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The Higgs mechanism allows massless fields to acquire mass through their coupling to a scalar field. But if the masses cannot be predicted because the couplings have to be fixed, what really is the utility of the Higgs mechanism? Instead of saying "Here are a priori couplings; the Higgs mechanism generates mass.", I could just as well say "Here are a priori masses. Period.".

I understand that the Higgs mechanism is crucial to electroweak unification, but I have the same question there. Why does electromagnetism and the weak force have to be unified? Even if the couplings of the photon, Z and W bosons became related on unification, this is still at the cost of introducing new parameters - so it's not really clear to me that something has been explained or tidied up.

Do either the Higgs mechanism or electroweak unification tell us something new? Do either make any predictions that don't come at the cost of extra parameters? (I'm not really challenging anything here; I'm sure the answer to both questions is 'yes' - I just want to fill the gaps in my understanding as I study the Standard Model)

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Dbrane, aside from "beauty", the electroweak unification is actually needed for a finite theory of weak interactions. The need for all the fields found in the electroweak theory may be explained step by step, requiring the "tree unitarity".

This is explained e.g. in this book by Jiří Hořejší:

http://www.amazon.com/dp/9810218575/
Google books:
http://books.google.com/books?id=MnNaGd7OtlIC&printsec=frontcover&hl=cs#v=onepage&q&f=false

The sketch of the algorithm is as follows:

Beta-decay changes the neutron to a proton, electron, and an antineutrino; or a down-quark to an up-quark, electron, and an anti-neutrino. This requires a direct four-fermion interaction, originally sketched by Fermi in the 1930s, and improved - including the right vector indices and gamma matrices - by Gell-Mann and Feynman in the 1960s.

However, this 4-fermion interaction is immediately in trouble. It's non-renormalizable. You may see the problem by noticing that the tree-level probability instantly exceeds 100% when the energies of the four interacting fermions go above hundreds of GeV or so.

The only way to fix it is to regulate the theory at higher energies, and the only consistent way to regulate a contact interaction is to explain it as an exchange of another particle. The only right particle that can be exchanged to match basic experimental tests is a vector boson. Well, they could also exchange a massive scalar but that's not what Nature chose for the weak interactions.

So there has to be a massive gauge boson, the W boson.

One finds out inconsistency in other processes, and has to include the Z-bosons as well. One also has to add the partner quarks and leptons - to complete the doublets - otherwise there are problems with other processes (probabilities of interactions, calculated at the tree level, exceed 100 percent). It goes on and on.

At the end, one studies the scattering of two longitudinally polarized W-bosons at high energies, and again, it surpasses 100 percent. The only way to subtract the unwanted term is to add new diagrams where the W-bosons exchange a Higgs boson. That's how one completes the Standard Model, including the Higgs sector. Of course, the final result is physically equivalent to one that assumes the "beautiful" electroweak gauge symmetry to start with.

It's a matter of taste which approach is more fundamental and more logical. But it's certainly true that the form of the Standard Model isn't justified just by aesthetic criteria; it can be justified by the need for it to be consistent, too.

By the way, 3 generations of quarks are needed for CP-violation - if this were needed. There's not much other explanation why there are 3 generations. However, the form of the generations is tightly constrained, too - by anomalies. For example, a Standard Model with quarks and no leptons, or vice versa, would also be inconsistent (it would suffer from gauge anomalies).

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  • $\begingroup$ Nice outline, Lubosh! I like you mentioned some difficulties. $\endgroup$
    – Vladimir Kalitvianski
    Feb 10, 2011 at 17:31
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    $\begingroup$ Now that is the kind of answer I was looking for :) I'm all for beauty and elegance too, but in the end I would only let an inconsistency like non-unitarity or anomalies make me add things to my theory. Thanks for the great answer again Lubos - I'll see if I can find a copy of the book you referred to. $\endgroup$
    – dbrane
    Feb 10, 2011 at 17:31
  • $\begingroup$ Thanks, @dbrane, and good luck in finding it for free (there are 3 copies left at amazon). The author was my QFT instructor in Prague. @Vladimir, right, there are problems, but the goal - and a successful one - of the procedure is to solve the difficulties. And indeed, they're solved at the end. $\endgroup$
    – Luboš Motl
    Feb 10, 2011 at 17:52
  • $\begingroup$ Great, I just discovered that there are physics books that you can't find in any of the Cambridge libraries. :/ $\endgroup$
    – dbrane
    Feb 10, 2011 at 18:05
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    $\begingroup$ It's because "Lubosh" is easier to type than "Luboš", at least on my computer. I'm lazy so I just call him "Dr. Motl". $\endgroup$
    – Carl Brannen
    Feb 11, 2011 at 0:06
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Electroweak unification means that there is a symmetry between electromagnetic and weak interactions -- you can use them interchangeably. In reality this is not the case -- $W$ and $Z$ bosons have mass, while photon haven't.

Higgs mechanism provides a way for a spontaneous symmetry breaking between those interactions: Lagrangian of standard model is electroweak symmetric, while the vacuum is not due to non-zero vacuum expectation value for the Higgs field.

