Photon-graviton mixing in an electromagnetic field

Einstein-Maxwell theory implies the mixing of photons with gravitons in an external electromagnetic field. This process and its possible observable consequences have been studied at tree level for many years. We use the worldline formalism for obtaining an exact integral representation for the one-loop corrections to this amplitude due to scalars and fermions. We study the structure of this amplitude, and obtain exact expressions for various limiting cases. 1 ar X iv :0 71 1. 09 92 v1 [ he pth ] 7 N ov 2 00 7 1 Photon-graviton mixing at tree level and one loop As has been realized already in the sixties [1] Einstein-Maxwell theory in a constant electromagnetic field contains a tree level vertex for photon-graviton conversion, κhμνF f α − 1 4 κhμF fαβ. (1) Here hμν denotes the graviton, fμν the photon, and F μν the external field. κ is the gravitational coupling constant. This vertex implies the possibility of photon-graviton oscillations [1, 2, 3, 4, 5, 6, 7] which are analogous to the better-known photon–axion oscillations in a field [8]. In the presence of an external field, the true eigenstates of propagation can be obtained by solving the following system of dispersion relations [3] ( ηk − kk i 2 κC i 2 κC k 2 4 (ηη+ηη−2ηη +...) )( aβ(k) hκλ(k) ) = 0 . (2) Here C denotes the Fourier transform of the vertex (1), C = (F · k)η + F k + F k − (F · k)η − (F · k)η . (3) For small deviations from the vacuum dispersion relations k = 0 this second order equation can be linearized. An efficient formalism for solving the system (3) under this condition was developed in [3]. Taking one–loop corrections into account, the dispersion relation matrix gets modified in the following way [9, 10], ( ηk − kk − Π̄ i 2 κC − Π̄ i 2 κC − Π̄ k2 4 (ηη+ηη−2ηη +...)−Π̄ )( aβ(k) hκλ(k) ) = 0 . (4)