Charged spinning fermionic configurations and a mass gap

Let us now write down expressions for some physically interesting parameters of the systems under consideration. The total dimensionless mass of the system can be found in the form

where the dimensionless $(^{t}_{t})$-component of the energy-momentum tensor (4) is

The total dimensionless angular momentum

where the dimensionless $(^{t}_{\varphi})$-component of the energy-momentum tensor (4) is

The occurrence of a nonzero angular momentum is due to the presence in the system of (i) a single fermion possessing an intrinsic angular momentum; and (ii) the crossed electric and magnetic fields. For this reason, in analogy to quantum particles possessing the quantum-mechanical spin, such configurations can be treated as spinning ones.

The total dimensionless Noether charge

where the dimensionless temporal component of the current density $\tilde{j}^{t}=U_{1}$. Note here that the normalization condition $Q_{\text{tot}}=1$ corresponds to one-particle solutions; in this case the coupling constant $Q$ will correspond to an electric charge of the system, and below we will be interested mostly in such configurations.

Magnetic moment of the system under investigation can be calculated in a standard way by considering the electric current flowing perpendicular to a meridional plane (see, e.g., the textbook Blokh ). As a result, one can obtain the following expression for the dimensionless magnetic dipole moment:

Finally, the dimensionless gyromagnetic ratio $g$, expressed in units of $q_{e}/\left(2M_{\text{tot}}\right)$ [where $q_{e}$ is the electric charge, see Eq. (22) below], is defined from the relation

where $\tilde{q}_{e}\equiv m\sqrt{\lambda}\,q_{e}$ is the dimensionless electric charge.

We will seek globally regular finite-energy nodeless solutions of the set of six partial differential equations (8)-(13). To do this, it is necessary to impose appropriate boundary conditions for the spinor and Maxwell fields. The behavior of solutions of Eqs. (8)-(13) in the vicinity of the boundaries of the domain of integration implies the following boundary conditions:

For numerical computations, it is convenient to introduce the compactified radial coordinate

in order to map the infinite interval $[0,\infty)$ to the finite region $[0,1]$. The results of numerical computations for axisymmetric systems presented below have been obtained using the Intel MKL PARDISO sparse direct solver and the CESDSOL library, and also verified for some particular cases using the package FIDISOL fidisol . These packages provide an iterative procedure for obtaining an exact solution starting from some approximate solution (an initial guess). As the initial guess, it is possible to use the solutions describing configurations in the absence of electric and magnetic fields Soler:1970xp . The equations (8)-(13) have been solved on a grid of $200\times 100$ points which covers the integration region $0\leq\bar{x}\leq 1$ [given by the compactified radial coordinate (21)] and $0\leq\theta\leq\pi$.

In contrast to the case of a linear spinor field, in the nonlinear case, there is a family of solutions, depending continuously on two parameters – the frequency $\tilde{\Omega}$ and the coupling constant $\tilde{Q}$, whose values completely determine all physical characteristics of the configurations under consideration. To illustrate this, Fig. 1 shows the spectrum of the total mass (14) of the systems under investigation as a function of $\tilde{\Omega}$ for some fixed values of $\tilde{Q}$. It is seen from these graphs that the behavior of the dependence $\tilde{M}_{\text{tot}}(\tilde{\Omega})$ is largely determined by the value of the coupling constant $\tilde{Q}$. Namely, the numerical calculations indicate that:

• (i)

  When $\tilde{Q}=0$, the total mass diverges as $\tilde{\Omega}\to 1$. In turn, for small $\tilde{\Omega}$, a rapid increase of $\tilde{M}_{\text{tot}}$ also occurs, and one may expect that in the limit $\tilde{\Omega}\to 0$ the total mass will tend to infinity as well. But in this limit a calculation of the mass using the integral (14) is a difficult technical problem, and we cannot verify this assumption by direct calculation.

• (ii)

  When $\tilde{Q}\neq 0$, the total mass in the limit $\tilde{\Omega}\to 1$ is already finite. In turn, there is some nonzero value $\tilde{\Omega}<1$ for which one can still perform numerical calculations. In doing so, one can observe that the mass demonstrates a rapid increase ($|\partial\tilde{M}_{\text{tot}}/\partial\tilde{\Omega}|\gg 1$); this can be regarded as an indication that there is some critical value $\tilde{\Omega}_{\text{crit}}$ for which the mass will eventually tend to infinity.

