Abstract
In this chapter we study the naturalness of the graviton mass and whether or not the specific structure of the graviton gets detuned in the full theory beyond the decoupling limit, and compare our conclusions to the estimations coming from the decoupling limit analysis. More concrete, we will address the two essential questions.
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Notes
- 1.
In the Euclidean version of massive gravity both the dynamical metric \({g}_{{\upmu }{\upnu }}\) and the reference metric \(\mathrm{f}_{{\upmu }{\upnu }}\) have to be ‘Euclideanized’, \({g}_{{\upmu }{\upnu }}\rightarrow {g}_\mathrm{ab}\) and \(\mathrm{f}_{{\upmu }{\upnu }}={\upeta }_{{\upmu }{\upnu }} \rightarrow {\updelta }_\mathrm{ab}\).
- 2.
As in de Rham et al. (2011b), this 2-parameter family of potential is the one for which there is no cosmological constant nor tadpole.
- 3.
Even if different momenta \(\mathrm{k}_\mathrm{j}\) contract one can always reexpress them as functions of \(\mathrm{k}_\mathrm{j}^2\), following a similar procedure to what is presented in Appendix A.3.
- 4.
At the quadratic level, the Fierz–Pauli term is undistinguishable from the ghost-free potential term \(\tilde{\mathcal {U}}_2\)
- 5.
This can be seen more explicitly, by writing the operator \(\mathrm{M}\) in terms of the background metric \(\bar{{\upgamma }}_\mathrm{ab}\), recalling that \(\bar{\mathrm{g}}_\mathrm{ab}=\bar{{\upgamma }}_\mathrm{ac}\bar{{\upgamma }}_\mathrm{bd}{\updelta }^\mathrm{cd}\) and the metric \(\mathrm{g}_\mathrm{ab}\) is given in terms of \(\bar{{\upgamma }}_\mathrm{ab}\) and the field fluctuation \(\mathrm{v}_\mathrm{ab}\) as in (6.44). Then it follows that symbolically,
where the two first terms in bracket arise from the transition to the ‘vielbein-inspired’ metric fluctuation and the last term is what would have been otherwise the standard linearized Einstein–Hilbert term on a constant background metric \(\bar{g}_\mathrm{ab}\). Written in this form, \(\mathrm{M}\) is manifestly conformally invariant.
- 6.
In particular, by dimensional analysis, one should think of the schematic form for the effective Lagrangian as containing a factor of \(1/{\upmu }^2\), where \({\upmu }\) carries units of \([\text {mass}]\).
- 7.
The only way to prevent \(\Xi \) from being \(\lesssim \)1 is to consider a region of space where some eigenvalues of the metric itself vanish, which would be for instance the case at the horizon of a black hole. However as explained in Deffayet and Jacobson (2012), Koyama et al. (2011a, b), Berezhiani et al. (2012), in massive gravity these are no longer coordinate singularities, but rather real singularities. In massive gravity, black hole solutions ought to be expressed in such a way that the eigenvalues of the metric never reach zero apart at the singularity itself. Thus we do not need to worry about such configurations here (which would correspond to \(\uplambda =1\) in what follows).
- 8.
The mass term can also arise in the form \((\uplambda _1/\uplambda _0) \, \mathrm{M}^2\) for the components \(\tilde{\mathrm{M}}^{00\mathrm{ii}}\), but the conclusions hereafter remain unchanged.
- 9.
Technically, in Euclidean space this means that the momentum is complex, or one could go back to the Lorentzian space-time for the purposes of this calculation, but these issues are irrelevant for the current discussion. Moreover, note that the on-shell condition is only being imposed for the external legs, and not for internal lines.
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Heisenberg, L. (2015). Renormalization Beyond the Decoupling Limit of Massive Gravity. In: Theoretical and Observational Consistency of Massive Gravity. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-18935-2_6
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