1 Introduction

One of the most attractive problems in cosmology is the nature of dark matter. Boson stars (BSs) are considered as a possible explanation for parts of the dark matter in the halo of galaxies [1,2,3]. In the 1950s, John Wheeler explored the classical field coupled to Einstein gravity and introduced the concept of geons [4, 5]. However, no stable geons are found in Einstein–maxwell theory. Later, Kaup et al. constructed a complex scalar field coupled to Einstein gravity theory, which was called Einstein–Klein–Gordon (EKG) theory. BSs described as macroscopic Bose–Einstein condensates are a kind of solutions for Einstein–Klein–Gordon theory. Heisenberg’s uncertainty principle provides a repulsive force against gravitational collapse mainly. They focus on spherical symmetry configuration, and it is likewise meaningful to understand the rotating boson stars. The cases of rotating boson stars were studied by Schunck and Mielke [6, 7], Yoshida and Eriguchi [8], Herdeiro and Radu [9]. Furthermore, the rotating axisymmetric solutions of BSs were generalized to the excited state case [10,11,12]. See Refs. [13, 14] for a review.

Besides Einstein gravity coupled to a single scalar field, it can also be coupled to several different fields. In Ref. [15], Deng and Huang dealt with the case of two scalar fields coupled to gravity, in which two different scalar fields coexist in the ground state. Then, Bernal et al. [16] constructed the model, which has two states, a ground state and a first excited state. They found that multistate boson stars have a higher critical mass than the boson stars with the ground state, and are more stable than excited cases. Later, more types of multistate boson stars are studied, including spherical symmetry configuration and axially symmetric configuration [17,18,19,20,21]. Furthermore, charged boson stars are the scalar field coupled to the electromagnetic field e.g. [22,23,24], fermion-boson stars consist of bosonic and fermionic matter which is approximately described as perfect fluids e.g. [14, 25, 26], and newtonian configurations of boson multistate were introduced in [27]. In addition to this, a lot of interesting studies on BSs have been shown in Refs. [28,29,30,31], and recent studies of boson stars [32,33,34,35,36,37,38,39,40,41,42,43,44] have also received a lot of attention.

Recently, the electron recoil excess events are observed by XENON1T. One of the explanations is solar axions coupled to electrons [45]. The axion is seen as a plausible candidate for dark matter. To solve the strong CP problem [46, 47], the quantum chromo dynamics (QCD) axion is studied [48,49,50,51]. In addition, axion like particles (ALPs) also plays an important role in string theory [52]. Guerra et al. studied the solutions of spherical symmetry axions boson stars (ABSs) [53]. This work was extended to rotating axion boson stars (RABSs) [54, 55]. When \(f_a\) is large, they suggested that the axion potential reduces to the massive, free, and complex scalar field, which matches the solution of the standard spinning mini-boson stars in the context of Einstein gravity coupled to free scalar field [7, 56].

The scope of this work is to construct the solutions of rotating hybrid axion-miniboson stars, which are a mixture of bosonic and axionic matter, minimally coupled to gravity, respectively. Using numerical calculation, we study the mass and the angular momentum for different frequencies.

This paper is organized as follows: in Sect. 2, we present the model of four-dimensional Einstein gravity minimally coupled to a massive free scalar field and a QCD axion field. In Sect. 3, boundary conditions of RHABSs are shown. In Sect. 4, we exhibit the numerical results and show the properties of \(^1S^2S\) state and \(^1S^2P\) state for different axion decay constants \(f_a\) and scalar field mass \(\mu _f\). In Sect. 5, conclusions and perspectives are given.

