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Flip Tanedo | USLHC | USA

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Who ate the Higgs?

While one of the priorities of the LHC is to find the Higgs boson (also see Aidan’s rebuttal), it should also be pointed out that we have already discovered three quarters of the Standard Model Higgs. Just don’t expect to hear about this in the New York Times; this isn’t breaking news—we’ve known about this “three quarters” of a Higgs for nearly two decades now. In fact, these three quarters of a Higgs live inside the belly of two beasts: the Z and W bosons!

What the heck do I mean by all this? What is “three quarters” of a particle? What does the Higgs have to do with the Z and the W? And to what extent have we or haven’t we discovered the Higgs boson? These are all part a subtle piece of the Standard Model story that we are now in an excellent position to decipher.

Goldstone Eaten

What we will find is that there’s not one, but four Higgs bosons in the Standard Model. Three of them are absorbed—or eaten—by the Z and W bosons when they become massive. (This is very different from the way matter particles obtain mass!) In this sense the discovery of massive Z and W bosons was also a discovery of these three Higgs bosons. The fourth Higgs is what we call the Higgs boson and its discovery (or non-discovery) will reveal crucial details about the limits of the Standard Model.

The difference between massless and massive vectors

In the not-so-recent past we delved into some of the nitty-gritty of vector bosons such as the force particles of the Standard Model. We saw that relativity forces us to describe these particles with four-component mathematical objects. But alas, such objects are redundant because they encode more polarization states than are physically present. For example, a photon can’t spin in the direction of motion (longitudinal polarization) since this would mean part of the field is traveling faster than the speed of light.

Now, what do we mean by polarization anyway? We’d previously seen that polarizations are different ways a quantum particle can spin. In fact, each polarization state can be thought of as an independent particle, or an independent “degree of freedom.” In this sense there are two photons: one which has a left-handed polarization and one with a right-handed polarization.

Because massive particles (which travel slower than light) can have a longitudinal polarization, they have an extra degree of freedom compared to massless particles. So repeat after me:

The difference between massless force particles (like the photon and gluon) and massive force particles (like the W and Z) is the longitudinal degree of freedom.

Since a “degree of freedom” is something like an independent particle, what we’re really saying is that the W and Z seem to have an “extra particle’s worth of particle” in them compared to the photon and gluon. We will see that this poetic language is also technically correct.

The mass of a force particle is important for large scale physics: the reason why Maxwell was able to write down a classical theory of electromagnetism in the 19th century is because the photon has no mass and hence can create macroscopic fields. The W and Z on the other hand, are heavy and can only mediate short-range forces—it costs energy for particles to exchange heavy force particles.

Massive vectors are a problem

The fact that the W and Z are massless is also important for the following reason:

In the early days of quantum field theory, massive vector particles didn’t seem to make any sense!

The details don’t matter, but the punchline is that the very mathematical consistency of a typical theory with massive vector particles breaks down at high energies. You can ask a well-posed physical question—what’s the probability of Ws to scatter off one another—and it is as if the theory itself realizes that something isn’t right and gives up halfway through, leaving your calculations in tatters. It seemed like massive vector particles just weren’t allowed.

If that’s the case, then how can the W and Z bosons be massive? Contrary to lyrics to a popular Lady Gaga song, the W and Z bosons were not “born this way.” Force particles naturally appear in theories as massless particles. From our arguments above, we now know that the difference between a massless and a massive particle is a single, extra longitudinal degree of freedom. Somehow we need to find extra longitudinal degrees of freedom to lend to the W and Z.

Technical remark & update (10 Oct): As a commenter has pointed out below, I should be more careful in how I phrase this. Theories of massive vectors (essentially nonlinear sigma models) only become non-unitary at tree-level so that we say they lose “perturbative unitarity.” This on its own is not a problem and certainly doesn’t mean that the they is “mathematically inconsistent” since they become strongly coupled and get large corrections from higher order terms. What we do lose is calculability and one has to wonder if there’s a better description of the physics at those scales. Many thanks to the ‘anonymous’ commenter for calling me out on this. :-)

 

Let them eat Goldstone bosons

Jeff Goldstone

Jeffrey Goldstone, image from his MIT webpage

Where can this extra degree of freedom come from? One very nice resolution to this puzzle is called the Higgs mechanism. The main idea is that vector particles can simply annex another particle to make up the “extra particle’s worth of particle” it needs to become massive. We’ll see how this works below, but what’s really fantastic is that this is one of the very few known ways to obtain a mathematically consistent theory of massive vector particles.

