# Gravitational lensing by a black hole in effective loop quantum gravity

Qi-Ming Fu<EMAIL_ADDRESS>Department of Physics, College of Sciences,
Northeastern University, Shenyang 110819, China  Institute of Physics, Shaanxi
University of Technology, Hanzhong 723000, China  Xin Zhang111Corresponding
author<EMAIL_ADDRESS>Department of Physics, College of Sciences,
Northeastern University, Shenyang 110819, China  Key Laboratory of Data
Analytics and Optimization for Smart Industry (Northeastern University),
Ministry of Education, China

###### Abstract

It is well known that general relativity is an effective theory of gravity at
low energy scale, and actually quantum effects cannot be ignored in the
strong-field regime. As a strong gravitational object, black hole plays a key
role in testing the quantum effects of gravity in the strong-field regime. In
this paper, we focus on black hole in effective loop quantum gravity and
investigate what the influences are of the quantum effects on the weak and
strong bending angles of light rays. We find that this black hole could be a
Schwarzschild black hole, a regular black hole, a one-way traversable
wormhole, or a two-way traversable wormhole for the different values of the
quantum parameter, and the strong bending angle for this compact object
exhibits two different divergent behaviors, i.e., the logarithmic divergence
and non-logarithmic divergence. There are a series of relativistic images on
both sides of the optical axis. Only the outermost one can be resolved as a
single image, and all others are packed together at the limiting angular
position. It is interesting to note that the angular separation between the
outermost relativistic image and the others initially increases and then
decreases as the quantum parameter increases, indicating that there is a
maximum in the angular separation. The maximum is reached after the black hole
becomes a wormhole, which can be taken as a signal for the formation of the
wormhole. Moreover, the limiting angular position decreases as the quantum
parameter increases but a little bounce occurs after the formation of the
wormhole, and the relative magnification in magnitudes first decreases and
then increases as the quantum parameter increases.

###### pacs:

95.30.Sf, 98.62.Sb, 97.60.Lf

##  I Introduction

It is widely known that general relativity (GR) has been very successful in
modern physics. It has passed an array of tests on solar systems [1] as well
as on binary pulsars [1, 2] and cosmology [3, 4]. Besides, the direct
detections of gravitational waves [5, 6, 7] and the image of the black hole
shadow [8] further confirm the validity of GR at the scale of high curvature,
such as near neutron stars or black holes. However, GR may not be the complete
theory of gravity, since its nonrenormalization, the disability of explaining
dark matter and dark energy, and so on. One of the most important issues is
the prediction of the spacetime singularity inside a black hole or at the
beginning of the universe, which suggests that the theory breaks down there.
Usually, it is believed that the singularity can be circumvented by some
quantum effects. To address the issues like this, one need to resort to the
quantum theory of gravity.

Although a well-developed quantum theory of gravity is still out of reach,
many efforts have been made to understand the spacetime singularity. One of
the most successful attempts is the phase space quantization, i.e., the so-
called polymerization schemes developed in loop quantum gravity (LQG), which
has been used to resolve the big-bang singularity [9, 10, 11, 12]. In this
quantization scheme, a small parameter called the polymer scale is introduced.
When approaching this scale, the quantum effects can no longer be neglected.
On the other hand, as an important candidate of the strong gravitational
regime, black hole plays a key role in understanding the nature of spacetime
at high-energy scales. It is natural to describe the black-hole spacetimes by
considering the quantum corrections, such as the polymerization schemes. Over
the past few years, a lot of effective polymerized black holes have been
constructed [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 27, 25, 26, 23], most
of which focus on the spherically symmetric cases. The common features of
these black holes are that the classical singularities inside them are
replaced by a transition surface, which connects a black hole to a white hole,
and that the spacetime is regular everywhere. Besides, since the resulting
quantum corrected spacetimes can still be described by an explicit line
element, one can study such nonsingular effective geometries by using the
well-developed mechanism for semiclassical black holes.

There have already been some investigations on various aspects of the
polymerized black holes in the literature. For example, the authors of Ref.
[28] studied the particle production by a regular LQG black hole, and showed
that the unitarity is recovered for the black hole evaporation. The relation
between the polymerized black holes and the non-singular black holes in
mimetic gravity was investigated in Ref. [29]. The exact solutions for the
static spherically symmetric black hole interior were obtained in Ref. [30].
In a status report [31], the authors gave a review about the possible
experimental or observational consequences of black holes in LQG. The greybody
factors for the scalar and fermion fields in the background spacetime of a
polymerized black hole were explicitly calculated in Ref. [32]. The
quasinormal modes of the polymerized spherically symmetric black holes were
studied in Refs. [33, 34]. In Ref. [35], the authors analyzed the
thermodynamic properties of the quantum corrected polymerized black holes and
investigated the effects of the corresponding quantum corrections as functions
of the masses of the black and white holes. The Hawking radiation spectra and
evaporation of spherically symmetric black holes in LQG were investigated in
Refs. [36, 37, 38, 39, 40, 41]. In Ref. [42], the authors constructed the
rotating polymerized black hole using the revised Newman-Janis algorithm and
investigated the shadow cast by this black hole. One can see Refs. [43, 44,
45, 46, 47, 48, 49] for more investigations on the polymerized black holes. In
this paper, we mainly focus on the polymerized black hole proposed in Refs.
[25, 26] and investigate the strong and weak deflection angles of light rays
by this black hole.

On the other hand, light rays will be bent as they pass through a massive
object, which is known as gravitational lensing, and is a prediction of GR.
Since the first observation of the deflection of starlight near the Sun in
1919, gravitational lensing by other astronomical objects has been observed
many times. In modern astronomy, gravitational lensing plays a key role in
many aspects in astrophysical and cosmological observations. Most studies of
gravitational lensing are in the weak deflection limit, and the bending angle
is at most a few arcseconds. However, when passing close to a black hole,
light rays may revolve many times around the black hole before arriving at the
observer. This phenomenon is known as the so-called strong gravitational
lensing, which was introduced a few decades ago for Schwarzschild spacetime
[50]. An analytical expression for the strong deflection angle of the
Schwarzschild black hole was given in Ref. [51]. Subsequently, the
observables, such as the positions, magnifications, and time delays of the
images for the Schwarzschild black hole in the strong deflection limit were
investigated systematically in Refs. [52, 53, 54, 55]. In Ref. [56], the
author introduced an analytical framework to investigate the strong
gravitational lensing effect for a generic static spherically symmetric
metric, and then applied it to Reissner-Nordström black hole and other
spherically symmetric objects with a photon sphere. Numerical investigations
on the gravitational lensing were performed in Refs. [53, 55, 57]. Other
interesting studies on gravitational lensing in spherically symmetric
spacetimes can be found in Refs. [58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70,
71, 72, 73, 74, 75, 76, 77, 80, 81, 60, 63, 78, 79, 82]. Besides,
gravitational lensing by a rotating black hole has been analyzed in Refs. [83,
84, 85, 86].

Although there are many studies on polymerized black holes, the deflection of
light rays by this type of black holes has not yet been studied. The
deflection angle of light rays can be used as an effective way to study black
holes and can provide us with valuable information about black holes. For
example, the mass of the black hole and the quantum parameters resulting from
quantum effects can be determined by analyzing the angular position of the
relativistic images. In addition, the angular separation between the outermost
relativistic image and the other images can present interesting features.
Thus, it is necessary to study the gravitational lensing effect of a
polymerized black hole in detail, and it is interesting to know what the
effects are of the quantum corrections on the deflection angle.

This paper is organized as follows. In Sec. II, we first give a brief review
of a LQG black hole and then analyze the deflection angle of light rays by
such a black hole. In Secs. III and IV, we calculate the bending angles of
light rays in the weak and strong deflection limits, respectively. The
observables of the relativistic images are investigated in Sec. V. Section VI
comes with the conclusion.

##  II Black holes in effective loop quantum gravity

Let us start with the quantum extension of the static and spherically
symmetric metric by solving the loop quantum gravity (LQG) effective equations
[25, 26, 42]:

$\displaystyle
ds^{2}=-\mathcal{A}(y)d\tau^{2}+\frac{dy^{2}}{\mathcal{A}(y)}+\mathcal{C}^{2}(y)d\Omega_{2}^{2},~{}$
(1)

where the metric functions in terms of the radial coordinate
$y\in(-\infty,\infty)$ are given by

$\displaystyle\mathcal{A}(y)$ $\displaystyle=$
$\displaystyle\left(1-\frac{1}{\sqrt{2A_{\lambda}(1+y^{2})}}\right)\frac{1+y^{2}}{\mathcal{C}^{2}(y)},$
(2) $\displaystyle\mathcal{C}^{2}(y)$ $\displaystyle=$
$\displaystyle\frac{A_{\lambda}}{\sqrt{1+y^{2}}}\frac{M_{B}^{2}(y+\sqrt{1+y^{2}})^{6}+M_{W}^{2}}{(y+\sqrt{1+y^{2}})^{3}},$
(3)

with $M_{B}$ and $M_{W}$ two Dirac observables in the model. The dimensionless
and non-negative parameter $A_{\lambda}$ is defined as
$A_{\lambda}\equiv(\lambda_{k}/(M_{B}M_{W}))^{2/3}/2$, where $\lambda_{k}$ is
a quantum parameter originated from holonomy modifications [25, 26]. An
important feature of this black hole is that inside the black hole, the areal
radius $\mathcal{C}$ reaches a minimum, which stands for a spacelike
transition surface that smoothly connects an asymptotically Schwarzschild
black hole to a white hole with mass $M_{B}$ and $M_{W}$, respectively [42].
In this paper, we focus on the physically interesting case with
$M_{B}=M_{W}=M$ corresponding to the symmetric bounce, i.e., the spacetime is
symmetric under $y\rightarrow-y$. Without loss of generality, we concentrate
on the branch of $y\geqslant 0$. After redefining two new coordinates
$r\equiv\sqrt{8A_{\lambda}}M_{B}y$ and
$t\equiv\tau/(\sqrt{8A_{\lambda}}M_{B})$, the metric (1) can be rewritten as
[42]

