# Wormhole solutions in $f(R,L_{m})$ gravity

Raja Solanki 0000-0001-8849-7688<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. Zinnat Hassan 0000-0002-6608-2075
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute of
Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. P.K. Sahoo 0000-0003-2130-8832
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute
of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. Faculty of Mathematics & Computer
Science, Transilvania University of Brasov, Eroilor 29, Brasov, Romania

###### Abstract

In this work, we intend to explore wormhole geometries in the framework of
$f(R,L_{m})$ gravity. We derive the field equations for the generic
$f(R,L_{m})$ function by assuming the static and spherically symmetric Morris-
Thorne wormhole metric. Then we consider two non-linear $f(R,L_{m})$ model,
specifically, $f(R,L_{m})=\frac{R}{2}+L_{m}^{\alpha}$ and
$f(R,L_{m})=\frac{R}{2}+(1+\lambda R)L_{m}$, where $\alpha$ and $\lambda$ are
free model parameters. We obtain the wormhole solutions by assuming three
cases, namely, a linear barotropic EoS, anisotropic EoS, and isotropic EoS
corresponding to model I. We observe that for both barotropic and anisotropic
cases, the corresponding wormhole solutions obey the flaring-out condition
under asymptotic background, while for the isotropic case, the shape function
does not follow the flatness condition. Also, we find that the null energy
condition exhibits negative behavior in the vicinity of the throat. Further,
we consider two different shape functions to investigate the behavior of model
II. We find some constraints on the model parameter for which the violation of
the null energy condition exhibits. Finally, we employ the volume integral
quantifier to calculate the amount of exotic matter required near the wormhole
throat for both models. We conclude that the modification of standard GR can
efficiently minimize the use of exotic matter and provide stable traversable
wormhole solutions.
Keywords: Wormholes, $f(R,L_{m})$ gravity, barotropic EoS, anisotropic EoS,
VIQ.

##  I Introduction

In 1916, the idea of a wormhole was first suggested by L. Flamm [1]. Later on,
Einstein and Rosen investigated the precise nature of a wormhole and made a
hypothetical bridge by taking into account the idea of Flamm, known as the
Einstein-Rosen bridge [2]. Wormholes are topologically framed hypothetical
structures that supply a subway for distinct space-times apart from each
other. A wormhole is supposed to be a tube-like structure that is
asymptotically flat on both ends and connected by a throat. Wormholes are
classified into two categories depending on the nature of the throat, namely
static and non-static wormholes. A wormhole with a constant radius of the
throat is referred to as a static wormhole, whereas a non-static wormhole
represents a variable radius. Fuller and Wheeler demonstrated that one could
not traverse through the Einstein-Rosen bridge, even a photon, since it would
collapse instantly upon formation [3]. Further, Morris et al. [4] proposed
that exotic forms of matter threaded through a wormhole might hold it open;
nevertheless, it remains to be seen whether such requirements are physically
viable. Finally, in [5], Morris and Thorne presented the static traversable
wormholes that define interstellar travel and exact solutions of general
relativity (GR). The authors show that a wormhole could be traversable if it
exhibits exotic matter with a minimal surface area satisfying the flare-out
condition. The matter content that violates null energy conditions (NEC) and
defined by $T_{\mu\nu}k^{\mu}k^{\nu}\geq 0$ for any null vector $k^{\mu}$
(where $T_{\mu\nu}$ is the stress-energy tensor) is called the exotic matter,
and hence it is an essential ingredient to construct a wormhole in GR; Indeed,
in classical GR, wormhole solutions violate all energy conditions [6].
Regardless, this type of hypothetical matter expresses unusuality in GR,
whereas from the perspective of quantum gravity, it can be found as a natural
consequence of fluctuations in the space-time topology [7]. Thus, minimizing
the violation of the energy conditions or reducing the quantity of exotic
matter at the throat is essential. In scalar-tensor theories, wormhole
solutions can be found for the scalar fields representing the function of
phantom fluid [8, 9, 10]. To bypass the undetected exotic matter, several
modified gravity theories have appeared in the literature, such as $f(R)$
gravity [11, 12, 13], brane-world [14, 15, 16], and curvature matter coupling
[17, 18]. The effective stress-energy tensor involving the components of the
geometry of space-time is now responsible for the violation of energy
conditions.
An extension of the $f(R)$ modified gravity that incorporates an explicit
coupling of the arbitrary function of the Ricci curvature $R$ with the matter
Lagrangian term $L_{m}$ proposed in [19]. Harko and Lobo further generalized
this case to arbitrary matter-geometry couplings [20]. The cosmological models
with non-minimal curvature-matter couplings have significant astrophysical and
cosmological applications [21, 22, 23, 24, 25]. Recently, Harko and Lobo
proposed [26] $f(R,L_{m})$ modified gravity that incorporates all curvature-
matter coupling theories, where $f(R,L_{m})$ is a generic function of the
Ricci curvature $R$ and the Lagrangian term $L_{m}$. In the $f(R,L_{m})$
modified gravity, the covariant divergence of the energy-momentum tensor does
not vanishes, an extra force orthogonal to four velocities arises, and the
motion of the test particle is non-geodesic. Further, the cosmological models
with $f(R,L_{m})$ gravity do not obey the equivalence principle, and that is
constrained by the solar system experimental tests [27, 28]. Recently, several
interesting cosmological and astrophysical works have been done in
$f(R,L_{m})$ gravity theory; for instance, one can check references [29, 30,
31, 32, 33].
The equation of state (EoS) parameter plays a vital role in cosmology and
astrophysics to describe the nature of the cosmic fluid. The fluid
characterized by the linear barotropic EoS ($p=\omega\rho$) with positive
energy density is found to be a viable candidate to describe cosmic evolution.
In the context of GR, wormhole geometry with phantom energy background has
been widely discussed [16, 34, 35]. Further, in the context of modified
gravity, such as $f(R)$, $f(R,T)$, and $f(Q)$ gravity, finding the exact
solutions of the corresponding field equations is quite tricky compared to GR.
This paper studies wormhole geometry in the context of $f(R,L_{m})$ gravity.
We developed the corresponding field equations for the presence of the
redshift function under the generic form of $f(R,L_{m})$ gravity. Then we
assumed two different $f(R,L_{m})$ models and investigated wormhole solutions
with various EoS relations and shape functions. This work is organized as
follows: In sec II, we present the fundamental formulations of $f(R,L_{m})$
gravity theory. In sec III, we derive the field equations for the generic
$f(R,L_{m})$ function corresponding to the static and spherically symmetric
Morris-Thorne wormhole metric. Further in sec IV, we consider a specific
$f(R,L_{m})$ model, namely, $f(R,L_{m})=\frac{R}{2}+L_{m}^{\alpha}$, where
$\alpha$ is a free model parameter. Then, we study wormhole solutions for a
linear barotropic EoS, an anisotropic EoS, and isotropic relation. In
addition, we consider another non-linear $f(R,L_{m})$ model, specifically,
$f(R,L_{m})=\frac{R}{2}+(1+\lambda R)L_{m}$ with two different shape functions
$b(r)=r_{0}+\gamma^{2}r_{0}\left(1-\frac{r_{0}}{r}\right)$ and
$b(r)=r\,e^{r_{0}-r}$, and then we analyzed energy conditions. Further, in sec
V, we have discussed VIQ to check the amount of exotic matter required for a
traversable wormhole. Finally, in the last section, we discuss our findings.

##  II  $f(R,L_{m})$ Gravity Theory

The generic action governing the dynamics of the universe in $f(R,L_{m})$
gravity read as

$S=\int{f(R,L_{m})\sqrt{-g}d^{4}x},$ (1)

where $R$ is the Ricci scalar corresponding to the metric $g_{\mu\nu}$ with
determinant $g$ and $L_{m}$ represents the matter Lagrangian.
The Ricci scalar curvature term $R$ can be obtained by the contraction of the
Ricci tensor $R_{\mu\nu}$ as

$R=g^{\mu\nu}R_{\mu\nu},$ (2)

where,

$R_{\mu\nu}=\partial_{\lambda}\Gamma^{\lambda}_{\mu\nu}-\partial_{\nu}\Gamma^{\lambda}_{\lambda\mu}+\Gamma^{\sigma}_{\mu\nu}\Gamma^{\lambda}_{\sigma\lambda}-\Gamma^{\lambda}_{\nu\sigma}\Gamma^{\sigma}_{\mu\lambda},$
(3)

with $\Gamma^{\alpha}_{\beta\gamma}$ denoting the components of the Levi-
Civita connection and can be calculated as

$\Gamma^{\alpha}_{\beta\gamma}=\frac{1}{2}g^{\alpha\lambda}\left(\frac{\partial
g_{\gamma\lambda}}{\partial x^{\beta}}+\frac{\partial
g_{\lambda\beta}}{\partial x^{\gamma}}-\frac{\partial
g_{\beta\gamma}}{\partial x^{\lambda}}\right).$ (4)

The following field equation acquired by varying the generic action (1)
corresponding to the metric tensor $g_{\mu\nu}$,

$f_{R}R_{\mu\nu}+(g_{\mu\nu}\square-\nabla_{\mu}\nabla_{\nu})f_{R}-\frac{1}{2}(f-f_{L_{m}}L_{m})g_{\mu\nu}\\\
=\frac{1}{2}f_{L_{m}}T_{\mu\nu}.$ (5)

Here $f_{R}\equiv\frac{\partial f}{\partial R}$,
$f_{L_{m}}\equiv\frac{\partial f}{\partial L_{m}}$, and $T_{\mu\nu}$ is the
energy-momentum tensor for the cosmic fluid, given by

$T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{m})}{\delta
g^{\mu\nu}}.$ (6)

Further, contraction of the field equation (4) gives following relation
between the energy-momentum scalar $T$, the Lagrangian term $L_{m}$, and the
Ricci scalar $R$ as

$Rf_{R}+3\square f_{R}-2(f-f_{L_{m}}L_{m})=\frac{1}{2}f_{L_{m}}T,$ (7)

where $\square
F=\frac{1}{\sqrt{-g}}\partial_{\alpha}(\sqrt{-g}g^{\alpha\beta}\partial_{\beta}F)$
for any scalar function $F$ .