2 The model setup

In this work, we consider the case of an axion field and a free complex massive scalar field, which is coupled to \((3+1)\)-dimensional Einstein gravity. The action is

$$\begin{aligned} S=\int \sqrt{-g}d^4x\left( \frac{R}{16\pi G}+\mathcal {L}_{m}\right) \, \end{aligned}$$
(1)

where the first term represents Einstein gravity, the second term \({{\mathcal {L}}}_{m}\) is marked as the matter Lagrangian about an axion field and a scalar field, and defined as follows

$$\begin{aligned} \mathcal {L}_{m}= & {} -\nabla _{\mu }\psi _a^*\nabla ^{\mu }\psi _a - U_a(|\psi _a|^2) -\nabla _{\mu }\psi _f^*\nabla ^{\mu }\psi _f\nonumber \\{} & {} -U_f(|\psi _f|^2). \end{aligned}$$
(2)

Here \(\psi _a\) and \(\psi _f\) represent the axion field and the free scalar field, respectively. \(U_a\) and \(U_f\) are the potentials of axion scalar field and free scalar field, respectively. By varying the action, we can derive the equation of motion. The field equations are

$$\begin{aligned} E_{\mu \nu }= & {} R_{\mu \nu }-\frac{1}{2}g_{\mu \nu }R-8\pi T_{\mu \nu }=0 \, \end{aligned}$$
(3)
$$\begin{aligned}{} & {} \times \Box \psi _a-\frac{\partial U_a}{\partial |\psi _a|^2}\psi _a=0 \, \Box \psi _f-\frac{\partial U_f}{\partial |\psi _f|^2}\psi _f=0 .\nonumber \\ \end{aligned}$$
(4)

Here

$$\begin{aligned} T_{\mu \nu }= & {} 2\nabla _{(\mu }\psi _a^{*}\nabla _{\nu )}\psi _a-g_{\mu \nu }(\nabla ^{\lambda }\psi _a^{*}\nabla _{\lambda }\psi _a+U_a) \nonumber \\{} & {} +2\nabla _{(\mu }\psi _f^{*}\nabla _{\nu )}\psi _f - g_{\mu \nu }(\nabla ^{\lambda }\psi _f^{*}\nabla _{\lambda }\psi _f + U_f) \end{aligned}$$
(5)

is the energy-momentum tensor associated with the axion field and the free scalar field.

We adopt the following ansatz for the metric and matter field [9, 57],

$$\begin{aligned} d s^{2}= & {} -e^{2 F_{0}(r, \theta )} d t^{2}+e^{2 F_{1}(r, \theta )}\left( d r^{2}+r^{2} d \theta ^{2}\right) \nonumber \\{} & {} +e^{2 F_{2}(r, \theta )} r^{2} \sin ^{2} \theta (d \varphi -W(r, \theta ) d t)^{2} \, \end{aligned}$$
(6)
$$\begin{aligned} \psi _a= & {} \phi _{a(n)}(r,\theta )e^{i(m_a\varphi -\omega _a t)},\nonumber \\{} & {} n=0,1,\ldots , \hspace{5pt} m_a=\pm 1,\pm 2, \cdots \, \end{aligned}$$
(7)
$$\begin{aligned} \psi _f= & {} \phi _{f(n)}(r,\theta )e^{i(m_f\varphi -\omega _f t)},\nonumber \\{} & {} n=0,1,\ldots , \hspace{5pt} m_f=\pm 1,\pm 2, \cdots . \end{aligned}$$
(8)

Here \(F_0\), \(F_1\), \(F_2\), W, \(\phi _{a(n)}\) and \(\phi _{f(n)}\) depend only on the radial coordinate r and the polar angle \(\theta \). The subscript n represents the node number (i.e., how many times matter field function cross zero). The state with \(n=0\) represents the ground state, and the states with \(n \ge 1\) are the excited states. m and \(\omega \) denote the azimuthal harmonic index and field frequency. For the solution of the RHABSs, we consider the minimal value of the azimuthal harmonic index (\(m_a=m_f=1\)). The frequency of the field is called the synchronized frequency when \(\omega _a=\omega _f=\omega \). The case of \(\omega _a\ne \omega _f\) is called nonsynchronized frequency.

