How Big is Big and How Small is Small

by Smith, Timothy Paul

  Figure 11.3 The nuclear force is repulsive at short distances, which keeps nucleons from being sucked into each other. It is attractive (positive) between 1 and 2 fm, which means nucleons are attracted to nearest neighbors. It vanishes beyond 2 fm.

  People often draw the nucleus to look like a bunch of grapes, a collection of spherical neutrons and protons clumped together, and this is about right. The nucleons like to stay about a 1 fm apart. What is missing from that image of grapes is the ethereal pions zipping back and forth. The exchange of pions bind nucleons into a nucleus much like a common currency binds an economy together.

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  What is smaller than a nucleon? The history of science has taught us to expect worlds within worlds: cells, atoms, nuclei and now what?

  Inside of a nucleon are quarks and gluons (see Figure 11.4). In some sense these particles are different than anything we have encountered so far, or rather, how we know about them is much different. We have seen the tracks of protons and pions as they streak across detectors leaving trails like meteors. But we have never seen a quark. We know a great deal about them, but our knowledge of these particles, which are less than 10−16 m across, is arrived at by indirect and subtle means.

  Figure 11.4 An image of a nucleus made of protons and neutrons with quarks inside of them. A nucleus appears as a collection of protons and neutrons. In the bottom right is a newly created pion. Inside of the nucleons are three quarks held together by gluons (drawn as springs). The pion contains a quark and antiquark. Courtesy of American Scientist, Nov/Dec 2010.

  The method works something like this. Physicists who study quarks generally are divided into two types: theorists and experimentalists. Theorists will develop models or theories about what is going on at small scales and high energies. For example they might predict that quarks gather at the center of a nucleon, or form an outer shell, or have excited orbits with a short lifetime. What makes their theories interesting is if they predict something macroscopic; if a theory makes no testable predictions it fails an important point of the scientific method.

  The experimentalist will try to measure that which a theory predicts. But experimentalists are limited as to what they can measure. They can collide particles and see what comes out, at least if whatever emerges comes out far enough to reach their detectors; a particle that exists only in the atom or a nucleus will not be seen at all. That means experimentalists can look at scattered fragments that travel at least a few millimeters. They can then measure the fragment’s angles, energies, mass, spins and charges, but that is about it.

  Ideally, competing theories will predict different distributions of angles or energies and the experiment will pick the winner. In reality it is hard to build the ideal experiment and often the results create more questions. Still, out of decades of data we have been able to tease out an understanding of what is inside of the nucleon. The successful theories of today are all based on the idea that inside of a nucleon there are quarks, and fleeting between them are gluons. The most successful theory we have is called quantum chromodynamics, or QCD for short.

  The force that holds quarks together, and therefore creates protons, pions, neutrons and many other particles, is the strong force, best described by QCD. What makes the strong force unique is that it has three different charges. Gravity has one charge: mass. Electromagnetism also has only one charge, even though it comes with both a plus and minus magnitude. But the strong force has three different charges. The names given to the charges are red, green and blue. They could have been named something unimaginative like q 1, q 2, q 3 or something whimsical like vanilla, chocolate and strawberry. However, red, green, blue is actually an inspired choice because it also tells us about neutrality. A particle with three quarks will have all three colors and therefore a total color-charge of white, which is to say neutral. Protons and neutrons are color neutral and so should have no strong force between them.

  The other piece of QCD is the gluons, the intermediate particles that are the carriers of the force. Gluons can also have color-charge, which means they act very differently than photons that are neutral. Since gluons can have color they can also interact with each other. Gluons stick to gluons. Their name is not accidental. Not only do they hold quarks together, they bind gluons together into a sticky web of interactions, the result of which is that we have never seen a free quark, and probably never will.

  But what if we tried to pry a quark free? What if we hit a nucleon very hard with an electron or something else? A quark can be pulled farther and farther out and the gluons will try harder and harder to contain that wayward quark. The tension in that web of gluons will rise and rise until either the quark is pulled back in or the gluon bundle snaps. When it breaks, like a rubber band snapping, more than enough energy is released to create two new particles, a quark–antiquark pair. The original nucleon gets the new quark and the escaping quark is paired with the new antiquark. The escaping pair most often is seen in the laboratory as a pion.

  Once there is enough energy to create new quarks the world becomes messy or fascinating, depending upon your point of view. This is because the newly created pair might not be of the same type you started with. If you started with a proton, you could end up with a neutron, a proton, or a more exotic and rarer particle.

  Quarks come in six different types, or flavors; and we still do not get to call them vanilla, chocolate and strawberry. Instead they are called: up, down, charmed, strange, top and bottom. The names sound whimsical, like the charges named after colors, but these flavor names are suppose to remind us that they come in pairs. The lightest and most common are the up and down quarks. The up quark is proton-like and the down quark is neutron like. A proton is made of two up quarks and one down quark, and is usually written as uud. The neutron is one up and two down quarks, or udd. The charm and strange quarks are heavier than the up and down quarks. Heavier still is the bottom quark. Finally, much heavier than any other quark is the top quark, with a mass almost 200 times greater than that of a nucleon.

