Feynman Path Integral Formulation

14 1 Continuum Formulationover the previous one, namely that it leads to considerably simpler Feynman rules,both for the graviton vertices and for the scalar-graviton vertices, which can beadvantageous when computing one-loop scattering amplitudes of scalar particles(Hamber and Liu, 1985). Even the original gravitational action has a simpler formin terms of the variables of Eq. (1.80) as shown originally in (Goldberg, 1958).Again, when performing Feynman diagram perturbation theory a gauge fixingterm needs to be added in order to define the propagator, for example of the form1K 2 (∂μ√ ggμν ) 2 . (1.81)In this new framework the bare graviton propagator is given simply byD μναβ (k)= η μαη νβ + η μβ η να − η μν η αβ2k 2 , (1.82)which should be compared to Eq. (1.77) (the extra factor of one half here is justdue to the convention in the choice of K). One notices that now there are no factorsof 1/(d − 2) for the graviton propagator in d dimensions. But such factors appearinstead in the expression for the Feynman rules for the graviton vertices, and such(d − 2) −1 pole terms appear therefore regardless of the choice of expansion field.For the three-graviton and two ghost-graviton vertex the relevant expressions arequite complicated. The three-graviton vertex is given byU(q 1 ,q 2 ,q 3 ) α1 β 1 ,α 2 β 2 ,α 3 β 3=− K [ ()q 2 2 (α 1q 3 β 1 )2η α2 (α 3η β3 )β 2− d−2 2 η α 2 β 2η α3 β 3()+q 1 (α 2q 3 β 2 )2η α1 (α 3η β3 )β 1− d−2 2 η α 1 β 1η α3 β 3()+q 1 (α 3q 2 β 3 )2η α1 (α 2η β2 )β 1− d−2 2 η α 1 β 1η α2 β 2+2q 3 (α 2η β2 )(α 1η β1 )(α 3q 2 β 3 ) + 2q1 (α 3η β3 )(α 2η β2 )(α 1q 3 β 1 ) + 2q2 (α 1η β1 )(α 3η β3 )(α 2q 1( β 2 ))+q 2 · q 3 2d−2 η α 1 (α 2η β2 )β 1η α3 β 3+ d−2 2 η α 1 (α 3η β3 )β 1η α2 β 2− 2η α1 (α 2η β2 )(α 3η β3 )β 1+q 1 · q 3 ( 2d−2 η α 2 (α 1η β1 )β 2η α3 β 3+ 2d−2 η α 2 (α 3η β3 )β 2η α1 β 1− 2η α2 (α 1η β1 )(α 3η β3 )β 2)+q 1 · q 2 ( 2d−2 η α 3 (α 1η β1 )β 3η α2 β 2+ 2d−2 η α 3 (α 2η β2 )β 3η α1 β 1− 2η α3 (α 1η β1 )(α 2η β2 )β 3)].The ghost-graviton vertex is given by(1.83)V (k 1 ,k 2 ,k 3 ) αβ,λμ = K [ −η λ(α k 1β) k 2μ + η λμ k 2(α) k 3β)], (1.84)and the two scalar-one graviton vertex is given byK2(p 1μ p 2ν + p 1ν p 2μ − 2d − 2 m2 η μν), (1.85)