The Theory of Almost Everything: The Standard Model, the Unsung Triumph of Modern Physics - Robert Oerter (2006)
Chapter 10. The Standard Model, At Last
Everything important is, at bottom, utterly simple.
—John Archibald Wheeler
“I want to know how God created this world. I am not interested in this or that phenomenon, in the spectrum of this or that element. I want to know His thoughts; the rest are details,” said Albert Einstein.1 To know God’s thoughts, the laws by which the universe runs—that is the way physicists view their goal. The details, the particular masses of individual particles, are important, but only as a means to a wider goal. Physicists want to know about the general principles, the overarching structure, the architecture of the universe.
During the course of the twentieth century, physicists painstakingly deduced bits of that structure. At the core is the relativistic quantum field: the idea that all interactions are fundamentally an exchange of particles, that these exchanges are completely random and unpredictable, and that we can discover only the likelihood of any process.
The Standard Model is built from relativistic quantum field theories we already know about. QED is incorporated into electroweak theory, and the whole is woven together with QCD to create a single theory whose essential elements can be written in a single equation. This equation is the simplicity at the bottom of it all, the ultimate source of all the complex behavior that we see in the physical world: atoms, molecules, solids, liquids, gases, rocks, plants, and animals.
In this chapter, we finally reach our goal: the Standard Model of elementary particles. It is not, of course, the ultimate goal. Gravitational interactions are left out, and without them we certainly can’t lay claim to a full understanding of the universe. Still, the Standard Model is an incredible achievement—a single theory that not only summarizes everything we know about matter and its interactions, but also answers fundamental questions about the symmetry and structure of the universe.
In the days of the Eightfold Way, only three flavors of quark—up, down, and strange—were needed to explain all of the then-known particles. By 2000, it was known that there are actually six quark flavors, and that they come in pairs, or families. One quark in each family has charge +2/3 and the other has charge -1/3. The masses also follow a clear pattern: The quarks in the second family are more massive than those of the first family, and those of the third family are more massive still.
The leptons, the lightweight particles, come in pairs as well. The electron interacts with its neutrino. The muon, as we have seen, is an overweight version of the electron. It has the same charge, the same spin, and has weak interactions but no strong interactions, just like the electron. The muon interacts with a second type of neutrino, known as the mu neutrino. Muons ignore electron neutrinos entirely, and electrons likewise ignore mu neutrinos. A third lepton, the tau, completes the pattern. The tau has a larger mass than the muon, but has the same charge and spin, lacks any strong interactions, and has its own associated neutrino. The existence of the tau neutrino was only experimentally confirmed in July 2000, although physicists had long been confident it would be found. It is the last but one of the predicted particles of the Standard Model to be found—only the Higgs remains.
Physicists thus found that Nature repeated herself, producing both quarks and leptons in three families.
This chart gives us a new sort of periodic table: the periodic table of the fermions. Only the particles of the first family appear in the atoms that make up the matter around us and in us. The other particles appear only fleetingly, the short-lived by-products of high-energy collisions.
It is striking that the number of quark and lepton families is the same. This is actually a crucial feature of the Standard Model. If there were different numbers of quark and lepton families, the model wouldn’t be renormalizable. The Standard Model is silent about why the fermions enjoy this family relationship, or why the number of families should be precisely three. A happy coincidence, or is there some deeper reason?
Now all the ingredients for a Theory of Almost Everything are in place. Quarks and gluons interact by way of QCD. Electrons, photons, and neutrinos interact by way of the unified electroweak theory. All that’s left is to cobble the two theories together into one relativistic quantum field theory that describes all the interactions of all the particles ever observed, plus the still unobserved Higgs.
According to the recipe for designing your own physics, begin by listing the particles of the theory. In addition to the three fermion families just listed, we need the following particles:
Of course, each particle has its antiparticle (except for those that are their own antiparticle). I won’t bother to list those.
Next, for each particle, we need the propagator that describes how the particle gets from place to place in the absence of interactions. The propagator is entirely determined by the mass and spin of the particle. In Feynman diagrams, we draw a straight line for the spin 1/2 fermions, a squiggly line for the spin 1 intermediate particles, and a dashed line for the spin 0 Higgs.
