Feynman Path Integral Formulation

1.4 Feynman Rules 15
where the p 1 , p 2 denote the four-momenta of the incoming and outgoing scalar field,
respectively. Finally the two scalar-two graviton vertex is given by
K 2 m 2 (
η
2(d − 2) μλ η νσ + η μσ η νλ − 2 )
d − 2 η μνη λσ , (1.86)
where one pair of indices (μ,ν) is associated with one graviton line, and the other
pair (λ,σ) is associated with the other graviton line. These rules follow readily
from the expansion of the gravitational action to order G 3/2 (K 3 ), and of the scalar
field action to order G (K 2 ), as shown in detail in (Capper et al, 1973). Note that
the poles in 1/(d − 2) have disappeared from the propagator, but have moved to
the vertex functions. As mentioned before, they reflect the kinematic singularities
that arise in the theory as d → 2 due to the Gauss-Bonnet identity. As an illustration,
Fig. 1.2 shows the lowest order diagrams contributing to the static potential between
two massive spinless sources (Hamber and Liu, 1995).
Fig. 1.2 Lowest order diagrams illustrating modifications to the classical gravitational potential
due to graviton exchange. Continuous lines denote a spinless heavy matter particle, short dashed
lines a graviton and the long dashed line the ghost loop. The last diagram shows the scalar matter
loop contribution.