$\displaystyle ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+C(r)(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),~{}$ (4)

where

$\displaystyle A(r)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{1}{B(r)}\\!=\\!\frac{\sqrt{8A_{\lambda}M^{2}+r^{2}}\big{(}\sqrt{8A_{\lambda}M^{2}+r^{2}}-2M\big{)}}{2A_{\lambda}M^{2}+r^{2}},~{}~{}~{}$
(5) $\displaystyle C(r)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!2A_{\lambda}M^{2}+r^{2}.$ (6)

Then, the radius of the horizon is $r_{h}=2M\sqrt{1-2A_{\lambda}}$. Near the
horizon $r=r_{h}+\epsilon$, $A(r)$ can be expanded as

$\displaystyle
A(r_{h}+\epsilon)=\frac{\sqrt{1-2A_{\lambda}}\epsilon}{(2-3A_{\lambda})M}+\mathcal{O}(\epsilon^{2}).$
(7)

Obviously, if $A_{\lambda}\ll\frac{1}{2}$, $A(r)\simeq\frac{\epsilon}{2M}$
near the horizon, which is the same as the Schwarzschild case. Besides, when
$r\rightarrow\infty$, we have $A(r)\rightarrow 1-\frac{2M}{r}$. Namely, the
spacetime reduces to the Schwarzschild black hole when it approaches infinity.
Thus, the metric (4) can not be distinguishable from the Schwarzschild black
hole at or beyond the horizon for $A_{\lambda}\ll\frac{1}{2}$.

It can be easily shown that the metric (4) represents (i) a Schwarzschild
black hole if $A_{\lambda}=0$ and $M\neq 0$, (ii) a regular black hole if
$0<A_{\lambda}<\frac{1}{2}$, (iii) a one-way traversable wormhole with a null
throat if $A_{\lambda}=\frac{1}{2}$, (vi) a traversable wormhole with a two-
way throat at $r=0$ if $A_{\lambda}>\frac{1}{2}$, and (v) a flat spacetime if
$M=0$. From the line element (4), the Lagrangian for the geodesics can be
expressed as

$\displaystyle
2\mathcal{L}=-A(r)\dot{t}^{2}+B(r)\dot{r}^{2}+C(r)(\dot{\theta}^{2}+\sin^{2}\theta\dot{\phi}^{2}),$
(8)

where the dot denotes the derivative with respect to an affine parameter along
the trajectory. Since the independence of the Lagrangian on $t$ and $\phi$,
one can obtain two conserved quantities:

$\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial\dot{t}}=-A(r)\dot{t}=-E,$
(9) $\displaystyle p_{\phi}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}}{\partial\dot{\phi}}=C(r)\dot{\phi}\sin^{2}\theta.$
(10)

The lightlike geodesics in the equatorial plane $\theta=\frac{\pi}{2}$ are
given by

$\displaystyle-A(r)\dot{t}^{2}+B(r)\dot{r}^{2}+C(r)\dot{\phi}^{2}=0.~{}$ (11)

Inserting the two conserved quantities into Eq. (11), the null geodesics can
be cast as

$\displaystyle\dot{r}^{2}+V(r)=0,$ (12)

where the effective potential is $V(r)\equiv\frac{L^{2}}{C(r)B(r)}-E^{2}$ and
$L\equiv C(r)\dot{\phi}$. Obviously, light rays can only exist in a region for
$V(r)\leq 0$. The effective potential and its derivatives with respect to the
radial coordinate $r$ can be calculated as

$\displaystyle V(r)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!E^{2}\left(b^{2}\frac{\left(8M^{2}A_{\lambda}-2M\sqrt{8M^{2}A_{\lambda}+r^{2}}+r^{2}\right)}{\left(2M^{2}A_{\lambda}+r^{2}\right){}^{2}}-1\right),$
(13) $\displaystyle V^{\prime}(r)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!-\frac{2b^{2}E^{2}r}{\left(2A_{\lambda}M^{2}+r^{2}\right)^{3}\sqrt{8A_{\lambda}M^{2}+r^{2}}}\bigg{(}2A_{\lambda}M^{2}\left(7\sqrt{8A_{\lambda}M^{2}+r^{2}}-15M\right)+r^{2}\left(\sqrt{8A_{\lambda}M^{2}+r^{2}}-3M\right)\bigg{)},$
(14) $\displaystyle V^{\prime\prime}(r)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{2b^{2}E^{2}}{\left(2A_{\lambda}M^{2}+r^{2}\right)^{4}\left(8A_{\lambda}M^{2}+r^{2}\right)^{3/2}}\bigg{(}32A_{\lambda}^{3}M^{6}\left(15M-7\sqrt{8A_{\lambda}M^{2}+r^{2}}\right)+44A_{\lambda}^{2}M^{4}r^{2}\bigg{(}11\sqrt{8A_{\lambda}M^{2}+r^{2}}$
(15) $\displaystyle-$
$\displaystyle\\!\\!24M\bigg{)}+3r^{6}\left(\sqrt{8A_{\lambda}M^{2}+r^{2}}-4M\right)+8A_{\lambda}M^{2}r^{4}\left(11\sqrt{8A_{\lambda}M^{2}+r^{2}}-30M\right)\bigg{)},$
$\displaystyle V^{\prime\prime\prime}(r)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!-\frac{24b^{2}E^{2}r}{\left(2A_{\lambda}M^{2}+r^{2}\right)^{5}\left(8A_{\lambda}M^{2}+r^{2}\right)^{5/2}}\bigg{(}16A_{\lambda}^{4}M^{8}\left(351M-160\sqrt{8A_{\lambda}M^{2}+r^{2}}\right)+40A_{\lambda}^{3}M^{6}r^{2}\big{(}32\sqrt{8A_{\lambda}M^{2}+r^{2}}$
(16) $\displaystyle-$
$\displaystyle\\!\\!67M\big{)}+72A_{\lambda}^{2}M^{4}r^{4}\left(7\sqrt{8A_{\lambda}M^{2}+r^{2}}-18M\right)+r^{8}\left(\sqrt{8A_{\lambda}M^{2}+r^{2}}-5M\right)+2A_{\lambda}M^{2}r^{6}\big{(}23\sqrt{8A_{\lambda}M^{2}+r^{2}}$
$\displaystyle-$ $\displaystyle\\!\\!75M\big{)}\bigg{)},$ $\displaystyle
V^{\prime\prime\prime\prime}(r)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\frac{24b^{2}E^{2}}{\left(2M^{2}A_{\lambda}+r^{2}\right){}^{6}\left(8M^{2}A_{\lambda}+r^{2}\right){}^{7/2}}\bigg{(}5r^{12}\left(\sqrt{8M^{2}A_{\lambda}+r^{2}}-6M\right)+80M^{2}r^{10}A_{\lambda}\left(4\sqrt{8M^{2}A_{\lambda}+r^{2}}-15M\right)$
(17) $\displaystyle+$
$\displaystyle\\!\\!256M^{12}A_{\lambda}^{6}\left(160\sqrt{8M^{2}A_{\lambda}+r^{2}}-351M\right)+1920M^{10}r^{2}A_{\lambda}^{5}\left(301M-136\sqrt{8M^{2}A_{\lambda}+r^{2}}\right)$
$\displaystyle+$
$\displaystyle\\!\\!128M^{8}r^{4}A_{\lambda}^{4}\left(5\sqrt{8M^{2}A_{\lambda}+r^{2}}+168M\right)+240M^{6}r^{6}A_{\lambda}^{3}\left(117\sqrt{8M^{2}A_{\lambda}+r^{2}}-280M\right)$
$\displaystyle+$
$\displaystyle\\!\\!60M^{4}r^{8}A_{\lambda}^{2}\left(87\sqrt{8M^{2}A_{\lambda}+r^{2}}-254M\right)\bigg{)},$

where the impact parameter is defined as
$b(r_{0})\equiv\frac{L}{E}=\sqrt{\frac{C(r_{0})}{A(r_{0})}}$, and $r_{0}$ is
the closest distance between the light ray and the black hole. Light rays with
the critical impact parameter $b_{m}\equiv b(r_{m})$ form unstable circular
orbits called photon sphere, where $r_{m}$ is the radius of the photon sphere,
which is determined by $V(r_{m})=V^{\prime}(r_{m})=0$ and
$V^{\prime\prime}(r_{m})\leqslant 0$. Light rays with the impact parameter
$b<b_{m}$ are captured by the black hole or the wormhole while light rays with
$b>b_{m}$ are deflected. The dimensionless effective potential $V(r)/E^{2}$
for the light ray with the critical impact parameter $b_{m}$ is plotted in
Fig. 1. Obviously, for $0<A_{\lambda}<\frac{225}{392}$, there are two unstable
photon spheres located at $r=\pm r_{m}$, respectively. For
$A_{\lambda}\geqslant\frac{225}{392}$, these two unstable photon spheres merge
into a marginally unstable photon sphere located at $r_{m}=0$. Besides, it
should be noted that the light orbits at $r=0$ satisfy $V^{\prime}(0)=0$ and
$V^{\prime\prime}(0)>0$ for $\frac{1}{2}\leqslant
A_{\lambda}<\frac{225}{392}$, however $V(0)\neq 0$, which indicates
$\dot{r}\neq 0$. Thus there is no antiphoton sphere at $r=0$ according to its
definition in Refs. [78, 79].

