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I want to understand the Higgs Boson. Not in terms of analogies and metaphors, but in terms of hard math.

Assume I have a solid background in pre-university maths and physics. (I have many bits and pieces of higher-level stuff, but let's ignore that knowledge because it is not comprehensive or cohesive.)

What sequence of things do I need to learn in order to properly understand the Higgs Boson?

spraff
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    Step 1: learn QFT. Step 2: learn about spontaneous symmetry breaking. Step 3: ??? Step 4: profit. – AccidentalFourierTransform Aug 28 '18 at 23:18
  • https://www.youtube.com/playlist?list=PLaNkJORnlhZlH9Klm3Eckk6o71IgMYdM6 (find the qft lectures he alludes to in the first minute and watch before if necessary) – bolbteppa Aug 28 '18 at 23:33
  • I have suggested a duplicate question on books for learning QFT which lists many resources (some free, I think). However the mathematics will also be a challenge and you'll need to gather those skills as you develop rest of the skills. Note this is a multi-year process even for the dedicated and those with an inclination towards it starting from your base. It's absolute torture for anyone else, IMO. – StephenG - Help Ukraine Aug 29 '18 at 00:54

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The basic answer is: learn quantum mechanics, then quantum field theory, then the standard model.

By the way, it's the Higgs field that does all the interesting things - the Higgs boson is, to our knowledge, the most useless aspect of the Higgs field. It does nothing of any importance, and yet, because the Higgs field is so central, the Higgs boson also ends up interacting with most other particles.

There is a question as to why the Higgs boson isn't rendered superheavy by virtual particles, and the specific value of its mass even suggests that it has been finetuned by an unknown mechanism. So the Higgs boson may yet be a window on important undiscovered physics. But for now, it's probably the Higgs field that you want to understand.

Mitchell Porter
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    I'll never get this deprecation towards the Higgs boson in favour of the Higgs field: you cannot have one without the other, so claiming that the latter is the interesting one is completely meaningless. – AccidentalFourierTransform Aug 29 '18 at 03:01
  • Actually you can have one without the other, it's called the vacuum state... Anyway, do you agree that the Higgs boson makes no contribution to the mass of things, even though this is what the media has told people? – Mitchell Porter Aug 29 '18 at 09:57
  • No, you cannot have one without the other. That is precisely the reason millions of dollars were spent in order to detect the particle: without it, the whole formalism is inconsistent. 2)"Giving mass" is a very loose notion. Strictly speaking, you could say that the mass was already there by itself, but the kinetic term for the gauge bosons lack the corresponding longitudinal d.o.f. You must introduce the Higgs boson to play that role, for otherwise the Lagrangian has incompatible terms. Whether the Higgs actively gives mass, or it just provides the missing d.o.f., is a matter of opinion.
    • – AccidentalFourierTransform Aug 29 '18 at 15:07
  • If you excuse the poor metaphor, the Higgs boson and the Higgs field are the two sides of the same coin, and you are claiming that the side with the number is the side with monetary value. In fact, it is the whole thing what has value, and claiming that the side with the number is the only important side is a meaningless statement, even if it plays the somewhat special role of making explicit what the coin is worth. – AccidentalFourierTransform Aug 29 '18 at 15:14
  • Am I missing something? The excitations of the Higgs field are three Goldstones and the Higgs boson. The Goldstones become components of massive gauge bosons, as you describe. The Higgs boson does not. The only thing that the Higgs boson does, to my knowledge, is unitarize WW scattering... – Mitchell Porter Aug 30 '18 at 06:06
  • Then there's the way that the fermions acquire mass from the Higgs, via yukawa couplings. One sometimes sees this illustrated in terms of chirality-switching fermions coupling to individual Higgs bosons emitted from the vacuum, but I'm not aware of any way to realize that picture formally. Actual derivations always seem to involve just the Higgs vev, a property of the field... – Mitchell Porter Aug 30 '18 at 06:11
  • Of course the Higgs field cannot exist without the possibility of individual Higgs bosons existing, but most of the time the field is in the vacuum state, and that is enough for it to perform its mass-giving function. This is why I (and others) would say that the Higgs boson (i.e. the Higgs particle) does nothing. – Mitchell Porter Aug 30 '18 at 06:13