# Effect of interparticle interaction on motility induced phase separation of
self-propelled inertial disks

Soumen De Karmakar<EMAIL_ADDRESS>Rajaraman Ganesh
<EMAIL_ADDRESS>Institute for Plasma Research, Bhat, Gandhinagar 382428,
Gujarat, India  Homi Bhabha National Institute, Training School Complex,
Anushaktinagar, Mumbai 400094, India

(August 29, 2024)

###### Abstract

Phase diagram of the phenomenon of motility induced phase separation (MIPS)
for a collection of self-propelled interacting disks is explored using
Langevin dynamics simulation with particular emphasis on disk wall softness
and the range of interaction amongst disks. We bring out important changes in
the MIPS phase diagram both due to softness and inertia of the disks.
Specifically, we show that overdamped softer disks phase separate while MIPS
becomes possible only for harder disks in the inertial limit. Unlike most of
the earlier studies on MIPS which consider hard-core disks, our findings may
be directly applicable to soft active matter for a range of biological
systems.

Motility induced phase separation (MIPS) is one of the fundamental properties
of self-propelled or active particles due to their persistent motionCates and
Tailleur (2015); Marchetti  _et al._ (2016). Fundamental criteria behind MIPS
is the blockage of free propulsion during collision. Unlike passive particles,
due to their persistent motion, two self-propelled particles tend to remain
together upon collision, until the self-propulsion direction of one of the
particle turns away from the other due to inherent fluctuation. Above a
minimum value of particle density and persistence time, MIPS is observed in
several experimental systemsButtinoni  _et al._ (2013); Theurkauff  _et al._
(2012); Palacci  _et al._ (2013) as well as in numerical simulationCates and
Tailleur (2015); Redner  _et al._ (2013a); Fily  _et al._ (2014); Bialké  _et
al._ (2013). In some recent studies, the complete phase diagram in the
density-persistence space for two dimensional self-propelled disks have been
obtained in numerical simulationFily  _et al._ (2014); Digregorio  _et al._
(2018); Klamser  _et al._ (2018). Properties of the different phases have been
studied to some extentRedner  _et al._ (2013a); Fily and Marchetti (2012). In
some recent work, various types of alignment mechanism of the motile particles
on MIPS have been studiedPu  _et al._ (2017); Theers  _et al._ (2018). Some of
the alignment mechanism favor MIPSSesé-Sansa  _et al._ (2018); Barré  _et al._
(2015); Peruani  _et al._ (2012), the others suppress MIPSvan der Linden  _et
al._ (2019); van Damme  _et al._ (2019).

Most of the studies, so far, have considered the overdamped limit of the self-
propelled particles. Inertia brings in important differences not only in the
dynamical properties, but also in the structural and steady state properties
such as active temperature and pressureScholz  _et al._ (2018); De Karmakar
and Ganesh (2020); Dauchot and Démery (2019); Das  _et al._ (2019); Caprini
and Marini Bettolo Marconi (2021). A recent numerical studyMandal  _et al._
(2019) of inertial self-propelled disks on MIPS found that MIPS vanishes at
large inertia. Moreover, unlike the overdamped self-propelled disks, where the
low and high density phases possess the same temperature, different
temperature was obtained in the two distinct phases in the inertial limit. The
low density phase remains at the higher temperature and the high density phase
acquires a low temperature.

Despite the important role of inter-particle interaction, most of the studies
of MIPS focus on either excluded volume interaction or sufficiently hard-core
and short ranged repulsion, e.g., Weeks-Chandler-Andersen (WCA)Pu  _et al._
(2017) type interaction, at the walls of the self-propelled particles. Redner
et at.Redner  _et al._ (2013b) demonstrated that a small attractive component
in the interaction potential modify the phases of the self-propelled disks,
from active gel-like state to motility induced clusters. Yan et al.Yan  _et
al._ (2016) demonstrated various collective states by changing the inter-
particle interactions between the Janus spheres. There are numerous examples
of biological active matter ranging from sub-cellular elements to
tissuesKlotsa (2019a), whose walls are sufficiently soft. Hence, the
interaction potential, which models those soft active materials, should have
much softer repulsion at the boundary of the particles compared to WCA type
potential. Furthermore, several propulsion mechanism produces the inter-
particle interaction ranging several particle diametersYan  _et al._ (2016);
Zhang  _et al._ (2021).

In this study, we incorporate finite size self-propelled disks in two
dimensions, interacting with the other disks through modified Yukawa potential
(see Fig. 1 and the corresponding expression given below). Choice of our
interaction potential enable us to control the softness or the stiffness at
the walls of the disks and the range of the interaction by tuning the
stiffness and the strength parameters of the interaction potential. Hence, we
are able to study interacting self-propelled disks, from much softer to a
sufficiently stiffer ones. Moreover, we have the control over the range of the
interaction, from a short range to relatively long range one. Furthermore, we
have considered finite inertia of the self-propelled disks. Controlling the
inertial parameter, we are able to move from a low inertia to sufficiently
large inertial domain. We demonstrate that softness of the interaction and the
inertia modify the phase diagram of our system in several important ways.

We perform the Langevin dynamics simulation of $N$ self-propelled inertial
disks (SPIDs) of uniform mass $m$, moment of inertia $I$, and diameter
$\sigma$. Due to their inherent motility, a self-propulsion direction
$\bm{n}_{i}=(\cos\theta_{i},\sin\theta_{i})$ is associated to each disk.
Dynamics of the center of mass velocities and the orientations
$\\{\textbf{v}_{i},\theta_{i}\\}$ are governed by $N$ translational and
rotational Langevin equations,

$\displaystyle m\dot{\textbf{v}}_{i}+\gamma\textbf{v}_{i}$
$\displaystyle=\textbf{F}_{i}+\textbf{F}^{a}_{i}+\sqrt{2\gamma^{2}D}\bm{\xi}_{i},$
(1) $\displaystyle I\ddot{\theta}_{i}+\gamma_{r}\dot{\theta}_{i}$
$\displaystyle=\sqrt{2\gamma_{r}^{2}D_{r}}\zeta_{i},$ (2)

respectively. The suffix $r$ denotes rotational parameters. $\gamma$,
$\gamma_{r}$ are dissipation coefficients. We assume diffusion coefficients,
$D$ and $D_{r}$, obey fluctuation-dissipation relation, and they are coupled
as $D=\sigma^{2}D_{r}$Löwen (2020). $\bm{\xi}_{i}$ and $\zeta_{i}$ are the
white Gaussian noises. The disks self-propel with a constant propulsion speed
$\textup{v}_{0}$ along the propulsion direction $\bm{n}_{i}(t)$. The
associated self-propulsion force is
$\textbf{F}_{i}^{a}(t)=\gamma\textup{v}_{0}\bm{n}_{i}(t)$. A disk interacts
with the other disks with purely repulsive pairwise force ($\textbf{F}_{ij}$)
such that the total conservative potential energy of the system is
$U=\displaystyle{\sum_{i<j}}V_{0}\frac{e^{-(r_{ij}-\sigma)/\lambda}}{r_{ij}}$.
By adjusting the interaction parameters $\lambda$ and $V_{0}$, the softness
and the range of interaction of the SPIDs is varied (see Fig. 1). We consider
$\sigma$, $1/D_{r}$, and the background thermal energy $k_{B}T$ as the unit of
normalization for length, time, and energy, respectively. $N$ normalized
dynamical equations for the self-propelled inertial disks are:

$\displaystyle M\dot{\textbf{v}}_{i}+\textbf{v}_{i}$
$\displaystyle=-\bm{\nabla}_{i}\Gamma\sum_{i<j}\frac{e^{-\kappa(r_{ij}-1)}}{r_{ij}}+\textup{P}_{e}\textbf{n}_{i}+\sqrt{2}\bm{\xi}_{i},$
(3) $\displaystyle J\ddot{\theta}_{i}+\dot{\theta}_{i}$
$\displaystyle=\sqrt{2}\zeta_{i}.$ (4)