##  III Wormhole Geometries in $f(R,L_{m})$ Gravity

We consider the static and spherically symmetric Morris-Thorne wormhole metric
[5, 6], given by

$ds^{2}=-U(r)dt^{2}+V(r)dr^{2}+r^{2}d\Omega^{2},$ (8)

Here, $d\Omega^{2}=d\theta^{2}+\text{sin}^{2}\theta d\phi^{2}$,
$U(r)=e^{2\Phi(r)}$ and $V(r)=\left(1-\frac{b(r)}{r}\right)^{-1}$. The
functions $b(r)$ and $\Phi(r)$ of the radial coordinate encode the information
about the shape of the wormhole and the gravitational redshift, and hence it
is known as the shape function and the redshift function, respectively. To
avoid the event horizon, $\Phi(r)$ must be finite everywhere. Moreover, to
have traversable wormhole geometry, the shape function $b(r)$ should obey the
following well-known constraints; specifically, the flaring-out condition
given by $(b-b^{\prime}r)/b^{2}>0$ [5], throat condition given by
$b(r_{0})=r_{0}$ ($r_{0}$ is the throat radius) with $b^{\,\prime}(r_{0})<1$,
and the asymptotic flatness condition, that is given by
$\frac{b(r)}{r}\rightarrow 0$ as $r\rightarrow\infty$. Also, another
significant criterion is the proper radial distance $l(r)$, represented as

$l(r)=\pm\int_{r_{0}}^{r}\frac{dr}{\sqrt{1-\frac{b(r)}{r}}},$ (9)

is needed to be finite everywhere. Here, the $\pm$ symbols indicate the upper
and lower portions of the wormhole, which are linked by the throat. Also, the
proper distance decreases from the upper universe $l=+\infty$ to the throat
and then from $l=0$ to $-\infty$ in the lower universe. Moreover, $l$ should
be greater than or equal to the coordinate distance $\mid l(r)\mid\geq
r-r_{0}$. The embedding surface of the wormhole can be obeyed by defining the
embedding surface $z(r)$ at a fixed $\theta=\pi/2$ and time
$t=\text{constant}$. Hence, equation (8) reduces to

$ds^{2}=\left(1-\frac{b(r)}{r}\right)^{-1}dr^{2}+r^{2}d\phi^{2}.$ (10)

The above metric can be embedded into three-dimensional Euclidean space with
cylindrical coordinates $r,\,\ \phi$ and $z$ as

$ds^{2}=dz^{2}+dr^{2}+r^{2}d\phi^{2}.$ (11)

Now, on comparing equations (10) and (11), we obtained the following slope
equation so that by integrating it, one can find the embedding surface $z(r)$,

$\frac{dz}{dr}=\pm\sqrt{\frac{r}{r-b(r)}-1}.$ (12)

Now corresponding to the metric (8), we obtained the non-vanishing components
of the Ricci tensor as

$R_{00}=e^{2\Phi}\left[\left(1-\frac{b}{r}\right)\left\\{\Phi^{\prime\prime}+\Phi^{\prime
2}+\frac{2\Phi^{\prime}}{r}\right\\}\right.\\\
\left.-\frac{(rb^{\prime}-b)}{2r^{2}}\Phi^{\prime}\right],$ (13)
$R_{11}=-\Phi^{\prime\prime}-\Phi^{\prime
2}+\frac{(rb^{\prime}-b)}{2r(r-b)}\left(\Phi^{\prime}+\frac{2}{r}\right),$
(14) $R_{22}=(b-r)\Phi^{\prime}+\frac{b^{\prime}}{2}+\frac{b}{2r},$ (15)
$R_{33}=\sin^{2}\theta\left\\{(b-r)\Phi^{\prime}+\frac{b^{\prime}}{2}+\frac{b}{2r}\right\\},$
(16)

Then by using equation (2), we obtained the Ricci curvature scalar $R$ for the
spherically symmetric configuration (8) as

$R=\frac{2b^{\prime}}{r^{2}}-2\left(1-\frac{b}{r}\right)\left\\{\Phi^{\prime\prime}+\Phi^{\prime
2}+\frac{\Phi^{\prime}}{r}\right\\}\\\
+\frac{\Phi^{\prime}}{r^{2}}\left(rb^{\prime}+b-2r\right).$ (17)

The stress-energy tensor corresponding to anisotropic fluid is defined by

$T_{\mu\,\nu}=\left(\rho+P_{t}\right)u_{\mu}\,u_{\nu}+P_{t}\,\delta_{\mu\,\nu}+\left(P_{r}-P_{t}\right)v_{\mu}\,v_{\nu},$
(18)

where, $u_{\mu}$ and $v_{\mu}$ represent the four-velocity vector and the
unitary space-like vector, respectively and they satisfy the conditions
$u_{\mu}u^{\nu}=-v_{\mu}v^{\nu}=-1$. Here, $\rho$ is the energy density,
$P_{r}$, and $P_{t}$ are the radial and tangential pressure, depending on the
radial coordinate $r$ only. In the case of a stress-energy tensor of the form
(18), the NEC can be defined as $\rho+p_{i}\geq 0$ (where $i$ is either $r$ or
$t$).
Now, inserting the metric (8) and the anisotropic fluid (18), into the
equations of motion (5), we obtained the following field equations, which read
as

$\left(1-\frac{b}{r}\right)\left[\left\\{\Phi^{\prime\prime}+\Phi^{\prime
2}+\frac{2\Phi^{\prime}}{r}-\frac{(rb^{\prime}-b)}{2r(r-b)}\Phi^{\prime}\right\\}F-\left\\{\Phi^{\prime}\right.\right.\\\
\left.\left.+\frac{2}{r}-\frac{(rb^{\prime}-b)}{2r(r-b)}\right\\}F^{\prime}-F^{\prime\prime}\right]+\frac{1}{2}\left(f-L_{m}f_{L_{m}}\right)=\frac{1}{2}f_{L_{m}}\rho,$
(19)
$\left(1-\frac{b}{r}\right)\left[\left\\{-\Phi^{\prime\prime}-\Phi^{\prime
2}+\frac{(rb^{\prime}-b)}{2r(r-b)}\left(\Phi^{\prime}+\frac{2}{r}\right)\right\\}F\right.\\\
\left.+\left\\{\Phi^{\prime}+\frac{2}{r}-\frac{(rb^{\prime}-b)}{2r(r-b)}\right\\}F^{\prime}\right]-\frac{1}{2}\left(f-L_{m}f_{L_{m}}\right)=\frac{1}{2}f_{L_{m}}p_{r},$
(20)
$\left(1-\frac{b}{r}\right)\left[\left\\{-\frac{\Phi^{\prime}}{r}+\frac{(rb^{\prime}+b)}{2r^{2}(r-b)}\right\\}F+\left\\{\Phi^{\prime}+\frac{2}{r}\right.\right.\\\
\left.\left.-\frac{(rb^{\prime}-b)}{2r(r-b)}\right\\}F^{\prime}+F^{\prime\prime}\right]-\frac{1}{2}\left(f-L_{m}f_{L_{m}}\right)=\frac{1}{2}f_{L_{m}}p_{t}.$
(21)

where $F=\frac{\partial f}{\partial R}$. Finally, we end up with a set of
three equations (19-21) involving six unknown quantities. Thus the above
system of equations is under-determined, and we need some extra constraints to
construct the wormhole solutions.

##  IV Wormhole Solutions for some specific Non-linear $f(R,L_{m})$ Models

The $f(R,L_{m})$ function that we have considered in our study is motivated by
the generic $f(R,L_{m})$ function; specifically,
$f(R,L_{m})=f_{1}(R)+f_{2}(R)G(L_{m})$ that represents arbitrary curvature-
matter coupling [36]. The models with minimal and non-minimal couplings have
received great attention from cosmologists in the recent past in the context
of different modified gravities such as $f(R,\mathcal{T})$,
$f(T,\mathcal{T})$, and $f(Q,\mathcal{T})$ gravity. The above generic
$f(R,L_{m})$ functions are attractive since they consist of minimal and non-
minimal coupling cases. We will explore wormhole geometry by incorporating
both minimal and non-minimal coupling cases.

###  IV.1 Model I

We presume the following minimal $f(R,L_{m})$ function to obtain the wormhole
solutions[36, 37]. The type of minimal coupling case we considered here is
motivated by an interesting work of Bose et al. [38] in the context of
$f(R,\mathcal{T})$ gravity.

$f(R,L_{m})=\frac{R}{2}+L_{m}^{\alpha}.$ (22)

where $\alpha$ is a free model parameter. In particular, for the case
$\alpha=1$, we retrieve the usual wormhole geometry of GR.
Therefore, we obtain the following field equations corresponding to this
specific $f(R,L_{m})$ function,

$\frac{b^{\prime}}{r^{2}}=(2\alpha-1)\rho^{\alpha},$ (23)
$-\frac{b}{r^{3}}+2\left(1-\frac{b}{r}\right)\frac{\Phi^{\prime}}{r}=\rho^{\alpha-1}\left\\{(1-\alpha)\rho+\alpha
p_{r}\right\\},$ (24)
$\left(1-\frac{b}{r}\right)\left[\Phi^{\prime\prime}+\Phi^{\prime
2}+\left\\{\frac{1}{r}-\frac{(rb^{\prime}-b)}{2r(r-b)}\right\\}\Phi^{\prime}\right.\\\
\left.-\frac{(rb^{\prime}-b)}{2r^{2}(r-b)}\right]=\rho^{\alpha-1}\left\\{(1-\alpha)\rho+\alpha
p_{t}\right\\}.$ (25)

In this work, we incorporate the constant redshift function
$\Phi(r)=\text{constant}$ to obtain the wormhole solution. Now in order to
find the analytic solutions of the obtained field equations, we need only one
extra ansatz. Therefore, we consider the following cases:

####  IV.1.1 Linear barotropic EoS

In this particular section, we are going to construct wormhole solutions by
considering the following linear barotropic EoS [39].

$p_{r}=\omega\rho$ (26)

where $\omega$ is the EoS parameter. In the context of late-time cosmic
acceleration, the $\Lambda$CDM description of the dark energy is characterized
by the EoS parameter $\omega=-1$ and is the most successful theory so far.
Another widely discussed time-dependent dark energy model is the quintessence
model characterized by the EoS parameter $\omega>-1$, whereas the least
theoretically understood dark energy is the phantom energy characterized by
$\omega<-1$. In Ref. [40], the authors investigated the asymptotically flat
phantom wormhole solutions. Also, it is mentioned in [41, 42] that finding
wormhole solutions with linear barotropic EoS in the framework of a non-linear
model is quite difficult in both teleparallel and symmetric teleparallel
gravity. However, we obtain the exact wormhole solution in the framework of
our non-linear $f(R,L_{m})$ model with linear barotropic EoS.
On solving the equations (23) and (24) by incorporating the relation (26), we
obtained the following first order differential equation

$b^{\prime}+\frac{(2\alpha-1)}{(1-\alpha+\alpha\omega)}\frac{b}{r}=0.$ (27)

Now on integrating the above equation by using the throat condition
$b(r_{0})=r_{0}$, we obtained the following expression for the shape function
$b(r)$ in terms of radial coordinate r

$b(r)=r_{0}\left(\frac{r_{0}}{r}\right)^{\frac{(2\alpha-1)}{(1-\alpha+\alpha\omega)}}.$
(28)

The behavior of the shape function given in equation (28) and the flaring-out
condition are presented in Figs. 1 and 2.

Figure 1: Shape function with $\omega=-2$ and $r_{0}=1$ ($p_{r}=\omega\rho$).
Figure 2: Flare-out condition with $\omega=-2$ and $r_{0}=1$
($p_{r}=\omega\rho$).

