RevModPhys.84.987

The gravitational cubic coupling they obtained the

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Unformatted text preview: e Lorentz minimal coupling is present and triggers a finite expansion of nonminimal terms. A salient feature of the Fradkin-Vasiliev construction is that there are now two independent expansion parameters: the pffiffiffiffiffiffiffi AdS mass parameter  $ jÃj and the dimensionless deformation parameter g :¼ ð‘p ÞðDÀ2Þ=2 that counts the order in the weak-field expansion, where the Planck length ‘p appears in front of the action through 1=‘DÀ2 and where one works p with dimensionless physical fields. At the cubic level and for any given triplet of spins fs; s0 ; s00 g, there appears a finite expansion in inverse powers of , where the terms with the highest negative power of  bring the highest number of (A)dS-covariant derivatives Xavier Bekaert, Nicolas Boulanger, and Per A. Sundell: How higher-spin gravity surpasses the spin- . . . acting on the weak fields. The highest power of 1= is proportional to s00 , so that for unbounded spins the FradkinVasiliev cubic Lagrangian is nonlocal. The massive parameter  simultaneously (i) sets the infrared cutoff via jÃj $ 2 and the critical masses M2 $ 2 for the dynamical fields, and (ii) dresses the derivatives in the interaction vertices thus enabling the Fradkin-Vasiliev mechanism. This dual role played by the cosmological constant is responsible for an exotic property of the Fradkin-Vasiliev cubic coupling. Exotic nonlocality of the Fradkin-Vasiliev Lagrangian.— In the physically relevant cases where one has a separation of length scales, i.e., ‘p ( ‘ ( À1 , where ‘ $ k’k=k@’k is some wavelength characterizing the physical system under consideration and where À1 denotes here a generic infrared scale, not necessarily related to the cosmological constant, two situations can arise for perturbatively local (cf. Sec. IV.C) Lagrangians having vertices Vn involving higher (n ! 3) derivatives of the fields: (a) Mild nonlocality: The theory is weakly coupled in the sense that Vn $ ð‘p =‘ÞnÀ2 ( 1. This situation arises for broken higher-spin symmetry, tensionful string sigma models, etc. (b) Exotic nonlocality: The theory is strongly coupled in the sense that the vertices Vn are proportional to ð‘ÞÀnþ2 ) 1. This is the situation for the FradkinVasiliev vertices. In the derivative expansion appearing within the Fradkin-Vasiliev mechanism, the terms involving the maximal number of derivatives are dominant since they contain the infrared cutoff instead of the ultraviolet one. Finally, we make a comment related to the fully nonlinear Vasiliev equations in order to show that the same behavior appears order by order in the weak-field expansion. In this ð1Þ theory, the first-order corrections T to the stress tensor 1 defined by T :¼ R À 2 g ðR À ÃÞ arise in an expansion P P of the form T ð1Þ ¼ 1¼0 pþq¼n Àn rp ’s rq ’s ; see n Kristiansson and Rajan (2003) for the scalar field contributions. One therefore sees the appearance of an infini...
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