$\bm{\nabla}_{i}$ is the gradient operator at the location of the $i$’th
particle $\bm{r}_{i}$. Inertial parameter $M=\frac{m/\gamma}{1/D_{r}}$ is the
ratio of inertial time scale ($m/\gamma$) to the persistence time scale
($\tau_{p}=1/D_{r}$). Rotational inertial parameter
$J=\frac{I/\gamma_{r}}{1/D_{r}}$ is the ratio of rotational inertial time
scale ($I/\gamma_{r}$) to the persistence time scale. Peclet number
$\textup{P}_{e}=\textup{v}_{0}\tau_{p}/\sigma$ is the ratio of persistence
length ($l_{p}=\textup{v}_{0}\tau_{p}$) to the the diameter of the self-
propelled disks. Reduced interaction parameters are
$\Gamma=\frac{V_{0}/\sigma}{k_{B}T}$, $\kappa=\sigma/\lambda$. All subsequent
results are provided in the reduced units.

We plot the interaction potential for $\Gamma=25$ in Fig. 1(a), and
$\kappa=14$ in Fig. 1(b) with the solid lines. For comparison, WCA interaction
with unit strength is plotted with the dashed lines.

Figure 1:  Plot of pairwise interaction ($U(r_{ij})$) for (a) $\Gamma=25.0$
and several values of $\kappa$, (b) $\kappa=14.0$ and several values of
$\Gamma$, shown in the legend. WCA potentialPu  _et al._ (2017); Mandal  _et
al._ (2019) with unit strength is shown by the dashed lines.

The softness of the interaction matches to that of WCA interaction softness
for $\kappa\approx 18$. For relatively low values of $\kappa$, the disks
become soft. We call $\kappa$ as the softness or the stiffness parameter.
Moreover, with decreasing $\kappa$, the range of interaction increases. For a
fixed stiffness, the strength of the potential $\Gamma$ dictates the effective
size of the disks.

We consider $N=48400$ SPIDs in a two-dimensional rectangular simulation box of
dimensional ratio $L_{x}/L_{y}=2/\sqrt{3}$ and perform Langevin dynamics
simulation with the upgraded GPU based Molecular Dynamics solver MPMDDe
Karmakar and Ganesh (2020). Packing fraction, $\phi=N\pi/4L_{x}L_{y}$, and
Peclet number are fixed at $\phi=0.5$ and $\textup{P}_{e}=75$, respectively,
unless, otherwise, specified. We have checked that the considered values of
$\phi$ and $\textup{P}_{e}$ are sufficient to produce MIPS in the overdamped
limit and for sufficiently stiff interaction parameter ($\kappa$), tending
towards WCA interactionMandal  _et al._ (2019). We consider doubly periodic
boundary conditions. Interaction cut-off is set at $r_{c}=8$. Integration time
step is fixed at $\Delta t=10^{-4}$, which gives good energy conservation in
the steady state for the whole parameter space that we have considered.

In Fig. 2, we plot the phase diagram of our system of inertial SPIDs in the
$\kappa-\Gamma$ space. The rotational inertial parameter $J$ is fixed at
$J=0.01$. $M$ is varied from $M=0.05$ (Fig. 3(a)) to $M=0.005$ (Fig. 3(b))
through $M=0.01$ (Fig. 3(c)) from left to right.

Figure 2:  Phase diagram in the $\kappa-\Gamma$ space for (a) $M=0.05$, (b)
$M=0.01$, and (c) $M=0.005$. The value of $J$ is fixed at $J=0.01$. The
magenta and blue dots denote the MIPS and homogeneous phases, respectively.

Blue and magenta dots indicate the homogeneous and phase separated states,
respectively. We observe a parabola shaped phase boundary separating the MIPS
and homogeneous regions over an order of magnitude variation in $M$. We
measure the phase separation from the two distinct peaks in the area fraction
distribution across the mean value, which is fixed at $\phi=0.5$. In Fig.
3(a), we plot the distribution of local area fraction for $\Gamma=25$, and
several values of $\kappa$, shown in the legend for $M=0.05$, $J=0.01$. The
single peaked distribution around $\phi=0.5$ becomes doubly peaked for
$\kappa\geq 12$. Unlike passive disks, the distribution is stretched along the
$\phi$ axis due to self-propulsion of the disks (not shown). Self-propulsion
makes the disks explore the extreme density ranges, overcoming the effective
size (see Fig. 5(a)) set by the interaction potential. In Figs. 3(b), (c), we
plot the configuration of the system in the homogeneous and the phase
separated phases at $\kappa=6$ and $\kappa=16$, respectively, corresponding to
Fig. 3(a). The color bars denote the local orientational orderBriand  _et al._
(2018); Bialké  _et al._ (2012) of the disks, given by

$q_{6}(i)=\frac{1}{6}\sum_{j=1}^{6}e^{i6\theta_{ij}}.$ (5)

$\theta_{ij}$ is the angle between the $i$ and $j$ disks with respect to some
arbitrary fixed axis. The ordered region in Fig. 3(c), which corresponds to
the high density peak in the local density distribution, is surrounded by low
density disordered region. The homogeneous phase in Fig. 3(b) has no order.
Near the phase boundary, in the homogeneous side of the phase diagram, we
observe few relatively high density scattered regions, which do not grow in
the simulation time to produce two distinct peaks in the distribution of local
area fraction (not shown). Rather, those regions vanish and emerge at some
other location with time, and maintain an overall dynamic homogeneous phase.

Figure 3:  (a) Distribution of local area fraction for various values of
$\kappa$, shown in the legend, and fixed $\Gamma=25$. Configuration plots of
the homogeneous (b), and phase separated (c) systems, corresponding to
$\kappa=6$ (soft) and $\kappa=16$ (stiff), respectively, of (a). Local
orientational order at the location of the disks are shown in the color bars
(see the main text for the definition of $q_{6}(i)$). The inertial parameter
values are fixed at $M=0.05$, and $J=0.01$.

We plot the distribution of the local area fraction for $J=0.0001$ and $J=0.1$
by the solid and dashed lines, respectively, in Fig. 4. Three different values
of $\Gamma$ are shown in the legend. The other parameter values are $M=0.05$
and $\kappa=12$. The dashed lines for $J=0.1$ are shifted along the $\phi$
axis for better visibility. Finite relaxation time of rotational fluctuations
of self-propulsion direction associated with the sufficiently large values of
$J$ increases the persistence of the disks. Higher persistence helps in MIPS.
Hence, the peaks are more prominent for $J=0.1$. For the remainder of the
work, we fix the value of $J$ at $J=0.01$. In this work we do not study the
effect of $J$ further.

Figure 4:  Distribution of local area fraction for $J=0.0001$ (solid lines)
and $J=0.1$ (dashed lines) for three different values of $\Gamma$, shown in
the legend. Inertial and softness parameters are fixed at $M=0.05$, and
$\kappa=12$, respectively. Dashed lines are shifted along the $\phi$ axis by
$1.5$ units, for better visibility.

As $M$ is reduced, the phase boundary moves towards the left along the
$\kappa$ axis (see Fig. 2). Hence, in the non-inertial or overdamped limit
($M\rightarrow 0$), relatively softer self-propelled disks phase separate into
high density motility induced clusters and a low density phase. At moderate
inertia, sufficiently stiffer disks phase separate into MIPS and low density
phases; softer disks do not phase separate. At large inertia, $M\geq 0.1$, we
do not observe any phase separation in the $\kappa-\Gamma$ plane, and the
system remains homogeneous. For a fixed $\kappa$, with increase in $\Gamma$,
the system re-enters the homogeneous phase. In the remainder of the work we
investigate the reason behind the above observations.

