On generalized ModMax model of nonlinear electrodynamics

A new generalized ModMax model of nonlinear electrodynamics with four parameters is proposed. The ModMax model and Born--Infeld-type electrodynamics are particular cases of the present model It is shown that a singularity of the electric field at the center of point-like charged particles is absent. We found corrections to Coulomb's law at $r\rightarrow\infty$ and obtain the total electrostatic and magnetic energies of point-like charges. Free electric and magnetic charges and their densities are obtained.

The duality-invariant conformal electrodynamics was introduced in [1] and it is described by the Lagrangian density where are Lorentz invariants with B, E being the magnetic induction and electric fields correspondingly, F µν = ∂ µ A ν − ∂ ν A µ ,F µν = (1/2)ǫ µναβ F αβ is the dual electromagnetic field, and γ is the dimensionless parameter. This model as well as Maxwell electrodynamics with the Lagrangian density L M = −F possess singularities in the centre of point-like charges. In addition, the electromagnetic energy of charges is infinite. To smooth singularities we propose the generalized ModMax model with the Lagrangian density where L is given by Eq. (1), β and λ have the dimensions of (length) 4 and σ is the dimensionless parameter. At σ = 1, λ = 0 we come to ModMax model, when γ = 0 one has the Born-Infeld-type model [12], [13], at σ = 1/2, γ = 0 we arrive at the generalized Born-Infeld model [14], and at σ = 1/2, γ = 0, λ = β one comes to Born-Infeld model (β = 1/b 2 ) [15]. At σ → ∞ the Lagrangian density (3) becomes Some exponential NED models were considered in [16], [17]. Thus, our model (3) allows us to consider different NED by fixing the parameters introduced. The Born-Infeld model is of interest because at the low energy D-brain dynamics is governed by Born-Infeld-type action [18], [19]. Making use of the Taylor series, at βL ≪ 1, βλG 2 ≪ 1, the Lagrangian density (3) becomes As a result, in the weak-field limit and small γ when the condition βL ≪ 1 is satisfied, the Lagrangian density (3) approaches to the ModMax model. At γ = 0 Eq. (5) corresponds to the Heisenberg-Euler-type electrodynamics [20]. Adding to Eq. (3) the source term A µ j µ and varying the action we obtain the Euler-Lagrange equations where Field equations (6) can be represented as Maxwell equations making use of definitions of the electric displacement and magnetic fields With help of Eq. (8) Euler-Lagrange equations (6) can be represented as Maxwell equations in Gaussian quantities Second pair of Maxwell equations follows from the Bianchi identity ∂ µF µν = 0, From Eq. (8) we obtain the relation According to the criterion of [21] the dual symmetry takes place if D · H = E · B. One can verify from Eq. (11) that the dual symmetry holds in two cases: σ = 1, λ = 0 which corresponds to ModMax model or for Born-Infeldtype model with σ = 1/2, λ = β. It is worth noting that the two-parametric generalized Born-Infeld model (σ = 1/2, λ = β) was considered and shown to be duality invariant in [4]. From Eq. (9), for the source of the point-like charged particle with the electric charge q, in Gaussian units, we obtain the equation as follows with the solution D = qr r 3 .
From Eqs. (8) and (13) one finds Introducing the dimensionless variables Eq. (14) takes the form It follows from Eq. (16) that at u → 0, v(0) = 1 for σ < 1, or in terms of electric fields Thus, the electric field of the point-like charged particle in the center is finite and possesses the maximum value for σ < 1. When σ increases the maximum value of electric fields increases, but if γ increases the electric field in the center decreases. For the cases σ = 1/2 (Born-Infeld-type model) and σ = 3/4 exact solutions to Eq. (16) and their asymptotic as The plots of function v(u) for σ = 1/4, σ = 1/2 and σ = 3/4 are given in Fig. (1). In the general case for σ < 1, the functions v(u) as u → 0 (r → 0) and Making use of Eqs. (15) and (19) we obtain the asymptotic value of the electric field as r → 0 and r → ∞ Equation (20) gives the correction to Coulomb's law as r → ∞. We have damping of the electric field because of parameter γ. Corrections to Coulomb's law for the case of Born-Infeld-type electrodynamics (σ = 1/2) and exponentiallike electrodynamics (4) (σ = ∞) are similar but with the opposite sign. At σ = 1, λ = 0, γ = 0, one has Maxwell's electrodynamics and we come to the Coulomb law E = q/r 2 as r → ∞, but the electric field at the origin is infinite.
