"The Double Simplex," hep-ph/0509037
QuickTime Movie of Wine & Cheese Seminar Slides (click through)
Any chart or mnemonic device should be an invitation to narrative, and that is what I intend for this representation of the particles and interactions. I want to represent what we know is true, what we hope might be true, and what we don't know--in other terms, to show the connections that are firmly established, those we believe must be there, and the open issues. I want also to show the aspect of play, of successive approximations, that animates the way scientists work.
I have also wanted to create an accurate representation of what we know that can be rendered on a page and built as a three-dimensional object that can be held, turned over in the hand, regarded from different perspectives, and peered into. Because science progresses, it is important to build a structure that can evolve as we learn more.
These web pages contain very skeletal depictions of a new way to envision the particles and interactions, a pair of interpenetrating tetrahedra that we might call the double simplex, in homage to the double helix that has just celebrated its fiftieth anniversary. The virtue of these prototypes is that they can be turned and zoomed on the computer screen, emulating the eventual pleasure of picking the model up and looking at it from every side. The shortcomings are plain: there are no labels or textures, planes are empty or opaque, and stick figures are more abstract than the real thing will be. But these computer graphics are quite useful for imagining what can be, and for seeing the outline of a narrative.
We begin with the empirical information that particles come in
two kinds: leptons and quarks. The leptons come in pairs, like the
electron and its neutrino. What leads us to recognize the pairs is
that a left-handed electron can be transformed into its neutrino
(and vice versa) by the action of the weak force particle the W
boson. I represent the pair as a dumbbell:
The quarks also come in flavor pairs, such as up and down, and I
represent them in the same way. But quarks also carry another kind
of charge, or attribute, the "color" charge of the strong
interactions, red or green or blue. Here are the quark
Again the slanted lines stand for the left-handed weak interaction. I array the pairs of quarks in a triangular formation to represent the color symmetry--or strong interaction, Quantum Chromodynamics--that links them together.
Do the quarks and leptons have anything to do with one another? We do not know for certain, but we think they must: all are structureless (at our current limits of resolution), spin-1/2 particles, they have the same pair structure, and we cannot build a consistent theory of the weak and electromagnetic interactions with only quarks or only leptons. If we join them together (regarding lepton number as a "fourth color," the basic structure is a tetrahedron decorated at each vertex with a dumbbell. (A tetrahedron is a three-dimensional simplex.)
The bright lines connecting different colors of quarks indicate the strong interactions mediated by gluons. If quarks and leptons are indeed related, the lines joining quarks and leptons also represent Interactions: new interactions that we have not observed directly, but that might help explain the excess of matter over antimatter in the Universe. Are these interactions present in Nature? That is a question that experiments searching for proton decay hope to answer. If they are present, why are they so different--so feeble--compared with the other interactions? We have ideas, but we cannot yet say for sure.
This is a good moment to peruse the first set of four graphics.
One of the striking features of our world is that left-handed and right-handed particles are different: the left-handed ones come in the pairs we have just seen, but their right-handed counterparts are immune from the familiar weak interaction. We do not know whether there is no right-handed weak interaction or whether--like the interaction that exchanges quarks and leptons--our instruments have just not been sensitive enough to detect it.
Neutrinos do not have strong or electromagnetic interactions, so
if there are no (observed) right-handed weak interactions, how
would we know that there is a right-handed neutrino? Experiments
that have detected neutrino oscillations tell us that a
right-handed neutrino must be there, but we don't know what it is.
We label it N, instead of Greek nu. I depict the right-handed
The line connecting them is dashed, to indicate that we don't know whether they form a (right-handed) weak-interaction pair, and it is slanted oppositely to the W-boson line that links left-handed particles. In the 3-dimensional model, the right-handed lines are orthogonal to the left-handed lines. The lines are solid, not dashed, in the computer graphics.
We do know about the right-handed quarks through their strong
and electromagnetic interactions, so here they are:
We join the right-handed quarks and leptons together in an inverted tetrahedron.
Now is a good moment to peruse the second set of graphics.
The next step is to join the left-handed and right-handed particles together in a pair of interpenetrating tetrahedra--the double simplex. This will allow us to explore possible interactions that link the left-handed particles to the right-handed particles.
Before we do that, let us take note of the fact in Nature we see three copies of the basic family structure. Besides the electron and its neutrino, we know the muon and its neutrino, and the tau and its neutrino. In addition to the up and down quarks, we know charm and strange, top and bottom. For the moment, the meaning of these repeated families is a mystery. I have chosen to illustrate the three families as side-by-side dumbbells (lines in the prototype). If we discover, or wish to conjecture, some sort of family symmetry, or find a common origin for the three families, we could arrange the trios of dumbbells differently, or decorate each dumbbell with three balls at each end, etc. The (W-boson) weak interactions mix particles of different generations. We show that by lines that connect the up quark not only to down, but also to strange and bottom. Lines of different thickness (or brightness) could show which are the major and minor couplings.
Now is the right moment to examine the third set of graphics.
An interaction that joins a left-handed particle to its right-handed counterpart endows the particle with mass. In the next set of prototypes, I've drawn the lines for the electron, for the down quarks, for the up quarks, and then for all at once. I've kept it to a single generation just to avoid clutter. We do not know what is the mechanism that gives mass to the quarks and leptons. I would represent that ignorance, in a richer graphic, by a cloud at the center of the double simplex. We have some ideas, of course; one possibility is that an unobserved particle called the Higgs boson gives mass to all the fermions, and determines the mixing among quark and lepton generations. We do not know whether this is so. In some theories, such as supersymmetry, one Higgs boson gives mass to the electron and the down quark, while another Higgs boson gives mass to the up quark. The double simplex allows a geometrical representation of this model because the electron-down and up lines pass through different points near the center of the figure. Other theories envisage different agents giving mass to different generations, and that could also be represented. In some theories, new interactions among new kinds of particles give rise to quark and lepton masses.
The next set of prototypes depicts the left-right interaction.
The decorated double simplex is a way of representing the 16-dimensional representation of the group SO(10) that unifies the strong, weak, and electromagnetic interactions. [We decompose SO(10) into SU(4) x SU(2) x S(2), representing the SU(4) part as the solid and the SU(2)s as the two kinds of decorations.] We are not assuming that SO(10) must be true, we need only to use the parts of it (it contains the normal standard model) and ask about the possibilities it suggests. In addition to the left-right connection we have just made, we could regard the inverted (top) tetrahedron as representing the left-handed charge-conjugate particles. The connections we discern then are all the gauge bosons of SO(10).
I see the role of the double simplex as inviting questions about what is and what might be. A natural question is whether what we know now might be part of a larger structure. A way to motivate such speculation is to note that the double simplex fits inside a cube, so there may be connections yet undreamed.
Another question is whether we already know all the particles we need to discern the patterns. Mendele'ev's periodic table was missing all the noble gases, because nineteenth-century chemistry could not discover them in reactions. Might there be forms of matter we have not perceived because they interact too feebly, or are too massive for us to produce?
See the last pair of prototypes.