Relativistic Nontopological Soliton Stars
in a U(1) Gauge Higgs Model

Nontopological solitons (NTSs) are localized solutions carrying a Noether charge in nonlinear field theories that have a continuous global symmetry. Rosen Rosen:1968mfz , in his pioneering work, showed that a self-interacting complex scalar field theory admits particlelike NTS solutions, and Coleman Coleman:1985ki proved the existence theorem of spherically symmetric NTS solutions with a conserved charge, he called them ‘Q-balls’ , in nonlinearly self-coupling complex scalar field theories with some conditions. Friedberg, Lee, and Sirlin Friedberg:1976me studied a coupled system of a complex scalar field and a real scalar field with a double-well potential, and showed the existence of the NTS solutions (see e.g. reviews Lee:1991ax ; Nugaev:2019vru and textbooks Shnir:2018yzp ). The NTS solutions in extended field theories including a gauge field were also studied in the works Lee:1988ag ; Shi:1991gh ; Gulamov:2015fya . The NTSs are interested as a possible candidate of dark matter Kusenko:1997si ; Kusenko:2001vu ; Fujii:2001xp ; Enqvist:2001jd ; Kusenko:2004yw , and as sources for baryogenesis Enqvist:1997si ; Kasuya:1999wu ; Kawasaki:2002hq .

Recently, NTS solutions were constructed in the theory that consists of a complex scalar field, a U(1) gauge field, and a complex Higgs scalar field with a Mexican hat potential which causes the spontaneous symmetery breaking Ishihara:2018rxg ; Ishihara:2019gim ; Ishihara:2021iag ¹¹1 NTSs in a generalized model are discussed in Refs.Forgacs:2020vcy ; Forgacs:2020sms . . In this model, there are interesting properties: the charges carried by two scalar fields are screening each other; and NTS solutions with infinitely large mass can exist Ishihara:2021iag . These would suggest that NTSs with astrophysical scale in this model can play important roles in cosmology. However, infinite mass should be prohibited if we take gravity produced by the NTS into account, namely, the mass should be limited by a maximum mass for self-gravitaing NTSs.

It is also investigated that localized objects made by self-gravitating complex scalar fields, so-called boson stars. In the model of a free massive complex scalar field with gravity, the gravitational mass of the localized solutions is quite small then the solutions are called mini boson stars Kaup:1968zz ; Ruffini:1969qy , while if the complex scalar field has nonlinear self-coupling, the mass of solution can be large Colpi:1986ye . Boson stars in various field models are studied in Refs.Jetzer:1989av ; Henriques:1989ar ; Henriques:1989ez (see also Jetzer:1991jr ; Schunck:2003kk ; Liebling:2012fv for review). Furthermore, self-gravity of the NTS is also studied in the Colman’s model Lynn:1988rb and the Friedberg-Lee-Sirlin’s model Lee:1986ts ; Friedberg:1986tq ; Lee:1986tr ; Kunz:2021mbm .

In this paper, we consider the coupled system of two complex scalar fields and a U(1) gauge field, which is studied in Refs.Ishihara:2018rxg ; Ishihara:2019gim ; Ishihara:2021iag , with Einstein gravity. The local U(1) symmetry of the system is spontaneously broken in a vacuum state where one of the scalar field has an expectation value. The system has two dimensionful parameters, the symmetry breaking scale, $\eta$, and the Plank scale, $M_{\text{P}}$, then the dimensionless parameter $\eta/M_{\text{P}}$ is an important quantity that characterizes the model.

Assuming spherically symmetry, stationary rotation of the phase of complex scalar fields, and static geometry, we derive a set of coupled ordinary differential equations. We obtain numerical solutions that describe self-gravitating objects, we call them ‘nontopological soliton stars (NTS stars)’ in this paper, and investigate properties of the solutions, especially, mass and radius that depend on $\eta/M_{\text{P}}$. It is interesting that NTS stars with mass of astrophysical scale are possible, e.g., the solar mass is possible for $\eta\sim$1GeV, and the NTS stars can be so compact that they have the innermost stable circular orbits for $\eta/M_{\text{P}}\ll 1$.

