eepl draft
Quasi-Compact Q-balls
D. Bazeia , M.A. Marques and R. Menezes , Departamento de F´ısica, Universidade Federal da Para´ıba, 58051-970 Jo˜ao Pessoa, PB, Brazil Departamento de Ciˆencias Exatas, Universidade Federal da Para´ıba, 58297-000 Rio Tinto, PB, Brazil
PACS – Extended classical solutions; cosmic strings, domain walls, texture
Abstract –This work deals with charged nontopological solutions that appear in relativistic mod-els described by a single complex scalar field in two-dimensional spacetime. We study a modelwhich supports novel analytical configurations of the Q-ball type, that engender double exponen-tial tails, in this sense being different from both the standard and compact Q-balls. The analyticalsolutions are also stable, both classically and quantum-mechanically, and the stability follows asin the case of the compact configurations.
1. Introduction.
The presence of Q-balls in relativis-tic models described by a single complex scalar field in(1 ,
1) spacetime dimensions was initiated in 1985 by Cole-man in Ref. [1] and was further investigated by several au-thors; see, e.g., Refs. [2–17] and references therein. Morerecently, Q-balls have also been studied with other moti-vations in Refs. [18–24]. In [18], for instance, the authorconsidered the possibility of supermassive compact objectsresiding in galactic centers be giant nontopological solitonsor Q-balls made of scalar fields. In [19] the investigationfocused on the linear perturbation of classically stable Q-balls solutions, and in [20] the authors investigated thelimiting case of a vanishing potential term, which yieldedan example of a hairy Q-ball. Moreover, in [21,22] the au-thors described the existence of models which support thepresence of composed soliton solutions, with vortex andQ-ball constituents. In [23] the investigation dealt withthe problem of classical stability of Q-balls, including thenonlinear evolution of classically unstable Q-balls, as wellas the behaviour of Q-balls in external fields. Also, in[24] the authors studied a model described by two scalarfields to describe nontopological soliton solutions consist-ing of two Q-ball components possessing opposite electriccharges.In 1998, it was shown by Kusenko and Shaposhnikov inRef. [6] that Q-balls can be produced in the early universein supersymmetric extensions of the standard model, andcan contribute to dark matter. The interesting perspectiveis related to the presence of Q-balls as dark matter can-didate under the Affleck-Dine mechanism, which is pos-tulated for explaining baryogenesis during the primordialexpansion, after the cosmic inflation. In this line, the re- views on dark matter [25] and on the origin of the matter-antimatter asymmetry [26] are of current interest. Theissue in general requires the presence of scalar and gaugefields and three space dimensions, and has been furtherstudied with several motivations. In particular, more re-cently a new type of charged Q-ball dark matter scenariohas been suggested; see, e.g., [27] and references therein.The charged solutions can carry both baryon and leptoncharges, and can be stable due to the baryonic component,so they can be a viable candidate for dark matter.In the simpler context with a single space dimension,global Q-balls were also studied by us in the recent works[28–30]. There, we focused mainly on the presence of ana-lytical solutions with distinct behavior. In [29], in particu-lar, we investigated Q-balls with the novel compact profile.The analytical results that we have found in [28–30] moti-vated us to further examine the subject to find other an-alytical configurations that are stable, but different fromthe standard and the compact Q-balls. In this sense, in thecurrent work we continue the search for analytical config-urations, but now we concentrate on the presence of solu-tions that are different from the standard and the compactconfigurations. As one knows, even though we are workingin (1 ,
1) spacetime dimensions, the mandatory presence ofnonlinearities and constraints complicates the investiga-tion of stable analytical solutions of the Q-ball type.In this work, we deal with these issues examining in thenext Sec. the basic properties of Q-balls, accounting forthe main characteristics of the standard and compact Q-balls. We then move on and in Sec. we introduce andsolve another model, and there we also investigate the Q-ball stability. The results show the presence of exact solu-p-1 a r X i v : . [ h e p - t h ] S e p . Bazeia et al. tions, which behave very much like the compact structuresthat we found in [29], although they are not compact atall. In Sec. we end the work, adding comments andsuggestions of new studies in the subject.
