  The Higgs trilinear coupling is a crucial tool to investigate the structure
of the Higgs sector and the nature of the electroweak phase transition, and to
search for indirect signs of New Physics. Classical scale invariance (CSI) is
an attractive concept for BSM model building, explaining the apparent alignment
of the Higgs sector and potentially relating to the hierarchy problem. A
particularly interesting feature of CSI theories is that, at one loop, they
universally predict the Higgs trilinear coupling to deviate by 67% from the SM
prediction at tree level. This result is however modified at two loops, and we
present here results from the first explicit computation of two-loop
corrections to the Higgs trilinear coupling in classically scale-invariant BSM
models. Taking as example a CSI variant of the Two-Higgs-Doublet Model, we show
that the inclusion of two-loop effects allows distinguishing different
scenarios with CSI, even though the requirement of correctly reproducing the
mass of the Higgs boson, as well as unitarity, severely restrict the possible
values of the Higgs trilinear coupling.
