# 6d Massive Gravity

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Massive gravity (with a Fierz-Pauli mass) in 4 dimensions is very well-studied, involving exotic phenomena like the vDVZ discontinuity and the Vainshtein effect that all have an elegant and physically transparent explanation in terms of an effective field theory of longitudinal modes, as explained by Arkani-Hamed, Georgi, and Schwartz. Is there any analogous work on six-dimensional massive gravity? (The right mass term would still be of Fierz-Pauli form, but the little group is bigger and so I would expect a more complicated set of longitudinal modes to think about.)

EDIT: added a bounty to renew interest.

This post has been migrated from (A51.SE)
asked Sep 24, 2011
retagged Mar 18, 2014
I am not sure if this is helpful, but 1010.0494 and 0902.0981 have good reviews of spinor helicity in six dimensions, which may help in efficient organization of the problem.

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## 1 Answer

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Unfortunately I am not really aware of any literature regarding this, however a generalization of the Arkani-Hamed et al paper to the case of d+1 dimensions should be rather straightforward.

Let us begin by noting a few basic facts about the group theory involved: The little group for a massless representation (this is the analog of helicity) of the Poincaré algebra is given by $SO(d-2)$, while the little group for a massive representation is given by $SO(d-1)$. The number of degrees of freedom corresponding to massive spin-2 is given by the symmetric traceless tensor of the little group, which has $\frac{d(d+1)}{2} -1$ degrees of freedom. In the massless case a similar argument leads to $\frac{(d-1)d}{2}-1$.

Now the point of the analysis by Arkani-Hamed et al is essentially to understand the theory in the UV, i.e. at energy scales much larger than the mass. To do this they try to decompose the massive representation in terms of massless ones, a straightforward counting exercise shows that in this case a massive spin-2 decomposes into a scalar, a helicity-1 vector and a helicity-2, exactly as in the 3+1d case. Using this knowledge it should be very easy to generalize the previous results, it is mostly a matter of carefully keeping track of the $d$'s. The vDVZ discontinuity, will still be there although the relative factor in the radiation-radiation and matter-matter interaction will depend on d, this can easily be seen by decomposing the tensor structure of the massive spin-2 propagator in terms of three massless one's corresponding to the helicities, a nice derivation for the $d=3$ case can be found in Zee's QFT book.

I hope that helps a bit...

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answered Nov 3, 2011 by (40 points)
Right, this much I had reasoned through already. But there's also a more complicated set of helicity states that could appear in the scattering amplitudes; e.g., the (Stückelberged-in, massless) vector part now transforms under $SO(4) \sim SU(2) \times SU(2)$, so one can consider scattering of a state carrying helicity $\pm 1$ under the first SU(2) factor with states carrying helicity $\pm 1$ under the second, and so on. It should be straightforward; I was just wondering if any reference has already worked it out. Thanks, though.

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