With auxiliary extra dimensions gabadadze 2009 de

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Unformatted text preview: vector and the rest ^ ^ scalars @Að@2 Þn . The scale Ã3 is carried only by the following terms: $ ^ ^ hð@2 Þn ; nÀ1 2nÀ2 MP m $ ^ ^ ð@AÞ2 ð@2 Þn : n 2n MP m (8.3) All other terms carry scales higher than Ã3 . It turns out that we can arrange to cancel all of the scalar self-couplings by appropriately choosing the coefficients of the higher order terms. We work with the form of the potential in Eq. (5.9) where indices are raised with the full metric, and ¨ the Stuckelberg formalism of Sec. VI.B. We do so because we eventually want to keep track of powers of h, so the form of ¨ the Stuckelberg replacement in Sec. VI.B is simpler. We are interested only in scalar self-interactions, so we may make the replacement (6.31) with the vector field set to zero, H ! 2@ @  À @ @ @ @ : (8.4) The interaction terms are a function of the matrix of second derivatives Å  @ @ . As reviewed in the Appendix, there is at each order in  a single polynomial in Å which is a total derivative. By choosing the coefficients (5.9) correctly, we can arrange for the  terms to appear in these total derivative combinations. The total derivative combinations have at each order in  as many terms as there are terms in the potential of Eq. (5.9), so all the coefficients must be fixed, except for one at each order which becomes the overall coefficient of the total derivative combination. The choice of coefficients in the potential (5.9) which removes the scalar self-interactions is, to fifth order (de Rham and Gabadadze, 2010a), c1 ¼ 2 c 3 þ 1 ; 2 c2 ¼ À3c3 À 1; 2 (8.5) 1 d1 ¼ À6d5 þ 16ð24c3 þ 5Þ; d2 ¼ 8 d 5 À 1 ð 6 c 3 þ 1 Þ ; 4 d3 ¼ d4 ¼ 1 3d5 À 16ð12c3 À6d5 þ 3c3 ; 4 (8.6) þ 1Þ; 7 f1 ¼ 32 þ 9c3 À 6d5 þ 24f7 ; 8 5 f2 ¼ À32 À 15c3 þ 6d5 À 30f7 ; 16 A. Tuning interactions to raise the cutoff $ 697 (8.2) f3 ¼ 3c3 À 3d5 þ 20f7 ; 8 1 f4 ¼ À16 À 3c3 þ 5d5 À 20f7 ; 4 (8.7) 3 f5 ¼ 16c3 À 3d5 þ 15f7 ; f6 ¼ d5 À 10f7 : At each order, there is a one-parameter family of choices that works to create a total derivative. Here c3 , d5 , and f7 are chosen to carry that parameter at order 3, 4, and 5, respectively. Note, however, that at order 5 and above (or D þ 1 and above if we were doing this in D dimensions), there is one linear combination of all the terms, the characteristic polynomial of h mentioned below Eq. (5.8) that vanishes identically. This means that one of the coefficients is redundant, Kurt Hinterbichler: Theoretical aspects of massive gravity 698 and we can, in fact, set d5 and its higher counterparts to any value we like without changing the theory. Thus there is only a two parameter family (D À 2 parameter in dimension D) of theories with no scalar self-interactions. This can be carried through at all orders, and at the end there will be no terms $ð@2 Þn . The only terms with interaction scales lower than Ã3 were ^ the scalar self-interactions ð@2 Þn , and terms with one vector...
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This document was uploaded on 09/28/2013.

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