From our investigation, we found that $\frac{b(r)}{r}\rightarrow 0$ as
$r\rightarrow\infty$ will satisfy only if (i) $\omega<-1$ with
$\alpha>-\frac{1}{\omega-1}$, and (ii) $-1<\omega<1$ with
$0<\alpha<-\frac{1}{\omega-1}$. As we are interested in exploring wormhole
geometry under the phantom scenario, we consider the range $\omega<-1$ with
$\alpha>-\frac{1}{\omega-1}$ in this analysis. By setting the appropriate
values from the above region, we have plotted the shape functions in Figs. 1
and 2. We found that the flaring-out condition is satisfied with asymptotic
flatness.
Now, by using the equation (28) in equations (23)-(25), one can acquire the
expressions for the energy density, radial pressure, and the tangential
pressure as

$\rho(r)=\left(-\frac{r_{0}\left(\frac{r_{0}}{r}\right)^{\frac{2\alpha-1}{\alpha(\omega-1)+1}}}{r^{3}(\alpha(\omega-1)+1)}\right)^{1/\alpha},$
(29)
$p_{r}(r)=\omega\left(-\frac{r_{0}\left(\frac{r_{0}}{r}\right)^{\frac{2\alpha-1}{\alpha(\omega-1)+1}}}{r^{3}(\alpha(\omega-1)+1)}\right)^{1/\alpha},$
(30)
$p_{t}(r)=-\frac{(\alpha(\omega-1)+2)\left(-\frac{r_{0}\left(\frac{r_{0}}{r}\right)^{\frac{2\alpha-1}{\alpha(\omega-1)+1}}}{r^{3}(\alpha(\omega-1)+1)}\right)^{1/\alpha}}{2\alpha}.$
(31)

Now we analyze the behavior of Null energy condition (NEC) in the vicinity of
the throat. The NEC corresponding to our linear barotropic EoS model read as

$\rho+p_{r}=(\omega+1)\left(-\frac{r_{0}\left(\frac{r_{0}}{r}\right)^{\frac{2\alpha-1}{\alpha(\omega-1)+1}}}{r^{3}(\alpha(\omega-1)+1)}\right)^{1/\alpha},$
(32)
$\rho+p_{t}=-\frac{(\alpha(\omega-3)+2)\left(-\frac{r_{0}\left(\frac{r_{0}}{r}\right)^{\frac{2\alpha-1}{\alpha(\omega-1)+1}}}{r^{3}(\alpha(\omega-1)+1)}\right)^{1/\alpha}}{2\alpha}.$
(33)

At the throat, the above expressions reduce to the following,

$\rho+p_{r}\mid_{r=r_{0}}=(\omega+1)\left(-\frac{1}{r_{0}^{2}(\alpha(\omega-1)+1)}\right)^{1/\alpha},$
(34)
$\rho+p_{t}\mid_{r=r_{0}}=-\frac{(\alpha(\omega-3)+2)\left(-\frac{1}{r_{0}^{2}(\alpha(\omega-1)+1)}\right)^{1/\alpha}}{2\alpha}.$
(35)

Figure 3: Energy density with $\omega=-2$ and $r_{0}=1$ ($p_{r}=\omega\rho$).
Figure 4: Radial NEC with $\omega=-2$ and $r_{0}=1$ ($p_{r}=\omega\rho$).

We have presented the behavior of the energy density and radial NEC in Fig. 3
and 4, respectively. We found that the energy density shows positive behavior,
whereas the radial NEC exhibits negative behavior. Also, one can observe that
the right-hand side of the (34) is a negative quantity corresponding to
positive values of $\alpha$ with $\omega<-1$, which confirms the violation of
NEC at the wormhole throat. Moreover, we have obtained the embedding surface
for this case by using the Eq. (12), which is shown in Fig.5.

Figure 5: Embedding diagram for the linear barotropic EoS model corresponding
to parameter values $\omega=-2$ and $\alpha=0.4$.

####  IV.1.2 Anisotropic EoS

In this subsection, we consider the following anisotropic EoS to construct
wormhole solutions [43].

$p_{t}=np_{r},$ (36)

where $n\neq 1$. On solving the equations (23)-(25) by incorporating the
relation (36), we obtained the following first-order differential equation

$b^{\prime}-\frac{(2n+1)(2\alpha-1)}{\left\\{2n(\alpha-1)+1\right\\}}\frac{b}{r}=0.$
(37)

Now on integrating the above equation by using the throat condition
$b(r_{0})=r_{0}$, we obtained the following expression for the shape function
$b(r)$

$b(r)=r_{0}\left(\frac{r}{r_{0}}\right)^{\frac{(2\alpha-1)(2n+1)}{[2n(\alpha-1)+1]}}.$
(38)

From our investigation, we found some constraints on parameters $\alpha$ and
$n$ to satisfy the asymptotic flatness condition corresponding to the shape
function obtained in the equation (38). We listed out the specific constraints
in Table 1.

$n$ | $n<-1$ | $n=-1$ | $-1<n\leq-\frac{1}{2}$ | $-\frac{1}{2}<n<0$ | $0<n<1$ | $n>1$
---|---|---|---|---|---|---
$\alpha$ | $\alpha\in\left(\frac{1}{n+1},\frac{2n-1}{2n}\right)$ | $\alpha<\frac{3}{2}$ |  $\alpha<\frac{2n-1}{2n}$ or $\alpha>\frac{1}{n+1}$ |  $\alpha>\frac{2n-1}{2n}$ or $\alpha<\frac{1}{n+1}$ | $\alpha\in\left(\frac{2n-1}{2n},\frac{1}{n+1}\right)$ | $\alpha\in\left(\frac{1}{n+1},\frac{2n-1}{2n}\right)$
Table 1: Table shows the acceptable ranges to satisfy the asymptotic flatness
condition.

Now we study the flaring-out condition for the shape function (38). By
choosing some appropriate values from Table 1, we have shown the behavior of
the flaring-out condition in Fig. 6. One can notice that the flaring-out
condition is satisfied at the wormhole throat.

Figure 6: Flare-out condition with $r_{0}=1$ ($p_{t}=np_{r}$). Here, we
plotted for the parameter values $(\alpha,n)$: $(1,-1.5)$, $(-1.4,-1)$,
$(4,-0.7)$, $(0.25,-0.1)$, $(0.3,0.5)$, $(0.54,2)$.

Now, by using the equation (38) in equations (23)-(25), one can acquire the
expressions for the energy density, radial pressure, and the tangential
pressure as

$\rho(r)=\left(\frac{(2n+1)r_{0}\left(\frac{r}{r_{0}}\right)^{\frac{(2\alpha-1)(2n+1)}{2(\alpha-1)n+1}}}{r^{3}(2(\alpha-1)n+1)}\right)^{1/\alpha},$
(39)
$p_{r}(r)=\frac{(\alpha-2)\left(\frac{(2n+1)r_{0}\left(\frac{r}{r_{0}}\right)^{\frac{(2\alpha-1)(2n+1)}{2(\alpha-1)n+1}}}{r^{3}(2(\alpha-1)n+1)}\right)^{1/\alpha}}{\alpha+2\alpha
n},$ (40)
$p_{t}(r)=\frac{(\alpha-2)n\left(\frac{(2n+1)r_{0}\left(\frac{r}{r_{0}}\right)^{\frac{(2\alpha-1)(2n+1)}{2(\alpha-1)n+1}}}{r^{3}(2(\alpha-1)n+1)}\right)^{1/\alpha}}{\alpha+2\alpha
n}.$ (41)

The NEC corresponding to our anisotropic EoS model read as

$\rho+p_{r}=\frac{2(\alpha+\alpha
n-1)\left(\frac{(2n+1)r_{0}\left(\frac{r}{r_{0}}\right)^{\frac{(2\alpha-1)(2n+1)}{2(\alpha-1)n+1}}}{r^{3}(2(\alpha-1)n+1)}\right)^{1/\alpha}}{\alpha+2\alpha
n},$ (42)
$\rho+p_{t}=\frac{(\alpha+(3\alpha-2)n)\left(\frac{(2n+1)r_{0}\left(\frac{r}{r_{0}}\right)^{\frac{(2\alpha-1)(2n+1)}{2(\alpha-1)n+1}}}{r^{3}(2(\alpha-1)n+1)}\right)^{1/\alpha}}{\alpha+2\alpha
n}.$ (43)

At the throat, the above expressions reduce to the following,

$\rho+p_{r}\mid_{r=r_{0}}=\frac{2(\alpha+\alpha
n-1)\left(\frac{2n+1}{r_{0}^{2}(2(\alpha-1)n+1)}\right)^{1/\alpha}}{\alpha+2\alpha
n},$ (44)
$\rho+p_{t}\mid_{r=r_{0}}=\frac{(\alpha+(3\alpha-2)n)\left(\frac{2n+1}{r_{0}^{2}(2(\alpha-1)n+1)}\right)^{1/\alpha}}{\alpha+2\alpha
n}.$ (45)

We have presented the behavior of the energy density and radial NEC in Fig. 7
and 8, respectively. We found that the energy density exhibits negative
behavior corresponding to the GR case i.e. $\alpha=1$, whereas it is positive
for other values of $\alpha$. Further, we found that NEC is violated in the
neighborhood of the throat corresponding to every chosen value of $\alpha$
satisfying the constraints obtained in Table 1.

Figure 7: Energy density with $r_{0}=1$ ($p_{t}=np_{r}$). Here, we plotted for
the parameter values $(\alpha,n)$: $(1,-1.5)$, $(4,-0.7)$, $(0.25,-0.1)$,
$(0.3,0.5)$. Figure 8: Radial NEC with $r_{0}=1$ ($p_{t}=np_{r}$). Here, we
plotted for the parameter values $(\alpha,n)$: $(1,-1.5)$, $(4,-0.7)$,
$(0.25,-0.1)$, $(0.3,0.5)$.

Further, we have investigated the 3D visualization of the surface sweep
through a $2\pi$ rotation around the $z$-axis for this case, is shown in Fig.
9.

Figure 9: Embedding diagram for the anisotropic EoS model corresponding to
parameter values $n=-1.5$ and $\alpha=0.85$.

####  IV.1.3 Isotropic EoS

Usually, the energy-momentum tensor of the asymptotically flat wormholes in
the GR is anisotropic i.e. $p_{r}\neq p_{t}$. We have already discussed the
wormhole solutions for the anisotropic EoS case. In this section, we shall
study wormhole solutions for the following isotropic relation

$p_{r}=p_{t}.$ (46)

Now inserting the Eqs. (24) and (25) in the above expression (46), we could
able to find the shape function

$b(r)=c\left(\frac{r}{c}\right)^{3},$ (47)

where $c$ is the integrating constant. Imposing the throat condition
$b(r_{0})=r_{0}$ to the above expression, we obtain the final version of the
shape function

$b(r)=r_{0}\left(\frac{r}{r_{0}}\right)^{3}.$ (48)

It is clear that the above expression is not asymptotically flat i.e.
$\frac{b(r)}{r}\nrightarrow 0$ as $r\rightarrow\infty$. Thus it can be
concluded that finding asymptotically flat wormhole solutions for isotropic
pressure under constant redshift function for the $f(R,L_{m})$ model (22) is
quite difficult.

