X l 1 l1 1 s 1 s s0

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Unformatted text preview: . More precisely, we write the requirement of gauge invariance of the cubic action Sð1Þ ½’s ; ’s0 Š under linearized spin-s gauge transformations ð0Þ ’1 ÁÁÁs ¼ [email protected]ð1 2 ÁÁÁs Þ : s ð0Þ Sð1Þ þ ð1Þ Sð0Þ ¼ 0; s s where Sð0Þ denotes the free part of the action, ð0Þ denotes the s free spin-s gauge transformations, and ð1Þ denotes the gauge s transformations taken at linear order in the fields f’s0 ; ’s g and linear in the spin-s gauge parameter 1 ÁÁÁsÀ1 . The above equation implies that Â1 ÁÁÁs is a conserved current: @1 Â1 ÁÁÁs ð’s0 ; ’s0 Þ % 0; so that the Lorentz-invariance condition (A2) in the S-matrix approach is indeed equivalent to the conservation of the Noether current in the Lagrangian approach. Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 q1 A1 ÁÁÁs ðp1 ; . . . ; pN ; qÞ ¼ 0; 1005 8 q: (A2) 2. Cubic vertices In the particular case where the Feynman diagram (A1) is a single straight line, i.e., it describes the free propagation of a single particle, then the modified Feynman diagram essentially is the tree-level process In momentum space, Z Sð1Þ ¼ dD qdD p1 dD p2 ðp1 þ p2 þ qÞÀ1 ÁÁÁs j1 ÁÁÁs0 j1 ÁÁÁs0 Âðp1 ;p2 ; qÞ’1 ÁÁÁs ðqÞ’1 ÁÁÁs0 ðp1 Þ’1 ÁÁÁs0 ðp2 Þ: The cubic vertex with the lowest number of derivatives is of the form À1 ÁÁÁs j1 ÁÁÁs0 j1 ÁÁÁs0 ðp1 ; p2 ; qÞ / À1 ÁÁÁs ðp1 ; p2 ; qÞ1 1 Á Á Á s0 s0 ; where there is an implicit symmetrization over all  indices and À1 ÁÁÁs ðp1 ; p2 ; qÞ / ðp1 À p2 Þ1 Á Á Á ðp1 À p2 Þs is the cubic vertex for a scalar particle coupled to a spin-s massless particle. This coupling is called minimal in the sense that it contains the minimal amount of derivatives and also because it corresponds to a coupling with the Berends– Burgers–van Dam conserved currents associated with the rigid symmetries ’s0 ðkÞ ¼ i1 ÁÁÁsÀ1 k1 Á Á Á ksÀ1 ’s0 ðkÞ (Berends, Burgers, and van Dam, 1986) [see also Bekaert, Joung, and Mourad (2009) for more detail]. In the low-energy limit q ! 0, the only surviving cubic interaction is indeed the minimal coupling with s derivatives. The Lorentz-invariance condition (A2) on the amplitude Aðp1 ; . . . ; pN ; q; Þ for the further emission (or absorption) of a soft massless spin-s particle implies the conservation law of order s À 1 on the N external momenta (1) where each inserted minimal vertex À1 ÁÁÁs ðpi ; Àpi À q; qÞ came up with a coupling constant gðsÞ [for more detail, see, e.g., i Weinberg (1995), Sec. 13.1, or Blagojevic (2002), Appendix G]. Equivalently, these conservation laws can be obtained from the Noether charges associated with the abovementioned rigid symmetries. APPENDIX B: WEINBERG-WITTEN THEOREM: A LAGRANGIAN REFORMULATION 1. Weinberg-Witten theorem Weinberg and Witten (1980) designed their no-go theorem to eliminate ‘‘emergent gravity’’...
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