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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
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
(iii) Consistency requires an inﬁnite tower of ﬁelds with
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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