3.4. Derivation of mPlanck/mproton

By the same token, if one of these GUTs is correct, it will provide a derivation of the 1, 2, 3 coupling constants at any scale from one unified constant U at the unification scale. Recall that the mass of the proton is fixed by the scale at which the SU(3) coupling diverges. Because of the slow variation of coupling with energy, this takes a large range of energy and leads to a large ratio of proton to unification mass.

We can run through a toy calculation as follows. Assuming the degrees of freedom are constant, the inverse couplings just depend linearly on the log of the energy scale, so (9) and (10) can be trivially integrated. Equating them at the unification scale MU, 1(MU) = 3(MU), yields

(11)

Naiively plugging in the standard model numbers (which give 2.1 for the denominator), and the values 1 (60)-1 and 3 _S 0.12 for the coupling constants at the Z scale, yields a mass ratio of MU / MZ exp[(60-8)/2.1] = 1011. This toy estimate is wrong in several details (most notably, not having included supersymmetry) but correctly illustrates the main point, that there exists an exact calculation that yields a large ratio of fundamental masses, roughly

(12)

The numerical factors here are just approximate, but are exactly computable within the framework of supersymmetric GUTs and yield a unification scale of MU 1016 GeV. In this framework, this is essentially the explanation of the ``weakness'' of gravity, the smallness of mproton/mPlanck. Since mPlanck 103 MU there are three of the nineteen orders of magnitude still to be accounted for, presumably by the final unification with gravity.

Formulae very similar to (12) have appeared for many years (see, for example, eq. (54) of Carr and Rees 1979). The rationale has always centered (as it does here) on the logarithmic divergences of renormalization but in the context of supersymmetry the derivation is much crisper - it comes in the framework of rigorous derivations in a well-motivated theory now being tested (Wilczek 1998). If this guess about unification is correct, we have most of the explanation of the large numbers of astrophysics, subject to the value of one independent, apparently arbitrary coupling-constant parameter (U or gU), a moderately small number (of the order of 1/25). The value of mproton / MU depends exponentially on U (and hence also on ). Changes of a few percent in the couplings lead to order-of-magnitude variations in the astrophysical Large Numbers, enough to cause qualitative change in the behavior of the astrophysical world. The fine structure constant thereby becomes a candidate for selective tuning connected to obtaining a suitable strength for gravitation!