The paper is organized as follows. In Sec.II, we present the model that has a symmetry breaking scale, and derive basic equations on the assumptions of the geometrical symmetries of the fields. In Sec.III, we solve the basic equations numerically, and present NTS star solutions. In Sec.IV, we investigate internal structures of the NTS stars: energy density, pressure, and charge density. In Sec.V, we study mass and radius of the NTS stars, and see that the maximum mass appears for each breaking scale. Paying attention to the NTS stars with maximum mass in various breaking scales, we investigate scale dependence of the maximum mass and the compactness in Sec.VI. Section VII is devoted to conclusions.

We consider the action

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

(1)

where $R$ is the scalar curvature with respect to a metric, $g_{\mu\nu}$, $g:={\rm det}(g_{\mu\nu})$, and $G$ is the gravitational constant. The matter Lagrangian, $\mathcal{L}_{m}$, is given by

	$\displaystyle\mathcal{L}_{m}=$	$\displaystyle-g^{\mu\nu}(D_{\mu}\psi)^{\ast}(D_{\nu}\psi)-g^{\mu\nu}(D_{\mu}\phi)^{\ast}(D_{\nu}\phi)-\frac{\lambda}{4}(\left|\phi\right|^{2}-\eta^{2})^{2}$		(2)
		$\displaystyle-\mu\left|\phi\right|^{2}\left|\psi\right|^{2}-\frac{1}{4}g^{\mu\alpha}g^{\nu\beta}F_{\mu\nu}F_{\alpha\beta},$		(3)

where it consists of a complex matter scalar field $\psi$, the field strength $F_{\mu\nu}:=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ of a U(1) gauge field $A_{\mu}$, and a complex Higgs scalar field $\phi$. The gauge field couples to the scalar fields through the gauge covariant derivative, $D_{\mu}:=\partial_{\mu}-ieA_{\mu}$. The parameter $e$ is the charge of the fields $\psi$ and $\phi$, $\lambda$ the Higgs-self coupling, $\mu$ the Higgs-scalar coupling, and $\eta$ the Higgs vacuum expectation value determining the scale of the symmetry breaking.

The Lagrangian (3) has local U(1) $\times$ global U(1) symmetry under the gauge transformation given by

	$\displaystyle\psi(x)\to\tilde{\psi}(x)=e^{i(\chi(x)-\gamma)}\psi(x),$		(4)
	$\displaystyle\phi(x)\to\tilde{\phi}(x)=e^{i(\chi(x)+\gamma)}\phi(x),$		(5)
	$\displaystyle A_{\mu}(x)\to\tilde{A}_{\mu}(x)=A_{\mu}(x)+e^{-1}\partial_{\mu}\chi(x),$		(6)

where $\chi(x)$ is an arbitrary function that depends on spacetime coordinate, and $\gamma$ is an arbitrary constant. Concering to this invariance, conserved currents,

$\displaystyle j^{\mu}_{\psi}:=ie\left(\psi^{\ast}(D^{\mu}\psi)-(D^{\mu}\psi)^{\ast}\psi\right),\quad j^{\mu}_{\phi}:=ie\left(\phi^{\ast}(D^{\mu}\phi)-(D^{\mu}\phi)^{\ast}\phi\right)$

(7)

and conserved charges

$\displaystyle Q_{\psi}:=\int d^{3}x\sqrt{-g}\rho_{\psi},\quad Q_{\phi}:=\int d^{3}x\sqrt{-g}\rho_{\phi},$

(8)

are defined, where $\rho_{\psi}:=j^{t}_{\psi}$ and $\rho_{\phi}:=j^{t}_{\phi}$ are charge densities induced by the complex scalar field $\psi$ and $\phi$, respectively. The integrations in (8) are performed on a timeslice $t=\text{\it const.}$

Owing to the potential term of the complex Higgs scalar field $\phi$ in (3), $\phi$ takes a nonzero vacuum expectation value $\eta$ in a vacuum state. As a result, the symmetry is spontaneously broken, then, the scalar field $\psi$ and the U(1) gauge field $A_{\mu}$ acquire masses, $m_{\psi}:=\sqrt{\mu}\eta$ and $m_{A}:=\sqrt{2}e\eta$, respectively, through interactions with the complex Higgs field. Simultaneously, a real scalar field as a fluctuation of the amplitude of $\phi$ around $\eta$ also acquires the mass $m_{\phi}:=\sqrt{\lambda}\eta$.

