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I am studying hadronic decay of the $W^-$ boson. I am aware that the decay into quarks from different generations is not possible and that charge as well as the total angular momentum have to be conserved. Why are we only considering the decay into two particles? For each decay that is possible, we could simply add a quark and its antiquark (or multiple of these pairs as long as the mass does not exceed the W boson mass) without changing the total charge and also the spin-projection onto the z axis of the two additional particles can cancel out.

What is the reason that this is still not possible?

My only guess is that it has something to do with addition of angular momentum. Is it not enough that the $z$-projection of the total spin remains conserved but does the square of the angular momentum operator (for the irreducible representation of the total angular momentum) also need to stay the same? Does that mean that the spin quantum numbers of the new particles always have to add up to exactly the spin of the initial particle?

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    $\begingroup$ Who says that?? $\endgroup$
    – Hyperon
    Commented Dec 28, 2022 at 20:23
  • $\begingroup$ @Hyperon I cannot find anything that even mentions the possibility of the boson decaying into an anti-up-quark, a down-quark and let's say 11 up-quark/anti-up-quark pairs. $\endgroup$
    – Takitoli
    Commented Dec 28, 2022 at 20:32
  • $\begingroup$ The "microscopic" vertex in the SM is, of course, $W \to q_1 \bar{q}_2$ (by the way, contrary to your claim, a quark and an antiquark from different generations can be produced because of the Cabibbo-Kobayashi-Maskawa mixing matrix). But, as quarks cannot exist as free particles (quark confinement), they form hadron jets, which are then observed in the detector with a large number of hadrons. $\endgroup$
    – Hyperon
    Commented Dec 28, 2022 at 20:51
  • $\begingroup$ @Hyperon Are all the constituents in those hadrons formed in the decay itself or are they from "the environment" (whatever that may be)? I have never quite understood exactly how the hadron jets come to be $\endgroup$
    – Takitoli
    Commented Dec 28, 2022 at 20:58
  • $\begingroup$ A quark from the primary vertex emitts a gluon that in turn produces another quark-antiquark pair. Repeating this also with these "new" quarks/antiquarks gives you a rough picture how jets are produced. You find all this in detail in textbooks on particle physics. $\endgroup$
    – Hyperon
    Commented Dec 28, 2022 at 21:08

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What is the reason that this is still not possible?

You have been misinformed. Hadronization of quarks is a rich process. People have anticipated and estimated rare decays $$ W^+\to \pi^+ \pi^0 \gamma,\\ \pi^+ \pi^+ \pi^-,\\ \pi^+ p\bar p, \\ \pi^+ K^+ K^- ,... $$ at BR of $10^{-6}$ or so... How do you justify something untrue?

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  • $\begingroup$ I see... I was really confused about sentences like "W bosons can decay [...] to a quark and antiquark of complementary types" from the Wikipedia article about W and Z bosons. I thought this excludes decays into more particles. $\endgroup$
    – Takitoli
    Commented Dec 28, 2022 at 23:35
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    $\begingroup$ At the fundamental level, that's what they do; but these quarks are never observed: they hadronize into any number of particles. $\endgroup$
    – Cosmas Zachos
    Commented Dec 28, 2022 at 23:39

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