So what are these extra particles?

Since particles with spin carry at least two degrees of freedom, this “extra longitudinal degree of freedom” can only come from a spin-less (or scalar) particle. Such a particle has to somehow be connected to the force particles that want to absorb it, so it should be charged under the weak force. (For example, neutrinos are uncharged under electromagnetism since they don’t talk to photons, but they are charged under the weak force since they talk to the W and Z bosons.)

Further, this particle has to obtain a vacuum expectation value (“vev”). Those of you who have been following along with our series on Feynman diagrams will already be familiar with this, though we’re now approaching the topic from a different direction.

In general, particles that can be combined with massless force particles to form massive force particles are called Goldstone bosons (or Nambu-Goldstone bosons including one of the 2008 Nobel prize winners) after Jeffrey Goldstone, pictured to the right. The Goldstone theorem states that

A theory with spontaneous symmetry breaking has massless scalar particle in its spectrum.

For now don’t worry about any of these words other than the fact that this gives a condition for which there must be scalar particles in a theory. We’ll get back to the details below and we’ll see that these scalar particles, the Goldstone bosons, are precisely the scalars which massless force particles can absorb in order to become massive.

So now we arrive at another aphorism in physics:

Force particles can eat Goldstone bosons to become massive.

In light of this terminology, perhaps a more appropriate cartoon of this is to draw the Goldstone particle as a popular type of fish-shaped cracker…

Force particles like Gold[stone]fishes because they're so delicious.

A W eating a Goldstonefish cracker... get it? (I hope we don't get sued for this.)

Technical remarks for experts: (corrected Oct 10 thanks to anon.) The “mathematical inconsistency” of a generic theory of massive vectors is the non-unitarity of tree-level WW scattering. This isn’t really an inconsistency since the theory of massive vectors has a cutoff; as one approaches the cutoff loop-level diagrams give large corrections to the amplitude and the theory becomes strongly coupled. While this isn’t a technical necessity for new physics, it is at least a very compelling reason to suspect that there is at least a better description.

In the Standard Model this is done perturbatively. The tree-level cross section for WW scattering increases with energy but is unitarized by the Higgs boson.

Saying that force particles are “born massless” is a particular viewpoint that lends itself to this UV completion by linearization of the nonlinear sigma model associated with a phenomenological theory of massive vectors. This isn’t the only game in town. For example, one can treat the ρ meson is a vector that can be understood as the massive gauge boson of a `hidden’ gauge symmetry in the chiral Lagrangian. The UV completion of such a theory is not a Higgs, but the appearance of the bound quarks that compose the ρ. The analogs of this kind of UV completion in the Standard Model are technicolorcomposite Higgs, and Higgs-less models.

Four Higgses: A different kind of redundancy

Okay, so we have three massive gauge bosons: the W+, W-, and Z. Each one of these has two transverse polarizations (right- and left-handed) in addition to a longitudinal polarization. This means we need three Goldstone bosons to feed them. Where do these particles come from? The answer should be no surprise, the Higgs.

Indeed, you might think I’m selling you the Standard Model like an informercial:

If you buy now, the Standard Model comes with not one, not two, not even three, but four—count them, four—Higgs bosons!

Four Higgs bosons?! That’s an awful lot of Higgs. But it turns out this is exactly what we have: we call them the H+, H-, H0, and h. As you can see, two of them are charged (you can guess these will be eaten by the Ws), two are uncharged. Here’s they are:

The Four Higgses of the Standard Model

"The Four Higgses of the Standard Model," biblical pun intended

Where did all of these Higgses come from? And why did our theory just happen to have enough of them? These four Higgses are all manifestations of a different kind of redundancy called gauge symmetry. The name is related to gauge bosons, the name we give to force particles.

