10f7 at each order there is a one parameter family of

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Unformatted text preview: ^ ^ [email protected] Þn . We succeeded in eliminating and the rest scalars @A the scalar self-interactions, but since these always came from ^ ^ combinations (A þ @ ) the terms @[email protected] Þn are automati A X ðnÞ , where the X ðnÞ are the functions cally of the form @   of @ @  described in the Appendix, which are identically ðn Þ conserved @ X ¼ 0. Thus, once the scalar self-interactions ^ ^ are eliminated, the @[email protected] Þn terms are all total derivatives and are also eliminated. Now the lowest interaction scale will be due to the terms in Eq. (8.3), $ ^ ^ [email protected] Þn ; nþ1 2nþ2 MP m $ ^ ^ [email protected] [email protected] Þn ; nþ2 2nþ4 MP m (8.8) which are suppressed by the scale Ã3 ¼ ðMP m2 Þ1=3 , so the cutoff has be raised to Ã3 , carried by the terms (8.8). This theory can, in fact, be resummed in an interesting way, using an action involving square roots (de Rham, Gabadadze, and Tolley, 2010; Hassan and Rosen, 2011b). The decoupling limit is now m ! 0; MP ! 1; Ã3 fixed; (8.9) and the only terms which survive are those in Eq. (8.3). To ¨ find these terms we must now go back to the full Stuckelberg replacement (6.31), and we must also expand the inverse metric and determinant in the potential of Eq. (5.9) in powers of h. The [email protected] Þn terms, up to quintic order in the decoupling limit, and up to total derivatives are (de Rham and Gabadadze, 2010a)  Z 1 ^ ð1 Þ ^ 4 1^ ; ^ h À h À4X ðÞ S ¼ d x h E 2 2  4 ð 6 c 3 À 1 Þ ð2 Þ ^ 16ð8d5 þ c3 Þ ð3Þ ^ þ X ðÞ þ X ðÞ Ã3 Ã6 3 3 1^ (8.10) h T  : þ MP  ð nÞ Here the X are the identically conserved combinations of ^ @ @  described in the Appendix. The [email protected] [email protected] Þ terms are found to cubic order by de Rham and Gabadadze (2010b). The terms with A’s can in any case be consistently set to zero at the classical level, since they never appear linearly in the Lagrangian, so we focus only on the terms involving h and . de Rham, Gabadadze, and Tolley (2010) used a nice trick to show that the decoupling limit Lagrangian (8.10) is exact to all orders in the fields, that is, there are no further terms [email protected] Þn for n ! 4. Properties of this Lagrangian, including its cosmological solutions, degravitation effects, and phenomenology are studied by de Rham et al. (2010). Spherical solutions are studied by Chkareuli and Pirtskhalava (2011). The cosmology of a covariantized version was studied by de Rham and Heisenberg (2011). Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 In terms of the canonically normalized fields (7.3), the gauge symmetries (6.34) of the full theory are 2 ^ ^ ^ ^ ^  @ A ; A ¼ @ à À m þ MP (8.11) 2 ^ ^ ^ h ¼ @  þ @  þ L ^h ; MP   (8.12) ^  ¼ ÀmÃ; (8.13) ^ ^ where we rescaled à ¼ ðmMP =2Þà and  ¼ ðMP =2Þ . In the decoupling limit (8.9), this gauge symmetry reduces to its linear form ^ ^ A ¼ @ Ã; (8.14) ^ ^ h ¼ @  þ @  ; (8.15)  ¼ 0: (8.16) The Lagrangian (8.10) should be inv...
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This document was uploaded on 09/28/2013.

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