A one-dimensional soliton system of gauged Q-ball and anti-Q-ball
aa r X i v : . [ h e p - t h ] J a n A one-dimensional soliton system of gauged Q-ball and anti-Q-ball
A. Yu. Loginov ∗ and V. V. Gauzshtein Tomsk Polytechnic University, 634050 Tomsk, Russia (Dated: January 3, 2019)The (1 + 1)-dimensional gauge model of two complex self-interacting scalar fields that interactwith each other through an Abelian gauge field and a quartic scalar interaction is considered. It isshown that the model has nontopological soliton solutions describing soliton systems consisting oftwo Q-ball components possessing opposite electric charges. The two Q-ball components interactwith each other through the Abelian gauge field and the quartic scalar interaction. The interplaybetween the attractive electromagnetic interaction and the repulsive quartic interaction leads tothe existence of symmetric and nonsymmetric soliton systems. Properties of these systems areinvestigated by analytical and numerical methods. The symmetric soliton system exists in thewhole allowable interval of the phase frequency, whereas the nonsymmetric soliton system existsonly in some interior subinterval. Despite the fact that these soliton systems are electrically neutral,they nevertheless possess nonzero electric fields in their interiors. It is found that the nonsymmetricsoliton system is more preferable from the viewpoint of energy than the symmetric one. Bothsymmetric and nonsymmetric soliton systems are stable to the decay into massive scalar bosons.
PACS numbers: 11.27.+d, 11.10.Lm, 11.15.-q
I. INTRODUCTION
There are many field models possessing global symme-tries and corresponding conserved Noether charges thatadmit the existence of nontopological solitons [1, 2]. Thedetermining property of a nontopological soliton is thatit is an extremum of the energy functional at a fixedvalue of the Noether charge. This feature of nontopolog-ical solitons leads to the characteristic time dependence ∝ exp ( − iωt ) of their fields. This nontrival time depen-dence of the soliton’s field allows to avoid severe restric-tions of Derrick’s theorem [3], so scalar nontopologicalsolitons can exist in any number of spatial dimensions.The simplest nontopological soliton, proposed in [4]and known as a Q-ball [5], has been found in a model ofa complex scalar field possessing a global U (1) symmetry.Q-balls can also exist in scalar field models possessing aglobal non-Abelian symmetry [6, 7]. They are present[8, 9] in the minimal supersymmetric extension of theStandard Model having flat directions in the interactionpotential of scalar fields. Q-balls are of great interest tocosmological models describing the evolution of the earlyUniverse [10, 11].There are also other types of nontopological solitons inglobal-symmetric field models. The most known of themis the nontopological soliton of the Friedberg-Lee-Sirlinmodel [12]. The model consists of two interacting scalarfields, one of which is real and the other is complex. Itpossesses a global U (1) symmetry and a renormalizableinteraction potential. Another example is the nontopo-logical soliton in the model of a massive self-interactingcomplex vector field [13].In all of the examples given above, the existence of non-topological solitons is due to a global invariance of the ∗ [email protected] corresponding Lagrangians, so the Noether charge of suchsolitons cannot be a source of a gauge field. At the sametime, nontopological solitons also exist in field modelspossessing a local gauge invariance, both Abelian [14–19] and non-Abelian [20, 21]. The nontopological soli-tons [14–19] possess a long-range gauge electric field, andNoether charges of these solitons are proportional to theirelectric charges. However, all these electrically chargednontopological solitons are three-dimensional ones. Thisis because any one-dimensional or two-dimensional fieldconfiguration with a nonzero electric charge possesses in-finite energy, as it follows from Gauss’s law and the ex-pression for the electric field energy density. Neverthe-less, there are electrically neutral low-dimensional solitonsystems that have a nonzero electric field in their inte-riors. In particular, the two-dimensional soliton systemsconsisting of vortex and Q-ball components interactingthrough an Abelian gauge field have been described in[22, 23].In the present paper, we research the (1 + 1)-dimensional gauge model of two complex self-interactingscalar fields interacting with each other through anAbelian gauge field and a quartic scalar interaction. Inparticular, it is found that symmetric and nonsymmetricsoliton systems exist in the model. The soliton systemsconsist of two Q-ball components having opposite elec-tric charges. The soliton systems are electrically neutralbut nevertheless possess nonzero electric fields in theirinteriors. The paper is structured as follows. In Sec. II,we describe briefly the Lagrangian and the field equa-tions of the model under consideration. By means ofthe Hamiltonian formalism and the Lagrange multipli-ers method, the time dependence is established for thesoliton system’s fields. Then, we give the ansatz usedfor solving the model’s field equations and establish thebasic relation for the nontopological soliton system. InSec. III, we derive the system of nonlinear differentialequations for the ansatz functions and the expressionsfor the electromagnetic current density and the energydensity