Why can't relativistic beams facilitate fusion? From this SE post, I see that the Coulomb force between two charged particles moving at relativistic speeds is balanced by the Lorentz force, and the overall force between the particles limits to zero as the particle's speed limits towards $c$.
This got me wondering: could this effect somehow be used to help overcome the Coulomb barrier in fusion reactions? For example, if a beam of deuterons could be accelerated to relativistic speeds, perhaps this effect could decrease the temperatures required to increase reaction cross sections to begin fusion.
I looked over the internet, and all I could find was this arXiv paper, which seems to suggest that this effect can be used to catalyze fusion.
What's wrong with this line of thinking? If this method was as simple as it sounds, then I imagine this would be a very active field of research - and yet, all I can find is a single, barely read, barely cited arXiv paper from 1997 over it. So, I'm guessing this idea, can't work, but I don't see why not.
The only thought I have is that possibly reaction rates may decrease due to time dilation, but again, I'm not extremely educated in special relativity.
 A: Here are two important differences between fission reactions and fusion reactions, from the perspective of an investor who would like to get more energy out of a machine than they put into it.
First, the magic thing about uranium fission is that fission interactions are triggered by fission decay products.  This means that you don't really have to do anything in order to extract energy from a uranium source.  In fact, if you can remove heat from the fission-produced neutrons without capturing them, they become even more effective at triggering new fission events. If you have enough uranium, with enough purity, you can just put it all in a pile.  Not only will the pile get hot all on its own, but there are certain ways of actively cooling the pile which cause it to generate heat more rapidly.  Fusion fuel doesn't have this effect. Every fusion reaction is equally hard-won.
Second, while fusion releases more energy per unit of fuel mass than does fission, fusion generates less energy per reaction.  The deuterium-tritium reaction
$$
\rm {^2H} + {^3H} \to {^4He} + n
$$
shares about 18 MeV between its output products, which is a factor of ten less than is released from a single uranium fission.  If your facilitating process costs more than a few MeV per fusion, other unavoidable inefficiencies mean you'll consume more energy than you produce.
A particle becomes relativistic when its kinetic energy is comparable to or larger than its rest mass.  A relativistic deuteron has a kinetic energy of many giga-eV. Relativistic beams of deuterium and/or tritium would have kinetic energies of many GeV per particle. If you want to spend one GeV to get 1.02 GeV back, the Carnot limits on the efficiency of heat engines tell you that you're going to have a bad time.
In a confined-plasma fusion device like ITER, the temperature corresponds to a typical kinetic energy measured in kilo-eV, so extracting energy from mega-eV reaction products is much less impossible.
A: 
This got me wondering: could this effect somehow be used to help
overcome the Coulomb barrier in fusion reactions? For example, if a
beam of deuterons could be accelerated to relativistic speeds, perhaps
this effect could decrease the temperatures required to increase
reaction cross sections to begin fusion.

Oh yes, for sure. In fact, fusion was discovered in 1932 by accelerating deuterons into a metal foil infused with deuterium. Every so often, one of the D's in the beam would hit one of the D's in the foil and hey presto, fusion. They noticed this by looking for the resulting alphas hitting a phosphor screen.
The issue is that "every so often", as in 10^-19 or so IIRC. You have to accelerate a whole bunch of D to get even a single reaction, so the energy balance is not exactly favorable. The main reason is that the D's will slow down very rapidly in the foil, and quickly be going too slow to fuse. See details here, the section on beam-beam devices (which maybe should be separated out?)