Finally, we need the interactions. If we had to consider all possible interactions of the 37 particles in the table, we’d be in trouble. Fortunately, the couplings of the intermediate particles with each other are fixed by the symmetries of Yang-Mills theory. The quarks, of course, interact with the gluons as required by color symmetry, and all the fermions (electrons, neutrinos, and quarks) interact with the intermediate particles of the electroweak theory (the W and Z° particles, and the photons). The Higgs, as we saw in the previous chapter, interacts with almost everything. It interacts with the intermediate particles, the leptons, the quarks, and, crucially, with itself, in just the right manner to produce the Mexican hat potential. These are the interactions that give all of the particles their mass after symmetry breaking occurs. If we knew the coupling constants for these interactions, we would know the masses. Unfortunately, the Standard Model tells us nothing about these coupling constants. The quark and lepton masses can’t be predicted from the Standard Model; they must be determined experimentally. (For more details of all these interactions, turn to Appendix C.)
There you have it. Knowing the propagators and the interactions of the Standard Model you can, in principle, calculate the probability of any process in the universe. (Well, any process not involving gravity.) Why quarks stick together in protons, why atoms bind into molecules, why my feet don’t pass through the floor beneath them—it’s all there. In principle. In practice, the calculation requires knowing how to translate these diagrams into equations. Below, in simplified notation, is the starting point: the Lagrangian function that summarizes all of the propagators and interactions we just listed.
This one equation describes everything we know about the way matter behaves, except, of course, for gravity. In the equation, G describes the gluon field and its interactions, W stands for the SU(2) field (describing the W+, W-, and Z° particles and their interactions with each other), F is the U(1) field, H is the Higgs, and fj is shorthand for all the fermions: electrons, neutrinos, and quarks.
There are 18 adjustable numerical parameters in total in the Standard Model. Think of the Standard Model as a machine with 18 knobs. If we twiddle the knobs just right, the machine spits out nearly perfect predictions about any (nongravitational) process in the universe. Just what are these magical numbers that encode everything we know about physics? First, the strengths of the color force and the electroweak force. Second, the Higgs couplings to the quarks and leptons that give them mass. Third, the numbers that specify the shape of the Mexican hat potential. Finally, there are a few additional parameters having to do with the interactions of the quarks: these are discussed in Appendix C.
Despite the beauty of the Higgs mechanism for spontaneous symmetry breaking and its spectacular success in predicting the W and Z0 masses, it must be admitted that our Standard Model machine, with its 18 knobs, is a bit unwieldy. Weinberg himself called it a “repulsive” model in 1971, and theorist Thomas Kibble remembers reading Weinberg’s paper and coming to the conclusion that, “It was such an extraordinarily ad hoc and ugly theory that it was clearly nonsense.”2 Is there rhyme or reason to the 18 parameters, the three generations of fermions, the odd combination SU(3) x SU(2) x U(1)? We’ll return to this question in Chapter 12.
The Standard Model summarizes and organizes everything we know about the particles that are the building blocks of our world. It makes testable, highly accurate predictions about reactions such as the nuclear reactions that cause the sun to shine. Apart from predicting the W and Z0 particles, which after all can only be produced using specialized machines costing hundreds of millions of dollars, what does the Standard Model tell us about the universe?
The Universe Is Left-Handed
We could call it the Lewis Carroll question: Is the looking-glass world different from our world? When Alice went through the mirror, she was surprised to find a very different world on the other side. Physicists were likewise surprised to find in 1956 that the mirror world is not, in fact, identical to ours. Everyday physical processes look essentially the same in the mirror world as in our world. If you filmed a game of billiards by pointing the camera at a mirror instead of directly at the table, the film would not reveal the trick. Right-handed players would appear to be playing left-handed, and vice versa. Nothing about the movement and collisions of the balls would look at all unusual. The same can be said of electromagnetic interactions or gravity. Until 1956, physicists had never encountered a process that was not also possible in the mirror world.
The process that first showed a violation of this mirror symmetry, or parity as it is called, was our old friend beta decay, in which a neutron decays into a proton, an electron, and an antineutrino. Ignoring the hard-to-detect antineutrino, here’s how the experiment might look in our world and in the mirror world.
