Anthropic estimates of the charge and mass of the proton

By combining a renormalization group argument relating the charge e and mass mp of the proton by e2lnmp≈−0.1π (in Planck units) with the Carter-Carr-Rees anthropic argument that gives an independent approximate relation mp∼e20 between these two constants, both can be crudely estimated. These equations have the factor of 0.1π and the exponent of 20 which depend upon known discrete parameters (e.g., the number of generations of quarks and leptons, and the number of spatial dimensions), but they contain no continuous observed parameters. Their solution gives the charge of the proton correct to within about 8%, though the mass estimate is off by a factor of about 1000 (16% error on a logarithmic scale). When one adds a fudge factor of 10 previously given by Carr and Rees, the agreement for the charge is within about 2%, and the mass is off by a factor of about 3 (2.4% error on a logarithmic scale). If this 10 were replaced by 15, the charge agrees within 1.1% and the mass itself agrees within 0.7%.