In the comments to the previous post, Monty asks a perfectly good question, which can be shortened to: “Is the Higgs boson really necessary?” The answer is a qualified “yes” — we need the Higgs boson, or something like it. That is, we can’t simply take the Standard Model as we know it and extend it to arbitrarily high energies without new physics kicking in.

The role of the Higgs field is to break the symmetry of the electroweak interactions, as discussed here. We think that there is a lot of symmetry underlying particle interactions, but that much of it is hidden from our low-energy view. In technical terms, the electroweak theory of Glashow, Weinberg and Salam posits an “SU(2)xU(1)” symmetry, that somehow gets broken down to “U(1).” That unbroken symmetry gives us electromagnetism, a force carried by a massless particle, the photon. The broken symmetries are still there, but their force-carrying particles become massive when the symmetry breaks — those are the W^{+}, W^{–}, and Z^{0} bosons.

There’s no question that something breaks the symmetry. The question that is worth asking is: “Can we imagine breaking the symmetry without introducing any new particles?”

Let’s first think about how the Higgs mechanism actually works. What we call the “Higgs field” is actually a collection of four fields that rotate into each other under the symmetry. But it’s hard to draw a picture of a four-dimensional field, so consider instead this picture of the potential for a two-dimensional field (φ_{1} and φ_{2}). Notice that there are two kinds of ways the field can oscillate: the flat direction around the circle, and the radial direction in which the potential is highly curved. For the realistic four-dimensional case, there would be three flat directions and one radial one.

All of these directions are important. At high temperatures in the early universe, the Higgs bounces around in its potential, and its average value is zero (near the origin). But at lower temperatures things settle down, and the Higgs can oscillate around some point in that circle of minimum energy. Here is a crucial point: vibrations of the field in each direction are associated with particles, and the curvature of the potential corresponds to the *mass* of the associated particle. So the flat directions are massless particles, and the curved radial direction is a massive one.

But massless particles are easy to produce, so why don’t we see all of these massless Higgs bosons? The answer is: we have! Due to their interactions with the force-carrying particles, the three kinds of particles that would be massless Higgs bosons get “eaten” by the three W and Z bosons, which in turn gives them mass. That’s the miracle of the Higgs mechanism: where you might expect three massless Higgs particles and three massless force particles, instead you just get three massive force particles. So that’s the most simple reason why we need a Higgs field or something like it: we have already observed it, in the form of the massive W and Z bosons.

So what about the radial vibrations of the Higgs field, the ones that have a large mass? Those are what we call the actual “Higgs boson,” and that’s what we’re looking for at particle accelerators.

You are welcome to imagine a theory that has the three massless Higgs particles that get eaten by the W’s and the Z, but nevertheless lacks the radial component that we’re still searching for. Indeed, you can easily construct such a theory by cranking up the mass of the Higgs — as the mass approaches infinity, the field doesn’t oscillate at all, and there’s no corresponding particle. This is known in the trade as a non-linear sigma model. So could we have a model like that explain electroweak symmetry breaking, and do without the visible Higgs boson?

No. The argument here is more subtle, but nonetheless airtight. It comes from the fact that a Higgsless version of the Standard Model would “violate unitarity,” which is a fancy way of saying it would give nonsensical predictions.

Think about the scattering of two W bosons. It’s easy to use Feynman diagrams to calculate the probability that two W’s that pass by each other will actually scatter. The problem is, the result is a quantity that gets bigger and bigger as the energy increases — without limit. In other words, the probability that two W’s will interact becomes larger than one! That can’t really happen.

The Higgs comes to the rescue. That increasing probability is what you would get if you only considered the force-carrying particles, not the Higgs. But if we allow for the Higgs, it will also contribute to W scattering. It’s contribution also grows with energy, but with the opposite sign of the troublesome contributions from the W’s and Z’s themselves! That’s the miracle of quantum mechanics — different contributions to the same final state can actually interfere with each other. So the Higgs can save the W bosons from the catastrophic result of getting probabilities that add up to more than one.

