Nodal compact $Q$-ball/$Q$-shell in the $\mathbb{C}P^{N}$ nonlinear sigma model

Compactons, i.e. field configurations that exist on finite size supports (”compact supports”), possess a distinct character among other solutions of standard field theory models. Namely, the field takes its vacuum values outside this support and the energy as well as the charge are always concentrated on the compact support. Initial study of such exotic configurations concerning the complex signum-Gordon model was presented in  Arodz and Lis (2008, 2009). Some preceding results involving real scalar fields were presented in Arodz (2002, 2004); Arodz et al. (2005). Such models possess standard kinetic terms and special V-shaped potential that gives rise to compact solutions. The “V-shaped character” of the potential means that the potential has at least one local minimum in the form of a spike. The first-order side derivatives at the minimum do not vanish at this point. Moreover, they are not equal to each other (frequently, e.g. for the signum-Gordon model, these derivatives differ from each other just by the overall sign).

A complex scalar field theory with some self-interactions has stationary soliton solutions called $Q$-balls Friedberg et al. (1976); Coleman (1985). $Q$-balls have attracted much attention in the studies of evolution of the early Universe Friedberg et al. (1987); Lee (1987). There is also a certain possibility that they survive the early phase of the Universe and constitute a major ingredient of dark matter  Kusenko (1997); Kusenko and Shaposhnikov (1998); Kusenko et al. (1998). The compact $Q$-balls in the $\mathbb{C}P^{N}$ model were presented in Klimas and Livramento (2017). Moreover, this model also supports the $Q$-shell compactons, even in the absence of an electromagnetic field (the case $N>3$). The gravitating compact $Q$-balls, i.e., the compact boson stars can harbor a Schwarzschild and a Reissner-Nordström type black hole Kleihaus et al. (2010); Kumar et al. (2014, 2015, 2016).

In the last few years, we made some efforts in the study of compact $Q$-balls in the nonlinear sigma model on a target space $\mathbb{C}P^{N}$, and also the boson stars corresponding to the model Klimas et al. (2019); Yanai (2019); Sawado and Yanai (2020, 2021); Klimas et al. (2021). In this paper, we explore excited compacton states, i.e. the nodal solutions of this model that differ from the standard solutions by the form of the radial function. The excited states of $Q$-balls or the boson stars are very important for theoretical study and also for astrophysical observations  Brihaye and Hartmann (2009); Bernal et al. (2010); Collodel et al. (2017); Alcubierre et al. (2018, 2019); Loginov and Gauzshtein (2020a, b); Almumin et al. (2021). The multi-state boson stars, which are superposed the ground and excited states of the boson star solutions, are considered for obtaining realistic rotation curves of spiral galaxies Bernal et al. (2010). We explore the multi-nodal, excited $Q$-ball solutions on the compact support. The point is that we impose the condition allowing to get second or more node points and then, formally, we are able to get the multi-nodal compacton with arbitrary node numbers. It requires that the radial function can change the sign at the node points. Our results have somewhat similarities with Loginov and Gauzshtein (2020a, b) which both are radially excited. A major difference is that our solutions are compactons that are nontrivial only on a certain compact support.

The discontinuous nature of the potential derivative at the minimum is responsible for the appearance of the signum function in the radial field equation. Consequently, the solution exhibits discontinuity of the second derivative at the node points. The first derivative of the radial function is continuous, however, not smooth at the node points. However, it turns out that the energy density of such field configurations is free from any difficulties. As a result, we obtain a large number of multi-nodal solutions for both the non-gauged and the gauged models.

The paper is organized as follows. In Section II we shall describe the model. The ansatz for the parametrization of the $\mathbb{C}P^{N}$ field is given in this section. Section III presents the solutions of the model. We give further analysis and discussion in Section IV. Conclusions and remarks are presented in the last Section.

