Determination of low energy parameters for NN-scattering at $N^4 LO$ in all partial waves with $j\le 5$

Abstract

The Variable S-matrix approach offers a unique way to extract low energy threshold parameters for a given NN potential. We extract those parameters for the np system from the NijmII and Reid93 potentials, to all partial waves with total angular momentum $j \le m5 $, using the generalized effective range expansion % $$ ({\bf f}^{sj})_{l,l'} k^l k^{l'} = -({{\bf a}^{sj}}^{-1})_{l,l'} + \frac12 ({\bf r}^{sj})_{l,l'} k^2 + ({\bf v_2}^{sj})_{l,l'} k^4 + + ({\bf v_3}^{sj})_{l,l'} k^6 + + ({\bf v_4}^{sj})_{l,l'} k^8 + ... - {\rm i} k^{l+l'+1} $$ % where $ {\bf f}^{sj} = ({\bf S}^{sj} - {\bf 1}) / (2 {\rm i} k) $ is the scattering amplitude and ${\bf S}^{sj}$ is the unitary S-matrix in coupled channel space with total spin $s$ and total angular momentum $j$. Our calculation includes all the relevant contributions of the full amplitude to order ${\cal O} (k^8) $ in the CM momentum. We also discuss the validity of the generalized effective range expansion in the region of analyticity $ k \le m_\pi /2$.