# The Theory of Almost Everything: The Standard Model, the Unsung Triumph of Modern Physics - Robert Oerter (2006)

### Chapter 9. The Weakest link

And so some day

The mighty ramparts of the mighty universe

Ringed round with hostile force

Will yield and face decay and come crumbling to ruins.

*—Lucretius, De Rerum Natura*

**A**t this point, it might seem like the story is over, the Theory of Almost Everything complete:

■ There are six flavors of quarks, each coming in three colors.

■ Ordinary matter contains only the two lightest quark flavors, up and down. Two ups and a down form a proton, two downs and an up form a neutron. Gluons bind the quarks together.

■ Neutrons and protons are themselves bound together in atomic nuclei by pion exchange, which is to say exchange of quark-antiquark pairs.

■ The electromagnetic force, in other words, photon exchange (or QED), holds electrons in atomic orbitals.

■ The Pauli exclusion principle determines how the electrons may be arranged in the orbitals, which in turn determines the atom’s chemical properties and its place in the periodic table.

■ The chemical properties of the atom determine all its interactions with other atoms. These interactions, together with the atomic masses, determine all the everyday properties of objects: density, texture, inflammability, frangibility, electrical conductivity, even color.

Apparently, this is everything we need to explain all everyday phenomena through the combination of QCD and QED. There is, however, one more piece to the puzzle. The missing piece has nothing to do with the strength of steel or the density of gold, but it is crucial for life on earth even to exist.

The story of the weak nuclear force (or weak force, for short) begins in the late nineteenth century with the discovery of radioactivity. One type of radioactivity resulted in the emission of beta rays, later identified as electrons. Where did the electrons come from? These were not the electrons in orbitals around the atom. Instead, they were produced by the nucleus, which changed its identity and became a different element in the process. In 1933, the caustic Italian physicist Enrico Fermi shocked the physics community by proposing that beta radiation occurred when a neutron decayed into an electron, a proton, and a new, never-detected particle that we now call an antineutrino. The shock was two-fold: First, predicting particles on purely theoretical grounds was still a recent and somewhat disreputable innovation (this was long before Gell-Mann, of course), and second, it was almost inconceivable that a neutron, at that time considered a fundamental particle, could decay.

To understand the need for the antineutrino, think about what would happen without it: The rest energy of the proton and electron add up to less than the rest energy of the neutron. This means that when the neutron decays there is some energy left over, which must go into the kinetic energy of the motion of the proton and electron. Taken together, conservation of energy and conservation of momentum completely fix the electron’s energy: All beta decays should produce electrons with the same energy. This is not what is found experimentally, however. The electrons from beta decay had a wide range of energies, from some minimum value up to the value expected from the argument above. In other words, the energies of the proton and the electron don’t add up to the initial rest energy of the neutron. To salvage the law of energy conservation, Fermi had to assume a third, unobserved particle carries away the missing energy.

The properties of the new particle could be easily deduced. The proton’s positive electric charge and the electron’s negative charge add up to zero, so the antineutrino must also have no electric charge—hence the name *neutrino,* or little neutral one. This is why the antineutrino was not observed—neutral particles don’t leave tracks in a cloud chamber. The neutron has spin 1/2, as do the electron and proton. The quantum rules for spin declare that two spin 1/2 particles can’t combine to produce spin 1/2, but the three spin 1/2 particles can do so. The neutrino must therefore have spin 1/2. (This also rules out the possibility that the missing energy is carried by photons, which have spin 1.) The mass of the neutrino can be inferred from the electron energies involved in beta decay. It was found to be nearly zero. The ever-present experimental uncertainties forbid the conclusion that the neutrino mass is exactly zero, but zero mass can be taken as a working hypothesis. Recent results (which will be revealed in Chapter 11) indicate that this hypothesis is probably wrong: Neutrinos most likely have mass, though it is much smaller than the electron mass. The particle in Fermi’s theory is called an antineutrino rather than a neutrino to preserve yet another conservation law: the conservation of lepton number, which was previously discussed. Both electric charge and lepton number are conserved in beta decay. In fact, lepton number (and electric charge) conservation holds true in all reactions that we know of: strong, weak, and electromagnetic.

