There are 3 generations of 4 fermions, each has a mass. One sets the scale of the theory, meaning we should express all the dimensionful quantities in terms of that mass. So, that gives 11 parameters.
Then there are the mixing angles in the CKM matrix. For three generations, that matrix is a 3x3 unitary matrix = 9 parameters. But, we can absorb arbitrary phases into each quark field, but should add one back for the global U(1) phase for 9-6+1 = 4 parameters, 3 mixing angles and 1 CP-violating phase. We're up to 15.
We have 3 forces in the Standard Model, each has a gauge coupling. 18.
There's the Higgs vev and mass, 20.
QCD admits a marginal topological quantity, θ_QCD. As far as anyone knows it's zero to many digits of precision. Still, it's in principle a free parameter of the Standard Model, so 21.
Then there's the Planck mass (or equivalently, G), 22, which is the number I remember.
I suppose now that we are certain neutrinos have mass I should include the parameters of the PMNS mixing matrix, if you're willing to count them in the Standard Model (they're not in the Glashow-Weinberg-Salaam work, for example). How many free parameters it contains depends on whether neutrinos are Majorana. If they're Dirac, it's 4 parameters (like the CKM matrix), if they're Majorana there are 2 additional CP-violating phases.
So, 26 or 28 continuous parameters in the Cosmic Control Booth according to the Standard Model. There's of course discrete kinds of questions you can ask, like "Why is the gauge group SU(3) x SU(2) x U(1)"; I'm not sure how to count such a choice. I think it's fair to ask, "Here's the physical theory with the matter content set, what are the free parameters?".
Then there are the mixing angles in the CKM matrix. For three generations, that matrix is a 3x3 unitary matrix = 9 parameters. But, we can absorb arbitrary phases into each quark field, but should add one back for the global U(1) phase for 9-6+1 = 4 parameters, 3 mixing angles and 1 CP-violating phase. We're up to 15.
We have 3 forces in the Standard Model, each has a gauge coupling. 18.
There's the Higgs vev and mass, 20.
QCD admits a marginal topological quantity, θ_QCD. As far as anyone knows it's zero to many digits of precision. Still, it's in principle a free parameter of the Standard Model, so 21.
Then there's the Planck mass (or equivalently, G), 22, which is the number I remember.
I suppose now that we are certain neutrinos have mass I should include the parameters of the PMNS mixing matrix, if you're willing to count them in the Standard Model (they're not in the Glashow-Weinberg-Salaam work, for example). How many free parameters it contains depends on whether neutrinos are Majorana. If they're Dirac, it's 4 parameters (like the CKM matrix), if they're Majorana there are 2 additional CP-violating phases.
So, 26 or 28 continuous parameters in the Cosmic Control Booth according to the Standard Model. There's of course discrete kinds of questions you can ask, like "Why is the gauge group SU(3) x SU(2) x U(1)"; I'm not sure how to count such a choice. I think it's fair to ask, "Here's the physical theory with the matter content set, what are the free parameters?".