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I was studying Particle Physics then suddenly I came up with a question that why only Baryons are made up of three quarks, at first I thought to to conserve Baryon number which is $\frac{1}{3}$ for Quarks and so the total comes to be '1' ($\frac{1}{3}$+$\frac{1}{3}$ + $\frac{1}{3}$ = 1 ), but I am not satisfied with my answer and I want a general answer which applies to all fundamental particles and provides a rule to figure out the number of quarks.

Why Baryons are made up of three quarks and why Mesons are made of only two quarks ?

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    $\begingroup$ They just are. The models and subsequent theories to describe them were constructed to accommodate this fact. No successful constituent model with different numbers has been constructed. By now, the fact is embedded in several ways in the theories, (including anomalies and neutral pion decay), but you may have to explain what you mean by "why"... $\endgroup$ Mar 23, 2023 at 17:36
  • $\begingroup$ In HSE you might ask "how did they conclude the baryons are made of three quarks and the mesons of a quark-antiquark pair?", which is a profoundly different question, with an astounding history of coincidences and lucky accidents... A deeper corollary question is "why then?". $\endgroup$ Mar 23, 2023 at 18:45
  • $\begingroup$ are you aware of the first data that showed symmetries among hadrons that led to the quark model? the figures shown here en.wikipedia.org/wiki/Quark_model are given by experimental data $\endgroup$
    – anna v
    Mar 23, 2023 at 19:25

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Quarks can only exist in bound states that have a net colour charge of zero. The reason for this is that the strong force between any particles with a net non-zero colour charge does not decrease with increasing distance. This means the energy required to separate any such pairs of particles quickly becomes large enough to generate new particle-antiparticle pairs.

The simplest way to create a particle with a net colour charge of zero is to combine the three colour charges or to combine a pair of colour and anti-colour charges. That is, combine either three quarks, or quark and anti quark pair, and this indeed what we observe in our particle colliders. Then we define the terms baryon to mean a three quark state and meson to mean a two quark state.

So the reason only baryons can have three quarks is because the term baryon is defined as a three quark state. If the state has three quarks we call it a baryon, and if it doesn't have three quarks we don't call it a baryon.

Though actually ... this is not quite true. Given our definition of the three quark state as a baryon it then makes sense to associate each quark with a baryon number of $+\tfrac13$ and each antiquark with a baryon number of $-\tfrac13$ just as you say in the question. Then it is possible to combine other numbers of quarks and antiquarks to create a particle with a zero colour charge and a total baryon number of one.

An example is the pentaquark, $qqqq\bar{q}$, which has a total baryon number of one. Wikipedia claims that the pentaquark is an exotic baryon on these grounds, though I confess I am unsure how reliable Wikipedia's claim is.

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  • $\begingroup$ Ummm the baryons had a definition long before the quark model was proposed. I suspect the OP is asking "why three and nothing else?"... $\endgroup$ Mar 24, 2023 at 16:32
  • $\begingroup$ @CosmasZachos right, but we now define it as I've described above. Actually I guess it could also be used for a pentaquark since that has a net baryon number of one. Maybe I should extend my answer to include that definition. $\endgroup$ Mar 24, 2023 at 16:34
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Well the development of quark model is itself a long journey, Gell-man and others proposed the quark model after seeing certain symmetries in particle systems, namely if you arrange particles according to their isospin and strangeness you get many symmetries Octet of Baryons: courtesy Wikipedia

Like in the arrangement you can see.

First the idea of nucleons was their in which the proton and neutron can be considered as two states of a single system, given their symmetries,

This could be explained by $SU(2)$ lie group which has isospin as it's generators but since we are adding the strangeness as well we moved to New symmetry group $SU(3)$ and it has total of 8 generators.

Namely $3 \otimes 3\otimes 3= 1 \oplus 8 \oplus 8 \oplus 10 $

Which we got through breaking it in symmetric and anti symmetric part, you see all the Baryons (known at that time) could be fit in to these representations moreover it predicted $\Omega$ Baryons.

The 8 represent octet of Baryons and the other 8 represent the resonance states and 10 represent the decuplet of Baryons and 1 represent the singlet,

See here to fit all this we needed to take the tensor product of three spaces, i.e. indicating a need of three particle system.

Physics boils down to the symmetries as with symmetry there are associated conserved quantities.

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  • $\begingroup$ Am leaving loose ends so that you try to look up for them. $\endgroup$
    – Pradyuman
    Mar 24, 2023 at 17:47

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