Singularity-Free Model of Electric Charge in Physical Vacuum: Non-Zero Spatial Extent and Mass Generation

Home , Geometrodynamics, Mass generation

Cent. Eur. J. Phys. • 11(3) • 2013 • 325-335 DOI: 10.2478/s11534-012-0159-z

Central European Journal of Physics

Singularity-free model of electric charge in physical vacuum: non-zero spatial extent and mass generation

Research Article

Vladimir Dzhunushaliev12∗, Konstantin G. Zloshchastiev3†

1 Department of Theoretical and Nuclear Physics, Kazakh National University, Almaty, 010008, Kazakhstan 2 Institut für Physik, Universität Oldenburg, Postfach 2503, D-26111, Oldenburg, Germany 3 School of Chemistry and Physics, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa

Received 17 August 2012; accepted 31 October 2012

Abstract: We propose a model of a spinless electrical charge as a self-consistent field configuration of the electromagnetic (EM) field interacting with a physical vacuum effectively described by the logarithmic quantum Bose liquid. We show that, in contrast to the EM field propagating in a trivial vacuum, a regular solution does exist, and both its mass and spatial extent emerge naturally from dynamics. It is demonstrated that the charge and energy density distribution acquire Gaussian-like form. The solution in the logarithmic model is stable and energetically favourable, unlike that obtained in a model with a quartic (Higgs-like) potential. PACS (2008): 11.27.+d, 11.15.Kc, 11.15.Ex, 03.75.Nt Keywords: particle model • electric charge • divergence-free solution • mass generation mechanism • superfluid vacuum © Versita sp. z o.o.

1. Introduction experimental evidence of either the internal structure or spatial extent of, e.g., the electron has been found down − to 10 16 cm. The mere postulate that a certain amount of matter with mass, charge and spin can be located in- Two of the oldest actual problems in fundamental particle side a set of zero spatial measure looks implausible to physics relate to finite self-energy and the possible ex- a physicist’s mind. This assumption, however, might be tendedness of electrically charged elementary particles. one of the reasons why unphysical divergences appear in This research direction is complicated by the fact that no quantum field theory (QFT). This difficulty already arises at the classical level: according to the standard theory ∗ E-mail: [email protected] of electromagnetism, the electrical field of a point charge † E-mail: [email protected]