###  IV.2 Model II

We presume the following non-minimal type $f(R,L_{m})$ function [31],

$f(R,L_{m})=\frac{R}{2}+(1+\lambda\,R)L_{m}$ (49)

where $\lambda$ is the coupling constant. The characteristics of Neutron stars
have been investigated by using NICER data in the framework of the above-
considered $f(R,L_{m})$ model [31]. The coupling constant presents large
values as compared to the weak-field limit. The same dependence appeared in
scalar-tensor theories, where this parameter varies according to the scalar
field mass, referred to as the chameleon mechanism [44]. It is quite
interesting to study the dependence of this coupling parameter in the context
of different astrophysical systems such as white dwarfs, black holes,
wormholes, and neutron stars. In particular, for the case $\lambda=0$, we
retrieve the usual wormhole geometry of GR. The cosmological implications of
the considered model have been tested in [29]. We will examine the
characteristics of the given model in the context of wormholes.
Therefore, we obtain the following field equations corresponding to this
specific $f(R,L_{m})$ function,

$\rho=\frac{\Phi^{\prime}\left(r\left(-b^{\prime}\right)+2r(r-b)\Phi^{\prime}-3b+4r\right)+2r(r-b)\Phi^{\prime\prime}(r)+r^{2}R}{2\left(\lambda\left(r\left(b^{\prime}-4\right)+3b\right)\Phi^{\prime}+2\lambda
r(b-r)\Phi^{\prime 2}+r\left(2\lambda(b-r)\Phi^{\prime\prime}+\lambda
rR+r\right)\right)},$ (50)
$P_{r}=\frac{1}{\mathcal{K}}\left[r\left(b^{\prime}\left((\lambda
rR+r)\Phi^{\prime}+4\lambda R+2\right)-r\left(2(\lambda rR+r)\Phi^{\prime
2}-4\lambda R\Phi^{\prime}(r)+r(\lambda
R+1)\left(2\Phi^{\prime\prime}+R\right)\right)\right)\right.\\\
\left.+b\left(r\left(\Phi^{\prime}\left(2(\lambda rR+r)\Phi^{\prime}-5\lambda
R-1\right)+2r(\lambda R+1)\Phi^{\prime\prime}\right)-4\lambda
R-2\right)\right],$ (51)
$P_{t}=\frac{1}{\mathcal{K}}\left[r\left(b^{\prime}\left(-\lambda
rR\Phi^{\prime}+2\lambda R+1\right)+2\lambda r^{2}R\left(\Phi^{\prime
2}+\Phi^{\prime\prime}\right)-r\left(rR(\lambda
R+1)+2\Phi^{\prime}\right)\right)\right.\\\
\left.+b\left(r\left(\Phi^{\prime}\left(-2\lambda rR\Phi^{\prime}+\lambda
R+2\right)-2\lambda rR\Phi^{\prime\prime}\right)+2\lambda R+1\right)\right],$
(52)

where

$\mathcal{K}=2r(\lambda
R+1)\left(\lambda\left(r\left(b^{\prime}-4\right)+3b\right)\Phi^{\prime}\right.\\\
\left.+2\lambda r(b-r)\Phi^{\prime
2}+r\left(2\lambda(b-r)\Phi^{\prime\prime}+\lambda rR+r\right)\right).$ (53)

####  IV.2.1  $b(r)=r_{0}+\gamma^{2}r_{0}\left(1-\frac{r_{0}}{r}\right)$

Here, we turn our attention to the model with
$b(r)=r_{0}+\gamma^{2}r_{0}\left(1-\frac{r_{0}}{r}\right)$ [45], where
$0<\gamma<1$ is particularly interesting to have wormhole solutions that
satisfy the flaring-out condition at wormhole throat. The stress-energy tensor
profile for this specific function under constant redshift function is given
by

$\rho=\frac{\gamma^{2}r_{0}^{2}}{r^{4}+2\gamma^{2}\lambda r_{0}^{2}},$ (54)
$P_{r}=\frac{1}{\mathcal{L}^{2}}\left[2\gamma^{2}\lambda
r_{0}^{3}\left(3\gamma^{2}r_{0}-2\left(\gamma^{2}+1\right)r\right)-r^{4}r_{0}\mathcal{M}\right],$
(55)
$P_{t}=\frac{r^{4}r_{0}\left(\gamma^{2}(r-2r_{0})+r\right)+4\gamma^{2}\lambda
r_{0}^{3}\mathcal{M}}{2\mathcal{L}^{2}},$ (56)

where $\mathcal{L}=\left(r^{4}+2\gamma^{2}\lambda r_{0}^{2}\right)$ and
$\mathcal{M}=\left(\gamma^{2}(r-r_{0})+r\right)$.
For the case of the NEC along the radial and tangential direction is provided
by

$\rho+P_{r}=-\frac{r_{0}\left(\gamma^{2}(r-2r_{0})+r\right)\left(r^{4}+4\gamma^{2}\lambda
r_{0}^{2}\right)}{\mathcal{L}^{2}},$ (57)
$\rho+P_{t}=\frac{\left(\gamma^{2}+1\right)rr_{0}\left(r^{4}+4\gamma^{2}\lambda
r_{0}^{2}\right)}{2\mathcal{L}^{2}}.$ (58)

One can find that at wormhole throat, the above equations reduce to

$\rho+P_{r}\mid_{r=r_{0}}=-\frac{\left(1-\gamma^{2}\right)\left(4\gamma^{2}\lambda+r_{0}^{2}\right)}{\left(2\gamma^{2}\lambda+r_{0}^{2}\right)^{2}},$
(59)
$\rho+P_{t}\mid_{r=r_{0}}=\frac{\left(\gamma^{2}+1\right)\left(4\gamma^{2}\lambda+r_{0}^{2}\right)}{2\left(2\gamma^{2}\lambda+r_{0}^{2}\right)^{2}}.$
(60)

For this specific case, we noticed that the RHS of Eq. (59) is a negative
quantity for any $\lambda>0$ and $0<\gamma<1$, which confirms the violation of
radial NEC at the throat. However, from Eq. (60), we can see that tangential
NEC is satisfied at the throat. The graphical representations of $\rho$ and
$\rho+P_{r}$ are shown in Fig. 10 and 11. Interestingly, energy density is
positive throughout space-time, whereas NEC is violated at the throat and its
neighborhood. Also, we observed that for substantial values of $\lambda$, the
radial NEC $\rho+P_{r}$ would be validated.

Figure 10: Energy density with $r_{0}=1$ and $\gamma=0.6$. Note that for
$\lambda=0$ corresponds to GR case. Figure 11: Radial NEC with $r_{0}=1$ and
$\gamma=0.6$. Note that for $\lambda=0$ corresponds to GR case.

####  IV.2.2 $b(r)=r\,e^{r_{0}-r}$

Considering the specific choice for the form function $b(r)=r\,e^{r_{0}-r}$
[46] where $0<r_{0}<1$ is particularly interesting to have wormhole solutions
that satisfy the condition $b^{{}^{\prime}}(r_{0})<1$. Taking this form into
account, the generalized field equations Eqs. (50-52) can be read as

$\rho=\frac{(r-1)e^{r_{0}}}{-e^{r}r^{2}+2\lambda re^{r_{0}}-2\lambda
e^{r_{0}}},$ (61)
$P_{r}=\frac{e^{r_{0}}\left(2\lambda\left(r^{2}-1\right)e^{r_{0}}-e^{r}r^{2}\right)}{\left(e^{r}r^{2}-2\lambda(r-1)e^{r_{0}}\right)^{2}},$
(62)
$P_{t}=\frac{e^{r_{0}}\left(e^{r}r^{3}-4\lambda(r-1)e^{r_{0}}\right)}{2\left(e^{r}r^{2}-2\lambda(r-1)e^{r_{0}}\right)^{2}}.$
(63)

Using Eqs. (61-63), one can obtain the NEC given by

$\rho+P_{r}=\frac{re^{r_{0}}\left(4\lambda(r-1)e^{r_{0}}-e^{r}r^{2}\right)}{\left(e^{r}r^{2}-2\lambda(r-1)e^{r_{0}}\right)^{2}},$
(64)
$\rho+P_{t}=\frac{(r-2)e^{r_{0}}\left(4\lambda(r-1)e^{r_{0}}-e^{r}r^{2}\right)}{2\left(e^{r}r^{2}-2\lambda(r-1)e^{r_{0}}\right)^{2}}.$
(65)

For this wormhole model, we assume the throat radius $r_{0}=0.5$ and hence the
flare-out condition $b^{{}^{\prime}}(0.5)=0.5<1$ obeyed at the throat. Also,
we can see from Eq. (64) that
$\rho+P_{r}\mid_{r=r_{0}}=-\frac{r_{0}\left(4\lambda+r_{0}^{2}-4\lambda
r_{0}\right)}{\left(2\lambda+r_{0}^{2}-2\lambda r_{0}\right)^{2}},$ which can
be violated for $0<r_{0}<1$ and $\lambda>\frac{r_{0}^{2}}{4(r_{0}-1)}$. Also,
we noticed that radial NEC could be validated at the throat for $0<r_{0}<1$
and
$\lambda<\frac{r_{0}^{2}}{2(r_{0}-1)}\lor\frac{r_{0}^{2}}{2(r_{0}-1)}<\lambda<\frac{r_{0}^{2}}{4(r_{0}-1)}$.
Further, we study tangential NEC at $r=r_{0}$ provided by
$\rho+P_{t}=-\frac{(r_{0}-2)\left(4\lambda+r_{0}^{2}-4\lambda
r_{0}\right)}{2\left(2\lambda+r_{0}^{2}-2\lambda r_{0}\right)^{2}}$. It is
observed that tangential NEC is violated for $0<r_{0}<1$ within the range
$\lambda<\frac{s^{2}}{2s-2}\lor\frac{r_{0}^{2}}{2r_{0}-2}<\lambda<\frac{r_{0}^{2}}{4r_{0}-4}$,
whereas satisfied within $\lambda>\frac{r_{0}^{2}}{4(r_{0}-1)}$. We have
presented the graphical behavior of energy density and NEC in Figs. 12 and 13
with some particular values discussed above. It is obvious that NEC is
violated, whereas energy density is respected in the vicinity of the throat.
However, one can notice that energy density is no longer validated far from
the throat.

Figure 12: Energy density with $r_{0}=0.5$. Note that for $\lambda=0$
corresponds to GR case. Figure 13: Radial NEC with $r_{0}=0.5$. Note that for
$\lambda=0$ corresponds to GR case.

##  V Amount of exotic matter

Now we will estimate the amount of exotic matter required for a wormhole to be
stable. To do this, we use the volume integral quantifier (VIQ) approach,
introduced by Visser et al. [47], which can quantify the average amount of
matter present in spacetime, violating NEC. The VIQ is defined as

$IV=\oint[\rho+P_{r}]dV$ (66)

where the volume can be read as $dV=r^{2}\,dr\,d\Omega$ with $d\Omega$ the
solid angle. Since $\oint
dV=2\int_{r_{0}}^{\infty}dV=8\pi\int_{r_{0}}^{\infty}r^{2}dr,$ we have

$IV=8\pi\int_{r_{0}}^{\infty}(\rho+P_{r})r^{2}dr.$ (67)

Now, the volume integral corresponding to a wormhole whose field varies from
the throat $r_{0}$ to a fixed radius $r_{1}$ with $r_{1}\geq r_{0},$ is given
as

$IV=8\pi\int_{r_{0}}^{r_{1}}(\rho+P_{r})r^{2}dr.$ (68)

Now with the help of the above Eq. (68), we have investigated the volume
integral and presented the nature of those in Figs. (14-16). It is evident
from the graphs that $IV\rightarrow 0$, as $r_{1}\rightarrow r_{0}$. Thus, we
can conclude that a small fraction of exotic matter can stabilize a
traversable wormhole. We found that one can minimize the total amount of
average null energy condition (ANEC) violating matter by taking suitable
wormhole geometry. One can check the references [49, 48] to review some
interesting applications of VIQ.