From the action (1), we can derive field equations

	$\displaystyle\frac{1}{\sqrt{-g}}D_{\mu}\left(\sqrt{-g}g^{\mu\nu}D_{\nu}\psi\right)-\mu\psi\left|\phi\right|^{2}=0,$		(9)
	$\displaystyle\frac{1}{\sqrt{-g}}D_{\mu}\left(\sqrt{-g}g^{\mu\nu}D_{\nu}\phi\right)-\frac{\lambda}{2}\phi(\left|\phi\right|^{2}-\eta^{2})-\mu\left|\psi\right|^{2}\phi=0,$		(10)
	$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}F^{\mu\nu})=j^{\nu}_{\psi}+j^{\nu}_{\phi},$		(11)
	$\displaystyle G_{\mu\nu}=8\pi GT_{\mu\nu},$		(12)

where $G_{\mu\nu}:=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R$ is the Einstein tensor and $T_{\mu\nu}$ is the energy-momentum tensor given by

	$\displaystyle T_{\mu\nu}:=$	$\displaystyle-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{m})}{\delta g^{\mu\nu}}$
	$\displaystyle=$	$\displaystyle 2(D_{\mu}\psi)^{\ast}(D_{\nu}\psi)-g_{\mu\nu}(D_{\alpha}\psi)^{\ast}(D^{\alpha}\psi)+2(D_{\mu}\phi)^{\ast}(D_{\nu}\phi)-g_{\mu\nu}(D_{\alpha}\phi)^{\ast}(D^{\alpha}\phi)$		(13)
		$\displaystyle-g_{\mu\nu}\left(\frac{\lambda}{4}(|\phi|^{2}-\eta^{2})^{2}+\mu|\psi|^{2}|\psi|^{2}\right)+\left(F_{\mu\alpha}F_{\nu}^{~{}\alpha}-\frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right).$		(14)

Here, we assume stationary and spherically symmetric fields in the form:

$\displaystyle\psi=e^{-i\omega t}u(r),\quad\phi=e^{-i\omega^{\prime}t}f(r),\quad A=A_{t}(r)dt,$

(15)

where we use a spherical coordinate $(t,r,\theta,\varphi)$. The parameters $\omega$ and $\omega^{\prime}$ are constant angular frequencies of the complex scalar fields. Owing to the gauge transformation (4)-(6), we can fix the variables as

$\displaystyle\phi(r)\to f(r),\quad\psi(t,r)\to e^{i\Omega t}u(r),\quad A_{t}(r)\to\alpha(r):=A_{t}(r)+e^{-1}\omega^{\prime},$

(16)

where $\Omega:=\omega^{\prime}-\omega$ is the parameter which characterizes the solution, and takes a positive value without loss of generality.

We also take static and spherically symmetric spacetime assumptions in the form

	$\displaystyle ds^{2}=$	$\displaystyle g_{\mu\nu}dx^{\mu}dx^{\nu}$
	$\displaystyle=$	$\displaystyle-\sigma(r)^{2}\left(1-\frac{2m(r)}{r}\right)dt^{2}+\left(1-\frac{2m(r)}{r}\right)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\varphi^{2},$		(17)

where $\sigma(r)$ and $m(r)$ are functions of $r$. Note that $\sigma(r)$ is dimensionless, and $m(r)$ has dimension of length. The Einstein equations reduce to

$\displaystyle G^{t}_{t}=8\pi GT^{t}_{t}$

$\displaystyle,\quad G^{r}_{r}=8\pi GT^{r}_{r},\quad G^{\theta}_{\theta}=G^{\varphi}_{\varphi}=8\pi GT^{\theta}_{\theta}=8\pi GT^{\varphi}_{\varphi}.$

(18)

Substituting assumptions (16) and (17) into (9)$-$(11), we obtain equations for the fields $\psi,\phi,$ and $A$ to be solved in the form:

