# Mass prediction

## Main Question or Discussion Point

I wonder if there are successful cases of particle mass prediction/calculation in Theoretical Particle Physics?

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arivero
Gold Member
Yukawa's pion was ok. It was expected to be of the order of the size of a nucleus and it was. Regretly the situation was complicated because pions, at the end, are not the full answer to short ranged strong force in nucleus.

I think that Witten settled the question of the difference between neutron and proton mass, but I am not sure.

In the semi-empirical spirit of hadron spectroscopy, of course flavour theory got to predict masses of the SU(3) decuplet, but I am not sure of the history.

W and Z were predicted from fermi interactions and fermi constant. W was easy, Z was a bit more in dispute. But W approx 80 GeV was already in textbooks in the early seventies.

Top failed. They kept increasing the prediction until it was finally found. But in the last years previous to discovery, QFT corrections were already predicting the right range.

arivero
Gold Member
And of course, string theory got to predict the masses of high spin hadrons. Only that at that time it was not string theory yet.

Really the hadron spectrum is not very hard to control in the big picture. String-Regee theory takes care of the mass of QCD excited states. The basic states are well organised via quark model and Gellmann-etc mass formulae. Even the decay constants seem to have some reasonable organisation, as you can be on this picture of mass against decay width:
http://dftuz.unizar.es/~rivero/research/nonstrong.jpg [Broken]

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apeiron
Gold Member
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?

arivero
Gold Member
We have a whole thread on particle numerology ("All the particle masses from..."). Inspect it to get some idea. Also, look for "trialogue" on fundamental constants, in the arxiv, for a discussion about what is a physical constant.

Problems for such approach are GIGO (measure how many garbage you put in and compare with the garbage, er, results, you get out) and the "law" of small integers (there are only a few of small integers, and a lot of mathematical results involving small integers, so the "birthday paradox" is bound to happen).

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?

tom.stoer
One should mention that these different predictions have rather different contexts.

W and Z in the el.-weak GSW model rely on fixing constants related to the "old" Fermi model. Low SU(3) hadrons use mainly group theoretic aspects w/o taking into account QCD. The difference is that hadrons are bound states but W and Z are (as far as we know today) elementary particles.

In the meantime lattice gauge calculations for (quenched QCD) are able to fit hadron masses within a few percent. Of course these calculations are not parameter-free.

The mass of the hydrogen atom in terms of the mass of the electron and proton.

The mass of the hydrogen atom in terms of the mass of the electron and proton.
Yes, something like this but for other particles. The Hydrogen is an exemplary case when knowing me, mp, and calculating E0 from some interaction, one can calculate MH.

It appears that we can calculate all known masses given the parameters of the standard model. I thought the OP was about Yukawa couplings.

It appears that we can calculate all known masses given the parameters of the standard model. I thought the OP was about Yukawa couplings.
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?

From quark masses and the strong interaction coupling constant?
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 :
http://arxiv.org/abs/hep-ph/9602240
(I was unaware of this particle paper before doing this search, and only use it to mention the use of Gell-Mann-Oakes-Renner relation, which is presented in nearly all QCD textbooks)

Thanks, Humanino, I will read it. 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?

Thanks, Humanino, I will read it.
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.

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?
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.

I like this introduction to lattice QCD :
http://arxiv.org/abs/hep-lat/0506036

tom.stoer
One 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.

Nevertheless the results are promising; a nice paper including results for hadron masses is http://de.arxiv.org/abs/0711.3091v2

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One 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.
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 subject

http://inside.hlrs.de/images/spring01_09/A8_09.jpg [Broken]
Original paper : Ab-initio Determination of Light Hadron Masses
Thanks 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