The strength of an interaction determines the time it takes for a particle to decay. The stronger the force, the faster the decay. Particles that decay via the strong force live about 10^{-24} seconds. In contrast, a particle that decays via the electromagnetic interaction lives about 10^{-16} seconds. This is 100 million times longer—a year compared to a blink of the eye. A particle decaying by way of the weak force typically lives about 10-^{8} seconds—another factor of 100 million longer. Weak though it is, life on earth would not be possible without this force. All life ultimately gets the energy it needs to exist from the sun. Plants trap the sun’s energy by photosynthesis, and animals eat plants (or eat other animals that eat plants). Where does the sun get its energy? What keeps it shining? The answer: nuclear fusion reactions. (This means solar energy is actually a form of nuclear power—don’t tell the environmentalists.)

One of the reactions that powers the sun involves the fusion of two protons. The fusion produces heavy hydrogen (a proton and a neutron stuck together), a positron, and a neutrino. The neutrino is a clue that the weak force is involved. A closer look at this reaction reveals what’s really going on. One of the protons changes into a neutron, plus a positron and a neutrino—just the opposite of what happens in beta decay. The other proton is just a bystander (although it provides some of the energy needed to make the reaction go). Without this inverse beta decay process the sun would not shine and life would never have arisen on earth.

**How to Design Your Own Universe**

How is it possible for a neutron to turn into a bunch of other particles? Maybe the neutron is really a proton and an electron locked together, and its decay simply involves the particles escaping? This plausible explanation was quickly ruled out. The size of the neutron was known from scattering experiments (recall how the Nerf ball could be reconstructed from the scattered BBs). The energy required to bind a proton and an electron in such a small region would give the neutron a mass (via *E* = *mc*^{2}) much greater than what was actually measured. No, the neutron must really change its character in some fundamental way. By now, we are used to the idea of such transformations. Similar processes occur in QED and QCD.

In 1933, when the theory of the weak nuclear force was being born, no one had ever considered such a possibility. Fermi, in his audacity, claimed, at a time when Gell-Mann was four years old and QED was still a puzzling mess, that the neutron transformed into a proton, an electron, and an antineutrino, like an alchemist’s mixture transforming into gold. Nowadays, physicists take this kind of transformation as a matter of course. In fact, all you need to do to invent a new theory of elementary particles is to decide which sorts of particles will transform into which other sorts of particles. The rest of the theory is determined by the structure of relativistic quantum field theory.

Here’s how to design your own physics. Start by listing all the particles that you want to have in your universe, and give their masses, charges, and spin values. For instance, suppose we want to describe a universe that just has electrons, neutrinos, and photons:

The rules of relativistic quantum field theory will guarantee that we also have the corresponding antiparticles: positrons and antineutrinos. The photon is its own antiparticle. To turn this into a relativistic quantum field theory, we need to build the Lagrangian function that is used in Feynman’s version of the least action principle. Two pieces are needed to build the Lagrangian: the propagator and the interactions. The propagator tells how a particle moves from point A to point B in the absence of interactions. It is completely determined by the particle’s mass and spin. In Feynman diagrams, we simply draw a line. Keep in mind, though, that the line is shorthand for the sum of *all possible paths* the particle could take from A to B. All that’s left to do to build our theory is to choose what interactions we want to have. We simply declare that whatever process we desire just happens—any particle can convert instantaneously into any other collection of particles. If we want electric charge to be conserved in our universe, we should choose interactions that respect the conservation law. In Feynman diagrams, the interactions are the junction points where several particle lines come together. In our example, let’s let the electron interact with the photon.

(This guarantees that all of QED will be included in our theory.) The neutrino has no charge, so we won’t allow it to interact with the photon. Next, suppose we let the electron and the neutrino interact directly:

(In our simplified universe, this is the only role for the weak force.) Finally, we need to specify the *strength* of the interactions, or, equivalently, the probability for the specified transformation to occur. Each interaction diagram has an associated *coupling constant,* the number that specifies the strength of the interaction. (For particles with spin, there may be more than one coupling constant corresponding to the ways particles with spins in different directions interact.) In our example, there are two: the electron-photon coupling constant, and the weak coupling constant that specifies the strength of the electron-neutrino interaction.

That’s all there is to it. Make up any particles you like, draw the diagrams for their interactions, and you have designed the physics (always excepting gravity) for your universe. You then apply the rules of relativistic quantum field theory to add up all the possible ways a process can take place. There’s one snag to the whole thing, however. If our theory is to have any hope of describing the real world, it had better be mathematically consistent. That is, it needs to be renormalizable. When the calculations tell us the answer is infinite, there must be a way to sweep the infinities under the rug. For some choices of particle interactions, the theory is renormalizable; for others it isn’t. With the electron-neutrino interaction that we chose earlier, there’s no way to hide all the infinities that show up, and so no way to make the theory mathematically consistent. To get a theory that describes the real world, we’ll have to do better.

