RevModPhys.84.671

On a b ghosts note that the lagrangian 710 is a higher

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Unformatted text preview: only 5 degrees of freedom. Following Creminelli et al. (2005), let us consider the stability of the classical solutions to Eq. (7.10) around a massive point source. We have a classical background ÈðrÞ, ^ which is a solution of the  equation of motion, and we expand the Lagrangian of Eq. (7.10) to quadratic order in the ^ fluctuation ’   À È. The result is schematically L’ $ Àð@’Þ2 þ ð@2 ÈÞ 2 2 ð@ ’Þ : Ã5 5 Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 (7.15) There is a four-derivative contribution to the ’ kinetic term, signaling that this theory propagates 2 linear degrees of freedom. As shown in Sec. 2 of Creminelli et al. (2005), one is stable and massless, and the other is a ghost with a mass of the order of the scale appearing in front of the higher derivative terms. So in this case the ghost has an r-dependent mass m 2 ð rÞ $ ghost Ã5 5 : @2 ÈðrÞ (7.16) This shows that around a flat background, or far from the source, the ghost mass goes to infinity and the ghost freezes, explaining why it was not seen in the linear theory. It is only around nontrivial backgrounds that it becomes active. Notice, however, that the backgrounds around which the ghost becomes active are perfectly nice, asymptotically flat configurations sourced by compact objects such as the Sun, and not disconnected in any way in field space (this is in contrast to the ghost in DGP, which occurs around only asymptotically de Sitter solutions). We are working in an effective field theory with a UV cutoff Ã5 ; therefore we should not worry about instabilities until the mass of the ghost drops below Ã5 . This happens at the distance rghost where @2 Èc $ Ã3 . For a source of mass M, 5 at distances r ) rV the background field is similar to ÈðrÞ $ ðM=MP Þð1=rÞ, so  1=3  1=5 M 1 M 1 ) rV $ : (7.17) rghost $ MP Ã5 MP Ã5 rghost is parametrically larger than the Vainshtein radius rV . As we see in Sec. VII.D, the distance rghost is the same distance at which quantum effects become important. Whatever UV completion takes over should cure the ghost instabilities that become present at this scale, so we will be able to consistently ignore the ghost. We see already that we cannot trust the classical solution even in regions parametrically farther than the Vainshtein radius. The best we can do is make predictions outside rghost , and we have more to say about this later. C. Resolution of the vDVZ discontinuity and the Vainshtein mechanism We are now in a position to see the mechanism by which nonlinearities can resolve the vDVZ discontinuity. This is known as the Vainshtein mechanism. It turns out to involve the ghost in a critical role. Far outside the Vainshtein radius, where the linear term of Eq. (7.10) dominates, the field has the usual Coulombic 1=r form. But inside the Vainshtein radius, where the cubic term dominates, it is easy to see by power counting that the field gets an r3=2 profile, 8 M >M 1; r ) rV ; < Pr ^  $ ...
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