High-Energy Collision Processes in Quantum Electrodynamics. I

We have made a systematic study of all two-body elastic scattering amplitudes in quantum electro-dynamics at high energies. In particular, we have calculated the high-energy behavior of the following processes: (1) Delbrück scattering, (2) electron Compton scattering, (3) photon-photon scattering, (4) electron-electron scattering, (5) electron-positron scattering, and (6) electron-proton scattering. The processes (1) and (2) are calculated up to the sixth order in the coupling constant $e$, the process (3) up to the eighth order, and the processes (4), (5), and (6) up to the fourth order. Our calculations show that all of these amplitudes are proportional to $s$, the square of the center-of-mass energy, as $s$ becomes large. In other words, we have found that, to these orders, ${\lim }_{s\to \infty }\frac{d\sigma }{\mathrm{dt}}$ exists and is nonzero for all $t\ne 0$, where $-t$ is the square of the momentum transfer. Furthermore, we found it meaningful to assign a factor (we call it the impact factor) to each particle. More precisely, for the high-energy scattering of $a+b\to a+b$, the imaginary coefficient of $s$ for the scattering amplitude is proportional to $\int d{\mathbf{q}}_{\perp }{\left[{({\mathbf{q}}_{\perp }+{\mathbf{r}}_1)}^2\right]}^{-1\mathrm{}}{\left[{({\mathbf{q}}_{\perp }-{\mathbf{r}}_1)}^2\right]}^{-1\mathrm{}}{\mathcal{I}}^a({\mathbf{r}}_1,{\mathbf{q}}_1){\mathcal{I}}^b({\mathbf{r}}_1,{\mathbf{q}}_{\perp })$, where 2r1 is the momentum transfer, and ${\mathcal{I}}^a({\mathbf{r}}_1,{\mathbf{q}}_{\perp })$ and ${\mathcal{I}}^b({\mathbf{r}}_1,{\mathbf{q}}_{\perp })$ are the impact factors of particles $a$ and $b$, respectively. The integration is over the two-dimensional transverse momentum of the virtual photons. The important point is that ${\mathcal{I}}^a \left({\mathcal{I}}^b\right)$ does not depend on what particle $b \mathrm{}\left(a\right)\mathrm{}$ is. We have explicitly found the impact factors for the photon (up to $e^4$) and for the electron, the positron, and the proton (up to $e^2$). In the case of Delbrück scattering, we have also taken care of all higher-order diagrams with an arbitrary number of photons exchanged between the virtual pair and the proton or nucleus. The coefficient of $s$ in this case can be expressed as the integral of the above-mentioned product ${\mathcal{I}}^a {\mathcal{I}}^b$ times modified photon propagators. The impact factor therefore appears to express an intrinsic property of a particle. Our result is consistent with neither the most straightforward interpretation of the Regge-pole model nor that of the droplet model. These inconsistencies are closely related to the nonplanar nature of the diagrams under consideration. Our results on Delbrück scattering are also qualitatively different from those of Bethe and Rohrlich based on the impact-parameter approximation.