2. Generalities.
In order to investigate the Q-balls inthe simplest scenario, we consider the action of a singlecomplex scalar field, ϕ , in (1 ,
1) Minkowski spacetime inthe usual form, with the Lagrangian L = 12 ∂ µ ¯ ϕ∂ µ ϕ − V ( | ϕ | ) , (1)with the overline on ϕ denoting the complex conjugationand V ( | ϕ | ) representing the potential. The field modulusis defined as | ϕ | = √ ¯ ϕϕ , and the model is invariant underrotations in the complex space of the field, ϕ → ϕ e iα , with α real constant. One may vary the action correspondingto the above Lagrangian to get the equation of motion¨ ϕ − ϕ (cid:48)(cid:48) + ϕ | ϕ | V | ϕ | = 0 , (2)with the dot and the prime denoting the derivative withrespect to time, t , and to the spatial coordinate, x , respec-tively, and V | ϕ | = dV /d | ϕ | . To search for Q-balls we takethe usual ansatz ϕ ( x, t ) = σ ( x ) e iωt . (3)In the above equation, ω represents an angular frequency.For simplicity, we only consider non-negative ω . Noticethat | ϕ ( x, t ) | = σ ( x ), which does not depend on the time.The presence of a phase invariance in the Lagrangiandensity is associated to a conserved Noether charge thathas the form Q = 12 i (cid:90) ∞−∞ dx ( ¯ ϕ ˙ ϕ − ϕ ˙¯ ϕ ) = ω (cid:90) ∞−∞ dx σ ( x ) . (4)The equation of motion (2) when combined with the time-dependent ansatz (3) becomes σ (cid:48)(cid:48) = V σ − ω σ. (5)As usual, the boundary conditions are σ (cid:48) (0) = 0 and σ ( ±∞ ) → . (6)One can show that the equation of motion (5) can berewritten as a simpler expression σ (cid:48)(cid:48) = U σ , (7)where U = U ( σ ) acts as an effective potential for the field σ . It is given by U ( σ ) = V ( σ ) − ω σ . (8)Notice that the effective potential depends explicitly on ω . Therefore, one has to be careful when choosing thepotential V ( | ϕ | ) because it has to lead to a U ( σ ) that allows for the presence of solutions compatible with theboundary conditions in Eq. (6). In this case, one can showthat, in order to attain this compatibility, ω must be inthe interval ω − < ω < ω + , (9)with ω + = V σσ (0) and ω − = (cid:112) V ( σ ) /σ , where σ de-notes the minimum of the ratio V ( σ ) /σ . Here, ω + and ω − are the upper and lower bound for the frequency, re-spectively.Invariance over spacetime translations leads to theenergy-momentum tensor T µν = 12 ∂ µ ¯ ϕ∂ ν ϕ + 12 ∂ µ ϕ∂ ν ¯ ϕ − η µν L . (10)The component T in the above equation is the energydensity, which is denoted by (cid:15) . It can be written as asum of the kinetic, gradient and potential energy densities, (cid:15) = (cid:15) k + (cid:15) g + (cid:15) p , which are respectively given by (cid:15) k = 12 | ˙ ϕ | , (cid:15) g = 12 | ϕ (cid:48) | and (cid:15) p = V ( | ϕ | ) . (11a)With the ansatz in Eq. (3), the energy density becomes (cid:15) = 12 ω σ + 12 σ (cid:48) + V ( σ ) . (12)By integrating it, we get the total energy of the Q-ball.One can also show that, by using the expression for theconserved charge in Eq. (4), the kinetic energy can bewritten as E k = 12 ωQ. (13)Regarding the other components of the energy-momentumtensor in Eq. (10) with the Q-ball ansatz we get T = T = 0 and the stress T = 12 ω σ + 12 σ (cid:48) − V ( σ ) . (14)As one knows, the energy-momentum tensor is conserved,obeying the equation ∂ µ T µν = 0. Since T = 0, we seethat T cannot depend on x . We define the charge densityas the function that is being integrated in Eq. (4), ρ Q = ωσ , and this allows us to write the energy associated toEq. (12) as E = ω (cid:90) ∞−∞ dx σ + 12 (cid:90) ∞−∞ dx σ (cid:48) + (cid:90) ∞−∞ dx V ( σ )= Q (cid:82) ∞−∞ dx σ + 12 (cid:90) ∞−∞ dx σ (cid:48) + (cid:90) ∞−∞ dx V ( σ ) . (15)The above expression is important because it allows us toperform variations in the energy keeping the charge fixed.At this point, we investigate how the Q-ball behaves undera rescale in the spatial coordinate, following a directionsimilar to the one in Refs. [31–34]. We then take x → z = λx , which makes σ → σ ( λ ) ( x ) = σ ( z ), and calculate thep-2uasi-Compact Q-ballsenergy of the rescaled solution using the