MUON AT THE SPEED OF LIGHT
From a distance we see a small single-prop airplane flying through the air—it appears to be moving in straight and true line on this clear, sunny day. But what about its propeller? The propeller is moving along the same straight line, but the tips of the propeller are executing a corkscrew-like motion. This is how any spinning object—in this case, the propeller—moves through space and time when it is also in uniform motion.
Recall that the muon has spin. We can never stop a muon from spinning. The spin of a muon is always “up” or “down” along any axis. So we find, upon very careful observation, that a muon (or an electron or a quark) traveling at a high speed moves through space-time like a propeller. The muon, like a corkscrew entering a wine bottle cork, or a drill bit drilling into wood, spins either clockwise or counterclockwise as it progresses through space. This is the “up” or “down” binary nature of the muon's spin combined with motion.
But time is frozen as the muon approaches the speed of light. How can it spin if time is frozen? The way to think of this is that the muon's path through space-time is like that of a corkscrew, either corkscrewing to the right or to the left. The two different corkscrewing paths are the two quantum states of spin, “up” or “down.” As the muon approaches the speed of light, these two quantum states become completely independent of one another—the muon has become schizophrenic—it has split into two different personalities altogether as it approaches the speed of light!
This splitting in two of the muon is the consequence of effectively turning off the mass of the muon by approaching the speed of light and by freezing time. The mass of the muon blends the two personalities into one, the usual muon that is heavy and at rest in our lab. But without mass, the muon always travels at the speed of light and then becomes one of two separate and different and independent entities, either a clockwise or counterclockwise rotating corkscrew, or propeller, or drill bit, or whatever metaphor you fancy—it's just one of two different possible massless muons.
If at this point you are starting to feel a bit uneasy, that somehow the elegant simplicity of mass as a mere “quantity of matter” is about to be lost forever, then we suggest that you open a fine bottle of Pinot Noir with a corkscrew wine bottle opener. Now, of course, your bottle opener will turn in one particular way as it penetrates downward into the cork in only one direction. My corkscrew turns clockwise as it descends deeper into the cork.
We have to be precise about what we mean by “clockwise”; that is, we define “clockwise” by looking down from above along the shaft of the corkscrew to the top of the wine bottle. And you can withdraw the corkscrew from the cork by turning it the other way (counterclockwise). And that is an important feature of a drill or a corkscrew—it will rotate one way as it goes in one direction, and the opposite way when it goes the opposite direction! As you contemplate a fresh glass of Pinot Noir, try to figure out if your corkscrew rotates clockwise or counterclockwise as it goes into the cork. To our knowledge, for no particular reason, all corkscrews are manufactured to turn clockwise (looking down from above) as they go into the cork. But there's no reason in principle why there cannot exist a counterclockwise rotating corkscrew. It's just a question of how they were fabricated. Call up the factory and order a dozen counterclockwise corkscrews. Maybe some corkscrews are counterclockwise, while most are clockwise—we're not sure. So, if there are clockwise and counterclockwise corkscrews, these are independent objects like the two pieces of the muon that become separate and independent when we turn off the muon's mass.
CHIRALITY
There's a fancier and more sophisticated way to describe this. We'll assume that you are right-handed (this is unfair to you southpaws, but it is just the way things are defined, so please accept our apology). As you rotate the “clockwise” corkscrew with your right hand by curling your fingers around the knob or handle on the corkscrew, in the manner shown in figure 5.14, you will see that the progression of the screw into the cork is pointed in the direction of your thumb. This is also true for most wood screws or metal screws when you are tightening them with a screwdriver. It's called the “right-hand rule.” The right-hand rule states that “the direction of progression of (most) screws is the direction of your thumb as you rotate the screwdriver by curling your fingers of your right hand around the handle.”
Figure 5.14. Corkscrew. A right-handed corkscrew will progress into the cork in the direction of the thumb of the right hand as the fingers curl around the handle of the screw.
