the neighborhood of the central point.

We could alternatively consider the crossed reaction ${\cal A}(+,-;+,-)$ which is obtained from ${\cal A}(+,+;+,+)$ by the exchange $s\leftrightarrow t$. This makes the quantum corrections somewhat different, with the corresponding kinematic factor being $$1 + \frac{\kappa^2 t}{16 \pi^2}
\left[ \ln\frac{-s}{t}\ln\frac{-u}{t}+ \frac{su}{2t^2} f \left(\frac{-s}{t},\frac{-u}{t}\right) \right]
= 1 +\frac{\kappa^2E^2\left(\frac{29}{10}\ln 2-\frac{67}{45}  \right)}{16\pi^2}$$ instead of the factor in Eq. \[running1\]. The quantum corrections in this channel differ from those of the original reaction and they are not accurately summarized by the same running coupling. This is a manifestation of the crossing problem.

Gravitational scattering of a massless scalar particle
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In renormalizable gauge theories, the running coupling applies universally to all processes. As mentioned above, this is because the running is tied to the renormalization of the gauge charge. General covariance requires that a valid definition of a running $G$ also be universally applicable. The gravitational coupling not only parameterizes the self interactions of gravitons, but it also describes the gravitational coupling of matter. In this section, we look at the effects of loops on the gravitational interactions of a scalar particle.

We consider a scalar particle that has only gravitational interactions. The scattering $\phi + \phi \to 