  We derive the constraints that can be imposed on the dimension-6 effective
theory extension of the Standard Model, using gluon fusion-initiated Higgs
boson pair production at the LHC. We use a realistic analysis focussing on the
$hh \rightarrow (b\bar{b}) ( \tau^+ \tau^- )$ final state, including
initial-state radiation and non-perturbative effects. We include the
statistical uncertainties on the signal rates as well as conservative estimates
of the theoretical uncertainties. We first consider a theory containing only
modifications of the trilinear coupling, through a $c_6 \lambda\, H^6/ v^2$
Lagrangian term, and then examine the full parameter space of the effective
theory, incorporating current bounds obtained through single Higgs boson
measurements. We also consider an alternative scenario, where we vary a smaller
sub-set of parameters. Allowing, finally, the values of the other coefficients
to vary within \textit{projected} experimental ranges, we find that the
currently unbounded parameter, $c_6$, could be constrained to lie within $|c_6|
\lesssim 0.6$ at 1$\sigma$ confidence, at the end of the high-luminosity run of
the LHC (14~TeV) in the full model, and to $-0.6 \lesssim c_6 \lesssim 0.5$ in
the alternative model. This study constitutes a first step towards the
inclusion of multi-Higgs boson production in a full fit to the dimension-6
effective field theory framework.