The mirror world experiment looks just the same, except that the spin of the neutron is reversed in the mirror world. If the electron is coming from the “north pole” of the neutron in our world, then it is coming from the “south pole” in the mirror world. Now, if parity was a symmetry of beta decay, these processes would have to be equally likely. A careful experiment done by Chien-Shiung Wu of Columbia University (using cobalt-60 rather than free neutrons) proved that north pole electrons were slightly more likely than south pole electrons. Mirror symmetry was violated in weak interactions. Wu’s experiment was quickly followed by others that confirmed parity violation, and in 1957, the two theorists who had suggested the possibility, Tsung-Dao Lee and Chen Ning Yang, became the youngest Nobel Prize winners ever, at the ages of 30 and 34.
By the time the Standard Model was being put together in the early 1970s (based, as we have seen, on another contribution of Yang’s, the Yang-Mills symmetry), parity violation was a long-accepted fact, and was therefore built into the theory in a very blatant way: All the neutrinos are left handed. The handedness of a particle is defined by its spin and its direction of motion. If you curl the fingers of your right hand in the direction of the spin and your thumb points in the direction of the particle’s motion, it is a right-handed particle. If you can do the same with your left hand, it is a left-handed particle:
Notice that to an observer who is moving faster than the particle, the handedness is reversed. In the following illustration, the jet is overtaking a left-handed particle. From the jet’s point of view, the particle is moving to the left. So, in this reference frame, the particle is right-handed.
How is it possible that the Standard Model has only left-handed neutrinos? Remember that neutrinos have no mass in the Standard Model, so they are always traveling at the speed of light. The jet can’t go faster than the speed of light, so it can never overtake the neutrino. There is no reference frame in which the neutrino looks right-handed. Later, we will consider how the Standard Model must be modified if neutrinos actually have mass.
Just three days before Wu announced her parity-violation results, Wolfgang Pauli wrote to a friend, “I do not believe that the Lord is a weak left-hander.”3 For once his intuition was wrong. The looking-glass world is not the same as our world.
Physics of the Large and the Small
The Standard Model contains three families of quarks and three families of leptons. Are there other, heavier families still to be discovered? Surprisingly, we can answer this question by studying the decays of the Z0. The Standard Model predicts that the Z0 can decay into a neutrino-antineutrino pair of any flavor. The more families there are, the more possible routes for the Z0 to decay. Every extra decay route increases the probability that the Z0 will decay, and so decreases its expected lifetime. So, a measurement of the lifetime of the Z0 tells us about how many fermion families there are. The answer is that there can be no more nor less than three families. It remains a mystery why nature cloned the electron family at all. But, if there are more fundamental particles yet to be discovered, they aren’t simply further duplicates of the families we already know.
Particle physicists weren’t surprised to learn from Z0 decays that there were only three lepton families; they had already been told the answer by the astrophysicists. To understand how the study of the skies got to the answer before the big accelerators did, we need to know a few things about the big bang origin of the universe. The motivation for the big bang model comes from the fact that everywhere we look galaxies are moving away from our galaxy. The farther the galaxy, the faster it is racing away. Now imagine running the movie of the universe backwards. At an earlier time, all the galaxies were closer together. Earlier yet, and there were no galaxies; a uniform hot gas filled the universe. About 93 percent of the atoms in the gas were hydrogen and about 7 percent were helium, with trace amounts of other elements: Those are the percentages we measure in the leftover gas clouds between galaxies today. Running the movie even farther back, the gas filling the universe becomes so dense and hot that atoms cannot exist; they are broken up into individual protons, neutrons, and electrons. We have reached a time only 1/1000 of a second after the big bang itself.
Still earlier, and the temperature is so high that neutrons and protons are torn apart into individual quarks. At this time, the entire currently observable universe, all the stars and galaxies that we can see, is packed into a ball that could fit comfortably between the current positions of our sun and the nearest stars. We have reached the quark era, when the universe was less than 10 microseconds old. The entire universe is filled with an incredibly hot, dense quark soup. All of the processes of the Standard Model are occurring at a frantic pace: Energy is constantly being converted from quarks to photons to electrons to neutrinos to Ws and Zs. Since particles of all types will be produced, the presence of extra particle families affects the energy balance of all the other particles. (Actually, it is only the presence of extra light particles, like neutrinos, that matters, since these are the easiest to produce.) As a result, changing the number of fermion families changes the numbers of neutrons and protons that are produced when the universe cools past the quark era. The neutron/proton ratio in turn affects the percentages of hydrogen and helium that get made when the universe cools still further. In this way, a fundamental fact of observational astronomy, the abundance of hydrogen and helium in the universe, is connected in a crucial way with a fundamental fact about the Standard Model, the number of fermion families. The astrophysical evidence implied the existence of only three fermion families even before measurements of the Z0 lifetime led to the same conclusion.