The key to building renormalizable theories can be stated in a word: symmetry. Specifically, the Yang-Mills type of symmetry that we used to create the color theory of quarks and gluons is precisely what is needed for renormalizability. That’s why the color theory is so useful (and will soon be revealed as an essential ingredient of the Standard Model)—it’s renormalizable. Here’s how it works: Quarks interact by exchanging a gluon. We say that the gluon *mediates* the strong interactions the same way the photon mediates the electromagnetic interaction. In contrast, the electron-neutrino interaction just proposed has no mediating particle. The electron and the neutrino just slam into each other.

In order to get a renormalizable theory, let’s replace the previous electron-neutrino interaction with a Yang-Mills type interaction. Yang-Mills symmetry requires that the interaction be mediated by a new particle; let’s just call it an X for now. In the basic interaction, an electron emits an X and turns into a neutrino, as in the top part of the illustration below. The bottom half of the diagram is just the same process in reverse.

The X must have a negative charge, because the electron has a negative charge and the neutrino is neutral. The presence of the mediating X particle softens the electron-neutrino collision, in a sense. Instead of having the two particles just slam into each other, we now have the interaction proceeding by a gentlemanly exchange of the X. By adding this single interaction (and the analogous ones required by symmetry—for example, a positron emits the antiparticle of the X and turns into an antineutrino), we get a renormalizable theory of electron-neutrino interactions.

Let’s try to build a realistic theory of the weak force using what we’ve learned so far. The particles that mediate weak interactions are called the W^{+} and W^{-}. (Pronounced “dubya-plus” and “dubya-minus,” these are each other’s antiparticles and are collectively called the W particle.) We use the Yang-Mills trick for interactions between electrons and neutrinos. Yang-Mills symmetry requires that we introduce a third intermediate particle, known as the Z^{0}(pronounced “zee-zero”). We can repeat the trick for the interactions of muons and neutrinos. Quarks, too, need to have weak interactions. The pion, for instance, decays into a muon and an antineutrino. We can include these easily enough: just decide what the coupling constants of the quarks with the W and Z^{0} should be and include the appropriate Feynman diagrams. At this point, it might seem like we’re done—we have a renormalizable theory of weak interactions. However, a complication crops up. The very Yang-Mills symmetry that guarantees renormalizability also forces the mediating particles to be massless. If they’re massless, though, why had they never been observed? A massless W particle should be like a photon with charge—photons are easy to detect, and charges are easy to detect. No such particle had ever been seen in experiments. Moreover, a massless particle produces a long-range force (electromagnetism is the prime example), while the weak force acts only at subatomic distances.

(Gluons are massless intermediate particles, and you might be wondering why gluons don’t suffer from the same objection. Recall that gluons are “colored” particles, and our study of the color force led to the conclusion that particles with color are never observed as free particles, what we called color confinement. For the same reason, there is no long-range color force: The gluons can’t break free of the protons and neutrons in which they are confined.)

This was the problem in early 1971: The weak interaction needed a massless intermediate particle like the W for renormalizability, but the W needed to have mass to describe the weak interaction correctly. Most particle physicists were not thinking in these terms at that time, but a Dutch theorist, Martinus Veltman, was on the scent. A graduate student of his, Gerard ’t Hooft, was just putting the finishing touches on the proof of renormalizability of Yang-Mills theories. Veltman called ’t Hooft into his office, and this was the conversation__ ^{1}__:

M.V.: I do not care what and how, but what we must have is at least one renormalizable theory with massive charged vector bosons, and whether that looks like Nature is of no concern, those are details that will be fixed later by some model freak...

G. ’t H.: I can do that.

M.V.: What do you say?

G. ’t H.: I can do that.

’t Hooft knew of a trick, called *spontaneous symmetry breaking,* which took a massless Yang-Mills theory, added an extra particle, and gave a new theory with intermediate particles that have mass (what Veltmann called “massive charged vector bosons”). What’s more, ’t Hooft had confidence based on his work on Yang-Mills theories that the resulting theory would be renormalizable.