The action of our model has the following form

$\displaystyle S=\int\sqrt{-g}d^{4}x\Bigl{[}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+4M^{2}g^{\mu\nu}\frac{D_{\mu}u^{\dagger}\cdot D_{\nu}u}{1+u^{\dagger}\cdot u}-4M^{2}g^{\mu\nu}\frac{\left(D_{\mu}u^{\dagger}\cdot u\right)\left(u^{\dagger}\cdot D_{\nu}u\right)}{\left(1+u^{\dagger}\cdot u\right)^{2}}-\mu^{2}V\Bigr{]}.$

(1)

$F_{\mu\nu}$ is the standard electromagnetic field tensor and the complex fields $u_{i}$ also are minimally coupled to the Abelian gauge fields $A_{\mu}$ through $D_{\mu}=\partial_{\mu}-ieA_{\mu}$. We employ the ‘V-shaped’ potential which is of the form

$$V=\sqrt{\frac{u^{\dagger}\cdot u}{1+u^{\dagger}\cdot u}}.$$

It is convenient to introduce the dimensionless coordinates

$\displaystyle x_{\mu}\to\frac{\mu}{M}x_{\mu}$

(2)

and also $A_{\mu}\to\mu/MA_{\mu}$. We also restrict $N$ to be odd, i.e., $N:=2n+1$. The solutions with vanishing magnetic field can be obtained within the ansatz

	$\displaystyle u_{m}(t,r,\theta,\varphi)=\sqrt{\frac{4\pi}{2n+1}}f(r)Y_{nm}(\theta,\varphi)e^{i\omega t}\,,$		(3)
	$\displaystyle A_{\mu}(t,r,\theta,\varphi)dx^{\mu}=A_{t}(r)dt$		(4)

which allows for reduction of the partial differential equations to the system of ordinary radial differential equations. $Y_{nm},-n\leq m\leq n$ are standard spherical harmonics and $f(r)$ is the matter profile function. Each $2n+1$ field $u=(u_{m})=(u_{-n},u_{-n+1},\cdots,u_{n-1},u_{n})$ is associated with one of $2n+1$ spherical harmonics for given $n$. The relation $\sum_{m=-n}^{n}Y_{nm}^{*}(\theta,\varphi)Y_{nm}(\theta,\varphi)=\dfrac{2n+1}{4\pi}$ is very useful for obtaining an explicit form of many inner products. For convenience, we introduce a new gauge field containing the gauge field $A_{t}$ and the constant $\omega$

$\displaystyle b(r):=\omega-eA_{t}(r)\,.$

(5)

Applying the ansatz (3) and (4) we get the dimensionless reduced Lagrangian of the $\mathbb{C}P^{N}$ model in the form

$\displaystyle\tilde{\mathcal{L}}_{\mathbb{C}P^{N}}=\frac{\kappa b^{\prime 2}}{2e^{2}}+\frac{4b^{2}f^{2}}{(1+f^{2})^{2}}-\frac{4f^{\prime 2}}{(1+f^{2})^{2}}-\frac{4n(n+1)f^{2}}{r^{2}(1+f^{2})}-V$

(6)

where for convenience we have introduced a dimensionless constant $\kappa:=\mu^{2}/M^{4}$. The potential simplifies to the following one

$$V=\frac{|f|}{\sqrt{1+f^{2}}}.$$

(7)

The potential (7) in the limit of small amplitude fields behaves as the potential of the signum-Gordon model i.e. $V\sim|f|$. Note that, in contradiction to our preceding publications, in this paper we do not restrict the sign of $f(r)$ to non-negative values – it can be any real number. The reduced matter field equation and Maxwell’s equations take the form of two coupled ordinary differential equations

	$\displaystyle f^{\prime\prime}+\frac{2}{r}f^{\prime}-\frac{n(n+1)f}{r^{2}}+\frac{1-f^{2}}{1+f^{2}}b^{2}f-\frac{2ff^{\prime 2}}{1+f^{2}}-\frac{1}{8}{\rm Sign}(f)\sqrt{1+f^{2}}=0\,,$		(8)
	$\displaystyle\kappa b^{\prime\prime}+\frac{2}{r}\kappa b^{\prime}-\frac{8e^{2}bf^{2}}{(1+f^{2})^{2}}=0\,.$		(9)

The presence of ${\rm Sign}(f)$ in the field equation is a direct consequence of the V-shaped character of the potential. In further part of the paper we solve these two coupled equations with fixed $\kappa$ (for simplicity we set $\kappa=1$).

