On non-topological solitons in Abelian gauge theories coupled to U(1)$\times$U(1) symmetric scalar fields

In a series of recent works Ishihara and Ogawa have investigated non-topological solitons (Q-balls) in a spontaneously broken Abelian gauge theory coupled to two complex scalar fields. The present paper extends their investigations to the most general U(1)$\times$U(1) symmetric quartic potential. Also a new class of charged Q-ball solutions with vanishing self-interaction terms is investigated and some of their remarkable properties is exhibited.

The term Q-ball denotes finite energy, non-radiating solutions in theories containing scalar fields with time-periodic phases and associated conserved charges. Their discovery goes back to the pioneering works of G. Rosen [1]. Prototype Q-balls appear in pure scalar field theories containing a complex scalar with a quartic potential coupled to a real one [2]. They have been shown to be stable, their stability being related to their conserved charge. For an excellent review see Ref. [3]. The term Q-ball comes from Ref. [4] where such scalar lumps with harmonic time dependent phases have been shown to occur in a scalar field theory containing a single complex scalar with a self interaction potential of at least degree 6. It turned out that similar lumps appear in gauge theories [5,6,7], for some detailed numerical investigations see Refs. [8,9,10] (with one complex, and one real scalar). For recent reviews, see Refs. [11,12].
Q-balls have gained large attention due to the possibility of their formation in the early universe [13], them being candidates as dark matter [14], their possible role in baryogenesis [15], and also their appearance in a large class of supersymmetric extensions of the Standard Model of Particle Physics [16,17].
The authors of Ref. [18] have found that for a spherically symmetric distribution of external charges coupled to an Abelian Higgs model, the Higgs field provides for perfect charge screening, cancelling out all long-range fields of the external charges. In Refs. [19,20], this observation has been tested on Q-balls in an Abelian Higgs model coupled to another charged, massive scalar field whose mass is provided by the Higgs mechanism. We note, that coupling the scalars this way is analogous to the much studied Higgs portal models [21,22] where a scalar dark sector is coupled exclusively to the Higgs fields of the Standard Model of Particle Physics.
In the present paper we extend the results of Refs. [19,20] to the case of the most general U(1)× U(1) symmetric scalar sector with quartic self interaction potentials. We show that the remarkably precise numerically observed cancellation of the charge contribution between the two charged scalar fields pointed out in Refs. [19,20] follows from Gauss' Theorem. It can therefore serve as an excellent test for the correctness of the numerical computations. In carrying out a detailed investigation of a larger phase-space which appears to be a natural setting for the models considered, an interesting subfamily of charged Q-balls is found where the quartic self-interaction terms are put to zero. This new family of charged Q-balls is a natural extension of previously considered ungauged Q-balls with vanishing potential in Ref. [23] and investigated in more detail in Ref. [24].

The model considered
Following Refs. [18,19,20], we shall consider an Abelian Higgs model containing two charged, complex scalar fields with an action given by where indices are raised and lowered by the Minkowski metric The interaction potential is given as which is the most general quartic, gauge invariant potential for the two complex scalar fields with a U(1)× U(1) symmetry. In the model defined by the action (1), there are two separately conserved U(1) currents: The conserved charges are given as It is convenient to adimensionalise the fields and the coordinates as φ → ηφ, ψ → ηψ, A µ → ηA µ and x µ → x µ /(eη) and in the action (1), which then becomes with where β 1,2 = λ 1,2 /e 2 , β 12 = λ ′ /e 2 , and µ = m 2 /(e 2 η 2 ).

