2.1 Overview of gravity Feynman rules

Scattering of gravitons in flat space may be described using Feynman diagrams [44Jump To The Next Citation Point45Jump To The Next Citation Point138Jump To The Next Citation Point]. The Feynman rules for constructing the diagrams are obtained from the Einstein–Hilbert Lagrangian coupled to matter using standard procedures of quantum field theory. (The reader may consult any of the textbooks on quantum field theory [107Jump To The Next Citation Point141Jump To The Next Citation Point] for a derivation of the Feynman rules starting from a given Lagrangian.) For a good source describing the Feynman rules of gravity, the reader may consult the classic lectures of Veltman [138Jump To The Next Citation Point].
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Figure 1: The Feynman propagator and three- and four-point vertices in Einstein gravity.

The momentum-space Feynman rules are expressed in terms of vertices and propagators as depicted in Figure 1View Image. In the figure, space-time indices are denoted by μi and νi while the momenta are denoted by k or ki. In contrast to gauge theory, gravity has an infinite set of ever more complicated interaction vertices; the three- and four-point ones are displayed in the figure. The diagrams for describing scattering of gravitons from each other are built out of these propagators and vertices. Other particles can be included in this framework by adding new propagators and vertices associated with each particle type. (For the case of fermions coupled to gravity the Lagrangian needs to be expressed in terms of the vierbein instead of the metric before the Feynman rules can be constructed.)

According to the Feynman rules, each leg or vertex represents a specific algebraic expression depending on the choice of field variables and gauges. For example, the graviton Feynman propagator in the commonly used de Donder gauge is:

1-[ ---2-- ]---i--- P μ1ν1;μ2ν2 = 2 ημ1μ2ημ2ν2 + ημ1ν2ην1μ2 − D − 2η μ1ν1ημ2ν2 k2 + i𝜖. (2 )
The three-vertex is much more complicated and the expressions may be found in DeWitt’s articles [44Jump To The Next Citation Point45Jump To The Next Citation Point] or in Veltman’s lectures [138Jump To The Next Citation Point]. For simplicity, only a few of the terms of the three-vertex are displayed:
μ1ν1,μ2ν2,μ3ν3 μ1ν1 μ2ν2 μ3ν3 μ3 ν3 μ1μ2 ν1ν2 G de Donder (k1,k2,k3) ∼ k1 ⋅ k2η η η + k1 k2 η η + many other terms, (3 )
where the indices associated with each graviton are depicted in the three-vertex of Figure 1View Image, i.e., the two indices of graviton i = 1,2,3 are μ ν i i.
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Figure 2: Sample gravity tree-level Feynman diagrams. The lines represent any particles in a gravity theory.
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Figure 3: Sample loop-level Feynman diagrams. Each additional loop represents an extra power of Planck’s constant.

The loop expansion of Feynman diagrams provide a systematic quantum mechanical expansion in Planck’s constant ¯h. The tree-level diagrams such as those in Figure 2View Image are interpreted as (semi)classical scattering processes while the diagrams with loops are the true quantum mechanical effects: Each loop carries with it a power of ¯h. According to the Feynman rules, each loop represents an integral over the momenta of the intermediate particles. The behavior of these loop integrals is the key for understanding the divergences of quantum gravity.

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