Not only was he right about renormalizability, but it turned out that the “model freaks” had been there already. Veltmann learned of a 1967 paper by Steven Weinberg that described a Yang-Mills theory using the symmetry group SU(2) x U(1) (pronounced “ess-you-two-cross-you-one”) that gave mass to the W particle via spontaneous symmetry breaking. It had been published and resoundingly ignored, as had an identical model developed independently by Abdus Salam, a theorist from Pakistan. According to Weinberg, from the paper’s publication until 1970, it was cited in physics journals only once.

The reason for this neglect was that in 1967 no one knew that Yang-Mills theories were renormalizable, much less the theories complicated by spontaneous symmetry breaking. With ’t Hooft’s proof of renormalizability, the Dutchman had the first complete theory of the weak interactions. In 1972, the year after’t Hooft’s paper, the Weinberg paper was cited 65 times, and the following year 165 times. As a bonus, the electromagnetic interactions were described by the same model. Weak and electromagnetic interactions had been unified into a single theory, which, as we will see, made some sharp, testable predictions, including the existence of three previously unsuspected particles. For their contribution, Weinberg, Salam, and Sheldon Glashow, who had worked on an earlier version of the unified theory, shared the Nobel Prize in 1979. Somewhat belatedly, Veltman and ’t Hooft were awarded the prize 20 years later.

**Breaking the Symmetry**

What does it mean to break a symmetry? Imagine a pencil perfectly balanced on its tip on a table. There is a rotational symmetry to the situation: rotate the picture by any angle about the vertical axis of the pencil, and it remains the same. The slightest air motion will make the pencil fall, though. Now the original symmetry is broken: There is a difference between the direction parallel to the pencil and the one perpendicular to the pencil. Imagine an ant crawling along the tabletop. Before the pencil falls, the ant can roam at will (as long as it avoids the point where the pencil tip touches the table). After the fall, the ant suddenly notices a difference: it’s easier to travel in one direction (parallel to the pencil) than in the other (perpendicular to the pencil, where the ant must climb over it). The symmetry breaking is spontaneous because any small push will cause the pencil to fall; the table doesn’t need to be slanted for it to happen.

To see what this has to do with elementary particles, we have to leave Feynman mode and enter Schwinger mode: stop thinking about particles and start thinking about fields. According to Schwinger a quantum field can be thought of as an infinite collection of harmonic oscillators, one at each point in spacetime. Suppose now that there are two identical quantum fields at each point. The directions in which these fields oscillate are perpendicular to each other. (Think: two different tracks on which an ant could roller-skate.) This construction forms a bowl at each point in spacetime.

The bowl is symmetric, so these two types of oscillation are indistinguishable: The two particles (the quanta of the field oscillations) are identical.

Now suppose that the bowl has a dimple in the bottom.

Since its shape resembles a sombrero, this is sometimes called the Mexican hat potential. When there is a lot of energy in the field, the dimple makes little difference, and the two directions are still identical. When there is little energy in the field, the situation is very different. As the energy is lowered, the field becomes trapped in the trough around the central hump (that is, it doesn’t have the energy to get over the hump). Picture the average field value as a ball resting on the central hump. Like the pencil balanced on its point, the ball will roll off the hump at the slightest disturbance, and will end up at some arbitrary point of the trough. The exact location is irrelevant, since the trough itself is still symmetric. But now when the field oscillates about the ball’s position, the same two directions look very different.

In the direction across the trough, oscillations occur just as if we had a harmonic oscillator. That is, this field oscillation looks like a normal particle with mass. But if the field oscillates the other direction, the trough is *flat.* A flat direction like this corresponds to a massless particle. Recall the skateboarder in the half-pipe: In the cross-pipe direction, there is no symmetry and the skateboarder oscillates back and forth; but in the direction along the pipe, the flat direction, the skateboarder moves smoothly without any difficulty.

To sum up: We started with two identical fields and a potential that is symmetric. Oscillations of the field at high enough energy to be above the hump in the potential are identical—the two particles represented by the field are indistinguishable. At low energy, though, the field must oscillate in the trough, where the original symmetry is broken. One of these oscillations represents a massive particle, the other a massless particle. Spontaneous symmetry breaking has transformed two identical particles into two apparently very different particles.

Now back to the Weinberg-Salam theory of electroweak interactions. Start with an SU(2) x U(1) Yang-Mills theory. It has four massless intermediate particles, similar to the gluons in QCD. Now add two new identical particles, together with the Mexican hat potential. These new particles are called Higgs particles, after the Scottish physicist Peter Higgs, who first introduced spontaneous symmetry breaking into elementary particle theory. Finally, let the Higgs particles interact with all of the other particles in the theory.