Sufficiently large density, persistence, and repulsion at the surface of the
self-propelled particles are the key ingredients for MIPS. During the
collision of two disks, surface interaction of the two disks provide repulsion
to block the free movement. Unlike passive disks, due to persistent self-
propulsion, two colliding disks form local cluster, hence, provide local
nucleation site for cluster growth. Within a persistence time
($\tau_{p}=1/D_{r}$), if other particles collide with the two-particle-
cluster, the cluster grows. For a sufficiently large cluster, disks at the
boundary move out of the cluster, when the rotational fluctuations take the
self-propulsion direction away form the cluster. A dynamical equilibrium
constraint the size of the cluster depending upon the density and the
persistence of the disks. During the cluster growth phase, many such clusters
are found to merge into a high density cluster, through several mechanism, and
the result is a complete phase separation of high density cluster and a low
density phase, known as MIPS in literatureCates and Tailleur (2015).

Figure 5:  Diagram depicting the collision process of two soft SPIDs with
large (left column) and small (right column) value of inertial parameter. Due
to the high kinetic energy, when two soft inertial disks (left column) with
large impact parameter $b$ collide, the shape is deformed and they move apart
with minor change in the propulsion directions. However, when two soft disks
with small inertial value $M$ collide with a large impact parameter $b$, the
shape is deformed but they stay together, due to their low kinetic energy,
during their persistence time. Hence, provide local nucleation sites for
cluster formation.

In the overdamped limit, due to negligible inertia, the momentum relaxation
time scale ($m/\gamma$) becomes very short. Consequently, the velocity of the
particles quickly relaxes towards the self-propulsion direction. Also, the
speed or the kinetic energy of the disks is small. On the other hand, in the
inertial limit, finite momentum relaxation time sets a natural time delay to
the velocity relaxation along the self-propulsion direction. The kinetic
energy of the inertial disks are much higher compared to the non-inertial
counterpart. Hence, when two disks with large value of inertial parameter $M$
collide, they bounces back and forth, and take finite time before they slow
down sufficiently to facilitate motility induced cluster formation. At large
inertia, when the momentum relaxation time scale is comparable to the
persistence time scale, nucleation sites with two disks form rarely. Hence, we
do not observe MIPS at large $M$ (e.g., $M\geq 0.1$) (not shown). However, due
to short relaxation time, instead of bouncing back, disks with low value of
$M$ slow down quickly. Stiffer walls at large $\kappa$ provide necessary
blockage on the free movement of the disks, without any significant
deformation in the shape of the disks during collision. Softer disks at small
$\kappa$ deform more. Head-on collision of two soft inertial disks still
hinder the propulsion of each other. However, when two soft inertial disks
approach each other with a large impact parameter ($0\ll b<\sigma/2$, see Fig.
5, where we draw a diagram, depicting collision process of two soft SPIDs with
large and small $M$), due to large kinetic energy and small stiffness, the
shape of the two disks deform, and move apart with minor change in the self-
propulsion directions, without forming local two-particle-cluster. Hence, at
relatively large inertia $M=0.05$, we do not observe phase separation for
relatively softer inertial disks in the intermediate $\Gamma$ range (e.g.,
$6\lesssim\kappa<12$, $10\lesssim\Gamma\lesssim 100$). However, in the small
inertial limit, due to low kinetic energy of the disks, softer disks can still
provide sufficient repulsion to the colliding disks with large impact
parameter ($0\ll b<\sigma/2$) (see Fig. 5). Hence, MIPS is observed for
sufficiently softer disks in the intermediate $\Gamma$ range (e.g,
$\kappa\simeq 8$, $10\leq\Gamma\leq 250$) with low inertial value, $M=0.005$.

Deep inside the homogeneous region of the phase diagram of our system of
inertial self-propelled disks, we plot the radial distribution function in
Fig. 6(a) and the diffusion coefficient with time in Fig. 6(b) for various
values of $\Gamma$, shown in the legend of RDF plot. The solid lines denote
the self-propelled disks with $\textup{P}_{e}=75$, and the dashed lines denote
the passive disks at $\textup{P}_{e}=0$Charan (2017). For better visibility,
the dashed lines in Fig. 6(a) are shifted by 12.5 units along the y-axis.
Inertial parameter is fixed at $M=0.05$, and the stiffness parameter is fixed
at $\kappa=8$. The other parameter values are considered as before.

Figure 6:  Plot of radial distribution function (a), and diffusion coefficient
with time (b) for various values of $\Gamma$ shown in the legend of (a). Solid
lines denote the self-propelled disks with $P=75$, and the dashed lines denote
the passive disks at $P=0$Charan (2017). For better visibility, the dashed
lines in (a) are shifted along y-axis by 12.5 units. The values of $M$ and
$\kappa$ are $0.05$ and $8$, respectively.

The values of the diffusion coefficient are orders of magnitude lower for the
passive disks, compared to their self-propulsive counterpart. Moreover, the
diffusion coefficient decreases (beyond the ballistic region) with time for
$\Gamma\geq 25$ for the passive disks. The first peak of the radial
distribution function of passive disks for $\Gamma=1$ is at $r\approx 1$, and
the location of the first peak increases with increasing $\Gamma$. The second
peak of the radial distribution of the passive disks starts to split into two
at $\Gamma=25$. Furthermore, at low stiffness with increasing $\Gamma$, the
interaction becomes increasingly longer in range (see Fig. 1). Hence, the
system of passive disks transforms from a homogeneous liquid at low $\Gamma$
to an ordered structure at large $\Gamma$, as observed earlier. On the other
hand, the system of self-propelled disks remain diffusive for all considered
values of $\Gamma$. Due to the persistence of the self-propulsive disks, the
location of the first peak is at a distance much less than the diameter of the
disks for $\Gamma=1$. The location of the first peak increases with $\Gamma$,
and nearly equals to that of the passive disks at $\Gamma=500$. The second
peak at $\Gamma=500$ for the self-propelled disks breaks into two. Moreover,
there is a density constraint. Hence, at large $\Gamma$, the system of self-
propelled disks forms homogeneous ordered state. But, unlike passive disks,
the ordered system of self-propelled disks at large $\Gamma$ are diffusive. At
larger $\kappa$, the RDF peaks of the self-propelled disks gradually become
more prominent (not shown). The location of the first peak at small values of
$\Gamma$ is at $r<1$ (not shown). Therefore, the size of the two-body-clusters
are so small that the other disks do not see them because of the constraint of
density in the persistence time scale. Hence, the two-body-clusters do not
grow much and we do not observe any phase separation at small $\Gamma$ (e.g.,
at the location of $\kappa=12$, $\Gamma\leq 10$, for $M=0.05$ in Fig. 2(a)).
At large values of $\Gamma$, the effective diameter of the self-propelled
disks (we denote the location of the first peak in the RDF as the effective
diameter of the disks) are sufficiently big. Hence, the system re-enters the
homogeneous ordered phase (e.g., at the location of $\kappa=12$, $\Gamma\geq
100$, for $M=0.05$ in Fig. 2(a)), due to fixed average density of the disks.
Furthermore, for softer disks at large $\Gamma$, due to relative long range
nature of the interaction, MIPS has not been observed (e.g., $\kappa<8$,
$\Gamma\gtrsim 100$, $M=0.05$). Similar explanation works for low inertial
disks. Hence, we get a parabola shaped phase boundary between homogeneous and
the MIPS state for all the inertial parameter values that we incorporate.