The energy-momentum tensor is given by From Eq. (21) we obtain the energy density In the case of pure electric field (B = 0) the energy density (22) becomes For the Born-Infeld-type electrodynamics with σ = 1/2 we can calculate the total electrostatics energy of point-like charged particles. Introducing the dimensionless variables we obtain from Eq. (14) equation Then, making use of Eqs. (23), (24) and (25), the total electrostatics energy of point-like charged particles (for σ = 1/2) is given by (26) Taking into account the asymptotic of hypergeometric function 2 F 1 (−1/2, 1/4; 5/4; −x 2 ) we obtain the total electrostatics energy of point-like charged particles For Born-Infeld electrodynamics (β = 1/b 2 ) at γ = 0 we come to the result obtained in [15].
In accordance with [15] we introduce the "free charge density" ρ f ree by the equation where E obeys Eq. (14). To calculate the distribution of the free charge one has to obtain ρ f ree from Eq. (29). We have exact solutions to Eq. (14) for σ = 1/2 and σ = 3/4. Making use of Eq. (15), from Eq. (18) we find E = q exp(−γ/2) where r 0 = 4 √ β √ q. When q is the charge of the electron, r 0 is the electron radius [15]. It follows from Eq. (30) that for r → ∞ the electric field E → (q/r 2 ) exp(−γ) and for r → 0 we have according to Eq. (20). From Eqs. (29) and (30) we obtain "free charge density" For Born-Infeld electrodynamics, in the case σ = 1/2, γ = 0, one comes from Eq. (32) to the result found in [15]. Now we can calculate free charges Making use of Eqs. (20) and (33) we find the free charge for any parameter σ q f ree = q exp(−γ).
Equation (34) shows that the free charge is less from q by the factor exp(−γ). From Eq. (30) we obtain the free electric charges, in the cases σ = 1/2 and σ = 3/4, inside the sphere r < r 0 q f ree (r 0 ) = 4π The plots of the functions q f ree (r 0 )/q f ree versus γ is presented in Fig. 2. It follows from Eq. (35) and the Fig. 2 that at γ = 0 q f ree (r 0 )/q f ree ≈ 0.71 for σ = 1/2 and q f ree (r 0 )/q f ree ≈ 0.85 for σ = 3/4. Therefore, 71% of the electron charge q f ree is contained in the electron radius sphere for σ = 1/2 and 85% of the electron charge is concentrated within the electron radius for σ = 3/4. When parameter γ increases more charge is inside the electron radius sphere. For the magnetic monopole we have equation where Q is the magnetic charge. For a magnetic monopole we have equation From Eq. (22) we obtain the magnetic energy density (E = 0) of the magnetic monopole The total magnetic energy is given by Integral (39) converges for 0 < σ < 1 and the energy of the magnetic monopole is finite within our model. In Table 1 we present the approximate values of the dimensionless energyĒ m = β 1/4 Q −3/2 exp(3γ/4)E m for σ = 0.1, 0.2, .., 0.7. Table 1 shows that with increasing parameter σ the magnetic energy of the monopole increasing. When parameter γ increases the magnetic energy E m decreases. From Eq. (8) we obtain the magnetic field of the monopole The "free magnetic charge density" η f ree is defined by the equation Making use of Eq. (41) one finds from Eq. (40) the "free magnetic charge density" Similar to Eq. (33) we obtain, by using Eq. (40), the free magnetic charge Thus, the equation for the free magnetic charge is similar to the free electric charge (34). From Eq. (40), one finds the free magnetic charge inside the sphere of the radius The plots of the functions Q f ree (r m )/Q f ree versus γ for σ = 1/8, σ = 1/2 and σ = 3/4 is presented in Fig. 3. According to Eq. (44) (1 + e −γ /(2σ)) σ−1 of the magnetic charge Q f ree is inside the sphere of the radius r m . The plot of the function Q f ree (r m )/Q f ree is in Fig. 4. Figure 4 shows that the ratio Q f ree (r m )/Q f ree increases with increasing parameters σ and γ. As a result, more magnetic charge is concentrated in the sphere of the radius r m with greater values of σ and γ. The generalised ModMax model proposed as compared to ModMax model (1) possesses the attractive features as follows.
• At some parameters β, λ, γ and σ we come to different models (including the Born-Infeld electrodynamics) discussed in the literature. • In the weak-field limit and γ = 0, the Lagrangian density leads to the Heisenberg-Euler electrodynamics.
• The electric field of the point-like charged particle in the origin is finite and possesses the maximum value (for σ < 1).
• The electric and magnetic energies of point-like charged particles is finite for some parameters σ.
Such properties take place also for the two-parametric duality invariant generalization of Born-Infeld electrodynamics found in [1], [4].
In addition, we calculated the free electric and magnetic densities in our model that allow us to study the distributes of the electric and magnetic charges in the space.