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Abstract
We revisit the problem of constraining the weak field limit of the gravitational lagrangian from S-matrix properties. From unitarity and Lorentz invariance of the S-matrix of massless gravitons, we derive on-shell gauge invariance to consist on the transverse part of the linearised diffeomorphisms group. Moreover, by looking to the interaction between sources, we conclude that there exist only two possible lagrangians that lead to a welldefined covariant interaction, corresponding to the weak field limits of General Relativity and Unimodular Gravity. Additionally, this result confirms the equivalence of the S-matrix of both theories around flat space-time.
Highlights
JHEP12(2018)106 bare cosmological constant in the gravitational lagrangian this is just a fine tuning problem, the fact that we need to adjust the value of the cosmological constant Λ with a precision of 120 orders of magnitude, makes this an unpleasant hierarchy problem much worse than the one of the Higgs boson mass
If we were deriving this S-matrix from General Relativity in its weak field version, the Fierz-Pauli lagrangian of metric perturbations hμν, we would have assumed that the gauge symmetry had to be the linearised diffeomorphisms group, corresponding to hμν → qμlν + qν lμ which looks similar to what we have found except for a little difference
Through this work we have studied how much a lagrangian formulation of a theory of gravitons is constrained by basic properties of the S-matrix
Summary
We will closely follow [35, 36] in what follows. we are interested in gravity and in gravitons, it will be useful to consider photons, since their construction is similar and they serve as a good simpler example of what we are studying. For a different momentum qμ we just need to act on eμ± with the rotation Rμ ν(q) which aligns the x3-axis with qμ ǫμ±(q) = Rμ ν (q)eν± From this construction, it is easy to see that the photon polarization satisfies the following properties (ǫμ±(q))∗ǫ±μ(q) = 1 ǫμ±(q)ǫμ±(q) = 0 (ǫμ±(q))∗ = ǫμ∓(q) ǫ0±(q) = qμǫμ± = 0. We will keep the full space-time greek indices thought, but we will resort to space indices later on this work It carries a Lorentz index, the photon polarization ǫμ±(q) is not a vector, since it does not transform homogeneously when acting with a Lorentz transformation upon it. Where |q| is the length of the space component of the momentum qi Note that if it where not because of the second term on the l.h.s., we would have found an homogeneous transformation, with the Lorentz transformation corresponding to a rotation around the x3 axis. In the case of higher spin particles, the corresponding transformation rules can be derived by considering the action of the operator Ξμν on the different pieces of the direct product
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