Gravitational Correction to Running of Gauge Couplings

We calculate the contribution of graviton exchange to the running of gauge couplings at lowest nontrivial order in perturbation theory. Including this contribution in a theory that features coupling constant unification does not upset this unification, but rather shifts the unification scale. When extrapolated formally, the gravitational correction renders all gauge couplings asymptotically free.

Figure 1

A typical Feynman diagram for a gravitational process contributing to the renormalization of a gauge coupling at one loop. Curly lines represent gluons. Double lines represent gravitons. The three-gluon vertex ■ is proportional to $g$, while the gluon-graviton vertex ● is proportional to $E/M_{\mathrm{P}}$.

Figure 2

When gravity is ignored, the three gauge couplings of a model theory evolve as the inverse logarithm of $E$ at one-loop order (dashed curves). Initial values at 100 GeV were set so that the curves exactly intersect at approximately ${10}^{16}\mathrm{GeV}$. When gravity is included at one loop (solid curves), the couplings remain unified near ${10}^{16}\mathrm{GeV}$, but evolve rapidly towards weaker coupling at high $E$.