Figure 1: Dimensionless effective potential $V/E^{2}$ in terms of the radial
coordinate $r$. The parameters are set to $M=1$, $A_{\lambda}=0$ for red line
(Schwarzschild metric), $A_{\lambda}=0.2$ for green dashed line (regular black
hole), $A_{\lambda}=\frac{1}{2}$ for black line (one-way traversable
wormhole), $A_{\lambda}=0.53$ for cyan dashed line (traversable wormhole ),
$A_{\lambda}=\frac{225}{392}$ for brown dotted line (traversable wormhole with
a marginally unstable photon sphere), and $A_{\lambda}=2$ for blue dotted
dashed line (traversable wormhole).

From Eq. (11), the deflection angle $\alpha(r_{0})$ can be computed to be

$\displaystyle\alpha(r_{0})=I(r_{0})-\pi,~{}$ (18)

where

$\displaystyle I(r_{0})\equiv
2\int^{\infty}_{r_{0}}\frac{dr}{\sqrt{\frac{C}{B}\left(\frac{C}{Ab^{2}}-1\right)}}.~{}$
(19)

##  III Weak deflection angle

In this section, we investigate the weak deflection angle of light rays in
this LQG black hole spacetime. By defining $x=r/2M$ and $T=t/2M$, the line
element (4) can be transformed into

$\displaystyle dS^{2}$ $\displaystyle=$
$\displaystyle(2M)^{-2}ds^{2}=-A(x)dT^{2}+B(x)dx^{2}$ (20) $\displaystyle+$
$\displaystyle C(x)\big{(}d\theta^{2}+\sin^{2}\theta d\phi^{2}\big{)},~{}$

where

$\displaystyle A(x)$ $\displaystyle=$
$\displaystyle\frac{1}{B(x)}=\frac{2\sqrt{2A_{\lambda}+x^{2}}(\sqrt{2A_{\lambda}+x^{2}}-1)}{A_{\lambda}+2x^{2}},~{}$
(21) $\displaystyle C(x)$ $\displaystyle=$
$\displaystyle\frac{A_{\lambda}}{2}+x^{2}.~{}$ (22)

Then, the integral (19) reduces to

$\displaystyle I(x_{0})$ $\displaystyle=$ $\displaystyle
2\int^{\infty}_{x_{0}}(A_{\lambda}+2x^{2})^{-1}\bigg{(}\frac{-2A_{\lambda}+\sqrt{2A_{\lambda}+x^{2}}-x^{2}}{\left(A_{\lambda}+2x^{2}\right){}^{2}}$
(23) $\displaystyle+$
$\displaystyle\frac{2A_{\lambda}-\sqrt{2A_{\lambda}+x_{0}^{2}}+x_{0}^{2}}{\left(A_{\lambda}+2x_{0}^{2}\right){}^{2}}\bigg{)}^{-1/2}dx,~{}$

and the impact parameter is given by

$\displaystyle\frac{b}{2M}=\left(\frac{A_{\lambda}}{2}+x_{0}^{2}\right)\bigg{(}2A_{\lambda}+x_{0}^{2}-\sqrt{2A_{\lambda}+x_{0}^{2}}\bigg{)}^{-1/2}.~{}$
(24)

After introducing a new variable $z=x_{0}/x$, the above integral can be
rewritten as

$\displaystyle I(x_{0})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int^{1}_{0}\frac{2x_{0}}{z^{2}}\left(A_{\lambda}+\frac{2x_{0}^{2}}{z^{2}}\right)^{-1}\\!\\!\bigg{(}\frac{2A_{\lambda}\\!-\\!\sqrt{2A_{\lambda}+x_{0}^{2}}\\!+\\!x_{0}^{2}}{\left(A_{\lambda}+2x_{0}^{2}\right){}^{2}}$
(25) $\displaystyle+$
$\displaystyle\frac{-2A_{\lambda}z^{2}+z\sqrt{2A_{\lambda}z^{2}+x_{0}^{2}}-x_{0}^{2}}{z\left(A_{\lambda}z^{2}+2x_{0}^{2}\right){}^{2}}\bigg{)}^{-1/2}dz,~{}$

which can not be integrated out analytically. However, in the weak field
limit, i.e., $x_{0}\gg 1$, the integrand can be expanded in terms of
$1/x_{0}$, which can be integrated out term by term as follows:

$\displaystyle\alpha(x_{0})$ $\displaystyle=$ $\displaystyle
I(x_{0})-\pi=\frac{2}{x_{0}}+\left(\pi\left(\frac{15}{16}-A_{\lambda}\right)-1\right)\frac{1}{x_{0}^{2}}$
(26) $\displaystyle+$
$\displaystyle\left(\left(\pi-\frac{25}{3}\right)A_{\lambda}+\frac{61}{12}-\frac{15\pi}{16}\right)\frac{1}{x_{0}^{3}}$
$\displaystyle+$ $\displaystyle\bigg{(}\frac{1}{64}A_{\lambda}\big{(}152\pi
A_{\lambda}-465\pi+928\big{)}+\frac{3465\pi}{1024}$ $\displaystyle-$
$\displaystyle\frac{65}{8}\bigg{)}\frac{1}{x_{0}^{4}}+\bigg{(}-\frac{1}{480}A_{\lambda}\big{(}24(135\pi-746)A_{\lambda}$
$\displaystyle-$ $\displaystyle
7875\pi+30940\big{)}+\frac{7783}{320}-\frac{3465\pi}{512}\bigg{)}\frac{1}{x_{0}^{5}}$
$\displaystyle+$ $\displaystyle\bigg{(}\frac{1}{512}A_{\lambda}\big{(}-3064\pi
A_{\lambda}^{2}+3(8601\pi-21376)A_{\lambda}$ $\displaystyle-$ $\displaystyle
30765\pi+86560\big{)}+\frac{310695\pi}{16384}-\frac{21397}{384}\bigg{)}\frac{1}{x_{0}^{6}}$
$\displaystyle+$ $\displaystyle\mathcal{O}\left(\frac{1}{x_{0}^{7}}\right).$

Besides, the inverse solution of Eq. (24) also can be obtained in the weak
field limit as

$\displaystyle\frac{1}{x_{0}}$ $\displaystyle=$
$\displaystyle\frac{2M}{b}+\frac{1}{2}\left(\frac{2M}{b}\right)^{2}+\frac{1}{8}(5-4A_{\lambda})\left(\frac{2M}{b}\right)^{3}$
(27) $\displaystyle+$
$\displaystyle\left(1-\frac{3A_{\lambda}}{2}\right)\left(\frac{2M}{b}\right)^{4}+\frac{3}{128}\big{(}64A_{\lambda}^{2}-168A_{\lambda}$
$\displaystyle+$ $\displaystyle
77\big{)}\left(\frac{2M}{b}\right)^{5}+\left(\frac{57A_{\lambda}^{2}}{8}-10A_{\lambda}+\frac{7}{2}\right)\left(\frac{2M}{b}\right)^{6}$
$\displaystyle+$
$\displaystyle\mathcal{O}\left(\left(\frac{2M}{b}\right)^{7}\right).$

Inserting the above relation into Eq. (25), the weak deflection angle as a
function of $b$ is found to be

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle\frac{4M}{b}+\frac{15\pi
M^{2}}{4b^{2}}-\frac{4\pi
A_{\lambda}M^{2}}{b^{2}}+\frac{(128-224A_{\lambda})M^{3}}{3b^{3}}$ (28)
$\displaystyle+$ $\displaystyle\mathcal{O}(b^{-4}).~{}$

It is obvious that the first two terms come from the weak deflection angle of
light rays by a spherically symmetric black hole [87], while the third term
originates from the quantum effects, which is of the same order as the charge
or spin of a black hole [88, 89]. Besides, the minus sign in front of
$A_{\lambda}$ indicates that the quantum effects make a negative contribution
to the weak deflection angle. What’ s more, the primary quantum correction to
the weak deflection angle from the LQG is larger than the non-loop quantum one
investigated in Ref. [90], where the primary correction is only at the order
of $b^{-3}$.

##  IV Strong deflection limit

In this section, we mainly focus on the bending angle in the strong deflection
limit, i.e., $x_{0}\rightarrow x_{m}$ or $b\rightarrow b_{m}$, for different
values of $A_{\lambda}$.

