  We present a phenomenological analysis of asymmetric bottom- and charm-quark
production within the LHCb acceptance relevant for $pp$ collisions at $\sqrt{s}
= 13 \, {\rm TeV}$. Predictions are provided for both anti-$k_t$ bottom- and
charm-jet pairs, which are kept differentially with respect to the invariant
mass of the jet pair. It is quantified how data in this region can provide
sensitivity to the couplings of the $Z$ boson to heavy quarks, and we
investigate what precision is needed to compete with LEP. We also discuss how
asymmetry and rate measurements can provide constraints on a particular class
of new-physics models, which contains gauge bosons with small/moderate
couplings to light/heavy quarks and masses of the order of $100 \, {\rm GeV}$.
Predictions are obtained including all relevant QCD and QED/weak contributions
up to next-to-leading order, which have been implemented in a Fortran code
which allows to directly compute the asymmetric cross sections. We provide all
relevant analytic formulas for our computations.
