Origin of Generalized Adler-Weisberger Sum Rule for the Nucleon and $Δ(1232)$

Details Origin of Generalized Adler-Weisberger Sum Rule for the Nucleon and $Δ(1232)$ Origin of Generalized Adler-Weisberger Sum Rule for the Nucleon and $Δ(1232)$

2008-12-20

arxiv.org/abs/0812.3982v1

Physics

High Energy Physics

High Energy Physics - Phenomenology

5 pages

Scientific paper

We derive a relation for the axial-coupling constants of the nucleon and spin-3/2 baryons including the $\Delta(1232)$ with the use of an $SU(2)_R\times SU(2)_L$ invariant Lagrangian. Therein the nucleon belongs to the fundamental representation of the chiral group, while the spin-3/2 baryons belong to higher-dimensional representations $(1,\half)\oplus (\half,1)$. The resulting relation reproduces the one derived from the generalized Adler-Weisberger (GAW) sum rule for the forward $\pi N$ scattering. We clarify the origin of the GAW sum rule for the nucleon and $\Delta$ by taking into account both the group-theoretical aspects and the spontaneous breaking of chiral symmetry. It turns out that the GAW sum rule for the nucleon and $\Delta$ is a consequence of the existence of interactions involving a derivative-pion-field and spontaneous chiral symmetry breaking. This implies that the couplings between $N$ and $\Delta$ in the GAW sum rule vanish with the change of the chiral condensate towards chiral restoration.

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