325 Singularity-free model of electric charge in physical vacuum: non-zero spatial extent and mass generation

in completely empty space (trivial vacuum) is described Intuitively, from a fundamental theory one would expect by the inverse square law, therefore the energy density of that spatial extent is not ab initio built-in, but naturally the electrical field integrated over the whole space turns emerges from dynamics. In this paper we propose a model out to be infinite. As a result, the total mass-energy of of a charged particle whose spatial extent, observable the point charge, together with its field, becomes infinite, charge and mass emerge as a result of the interaction of therefore, such a system would be impossible to move. At the EM field with the physical vacuum. For simplicity we the quantum level, this problem manifests itself in ultra- neglect internal degrees of freedom, such as spin, isospin, violet divergences appearing in loop diagrams. In some etc., so that we can assume spherical symmetry where pos- theories, these divergences can be removed by means of sible. The resulting solution describes a charged object regularization and renormalization procedures. This can which does not have a boundary in a classical sense; its be very useful for doing specific computations, but does stability is supported not by surface tension but by non- not shed much light upon the essence of the problem [1–5]. linear quantum effects in the bulk. This makes our model Theoretical attempts towards better understanding should more realistic from the quantum-mechanical point of view, not, however, be abandoned. since the actual observability of a definite boundary with smooth surface would be as contradictory to the quantum Historically the first effort to address this problem was uncertainty principle as the notion of a smooth trajectory probably a model which described the electron as a ball or worldline in the quantum realm. This can be shown by with spatially distributed electrical charge. That model performing a simple Gedankenexperiment: making an ex- conflicted with relativity, however, because the ball was tended object with a definite boundary propagate through assumed to be absolutely rigid. The description of spin space and measuring the velocity and position uncertain- was also not clear. Similar difficulties were found in mod- ties on its surface. In turn, it means that the surface ten- els proposed by Abraham and Lorentz [6–8], although work sion is a well-defined notion only in the classical limit, in that direction continues [9, 10]. Dirac’s shell model of but for more fundamental purposes it must be used with the extended electron, together with its subsequent modi- utmost care. fications and variations [11–14], provides another notable The structure of the paper is as follows. The phenomeno- research direction. Yet another model of an extended elec- logical approach to physical vacuum is described in the trical charge was the Einstein’s wormhole approach. In next section, the main equations of the model can be found his model of an electron the electrical flux lines enter one in Section3, the regular solution and its properties are side of the wormhole and exit from another, resulting in analyzed, both analytically and numerically, in Section4. the front side looking like a negative charge and the rear A comparison between our solution and its Higgs-type like a positive charge. The wormhole models were crit- (quartic) counterpart is done in Section 5, and conclu- icized by Wheeler for issues of stability, non-quantized sions are drawn in Section 6. charge, wrong mass-charge ratio and spin [15]. Numerous attempts towards finding regular particle-like solutions were made in conventional and nonlinear electrodynam- 2. Physical vacuum ics [16], both with and without engaging general relativity [17–28]. Another interesting approach is wave-corpuscle mechanics (for a review see [29]). The general mathemat- As mentioned above, the Coulomb divergence problem es- ical formalism used therein formally resembles the one sentially means that one cannot find regular particle-like used in this paper. The important difference, however, solutions of the Maxwell field in empty space, not even in is about underlying physics: the origin and explicit form general relativity [44]. From a quantum physicist’s point of their nonlinear self-interacting wave term G(|ψ|2) are of view, however, this problem is not as severe as it looks not specified on physical grounds and a fully satisfactory to a non-quantum theorist, because the notion of abso- particle-like solution is not given. lutely empty space (or “mathematical vacuum”) cannot be realized in nature anyway. This is because the existence A popular approach to the classical electron model was of such space seriously contradicts quantum-mechanical made using the Einstein-Dirac [18] and Kerr-Newman laws. According to the latter, the genuine (physical) vac- (KN) solutions [30–42] (an extensive bibliography can be uum must be a non-trivial quantum medium which acts as found in [43]). While the original KN solution does have a non-removable background and affects particles propa- the correct gyromagnetic ratio, it also contains a naked gating through [1, 45]. singularity and thus requires an additional mechanism to At this time, no commonly accepted theory of physical circumvent the regularization problem; the story is far from vacuum exists. The amount of experimental and observa- being complete yet. tional data is still too far from conclusively identifying a

