1 Introduction

Wormholes are fascinating hypothetical structures in space that have captured the imagination of scientists and science-fiction writers alike. The concept of a wormhole was first introduced in 1916 by the German physicist Ludwig Flamm. However, it wasn’t until 1935 that A. Einstein and N. Rosen developed the theory of wormholes as part of their work on general relativity, also known as an Einstein–Rosen bridge [1], which is a theoretical tunnel or shortcut that connects two separate points in spacetime. These tunnels can be used to travel vast distances through space or even time, bypassing the need to traverse the vast distances of the universe at sub-light speeds. However, such a Einstein–Rosen wormhole would not be traversable for matter or for light [2, 3]. Later, M. Morris and K. Thorne [4] found that creating or finding a humanly traversable wormhole would require the existence of exotic matter with negative energy density, which violates null energy conditions (NEC) [5]. Several types of such traversable wormholes have actually been discovered in [6,7,8,9], and a phantom scalar field is selected as the potential candidate for such exotic matter, which has a reversed sign in its kinetic term.

Boson stars (BSs) originated from the study of geons (gravitational electromagnetic entity), which were proposed by Wheeler [10, 11]. Geons are considered to be stable, particle-like solutions without horizons in the context of gravity coupled to a classical electromagnetic field and gravitational wave. However, geons consisting of massless vector fields have not been found. Later, Kaup et al. obtained Klein-Gordon geons (i.e., boson stars) by replacing the massless vector field with a massive complex scalar field [12]. Ruffini also independently studied boson stars by considering quantized real scalar fields [13]. The original boson stars are spherically symmetric and composed of free scalar fields with fundamental configurations. It is generalized to the cases of rotation [14,15,16], the excited BSs [17,18,19], static multipolar BSs [20, 21], the construction of vector boson stars (Proca stars) – see e.g. [22,23,24,25,26] and different kinds of multi-field or multi-state configuration [17, 27,28,29,30,31,32,33,34,35,36,37,38]. There are other generalizations as well, see the reviews [39, 40].

Previously, most of the research on boson stars focused on topologically trivial spacetime. In [41, 42], the investigation focused on mixed systems that included boson stars with wormholes in the core of the star. The bosonic matter that comprised these systems consisted of an ordinary complex boson field with self-interaction, which could exhibit symmetric or asymmetrical distribution across the two asymptotically flat regions. As a result, the wormhole spacetime could be either symmetric or asymmetric. Furthermore, the rotation of the scalar field induces a symmetric and asymmetric rotating wormhole spacetime as well [43, 44]. In these works, both the symmetric and asymmetric wormhole solutions the complex scalar field exhibit the same sign difference with respect to the two asymptotically flat spacetime regions connected by the wormhole. Thus, it is an interesting question whether there exist complex scalar fields with opposite sign differences at the two ends of a wormhole. In this paper, we reinvestigate the model of bosonic matter in the form of a free complex scalar field with a nontrivial wormhole spacetime topology supported by a free phantom field. A new kind of boson star with wormhole solutions is obtained, in which the complex scalar field could possess full parity-odd symmetry with respect to the two asymptotically flat spacetime regions. In particular, the scalar field that form two boson stars with respect to the two spacetime regions has opposite phase differences in the limit of vanishing throat size. It is noteworthy that this new kind of solution still possesses symmetric wormhole spacetime structure.

The paper is organized as follows. In Sect. 2, we review the model of a complex scalar field and a phantom scalar field field coupled with gravity. We present the boundary conditions and quantities of interest in Sect. 3, discuss the numerical results in Sect. 4. The last section is devoted to conclusion.

