How to identify a hard high energy process from a soft one?

A hard process can be described as a hard scattering, and a soft as a soft scattering.

The measurement of hard scattering processes, meaning those with energy scales of more than a few GeV, is the main method by which physics is being explored and extended by the experiments at the Large Hadron Collider

within the article:

“Hardness” is usually specified in terms of the momentum component that is transverse to the beam, p_T.

High transverse momenta flag events where deep inelastic processes can take place, exploring the structure of the interaction.

Typical lepton and photon transverse momentum thresholds applied in measurements of hard processes are around 20 GeV.

In your process:

$$gg to h^0 to Z^0Z^0 to mu^+mu^-q bar q$$

one measures a muon pair and two jets with a large invariant mass, since one is looking for the h0. This means that there are large angles between the four measurered four vectors, high transverse momenta to the beam.

Soft are the processes with small angles to the beam ( small p_T) and small energies.