That expanding the cubic term yields new

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Unformatted text preview: khalava (2011). E. Quantum corrections in the Ã3 theory As in Sec. VII.D, we expect quantum mechanically the presence of all operators with at least two derivatives per , now suppressed by the cutoff Ã3 (we ignore for simplicity the scalar-tensor interactions), $ ^ @q [email protected] Þp Ã3pþqÀ4 3 : (8.32) These are in addition to the classical Galileon terms in Eq. (8.18), which have fewer derivatives per  and are of the form $ ^ ^ ð@Þ2 [email protected] Þp Ã3p 3 : (8.33) Kurt Hinterbichler: Theoretical aspects of massive gravity An analysis similar to that of Sec. VII.D shows that the terms (8.32) become important relative to the kinetic term at the radius r $ ðM=MPl Þ1=3 ð1=Ã3 Þ. This is the same radius at which classical nonlinear effects due to Eq. (8.33) become important and alter the solution from its Coulomb form. Thus we must instead compare the terms (8.32) to the classical nonlinear Galileon terms (8.33). We see that the terms (8.32) are all suppressed relative to the Galileon terms (8.33) by powers of @=Ã3 , which is $1=Ã3 r regardless of the nonlinear solution. Thus, quantum effects do not become important until the radius 1 ; (8.34) rQ $ Ã3 which is parametrically smaller than the Vainshtein radius (8.22). This behavior is much improved from that of the Ã5 theory, in which the Vainshtein region was swamped by quantum correction. Here there is a parametrically large intermediate classical region in which nonlinearities are important but quantum effects are not, and in which the Vainshtein mechanism should screen the extra scalar. In this region, GR should be a good approximation; see Fig. 2. As in the Ã5 theory, quantum corrections are generically expected to ruin the various classical tunings for the coefficients, but the tunings are still technically natural because the corrections are parametrically small. For example, cutting off ^ loops by Ã3 , we generate the operator $ð1=Ã2 ÞðhÞ2 , which 3 ^ corrects the mass term. The canonically normalized  is related to the original dimensionless metric by h $ ^ ð1=Ã3 Þ@@, so the generated term corresponds in unitary 3 2 gauge to Ã4 h2 ¼ Mp m2 ðÃ3 =Mp Þh2 , representing a mass 3 2 $ m2 ðà =M Þ. This mass correction is paracorrection m 3 p metrically smaller than the mass itself and so the hierarchy m ( Ã3 is technically natural. This correction also ruins the Fierz-Pauli tuning, but the pathology associated with the detuning of Fierz-Pauli, the ghost mass, is m2 $ g m2 =ðm2 =m2 Þ $ Ã2 , safely at the cutoff. 3 We mention another potential issue with the Ã3 theory. It was found by Nicolis, Rattazzi, and Trincherini (2008) that Lagrangians of the Galileon type inevitably have superluminal propagation around spherical background solutions. No matter what the choice of parameters in the Lagrangian, if the solution is stable, then superluminality is always present at distances far enough from the source [see also Osipov and 701 Rubakov (2008)]. It was argued that such s...
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This document was uploaded on 09/28/2013.

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