Resonant phenomena in antihydrogen-hydrogen scattering

We present a treatment of cold hydrogen-antihydrogen collisions based on the asymptotic properties of atom-antiatom interactions. We derive general formulas for the elastic and inelastic cross sections and for the scattering lengths and analyze their sensitivity to the parameters characterizing the inelasticity of the collision process. Given the inelasticity, we obtain bounds for the complex scattering length. We investigate the influence of strong nuclear forces and the isotope effects in $\text{H}\overline{\text{H}}$ and $\text{D}\overline{\text{H}}$ collisions and demonstrate enhancement of these effects due to the presence of the near-threshold narrow $\text{H}\overline{\text{H}}$ $\left(\text{D}\overline{\text{H}}\right)$ states. The values of the elastic and inelastic cross sections with simultaneous account of rearrangement and strong forces are presented. General expressions for the (complex) energies of the near-threshold $\text{H}\overline{\text{H}}$ states are obtained.

Figure 1

(Color online) Elastic cross section for $\text{H}-\overline{\text{H}}$ collisions as a function of phase ${\delta }_R$, calculated at $E\to 0$ according to ${\sigma }_{el}=4\pi {\left|a\right|}^2$ with $a$ given by Eq. (7).

Figure 2

(Color online) Annihilation cross section for $\text{H}-\overline{\text{H}}$ collisions as a function of phase ${\delta }_R$, calculated at the energy $E={10}^{-6}\text{a}\text{.u}\text{.}$ according to ${\sigma }_{in}=4\pi \text{Im}\left(a\right)/k$, with $a$ given by Eq. (7).

Figure 3

(Color online) Ratio of the elastic to the annihilation cross section as a function of phase ${\delta }_R$, calculated at the energy $E={10}^{-6}\text{a}\text{.u}\text{.}$, with $a$ given by Eq. (7).

Figure 4

(Color online) Complex scattering length $a$ as a function of the short-range phase shift ${\delta }_R$, for the fixed value of ${\delta }_I$. The square indicates the complex value of the scattering length obtained in model I; circle—in model II; and triangle—in model III. The value of $a\simeq 5-i5\text{a}\text{.u}\text{.}$ marked by the star corresponds to the limiting case of strong absorption ${\delta }_I\gg 1$.

Figure 5

(Color online) Elastic cross section as a function of the real part $v$ of the optical potential (24), calculated at the energy $E={10}^{-6}\text{a}\text{.u}\text{.}$

Figure 6

(Color online) Inelastic cross section as a function of the real part $v$ of the optical potential (24), calculated at the energy $E={10}^{-6}\text{a}\text{.u}\text{.}$

Figure 7

(Color online) Evolution of the poles of the $S$-matrix in the complex $k$-plane as a function of the real part of the short-range phase shift, $-\pi /2\le {\delta }_R\le \pi /2$. The solid line corresponds to the fixed inelasticity parameter ${\delta }_I=0.3$ (model I), the dash-dotted line corresponds to the inelasticity parameter ${\delta }_I=0.41$ (model II), and the dashed line corresponds to the inelasticity parameter ${\delta }_I=0.7$ (model III). An empty square marks the virtual state, a star marks the point where the $S$--matrix pole crosses the real axis of the complex $k$-plane, and a circle marks the position of the bound state corresponding to the “true” value of the phase-shift ${\delta }_R$, deduced for each model from the corresponding scattering length. In calculations, the diverging character of the virtual states was tempered using the technique of exterior complex scaling [37]. The arrow shows the direction of the pole evolution with increasing ${\delta }_R$, starting from ${\delta }_R=-\frac{\pi }{2}$.