  We study scattering amplitudes in massless non-abelian gauge theory where all
outgoing gluons have positive helicity. It has been argued recently by Costello
that for a particular fermion representation (8 fundamentals plus one
antisymmetric-tensor representation in $SU(N)$) the one-loop amplitudes vanish
identically. We show that this vanishing leads to previously-observed
identities among one-loop color-ordered partial amplitudes. We then turn to two
loops, where Costello has computed the all-plus amplitudes for this theory, as
rational functions of the kinematics for any number of gluons using the
celestial chiral algebra (CCA) bootstrap. We show that in dimensional
regularization, these two-loop amplitudes are not rational, and they are not
even finite as $\epsilon\to0$. However, the finite remainder for four gluons
agrees with the formula by Costello. In addition, we provide a mass regulator
for the infrared-divergent loop integrals; with this regulator, the CCA
bootstrap formula is recovered exactly. Finally, we use the CCA bootstrap to
compute the double-trace terms in the theory at two loops for an arbitrary
number of gluons.
