  The Higgs associated production cross section at an $e^+ e^-$ collider is
indirectly sensitive to the Higgs self-coupling, $h^3$, at next-to-leading
order (NLO). Utilizing this, a new indirect method is proposed for constraining
deviations in the self-coupling below the di-Higgs production threshold in
certain models. Although this indirect constraint is model-dependent, making it
valid only under specific assumptions, meaningful indirect constraints on the
self-coupling may be realized. Specific realistic scenarios where the indirect
constraint applies are discussed and in particular it is shown that in the
well-motivated class of two Higgs-doublet models there exist regions of
parameter space in which the NLO effects from a modified self-coupling dominate
over the LO modifications, demonstrating a concrete scenario in which large
modifications of the Higgs self-coupling may be indirectly constrained using
the proposed method. Other models, such as strongly coupled scenarios, are also
discussed. The indirect method would give valuable constraints on deviations in
the Higgs self-coupling, and would be complementary to the direct measurements
possible with di-Higgs production at other colliders, providing precious
additional information in the effort to unravel the properties of the Higgs
boson.