above equationto get E ( λ ) = Q (cid:82) ∞−∞ dx σ ( λ )2 + 12 (cid:90) ∞−∞ dx σ ( λ ) (cid:48) + (cid:90) ∞−∞ dx V ( σ ( λ ) )= λQ (cid:82) ∞−∞ dz σ ( z ) + λ (cid:90) ∞−∞ dz σ z ( z )+ 1 λ (cid:90) ∞−∞ dz V ( σ ( z )) , (16)where σ z ( z ) = dσ ( z ) /dz . The stability against contrac-tions and dilations in the solutions requires that λ = 1minimizes E ( λ ) . This requirement is satisfied by the stress-less condition, i.e., T = 0. Therefore, we can write12 σ (cid:48) = U ( σ ) , (17)with U ( σ ) being the effective potential described byEq. (8). It is straightforward to show this first-order equa-tion is compatible with the equation of motion (7).The stressless condition also has consequences on theenergy density (12); it allows us to relate its contributionsby (cid:15) p = (cid:15) k + (cid:15) g . The same is valid for the correspondingenergies. In this case, the total energy is E = 2( E k + E g ) . We note that the stressless condition only ensures stabil-ity under contractions and dilations. However, the stabil-ity of Q-balls may be also investigated in other directions;we consider the two following possibilities [16]:(i) The quantum mechanical stability, which is the sta-bility against decay into free particles. From Eq. (9), ω must be in a specific range of values in orderto get Q-balls solutions. The Q-ball is quantum-mechanically stable if the ratio between the energyand the charge, for any ω allowed in the aforemen-tioned range, satisfies E/Q < ω + .(ii) The classical stability, which is the one associatedto small perturbations of the field. The Q-ball isclassically stable if dQ/dω <
0, i.e., the charge Q ismonotonically decreasing with ω .There is a third type of stability, which is against fission.However, as shown in Ref. [16], classically stable Q-ballsare also stable against fission.The form of the solution depends on the potential V ( | ϕ | )that one considers in the equation of motion (5). Below,we review the standard Q-ball, driven by the | ϕ | potentialinvestigated in Ref. [28], and the compact Q-ball describedin Ref. [29], whose solution and energy density exists onlyin a compact space. Then, we introduce a new model,which is not compact but attains some properties of thecompact structure. For simplicity, we work with dimen-sionless units. Our first example is with themodel, driven by the | ϕ | potential V ( | ϕ | ) = 12 | ϕ | − | ϕ | + 14 a | ϕ | , (18) with a being a real and positive parameter, a ≥
0. Thismodel was studied in Refs. [2, 3, 28]. As one knows, dueto the presence of time in the ansatz (3), this potentialis influenced by the angular frequency in the equation ofmotion. This gives rise to an effective potential that canbe calculated from Eq. (8); it is given by U ( σ ) = 12 (1 − ω ) σ − σ + 14 a σ . (19)One may use the first order equation (17) to show thatthe solution has the form σ s ( x ) = (cid:114) − ω a (cid:20) tanh (cid:18) (cid:112) − ω x + b (cid:19) − tanh (cid:18) (cid:112) − ω x − b (cid:19)(cid:21) , (20)where b = arctanh (cid:16) (cid:112) (1 − ω ) a/ (cid:17) /
2. The above ex-pression obeys the boundary conditions (6) for severalvalues of ω according to Eq. (9). For a ≥ /
9, we have ω − = (cid:112) − / (9 a ) and ω + = 1. This solution is bellshaped and goes asymptotically to zero exponentially. Infact, one can show that σ s ( x ) ∝ e − √ ω − ω | x | (21)for x → ±∞ . One can use the exact solution above tocalculate the conserved charge in Eq. (4), which yields to Q s = 4 ω √ − ω a (2 b coth(2 b ) − . (22)This charge goes to infinity as ω → ω − for a > /
9. Thecase a = 2 / Q s → ω → ω − . Inthe other side of the angular frequency range, for ω → ω + ,the charge vanishes. We have shown in the recent inves-tigation [28] that the solution (20) exhibits quantum me-chanical stability for a > . a > . A Q-ball with non-standard be-havior, which engenders a compact profile, was presentedin Ref. [29], motivated by the work [35]. It arises with thepotential V ( | ϕ | ) = 12 | ϕ | − | ϕ | − /r + 14 a | ϕ | − /r , (23)where r is a real parameter such that r > a ≥
