16

A Beautiful Answer

Why are protons so light? Because we understand how the mass of a proton arises, we can give a beautiful answer to that question. The answer removes a major barrier to a unified theory of the forces, and encourages us to seek such a theory.

 

 

LET’S BRIEFLY RECALL how the proton got its mass, with an eye toward finding something in the process that makes the mass small. (This recapitulates part of Chapter 10.)

A proton’s mass is a compromise between two conflicting effects. The color charge carried by quarks disturbs the gluon fields around them. The disturbance is small at first but grows as you get farther from the quark. These disturbances in the gluon field cost energy. The stable states will be those with the smallest possible energy, so we have to cancel these costly disturbances. The disturbing influence of the quark’s color charge can be nullified by an antiquark of the opposite charge nearby, or—the way it’s done in protons—by two additional quarks with complementary colors. Put the nullifying quarks positioned right on top of the original quark, and there’d be no disturbance left. That, of course, would lead to the (null) disturbance with the lowest possible energy (zero).

Quantum mechanics, however, imposes a different energetic cost, which forces a compromise. Quantum mechanics says that a quark (or any other particle) does not have a definite position. It has a spread of possible positions, described by its wave function. We sometimes speak of “wavicles” instead of particles, to emphasize that fundamental aspect of quantum theory. To force a waviquark into a state with a small spread in positions, we must allow it a large energy. In short, it takes energy to localize quarks. The complete nullification we considered in the previous paragraph would require that the nullifying quarks have precisely the same positions as the original quark. That won’t fly, because its cost in localization energy is prohibitive.

So there must be a compromise. In the compromise solution, there will be some residual energy from the not-completely-canceled disturbance in the gluon fields, and some residual energy from the not-quite-completely-unlocalized positions of the quarks. From the total E of these energies the proton mass arises, according to Einstein’s second law m = E/c².

In this account, the newest and trickiest element is the way the disturbance in the gluon field grows with distance. It is closely related to asymptotic freedom, a discovery that recently got three lucky people Nobel Prizes. Asymptotic freedom is a subtle feedback effect from virtual particles, as I explained earlier. It can be thought of as a form of “vacuum polarization,” in which the entity we call empty space, the Grid, antiscreens an imposed charge. The Grid Strikes Back, The Runaway Grid, Grid Gone Wild—it has the makings of a thinking person’s horror movie.

But the reality is subdued. Antiscreening builds up gradually, especially at first. If the seed (color) charge is small, its effect on the Grid starts out small. The Grid itself, by antiscreening, builds up the effective charge, so the next step in the buildup is a little quicker, and so on. Eventually, the disturbance grows large and threatening and must be canceled off. But that might take a while—that is, you might have to get rather far from the seed quark before it happens.

If the disturbance is slow to build, then the pressure to localize the nullifying quarks is correspondingly mild. We don’t have to localize very strictly. Thus the energies involved in both the disturbance and the localization are small—and therefore so is the mass of the proton.

And that’s why protons are so light!

What I’ve just given you is what we call a hand-waving explanation . You couldn’t see me, but while I was typing it I kept interrupting myself to sketch out clouds with my hands, showing my Italian side. Feynman was famous for his hand-waving arguments. Once he explained his theory of superfluid helium to Pauli using such arguments. Pauli, a tough critic, was unconvinced. Feynman kept at it, and Pauli stayed unconvinced, until Feynman, exasperated, asked, “Surely you can’t believe that everything I’ve said is wrong?” To which Pauli replied, “I believe that everything you’ve said is not even wrong.”

To make an explanation that might be wrong we have to get much more specific. When we say protons are light, how light is light? What are the numbers? Can we really explain the ridiculous feebleness of gravity, which, you’ll remember, involved fantastically small numbers?

Pythagoras’ Vision, Planck’s Units

Suppose you had a friend in the Andromeda galaxy whom you could contact only by text-messaging. How would you transmit your vital statistics—your height, weight, and age? This friend doesn’t have access to Earth’s rulers or scales or clocks, so you can’t just say, “I’m so-and-so inches tall, such-and-such pounds, and this-and-that years old.” You need universal measures.