Next, we specify the axion field potential and the free scalar field potential

$$\begin{aligned}{} & {} U_a=\frac{2\mu _a^2f_a^2}{B}\left[ 1-\sqrt{1-4B sin^2 \left( \frac{\phi _a}{2f_a}\right) }\right] \, \end{aligned}$$
(9)
$$\begin{aligned}{} & {} U_f=\mu _f^2\psi _f^2 . \end{aligned}$$
(10)

Here \(B=Z/(1+Z)^2 \approx 0.22\), \(Z \equiv \frac{m_u}{m_d} \approx 0.48\), \(m_u\) and \(m_d\) are masses of the up and down quarks, respectively. \(\mu _f\) is the free scalar field mass. The axion potential has two free parameters \(\mu _a\) and \(f_a\), which are the axion mass and decay constant, respectively. We expand the axion potential around \(\phi = 0\) and obtain the form as follows

$$\begin{aligned} U_a(\phi _a)=\mu _a^2\phi _a^2-\left( \frac{3B-1}{12}\right) \frac{\mu _a^2}{f_a^2}\phi _a^4+\cdots . \end{aligned}$$
(11)

According to the above equation, the \(f_a\) can be considered as a measure of self-interaction of the axion field. If \(f_a \gg \phi _a\), the axion potential tends to the free scalar potential. The model will reduce to the rotating multistate boson stars in Ref. [18].

3 Boundary conditions

In order to solve Eqs. (3) and (4), boundary conditions are necessary. The metric functions \(F_0(r,\theta )\), \(F_1(r,\theta )\), \(F_2(r,\theta )\), \(W(r,\theta )\) and the field functions \(\phi _{a}(r,\theta )\) and \(\phi _{f}(r,\theta )\) need to be specified. For asymptotically flat solutions, the functions must be

$$\begin{aligned} F_0=F_1=F_2=W=\phi _{a}=\phi _{f}=0 \hspace{5pt} \end{aligned}$$
(12)

at infinity \(r \rightarrow \infty \). For axial symmetry,

$$\begin{aligned} \partial _\theta F_0=\partial _\theta F_1=\partial _\theta F_2 =\partial _\theta W=\phi _{a}=\phi _{f}=0 \end{aligned}$$
(13)

on the axis (\(\theta =0,\pi \)). Due to the symmetry of ansatz function about \(\theta =\pi /2\) plane, we only consider the range \(\theta \in [0,\pi /2]\) for all solutions [9]. If the matter field are symmetric for \(\theta =\pi /2\) plane, we have

$$\begin{aligned} \partial _\theta F_0=\partial _\theta F_1=\partial _\theta F_2 = \partial _\theta W=\partial _\theta \phi _{a} =\partial _\theta \phi _{f} = 0 . \end{aligned}$$
(14)

If the matter field are anti-symmetric for \(\theta =\pi /2\) plane, we have

$$\begin{aligned} \partial _\theta F_0=\partial _\theta F_1=\partial _\theta F_2 = \partial _\theta W=\phi _{a} = \phi _{f} = 0 . \end{aligned}$$
(15)

At the origin we require

$$\begin{aligned} \phi _{a}&=\phi _{f} = 0 \ , \nonumber \\ \partial _r F_0=\partial _r F_1&=\partial _r F_2=\partial _r W = 0 \ . \end{aligned}$$
(16)

To compute the ADM mass M and angular momentum J, we expand \(g_{tt}\) and \(g_{t\phi }\) at \(r\rightarrow \infty \) as follows

$$\begin{aligned} g_{tt}= -1+\frac{2GM}{r}+\cdots \, \nonumber \\ g_{\varphi t}= -\frac{2GJ}{r}\sin ^2\theta + \cdots . \end{aligned}$$
(17)

4 Numerical results

On the one hand, to simplify the form of equations, we use natural units set by \(\mu _a\) and G as follows

$$\begin{aligned} r \rightarrow r\mu _a , \hspace{5pt} \phi \rightarrow \phi M_{PI} , \hspace{5pt} \omega \rightarrow \omega /\mu _a , \hspace{5pt} \mu _f \rightarrow \mu _f/\mu _a .\nonumber \\ \end{aligned}$$
(18)

Here \(M_{PI}^2=G^{-1}\) is the Plank mass. We set \(G = c = \mu _a = 1\) and \(\mu _f=0.93\) in our models. On the other hand, it’s convenient that we transform the radial coordinate \([0,\infty )\) to [0, 1] as follows

$$\begin{aligned} x=\frac{r}{1+r} . \end{aligned}$$
(19)

All equations are handled by finite element methods. The computation has \(200 \times 120\) grid points in the integration region \(0 \le x \le 1\) and \(0 \le \theta \le \pi /2\). The relative error for the numerical solutions is less than \(10^{-5}\).

