ar X iv : h ep - p h / 93 10 25 9 v 1 1 1 O ct 1 99 3 SNUTP 93 - 71 Factorization in graviton interactions

The study of factorization in the linearized gravity is extended to the graviton scattering processes with a massive scalar particle, with a massless vector boson and also with a graviton. Every transition amplitude is shown to be completely factorized and the physical implications of their common factors are discussed. Pacs numbers : 04.60.+n, 12.25.+e Typeset using REVTEX 1 Gravitational gauge invariance and graviton transversality force transition amplitudes of four-body graviton interactions to be factorized. Yang-Mills theories [1] also exhibit a similar factorization property. In our previous work, we have shown that the transition amplitudes of ge → γe [2] and ge → ge [3] processes are completely factorized into a kinematical factor, a fermion-QED Compton scattering form, and another gauge invariant term. In particular, the transition amplitude of graviton scattering with a massive fermion reads Mgf = − κ 2 F [ (ǫ1 ·ǫ2) + (p1 ·ǫ1)(p2 ·ǫ2) (p1 ·k1) + (p2 ·ǫ1)(p1 ·ǫ2) (p1 ·k2) ] × ū(p2) [ 6ǫ2( 6p1+ 6k1 +m)6ǫ1 2(p1 ·k1) + 6ǫ1( 6p1+ 6k2 +m)6ǫ2 2(p1 ·k2) ] u(p1), (1) where all particles are assumed to be incoming, the ǫμ1ǫ ν 1 (ǫμ2ǫ ν 2 ) and k 1 (k μ 2 ) are the wave tensor and four-momentum of the initial(final) graviton, and p1(p2) is the four-momentum of the initial(final) fermion, respectively. The coupling is denoted by κ = √ 32πGN and the overall kinematical factor F is given by F = (p1 ·k1)(p1 ·k2) (k1 ·k2) . (2) The first bracket term in Eq. (1) is none other than the photon-scalar scattering amplitude and the second one is the photon-fermion scattering amplitude. In this paper, we extend the study of factorization in the linearized gravity to the graviton scattering processes with a massive scalar, a massless vector boson, and a graviton itself. Let us now introduce a manifestly gauge invariant four-vector ǫ̃i(i = 1, 2) defined as ǫ̃i = ǫi − (p1 ·ǫi) (p1 ·ki) ki. (3) As a result, the expression of photon-scalar scattering amplitude is converted into a greatly simplified form: ǫ1 ·ǫ2 + (p1 ·ǫ1)(p2 ·ǫ2) (p1 ·k1) + (p2 ·ǫ1)(p1 ·ǫ2) (p1 ·k2) = ǫ̃1 · ǫ̃2. (4)