Q-balls without a potential

We study non-topological Q-ball solutions of the (3+1)-dimensional Friedberg-Lee-Sirlin two-component model. The limiting case of vanishing potential term yields an example of hairy Q-balls, which possess a long range massless real field. We discuss the properties of these stationary field configurations and determine their domain of existence. Considering Friedberg-Lee-Sirlin model we present numerical evidence for the existence of spinning axially symmetric Q-balls with different parity. Solution of this type exist also in the limiting case of vanishing scalar potential. We find that the hairy Q-balls are classically stable for all range of values of angular frequency.

type were constructed in Abelian gauge models with local U(1) symmetry [9,10], in non-Abelian gauge theories [1,11,12] and other models. An interesting realization of the Q-balls exists in condensed matter systems where they appear in the Bose-Einstein condensate [13], or in the superfluid 3 He-B [14].
Spherically symmetric Q-balls exist only within a certain angular frequency range, which is determined by the explicit structure of the potential. Notably, in the Friedberg-Lee-Sirlin model, which describes dynamics of a real self-interacting scalar field, coupled to a complex scalar field, the lower critical frequency is zero. Typically, there are two branches of Q-ball solutions, which are represented by two curves of the dependencies of the energy of the configuration on its charge [1]. Solutions are stable along the lower branch, when their mass is smaller than the mass of free charged quanta of scalar excitations. In the simplest case the Q-balls are spherically symmetric, however there are generalized spinning axially symmetric solutions with non-zero angular momentum [16,17]. The energy density of these spinning Q-balls is of toroidal shape.
There are some important differences between the Q-ball solutions of the Coleman model [2] with a single complex field and sextic potential, and the corresponding solutions of the renormalizable Friedberg-Lee-Sirlin model [1]. An interesting feature of Q-balls in the Friedberg-Lee-Sirlin model is that in 3+1 dimensions these localized configurations with finite energy may also exist in the limiting case of vanishing scalar potential [15]. It was pointed out that in such a limit the Q-balls are stabilized by the gradient terms in the energy functional. Further, the corresponding real scalar component becomes massless, it possess Coulomb-like asymptotic tail.
However, the paper [15] contains only qualitative discussion of the corresponding solutions, they were not constructed explicitly, further the authors do not study the frequency dependence of these solitons and their stability.
The main purpose of this work is to construct explicit examples of stationary solutions of the Friedberg-Lee-Sirlin model in the limit of vanishing potential that have not been studied so far, fully investigate properties of these Q-balls and determine their domains of existence.
We also consider spinning configurations with non-zero angular momentum with both even and odd parity and address the issue of classical stability of these solutions.