	$\displaystyle u^{\prime\prime}+\left(\frac{2}{r}\left(1+\frac{m-rm^{\prime}}{r-2m}\right)+\frac{\sigma^{\prime}}{\sigma}\right)u^{\prime}+\left(1-\frac{2m}{r}\right)^{-1}\left(\frac{(e\alpha-\Omega)^{2}u}{\sigma^{2}(1-2m/r)}-\mu f^{2}u\right)=0,$		(19)
	$\displaystyle f^{\prime\prime}+\left(\frac{2}{r}\left(1+\frac{m-rm^{\prime}}{r-2m}\right)+\frac{\sigma^{\prime}}{\sigma}\right)f^{\prime}$		(20)
	$\displaystyle\hskip 56.9055pt+\left(1-\frac{2m}{r}\right)^{-1}\left(\frac{e^{2}f\alpha^{2}}{\sigma^{2}(1-2m/r)}-\frac{\lambda}{2}f(f^{2}-1)-\mu fu^{2}\right)=0,$		(21)
	$\displaystyle\alpha^{\prime\prime}+\left(\frac{2}{r}-\frac{\sigma^{\prime}}{\sigma}\right)\alpha^{\prime}+\left(1-\frac{2m}{r}\right)^{-1}\left(-2e^{2}f^{2}\alpha-2e(e\alpha-\Omega)u^{2}\right)=0,$		(22)

where the prime denotes the derivative with respect to $r$. Here and hereafter, $r$, $u(r)$, $f(r)$, $\alpha(r)$, $m(r)$, and $\Omega$ are normalized by $\eta$.

As for the Einstein equations, we solve the time-time component of (18) and the combination

$\displaystyle G^{r}_{r}-G^{t}_{t}=8\pi G(T^{r}_{r}-T^{t}_{t}).$

(23)

The rests are guaranteed by the Bianchi identity. Explicit forms of the energy-momentum tensor and the charge densities are given in Appendix. Then, we have

	$\displaystyle\frac{2m^{\prime}}{r^{2}}-8\pi G\eta^{2}\biggl{(}\frac{e^{2}f^{2}\alpha^{2}}{\sigma^{2}(1-2m/r)}$	$\displaystyle+\frac{(e\alpha-\Omega)^{2}u^{2}}{\sigma^{2}(1-2m/r)}+\left(1-\frac{2m}{r}\right)(f^{\prime 2}+u^{\prime 2})$		(24)
		$\displaystyle+\frac{\lambda}{4}(f^{2}-1)^{2}+\mu f^{2}u^{2}+\frac{1}{2\sigma^{2}}\alpha^{\prime 2}\biggr{)}=0,$		(25)

and

	$\displaystyle\frac{(1-2m/r)\sigma^{\prime}}{r\sigma}-8\pi G\eta^{2}\bigg{(}\frac{e^{2}f^{2}\alpha^{2}}{\sigma^{2}(1-2m/r)}$	$\displaystyle+\frac{(e\alpha-\Omega)^{2}u^{2}}{\sigma^{2}(1-2m/r)}$		(26)
		$\displaystyle+\left(1-\frac{2m}{r}\right)(f^{\prime 2}+u^{\prime 2})\bigg{)}=0.$		(27)

The dimensionless parameter $G\eta^{2}=\left(\eta/M_{\text{P}}\right)^{2}$ represents the symmetry breaking scale with respect to the Planck mass, $M_{\text{P}}$. In the limiting case of $G\eta^{2}\to 0$, the gravitational field decouples with the matter fields, where the matter system (19)-(22) with $\sigma(r)=1$ and $m(r)=0$ are studied in Refs.Ishihara:2018rxg ; Ishihara:2019gim ; Ishihara:2021iag .

