An N/D study of the S ₁₁ channel πN scattering amplitude *

Abstract

Extensive dynamical calculations are conducted in the study of channel low energy scatterings, based on various phenomenological model inputs of left cuts at the tree level. The subtleties of the singular behavior of the partial wave amplitude, at the origin of the complex plane, are analysed in detail. Furthermore, it is found that the dispersion representation for the phase shift, , must be modified in the case of scatterings. An additional contribution from the dispersion integral exists, which approximately cancels the contribution of the two virtual poles located near the end points of the segment cut, induced by channel nucleon exchanges. With limited reliance on the details of the dynamical inputs, the subthreshold resonance survives.

The $N/D$ study on the singularity structure of $πN$ scattering amplitudes

“…It is pointed out that in the 2 ,2 = 11 channel partial wave scattering amplitude, there exist two virtual poles, according to the Ref. [15] . In fact, the existence of virtual poles are quite universal, but different channels behaves rather differently.…”

“…In Ref. [15], The authors do several calculations with various Im ( ) as the input of / method. Different input lead the unitary amplitudes ( ) possessing different .ℎ.…”

“…The phase shift and its PKU decomposition are plotted in Ref. [15], and it tells us the background cut ( .ℎ. .)…”

“…One way is like the Ref. [15], and let the auxiliary function ˜ ( ′ , ) in eqs. ( 6) and ( 7) formally be written as ˜ ( ′ , ) = ( ′ ) − ( ), where is taken as the O ( 1 ) partial wave amplitude plus the O ( 2 ) part of +− ..…”

“…The definition can be found in Ref. [15] Another ways is to introduce intermediate function ¯ ( ) and the partial wava amplitude ( ) = ¯ ( )/ 2 , where ¯ ( ) can be written as :…”