**5. Concluding Remarks**

In this paper we have addressed two longstanding questions in particle physics: (1) Why do the elementary fermionic particles that are so far observed have such low mass-energy compared to the Planck scale? (2) What mechanical energy may be counterbalancing the divergent electrostatic and strong force energies of point-like charged fermions in the vicinity of the Planck scale? Using a hitherto unrecognised mechanism extracted from the well known Hehl–Datta equation, we have presented numerical estimates suggesting that the torsion contributions within the Einstein–Cartan–Sciama–Kibble extension of general relativity can address both of these questions in conjunction.

The first of these problems, the Hierarchy Problem, can be traced back to the extreme weakness of gravity compared to the other forces, inducing a difference of some 17 orders of magnitude between the electroweak scale and the Planck scale. There have been many attempts to explain this huge difference, but none is simpler than our explanation based on the spin induced torsion contributions within the ECSK theory of gravity. The second problem we addressed here concerns the well known divergences of the electrostatic and strong force self-energies of point-like fermions at short distances. We have demonstrated above, numerically, that torsion contributions within the ECSK theory resolves this difficulty as well, by counterbalancing the divergent electrostatic and strong force energies close to the Planck scale.

It is widely accepted that in the standard model of particle physics charged elementary fermions acquire masses via the Higgs mechanism. Within this mechanism, however, there is no satisfactory explanation for how the different couplings required for the fermions are produced to give the correct values of their masses. While the Higgs mechanism does bestow masses correctly to the heavy gauge bosons and a massless photon, and while our demonstration above does not furnish a fundamental explanation for the fermion masses either, we believe that what we have proposed in this paper is worthy of further research, since our proposal also offers a possible resolution of the Hierarchy Problem.

In Reference [33], Singh points out that there appears to be a symmetry between small and large masses for spin-torsion coupling and energy-curvature coupling. We have noted that there also appears to be a symmetry in that the energy-curvature coupling is effectively infinite while the spin-torsion coupling is very short-ranged near the Planck length.

One may wonder why a gravitational coupling would be involved in the torsion term involving spin-squared, but we suspect it has more to do with Planck length than gravity. The torsion term with our *r<sup>t</sup>* cancellation length can be simplified to

$$-\frac{3\,\pi G\hbar^2}{c^2r\_t^3} = -\frac{3\,\pi(l\_P)^2\hbar c}{(\sqrt{\frac{3\pi}{a}}l\_P)^3} = -\frac{a\hbar c}{\sqrt{\frac{3\pi}{a}}l\_P} \approx -2.479 \times 10^{15}\,\text{GeV}.\tag{54}$$

Instead of gravitational coupling, now the term has become simple and involves only constants and the Planck length.

Needless to say, the geometrical cancellation mechanism for divergent energies we have proposed here also dispels the need for mass-renormalisation, since we have obtained finite solutions for *r<sup>x</sup>* taming the infinities. Thus, both classical and quantum electrodynamics appear to be more complete with torsion contributions included.

**Author Contributions:** The authors C.F.D.III and J.C. contributed to the article equally. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors wish to thank Tejinder P. Singh for encouragement and discussions concerning the significance of torsion.

**Conflicts of Interest:** The authors declare no conflict of interest.

### **Appendix A. Calculations of Fermion Radii using Wolfram Mathematica**

In this appendix we explain how we used the arbitrary-precision in Mathematica to solve the numerical equations out to 24 significant figures. Each equation displayed below—derived from our central Equation (33)—is simplified so that only the numerical factors have to be used, since the dimensional units cancel out, leaving lengths in meters. For decimal factors, the numbers must be padded out to 26 digits with zeros. Then the numerical part of electrostatic energy density is defined as *A* and the numerical part of spin energy density is defined as *B*, just as in Equation (44) above. These are then used throughout to perform the calculations. For the values of various physical constants involved in the calculations we have used the 2014 CODATA values, Reference [34] and values from the Particle Data Group, Reference [35].