2 The model

We consider the Einstein–Hilbert action including the Lagrangian for the complex scalar field and the phantom scalar field field

$$\begin{aligned} S=\int \sqrt{-g}d^4x\left( \frac{R}{2\kappa }+\mathcal {L}_{p}+\mathcal {L}_{m}\right) , \end{aligned}$$
(1)

where R is the Ricci scalar. The term \(\mathcal {L}_{p}\) and \(\mathcal {L}_{m}\) are the Lagrangians defined by with

$$\begin{aligned} \mathcal {L}_{m}= & {} -\nabla _a\Psi ^*\nabla ^a\Psi - \mu _0^2\Psi \Psi ^*, \end{aligned}$$
(2)
$$\begin{aligned} \mathcal {L}_{p}= & {} \nabla _a\Phi \nabla ^a\Phi . \end{aligned}$$
(3)

Here \(\Psi \) and \(\Phi \) represent the complex scalar field and the phantom field, respectively. By varying the action (1) with respect to the metric, we can obtain the Einstein equations

$$\begin{aligned} R_{\mu \nu }-\frac{1}{2}g_{\mu \nu }R-\kappa T_{\mu \nu }=0 \, \end{aligned}$$
(4)

with stress-energy tensor

$$\begin{aligned} T_{\mu \nu } = g_{\mu \nu }({{{\mathcal {L}}}}_m+{{{\mathcal {L}}}}_p) -2 \frac{\partial ({{{\mathcal {L}}}}_m+{{{\mathcal {L}}}}_p)}{\partial g^{\mu \nu }} \, \end{aligned}$$
(5)

and the matter field equations by varying with respect to the scalar field and phantom field

$$\begin{aligned} \Box \Psi -\mu _0^2\Psi =0, \end{aligned}$$
(6)

and

$$\begin{aligned} \Box \Phi =0. \end{aligned}$$
(7)

We consider the general static spherically symmetric solution with a wormhole, and adopt the Ansätzes as follows, see e.g. [42]:

$$\begin{aligned} ds^2 = -e^{A} dt^2 +B e^{-A} \left[ d r^2 + h (d \theta ^2+\sin ^2 \theta d\varphi ^2) \right] , \end{aligned}$$
(8)

where A and B are functions of radial coordinate r, \(h=r^2+r_0^2\) with the throat parameter \(r_0\). and r ranges from positive infinity to negative infinity. The two limits \(r\rightarrow \pm \infty \) correspond to two distinct asymptotically flat spacetime. Furthermore, we assume stationary complex scalar field and phantom field in the form

$$\begin{aligned} \Psi&=\psi (r)e^{i\omega t}, \quad \Phi&=\phi (r). \end{aligned}$$
(9)

Here, \(\psi \) is only a real function of the radial coordinate r, and the constant \(\omega \) is referred to as the synchronization frequency. Moreover, the phantom field \(\Phi \) is also a real function and is independent of the time coordinate t.

Substituting the above Ansätzes (8) and (9) into the Einstein equations (4) leads to the following field equations

$$\begin{aligned}{} & {} 4 \kappa e^{-2 A} \psi ^2 B^2 \left( \mu _0^2 e^{A}-\omega ^2\right) +\frac{3 r B'}{h}+B''-\frac{B'^2}{2 B}=0, \nonumber \\ \end{aligned}$$
(10)
$$\begin{aligned}{} & {} A''+\frac{2 r A'}{h}+\frac{A' B'}{2 B}+2 \kappa e^{-2 A} \psi ^2 B \left( \mu _0^2 e^{A}-2 \omega ^2\right) =0, \nonumber \\ \end{aligned}$$
(11)
$$\begin{aligned}{} & {} \frac{e^{2 A} \left( -B^2 \left( h^2 A'^2+4 r_0^2\right) +4 r h B B'+h^2 B'^2\right) }{2 h^2 B^2}\nonumber \\{} & {} \quad +\kappa \left( 2 \psi ^2 B \left( \mu _0^2 e^{A}-\omega ^2\right) +e^{2 A} \left( \phi '^2-2 \psi '^2\right) \right) =0. \nonumber \\ \end{aligned}$$
(12)

Variation of the action with respect to the complex scalar field and to the phantom field leads to the equations

$$\begin{aligned} \left( \omega ^2 e^{-2 A} \psi B-\mu _0^2 e^{-A} B\right) +\frac{(h \sqrt{B} \psi ')'}{\sqrt{B} h}= & {} 0, \end{aligned}$$
(13)
$$\begin{aligned} (h \sqrt{B} \phi ')'= & {} 0. \end{aligned}$$
(14)