In 1899 and 1900, Max Planck was deeply immersed in the research that inaugurated the quantum theory. The climax came in December 1900, when he introduced the famous constant h—Planck’s constant—that appears in the fundamental equations of quantum mechanics we use today. Just before that, he gave an address to the august Prussian Academy of Sciences in Berlin, in which he posed essentially the question above. (Though he didn’t phrase it in terms of text-messaging.) He called it the challenge of defining absolute units. What excited Planck about his research was not any sense that he might unlock the secrets of the atom, overthrow classical logic, or level the foundations of physics. All that came much later, and from others. What excited Planck was that he saw a way to solve the problem of absolute units.

The problem of absolute units might sound academic, but it is close to the hearts of philosophers, mystics, and philosophically minded scientific mystics.

The manifesto of twentieth- (and twenty-first-) century post-classical physics was issued long before Planck, in around 600 BCE, when Pythagoras of Samos proclaimed an awesome vision. By studying the notes sounded by plucked strings, Pythagoras discovered that the human perception of harmony is connected to numerical ratios. He examined strings made of the same material, having the same thickness, and under the same tension, but of different lengths. Under these conditions, he found that the notes sound harmonious precisely when the ratio of the lengths of string can be expressed in small whole numbers. For example, the length ratio 2:1 sounds a musical octave, 3:2 a musical fifth, and 4:3 a musical fourth. The maxim “All things are number” sums up his vision.

At this remove, it’s hard to be sure exactly what Pythagoras had in mind. Probably part of it was a form of atomism, based on the idea that you could build up shapes from numbers. Today’s terminology of squares and cubes of numbers descends from that shape building. Our construction of “Its from Bits” richly fulfills the promise that “Some important things are number.” In any case, if we take it literally, Pythagoras’s maxim surely goes too far. Abstract numbers such as “3” don’t have a length, a mass, or a duration in time. Numbers by themselves can’t provide physical units for measurement; they can’t make rulers or scales or clocks.

Planck’s problem of absolute units takes aim at precisely this issue. In this digital age we are used to the idea that information, as it appears in text-messaging, can be encoded in a sequence of numbers (indeed, 1s and 0s). So Planck was asking, in effect: Are numbers sufficient, if not to construct then at least to describe every physically meaningful aspect of a material body—in other words, “all things” about it? Specifically, can we convey measures of length, mass, and time using just numbers?

Planck noted that although the Andromedans wouldn’t have access to our rulers, scales, or clocks, they would have access to our physical laws, which are the same as theirs. They could measure, in particular, three universal constants:

 

c: The speed of light.

G: Newton’s gravitational constant. In Newton’s theory, this is a measure of the strength of gravity. To be precise, in Newton’s law of gravity, the gravitational force between bodies of masses m₁, m₂ separated by distance r is Gm₁m₂/r ².

h: Planck’s constant.

 

(Actually Planck used a slightly different quantity from the modern h, which he hadn’t invented yet.)

From these three quantities, by taking powers and ratios, one can manufacture units of length, mass, and time. They are called Planck units. Here they come:

L_P : The Planck length. Algebraically, it is. Numerically, it is 1.6 × 10^-33 centimeter.

M_P : The Planck mass. Algebraically, it is. Numerically, it is 2.2 × 10^-5 gram.

T_P : The Planck time. Algebraically, it is. Numerically, it is 5.4 × 10^-44 second.

Obviously Planck units are not very handy for everyday use. The length and times are ridiculously tiny, even for doing subatomic physics. The Planck length, for example, is 1/100,000,000,000,000, 000,000 (10^-20) times the size of a proton. The Planck mass, 22 micrograms, is not entirely impractical. Vitamin dosages, for example, are often measured in micrograms. So you might go to your health food store and look for pills with a Planck mass of vitamin B12. For fundamental physics, however, the Planck mass is ridiculously big; it is roughly the mass of 10,000,000,000,000,000,000 ( 10¹⁹) protons.

