  We discuss some consistency tests that must be passed for a successful
explanation of a diphoton excess at larger mass scales, generated by a scalar
or pseudoscalar state, possibly of a composite nature, decaying to two photons.
Scalar states at mass scales above the electroweak scale decaying significantly
into photon final states generically lead to modifications of Standard Model
Higgs phenomenology. We characterise this effect using the formalism of
Effective Field Theory (EFT) and study the modification of the effective
couplings to photons and gluons of the Higgs. The modification of Higgs
phenomenology comes about in a variety of ways. For scalar $0^+$states, the
Higgs and the heavy boson can mix. Lower energy phenomenology gives a limit on
the mixing angle, which gets generated at one loop in any such theory
explaining the diphoton excess. Even if the mixing angle is set to zero, we
demonstrate that a relation exists between lower energy Higgs data and a
massive scaler decaying to diphoton final states. If the new boson is a
pseudoscalar, we note that if it is composite, it is generic to have an excited
scalar partner that can mix with the Higgs, which has a stronger coupling to
photons. In the case of a pseudoscalar, we also characterize how lower energy
Higgs phenomenology is directly modified using EFT, even without assuming a
scalar partner of the pseudoscalar state. We find that naturalness concerns can
be accommodated, and that pseudoscalar models are more protected from lower
energy constraints.
