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**Unformatted text preview: **theories where the graviton
is a bound state of particles with spin one or lower. Its proof Xavier Bekaert, Nicolas Boulanger, and Per A. Sundell: How higher-spin gravity surpasses the spin- . . . 1006 involves S-matrix manipulations which will be discussed in
more detail in the next section on its reﬁned version. If one
assumes locality, then it becomes surprisingly easy to prove
the Lagrangian version of the Weinberg-Witten theorem. Let
½s denote the integer part of the spin s.
Lemma: Any local polynomial which is at least quadratic
in a spin-s massless ﬁeld, nontrivial on shell and gauge
invariant, must contain at least 2½s derivatives.
Proof: Corollary 1 of Bekaert and Boulanger (2005) states
that, on shell, any local polynomial which is gauge invariant
may depend on the gauge ﬁelds only through the Weyl-like
tensors. The latter tensors contain ½s derivatives, thus the
lemma follows.
j
A straightforward corollary of this lemma is a version of
the Weinberg-Witten theorem.
Weinberg-Witten theorem (Lagrangian formulation):
(i) Any perturbatively local theory containing a charge
current J which is nontrivial, Lorentz covariant, and gauge
invariant, forbids massless particles of spin s > 1=2.
(ii) Any perturbatively local theory containing a Lorentzcovariant and gauge-invariant energy-momentum tensor T
forbids massless particles of spin s > 3=2.
Proof: In the free limit, any Noether current in a perturbatively local theory must be a quadratic local polynomial. For
massless ﬁelds of spin s > 1=2, the lemma implies that this
polynomial must contain at least two derivatives (or four
derivatives if s > 3=2). However, the charge current contains
one derivative and the energy-momentum tensor contains two
derivatives.
j
The lower bound s > 3=2 of this version is slightly weaker
than the lower bound s > 1 of the original Weinberg-Witten
theorem (Weinberg and Witten, 1980). Anyway the case
s ¼ 3=2 is low spin and thereby is not a main concern of
this paper.
2. Reﬁnement of Weinberg-Witten theorem Porrati (2008) takes gauge invariance into account in order
to still use Weinberg-Witten’s argument but in a context
where the stress-energy tensor need not be gauge invariant
(or Lorentz covariant, which is the same in a secondquantized setting) any more.
In the original work (Weinberg and Witten, 1980) a particular matrix element was considered: elastic scattering of a
spin-s massless particle off a single soft graviton. The initial
and ﬁnal polarizations of the spin-s particle are identical,
say þ s, its initial momentum is p and its ﬁnal momentum is
p þ q. The graviton is off shell with momentum q. The
matrix element is
hþs; p þ qjT j þ s; pi: (B1) In the soft limit q ! 0 the matrix element is completely
determined by the equivalence principle, as recalled above
when reviewing Weinberg’s low-energy theorem. Using the
relativistic normalization for one-particle states hpjp0 i ¼
2p0 ð2Þ3 3 ðp À p0 Þ, we...

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