In this work, we demonstrate that relatively softer disks phase separate due
to motility in the inertia-less limit. With increasing inertia, MIPS is
observed for self-propelled disks with much stiffer interaction. Moreover,
stronger repulsion at larger interaction strength increases the effective
diameter of the disks. Hence, the system re-enters the homogeneous ordered
state at large interaction strength. Furthermore, long range interaction helps
to homogenize the system of self-propelled disks. Hence, long range
interactions diminishes MIPS phase space.

It would be interesting to study the structural and dynamical properties of
the high and low density phases separately in the soft interaction limit.
Janus particles typically has size of $\sigma\sim$
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and move at a speed of $v\sim$
$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}\sec^{-1}$ in a background
like water at room temperatureBechinger  _et al._ (2016). Hence, Reynolds
number ($\textup{Re}=$ inertia forces/viscous forces)Klotsa (2019b) becomes
$\textup{Re}\sim 10^{-3}$, which is an overdamped limit. Re of the system can
be increased few orders of magnitude either by decreasing the background
density, e.g., micron sized particles in activated complex plasma move at high
speed in the low density backgroundBechinger  _et al._ (2016); Nosenko  _et
al._ (2020); Arkar  _et al._ (2021), or by increasing the propulsion speed of
the motile particles by several mechanisms(Dietrich  _et al._ , 2020; Bourgoin
_et al._ , 2020). Consequently, inertial self-propelled system can be
realized. We believe that the present work should be able to bring attention
to the novel active matter category of activated complex plasma that could
find numerous applications of active materials in the less explored plasma
environment. Furthermore, biological materials, e.g., cells, tissues, etc.,
are much softer in nature compared to artificial self-propelled Janus
particles. Our work should help further in understanding the properties of
biological active matter.

One of the authors (S. D. K.) would like to thank Devshree Mandal for her help
in generating the diagram in Fig. 5. Numerical simulations for this work has
been performed on GPU nodes of the ANTYA cluster at Institute for Plasma
Research, India.

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# Generalized BF state in quantum gravity

Shinji Yamashita111 Email<EMAIL_ADDRESS>, Satoshi Yajima, and Makoto
Fukuda
Department of Physics, Kumamoto University, Kumamoto 860-8555, Japan

###### Abstract

The BF state is known as a simple wave function that satisfies three
constraints in canonical quantum gravity without a cosmological constant. It
is constructed from a product of the group delta functions. Applying the
chiral asymmetric extension, the BF state is generalized to the state for real
values of the Barbero–Immirzi parameter.

##  1 Introduction

In modern canonical quantum gravity, the connections with the Barbero–Immirzi
parameter $\beta$ play the role of fundamental variables [1, 2, 3]. Wave
functions are required to solve the three constraints, i.e., Gauss,
diffeomorphism, and Hamiltonian constraints.

The Chern–Simons (CS) state, which is also called the Kodama state, is known
as an exact solution of these three constraints with a cosmological constant
[4]. In this case, the configuration variable is a complex
$sl(2,{\mathbb{C}})$-valued connection, which takes a left- or right-handed
form, namely, $\beta=\pm{\rm i}$ for the Lorentzian case. However, loop
quantum gravity (LQG) proposes that the Barbero–Immirzi parameter takes real
values for several technical reasons. The real value of $\beta$ gives a real
$su(2)$-valued connection, but it makes the Hamiltonian constraint more
complicated.

The generalization of the CS state to real values of $\beta$ was achieved by
Randono [5, 6, 7]. The generalized states are parameterized by the Levi-Civita
curvature, and solve some difficulties of the ordinary CS state, e.g., the
normalizability and the reality conditions.

On the other hand, the wave function without a cosmological constant was found
by Miković [8, 9]. It is called the BF state here. This state is given as a
product of the group delta functions of the curvature, and constructed from
the left-handed complex connection as well as the ordinary CS state. In this
paper, the generalization of the BF state is considered as an analog of the
generalization of the CS state. We would like to emphasize that the process of
the generalization follows Refs. [5, 6, 7]. Specifically, it is carried out
via the chiral asymmetric extension.

In Sect. 2, we briefly review the BF state for $\beta={\rm i}$. Then, using
the chiral asymmetric model, the BF state is extended to the case of generic
purely imaginary values of $\beta$. In Sect. 3, the BF state is extended
further to the case of generic real values of $\beta$. This state is expressed
in terms of the real $su(2)$-valued connection and the Levi-Civita curvature.
Making use of the appropriate inner product, three constraints with real
values of $\beta$ are solved. In Sect. 4, we present the conclusions and
discuss the results.

##  2 BF state and chiral asymmetric extension

###  2.1 BF state

The three constraints of canonical quantum gravity without a cosmological
constant can be derived from the Holst action [10]

$S_{\rm H}=\frac{1}{4k}\int\left(\epsilon_{IJKL}\,e^{I}\wedge
e^{J}\wedge\Omega^{KL}-\frac{2}{\beta}\,e^{I}\wedge
e^{J}\wedge\Omega_{IJ}\right)\ ,$ (1)

where $k=8\pi G$, $e^{I}$ is the tetrad, and $\Omega={\rm
d}\omega+\omega\wedge\omega$ is the curvature of the spin connection
$\omega^{IJ}$. Capital Latin indices $I,J,\dots$ are used as Lorentz indices.
Performing the Legendre transformation, one obtains

$S_{\rm H}=\frac{1}{k\beta}\int{\rm
d}^{4}x\left(E_{i}^{a}\dot{A}^{(\beta)}{}_{a}^{i}+\lambda^{i}G_{i}+N^{b}V_{b}+NH\right)\
,$ (2)

where $\dot{A}^{(\beta)}{}_{a}^{i}={\cal L}_{t}A^{(\beta)}{}_{a}^{i}$ ,
$\lambda^{i},N^{a}$, and $N$ are Lagrange multipliers, and $G_{i},V_{b}$, and
$H$ are Gauss, diffeomorphism, and Hamiltonian constraints respectively.
Letters $i,j,\dots$ and $a,b,\dots$ denote 3D internal and spatial indices,
respectively. The configuration variable
$A^{(\beta)}{}_{a}^{i}=\Gamma_{a}^{i}+\beta K_{a}^{i}$ is constructed from the
Levi-Civita spin connection $\Gamma_{a}^{i}$ and the extrinsic curvature
$K_{a}^{i}$. Choosing the time gauge $e_{a}^{I}|_{I=0}=0$, the canonical
momentum variable can be written as $E_{i}^{a}=\det(e_{b}^{j})e_{i}^{a}$.

For $\beta={\rm i}$, the action (1) can be written only with the left-handed
variables:

$S_{\rm H}^{(+)}=\frac{{\rm i}}{k}\int\Sigma^{(+)IJ}\wedge\Omega_{IJ}^{(+)}\
,$ (3)

where

$\Sigma^{(+)IJ}=\frac{1}{2}\left(e^{I}\wedge e^{J}-\frac{{\rm
i}}{2}\epsilon^{IJ}{}_{KL}e^{K}\wedge e^{L}\right)\ ,$ (4)

and

$\Omega_{IJ}^{(+)}=\frac{1}{2}\left(\Omega_{IJ}-\frac{{\rm
i}}{2}\epsilon_{IJ}{}^{KL}\Omega_{KL}\right)\ .$ (5)

The sign $(+)$ explicitly denotes that the variable is left-handed, namely,
$\beta={\rm i}$. The three constraints are