  QCD and the color rules tell us that a particle can have three quarks or one quark and one antiquark. Now, with six flavors one can construct 216 different combinations of three quarks, a few of which are shown in Figure 11.5. Most of these combinations are massive, exotic and short-lived. For example, the combination of up, down and strange (uds) may form the particle called the Λ0, which decays into a proton and a pion in about 2 × 10−10 s. Heavier combinations disintegrate even faster. Particles that contain top quarks hardly live long enough to justify being called particles before they decay into lighter quarks. When we list these quark combinations, these exotic particles, it sounds like the whole greek alphabet:

  This sort of physics was the bread and butter of large accelerators from the 1950s to the 1990s, when the heaviest quark, the top-quark, was finally discovered and measured. But when talking about the sizes and shapes of things in nature, what is new in the last few years is the role of the gluons. However, we have one other loose end to deal with first.

  When describing extraction of a quark and creation of a pion, something that sounds like it was out of science fiction slipped into the description: antiquarks. Antiquarks are like quarks, except they are made of antimatter. Antimatter is the stuff that is suppose to power starships or is at the heart of futuristic science fiction weapons. Actually antimatter is real; it is just very fleeting. It tends to do what the science fiction stories tell us; when it meets matter they both annihilate and release energy. It is just that in the real world these moments of annihilation take place a quark at a time.

  Figure 11.5 A collection of particles made of quarks. Flavors are labeled. The color charges always add up to white, or neutral.

  Quarks have color: red, green and blue. Antiquarks have anticolor: antired, antigreen and antiblue. These are not new color-charges, it is just a way of writing negative color. So if we combine a quark and an antiquark with the right colors—let us say blue and antiblue—they would attract ea
ch other, and because they have opposite color charge, they would add up to white. That is in fact what is happening in a pion and we can now revisit nuclear forces, but from a quark point of view.

  For example, consider a proton interacting with a neutron. In the original proton there are three quarks (uud). At some time, due to internal quantum fluctuations, an up quark is ejected. As it moves outward a down-antidown pair of quarks is produced, with the down quark staying in the nucleon and the antidown joining the ejected up quark. The original nucleon is now a neutron, since it contains udd. The pair are a pion, in this case a π + because of the combination of up-antidown (u) and the electric charge. Finally the pion is absorbed by a neutron, which then becomes a proton. In this whole process electric charges, color charge and flavor are preserved, as are energy and momentum. In fact the whole process moves momentum from one particle to the other, which is what keeps them bound together.

  A tool for visualizing this is a Feynman diagram, named after the theorist Richard Feynman (1918–1988) who first used them. In truth, a Feynman diagram will tell us exactly the equations to solve. Here I am only using them as a cartoon. In these figures a line represents a particle. As it moves to the right, time is passing. If they are separated up and down, they are separated spatially. Line intersections or vertices are where and when particles are interacting, either creating or absorbing those intermediate particles. All of these dynamics happen within a nucleus, in fact between near neighbors, but outside of the nucleon. So Figure 11.6 shows what the world between 1 and 2 fm looks like.

  Figure 11.6 The interaction between a proton and neutron in terms of pions and quarks (A) We understand the nuclear force as the exchange of pions. (B) We can also look inside the nucleons and pions and view the force as an exchange of quarks. The gray area is just to emphasize which quarks are grouped together to form nucleons and the pion.

  We can now turn to the world inside a nucleon. Quarks are held together by gluons to form nucleons, pions and other particles. After having described the way pions hold nucleons together, the way gluons work is analogous. When we look at the world at 10−16 m—that is, inside a nucleon—we see three quarks swirling about each other, bound together by a mesh of gluons. Now if we closely watch the quarks and gluons we can follow their color charge, as seen in Figure 11.7. Imagine we have an up quark with a red color charge interacting with a down quark with a green color charge. The up (r) quark may emit a gluon with red-antigreen (rḡ) color charge. That up quark is now left with a green charge. Soon the gluon is absorbed by the down (g) quark that now takes the red color charge. At all times color charge is conserved, and therefore the quarks are bound.

  Figure 11.7 The interaction between quarks in terms of gluons and color charge. (A) Two quarks interact via a gluon. (B) A red quark could emit a red-antigreen gluon and then becomes green. That gluon could then be absorbed by a green quark, which then will have a red color charge.

  So let us step back once again for a moment. Inside of a nucleon there are always at least three quarks. The quarks are buzzing around like electrons in an atom. Between them gluons are shooting back and forth. By the fact that gluons have color charge the gluons can interact with not only quarks, but with other gluons. There is also enough energy for quark–antiquark pairs to be produced all the time, sometimes with exotic heavy quark flavors. And every once in a while a pion will pop off the surface of the nucleon and be absorbed by a neighbor. I imagine it like a boiling pot of tomato sauce that generally stays in the pot; occasionally a small droplet will spatter out and cover the stove top with red. Nucleons contain a whirl of activity.

