Quantum Discontinuity for Massive Gravity with a Cosmological Term

We report on the recent work on the van Dam-Veltman-Zakharov discontinuity for massive and partially massless gravitons in A(dS) space at one-loop. The question of whether the graviton has a small non-zero mass or exactly zero mass has been addressed by van Dam and Veltman, and Zakharov[l]. The difference is physically detectible, e.g. by effect on bending of light by the sun. The case for non-zero A has been studies by references [2] and [3] for de-Sitter and Anti-deSitter space respectively. In the latter it was found that the massive graviton has a smooth limit. An interesting phenomenon [4] that occurs in (A)dS space is partial masslessness with reduced degrees of freedom Theory M < |A M = |A M -0 D.O.F. 5 4 2 Unitary? No Yes Yes All the above analysis is classical. One might ask the question what happens when loop effects are taken into account. It was shown [5] that the discontinuity persists in the flat space case. In [6] it was shown that this is also true in the case A ^ O . The starting point of the analysis is our starting point will be the action } -f SM[/V] + ^[h T] where SL the Einstein-Hilbert action SE = -^^ fMdx^(R 2A), linearized about a background metric g^ according to g^ = g^v -f /c/i^. If the background metric is Einstein, then the linearized action for h^v is n 2R^a) h<>« V'/iPM V"/#] (1) where h^v = h^ — ̂ g^h^. The linearized Lagrangian SL has a diffeomorphism symmetry which is broken when we add the Pauli-Fierz spin-2 mass term 1 [email protected] CP624, Cosmology and Elementary Particle Physics, edited by B. N. Kursunoglu et al. © 2002 American Institute of Physics 0-7354-0073-3/027$ 19.00 344 The source term for TMi/ (assumed conserved) is given by In order to neatly compare the massive (and partially massless) case with the massless one, we use the Stiickelberg formalism to restore gauge symmetry. This is done by introducing an auxiliary vector field V^ and then making a change of field integration variables h^v -> h^ — ̂ V(^K)The gauge invariances are then massve v -> -> -f -h partially massless Note that the parameters £M and a correspond to diffeomorphisms and Weyl rescalings respectively. For gauge-fixing we use a lagrangian quadratic in the constraint with the additional constraint h = 0 for the partially massless case. We decompose the metric fluctuation into traceless and scalar parts given respectively by ̂ = = h so that