These five equations are divided into three groups: three of these Eqs. (10), (11) and (13) are solved together, yielding a coupled system of three ODEs on the three unknown functions A, B and \(\psi \). After obtaining the solution for these three functions, ones can solve Eq. (14) to determine the value of \(\phi \). The remaining one Eq. (12) is treated as the constraint and used to check the numerical accuracy of the method. Especially, The above derivative in Eq. (14) happens to be zero, which means it can be equal to a constant. Integrating the last equation, ones can leads to

$$\begin{aligned} \phi ' = \frac{\sqrt{{\mathcal {D}}}}{h \sqrt{B}}. \end{aligned}$$
(15)

Here the constant of integration \({\mathcal {D}}\) denotes the scalar charge of the phantom field. By taking Eq. (15) into Eq. (12) the scalar charge \({\mathcal {D}}\) can be expressed into the following form

$$\begin{aligned} {{\mathcal {D}}}= & {} -\frac{h^2 B}{\kappa e^{2 A} }\left( \frac{e^{2 A} \left( -B^2 \left( h^2 A'^2+4 r_0^2\right) +4 r h B B'+h^2 B'^2\right) }{2 h^2 B^2}\right. \nonumber \\{} & {} \left. +\kappa \left( 2 \psi ^2 B \left( \mu _0^2 e^{A}-\omega ^2\right) -2 e^{2 A} \psi '^2\right) \right) . \end{aligned}$$
(16)

The system of coupled non-linear Eqs. (10), (11) and (13) presents a formidable challenge in seeking a general analytical solution, necessitating the use of numerical methods to obtain the solutions to differential equations. However, prior to numerically solving the aforementioned equations, it is worthwhile to investigate two special solution cases. Firstly, when the complex scalar field vanishes, one can derive the solution for an Ellis wormhole. Secondly, as the throat size \(r_0\) approaches zero, two mini-boson stars emerge about the \(r_0=0\) symmetry and correspond to two distinct asymptotically flat spacetimes. These two special cases provide valuable insight into the behavior of the system and can inform subsequent numerical analyses.

3 Boundary conditions and quantities of interest

To solve the coupled non-linear Eqs. (10), (11) and (13) in asymptotically flat spacetime, we give appropriate boundary conditions for the functions as follows

$$\begin{aligned} \psi =A=0, \hspace{5pt} B=1, \end{aligned}$$
(17)

at infinity (\(r \rightarrow \infty \)). At the origin (\(r=0\)), the symmetry of functions can be used to distinguish between two types of solutions. For even parity solutions, we require the following boundary conditions

$$\begin{aligned} \partial _r \psi =\partial _r A=\partial _r B= 0. \end{aligned}$$
(18)

The above boundary conditions were employed in [42] to investigate the scenario of symmetric solutions. Considering that the complex scalar with parity-odd symmetry, we impose

$$\begin{aligned} \psi&= 0, \nonumber \\ \partial _r A&=\partial _r B= 0. \end{aligned}$$
(19)

The ADM mass M is the key quantities we are interested in, which is encoded in the asymptotic expansion of metric components

$$\begin{aligned} g_{tt}= -1+\frac{2 M}{r}+\cdots . \end{aligned}$$
(20)

The action of the complex scalar field is invariant under the U(1) transformation \(\psi \rightarrow e^{i\alpha }\psi \) with a constant \(\alpha \). According to Noether’s theorem, there is a conserved current corresponding to the complex scalar field:

$$\begin{aligned} J^{\mu } = -i\left( \psi ^*\partial ^\mu \psi - \psi \partial ^\mu \psi ^*\right) , \quad J^\mu _{\,\,; \mu } =0 . \end{aligned}$$
(21)

Integrating the timelike component of the above conserved currents on a spacelike hypersurface \({\mathcal {S}}\), ones could obtain Noether charge:

$$\begin{aligned} Q= & {} \int _{{\mathcal {S}}}J_S^t \nonumber \\= & {} - \int J^t \left| g \right| ^{1/2} dr d\Omega _{2} \nonumber \\= & {} 8 \pi \omega \int _0^{\infty } |g| ^{1/2} \frac{\psi ^2}{e^{A}} \,dr . \end{aligned}$$
(22)

The global charge Q is equivalent to the number of particles in the complex scalar field.

