# Why are delta resonances close in width

• I
The width of Δ resonances is around 110...120 MeV.
All four of them. With modest differences. The difference in width between Δ0 and Δ++ is estimated from 5 to 9 MeV.
Why?
Δ+ and Δ0 resonances have two options to decay.
Δ+→p+π0
Δ+→n+π+
and correspondingly
Δ0→p+π-
Δ0→n+π0
In contrast, Δ++ and Δ- have only one option each:
Δ++→p+π+
Δ-→n+π-
So, from pure consideration of statistics/phase space, shouldn't Δ++ and Δ- live twice as long as Δ0 and Δ+, with widths of 50...60 MeV range?

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Orodruin
Staff Emeritus
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You are implicitly assuming that Δ+→p+π0 should have the same decay rate as Δ++→p+π+. This is not the case.

You can understand this better from a symmetry point of view. In each case you have a state in an isospin 3/2 representation decaying and so this symmetrises the final state isospin. The Δs decay into a particular isospin combination in the final state, it is just that in the case of the ++ and -, this state is a set of physical states while in the + and 0, the state is a linear combination of the physical states and so when you project out the physical states, each channel obtains a factor of 1/2.

mfb
Orodruin
Staff Emeritus