In 2005, Sundance O. Bilson-Thompson wrote hep-ph/0503213v2, an arXiv paper titled “A topological model of composite preons”. The paper gave a preon model of quarks, leptons, and gauge bosons. That is, it modeled these particles as composite particles made up of preons. The preons he used were elements of the braid group B_3. A later paper by Sundance O. Bilson-Thompson, Fotini Markopoulou, and Lee Smolin showed that these states can be obtained from background independent models of quantum spacetime that “propagate coherently as they can be shown to be noiseless subsystems of the microscopic quantum dynamics”.
Since I play with a similar preon model, I thought I would comment on this theory from my perspective. I’ve given links to a few of the technical terms, but this post is not going to make a lot of sense to those not already playing with particle physics.
In his original paper, Bilson-Thompson called his idea the “Helon Model”. It is based on an earlier preon model by Harari and Shupe that was called the “Rishon Model”. See Wikipedia on Rishon model, J. D. Shelton’s short description, Harari’s original paper (scanned to acrobat), or, if you have Phys. Lett. B access, Shupe’s original article.
The Rishon Model described the fundamental fermions of the first generation, that is, the electron and its neutrino, the up quark, and the down quark, (as well as their antiparticles) as being composites made up of three preons each. Just like the quarks, the preons com in three colors (or precolors), and the observed fermions are made up of color singlets in the same manner as the baryons. In addition to color, the preons also come with electric charges of 1/3 or 0. These preons are Dirac spinor particles, so they come in four varieties. Such a preon can have spin up or down, and can be a particle or antiparticle. The neutral preons are called V and the charged ones are called T.
With these particles, the structure of a generation of elementary fermions can be constructed by assuming only two requirements are necessary to produce a bound state: (1) It must be a precolor singlet. (2) It must be composed purely of 3 preons or 3 anti preons. The resulting particle content is as follows (for the first generation):
TTV up quark, red
TVT up quark, green
VTT up quark, blue
VVT anti down quark, red
VTV anti down quark, green
TVV anti down quark, blue
VVV electron neutrino
In the above, note that while the three preons included are all particles, the resulting fermions are half particles (the up quark and electron neutrino), and half anti-particles (the down quark and positron). If you change the Ts and Vs to antiparticles, you get the down quark, electron, anti up quark, and anti electron neutrino.
Bilson-Thomposon’s Changes to the Rishon Model
Preon models are supposed to use the traditional point particles of quantum field theory. Bilson-Thompson’s model is not of point particles, but instead is a topological model. He wrote:
The reader should note the subtle yet important distinction that this is not a preon model per se, based upon point-like particles, but rather a preon-inspired model, which may be realised as a topological feature of some more comprehensive theory.
The primary disadvantage of a topological model is that it is difficult, or impossible, to calculate useful results from it. And in fact there have been no predictions of particle properties to come out of this work. But particle mass predictions were difficult for the Rishon model that preceded this.
The Helon Model is built in two stages. The first stage consists of a ribbon. The topology of a ribbon consists of how many twists are present in it. Twists are either by the “U” or , the “E”. What is important is the topological connections on the ribbon, so these twists commute. Twists combine in pairs. One might think of these as consecutive twists in the same ribbon. Combining two of these twists, there are only three possibilities, UU, EE, and UE = EU. These three cases are called “Helons” and are labeled as follows:
The twists commute, and since electric charge is represented by a commutative group U(1), it is no surprise that the electric charge is proportional to the net twist. The proportionality constant is , so has an electric charge of Q = +1/3, while has an electric charge of Q = -1/3, and finally has charge Q = 0.
Because of the properties of 3-dimensions, it is difficult to come up with a point particle model of the ribbon that works for a stationary massive particle. The path of a stationary massive particle doesn’t admit the concept of twist. However, the handed fermions are massless (and therefore travel at speed c) and for them it is natural to draw paths that can wind around one another. Bilson-Thompson’s model is not of the fermions themselves, but of their handed states, and so if two massless handed states are moving in the +z direction, we can draw them as:
The second stage in the Helon scheme is to combine three helons to form a fermion. Unlike the first stage, in this second stage, the three helons are not connected to each other, but can braid together in various ways. We can illustrate the topological effect by drawing the paths of six particles moving in 3 dimensions. We now have six point particles. They are combined into three pairs. Each pair is drawn with the same color. The elementary fermions are braids of three ribbons, with the electric charge given by the sum of the electric charges of the Helons, and the color charge is given by the distinctions between the mixed states (as in the Rishon model). For example, a green anti down quark works out as:
The above example is “braided”, that is, the three ribbons cross over or under each other. These objects form the fermions. Each of the fermions has a particular braid that corresponds to it. The gauge bosons, on the other hand, are triplets that have no crossings. So the gauge bosons are simpler:
The triplets of Helons define particle interaction by topological conjunction or cutting. In the words of Bilson-Thompson:
A triplet of helons may split in half, in which case a new connection forms at the top or bottom of each resulting triplet. The reverse process may also occur when two triplets merge to form one triplet, in which case the connection at the top of one triplet and the bottom of the other triplet “annihilate” each other.
If one examines the helon forms for the up and anti-down quark of the same color, one finds that if one connects these, and possibly allows twists to move up and down the now connected ribbons, one obtains the helon model of the W+. This is how the model works topologically.