Figure 14: Profile of VIQ with $\alpha=2$, $\omega=-2$, and $r_{0}=1$. Figure
15: Profile of VIQ with $\alpha=4$, $n=-0.7$ and $r_{0}=1$. Figure 16: Profile
of VIQ with $\gamma=0.6$, $\lambda=2$ and $r_{0}=1$.

##  VI Conclusions

A wormhole is a short-cut that act as a subway for distinct regions of
spacetime or distinct spacetimes apart from each other. The theoretical
solution of field equations of GR that violates energy conditions gave rise to
the concept of a wormhole. Nowadays, wormhole solutions in the context of
modified gravity theories have attracted the attention of researchers.
Modified gravity theories play a significant role in bypassing the necessity
of undetected exotic matter in constructing wormholes. In this paper, we have
examined different wormhole solutions in the $f(R,L_{m})$ gravity. This theory
has been investigated to analyze some dark energy candidates by utilizing
dynamical system technique [50]. Further, hydrostatic equilibrium
configurations of neutron stars in the context of $f(R,L_{m})$ gravity is
obtained in [51]. Moreover, $f(R,L_{m})$ gravity describes pulsars as massive
as PSR J2215+5135 [52] with the help of an equation of state for nuclear
matter. These attractive features of $f(R,L_{m})$ gravity motivated us to
study wormholes in this gravity.
In this manuscript, we attempted to explore wormhole geometry in the framework
of $f(R,L_{m})$ gravity. We have derived the field equations for the wormhole
metric (8) in the $f(R,L_{m})$ gravity background. Then we have considered two
different $f(R,L_{m})$ models, such as,
$f(R,L_{m})=\frac{R}{2}+L_{m}^{\alpha}$ (model I) and
$f(R,L_{m})=\frac{R}{2}+(1+\lambda\,R)L_{m}$ (model II), where $\alpha$ and
$\lambda$ are arbitrary model parameters. Firstly, we have studied the
wormhole solutions for three different cases (i) a linear barotropic EoS, (ii)
an anisotropic EoS, and (iii) isotropic EoS, under model I. For the linear
barotropic case, we calculated the shape function $b(r)$ and obtained the
possible regions where the flatness condition is satisfied. We found the
flaring-out condition is satisfied for any $\omega<-1$ corresponding to
$\alpha>-\frac{1}{\omega-1}$ under the asymptotic background. Further, we have
investigated the energy conditions and noticed that energy density is positive
in the entire space-time. NEC is violated in the neighborhood of the throat
for increasing values of $\alpha$ under the phantom region. However, the
violation of NEC may not be maintained for very large values of $\alpha$.
Again, for the anisotropic case, we have obtained the possible ranges of
$\alpha$ and $n$ using the asymptotically flatness condition, which is listed
in Table 1. By setting the appropriate values from the Table 1, we have
studied the behavior of the shape functions and energy conditions. We found
that the flaring-out condition is satisfied at the wormhole throat. The energy
density exhibit negative behavior corresponding to the GR case, i.e.,
$\alpha=1$, while it is positive for other values of $\alpha$. The NEC is
violated in the neighborhood of the throat corresponding to every chosen value
of $\alpha$ satisfying the constraints obtained in Table 1. Further, we have
investigated wormholes for the isotropic relation and observed that the
obtained shape function is not asymptotically flat. It seems that finding
asymptotically flat wormhole solutions for isotropic pressure under constant
redshift function in $f(R,L_{m})$ scenario is quite difficult.
Further, finding the wormhole solutions for model II in an analytic way is
quite tricky due to the complexity of the field equations. Therefore, we have
studied wormhole solutions with two different shape functions, i.e.,
$b(r)=r_{0}+\gamma^{2}r_{0}\left(1-\frac{r_{0}}{r}\right)$ and
$b(r)=r\,e^{r_{0}-r}$ in order to investigate model II. For
$b(r)=r_{0}+\gamma^{2}r_{0}\left(1-\frac{r_{0}}{r}\right)$ case, we found that
radial NEC is violated, and tangential NEC is satisfied near the throat within
$\lambda>0$ and $0<\gamma<1$. Energy density is positive in the entire
spacetime. Again, for $b(r)=r\,e^{r_{0}-r}$, we investigated the NEC
thoroughly and obtained some validity ranges for NEC. It is observed radial
NEC is violated within $0<r_{0}<1$ and $\lambda>\frac{r_{0}^{2}}{4(r_{0}-1)}$
whereas, tangential NEC is violated for $0<r_{0}<1$ within the range
$\lambda<\frac{r_{0}^{2}}{2r_{0}-2}\lor\frac{r_{0}^{2}}{2r_{0}-2}<\lambda<\frac{r_{0}^{2}}{4r_{0}-4}$.
In this case, we found that energy density is positive in the neighborhood of
the throat. Moreover, we noticed that the value of the coupling constant
obtained in our study is consistent with the values obtained in [30, 31].
According to the Ref. [31], the best result can be achieved when the coupling
parameter $\lambda$ is around $30$. Further, in case of white dwarf system the
consistent value of $\lambda$ lies between $0$ and $1$ [30]. However, in our
study, we noticed that for a traversable wormhole solution, $\lambda$ is a
non-negative value and that is consistent with values obtained in the above
results.
In addition, we have examined the VIQ to study the amount of exotic matter
required at the throat for a traversable wormhole. In our analysis, we found
that a small amount of exotic matter is necessary for a traversable wormhole.
We can conclude that the modification of standard GR can efficiently minimize
the use of exotic matter and provide a stable traversable wormhole solution.
Although wormholes have not been detected yet, in this study, we have
investigated the possible existence of wormhole geometries in the context of
$f(R,L_{m})$ gravity. In this study, we have considered the constant redshift
function, i.e., $\Phi(r)=\text{constant}$. It would be interesting to
investigate wormholes with non-constant redshift functions in this modified
gravity in the near future.

## Data Availability Statement

There are no new data associated with this article.

## Acknowledgments

RS acknowledges University Grants Commission (UGC), New Delhi, India, for
awarding a Senior Research Fellowship (UGC-Ref. No.: 191620096030). ZH
acknowledges the Department of Science and Technology (DST), Government of
India, New Delhi, for awarding a Senior Research Fellowship (File No.
DST/INSPIRE Fellowship/2019/IF190911). P.K.S. acknowledges the National Board
for Higher Mathematics (NBHM) under the Department of Atomic Energy (DAE),
Govt. of India for financial support to carry out the Research project No.:
02011/3/2022 NBHM(R.P.)/R&D II/2152 Dt.14.02.2022 and Transilvania University
of Brasov for Transilvania Fellowship for Visiting Professors. We are very
much grateful to the honorable referees and to the editor for the illuminating
suggestions that have significantly improved our work in terms of research
quality, and presentation.

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# A Generic Method for Fine-grained Category Discovery in Natural Language
Texts

Chang Tian†, Matthew B. Blaschko†, Wenpeng Yin, Mingzhe Xing,
Yinliang Yue, Marie-Francine Moens†
† KU Leuven
<EMAIL_ADDRESS>
###### Abstract

Fine-grained category discovery using only coarse-grained supervision is a
cost-effective yet challenging task. Previous training methods focus on
aligning query samples with positive samples and distancing them from
negatives. They often neglect intra-category and inter-category semantic
similarities of fine-grained categories when navigating sample distributions
in the embedding space. Furthermore, some evaluation techniques that rely on
pre-collected test samples are inadequate for real-time applications. To
address these shortcomings, we introduce a method that successfully detects
fine-grained clusters of semantically similar texts guided by a novel
objective function. The method uses semantic similarities in a logarithmic
space to guide sample distributions in the Euclidean space and to form
distinct clusters that represent fine-grained categories. We also propose a
centroid inference mechanism to support real-time applications. The efficacy
of the method is both theoretically justified and empirically confirmed on
three benchmark tasks. The proposed objective function is integrated in
multiple contrastive learning based neural models. Its results surpass
existing state-of-the-art approaches in terms of Accuracy, Adjusted Rand Index
and Normalized Mutual Information of the detected fine-grained categories.
Code and data will be available at https://github.com/XX upon publication.

A Generic Method for Fine-grained Category Discovery in Natural Language Texts

Chang Tian†, Matthew B. Blaschko†, Wenpeng Yin, Mingzhe Xing, Yinliang Yue,
Marie-Francine Moens† † KU Leuven<EMAIL_ADDRESS>

##  1 Introduction

Fine-grained analysis has drawn much attention in many artificial intelligence
fields, e.g., Computer Vision (Chen et al., 2018; Wang et al., 2024a; Park and
Ryu, 2024) and Natural Language Processing (Ma et al., 2023; Tian et al.,
2024; An et al., 2024), because it can provide more detailed features than
coarse-grained data.

Figure 1: A fine-grained intent detection example. Left: This panel
illustrates the label hierarchy, transitioning from coarse-grained to fine-
grained granularity. Right: This example demonstrates intent detection in a
conversation about car choices, showing how coarse-grained analysis alone can
lead to incorrect recommendations by a life assistant due to a lack of fine-
grained analysis.

For instance, as illustrated in Figure 1, solely based on coarse-grained
analysis, the chatbot might incorrectly recommend a roadster, which is
unsuitable for field adventures. Detecting the fine-grained intent would allow
the chatbot to recommend an off-road vehicle that aligns with the user’s
requirements. However, annotating fine-grained categories can be labor-
intensive, as it demands precise expert knowledge specific to each domain and
involved dataset. Addressing this challenge, An et al. (2022) recently
introduced Fine-grained Category Discovery under Coarse-grained Supervision
(FCDC) for language classification tasks (details in Section 3). FCDC aims to
reduce annotation costs by leveraging the relative ease of obtaining coarse-
grained annotations, without requiring fine-grained supervisory information.
This approach has sparked significant research interest in the automatic
discovery of fine-grained language categories. (Ma et al., 2023; An et al.,
2023a; Vaze et al., 2024; Lian et al., 2024).

Existing methods for addressing FCDC are typically grouped into three groups
(An et al., 2024): language models, self-training methods, and contrastive
learning methods. Language models (Devlin et al., 2019a; Touvron et al.,
2023), including their fine-tuned versions with coarse labels, generally
perform poorly on this task due to a lack of fine-grained supervision. Self-
training methods (Caron et al., 2018; Zhang et al., 2021) and their variants
often employ clustering assignments as fine-grained pseudo labels, filtering
out some noisy pseudo labels, and training with these labels. Dominant
contrastive learning methods (Chen et al., 2020; Mekala et al., 2021; An et
al., 2022, 2023a) typically identify positive and negative samples for a given
query by measuring their semantic distances. The contrastive loss ensures that
the query sample moves closer to positive samples and further away from
negative samples. So these methods form clusters of samples in the embedding
space, with each cluster representing a discovered fine-grained category,
without requiring fine-grained category supervision.

However, past methods did not utilize comprehensive semantic similarities
(CSS) in the logarithmic space to guide sample distributions in Euclidean
space. We define CSS as the fine-grained semantic similarities measured by
bidirectional KL divergence in the logarithmic space between the query sample
and each available positive or negative sample, as illustrated in Figure 2.
Although An et al. (2024) recently explored similarities measured by rank
order between the query sample and positive samples, they ignore similarities
with negative samples.