For any rotational motion that is also accompanied by a linear progression, we say the system has “chirality.” Our corkscrew in the above example has “right-handed chirality,” and we'll call it chirality “R.” But, as we said, we can always manufacture a corkscrew that advances into the cork as we rotate counterclockwise. The progression into the bottle as we turn the handle with our left hand would then point in the direction of the left-hand thumb. This is a corkscrew with the opposite chirality, a “left-handed chirality” corkscrew, and we'll call it chirality “L.” Likewise, we can have an ordinary wood screw that is “left-handed” and requires rotating the screwdriver in the opposite way to drive the screw into a block of wood.
THE SPACE-TIME PICTURE WITH CHIRALITY
The approximately massless muon, traveling at almost the speed of light progresses through space as much as it is progressing through time, either with chirality L or chirality R.
So now we can depict our massless muon as it travels at the speed of light. It is either a right-handed, R, or a left-handed, L, particle. If the spin of a particle is pointed along the eastern direction as it moves east, then it is R; and if the spin is still pointing east but the particle is moving west, it is L. The R muon state is completely independent of the L state of the muon—the muon has essentially broken apart into two distinct particles, L and R.
Of course, this ambidextrous L and R quality of very fast particles like high-energy muons comes from the quantum phenomenon of spin. But the two spin states of the resting or slowly moving muon are easily related: we can simply rotate the muon and one spin (e.g., “up”) flips into the other (“down”). And, the resting muon has no chirality—it is sitting still, so there is no “progression through space” associated with the spin. But, as the muon travels near at the speed of light, we cannot rotate one chirality state into another anymore (we would have to stop the muon to do this). For a muon traveling east, the two spin states of the muon that were “up” or “down” have now become “spin pointing east” (R chirality) and “spin pointing west” (L chirality) and are now two independent particles. Note that chirality, L or R, is the combination of the direction of the motion and the direction of the spin. Chirality involves both of these concepts, linear motion and spin, combined together at the same time.
Figures 5.15a–d. Motion of Muon through Space-Time as chiral. Figure 5.15 a. A right-handed particle has its spin (as defined by the right-hand rule) aligned with the direction of motion. As the particle approaches the speed of light we call this “right-handed chirality.” Here the particle is moving “east” while the spin is also pointed “east.”
Figure 5.15b. A left-handed particle has its spin (as defined by the right-hand rule) aligned opposite to the direction of motion. As the particle approaches the speed of light we call this “left-handed chirality.” Here the particle is moving “east” while the spin is pointed “west.”
Figure 5.15c. A right-handed particle has its spin (as defined by the right-hand rule) aligned with the direction of motion. Here the particle is moving “west” while the spin is also pointed “west.”
Figure 5.15d. A left-handed particle has its spin (as defined by the right-hand rule) aligned opposite to the direction of motion. Here the particle is moving “west” while the spin is pointed “east.”
THE FORCES OF NATURE KNOW ABOUT CHIRALITY
The forces of nature respect certain laws that govern everything. Perhaps one of the most important
of these is the law of conservation of energy. The total energy of a system before an interaction occurs is the same as the total energy afterward. Likewise, you are probably familiar with the conservation of momentum, which is why it's hard to stop on a slippery surface, and the conservation of angular momentum, by which gyroscopes always like to point in the same direction and by which it is even possible to ride a bicycle or a motorcycle.
Now we learn a new and somewhat obscure conservation law: it is a stunning fact that all known fundamental (gauge) forces among elementary particles in nature share a special property. They also “conserve chirality.”
For example, if a photon interacts with an R particle, the particle will remain R (see figure 5.16a). A photon cannot convert R particles into L (or vice versa). This means that the R particle is really quite independent of the L particle as far as electromagnetism is concerned (so, too, the strong force of quarks, where we replace the photon by the “gluon,” and likewise for the weak force, which we'll soon discuss in detail). The interaction strength of the force of electromagnetism, i.e., the electric charge, of an R muon is exactly the same as that of an L muon, even though a photon only makes R go to R, and L go to L.