326 Vladimir Dzhunushaliev, Konstantin G. Zloshchastiev

single model. One of the candidate theories lies within vacuum the parameter β˘ is related to the quantum (non- the framework of the superfluid vacuum approach [46–51]. thermal) temperature which is conjugated to the Everett This theory is based on the idea [52–54] that a phys- information entropy [59], whereas a˘ can be related to the ical vacuum can be viewed as some sort of background characteristic inhomogeneity scale of the superfluid [60]. superfluid condensate described by the logarithmic wave The potential (2) is regular in the origin - while the loga- equation (the latter was studied previously on grounds rithm itself diverges there, the factor |Ψ|2 recovers reg- | | a−3/2; of the dilatation covariance [55] or separability [56–58]). ularity. There is a local maximum at Ψ max = ˘ It was shown that small fluctuations of the logarithmic i.e., it always has the (upside-down) Mexican-hat shape condensate obey the Lorentz symmetry and can be inter- if plotted as a function of Ψ, (see Fig. 1). In what follows preted not only as the relativistic particle-like states but we call this potential logarithmic - due to the property also as the gravitational ones [54] depending on a type of dV /d|Ψ|2 ∝ ln (a˘3|Ψ|2) which yields the logarithmic term mode. In this approach, therefore, the Lorentz symmetry in the corresponding field equation. is not an exact symmetry of nature, but rather pertains to 0 small fluctuations of the physical vacuum, and thus gets deformed at high energies and/or momenta. 0.25 As long as the superfluid vacuum approach must be fully 0.5 consistent and applicable to reality, one would expect that the behaviour of the conventional Maxwell field becomes 0.75 regular in the presence of particle-like solutions, when 1 the empty space is replaced by the logarithmic vacuum 1.25 condensate. This behaviour is going to be the main sub- ject of the current study. One would also expect that the 1.5 spatial extent mentioned above must appear naturally in the approach. This has already been shown at the non- 2 1 0 1 2 relativistic level in [59], so in the current study we will investigate the relativistic case. Figure 1. The field-theoretical potential (2) (in units of β˘a˘3) ver- − / The effective low-energy Lagrangian for the Maxwell field sus Re(Ψ) (in units of a˘ 3 2). Vertical lines conditionally interacting with small fluctuations of the physical vacuum represent inequality (3). In an approximation when the symmetry-breaking energy scale is much less than the was proposed in [54], along with a mass generation mech- vacuum one (& 10 TeV) these walls can be assumed infi- anism which was analogous to the Higgs one. In this nite. paper we propose a different mass generation mechanism which uses the same Lagrangian, except that the scalar We emphasize that, according to this approach, the La- potential is assumed to be “upended”. As will be shown grangian (1) is an approximate one, thus it is not valid below, the latter solves the issue of a wrong sign in the for arbitrarily large (or short) scales of energy (or length) quadratic (mass) term of the potential at energies above whichmeans that the Ψ field cannot take arbitrarily large the symmetry-breaking scale - when symmetry is unbro- values: ken and false vacuum is stable. In the case of three spatial R |Ψ| 6 |Ψc| < ∞; (3) dimensions the action is proportional to d4x L where, | c| | | c ~ where Ψ =E lim→E Ψ is some limit value corresponding adopting the natural units = = 0 = 1 and metric 0 − −− E signature (+ ), we assume to the cutoff energy scale 0 which is also the characteristic energy scale of the vacuum. In the effective theory the a | | 1 µν ˘ 2 appearance of upper bound for Ψ will be shown in Sec- L = − Fµν F + |DµΨ| − V (|Ψ|2); (1) 4 2 tion 4.2 below, whereas in a full theory it could result from, for instance, the normalization condition for Ψ, similarly where Dµ = ∂µ+igAµ, Fµν = ∂µAν −∂ν Aµ, and the vacuum- to the one in the theory of Bose-Einstein condensation. induced field potential is defined as (up to an additive The constraint (3) also means that the potential (2) does constant) not have to be bounded from below, as in a standard relativistic QFT. Alternatively, one can take the potential (2) − − with an opposite sign, thus making it bounded from below V (|Ψ|2) = −β˘ 1 |Ψ|2 ln (a˘3|Ψ|2) − 1 + a˘ 3 ; (2) at positive β˘, but treat Ψ as a phantom field. Further, performing the rescaling where a˘ and β˘ are parameters of dimensionality length √ / in adopted units. In the underlying theory of superfluid ψ = a˘Ψ; β = a˘β;˘ a = a˘2 3; (4)

327 Singularity-free model of electric charge in physical vacuum: non-zero spatial extent and mass generation

ν we can rewrite (1) and (2) in a more regular form: where the current j is deﬁned as

