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**Unformatted text preview: **conventions, the Lagrangian is a
polynomial in the length parameter ‘). A weaker form of
locality is the requirement that the Lagrangian L is perturbatively local in the sense that it admits a power series
expansion in the ﬁelds and all their derivatives (so, in our
conventions, each vertex must admit a power series expansion
in the length scale ‘). Strictly speaking, this weak form of
locality is rather a mild form of nonlocality because it is
obviously not equivalent to the genuine requirement of locality. Nevertheless, it guarantees that somehow the nonlocality
(if any) is under control: at each order in the length scale, the
theory is local; the bound on the total number of partial
derivatives is controlled by the power of ‘. Concretely, this
means that there is no strong nonlocality (such as inverse
powers of the Laplacian) and that, perturbatively, it can be
treated as a local theory. Effective Lagrangians provide standard examples of perturbatively local theories.
We note that if at the cubic level one forgoes the assumption of perturbative locality, then the higher-spin gravitational
minimal coupling around ﬂat space would become automatically consistent. Remember that, in the early attempts to
minimally couple higher-spin particles around ﬂat space
(Berends et al., 1979; Aragone and Deser, 1980; Aragone
and La Roche, 1982), the problem was that the higher-spin
variation of the cubic Lagrangian creates terms " Smin $
R
" Á ðW@’ þ @W’Þ proportional to the spin-two linearized
Weyl tensor W , where " is the higher-spin gauge parameter. Xavier Bekaert, Nicolas Boulanger, and Per A. Sundell: How higher-spin gravity surpasses the spin- . . . 998 These terms cannot be compensated by an appropriate
local gauge transformation for the spin-two ﬁeld, since the
linearized Weyl tensor (or its symmetrized and traceless
derivative) does not vanish on shell. However, if one deals
with wildly nonlocal operators and inserts the formal object
‘‘h=h’’ in front of the Weyl tensor, one can compensate
R
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the terms " Á ðh hW@’ þ @ h hW’Þ by appropriate nonlocal spin-two gauge transformations of the form h $
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h @ ð"@’ þ @"’Þ, using the fact that, contrary to the Weyl
tensor, the d’Alembertian of the Weyl tensor is proportional
to the ﬁeld equations for the spin-two ﬁeld. Schematically,
hW $ @C where C denotes the (linearized) Cotton tensor
which is itself a linear combination of the curl of the (linearized) Einstein tensor.
(ii) Higher-spin vertices are higher derivative
The higher-derivative property has been observed in all
known examples of higher-spin cubic couplings. A summary
of the general situation at the cubic level and in ﬂat space is as
follows.
Cubic interactions (Metsaev, 2006): In ﬂat space, the total
number n of derivatives in any consistent local cubic vertex of
type s-s0 -s00 (with s s0 s00 ) is bounded by
s0 þ s00 À s n s þ s0 þ s00 : Therefore, the ve...

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