Our work follows rotating boson stars in the first excited state and rotating multistate boson stars. Some properties of rotating boson stars deserve to be introduced. The ground state is the \(^1S\) state, which has no nodes. By setting different boundary conditions (14) and (15), there are two types of excited solutions. The \(^2S\) state has a node along the radial direction, and the scalar field is symmetric for \(\theta =\pi /2\) plane. The \(^2P\) state has a node along the angular direction, and the scalar field is anti-symmetric for \(\theta =\pi /2\) plane. For the rotating multistate boson stars (RMSBSs) without self-interaction, the scalar fields are not all in the same state, instead of existing in different states [18]. In the case of the RHABSs, we also do not find nontrivial solutions in which two scalar fields are in the same state. According to Eq. (11), when \(f_a \gg \phi _a\), the RHABSs reduce to rotating multistate boson stars. Here, we follow a similar conception. We assume that the axion field exists in the ground state and the free scalar field exists in the first excited state for this model. If the free scalar field is in the \(^2S\) state, which has a radial node \(n_r=1\), the coexisting state is called the \(^1S^2S\) state. If the free scalar field is in the \(^2P\) state, which has an angular node \(n_{\theta }=1\), the coexisting state is called the \(^1S^2P\) state.

In order to make RHABSs present more branches and more complex structures, we want to make \(f_a\) smaller. If \(f_a \rightarrow \infty \), the axion field reduces to a free complex scalar field. When \(f_a=1\), \(f_a\) is enough large, the behavior of rotating axion boson stars is similar to mini-boson stars [54]. Thus, we choose the axion stars with \(f_a=1\) to represent mini-boson stars. When \(f_a \rightarrow 0\), the numerical results are so complex that this limit cannot be reached. By choosing axion decay constant \(f_{a}=\{1, 0.025, 0.015, 0.009\}\), we expect that the RHABSs present more branches and more complex structures.

4.1 \(^1S^2S\) state

Fig. 1
figure 1

Left: The distribution of the free scalar field \(\phi _f\) as a function of x and \(\theta \) for the case of synchronized frequency \(\omega =0.858\). Semi-translucent surface represents \(f_a=0.009\) and the opaque surface represents \(f_a = 1\). Right: At \(\theta =\pi /2\), the distribution of the free scalar field \(\phi _f\) with the same synchronized frequency \(\omega =0.858\) for \(f_{a}=\{1, 0.025, 0.015, 0.009\}\)

Full size image

The distribution of the free scalar field \(\phi _f\) for different axion decay constants \(f_a\) are presented in Fig. 1. In the left panel, we show the two-dimensional distribution of the free scalar field \(\phi _f\). The semi-translucent surface indicate \(f_a = 0.009\), and the opaque surface indicate \(f_a = 1\). In the right panel, we plot the radial distribution of the free scalar field \(\phi _f\) at \(\theta =\pi /2\) for four different axion decay constant \(f_a\). For lower \(f_a\), the maxima and minima of the free scalar field \(\phi _f\) become higher. Meanwhile, we observe that the scalar field \(\phi _f\) has no nodes along the angular \(\theta \) direction. Along the radial r direction, the scalar field \(\phi _f\) has a node. Thus, the free scalar field is in the \(^2S\) state as we said in the Sect. 4. Next, we show the mass M versus the synchronized frequency \(\omega \) as well as the nonsynchronized frequency \(\omega _f\) (\(\omega _a=0.8\)). These behaviors are shown in Figs. 2 and 3. And then, we study the existence domain of the synchronized frequency \(\omega \) for different \(f_a\) and \(\mu _f\). For the RHABSs in the \(^1S^2S\) state, the relationship between mass M and angular momentum J will be discussed later in 4.2.

