**4. Possible Solution of the Hierarchy Problem**

As alluded to in the introduction, the Hierarchy Problem refers to the fact that gravitational interaction is extremely weak compared to the other known interactions in Nature. One way to appreciate this difference is by combining the Newton's gravitational constant *G* with the reduced Planck's constant *h*¯ and the speed of light *c*. The resulting mass scale is the Planck mass, *mP*, which some have speculated to be associated with the existence of smallest possible black holes [7]. If we compare the Plank mass with the mass of the top quark (the heaviest known elementary particle),

$$\begin{aligned} m\_P &= \sqrt{\frac{\hbar c}{G}} \approx 2.1765 \times 10^{-8} \text{ kg/s} \\ m\_t &= \frac{173.21 \text{ GeV}}{c^2} \approx 3.1197 \times 10^{-25} \text{ kg/s} \end{aligned}$$

then we see that there is some 17 orders of magnitude difference between them. This illustrates the enormous difference between the Planck scale and the electroweak scale. Many solutions have been proposed to explain this difference, such as supersymmetry and large extra dimensions, but none has been universally accepted, for one reason or another. Furthermore, recent experiments performed with the Large Hadron Collider are gradually ruling out some of these proposals. Regardless of the nature of any specific proposal, it is clear from the above values that predictions of numbers with at least 17 significant figures are necessary to successfully explain the difference between *m<sup>P</sup>* and *m<sup>t</sup>* .

We saw from our numerical demonstration in the previous section that within the ECSK theory minute changes in length can induce sizable changes in the observed masses of elementary particles, and that we do have numbers at our disposal with more than 17 significant figures for producing those masses. Moreover, all length changes occurring in our demonstration are taking place close to the Planck length. Thus, since we are "cancelling out" near the Planck length to obtain masses down to the electroweak scale, ours is clearly a possible mechanism for resolving the Hierarchy Problem. We can appreciate this fact by simplifying our central equation by setting ¯*h* = *c* = 1, which then reduces to

$$\frac{\mathfrak{a}}{r\_{\mathfrak{X}}} - \frac{\mathfrak{Z}\pi G}{r\_{\mathfrak{X}}^3} = m\_{\mathfrak{X}}\,. \tag{51}$$

It is now easy to see from this equation that the observed mass-energy only depends on the coupling constants and the radii (geometry). Moreover, it is confined entirely within a volume close to the Planck volume, as we saw in our calculations in the previous section. Thus, we are led to

Planck Scale =⇒ Electroweak Scale.

In other words, there is no hierarchy problem in the ECSK theory, because Planck scale physics is producing the electroweak scale physics in the form of the mass-energy of fermions as a byproduct of the very geometry of spacetime.

Within the ECSK theory, which extends general relativity to include spin-induced torsion, gravitational effects near micro scales are not necessarily weak. On the other hand, since torsion is produced in the ECSK theory by the spin density of matter, it is confined to that matter, and thus is a very short range effect, unlike the infinite range effect of Einstein's gravity produced by mass-energy. In fact, the torsion field falls off as 1/*r* 6 , as shown in the calculations of Section 3, since it is produced by spin density squared, confined to the matter distribution [9].

To compare the strengths of gravitational and torsion effects at various scales, we may define a mass-dependent dimensionless gravitational coupling constant, *Gm*2/(*hc*¯ ), and evaluate it for the electron, top quark and Planck masses:

$$\begin{split} \alpha\_{\text{G}\_{\text{t}}} &= \frac{Gm\_{\text{c}}^{2}}{\hbar c} \approx 1.7517 \times 10^{-45} \text{ } \\ \alpha\_{\text{G}\_{\text{t}}} &= \frac{Gm\_{\text{t}}^{2}}{\hbar c} \approx 1.1620 \times 10^{-36} \text{ } \\ \alpha\_{\text{G}\_{\text{P}}} &= \frac{Gm\_{\text{P}}^{2}}{\hbar c} = 1 \text{ } \\ \alpha\_{\text{c}} &= \frac{\text{e}^{2}}{4\pi\hbar c} \simeq 7.2973 \times 10^{-3} \text{ } \end{split}$$

Here *α<sup>e</sup>* is the electromagnetic coupling constant, or the fine structure constant. From these values we see that near the Planck scale the gravitational coupling is very strong compared to the electromagnetic coupling. However, as we noted above and in Section 3, near the Planck scale torsion effects due to spin density are also very strong, albeit with opposite polarity compared to that of Einstein's gravity, akin to a kind of "anti-gravity" effect of a very short range.

For our demonstration above we have used electrostatic energy density and spin density for matter in a static approximation, for which the field equation within the ECSK theory reduces to *G* <sup>00</sup> = *T* <sup>00</sup>. A numerical estimate for *G* <sup>00</sup> from the contributions of the electrostatic energy and spin density parts of *T* <sup>00</sup> at our cancellation-radius gives

$$G\_{stat}^{00} = \frac{8\pi G}{c^4} \frac{a\hbar c}{r\_t^4} \approx +4.209 \times 10^{62} \text{ m}^{-2} \tag{52}$$

and

$$\mathbf{G}\_{\rm spin}^{00} = -\frac{8\pi G}{c^4} \frac{3}{c^2 r\_t^6} \approx -4.209 \times 10^{62} \,\mathrm{m}^{-2}.\tag{53}$$

Evidently, these field strengths at the cancellation radius are quite large even for a single electron. Fortunately they are never realised in Nature, because, as we can see, they cancel each other out to produce *G* 00 *net* = 0. On the other hand, if we use only the mass-energy density for electron at the cancellation radius, then we obtain *G* 00 *mass* <sup>≈</sup> 3.0674 <sup>×</sup> <sup>10</sup><sup>43</sup> <sup>m</sup>−<sup>2</sup> , which is again some 19 orders of magnitude off the mark. What is more, the latter field strength does not fall off as fast as that due to the spin-induced torsion field. Thus, it is reasonable to conclude that without the cancellation of divergent energies due to the spin self-interaction we have explored here, our universe would be highly improbable.

While there have been other approaches to the hierarchy problem from the viewpoint of the ECSK theory [28–32], our partial solution to the problem is simpler. We have fermions with near Planck scale radii (size) producing rest mass energy in the electro-weak scale. While we have not explained why higher generation fermions do not exist, given that as an assumption, our solution is more complete.
