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I would particularly like to know why this process is considered the main search mode for Tevatron but useless for search at LHC.

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Short answer: at high enough $\sqrt{s}$, gluon fusion (gg -> H) and weak-boson fusion (qq -> H qq with two vectors as intermediary states) dominate over the Higgsstrahlung channel. For the first process, you need a loop of tops (costly in terms of energy) and the second one needs two massive gauge bosons (same story). I'm guessing that at Tevatron, these two modes were suppressed. –  Vibert Feb 10 '13 at 11:54

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What you're asking about are really the Higgs production mechanisms, not their search channels. Since the two colliders have different particles in their beams, they have different dominant Higgs production mechanisms: The Tevatron collided protons with anti-protons ($p \overline{p}$) while the LHC collides protons with other protons ($pp$). The first order Higgs Strahlung Feynman diagram shows a quark colliding with an anti-quark to produce a $W^{\pm}$ or $Z^0$ boson, which then radiates a Higgs. Since a $p$ is made of quarks (not anitquarks) and gluons, the LHC has a much lower probability of creating a Higgs through the Higgs Strahlung channel compared to the Tevatron, which had anti-quarks in abundance due to the $\overline{p}$ beam.

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