At this point, I need to make a confession. There is a way to avoid the conclusion that the three families that we know are the only ones that exist. The Standard Model assumes that all neutrinos are massless, like photons. Recent experiments show, though, that this assumption is almost certainly wrong. These experiments take us beyond the Standard Model, however, so I leave discussion of them to Chapter 11. But both of the arguments “proving” that only three families exist rely on the assumption that neutrinos are, if not strictly massless, at least very much lighter than an electron. So we can’t rule out the possibility of a fourth family with a heavy neutrino.
The convergence of results from astrophysics and elementary particle physics is part of a surprising synergy that has developed in the past 20 years where the physics of the very large (the universe, galaxies, supernovas) and the very small (the Standard Model) come together to produce new results. The Standard Model brings us full circle, to a point where particle physics and astrophysics are no longer separate disciplines. New ideas in one area can have a profound impact on the other area. This was certainly the case when a young particle physicist at SLAC developed a twist on the big bang model known as the inflationary universe.
Blowing Up the Universe
In 1979, Alan Guth was a postdoctoral fellow still looking for a tenure-track job. Together with Henry Tye, he began to consider what effect the Higgs field would have on the picture of the early universe derived from the big bang model. The effect, he discovered, was profound, and solved at one blow many of the outstanding difficulties of the big bang model. (I will describe an improvement of Guth’s model, the new inflationary universe model, developed by Andrei Linde and independently by Paul Steinhardt, rather than Guth’s original theory.)
In the very early universe, the temperature must have been so high that spontaneous symmetry breaking could not occur. The Higgs field had enough energy to keep it above the hump in the Mexican hat potential. If we represent the average value of the Higgs field by a ball rolling in the “bowl” (the potential), it starts off perched on top of the hump. As the universe expands and cools, the energy in the Higgs field drops below the top of the hump. Now the situation is unstable, like the pencil balanced on its point. The ball wants to roll down into the trough. Since the cooling happens so quickly, though, it remains momentarily at the center, creating a situation known as supercooling. A similar situation occurs when very pure water is cooled below its freezing point. Not having any impurities to start the formation of ice crystals, the water remains liquid until it is disturbed by some external influence, at which point it all freezes very rapidly. In the supercooled early universe, the Higgs field is disturbed by quantum fluctuations and the ball begins to roll off the hump. Here’s where things start to get interesting. According to Einstein’s theory of general relativity, the rate of expansion of the universe depends on the state of the matter and energy in the universe. Guth discovered that the transition out of the supercooled state caused an exponential expansion of the universe. The size of any given region of the universe would double in the unimaginably tiny time of 10-37 second. The universe might go through 100 such doublings in the first moments of its existence. At the end of this inflationary period, the currently observable universe would have been only about a 100 miles across.
The word “about” in the previous sentence should be taken very liberally. The predicted size depends on many assumptions and details about the inflationary model chosen. The exponential expansion is so great, though, that we can say with confidence that before inflation, the entire currently observable universe—all the stars, gas, and dust, not only in our galaxy, but in all the galaxies from us to the farthest visible quasars—was crammed into a space much smaller than a proton.
Finally, after the ball rolls off the hump, the field oscillates in the trough of the Mexican hat, and the oscillations gradually diminish as the remaining energy in the Higgs field gets converted to all of the various Standard Model particles. Eventually, the energy level drops until the field value comes to rest in its new stable position. The particles created from the Higgs energy go on to become the protons, neutrons, electrons, and photons that make up everything in the universe. This gives a new meaning to Leon Lederman’s term for the Higgs: the God Particle.