  We have only touched the tip of the iceberg. There are a great number of combinations of flavors and colors as well as spins, antiquarks and excited states. Also it has become more and more apparent that gluons are not just the cement that holds quarks together, but that they also have an important role in creating the macroscopic particles and determining the properties of those particles. But in this book we are concentrating upon the sizes and shapes of things. So I will only add one more tidbit about neutrons.

  Two down quarks and one up quark swirl about inside a neutron. The question is how do they orbit each other? We would expect all three to occupy an orbit much like the ground state of the atom. It will be spherically symmetric, only much smaller, since the strong force binds them tightly. At first we would expect all three quarks to be on top of each other. However, there is a small correction. The two down quarks end up spin-aligned which gives them a slight magnetic repulsion, pushing them out. At the same time the up quark is magnetically attracted towards the other two, pushing it towards the middle. These are small changes in their configuration, with the up quark’s orbit radius just under 0.4 fm and the two down quarks at just over 0.4 fm. This means that the outside of a neutron is slightly negative and the inside is a bit positive (see Figure 11.8).

  Figure 11.8 The electric charge distribution of a neutron. The total electric charge is zero – a neutron is neutral. However, the outside is negative and the inside is positive. Courtesy of the BLAST Collaboration/MIT Bates Laboratory.

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  The biggest accelerator on Earth, the Large Hadron Collider at CERN (Organisation Européenne pour la Recherche Nucléaire or European Organization for Nuclear Research), is located under the Swiss–French border near Geneva. It has an energy of about 8 TeV (T = tera = 1012). That means that the particles in the beam have a wavelength of about 5 × 10−19 m. This means that our present experimental limit is about 10−18 m. So when we look at the world at that scale what types of structures do we see? Actually the smallest structure we have seen is the orbits of the quarks in the neutron and that was observed at 10−16 m. These high-energy machines are not looking for structure as much as for new forms of matter: heavy quarks, Higgs bosons and so forth.

  So our search downward ends at 10−18 m; we have hit bottom. But we have not measured such obvious things as the size of an electron or a quark. In fact, according to QCD, quarks and electrons are point-like. Their diameters are zero. That is something hard to think about. These are not dots that shrink until they are gone, because they are still there. Trying to imagine something with a zero radius is as taxing as imaging something with an infinite radius. But are they really point-like? We do not know and can only say that from an experimental point of view they are smaller than 10−18 m. We also do not know if QCD is the last word in particle physics. It is very successful: it explains most of particle physics and probably most of nuclear physics as well. However, we are still allowed to ask the question, “What might be smaller?” A dimension of 10−18 m only marks the line between what we know and what we suspect. Physics does put limits upon what happens at smaller scales, but without sharp details. And we, being humans, still ask the questions.

  I made the statement that according to QCD quarks and electrons are point-like and that all experiment so far supports that. But this is like standing at the end of the Pan-American highway in Prudhoe Bay, Alaska, and saying, “Based upon what I can see, Alaska goes on forever.” Alaska is huge, but it does not go on forever. Electrons and quarks are tiny and must certainly be smaller than 10−18 m across for QCD to be so successful, but that does not mean that there is nothing else smaller in nature.

  The standard model of particle physics is our best description of nature at this scale. It combines our best explanation of the strong, weak and electromagnetic forces, so QCD is one of its components. The standard model incorporates everything particle physicists think about except gravity. The outstanding question in the field over the last decade is not whether this model works or is right, but rather whether it is the last word in particle physics.

  I was once at a talk about the problems of the standard model—these talks are almost always called “Physics Beyond the standard model’—when a senior professor from MIT leaned over and whispered to me, “The problem with the standard model is that there are no problems with the standard model.” People
have spent years looking for holes in it and the best we can say is that it does not feel right. It is too gangly, too unwieldy; it has too many parts; it lacks elegance and aesthetics.

  And then there is gravity. The standard model does not even attempt to explain gravity. Also, our best theory of gravity—general relativity—cannot simply be appended to the standard model. There are a number of candidate theories and models for higher energy, smaller scales and more exotic particles, but at this time we do not have the data or tools to move forward with certainty.

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  Instead of trying to see clearly the next smallest structure (if there is one), we can try to turn the problem around and ask that very fundamental question, “Is there a lower limit of nature?” Surprisingly enough, if quantum mechanics is still the law of nature at ever-shrinking scales, there is a bottom. There is the Planck length.

  We mentioned the Planck length in Chapter 5 when describing the Planck constant, but only briefly. Max Planck proposed the Planck length as

  It is a combination of the Planck constant (h or ), which is at the heart of quantum mechanics, Newton’s gravitational constant (G), which is the key to both Newton’s law of gravity and Einstein’s theory of general relativity, and the speed of light (c), famous from special relativity. Anyone, anywhere in the universe could measure these three constants and construct the Planck length. It is truly universal. It does not depend upon the length of a platinum–iridium bar in Paris or the size of our home planet or the choice of a spectral line. The Planck length is set by nature, not by mutual agreement, which makes it unique. But is it significant?