ν µ ∗ ∗ µ 1 µν 1 2 j = ig [(D ψ) ψ − ψ (D ψ)] : (10) L = − Fµν F + |Dµψ| − V (ψ); (5) 4 2 V ψ −β−1 |ψ|2 a3|ψ|2 − 1 : We look for a solution in the electrostatic form ( ) = ln ( ) 1 + a3 (6) h i ~ −iEt Aµ = φ(r); 0 ; ψ(t; r) = e ψ(r); (11) Note that by expanding the potential in the vicinity of |ψ|2 ε/a3 ε = , being a non-negative dimensionless num- with E being a real-valued constant. Then the equations ber, one arrives at the following perturbative expression of motion become simply (up to an additive constant): ψ00 2 ψ0 − E − gφ 2 ψ − β−1ψ a3ψ2 ; λ + = ( ) ln( ) (12) eff 4 1 2 2 2 3 3 r V (ψ) ≈ |ψ| + mb|ψ| + O (|ψ| − ε/a ) ; (7) 4! 2 φ00 2 φ0 − g E − gφ ψ2; + r = 2 ( ) (13) λ − a3/εβ where eff = 12 is the effective quartic coupling p r and mb = 2(1 − ln ε)/β. If the radicand of mb is non- where primes indicate derivatives with respect to . Intro- negative then mb can be interpreted as the mass of an ducing the quantities effective scalar particle (before the symmetry breaking). r x √ ; ψ˜ a3/2ψ a3/2 ; φ˜ Indeed, one can always quantize the approximate model = β = = ˘ Ψ = by analogy with a quartic (hence renormalizable) scalar p q −gφ β −gφ aβ; QFT in the vicinity of a non-trivial vacuum represented by = = ˘˘ (14) s the ground-state solution of the original model (6) which r p q β β˘ will be discussed in the following sections. To date, a E˜ = E β = E a˘β;˘ g˜ = g = g ; (15) a3 a˘ number of different quantization approaches have been developed for such cases - see, e.g., works [61–63] and we obtain references therein. We have now expressed the physical parameters of the 00 2 0 2 ψ˜ + ψ˜ = − E˜ + φ˜ ψ˜ − ψ˜ ln(ψ˜ 2); (16) scalar sector, such as mass and coupling, in terms of the x primary parameters of our theory. If the value of ε is 00 2 0 2 2 φ˜ + φ˜ = 2g˜ E˜ + φ˜ ψ˜ ; (17) close to one (or, at least, less than the base of natural x logarithm) then it is indeed important that the potential (6) where primes indicate derivatives with respect to x when has the upside-down Mexican hat shape (cf. a˘ > 0 and applied to tilded quantities. One can see that equations β˘ > 0) otherwise the quadratic term would appear with a depend on only one parameter g˜, whereas E˜ = E˜(g˜) can wrong sign. Also the effective quartic coupling turns out to be treated as an eigenvalue at a given g˜. The full the- be negative in this case which is a remarkable difference ory of physical vacuum would provide the value of E˜ (or, from the standard Higgs potential and thus it can serve equivalently, ψ˜(0), see the analytical solution section be- for experimental testing. In any case, the interpretation low) as a function of the primary parameters (from, e.g., based on (7) is only approximate (for instance, such series some sort of normalization condition, cf. [59]) but in the expansion does not converge to (6) for very small |ψ|), approximate theory E˜ stays a free parameter which can therefore, in what follows we will be working with the be fixed only from external considerations. exact expression for V . Further, for a given solution, the energy density is defined as 3. Field equations 1 D ψ∗D0ψ 1 D ψ∗Diψ V ψ 1 |E|~ 2; = 0 + i + ( ) + (18) 2 2 2 The field equations corresponding to the Lagrangian (5) i ; ; E −∇φ are given by where = 1 2 3 and = is the electric field strength. Substituting the ansatz (11) we obtain µν ν ∂µF j ; = (8) 2 βa3 1 E˜ φ˜ ψ˜ 2 1 ψ˜02 − ψ˜ 2 ψ˜ 2 µ ∂ V µ − = + + ln + DµD ψ DµD β 1 a3|ψ|2 ψ 2 2 + ∂ψ∗ = + ln ( ) = 1 02 ; + φ˜ : (19) = 0 (9) 2g˜2

328 Vladimir Dzhunushaliev, Konstantin G. Zloshchastiev

∞ R r → ∞ r2E → q The total energy can be calculated as W = 4π r2(r)dr and in the limit we have hence 0 so we obtain √ ∞ π β Z W 4 W˜ g ; q g E − gφ ψ2r2dr: = a3 (˜) (20) = 2 ( ) (28) 0 where we denoted the dimensionless total energy Thus, we arrive at the following relation between the bare ∞  Z 2 and observable charges 1 02 W˜ (g˜) =  E˜ + φ˜ ψ˜ 2 + ψ˜ + 2 r 0 gβ β  q 2 I g g I g ; 0 !2 ˜ ˜ ˜ φ˜ = a3 ( ) = 2 a3 ( ) (29) −2ψ˜ 2 ln ψ˜ 2 +  x2dx: (21) g˜ where we denoted

∞ For an observer in the reference frame associated with the Z W /c2 2 2 center of mass of a localized solution the quantity I(g˜) = x E˜ + φ˜ ψ˜ dx: (30) is equivalent to the rest mast of a corresponding particle. 0

4. Particle-like solution and its Expression (26) shows us that at large distance we recover the Coulomb potential while the ﬁeld ψ˜ decreases expo- properties nentially. Thus, the ﬁeld ψ˜ is in fact unobservable, unless very short length scales are probed. From the asymptotics 4.1. Asymptotic behaviour of the solution one can infer that the charge radius of the solution is determined by the parameter β: To search for the regular solution, the functions ψ˜(x) and q φ˜(x) should have the following behaviour near the origin: p size ∼ β ∼ a˘β;˘ (31)

