Main Question or Discussion Point
I wonder if there are successful cases of particle mass prediction/calculation in Theoretical Particle Physics?
What kind of quantities or constants would it be most natural for particles masses to depend on when people are doing "particle numerology"?
So pi, e, phi, etc....a set of mathematical constants....
Or physical constants like c, k, g, etc?
What is the right way to think about the choices that could be made?
No, I did not mean Yukawa but the Standard Model first of all. For example, a pi-meson mass, what from it is calculated? From quark masses and the strong interaction coupling constant?It appears that we can calculate all known masses given the parameters of the standard model. I thought the OP was about Yukawa couplings.
Yes, at least in principle. For instance lattice methods (including chiral extrapolation to physical quark masses) adjust the quark masses (Yukawa couplings) to the necessary amount of measured masses in the spectrum (for instance, one needs 3 hadron masses to adjust 3 quark masses) and correctly reproduce the full hadronic spectra of masses with other hadrons. Alternatively, one can attempt to calculate directly the pion mass from the Gell-Mann-Oakes-Renner relation, including some modeling for the quark condensate. The following link is the first result of "pion mass formula" from google :From quark masses and the strong interaction coupling constant?
I did not read it myself further than the abstract and I do not know whether it is worth reading. On this matter, I guess a QCD textbook would be more suited.Thanks, Humanino, I will read it.
I do not have time right now to make a decent description. Lattice QCD is merely a (non-perturbative) brute force computation of the path integral. I use "brute force" in parenthesis because quite some technical tricks are necessary to make it manageable, even with powerful supercomputers. The renormalisation amounts to taking the continuum limit, since the regulator is the lattice itself (it introduces a momentum cutoff at the lattice spacing). To compute bound state properties, one has to choose an operator with the appropriate quantum numbers and we get as a result mostly the propagator for the corresponding state.By the way, in the lattice calculations (numerical approach, I guess), what is solved? Equations for bound states? Do these calculations involve renormalizations, counter-terms?
The quenched approximation was a major limitation in the past. There are several ways to include the fermion determinant. I am not sure this technical discussion is appropriate here, but I can dig references if you want. In any case, there is a popular paper on the subjectOne should mention that (as far as I know due restricted computing power) still most lattice calculations must be restricted to the "quenched approximation". That means in the path integral the fermion determinant is fixed to One = the quarks are somehow static instead of dynamic; virtual quark-antiquark loops are suppressed. So the quark content of the hadron under investigation is fixed upfront.
See also : Colloquium on the calculationThanks to continuous progress [...] lattice QCD calculations can now be performed with[out the] neglect [of] one or more of the ingredients required for a full and controlled calculation. The five most important of those are, in the order that they will be addressed below:
- inclusion of fermion determinant
- determination of the light ground-state (Three fix the masses of u, d and s)
- Large volumes
- Controlled interpolations to physical mass
- Controlled extrapolations to the continuum