Despite their impracticality, Planck was proud that his units are based on quantities that appear in (presumably) universal physical laws. They are, in his terms, absolute units. You can use them to solve that pressing problem of text-messaging your vital statistics to a friend in Andromeda. You just express your length, mass, and duration in time (that is, your age) as—big!—multiples of the appropriate Planck units.

Over the twentieth century, as physics developed, Planck’s construction took on ever greater significance. Physicists came to understand that each of the quantities c, G, and h plays the role of a conversion factor, one you need to express a profound physical concept:

• Special relativity postulates symmetry operations (boosts, a.k.a. Lorentz transformations) that mix space and time. Space and time are measured in different units, however, so for this concept to make sense, there must be a conversion factor between them, and c does the job. Multiplying a time by c, one obtains a length.

• Quantum theory postulates an inverse relation between wavelength and momentum, and a direct proportionality between frequency and energy, as aspects of wave-particle duality; but these pairs of quantities are measured in different units, and h must be brought in as a conversion factor.

• General relativity postulates that energy-momentum density induces space-time curvature, but curvature and energy density are measured in different units, and G must be brought in as a conversion factor.

 

Within this circle of ideas, c, h, and G attain an exalted status. They are the enablers of profound principles of physics that couldn’t make sense without them.

Unification Scorecard

With the help of Planck’s units, we can assess how well our understanding of the origin of the proton’s mass accounts for the feebleness of gravity, and whether it removes the barrier to unification that gravity’s feebleness seemed to present.

If we are going to produce a unified theory in which special relativity, quantum mechanics, and general relativity are primary components, then we should find that the most basic, underlying laws of physics appear natural when expressed in Planck units. No very large or very small numbers should occur in them.

The root of the our trouble with the apparent feebleness of gravity is that the proton mass is very small in Planck units. But we’ve come to understand that the proton mass is not a direct reflection of the most basic laws of physics. It comes from a compromise between gluon field energy and quark localization energy. The basic physics behind the proton’s mass—the phenomenon that gets the process going—is the underlying basic unit of color charge. The strength of that seed (color) charge determines how fast the growing bloom of gluon field energy becomes threatening; and thus how much of a hit, in quantum localization energy, quarks must take to cancel it; and thus the value of proton mass, according to Einstein’s second law.

Is it possible that a reasonable seed charge leads to the actual, very small—in Planck units—value of the proton mass? To answer this question, of course, we have to specify what we consider a reasonable value of the seed charge. To measure the strength of the basic seed charge, we need to consider the basic physical effects it causes. We could consider any of several effects: the force it generates, the potential energy, or (for experts) the cross-section. As long as we measure everything at the Planck distance using Planck units, we’ll get similar answers whatever measure we use. Since it’s the most vivid and familiar effect, let’s focus on the force.

So according to Planck, the seed charge is reasonable if it leads to a force between quarks separated by a Planck length that is neither terribly small nor terribly large when measured in Planck units. Of course he would say that. The point is not the prestige of Planck’s authority but the ideal his units embody: the ideal that special relativity, quantum mechanics, and gravity (general relativity) can be unified with the other interactions. We’re turning it around and asking if, by assuming that ideal, we are led to a consistent understanding of why protons are light, and thus also of why gravity is feeble in practice.

Finally, then, it all boils down to a very concrete numerical question: Is the magnitude of the seed strong force between quarks, at the Planck length, expressed in Planck units, close to 1?

To answer that question we must extrapolate the laws of physics we know down to distances far smaller than where they have been checked experimentally. The Planck length is very small. Many things could go wrong. Nevertheless, in the spirit of our Jesuit Credo, “It is more blessed to ask forgiveness than permission,” let’s just do it.

The required calculation is actually quite a simple one by the standards of modern theoretical physics. We’ve discussed all the necessary ideas in words. It breaks my heart not to display the algebra, but I’m a merciful man, and besides my publisher warned me against it. So I’ll just state the result:

We find that the seed strong force between quarks, at the Planck scale, measured in Planck units, is about 1/25. That’s quite an improvement over the 1/10,000,000,000,000,000,000,000,000,000, 000,000,000,000 discrepancy we thought we had!