When we described vector particles, we said that our mathematical structure was redundant: our four-component objects have too many degrees of freedom than the physical objects they represented. One redundancy came from the restriction that massless particles can have no longitudinal polarization. This brings us down from 4 degrees of freedom to 3. However, we know that massless particles only have two polarizations—we have to remove one more polarization. (Similarly for massive particles, which have 3, not 4, degrees of freedom.) This left-over redundancy is precisely what we mean by gauge symmetry.

For those with some calculus-based physics background: this is related to the fact that the electromagnetic field can be written as derivatives of a potential. This means the potential is defined up to an constant. This overall constant (more generally, a total derivative) is a gauge symmetry. To connect to the quantum picture, we previously mentioned that the vector potential is the classical analog of the 4-vector describing the photon polarization.

Technical remark: in some sense, this gauge symmetry is not a ‘symmetry’ at all but an overspecification of a physical state such that distinct 4-vectors may describe identical state. (Compare this to a symmetry where different states yield the same physics.)

Gauge symmetry doesn’t just explain the redundancy in the vector particles, but it also imposes a redundancy in any matter particles that are charged under the associated force. In particular, the gauge symmetry associated with the weak force requires that the Higgs is described by a two component complex-valued object. Since a complex number contains two real numbers, this means the Higgs is really composed of four distinct particles—the four particles we met above.

Now let’s get back to the statement of Goldstone’s theorem that we gave above:

A theory with spontaneous symmetry breaking has massless scalar particle in its spectrum.

We’re already happy with the implications of having a scalar. Let’s unpack the rest of this sentence. The hefty phrase is “spontaneous symmetry breaking.” This is a big idea that deserves its own blog post, but in our present case (the Standard Model) we’ll be “breaking” the gauge symmetry associated with the W and Z bosons.

What happens is that one of the Higges (in fact, this is “the Higgs,” the one called h) gets a vacuum expectation value. This means that everywhere in spacetime there Higgs field is “on.” However, the Higgs carries weak charge—so if it is “on” everywhere, then something must be ‘broken’ with this gauge symmetry… the universe is no longer symmetric since there’s a preferred weak charge (the charge of the Higgs, h).

For reasons that we’ll postpone for another time, Goldstone’s theorem then implies that the other Higgses serve as Goldstone bosons. That is, the H+H-, and H0 can be eaten by the W+W-, and Z respectively, thus providing the third polarization required for a massive vector particle (and doing so in a way that is mathematically consistent at high energies).

Three of the four Higgses are Goldstones and are eaten by the W and Z.

Epilogue

There are still a few things that I haven’t told you. I haven’t explained why there was exactly one Goldstone particle for each heavy force particle. Further, I haven’t explained why it turned out that each Goldstone particle had the same electric charge as the force particle that ate it. And while we’re at it, I haven’t said anything about why the photon should be massless while the W and Z bosons gain mass—they’re close cousins and you may wonder why the photon couldn’t have just gone off and eaten the h.

Alas, all of these things will have to wait for a future post on what we really mean by electroweak symmetry breaking.

What we have done is shown how gauge symmetry and the Higgs are related to the mass of force particles. We’ve seen that the Higgs gives masses to vector bosons in a way that is very different from the way it gives masses to fermions. Fermions never “ate” any part of the Higgs but bounced off its vacuum expectation value, while the weak gauge bosons feasted on three-fourths of the Higgs! This difference is related to the way that relativity restricts the behavior of spin-one particles versus spin–one-half particles.

Finally, while we’ve shown that we’ve indeed discovered “3/4th of the Standard Model Higgs,” that there is a reason why the remaining Higgs is special and called the Higgs—it’s the specific degree of freedom which obtains the vacuum expectation value which breaks the gauge symmetry (allowing its siblings to be eaten). The discovery of the Higgs would shed light on the physics that induces this so-called electroweak symmetry breaking, while a non-discovery of the Higgs would lead us to consider alternate explanations for what resolves the mathematical inconsistencies in WW scattering at high energies.