At least, it can do that if its mass is low enough that it kicks in in time. Running the numbers, we find that the Higgs mass has to be lower than 800 GeV or so in order for W scattering to make sense. That’s why the Large Hadron Collider is built to look at energies of up to 1000 GeV; it’s important to make sure we should be able to find the Higgs. (Although there are still no guarantees.)

The above argument is airtight, but its conclusion is that *something* needs to happen before 800 GeV. That something might be the Higgs, or it might be something even more exotic. It’s the closest we have to a “no-lose” theorem in physics — either the Higgs boson will be there, or something more exciting. The experiments will have the final say, as they tend to do.

6th paragraph – “But massive particles are easy to produce” should presumably read “massless particles”.

Regarding the Higgs potential, this is the way I’ve usually read it, but I’ve also seen discussions of it in terms of an effective potential having phi and T dependence, so that the effective mass is positive above the transition temperature and negative below it. Are these essentially equivalent descriptions?

Thanks; that’s very technical but still very clear. Except for

But it’s not your responsibility to teach me that basic notation.“SU(2)xU(1)” symmetry, that somehow gets broken down to “U(1).”I had a couple of quibbles about wording. At one point you say “massive” where I think you mean “massless.” The other one? I forget…

James—you are correct that it has both a phi and a T dependence. Sean shows the potential at T=0. At higher temperature, the quadratic term (curvature at the origin) goes from being negative to being positive, changing the picture above into a bowl with a minimum at phi=0. So the symmetry is not broken at high temperature and broken at low temperature (just like the rotational symmetry of a ferromagnet).

Fixed the massive/massless snafu, thanks. Marc is completely right about the potential. Note that there is some issue about what you exactly mean by “the potential” when you are at nonzero temperature; one usually means “the effective potential,” which controls the

averagevalue of the field.Nice, nice little summary. Thank you.

“Low Math, Meekly Interacting” Wonderful! I love it!

Sean, I’ve heard lots about unitarity violation and the Higgs, but I haven’t seen it spelled out in any text. Could you point us to a good pedagogical text or review article? I suppose I could try to work it out myself, but eh, I’m a lazy grad student. If you can, thanks!

An excellent, concise post.

Lord Kelvin also thought his calculation of the age of the Earth was airtight.

No, he didn’t. He included a caveat about the possible discovery of a new source of energy. As we now know, he was right and radioactivity, unknown at the time of his prediction, is responsible for keeping the Earth warm and hence not taking this into account was the reason why he underestimated the age of the Earth.

Lord Kelvin also thought his calculation of the age of the Earth was airtight.But Kelvin didn’t conclude, “EITHER the earth is

nMYr old OR there is a hitherto unknown factor which permits it to be much older and whose discovery would be just as exciting as empirical demonstration of my conclusions”, which would be the analogy to what Sean is saying here. Kelvin’s mistake was to discount the possibility of unknown unknowns.Philip: “No, he didn’t. He included a caveat about the possible discovery of a new source of energy.”

First time I hear about this caveat, can you back your claim by some kind of evidence?

If true it shows that after all he wasn’t as blind about his potential ignorance as certain the-laws-underlying-the-physics-of-everyday-life-are-completely-understood modern day physicists 😛

@AI:

http://en.wikipedia.org/wiki/Lord_Kelvin#Age_of_the_Earth:_Geology_and_theology

Check the references etc

@Chris Winter: Check out http://www.amazon.com/Deep-Down-Things-Breathtaking-Particle/dp/080187971X . It’s a book written for lay people about particle physics that’s not afraid to include and explain a formula or two, including those describing symmetries.

Maybe this is a dumb question, but why can’t two particles interact with > 1 probability?

Couldn’t two particles affect each other more than once simultaneously in 5-space?

Re: 9-13:

Radioactivity is irrelevant to why Kelvin’s geochronology was off by two orders of magnitude.

It was his assumption that the earth did not convect that was his main mistake (convection drastically elevates the effective thermal conductivity). John Perry pointed this out in the 1890’s in a number of nature papers, but ignored for 60 years until evidence of mantle convection became indisputable.

