  The recent experimental condensation of ultracold atoms in a triangular
optical lattice with negative effective tunneling energies paves the way to
study frustrated systems in a controlled environment. Here, we explore the
critical behavior of the chiral phase transition in such a frustrated lattice
in three dimensions. We represent the low-energy action of the lattice system
as a two-component Bose gas corresponding to the two minima of the dispersion.
The contact repulsion between the bosons separates into intra- and
inter-component interactions, referred to as $V_{0}$ and $V_{12}$,
respectively. We first employ a Huang-Yang-Luttinger approximation of the free
energy. For $V_{12}/V_{0} = 2$, which corresponds to the bare interaction, this
approach suggests a first order phase transition, at which both the U$(1)$
symmetry of condensation and the $\mathbb{Z}_2$ symmetry of the emergent chiral
order are broken simultaneously. Furthermore, we perform a renormalization
group calculation at one-loop order. We demonstrate that the coupling regime
$0<V_{12}/V_0\leq1$ shares the critical behavior of the Heisenberg fixed point
at $V_{12}/V_{0}=1$. For $V_{12}/V_0>1$ we show that $V_{0}$ flows to a
negative value, while $V_{12}$ increases and remains positive. This results in
a breakdown of the effective quartic field theory due to a cubic anisotropy,
and again suggests a discontinuous phase transition.
