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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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