Hadron Mass Spectrum: Can We Calculate It?

[jostpuur]

Why have I not seen equations that would tell the masses of hadrons as functions of other constants, like masses of quarks, and some coupling constants? After all, aren't the different hadrons merely different excitation states of some bound systems, and in principle it should be possible to calculate the excitation energies (particle masses) like the energies of the hydrogen atom?

Is it because the equations are too difficult, or because physicists don't know what kind of equation to write down? The standard model is supposed to be the theory of particle interactions, so it sounds like we should know what equation to use for the hadron masses, but is that really the case?

 

[BenTheMan]

The answer is (I think) that QCD is strongly coupled, which means that we don't really know how to do calculations with it. The problem is that we don't know how to deal with strongly coupled theories, except numerically. So some people ARE trying to calculate hadron spectra, but they have to use large computers and they can only calculate simple things.

The situation is analagous to using a Taylor series to approximate a function. If you terminate the Taylor series at some finite order, you introduce SOME error, but you generally can be assured that that error is small because the thing you were expanding in is small as well. If the thing that you are expanding in ISN'T small, then you have to keep more and more terms to get a small error. The case with QCD is that the thing you are expanding in is large, so each term is larger than the previous one.

This is called perturbation theory, and it cannot be applied when the theory is strongly coupled, like QCD. It cannot be applied because the coupling constant (the thing that you're expanding in) is larger than 1.

That being said, there are some loose relationships among mesons, I think, but I can't remember them. I couldn't find them on google either. I will try to look them up later when I get to my office.

 

[arivero]

That being said, there are some loose relationships among mesons, I think, but I can't remember them. I couldn't find them on google either. I will try to look them up later when I get to my office.

Well, basically all you get to link mesons are Gell-Mann-Okubo formulae and then some chiral perturbation theory.

In the empirical side, the thread "All the lepton masses..." also lists some suggestions for mesons. I took care to upload the more amusing ones to the arxiv, as footnotes or as part of other papers.

Of course, a hot issue is if string theory can be useful to overrun some of the non-perturbative barriers of QCD, but the AdS/CFT duality does not apply directly to the topic of meson masses.

 

[jostpuur]

The perturbation theory is supposed to approximate some precise theory. So there exists a precise theory which in principle should tell the hadron masses, but the perturbation theory remains as the only way to deal with it? It sounds strange... I see that perturbation theory could be the only way to do some calculations, but one might think that if the precise theory exists, we could just make some big computer solve things with it?

 

[arivero]

The perturbation theory is supposed to approximate some precise theory. So there exists a precise theory which in principle should tell the hadron masses, but the perturbation theory remains as the only way to deal with it? It sounds strange... I see that perturbation theory could be the only way to do some calculations, but one might think that if the precise theory exists, we could just make some big computer solve things with it?

In this case the precise theory is an approximate one :-(

I happen to have in the night table, over the last Harry Potter, the red book of Donoghue et al "Dynamics of the Standard Model". It has some hints towards the so-called "chiral perturbation theory". Also, there is a Russian topcited paper about the topic. Ugh, my memory is as bad as BenTheMan's

 

[humanino]

Of course, a hot issue is if string theory can be useful to overrun some of the non-perturbative barriers of QCD, but the AdS/CFT duality does not apply directly to the topic of meson masses.

Those people making AdS/QCD models actually compute the energy-momentum tensor of partons in mesons. This is also possible in principle for hadrons just more difficult and has not made it to the publication level so far.

edit
What is this doing in BTSM ?

 

[jostpuur]

What is this doing in BTSM ?

The original question was about that does the standard model give the principles that determine hadron masses, or does it not give them. Since I did not know the answer, I could not know where the question belongs! (Actually I still don't know, because the answers were technical, and I didn't understand their content.) Fortunately this is probably not a very serious matter

 

[humanino]

The original question was about that does the standard model give the principles that determine hadron masses, or does it not give them.

I think Ben gave a satifactory answer. We do have the right equations (QCD) within the SM, but we do not know how to calculate the solutions.

This rigorously posed problem, commonly referred to as the mass-gap problem, is so important and has so many ramifications that the Clay mathematical institute has classified it among the "Millenium problems" awarding it $1M. But of course, it is merely an "engineering calculation problem"...

 

[arivero]

Those people making AdS/QCD models actually compute the energy-momentum tensor of partons in mesons. This is also possible in principle for hadrons just more difficult and has not made it to the publication level so far.

edit
What is this doing in BTSM ?

I should be reporting this
(Indeed I expect that at some moment the mods will move it to Nuclear, SM &c)

Now, to keep answering the question. The OP poster could enjoy to check spires for
QCD and Resonance Physics. Sum Rules.
by Mikhail A. Shifman, A.I. Vainshtein, Valentin I. Zakharov (Moscow, ITEP) .
http://www.slac.stanford.edu/spires/find/hep/www?j=NUPHA,B147,385
and then click "Cited 3340 times" to peek what the bussiness of mass formulae is about.

Also, we should reminder that for charmed and bottomed mesons the masses of the constituent quarks become a relevant component of the mass of the particle.

As for Gell-Mann-Okubo for the octet of mesons, I explained them / referred to them in the post
https://www.physicsforums.com/showpost.php?p=1451883&postcount=373
but the reader should be alerted that this is one of the borderline threads in PF. It has a lot of references to published literature, but also a lot of speculation.

 

[Jim Kata]

I'm jumping on. I heard Skyrmions are used to explain hadrons in the standard model. How does this work, in theory.

 

[humanino]

I'm jumping on. I heard Skyrmions are used to explain hadrons in the standard model. How does this work, in theory.

I was hoping someone more competent than me would provide an answer. The idea has a long history, and relies on topological charges of "knotted" configurations of a mesonic "fluid". Skyrme held the idea that bosonic fields are no less fundamental than fermionic ones, and that they should in some sens be interchangeable, not in a supersymmetric scheme (not written at that time) but in a "bootstrap" model (a sort of "nuclear democracy"). Skyrme modified the non-linear sigma model and proposed a model for the nucleon (called topological skyrmion), and Witten & Co computed static properties of the nucleon with this lagrangian.

Basically, when you hold the quarks together in a MIT-bag, chiral symmetry breaking occurs rather inelegantly right at the surface of the bag where quarks are "reflected". In the non-linear sigma model, the Goldstone boson nature of the pion occurs naturally.

 

[humanino]

Since I can not really do justice do the entire story of Skyrmion, here is an entry point paper :
What exactly is a Skyrmion?[/color]

 

[Gokul43201]

And are these skyrmions topologically related to Solid State skyrmions (spin excitations of a Fermi liquid/gas)?