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In the textbook, I am following it describes flavour space with basis up, down and strange quarks. I am not sure why we did not choose up, charm and top as the basis and why only three bases can define the complete flavour symmetry.

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  • $\begingroup$ You are egregiously misusing "flavor space" and "flavor symmetry", to the point of obscurity. You probably mean "family/generation structure" which your alternate description typifies by the charge 2/3 quark representatives of each. Are you trying to actually understand the somewhat artificial family classification of quarks? You must reorganize and clarify your question. $\endgroup$ – Cosmas Zachos Oct 19 at 13:35
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The answer to this is probably historical.

The up, down, and strange quarks form a basis of states in an $SU(3)$ algebra. This lead to the concept of flavour symmetry and the what is called the eight-fold way.

The particles of the this eight-fold way are built from these three quarks. This was earlier on in the formulation of baryon states.

Later on, it was discovered that a separate $SU(3)$ symmetry, associated with quantum chromodynamic degrees of freedom, exists which describes the colours of quarks. In this case, each flavor of quark can exist in three colors. Other flavors of quarks (charm, bottom, and top) were discovered subsequently. They are much heavier than u, d, and s and so do not fit easily into a generalisation $SU(n )$ of $SU(3)$ with $n > 3$.

But the approximate $SU(3)$ symmetry of particles containing up, down, and strange quarks remains a useful guide to the properties of the strong interactions.

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The symmetries were discovered in experiments with leptons and protons in a historical train over many years. To start with , the discovery of quarks happened when the observed particles could be assigned into an SU(3) group, the eightfold way:

dec

The baryon decuplet.

The the $Ω^-$ was predicted from these symmetries.

Note that these beautiful data in the quark model have as valence quarks the u, down , and strange, and they are low mass. So the basis for the quark model was chosen with these low mass quarks.

What you propose would not be consistent with the data of the eightfold way, the observed particles.

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