Later Progess with the Helon Model and Remaining Issues
The Bilson-Thompson paper originally came out in October 2005. Version 2 came out in October 2006 and added hypercharge. In between these papers, the paper by Bilson-Thompson, Markopoulou, and Smolin came out putting the Helon model in terms of quantum gravity. From that paper, the heart of the technique was:
To obtain these results we find very helpful a new point of view about how the low energy limit of a quantum gravity theory may be expected to emerge . The idea is to study the low energy limit of a background independent quantum theory of gravity by asking how the states of elementary particles remain coherent when they are continually in interaction with the quantum fluctuations of the microscopic theory. The answer is that they are protected by symmetries in the dynamics. We can then analyze the low energy physics in terms of the symmetries that control the low energy coherent quantum states rather than in terms of emergent classical geometry. In  it was shown that one can apply to this the technology of noiseless subsystems, or NS, from quantum information theory. In this framework, subsystems which propagate coherently are identified by their transforming under symmetries that commute with the evolution. These protect the subsystems from decoherence.
While this is progress in terms of making the model background independence, this sort of thing is difficult to translate into a preon (point) particle model. My opinion is that while this second paper is a technical tour de force, it makes only negative progress in terms of obtaining a theory in which it is possible to make calculations.
Unresolved Problems for the Helon Model
As Bilson-Thompson stated in his original paper, the Helon model is not a complete theory. The remaining difficulties he listed as “the origin of spin, the origin of mass, and the nature of Cabbibo-mixing.” In addition, various problems come up when the theory is compared with what we know from the rest of elementary particle physics.
The origin of spin: Spin is tightly connected to the Poincare symmetry of spacetime. I think any real particle theory cannot ignore spin. One can imagine that this could be treated as an attribute that arises from some sort of complicated interactions, but it doesn’t give one a comfortable sense of a toy theory when there is no sketch of how this comes about.
If we assign spin-1/2 to the helons, it “would necessitate further assumptions to explain why helons never combine to form spin-3/2 states.” Of the attributes of particles, spin is one of those that is likely to be present in all versions of our theory as it is one of the few quantum numbers that is absolutely always conserved, and is present in all observed particles.
The mixing angles:Presumably the mixing angles can only be explained by a non topological theory. While the Helon model may (or may not) be an advantage over the Rishon model in terms of reducing the number of arbitrary requirements, it is certainly not an advantage in terms of papers that compute mixing angles (and masses too). Point particles are easier to compute with.
The origin of mass: Mass is left unexplained in that the Helon model does not include braids for the graviton nor Higgs. One supposes that if these fell easily out of the theory the author would have included them, and so the theory may have problems here.
Linear superposition: The Helon model is one of bound states of more elementary particles. Given a set of elementary particles, in quantum mechanics the bound states are typically linear combinations of simple combinations of those particles. The Helon model assumes that the observed fermions just happen to not require any such linear combinations. This is a heck of a coincidence.
The baryons are an example of what happens when a force ties three particles together. The majority of the resulting states are linear superpositions of the simple products such as . The force that holds these particles together is what is left over when the Helon force is applied to particles that are not quite identical (i.e. the mixed states of T and V), and yet in the Helon model there seems to be no need for linear superposition.
The two places where linear superposition almost certainly needs to be applied to the elementary fermions are the mixtures across generations (i.e. the mixtures between particles which have the same quantum numbers other than the generation numbers that the Helon model largely ignores anyway), and the mixtures within spin. For example, the splitting into left and right handed states needs to occur in a manner that allows linear superposition.
Too many neutrinos / generations: The Helon model has no way of limiting the number of generations, and no way to assign extremely high masses to the neutrinos of generations beyond the third. And yet it is known that there are only three neutrinos up to masses of many eV. This is considered strong evidence that there are exactly three generations, but the Helon model allows an infinite number of generations.
I attended last year’s American Physical Society meeting at Dallas. When Smolin gave a talk on deriving the Helon model from quantum gravity, the small room was jam packed. At the end of the talk, most of the buzz in the audience was on this topic.
What Causes all that braiding: If the Helon model is to be taken literally, the two ends of a ribbon are tightly bound together. To put this into a preon model, one needs a force to keep the ends together. This would require the exchange of a gauge boson. The same applies at the second stage binding. This amounts to the question, “what binds the Helons together?”
Since the color force in the Helon / Rishon model amounts to what is left over when the T and V states are mixed in color singlets, one would suppose that what we know about the color force should be able to be applied to the Helon model as well.
Topology in massless states: The Helon model is a model of the left and right handed states of the fermions which are supposed to be massless, and therefore to travel at the speed of light. Things that travel at the speed of light are supposed to be frozen by time dilation. The irony here is that Smolin’s interest in the Helon model is apparently related to the objective of obtaining background independence. Instead, Helons seem to violate Einstein’s special relativity, the theory that started background independence whenb it destroyed the old aether theories.
Since both the photon and the fermions are to be made from the same preons, a conclusion is that the speed of light (and the maximum speed of matter) is less than the speed of its preon components. This is no surprise to the readers of this blog! Exactly this was the subject of this blog’s previous post; in fact the speed of gravity has not yet been measured and there is at least some very early evidence that it is not c. This is a small amount of experimental wiggle room; perhaps it is enough to allow Nature to bind matter even at the speed of light.
Those of you who wish to write comments (or blog posts) using latex, may find it useful to include the magic sequence ” &bg=ffffff&fg=000000&s=1″ at the end of your latex. For example, if one enters:
(note that “&” and “&” amount to the same thing), the rendering is: .