See:

http://www.geosociety.org/gsatoday/archive/17/1/pdf/i1052-5173-17-1-4.pdf

for a modern review,

http://en.wikipedia.org/wiki/John_Perry_(engineer)

for background on Perry,

or

Nature 51, 341-342 (7 February 1895)

and

Nature 51, 582-585 (18 April 1895)

For the original articles

Modern estimates put radioactive decay at about 20-40% of the current oceanic crustal heat flow, with the rest being primordial heat.

max– I don’t know of any particularly good review, and I’m away from my books right now. You can find the relevant Feynman diagrams in this talk:

http://lhc.fuw.edu.pl/kalinWW.pdf

From there it’s just a matter of calculation…

My understanding is that while the SM Higgs has all these wonderful properties, it introduces new problems of its own, i.e. keeping the Higgs mass big enough to have evaded observation so far, but not absurdly big, requires considerable fine tuning, and maybe SUSY (or something) comes to the rescue. Even with this liability, other current ideas about EW symmetry breaking are themselves plagued with difficulties at least as bad.

I try my best to understand the pros and cons of the above, but tend to get stuck on the basics (I’m not sure I understand the strict meaning of “quartic coupling” sufficiently, nor why the nature of these couplings naturally seems to make the Higgs mass blow up unless exquisite radiative corrections or other mechanisms fix the problem).

If it’s not an imposition on the author or his readership, any help?

For a graduate student that wants to see how unitarity requires the Higgs, a good reference is

http://www.amazon.com/Introduction-Electroweak-Unification-Standard-Unitarity/dp/9810218575

Note that this is only “tree-level” unitarity, meaning that if you only look at the lowest order diagrams (the trees) that unitarity is not preserved. The loophole is that higher order diagrams may restore unitarity. As the energies get higher, this requires higher order diagrams to become more important than low order diagrams, a sign that your perturbation theory is not worth much anymore in the fields you have chosen to expand it. The same thing happens at really low energies with QCD: it becomes perturbatively non-unitary and the best perturbative description we have is in terms of hadrons.

You can still argue that quarks are gluons are not irrelevant at low-energies. Lattice QCD does provide a way of doing some calculations in the non-perturbative regime. As far as I am aware, this is still a viable “out” from the Higgs theorem. We may not see anything “new”, but interesting strong dynamics.

By “new” I am referring to new fundamental fields. In much the same way that we don’t consider the proton to be a “new field” at low energies, but a bound state of some messy QCD stuff. In terms of doing computations at these energies, it is certainly convenient to treat the proton as a new field and thus restore perturbative unitarity.

Anonymous (#15)– “Probability greater than 1” doesn’t mean that there is more than one interaction — it means that some kind of interaction happens more than 100% of the time. That’s just not possible.

LMMI– No question that the Higgs introduces other problems, especially with fine-tuning. Supersymmetry and other approaches aim to solve those problems. But the breakdown of unitarity isn’t just an annoyance; it’s a serious flaw that simply has to be addressed somehow. It might be the Higgs, or something else.

We don’ need no steenking Heeeggs!!

#21 Brian Too —

Speak for yourself, paleface! Maybe most people don’t need no “stinking Higgs”, but there are some nice people constituting the particle physics priesthood who really need it. They are the closest the media establishment can come to ordaining science in place of religion to “explain” (haha!) the universe. The masters of the world will look pretty stupid (or even more stupid) if all the Nova programs, Time magazine covers and all imitations are discovered to be erroneous, while the illiterate telepreachers blame gay atheistic eggheads for tinkering with God’s handiwork.

Thanks Sean and kiwidamien! Looks like I’ve got some homework to do.

Great post Sean. This maybe a bit OT but since this wonderful article by David Kaiser

on connections between Higgs fields and Brans-Dicke field and how both ideas were born

almost simulataneosly but without people knowing of each other’s work

http://web.mit.edu/dikaiser/www/Kaiser.WhoseMass.pdf

Dear Sean,

Nice article! I have two questions.

1. Can Higgs really serve as inflation field??

2. At finite temperature will the behavior of Higgs change??

Thanks.