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21 Responses to “Who ate the Higgs?”

  1. anonymous says:

    Hi Flip,

    The SM without the Higgs is not “mathematically inconsistent”. What is a problem is tree level unitarity – the theory becomes strongly coupled but it is not ill-defined. You just don’t have a small parameter to do a perturbative expansion any more (Like chiral perturbation theory close to the cut-off).

    Cheers!

    • Leo says:

      hi anonymous,
      I don’t think I agree with this, because such a breakdown of unitarity signals the emergence of new physical degrees of freedom. In other words, the effective theory breaks down at some energy where there are particles that have been integrated out already. In the case of the
      SM, this is the Higgs particle. The non-unitarity signals the presence of a new particle, similar to the presence of the electroweak bosons in a 4-Fermi coupling.
      cheers

    • Flip Tanedo says:

      Hi anon. Yes, you’re right—I should be more careful when I say this. The theory is not _perturbatively_ unitary, but as you say it becomes strongly coupled and incalculable. I’m updating the relevant technical remarks.

    • Flip Tanedo says:

      @Leo: I think anon.’s point was that unitarity breaks down only at tree level. Before you reach non-unitary scattering the theory becomes strongly coupled and loop corrections become sizable. While this theory is incalculable, anon. is right that it’s not “mathematically inconsistent.”

    • leo says:

      @Flip:
      Just to make sure I understand this correctly: Is the argument that the SM is consistent, or that we cannot tell weather it is? The fact that it is incalculable suggests the latter to me. In addition, could you tell me which perturbation expansion breaks down? It seems to me that it would be a (p^2/M^2) expansion with M somewhere around the Higgs scale (ie, the description as an effective theory fails).

      Since the SM has all the properties of an EFT (most general given symmetries, Neutrino masses as the first 1/M correction), I would argue that the breakdown of the 1/M expansion does signal a mathematical inconsistency: You have not included states (elementary or bound, whatever) of mass > M which you should at that point. The breakdown of tree-level unitarity is a symptom that you are missing something, but the breakdown of the EFT is the real inconsistency

    • Flip Tanedo says:

      @ Leo: I think the correct statement is that the SM is not necessarily inconsistent, but very plausibly breaks down in favor of a UV completion.

    • renormalisable? says:

      I think there is still one more important point being missed here which is to do with whether the theory is renormalisable. To highlight this lets look at two separate cases: with and without a Higgs boson.

      With the higgs boson, the theory is renormalisable and so UV complete. However if the Higgs mass is too large the breakdown of tree unitarity signals that the theory becomes strongly coupled and so not calculable, but not inconsistent. All loop orders become relevant, keeping the theory unitary

      Without the Higgs boson, the theory is indeed like a non-linear sigma model but this is non-renormalisable, So it is not a complete theory, only an effective theory up to a cut-off. The breakdown of tree unitarity in this scenario actually tells us where this cut-off is and above the cut-off we would need new physics which ultimately must be a renormalisable unitary theory.

      The only caveat to the above is if the non-linear sigma model is asymptotically safe, in which case the theory could be non-perturbatively renormalisable and hence UV complete. Here the breakdown of tree unitarity again simply informs us when perturbation theory breaks down (strong coupling). This idea has received some attention recently by Roberto Percacci.

  2. gunn says:

    The fine Dirac equation with two nonzero massive members is enough for the description of fermions with nonzero mass. Bosons locally submit to analog of the equation of Klein-Gordon with nonzero masses. Higgs aren’t necessary: http://arxiv.org/abs/physics/0302013

  3. jal says:

    I hope you will address the mass mechanism for the proton and quarks when you are finished the the ws’ and z.
    jal

    • Flip Tanedo says:

      Hi jal: At the level at which I think is appropriate for the blog, the quarks get masses from “bouncing off the Higgs vev,” as discussed in a previous post (linked in the main text). The proton mass is a whole different story and has to do with strong coupling… I may have referred to it at some point in the past when I mentioned confinement. The main idea is that the proton mass is the binding energy of the proton. Unlike the hydrogen atom, which has a negative binding energy because the energy decreases as you pull the electron away, a proton has a positive binding energy because the chromodynamic energy increases as you pull apart two point charges. Maybe in the future I’ll delve a bit more into this, but I’m trying to get to supersymmetry, extra dimensions, dark matter, and all of that stuff. :-)

  4. Alejandro Rivero says:

    A similar question, used very effectively by Dam and Veltman (and in a minor degree by Case Gasiorowicz time ago), is the difference between massless and massive bosons in the limit of small mass. I have seen textbooks arguing on this already at the level of field quantisation of free particles, and sometimes they argue exactly in the reverse direction: the massive vector particle, as mass goes to zero, spits out an scalar.

  5. ohwilleke says:

    The precision electroweak fits seem to be optimized by a Higgs boson mass of 92 GeV (which, of course, has already been ruled out by direct detection).

    Would it perhaps make sense to simply use a 92 GeV Higgs mass as a ghost number for calculations and refine that value as necessary to make precision electroweak fits better without actually trying to hard to look for the damn thing?

    After all, if the value makes the equations work right, we don’t really need the actual boson at all.

  6. fluidic says:

    Few things are in order. I have dearly noticed that results and data from several LHC experiments are being interpreted differently, by different scientists and physicists at CERN and outside CERN, so we feel there is not much sense anymore.

    – Science is not telling stories and making up a mind boggling scenario or beautiful “what i think” interpretation of an outcome. Look into this blog post <<>>

    We cannot continue to play with the minds of people outside CERN and outside physics and science by trying to interpret things, predict things, guess what nature is hiding, find a nice story that explains some behavior in the data, etc…

    When we see the CERN community divided along the lines of whether the higgs boson can make the 5Billion Euros spent or whether we have to allow ourselves the pleasure of celebrating 30-40 yrs of looking for this monster creation and declaring victory for NOT finding traces for it anywhere and that the 2D hilbert space is some good abstract mathematical tool to continue using for spin up and spin down “2D” to satisfy Heisenberg’s uncertainty principle of the never seen electron “spin inside” the atom, but surely well observed outside the atom …

    There are dozens and dozens of reasons that violate and crack the validity of the standard model.

    anyway!

    • Flip Tanedo says:

      Hello fluidic. I should address a few things.

      * First, what do you mean by the LHC results being interpreted differently? Thus far the results have not found any conclusive evidence of deviations from the Standard Model. People are free to speculate that “hints” (deviations that are not of statistical significance) may suggest one thing or another, but to the best of my knowledge there have been no serious claims (speculative claims perhaps, but this is different) that the LHC sees anything other than the Standard Model.

      * I do not understand your conclusion that “there is not much sense” in the interpretation of physics at CERN. We always have imperfect data. If you bump into something in the dark—what is it? Maybe it’s a tree. Maybe it’s a lamp post. Maybe it’s a tree that happens to be shaped like a lamp post. These are all valid speculations that are consistent with the data. They encourage us to look for cross checks (maybe there are leaves) to test hypotheses. This is the scientific method.

      * You also misinterpret the purpose of this blog post. I’ve “told a story” about the Standard Model. I’m not saying that the Standard Model is the ultimate theory (this is highly unlikely), but I’ve presented the “straw man” model that physicists use. The Standard Model is nice because we expect the “true” theory of electroweak interactions to reduce to the Standard Model at low energies in order to explain its remarkable consistency with experiments. I’ve even made a point to comment that the search for the Higgs is a test of the Standard Model and that its non-discovery in the expected range would be a strong signal that there must be something new.

      * I do not understand your comment about the CERN community being divided about anything. I suspect that you may projecting interpretations of other things that you have read into this blog post. The scientific method is coming up with hypotheses to explain observed data and then taking more data to test these hypotheses further. Indeed, it is better to have many hypotheses that one can test. In practice we must also input some theoretical bias to focus on hypotheses which are sufficient ‘elegant’ or at least theoretically motivated. You seem to interpret the multiplicity of hypotheses as an inconsistency in the scientific method—this is quite the opposite, it is a necessary step in the scientific method.

