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Unformatted text preview: ia, M. Henneaux, C. Iazeolla, V. Mathieu, K. Meissner, R. Metsaev, J. Mourad, D. Polyakov, M. Porrati, A. Sagnotti, E. Sezgin, E. Skvortsov, D. Sorokin, Ph. Spindel, M. Taronna, M. Tsulaia, M. A. Vasiliev, Y. Zinoviev, and Xi Yin for various discussions over the years. APPENDIX A: WEINBERG LOW-ENERGY THEOREM: S MATRIX AND LAGRANGIAN DICTIONARY Weinberg (1964) obtained stringent constraints on S-matrix elements by considering the effects tied to the emission of soft massless quanta. Consider an S-matrix element with N external particles of momenta p (i ¼ 1; 2; . . . ; N ) corresponding to the Feynman i diagram where all external momenta pi are on their respective mass shells. For simplicity, all momenta are taken to be ingoing and the polarizations of these particles are left implicit in A. 1. Emission of a massless particle: Lorentz versus gauge invariances The amplitude for the further emission (or absorption) from any leg of a single massless spin-s particle of momentum q and polarization 1 ÁÁÁs ðqÞ is denoted by Aðp1 ; . . . ; pN ; q; Þ: In general, the line of this extra particle can be attached to any other line, either internal or external. In relativistic quantum field theory, the polarizations are not Lorentz-covariant objects: under Lorentz transformations, one has 1 ÁÁÁs ðqÞ ! 1 ÁÁÁs ðqÞ þ sqð1 2 ÁÁÁs Þ ðqÞ Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 Xavier Bekaert, Nicolas Boulanger, and Per A. Sundell: How higher-spin gravity surpasses the spin- . . . for some symmetric tensor  where the round brackets denote complete symmetrization over the indices. This property is well known for massless particles and is the counterpart of gauge invariance in the Lagrangian approach. Lorentz invariance of the S matrix and the decoupling of spurious degrees of freedom thus require the condition so À1 ÁÁÁs ðp1 ; p2 ; qÞ :¼ A1 ÁÁÁs ðp1 ; p2 ; qÞ is the part of the cubic vertex which corresponds to the Noether current in the Lagrangian approach. The conservation of the Noether current in the Lagrangian approach is equivalent to the Lorentz-invariance condition (A2) in the S-matrix approach. We see this in more detail by considering a cubic vertex of type s-s0 -s0 with s Þ s0 . The massless particle of spin s is of arbitrary momentum q (so off shell) while the two particles of spins s0 are on shell with respective momenta p1 and p2 . Writing explicitly the polarizations ð1Þ ðp1 Þ and ð2Þ ðp2 Þ of the two spin-s0 particles, the cubic vertex takes the form À1 ÁÁÁs ðp1 ; p2 ; qÞ ¼ À1 ÁÁÁs j1 ÁÁÁs0 j1 ÁÁÁs0 ðp1 ; p2 ; qÞ Â ð11ÞÁÁÁs0 ðp1 Þð21ÞÁÁÁs0 ðp2 Þ:   In the Lagrangian language, the cubic interaction term corresponding to the cubic vertex is, without loss of generality, of the form Sð1Þ ½’s ; ’s0 Š :¼ Z dD x L ð 1 Þ ; Lð1Þ :¼ ’1 ÁÁÁs Â1 ÁÁÁs ð’s0 ; ’s0 Þ; where Â1 ÁÁÁs is bilinear in ’s0...
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