Tetraquarks are 4 particle quark states since 3^4 = 3×3×3×3 = 81. In trying to organize the 6 known quarks, charm, truth, beauty, up, down, and strange in the weight diagram of SU(3), does the description of and the properties of the X(3872), X(4140), X(5568), X(6900) exotic Hadron tetraquarks fit into the (2,2) + (2,2) + (2,2) = 27 + 27 + 27 = 81 highest weight representation of SU(3) and the (2,2) highest weight representation of SU(3) be a basis from which one could make predictions about the existence of undiscovered tetraquarks?
See B. Hall, Lie Groups, Lie Algebras and Representations, p 151 for a description of the (2,2) highest weight diagram.
Tetraquarks aren't described by representations of a flavor group, but as four quark states of the SU(3) color group, just like baryons are described as three quark states of the SU(3) color group and mesons as two quark states of the same color group.
I don't think that answers the question. SU(3) color deals with the gluon force that binds quarks together. SU(3) flavor deals with three of the six quark flavors, charm, truth, beauty, up, down, and strange and the flavor changing force or the weak force. Just because I mention the 6 quark flavors it does not mean I am referring to tetraquarks as representations of SU(3) flavor.
Once more: The force that binds quarks in bound states is the strong force, mediated by gluons. Flavor doesn't mediate any particular force.
This is not take the subject away from tetraquarks, but
The dimension of the Cartan subalgebra of SU(3) + SU(2) + U(1) is 4. There are 24 standard model particles.
The 6 quarks
The 6 leptons electron, muon, tau, electron neutrino, muon neutrino, and tau neutrino.
The 8 gluons
The W+, W-, Z and the photon.
Think the F4 Lie Algebra which has 48 roots on its weight lattice.
I am going on the assumption that tetraquarks exists and that the
X(3872), X(4140), X(5568), X(6900) should be fit into the proper framework. Have any tetraquarks been observed at SLAC?
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