Fig. 2
figure 2

The mass M of the RHABSs as a function of the synchronized frequency \(\omega \) (\(\omega _a=\omega _f=\omega \)) for \(f_a=\{1, 0.025, 0.015, 0.009\}\). The black dotted line indicates the RABSs in the \(^1S\) state, The red dotted line indicates the rotating boson stars in the \(^2S\) state, and the blue line represents the RHABSs, respectively

Full size image

In the case of synchronized frequency in Fig. 2, we study the mass M of RHABSs versus the synchronized frequency \(\omega \) with the \(f_a=\{1, 0.025, 0.015, 0.009\}\), respectively. RHABSs are self-gravitational bound state. In order to ensure that boson stars are bound states, the condition \(\omega _a \le \mu _a\) and \(\omega _f \le \mu _f\) is necessary. The black dotted line represents the RABSs in the ground state. The red dotted line represents the mini-boson stars in the first excited state. The blue line represents the RHABSs consisting of the axion field in the ground state and the free scalar field in the first excited state. When \(f_a=1\), the black dotted line starts at \(\omega _a=1\), as the frequency \(\omega _a\) decreases, the mass M gradually increases to a maximum, and then the black dotted line spirals to the center. The behavior of the black dotted line is similar to rotating mini-boson stars without self-interaction. As the axion decay constant decreases, the behavior of the RABSs becomes more and more complex, the RABSs allows for the solution with lower synchronized frequency, and even there are more branches. The red dotted line starts at \(\omega _f=0.93\) and also spirals to the center. The blue line connects the black dotted line to the red dotted line. As the synchronized frequency \(\omega \) decreases, the mass M of RHABSs monotonically increases. The RHABSs have a higher mass than the RABSs in the \(^1S\) state but a lower mass than the rotating mini-boson stars in the \(^2S\) state. When \(f_a=1\), the case is similar to the \(^1S^2S\) state of the RMSBSs consisting of two free scalar fields in Ref. [18]. As the axion decay constant decreases, the existence domain of synchronized frequency is expanded for the RHABSs, and the RHABSs have lower ADM mass. We also observe that, the axion field vanishes when synchronized frequency \(\omega \) tends to its minimum, and there exist only a single free scalar field in the first excited state \(^2S\). On the contrary, the free scalar field vanishes when synchronized frequency \(\omega \) tends to their maxima, and there exists only a single axion field in the ground state \(^1S\).

Fig. 3
figure 3

The mass M of the RHABSs as a function of the nonsynchronized frequency \(\omega _f\) with \(f_a=\{1, 0.025, 0.015, 0.009\}\) for the fixed parameter \(\omega _a=0.8\). The black dotted line indicates the RABSs in the \(^1S\) state. The red dotted line indicates the rotating boson stars in the \(^2S\) state. The blue line represents the RHABSs, respectively. The intersection of the green dashed lines represents the horizontal and vertical coordinates of the point where the mass M of the RABSs is equal to the minimum mass \(M_{min}\) of the RHABSs

Full size image

In the case of nonsynchronized frequency, in Fig. 3, we show the mass M of RHABSs versus the nonsynchronized frequency \(\omega _f\) with the \(f_a=\{1, 0.025, 0.015, 0.009\}\). Here, we set the axion field frequency of the RHABSs \(\omega _a=0.8\). The black dotted line and the red dotted line are completely the same as Fig. 2. For the blue line, as the decrease of the nonsynchronized frequency \(\omega _f\), the RHABSs in the \(^1S^2S\) state reduce to the mini-boson stars in the first excited state \(^2S\). When the nonsynchronized frequency \(\omega _f\) approach their maxima, the RHABSs in the \(^1S^2S\) state reduce to the RABSs in the ground state \(^1S\). In addition, the blue line crosses the black dotted line for lower \(f_a\). This is different from the RMSBSs with nonsynchronized frequency [18, 19]. The explanation is that the red and blue lines are functions of \(\omega _f\), while the black line is a function of \(\omega _a\). Therefore, the blue line does not end in the black line for the case of nonsynchronized frequency. When \(\omega _f\) tends to its maximum, only a single RABSs in the ground state remains, which corresponds to the fuchsia point (\(\omega _a=0.8\)) on the black line. The cases of synchronized and nonsynchronized have the same feature. As \(f_a\) decreases, the shape of the curve becomes more bent such that the behavior is similar to the mini-boson stars in the first excited state \(^2S\) in Figs. 2 and  3.

