RevModPhys.84.671

Hence strongly coupled when e 5 thus 5 is the maximal

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Unformatted text preview: tives per field, the so-called Galileon terms (Nicolis, Rattazzi, and Trincherini, 2008) which change by a total derivative under the Galileon symmetry (Nicolis, Rattazzi, and Trincherini, 2008). However, there is a nonrenormalization theorem that says these are not generated at any loop by quantum corrections (Hinterbichler, Trodden, and Wesley, 2010), so we need not include them. We encounter them later when we raise the cutoff to Ã3 . Kurt Hinterbichler: Theoretical aspects of massive gravity 696 To compare with Eq. (7.19), we go back to the original ^ normalization for the fields by replacing  $ m2 MP  and recall that @ @  comes from an h to find that in unitary gauge the coefficients cp;q are similar to À p 2 cp;q $ Ã5 3pÀqþ4 MP m2p ¼ ðm16À4qÀ2p MPpÀqþ4 Þ1=5 : (7.21) This comparison is possible because the operations of taking the decoupling limit and computing quantum corrections should commute. Notice that the term with p ¼ 2, q ¼ 0 is a mass term, 2 $ðMP m4 =Ã2 Þh2 , corresponding to a mass correction m2 ¼ 5 2 m ðm2 =Ã2 Þ. This is down by a factor of m2 =Ã2 from the tree 5 5 level mass term. Thus a small mass graviton m ( Ã5 is technically natural, and there is no quantum hierarchy problem associated with a small mass. This is in line with the general rule of thumb that a small term is technically natural if a symmetry emerges as the term is dialed to zero. In this case, it is the diffeomorphism symmetry of GR which is restored as the mass term goes to zero. The quantum mass correction will also generically ruin the Fierz-Pauli tuning, but its coefficient is small enough that ghosts and tachyons associated with the tuning violation are postponed to the cutoff; indeed the resulting ghost mass, using the relations in Sec. II, is $Ã5 . It is important that there are no nonparametric modifications to the kinetic structure of the Einstein-Hilbert term, even though the lack of gauge symmetry suggests that we are free to make such modifications. Suppose we try to calculate the mass correction directly in unitary gauge. The graviton mass term contributes no vertices but alters the propagator so that its high energy behavior is $k2 =m4 (the next leading terms are similar to 1=m2 and then 1=k2 ). At one loop, there are two one-particle irreducible diagrams correcting the mass: ^ one containing two cubic vertices ð1=MP [email protected] h3 from the Einstein-Hilbert action and two propagators, and another 2 ^ containing a single quartic vertex ð1=MP [email protected] h4 from the Einstein-Hilbert action and a single propagator. Cutting off the loop at the momenta kmax $ Ã5 , the first diagram gives 2 the largest naive correction m2 $ ð1=MP m8 ÞÃ12 $ Ã2 . (The 5 5 second diagram gives a smaller correction.) This is at the cutoff, dangerously higher than the small correction m2 $ ¨ m2 ðm2 =Ã2 Þ we found in the Stuckelberg formalism. 5 This means that there must be a nontrivial cancellation of this leading divergence in unitary g...
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