RevModPhys.84.671

Vector and one in a canonical massless scalar6 note

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Unformatted text preview: scalar potential exactly accounts for the discrepancy between the massless limit of massive gravity and massless gravity. ¨ As a side note, one can see from this Stuckelberg trick that violating the Fierz-Pauli tuning for the mass term leads to a ¨ ghost. Any deviation from this form, and the Stuckelberg scalar will acquire a kinetic term with four derivatives $ðhÞ2 , indicating that it carries 2 degrees of freedom, one of which is ghostlike (de Urries and Julve, 1995, 1998). The Fierz-Pauli tuning is required to exactly cancel these terms, up to total derivative. 6 Ordinarily the Maxwell term would come with a 1 and the scalar 4 kinetic term with a 1 , but we leave different factors here just to 2 avoid unwieldiness. Kurt Hinterbichler: Theoretical aspects of massive gravity Returning to the action for m Þ 0 (and a not necessarily conserved source), we now know to apply the transformation h ¼ h0 þ  2  ; DÀ2 which yields S¼ Z 1 dD xLm¼0 ðh0 Þ À m2 ðh0 h0 À h02 Þ  2   1 DÀ1 D  hþ m2  À F F þ 2 2 DÀ2 DÀ2 0  0  À 2mðh @ A À h @ A Þ DÀ1 2 0 ðm h  þ 2m@ A Þ þ h0 T   DÀ2 2 2 2 T À A @ T  þ 2 @@T: þ DÀ2 m m (4.29) þ2 The gauge symmetry reads h0 ¼ @  þ @  þ  2 mà ; DÀ2 (4.30) A ¼ Àm ; A ¼ @ Ã;  ¼ ÀmÃ: (4.31) We can go to a Lorentz-like gauge, by imposing the gauge conditions (Huang and Parker, 2007; Nibbelink, Peloso, and Sexton, 2007) 685 S þ SGF1 þ SGF2 Z 1 ¼ dD x h0 ðh À m2 Þh0 2  1 À h0 ðh À m2 Þh0 þ A ðh À m2 ÞA 4 DÀ1 ðh À m2 Þ þ h0 T  þ2  DÀ2 2 2 2 T À A @ T  þ 2 @@T: þ DÀ2 m m (4.36) The propagators of h0 , A , and  are now, respectively,    Ài 1 1 ð    þ     Þ À   ; DÀ2 p2 þ m 2 2 Ài 1 DÀ2 Ài ; ; (4.37) 2 p2 þ m 2 4ð D À 1Þ p2 þ m 2 which all behave as $1=p2 for high momenta, so we may now apply standard power counting arguments. With some amount of work, it is possible to find the gauge invariant mode functions for h0 , A , and , which can then  be compared to the unitary gauge mode functions of Sec. II.B. In the massless limit, there is a direct correspondence;  is gauge invariant and its 1 degree of freedom is exactly the longitudinal mode (2.20), the A has the usual Maxwell gauge symmetry and its gauge invariant transverse modes are exactly the vector modes (2.21), and finally the h0 has  the usual massless gravity gauge symmetry and its gauge invariant transverse modes are exactly the transverse modes of the massive graviton. V. NONLINEAR INTERACTIONS @ h0  À 1 0 2@ h @ A þ m  þ mA ¼ 0;  10 DÀ1 h þ2  ¼ 0: 2 DÀ2 (4.32) (4.33) The first condition fixes the  symmetry up to a residual transformation satisfying ðh À m2 Þ ¼ 0. It is invariant under à transformations, so it fixes none of this symmetry. The second condition fixes the à symmetry up to a residual transformation satisfying ðh À m2 Þà ¼ 0. It is invaria...
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