Table 1 The existence domain of the synchronized frequency \(\omega \) depends on the axion decay constant \(f_a\) and the free scalar field mass \(\mu _f\) for the \(^1S^2S\) state. The rows represent different axion decay constant \(f_a\). The columns represent the free scalar field mass \(\mu _f\)
Full size table

Table 1 represents the existence domain of the synchronized frequency \(\omega \) depends on the axion decay constant \(f_a\) and the free scalar field mass \(\mu _f\) for the \(^1S^2S\) state. As the decrease of the decay constant \(f_a\), the synchronized frequency \(\omega \) overall increases. With the mass \(\mu _f\) decreasing, the synchronized frequency \(\omega \) overall decreases. This means that the smaller the scalar field mass \(\mu _f\) is, the more the \(M-\omega \) curve shifts to the left in the Fig. 2. In addition, since the minimum of \(\omega \) represents the RABSs in the first excited state, the axion decay constant has no effect on the minimum of \(\omega \).

4.2 \(^1S^2P\) state

Fig. 4
figure 4

For the \(^1S^2P\) state, the distribution of the axion field \(\phi _f\) as a function of x and \(\theta \) where the axion decay constant \(f_a=1\) (left panel) with the same synchronized frequency \(\omega =0.842\), and the axion field \(\phi _a\) where the axion decay constant \(f_a=0.009\) (right panel) with the same synchronized frequency \(\omega =0.842\), the free scalar field \(\phi _f\) as a function of x with \(f_a=1\) and \(f_a=0.009\) at \(\theta =\pi /4\) (bottom panels)

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In this subsection, we will show the properties of the RHABSs with the coexisting state \(^1S^2P\). We exhibit the effect of the axion decay constant \(f_a\) on the two-dimensional distribution of the free scalar field \(\phi _f\) in the left panel of Fig. 4. In the right panel, we plot the radial distribution of the free scalar field \(\phi _f\) at \(\theta =\pi /4\) for four different axion decay constant \(f_a\). For lower axion decay constant \(f_a\), the maxima of the free scalar field \(\phi _f\) becomes higher, and the distribution of the free scalar field becomes steeper. We observe that the free scalar field in the first excited state \(^2P\) has an angular node in the \(\theta =\pi /2\). Next, the mass M of RHABSs in the \(^1S^2P\) state as a function of the synchronized frequency \(\omega \) as well as the nonsynchronized frequency \(\omega _f\) are exhibited in Figs. 5 and 6, respectively. We also show the table of the existence domain of the synchronized frequency for the \(^1S^2P\) state. Furthermore, we discuss the relationship between mass M and angular momentum J for the case of the \(^1S^2P\) state.

Fig. 5
figure 5

The mass M of the RHABSs as a function of the synchronized frequency \(\omega \) for \(f_a=\{1, 0.025, 0.015, 0.009\}\) for the case of \(^1S^2P\) state. The black dotted line indicates the RABSs in the \(^1S\) state, The red dotted line indicates the rotating boson stars in the \(^2P\) state, and the blue represents the coexisting state of the ground state and the first excited state, respectively. All the above solutions have \(m_f=1\)

