More on Heisenberg’s model for high energy nucleon-nucleon scattering

We revisit Heisenberg’s model for nucleon-nucleon scattering which admits a saturation of the Froissart bound. We examine its uniqueness, and find that up to certain natural generalizations, it is the only action that saturates the bound. We find that we can extract also subleading behavior for ${\sigma }_{\mathrm{tot}}(s)$ from it, though that requires a knowledge of the wave function solution that is hard to obtain, and a black-disk model allows the calculation of ${\sigma }_{\text{elastic}}(s)$ as well. The wave-function solution is analyzed perturbatively, and its source is interpreted. Generalizations to several mesons, the addition of vector mesons, and curved space regimes are also found. We discuss the relations between Heisenberg’s model and holographic models that are dual to QCD-like theories.

Figure 1

The total cross section as a function of $\sqrt{s}$ for $pp$ and $p\overline{p}$ scattering.

Figure 2

On the right side we have the antipodal setup of the Sakai-Sugimoto model where $u_0=u_{\mathrm{\Lambda }}$. On the left side we have the generalized nonantipodal setup.

Figure 3

$\varphi (t)$ as a function of $t$ for $\frac{m}{l}=0.1$.

Figure 4

$\varphi (t)$ as a function of $t$ for $\frac{m}{l}=1$.

Figure 5

$\varphi (t)$ as a function of $t$ for $\frac{m}{l}=5$.