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Section 2: Forces and Fundamental Interactions

Two examples of a scattering cross section.

Figure 4: Two examples of a scattering cross section.

Source: © David Kaplan. More info

A way to measure the fundamental forces between particles is by measuring the probability that the particles will scatter off each other when one is directed toward the other at a given energy. We quantify this probability as an effective cross sectional area, or cross section, of the target particle. The concept of a cross section applies in more familiar examples of scattering as well. For example, the cross section of a billiard ball (See Figure 4) is area at which the on coming ball's center has to be aimed in order for the balls to collide. In the limit that the white ball is infinitesimally small, this is simply the cross sectional area of the target (yellow) ball.

The cross section of a particle in an accelerator is similar conceptually. It is an effective size of the particle—like the size of the billiard ball—that not only depends on the strength and properties of the force between the scattering particles, but also on the energy of the incoming particles. The beam of particles comes in, sees the cross section of the target, and some fraction of them scatter, as illustrated in the bottom of Figure 4. Thus, from a cross section, and the properties of the beam, we can derive a probability of scattering.

Figure 5: This movie shows the simplest way two electrons can scatter.

Source: © David Kaplan. More info

The simplest way two particles can interact is to exchange some momentum. After the interaction, the particles still have the same internal properties but are moving at different speeds in different directions. This is what happens with the billiard balls, and is called "elastic scattering." In calculating the elastic scattering cross section of two particles, we can often make the approximation depicted in Figure 5. Here, the two particles move freely toward each other; they interact once at a single point and exchange some momentum; and then they continue on their way as free particles. The theory of the interaction contains information about the probabilities of momentum exchange, the interactions between the quantum mechanical properties of the two particles known as spins, and a dimensionless parameter, or coupling, whose size effectively determines the strength of the force at a given energy of an incoming particle.

Such an approximation, that the interaction between particles happens at a single point in space at a single moment in time, may seem silly. The force between a magnet and a refrigerator, for example, acts over a distance much larger than the size of an atom. However, when the particles in question are moving fast enough, this approximation turns out to be quite accurate—in some cases extremely so. This is in part due to the probabilistic nature of quantum mechanics, a topic treated in depth in Unit 5. When we are working with small distances and short times, we are clearly in the quantum mechanical regime.

We can approximate the interaction between two particles as the exchange of a new particle between them called a force carrier. One particle emits the force carrier and the other absorbs it. In the intermediate steps of the process—when the force carrier is emitted and absorbed—it would normally be impossible to conserve energy and momentum. However, the rules of quantum mechanics govern particle interactions, and those rules have a loophole.

The loophole that allows force carriers to appear and disappear as particles interact is called the Heisenberg uncertainty principle. German physicist Werner Heisenberg outlined the uncertainty principle named for him in 1927. It places limits on how well we can know the values of certain physical parameters. The uncertainty principle permits a distribution around the "correct" or "classical" energy and momentum at short distances and over short times. The effect is too small to notice in everyday life, but becomes powerfully evident over the short distances and times experienced in high-energy physics. While the emission and absorption of the force carrier respect the conservation of energy and momentum, the exchanged force carrier particle itself does not. The force carrier particle does not have a definite mass and in fact doesn't even know which particle emitted it and which absorbed it. The exchanged particles are unobservable directly, and thus are called virtual particles.

Physicists like to draw pictures of interactions like the ones shown in Figure 6. The left side of Figure 6, for example, represents the interaction between two particles through one-particle exchange. Named a Feynman diagram for American Nobel Laureate and physicist Richard Feynman, it does more than provide a qualitative representation of the interaction. Properly interpreted, it contains the instructions for calculating the scattering cross section. Linking complicated mathematical expressions to a simple picture made the lives of theorists a lot easier.

Even more important, Feynman diagrams allow physicists to easily organize their calculations. It is in fact unknown how to compute most scattering cross sections exactly (or analytically). Therefore, physicists make a series of approximations, dividing the calculation into pieces of decreasing significance. The Feynman diagram on the left side of Figure 6 corresponds to the first level of approximation—the most significant contribution to the cross section that would be evaluated first. If you want to calculate the cross section more accurately, you will need to evaluate the next most important group of terms in the approximation, given by diagrams with a single loop, like the one on the right side of Figure 6. By drawing every possible diagram with the same number of loops, physicists can be sure they haven't accidentally left out a piece of the calculation.

Feynman diagrams are far more than simple pictures. They are tools that facilitate the calculation of how particles interact in situations that range from high-energy collisions inside particle accelerators to the interaction of the constituent parts of a single, trapped ion. As we will see in Unit 5, one of the most precise experimental tests of quantum field theory compares a calculation based on hundreds of Feynman diagrams to the behavior of an ion in a trap. For now, we will focus on the conceptually simpler interaction of individual particles exchanging a virtual force carrier.

Feynman diagram representing a simple scattering of two particles (left) and a more complicated scattering process involving two particles (right).

Figure 6: Feynman diagram representing a simple scattering of two particles (left) and a more complicated scattering process involving two particles (right).

Source: © David Kaplan. More info


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