proton collision

[JohnDenker]

Let's consider example 14.13 ("proton collision") in section 14.8 on page 372.

Advantage:

The idea behind this example is highly valuable. It is a great way to show the power of spacetime and 4-vectors.

First problem:

Although idea is excellent, the execution is flawed. It starts by saying:

Two protons [...] collide to produce a particle that has a mass 300 times the mass of each proton.

It soon becomes clear we are talking about a single reaction product, with no other products.

I don't think that is possible.

• The element Rg has very nearly the desired mass; however it cannot be produced in a p-p collision, because that would violate conservation of baryon number.
• Maybe I'm overlooking something, but I don't see how to produce anything else either, anything that would qualify as a single bound particle.

  Second problem:

The example concludes by saying:

It is therefore not unreasonable that the energy of the proton is on the order of microjoules.

I reckon the energy in question is something like 85 TeV. That seems like playing basketball on Pluto. It seems unreasonable, compared to (say) the CERN LHC, which in 2015 was upgraded to 6.5 TeV per beam. It's going to be a long time before anybody gets to 85 TeV.

Third problem:

An important part of the conceptual development depends on checkpoint 14.24. This is the last sentence in the chapter, and it sits in a place where it could easily be overlooked.

I'm not saying this is the most important result in the world, but it is amusing and it has a favorable cost/benefit ratio. For anyone who followed example 14.13, the incremental cost of redoing the analysis in the "collider" geometry is small, and the result is interesting.

Fourth problem:

Last but not least, the section title refers to Conservation of Energy but that seems unhelpful. Certainly it is inconsistent with the spacetime viewpoint. The thing that makes this example interesting is the fact that you have to conserve both energy and momentum. In other words, the 4-momentum is conserved. The 4-vector is conserved, no matter what frame (if any!) you choose. In any particular frame, each of the four components is separately conserved.

Suggestions:

• Choose a reaction that occurs on a reasonable energy-scale and satisfies the conservation laws. Pick a specific reaction that actually occurs. Note that producing a particle/antiparticle pair helps satisfy the conservation laws.
  • A nice simple example is p+p --> p+p+p+pbar
  • If you want to get fancy you can produce something heavier, perhaps a charmed Omega (and its antiparticle).
• Restate the question to ask for the minimum energy. Explain that this occurs when all the products move off together in a bundle. This is easy to understand in the center-of-mass frame. It is a loose bundle -- _not_ a single bound particle.
• Draw the spacetime diagrams to show what's going on. See e.g. https://www.av8n.com/physics/spacetime-welcome.htm#sec-bevatron-vector and https://www.av8n.com/physics/spacetime-welcome.htm#sec-bevatron-graphical

• The third problem could be alleviated by rewording a couple of the existing exercises, so that they touch on the same point. However there is still some chance that the point will get dropped, because usually only a subset of the exercises gets assigned. Perhaps it would help to add a short exercise that focuses tightly on this issue:

  Explain why the CERN LHC is a collider ... even though achieving a collision (as opposed to a miss) is verrrry difficult and reduces the reaction rate; it would be much simpler to use a single beam and let it impinge on a large dense target at rest in the lab frame.

That still doesn't guarantee that the exercise gets assigned. Perhaps the badge system discussed in item #198 would help.

• Change the section title to Conservation of 4-Momentum. Emphasize that the vector is conserved. It is conserved no matter what frame (if any) you choose. In any particular frame, this means that each of the four components is separately conserved. In any particular frame, the timelike piece is conservation of energy, while the spacelike piece is conservation of momentum.
  Still, the idea of conservation is frame-independent, which is important because the calculation in example 14.13 requires several switches from one frame to another. Frame-switching seems routine if you already know how to do it, but students were not born knowing how to do it.

[JohnDenker]

Additional suggestion:

• Change the section title to Conservation in Spacetime or something like that.

Rationale: Most of the calculational effort in example 14.13 revolves around conservation of the [energy,momentum] 4-vector ... but before you can even begin that calculation, you need to write down a reaction that conserves charge, conserves baryon number, conserves various types of lepton number, et cetera. It's supposed to be a book about principles, and this is a golden opportunity to mention and reinforce some of the most fundamental principles in all of physics.