RevModPhys.84.671

And mukhanov 2010 in which they make the inverse

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Unformatted text preview: coefﬁcients C1 , C2 , etc. ¨ VII. STUCKELBERG ANALYSIS OF INTERACTING MASSIVE GRAVITY ¨ In this section, we set D ¼ 4 and apply the Stuckelberg analysis to the massive GR action (5.1) in the case of a ﬂat absolute metric. The mass term reads Smass M2 m2 Z 4  ¼À P d x  ðh h À h h Þ: 24 (7.1) ¨ The Stuckelberg analysis instructs us to make the replacement (6.18), h ! H ¼ h þ @ A þ @ A þ @ A @ A þ 2@ @  þ @ @ @ @  Á Á Á : (7.2) The extra terms with h in the ellipsis will not be important for this theory, as we will see. At the linear level, this replacement is exactly the linear ¨ Stuckelberg expansion of Sec. IV. We have to canonically normalize the ﬁelds here to match the ﬁelds of the linear analysis. Using a hat to signify the canonically normalized ﬁelds with the same coefﬁcients as used in Sec. IV (although there we omitted the hats), we have ^ h ¼ 1MP h; 2 ^ A ¼ 1mMP A; 2 ^  ¼ 1m2 MP : 2 (7.3) \$ ^ [email protected] Þ3 ; Ã5 5 693 Ã5 ¼ ðMP m4 Þ1=5 : (7.6) In terms of the canonically normalized ﬁelds (7.3), the gauge symmetries (6.19) read 2 ^ ^ ^ h ¼ @  þ @  þ L ^h ; MP   2 ^ ^ ^ ^ ^ A ¼ @ Ã À m þ  @ A MP 2 ^ ^ ^ A A @ @  À Á Á Á ; À 2 mMP ^  ¼ ÀmÃ; (7.7) ^ ^ where we rescaled Ã ¼ ðmMP =2ÞÃ and  ¼ ðMP =2Þ . Finally, note that since the scalar ﬁeld  always appears ¨ with at least two derivatives in the Stuckelberg replacement (7.2), the resulting action is automatically invariant under the global Galilean symmetry ðxÞ ! c þ b x ; (7.8) where c and b are constants. In addition, the action is automatically invariant under global shifts in A ! A þ c for constant c . It will persist even in limits where the gauge symmetries on A and  no longer act. A. Decoupling limit and breakdown of linearity As seen in Sec. IV.B, the propagators have all been made to go as \$1=p2 , so normal power counting applies, and the lowest scale Ã5 is the cutoff of the effective ﬁeld theory. To focus in on the cutoff scale, we take the decoupling limit m ! 0; MP ! 1 ; T ! 1; Ã5 ; T fixed: MP (7.9) (7.5) All interaction terms go to zero, except for the scalar cubic term (7.6) responsible for the strong coupling, which we calculate using the replacement H ¼ 2@ @  þ @ @ @ @  since we do not need the vector and tensor terms. As discussed in Sec. IV.B, we must also do the conformal transformation h ¼ h0 þ m2  . This will  diagonalize all the kinetic terms (except for various cross terms proportional to m which are eliminated with appropriate gauge ﬁxing terms, as discussed in Sec. IV.B, and which go to zero anyway in the decoupling limit). After all this, the Lagrangian for the scalar reads, up to a total derivative, Z 2 ^ ^ ^ ^ S ¼ d4 x À 3ð@Þ2 þ 5 ½ðhÞ3 À ðhÞð@ @ Þ2  Ã5 1^ þ T: (7.10) MP The larger , the smaller the scale, since m < MP . We have n þ nA þ nh ! 3, since we are only considering int...
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