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I am just looking on the TopQuark production via proton antiproton collision and strong interaction. There seem to be three basic possibilities.

  1. $q + \bar q \rightarrow Gluon \rightarrow t +\bar t $

    This method is clear

  2. $Gluon + Gluon \rightarrow Gluon \rightarrow t + \bar t$

    Why do I need two Gluons to begin with? Why can't I just take one?

  3. What is the particle going straight up/down in this image? enter image description here

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You need the two initial gluons in your #2 because each one comes from the sea of one of the participating particles (i.e. the proton and the anti-proton). There isn't enough energy in a (anti-)proton to produce a top quark, but the energetic collision supplies enough.

The vertical line in your #3 is a space-like (anti-)quark line. It's meaning is very clear in the connection between quantum field theories and Feynman diagrams, but about the best you can say without the math is that it represents a virtual fermion. You may recall that anti-particles are mathematically equivalent to particles moving backward in time, With allows an interpretation in which it takes two interaction with the gluon to "reverse" it and then you have a single particle that comes from the in as an anti-particle, gets turned around and heads back to the future as a particle.

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2) A collision is taking place between two protons. One gluon comes from one proton, the other from the other proton. You cannot "just take one" gluon, because if you did, you would be describing a process in which one proton spontaneously radiates, rather than what you are interesting in: a proton-proton collision. And one proton cannot spontaneously radiate a gluon that decays to two top quarks due to energy conservation.

3) The top quark

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In general in order to have an interaction a scattering is needed. One scatters one particle against the other. Otherwise momentum would not be conserved.

It is the same with decays, at least two particles are necessary for conservation of momentum.

3) is a diagram for 2). i.e. gluon gluon to top antitop. What is exchanged is a virtual top .

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