**Unformatted text preview: **turbation theory of higher-spin gravity is
more similar in spirit to that of open string theory.
22 Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 In other words, in ﬂat spacetime there are severe no-go
theorems forming a spin-two barrier that cannot be surpassed
in the sense that massless particles of spins s > 2 cannot
interact with massless particles of spins s 2 provided the
lower-spin sector contains ﬁnite minimal spin-two couplings.
Thus, if one wishes to proceed in seeking strictly massless
higher-spin gauge theories (with Ã ¼ 0) then one is forced
toward unnatural theories without any minimal spin-two
couplings, whereas if one switches on a ﬁnite Ã then one is
naturally led into the realms of higher-spin gravity.
B. The need for a complete theory We now emphasize the need for a complete theory of
higher-spin gravity already at the classical level, i.e., a consistent action principle, or alternatively, set of equations of
motion, that contains a complete set of strongly coupled
derivative corrections.
To this end, we return to the Fradkin-Vasiliev cancellation
mechanism within the Fronsdal program: in the presence of a
nonvanishing cosmological constant Ã, the Lorentz minimal
cubic coupling (two derivatives) for a spin-s ﬁeld becomes
embedded into the Fradkin-Vasiliev quasiminimal vertex
terminating in the non-Abelian-type 2-s-s vertex (2s À 2
derivatives) that remains consistent in the Ã ! 0 limit
(Boulanger, Leclercq, and Sundell, 2008); this ‘‘top vertex’’
is thus the seed from which the subleading powers in Ã are
grown by imposing Abelian spin-s gauge invariance. The
crux of the matter, however, is that the cubic piece of a
complete action (consistent to all orders) may in principle
contain additional nonminimal interactions with more derivatives that are strongly coupled in the Ã expansion.
Applying dimensional analysis one arrives at the following
problem: for Ã < 0 the on-shell amplitude (Witten diagram)
with three external massless gauge bosons need not vanish,
and since Ã now sets both the infrared cutoff (assuming the
free theory to consist of standard tachyon-and-ghost free
Fronsdal kinetic terms) and the mass scale for higherderivative vertices, the contributions to the amplitude from
vertices with n derivatives grow similar to the nth power of a
large dimensionless number. Thus, although the top (highestderivative) vertex dominates the terms with fewer derivatives
inside the quasiminimal coupling (including the Lorentz
minimal coupling), it will in its turn be washed out by any
genuinely nonminimal interaction, whose couplings (overall
normalization in units of Ã) must hence be determined in
order to estimate the three-particle amplitude.
Toward this end one may in principle work within a slightly
reﬁned Fronsdal program as follows: (i) ﬁx a free Fronsdal
action; (ii) parametrize all consistent cubic vertices including
a nonlocal Born-Infeld tail, that is, a strongly coupled expansion in terms...

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