We propose a method (STAR) for detecting fine-grained clusters of semantically
similar texts through a novel objective function, with the core component
considering CSS. This component guides sample distributions in the Euclidean
space based on the magnitude of CSS in the logarithmic space. Large semantic
differences (low similarity) in the logarithmic space between the query sample
and an available sample push the query sample further away in Euclidean space,
while small semantic differences bring the query sample closer to the
available sample. Thus, samples form distinguishable fine-grained clusters in
Euclidean space, with each cluster representing a discovered category.

Additionally, clustering inference used by previous works (An et al., 2022,
2023a, 2024) can not support real-time scenarios, so we propose a variant
inference mechanism utilizing approximated fine-grained cluster centroids,
delivering competitive results for the tasks considered.

Our main contributions in this work can be summarized as follows:

  * •

Method: STAR enhances existing contrastive learning methods by leveraging
comprehensive semantic similarities in a logarithmic space to guide sample
distributions in the Euclidean space, thereby making fine-grained categories
more distinguishable.

  * •

Theory: We interpret STAR from the perspectives of clustering and generalized
Expectation Maximization (EM). Also, we conduct loss and gradient analyses to
explain the effectiveness of using CSS for category discovery.

  * •

Experiments: Experiments on three text classification tasks (intent detection
(Larson et al., 2019), scientific abstract classification (Kowsari et al.,
2017), and chatbot query (Liu et al., 2021)) demonstrate new state-of-the-art
(SOTA) performance compared to 22 baselines, validating the theoretical
method.

Figure 2: Visualization of comprehensive semantic similarities (CSS). The wavy
line indicates the bidirectional KL divergence between two samples.

##  2 Related Work

###  2.1 Fine-grained Category Discovery

Fine-grained data analysis is crucial in Natural Language Processing (Guo et
al., 2021; Ma et al., 2023; Tian et al., 2024) and Computer Vision (Pan et
al., 2023; Wang et al., 2024b). However, effectively discovering fine-grained
categories from coarse-grained ones remains challenging (Mekala et al., 2021).
Traditional category discovery methods often assume that known and discovered
categories are at the same granularity level (An et al., 2023b; Vaze et al.,
2024).

To discover fine-grained categories under the supervision of coarse-grained
categories, (An et al., 2022) introduced the FCDC task. Self-training
approaches, such as Deep Cluster (Caron et al., 2018; An et al., 2023a), use
clustering algorithms to detect the fine-grained categories, assign pseudo
labels to the clusters and their samples, and then train a classification
model with these pseudo labels. Its variant, Deep Aligned Clustering (Zhang et
al., 2021), devises a strategy to filter out inconsistent pseudo-labels during
clustering. Contrastive learning has become prevalent in FCDC tasks; (Bukchin
et al., 2021; An et al., 2022) developed angular contrastive learning tailored
for fine-grained classification. An et al. (2022) proposed a weighted self-
contrastive framework to enhance the model’s discriminative capacity for
coarse-grained samples. (Ma et al., 2023) and (An et al., 2023a) used noisy
fine-grained centroids and retrieved neighbors as positive pairs,
respectively, applying constraints to filter noise. An et al. (2024) advanced
this approach with neighbors that are manually weighted as positive pairs.
However, previous efforts have not leveraged comprehensive semantic
similarities to guide sample distributions and thereby to enhance fine-grained
category discovery.

###  2.2 Neighborhood Contrastive Learning

Contrastive learning enhances representation learning by bringing the query
sample closer to positive samples and distancing it from negative samples
(Chen et al., 2020). Prior research has focused on constructing high-quality
positive pairs. (He et al., 2020) utilized two different transformations of
the same input as query and positive sample, respectively. Li et al. (2020)
introduced the use of prototypes, derived through clustering, as positive
instances. Additionally, An et al. (2022) employed shallow-layer features from
BERT as positive samples and introduced a weighted contrastive loss. This
approach primarily differentiates data at a coarse-grained level, and the
manually set weights limit its broader applicability.

To circumvent complex data augmentation, neighborhood contrastive learning
(NCL) was developed, treating the nearest neighbors of queries as positive
samples (Dwibedi et al., 2021). Zhong et al. (2021) extended this by utilizing
k-nearest neighbors to identify hard negative samples, while Zhang et al.
(2022) selected a positive key from the k-nearest neighbors for contrastive
representation learning. However, these approaches often deal with noisy
nearest neighbors that include false-positive samples. (An et al., 2023a)
addressed this by proposing three constraints to filter out uncertain
neighbors, yet they overlooked semantic similarities between query sample and
each available sample. An et al. (2024) represented semantic similarities
using rank order among positive samples but neglected similarities among
negative samples. In contrast, STAR uses comprehensive semantic similarities
to guide sample distributions in the Euclidean space, offering richer features
and a superior approach to pure contrastive learning.

##  3 Problem Formulation

Given a set of coarse-grained categories
$Y_{coarse}=\\{C_{1},C_{2},\dots,C_{M}\\}$ and a coarsely labeled training set
$D_{train}=\\{(x_{i},c_{i})\mid c_{i}\in Y_{coarse}\\}_{i=1}^{N}$, where $N$
denotes the number of training samples, the task of FCDC involves developing a
feature encoder $F_{\theta}$. This encoder maps samples into a feature space,
further segmenting them into distinct fine-grained categories
$Y_{fine}=\\{F_{1},F_{2},\dots,F_{K}\\}$, without any fine-grained supervisory
information. Here, $Y_{fine}$ represents sub-classes of $Y_{coarse}$. Model
effectiveness is evaluated on a testing set $D_{test}=\\{(x_{i},y_{i})\mid
y_{i}\in Y_{fine}\\}_{i=1}^{L}$, with $L$ as the number of test samples,
utilizing features extracted by $F_{\theta}$. For evaluation consistency and
fairness, only the number of fine-grained categories $K$ is used, aligning
with methodologies established in previous research (Ma et al., 2023; An et
al., 2022, 2023a).

##  4 Method

Figure 3: STAR-DOWN integrates the baseline DOWN with the STAR method (shown
in the red dashed box). In the visual representation, colors differentiate
samples, squares represent features extracted by the Encoder, and circles
denote features extracted by the Momentum Encoder. Unidirectional arrows
indicate proximity, while bidirectional arrows signify distance between
samples.

STAR leverages comprehensive semantic similarities and integrates seamlessly
with contrastive learning baselines by modifying the objective function. We
have developed variants for three baselines: PseudoPrototypicalNet (PPNet)
(Boney and Ilin, 2017; Ji et al., 2020), DNA (An et al., 2023a), and DOWN (An
et al., 2024). This section focuses on STAR-DOWN because DOWN outperforms
other baselines, with additional method variants detailed in Appendix

DOWN involves three steps: pre-training with coarse-grained labels (Section
4.1), retrieving and weighting nearest neighbors (Section 4.2), and training
with a contrastive loss. STAR-DOWN follows the same first two steps but
replaces the third with a novel objective function (Section 4.3). Like DOWN,
STAR-DOWN iterates the last two steps until the unsupervised metric, the
silhouette score of the clustering into fine-grained clusters, does not
improve for five consecutive epochs. The detailed algorithm is provided in
Appendix

###  4.1 Multi-task Pre-training

As illustrated in Figure 3, the baseline DOWN (An et al., 2024) utilizes the
BERT Encoder $F_{\theta}$ to extract normalized feature embeddings
$q_{i}=F_{\theta}(x_{i})$ for input $x_{i}$, where $\theta$ represents the
Encoder parameters. To ensure effective initialization for fine-grained
training, DOWN pre-trains the Encoder on the coarsely labeled train set
$D_{train}$ with labels $Y_{coarse}$. DOWN utilizes the sum of a cross-entropy
loss $L_{\text{ce}}$ and a masked language modeling loss $L_{\text{mlm}}$ for
multi-task pre-training of the Encoder (detailed in Appendix

###  4.2 Neighbors Retrieval and Weighting

In Figure 3, the Momentum Encoder $F_{\theta_{k}}$ with parameters
$\theta_{k}$ extracts and stores gradient-free normalized neighbor features
$h_{i}=F_{\theta_{k}}(x_{i})$ in a dynamic data queue $Q$. To ensure
consistency between the outputs of $F_{\theta_{k}}$ and $F_{\theta}$,
$F_{\theta_{k}}$’s parameters are updated via a moving-average method (He et
al., 2020): $\theta_{k}\leftarrow m\theta_{k}+(1-m)\theta$, where $m$ is the
momentum coefficient. For each query feature $q_{i}$, in order to facilitate
semantic similarity capture and fine-grained clustering, its top-k nearest
neighbors $N_{i}$ are determined from $Q$ using cosine similarity (Sim):
$N_{i}=\\{h_{j}\mid h_{j}\in\operatorname{argtopK}_{h_{l}\in
Q}(\operatorname{Sim}(q_{i},h_{l}))\\}$, where
$\text{Sim}(q_{i},h_{l})=\frac{q_{i}^{\mathrm{T}}h_{l}}{\|q_{i}\|\cdot\|h_{l}\|}$
is the cosine similarity function.

To counteract potential false positives in $N_{i}$, DOWN utilizes a soft
weighting mechanism based on neighbor rank to balance information utility
against noise, with weights $\omega_{j}$ of neighbor $h_{j}$ calculated as:
$\omega_{j}=\phi\cdot\alpha^{-\frac{l_{ij}}{k}}$, where $\phi$ is a
normalizing constant for weights, $\alpha$ serves as the exponential base, $k$
is the retrieved neighbor count, and $l_{ij}$ denotes the rank of $h_{j}$ as a
neighbor to $q_{i}$.

To align with the model’s evolving accuracy in neighbor retrieval during
training, DOWN periodically decreases $\alpha$ every five epochs, the values
for $\alpha$ in $\omega_{j}$ are: $\alpha_{\text{set}}=\\{150,10,5,2\\}$. The
$\omega_{j}$ of each positive sample $h_{j}$ is used in Eqs. 3 and 4.

###  4.3 Training

####  4.3.1 Objective Function

Given a training batch $N_{train}\in D_{train}$, where $Y_{c}$ is the set of
coarse-grained labels of $N_{train}$, DOWN trains the model using the loss:

$L_{\text{train}}=L_{\text{ce}}+L_{\text{DOWN}},$ (1)

$L_{\text{DOWN}}=\frac{1}{|N_{train}|}\sum_{q_{i}\in N_{train}}L_{1}^{i}.$ (2)

As shown in Eq. 3, DOWN uses a conventional contrastive objective function in
the Euclidean space, while STAR-DOWN introduces a novel objective function in
Eq. 4, leveraging CSS in a logarithmic space to guide sample distributions in
the Euclidean space, the temperature $\tau$ is a fixed constant in Eq. 3 and
Eq. 4:

$\displaystyle L_{1}^{i}=-\sum_{h_{j}\in
N_{i}}\omega_{j}\cdot\log\frac{\exp(q_{i}^{\mathrm{T}}h_{j}/\tau)}{\sum\limits_{h_{k}\in
Q}\exp(q_{i}^{\mathrm{T}}h_{k}/\tau)}.$ (3) $\displaystyle L_{2}^{i}=$
$\displaystyle-\gamma\sum_{h_{j}\in
N_{i}}\omega_{j}\cdot\log\frac{\exp(-d_{KL}(q_{i},h_{j})/\tau)}{\sum\limits_{h_{k}\in
Q}\exp(-d_{KL}(q_{i},h_{k})/\tau)}$ (4) $\displaystyle-\sum_{h_{j}\in
N_{i}}\omega_{j}\cdot\log\frac{\exp(q_{i}^{\mathrm{T}}h_{j}/\tau)}{\sum\limits_{h_{k}\in
Q}B^{d_{KL}(q_{i},h_{k})}\cdot\exp(q_{i}^{\mathrm{T}}h_{k}/\tau)}.$

During training, STAR-DOWN optimizes the following objective function:

$L_{\text{train}}=L_{\text{ce}}+L_{\text{STAR}},$ (5)

$L_{\text{STAR}}=\frac{1}{|N_{train}|}\sum_{q_{i}\in N_{train}}L_{2}^{i}.$ (6)

As shown in Eq. 4, the term $d_{KL}(q_{i},h_{k})$ in $L_{2}^{i}$ represents
the bidirectional KL divergence in a logarithmic space between the query
sample embedding $q_{i}$ and the data queue sample embedding $h_{k}$ (detailed
in Appendix $B$ is a trainable scalar representing the exponential base.