Thus we’ve explained the (apparent) feebleness of gravity starting from fundamental, new, yet firmly based physics. And we’ve overcome a major obstacle blocking the path toward a unified theory of the forces.

Next Steps

I hope you’ll agree that it’s a pretty story, and it hangs together. Declarations of “mission accomplished” have been based on much less.

But it’s unnerving to draw grand conclusions from such a narrow base. It’s like constructing an inverted pyramid balanced on one point. To firm it up, we need a broader base.

There’s no more convincing way to demonstrate that you’ve cleared a hurdle than to finish the course. The path to unification opens before us. Let’s follow it.

 

Plate 1 Photograph taken at the Large Electron-Positron collider (LEP) that operated at CERN, near Geneva, through the 1990s. The jets of particles that emerge from this collision follow the flow patterns predicted theoretically for a quark, an antiquark, and a gluon. Jets give operational meaning to those entities, which cannot be observed as particles in the usual sense.

Plate 2 A two-jet process, which we interpret as manifestation of a quark and an antiquark.

Plate 3 Symmetry in science, art, and reality, illustrated through icosahedra. a. An icosahedronhas 20 equal sides; all are equilateral triangles. An icosahedron supports 59 distinct symmetryoperations; that is, there are 59 distinct rotations that take an icosahedron into itself (while interchanging some of the sides). This compares with 2 distinct symmetry operations for an equilateral triangle. Metaphorically, the symmetry of QCD stands to that of QED as the symmetry of an icosahedron stand to that of a single triangle. b. Enormous symmetry allows one to specify elaborate structures using simple components, a feature viral DNA (or RNA) exploits. Shown here is a virus for the common cold. Note the similarity to a.!

Plate 4 Deep structure of the quantum Grid. This is a typical pattern of activity in the gluon fields of QCD. These patterns of activity are at the heart of our successful computation of hadron masses, as discussed in Chapter 9, so we can be confident that they correspond to reality. This beautiful image was computed by Derek Leinweber, University of Adelaide.

Plate 5 End result of a heavy ion collision—a miniature version of the big bang.

Plate 6 A disturbance in the Grid. At the left, a quark and an antiquark have been injected. They soon establish a dynamic equilibrium, with the energy of the disturbance confined to a small spatial region, moving through time. The Grid fluctuations have been averaged over, leavingonly the net distribution of excess energy. By taking a slice, we find the energy distribution inside the particle being reproduced: in this case, a π meson. The total energy gives the mass of the π meson, according to Einstein’s second law.

 

Plate 7 The Siren’s alluring song asks us to leave comfortable certainty behind and to meet her on a dubious shore. In return, she promises beauty and illumination. Is she teaching, or teasing?

Plate 8 View of the LHC from the air. The Jura Mountains and Lake Geneva frame a mystic scene. Some image processing has been committed here; in reality the machine is underground.

Plate 9 The ATLAS detector for the LHC, in an early stage of construction. In the final, operational form of the detector, this gigantic framework gets densely packed with magnets, sensors, and ultrafast electronics. This is what it takes to make a camera capable of resolving times of order 10^-27 second and distances of 10^-17 centimeter!

Plate 10 Darkness visible. Dark matter does not emit light; it’s “seen” only through its gravitationalinfluence on the motion of ordinary matter. Through image processing we can let our eyes see the world as gravitons do. This ROSAT picture shows confined hot gas highlighted in false purple color. It provides clear evidence for gravity exceeding that exerted by the galaxies inside. The extra gravity is attributed to dark matter. Ideas to improve the equations of physics predict new forms of matter whose properties make them good dark matter candidates. Soon we may learn which, if any, of those ideas correspond to reality.

Plate 11 The author with noted blogger Betsy Devine inside a piece of the other main detector for the LHC, whose acronym is CMS - for Compact (!) Muon Solenoid.