the spin of Δ(1232) is 3/2 ,but its decay product,like nuclear is 1/2,pion is 0, so the spin isn't consistent before and after decay reaction .Is some of the spin angular momentun translate to orbital angular momentum?So what is the trajectory of a free particle that carries orbital angular momentum?
The total angular momentum is conserved, not spin and orbital angular momentum separately. In the rest frame of the Δ, the total angular momentum is its spin angular momentum; if the decay products are pions, they will carry this as orbital angular momentum, that's all.
It doesn't decay into one particle, however.
You are right ,the branching fractions are estimates .but could you share me some Information on recent advances in Δ experiment research? in addition ,I wonder what is the trajectory of a free pion which after decay that carries orbital angular momentum in fact but not in theory.
There's nothing particularly new to report about these resonances. The statement about the pion is wrong: it carries orbital angular momentum in theory and in fact. Its trajectory is described in terms of the cross section for observing it-more precisely its own decay products. The calculation is standard and described in any textbook on particle physics.
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