$\displaystyle G_{i}^{(+)}$ $\displaystyle=$
$\displaystyle(D_{a}^{(+)}E^{(+)a})_{i}=\partial_{a}E^{(+)}{}_{i}^{a}+\epsilon_{ij}{}^{k}A^{(+)}{}_{a}^{j}E^{(+)}{}_{k}^{a}\
,$ (6) $\displaystyle V_{b}^{(+)}$ $\displaystyle=$ $\displaystyle
E^{(+)}{}_{i}^{a}F^{(+)}{}_{ab}^{i}\ ,$ (7) $\displaystyle H^{(+)}$
$\displaystyle=$ $\displaystyle-\frac{{\rm
i}}{2\sqrt{|\det(E^{(+)})}|}\epsilon^{ijk}E^{(+)}{}_{i}^{a}E^{(+)}{}_{j}^{b}F^{(+)}_{ab\,k}\
,$ (8)

where
$E^{(+)}{}_{i}^{a}=\epsilon^{abc}\epsilon_{ijk}\Sigma^{(+)}{}_{bc}^{jk}$, and
$F^{(+)}{}_{ab}^{i}$ is the curvature of the connection
$A^{(+)}{}_{a}^{i}=\Gamma_{a}^{i}+{\rm i}K_{a}^{i}$. The wave function has to
satisfy the quantized constraints, which are formally written as

$\hat{G}^{(+)}_{i}\Psi=\hat{V}^{(+)}_{b}\Psi=\hat{H}^{(+)}\Psi=0\ .$ (9)

In Ref. [8], it is suggested that the product of the group delta functions

$\Psi_{\rm BF}(A^{(+)})=\prod_{x\in\Sigma}\prod_{a,b}\delta\left({\rm
e}^{F^{(+)}_{ab}(x)}\right)$ (10)

is a solution of the constraints. This state is originally derived from the
formal integral $\int{\cal DB}\,\exp[\,{\rm i}S_{\rm BF}\,]=\delta({\rm
e}^{F})$, where $S_{\rm BF}=\int_{\Sigma}{\rm Tr\,}(B\wedge F)$ is the $SU(2)$
BF action in 3D Euclidean space $\Sigma$. Thus let us call state (10) the BF
state. The group delta function has the following properties:

$\displaystyle\delta\left(g\,{\rm e}^{F^{(+)}_{ab}}g^{-1}\right)$
$\displaystyle=$ $\displaystyle\delta\left({\rm e}^{F^{(+)}_{ab}}\right)\ ,$
(11) $\displaystyle F^{(+)}_{ab}\delta\left({\rm e}^{F^{(+)}_{ab}}\right)$
$\displaystyle=$ $\displaystyle 0\ ,$ (12)

where $g$ is an element of the gauge group. Therefore the state $\Psi_{\rm
BF}(A^{(+)})$ is gauge invariant and $\hat{V}^{(+)}_{b}\Psi_{\rm
BF}(A^{(+)})=\hat{H}^{(+)}\Psi_{\rm BF}(A^{(+)})=0$. This state is proposed as
a tool to construct a flat vacuum state [9].

###  2.2 Chiral asymmetric extension

Following the strategy of Ref. [6], we first consider the chiral asymmetric
model with purely imaginary values of $\beta$. The left-handed action (3) is
extended to the chiral asymmetric one as follows:

$\displaystyle S$ $\displaystyle=$ $\displaystyle\alpha^{(+)}S_{\rm
H}^{(+)}+\alpha^{(-)}S_{\rm H}^{(-)}$ (13) $\displaystyle=$
$\displaystyle\frac{1}{4k}\int\biggl{[}\left(\alpha^{(+)}+\alpha^{(-)}\right)\epsilon_{IJKL}\,e^{I}\wedge
e^{J}\wedge\Omega^{KL}+2{\rm
i}\left(\alpha^{(+)}-\alpha^{(-)}\right)e^{I}\wedge
e^{J}\wedge\Omega_{IJ}\biggr{]}.$

Here $\alpha^{(+)}$ and $\alpha^{(-)}$ are mixing parameters of the left- and
right-handed components. The sign $(-)$ means right-handed, i.e., $\beta=-{\rm
i}$. To identify the action (13) with (1), the following identities are
obtained:

$\alpha^{(+)}+\alpha^{(-)}=1\ ,\hskip
14.22636pt\alpha^{(+)}-\alpha^{(-)}=\frac{{\rm i}}{\beta}\ .$ (14)

Note that, in the case of the left-handed action, these parameters take
$\alpha^{(+)}=1$ and $\alpha^{(-)}=0$. One can find that imaginary $\beta$
controls the degree of the chiral asymmetry. In this model, the Poisson
brackets of the canonical variables $(A^{(+)},E^{(+)})$ and
$(A^{(-)},E^{(-)})$ are

$\left\\{A^{(\pm)}{}_{a}^{i}(x),E^{(\pm)}{}_{j}^{b}(y)\right\\}=\pm\frac{{\rm
i}k}{\alpha^{(\pm)}}\,\delta_{j}^{i}\delta_{a}^{b}\delta^{3}(x-y)\ .$ (15)

Here $E^{(+)}{}_{i}^{a}=E^{(-)}{}_{i}^{a}=\det(e)e_{i}^{a}$ in the time gauge
$e_{a}^{0}=0$; nevertheless, these variables are treated independently of each
other. Each of the three constraints separates into left- and right-handed
components independently. Therefore, the extended wave function is given by

$\Psi(A^{(+)},A^{(-)})=\prod_{x\in\Sigma}\prod_{a,b}\delta\left({\rm
e}^{\alpha^{(+)}F_{ab}^{(+)}(x)}\right)\delta\left({\rm
e}^{-\alpha^{(-)}F_{ab}^{(-)}{(x)}}\right)\ .$ (16)

##  3 Generalized BF state

###  3.1 Real values of $\beta$

To consider the extended BF state for generic real values of $\beta$, new
configuration variables are introduced:

$\displaystyle A^{(-\frac{1}{\beta})}{}_{a}^{i}$ $\displaystyle=$
$\displaystyle\alpha^{(+)}A^{(+)}+\alpha^{(-)}A^{(-)}=\Gamma_{a}^{i}-\frac{1}{\beta}K_{a}^{i}\
,$ (17) $\displaystyle A^{(\beta)}{}_{a}^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha^{(+)}-\alpha^{(-)}}\left(\alpha^{(+)}A^{(+)}-\alpha^{(-)}A^{(-)}\right)=\Gamma_{a}^{i}+\beta
K_{a}^{i}\ .$ (18)

The corresponding momentum variables are

$\displaystyle C_{i}^{a}$ $\displaystyle=$ $\displaystyle\frac{1}{2{\rm
i}}\left(E^{(+)}{}_{i}^{a}-E^{(-)}{}_{i}^{a}\right)=\epsilon^{abc}e_{bi}e_{c}^{0}\
,$ (19) $\displaystyle E_{i}^{a}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(E^{(+)}{}_{i}^{a}+E^{(-)}{}_{i}^{a}\right)=\det(e)e_{i}^{a}\
.$ (20)

One can obtain the Poisson bracket relations as follows:

$\displaystyle\left\\{A^{(-\frac{1}{\beta})}{}_{a}^{i}(x),C_{j}^{b}(y)\right\\}$
$\displaystyle=$ $\displaystyle k\delta_{j}^{i}\delta_{a}^{b}\delta^{3}(x-y)\
,$ (21) $\displaystyle\left\\{A^{(\beta)}{}_{a}^{i}(x),E_{j}^{b}(y)\right\\}$
$\displaystyle=$ $\displaystyle
k\beta\delta_{j}^{i}\delta_{a}^{b}\delta^{3}(x-y)\ .$ (22)

To construct the generalized BF state, we attempt to define the extended BF
(EBF) action

$S_{\rm EBF}=\int{\rm Tr\,}\left[\alpha^{(+)}e^{(+)}\wedge
F^{(+)}-\alpha^{(-)}e^{(-)}\wedge F^{(-)}\right]\ ,$ (23)

where $e^{(\pm)}$ are triads playing the role of the $B$ field of the BF
action and are written as

$e^{(\pm)}{}_{a}^{i}=e_{a}^{i}=\frac{1}{2\sqrt{|\det(E)|}}\epsilon_{abc}\epsilon^{ijk}E_{j}^{b}E_{k}^{c}\
.$ (24)

The action $S_{\rm EBF}$ is expressed in terms of the variables
$A^{(-\frac{1}{\beta})}$ and $A^{(\beta)}$ as

$\displaystyle S_{\rm EBF}$ $\displaystyle=$ $\displaystyle\frac{{\rm
i}}{\beta}\int{\rm Tr\,}\left[e\wedge\left(F-\left(1+\beta^{2}\right)K\wedge
K\right)\right]$ (25) $\displaystyle=$ $\displaystyle\frac{{\rm
i}}{\beta}\int{\rm
Tr\,}\left[e\wedge\left(\left(1+\frac{1}{\beta^{2}}\right)R-\frac{1}{\beta^{2}}F-\beta{\rm
d}_{\Gamma}K\right)\right]\ ,$

where $F$ and $R$ are the curvatures of the connections $A^{(\beta)}$ and
$\Gamma$ respectively, and ${\rm d}_{\Gamma}K={\rm d}K+[\Gamma,K]$. The last
term vanishes for the torsion-free condition ${\rm d}_{\Gamma}e=0$.