The dimensionless (reduced) Hamiltonian of the model reds

$\displaystyle\mathcal{H}_{\mathbb{C}P^{N}}=\frac{4b^{2}f^{2}}{(1+f^{2})^{2}}+\frac{4f^{\prime 2}}{(1+f^{2})^{2}}+\frac{4n(n+1)f^{2}}{r^{2}(1+f^{2})}+\frac{\kappa b^{\prime 2}}{2e^{2}}+V\,$

(10)

and it has interpretation of radial profile function of the energy density. Integrating the energy density over whole space one gets the energy of the compacon

$\displaystyle E=4\pi\int r^{2}dr\biggl{[}\frac{4b^{2}f^{2}}{(1+f^{2})^{2}}+\frac{4f^{\prime 2}}{(1+f^{2})^{2}}+\frac{4n(n+1)f^{2}}{r^{2}(1+f^{2})}+\frac{\kappa b^{\prime 2}}{2e^{2}}+V\biggr{]}\,.$

(11)

Now we shall look at the question of Noether charges. The action with the covariant derivative (1) is invariant under the following local $U(1)^{N}$ symmetry

	$\displaystyle A_{\mu}(x)\to A_{\mu}(x)+e^{-1}\partial_{\mu}\Lambda(x)$
	$\displaystyle u_{i}\to\exp[iq_{i}\Lambda(x)]u_{i},~{}~{}~{}~{}i=1,\cdots,N\,.$		(12)

where $q_{i}$ are some real numbers. The following Noether current is associated with the invariance of the action (1) under transformations (12)

$\displaystyle J^{(i)}_{\mu}=-i\frac{4M^{2}}{\left(1+u^{\dagger}\cdot u\right)^{2}}\sum_{j=1}^{N}\Big{[}u_{i}^{*}\Delta_{ij}^{2}D_{\mu}u_{j}-D_{\mu}u_{j}^{*}\Delta_{ji}^{2}u_{i}\Big{]}\,.$

(13)

Making use of the ansatz (3),(4) we find the following form of the Noether currents

	$\displaystyle J_{t}^{(m)}(r,\theta)=\frac{(n-m)!}{(n+m)!}\frac{8bf^{2}}{(1+f^{2})^{2}}\bigl{(}P^{m}_{n}(\cos\theta)\bigr{)}^{2}\,,$		(14)
	$\displaystyle J_{\varphi}^{(m)}(r,\theta)=\frac{(n-m)!}{(n+m)!}\frac{8mf^{2}}{(1+f^{2})^{2}}\bigl{(}P^{m}_{n}(\cos\theta)\bigr{)}^{2}$		(15)

and $J_{r}^{(m)}=J_{\theta}^{(m)}=0$ for $m=-n,-n+1,\cdots,n-1,n$. Note that both non vanishing currents do not depend on variables $t$ and $\varphi$. Hence, the conservation of currents is explicit after writing the continuity equation in the form

$\displaystyle\frac{1}{\sqrt{-g}}\partial_{\mu}\Big{(}\sqrt{-g}g^{\mu\nu}J_{\nu}^{(m)}\Big{)}=\partial_{t}J_{t}^{(m)}+\frac{1}{r^{2}\sin^{2}\theta}\partial_{\varphi}J_{\varphi}^{(m)}=0.$

(16)

The corresponding conserved Noether charge reads

	$\displaystyle Q^{(m)}$	$\displaystyle:=\frac{1}{2}\int_{\mathbb{R}^{3}}d^{3}x\sqrt{-g}J^{(m)}_{t}(x)$
		$\displaystyle=\frac{16\pi}{2n+1}\int r^{2}dr\frac{bf^{2}}{(1+f^{2})^{2}}\,.$		(17)

Owing to our ansatz, the charge does not depend on index $m$, which means that the symmetry of the solutions is reduced to the $U(1)$ symmetry. However we shall keep the index $m$ for completeness.