in terms of these functions. Then, some gen-eral properties of the soliton system are established, itsasymptotic properties are researched, and the virial re-lation for the soliton system is derived. In Sec. IV, westudy properties of the soliton system in the thick-walland thin-wall regimes and establish its stability to decayinto free massive scalar bosons. In Sec. V, we briefly de-scribe the procedure for numerical solving of a boundaryvalue problem and discuss possible types of soliton so-lutions of the problem. The dependences of the energyand the Noether charge on the phase frequency are pre-sented for both (symmetric and nonsymmetric) types ofthe soliton solutions. Then, we show the dependences ofthe symmetric soliton system’s energy and the energy dif-ference between the symmetric and nonsymmetric solitonsystems on the Noether charge. After that, we presentthe numerical results for the ansatz functions, the en-ergy density, the electric charge density, and the electricfield strength for the symmetric and nonsymmetric soli-ton systems.Throughout the paper the natural units c = 1, ~ = 1are used. II. THE LAGRANGIAN AND THE FIELDEQUATIONS
The (1 + 1)-dimensional gauge model we are interestedin is described by the Lagrangian density L = − F µν F µν + ( D µ φ ) ∗ D µ φ + ( D µ χ ) ∗ D µ χ − V ( | φ | ) − U ( | χ | ) − W ( | φ | , | χ | ) . (1)It describes the two complex scalar fields φ and χ that minimally interact with the Abelian gauge field A µ through the covariant derivatives: D µ φ = ∂ µ φ + ieA µ φ, D µ χ = ∂ µ χ + iqA µ χ. (2)The scalar fields interact with each other and self-interact. The self-interaction potentials of the scalarfields have the form V ( | φ | ) = m φ | φ | − g φ | φ | + h φ | φ | , (3) U ( | χ | ) = m χ | χ | − g χ | χ | + h χ | χ | , (4)whereas the interaction potential is W ( | φ | , | χ | ) = λ | φ | | χ | . (5)We suppose that the self-interaction potentials V and U admit the existence of usual non-gauged nontopologicalsolitons (Q-balls) formed from the scalar fields φ and χ ,respectively. We also suppose that the potentials V and U possess global minima at φ = 0 and χ = 0, respec-tively. Then the parameters of the potentials satisfy thecondition m i h i g i > , (6)where the index i = ( φ, χ ).The Lagrangian (1) is invariant under the local gaugetransformations. At the same time, it is also invariantunder the two independent global gauge transformations: φ ( x ) → φ ′ ( x ) = exp ( − iα ) φ ( x ) , (7a) χ ( x ) → χ ′ ( x ) = exp ( − iβ ) χ ( x ) . (7b)The Noether currents corresponding to transformations(7) are written as j µφ = i (cid:2) φ ∗ D µ φ − ( D µ φ ) ∗ φ (cid:3) , (8a) j µχ = i (cid:2) χ ∗ D µ χ − ( D µ χ ) ∗ χ (cid:3) . (8b)The presence of the two separately conserved Noethercharges Q φ = R j φ dx and Q χ = R j χ dx is the result ofthe structure of the interaction potential W and the neu-trality of the Abelian gauge field A µ .The field equation of the model are obtained by varyingthe action S = R L d x in the corresponding fields: D µ D µ φ + ∂V∂ | φ | φ | φ | + ∂W∂ | φ | φ | φ | = 0 , (9) D µ D µ χ + ∂U∂ | χ | χ | χ | + ∂W∂ | χ | χ | χ | = 0 , (10) ∂ µ F µν = j ν , (11)where the electromagnet current j ν is written in termsof two Noether currents (8) j ν = ej νφ + qj νχ . (12)The symmetric energy-momentum tensor of the model iswritten as T µν = − F µλ F λν + 14 g µν F λρ F λρ + ( D µ φ ) ∗ D ν φ + ( D ν φ ) ∗ D µ φ + ( D µ χ ) ∗ D ν χ + ( D ν χ ) ∗ D µ χ − g µν (cid:2) ( D µ φ ) ∗ D µ φ + ( D µ χ ) ∗ D µ χ − V ( | φ | ) − U ( | χ | ) − W ( | φ | , | χ | )] , (13)so we have the following expression for the energy density T = E = 12 E x + ( D t φ ) ∗ D t φ + ( D x φ ) ∗ D x φ + ( D t χ ) ∗ D t χ + ( D x χ ) ∗ D x χ + V ( | φ | ) + U ( | χ | ) + W ( | φ | , | χ | ) . (14)By analogy with nontopological solitons, we find a so-lution of model (1) that is an extremum of the energyfunctional E = R E dx at a fixed value of the Noethercharge Q χ = R j χ dx . Such a solution is an unconditionalextremum of the functional F = Z E dx − ω Z j χ dx = E − ωQ χ , (15)where ω is the Lagrange multiplier. To determine thetime dependence of the soliton solution, we will use theHamiltonian formalism. We adopt the axial gauge inwhich the spatial component of the gauge potential van-ishes: A x = A = 0. In this case, the gauge model isdescribed in terms of the eight canonically conjugatedfields: φ , π φ = ( D φ ) ∗ , φ ∗ , π φ ∗ = D φ , χ , π χ = ( D χ ) ∗ , χ ∗ , and π χ ∗ = D χ . Then, the Hamiltonian density hasthe form H = π φ ∂ t φ + π φ ∗ ∂ t φ ∗ + π χ ∂ t χ + π χ ∗ ∂ t χ ∗ − L = −
12 ( ∂ x A ) + π φ π φ ∗ + π χ π χ ∗ + ∂ x φ ∗ ∂ x φ + ∂ x χ ∗ ∂ x χ + ieA { φ ∗ π φ ∗ − φπ φ } + iqA { χ ∗ π χ ∗ − χπ χ } + V ( | φ | ) + U ( | χ | ) + W ( | φ | , | χ | ) , (16)where the time component A is determined in terms ofthe canonically conjugated fields by Gauss’s law ∂ x A + ie { φ ∗ π φ ∗ − φπ φ } + iq { χ ∗ π χ ∗ − χπ χ } = 0 . (17)Note that energy density (14) does not coincide withHamiltonian density (16): H − E = − ( ∂ x A ) + ieA { φ ∗ π φ ∗ − φπ φ } + iqA { χ ∗ π χ ∗ − χπ χ } . (18)However, the integral of Eq. (18) over the space dimen-sion vanishes for field configurations possessing finite en-ergy and satisfying Gauss’s law (17). So, for such config-urations E = Z E dx = H = Z H dx. (19)It can be shown that the field equations (9) and (10)can be rewritten in the Hamiltonian form: ∂ t φ = δHδπ φ = δEδπ φ , ∂ t π φ = − δHδφ = − δEδφ , (20) ∂ t χ = δHδπ χ = δEδπ χ , ∂ t π χ = − δHδχ = − δEδχ . (21)Further, the first variation of the functional F vanishesfor the soliton solution: δF = δE − ωδQ χ = 0 , (22)where the first variation of the Noether charge Q χ can beexpressed in terms of the canonically conjugated fields δQ χ = − i Z ( π χ δχ + χδπ χ − c.c.) dx. (23) From Eqs. (20), (21), (22), and (23), we obtain the fol-lowing Hamilton field equations: ∂ t χ = δEδπ χ = ω δQ χ δπ χ = − iωχ, (24) ∂ t χ ∗ = δEδπ χ ∗ = ω δQ χ δπ χ ∗ = iωχ, (25)while time derivatives of the other model’s fields are equalto zero. Thus, in the adopted gauge A x = 0, only thescalar field χ has nontrivial time dependence, whereasthe model’s fields φ and A do not depend on time: φ ( x, t ) = f ( x ) , (26a) χ ( x, t ) = s ( x ) exp ( − iωt ) , (26b) A µ ( x, t ) = ( a ( x ) , . (26c)From extremum condition (22), it follows that the solitonsolution satisfies the important relation dEdQ χ = ω, (27)where the Lagrange multiplier ω is some function of theNoether charge Q χ . Note that unlike Eqs. (26), relation(27) is gauge-invariant. Just as in the case of non-gaugednontopological solitons [1], relation (27) plays the pri-mary role in the determining of properties of the gaugednontopological soliton system. III. SOME PROPERTIES OF THE SOLUTION
In Eqs. (26), f ( x ) and s ( x ) are some complex func-tions of the real argument x . Substituting Eqs. (26) intofield equations (9) – (11), we obtain the system of or-dinary nonlinear differential equations for the functions a ( x ), f ( x ), and s ( x ): a ′′ ( x ) − (cid:16) e | f ( x ) | + q | s ( x ) | (cid:17) a ( x ) (28)+2 qω | s ( x ) | = 0 ,f ′′ ( x ) − (cid:16) m φ − e a ( x ) (cid:17) f ( x ) (29)+ (cid:16) g φ | f ( x ) | − h φ | f ( x ) | − λ | s ( x ) | (cid:17) f ( x ) = 0 ,s ′′ ( x ) − (cid:16) m χ − ( ω − qa ( x )) (cid:17) s ( x ) (30)+ (cid:16) g χ | s ( x ) | − h χ | s ( x ) | − λ | f ( x ) | (cid:17) s ( x ) = 0 . From Eq. (29), it follows that the real and imaginaryparts of f ( x ) satisfy the same differential equation,whereas Eq. (30) leads us to the same conclusion for thefunction s ( x ). This in turn means that the functions f ( x ) and s ( x ) can be written as f ( x ) = exp ( iα ) | f ( x ) | and s ( x ) = exp ( iβ ) | s ( x ) | , where α and β are real con-stant phases. These phases, however, can be gauged awayby global gauge transformations (7). Thus we can sup-pose without loss of generality that f ( x ) and s ( x ) arereal functions of x . Substituting Eqs. (26) into Eq. (12)and Eq. (14), we obtain the electromagnetic current den-sity and the energy density in terms of the real functions a ( x ), f ( x ), and s ( x ): j µ = (cid:0) qωs − (cid:0) q s + e f (cid:1) a , (cid:1) , (31) E = a ′ f ′ + s ′ + ( ω − qa ) s + e a f + V ( f ) + U ( s ) + W ( f, s ) . (32)The finiteness of the soliton system’s energy E = R E dx leads to the following boundary conditions for the func-tions a ( x ), f ( x ), and s ( x ): a ′ ( x ) −→ x →−∞ , a ′ ( x ) −→ x →∞ , (33a) f ( x ) −→ x →−∞ , f ( x ) −→ x →∞ , (33b) s ( x ) −→ x →−∞ , s ( x ) −→ x →∞ . (33c)Let us discuss some general properties of the solitonsystem. The invariance of the Lagrangian (1) under thecharge conjugation leads to the invariance of system (28)– (30) under the discrete transformation ω, a , f, s −→ − ω, − a , f, s. (34)From Eqs. (31), (32), and (34), it follows that the energy E is an even function of ω , whereas the Noether charges Q φ and Q χ are odd functions of ω : E ( − ω ) = E ( ω ) , (35) Q φ,χ ( − ω ) = − Q φ,χ ( ω ) . (36)The Lagrangian (1) is also invariant under the paritytransformation. It follows that system (28) – (30) is in-variant under the space inversion: x → − x . Thus, if a ( x ), f ( x ), and s ( x ) is a solution of Eqs. (28) – (30),then a ( − x ), f ( − x ), and s ( − x ) is also a solution. Thisfact, however, does not mean that a ( x ), f ( x ), and s ( x )must be even functions of x . Indeed, we shall see laterthat system (28) – (30) together with boundary condi-tions (33) has nonsymmetric soliton solutions.Eq. (28) can be written as a ′′ = − j , where j is electriccharge density (31). Integrating this equation over x ∈ ( −∞ , ∞ ) and taking into account boundary conditions(33), we conclude that the total electric charge of a fieldconfiguration with a finite energy vanishes: Q = eQ φ + qQ χ = 0 . (37)Substituting the power expansions for the functions a ( x ), f ( x ), and s ( x ) into Eqs. (28) – (30), we obtainthe asymptotic form of the solution as x → a ( x ) = a + a x + a x + O (cid:0) x (cid:1) , (38a) f ( x ) = f + f x + f x + O (cid:0) x (cid:1) , (38b) s ( x ) = s + s x + s x + O (cid:0) x (cid:1) , (38c) where the next-to-leading coefficients a = 2 a (cid:0) e f + q s (cid:1) − qωs , (39a) f = f (cid:0) m φ − g φ f + h φ f − e a + λs (cid:1) , (39b) s = s (cid:0) m χ − ( ω − qa ) − g χ s + h χ s + λf (cid:1) (39c)are determined in terms of the three leading coefficients a , f , s and the model’s parameters. The next coeffi-cients a n , f n , s n , where n = 3 , , , . . . are determinedby the six leading coefficients a , f , s , a , f , s , andthe model’s parameters. It can be easily shown that ifthe coefficients a , f , and s vanish, all the other co-effients with an odd n also vanish, and we have an evensolution of Eqs. (28) – (30).Linearization of Eqs. (28) – (30) at large x togetherwith corresponding boundary conditions (33) lead us