      * I do not understand your comments about 2D Hilbert space at all.

      * There’s some questionable PR about the LHC being built to discover the Higgs. A more accurate statement is that the LHC was built to discover the mechanism for electroweak symmetry breaking. The way in which nature breaks electroweak symmetry may be a Higgs, it may be something else Higgs-like, it may be nothing like a Higgs… but we have very good reasons to expect the physics of electroweak symmetry breaking to become manifest at scales accessible to the LHC. As mentioned above, *not* discovering the Higgs in the 100 – 200 GeV mass range would be a very bold statement about what’s actually happening.

      * Regarding your statement that there are dozens of reasons why the Standard Model should fail… I AGREE. This is the whole point of why the LHC is interesting. We know that Dark Matter exists and is not described by the Standard Model. If it’s a thermal relic, then it should be able to be produced at the LHC so that we may study “dark matter in the lab.” Properties of QCD at high densities and temperatures are not well understood within perturbative methods—so the heavy ion collisions can shed a lot of light about what kinds of descriptions are relevant in such processes. Other things not included in the Standard Model: inflation, quantum gravity, satisfactory explanations for the electroweak symmetry breaking scale, the origin of CP violation in the universe, etc. etc. etc. THIS IS WHY PARTICLE PHYSICS IS INTERESTING—there are plenty of well defined open questions.

      I think the main point is that science is not about lecturing, it’s about curiosity and discovery. We have theories about how the world works—and we test and re-test them over and over. Either the models continue to pass the tests and we suspect they are reliable descriptions of nature at that scale, or they break down (either through experimental disagreement or theoretical inconsistency). When they break down we have to figure out why they break down (maybe the original idea was just wrong, maybe something more subtle is happening) and continue the process.

      Best,
      F

  7. fluidic says:

    between <<>> i was referencing the eating of 3 bosons by the W and Z bosons and that the 4th boson is the higgs boson whose discovery will reveal crucial details about limits of the standard model, which was quoted from this blog post. sorry! looks like the system also ate my bosons between the <<>> too!!!

  8. fluidic says:

    Thank you Flip for your beautiful articulation. In fact you have completed many of what i needed to say in this blog. I agree with your arguments.

    However, I believe when higgs mechanism fails, and most likely, it will, SSB, Electroweak SB, W and Z, Goldstones bosons, all this theory of the electroweak forcesa and interactions will follow. disaster for physicists!

    If we have not been able to just simply see the electron inside the atom or detect how it moves, orients itself, rotates, spins, where does it exactly dwell (no probabilistics in nature) then we are not too sure about anything, because we can be always very smart to only setup epxeriments to show us exactly what we expect our theory to see! Einstein’s great imagination.

    good luck Flip.

  9. fluidic says:

    Sorry Flip there is something else I would like you to think of.

    When clouds start spitting raindrops, where there rain inside these clouds? NO. there were only tiny particles of H2O hanging in there waiting there turn to assemble to form a raindrop and reach critical mass to fall as raindroplets. CERN most intelligent scientists will shortly discover after the failure of higgs mass absorption mechanism, that matter is continuous. the plasma state discovered by CERN scientists in 2000 is a manifestation of matter’s fluidity and plasma nature. NO disconinuities exists in matter, but particles are formed on exit or escape just like nature shows us when raindrops form on exit from clouds due to incoming or changing of surrounding currents or fields, to cooler, more humid lower pressure etc…

    best of luck Flip.

  10. Yoron says:

    There are some things I don’t get here. You speak about polarizations as ‘degrees of freedom’? And if I got you right some polarizations are not allowed as they would ‘spin faster than light’?. But ‘spin’ is already spinning faster than light as I understands it, if treated classically? Although I enjoy your writing and find it very good I get stuck on that one.

    And ‘mass’ would then be ‘bosons’ getting eaten by ‘bosons’? As ‘particles’ or as waves? As waves they can reinforce and quench each other, as ‘particles’ they can interact and as bosons also get superimposed on each other? But how do they ‘eat each other’?

  11. Martin F says:

    This answers this question I believe.

    http://arxiv.org/abs/1401.6924

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