S0 s00 is bounded by s0 s00 s n s s0 s00 therefore

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Unformatted text preview: rtex contains at least s00 derivatives. In other words, the value of the highest spin involved (s00 ) gives the lowest number of derivatives that the cubic vertex must contain. Note that this proposition applies to low and high spins. Examples of type 1-1-1 and 2-2-2 vertices are the cubic vertices in Yang-Mills and Einstein-Hilbert actions; they contain one and two derivatives, respectively. Examples of 2-s-s vertices are, for low spins, the Lorentz minimal coupling (s 3=2) where the energy-momentum tensor involves two derivatives (also for s ¼ 2) and, for high spins (s > 2) the higher-derivative nonminimal coupling mentioned before. The following two exotic properties of higher-spin particles are straightforward corollaries of results presented so far. Higher-derivative property: In flat space, local cubic vertices including at least one massless particle of spin strictly higher than two contains three derivatives or more. Low-spin coupling: In flat space, massless higher-spin particles couple nonminimally to low-spin particles. In (A)dS, they couple quasiminimally, thereby restoring Weinberg’s equivalence principle (gravitational coupling) and the conventional definition of electric charge (electromagnetic coupling). (iii) Consistency requires an infinite tower of fields with unbounded spin A local cubic vertex is said to be perturbatively consistent at second order if it admits a local, possibly null, quartic continuation such that the resulting Lagrangian incorporating the cubic and associated quartic vertices (with appropriately modified gauge transformation laws) is consistent at second order in the perturbative coupling constant. Note that the assumption of (perturbative) locality is crucial here. If this assumption is dropped, then consistency is automatic beyond the cubic level [see, e.g., the general theorem by Barnich and Henneaux (1993)] in the sense that any cubic vertex can be completed by nonlocal quartic Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 vertices, etc. It is the assumption of (perturbative) locality that imposes very strong constraints on the set of possibilities. In the local, non-Abelian deformation problem, a necessary requirement for the consistency of cubic vertices to extend to the quartic level is the closure of the algebra of gauge symmetries (at lowest order and possibly on shell). This imposes stringent constraints on the algebra in (A)dS spacetime (Fradkin and Vasiliev, 1987a): the presence of at least one higher-spin gauge field requires for consistency at quartic order an infinite tower of gauge fields with unbounded spin (more precisely the minimal spectrum seems to be a tower including all even spins). At the cubic level, the coupling constants of each cubic vertex are independent from each other. Another constraint coming from the consistency at the quartic level is that the coupling constants of the cubic vertices are expressed in terms of a single one. Surprisingly, similar resu...
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