  Could new physics first manifest itself in Higgs self-coupling measurements?
In other words, how large could deviations in the Higgs self-coupling be, if
other Higgs and electroweak measurements are compatible with Standard Model
predictions? Using theoretical arguments supported by concrete models we derive
a bound on the ratio of self-coupling to single-Higgs coupling deviations in
ultraviolet completions of the Standard Model where parameters are not
fine-tuned. Broadly speaking, a one-loop hierarchy is allowed. We thus stress
that self-coupling measurements at the LHC and future colliders probe uncharted
parameter space, presenting discovery potential even in the absence of emerging
hints in single-Higgs coupling measurements. For instance, if other observables
show less than two-sigma deviations by the end of the LHC programme, the Higgs
self-coupling deviations could still exceed 200% in the models discussed,
without introducing fine-tuning of ultraviolet parameters.
