At radii larger than the vainshtein radius so c mmp 2

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Unformatted text preview: f the Vainshtein radius. Below the Vainshtein radius (high energies), a new degree of freedom, the ghost (analogous to the physical Higgs in the standard model), kicks in. Much below the Vainshtein radius, everything is again linear and weakly coupled, with the difference that there are now 2 active degrees of freedom, so one can think of this as a classical UV completion of the effective theory. Rev. Mod. Phys., Vol. 84, No. 2, April–June 2012 695 Of course, this ghostly mechanism for restoring continuity with GR relies on an instability, which would become apparent were we to investigate small fluctuations beyond the gross-scale features described here. Furthermore, as we see in the next section, the ghost issue is moot, since the classical mechanism described in this section occurs outside the regime of validity of the quantum effective theory and is swamped by unknown quantum corrections. D. Quantum corrections and the effective theory Quantum mechanically, massive gravity is an effective field theory, since there are nonrenormalizable operators suppressed by the mass scale Ã5 . The amplitude for  !  scattering at energy E, coming from the cubic coupling in Eq. (7.10), is similar to A $ ðE=Ã5 Þ10 . This amplitude should correspond to the scattering of longitudinal gravitons. The wave function of the longitudinal graviton (2.20) for a large boost is proportional to mÀ2 , while the largest term at high momentum in the graviton propagator (2.44) is proportional to mÀ4 , so naive power counting suggests that the amplitude at energies much larger than m is similar to A $ 2 E14 =MP m12 . However, as recognized by Arkani-Hamed, Georgi, and Schwartz (2003) and calculated explicitly by Aubert (2004), there is a cancellation in the diagrams so ¨ that the result agrees with the result of the Stuckelberg description. We encounter these kinds of cancellations again ¨ in loops, and part of the usefulness of the Stuckelberg description is that they are made manifest. The amplitude becomes of the order of 1 and hence strongly coupled when E $ Ã5 . Thus Ã5 is the maximal cutoff of the theory. We expect to generate all operators compatible with the symmetries, suppressed by appropriate powers of the cutoff. In the unitary gauge, there are no symmetries, so we generate all operators of the form cp;q @q hp : (7.19) We determine the scales in the coefficient cp;q . ¨ After Stuckelberging, the decoupling limit theory contains ^ only the scalar  and the single coupling scale Ã5 . In ^ ^ addition, there is the Galileon symmetry  !  þ c þ c x . Quantum mechanically, we expect to generate in the quantum effective action all possible operators with this symmetry, suppressed by the appropriate power of the cutoff ^ Ã5 . The Galileon symmetry forces each  to carry at least two derivatives,13 so the general term we can have is $ 13 ^ @q ð@2 Þp Ã3pþqÀ4 5 : (7.20) Actually, there are a finite number of terms which have fewer than two deriva...
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