More on Violation of Angular-Momentum Selection Rules in Quantum Gravity

In a recent paper, we noticed that, in the s-channel scattering mediated by a graviton in the linearized Einstein theory, angular momentum seems not to be conserved in the graviton coupling to massive matter fields, due to the presence of a scalar component in the partial-wave expansion of the relevant amplitudes. The present Letter answers to some criticism by Han and Willenbrock, claiming that there is no problem in propagating a scalar component through a graviton exchange. We show explicitly that the partial-wave decomposition of the scattering amplitude contrasts with the rotation-group representations propagating through the graviton exchange in the considered processes. This phenomenon is crucially connected to the critical dependence of the trace of the graviton propagator on the effective dimension of the problem. In a recent work [1], we showed that angular-momentum selection rules are violated in the linearized limit of quantum gravity in the Einstein theory. In particular, we considered the scattering amplitudes of either massive fermion or massive gauge-boson pairs mediated by a graviton in the s-channel, and computed their non vanishing interferences with the amplitudes where a scalar field replaces the graviton. Since in the s-channel scattering of real particles only graviton components of angular momentum J = 2 propagate [2], a non vanishing interference with the scalar (J = 0) exchange 1 amplitude seems to point to a violation of the angular-momentum conservation in the graviton-matter vertex. Our statement was not based on special assumptions. It was derived in a straightforward way from the Feynman rules of the Einstein theory minimally coupled to massive fermion and vector-boson fields. In a following paper by Han and Willenbrock [3], it has been argued that our conclusion would be incorrect, since it is based on the wrong assumption that no J = 0 component is exchanged in the graviton propagator. The authors stress that in the s-channel amplitude mediated by a graviton, also an SO(3, 1) scalar component can propagate. The latter couples to the trace of the energy momentum tensor of the matter fields Tμν (that is non vanishing for massive fields). They conclude that there is no violation of angular-momentum selection rules in quantum gravity. In our paper [1], the crucial statement that only J = 2 graviton components contribute to the considered amplitudes was based on a previous work by van Dam and Veltman [2], and was not proved explicitly. The purpose of the present Letter is to clarify this issue, by showing that there are processes where there is a J = 0 component in the partial-wave decomposition of the s-channel amplitude, and, on the other hand, no J = 0 component propagates in the virtual graviton state. This fact will be shown to be crucially connected to the critical dependence of the trace of the graviton propagator tensor on the effective dimension of the analyzed problem. Let us consider the general form of the graviton propagator in the linearized Einstein theory. When contracted with conserved energy momentum tensors (i.e., for qμTμν = 0, with qμ the momentum flowing in the propagator), terms proportional to the momentum are not relevant, and the effective massless graviton propagator becomes∗ Gμναβ(q) = i Pμναβ q2 + iǫ ; Pμναβ = 1 2 (ημαηνβ + ημβηνα − ημνηαβ) , (1) The s-channel p1p1 → p2p2 scattering amplitude mediated by a graviton is then given by A = −i M P ( T f† μν Gμναβ(q) T i αβ ) (2) where T i(f) μν are the matrix elements of the conserved energy momentum tensor of the initial (final) states, and MP is the Planck mass. In this Letter, indices (μ, ν, α, β) are contracted according to the Minkowski metric ημν = Diag(1,−1,−1,−1).