Fig. 1
figure 1

The radial distribution of the complex scalar field. Left: the distribution with several values of the frequency \(\omega \) for \(r_0=1\). Right: the distribution with several values of the throat size \(r_0\) for \(\omega =0.95\)

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Fig. 2
figure 2

The ADM mass M and Noether charge Q as a function of the frequency \(\omega \) with several values of the throat size \(r_0\)

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4 Numerical results

In this work, all the numbers are dimensionless as follows

$$\begin{aligned} r \rightarrow r\mu _0 , \hspace{5pt} \phi \rightarrow \phi \kappa ^{-1/2}, \hspace{5pt} \omega _i \rightarrow \omega _i/\mu _0 . \end{aligned}$$
(23)

For convenience, we set \( \mu _0 = 1\) and \(\kappa =0.2\). We transform the radial coordinates by the following equation

$$\begin{aligned} x= \frac{2}{\pi }\arctan (r) , \end{aligned}$$
(24)

The use of Eq. (24) map the infinite region \((-\infty ,\infty )\) to the finite region \((-1,1)\). Next, what we need is to discretize the differential equations on a grid. This allows the ordinary differential equations to be approximated by algebraic equations. The grid with 500 points covers the integration region \(-1 \le x \le 1\). In our work, relative errors are less than \(10^{-6}\).

First, as an example, we plot in Fig. 1 two typical profiles of our numerical results for the scalar field \(\psi \) as a function of x. In the left panel, the throat size \(r_0 =1\) is fixed and the curves with several values of the frequency \(\omega \) are shown. We can see that the scalar field \(\psi \) has an antisymmetric structure about the origin \(x=0\). \(\psi _{\max }\) (the maximum value of the field function \(\psi \)) increases as the frequency \(\omega \) decreases. Moreover, the extreme points converge towards the origin \(x=0\) as the frequency decreases. In the right panel, the frequency \(\omega = 0.95\) is fixed and curves for different values of throat size \(r_0\) are displayed. In particular, we represent the two boson star solutions with black dashed lines, and these two solutions form an anti-symmetric structure at the two ends of the wormhole. Ones can see that at positions where x is large, the curves for different values of \(r_0\) do not differ much and basically overlap. In the region where x is small, the value of the scalar field increases as \(r_0\) decreases. When the limit \( r_0 \rightarrow 0\), the solutions that are very close to two boson stars at the two ends of the wormhole except near the origin.

4.1 Global charges of the solutions

In this subsection, we will initially examine the global charges of the resulting wormhole solutions. In Fig. 2, we exhibit the mass M and Noether charges Q versus the frequency \(\omega \) with the different values of throat size \(r_0\), the black curve corresponds to the boson star solution. It is interesting that the curve does not form a spiral. Instead, When \(r_0\) starts to increase from 0 value, the curve will first form a loop structure. Then, as \(r_0\) continues to increase, the multi-valued curve will become a single-valued curve. In the first branch, when the frequency is fixed, we find that as \(r_0\) increases, both the mass M and charge Q also increase, which have similar behavior.

Fig. 3
figure 3

The scalar charge \({\mathcal {D}}\) of the phantom field as a function of the frequency \(\omega \) with several values of the throat size \(r_0\)

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In Fig. 3, we exhibit the scalar charge \({\mathcal {D}}\) versus the frequency \(\omega \) with the different values of throat size \(r_0\). We can see that as \(r_0\) increases, the scalar charge also increases, which means that more phantom matter is needed. For curves with larger \(r_0\) values, the charge decreases monotonically with decreasing frequency \(\omega \).