II. SPHERICALLY SYMMETRIC SOLUTIONS
The 3+1 dimensional Friedberg-Lee-Sirlin model describes a real self-interacting scalar field ξ, coupled to a complex scalar field φ: where κ is the coupling constant. The potential of the real scalar field is thus, ξ → 1 in the vacuum and the complex field φ becomes massive due to the coupling with its real partner. Thus, the parameters µ and m corresponds to the mass of the real and complex components, respectively.
Similar to the Coleman model with a non-renormalizable sextic potential [2], the model Noether current, associated with this symmetry, is and the conserved charge is First, we consider the usual spherically symmetric parametrization of the fields where X(r) and Y (r) are real functions of radial variable and ω is the frequency of stationary rotation.
Substitution of this ansatz into the stationary energy functional gives and the charge of the spherically symmetric Q-ball is  The field equations of the model become Like other Q-ball dynamical equations [1,16,[19][20][21], this system effectively describes a a unit mass pseudo-particle moving in two dimensional plane parameterized by the "coordinates" X(r), Y (r) and "time" r, in direction Y in the effective potential Non-topological soliton solution may exist when the trivial configuration X = Y = 0 corresponds to a local maximum of the effective potential. This restriction corresponds to the upper bound on the angular frequency ω + = m, the Q-ball continuously approaches perturbative solutions as ω approaches this critical value [1]. Thus, the upper bound of the angular frequency of the Q-ball corresponds to the mass of the complex component m, the localized soliton configuration with finite energy may exist as its real component X(r) becomes massless [15]. Hereafter we fix m = 1, without loss of generality. Note that, unlike the Q-balls in the non-renormalizable model with single complex field and sextic potential [2], there is no lower bound on the frequency, the solutions of the model (1) exists for all non-zero values of the angular frequency ω. As ω decreases, the characteristic size of the configuration is increasing.
The vacuum boundary conditions on the spacial infinity are X → 1, Y → 0 as r → ∞ and the condition of regularity at the origin is Imposing the boundary conditions we can find numerical solutions of the system of coupled ordinary differential equations (8). In our calculations we used the usual shooting algorithm, based on Dormand-Prince 8th order method with adaptive stepsize. The relative errors of calculations are lower than 10 −10 .
In Fig. 1 we displayed the corresponding solutions at m = 1 and µ 2 = 0.25. The parameter µ yields the mass of the excitations of the real component X, it approaches the vacuum asymptotic value as X ∼ 1 − e − √ µ 2 −ω 2 r . Setting µ = 0 changes the asymptotic behavior, in such a case the real massless field X(r) decays as [15] This is a long-range Coulomb asymptotic with a scalar charge C, see Fig. 1, bottom plots.
The charge and the energy of the Q-balls are given by (6) exhibits the E(Q) curves of the configurations at µ 2 = 0.25 and in the massless case µ = 0.
In Fig. 3 we also indicate the energy of Q free scalar quanta E = mQ, this is a straight line separating the stability region, the configuration is classically stable below this line. Indeed, for µ = 0, there are two branches of E(Q) curves with a sharp cusp at ω = ω cr (see Fig. 3, left plot). The lower in energy branch corresponds to the values of the frequency ω < ω cr whereas the upper branch corresponds to ω > ω cr . As the real component remains massive, the configurations on the upper branch are unstable [1]. We observe that decrease of the mass parameter µ shifts the critical value ω cr towards the upper limit ω + .
The situation changes dramatically in the massless limit µ = 0, the stable branch extends all the way up to the critical value ω + = m, here both components of the Q-ball approach the corresponding vacuum values and both the energy and the charge of the configuration tend to zero, see Fig. 2 Note that the scalar charge C, which corresponds to the Coulomb asymptotic tail of the hairy Q-ball (10), is not a constant. The configuration is not rigid, its characteristic size varies with the angular frequency ω. Indeed, numerical calculations show that the value of the charge C monotonically decreases, as ω increases, see Fig. 4. However, the relation C = Q/2, which follows from some qualitative arguments [15], holds only for large Q-balls, i.e. for small values of the angular frequency ω.

