  Common lore has it that naturalness of electroweak breaking in the SM
requires new physics (NP) at Lambda < 2-3 TeV, hopefully within the reach of
LHC. Moreover the Higgs should be light (m_h < 219 GeV) to pass electroweak
precision tests (EWPT). However one should be prepared for "unexpected"
(although admittedly unpleasant) results at LHC, i.e. no NP and/or a heavy
Higgs. We revisit recent attempts to accommodate this by modifying the SM Higgs
sector (using 2-Higgs-doublet models). We find that these models do not improve
the naturalness of the SM, and so they do not change the expectations of
observing NP at LHC. We also stress that a heavy SM Higgs would not be evidence
in favour of a modified Higgs sector, provided certain higher order operators
influence EWPT. On the other hand, we show that NP can escape LHC detection
without a naturalness price, and with the pure SM as the effective theory valid
at LHC energies, simply if the cut-off for top loops is slightly lower than for
Higgs loops.