x2 ψ x ψ ψ O x4 ; ˜( ) = ˜0 + ˜2 + ( ) (22) 2 which essentially means that the combination of parame- x2 ters β = a˘β˘ must have an extremely small value for the φ x φ φ O x4 : ˜( ) = ˜0 + ˜2 + ( ) (23) 2 known elementary particles. For instance, if one takes the values of the classical radius e2/m as conservative esti- When substituting this into (16) and (17) we obtain the mates, then for the electron and muon one would obtain β < −26 2 β < −32 2; solution constraints of (e) 10 cm and (µ) 10 cm although it is not entirely clear whether the classical radius ψ˜ 2 should be analogous to the “smearing” size which is our ψ − 0 E φ ψ2 ; ˜2 = ˜ + ˜0 + ln ˜0 (24) deﬁnition of size here. 3 φ 2 g2 E φ ψ2: ˜2 = ˜ ˜ + ˜0 ˜0 (25) 4.2. Approximate analytical solution 3 While the exact expression for a full analytical solution is Further, the asymptotic behaviour at x → ∞ is given by unknown, it is possible to solve the system (16) and (17) using the approximation of weak EM coupling 1 −E˜ 2−x2 q˜ ψ˜ → 2 (3 ) O g2 ; φ˜ → − ; e 1 + (˜ ) x (26) g˜2 1; (32) where q˜ is some constant to be determined. It is instruc- tive to relate the bare charge g to the observable one which is equivalent to g2 a3/β or g2 a/˘ β˘. The q = q/g˜ . By integrating (13) we obtain observational constraints suggest that this approximation might have a good chance to be valid for the known el- r a Z ementary particles - unless ˘ turns out to be very small. r2E = 2g (E − gφ) ψ2r2dr; (27) On the other hand, as long as our approach is an eﬀective 0 one it has certain applicability conditions - and one of

329 Singularity-free model of electric charge in physical vacuum: non-zero spatial extent and mass generation

them is that the vacuum eﬀects predominate the electro- which can also be written in terms of the observable magnetic ones. Therefore, large values of g˜ might push charge q and rest mass W : our approach outside its applicability range and thus the h √ i corresponding approximation is not very interesting from W ≈ W β−1a3q2 E˜ 2−3 E˜ 2−13/12 ; (0) 1 + 4 2 e E˜ 2−1/2 (41) the physical point of view. Thus, imposing the boundary conditions where √ 3/2 π β −E2 0 W 3 E2 − 1 3 ˜ ; φ˜ < ∞; φ˜ ∞ ; ψ˜ ∞ < ∞; ψ˜ ; (0) = ˜ e (42) (0) (+ ) = 0 (+ ) (0) = 0 (33) 2 a3 2

one obtains a solution which is regular for 0 6 r 6 +∞ so one can see that the obtained formula does not contain (see appendix for the details of derivation): any divergences. It is also apparent that mass W does not vanish when √ 1 −E˜ 2 charge is set to zero, which indicates that the theory is φ˜ = − πg˜2E˜ e3 erf(x) + O(g˜4); (34) x also capable of incorporating non-charged particles into 2 1 −E˜ 2−x2 1 −E˜ 2−x2 the scheme, by taking the corresponding limit. In fact, the ψ˜ = e 2 (3 ) 1 + g˜2E˜ 2e3 4 mass formula (41) implies that for an electrically charged √ W π x2 particle with mass there can exist not only an antipar- 1 + 2x2 + 1 e erf(x) + O(g˜4); (35) 2x ticle of the same mass but also a neutral particle of related W W/W mass (0). It is interesting that the ratio (0) grows |E| where dimensionless energy E˜ can be also expressed via exponentially with growing ˜ , which results in two pos- the boundary value sible scenarios: (i) the mass of a neutral partner is very small (yet non-zero) as compared to the mass of a charged q |E˜| |ψ˜ | E − ψ2 − 1 g2ψ2 O g4 ; one: this happens if 1 (or, equivalently, 0 1); ˜ = 3 ln (˜0 ) 1 ˜ ˜0 + (˜ ) (36) 2 (ii) if |E˜| is of order one or less then both masses would be of the same order of magnitude. The possible phenomeno- with the square root being defined up to a sign. Of course, logical implications of this mechanism are discussed in the ψ˜ this formula is valid only if the magnitude of is bounded conclusion. from above: |ψ˜| |ψ˜ | 3/2; 6 0 6 e (37) 4.3. Numerical solution and stability which a posteriori affirms the condition of applicability (3), although this upper bound does not necessarily saturate While we have managed to find the approximate regular / the critical value there: |Ψc| > (e/a˘)3 2. solution analytically, it is important to check that a reg- g Further, one can check that the electric part indeed has ular solution exists for non-small ˜’s and that terms with g the Coulomb behaviour at large r, and then the effective higher-order powers of ˜ will not introduce any spatial charge can be computed as singularities. For this purpose we solve equations (16) and (17) numerically. For the computations we choose √ g 1 2 3−E˜ 2 4 ˜ = 1 and the following boundary conditions q˜ = πg˜ E˜ e + O(g˜ ); (38) 2 0 ψ˜(0) = 0:1; ψ˜ (0) = 0; therefore, the observable charge, 0 φ˜(0) = −0:1; φ˜ (0) = 0; (43) p 2 q q/g ≈ 1 πβE 3−E˜ g ≈ = ˜ / ˜ e ˜ 2a3 2 E˜ √ and is treated as an eigenvalue. It should be noted ≈ 1 πβ˘E˜ 3−E˜ 2 g; that φ˜(x) must be always taken as non-positive on the a e (39) 2˘ positive semi-axis of x, due to the asymptotic require- depends on the whole combination of parameters de- ments (26). The numerical solution is presented in Fig 2, scribing the interaction of the electromagnetic field with and in Fig. 3 the corresponding profiles of the electric E˜ −dφ/dx˜ x2E˜ a physical vacuum. The dimensionless total energy (21) field = and are given. From these one turns out to be sees that the electric field is regular at the origin and asymptotically displays Coulomb behaviour. The profile √ 3 3−E˜ 2 2 of the dimensionless energy density is shown in Fig. 4. W˜ = π e 2E˜ − 1 + 16 The direct stability analysis of the solution is complicated 2 g˜ −E˜ 2 by the fact that the perturbed electric field becomes time- + √ E˜ 2 12E˜ 2 − 13 e3 + O(g˜4); (40) 3 2 dependent, which leads to the appearance of a magnetic