Fig. 4
figure 4

Geometric properties of throats: two dimensional view of the isometric embedding of the equatorial plane of symmetric solutions with throat parameter \(r_0=1\) and several values of the complex scalar frequency \(\omega \)

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Fig. 5
figure 5

Geometric properties of throats: isometric embeddings of the equatorial plane of symmetric solutions with throat parameter \(r_0=1\). a Three dimensional plot with three throats for \(\omega =0.15\). b Three dimensional plot with a single throat for \(\omega =0.4\)

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4.2 Geometric properties of wormhole

To explain that the metric (8) does indeed describe a wormhole, one can make use of geometrical embedding diagrams. It is helpful to consider a two-dimensional space at a fixed time and \(\theta \). The resulting two-dimensional spatial hypersurface of the wormhole spacetime can then be embedded in a three-dimensional Euclidean space, where the embedding diagram can be used to visualize the wormhole geometry. This technique allows us to better understand the topology and properties of the wormhole solution.

To characterize the geometry of the throat, we begin by examining the embeddings of planes with \(\theta =\pi /2\). Using cylindrical coordinates \((\rho ,\varphi ,z)\), The metric on this plane can be expressed by the following formula

$$\begin{aligned} ds^2&= B e^{-A} d r^2 + B e^{-A} h d\varphi ^2 \, \end{aligned}$$
(25)
$$\begin{aligned}&= d \rho ^2 + dz^2 + \rho ^2 d \varphi ^2 . \end{aligned}$$
(26)

Comparing the two equations above, we then obtain the expression for \(\rho \) and z,

$$\begin{aligned} \rho (r)= & {} \sqrt{ B(r) e^{-A(r)} h(r) },\quad \nonumber \\ z(r)= & {} \pm \int \sqrt{ B(r) e^{-A(r)} - \left( \frac{d \rho }{d r} \right) ^2 } d r . \end{aligned}$$
(27)

Here \(\rho \) corresponds to the circumferential radius, which corresponds to the radius of a circle located in the equatorial plane and having a constant coordinate r. The function \(\rho (r)\) has extreme points, where the first derivative is zero. When the second derivative of the extreme point is greater than zero, we call this point an throat, which corresponds to a minimal surface. When the second derivative of the extreme point is less than zero, we call this point a equator, which corresponds to a maximal surface.

To illustrated geometric properties of throats, In Fig. 4, we show two dimensional view of the isometric embedding of the equatorial plane of symmetric solutions with throat parameter \(r_0=1\) and several values of the complex scalar frequency \(\omega \). When the frequency is high, the wormhole has only one throat with a small radius. As the frequency increases, the radius of the throat also increases. At near \(\omega =0.26\), two new throats begin to appear, and the wormhole then has three throats. As the frequency further increases, the radii of the three throats will first increase and then decrease. In particular, the middle throat will become less and less visible, although it always exists, but tends to disappear. In order to get further insight into the geometry of the wormholes, we present two three-dimensional plot in Fig. 5, including one with three throats and two equators with \(\omega =0.15\), and the other with only one throat with \(\omega =0.4\).

5 Conclusions

In this paper, we reconsidered the model of a complex boson field and a phantom field minimally coupled to Einstein gravity and investigated the domain of existence and physical characteristics of wormhole solutions embedded in a complex bosonic matter field with parity-odd symmetry. We have presented the relationship between the conserved charges and frequency, which depends on the throat size. Furthermore, we have compared our results to those of the single boson star solution. Surprisingly, the curve does not exhibit a spiral shape. Instead, as the throat size \(r_0\) increases from zero, the curve initially forms a loop structure. As \(r_0\) continues to increase, the multi-valued curve gradually becomes a single-valued curve. Unlike the even-symmetric case studied previously, the solutions with a complex odd-symmetric bosonic matter field cannot degenerate into boson stars. One noteworthy aspect is that solutions are discontinuous in the limit of zero radius of the throat. Although there will appear two phases with opposite signs, resembling a pair of bosonic stars, the configuration of these complex scalar field solutions is distinct from the dipolar configurations of boson stars studied in [20]. In the latter case, the solutions remain continuous and smooth throughout the entire space.

There are some interesting extensions of our work which we plan to investigate in future projects. First, in order to achieve a wormhole solution in general relativity, the presence of exotic matter is required. However, under modified gravity models, it is possible to construct wormhole solutions without the need for exotic matter. Therefore, studying the existence of such solutions under modified gravity is interesting. Furthermore, we are planning to investigate the stability of these solutions. Finally, considering that boson stars have both ground state and excited state solutions, we can explore the excited state solutions in the future.