tothe asymptotic form of the solution as x → ±∞ : f ( x ) ∼ f ±∞ exp ( ∓ e m φ ± x ) , (40a) s ( x ) ∼ s ±∞ exp ( ∓ e m χ ± x ) , (40b) a ( x ) ∼ a ±∞ + a ±∞ e f ±∞ e m φ ± × exp ( ∓ e m φ ± x ) − ( ω − qa ±∞ ) × qs ±∞ e m χ ± exp ( ∓ e m χ ± x ) , (40c)where the mass parameters e m φ ± and e m χ ± are defined bythe relations: e m φ ± = m φ − e a ±∞ , (41) e m χ ± = m χ − ( ω − qa ±∞ ) . (42)From Eqs. (41) and (42), we obtain the upper boundarieson the absolute values of a ( ±∞ ) = a ±∞ and ω : | a ( ±∞ ) | < m φ e , | ω | < m χ + qe m φ . (43)From Eqs. (38) – (40), it follows that there may betwo types of solutions: the symmetric one for which f ( − x ) = f ( x ), s ( − x ) = s ( x ), a ( − x ) = a ( x ) and thenonsymmetric one that does not possess this property.For a symmetric solution, the series coefficients a n , f n ,and s n with an odd n vanish, and so in Eqs. (40) – (42),the asymptotic parameters corresponding to x → −∞ are equal to those corresponding to x → ∞ .If the values of the model’s parameters are fixed, thenthe behavior of a nonsymmetric solution f ( x ), s ( x ), a ( x ) as x → a , f , s , a , f , and s in Eqs. (38). The behavior of thenonsymmetric solution as x → ±∞ is also determinedby the six parameters in Eqs. (40), namely a −∞ , f −∞ ,and s −∞ as x → −∞ and a ∞ , f ∞ , and s ∞ as x → ∞ .Thus we have twelve free parameters in all. The continu-ity condition for f ( x ), s ( x ), a ( x ) and their derivatives f ′ ( x ), s ′ ( x ), a ′ ( x ) at arbitrary x < x > S = R L dxdt . At the same time,the Lagrangian density (1) does not depend on time inthe case of field configurations (26). It follows that anysolution of Eqs. (28) – (30), satisfying boundary condi-tions (33), is an extremum of the Lagrangian L = R L dx .Let a ( x ), f ( x ), and s ( x ) be a solution of system (28)– (30), satisfying boundary conditions (33). After thescale transformation of the solution’s argument x → λx ,the Lagrangian L becomes a function of the scale param-eter λ . The function L ( λ ) has an extremum at λ = 1,so its derivative with respect to λ vanishes at this point: dL/dλ | λ =1 = 0. From this equation, we obtain the virialrelation for the soliton system: E ( E ) + E ( P ) − E ( G ) − E ( T ) = 0 , (44)where E ( E ) = Z a ′ dx (45)is the electric field’s energy, E ( G ) = Z (cid:0) f ′ + s ′ (cid:1) dx (46)is the gradient part of the soliton’s energy, E ( T ) = Z (cid:16) ( ω − qa ) s + e a f (cid:17) dx (47)is the kinetic part of the soliton’s energy, and E ( P ) = Z ( V ( f ) + U ( s ) + W ( f, s )) dx (48)is the potential part of the soliton’s energy.The obvious equality E = E ( E ) + E ( T ) + E ( G ) + E ( P ) and virial relation (44) lead to the following representa-tions for the soliton system’s energy: E = 2 (cid:16) E ( T ) + E ( G ) (cid:17) , (49) E = 2 (cid:16) E ( P ) + E ( E ) (cid:17) . (50)Integrating the term a ′ / E = 12 ωQ χ + E ( G ) + E ( P ) , (51)which, in turn, leads to the relation between the Noethercharge Q χ , the electric field’s energy E ( E ) , and the ki-netic energy E ( T ) : ωQ χ = 2 (cid:16) E ( E ) + E ( T ) (cid:17) . (52) IV. THE THICK-WALL AND THIN-WALLREGIMES OF THE SOLITON SYSTEM
In this section, we research properties of the symmetricsoliton solution in two extreme regimes. In the thick-wallregime, the mass parameters e m φ and e m χ tend to zero,leading to a spatial spreading of the soliton system. Thisfact and Eqs. (41) and (42) lead to the limiting valuesof the potential a ( ∞ ) and the phase frequency ω in thethick-wall regime: | a ( ∞ ) | = m φ e , ω tk = sgn ( a ( ∞ )) (cid:16) m χ + qe m φ (cid:17) . (53)In the thick-wall regime, where e m φ ≈ e m χ →
0, we under-take the following scale transformation of the fields andthe x -coordinate: f ( x ) = ∆ ¯ f (¯ x ) , s ( x ) = ∆¯ s (¯ x ) ,a ( x ) = m φ e + ∆ m φ ¯ a (¯ x ) , x = ∆ − ¯ x, (54)where the scale factor ∆ is defined as∆ = m φ − e a ( ∞ ) ≈ m χ − ( ω − qa ( ∞ )) ≈ κ (cid:0) ω − ω (cid:1) . (55)In Eq. (55), the factor κ is expressed in terms of the scalarparticles’ masses and the gauge coupling constants: κ = e (cid:20) m φ m χ ( em φ + qm χ ) ( em χ + qm φ ) (cid:21) . (56)Let us consider the functional F , which has been de-fined in Eq. (15). This functional is related to the en-ergy functional by means of Legendre transformation: F ( ω ) = E ( Q χ ) − ωQ χ . On field configuration (54), thefunctional F can be written as F ( ω ) = ∆ ¯ F + O (cid:0) ∆ (cid:1) , (57)where the functional ¯ F does not depend on ω :¯ F = Z h ¯ f ′ (¯ x ) + ¯ s ′ (¯ x ) + ¯ f (¯ x ) + ¯ s (¯ x ) (58) − g φ f (¯ x ) − g χ s (¯ x ) + λ ¯ f (¯ x ) ¯ s (¯ x ) i d ¯ x. In the thick-wall regime, the phase frequency ω tends tothe limiting value ω tk , so the parameter ∆ vanishes, andit is possible to ignore the higher-order terms in ∆ inEq. (57). Using known properties of Legendre transfor-mation, we obtain sequentially Q χ ( ω ) = − dF ( ω ) dω = 3 ¯ F κ ω (cid:0) ω − ω (cid:1) , (59) E ( ω ) = F ( ω ) − ω dF ( ω ) dω = ¯ F κ (cid:0) ω + ω (cid:1) (cid:0) ω − ω (cid:1) . (60)From Eqs. (59) and (60), we obtain the dependence ofthe energy E on the Noether charge Q χ in the thick-wallregime: E ( Q χ ) = ω tk Q χ −
