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Just over two-thirds of the time, a $W$ boson decays into quarks, usually an up quark and a down antiquark... Right?

Since quarks hate being alone, what happens next?

Does the up quark pull a single other quark (or, rather, antiquark) from the 'sea' and become a meson, that then decays?

And, then, the down antiquark does the same thing?

Is there any chance a proton or neutron is created?

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The quarks and gluons undergo a process called "hadronization". Just after the decay, the quarks can be free (at very high energies/ very short scales) but as the distance increases, the strong force starts to increase. At one point it becomes so strong that new quarks and generated from the vacuum and form mesons.

You can look at the Lund string model which is actually used in Monte Carlo methods in (eg Pythia) to generate such processes. Baryons also form just is difficult to visualize how. Any in jets mesons are much more prevalent than baryons.

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AngryBach's answer is correct, but I want to address what seems to be a misconception in the question: the idea that a single quark can pull another single quark from the vacuum in order to make itself color-neutral.

There are never any lone quarks in the hadronization process. The $u$ and $\bar d$ are produced in a color-singlet state, confined together in a flux bag, making them basically a ridiculously excited state of a pion. They fly apart, and the bag confining them stretches until pair-production of additional valence quarks allows it to split into color-neutral pieces separated by vacuum. All of the quarks are produced in pairs, and it is not really the $u\bar d$ quarks that trigger pair production; it would be more accurate to say that the flux tube does it.

Quarks are never alone in vacuum. When they're said to be "free", it really means they are confined in a region large enough, and/or the time scale under consideration is short enough, that they don't notice their confinement.

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