0. Inthis potential, one notices the presence of fractional pow-ers of | ϕ | . As far as we know, the appearance of fractionalpowers in the scalar field was explored before in [36], withthe aim to generate stable kinklike configuration in arbi-trary dimensions, circumventing the Derrick-Hobart scal-ing theorem [37, 38]. In particular, in the case of a singlespatial dimension, one has shown in [36] that the presenceof fractional power modifies the kink profile, inducing aninternal structure to the field configuration. Interestingly,p-3. Bazeia et al. this kind of internal structure was found experimentally in[39], in the study of domain walls in the micrometer-sizedFe Ni magnetic material in the presence of constrainedgeometries.We go on and use the above potential (23) taking theusual procedure to write the effective potential (8) in theform U ( σ ) = 12 (1 − ω ) σ − σ − /r + 14 a σ − /r . (24)This effective potential admits a minimum at σ = 0 , ∀ ω ,and a (non-minimum) zero that depends on ω , at σ (cid:54) = 0.So, there is a non-topological solution that connects thesepoints. Here, the condition (9) is still valid, and gives ω − = (cid:112) − / (9 a ) and ω + = ∞ in this case. Again, wetake a ≥ / ω is real. Then, the solutionsof Eq. (5) are valid for any ω > ω − .One may use the first order equation (17) to calculatethe solution, which admits different forms depending onthe value of ω . First, we consider the case ω − < ω <
1, inwhich we get σ c ( x ) = (cid:16) − ω a (cid:17) − r/ (cid:104) coth (cid:16) √ − ω r x + b (cid:17) − coth (cid:16) √ − ω r x − b (cid:17)(cid:105) − r , | x | ≤ x c , , | x | > x c , (25)where x c = 2 rb/ √ − ω delimits the compact space − x c ≤ x ≤ x c , in which the solution lives, and b is aparameter that depends on a and ω as b = arccoth (cid:32) √ (cid:112) − − ω ) a (cid:112) a (1 − ω ) (cid:33) . (26)The solution has a different expression for ω = 1, givenbelow σ c ( x ) = (cid:40)(cid:16) a − x r (cid:17) r , | x | ≤ x c , , | x | > x c , (27)where x c now changes to x c = 3 r (cid:112) a/ ω >
1, one can show that σ ( x ) is written by σ c ( x ) = (cid:16) ω − a (cid:17) − r/ (cid:104) cot (cid:16) √ ω − r x + d (cid:17) − cot (cid:16) √ ω − r x − d (cid:17)(cid:105) − r , | x | ≤ x c , , | x | > x c , (28)where x c = 2 rd/ √ ω − d is given by d = arccot (cid:32) √ (cid:112) ω − a (cid:112) a ( ω − (cid:33) . (29)Regardless the value of ω , the solution vanish outside acompact interval of the real line. This is a different be-havior from the solution of the standard case described inEq. (20), which only vanish asymptotically, going to zero with an exponential tail. Therefore, compact Q-balls donot have a tail, so they seem to behave as hard chargedballs.An interesting fact is that, even though the solutionhas different expressions, depending on the value of theangular frequency, as in Eqs. (25), (27) and (28), one cancalculate the conserved charge in Eq. (4) to show that itobeys the following single expression Q c = 2 r ω √ π r +1 a r +1 / (cid:16) (cid:16) (cid:112) − − ω ) a/ (cid:17)(cid:17) r +1 Γ(2 r + 1)Γ (cid:0) r + (cid:1) F , r + 1; 2 r + 32 ; 9(1 − ω ) a/ (cid:16) (cid:112) − − ω ) a/ (cid:17) , (30)where Γ( z ) is the Gamma function and F ( a, b ; c ; z ) isHypergeometric function. Notice that the above expres-sion depends on r , a and ω . Similarly to what happenswith the charge in Eq. (22) for the standard case, we have Q c → ∞ in the limit ω → ω − for a > /
9, and Q c → ω → ω − for a = 2 /
9. The limit ω → ω + leads to nullcharge; this does not depend on the a chosen.Regarding the stability of the compact Q-balls, it wasshown in Ref. [29] that they are quantum mechanically sta-ble, because ω + is infinity. This means that the inequality E c /Q c < ω + = ∞ is satisfied for a ≥ /
9. Therefore, com-pact Q-balls never decay into free particles, for r >
2. Onthe other hand, the classical stability is not ensured andmust be investigated carefully. In the case of r = 3, it wasshown in [29] that the compact Q-ball is classically sta-ble for a > . ω .