154 1¯ F κ ω Q χ + O (cid:0) Q χ (cid:1) . (61)We see from Eqs. (37), (59), (60), and (61) that the en-ergy E and the Noether charges Q φ and Q χ of the solitonsystem tend to zero in the thick-wall regime. Further,Eq. (61), basic relation (27), and the inequality ω < ω lead to the conclusion that E ( Q χ ) < ω tk Q χ for all val-ues of Q χ . From Eqs. (36), (37), and (53) it follows that ω tk Q χ is equal to m φ | Q φ | + m χ | Q χ | , which, in turn,is the rest energy of the neutral plan-wave configurationformed from the charged scalar φ and χ -particles. Hence,the symmetric soliton system is stable to decay into thescalar φ and χ -particles.The second extremal regime of the symmetric solitonsystem is the thin-wall regime in which the absolute valueof the phase frequency tends to some minimum value ω tn .In the thin-wall regime, the spatial size of the soliton sys-tem increases indefinitely, with the result that its energy E and Noether charges Q φ and Q χ also tend to infinity.In the thin-wall regime, when the spatial size of the soli-ton system L → ∞ , the gradient operator gives a factorproportional to L − . Therefore, we can ignore the elec-tric field’s energy (45) and the gradient energy (46) incomparison with the kinetic energy (47) and the poten-tial energy (48). Then, from Eq. (44) it follows that thefollowing limiting relation holds in the thin-wall regime:lim ω → ω tn E ( T ) E ( P ) = 1 , (62)and, as a consequence,lim ω → ω tn E ( T ) E = lim ω → ω tn E ( P ) E = 1 . (63)Further, electric charge density (31) tends to zero in thethin-wall regime, since the soliton system’s electric chargeis strictly equal to zero, whereas its spatial size tends toinfinity. Then, using Eqs. (31), (47), and (63), we obtainthe limiting relationlim ω → ω tn E ( T ) Q χ = lim ω → ω tn EQ χ = ω tn , (64)which is consistent with basic relation (27) and Eq. (52). V. NUMERICAL RESULTS
The system of differential equations (28) – (30) withboundary conditions (33) is the mixed boundary valueproblem on the infinite interval x ∈ ( −∞ , ∞ ). Thisboundary value problem can be solved only by numer-ical methods. In this paper, the boundary value problem was solved using the Maple package [25] by the methodof finite differences and subsequent Newtonian iterations.Equations (27), (37), and (44) were used to check the cor-rectness of numerical solutions.Let us discuss possible types of solutions of the bound-ary value problem. If the quartic coupling constant λ and the electromagnetic coupling constants e and q areset equal to zero, then the Lagrangian (1) will describethe system of two self-interacting complex scalar fieldsthat, however, do not interact with each other. In thiscase, the boundary value problem has the solution de-scribing a system of two non-interacting non-gauged one-dimensional Q-balls. Generally, these two Q-balls havedifferent shapes and can be at an arbitrary distance fromeach other, so the solution will not be symmetric. How-ever, the situation changes when the electromagnetic in-teraction is turned on. In this case, from Eq. (37) itfollows that the electric charges of two Q-ball compo-nents are equal in magnitude, but opposite in sign. It isimportant to note that the electric charges of two gaugedQ-balls are conserved separately owing to the neutralityof the Abelian gauge field. Since the opposite electriccharges attract each other, the initially nonsymmetricsoliton system transits to a symmetric one. Now we turnon the quartic interaction between the two complex scalarfields φ and χ by letting the coupling constant λ be somepositive value. From Eq. (5), it follows that the energy ofthe quartic interaction increases with the increase of over-lap between the Q-ball components of the soliton systemand is negligible at large separations between the Q-ballcomponents. Such a behavior of the quartic interactioncorresponds to a short-range repulsive force between theQ-ball components, while the electromagnetic long-rangeattractive force results in the confinement of the the Q-ball components. One would expect that for a sufficientlylarge positive coupling constant λ , the action of these op-posite forces leads to an equilibrium nonsymmetric soli-ton configuration, which is the solution of boundary valueproblem (28) – (30), and (33). Indeed, we shall see laterthat such a nonsymmetric soliton solution really exists.The system of differential equations (28) – (30) de-pends on the ten dimensional parameters: ω , e , q , m φ , m χ , g φ , g χ , h φ , h χ , and λ . It is readily seen, however,that the dimensionless functions a ( x ), f ( x ), and s ( x )can depend only on nine independent dimensionless com-binations of these parameters. Therefore without lossof generality, we can choose the mass m φ of the scalar φ -particle as the energy unit. We consider a generalcase in which the corresponding dimensionless param-eters are values of the same order: ˜ e = e/m φ = 0 . q = q/m φ = 0 .
2, ˜ m χ = m χ /m φ = 1 .
25, ˜ g φ = g φ /m φ = 1,˜ g χ = g χ /m φ = 1 .
5, ˜ h φ = h φ /m φ = 0 .
22, ˜ h χ = h χ /m φ =0 .
31, and ˜ λ = λ/m φ = 0 . E = m − φ E and Noether charge Q χ on the dimensionless phase frequency ˜ ω = m − φ ω . Themost striking feature of Figs. 1 and 2 is the coexistence of ΩŽ E ŽŽ FIG. 1. The dependence of the dimensionless soliton energy˜ E = m − φ E on the dimensionless phase frequency ˜ ω = m − φ ω .The solid curve corresponds to the symmetric soliton system,and the dashed curve corresponds to the nonsymmetric one. the symmetric and nonsymmetric soliton solutions. In-deed, it has been found numerically that the symmetricsoliton solution exists in the range from the minimumvalue ˜ ω min = 1 . ω tk = 2 . ω lb = 1 .