###  IV.1 Case of $0\leqslant A_{\lambda}<\frac{225}{392}$

For $0\leqslant A_{\lambda}<\frac{225}{392}$, there are two unstable photon
spheres at $x=\pm x_{m}$, respectively. Without loss of generality, we focus
on the unstable photon sphere at $x=x_{m}$ and investigate the deflection
angle of light rays in the strong deflection limit. After redefining a new
variable $\bar{z}$ as

$\displaystyle\bar{z}=1-\frac{x_{0}}{x},~{}$ (29)

the integral (19) can be rewritten as

$\displaystyle I(x_{0})=\int^{1}_{0}R(\bar{z},x_{0})f(\bar{z},x_{0})d\bar{z},$
(30)

where

$\displaystyle R(\bar{z},x_{0})$ $\displaystyle=$
$\displaystyle\frac{2x_{0}\sqrt{ABC_{0}}}{C(1-\bar{z})^{2}},$ (31)
$\displaystyle f(\bar{z},x_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{A_{0}-\frac{C_{0}}{C}A}}.$ (32)

It can be easily shown that the function $R(\bar{z},x_{0})$ is regular for all
values of $\bar{z}$ and $x_{0}$, while $f(\bar{z},x_{0})$ diverges when
$\bar{z}\rightarrow 0$. Then, following Ref. [56], the integral $I(x_{0})$ can
be divided into two parts:

$\displaystyle I(x_{0})=I_{D}(x_{0})+I_{R}(x_{0}),$ (33)

i.e., the divergent part $I_{D}(x_{0})$ and the regular part $I_{R}(x_{0})$.
The divergent part including the divergence is given by

$\displaystyle
I_{D}(x_{0})=\int^{1}_{0}R(0,x_{m})f_{D}(\bar{z},x_{0})d\bar{z},$ (34)

where $f_{D}(\bar{z},x_{0})$ is the series expansion of the argument of the
square root in $f(\bar{z},x_{0})$ up to the second order in $\bar{z}$:

$\displaystyle
f_{D}(\bar{z},x_{0})=\frac{1}{\sqrt{c_{1}(x_{0})\bar{z}+c_{2}(x_{0})\bar{z}^{2}}},$
(35)

with the expansion coefficients $c_{1}$ and $c_{2}$ given by

$\displaystyle c_{1}(x_{0})$ $\displaystyle=$ $\displaystyle
x_{0}\left(\frac{A_{0}C_{0}^{\prime}}{C_{0}}-A_{0}^{\prime}\right),$ (36)
$\displaystyle c_{2}(x_{0})$ $\displaystyle=$
$\displaystyle\frac{x_{0}}{2C_{0}^{2}}\big{(}-2x_{0}A_{0}C_{0}^{\prime
2}-C_{0}^{2}(2A_{0}^{\prime}+x_{0}A_{0}^{\prime\prime})$ (37) $\displaystyle+$
$\displaystyle
C_{0}(2x_{0}A_{0}^{\prime}C_{0}^{\prime}+A_{0}(2C_{0}^{\prime}+x_{0}C_{0}^{\prime\prime}))\big{)},$

where the functions with subscript $0$ denote that they are computed at
$x_{0}$.

After substracting the divergent part, one can obtain the regular part:

$\displaystyle I_{R}(x_{0})=\int^{1}_{0}g_{R}(\bar{z},x_{0})d\bar{z},~{}$ (38)

where $g_{R}(\bar{z},x_{0})\equiv
R(\bar{z},x_{0})f(\bar{z},x_{0})-R(0,x_{m})f_{D}(\bar{z},x_{0})$. Then, the
deflection angle of light rays in the strong field limit can be expressed as

$\displaystyle\alpha(b)\\!=\\!-\bar{a}\ln\left(\frac{b}{b_{m}}-1\right)\\!+\\!\bar{u}\\!+\\!\mathcal{O}[(b-b_{m})\ln(b-b_{m})],~{}$
(39)

where

$\displaystyle\bar{a}$ $\displaystyle\equiv$
$\displaystyle\frac{R(0,x_{m})}{2\sqrt{c_{2}(x_{m})}},~{}$ (40)
$\displaystyle\bar{u}$ $\displaystyle\equiv$
$\displaystyle\bar{a}\ln\bar{\delta}+I_{R}(x_{0})-\pi,$ (41)
$\displaystyle\bar{\delta}$ $\displaystyle\equiv$ $\displaystyle
x_{m}^{2}\left(\frac{C^{\prime\prime}(x_{m})}{C(x_{m})}-\frac{A^{\prime\prime}(x_{m})}{A(x_{m})}\right).~{}$
(42)

After inserting Eqs. (21) and (22) into Eqs. (40) and (42), the coefficients
in the deflection angle (39) can be calculated as

$\displaystyle\bar{a}(x_{m})$ $\displaystyle=$
$\displaystyle\frac{\left(2A_{\lambda}+x_{m}^{2}\right){}^{3/4}\left(A_{\lambda}+2x_{m}^{2}\right)}{\sqrt{4x_{m}^{6}\sqrt{2A_{\lambda}+x_{m}^{2}}+12A_{\lambda}x_{m}^{4}+3A_{\lambda}^{2}x_{m}^{2}\left(4-3\sqrt{2A_{\lambda}+x_{m}^{2}}\right)+A_{\lambda}^{3}\left(14\sqrt{2A_{\lambda}+x_{m}^{2}}-15\right)}},$
(43) $\displaystyle\bar{\delta}(x_{m})$ $\displaystyle=$
$\displaystyle-\frac{2x_{m}^{2}\left(4x_{m}^{6}\sqrt{2A_{\lambda}+x_{m}^{2}}+12A_{\lambda}x_{m}^{4}+3A_{\lambda}^{2}x_{m}^{2}\left(4-3\sqrt{2A_{\lambda}+x_{m}^{2}}\right)+A_{\lambda}^{3}\left(14\sqrt{2A_{\lambda}+x_{m}^{2}}-15\right)\right)}{\left(2A_{\lambda}+x_{m}^{2}\right){}^{3/2}\left(A_{\lambda}+2x_{m}^{2}\right){}^{2}\left(-2A_{\lambda}+\sqrt{2A_{\lambda}+x_{m}^{2}}-x_{m}^{2}\right)}.$
(44)

Expanding the integrand of the regular part (38) in powers of $x_{0}-x_{m}$:

$\displaystyle
I_{R}(x_{0})=\sum^{\infty}_{j=0}\frac{1}{j!}(x_{0}-x_{m})^{j}\int^{1}_{0}\frac{\partial^{j}g}{\partial
x^{j}_{0}}\Bigg{|}_{x_{0}=x_{m}}d\bar{z},$ (45)

and focusing on the primary term $j=0$, the regular part can be approximated
as

$\displaystyle I_{R}(x_{0})\sim
I_{R}(x_{m})=\int^{1}_{0}g(\bar{z},x_{m})d\bar{z},$ (46)

which can be calculated numerically. Figure 2 shows the deflection angle of
light rays (39) in terms of $b$ for several values of $A_{\lambda}$. It can be
seen that the strong deflection angle is an excellent approximation for
photons passing close to the photon sphere, as shown in Fig. 2. The weak
deflection angle (28) is also a good approximation to the exact deflection
angle (18). Besides, for $A_{\lambda}=0$, the strong deflection angle of light
rays by Schwarzschild black hole is recovered [56]:

$\displaystyle\alpha(b)$ $\displaystyle=$
$\displaystyle-\ln\left(\frac{b}{3\sqrt{3}M}-1\right)+\ln 6+0.9496-\pi$ (47)
$\displaystyle+$ $\displaystyle\mathcal{O}[(b-b_{m})\ln(b-b_{m})].$

Figure 2: Plots of the exact deflection angle (18) (the black full lines), the
corresponding strong deflection limit (39) (the red dashed lines) and weak
deflection limit (28) (the blue dashed lines) as a function of the impact
parameter $b$ for some values of $A_{\lambda}$.

###  IV.2 Case of $A_{\alpha}>\frac{225}{392}$

In the case of $A_{\alpha}>\frac{225}{392}$, two unstable photon spheres merge
into one unstable photon sphere located at the wormhole throat $x=0$. Defining
a new radial coordinate $\rho=\sqrt{\frac{A_{\lambda}}{2}+x^{2}}$, the line
element (20) can be transformed into

$\displaystyle
dS^{2}\\!=\\!-\tilde{A}(\rho)dT^{2}\\!+\\!\tilde{B}(\rho)d\rho^{2}\\!+\\!\tilde{C}(\rho)(d\theta^{2}\\!+\\!\sin^{2}\theta
d\phi^{2}),$ (48)

where

$\displaystyle\tilde{A}(\rho)$ $\displaystyle=$ $\displaystyle
1+\frac{3A_{\lambda}}{2\rho^{2}}-\frac{\sqrt{6A_{\lambda}+4\rho^{2}}}{2\rho^{2}},$
(49) $\displaystyle\tilde{B}(\rho)$ $\displaystyle=$
$\displaystyle\frac{2\rho^{2}}{(2\rho^{2}-A_{\lambda})\tilde{A}(\rho)},$ (50)
$\displaystyle\tilde{C}(\rho)$ $\displaystyle=$ $\displaystyle\rho^{2}.$ (51)

Since the variable $\bar{z}$ defined in Eq. (29) would not work well in this
case, we introduce a new definition $\tilde{z}=1-\frac{\rho_{0}}{\rho}$. Then,
the integral (19) can be expressed as

$\displaystyle
I(\rho_{0})=\int^{1}_{0}\tilde{R}(\tilde{z},\rho_{0})\tilde{f}(\tilde{z},\rho_{0})d\tilde{z},~{}$
(52)

where

$\displaystyle\tilde{R}(\tilde{z},\rho_{0})$ $\displaystyle=$
$\displaystyle\frac{\rho_{0}\sqrt{\tilde{C}(\rho_{0})}}{\tilde{C}(\rho)(1-\tilde{z})^{2}},$
(53) $\displaystyle\tilde{f}(\tilde{z},\rho_{0})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\tilde{A}(\rho)\tilde{B}(\rho)}{\tilde{A}(\rho_{0})-\frac{\tilde{C}(\rho_{0})}{\tilde{C}(\rho)}\tilde{A}(\rho)}},$
(54)

and $\rho_{0}=\sqrt{\frac{A_{\lambda}}{2}+x_{0}^{2}}$. After inserting
$\tilde{A}(\rho)$, $\tilde{B}(\rho)$ and $\tilde{C}(\rho)$, the above
expressions can be simplified to