The first term in $L_{2}^{i}$ minimizes the KL divergence between query
samples and positive samples (retrieved neighbors in Section 4.2) while
increasing it for negative samples in the logarithmic space, with $\gamma$ as
a balancing hyperparameter. The second term in $L_{2}^{i}$ uses CSS in the
logarithmic space, denoted by $B^{d_{KL}(q_{i},h_{k})}$, to guide query sample
distribution in the Euclidean space. $q_{i}^{\mathrm{T}}h_{k}$ quantifies the
cosine similarity between normalized $q_{i}$ and $h_{k}$, equivalent to the
negative Euclidean distance (detailed in Appendix . The value of the trainable
scalar $B$ is updated during loss backpropagation, so
$B^{d_{KL}(q_{i},h_{k})}$ is fully trainable and can integrate with
contrastive learning methods, making the STAR method _generic_.

####  4.3.2 Loss Analysis

Since STAR-DOWN discovers fine-grained categories in the Euclidean space, we
analyze the second term $L_{2-2}^{i}$ of the loss $L_{2}^{i}$, which optimizes
sample distributions in the Euclidean space:

$\displaystyle L_{2-2}^{i}=$ $\displaystyle-\sum_{h_{j}\in
N_{i}}\omega_{j}\cdot\log\frac{\exp(q_{i}^{\mathrm{T}}h_{j}/\tau)}{\sum\limits_{h_{k}\in
Q}B^{d_{KL}(q_{i},h_{k})}\cdot\exp(q_{i}^{\mathrm{T}}h_{k}/\tau)}$ (7)
$\displaystyle=$ $\displaystyle\sum_{h_{j}\in
N_{i}}\omega_{j}\cdot(\log\sum\limits_{h_{k}\in
Q}B^{d_{KL}(q_{i},h_{k})}\cdot\exp(q_{i}^{\mathrm{T}}h_{k}/\tau)$
$\displaystyle-(q_{i}^{\mathrm{T}}h_{j}/\tau)).$

In the loss $L_{2-2}^{i}$, $B^{d_{KL}(q_{i},h_{k})}$ uses CSS in the
logarithmic space to guide sample distributions in the Euclidean space. A
large $d_{KL}(q_{i},h_{k})$ (low semantic similarity) causes $q_{i}$ to
distance itself from $h_{k}$ in the Euclidean space, reducing
$q_{i}^{\mathrm{T}}h_{k}$, while a small $d_{KL}(q_{i},h_{k})$ allows $q_{i}$
to remain relatively close to $h_{k}$ compared to negative samples. This
results in the formation of compact fine-grained clusters, with each cluster
representing a discovered category. We also analyze the STAR method from the
perspectives of gradient, clustering, and generalized EM. Detailed analyses
are provided in Appendix

###  4.4 Inference

Previous methods (An et al., 2023a, 2024) use clustering inference on sample
embeddings from $F_{\theta}$ extracted from $D_{test}$, which is unsuitable
for real-time tasks, such as intent detection, which require immediate
response and can not wait to collect enough test samples for clustering. We
introduce an alternative, centroid inference, suitable for both real-time and
other contexts. Using $F_{\theta}$, we derive sample embeddings from
$D_{train}$ and assign fine-grained pseudo labels through clustering. For each
fine-grained cluster, only the embeddings of samples from the predominant
coarse-grained category (the category with the most samples in this fine-
grained cluster) are averaged to form centroid representations. These
approximated centroids are used to determine the fine-grained category of each
test sample based on cosine similarity. A visual explanation is in Appendix

##  5 Experiments

###  5.1 Experimental Settings

####  5.1.1 Datasets

Dataset | $\lvert\mathcal{C}\rvert$ | $\lvert\mathcal{F}\rvert$ | # Train | # Test
---|---|---|---|---
CLINC | 10 | 150 | 18000 | 1000
WOS | 7 | 33 | 8362 | 2420
HWU64 | 18 | 64 | 8954 | 1031
Table 1: Statistics of datasets (An et al., 2023a). #: number of samples.
$\lvert\mathcal{C}\rvert$: number of coarse-grained categories.
$\lvert\mathcal{F}\rvert$: number of fine-grained categories.

We conduct experiments on three benchmark datasets: CLINC (Larson et al.,
2019), WOS (Kowsari et al., 2017), and HWU64 (Liu et al., 2021). CLINC is an
intent detection dataset spanning multiple domains. WOS is used for paper
abstract classification, and HWU64 is designed for assistant query
classification. Dataset statistics are provided in Table 1.

####  5.1.2 Baselines for Comparison

We compare our methods against the following baselines. Language models: BERT
(Devlin et al., 2019b), BERT with coarse-grained fine-tuning, Llama2 (Touvron
et al., 2023), Llama2 with coarse-grained fine-tuning and GPT4 (Achiam et al.,
2023). Self-training baselines: DeepCluster (DC) (Caron et al., 2018),
DeepAlignedCluster (DAC) (Zhang et al., 2021), and PseudoPrototypicalNet
(PPNet) (Boney and Ilin, 2017; Ji et al., 2020). Contrastive learning
baselines: SimCSE (Gao et al., 2021), Ancor (Bukchin et al., 2021), Delete (Wu
et al., 2020), Nearest-Neighbor Contrastive Learning (NNCL) (Dwibedi et al.,
2021), Contrastive Learning with Nearest Neighbors (CLNN) (Zhang et al.,
2022), Soft Neighbor Contrastive Learning (SNCL) (Chongjian et al., 2022),
Weighted Self-Contrastive Learning (WSCL) (An et al., 2022), Denoised
Neighborhood Aggregation (DNA), and Dynamic Order Weighted Network (DOWN) (An
et al., 2023a, 2024). We also explore variants incorporating the cross-entropy
loss (+CE).

####  5.1.3 Evaluation Metrics

To evaluate the quality of the discovered fine-grained clusters, we use the
Adjusted Rand Index (ARI) (Hubert and Arabie, 1985) and Normalized Mutual
Information (NMI) (Lancichinetti et al., 2009). For assessing classification
performance, we use clustering Accuracy (ACC) (kuh, 1955; An et al., 2023a).
Detailed descriptions of these metrics are provided in Appendix

####  5.1.4 Implementation Details

To ensure fair comparisons with baselines, we use the BERT-base-uncased model
as the backbone for all STAR method variants. We adhere to the hyperparameters
used by the integrated baselines to demonstrate the effectiveness of our STAR
method. The learning rate for both pre-training and training is $5e^{-5}$,
using the AdamW optimizer with a 0.01 weight decay and 1.0 gradient clipping.
The momentum coefficient $m$ is set to 0.99. The batch size for pre-training,
training, and testing is 64. The temperature $\tau$ is set to 0.07. The number
of neighbors $k$ is set to {120, 120, 250} for the CLINC, HWU64, and WOS
datasets, respectively. Epochs for pretraining and training are set to 100 and
20, respectively. Further details are provided in Appendix

####  5.1.5 Research Questions

The following research questions (RQs) are investigated: 1\. What is the
impact of STAR method on FCDC tasks? 2\. What are the effects of the proposed
real-time centroid inference compared to traditional clustering inference? 3\.
How does each component of the STAR method affect performance? 4\. How can we
effectively and efficiently set the base for the exponential function in the
STAR method?

###  5.2 Result Analysis (RQ1)

Methods | HWU64 | CLINC | WOS
---|---|---|---
ACC | ARI | NMI | ACC | ARI | NMI | ACC | ARI | NMI
BERT (Devlin et al., 2019b) | 33.52 | 17.04 | 56.90 | 34.37 | 17.61 | 64.75 | 31.97 | 18.36 | 45.15
BERT + CE | 37.89 | 33.68 | 74.63 | 43.85 | 32.37 | 78.58 | 38.29 | 36.94 | 64.72
Llama2 (Touvron et al., 2023) | 19.27±1.21 | 5.21±0.46 | 44.34±0.85 | 20.77±2.61 | 5.83±1.52 | 49.7±3.68 | 9.85±1.14 | 1.26±0.75 | 18.27±2.28
Llama2 + CE | 32.40±5.46 | 17.32±5.95 | 57.53±5.78 | 45.69±6.85 | 29.38±6.55 | 72.66±7.13 | 18.51±1.50 | 7.8±1.18 | 29.66±3.23
GPT4 (Achiam et al., 2023) | 10.77±1.86 | 0.14±0.05 | 35.17±3.68 | 9.56±2.12 | 0.11±0.06 | 46.69±3.24 | 7.56±1.51 | 0.15±0.04 | 27.78±2.98
DC (Caron et al., 2018) | 18.05 | 43.34 | 29.74 | 26.40 | 12.51 | 61.26 | 29.17 | 13.98 | 53.27
DAC (Zhang et al., 2021) | 29.14 | 12.89 | 52.99 | 29.16 | 14.15 | 62.78 | 28.47 | 15.94 | 43.52
DC + CE | 41.73 | 27.81 | 66.81 | 30.28 | 13.56 | 62.38 | 38.76 | 35.21 | 60.30
DAC + CE | 42.19 | 28.15 | 66.50 | 42.09 | 28.09 | 72.78 | 39.42 | 33.67 | 61.60
PPNet (Ji et al., 2020) | 58.36±2.51 | 47.63±1.96 | 79.75±1.02 | 70.15±1.86 | 59.31±0.96 | 85.08±0.81 | 62.59±1.41 | 50.81±1.21 | 72.19±0.68
STAR-PPNet (ours) | 63.19±2.38 | 52.21±1.33 | 81.66±1.21 | 73.21±1.97 | 61.87±0.79 | 86.16±0.47 | 66.15±1.33 | 53.61±1.24 | 73.82±0.74
Delete (Wu et al., 2020) | 21.30 | 6.52 | 44.13 | 47.11 | 31.28 | 73.39 | 24.50 | 11.68 | 35.47
SimCSE (Gao et al., 2021) | 24.48 | 8.42 | 46.94 | 40.22 | 23.57 | 69.02 | 25.87 | 13.03 | 38.53
Ancor + CE | 32.90 | 30.71 | 74.73 | 44.44 | 31.50 | 74.67 | 39.34 | 26.14 | 54.35
NNCL (Dwibedi et al., 2021) | 32.98 | 30.02 | 73.24 | 17.42 | 13.93 | 67.56 | 29.64 | 28.51 | 61.37
SimCSE + CE | 34.04 | 31.81 | 74.86 | 52.53 | 37.03 | 77.39 | 41.28 | 34.47 | 61.62
Delete + CE | 35.13 | 31.84 | 74.88 | 47.87 | 33.79 | 76.25 | 41.53 | 33.78 | 61.01
CLNN (Zhang et al., 2022) | 37.21 | 34.66 | 75.27 | 19.96 | 14.76 | 68.30 | 29.48 | 28.42 | 60.99
Ancor (Bukchin et al., 2021) | 37.34 | 34.75 | 74.99 | 45.60 | 33.11 | 75.23 | 41.20 | 37.00 | 65.42
SNCL (Chongjian et al., 2022) | 42.32 | 38.17 | 76.39 | 55.01 | 45.64 | 82.93 | 36.27 | 33.62 | 62.35
WSCL (An et al., 2022) | 59.52 | 49.34 | 79.31 | 74.02 | 62.98 | 88.37 | 65.27 | 51.78 | 72.46
DNA (An et al., 2023a) | 70.81 | 59.66 | 83.31 | 87.66 | 81.82 | 94.69 | 74.57 | 63.30 | 76.86
STAR-DNA (ours) | 75.79±0.93 | 65.27±1.12 | 85.34±0.36 | 89.25±0.17 | 83.47±0.27 | 95.11±0.05 | 77.19±0.81 | 64.97±0.75 | 77.91±0.76
DOWN (An et al., 2024) | 78.92 | 68.17 | 86.22 | 91.79 | 86.70 | 96.05 | 80.00 | 67.09 | 78.87
STAR-DOWN (ours) | 80.31±0.26 | 70.22±0.59 | 87.28±0.31 | 92.45±0.38 | 87.05±0.17 | 96.20±0.07 | 81.98±0.67 | 69.27±0.60 | 79.99±0.40

