Strong-coupling solutions in an isotopic eikonal problem

An eikonal representation for small-angle $\mathrm{NN}$ scattering is constructed in the strong-coupling limit of the simplest SU(2) massive-vector-meson exchange model. One finds that multiple charged exchanges corresponding to net isospin transfers different from zero tend to be washed out in this limit. The isotopic triplet and singlet amplitudes have the form $T_{\mathrm{eik}}^{\mathrm{}\left(I\right)\mathrm{}}\mathrm{}(s,t)\mathrm{} = \left(\frac{\mathrm{is}}{2m^2}\right)\int d^{2}b{e}^{i\overrightarrow{q}·\overrightarrow{b}}[1-\exp (i{\chi }_0^{\mathrm{}\left(s\right)\mathrm{}} + i〈{\overrightarrow{\tau }}_1·{\overrightarrow{\tau }}_2〉 = {\chi }_0^{\mathrm{}\left(V\right)\mathrm{}}\left)\right]\exp \left[Q^{\mathrm{}\left(I\right)\mathrm{}}\right(b,\mu ,q,s\left)\right]$, where ${\chi }_0^{\mathrm{}\left(s\right)\mathrm{}}\mathrm{}\left(b\right)\mathrm{}$ is the typical eikonal function of isoscalar exchange, ${\chi }_0^{\mathrm{}\left(V\right)\mathrm{}}\mathrm{}\left(b\right)\mathrm{}$ is the same function with coupling and mass parameters of the isovector exchange, and $〈{\overrightarrow{\tau }}_1·{\overrightarrow{\tau }}_2〉 = 2I(I+1\mathrm{})-3\mathrm{}$. Very crude estimates, especially for the singlet case, suggest that $Q^{\mathrm{}\left(1\right)\mathrm{}}\simeq -\ln (1+{\phi }^2)$, and $Q^{\mathrm{}\left(0\right)\mathrm{}}\simeq -\left(\frac{1}{2}\right)\ln [1+{\left(4\mathrm{}\phi \right)}^2]$, with $\phi \simeq \left(\frac{1}{\pi }\right)\ln \left(\mu b\right)$, $1>\frac{q}{\sqrt{s}}>\mu b$, and $\phi \simeq \frac{q}{\mu b{\left(2\mathrm{}\pi s\right)}^{\frac{1}{2}}}$, $1>\mathrm{}\mu b>\mathrm{}\frac{q}{\sqrt{s}}$. A qualitative discussion is given of the expected small effect of multiple isovector-meson exchange on form-factor damping at large momentum transfers.