We require regularity of the spherically symmetric fields at the origin described by

$\displaystyle\sigma^{\prime}=0,\quad m=0,\quad u^{\prime}=0,\quad f^{\prime}=0,\quad\alpha^{\prime}=0.$

(28)

In addition, we assume that the solutions are localized in finite regions. Therefore, for the matter fields, we assume

$\displaystyle u=0,\quad f=1,\quad\alpha=0,$

(29)

at spatial infinity. The energy-momentum tensor $T_{\mu\nu}$ of the matter fields satisfying (28) and (29) is localized in a neighborhood of the origin with a quick radial decay, then we require that the gravitational fields satisfy

$\displaystyle\sigma=1,\quad m=m_{\infty}=\text{const}.$

(30)

at spatial infinity.

In this section, we obtain solutions by solving the set of equations (19), (21), (22), (25), and (27), numerically, and study properties of the numerical soutions. In this article, we fix the coupling constants as $e=1.0$, $\mu=1.4$, and $\lambda=1.0$, for an example. On the other hand, we consider various symmetry breaking scales. For the Planck breaking scale, $\eta/M_{\text{P}}=1$, we have $m_{\psi}\sim m_{\phi}\sim m_{A}\sim 10^{19}\text{GeV}$, and for the lower breaking scale, $\eta/M_{\text{P}}=10^{-3}$, $m_{\psi}\sim m_{\phi}\sim m_{A}\sim 10^{16}\text{GeV}$, respectively.

At a large distance, $u(r)$ should decrease quickly so that the energy is localized in a compact region. The boundary conditions at the spatial infinity, (29) and (30), require $f=1,\alpha=0$, and $\sigma=1,m=\text{const}$, then the equation (19) for $u(r)$ reduces to

$\displaystyle u^{\prime\prime}-(\mu-\Omega^{2})u=0.$

(31)

Then, $\Omega$ is bounded above by $\Omega_{\text{max}}:=\sqrt{\mu}$. There is also the lower bound $\Omega_{\text{min}}$, which depends on $\eta/M_{\text{P}}$, for existence of a solution, and, as is seen later, the solution has the maximum mass for $\Omega$ near $\Omega_{\text{min}}$. We can find numerical solutions for the parameter $\Omega$ in the range

$\displaystyle\Omega_{\text{min}}\leq\Omega<\Omega_{\text{max}}.$

(32)

Typical behaviors of the fields obtained by numerical calculations are shown as functions of $r$ in Fig.1 for the Planck breaking scale case, and in Fig.2 for the lower breaking scale case. They show that the matter fields and the gravitational field are localized in a finite region in each case. Thus, they represent nontopological solitons with self-gravity, we call them nontopological soliton star (NTS star) solutions.

In the cases (a) and (b) in the Planck breaking scale, and (d) in the lower breaking scale, the function $u$ has Gaussian forms, while $f\sim 1$ and $\alpha\sim 0$ almost everywhere. In these cases, the scalar field $\phi$ and the gauge field $A$ are not excited anywhere, and only the scalar field $\psi$, which has the mass $m=m_{\psi}$ by the Higgs mechanism, becomes a source of gravity. The behavior that the massive scalar field with gravity yields compact objects is just the same as ‘mini boson star’ solutions Kaup:1968zz ; Ruffini:1969qy . On the other hand, in the cases (e) and (f) in the lower breaking scale, $\psi,\phi$, and $A$ are exited inside the stars. In these cases, interaction between the scalar fields and the gauge field plays an important role for appearance of solutions Ishihara:2018rxg ; Ishihara:2019gim ; Ishihara:2021iag .

As for the gravity, in the cases (a), (b), (d), and (e), the lapse function, $\sigma$, is almost constant. It means that the gravity is weak so that the the Newtonian description is possible. In contrast, in the cases (c) and (f), $\sigma$ varies significantly from $r=0$ to infinity, i.e., it means the gravity requires relativistic description. Paying attention to the scale of horizontal axises, we see the size of the NTS stars depend on $\Omega$. The dependence is discussed later in detail.

We see in Figs.1, 2, 3, that the size of NTS solutions much depend on the parameter $\Omega$. We study total mass and radius of the solutions in this section.

As shown in the previous section, in each breaking scale, there exists the maximum NTS star that has maximum gravitational mass. The mass, the surface radius, and the compactness of the maximum NTS star much depend on the breaking scale. Here, we show how these properties of the maximum NTS star depend on the breaking scale.