409 to the right one ˜ ω rb = 1 . ω i1 and ˜ ω i2 in Figs. 1 and 2, respectively. These intersectionpoints are slightly different: ˜ ω i1 = 1 . ω i2 = 1 . Z ˜ ω rb ˜ ω lb [ Q χ a (˜ ω ) − Q χ s (˜ ω )] d ˜ ω = 0 , (65)and Z ˜ ω rb ˜ ω lb h ˜ E a (˜ ω ) − ˜ E s (˜ ω ) i d ˜ ω = 0 , (66)which were checked numerically.When ˜ ω tends to its minimal value ˜ ω tn , the symmet-ric soliton system goes into the thin-wall regime. In thisregime, the energy ˜ E , the Noether charges Q χ and Q φ ,and the effective spatial size L of the symmetric solitonsystem increase indefinitely. In particular, we found nu-merically that ˜ E , Q χ , Q φ , and L increase logarithmically ΩŽ Q Χ FIG. 2. The dependence of the soliton Noether charge Q χ on the dimensionless phase frequency ˜ ω = m − φ ω . The solidcurve corresponds to the symmetric soliton system, and thedashed curve corresponds to the nonsymmetric one. as ˜ ω → ˜ ω tn : ˜ E ∼ − ˜ ω tn B ln (˜ ω − ˜ ω tn ) , (67) Q χ ∼ − B ln (˜ ω − ˜ ω tn ) , (68) Q φ ∼ B qe ln (˜ ω − ˜ ω tn ) , (69) L ∼ − C ln (˜ ω − ˜ ω tn ) , (70)where B and C are some positive constants, and the lim-iting thin-wall phase frequency ˜ ω tn = 1 . ω tn is slightly less thanthe minimal value ˜ ω min = 1 . E , Q χ , Q φ , and L is similar tothat of the corresponding values of the one-dimensionalnon-gauged Q-ball, as it follows from Eqs. (67) – (70)and (A9) – (A11).When ˜ ω tends to its maximal value ˜ ω tk , the symmet-ric soliton system goes into the thick-wall regime. Inthis regime, the soliton system is spread out over one-dimensional space, while the amplitudes of the scalarfields φ and χ tend to zero as (˜ ω tk − ˜ ω ) / in accor-dance with Sec. IV. It was found numerically that in thethick-wall regime, ˜ E , Q χ , and Q φ also tend to zero as(˜ ω tk − ˜ ω ) / , whereas the effective spatial size L divergesas (˜ ω tk − ˜ ω ) − / : ˜ E ∼ b ˜ ω tk (˜ ω tk − ˜ ω ) , (71) Q χ ∼ b (˜ ω tk − ˜ ω ) , (72) Q φ ∼ − b qe (˜ ω tk − ˜ ω ) , (73) L ∼ c (˜ ω tk − ˜ ω ) − / . (74)From Eqs. (71) – (74) and (A5) – (A7), it follows thatthe behavior of E , Q χ , Q φ , and L is similar to that of the Q Χ E ŽŽ FIG. 3. The dependence of the dimensionless energy ˜ E = m − φ E of the symmetric soliton system on the Noether charge Q χ (solid curve). The dash-dotted line is the straight line˜ E = ˜ ω tk Q χ = (1 + m χ /m φ ) Q χ . corresponding values of the one-dimensional non-gaugedQ-ball in the thick-wall regime.Figure 3 shows the dependence of the dimensionless en-ergy ˜ E of the symmetric soliton system on the Noethercharge Q χ . We see that the dependence ˜ E ( Q χ ) is anincreasing convex ( d ˜ E/dQ χ > , d ˜ E/dQ χ <
0) functionoutgoing from the coordinate origin. It follows that thesymmetric soliton system is stable to decay into severalsmaller ones. We also see that in accordance with Sec. IV,the curve ˜ E ( Q χ ) lies below the straight line ˜ E = ˜ ω tk Q χ for all positive Q χ . From this, it follows that the symmet-ric soliton system is stable to decay into massive scalar φ and χ -bosons.In Fig. 4, we can see the dependence of the energy dif-ference ∆ ˜ E = ˜ E s − ˜ E a between the symmetric and non-symmetric soliton solutions on the Noether charge Q χ .From Fig. 4, it follows that the energy of the symmetricsoliton system slightly exceeds the energy of the nonsym-metric one in the whole range of the Noether charge Q χ for which the existence of the nonsymmetric soliton sys-tem is possible. It follows that the nonsymmetric solitonsystem is more preferable from the viewpoint of energyas compared to the symmetric one. Note, however, thatthe difference ∆ ˜ E is rather small and is of the order of0 .
1% of the soliton system’s energy. As well as the sym-metric soliton system, the nonsymmetric one is stable todecay into massive scalar bosons. Note, however, thatthe symmetric soliton system is unstable to the transi-tion into the nonsymmetric one because of the possibilityof tunneling under the potential barrier.Figure 5 presents the nonsymmetric soliton solutioncorresponding to the dimensionless phase frequency ˜ ω =1 .
