7.2. Vacuum polarization and the value of
Zeldovich (1968), having demonstrated that the energy density of the vacuum was infinite at the one-loop level, suggested that after the removal of divergences, the `regularized' vacuum polarization contributed by a fundamental particle of mass m would be described by the expression
One can arrive at this result by means of the following argument: the vacuum consists of virtual particle-antiparticle pairs of mass m and separation = /mc. Although the regularized self-energy of these pairs is zero, their gravitational interaction is finite and results in the vacuum energy density vac vac c2 ~ (G m2 / ) / 3 = Gm6 c4 / 4 corresponding to (81). (In terms of Feynman diagrams this corresponds to the energy associated with the two-loop vacuum graph shown in figure 13.) Substituting m me(mp) we find that the electron (proton) mass gives too small (large) a value for . On the other hand, the pion mass gives just the right value  (13)
Finally, a small value of can be derived from dimensionless fundamental constants of nature using purely numerological arguments. For instance, the fine structure constant e2 / c 1/137 when combined with the Planck scale P, suggests the relation 
Or, when expressed in terms of = (8 G / 3 H02) we get h2 = 0.335, in excellent agreement with observations. In principle, could be some other fundamental constant, such as the `string constant' associated with superstring theory, which might enter into exponentially small expressions for of this type.
13 The large difference between obtained using (81) for the proton and its observed value prompted Zeldovich to suggest that Fermi's weak interaction constant GF might play a role in determining the vacuum energy, so that
Although this leads to some improvement, for the proton is still several orders of magnnitude larger than its observed value. Back.