The same applies for fermions -- you cannot introduce mass terms for quarks and leptons in the Lagrangian, while preserving the electroweak symmetry. But it is possible to introduce electroweak symmetric Yukawa terms, coupling Higgs field to the fermions.

Edit:
I don't think that Higgs mechanism can "tell us something new". It is just a simplest way to ensure spontaneous symmetry breaking. While electroweak unification means that those interactions are gauge interactions and establishes the gauge symmetry itself. The classification of fermions into three generations is also done from the "electroweak point of view".

Of course you can argue that this systematization or classification is not "something new". But from such a point of view one can criticise almost every theoretical construction that attempts to predict results of future experiments.

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    $\begingroup$ Thanks, but you haven't answered my question, you've just made it a bit more explicit. Introducing mass terms breaks electroweak symmetry. So instead, we impose electroweak symmetry on a massless theory and use the Higgs mechanism to generate back the masses. My question is, why not assume there never was an electroweak symmetry to start with, and all along work with a massive theory without any symmetry breaking? That's one less scalar field and one less particle to look for. $\endgroup$
    – dbrane
    Feb 10, 2011 at 16:14
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    $\begingroup$ Of course, it's great that this leads to the prediction of a new particle. But I can't, at the moment, see why Nature would want to burden itself with an additional scalar field and associated particle when it could do everything as an SSB-less theory with the same number of free parameters. $\endgroup$
    – dbrane
    Feb 10, 2011 at 16:16
  • $\begingroup$ Very good reasoning! I support it with two my hands! $\endgroup$
    – Vladimir Kalitvianski
    Feb 10, 2011 at 16:17
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It is true that recognizing that the data have an SU2xU1 symmetry one has a number of parameters and one may say that problem number one has been reduced to problem number two which has the same number of unknowns.

Let me give the often quoted example of the epicycles in the geocentric system and the ellipses in the heliocentric one. The number of parameters are the same, and if you go to a planetarium program and go to the geocentric frame, the epicycles will appear in all their glory. Nevertheless there is nobody now who would ask, "what is the use of ordering the data in the heliocentric system".

Asking "why should electromagnetism be unified with weak theory" is a bit like asking "why have a heliocentric system".The answer is that in both cases, the data fall into an ordered form effortlessly. And then we were led to higher symmetries (SU3xSU2xU1) and more inclusive theories.

I should have added that the clarity introduced by symmetries/order once manifested lead to calculable theories which can describe the data and make predictions for new observations. The Higgs is one such prediction coming out from the theories describing the quark model symmetries. If it is not found, a rethink will be in order, (though there are papers which claim that a Higgs mechanism can be a composite manifestation, not necessarily one particle).

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  • $\begingroup$ "effortlessly"? How much fog they supply to hide failures of this approach! $\endgroup$
    – Vladimir Kalitvianski
    Feb 10, 2011 at 16:36
  • $\begingroup$ @Anna: No, it's not the same because requiring unification introduces a new physical particle. It's not at all like looking at things from a more natural frame of reference (as in heliocen. vs. geocen.) because there the two descriptions are equally valid in that they don't differ in predictions - but differ in how they fit in naturally with known theories of celestial mechanics. $\endgroup$
    – dbrane
    Feb 10, 2011 at 16:36
  • $\begingroup$ @dbrane Well, the heliocentric system introduces a force, that was not present in the geocentric view of the world, no? $\endgroup$
    – anna v
    Feb 10, 2011 at 16:39
  • $\begingroup$ But the geocentric view involves forces too of course - only they are much more complicated to calculate because of the "unnatural" choice of coordinates. And again, in the end, the predictions are the same. $\endgroup$
    – dbrane
    Feb 10, 2011 at 16:42
  • $\begingroup$ To Anna: I disagree with your analogy with celestial mechanics. And you speak of introducing a "new physical particle" as a human intellect achievement whereas it is a fix of worthless (massless) theory. I can remind you bare particles, virtual particles, ghosts - all invented to stay in this failed approach. I would love to learn how physical electron looks like. Nobody can describe it yet. $\endgroup$
    – Vladimir Kalitvianski
    Feb 10, 2011 at 16:42
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In a phenomenological approach one introduces masses without problem.

It is in the "local gauge invariance" approaches where one needs a fix because the gauge approach does not give masses, unfortunately. Technically it is is implemented as "coupling" to Higgs in some way.

I never considered the "local gauge invariance principle" as reasonable or physical. It is one of many blind ways to "construct" interacting theories; it does not guarantee anything physical and does not save us from infinities in calculations. It's been done by analogy with QED which itself has problems. This way needs fixes.

I would say the Higgs is a price for choosing a "mathematical" rather than physical approach to constructing theories. It is a wrong guess, in my opinion. There are more physical ideas on this subject but they remain unknown and thus unexploited.

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    $\begingroup$ what is this "phenomenological approach" you refer to? $\endgroup$
    – user346
    Feb 10, 2011 at 16:23
  • $\begingroup$ Do you remember the Schroedinger equation with $m_e$ or Dirac equation with $m_e$? Something like this, if you like. $\endgroup$
    – Vladimir Kalitvianski
    Feb 10, 2011 at 16:33

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