• (iii)

  A distinctive feature of the configurations from (i) and (ii) is the presence of a minimum of the mass for all values of $\tilde{Q}$ lying in the interval $-0.27\lesssim\tilde{Q}\leq 0$. This minimum corresponds to the presence in the system of a mass gap where $\partial\tilde{M}_{\text{tot}}/\partial\tilde{\Omega}=0$. In turn, for $\tilde{Q}\lesssim-0.27$, such a mass gap is already absent: the total mass demonstrates a gradual decrease as $\tilde{\Omega}$ increases, and eventually $\tilde{M}_{\text{tot}}$ reaches some minimum value as $\tilde{\Omega}\to 1$.

• (iv)

  The dependence of the total mass on the coupling constant $\tilde{Q}$ for different values of $\tilde{\Omega}$ corresponding to the location of the mass gap is shown in the inset of Fig. 1. It is seen from this inset that for the system with $\tilde{Q}=0$ (and correspondingly without the magnetic field), $\tilde{\Omega}\approx 0.936$ (cf. Ref. Ranada:1973hna ). On the other hand, there exists a maximum possible value of the coupling constant $|\tilde{Q}|\approx 0.27$ for which the frequency $\tilde{\Omega}\to 1$. For larger (modulus) values of $\tilde{Q}$ the curve $\tilde{M}_{\text{tot}}(\tilde{\Omega})$ has no minimum already, that is, the derivative $\partial\tilde{M}_{\text{tot}}/\partial\tilde{\Omega}$ is nowhere equal to zero, and correspondingly the mass gap is absent.

• (v)

  There is some critical value of the coupling constant $\tilde{Q}_{\text{crit}}$ approaching that the interval $\tilde{\Omega}_{\text{crit}}\leq\tilde{\Omega}\leq 1$ (where the solutions do exist) becomes narrower, and eventually $\tilde{\Omega}_{\text{crit}}\to 1$ and the complete set of solutions with different $\tilde{\Omega}$ degenerates to the only solution with $\tilde{\Omega}=1$. Numerical calculations show that $\tilde{Q}_{\text{crit}}\approx-0.3119$, and in this limit the total mass $\tilde{M}_{\text{tot}}\approx 97.13$.

• (vi)

  For all these spinning systems, a straightforward computation shows that the total angular momentum $J_{\text{tot}}$ from Eq. (16) and the total Noether charge $Q_{\text{tot}}$ from Eq. (18) are related by $J_{\text{tot}}=\frac{1}{2}Q_{\text{tot}}$, although the angular momentum density and the Noether charge density are not proportional.

• (vii)

  From the form of Eqs. (8)-(13), it is evident that the solutions under consideration are invariant with respect to a change in the sign $\tilde{Q}\to-\tilde{Q},\tilde{\phi}\to-\tilde{\phi},\tilde{\sigma}\to-\tilde{\sigma}$, that is, they may describe systems with the coupling constant opposite in sign and the same physical characteristics.

Note here that, in order to apply the results obtained above for a description of one particle, it is necessary to normalize the solutions so that the Noether charge $Q_{\text{tot}}$ from Eq. (18) would be equal to 1. In this case the coupling constant $Q$ will correspond to an electric charge, i.e., $Q=q_{e}$. This one-particle condition can be fulfilled by a suitable choice of the free system parameters $\lambda$ and $m$. However, in doing so, one should bear in mind that in this case there will be its own particular set of the parameters $\lambda$ and $m$ for every point in the $\left[\tilde{M}_{\text{tot}}-\tilde{\Omega}\right]$-plane, i.e., different points of the plane will correspond to different models.

We conclude this subsection with the expression for the effective radial pressure $p_{r}\equiv-T_{r}^{r}$. Using the energy-momentum tensor (4) and the Dirac equations (8)-(11), it can be shown that the radial pressure contains the terms

This implies the following physical meaning of the nonlinearity parameter: the case of $\lambda>0$ corresponds to the attraction and the case of $\lambda<0$ to the repulsion. Correspondingly, for the configurations considered above, the attraction of the spinor field related to the choice of positive values of the nonlinearity parameter provides a counter-balance to the effective repulsion due to the presence of the electric charge.

For completeness of analysis of the numerical solutions obtained above, let us write down analytical expressions for asymptotic solutions. The Maxwell equations (12) and (13) have the following asymptotic ($x\to\infty$) behavior of the electric and magnetic fields:

Using these expressions, the numerical values of the electric charge $\tilde{q}_{e}$ and of the magnetic moment $\tilde{\mu}_{m}$ can be found in the form

Note that the value of $\tilde{\mu}_{m}$ calculated using the above formula coincides with that of obtained using Eq. (19).

