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same result had already been obtained in various particular
instances, where it was shown that the Lorentz minimal
coupling prescription applied to free higher-spin gauge ﬁelds
enters in conﬂict with their Abelian gauge symmetries
(Aragone and Deser, 1979; Berends et al., 1979; Aragone
and La Roche, 1982; Boulanger and Leclercq, 2006). The
complete no-go result ruling out the Lorentz minimal coupling of type 2-s-s in the Lagrangian approach is given in
Boulanger, Leclercq, and Sundell (2008).
In between the Lagrangian and the S-matrix approaches
lies the light-cone approach where all local cubic vertices in
dimensions from four to six have been classiﬁed [see, e.g.,
Metsaev (2006) and references therein] and where the same
negative conclusions concerning the Lorentz minimal coupling of higher-spin gauge ﬁelds to gravity had already been
reached and stated in complete generality.
12 For a pedagogical essay, see, e.g., Loebbert (2008). Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012 This being said, consistent cubic vertices between spin-two
and higher-spin gauge ﬁelds do exist, even in Minkowski
spacetime (Boulanger and Leclercq, 2006; Metsaev, 2006;
Boulanger, Leclercq, and Sundell, 2008). Instead of describing Lorentz’s minimal coupling, they contain more than two
derivatives in total. As one can see, the generalized WeinbergWitten theorem does not by itself forbid such type 2-s-s
interactions. The crux of the matter is to combine this theorem with the Weinberg equivalence principle.
Together, the Weinberg equivalence principle and the generalized Weinberg-Witten theorem prohibit the cross couplings of massless higher-spin particles with low-spin
particles in ﬂat spacetime (Porrati, 2008). The argument
goes as follows: elementary particles with spin not greater
than 2 are known to couple minimally to the graviton at low
energy. Therefore (Weinberg’s equivalence principle) all particles interacting with low-spin particles must also couple
minimally to the graviton at low energy, but [generalized
Weinberg-Witten theorem (Porrati, 2008) and identical results presented in Metsaev (2006) and Boulanger, Leclercq,
and Sundell (2008)] massless higher-spin particles cannot
couple minimally to gravity around the ﬂat background.
Consequently, at low energies massless higher-spin particles
must completely decouple from low-spin ones. Hence, if the
same Lagrangian can be used to describe both the low-energy
phenomenology and the Planck-scale physics, then no higherspin particles can couple to low-spin particles (including
spin two) at all.
E. Velo-Zwanziger difﬁculties In this section, we stress that, contrary to widespread
prejudice, the Velo-Zwanziger difﬁculties do not constitute
a serious obstruction to the general program of constructing
consistent interactions involving higher-spin ﬁelds. The observed pathologies are nothing but symptoms of nonintegrability in the sense of Cartan of the differential equations
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