330 Vladimir Dzhunushaliev, Konstantin G. Zloshchastiev

0.10 0.08

0.05 0.06

0.00 0.04

- 0.05 0.02

2 4 6 8 2 4 6 8 10

Figure 2. Profiles of ψ˜ x (top curve) and the electrostatic poten- ( ) Figure 4. Profile of dimensionless energy density βa3(x), using the tial φ x (bottom curve), computed for eigenvalue E ˜( ) ˜ = same E˜ eigenvalue as in previous figures. 2:8436935588.

2 4 6 8 energy of the whole conﬁguration is smaller than the sum

-0.005 of the energies of the separate electric and scalar ﬁelds. Evaluating the binding energy on approximate solu-

-0.010 tion (34), (35), we ﬁnd that it is negative-deﬁnite

√ 2 −5/2 3−E˜ 2 4 -0.015 ∆W˜ = −2 π g˜E˜ e + O(g˜ ); (45)

- 0.020 which means that the creation of the regular electric charge in the logarithmic model is energetically - 0.025 favourable. Another way to study the stability of the solution is to write it as a solution of the Schrödinger equation for a Figure 3. Profiles of the electric field E˜(x) (top) and x2E˜(x) (bottom curve). fictitious particle

− V x ε ; ∆Ψ + eff( )Ψ = Ψ (46) field such that this system cannot be regarded as spher- d2/dx2 ε E2 ically symmetric anymore. It is, however, still possible to where ∆ = , = ˜ and the effective potential is use the energy-based arguments as well as to investigate derived as the behaviour of an effective Schroedinger equation po- V x − Eφ x − φ2 x − ψ2 x ; tential. Let us consider the dimensionless total energy W˜ eff( ) = 2˜ ˜( ) ˜ ( ) ln [˜ ( )] (47) given by (21), as well as the energy of the field ψ˜ alone, W W | ˜ ψ = ˜ φ→0, and the energy of the electric field alone, where the tilded potentials are given by our regular so- W W | ˜ φ = ˜ ψ→0. Then we can define the dimensionless lution. According to (26), the asymptotic behaviour of the V ∝ x2 binding energy as solution implies that the effective potential eff at large x, (see also Fig. 5), and thus the “particle” is al- ∞ x Z ways localized in a finite region of . With respect to 1 ∆W˜ = W˜ −(W˜ φ +W˜ ψ ) = φ˜ φ˜ + 2E˜ ψ˜ 2x2dx; (44) the solution itself this means that it cannot spread or be 2 0 destroyed when subjected to small perturbations. where all the potentials are assumed to be given by our 5. Logarithmic versus quartic po- regular solution. One considers the following two possi- bilities. If binding energy is positive then it is necessary to tential add a certain amount of energy to create a regular electric charge when coupling to the ψ field. In the opposite case One may wonder whether the singularity-free solution ex- binding energy gets released during the process, and the ists when the scalar sector of our model is controlled not