Spherically symmetric Q-ball solutions
We shall seek spherically symmetric Q-ball solutions of the model defined by the action (1). The simplest spherically symmetric, purely electric Ansatz leading to non-trivial finite energy solutions can be written in the Lorenz gauge as where r, ϑ, ϕ are spherical coordinates, and all other vector potential components vanish. It is convenient to gauge transform (7) to a simpler form where one of the scalars is time-independent, leading to the Ansatz also employed in Ref. [19,20]: which shall be also used in this paper. The reduced action of the configuration within Ansatz (8) can be written as with the effective potential is given by The spherically symmetric field equations resulting from the variation of the reduced action (9) are given as: The boundary conditions required for the solution of Eqs. (11)(12)(13) are derived, on one hand, from regularity of the fields at the origin, and from the requirement of vanishing energy density at infinity, i.e., approaching the vacuum manifold of the theory, The energy of a field configuration defined by the Ansatz (8)is expressed as where The electric charges of φ resp. ψ are written as where In order to show the "perfect" charge screening in this setting, integrate Eq. (13) from zero to ∞, which yields taking into account that α decays exponentially: As it has been analysed in the literature Q-balls typically exist within a frequency interval where ω min and ω max are determined by the parameters of the theory. In the present case these parameters are β 1 , β 2 , β 12 , µ and the charges q i . We also note that one can set for example β 12 = 1 without loosing generality. The maximal frequency is determined by the demanding f 2 → 0 for large radii, to ensure the finiteness of the energy. Since The value of ω min is determined as follows [19,20]: The effective potential, U eff (10) has critical points corresponding to the "false" vacuum f 1 = 1, f 2 = α = 0 and to a "true" vacuum Near the minimal frequency, ω ≈ ω min , the Q-ball tends to a homogenous ball filled by the false vacuum. The minimal frequency is determined by the condition the values of the effective potential are the same for the false resp. true vacua. For general values of the parameters of the theory this leads to a somewhat complicated algebraic equation of order 5 for ω min (β 1 , β 2 , β 12 , µ, q 1 , q 2 ). In the special case when β 2 = 0 and µ = β 12 one obtains an easily solvable cubic equation and finds ω min = √ 2 q √ 2β 1 µ − q 2 β 1 (q = q 2 /q 1 ). Let us note another interesting special case considered in our paper: for β 1 = 0 it is found that ω min = 0.
Yet another constraint for the existence of a Q-ball solution has been exhibited for the case β 2 = 0, µ = β 12 in Refs. [19,20], where it has been shown (in the case q 1 = q 2 = 1) that is necessary for the existence of a Q-ball solution, arising as a condition for the existence of the "true" vacuum with (f 0 1,2 ) 2 ≥ 0. By numerical evaluation of ω min (β 1 , β 2 , β 12 , µ, q 1 , q 2 ), we have found that for fixed values of β 1 , β 12 , and µ, for values satisfying the criterion (20), ω min is an increasing function of β 2 . We have also found ω min to be an increasing function of µ. This indicates that a non-zero β 2 decreases the domain of existence of the solutions (by increasing ω min ).
We also note that from a standard Derrick-type scaling argument (r → λr, Refs. [25,26]) one obtains from the effective action S eff = I 1 − I 3 the virial relation where An interesting consequence of the virial relation (22) The reason that the above expression is asymmetric between Q ψ and Q φ result has been obtained is due to the gauge choice implicit in Ansatz (8). A more symmetric form is readily found using Ansatz (7):

Numerical solutions
Q-ball solutions of the model defined by the action (1) conforming to the Ansatz (8) have been obtained by the numerical solution of the radial equations (11-13), using the Colnew package [27,28] implementing collocation on Gaussian points. We have used an interval 0 ≤ r ≤ r max , which was large enough for the radial functions to reach their limiting values within numerical precision. In Fig. 1a, an example of a Q-ball in the full non-linear model is shown. The visual appearance of solutions with and without the quartic self-interaction term of the field ψ are almost identical. In Fig. 1b, the energy and charge distributions of the solution of the non-linear model is displayed. The cancellation of the charge densities due of the two fields is again local. We have chosen the parameters β 1 = 0.5, β 12 = µ = 1.4 as in Ref. [19,20] 1 , and considered the numerically available frequency range 1.174 ≤ ω ≤ 1.183. For each value of the frequency ω, we calculated the Q-ball solutions for a range of the parameter β 2 , starting from β 2 = 0, increasing β 2 to upper values of the order 0.1. In these ranges, we have found that the energy E, and the magnitude of the charges |Q φ | and |Q ψ | were monotonously increasing functions of the parameter β 2 . The frequency dependence of the energy and charges is depicted in Fig. 2. In Fig. 3(a)-(c) we present the change of solutions when approaching the lower limiting frequency (see also Refs. [19,20]). Upon approaching the minimal frequency, the difference between the effective potential U eff of the false vacuum at the origin and the real one becomes smaller, and, therefore, the due to a Derrick-type argument, the gradient terms must become small as well, and the Q-ball expands.  The behaviour for large β 2 is similar to the one for the frequency approaching the minimal one, see Fig. 4. The core of the Q-ball expands, and the energy and the charge diverge. The energy E and the charge Q φ are depicted as a function of β 2 in Fig. 5. Stability of the theory requires β 2 ≥ 0, and for β 2 = 0, the solutions of Refs. [19,20] are recovered.
Varying the parameter µ is quite similar to the above ones, there is a lower limit due to the exponent of the radial decay of the profile function f 2 , µ min = ω 2 , and upon aproaching it, the tail of the solution in f 2 expands, and not the core; see Fig. 6a. There is also an upper limit, µ = µ max , where the core expands, depicted in Fig. 6b. The dependence of the integrated quantities E, Q φ and Q ψ are depicted in Fig. 7 In agreement with our analytical result (18) we have found, that the total charges Q φ and Q ψ cancel each other for all solutions considered within numerical precision. In addition, we have considered the precision of the cancellation of the charges as a function of the radius, for various parameters. In Fig. 8 we have depicted the charge remaining after the cancellation for different frequencies. It can be seen, that the precision of the (local) cancellation decreases with lower frequencies, and the remaining charge is sharply peaked, and the peak moves to larger radii when the frequency ω approaches the lower limit, ω min . Upon approaching the upper limit ω max = √ µ, the remaining charge becomes smaller, but remains there for larger radii, in a less peaked shape. The dependence on other parameters can be explained by considering what happens when changing the given parameter. When the parameter increases ω min , the same happens (for fixed frequency) as when ω approaches ω min ; this is the case for increased mu and β 2 . On the contrary, upon lowering µ, the solutions behave as if ω was increased.