Figure 5.16a. A photon is emitted from an incoming right-handed chirality muon. The muon recoils (changes direction) and its spin simultaneously flips, so that the outgoing muon is also right-handed chirality. Note that the initial spin is +1/2. The emitted photon has spin +1, and the outoing muon has spin –1/2, so the total spin angular momentum is conserved since +1/2 = +1 + (–1/2).
Figure 5.16b. A photon is emitted from an incoming left-handed chirality muon. The muon recoils and its spin flips, so that the outgoing muon is also left-handed chirality. Note that the initial spin is –1/2. The emitted photon has spin -1, and the outoing muon now has spin +1/2, so the total spin angular momentum is conserved since: –1/2 = –1 + 1/2.
RESTORING MASS TO THE MUON
Now let's turn the muon mass back on. Of course, our muon always had a mass, but we made its effects unnoticeable by accelerating the muon up to the speed of light. Mass permits a particle, such as our muon, to travel at any speeds less than c, or to sit still at rest.
Figure 5.17. A muon at rest has a spin that points in the space direction, but it has no spatial velocity. Therefore, it has no definite chirality. The chirality is meaningful as a symmetry only when the muon is massless, travels at the speed of light, whence the R and L muons become distinct particles. Chirality then becomes conserved.
In figure 5.17 we see a massive muon at rest. It simply moves forward through time, with no progression through space. It has spin, but it has no chirality, because there's no direction in space of velocity to compare the spin direction to. Somehow our two independent chiral states of the muon, L and R, must still be there, but they are now blended together to make a resting muon. How?
Nature does this trick by the effect of mass. But mass in particle physics has a new meaning. Mass makes an L chiral particle oscillate into a R chiral particle and back again, without changing the direction of the spin (recall that spin is conserved, the conservation of angular momentum, and the resting muon isn't interacting with anything that can flip the spin around). Mass changes chirality from L to R or vice versa.
Figure 5.18. At the quantum level, a muon at rest has a definite spin, but it is oscillating rapidly between L and R chirality.
So, the resulting motion of the resting or slow, massive muon, in terms of its two chiral parts, is actually a kind of forced march. The muon goes R-L-R-L-R…, to the grunting call of the drill sergeant, called “mass,” even though it is sitting still, simply moving forward in time. Each time the muon oscillates, it flips from being an R muon to an L muon back to an R muon again, over and over. The chirality is not conserved, that is, it is not always the same for particles that have mass. L and R are blended into one marching particle that hobbles through space and time oscillating between its schizophrenic parts of left- and right-handed chirality.
In this R-L-R-L oscillation the direction of the spin stays the same (the spin is “conserved”), and the electric charge remains the same since L and R muons both have the same electric charges. The R muon and L muon oscillate back and forth between one another, so on average the muon can sit still in space, or move along very slowly with a definite value of its spin in the eastward direction. We have a name for this rapid oscillation of the muon between its two chiralites: we call it “Zitterbewegung.”5
So the ancient concept of mass is now seen to be richer in the fundamental world of the elementary particles out of which we are built. It now involves “zitter”—the rapid oscillation between L and R. As the muon approaches the speed of light, time becomes frozen, and the muon becomes frozen into either an independent L or R state.
But now let's reconsider the conservation of electric charge. The chirality was conserved by the interaction of the muon with the photon, even though chirality flips L-R-L-R…due to mass. But this means that we can make a massive muon (or a massive electron, or a massive proton, or massive quarks inside the proton) out of two different pieces, L and R, only if the two pieces have the same electric charge. Electric charge cannot be created out of nothing or made to vanish into nothing. That is, electric charge must be conserved in nature: this is a fundamental law of nature.
If the R muon had an electric charge, which we take to be –1 (in some units; this is a standard definition of the electric charge), and if the L muon had, let's say, 0 electric charge, then our R muon could not possibly turn into an L muon because it would be converting charge –1 into charge 0. That would violate the conservation of electric charge symmetry. That is not permitted!