331 Singularity-free model of electric charge in physical vacuum: non-zero spatial extent and mass generation

3,0 25

2,5

2,0

1,5

10 1,0

0,5

1 2 3 4 5

-4 -2 0 2 4

Figure 5. Effective potential (47) versus x at g˜=1, using the same E˜ eigenvalue as in previous figures. Figure 6. The profiles of the dimensionless energy W˜ (g˜) (H-curve) and the eigenvalues E˜(g˜) ( - curve).

by the logarithmic potential (6) but by a more orthodox one, such as the Higgs-type (quartic) potential: 1.0

2 κ m 0.5 VH (ψ) = − ψ4 + ψ2: (48) 4 2

Corresponding dimensionless equations with the ansatz 10 20 30 40 50 (11) are -0.5

2 ψ˜ 00 2 ψ˜ 0 E˜ φ˜ − λψ˜ 2 ψ;˜ + x = + + 1 (49) -1.0 φ˜ 00 2 φ˜ 0 g2 E˜ φ˜ ψ˜ 2: + x = 2˜ + (50) Figure 7. The proﬁles of ψ˜(x) (top) and φ˜(x) (bottom) for the quartic model. Eigenvalue ψ˜(0) = 1:12345, g˜ = 0:1 and parame- where we introduced following dimensionless quantities ters E˜ = 0:1; λ = 1:0 have been used.

ψ E x mr ψ˜ E˜ = ; = ψ ; = m ; (0) The asymptotic behaviour for the functions ψ and φ at gφ ψ(0)2κ large x is given by φ˜ = − ; λ = : (51) m m2 √ −x E˜ e +1 ψ˜ x → ψ∞ ; The boundary conditions are ( ) x2 (53) q φ˜ x → − ˜ ; 0 0 ( ) x (54) ψ˜ (0) = 0; φ˜(0) = −1:04; φ˜ (0) = 0: (52) where ψ∞ is some constant. This still looks very similar As in the logarithmic model, the regular solution exists to what we had earlier in the logarithmic case, however if only if the potential (48) opens down, i.e., when κ > 0. we study the stability of this solution then differences do The profiles of ψ˜(x) and φ˜(x) are presented in Fig. 7. For arise. At first, if one computes the binding energy simi- technical reasons in this case an eigenvalue is ψ˜(0) not larly to (44) then it turns out to be positive, therefore, the E˜. In Fig. 8 the profile of the electric field E is shown. creation of the regular electric charge by coupling elec- In order to show that the electric field asymptotically has trical field to the quartic scalar one is energetically un- Coulomb behaviour we also present the profile x2E(x) in favourable. Further, if one computes the fictitious-particle Fig. 8. From these figures one can see that the qualitative potential for this solution, cf. (47), behaviour of the potential φ(x) and the electric field E(x) V x − Eφ x − φ2 x − λψ2 x ; are the same as for the logarithmic potential. eff( ) = 2˜ ˜( ) ˜ ( ) ˜ ( ) + 1 (55)

332 Vladimir Dzhunushaliev, Konstantin G. Zloshchastiev

then one finds that it approaches a constant at large x static field yields the regular wave functions (the hydro- (see also Fig. 9). The “particle” is not, therefore, neces- gen atom being an example), but the Dirac/Schrödinger sarily localized in a finite region of x. With respect to equation coupled to Maxwell equations does not lead to the solution itself, this means that the latter can spread a regular stationary solution. The reason is that the or become unstable against small perturbations. Dirac/Schrödinger equation ab initio describes a point- like particle which might be a good approximation for 10 20 30 40 50 long-wavelength measurements, but in higher-energy and -0.02 shorter-length regimes this approximation eventually becomes too crude, since it neglects internal structure and -0.04 non-zero spatial extent. Among other things, this leads to -0.06 the densities of energy and charge becoming infinite at the particle’s position. Here we have shown that by intro- -0.08 ducing an additional player on scene, the physical vac- -0.10 uum condensate, one can obtain a regular solution, thus endowing particles with internal structure and spatial ex- -0.12 tent. The solution turns out to be stable and energetically favourable. Using its features, some observational constraints for the parameters of the theory have been de- Figure 8. Electric field versus x for the quartic model. The top curve rived. We also specified the conditions under which our is 10 E˜(x), the bottom one is x2E˜(x). model can be (approximately) interpreted in terms of a scalar particle and those under which it cannot.