Compactons in our model are extended objects with the spherical shape surface (the border of the compacton), such that the matter profile function satisfies the boundary conditions

$\displaystyle f(R)=0,~{}~{}f^{\prime}(R)=0,~{}~{}~{}~{}r=R\textrm{~{}:~{}the~{}compacton~{}radius}$

(18)

Looking at typical numerical solutions $f(r)$ one can see that in many cases it oscillates. Usually, the first zero of the function $f$ is chosen as the radius of the compacton border. In such a case the first zero corresponds with the local minimum of $f(r)$. Mathematically, it is not obvious that the choice of the first zero as the compacton radius is a unique possibility.

Indeed, for an oscillating function the choice of some other minima as the compacton border leads to certain solutions with the profile function having some extra zeros. The number of such zeros can label different types of compact solutions. The purpose of this paper is to find the plausible answer for two questions. First, whether the solution exists for the choice of the second zero (minimum of maximum) of $f(r)$? Second, if a solution exists, what is his form and properties? We shall answer these questions in the following part of the paper.

In this section, we shall give a comment on discontinuities associated with the change of the sign of the profile $f$ in Eq.(8). In particular, we are interested in the fact whether it has some consequences on the energy density. Continuity of the energy density requires that both $f$ and $f^{\prime}$ are continuous. We do not expect discontinuity in gauge function $b(r)$ because the Maxwell equations do not contain any signum function. In this section, we study how the behavior of the radial function $f(r)$ around the nodal points.

First, we numerically check the continuity condition. From the equation (8), we define a function

$\displaystyle F(r):=f^{\prime\prime}(r)+\frac{2}{r}f^{\prime}(r)-\frac{n(n+1)f(r)}{r^{2}}+\frac{(1-f(r)^{2})b^{2}f(r)}{1+f(r)^{2}}-\frac{2f(r)f^{\prime}(r)^{2}}{1+f(r)^{2}}.$

(25)

which is equal to l.h.s. of the matter field equation minus the potential derivative. Then we substitute a solution into (25) and evaluate $F(r)$. In Fig.12 we plot $F(r)$ in vicinity of the nodal point $f\sim 0$. For the correct solution, $F(r)$ should behave like the signum function. This is exactly what we can see in Fig.12(b). At both open segments separated by the nodal point, $F(r)$ is continuous, hence we impose $F(r)=0$ at the nodal point. With this appropriate definition, one gets the signum function. Note, that from a physical point of view $F(r)=0$ at a certain segment of space is consistent with the vacuum solution $f(r)=0$. This simple analysis proves that $f(r)$ is the solution of (8). Next, we examine the series expansion of solutions at both sides of the nodal point $r=R_{\mathrm{nd}}$

$\displaystyle f(r)=\sum_{k=0}^{\infty}F_{k}(R_{\mathrm{nd}}-r)^{k},~{}~{}~{}b(r)=\sum_{k=0}^{\infty}B_{k}(R_{\mathrm{nd}}-r)^{k}.$

(26)

Since the value of $f$ is zero at the nodal points, $F_{0}=0$, then the expansions can be written as

	$\displaystyle f_{+}(r)$	$\displaystyle=F_{1}^{+}(r-R_{\mathrm{nd}})+\left(\frac{F_{1}^{+}}{R_{\mathrm{nd}}}+\frac{1}{16}\right)(r-R_{\mathrm{nd}})^{2}+O((r-R_{\mathrm{nd}})^{3})$		(27)
	$\displaystyle f_{-}(r)$	$\displaystyle=F_{1}^{-}(r-R_{\mathrm{nd}})+\left(\frac{F_{1}^{-}}{R_{\mathrm{nd}}}-\frac{1}{16}\right)(r-R_{\mathrm{nd}})^{2}+O((r-R_{\mathrm{nd}})^{3}).$		(28)

$f_{+}$ stands for the expansion where $f(r)$ is positive, and $f_{-}$ stands for the expansion where $f(r)$ is negative. Thus, even when the first-order coefficients match, $F_{1}^{+}=F_{1}^{-}$, the left and right second-order coefficients (and further) are different from each other.

The point is, therefore, whether the first-order derivatives coincide or not. In the case of their equality, the energy density, which contains the field and its first derivative, becomes continuous (but not necessarily differentiable).

