V | 0.50 | $18.4^{+0.3}_{-0.3}$
32 | 110.81476720 | -73.45898890 | 23.29 | 23.33 | 23.65 | 24.26 | 24.72 | 25.14 | K0V | 0.13 | $18.5^{+1.4}_{-0.3}$
33 | 110.72384130 | -73.47836710 | 26.10 | 25.76 | 25.97 | 26.51 | 26.98 | 27.40 | K3V | 0.30 | $19.8^{+0.4}_{-0.4}$
34 | 110.63281260 | -73.49283500 | 25.68 | 25.16 | 25.40 | 25.96 | 26.50 | 26.94 | M0V | 0.67 | $18.9^{+0.3}_{-0.3}$
35 | 110.72055620 | -73.47503090 | 24.00 | 23.91 | 24.23 | 24.81 | 25.33 | 25.72 | K1V | 0.10 | $18.7^{+0.3}_{-0.3}$
36 | 110.66272890 | -73.48596090 | 25.92 | 25.65 | 25.85 | 26.44 | 26.96 | 27.41 | K5V | 0.06 | $19.8^{+0.3}_{-0.3}$
37 | 110.65067740 | -73.49085180 | 26.12 | 25.79 | 25.96 | 26.52 | 27.01 | 27.31 | K7V | 0.34 | $19.7^{+0.5}_{-0.3}$
38 | 110.84558040 | -73.44554930 | 24.30 | 24.28 | 24.61 | 25.22 | 25.65 | 26.09 | G3Va | 0.17 | $19.3^{+1.6}_{-0.4}$
39 | 110.80421890 | -73.45313020 | 24.13 | 24.09 | 24.44 | 24.98 | 25.48 | 25.86 | G3Va | 0.23 | $19.1^{+0.4}_{-0.4}$
40 | 110.67477770 | -73.47867250 | 25.45 | 25.14 | 25.37 | 25.95 | 26.45 | 26.82 | K3V | 0.08 | $19.4^{+0.3}_{-0.3}$
41 | 110.63467750 | -73.48880260 | 24.35 | 24.07 | 24.36 | 24.93 | 25.44 | 25.81 | K3V | 0.37 | $18.3^{+0.3}_{-0.3}$
42 | 110.79796380 | -73.45281710 | 26.56 | 25.94 | 26.08 | 26.73 | 27.12 | 27.56 | M1.5V | 0.33 | $19.1^{+0.3}_{-0.7}$
43 | 110.64595860 | -73.48242080 | 22.72 | 22.87 | 23.24 | 23.86 | 24.43 | 24.81 | G1V | 0.15 | $18.7^{+0.3}_{-0.5}$
44 | 110.83642120 | -73.44363830 | 25.95 | 25.57 | 25.77 | 26.27 | 26.58 | 27.00 | K7V | 0.47 | $19.1^{+0.4}_{-1.1}$
45 | 110.76016130 | -73.45909700 | 25.50 | 25.19 | 25.45 | 25.99 | 26.40 | 26.89 | K3V | 0.07 | $19.4^{+0.3}_{-0.4}$
46 | 110.72820980 | -73.46594990 | 24.27 | 24.38 | 24.58 | 25.10 | 25.70 | 26.12 | G3Va | 0.54 | $19.4^{+1.5}_{-0.3}$
47 | 110.78956710 | -73.45340770 | 26.71 | 26.31 | 26.46 | 27.05 | 27.36 | 27.61 | K7V | 0.58 | $19.2^{+0.8}_{-1.1}$
48 | 110.83748180 | -73.44288410 | 25.02 | 24.67 | 24.87 | 25.44 | 25.91 | 26.31 | K7V | 0.28 | $18.6^{+0.4}_{-0.3}$
49 | 110.65875080 | -73.47794840 | 26.18 | 25.71 | 25.84 | 26.43 | 26.88 | 27.12 | M1V | 0.36 | $18.7^{+0.6}_{-0.6}$
50 | 110.78230860 | -73.45339920 | 26.37 | 25.93 | 26.00 | 26.69 | 27.03 | 27.62 | M0V | 0.39 | $19.4^{+0.3}_{-0.3}$
51 | 110.78782340 | -73.45101940 | 23.53 | 23.51 | 23.82 | 24.41 | 24.94 | 25.31 | G3Va | 0.18 | $18.5^{+0.4}_{-0.4}$
