  We derive constraints on the sign of couplings in an effective Higgs
Lagrangian using prime principles such as the naturalness principle, global
symmetries, and unitarity. Specifically, we study four dimension-six operators,
O_H, O_y, O_g, and O_gamma, which contribute to the production and decay of the
Higgs boson at the Large Hadron Collider (LHC), among other things. Assuming
the Higgs is a fundamental scalar, we find: 1) the coefficient of O_H is
positive except when there are triplet scalars, resulting in a reduction in the
Higgs on-shell coupling from their standard model (SM) expectations if no other
operators contribute, 2) the linear combination of O_H and O_y controlling the
overall Higgs coupling to fermion is always reduced, 3) the sign of O_g induced
by a new colored fermion is such that it interferes destructively with the SM
top contribution in the gluon fusion production of the Higgs, if the new
fermion cancels the top quadratic divergence in the Higgs mass, and 4) the
correlation between naturalness and the sign of O_gamma is similar to that of
O_g, when there is a new set of heavy electroweak gauge bosons. Next
considering a composite scalar for the Higgs, we find the reduction in the
on-shell Higgs couplings persists. If further assuming a collective breaking
mechanism as in little Higgs theories, the coefficient of O_H remains positive
even in the presence of triplet scalars. In the end, we conclude that the gluon
fusion production of the Higgs boson is reduced from the SM rate in all
composite Higgs models. Our study suggests a wealth of information could be
revealed by precise measurements of the Higgs couplings, providing strong
motivations for both improving on measurements at the LHC and building a
precision machine such as the linear collider.
