(Submitted on 30 Oct 2012 (v1), last revised 28 Mar 2013 (this version, v2))
Abstract: We show that a longitudinal gauge degree of freedom for a vector field is equivalent to a Pais-Uhlenbeck scalar field. With the help of this equivalence, we can determine natural interactions of this field with scalars and fermions. Since the theory has a global U(1) symmetry, we have the usual conserved current of the charged fields, thanks to which the dynamics of the scalar field is not modified by the interactions. We use this fact to consistently quantize the theory even in the presence of interactions. We argue that such a degree of freedom can only be excited by gravitational effects like the inflationary era of the early universe and may play the role of dark energy in the form of an effective cosmological constant whose value is linked to the inflation scale.
Who ate the Higgs? - Quantum Diaries
www.quantumdiaries.org/2011/10/10/who-ate-the-higgs/
[PDF]Particles with spin one
www.staff.science.uu.nl/~wit00103/ftip/Ch04.pdf
Utrecht University
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Polarization vectors of massive and massless particles
physics.stackexchange.com/.../polarization-vectors-of-m...
Stack Exchange
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[PDF]W-Boson Decay
www.hep.phy.cam.ac.uk/.../partIIIparticles/Han...
University of Cambridge
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Many-Body Problems and Quantum Field Theory: An Introduction
https://books.google.com/books?isbn=3540213201
Philippe-André Martin, Francois Rothen - 2004 - Science
The Higgs Field and Particle There is one final step. ... corresponding to particles of spin 1, has three internal degrees of freedom (the components ... the gauge field thus acquires a supplementary degree of freedom (a longitudinal component) ...How many degrees of freedom does the photon have in 2+1 ...
www.researchgate.net/.../How_many_degrees_of_freedom...
ResearchGate
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Particle Accelerator Physics - Page 294 - Google Books Result
https://books.google.com/books?isbn=3540490434
Helmut Wiedemann - 2007 - Science
... of a particle in all three degrees of freedom or in six-dimensional phase space. ... but the physics leading to damping in the longitudinal degree of freedom is ...The Reference Frame: Who ate the Higgs?
motls.blogspot.com/.../who-ate-higgs.html
Lubos Motl's Reference Frame
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Cracking the Quantum Code of the Universe
https://books.google.com/books?isbn=0199915520
John Moffat - 2014 - Science
The degrees of freedom are the two transverse and one longitudinal degree of ... and ferromagnetism in solid-state physics could apply to relativistic particle ...[PDF]chapter 6
www.cns.gatech.edu/FieldTheory/06-Ghoulies.pdf
Flip Tanedo | USLHC | USA
View Blog | Read Bio
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.
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.
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 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.
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.
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
So now we arrive at another aphorism in physics:
![Force particles like Gold[stone]fishes because they're so delicious.](http://www.quantumdiaries.org/wp-content/uploads/2011/10/Goldstonefish-300x174.png)
Indeed, you might think I’m selling you the Standard Model like an informercial:
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.
Now let’s get back to the statement of Goldstone’s theorem that we gave above:
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).
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.
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.
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
Jeffrey Goldstone, image from his MIT webpage
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…
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 technicolor, composite Higgs, and Higgs-less models.
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 technicolor, composite 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," biblical pun intended
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.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.)
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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