Further, we have established, both numerically and an- 2 4 6 8 10 alytically, that the mass and spatial extent of a charged

-0.2 particle emerge due to interaction of the EM ﬁeld with vacuum. It has been demonstrated that the average charge -0.4 radius becomes non-zero, and the charge density acquires a Gaussian-like form. Looking at the form of the analytical - 0.6 solution from Section 4.2, one can infer that it describes

-0.8 the object without border in a classical sense, therefore, its stability is supported not by surface tension but by -1.0 nonlinear quantum effects in the bulk, similarly to the non- relativistic case [59]. Due to non-singular behaviour of the solution at the origin, the derivation of self-energy turned out to be entirely divergence-free. Figure 9. Effective potential (55) versus x at g˜ = 1 for the quartic model. The derived mass formula (41) suggested that for an electrically charged massive elementary particle there exists not only an antiparticle of the same mass (in the leading- order approximation with respect to the Planck constant, 6. Conclusion at least) but also a neutral particle of related mass. This might explain, at least qualitatively, why an electrically The classical model of a spinless electrical charge is charged elementary particle is very often accompanied by described as a self-consistent configuration of the EM a single neutral particle of a similar kind, but not vice field interacting with fluctuations of a nontrivial phys- versa. Indeed, such “mass pairing” feature has been ob- ical vacuum effectively represented by the logarithmic served (provided one disregards the influence of inter- Bose-Einstein condensate. We have shown that a reg- nal degrees of freedom such as spin, isospin, etc.) not ular solution does exist - as opposed to the case of only for elementary particles such as leptons and weak the EM field propagating in absolutely empty space. bosons, but also for stable composite ones such as nu- In this regard we recall the state of affairs in quan- cleons. (Quarks might not fit this scheme since they are tum mechanics: the Dirac/Schrödinger equation without confined inside hadrons). We presented some arguments any external potential has the de Broglie wave solution; for why the rest mass of a neutral partner can sometimes the Dirac/Schrödinger equation with the external electro- be so much smaller, yet still non-vanishing, than the mass

333 Singularity-free model of electric charge in physical vacuum: non-zero spatial extent and mass generation

of the charged one (leptons), and sometimes they are of By solving them we obtain the same order of magnitude (weak bosons or nucleons). c Finally, we have compared the logarithmic vacuum model φ˜ ξ c − 1 = Λ + 2 x + with one based on a Higgs-type (inverted quartic) poten- √ ξ π tial. It turns out that the corresponding regular solution is 3−(E˜+Λ)2 2 − (E˜ + Λ)e erf(x) + O(ξ ); (A6) unstable and energetically unfavourable, in contrast with 2x (3−(E˜+Λ)2−x2)/2; the logarithmic case. Ψ˜ 0 = e (A7) 1 E 2 (3−(E˜+Λ)2−x2) Ψ˜ 1 = (˜ + Λ) e 4 √ Acknowledgments π x2 1 + (2x2 + 1)e erf(x) + 2x Both this work and visit of V. D. to the University of Wit- −c E c x − 1 2(˜ + Λ) + 3 + watersrand were supported by a grant from the National 2x √ Research Foundation of South Africa. Authors are grate- π 2 c x2 − x − x ; + 4 (2 1)erfi( ) e (A8) ful to Alexander Avdeenkov for supporting their visit to 2x the National Institute of Theoretical Physics (Stellen- bosch node). V. D. acknowledges also the support from where ci and Λ are integration constants whose values the Research Group Linkage Program of the Alexander must be fixed by means of the boundary conditions. Impos- von Humboldt Foundation. K. Z. is grateful to Alessandro c c c c − /ξ ing (33), one obtains: 1 = 3 = 4 = 0 and 2 = Λ . Sergi for supporting his visit to the University of KwaZulu- Then, after making the energy redefinition E˜ +Λ → E˜ (al- Natal (Pietermaritzburg campus). The proofreading of the c ternatively, one can set Λ = 2 = 0 from the beginning), manuscript by P. Stannard is greatly acknowledged. one eventually arrives at expressions (34) and (35).

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