5, whereas Fig. 6 presents the energy and electric chargedensities and the electric field strength corresponding to
10 12 14 16 Q Χ D E ŽŽ FIG. 4. The dependence of the energy difference ∆ ˜ E =˜ E s − ˜ E a between the symmetric and nonsymmetric solitonsolutions on the Noether charge Q χ . Fig. 5. The nonsymmetric character of the soliton systemis obvious from Figs. 5 and 6. The most interesting fea-ture of the nonsymmetric soliton system is the presenceof the unidirectional electric field in its interior, as for aplane capacitor. From Fig. 5, it follows that the chargedscalar φ and χ -particles can acquire the energy equal to − e ∆ a ≈ . m φ in the electric field of the nonsymmet-ric soliton system. Note that this energy is comparablewith the scalar particles’ masses. Lighter particles (e.g.light charged fermions) passing through the interior ofthe nonsymmetric soliton system can be accelerated torelativistic velocities and energies.Similar to Figs. 5 and 6, Figs. 7 and 8 give informa-tion about the symmetric soliton solution. From Fig. 8, - - - x Ž FIG. 5. The nonsymmetric numerical solution for f (˜ x ) (solidcurve), s (˜ x ) (dashed curve), and ˜ ea (˜ x ) (dotted curve). Thedimensionless phase frequency ˜ ω = 1 . - - - x Ž - FIG. 6. The dimensionless versions of the energy density˜ E = m − φ E (solid curve), the scaled electric charge density˜ e − ˜ j = ˜ e − m − φ j (dashed curve), and the scaled electricfield strength ˜ e − ˜ E x = ˜ e − m − φ E x (dotted curve), corre-sponding to the nonsymmetric solution in Fig. 5. it follows that the energy and electric charge densitiesare symmetric with respect to the center of the solitonsystem, while the electric field strength is antisymmet-ric. For positive ω , it is directed from the soliton sys-tem’s center, so it attracts negatively charged particlesand repels positively charged ones. For negative ω , itis directed to the soliton system’s center, so the rolesof negatively and positively charged particles are inter-changed. For positive (negative) ω , the form of the elec-tromagnetic potential a corresponds to a potential wellfor negatively (positively) charged particles. It followsthat bound fermionic and bosonic states can exist in theelectric field of the symmetric soliton system. - - - x Ž FIG. 7. The symmetric numerical solution for f (˜ x ) (solidcurve), s (˜ x ) (dashed curve), and ˜ ea (˜ x ) (dotted curve). Thedimensionless phase frequency ˜ ω = 1 . - - - x Ž - FIG. 8. The dimensionless versions of the energy density˜ E = m − φ E (solid curve), the scaled electric charge density˜ e − ˜ j = ˜ e − m − φ j (dashed curve), and the scaled electricfield strength ˜ e − ˜ E x = ˜ e − m − φ E x (dotted curve), corre-sponding to the symmetric solution in Fig. 7. VI. CONCLUSION
In the present paper, the one-dimensional nontopolog-ical soliton system consisting of two self-interacting com-plex scalar fields has been investigated. The scalar fieldsinteract with each other through the Abelian gauge fieldand the quartic scalar interaction. The finiteness of theenergy of the one-dimensional soliton system leads to itselectric neutrality, so its two scalar components have op-posite electric charges. The neutrality of the Abeliangauge field leads to the separate conservation of the elec-tric charges of these scalar components. The interplaybetween the attractive electromagnetic interaction andthe repulsive quartic interaction leads to the existence ofsymmetric and nonsymmetric soliton systems.The symmetric soliton system exists in the whole al-lowable interval of the phase frequency ω . When ω tendsto its minimal (maximal) value, the symmetric solitonsystem goes into the thin-wall (thick-wall) regime. Inthe thin-wall regime, the energy, the Noether charges,and the spatial size of the symmetric soliton system tendto infinity. In the thick-wall regime, the spatial size ofthe symmetric soliton system also tends to infinity, butthe energy and the Noether charges tend to zero. Incontrast to this, the nonsymmetric soliton system existsonly in some interior subinterval between the minimaland maximal allowable phase frequencies ω tn and ω tk . Itfollows that there exists an interval of the Noether charge Q χ (and, consequently, an interval of the Noether charge Q φ = − qe − Q χ ), where the symmetric and nonsymmet-ric soliton systems coexist. In all this interval, the energyof the nonsymmetric soliton system turns out to be lessthan that of the symmetric soliton system, so the sym-metric soliton system can turn into the nonsymmetric0one through quantum tunneling. Both symmetric andnonsymmetric soliton systems are stable to decay intomassive scalar φ and χ -bosons.Despite the fact that the soliton system is electricallyneutral, it nevertheless possesses a nonzero electric fieldin its interior. Note that the electric fields of the sym-metric and nonsymmetric soliton systems are essentiallydifferent. The electric field of the nonsymmetric solitonsystem is unidirectional in its interior, like the electricfield of a plane capacitor. It can accelerate light particlesup to relativistic velocities and energies. In contrast, theelectric field of the symmetric soliton system correspondsto the electromagnetic potential of a potential well. Insuch an electric field, the existence of bound bosonic andfermionic states is possible.It is known [1, 12] that the field configuration of a non-topological soliton composed only of scalar fields can bedescribed in terms of a mechanical analogy. For the one-dimensional case, it corresponds to the motion of a par-ticle with the unit mass in the time x in the conservativeforce field of a certain potential. The dimension of spacein which the particle moves is equal to the number ofscalar fields constituting the nontopological soliton. Us-ing this analogy, one can easily explain the behavior ofthe pure scalar nontopological soliton both in the thin-wall and in the thick-wall regimes. Moreover, one caneasily determine whether a soliton solution can exists forany values of the model’s parameters. At the same time,system of differential equations (28) – (30) describing thesoliton system of the present paper has no interpretationin terms of any mechanical analogy. For this reason, theexistence of the soliton system should be established forany given set of the model’s parameters by means of nu-merical methods.Finally, let us stress the specific character of the (1+1)-dimensional electromagnetic field. Its characteristic fea-ture is the absence of nondiagonal terms of the electro-magnetic stress-energy tensor. This is because the mag-netic field does not exist in (1 + 1)-dimensions, so thePoynting vector vanishes there. Therefore, the (1 + 1)-dimensional electromagnetic field can not transfer anyenergy or momentum. Instead, the scalar fields’ kineticenergy can transform to the one-dimensional electricfield’s energy, which, in turn, can transform back to thescalar fields’ energy. Note also that in (1+1)-dimensions,the potential energy of two oppositely charged particlesis proportional to the distance between them, so the elec-tromagnetic interaction is confining there. Thus, we canconclude that the (1 + 1)-dimensional electromagnetic in-teraction is similar to an elastic string. The only differ-ence is that there is no energy and momentum transferin the one-dimensional electric field, whereas in the elas-tic string waves can transfer energy and momentum. Thebehaviour of the (1+1)-dimensional electromagnetic fieldis completely determined by Gauss’s law, which is not adynamic field equation but is the condition imposed on aninitial field configuration. Indeed, in the adopted gauge A x = 0, Gauss’s law does not contain time derivatives of the electromagnetic potential A . In this connection, itcan be said that the (1 + 1)-dimensional electromagneticfield is not a dynamic one. ACKNOWLEDGMENTS
The research is carried out at Tomsk Polytechnic Uni-versity within the framework of Tomsk Polytechnic Uni-versity Competitiveness Enhancement Program grant.
