Creating even one Higgs boson requires a tremendous amount of energy. The Higgs boson is 125 times heavier than a proton. At a proton accelerator and collider like the LHC, most of the energy in creating a Higgs boson comes from the kinetic energy of the protons. Each proton needs -62.5 GeV of kinetic energy to produce a Higgs boson, which requires them to collide at 99.999991% the speed of light. This is the minimum energy to create a Higgs boson, but it doesn't guarantee that one will be produced. The likelihood of one being produced is once in every 10 billion proton collisions.
The Standard Model predicts a few ways that Higgs bosons could be produced, but they are all extremely rare. The most common modes of production are shown in the Feynman diagrams below (Fig 1). The left is by far the most likely, where gluons from inside the colliding protons annihilate into a loop of virtual top and bottom quarks, which then produce a Higgs boson. Since the coupling of particles to the Higgs is proportional to their mass, these heavy quarks are the most likely to produce a Higgs boson. In the right diagram, two fermions emit W or Z bosons, which annihilate into a Higgs boson. This process works for any two fermions.
The Higgs boson has too short of a lifespan to be directly observed, so we have to observe it through its decay products. The most common decay mechanism for the Higgs boson is a virtual loop of top and bottom quarks, which then produce a pair of gamma rays that hit a detector. Computer algorithms at CERN analyze the particles produced in proton-proton collisions and only keeps the events with decay products of interest to a Higgs boson production. From here, statistical analysis is needed for physicists to be confident that the decay products are produced by a Higgs boson and not by random chance of other Standard Model processes.
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