Guth, followed by Linde and Steinhardt, went on to show how inflation solved several puzzles of the standard big bang scenario. Why is the observable universe so uniform? Because it started out smaller than a proton, and such a small region must have been at a uniform temperature. Why is the universe so nearly flat? Any curvature would have been smoothed out by the expansion. Why have magnetic monopoles never been observed? (Magnetic monopoles—magnetic particles with a “north” but no “south” pole or vice versa—are predicted by grand unified theories, which we will encounter in the next chapter.) Any monopoles that existed before inflation would be so spread out by the expansion that we would have very little chance of encountering one.
Attractive as it is for solving cosmological difficulties, inflation has its own problems. More modern views of the inflationary universe require the hump in the Mexican hat potential to be low and flat. Exactly how flat it needs to be depends on the details of the theory under consideration, but it often ends up so flat that it is useless for spontaneous symmetry breaking. Some physicists have suggested that inflation is not caused by the Higgs boson at all, but by some other particle with a Mexican hat potential, which they call the inflaton. This of course raises more questions: Why is there an inflaton? Why is its potential so flat? Answers will come, if at all, only from theories that go beyond the Standard Model.
Is it unconscionable hubris for physicists to claim they know what the universe was like 14 billion years ago, a fraction of a second after its creation? The answer hinges on another question: how well do we know the limits of our theory? The early universe discussion relied on the coupling of the big bang model of general relativity with the Standard Model. The Standard Model works flawlessly (except for those neutrino masses) at the energies attainable at current accelerators—up to about 200 billion electron-volts. In the big bang model, this is the average energy about a billionth of a second after the big bang. We have no reason to doubt the predictions of the Standard Model at these early times.
How about general relativity? Its fundamental parameters are the gravitational constant, G, and the speed of light, c. No one knows the limits of validity of general relativity; no experiment has ever been done that is in conflict with the theory, but we might expect that it will break down when quantum effects become important. We can combine the constants G and c with Planck’s constant, ℏ, from quantum mechanics to get an energy:
If this energy, called the Planck energy, is truly the limit of validity of general relativity, then we should be able to trust the big bang model as early as the Planck time: 10-43 seconds after the big bang.
As far as we know, then, we are working within the limits of validity of the theories when we describe the early universe. What’s more, we can make predictions (although they are really retrodictions about things that happened long ago) using this picture: the primordial abundance of deuterium, helium (3He and 4He), and lithium, all of which accord well with observations. The Standard Model-big bang model combination allows us to travel in our imagination confidently back to the first milliseconds of the universe’s existence.
Any physical theory has its limits. Maxwell’s equations work beautifully to describe electromagnetic phenomena, but only up to a certain point. When dealing with the physics of the very small, electrons in atoms or photons in high-energy scattering experiments, Maxwell’s equations fail; they must be replaced by the equations of the Standard Model. Sometimes, it is a new experiment that hurls the experimenter beyond the limits of validity of the theory, as when the cosmic ray experiments of the 1950s and 1960s began turning up unexpected new particles. Sometimes, though, it is the theory itself that tells us its limitations. If we use the equations of general relativity to push the big bang model back past the Planck time, we find that the temperature everywhere in the universe rises toward infinity. Surely, this is a sign that the theory has broken down, that it is incapable of making correct predictions in that extreme environment so far beyond our experience.
Does the Standard Model give us any hints about its limits of validity? A careful look at the behavior of the three forces in the theory reveals a surprising fact. Remember that the strength of a force depends on the energy scale of the particular experiment. The strength of the electromagnetic force increases as you go to higher energies, but the strength of the strong force decreases. The weak force likewise changes with the energy scale. Plotting the strength of the force as a function of the energy, we find that the three curves meet very nearly at a single point (α1 is the electromagnetic coupling constant, α2 is the weak coupling constant, and α3 is the strong (color) coupling constant).
It appears that the curves meet at energy scale of around 1015 GeV. Is this pure coincidence? Or is Nature trying to tell us that the three forces are really different aspects of a single, unified force that our accelerators are just too feeble to see? This unification energy scale is a hundred trillion times larger than current accelerators can reach; as Leon Lederman says, this “puts it out of the range of even the most megalomaniacal accelerator builder.”4 However, theories that unify the three forces at this energy also leave traces at lower energies, traces that might show up soon in accelerator experiments or through astronomical observations. Let’s take a look at some experiments that might already be showing us where the Standard Model breaks down.