The uniting of L and R parts to make a massive muon works just fine: the electric charge is the same for L and R, so it is conserved. It would appear that there is “no problemo” for a muon, or an electron, or quarks, etc. to oscillate rapidly between L and R states, ergo, to have mass.
HOLD ON A MINUTE!
Hamlet has been known to say that there's more than what we dream of in our philosophy. This is one of the most profound notions in all of Shakespeare, and it captures why physics is such a fascinating subject. Physics is the ultimate philosophy about nature and reality. And whenever we think we've gotten close to understanding it all, Hamlet pops up and reminds us that there is much more.
Let's return to that peculiar radioactive decay of the muon. That guy Lederman and his colleagues showed that the muon decay violates parity. That means it looks different in the mirror world of Alice than it does in our world. But remember, left and right are always swapped in the mirror world—L becomes R in the mirror world and vice versa. Therefore all chiralities are flipped in the mirror. So what is the parity violation effect of Lederman et al. telling us?
When we dissect the muon decay, we find that it is only when the muon has flipped into its L chiral part that it decays. The short of it is: the weak interactions only involve the left-handed L leptons and L quarks (this also implies that weak interactions only involve the right-handed, R, anti-leptons and antiquarks). This is the reason why Lederman et. al. observed parity violation in the weak interactions.
The R chiral part of the muon does not feel the weak force. By itself, the R muon would be stable. But the muon is not stable because, as we have just seen, it always oscillates back and forth between the two chiralities, R and L, due to its mass. So the muon at some point will always flip into its L chiral part, and then it can decay.
The weak force “knows” something about the L particles that it doesn't share with the R particles. The weak force involves only L but not R. But the weak force also has a conserved charge, just like electromagnetism. Only the L muon has this weak charge—the R muon has no weak charge. Sometimes we say that the R particles are “sterile” under the weak force.
Alas, therefore, the L-R-L-R…oscillation of a massive muon ought not to be allowed! It would violate the conservation of the weak charges. Yet it does
happen! How can we, therefore, combine two totally different things together, L muons (with weak charges) and R muons (with no weak charges) to make the whole muon that sits at rest on a table? That is, how can the muon, or the electron, or the quarks, etc. have mass?
Please reread the last few paragraphs. It's really simple logic, but there are a few things to keep track of. But we're almost starting to glimpse the Higgs boson. It's a short climb to get there. Do you like rainbow trout?
We're going to go fishing for rainbow trout. There's a beautiful mountain lake, high in the Rockies. It's a bit of climb to get there, but it's well worth it. Not only will the exercise be good for us and the fish delectable, but we'll see the most beautiful pinnacles in nature. Are you ready? Got your hiking boots and your fishing rod? Let's go.
As we have seen, the muon decomposes into two independent components when it travels nearly at the speed of light, where the effect of mass becomes irrelevant. It's as though we have banished the muon mass altogether. One of these pieces is R (the spin is pointing in the direction of motion) and the other is L (the spin pointing opposite to the direction of motion). The mass of the muon creates the union of these two pieces into one. A slowly moving muon (much less than the speed of light and perhaps at rest) marches through space-time, oscillating between L and R like the drone of a military marching drill,…L-R-L-R…. At the speed of light, time becomes frozen for our muon, the oscillation stops, and the muon can become one of either the pure L or pure R state.
All matter particles, the electron, the tau, the quarks, even the lowly neutrinos that have miniscule masses, oscillate in this fashion. Mass, which was the measure of the “quantity of matter,” inherited from antiquity, now has a deeper meaning in particle physics. L and R are like the “atoms” of the phenomenon of mass. Only in the utopian world, in which all particles have zero mass and always travel at the speed of light, is the union of L and R broken apart such that particles decompose into their two completely independent entities. In the massless world the L and R components of every quark and lepton take on their own unique identities.
Beyond the God Particle Page 13