**Unformatted text preview: **t, it follows that one must send ÃQCD to zero. In this
limit, the size of a resonance grows indeﬁnitely, however, and
it becomes undetectable to an observer of ﬁxed size, since the
observer lives within the resonance’s Compton wavelength.15
B. Asymptotic states and conserved charges The S-matrix theorems concern only particles that appear
as asymptotic states. Moreover, within the perturbative approach, these asymptotic states are assumed to exist at all
energy scales. Thus, an intriguing possibility is that there
exists nonperturbatively deﬁned higher-spin gauge theories in
ﬂat spacetime with mass gaps and conﬁnement. We are not
aware of any thorough investigations of such models and
mechanisms so far, although Vasiliev’s higher-spin gravities
in four-dimensional anti–de Sitter spacetime have been conjectured to possess a perturbatively deﬁned mass gap, resulting from dynamical symmetry breaking induced via radiative
corrections (Girardello, Porrati, and Zaffaroni, 2003), as we
comment on below.
As far as conﬁnement is concerned,16 one may ask whether
the higher-spin charges of asymptotic states might all vanish,
such as for color charges in QCD. Incidentally, Weinberg
pointed out in his book (Weinberg, 2000), p. 13, that some
subtleties arise in the application of the Coleman-Mandula
theorem in the presence of infrared divergences, but that there
is no problem in non-Abelian gauge theories in which all
massless particles are trapped—symmetries if unbroken
would only govern S-matrix elements for gauge-neutral
bound states.
14 Strictly speaking, one can arguably refer to the proton as stable
while already the neutron is metastable while all other massive
excitations are far more short lived.
15
We thank one of the referees for this comment.
16
This way out was brieﬂy mentioned in the conclusions of
Bekaert, Joung, and Mourad (2009). 994 Xavier Bekaert, Nicolas Boulanger, and Per A. Sundell: How higher-spin gravity surpasses the spin- . . . C. Lorentz minimal coupling To reiterate slightly, the S-matrix no-go theorems17 for
´
higher-spin interactions are engineered for Poincare-invariant
relativistic quantum ﬁeld theories aimed at describing physics
at intermediate scales lying far in between the Planck
and Hubble scales. In Lagrangian terms, the generalized
Weinberg-Witten theorem can essentially be understood as
resulting from demanding compatibility between linearized
gauge symmetries and the Lorentz minimal coupling in the
absence of a cosmological constant. This compatibility requires consistent cubic vertices with one and two derivatives
for fermions and bosons, respectively. Vertices with these
numbers of derivatives have the same dimension as the ﬂatspace kinetic terms. If consistent, they therefore do not
introduce any new mass parameter. Hence it is natural to
extrapolate the Lorentz minimal coupling to all scales. In
doing so, however, one needs to keep in mind not only the
barrier for quantum ﬁelds in th...

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