In Fig.10, we plot the mass, $M_{*}$, and the surface radius, $R_{*}$, of the maximum NTS stars for the various values of the breaking scale $\eta/M_{\text{P}}$. We observe that the both $M_{*}$ and $R_{*}$ obey power laws of $\eta/M_{\text{P}}$ with two different power indices as

$\displaystyle\frac{M_{*}}{M_{\text{P}}}\propto\begin{cases}&\displaystyle\left(\frac{\eta}{M_{\text{P}}}\right)^{-2}\quad\text{for}~{}\eta<\eta^{M}_{\text{cr}},\cr&\displaystyle\left(\frac{\eta}{M_{\text{P}}}\right)^{-1}\quad\text{for}~{}\eta>\eta^{M}_{\text{cr}},\end{cases}$

(36)

and

$\displaystyle\frac{R_{*}}{R_{\text{P}}}\propto\begin{cases}&\displaystyle\left(\frac{\eta}{M_{\text{P}}}\right)^{-2}\quad\text{for}~{}\eta<\eta^{R}_{\text{cr}},\\ &\displaystyle\left(\frac{\eta}{M_{\text{P}}}\right)^{-1}\quad\text{for}~{}\eta>\eta^{R}_{\text{cr}},\end{cases}$

(37)

where the critical values $\eta^{M}_{\text{cr}}$ and $\eta^{R}_{\text{cr}}$ are order of $0.1M_{\text{P}}$, and the ratio is $\eta^{M}_{\text{cr}}/\eta^{R}_{\text{cr}}\sim 3.3$. The power index in $\eta\gg\eta_{\text{cr}}$ is the same as mini boson stars studied in Kaup:1968zz ; Ruffini:1969qy , and the one in $\eta\ll\eta_{\text{cr}}$ is the same as soliton stars studied in Friedberg:1986tq ; Lee:1986tr .

We consider simple model formulae for $M_{*}$ and $R_{*}$ shown in Fig.10 as

	$\displaystyle M_{*}$	$\displaystyle=\frac{M_{\text{cr}}}{2}\Big{(}(\eta/\eta^{M}_{\text{cr}})^{-2}+(\eta/\eta^{M}_{\text{cr}})^{-1}\Big{)},$		(38)
	$\displaystyle R_{*}$	$\displaystyle=\frac{R_{\text{cr}}}{2}\Big{(}(\eta/\eta^{R}_{\text{cr}})^{-2}+(\eta/\eta^{R}_{\text{cr}})^{-1}\Big{)},$		(39)

where $M_{\text{cr}}$ and $R_{\text{cr}}$ are constants. If we can extrapolate (38) and (39) for much lower breaking scale than the cases calculated in the present paper, typical scales of $M_{*}$ and $R_{*}$ for the maximum NTS stars are listed in Table.1. We see that the maximum NTS star would be an astrophysical scale for $\eta\lesssim$ 100GeV.

According to the model functions (38) and (39), we have

$\displaystyle C_{*}=\frac{2GM_{*}}{R_{*}}$

$\displaystyle=\frac{2GM_{\text{cr}}}{R_{\text{cr}}}\frac{\eta^{M}_{\text{cr}}}{\eta^{R}_{\text{cr}}}\frac{(\eta^{M}_{\text{cr}}+\eta)}{(\eta^{R}_{\text{cr}}+\eta)},$

(40)

then $C_{*}$ takes the different constant values for $\eta\ll\eta_{\text{cr}}$ and $\eta\gg\eta_{\text{cr}}$, respectively, and the ratio of them becomes

$\displaystyle\frac{C_{*}(\eta\ll\eta_{\text{cr}})}{C_{*}(\eta\gg\eta_{\text{cr}})}=\frac{\eta^{M}_{\text{cr}}}{\eta^{R}_{\text{cr}}}\sim 3.3.$

(41)

In Fig.11, we depict the compactness of the maximum NTS stars, $C_{*}$, by the use of the numerical values of $M_{*}$ and $R_{*}$ as a function of the breaking scale. From numerical results we see $C_{*}\sim 0.553$ for $\eta/M_{\text{P}}\lesssim 10^{-2}$ and $C_{*}\sim 0.167$ for $\eta/M_{\text{P}}\gtrsim\eta_{\text{cr}}$. Therefore, the maximum NTS stars in $\eta\ll\eta_{\text{cr}}$ are relativistic self-gravitating objects that have innermost stable circular orbits while no photon sphere.