Stability of the solutions
A detailed analysis of the stability of Q-balls is still a challenging subject, with some open questions (see the Review [12]). However, an approximate indication of stability can be obtained by comparing the energy E of the solution to the energy E free of free ψ particles with the same charge [3]; if the energy of free ψ particles is larger, it is energetically not favourable for the soliton to fall apart into free particles, where In addition, if the ratio E/E free is not only below 1, but a decreasing function of N, it is also not favourable for the Q-ball to split into smaller Q-balls.
The ratio E/E free is depicted in Fig. 9a as a function of ω. The behaviour for nonzero quartic coupling β 2 is the same as found in Refs. [19,20] for β 2 = 0, i.e., below a critical frequency ω = ω cr , E/E free < 1, indicating solutions stable against falling apart into free particles (with arbitrarily large energy and charge, when approacting ω min . At β 1 = 0.5, β 12 = µ = 1.4 we have found that for β 2 = 0.25, ω cr = 1.1810 and for β 2 = 0, ω cr = 1.1795.   Note, that on the branch of the figure correspondig to ω < ω cr , the function is monotonously decreasing, indicating that the corresponding Q-balls are also stable against falling apart into smaller ones.

The effect of varying the charges
We have also considered the effect of varying the charges q 1 resp. q 2 of the fields φ and ψ. In this paper we restrict ourselves to the case β 2 = 0 and β 12 = µ.
The limiting cases are quite remarkable here. Considering q = q 2 /q 1 > 1, the positivity condition (20) is modified as β 1 < µq 2 1 /2. Increasing the relative charge q, while keeping β 1 and µ fixed, at q 1 = 2β 1 /µ the positivity condition becomes violated, ω min = ω max = √ µ is reached, and the solutions cease to exist. For this reason the solutions of Ref. [8] at q 1 = 0 thus cannot be reached from the solutions considered here by increasing the charge ratio q continuously; they seem to belong to a different family of solutions, with the electromagnetic field massless, and α approaching its limit at r → ∞ as ∝ Q ψ /r. We have found numerical evidence (see Fig. 10) showing that this family of solutions can also be deformed to q 1 = 0, at least for small values. Their detailed investigation will be part of a further study.
The limit q 2 → 0 is, on the other hand, quite simple. In this case, as q 2 decreases, the sum of the local charges Q φ + Q ψ becomes smaller, and at q 2 = 0 the purely scalar solutions of Ref. [2] are recovered, (obviously with α = 0), see Fig. 11.

The limit of vanishing quartic couplings
Another interesting limiting case is β 1,2 → 0. For this to be a regular limit, the boundary conditions f 1 → 1, f 2 , α → 0 as r → ∞ are kept. We have constructed such solutions, parameterised by a frequency 0 < ω < √ µ. For an example of such a solution, see Fig. 12. In the small frequency limit, the charges and the energy of the solutions increases, and E/E free , the total energy over the mass of free particles of type ψ, decreases. Also, interestingly, as ω approaches ω max , unlike in the other cases, charge and energy do not diverge, instead, the solutions approach the vacuum. Also, instability due to E/E free > 1 and the cusp on the E/E free -N curve has not been observed here (see Fig. 13).

Conclusions
In the present paper, we have extended the investigations of Ref. [19,20] to the case of the most general U(1)×U(1) potential, when both scalar fields of the model are self-interacting. We have found, that in this model, the perfect (local) charge screening of the solutions discussed in Refs. [18,19,20] persist, the charge densities due to the massive self-interacting scalar field ψ 2 and the Higgs-field φ, also responsible for the mass of ψ cancel each other. The main effect of the self-interaction of the second scalar field upon the solutions is a larger energy, charge, and, in most cases, a narrower domain of existence. We have also exhibited a new family of solutions where both quartic scalar self-interactions terms are absent, and found that some of their properties differ qualitatively from the generic case.

A Supplementary material: numerical data
In Tables 1-4 we have collected the data of a number of Q-ball solutions. The parameters f 1,2 (0) and α(0) are given for reproducibility. The energy and the charges are defined as the integrals of the respective densities from 0 to 200. In addition, we present some more figures depiciting the chage of profile functions with various parameters approaching their extreme values. Fig. 14 is the reconstruction of a solution considered in Refs. [19,20]. Fig. 15 shows how the Q-ball deforms as ω → ω min , whereas Fig.  15 what happens as ω → ω max . Similarly, Fig. 17 shows the limit µ → µ min = ω 2 , and Fig. 18 the one µ → µ max .   Table 2: Properties of Q-balls for β 1 = 0.5, β 12 = µ = 1.4 and ω = 1.177  Table 3: Properties of Q-balls for β 1 = 0.5, β 12 = µ = 1.4 and ω = 1.179  Table 4: Properties of Q-balls for β 1 = 0.5, β 12 = µ = 1.4 and ω = 1.181 A typical solution for β 2 is shown in Fig. 14, and the energy and charge distributions of the same solution are shown in Fig. 14b. This solution is one of the family constructed in Refs. [19,20].