I have the Particles Physics Booklet and I noticed that the resonances that decay into a nucleon and pion are indicated by an abbreviation. For example $P_{33}$ is associated to the $\Delta (1232)$ resonance. What is the meaning of this symbol?

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What is the meaning of which symbol? – dmckee Mar 21 '11 at 20:41
Of the symbol $P_{33}$ – Andrea Amoretti Mar 21 '11 at 21:19

$\Delta$ represents the constituents (up and down quarks, three in total) in the spin and isospin 3/2 state, the number (1232) represents its the mass of the resonance in MeV.

As for the notation $P_{33}$ -- this is an example of the notation $L_{2I,2J}$. $I$ stands for the isospin ($I=3/2$ in this case) and $J$ stands for the total angular momentum ($J=3/2$) of the resonance.

The letter $L$ takes the 'values' $S$, $P$, $D$, $F$, $G$, $H$,.. (which corresponds to $L=0,1,2,...$

It represents the angular momentum state of the nucleon-pion pair to which the resonance decays.

http://jazz.physik.unibas.ch/~krusche/resonance_basics.html

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For baryons, the notation $P_{33} = L_{2I\;2J}$ indicates the symmetry of this set of particles.

"L" gives the angular momentum. This counts up as "S", "P", "D", "F". In the case of $P_{33}$ we have a P which indicates the angular momentum is 1 (or $\hbar$ if you prefer).

The "I" values is 3/2. This is the "isospin" quantum values. The 3/2 says that there are four values possible, 3/2, 1/2, -1/2, and -3/2. Isospin is the symmetry that relates up and down quarks. Up quarks have charge +2/3 while down quarks have charge -1/2. So the particle comes in four isospin states: $\Delta^{++},\Delta^+,\Delta^0,\Delta^-$.

The "J" is for total angular momentum. So if you measured the total angular momentum you could get +3/2, +1/2, -1/2, or -3/2.

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