Jarek 31 said:
This question requires rho(r) energy density function: asymptotically rho~1/r^4, but regularized: going down to r=0 (not some subjective cutoff) and integrating at most to 511keV mass of electron.
Maybe equal to 511keV (like in Faber) - if below, where is the difference?
At least approximate answer - the complete one should also include angular dependence from energy of magnetic field from electron dipole moment.
Claiming Standard Model is practically complete, shouldn't we know answer to such a basic question?
Maybe e.g. lattice QED could allow to answer it?
The problem is that the Standard Model (or QED as part of it to keep it simple), based on relativistic QFT, is much less complete than classical electrodynamics with classical point particles, because in fact there is no fully satisfactory description of it, and it's precisely related to the question you address here, i.e., the energy of the electromagnetic field of a point particle, which takes into account the interaction of the particle with its own electromagnetic field.
The best one can come up with still is the Lorentz-Abraham-Dirac equation. This can be done in a modern way using a pretty simple regularization scheme, invented by Lechner et al, which just introduces some length scale ##\epsilon## into the retarded propagator of the D'Alembert operator,
$$D(x)=\frac{1}{2 \pi} \delta(x^2), \quad x^2=x \cdot x=(x^0)^2-\vec{x}^2,$$
defining its regularized form by
$$D_{\epsilon}(x)=\frac{1}{2 \pi} \delta(x^2-\epsilon^2).$$
You calculate the em. field of the charge as a functional of its time-like worldline and then the self-force of this regularized particle along its world line. You end up with an equation, where the piece that is diverging in the physical limit ##\epsilon \rightarrow 0## can be lumped to the left-hand side of the equation of motion, thus providing an inifinite contribution to the particle mass. Assuming also an infinite bare mass for the particle, this diverging part cancels resulting in the physical (renormalized) mass of the particle. Then you can make ##\epsilon \rightarrow 0##, and you end up with the LAD equation (working in Heaviside-Lorentz units with ##c=1##),
$$m \ddot{x}^{\mu}=K_{\text{ext}}^{\mu}+\frac{q^2}{6 \pi} \left (\dddot{x}^{\mu} + \ddot{x}^2 \dot{x}^{\mu} \right).$$
The trouble with this equation is that it is obviously of 3rd order in the derivatives (the dot stands for derivatives wrt. proper time), and this gives rise to some unphysical solutions, with the acceleration going to infinity at the remote past ("self-acceleration and runaway solutions") which have to be excluded by boundary conditions concerning the motion for ##\tau \rightarrow \pm \infty##. Basically one has to assume that the acceleration goes to ##0## in the infinite past and future. Even excluding the runaway solutions there is also preacceleration, i.e., assuming an external force acting only within a limited space-time region, admitting the boundary conditions concerning the acceleration to be fulfilled, the acceleration of the particle sets in before the external force is actually supposed to be acting. For not too large external forces this violation of causality is limited to the time scale given by ##q^2/(6 \pi)## in the LAD equation, but it's nevertheless a violation of causality. So there's strictly speaking no satisfactory self-consistent model for point charges, except for the free particle, where the boundary conditions exclude the self-acceleration and runaway solutions, leading to the "trivial" solution ##\dot{x}^{\mu}=0## as it should be for a free particle with the (boosted) Coulomb field as the particle's field, which is of course having the infinite self-energy of the particle which has to be considered to be absorbed in the particle's finite physical mass.
It's also clear, however, that the time scale ##q^2/(6 \pi)## related to the socalled classical electron (if we consider electrons as the charged particles) radius up to some factor, is way smaller than the Compton wavelength (even for the lightest known charged particle, the electron), and this indicates indeed that the classical point-particle limit must break down way before we can resolve these small length and/or time scales such as the classical electron radius. That justifies the approximation of the LAD equation by the socalled Landau-Lifshitz equation, which has no preacceleration behavior.
I've started to write all this down in my SRT FAQ article (but it's not finished yet):
https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf
For the complete discussion see the quoted textbook by Lechner and the also quoted paper as well as the paper by Nakhleh.
The more classical regularization is to consider charged particles of finite extent, including the introduction of Poincare stresses to keep this "particle" stable, leading to a differential-difference equation, which is nicely discussed (in the non-relativistic limit though) in
https://doi.org/10.1119/1.3269900
