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**Unformatted text preview: **te derivative tail in the standard ﬁeld equations already at ﬁrst order in
the weak-ﬁeld expansion (Sezgin and Sundell, 2002a). This
leads to tree-level amplitudes depending on the following two
dimensionless scales: (i) the weak-ﬁeld expansion coupling
g ¼ ð‘p ÞðDÀ2Þ=2 that can always be taken to obey g ( 1;
and (ii) the derivative-expansion coupling ð‘ÞÀnþ2 where ‘
is the characteristic wavelength. Thus the tails are strongly
coupled around solutions that are close to the AdSD solution
since here ‘ ( 1.
C. Main lessons The ﬁrst important lesson which one can draw from the
previous discussions is that, contrary to widespread prejudices, many doors are left open for massless higher-spin particles. The second important lesson is that interactions for
higher-spin gauge ﬁelds exist but are rather exotic. Some of
their properties clash with standard lores inherited from lowspin physics, and indeed, there is no fundamental reason to
expect that higher-spin ﬁelds must behave as their low-spin
companions.
Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 997 Some model-independent features of non-Abelian higherspin gauge theories seem to emerge from all known no-go
theorems and yes-go examples. It appears that most of the
exotic properties of higher-spin ﬁelds can roughly be explained by mere dimensional arguments. As done in the
previous section, we introduce a parameter ‘ with the dimension of a length and rescale all objects in order to work with
dimensionless Lagrangian L and ﬁelds ’. The action takes
R
the form S ¼ ‘ÀD dD xLð’; ‘@’; ‘2 @2 ’; . . .Þ where each
derivative is always multiplied by a factor of ‘. The
Lagrangian counterpart of Feynman rules in S-matrix arguments is the weak-ﬁeld expansion, i.e., the ﬁelds ’ are
perturbations around some background for which the
higher-spin Lagrangian L (if any) admits a usual perturbative
power expansion in terms of these ﬁelds ’. Around a stable
vacuum solution, this expansion starts with a quadratic
kinetic term Lð0Þ with at most two derivatives and it goes
on with vertices of various homogeneity degrees in ’: a cubic
vertex Lð1Þ , a quartic vertex Lð2Þ , etc.
In the following we present four general facts (of which
there is no proof in full generality but no counterexample has
ever been found) that seem to capture universal properties of
any massless higher-spin vertex.
(i) Higher-spin vertices are local order by order in some
length scale
A function of the ﬁeld and its derivatives (treated as
independent variables) is said to be local if it depends only
on a ﬁnite number of derivatives @’; @2 ’; . . . :; @k ’ (for some
ﬁxed integer k) and, moreover, if it depends only polynomially on these derivatives.
In the Lagrangian framework, the strong form of locality is
the condition that the Lagrangian L must be a local function
of the ﬁeld ’, i.e., the total number of derivatives is bounded
from above (so, in our...

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