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Due to some recent "discoveries" (yet to be completely confirmed), we have hinted the existence of tetraquark particles.

Why QCD exotics like multiquark states beyond n=2,3 (mesons, protons,neutrons,...) like tetraquarks, pentaquarks,...or even glueballs could be useful for new physics? And more generally: Quark spectroscopy and "molecular" QCD/quark chemistry...What could they be useful for?

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As I always say to these sorts of questions: define "useful." If you mean "can I make money off it" the answer is clearly no, not in the forseeable future anyway. But the more test cases we have for our ability to do calculations in a strongly coupled field theory the better! I guess the main application (beyond their intrinsic interest) will be helping us understand the QCD phase transition in the early universe. There are still mysteries about how the universe made the transition from free quarks and gluons to colour-bound hadrons. –  Michael Brown Jul 13 '13 at 1:02

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In my point of view there is a diversity of reasons, why such models are investigated for some years.

  1. First of all the Extension of known Hadronic spectrum domninated by the usual mesons $\left(S=0, 1, 2,\ldots\right)$ (as $\left|q\overline{q}\right\rangle$ states) and baryons $\left(S=\frac{1}{2}, \frac{3}{2},\ldots\right)$ (as $\left|qqq\right\rangle$ states) to new exotic states whose are impossible for the previously mentioned hadrons.

  2. Get a deeper insight into hadronic structure

  3. The two and three quark structure all have a unique color structure. Multi-qark systems have different color structures and understanding the reason for the different color strucure is basic step in the general reason for color abundance in QCD.

  4. The experimental search for exotic hadronic states is the only way to falsificate corresponding theoretical models, like MIT-Bag model, etc. It is the same reason why they built the LHC to search for Higgs as the last missing SM-Model or to look for physics beyond the Standard Model like supersymmetric extensions of the Standard Model which might resolve open questions arising within the Standard Model.

  5. As so often in physics in the end your particular model of a system might not apply in the sense you might have guessed before. For example, long before QCD was established the appropriate theory of strong interactions the Regge model with strings was favored. This string model turned out not to be consistent with experiment, so it was thrown over in regard of the strong interactions and instead the String Model became the basic ingredient in String Theory which is now a possible candidate for Grand Unifying Theory of Nature. The main reason why it is still investigated, is the same why multi-quark states are still searched for nobody has until now proven that it is wrong. As Feynman said:

" It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong."

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except I think that the Regge model was not falsified. Just that the quark model and SU(3)xSU(2)xU(1) were so elegant and made so many successful predictions that the Regge model fell by the way. Nobody was interested in higher spin hadronic resonances any more, which was the success of the Regge model. –  anna v Jul 13 '13 at 9:55
    
String model for strong interaction proved not useful and thus was thrown over also I think there were inconsistencies due to high spin representations in the Regge model as I remember. –  Hansenet Jul 13 '13 at 21:36

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