$\displaystyle\tilde{R}(\tilde{z},\rho_{0})$ $\displaystyle=$ $\displaystyle
2,$ (55) $\displaystyle\tilde{f}(\tilde{z},\rho_{0})$ $\displaystyle=$
$\displaystyle\frac{2\rho_{0}^{2}}{\sqrt{\tilde{G}(\tilde{z},\rho_{0})}},~{}$
(56)

where

$\displaystyle\tilde{G}(\tilde{z},\rho_{0})$ $\displaystyle=$
$\displaystyle\left(2\rho_{0}^{2}-(\tilde{z}-1)^{2}A_{\lambda}\right)\bigg{[}2\rho_{0}^{2}-\sqrt{6A_{\lambda}+4\rho_{0}^{2}}$
(57) $\displaystyle+$ $\displaystyle
3A_{\lambda}-\Big{(}(\tilde{z}-1)^{2}\big{(}(\tilde{z}-1)\sqrt{6(\tilde{z}-1)^{2}A_{\lambda}+4\rho_{0}^{2}}$
$\displaystyle+$ $\displaystyle
3(\tilde{z}-1)^{2}A_{\lambda}+2\rho_{0}^{2}\big{)}\Big{)}\bigg{]}.$

Obviously, $\tilde{R}(\tilde{z},\rho_{0})$ is regular and
$\tilde{f}(\tilde{z},\rho_{0})$ is divergent when $\tilde{z}\rightarrow 0$. To
solve the integral (52), we also split it into two parts
$I(\rho_{0})=I_{d}(\rho_{0})+I_{r}(\rho_{0})$. The divergent part
$I_{d}(\rho_{0})$ is given by

$\displaystyle I_{d}(\rho_{0})$ $\displaystyle=$
$\displaystyle\int^{1}_{0}2\tilde{f}_{d}(\tilde{z},\rho_{0})d\tilde{z},$ (58)
$\displaystyle\tilde{f}_{d}(\tilde{z},\rho_{0})$ $\displaystyle=$
$\displaystyle\frac{2\rho_{0}^{2}}{\sqrt{c_{3}(\rho_{0})\tilde{z}+c_{4}(\rho_{0})\tilde{z}^{2}+c_{5}(\rho_{0})\tilde{z}^{3}}},~{}$
(59)

where $\rho_{m}=\sqrt{\frac{A_{\lambda}}{2}+x_{m}^{2}}$, and the expansion
coefficients of the function $\tilde{G}(\tilde{z},\rho_{0})$ up to the third
order in $\tilde{z}$ are given by

$\displaystyle c_{3}(\rho_{0})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!2\rho_{0}^{3}\left(A_{\lambda}-2\rho_{0}^{2}\right)\left(\tilde{A}_{0}^{\prime}-\frac{\tilde{A}_{0}\tilde{C}_{0}^{\prime}}{\tilde{C}_{0}}\right),$
(60) $\displaystyle c_{4}(\rho_{0})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!-\frac{\rho_{0}^{3}}{\tilde{C}_{0}^{2}}\bigg{(}\tilde{C}_{0}^{2}\left(2\tilde{A}_{0}^{\prime}\left(A_{\lambda}+2\rho_{0}^{2}\right)\\!+\\!\rho_{0}\tilde{A}_{0}^{\prime\prime}\left(2\rho_{0}^{2}-A_{\lambda}\right)\right)$
(61) $\displaystyle+$
$\displaystyle\\!\\!\tilde{C}_{0}\Big{(}\tilde{A}_{0}\left(\rho_{0}\tilde{C}_{0}^{\prime\prime}\left(A_{\lambda}-2\rho_{0}^{2}\right)-2\tilde{C}_{0}^{\prime}\left(A_{\lambda}+2\rho_{0}^{2}\right)\right)$
$\displaystyle+$
$\displaystyle\\!\\!2\rho_{0}\tilde{A}_{0}^{\prime}\tilde{C}_{0}^{\prime}\left(A_{\lambda}-2\rho_{0}^{2}\right)\Big{)}$
$\displaystyle+$
$\displaystyle\\!\\!2\rho_{0}\tilde{A}_{0}\tilde{C}_{0}^{\prime
2}\left(2\rho_{0}^{2}-A_{\lambda}\right)\bigg{)},$ $\displaystyle
c_{5}(\rho_{0})\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!-\frac{\rho_{0}^{5}}{3\tilde{C}_{0}^{3}}\bigg{(}\tilde{C}_{0}^{3}\big{(}-\tilde{A}_{0}{}^{(3)}\left(A_{\lambda}-2\rho_{0}^{2}\right)+12\rho_{0}\tilde{A}_{0}^{\prime\prime}$
(62) $\displaystyle+$
$\displaystyle\\!\\!12\tilde{A}_{0}^{\prime}\big{)}-\tilde{C}_{0}^{2}\Big{(}-3\tilde{A}_{0}^{\prime\prime}\tilde{C}_{0}^{\prime}\left(A_{\lambda}-2\rho_{0}^{2}\right)$
$\displaystyle+$
$\displaystyle\\!\\!3\tilde{A}_{0}^{\prime}\left(8\rho_{0}\tilde{C}_{0}^{\prime}-\tilde{C}_{0}^{\prime\prime}\left(A_{\lambda}-2\rho_{0}^{2}\right)\right)$
$\displaystyle+$
$\displaystyle\\!\\!\tilde{A}_{0}\left(-\tilde{C}_{0}{}^{(3)}\left(A_{\lambda}-2\rho_{0}^{2}\right)+12\rho_{0}\tilde{C}_{0}^{\prime\prime}+12\tilde{C}_{0}^{\prime}\right)\Big{)}$
$\displaystyle+$
$\displaystyle\\!\\!6\tilde{C}_{0}\tilde{C}_{0}^{\prime}\Big{(}\tilde{A}_{0}\left(4\rho_{0}\tilde{C}_{0}^{\prime}-\tilde{C}_{0}^{\prime\prime}\left(A_{\lambda}-2\rho_{0}^{2}\right)\right)$
$\displaystyle-$
$\displaystyle\\!\\!\tilde{A}_{0}^{\prime}\tilde{C}_{0}^{\prime}\left(A_{\lambda}-2\rho_{0}^{2}\right)\Big{)}+6\tilde{A}_{0}\tilde{C}_{0}^{\prime
3}\left(A_{\lambda}-2\rho_{0}^{2}\right)\bigg{)},$

where the functions with subscript $0$ denote that they are computed at
$\rho_{0}$.

Expanding $c_{3}(\rho_{0})$, $c_{4}(\rho_{0})$, and $c_{5}(\rho_{0})$ in
powers of $\rho_{0}-\rho_{m}$:

$\displaystyle c_{3}(\rho_{0})\\!$ $\displaystyle=$
$\displaystyle\\!\left(28\sqrt{2}A_{\lambda}^{3/2}-30A_{\lambda}\right)(\rho_{0}-\rho_{m})+\bigg{(}44A_{\lambda}$
(63) $\displaystyle-$
$\displaystyle\\!\frac{39\sqrt{A_{\lambda}}}{\sqrt{2}}\bigg{)}(\rho_{0}-\rho_{m})^{2}+\mathcal{O}\big{(}(\rho_{0}-\rho_{m})^{3}\big{)},~{}$
$\displaystyle c_{4}(\rho_{0})\\!$ $\displaystyle=$
$\displaystyle\\!\left(28A_{\lambda}^{2}-15\sqrt{2}A_{\lambda}^{3/2}\right)+\bigg{(}\frac{153A_{\lambda}}{4}$
(64) $\displaystyle-$
$\displaystyle\\!30\sqrt{2}A_{\lambda}^{3/2}\bigg{)}(\rho_{0}-\rho_{m})+\mathcal{O}\big{(}(\rho_{0}-\rho_{m})^{2}\big{)},$
$\displaystyle c_{5}(\rho_{0})\\!$ $\displaystyle=$
$\displaystyle\\!\left(\frac{231A_{\lambda}^{3/2}}{4\sqrt{2}}-52A_{\lambda}^{2}\right)+\mathcal{O}\big{(}(\rho_{0}-\rho_{m})\big{)},~{}$
(65)

one can easily see that $\tilde{f}_{d}(\tilde{z},\rho_{0})$ diverges as
$\tilde{z}^{-1}$ in the strong deflection limit $\rho_{0}\rightarrow\rho_{m}$.