3. Novel structures.
In this section, we get inspira-tion from the lumplike structures recently introduced inRef. [40] and present a new model which supports Q-ballswith new features that make them special. We considerthe potential V ( | ϕ | ) = 12 | ϕ | (cid:0) a + ln | ϕ | (cid:1) . (31)Here, a is a real non-negative parameter that is squaredfor convenience. Potentials with a logarithmic term ap-peared before in Refs. [41, 42]; in particular in [42] the au- Fig. 1: The effective potential in Eq. (32) depicted for a = 1,and ω = 1 . , . .
5. The thickness of the lines increaseswith ω . p-4uasi-Compact Q-ballsthor investigated the formation of Q-balls in the presenceof the so-called thermal logarithmic potential. The resultindicates that the growth of the field fluctuations is fastenough to create Q-balls, despite the shrinking instabilitywhich is due to the decreasing temperature, etc; see alsoRef. [43] for other results related to [42]. We further noticethe work [44], in which the thermal logarithmic potentialis also investigated, but now with variational estimationinstead of lattice simulation. The study led to analyti-cal results on the Q-balls properties such as radio, energyand stability, without the need to solve the nonlinear fieldequation numerically.As usual in the search for Q-balls, the above potential(31) is influenced by the contribution of the angular fre-quency in the equation of motion. This gives rise to aneffective potential – see Eq. (8) – that has the form U ( σ ) = 12 σ (cid:0) a − ω + ln σ (cid:1) . (32)This effective potential admits a minimum at σ = 0 anda neighbor zero at σ (cid:54) = 0. As usual, the allowed rangefor ω is obtained through Eq. (9). Here, we get ω − = a and ω + = ∞ . It is worth commenting, though, that ω + isinfinite as in the compact Q-ball for a different reason: thishappens here due to the presence of the logarithmic termin the above potential, not because of the unusual powerthat appeared in the potential (24). In Fig. 1, we displaythe behavior of the above effective potential for severalvalues of the angular frequencies and a = 1. We can seethat, as ω increases, the zero of the potential approachesto the minimum at the origin. Meanwhile, the minimumoutside the interval where the solution exists goes deeperand farther. Fig. 2: (Left panel) The solution (34) depicted for a = 1, and ω = 1 . , . .
5, with the thickness of the lines increas-ing with ω . (Right panel) The standard solution (20) (dottedline) and the quasi-compact solution (34) (solid line) are showntogether, for comparison. According to the boundary conditions in Eq. (6) and thefirst order equations (17), the zero of the potential whichis in the neighborhood of the minimum of the potentialgives the amplitude of the solution, which we denote by A .In this case, one can show that A = exp (cid:0) − (cid:112) ( ω − a ) (cid:1) .Thus, the amplitude of the solution starts at the maximumvalue A max = 1, in the limit ω → a , and gets smaller as ω increases, as expected from the behavior of the effectivepotential (31). To calculate the solution, we use the firstorder equation (17), which becomes σ (cid:48) = σ (cid:0) a − ω + ln σ (cid:1) . (33)It admits an analytical solution satisfying the boundaryconditions (6), with the form σ q ( x ) = e −√ ω − a cosh( x ) . (34)This solution presents a double exponential tail, i.e., itdecays asymptotically as σ q ∝ e −√ ω − a e | x | , (35)for x → ±∞ . This solution is depicted in the left panelin Fig. 2 and, even though it vanishes faster than thestandard solution in Eq. (20) as x increases, see Eq. (21),it is not compact as the solutions in Eqs. (25), (27) and(28), which do not exhibit a tail.The above solution represents a quasi-compact Q-ball,because of the decay as the exponential of an exponential,engendering a tail suppression which is much stronger thanthe standard Q-balls. In order to further emphasize thischaracteristic, we also display in the right panel in Fig.2 the standard and the quasi-compact solutions together.They are shown with the same amplitude and the same ω , but with distinct values of a , since a in the standardmodel controls the fourth-order power of the scalar field,and in the quasi-compact model it has a very differentmeaning, contributing to the second-order power of thescalar field. In the right panel in Fig. 2 we used ω = 0 . a = 0 . a = 0 .
