By combining a renormalization group argument relating the charge e and mass m _ { p } of the proton by e ^ { 2 } \ln { m _ { p } } \approx - 0.1 \pi ( in Planck units ) with the Carter-Carr-Rees anthropic argument that gives an independent approximate relation m _ { p } \sim e ^ { 20 } between these two constants , both can be crudely estimated .
These equations have the factor of 0.1 \pi 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 % .