Subtracting the divergent part from $I(\rho_{0})$, the regular part can be
obtained as

$\displaystyle
I_{r}(\rho_{0})=\int^{1}_{0}\tilde{g}(\tilde{z},\rho_{0})d\tilde{z},$ (66)

where $\tilde{g}(\tilde{z},\rho_{0})\equiv
2\left(\tilde{f}(\tilde{z},\rho_{0})-\tilde{f}_{d}(\tilde{z},\rho_{0})\right)$.
Then, the deflection angle of light rays in the strong deflection limit
$\rho_{0}\rightarrow\rho_{m}$ is

$\displaystyle\alpha(b)\\!=\\!-\tilde{a}\ln\left(\frac{b}{b_{m}}-1\right)\\!+\\!\tilde{u}\\!+\\!\mathcal{O}[(b-b_{m})\ln(b-b_{m})],~{}$
(67)

where

$\displaystyle\tilde{a}$ $\displaystyle\equiv$
$\displaystyle\frac{4\rho_{m}^{2}}{\sqrt{c_{4}(\rho_{m})}},$ (68)
$\displaystyle\tilde{u}$ $\displaystyle\equiv$
$\displaystyle\tilde{a}\ln\tilde{\delta}+I_{r}(\rho_{0})-\pi,$ (69)
$\displaystyle\tilde{\delta}$ $\displaystyle\equiv$ $\displaystyle
2\rho_{m}\left(\frac{\tilde{C}^{\prime}(\rho_{m})}{\tilde{C}(\rho_{m})}-\frac{\tilde{A}^{\prime}(\rho_{m})}{\tilde{A}(\rho_{m})}\right).$
(70)

By inserting $\tilde{A}(\rho)$, $\tilde{B}(\rho)$ and $\tilde{C}(\rho)$, the
parameters $\tilde{a}$ and $\tilde{\delta}$ reduce to

$\displaystyle\tilde{a}(\rho_{m})\\!$ $\displaystyle=$
$\displaystyle\\!\rho_{m}\sqrt{6A_{\lambda}+4\rho_{m}^{2}}\bigg{(}18A_{\lambda}^{2}-3\rho_{m}^{2}\sqrt{6A_{\lambda}+4\rho_{m}^{2}}$
(71) $\displaystyle-$
$\displaystyle\\!6A_{\lambda}\left(\sqrt{6A_{\lambda}+4\rho_{m}^{2}}-3\rho_{m}^{2}\right)+4\rho_{m}^{4}\bigg{)}^{-1/2},$
$\displaystyle\tilde{\delta}(\rho_{m})\\!$ $\displaystyle=$
$\displaystyle\\!4\left(6A_{\lambda}-\frac{3\sqrt{2}\left(2A_{\lambda}+\rho_{m}^{2}\right)}{\sqrt{3A_{\lambda}+2\rho_{m}^{2}}}+2\rho_{m}^{2}\right)\big{(}3A_{\lambda}$
(72) $\displaystyle-$
$\displaystyle\\!\sqrt{6A_{\lambda}+4\rho_{m}^{2}}+2\rho_{m}^{2}\big{)}^{-1}.$

Expanding $I_{r}(\rho_{0})$ in powers of $\rho_{0}-\rho_{m}$:

$\displaystyle
I_{r}(\rho_{0})=\sum^{\infty}_{j=0}\frac{1}{j!}(\rho_{0}-\rho_{m})^{j}\int^{1}_{0}\frac{\partial^{j}\tilde{g}}{\partial\rho^{j}_{0}}\Bigg{|}_{\rho_{0}=\rho_{m}}d\tilde{z},$
(73)

and focusing on the term of $j=0$, the regular part can be approximated in the
strong deflection limit as

$\displaystyle I_{r}(\rho_{0})\sim
I_{r}(\rho_{m})=\int^{1}_{0}\tilde{g}(\tilde{z},\rho_{m})d\tilde{z},$ (74)

which can be calculated numerically. Figure 3 shows the deflection angle of
light rays (67) in terms of $b$ for the case of $A_{\lambda}>\frac{225}{392}$.
Obviously, the strong and weak deflection angles also approximate well to the
exact deflection angle, as can be seen in the left panel of Fig. 3. The middle
and right panels of Fig. 3 show that the critical impact parameter has a
minimum $b_{m}=0.6495$ at $A_{\lambda}=1.125$, which indicates that the radius
of the photon sphere also reaches a minimum there.

Figure 3: Plots of the exact deflection angle (18) (the black full lines), the
corresponding strong deflection limit (67) (the red dashed lines) and weak
deflection limit (28) (the blue dashed lines) as a function of the impact
parameter $b$ for some values of $A_{\lambda}$.

###  IV.3 Case of $A_{\alpha}=\frac{225}{392}$

In the case of $A_{\lambda}=\frac{225}{392}$, the unstable photon sphere is
located at $x=x_{m}=0$ or $\rho=\rho_{m}=\sqrt{A_{\lambda}/2}$, which is
marginally unstable since the light rays satisfy
$V(x_{m})=V^{\prime}(x_{m})=V^{\prime\prime}(x_{m})=V^{\prime\prime\prime}(x_{m})=0$
and $V^{\prime\prime\prime\prime}(x_{m})<0$.

When $A_{\lambda}=\frac{225}{392}$, $\tilde{f}(\tilde{z},\rho_{m})$ reduces to

$\displaystyle\tilde{f}(\tilde{z},\rho_{m})$ $\displaystyle=$
$\displaystyle\sqrt{\frac{15}{\tilde{z}(2-\tilde{z})}}\Big{(}4-(\tilde{z}-1)^{2}\big{(}45(\tilde{z}-1)^{2}$
(75) $\displaystyle+$ $\displaystyle
28\sqrt{3(\tilde{z}-2)\tilde{z}+4}(\tilde{z}-1)+15\big{)}\Big{)}^{-1/2},$

and Eqs. (63), (64) and (65) become

$\displaystyle c_{3}(\rho)$ $\displaystyle=$
$\displaystyle\frac{855}{196}(\rho_{0}-\rho_{m})^{2}+\mathcal{O}\big{(}(\rho_{0}-\rho_{m})^{3}\big{)},$
(76) $\displaystyle c_{4}(\rho)$ $\displaystyle=$
$\displaystyle\frac{38475}{10976}(\rho_{0}-\rho_{m})+\mathcal{O}\big{(}(\rho_{0}-\rho_{m})^{2}\big{)},$
(77) $\displaystyle c_{5}(\rho)$ $\displaystyle=$
$\displaystyle\frac{192375}{307328}+\mathcal{O}\big{(}(\rho_{0}-\rho_{m})\big{)}.$
(78)

Then, from Eq. (59), one can conclude that $\tilde{f}(\tilde{z},\rho_{m})$
diverges as $\tilde{z}^{-2/3}$ in this case, and the divergent part of
$\tilde{f}(\tilde{z},\rho_{m})$ becomes

$\displaystyle\hat{f}_{d}(\tilde{z},\rho_{m})\equiv\sqrt{\frac{10}{19\tilde{z}^{3}}},$
(79)

which can be integrated out as

$\displaystyle I_{d}$ $\displaystyle=$
$\displaystyle\int^{1}_{0}2\hat{f}_{d}(\tilde{z},\rho_{0})d\tilde{z}=\left(\frac{4\sqrt{\frac{10}{19}}}{\sqrt{\tilde{z}}}\right)\Bigg{|}_{\tilde{z}=0}-4\sqrt{\frac{10}{19}},$
(80) $\displaystyle=$
$\displaystyle\left(\frac{4\sqrt{\frac{10}{19}}}{\sqrt{\frac{\rho_{0}}{\rho_{m}}-1}}\right)\Bigg{|}_{\rho_{0}=\rho_{m}}-4\sqrt{\frac{10}{19}}.$

From the expression of the impact parameter in terms of the new radial
coordinate, i.e., $b=\sqrt{\frac{\tilde{A}(\rho_{0})}{\tilde{C}(\rho_{0})}}$,
one can obtain

$\displaystyle
b-b_{m}=\frac{133\sqrt{\frac{3}{5}}}{16}(\rho_{0}-\rho_{m})^{2}+\mathcal{O}\big{(}(\rho_{0}-\rho_{m})^{3})\big{)}.$
(81)

Then, the divergent term $I_{d}$ as a function of the impact parameter $b$ can
be expressed as

$\displaystyle
I_{d}=\frac{2\sqrt[4]{\frac{6}{19}}\sqrt{5}}{\sqrt[4]{\frac{b}{b_{m}}-1}}-4\sqrt{\frac{10}{19}}.$
(82)

Subtracting the divergent part, the regular part is

$\displaystyle
I_{r}(\rho_{m})=\int^{1}_{0}2\big{(}\tilde{f}(\tilde{z},\rho_{m})-\hat{f}_{d}(\tilde{z},\rho_{m})\big{)}d\tilde{z},$
(83)

which can be calculated numerically. Now, the deflection angle of light rays
in the strong deflection limit $\rho_{0}\rightarrow\rho_{m}$ can be given by

$\displaystyle\alpha=\frac{\hat{c}}{\sqrt[4]{\frac{b}{b_{m}}-1}}+\hat{d}+\mathcal{O}[(b-b_{m})\ln(b-b_{m})],~{}$
(84)

where

$\displaystyle\hat{c}$ $\displaystyle=$ $\displaystyle
2\sqrt[4]{\frac{6}{19}}\sqrt{5}\sim 3.35247,$ (85) $\displaystyle\hat{d}$
$\displaystyle=$ $\displaystyle-4\sqrt{\frac{10}{19}}+I_{r}-\pi\sim-3.24981.$
(86)

Figure 4 shows that the deflection angles in the strong and weak deflection
limits also approximate well to the exact deflection angle.

Figure 4: Plots of the exact deflection angle (18) (the black full lines), the
corresponding strong deflection limit (84) (the red dashed lines) and weak
deflection limit (28) (the blue dashed lines) as a function of the impact
parameter $b$ for some values of $A_{\lambda}$.