Table 2: The average performance (%) in terms of Accuracy (ACC), Adjusted Rand
Index (ARI), and Normalized Mutual Information (NMI) on three datasets for the
FCDC language task. To ensure fair comparisons with previous works (An et al.,
2022, 2023a, 2024) and demonstrate the effectiveness of STAR, we use the same
clustering inference mechanism and also average the results over three runs
with identical common hyperparameters. Some baselines results are cited from
aforementioned previous works, where standard deviations are not originally
provided.

Methods | HWU64 | CLINC | WOS
---|---|---|---
| ACC | ARI | NMI | ACC | ARI | NMI | ACC | ARI | NMI
STAR-DOWN (clustering) | 80.31±0.26 | 70.22±0.59 | 87.28±0.31 | 92.45±0.38 | 87.05±0.17 | 96.20±0.07 | 81.98±0.67 | 69.27±0.60 | 79.99±0.40
STAR-DOWN (centroid) | 79.44±0.51 | 69.13±0.75 | 86.97±0.40 | 92.60±0.45 | 87.16±0.53 | 96.21±0.09 | 81.89±0.53 | 69.05±0.39 | 79.78±0.32

Table 3: Comparison of clustering and centroid inference mechanisms.
"Clustering" clusters test set sample embeddings to determine each sample’s
fine-grained category, while "Centroid" infers the category by comparing each
test sample’s cosine similarity to fine-grained centroids.

As shown in Table 2, STAR method variants outperform SOTA methods across all
datasets and metrics, validating the effectiveness of the STAR method in FCDC
tasks. Language models like BERT, Llama2 and GPT4 (Devlin et al., 2019b;
Touvron et al., 2023; Achiam et al., 2023) (GPT4 prompt in Appendix ) perform
poorly on the FCDC task due to the lack of fine-grained supervisory
information. Self-training methods like DC, DAC, and PPNet (Caron et al.,
2018; Zhang et al., 2021; Ji et al., 2020) also struggle because they rely on
noisy fine-grained pseudo-labels and overlook comprehensive semantic
similarities (CSS). Contrastive learning methods such as SNCL (Chongjian et
al., 2022) and WSCL (An et al., 2022) perform better by leveraging positive
pairs. DNA (An et al., 2023a) and DOWN (An et al., 2024) further enhance
feature quality by filtering false positives and weighting them by rank.
However, these methods still do not use CSS for sample distributions.
Integrating the STAR method with existing baselines enhances performance
across all datasets, consistently improving sample distributions in Euclidean
space.

The superior performance of STAR is attributed to three factors: First,
bidirectional KL divergence measures CSS, pushing negative samples further
away and relatively bringing positive samples closer based on CSS magnitude,
making fine-grained clusters easier to distinguish. Second, the base $B$ of
the exponential in Eq. 4 is a trainable scalar, balancing CSS magnitude and
semantic structure. Third, STAR variants iteratively bootstrap model
performance in neighborhood retrieval and representation learning through a
generalized EM process (detailed in Appendix ).

###  5.3 Inference Mechanism Comparison (RQ2)

Previous methods (Chongjian et al., 2022; An et al., 2023a, 2024) perform a
nearest neighbor search over the examples of the found fine-grained clusters
for fine-grained category prediction (we refer to this technique as cluster
inference). We speed up this process making it better suitable for real-time
tasks by developing a centroid inference mechanism (see Section 4.4). Results
in Table 3 demonstrates that results of centroid inference are competitive
with cluster inference. When results are of the former are lower, this is due
to two factors: clustering inference leverages inter-relations among test set
samples for richer features, while centroid inference depends on centroids
derived from noisy samples with fine-grained pseudo-labels. Despite these
issues, centroid inference remains a viable option for real-time applications,
balancing immediate analytical needs with slight performance trade-offs.

###  5.4 Ablation Study (RQ1 & RQ3)

Methods | ACC | ARI | NMI
---|---|---|---
ours | 80.31±0.26 | 70.22±0.59 | 87.28±0.31
w/o CE | 78.61±0.44 | 67.32±0.86 | 85.62±0.36
w/o KL loss | 78.97±0.32 | 68.03±0.36 | 85.81±0.16
w/o KL weight | 79.26±0.42 | 68.86±0.37 | 86.21± 0.07
w/o KL weight and loss | 78.96±0.15 | 68.21±0.22 | 86.32±0.10

Table 4: Results (%) of the ablation study for STAR-DOWN on the HWU64 Dataset.

We examine the impact of various components of the STAR method in STAR-DOWN,
as detailed in Table 4. Our results yield the following insights. (1)
Excluding coarse-grained supervision information during training (w/o CE)
reduces model performance, as this information is crucial for effective
representation learning. (2) Omitting the first loss term (w/o KL loss) from
Eq. 4 diminishes performance. The KL loss term aligns the KL divergence
between data samples and the query with their semantic similarities. Without
it, $B^{d_{KL}(q_{i},h_{k})}$ fails to guide the query sample distribution
based on semantic similarities in Eq. 4. (3) Removing the KL weight
$B^{d_{KL}(q_{i},h_{k})}$ from Eq. 4 (w/o KL weight) reduces effectiveness.
The loss no longer utilizes fine-grained semantic similarities measured by
$B^{d_{KL}(q_{i},h_{k})}$ in the logarithmic space to direct the query sample
distribution in comparison to all samples. (4) Eliminating both the KL loss
term and the KL weight in Eq. 4 leads to a performance decline. This omission
prevents the optimization of the query sample towards positive samples in the
logarithmic space and fails to leverage fine-grained semantic similarities in
logarithmic space to influence the distribution of query samples relative to
all samples in the Euclidean space.

###  5.5 Exponential Base Impact (RQ4)

In the STAR method’s loss equation (Eq. 4), $B^{d_{KL}(q_{i},h_{k})}$
modulates the distribution of $q_{i}$ and $h_{k}$ in the Euclidean space based
on their semantic similarity in the logarithmic space, as quantified by the
bidirectional KL divergence. The base $B$ is used to enhance semantic
differences, improving the discriminability of fine-grained categories. We
experimented with multiple constant values and a trainable configuration for
$B$, with multiple STAR-DOWN results presented in Table 5. The multiple STAR-
DOWN methods with various base values consistently outperform the DOWN method
(Table 2), demonstrating the effectiveness and robustness of the STAR method
regardless of the base value $B$.111For interesting form similarities to
physical laws within the STAR method, see Appendix . Notably, base values that
are either too low (e.g., $e$) or too high (e.g., 66) disrupt the semantic
representation by inadequately or excessively emphasizing semantic
similarities in the logarithmic space. To set base value conveniently, we set
$B$ as a trainable scalar, achieving favorable outcomes as indicated in Table
5.

Base value | ACC | ARI | NMI
---|---|---|---
trainable $B$ (ours) | 80.31±0.26 | 70.22±0.59 | 87.28±0.31
$e$ | 79.96±0.12 | 68.89±0.55 | 86.66±0.10
10 | 80.22±0.27 | 69.61±0.65 | 87.08±0.30
16 | 80.73±0.32 | 70.14±0.58 | 87.25±0.36
66 | 80.57±0.38 | 70.20±0.52 | 87.07±0.15

Table 5: Averaged results (%) and their standard deviations over three runs of
multiple STAR-DOWN methods with five different base values on the HWU64
dataset. To set base value conveniently, we set $B$ as a trainable scalar.

###  5.6 Inference of Category Semantics

Prior works (An et al., 2023a, 2024) only discovered fine-grained categories
and assigned them numeric indices without elucidating the categories
semantics, thus constraining their broader application. We propose utilizing
the commonsense reasoning capabilities of large language models (LLMs) to
infer the semantics of these categories. Specifically, we employ a trained
encoder, $F_{\theta}$, to extract embeddings from all train set samples and
cluster these embeddings to assign fine-grained pseudo-labels to each train
set sample. For each fine-grained category indicated by a specific pseudo-
label, we aggregate all predicted samples from the training set and use an LLM
to deduce the category semantics. Details on the LLM prompt are provided in
Appendix .

###  5.7 Visualization

Figure 4: The t-SNE visualization of sample embeddings from STAR-DOWN method
on the HWU64 dataset, with different colors representing different coarse-
grained categories. The distinct clusters represent the discovered fine-
grained categories.

We visualize the sample embeddings of STAR-DOWN in Figure 4. The results
demonstrate that our method forms distinguishable clusters for fine-grained
categories, proving STAR’s effectiveness in separating dissimilar samples and
clustering similar ones. Additionally, we visualize the generalized EM
perspective of STAR-DOWN in Appendix.

##  6 Conclusion

We propose the STAR method for fine-grained category discovery in natural
language texts, which utilizes comprehensive semantic similarities in the
logarithmic space to guide the distribution of textual samples, including
conversational intents, scientific paper abstracts, and assistant queries, in
the Euclidean space. STAR pushes query samples further away from negative
samples and brings them closer to positive samples based on the comprehensive
semantic similarities magnitude. This process forms compact clusters, each
representing a discovered category. We theoretically analyze the effectiveness
of STAR method. Additionally, we introduce a centroid inference mechanism that
addresses previous gaps in real-time evaluations. Experiments on three natural
language benchmarks demonstrate that STAR achieves new state-of-the-art
performance in fine-grained category discovery tasks for text classification.

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