Sm is the 4d matter action and the factor of 2 in

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Unformatted text preview: Nicolis and Rattazzi, 2004; Gabadadze and Iglesias, 2006) and are suppressed by the scale Ã3 ¼ ðM4 m2 Þ1=3 (in fact, this was where the Galileons were first uncovered). In this sense, DGP is analogous to the nicer Ã3 theories of Sec. VIII. The theory is free of ghosts and instabilities around solutions connected to flat space (Nicolis and Rattazzi, 2004), but changing the asymptotics to the selfaccelerating de Sitter brane solutions flips the sign of the kinetic term of the longitudinal mode, so there is a massless ghost around the self-accelerating branch (Koyama, 2007). Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 C. Resonance gravitons Ài pffiffiffiffiffiffi : p þ m p2 (9.51) 2 Setting z ¼ Àp2 , the propagator has a branch cut in the z plane from ð0; 1Þ, with discontinuity 2m À pffiffiffi : zð z þ m 2 Þ (9.52) A branch cut can be thought of as a string of simple poles, in the limit where the spacing between the poles and their R1 residues both go to zero. The function fðzÞ ¼ À1 dðÞ=ðz À Þ has a cut along the real axis everywhere that  is nonzero, with discontinuity À2iðzÞ. We can see this by noting Z1 1 1 À disc fðzÞ ¼ dðÞ z À  þ i z À  À i À1 Z1 ¼ dðÞ½À2iðz À ފ: À1 Using all this, and the fact that analytic functions are determined by their poles and cuts, we can write the propagator in the spectral form Z1 Ài Ài ds 2 ðsÞ; pffiffiffiffiffiffi ¼ 2 þ m p2 p þs 0 p m ðsÞ ¼ pffiffiffi > 0: (9.53)  sð s þ m 2 Þ The spectral function is greater than zero, so this theory contains a continuum of ordinary (nonghost, nontachyon) gravitons, with masses ranging from 0 to 1. This is what would be expected from dimensionally reducing a noncompact fifth dimension. The Kaluza-Klein tower has collapsed down into a Kaluza-Klein continuum. In the limit m ! 0, where the action becomes purely four dimensional, the spectral function reduces to a delta function, ðsÞ ! 2ðsÞ; (9.54) Ài=p2 representing a single and the propagator reduces to massless graviton, vector, and scalar, as can be seen from Kurt Hinterbichler: Theoretical aspects of massive gravity 706 Eq. (4.37) (the extra factor of 2 is taken care of by noting that the integral is from 0 to 1, so only half of the delta function actually gets counted). This theory therefore contains a vDVZ discontinuity. The potential of a point source of mass M sourced by this resonance graviton displays an interesting crossover behavior. Looking back at pffiffiffiffiffiffi (3.11) with the momentum Eq. space replacement m2 ! m p2 , and using the relation  ¼ Àh00 =M4 for the Newtonian potential, we have À 2 M Z d 3 p i p Áx À1 e 2 3 2 þ mjpj 3M 4 ð 2 Þ p   2 M r r ci ¼ sin 2 3 M 4 2 2 r r0 r0     1 r r þ cos  À 2si ; 2 r0 r0 ðrÞ ¼ (9.55) X. CONCLUSIONS AND FUTURE DIRECTIONS where siðxÞ  Z x dt 0 t sint; ciðxÞ  þ lnx þ Z x dt 0 t (9.56) ðcost À 1Þ; % 0:577 . . . is the Euler-Mascer...
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This document was uploaded on 09/28/2013.

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