##  V Observables in the strong deflection limit

In the previous section, we have obtained the deflection angles as a function
of the impact parameter. In this section, we will investigate the observables
of this lensing system. We consider a beam of light emanated from a source
(S), deflected by a lens (L) of mass $M$, and then arriving at an observer
(O). Both the source and the observer are in asymptotically flat spacetime.
$D_{\text{LS}}$ stands for the distance between the lens L and the source S,
and $D_{\text{OS}}=D_{\text{OL}}+D_{\text{LS}}$ is the distance between the
observer O and the source S. Defining the optical axis as the line connecting
the lens and the observer, and focusing on a small-angle approximation, i.e.,
the source, lens, and observer are highly aligned, the lens equation can be
written as

$\displaystyle\beta=\theta-\frac{D_{\text{LS}}}{D_{\text{OS}}}\alpha^{*},~{}$
(87)

where $\beta$ and $\theta$ denote the angular position of the source and image
with respect to the optical axis, respectively. $\alpha^{*}$ is the effective
deflection angle defined by $\alpha^{*}\equiv\alpha-2n\pi$ with
$n\in\mathbb{N}$. The deflection angle $\alpha(\theta)$ can be expanded around
$\theta=\theta_{n}^{0}$ as

$\displaystyle\alpha(\theta)=\alpha(\theta_{n}^{0})+\frac{d\alpha}{d\theta}\bigg{|}_{\theta=\theta_{n}^{0}}(\theta-\theta_{n}^{0})+\mathcal{O}\big{(}(\theta-\theta_{n}^{0})^{2}\big{)},~{}$
(88)

where $\theta_{n}^{0}$ is determined by

$\displaystyle\alpha(\theta_{n}^{0})=2n\pi.~{}$ (89)

###  V.1 $A_{\lambda}\neq\frac{225}{392}$

In the case of $A_{\lambda}\neq\frac{225}{392}$, both of the deflection angles
(39) and (67) are logarithmic divergence in the strong deflection limit, and
they can be rewritten as a function of $\theta$:

$\displaystyle\alpha(\theta)$ $\displaystyle=$
$\displaystyle-\tilde{\bar{a}}\ln\left(\frac{\theta}{\theta_{\infty}}-1\right)+\tilde{\bar{u}}$
(90) $\displaystyle+$
$\displaystyle\mathcal{O}\left(\left(\frac{\theta}{\theta_{\infty}}-1\right)\ln\left(\frac{\theta}{\theta_{\infty}}-1\right)\right),~{}$

where $\tilde{\bar{X}}$ denotes $\bar{X}$ or $\tilde{X}$, and we have used the
first-order approximations $\theta\approx b/D_{\text{OL}}$ and
$\theta_{\infty}\approx b_{m}/D_{\text{OL}}$. From the above expression, one
can easily obtain

$\displaystyle\frac{d\alpha}{d\theta}\bigg{|}_{\theta=\theta_{n}^{0}}=-\frac{\tilde{\bar{a}}}{\theta_{n}^{0}-\theta_{\infty}}.~{}$
(91)

Solving Eq. (90) with $\alpha(\theta_{n}^{0})=2n\pi$, we find

$\displaystyle\theta_{n}^{0}=\left(1+\text{e}^{\frac{\tilde{\bar{u}}-2n\pi}{\tilde{\bar{a}}}}\right).~{}$
(92)

Then, from Eqs. (88), (91) and (92), the effective deflection angle
$\alpha^{*}$ can be calculated as

$\displaystyle\alpha^{*}(\theta_{n})=\frac{\tilde{\bar{a}}}{\theta_{\infty}\text{e}^{\frac{\tilde{\bar{u}}-2n\pi}{\tilde{\bar{a}}}}}(\theta_{n}^{0}-\theta_{n}).~{}$
(93)

After inserting the effective deflection angle (93) into the lens equation
(87), the angular position of the $n$-th image can be calculated as

$\displaystyle\theta_{n}(\beta)=\theta_{n}^{0}+\frac{\theta_{\infty}\text{e}^{\frac{\tilde{\bar{u}}-2n\pi}{\tilde{\bar{a}}}}D_{\text{OS}}(\beta-\theta_{n}^{0})}{\tilde{\bar{a}}D_{\text{LS}}},$
(94)

where the second term is a small correction to $\theta_{n}^{0}$. In the case
of the observer, the lens, and the source perfectly aligned in a line, i.e.,
$\beta=0$, the point images become an infinite series of concentric rings with
angular position

$\displaystyle\theta_{\text{E}n}\equiv\theta_{n}(0)=\left(1-\frac{\theta_{\infty}\text{e}^{\frac{\tilde{\bar{u}}-2n\pi}{\tilde{\bar{a}}}}D_{\text{OS}}}{\tilde{\bar{a}}D_{\text{LS}}}\right)\theta_{n}^{0},$
(95)

which are the so-called Einstein rings.

With the approximation of $\theta_{n}\sim\theta_{n}^{0}$, the magnification of
the $n$-th relativistic image can be given by

$\displaystyle\mu_{n}\equiv\frac{1}{\frac{\beta}{\theta}\frac{d\beta}{d\theta}}\Bigg{|}_{\theta=\theta_{n}^{0}}\simeq\frac{\theta_{\infty}^{2}D_{\text{OS}}}{\tilde{\bar{a}}\beta
D_{\text{LS}}}\text{e}^{\frac{\tilde{\bar{u}}-2n\pi}{\tilde{\bar{a}}}}\left(1+\text{e}^{\frac{\tilde{\bar{u}}-2n\pi}{\tilde{\bar{a}}}}\right).$
(96)

Obviously, the magnification gets exponentially suppressed with increasing
$n$, which means that the first image is the brightest one and the brightness
of other relativistic images are highly demagnified unless the lens is highly
aligned with the source, i.e., $\beta\sim 0$.

In the simplest situation, we consider that only the outermost relativistic
image 􏰆$\theta_{1}$ can be resolved separately and all other relativistic
images are packed together at the limiting angular position
$\theta_{\infty}=b_{m}/D_{\text{OL}}$. Then, the angular separation $s$
between the outermost one and the others can be defined by

$\displaystyle
s\equiv\theta_{1}-\theta_{\infty}\sim\theta_{1}^{0}-\theta_{\infty}^{0}=\theta_{\infty}\text{e}^{\frac{\tilde{\bar{u}}-2\pi}{\tilde{\bar{a}}}},~{}$
(97)

and the quotient of the flux of the outermost relativistic image to that of
all other relativistic images is

$\displaystyle
r\equiv\frac{\mu_{1}}{\sum^{\infty}_{n=2}\mu_{n}}\sim\frac{\big{(}\text{e}^{\frac{4\pi}{\tilde{\bar{a}}}}-1\big{)}\big{(}\text{e}^{\frac{2\pi}{\tilde{\bar{a}}}}+\text{e}^{\frac{\tilde{\bar{u}}}{\tilde{\bar{a}}}}\big{)}}{\text{e}^{\frac{4\pi}{\tilde{\bar{a}}}}+\text{e}^{\frac{2\pi}{\tilde{\bar{a}}}}+\text{e}^{\frac{\tilde{\bar{u}}}{\tilde{\bar{a}}}}},~{}$
(98)

where the cumulative flux from all other relativistic images except the
outermost one is

$\displaystyle\sum^{\infty}_{n=2}\mu_{n}\sim\frac{\theta_{\infty}^{2}D_{\text{OS}}\big{(}\text{e}^{\frac{4\pi}{\tilde{\bar{a}}}}+\text{e}^{\frac{2\pi}{\tilde{\bar{a}}}}+\text{e}^{\frac{\tilde{\bar{u}}}{\tilde{\bar{a}}}}\big{)}\text{e}^{\frac{\tilde{\bar{u}}-4\pi}{\tilde{\bar{a}}}}}{\beta\tilde{\bar{a}}D_{\text{LS}}\big{(}\text{e}^{\frac{4\pi}{\tilde{\bar{a}}}}-1\big{)}}.$
(99)

###  V.2 $A_{\lambda}=\frac{225}{392}$

In the case of $A_{\lambda}=\frac{225}{392}$, the deflection angle as a
function of $\theta$ can be derived as

$\displaystyle\alpha(\theta)=\frac{\hat{c}}{\left(\frac{\theta}{\theta_{\infty}}-1\right)^{1/4}}+\hat{d},~{}$
(100)

which yields

$\displaystyle\frac{d\alpha}{d\theta}\bigg{|}_{\theta=\theta_{n}^{0}}=-\frac{\hat{c}}{4\theta_{\infty}}\left(\frac{\theta_{n}^{0}}{\theta_{\infty}}-1\right)^{-5/4}.$
(101)

Then, from Eqs. (89) and (100), one can obtain

$\displaystyle\theta_{n}^{0}=\left[1+\left(\frac{\hat{c}}{2n\pi-\hat{d}}\right)^{4}\right]\theta_{\infty},$
(102)

and the effective deflection angle can be calculated as

$\displaystyle\alpha^{*}(\theta_{n})=\frac{(2n\pi-\hat{d})^{5}}{4\theta_{\infty}\hat{c}^{4}}(\theta_{n}^{0}-\theta).$
(103)

Inserting the effective deflection angle into the lens equation (87), the
angular positions of the relativistic images are solved as

$\displaystyle\theta_{n}(\beta)=\theta_{n}^{0}+\frac{4\hat{c}^{4}D_{\text{OS}}\theta_{\infty}(\beta-\theta_{n}^{0})}{D_{\text{LS}}(2n\pi-\hat{d})^{5}}.~{}$
(104)

Then, the relativistic Einstein rings are given by

$\displaystyle\theta_{\text{E}n}(0)=\left[1-\frac{4\hat{c}^{4}D_{\text{OS}}\theta_{\infty}}{D_{\text{LS}}(2n\pi-\hat{d})^{5}}\right]\theta_{n}^{0}.$
(105)

With the approximation of $\theta_{n}\sim\theta_{n}^{0}$, the magnification of
the $n$-th image is

$\displaystyle\mu_{n}\simeq\frac{4\theta_{\infty}^{2}\hat{c}^{4}D_{\text{OS}}F_{n}}{\beta
D_{\text{LS}}},$ (106)

where

$\displaystyle
F_{n}=\frac{1+\left(\frac{\hat{c}}{2n\pi-\hat{d}}\right)^{4}}{(2n\pi-\hat{d})^{5}}.$
(107)

The separation of the angular positions between the outermost relativistic
image and the others is

$\displaystyle
s=\theta_{1}-\theta_{\infty}=\left(\frac{\hat{c}}{2\pi-\hat{d}}\right)^{4}\theta_{\infty},~{}$
(108)

and the luminosity ratio of the flux of the outermost one to that of all
others is

$\displaystyle
r=\frac{\mu_{1}}{\sum^{\infty}_{n=2}\mu_{n}}\sim\frac{F_{1}}{\sum^{\infty}_{n=2}F_{n}}=9.95368,~{}$
(109)

where

$\displaystyle F_{1}\sim 1.28957\times 10^{-5},$ (110)

and

$\displaystyle\sum^{\infty}_{n=2}F_{n}\sim 1.29557\times 10^{-6}.$ (111)

From the observation of the relativistic images, one can obtain the limiting
angular position $\theta_{\infty}$, the angular separation $s$, and the flux
ratio $r$. By inverting Eqs. (97) and (98) or Eqs. (108) and (109), one then
obtains the coefficients of the strong deflection angle, which contain the
information about the parameters of the lensing black hole.