We studied the coupled system of field theory that consists of a complex scalar field, a U(1) gauge field, a complex Higgs scalar field that causes a spontaneous symmetry breaking, and Einstein gravity. The system has a dimensionless parameter, $\eta/M_{\text{P}}$, which represents the ratio of the symmetry breaking scale to the Plank scale. We obtained numerical solutions that describe nontopological soliton stars, parametrized by the angular phase velocity of the complex scalar field, $\Omega$. The solutions have a variety of properties depending on the parameters $\eta/M_{\text{P}}$ and $\Omega$.

In the case of the large breaking scale, $\eta/M_{\text{P}}\gtrsim 0.1$, the solutions are almost determined by the gravitational field and a scalar field that acquires its mass by the Higgs mechanism. Then, the solutions are almost same as the mini boson stars obtained in the system of the gravitational field and a massive complex scalar field. On the other hand, in the case of small breaking scale, $\eta/M_{\text{P}}\ll 1$, the solutions are classified into three types: mini boson stars, matter-interacting NTS stars, and gravitating NTS stars. For the first type, gravity and a scalar field contribute the solutions as same as the large breaking scale case. In the second type, interactions between matter fields are important as in the case of nontopological solitons discussed in Refs.Ishihara:2018rxg ; Ishihara:2019gim ; Ishihara:2021iag . In the last one, the both matter interaction and self-gravity are important, and NTS star solutions in this type can have much larger mass than other types. In the cases of mini boson stars and matter-interacting NTS stars, the gravity is weak because the lapse function is almost constant everywhere, while in the case of gravitating NTS stars, gravity requires relativistic description where the lapse varies significantly.

We found that the maximum mass, which depend on the breaking scale, obeys a double power law: $M_{*}\propto\eta^{-1}$ for $\eta\gtrsim\eta_{\text{cr}}$ and $M_{*}\sim\eta^{-2}$ for $\eta\lesssim\eta_{\text{cr}}$, where $\eta_{\text{cr}}\sim M_{\text{P}}/3$ . If we can extrapolate this for much lower breaking scale, the maximum mass of the NTS star can be astrophysical scale, the solar mass for $\eta\sim$1GeV and the claster of galaxies scale for $\eta\sim$1eV.

We studied the compactness, the gravitational radius over the radius of the NTS stars. The compactness of NTS stars with maximum mass, $C_{*}$, takes the value $C_{*}\sim 0.167$ for $\eta\gg M_{\text{P}}$ and $C_{*}\sim 0.553$ for $\eta\ll M_{\text{P}}$, and $C_{*}$ change in its value quickly around $\eta\sim 0.1M_{\text{P}}$. It means the NTS stars with maximum mass in the lower breaking scale are relativistically compact object that have the innermost stable circular orbits. Therefore, the NTS stars in the case $\eta\ll M_{\text{P}}$ can be seeds of supermassive black holes.

It is an important issue to clarify the stability of the NTS stars. We would expect that the NTS stars with maximum mass evolve to black holes if they become unstable. Linear perturbation of the NTS stars would be our next work.

In this paper, we construct NTS star solutions whose internals are filled by kinetic energy of the scalar fields. These are self-gravitating solutions of dust balls Ishihara:2021iag . There are other types of NTSs, potential balls and shell balls, in the model without gravity Ishihara:2021iag . If we take gravity into account, a self-gravitating potential ball would be a ‘gravastar’ Mazur:2001fv ; Mazur:2004fk that join de Sitter and Scwarzshild spacetimes by a spherical shell, and a self-gravitating shell ball would join Minkowski and Scwarzshild spacetimes Kleihaus:2009kr . It is also interesting to construct these solutions.

We would like to thank K.-i. Nakao, H. Yoshino, and M. Minamitsuji for valuable discussion. This work was partly supported by Osaka City University Advanced Mathematical Institute: MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849.