Appendix A: The one-dimensional non-gaugedQ-ball
Here we collect formulae concerning the one-dimensional non-gauged Q-ball in the model of a self-interacting complex scalar field with the six-order self-interaction potential V ( | φ | ) = m | φ | − g | φ | / h | φ | /
3. Note that an analytical Q-ball solution existsonly in the (1 + 1)-dimensional case [1], where it can bewritten as φ ( t, x ) = 2 √ g p m − ω × (cid:18) − m − ω m − ω (cid:19) × cosh (cid:16) p m − ω ( x − x ) (cid:17)! − × exp ( − iω ( t − t )) . (A1)In Eq. (A1), the squared phase frequency ω ∈ (cid:0) ω , m (cid:1) , where ω = m (cid:18) − g m h (cid:19) . (A2)The Noether charge and the energy of the one-dimensional Q-ball can be expressed in a rather compactform: Q = 4 ω r h arctanh (cid:20) m − ω m − ω (cid:21) − (cid:20) m − ω m − ω − (cid:21) ! , (A3)and E = ωQ − r h (cid:0) ω − ω (cid:1) × arctanh " p m − ω − p ω − ω p m − ω + p ω − ω + r h q ( m − ω ) ( m − ω ) . (A4)1Let us present the expressions of the Noether charge Q and the energy E in two extreme regimes. In the thick-wall regime, the squared phase frequency tends to itsmaxmum value: ω → m . Using Eqs. (A3) and (A4),we obtain the expressions of the soliton’s Noether chargeand energy in the thick-wall regime: Q = sgn ( ω ) 2 √ m / p h ( m − ω ) √ δ × (cid:18) − m − ω m ( m − ω ) δ + O (cid:0) δ (cid:1)(cid:19) , (A5)and E = 2 √ m / p h ( m − ω ) √ δ (A6) × (cid:18) − m − ω m ( m − ω ) δ + O (cid:0) δ (cid:1)(cid:19) , where the variable δ = m − | ω | . Furthermore, Eq. (A1)leads to the soliton’s width at half-height in the thick-wall regime: L = cosh − (7) √ √ mδ + 14 √ × √ mm − ω + cosh − (7) m / ! √ δ + O (cid:16) δ (cid:17) . Using Eqs. (A5) and (A6), we obtain the dependence of E on Q in the thick-wall regime: E = m | Q | − h m − ω m | Q | + O (cid:16) | Q | (cid:17) . (A8)From Eqs. (A5) – (A7), it follows that in the thick-wallregime, the soliton’s Noether charge and energy vanish as √ δ , whereas the soliton’s effective size diverges as 1 / √ δ . In the thin-wall regime, the squared phase frequencytends to its minimum value: ω → ω . In this regime,the Noether charge, the energy, and the width at half-height of the one-dimensional Q-ball behave as follows: Q = sgn ( ω ) r h ω tn " ln (cid:0) m − ω (cid:1) ω tn ¯ δ ! − s ω tn ¯ δm − ω + O (cid:0) ¯ δ (cid:1) , (A9) E = r h ω " ln (cid:0) m − ω (cid:1) ω tn ¯ δ ! (A10)+ m ω − − s ω tn ¯ δm − ω + O (cid:0) ¯ δ (cid:1) , and L = 2 − (cid:0) m − ω (cid:1) − / ln (cid:0) m − ω (cid:1) ω tn ¯ δ ! +4 √ p ω tn ¯ δm − ω + O (cid:0) ¯ δ (cid:1) , (A11)where the variable ¯ δ = | ω | − ω tn . From Eqs. (A9) and(A10), we obtain the dependence of E on Q in the thin-wall regime: E = ω tn | Q | + r h (cid:0) m − ω (cid:1) + O exp − r h | Q | ω tn !! . (A12)From Eqs. (A9), (A10), and (A11), it follows that theNoether charge, the energy, and the effective size ofthe one-dimensional Q-ball logarithmically diverge in thethin-wall regime. [1] T. D. Lee and Y. Pang, Phys. Rep. , 251 (1992).[2] E. Radu and M. Volkov, Phys. Rep. , 101 (2008).[3] G. H. Derrick, J. Math. Phys. , 1252 (1964).[4] G. Rosen, J. Math. Phys. (N.Y.) , 996 (1968).[5] S. Coleman, Nucl. Phys. B , 263 (1985).[6] A. Safian, S. Coleman, and M. Axenides, Nucl. Phys. B , 498 (1988).[7] A. Safian, Nucl. Phys. B , 403 (1988).[8] A. Kusenko, Phys. Lett. B , 108 (1997).[9] A. Kusenko, M. Shaposhnikov, and P. Tinyakov, Pis’maZh. Exp. Teor. Fiz. , 229 (1998), [JETP Lett. , 247(1998)].[10] A. Kusenko and M. Shaposhnikov, Phys. Lett. B , 46(1998).[11] K. Enquist and A. Mazumdar, Phys. Rep. , 99 (2003).[12] R. Friedberg, T. D. Lee, and A. Sirlin, Phys. Rev. D , 2739 (1976).[13] A. Yu. Loginov, Phys. Rev. D , 105028 (2015).[14] K. Lee, J. A. Stein-Schabes, R. Watkins, and L. M.Widrow, Phys. Rev. D , 1665 (1989).[15] C. H. Lee and S. U.Yoon, Mod. Phys. Lett. A , 1479(1991).[16] K. N. Anagnostopoulos, M. Axenides, E. G. Floratos,and N. Tetradis, Phys. Rev. D , 125006 (2001).[17] T. S. Levi and M. Gleiser, Phys. Rev. D , 087701(2002).[18] H. Arodz and J. Lis, Phys. Rev. D , 045002 (2009).[19] I. E. Gulamov, E. Y. Nugaev, and M. N. Smolyakov,Phys. Rev. D , 085006 (2014).[20] R. Friedberg, T. D. Lee, and A. Sirlin, Nucl. Phys. B , 1 (1976).[21] R. Friedberg, T. D. Lee, and A. Sirlin, Nucl. Phys. B , 32 (1976).[22] A. Yu. Loginov, Phys. Lett. B , 340 (2018).[23] A. Yu. Loginov and V. V. Gauzshtein, Phys. Lett. B ,112 (2018). [24] V. Rubakov, Classical Theory of Gauge Fields (PrincetonUniversity Press, Princeton, 2002).[25]