At low energy, spontaneous symmetry breaking occurs. Using the Schwinger picture, the quantum field associated with the Higgs particle looks like a Mexican hat at each point in space. When the energy in the Higgs field gets lower than the hump of the Mexican hat potential, the Higgs field rolls off the hump and ends up in the trough. This subtle shift occurs *at* every *point in the entire universe.* Now all the interactions between the particles cause something very odd to occur. We expect to end up with one massless Higgs and one massive Higgs particle as described earlier. Instead, we find only one massive Higgs particle—the other Higgs particle disappears from the theory completely. What’s more, three of the four intermediate particles become massive, while the fourth remains massless. This is just what we wanted. The massless particle is the photon, and has exactly the right properties for the electromagnetic interaction. Because of their interactions with the Higgs, the other three particles get masses. Physicists like to say that the intermediate particles “eat the extra Higgs particle” and gain weight. What’s actually happening is this: The Higgs Field got shifted when it rolled down into the trough. The intermediate particles are connected to the Higgs by the interactions we gave them. Wherever they go, they encounter the shifted Higgs field. Like someone trying to run through waist-deep water, the intermediate particles have to move against the drag of the Higgs field. The now-massive intermediate particles are the W^{+} and W^{-}, and the Z^{0}. This was the accomplishment that won two Nobel prizes: on the one hand, the intermediate particles have picked up the mass that is necessary to explain the short-range nature of the weak force; on the other hand, because the masses arise from spontaneous symmetry breaking, the theory remains renormalizable.

The masses of the W and Z^{0} particles could be calculated from the theory, but they were beyond the reach of the particle accelerators of the early 1970s, so it was not surprising that the particles had never been detected. Could they be found? The theorists had thrown the gauntlet, and experimentalists were not slow to take up the challenge. The Italian Carlo Rubbia convinced CERN, the European particle physics laboratory, that the existing proton accelerator could be converted to a machine that stored both protons and antiprotons. The antiprotons would travel the opposite direction of the protons, and when the beams were brought together, the resulting head-on collisions between protons and antiprotons would have enough energy to produce the massive intermediate particles. The Europeans were in a race with Fermilab in Illinois, which was struggling to get its Tevatron accelerator functional. A win would mean a lot to the Europeans, who had lagged behind the United States in high-energy physics for almost 50 years. Rubbia delivered, and by January 1983, the discovery of the W was announced. The Z° was found later the same year. Both masses were just where they were supposed to be. Rubbia and Simon Van der Meer shared the Nobel Prize in 1984.

Finally, and surprisingly, the Higgs particle gives mass to the electrons and (if you include them in the model) the quarks. Since the Higgs interacts with these particles, they experience the same sort of drag effect that gave mass to the W and Z^{0}. That is, starting with a theory of entirely massless particles—massless electrons, massless quarks, massless neutrinos, and massless intermediate particles—and adding in the Higgs particle with its Mexican hat potential, spontaneous symmetry breaking yields a theory with massive electrons, massive quarks, massive W^{+}, W^{-}, and Z^{0} particles, and a massless photon. Just what we see in the real world. Leon Lederman calls it the God Particle: Without the Higgs there would be nothing in the universe but massless particles. A massless electron could never be captured by a proton to form an atom. Without atoms, no galaxies, no stars, no earth would form. The universe would be very boring. Only by breaking the symmetry do we get massive particles that can become the constituents of matter.

The amazing electroweak unification was accomplished at this cost: Weinberg and Salam needed to invent four new particles. The three massive intermediate particles have since been found, with the masses and properties predicted. The remaining particle predicted by unification is the Higgs particle, which has not yet been detected in any experiment. Is its mass simply too large for the reach of today’s accelerators? Or is there some deeper problem with the Standard Model?

In August of 2000, physicists working at the Large Electron Positron Collider (LEP) at CERN saw a few events that looked like Higgs events are expected to look. LEP was due to be shut down in order to be upgraded to much higher energy. The CERN directors faced an agonizing decision: to delay the upgrade in order to look for enough Higgs-like events to declare that they had discovered the elusive particle, or to shut down, perform the upgrade, and look for the Higgs with the improved machine. After a one-month delay during which additional events failed to materialize, the decision was made to do the upgrade. Scheduled to begin operation in 2007, the new Large Hadron Collider will reach beam energies of several trillion electron-volts. The mysteries of the Higgs may soon be solved.