Elastic scattering of virtual photons via a quark loop in the double-logarithmic approximation

We calculate the amplitude of elastic photon-photon scattering via a single quark loop in the double-logarithmic approximation, presuming all external photons to be off-shell and unpolarized. At the same time we account for the running coupling effects. We consider this process in the forward kinematics at arbitrary relations between $t$ and the external photon virtualities. We obtain explicit expressions for the photon-photon scattering amplitudes in all double-logarithmic kinematic regions. Then we calculate the small-$x$ asymptotics of the obtained amplitudes and compare them with the parent amplitudes, thereby fixing the applicability regions of the asymptotics, i.e., fixing the applicability region for the nonvacuum Reggeons. We find that these Reggeons should be used at $x<{10}^{-8}$ only.

Figure 1

Integration region for moderately virtual photons.

Figure 2

Integration region for deeply virtual photons.

Figure 3

Infrared evolution equation for the amplitude $A_{\gamma \gamma }$. The dashed lines denote the external photons, whereas the straight lines correspond to quarks. The blobs stand for amplitudes calculated in DLA. The letters on the blobs denote the IR cutoffs for the involved amplitudes.

Figure 4

Infrared evolution equation for the amplitude $A_{\gamma q}$.

Figure 5

Dependence of $R_{\gamma }$ on $x$.

Figure 6

Dependence of $R_{\gamma }^{(M)}$ on $x$.

Figure 7

Dependence of $R_{\gamma }^{(D)}$ on $x$.