Sum of momenta of two protons in LHC moving in opposite directions is assumed to be zero but it can't be exactly zero so how close does it get to zero?
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$\begingroup$ Note that the accelerator itself is a really good energy filter - every proton bunch in the machine has the same energy or they would not be circulating nicely. $\endgroup$– Jon CusterApr 14, 2021 at 0:38
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$\begingroup$ If they have exactly the same energy the length of a bunch would be zero. But the length is finite albeit small ($5\ \mu m$ I think) so there must be relative motion between protons withing a bunch. The more sluggish ones are accelerated by the radiofrequency resonators and the faster ones are slowed down to keep them in the bunch but how close can their energies be brought together? $\endgroup$– Some StudentApr 14, 2021 at 0:50
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1 Answer
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The sum of the momenta of the two protons is a frame-dependent quantity. In the center of mass frame, their momenta sum to zero. In, for instance, a frame which is boosted parallel to the momentum of one of the protons, relative to the center of mass frame, the sum of their momenta is nonzero.
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$\begingroup$ I forgot about the reference frame. What about reference frame of a detector? $\endgroup$ Apr 14, 2021 at 0:14
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$\begingroup$ The detector's reference frame, commonly called the "lab frame" is indeed a valid frame. It may or may not be the same as the center of mass frame. $\endgroup$ Apr 14, 2021 at 0:18