Field Theory of the Two-Pion System (SandDWaves)

This is a sequel to a previous paper, designated as (I), which dealt primarily with the two-pion $P$ wave in the $\lambda {\phi }^4$ model. Here, as in (I), we make use of a recently developed series expansion, applying the procedure to $S$ and $D$ waves. With an input of $\left(\pi \mathrm{mass}\right)=1\mathrm{}$, $\left(\rho \mathrm{mass}\right)=5.53\mathrm{}$, our results for the isospin-$T$, spin-$J$ scattering lengths ${a_J}^T$ and the Chew-Mandelstam coupling constant $\lambda$ are: ${a_0}^0=-0.72\mathrm{}$, ${a_0}^2=-0.40\mathrm{}$, ${a_1}^1=+0.032\mathrm{}$, ${a_2}^0=+0.0032\mathrm{}$, ${a_2}^2=0.0026\mathrm{}$, and $\lambda =+0.22\mathrm{}$. The results for ${a_0}^{0,2}$ and $\lambda$ are improved here [as compared with those in (I)] by a higher order correction, but with little change; ${a_1}^1$ is as quoted in ($I$); the ${a_2}^{0,2}$ are new results. We also calculate the $S$-wave phase shifts ${{\delta }_0}^T$ and find that ${{\delta }_0}^0$ and ${{\delta }_0}^2$ go through $-\frac{1}{2}\pi$ on the way down at energies of 760 and 630 MeV, respectively. We finally calculate the $D$-wave phase shifts: The $f_0$ resonance is found at 1950 MeV, and a corresponding $T=2\mathrm{}$, $D$-wave resonance is found at 1520 MeV. The substantial discrepancy with experiment for the $f_0$ is attributed to the need for considering more terms in our expansion as energy increases, as well as to a diminished reliability of the $\lambda {\phi }^4$ model at high energy.