Away in eq 41 we have scaled m1 in order to

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Unformatted text preview: ins preserved. Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 Rather than unitary gauge, we can instead fix a Lorentzlike gauge for the action (4.3), @ A þ m ¼ 0: (4.9) This gauge fixes the gauge freedom up to a residual gauge parameter satisfying ðh À m2 ÞÃ ¼ 0. We can add the gauge fixing term SGF ¼ Z 1 dD x À ð@ A þ mÞ2 : 2 (4.10) As in the massless case, quantum mechanically this term results from the Fadeev-Popov gauge fixing procedure. Adding the gauge fixing term diagonalizes the Lagrangian, S þ SGF ¼ Z 1 1 dD x A ðh À m2 ÞA þ ðh À m2 Þ 2 2 1 þ A J  À @ J  ; (4.11) m and the propagators for A and  are, respectively, Ài ; p2 þ m 2 Ài ; p2 þ m 2 (4.12) which are similar to $1=p2 at high momenta. Thus we have managed to restore the good high energy behavior of the propagators. It is possible to find the gauge invariant mode functions for A and , which can then be compared to the unitary gauge mode functions of the massive photon. In the massless limit, there is a direct correspondence;  is gauge invariant and becomes the longitudinal photon, A has the usual Maxwell gauge symmetry and its gauge invariant transverse modes are exactly the transverse modes of the massive photon. ¨ B. Graviton Stuckelberg and origin of the vDVZ discontinuity Now consider massive gravity, S¼ Z (4.7) and the gauge symmetry is 683 1 dD xLm¼0 À m2 ðh h À h2 Þ þ h T  ; 2 (4.13) where Lm¼0 is the Lagrangian of the massless graviton. We want to preserve the gauge symmetry h ¼ @  þ @  ¨ present in the m ¼ 0 case, so we introduce a Stuckelberg field A patterned after the gauge symmetry, h ! h þ @ A þ @ A : (4.14) The Lm¼0 term remains invariant because it is gauge invariant and Eq. (4.14) looks like a gauge transformation, so all that changes is the mass term, S¼ Z 1 1 dD xLm¼0 À m2 ðh h À h2 Þ À m2 F F 2 2 2   À 2m ðh @ A À h@ A Þ þ h T  À 2A @ T  ; (4.15) where we have integrated by parts in the last term, and where F  @ A À @ A . Kurt Hinterbichler: Theoretical aspects of massive gravity 684 There is now a gauge symmetry h ¼ @  þ @  ; A ¼ À ; S¼ (4.16) and fixing the gauge A ¼ 0 recovers the original massive gravity action (as in the vector case, this is a gauge condition for which it is permissible to substitute back into the action, because the potentially lost A equation is implied by the divergence of the h equation). At this point, we might consider scaling A ! mÀ1 A to normalize the vector kinetic term, and then take the m ! 0 limit. In this limit, we end up with a massless graviton and a massless photon, for a total of 4 degrees of freedom (in four dimensions). So at this point, m ! 0 is still not a smooth limit, since we would be losing one of the original 5 degrees of freedom. We go one step further and introduce a scalar gauge ¨ symmetry, by introducing another Stuckelberg field , A...
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