This is what our numerical results suggest. From Eq.(8), one can verify indirectly that the profile $f$ is regular (i.e., continuous and differentiable) at the point of $f\sim 0$. Equation (8) can be cast in the form

$\displaystyle\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}f^{\prime}\right)=\frac{n(n+1)f}{r^{2}}-\frac{(1-f^{2})b^{2}f}{(1+f^{2})}+\frac{2ff^{\prime 2}}{(1+f^{2})}+\frac{1}{8}{\mathrm{Sign}}(f)\sqrt{1+f^{2}}.$

(29)

Integrating over the segment $[r_{-},r_{+}]$ that contains the nodal point $r=R_{\mathrm{nd}}:f(r)=0,r_{-}<r<r_{+}$, one gets

$\displaystyle\Bigl{[}r^{2}f^{\prime}\Bigr{]}_{r_{-}}^{r_{+}}=\int_{r_{-}}^{r_{+}}\Omega(f,f^{\prime})r^{2}dr$

(30)

Most of terms in $\Omega$ contain $f$ which means that the integral containing such terms vanishes in the limit $r_{\pm}\rightarrow R_{\mathrm{nd}}$. It is enough to examine the term Sign$(f)$. In vicinity $r\rightarrow R_{\mathrm{nd}}$, the integrand signum function is an odd functional, hence the value of the integral is expected to vanish. We conclude that the first derivative $f^{\prime}$ on the zero crossing point is continuous.

We have presented in this paper several nodal compact $Q$-ball ($Q$-shell) solutions in the $\mathbb{C}P^{N}$ nonlinear sigma models. There are an infinite number of radii that satisfy the compacton condition such that the field configuration is zero at the compacton radius. They differ by the number of nodes. For a given solution we choose the number of nodes. The presence of the signum function in the field equation is a consequence of a V-shaped potential in the Lagrangian. The nodal solutions are such that the profile function changes its sign. Each point where it takes place is a node of the solution. At the last point (radius) the field satisfies the compacton condition. This is the compacton border where the field matches the vacuum solution $f=0$. We have obtained new solutions with a given number of nodes for both non-gauged and gauged models.

When smoothly varying parameters in the gauged case we observe that the nodal $Q$-ball smoothly connects the $Q$-shell with the same number of nodes. For the gauged solutions we have observed that with increasing the charge (or the radius) the nodal $Q$-shell transforms into the field configuration with some local minima which substitute the nodes. Such minima are localized under the $r$ axis. In other words, the profile function has no sign change for such configurations. We have denoted it by k local minimum: $f^{[k-\ell m]}(r)$.

We have looked also at the energy and charge for nodal compactons. Our results show that for fixed Noether charge $Q$ the energy of the solution $E$ increases together with the number of nodes. Looking at the energy and charge densities of $Q$-balls or $Q$-shells we found that they do not change qualitatively in their form even when the nodal solution transforms into the solutions with some local minima. This clearly means that the $n$-th $Q$-ball, and $n$-th $Q$-shell with k-local minima belong to the same class.

The first derivative of the profile function $f^{\prime}$ is non-differentiable at the nodal points. It results in the appearance of discontinuity of the second derivative of $f$. We have checked that under continuity of $f^{\prime}$ at the nodal points the energy and the charge are continuous function, even for nodal solutions. These conditions were investigated both numerically and analytically.

Our new solution has possible applications to boson stars with non-trivial excitations just by implementing gravity into the equation. This solution can be seen as the weak gravity limit of true gravitating boson star solutions. Extending a class of solutions by the inclusion of excitations we naturally offer a variety of solutions that can be useful for the description of several astronomical phenomena. Alternatively, our results can also be applied to a phenomenon of evaporation of $Q$-balls and to the production of fermions. These models will be discussed in our subsequent papers in order.

Acknowledgment

The authors would like to thank Yves Brihaye who gave the initial birth of this peculiar setup of the solutions. We also appreciate Yuki Amari, Luiz Agostinho Ferreira, Atsushi Nakamula and Kouichi Toda for valuable discussions. Discussions during the YITP workshop YITP-W-20-03 on “Strings and Fields 2020” and YITP workshop YITP-W-21-04 on “Strings and Fields 2021” have been useful to complete this work. N.S. was supported in part by JSPS KAKENHI Grant Number JP B20K03278(1).