###  V.3 Applications: Black holes in our Galaxy and M87

To evaluate the observables defined above, we respectively take the black
holes with mass $M=4.3\times 10^{6}M_{\odot}$ and $6.5\times 10^{9}M_{\odot}$
at the centers of our Galaxy [91] and $M87$ [92] as examples and apply the
strong deflection limit. The distances between the observer and the lens are
taken as $D_{OL}=8.35$ kpc and $16.8$ Mpc, respectively. The numerical
estimates for the observables with different $A_{\lambda}$ are given in Tables
1 and 2. It can be seen that the limiting angular position $\theta_{\infty}$
decreases rapidly with increasing $A_{\lambda}$ until $A_{\lambda}\sim 0.7$
for both black holes, indicating that the limiting relativistic images are
closer to the optical axis than the Schwarzschild case, and then there is a
little bounce at $A_{\lambda}=1.125$, as shown in the left panels of Figs. 5
and 6. Since the independences of the black hole mass $M$ and the distance
$D_{OL}$, the relative magnifications are the same for both black holes as
shown in Tables 1 and 2 or the right panels of Figs. 5 and 6. Obviously, the
relative magnifications in magnitudes $r_{m}$ first decrease and then increase
as $A_{\lambda}$ increases, which suggests that the outermost relativistic
image first becomes fainter and then brighter compared to the others. Besides,
we also find that there is a maximum of the observable $s$ at
$A_{\lambda}=0.58$ for both black holes, as shown in the middle panels of
Figs. 5 and 6, which means that the angular separation between the outermost
relativistic image and the others first becomes wider and then decreases with
increasing $A_{\lambda}$. This feature can be easily recognized in the
astronomical observation. What’s more, since the lensing parameters are
associated with a wormhole when $A_{\lambda}\geqslant\frac{1}{2}$, the maximum
of the angular separation $s$ signals that a wormhole has formed. Recently,
from the image of the black hole shadow [42], the perihelion advance, and
Shapiro time Delay [93], the tightest constraint on $A_{\lambda}$ is
$0<A_{\lambda}<4\times 10^{-6}$. With the maximum $A_{\lambda}=4\times
10^{-6}$, the observables for the black hole Sgr$\text{A}^{\ast}$ are
$\theta_{\infty}=26.4232322$, $s=0.0331672$ and $r_{m}=6.8218757$, which
correspond to a $4.46\times 10^{-6}$ deviation from the case of the
Schwarzschild black hole at best.

Table 1: Numerical estimations for the main observables for the black hole at the center of our Galaxy. Observable | Schwarzschild | Loop Quantum Black Hole
---|---|---
| ($A_{\lambda}=0$) | $A_{\lambda}=0.1$ | $A_{\lambda}=0.3$ | $A_{\lambda}=0.5$ | $A_{\lambda}=\frac{225}{392}$ | $A_{\lambda}=0.7$ | $A_{\lambda}=0.9$ | $A_{\lambda}=1.1$ | $A_{\lambda}=1.2$
$\theta_{\infty}$ ($\mu$arcsec) | 26.42 | 24.57 | 20.18 | 13.98 | 10.55 | 7.64 | 6.75 | 6.60 | 6.61
$s$ ($\mu$arcsec) | 0.0331 | 0.0375 | 0.0529 | 0.1252 | 0.3961 | 0.0343 | 0.0014 | 0.0002 | 0.0001
$r_{m}$ (magnitude) | 6.82 | 6.65 | 6.18 | 4.77 | 2.49 | 5.54 | 8.10 | 9.51 | 10.02

Figure 5: Plots of the observables $\theta_{\infty}$ (left), $s$ (middle), and $r$ in magnitudes ($r_{m}=2.5\log r$, right) as a function of $A_{\lambda}$ for the supermassive black hole at the center of our Galaxy. Table 2: Numerical estimations for the main observables for the black hole at the center of M87. Observable | Schwarzschild | Loop Quantum Black Hole
---|---|---
| ($A_{\lambda}=0$) | $A_{\lambda}=0.1$ | $A_{\lambda}=0.3$ | $A_{\lambda}=0.5$ | $A_{\lambda}=\frac{225}{392}$ | $A_{\lambda}=0.7$ | $A_{\lambda}=0.9$ | $A_{\lambda}=1.1$ | $A_{\lambda}=1.2$
$\theta_{\infty}$ ($\mu$arcsec) | 19.85 | 18.46 | 15.16 | 10.50 | 7.92 | 5.74 | 5.07 | 4.96 | 4.97
$s$ ($\mu$arcsec) | 0.0249 | 0.0281 | 0.0398 | 0.0941 | 0.2975 | 0.0258 | 0.0011 | 0.0001 | 0.00009
$r_{m}$ (magnitude) | 6.82 | 6.65 | 6.18 | 4.77 | 2.49 | 5.54 | 8.10 | 9.51 | 10.02

Figure 6: Plots of the observables $\theta_{\infty}$ (left), $s$ (middle), and
$r$ in magnitudes ($r_{m}=2.5\log r$, right) as a function of $A_{\lambda}$
for the supermassive black hole at the center of M87.

##  VI Conclusion

In this paper, we investigated the deflection angles of light rays by a black
hole in effective loop quantum gravity (LQG) in the strong and weak deflection
limits. It was found that the weak deflection angle is consistent with the
Schwarzschild black hole at the lowest order, and the corrections originating
from quantum effects make a negative contribution to the weak deflection
angle. The primary quantum correction is of the same order as the spin or
charge of a black hole.

Then, we investigated the bending angle in the strong deflection limit. It was
found that the LQG black hole reduces to a Schwarzschild black hole for
vanishing $A_{\lambda}$, a regular black hole for $0<A_{\lambda}<\frac{1}{2}$,
a one-way traversable wormhole with a null throat for
$A_{\lambda}=\frac{1}{2}$, and a traversable wormhole with two-way throat for
$A_{\lambda}>\frac{1}{2}$. Besides, if $M=0$, the LQG black hole disappears
and the spacetime is flat as expected, while the spacetime of the black-bounce
black hole reduces to an Ellis wormhole with vanishing $M$, which seems
unnatural [78, 94]. There are two unstable photon spheres for
$0<A_{\lambda}<\frac{225}{392}$ located at $r=\pm r_{m}$, respectively. When
$A_{\lambda}=\frac{225}{392}$, these two unstable photon spheres merge into
one marginally unstable photon sphere. It was found that the behavior of the
strong deflection angle exhibits logarithmic divergence in terms of the impact
parameter for $A_{\lambda}\neq\frac{225}{392}$ and nonlogarithmic divergence
for $A_{\lambda}=\frac{225}{392}$.

Next, we calculated the observables of this lensing system, such as the
angular positions of the relativistic images, the angular separation between
the outermost relativistic image and the others, and also their flux ratio.
These observables were estimated numerically, using the black holes at the
centers of our Galaxy and M87 as examples. We showed that the observables for
both black holes have the similar behaviors. To be specific, the limiting
angular position $\theta_{\infty}$ and the relative magnification in
magnitudes $r_{m}$ decrease with $A_{\lambda}$ for $0\leqslant
A_{\lambda}\leqslant\frac{225}{392}$. However, when
$A_{\lambda}>\frac{225}{392}$, there is a little bounce at $A_{\lambda}=1.125$
in the limiting angular position, and the relative magnification in magnitudes
$r_{m}$ increases with $A_{\lambda}$, suggesting that the outermost
relativistic image gradually becomes brighter relative to the others. Besides,
it is interesting to find that there is a maximum in the angular separation at
$A_{\lambda}=0.58$, which means that the angular separation first becomes
wider and then decreases with increasing $A_{\lambda}$. The maximum is reached
after the wormhole is formed, which can be considered as a signal for the
formation of a wormhole. It is expected that these features will be used to
distinguish LQG from general relativity.

###### Acknowledgements.

This work was supported by scientific research programs funded by the National
Natural Science Foundation of China (Grants No. 11975072, 11835009, 11875102,
and 11690021), the Liaoning Revitalization Talents Program (Grant No.
XLYC1905011), the Fundamental Research Funds for the Central Universities
(Grant No. N2005030), the National Program for Support of Top-Notch Young
Professionals (Grant No. W02070050), the National 111 Project of China (Grant
No. B16009), and Research Foundation of Education Bureau of Shaanxi Province,
China (Grant No. 19JS009).

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