Full size image

In Fig. 5, we show the mass M of RHABSs versus the synchronized frequency \(\omega \) for the \(^1S^2P\) state. The black dotted line (RABSs in the ground state \(^1S\)) are the same as Fig. 2. The red dotted line represents the rotating boson stars in the \(^2P\) state, which starts at \(\omega =0.93\), gradually increases as the frequency decreases, and then decreases as the synchronized frequency decreases. The blue line represents the RHABSs in the \(^1S^2P\) state. The \(^1S^2P\) state has similar behavior to the \(^1S^2S\) state. The mass M of RHABSs decreases with the synchronized frequency \(\omega \) increasing. When the synchronized frequency \(\omega \) increases to maximum, the mass of RHABSs would be minimum, which is completely provided by the RABSs. On the contrary, when the synchronized frequency \(\omega \) decreases to a minimum, the mass of RHABSs would be maximum, which is completely provided by the mini-boson stars in the \(^2P\) state. In addition, the \(^1S^2P\) state has a lower maximum mass than the \(^1S^2S\) state.

Fig. 6
figure 6

The mass M of the RHABSs as a function of the nonsynchronized frequency \(\omega _f\) (\(\omega _a=0.8\)) for \(f_a=\{1, 0.025, 0.015, 0.009\}\) at \(^1S^2P\) state. The black dotted line indicates the RABSs in the \(^1S\) state, The red dotted line indicates the rotating boson stars in the \(^2P\) state, and the blue represents the multistate of the axion field and the free scalar field, respectively. All the above solutions have \(m_f=1\)

Full size image

In Fig. 6, we show the mass M of RHABSs versus the nonsynchronized frequency \(\omega _f\). The black dotted line and the red dotted line are the same as Fig. 5. We fix the axion field frequency \(\omega _a=0.8\) and change the free scalar field frequency \(\omega _f\). Similarly, the blue line doesn’t end on the black dotted line. For low \(f_a\), both \(^1S^2S\) state and \(^1S^2P\) state crosses the black dotted line in the case of nonsynchronized frequency. The mass M of RHABSs decreases with the nonsynchronized frequency \(\omega _f\) increasing. When the nonsynchronized frequency \(\omega _f\) increases to maximum, the RHABSs reduce to the RABSs with \(\omega =0.8\).

For the two types of hybrid solutions we obtained, we have found only one family of RHABSs between the ground state and the first excited state. Generally speaking, we believe RHABSs have more curves from other branches of the RABSs. However, we did not find these curves until now. But this does not mean that any new family of RHABSs can’t exist, only that it’s difficult for us to find another family of RHABSs.

Table 2 The existence domain of the synchronized frequency \(\omega \) depends on the axion decay constant \(f_a\) and the mass \(\mu _f\) for the \(^1S^2P\) state. The rows represent different axion decay constant \(f_a\). The columns represent different free scalar field mass \(\mu _f\)
Full size table

In Table 2, for the RHABSs with the \(^1S^2P\) state, the existence domain of the synchronized frequency \(\omega \) for different \(f_a\) and \(\mu _f\) are shown. We can see that as the mass \(\mu _f\) decreases, the synchronized frequency \(\omega \) overall decreases. This means that the \(M-\omega \) curve will shift to the left in the Fig. 5. As the axion decay constant \(f_a\) decreases, the upper limit of the synchronized frequency \(\omega \) becomes higher for the fixed mass \(\mu _f\), the existence domain of the synchronized frequency \(\omega \) would be extended. Likewise, the minimum of the synchronized frequency \(\omega \) is almost unchanged.

Then, we consider the relationship between the mass M and the angular momentum J. In Figs. 7 and  8, we exhibit the \(M-J\) curves for the \(^1S^2S\) state and \(^1S^2P\) state. The black line (\(^1S^2S\) state at the synchronized frequency), the red line (\(^1S^2S\) state at nonsynchronized frequency), the blue line (\(^1S^2P\) state at the synchronized frequency) and the green line (\(^1S^2P\) state at nonsynchronized frequency) are almost straight. As the angular momentum of the RHABSs increases, the mass of the RHABSs increases proportionally. In addition, the curves of the \(^1S^2S\) state and the \(^1S^2P\) state end at the same point in the case of synchronized frequency. Similarly, the curves of the \(^1S^2S\) state and the \(^1S^2P\) state also end at the other point in the case of nonsynchronized frequency. This is because when the synchronized frequency and the nonsynchronized frequency approach their maxima, the RHABSs with the \(^1S^2S\) state and the \(^1S^2P\) state reduce to the same RABSs in the ground state.

