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In the Standard Model, we have these free parameters:

  • 6 quark masses

  • 6 lepton masses

  • 3 coupling constants

  • Higgs mass

  • Higgs VEV

  • 4 CKM parameters

  • 4 PMNS parameters

  • QCD vacuum angle

That's 26 in total. For gravity, we also need the cosmological constant $\Lambda$.

So that would make... 27 parameters.

Is that all? I'm assuming quantum theory + Poincarè invariance + $U(1) \times SU(2) \times SU(3)$ gauge symmetry + general covariance on a 4-dimensional Lorentzian manifold.

Does string theory reduce this number? What about loop quantum gravity? Do we have any theory which reduces it?

What about strong constraints on some of these parameters determined by some others?

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  • $\begingroup$ Actually, $\hbar$ and $M_P$ (i.e. $G$) just specify a system of units. They are not really physical parameters, you can work in a system of units where they are both 1. They don't need to me included here. $\endgroup$
    – innisfree
    Commented Oct 19, 2016 at 0:33
  • $\begingroup$ Oh yes, that's right, I'll edit it. $\endgroup$
    – user20250
    Commented Oct 19, 2016 at 0:34
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    $\begingroup$ The dimensionality of spacetime isn't really a "free parameter" as usually defined. First of all, the phrase "free parameter" usually only includes parameters than can range over continuous values, because if you include discrete parameters, it's ambiguous what counts. You need to specify all the representations that the various matter fields live it, which would require at least 18 more (discrete) parameters. And would the gauge group count as another parameter? $\endgroup$
    – tparker
    Commented Oct 19, 2016 at 1:05
  • $\begingroup$ Second, for dimensionalities other than 4, the field content could be totally different, as we'd need to include more or fewer renormalizable interactions, the identity $\mathfrak{so}_4 \cong \mathfrak{su}_2 \times \mathfrak{su}_2$ wouldn't allow us to consider left- and right-handed spinors separately, the gamma matrices would need to be modified, etc. $\endgroup$
    – tparker
    Commented Oct 19, 2016 at 1:08
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    $\begingroup$ I see, it's a valid argument. To avoid these ambiguities and complications, I will edit to assume the dimensionality. $\endgroup$
    – user20250
    Commented Oct 19, 2016 at 1:19

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I'm not qualified to comment on string theory or loop quantum gravity, but I believe that part of the motivation for Grand Unified Theory is that embedding the gauge group $SU(3) \times SU(2) \times U(1)$ into a larger (simple or semisimple) gauge group (e.g. SU(5) or SO(10)) may reduce the number of independent parameters. For example, doing so would replace the three gauge couplings by a single one, while also giving a natural explanation for the representations of the mass fields, the number of generations, etc. Unfortunately, no one has figured out a concrete way to do this without violating observed data - for example, most GUTs predict that the proton is unstable, with a lifetime that is enormously long but still short enough to have been ruled out experimentally.

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