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I am interested in finding out, why are collisions at 14TeV done in the LHC, instead of some other energy?

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Note that this years running is at $4\text{ TeV}$ per beam, not $7$, leading to a current CoM energy of $8\text{ TeV}$. –  dmckee Jul 6 '12 at 12:17
    
Hi Poonam, and welcome to Physics Stack Exchange! We generally prefer that you ask one question per post. To help you out, I removed the extra questions that were separate from your main question. Feel free to post them individually as separate posts, but do check our higgs-boson questions first to see if some of what you want to know has already been answered. –  David Z Jul 6 '12 at 17:28

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This year, the LHC is running at $8\text{ TeV}$. The energy of the beam is limited by the current in the magnets that steer the beam. The faster the particles travel, the higher the magnetic field has to be to turn them.

If the magnets were not the limiting factor, like at LEP, we would be limited by Synchrotron-radiation losses.

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The actual effective collision energy of the LHC is less than the beam energy, because the machine is not really colliding the protons, but only their constituents, the quarks and gluons. Imagine you are shooting two shotguns at the same point. Instead of the two shells hitting each other, you will, at most, get occasional collisions of the pellets that are inside the shells. The pellets do, of course, carry only a fraction of the kinetic energy of the shells in total. Very much the same is true for a hadron collider.

Moreover, how much of the energy in the quarks is available in the collision to produce new particles, depends on how close they actually get to each other. In a perfect head-on collision, all of the energy is available in the center of mass system. If there is a slight offset between the colliding particles, we are talking about a soft collision and far less energy is available for the actual particle-particle interaction that creates the new particles we are interested in.

To make sure that we get a lot of particles like the Higgs, which have a large rest mass, and therefor a high threshold energy for their creation, the machine has to have as high a beam energy as possible, because we are losing most of it in the partition due to the constituents and the less then head-on collision events. As others have pointed out, the maximum energy is basically limited by the size of the tunnel, and the field in the magnets. That's why the US tried to build a much larger machine in a much larger tunnel in the 1980s-1990s. That machine was called the SSC and it would have had a beam energy of 20TeV per proton. If the program hadn't been cancelled due to cost overruns and other problems, it would have been the far more capable accelerator. Sadly, sometimes "better" is too expensive, so now we have to do with the LHC for the time being.

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This is not really true. Though these events are rare (very rare!), there are $x_{1,2} > 1$ events; that is you can occasionally get parton collision energies to approach the arbitrarily close to the beam energy (and for the momentum transfer to exceed the beam momentum). If you are not familiar with this concept the $x$ there is "Bjorken scaling variable" also known as the light-cone momentum fraction. –  dmckee Sep 13 at 23:21
    
@dmckee: I theory you are right, but in practice the cross section of these processes is too small to be useful. As I said... "the actual EFFECTIVE collision energy"... and that's all that matters for the machine design. –  CuriousOne Sep 14 at 19:04

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