7, for the standard and thequasi-compact solutions, respectively.We can use the analytical solution given above to cal-culate the energy density in Eq. (12), which leads to (cid:15) q = (cid:0) ω cosh ( x ) − a sinh ( x ) (cid:1) e − √ ω − a cosh( x ) . (36)We are also able to calculate the charge density analyti-cally; it is ρ Q q ( x ) = ωσ q ( x ), which is the function that isbeing integrated in Eq. (4). It has the form ρ Q q = ω e − √ ω − a cosh( x ) . (37)In Fig. 3, we display the energy density in Eq. (36) andthe charge density above. Notice that they are both bell-shaped and present a double exponential tail. So, theybehave similarly to the solution (34).One may integrate the quantities in Eqs. (36) and (37)all over the space to calculate the energy E and the charge Q of the Q-ball, respectively. Unfortunately, we have beenunable to find the analytical expressions for them, so wemust proceed using numerical methods. Notice that theydepend on the parameters a and ω , with the latter re-stricted by the range informed in Eq. (9). To illustratetheir behavior, we plot them for a = 1, as a function of ω ,in Fig. 4.p-5. Bazeia et al. Fig. 3: The energy density in Eq. (36) (left) and the chargedensity in Eq. (37) (right) depicted for a = 1, and ω = 1 . , . .
5. The thickness of the lines increases with ω .Fig. 4: The energy (left) and charge (right) associated toEqs. (36) and Eq. (37) displayed as functions of ω for a = 1. We now focus on the stability of the new Q-balls presentin the model described by the potential (31). As we dis-cussed in Sec. , both the energy and charge of the solutionsplay an important role in the stability of the Q-ball. Aswe commented before, the ratio
E/Q is associated to thequantum mechanical stability of the Q-ball, which is re-lated to the stability against decay into free particles. Onthe other hand, the behavior of the charge Q with respectto ω dictates the classical stability of the Q-ball, againstsmall fluctuations in the field.In order to investigate the quantum mechanical stabil-ity, we must remember that ω + = ∞ . So, the condition E q /Q q < ω + is always satisfied and the Q-ball is quan-tum mechanically stable regardless the value of a . Thisfeature is the same of the compact Q-ball. Here, however,the Q-ball is not compact: it presents a tail that decaysextremely fast, but not enough to become compact. So,we have found a Q-ball that presents a double exponentialtail whose qualitative behavior of the quantum mechan-ical stability is the same of the compact Q-ball. Thesenew structures never decay into free particles, and this isanother motivation to call them quasi-compact Q-balls.The other type of stability that we have to study is theclassical one. To do that, we plot the charge for a = 0 interms of ω in Fig. 5. In this case, the charge presents ahole near ω = 0; it is not monotonically decreasing with ω ,which makes the Q-ball unstable under small fluctuations.We continue the investigation increasing a until we arrive Fig. 5: The charge associated to Eq. (37) as function of ω for a = 0 (left) and a = 0 . at a = 0 . ω inflec ≈ .
21 and Q inflec ≈ . a , we have dQ/dω ≤
0. Therefore, for a > . a >
0, we conclude this new structure is stable for a > .
4. Conclusion.
We have introduced and investigated anew model of Q-balls, whose solution presents a doubleexponential tail. In this way, it is different from both thestandard and the compact Q-balls. The associated poten-tial presents infinite second derivative at the minimum,but this does not make the solution compact, as it hap-pens in the model investigated in Ref. [29]. Nevertheless,it leads to ω + = ∞ , which is the upper limit for the an-gular frequency. This property is interesting, because itensures the quantum mechanical stability as in the com-pact Q-ball. So, we were able to introduce Q-balls thatare not compact but present the same quantum mechani-cal stability of the compact solutions found in [29], so wecalled them quasi-compact Q-balls. We also investigatedtheir classical stability and concluded that they are clas-sically stable for a > . , U (1)symmetry in a way capable of adding internal structuresto the nontopological solutions. These and others relatedissues are currently under consideration, and we hope toreport on them in the near future. ∗ ∗ ∗ The authors are grateful to Conselho Nacional de De-senvolvimento Cient´ıfico e Tecnol´ogico (CNPq) for partialfinancial support.