In turn, the asymptotic behavior of the spinor fields follows from the Dirac equations  (8)-(11),

The form of the functions $f(x)$ and $g(x)$ appearing here depends on the value of $\tilde{\Omega}$. Namely, for $0<\tilde{\Omega}<1$, we have

where $f_{\infty}$ is an integration constant. In the case of $\tilde{\Omega}=1$, we have

Notice here that regular solutions with $\tilde{\Omega}=1$ are only possible in the presence of the charge.

The freedom in the choice of values of the parameters $\lambda$ and $m$ enables us to model various objects. In doing so, a choice of a specific value of $\tilde{\Omega}$ can be made on the basis of a physically reasonable assumption that an energetically stable system must possess a minimum energy (or, equivalently, a minimum mass $\tilde{M}_{\text{tot}}$). Consistent with this, consider, for example, the case where the coupling constant $\tilde{Q}$ is understood to be so chosen that at a minimum of the curve $\tilde{M}_{\text{tot}}(\tilde{\Omega})$ the electric charge of the system $q_{e}$ would be equal to the charge of an electron. The corresponding dependencies $\tilde{M}_{\text{tot}}(\tilde{\Omega})$ and $\tilde{q}_{e}(\tilde{\Omega})$ are shown in the left panel of Fig. 2. It is seen that the minimum of the mass curve (the maximum of the charge curve) is located at $\tilde{\Omega}\approx 0.937$, and the total mass and Noether charge are

This yields the dimensional mass $M_{\text{tot}}=1.027\,m\,Q_{\text{tot}}$ (see the middle panel of Fig. 2). For a normalized solution, $Q_{\text{tot}}=1$; correspondingly, there is a mass renormalization of $2.7\%$. Then, in order to make the total mass of the system $M_{\text{tot}}$ equal to the electron mass $m_{e}$, it is necessary to take $m=m_{e}/1.027$. This in turn leads to the corresponding renormalization of the magnetic moment and change in the value of the gyromagnetic ratio $g$, whose graph is shown in the right panel of Fig. 2. For the case under consideration, $g\approx 2.083$. Thus we have a configuration with the mass $M_{\text{tot}}=0.511\,\text{MeV}$ and charge $q_{e}=-0.3028$ equal to the mass and charge of an electron, but with $g>2$.

Also, for a normalized solution, i.e., when $Q_{\text{tot}}=1$, the quantum-mechanical angular momentum, which is determined using the operator of the total angular momentum,

(here $\hat{L}_{z}=-\imath\partial_{\varphi}$ is the operator which projects the orbital angular momentum on the $z$-axis and $\hat{S}_{z}$ is the operator which projects the spin on the $z$-axis), defines the value of the total angular momentum as an eigenvalue $M_{z}$,

For the spinor (5), this eigenvalue is $M_{z}=1/2$, and it coincides with that calculated using the integral formula (16) (cf. the value of $J_{\text{tot}}$ from the middle panel of Fig. 2). It is worth noting that for the solutions that are not normalized to unity this coincidence no longer occurs.

The characteristic size of such a charged configuration supported by the spinor field can be estimated from the asymptotic behavior of the field (24) as

For $\tilde{\Omega}\approx 0.937$, this yields $r_{\text{ch}}\sim 10^{-10}\,\text{cm}$, a value that is comparable in order of magnitude to the electron Compton wavelength.

Note that the spinor functions $\tilde{X},\tilde{Y},\tilde{V},$ and $\tilde{W}$ appearing in the Dirac equations (8)-(11) are neither even nor odd functions with respect to the equatorial plane $\theta=\pi/2$. Nevertheless, the system possesses a $\mathbb{Z}_{2}$ symmetry with respect to this plane; this can be shown by considering the corresponding combinations of the spinor functions (see Appendix A). To demonstrate this fact in a pictorial way, the upper row of Fig. 3 shows the graphs of the functions $P_{+}$ and $P_{-}$ from Eqs. (33) and (34), respectively. Also, this figure shows the corresponding $\mathbb{Z}_{2}$-symmetric distributions of the components of the energy-momentum tensor (4) and current density, as well as the electric and magnetic field strengths defined by the expressions (7). The structure of the magnetic field strength corresponds to an axially symmetric dipole field sourced by the current associated with the spinor field given on the right-hand side of Eq. (3). The radial distribution of the current and the magnitude of the magnetic field are determined by the value of the coupling constant $Q$. In turn, the structure of the electric field strength corresponds to a negative charge with force lines directed toward the center of the configuration.

In conclusion, note that by choosing $m=m_{\mu}/1.027$, where $m_{\mu}$ is the muon mass, we get characteristics typical for a muon/antimuon.