2010 used a nice trick to show that the decoupling

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Unformatted text preview: ariant under the decoupling limit gauge symmetries (8.16). Indeed, the identity ðn Þ @ X ¼ 0 ensures that it is. The scalar  is gauge invariant in the decoupling limit, but the fact that it always comes with two derivatives means that the global Galileon symmetry (7.8) is still present, as is the shift symmetry on A . Note that for the specific choices c3 ¼ 1=6 and d5 ¼ À1=48, all the interaction terms disappear. This could mean that the theory becomes strongly coupled at some scale larger than Ã3 , or there could be no lowest scale, since there are scales arbitrarily close to but above Ã3 . In the latter case, the theory would have no nonlinear behavior, and so no mechanism to recover continuity with GR, and it would therefore be ruled out observationally. B. The appearance of Galileons and the absence of ghosts We partially diagonalize the interaction terms in Eq. (8.10) by using the properties (A.18). First we perform the conformal transformation needed to diagonalize the linear terms, ^ ^ ^ h ! h þ  , after which the Lagrangian takes the form Z 1^ ^ S ¼ d4 x h E ; h 2   1^ 4ð6c3 À 1Þ ^ ð2Þ 16ð8d5 þ c3 Þ ^ ð3Þ X þ X  À h 2 Ã3 Ã6 3 3 þ 1^ 6 ð 6 c3 À 1 Þ ^ 2 ^ ^ ð@Þ h h T  À 3ð@Þ2 þ MP  Ã3 3 þ 16ð8d5 þ c3 Þ ^ 2 ^ 2 1^ ^ ð@Þ ð½ÅŠ À ½Å2 ŠÞ þ T: MP Ã6 3 (8.17) ^ ^ Here the brackets are traces of Å  @ @  and its powers (the notation is explained at the end of the Introduction). The cubic h couplings can be eliminated with a field redefinition 2 ð 6 c3 À 1 Þ ^^ ^ ^ @ @ ; h ! h þ Ã3 3 Kurt Hinterbichler: Theoretical aspects of massive gravity after which the Lagrangian reads S¼ Z 1^ 8ð8d5 þ c3 Þ ^ ^ ð3Þ ^ d4 x h E ; h À h X  2 Ã6 3 þ 1^ 6ð6c3 À 1Þ ^ 2 ^ ^ ð@Þ h h T  À 3ð@Þ2 þ MP Ã3 3 À4 À ð6c3 À 1Þ2 À 4ð8d5 þ c3 Þ ^ 2 ^ 2 ^ ð@Þ ð½ÅŠ À ½Å2 ŠÞ Ã6 3 40ð6c3 À 1Þð8d5 þ c3 Þ ^ 2 ^ 3 ^ ^ ð@Þ ð½ÅŠ À 3½Å2 Š½ÅŠ Ã9 3 1^ 2 ð 6 c3 À 1 Þ ^ ^^ þ 2½Å3 ŠÞ þ @ @ T  : T þ MP Ã3 MP 3 (8.18) There is no local field redefinition that can eliminate the h quartic mixing (there is a nonlocal redefinition that can do it), so this is as unmixed as the Lagrangian can get while staying local. The scalar self-interactions in Eq. (8.18) are given by the following four Lagrangians: L2 ¼ À1ð@Þ2 ; 2 L3 ¼ À1ð@Þ2 ½ÅŠ; 2 L4 ¼ À1ð@Þ2 ð½ÅŠ2 À ½Å2 ŠÞ; 2 L5 ¼ À1ð@Þ2 ð½ÅŠ3 2 (8.19) À 3½ÅŠ½Å Š þ 2½Å ŠÞ: 2 3 These are known as the Galileon terms (Nicolis, Rattazzi, and Trincherini, 2008) [see also Sec. II of Hinterbichler, Trodden, and Wesley (2010) for a summary of the Galileons]. They share two special properties: their equations of motion are purely second order (despite the appearance of higher derivative terms in the Lagrangians), and they are invariant up to a total derivative under the Galilean symmetry (7.8), ðxÞ ! ðxÞ þ c þ b x...
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