52 | 110.85693660 | -73.43664670 | 26.00 | 25.70 | 25.91 | 26.42 | 26.84 | 27.40 | K3V | 0.20 | $19.9^{+0.3}_{-0.4}$
53 | 110.66720140 | -73.47507180 | 25.12 | 24.92 | 25.20 | 25.63 | 26.17 | 26.57 | K3V | 0.84 | $19.2^{+0.3}_{-0.4}$
54 | 110.81224460 | -73.44403790 | 23.88 | 23.77 | 24.05 | 24.59 | 25.04 | 25.46 | K2V | 0.26 | $18.4^{+0.3}_{-0.3}$
55 | 110.66261310 | -73.47706440 | 24.46 | 24.33 | 24.63 | 25.23 | 25.77 | 26.14 | K2V | 0.07 | $19.0^{+0.3}_{-0.3}$
56 | 110.78046660 | -73.45030070 | 26.49 | 26.12 | 26.31 | 26.82 | 27.17 | 27.61 | K7V | 0.25 | $19.8^{+0.3}_{-0.4}$
57 | 110.76385480 | -73.45338740 | 25.74 | 25.40 | 25.62 | 26.17 | 26.64 | 27.07 | K3V | 0.27 | $19.5^{+0.4}_{-0.4}$
58 | 110.79180220 | -73.44578380 | 23.77 | 23.71 | 24.06 | 24.67 | 25.13 | 25.57 | G3Va | 0.37 | $18.7^{+0.4}_{-0.4}$
59 | 110.77367240 | -73.44919120 | 25.65 | 25.25 | 25.47 | 26.12 | 26.56 | 27.03 | K7V | 0.67 | $19.3^{+0.4}_{-0.4}$
60 | 110.77943010 | -73.44784300 | 26.23 | 25.88 | 26.04 | 26.55 | 26.93 | 27.25 | K7V | 0.50 | $19.4^{+0.4}_{-0.8}$
61 | 110.62570660 | -73.47884900 | 22.68 | 22.76 | 23.14 | 23.70 | 24.24 | 24.64 | G2V | 0.13 | $18.2^{+0.6}_{-0.3}$
62 | 110.84937560 | -73.43260770 | 22.94 | 23.00 | 23.39 | 24.10 | 24.47 | 24.88 | G2V | 0.83 | $18.7^{+0.4}_{-0.5}$
63 | 110.75095580 | -73.45261390 | 25.81 | 25.45 | 25.65 | 26.19 | 26.60 | 27.07 | K7V | 0.17 | $19.2^{+0.4}_{-0.3}$
64 | 110.83870650 | -73.43454150 | 25.64 | 25.39 | 25.56 | 26.09 | 26.52 | 26.96 | K3V | 0.33 | $19.5^{+0.3}_{-0.4}$
65 | 110.67197890 | -73.46799340 | 26.70 | 26.22 | 26.38 | 26.86 | 27.31 | 27.90 | M0V | 0.36 | $19.6^{+0.4}_{-0.6}$
66 | 110.61303870 | -73.47875580 | 24.38 | 24.03 | 24.27 | 24.85 | 25.37 | 25.77 | K3V | 0.22 | $18.3^{+0.3}_{-0.4}$
67 | 110.79838860 | -73.43960650 | 26.59 | 26.07 | 26.24 | 26.82 | 27.23 | 27.61 | M0V | 0.37 | $19.1^{+0.5}_{-0.7}$
68 | 110.64934660 | -73.46977780 | 25.55 | 25.21 | 25.47 | 26.01 | 26.55 | 26.97 | K3V | 0.16 | $19.4^{+0.3}_{-0.3}$
69 | 110.81135150 | -73.43491940 | 25.24 | 24.98 | 25.16 | 25.76 | 26.18 | 26.65 | K3V | 0.11 | $19.2^{+0.3}_{-0.3}$
70 | 110.65634220 | -73.46544270 | 25.81 | 25.33 | 25.53 | 26.04 | 26.42 | 26.78 | M1V | 0.31 | $18.1^{+0.7}_{-0.8}$
71 | 110.61594180 | -73.47252690 | 25.59 | 25.32 | 25.55 | 26.07 | 26.55 | 26.83 | K3V | 0.30 | $19.5^{+0.3}_{-0.4}$

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