Nucleon Resonances
Nucleon resonances are excited states of the nucleon. States with isospin
I=3/2 are usually called `Delta'-resonances, states with isospin I=1/2 are
called N*-states. The states are characterized in the following
nomenclature: L2I2J(W). L is the orbital angular momentum of the
nucleon - pion (or other pseudoscalar meson) pair from the decay of the
resonance in spectroscopical notation (S,P,D,F corresponding to l=0,1,2,3).
I and J are isospin and total angular momentum of the resonance and W stands
for its mass in units of MeV/c2. As an example, the first excited
state of the nucleon, the well-known `Delta'-resonance is notated:
P33(1232) corresponding to an I=3/2, J=3/2 state at
1232 MeV/c2 decaying into a nucleon - pion pair with relative orbital
angular momentum of l=1. The nucleon ground state is P11(938).
Resonances with odd l (P,F,...) have positive, resonances with even l (S,D,...)
have negative parity [the meson - nucleon relative angular momentum contributes
as usual a factor (-1)l to the parity but the decay is via emission
of a pseudoscalar meson with negative intrinsic parity].
All nucleon resonances can decay to the nucleon ground state via the strong
interaction by the emission of mesons. The ususal spin, isospin selection
rules apply of course, so that e.g. `Delta'-resonances (I=3/2) can decay to
the nucleon ground state (I=1/2) via the emission of pions (I=1) but not via the
emission of eta-mesons (I=0) while both decays are possible for the I=1/2
N*-resonances. Due to the decay via the strong interaction the
lifetimes of the resonances are very small (typically 10-23s) and
their widths are correspondingly large. Typical widths of nucleon resonances
are on the order of 100 - 300 MeV/c2 so that most resonances
overlap with their neighbors.
A schematical `level scheme' of the known low lying nucleon resonances is shown
in the above picture. The decays of the resonances via emission of pions or eta
mesons are indicated via the arrows (full: pions, dashed: etas). The widths of
the arrows is proportional
to the branching ratios. All of these resonances have large branching ratios
into N pi, but only the S11(1535) has a significant branching ratio
(~50%) into N eta. This behavior is not yet understood and gave rise to many
discussion about the strucuture of this state. The resonances in the dashed
rectangular (S11, D13, P11),
which all overlap, form the so-called second resonance region of the nucleon.
In the framework of the constituent quark model the simplest excitations of the
nucleon are described by a spin-flip or an orbital excitation of one quark.
The wavefunctions are again composed of space, spin, flavor and color part
|B>=|B>space x |B>spin x |B>flavor
x |B>color
where the total wavefunction and the color part must be antisymmetric. In the
simplest model the constituent quarks are put in a harmonic oscillator
potential with additional spin-spin and spin-orbit dependent interactions
arising from gluon exchange terms. The schematical low energy level scheme
in the framework of such models is shown in the picture.
The P33(1232) is produced via the parallel alignment
of all three quark spins. This state would be degenerate with the nucleon
ground state if only the oscillator potential is considered but the two states
are split by the spin-spin interaction which turns out to be repulsive for
parallel spins. The other low lying resonances arise from the excitation of one
quark into the L=1 p-shell. The L=1 multiplet of the isopspin I=1/2
(N*) resonances is also shown in the picture. Again the states are
split into two groups via the spin - spin interaction and the two groups are
furthermore split by spin - orbit terms.
In this picture the observed D13(1520) and S11(1535)
are assigned to the S=1/2 multiplet and the S11(1650), the
D13(1700) and the D15(1675) to the S=3/2 multiplet.
In addition to this (N*) resonances two isospin 3/2 resonances
with S=1/2, L=1, J=1/2,3/2 complete the set of states with one oscillator
quantum. These basic configurations are schematically depicted below (keep
in mind that the appropriate symmetries must be chosen for the flavor and
color parts).
An excited P11
state should correspond to a 1s --> 2s transition. This would be expected
at a higher excitation energy than the actually observed P11(1440)
(`Roper'-resonance). The structure of this resonance is therefore controversely
discussed in the literature.
In more realistic models resonances with the same quantum numbers like the
two S11 states are of course mixtures of these basic configurations.
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