One can propose an extended BF state defined in the following form:

$\displaystyle\Psi(A^{(\beta)},A^{(-\frac{1}{\beta})})$ $\displaystyle=$
$\displaystyle\int{\cal D}e\ \exp\left[\frac{{\rm
i}}{\alpha^{(+)}-\alpha^{(-)}}\ S_{\rm EBF}\right]$ (26) $\displaystyle=$
$\displaystyle\prod_{x\in\Sigma}\prod_{a,b}\delta\left(\exp\left[\left(1+\frac{1}{\beta^{2}}\right)R_{ab}(x)-\frac{1}{\beta^{2}}F_{ab}(x)\right]\right)\
.$

Note that when $\beta={\rm i}$, state (26) keeps the ordinary form (10). This
state has a problem. Due to the gauge fixing $e_{a}^{0}=0$, the wave function
should satisfy the additional constraint:

$\hat{C}_{i}^{a}\Psi=-{\rm i}k\frac{\delta}{\delta
A^{(-\frac{1}{\beta})}{}_{a}^{i}}\Psi=0\ .$ (27)

This equation implies that the wave function does not depend on the variable
$A^{(-\frac{1}{\beta})}$. However, the connection $\Gamma$ included in the
curvature $R$ is the explicit function of both $A^{(\beta)}$ and
$A^{(-\frac{1}{\beta})}$, namely,

$\Gamma_{a}^{i}=\frac{A^{(\beta)}{}_{a}^{i}+\beta^{2}A^{(-\frac{1}{\beta})}{}_{a}^{i}}{1+\beta^{2}}\
.$ (28)

To avoid this problem, we regard the connection $\Gamma$ as the explicit
variable of $E$, instead of $A^{(\beta)}$ and $A^{(-\frac{1}{\beta})}$. This
can be done via the torsion-free condition ${\rm d}_{\Gamma}e=0$. Taking this
modification into account, the extended BF state is redefined:

$\Psi_{R}(A^{(\beta)})=\prod_{x\in\Sigma}\prod_{a,b}\delta\left(\exp\left[\left(1+\frac{1}{\beta^{2}}\right)R_{ab}(x)-\frac{1}{\beta^{2}}F_{ab}(x)\right]\right)\
.$ (29)

Although the state $\Psi_{R}(A^{(\beta)})$ has the same form as (26), it is
the explicit function of $A^{(\beta)}$ only, and is parameterized by the Levi-
Civita curvature $R$. It is an analog of the fact that the ordinary wave
function $\Psi_{p}(x)=\exp[\,-{\rm i}\,(Et-{\bf p}\cdot{\bf x})\,]$ can be
regarded as the position function parameterized by the momentum.

###  3.2 Constraints and inner products

Here, we confirm whether state (29) satisfies the three constraints with real
values of $\beta$. The Gauss constraint requires the wave function to be
invariant under the $SU(2)$ gauge transformation. The state
$\Psi_{R}(A^{(\beta)})$ is gauge invariant because of the property of the
group delta function (11).

The simple inner product between two states can be supposed as

$\displaystyle\langle\,\Psi_{R^{\prime}}|\Psi_{R}\,\rangle$ $\displaystyle=$
$\displaystyle\int{\cal DA}\
\Psi_{R^{\prime}}^{\dagger}(A^{(\beta)})\Psi_{R}(A^{(\beta)})$ (30)
$\displaystyle=$
$\displaystyle\prod_{x}\prod_{a,b}\delta\left(\exp\left[\left(1+\frac{1}{\beta^{2}}\right)\left(R_{ab}-R^{\prime}_{ab}\right)\right]\right)$
$\displaystyle\equiv$ $\displaystyle\delta\left(R-R^{\prime}\right)\ .$

Here ${\cal DA}$ is the appropriate measure of the connection $A^{(\beta)}$
normalized such that $\int{\cal DA}=1$. This inner product is too sensitive.
When $R^{\prime}$ takes a different value from $R$, it vanishes, even if $R$
and $R^{\prime}$ are in the equivalence class of $SU(2)$ gauge and
diffeomorphism transformations. To make the inner product more convenient, the
following new inner product is introduced:

$\displaystyle(\,\Psi_{R^{\prime}}|\Psi_{R}\,)$ $\displaystyle=$
$\displaystyle\int{\cal D}\phi\ \langle\,\Psi_{R^{\prime}}|\ {\cal
U}(\phi)\,|\Psi_{R}\,\rangle$ (31) $\displaystyle=$ $\displaystyle\int{\cal
D}\phi\ \delta\left(R-\phi R^{\prime}\right)\ .$

Here ${\cal U}(\phi)$ is an operator of the gauge and diffeomorphism
transformations. The integral $\int{\cal D}\phi$ is over all of both
transformations. This construction of the inner product is an analogy of LQG
[11]. One can find that the dual state

$(\,\Psi_{R^{\prime}}|=\int{\cal D}\phi\ \langle\,\Psi_{R^{\prime}}|\ {\cal
U}(\phi)=\int{\cal D}\phi\ \langle\,\Psi_{\phi R^{\prime}}|$ (32)

is a solution of the Gauss and diffeomorphism constraints.

Finally, we consider the Hamiltonian constraint

$H=-\frac{\beta}{2\sqrt{|\det(E)|}}\epsilon^{ijk}E_{i}^{a}E_{j}^{b}\left[F_{abk}-\left(1+\beta^{2}\right)\epsilon_{klm}K_{a}^{l}K_{b}^{m}\right]\
.$ (33)

Performing a similar calculation to (25), the smeared Hamiltonian constraint
is deformed as

$\displaystyle H(N)$ $\displaystyle=$ $\displaystyle\int{\rm d}^{3}x\ NH$ (34)
$\displaystyle=$ $\displaystyle-\int{\rm d}^{3}x\
\frac{N\beta}{2\sqrt{|\det(E)|}}\epsilon^{ijk}E_{i}^{a}E_{j}^{b}\left[\left(1+\frac{1}{\beta^{2}}\right)R_{abk}-\frac{1}{\beta^{2}}F_{abk}\right]\
.$

Therefore, if the Levi-Civita curvature operator $\hat{R}$ can be well
defined, i.e., $\hat{R}\Psi_{R}(A^{(\beta)})=R\Psi_{R}(A^{(\beta)})$, then the
state $\Psi_{R}(A^{(\beta)})$ will satisfy

$\int{\rm d}^{3}x\
\chi^{abk}\left[\left(1+\frac{1}{\beta^{2}}\right)\hat{R}_{abk}-\frac{1}{\beta^{2}}\hat{F}_{abk}\right]\
\Psi_{R}(A^{(\beta)})=0\ ,$ (35)

where $\chi$ is a test function. According to Ref. [6], the Levi-Civita
curvature operator $\hat{R}$ is defined as follows:

$\int{\rm d}^{3}x\ \chi^{abk}\hat{R}_{abk}=\int{\cal D}\phi{\cal
D}R^{\prime}\left[\int{\rm d}^{3}x\ \chi^{abk}\left(\phi
R^{\prime}_{abk}\right)\right]|\Psi_{\phi
R^{\prime}}\,\rangle\langle\,\Psi_{\phi R^{\prime}}|\ ,$ (36)

where the integral $\int{\cal D}R^{\prime}$ is over the Levi-Civita curvature
$R^{\prime}$ modulo the equivalence class of the gauge and diffeomorphism
transformations. The action of this operator on the state $|\Psi_{R}\,\rangle$
becomes

$\displaystyle\int{\rm d}^{3}x\ \chi^{abk}\hat{R}_{abk}|\Psi_{R}\,\rangle$
$\displaystyle=$ $\displaystyle\int{\cal D}\phi{\cal D}R^{\prime}\
\delta(R-\phi R^{\prime})\left[\int{\rm d}^{3}x\ \chi^{abk}\left(\phi
R^{\prime}_{abk}\right)\right]|\Psi_{\phi R^{\prime}}\,\rangle$ (37)
$\displaystyle=$ $\displaystyle\int{\rm d}^{3}x\
\chi^{abk}R_{abk}|\Psi_{R}\,\rangle\ .$

With this operator, one obtains

$\hat{H}(N)\Psi_{R}(A^{(\beta)})=0\ .$ (38)

Consequently, the state $\Psi_{R}(A^{(\beta)})$ satisfies all three
constraints.

##  4 Conclusions and discussion

In this paper, we have constructed the generalized BF state for real values of
$\beta$. This has been done via the chiral asymmetric extension. The
generalized state is an explicit function of the connection $A^{(\beta)}$, and
is parameterized by the Levi-Civita curvature $R$ as well as in the
generalized CS state. It is gauge invariant and solves all constraints with
the appropriate inner product and the operator.

This state would be associated with the space such that
$(1+\beta^{2})R_{ab}-F_{ab}=0$. It contains a special case, i.e., a flat 3D
space $R=F=0$. More discussions are necessary to obtain further specific
interpretations. Problems with the connection with generic $\beta$ may arise,
because this connection is not a pull-back of a space-time connection [12].

It would be interesting to consider a loop representation of the state
$\Psi_{R}(A^{(\beta)})$:

$\displaystyle\Psi_{R}(\gamma)$ $\displaystyle=$
$\displaystyle\langle\,\gamma|\Psi_{R}\,\rangle=\int{\cal DA}\
\langle\,\gamma|A^{(\beta)}\,\rangle\langle\,A^{(\beta)}|\Psi_{R}\,\rangle$
(39) $\displaystyle\sim$ $\displaystyle\int{\cal DA}\
W(A^{(\beta)},\gamma)\prod_{x}\prod_{a,b}\delta\left(\exp\left[F_{ab}-(1+\beta^{2})R_{ab}\right]\right)\
.$

Here $W(A^{(\beta)},\gamma)$ is a spin network with a graph $\gamma$, which is
a generalized Wilson loop constructed from holonomy edges and invariant
tensors. The part

$\int{\cal DA}\
\prod_{x}\prod_{a,b}\delta\left(\exp\left[F_{ab}-(1+\beta^{2})R_{ab}\right]\right)$
(40)

looks like a generating functional of the spin foam model with a source term,
which is known as the Freidel–Krasnov (FK) model [13]. Let us consider a
discretized 3D space with a triangulation $\Delta$. The corresponding dual
cell $\Delta^{*}$ has vertices $v$, edges $l$, and faces $f$. In the FK model,
the discretized generating functional is given by

$\displaystyle Z_{\rm FK}[J]$ $\displaystyle=$ $\displaystyle\int{\cal DADB}\
\exp\left[{\rm i}\int{\rm Tr\,}(B\wedge F+B\wedge J)\right]$ (41)
$\displaystyle=$ $\displaystyle\int\prod_{l}{\rm d}g_{l}\
\prod_{f}\sum_{\Lambda_{f}\in{\rm Irrep}}\dim(\Lambda_{f})\,{\rm
Tr\,}\left[R^{(\Lambda_{f})}\left(g_{l_{1}}{\rm e}^{J_{v_{1}}}\cdots
g_{l_{n}}{\rm e}^{J_{v_{n}}}\right)\right],$

where $J$ is a source term for $B$, $R^{(\Lambda)}(g_{l})$ is a representation
of the group element $g_{l}=\exp[\int_{l}A]$ with a spin label $\Lambda$,
vertices $v_{1},\cdots,v_{n}$ and edges $l_{1},\cdots,l_{n}$ are associated
with the $n$-polygonal $\partial f$, and the sum is taken over all irreducible
representations. Thus the loop representation (39) is expressed as

$\displaystyle\Psi_{R}(\gamma)$ $\displaystyle=$
$\displaystyle\int\prod_{l}{\rm d}g_{l}\
W(A_{l}^{(\beta)},\gamma)\prod_{f}\sum_{\Lambda_{f}\in{\rm
Irrep}}\dim(\Lambda_{f})\,{\rm Tr\,}\left[R^{(\Lambda_{f})}\left(g_{l_{1}}{\rm
e}^{r_{v_{1}}}\cdots g_{l_{n}}{\rm e}^{r_{v_{n}}}\right)\right],$ (42)

where $r=-(1+\beta^{2})R$. The limit $R\to 0$ is consistent with the spin
network invariant $\Psi_{R=0}(\gamma)$ in Ref. [8].

## Acknowledgements

The authors would like to thank H. Taira, T. Oka, and K. Eguchi for helpful
discussions.

## References

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  * [3] A. Ashtekar and J. Lewandowski, Background independent quantum gravity: a status report, Class. Quantum Grav. 21, R53 (2004), arXiv:gr-qc/0404018.
  * [4] H. Kodama, Holomorphic wave function of the universe, Phys. Rev. D 42, 2548 (1990).
  * [5] A. Randono, A generalization of the Kodama state for arbitrary values of the Immirzi parameter, arXiv:gr-qc/0504010.
  * [6] A. Randono, Generalizing the Kodama state I: Construction, arXiv:gr-qc/0611073.
  * [7] A. Randono, Generalizing the Kodama state II: Properties and physical interpretation, arXiv:gr-qc/0611074.
  * [8] A. Miković, Quantum gravity vacuum and invariants of embedded spin networks, Class. Quantum Grav. 20, 3483 (2003), arXiv:gr-qc/0301047.
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Louis Gabarraa,b, on behalf of the Euclid Consortium

# Construction of spectral libraries of star-forming galaxies at $0.3\leq
z\leq 2.5$

###### Abstract

Spectral energy distribution (SED) models of galaxies at $0.3\leq z\leq 2.5$
were constructed to be processed by the Euclid spectroscopic channel simulator
in order to investigate the Euclid Near-Infrared Spectrometer and Photometer
(NISP) capabilities.

We developed a solid methodology to build a realistic and representative
synthetic SED library of star-forming galaxies, including reliable emission
line fluxes and widths. The construction of the SEDs consists in computing the
continuum using the Bruzual & Charlot (2003) models, calling out SED fitting
parameters available in publicly released multi-wavelength catalogues.
Emission lines were added using empirical and theoretical relations.