Fig. 7
figure 7

Left: The mass M of the RHABSs versus the angular momentum J for the synchronized frequency \(\omega \) and the nonsynchronized frequency \(\omega _f\) with \(f_a=1\). Right: The mass M of the RHABSs versus the angular momentum J with \(f_a=0.025\)

Full size image
Fig. 8
figure 8

Left: The mass M of the RHABSs versus the angular momentum J for the synchronized frequency \(\omega \) and the nonsynchronized frequency \(\omega _f\) with \(f_a=0.015\). Right: The mass M of the RHABSs versus the angular momentum J with \(f_a=0.009\)

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5 Conclusions

In this article, we constructed a family solution of rotating hybrid axion-miniboson stars composed of a free scalar field and an axion field. Due to the different boundary conditions we have chosen, we obtained two types of solutions, including \(^1S^2S\) state and \(^1S^2P\) state. We analyzed the influence of the axion decay constant, scalar field mass, and matter field frequency. There is an existence domain of the frequency for the RHABSs, including the case of synchronized frequency and nonsynchronized frequency. As the axion decay constant decreases, the existence domain of the RHABSs is extended, and the RHABSs have lower ADM mass and angular momentum. Moreover, with the decreasing of frequency, the RHABSs reduce to a single free scalar boson star in the first excited state. On the contrary, with the increase of frequency, the RHABSs reduce to an axion star in the ground state.

In this article, we only consider numerical solutions for rotating hybrid axion-miniboson stars and have not yet studied the stability of RHABSs. We know that the rotating boson stars without self-interaction are dynamical unstable, even if the BSs exist in the ground state [39, 40]. By adding an axionic potential, the instability of boson star with \(m=1\) can be quenched in some region of parameter space [43]. In addition, a dynamical unstable excited BS is stabilized by adding a sufficiently large spherical BS [16, 42]. Thus, we think that the superposition of a stable axion stars in the ground state and an unstable mini-boson star in the excited state may be a stabilization mechanism of the RHABSs. In the future, we will further investigate the stability of RHABSs.

We assume the configurations with the axion field in the ground state and the free scalar field in the first excited state. Actually, we can also consider the possibility that both fields are in the ground state, or the free scalar field is in the ground state and the axion field is in the first excited state. As for the former, two different fields can both be in the ground state. However, when the axion decay constant is large, the axion field reduces to a free scalar field, and boson stars consisting of two identical fields in the same state are trivial [16]. For comparison with the rotating multistate boson stars [18], we consider one field is in the ground state and the other field is in the first excited state. As for the latter, RHABSs with the axion field in the ground state may be more stable, as we have said above. Meanwhile, the behavior of the excited axion field is relatively complicated. It’s more difficult for us to construct those solutions with the axion field in the excited state. Therefore, we first study the configurations with the axion field in the ground state and the free scalar field in the first excited state. We will study other more complex configurations as further work.

Another noteworthy point is that we choose free scalar field mass \(\mu _f=0.93\) instead of 1. This is because we do not find the kind of solution with \(\mu _f=1\) in the case of synchronized frequency. On the contrary, in the case of nonsynchronized, there exists the type of solution with \(\mu _f=1\). Here, we do not show the result in this article. Similar results have been shown in previous work [19]. In order to compare synchronized frequency with nonsynchronized frequency, we uniformly set \(\mu _f = 0.93\).

We will continue to work on the expansion of our research. Firstly, we have studied the RHABSs. Next, we will investigate the case of the double axion field, where one axion field exists in the ground state, and the other axion field exists in the first excited state. We will further study the difference between RMSBSs and RHABSs in the cases of low \(f_a\). Finally, Proca stars are very interesting. Rotating Proca stars have some similar properties of Kerr Black holes [30, 58]. Recently, a merger of Proca stars is seen as an explanation for the events of GW190521 [59]. We intend to construct the excited Proca stars and the excited Kerr BHs with Proca hair in future work.