III. SPINNING SOLUTIONS
A generalization of the fundamental spherically symmetric Q-ball can be constructed when we consider spinning axially symmetric configuration [16,17]: which generalizes the ansatz (5). Here n ∈ Z is a rotational quantum number, the angular momentum of the stationary spinning Q-ball is classically quantized as [16] The real functions X(r, θ), Y (r, θ) depend on the polar angle θ and radial variable r. The energy of the configuration then read The corresponding field equations are Note that these equations are symmetric with respect to reflections in the θ = π/2 plane.
To find numerical solutions of these coupled partial differential equations we used the software package CADSOL based on the Newton-Raphson algorithm [22]. The numerical calculations are mainly performed on an equidistant grid in spherical coordinates r and θ.
Typical grids we used have sizes 70 × 60. As before, we map the infinite interval of the variable r onto the compact radial coordinate x = r/r 0 1+r/r 0 ∈ [0 : 1]. Here r 0 is a real scaling constant, which typically is taken as r 0 = 4 − 6. For spinning Q-balls the component Y (r, θ) must vanish at the origin, both in the massive and in the massless cases. The restriction of regularity also yields ∂ r X(r θ) = 0 as r → 0.
The spinning Q-balls correspond to stationary points of the action functional, they exist only for a restricted frequency range. Previously they have been constructed only for the model with a single complex field and sextic potential [16][17][18]. A peculiar feature of these Q-balls is that for a non-zero rotational quantum number n there are two different solutions We observe that, similar to the spinning Q-balls in the single component model [16][17][18] for each value of integer winding number n, there are two types of solutions possessing different parity, so called parity-even and parity-odd Q-balls. Indeed, the equations (14) in the limiting case of ω 2 ∼ m 2 (small Q-balls) can be linearized. Then the second of these equations is reduced to the standard harmonic equation and the solution are associated with the usual spherical harmonics Here P n l (cos θ) are the associated Legendre polynomials of degree l and order n. Thus, the spherically symmetric fundamental Friedberg-Lee-Sirlin Q-ball corresponds to the spherical harmonic Y 0 0 , and there are two spinning configuration in the sector n = 1, the parity-even solution Y 1 1 and parity-odd solution Y 1 2 , respectively. Further, this observation suggests that the equations (14) also support solutions which correspond to higher energy angular excitations of the fundamental Q-ball [20]. In Figs. 6,7 we displayed the fields of the spinning parity-even and parity-odd Q-balls, both in the massive and massless cases. For parity-even solutions the spinning component Y (r, θ) is maximal in the symmetry plane, as µ = 0 the massless component X(r, θ) is minimal at the origin, it decays asymptotically, according to (10) (see Fig. 6, bottom left plot). If the mass of the X(r, θ) component is non-zero, it decays exponentially, as it is seen in Fig 6, upper left plot. We also observe that in the latter case the minimum of this component is shifted to the x − y plane. The energy density distributions of the rotating even-parity Q-balls is torus-shaped in both cases. However, as the component X(r, θ) is massive, for the same value of the frequency ω, both the energy and the amplitudes of the fields are much larger.  Considering the frequency dependence of rotating Q-balls, we found that it is qualitatively the same as in the case of the fundamental n = 0 solutions, see Figs. 2,3 above.
In Fig. 8, left plot, we show the charge and the energy of the parity-even Q-balls as functions of the angular frequency ω. We observe that the solutions exist for all allowed range of values of ω restricted from above by the mass of the complex field m = 1. As ω approaches this upper limit, the amplitude of both components is decreasing. As the real component X(r, θ) remains massive, it decays exponentially. In such a case we observe the same usual pattern as in the model with single component complex field and a polynomial self-interaction potential [16][17][18]. Both the energy and the charge of the configuration of these Q-balls are minimal at some critical value ω cr and the curve E(Q) shows the same cusp structure as displayed in Fig. 3, left plot. Thus, there are two branches of E(Q) curves, the existence of two different solutions with the same value of charge Q indicates that the more energetic configurations on the upper branch are unstable.
The situation is different for the spinning Q-balls with massless long-range component X(r, θ). As for the spherically symmetric n = 0 configurations, both the charge and the energy monotonically depend on ω, see Fig. 3, right plot. Thus, there is no critical frequency and only one branch of classically stable solutions exist for all range of values of ω. The scalar charge C of the spinning Q-balls with massless hair also depends on the angular frequency.
Numerical calculations show that dependency of its value per rotational quantum number n is identical with the dependency of the scalar charge of the spherically symmetric hairy Q-ball displayed in Fig. 4 above.

IV. CONCLUSIONS
The objective of this work was to investigate properties of new Q-ball solutions with long-range massless scalar hair, existence of which was conjectured in the pioneering study [15]. The work here should be taken further by considering gauged spinning Q-balls with massless real scalar component, another interesting direction is to investigate properties of self-gravitating spinning Q-balls without potential. We hope we can address these issues in our future work.