##  1 Context

Euclid will provide tens of millions of spectra in a redshift range yet under-
sampled. This redshift range includes the cosmic noon at $z\approx 2$, when
galaxies were particularly prolific in producing stars. Therefore, beyond the
primary cosmological science of the mission, an unprecedented amount of data
will be available to revolutionise galaxy evolution study. The analysis will
provide crucial information through line diagnosis and scaling relations. In
order to be prepared when the huge amount of data will be released, an
efficient strategy to fine tune the data reduction and processing pipelines is
to work with simulations. We present in this work a robust methodology to
build a spectral library of star-forming galaxies (SFGs) at $0.3\leq z\leq
2.5$ that can be processed through the Euclid spectroscopic channel simulator.

##  2 Method

###  2.1 Starting from publicly released photometric catalogues

To construct the spectral library, we started from publicly released
catalogues from the Cosmic Assembly Near-infrared Deep Extragalactic Legacy
Survey (CANDELS), namely the BARRO2019 catalogue ([2]; B19 hereafter) covering
the Great Observatories Origins Deep Survey North field (GOODS-N) and the
COSMOS2015 catalogue ([6]; L16 hereafter) covering the COSMOS field. Both
catalogues are well furnished with multi-wavelength observation from ground
and space-based telescopes. The coverage from the UV to the far infrared
allowed inferring robust SED fitting parameters using the Lephare software,
e.g, $z$, star-formation rate (SFR), total stellar mass ($M_{\star}$). We
selected sources within the NISP photometric sensitivity and identified as
SFGs using colour-colour diagrams (see [3] and Euclid Collaboration: Gabarra
et al., in prep., for further details). We present the selected sources from
the COSMOS2015 catalogue in Table 1.

Table 1: Description of the SED-fitting parameters of the 24 259 sources
selected from the L16 catalogue that covers the COSMOS field. The selection
criteria account for the redshift range of interest, the magnitude limit of
the NISP photometric channel, i.e., $H_{\rm E}\leq 24\,\text{mag}$, and on the
star-formation activity following colour-colour diagrams (see Euclid
Collaboration: Gabarra et al., in prep).

Sources selected from the COSMOS2015 catalogue covering the COSMOS field
---
Redshift | Number | $H$ |  $\operatorname{log_{10}}$($M_{\star}$) |  $\operatorname{log_{10}}$(SFR) |  $\operatorname{log_{10}}$(Age) | Re
bins | sources | [AB mag] | [$M_{\odot}$] | [$M_{\odot}\,{\rm yr}^{-1}$]) | [yr] | [arcsec]
$0.3\leq z\leq 0.5$ | 7356 | 17–23.5 | 8.0–11.7 |  $-$19.91–2.28 | 8.0–10.0 | 0.01–14.3
$0.5<z\leq 1.0$ | 14 603 | 17.3–24 | 8.6–11.7 |  $-$19.49–2.83 | 8.0–9.9 | 0.02–13.4
$1.0<z\leq 1.5$ | 1830 | 18.7–23.6 | 8.8–11.8 |  $-$10.35–2.95 | 8.0–9.7 | 0.03–6.3
$1.5<z\leq 2.0$ | 362 | 19.8–23.8 | 9.4–11.5 |  $-$9.58–2.98 | 8.0–9.6 | 0.0–12.8
$2.0<z\leq 2.5$ | 108 | 20.5–23.5 | 9.4–11.5 |  $-$3.01–2.71 | 8.0–9.4 | 0.1–14.2

.

Figure 1: _Left_ : The $M_{\star}$-metallicity relation (MZR) as presented in
[7] in the redshift bin $1.5\leq z\leq 2.0$ with the predicted metallicity
indicated on the right vertical axis. The left vertical axis indicates the
corresponding ratio $N_{2}$ predicted using the linear calibrations presented
in [1]. _Right_ : $N_{2}$-BPT diagram as presented in [5] for sources between
$1.5\leq z\leq 2.0$. See Euclid Collaboration: Gabarra et al., in prep, for
further details.

###  2.2 Continuum reconstruction and emission lines flux predictions

We reconstructed the continuum using the Bruzual & Charlot (2003) models. We
added the nebular emission Balmer, Paschen, [N ii]$\lambda\lambda 6584,6549$,
[O ii]$\lambda\lambda$3727,3729, [O iii]$\lambda\lambda$5007,4959, [S
ii]$\lambda\lambda$6731,6717, and [S iii]$\lambda\lambda$9531,9069 lines
making use of calibrations available in the literature. We refer to empirical
relations for emission line analysis such as H$\alpha$ luminosity-SFR [4], the
mass-metallicity relation, the Baldwin-Phillips-Terlevich diagram (BTP, see
Fig. 1), and photoionisation models. The emission line fluxes are then
transformed into Gaussian profiles and integrated to the continuum.

The Euclid spectroscopy will be performed in slitless configuration. This can
be thought as if the slit were the galaxy itself. A strong effect of the
galaxy shape on the quality of the extracted spectra is therefore expected.
The shape of the galaxy will mainly affect the signal to noise ratio and the
full width at half maximum (FWHM) of the emission lines (see Euclid
Collaboration: Gabarra et al., in prep, for details). This aspect led us to
take a special care in modelling the shape of the emission lines, considering
the velocity dispersion. To account for it, we applied the velocity dispersion
to stellar mass relation proposed by Bezanson et al. (2018).

##  3 Results and conclusion

Spectroscopic data are available in the B19 and L16 catalogues for some of the
emission lines predicted in this work. We present in Fig. 2 a comparison
between our predictions and observation. In Fig. 3, we present an example of a
spectrum for a source from the B19 catalogue. We indicate on the figure the
magnitudes from observation within the SHARDS and HST surveys.

We presented the construction of spectroscopic data starting from publicly
released catalogues from the COSMOS and CANDELS/GOODS-N fields. The SEDs are
reconstructed based on stellar population models using the software GALAXEV,
and we calculated the nebular emission line fluxes using observed scaling
relations and photoionisation models. The constructed SEDs have been compared
and validated using photometric and spectroscopic data available in the
catalogues. The good agreement between the predictions and the observations
served as a validation test to attest for the realistic emission line flux
estimations.

Figure 2: Comparison of the predicted fluxes to the observed fluxes coming
from publicly released data of the near-infrared spectroscopic surveys 3D-HST
(purple), FMOS (green) and LEGA-C (cyan). The contours correspond to iso-
proportions of the distribution density starting at 20% with a 10% step. See
Euclid Collaboration: Gabarra et al., in prep, for further details. Figure 3:
Examples of photometric evaluation of the incident spectra created in this
work. The source is from the B19 catalogue and the continuum is compared to
observational data from the SHARDS and HST surveys photometry (grey and black
triangles). The zoom-in is made on the H$\alpha$-[N
ii]$\lambda\lambda$6584,6549 triplet. The coloured diamonds indicate the
fluxes obtained with the Euclid filters.

Acknowledgements. The author thanks Chiara Mancini, Lucia Rodríguez-Muñoz,
Giulia Rodighiero, Chiara Sirignano, Marco Scodeggio and Margherita Talia for
their help in shaping this work. The Euclid Consortium acknowledges the
European Space Agency and a number of agencies and institutes that have
supported the development of Euclid, in particular the Academy of Finland, the
Agenzia Spaziale Italiana, the Belgian Science Policy, the Canadian Euclid
Consortium, the French Centre National d’Etudes Spatiales, the Deutsches
Zentrum für Luft- und Raumfahrt, the Danish Space Research Institute, the
Fundação para a Ciência e a Tecnologia, the Ministerio de Ciencia e
Innovación, the National Aeronautics and Space Administration, the National
Astronomical Observatory of Japan, the Netherlandse Onderzoekschool Voor
Astronomie, the Norwegian Space Agency, the Romanian Space Agency, the State
Secretariat for Education, Research and Innovation (SERI) at the Swiss Space
Office (SSO), and the United Kingdom Space Agency. A complete and detailed
list is available on the Euclid web site (http://www.euclid-ec.org)..

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