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# VIRTUAL PROCESSES

Most processes in quantum field theory occur via virtual particles. One can think of them as short-lived imposters of real particles. They act as a kind of currency, allowing real particles to exchange momentum, energy, and charges. They explain how particles decay, scatter off one another, resonantly produce other particles, form bound states, and ultimately explain how long-range forces arise. All these processes can be calculated from Feynman diagrams, invented by Richard Feynman in 1949. External lines in the Feynman diagram of Figure 1(a) represent real particles, and internal lines represent virtual particles. In this diagram, a pair of particles annihilate each other, producing the wavy virtual particle, which then turns into another pair of particles. There are also diagrams where virtual particles produce other virtual particles, which loop around before being reabsorbed, as in Figure 1(b).

FIGURE 1

The most obvious difference between virtual and real particles is their energy. In special relativity, the energy of real particles is determined by their mass and momentum, which is called being on mass shell. For massive particles viewed in their rest frame (where their momentum is zero), this relation reduces to E = mc2. Virtual particles have the same mass as their real cousins, but E is not fixed to being mc2, and they are said to be off mass shell. They get away with having the "wrong energy" because of the uncertainty inherent in quantum mechanics. Still, the further off mass shell the virtual particle, the smaller its contribution to a process, and the more ephemeral its existence.

## Scattering

Real particles scatter via the exchange of momentum through virtual particles. Thus, virtual particles are carriers of force. If the real particles just change direction, the scattering is called elastic, but if they change into other particles, it is called inelastic.

Virtual gluons mediate the strong force, virtual photons mediate the electromagnetic force, and virtual W and Z bosons mediate the weak force. Thus, an electron and a positron can scatter electromagnetically via a virtual photon. In Figure 2(a), they change direction after the virtual photon is emitted or absorbed. In Figure 2(b), they annihilate into a virtual photon, which then produces another electron and positron. Since the initial and final particles are the same, the diagrams contribute coherently.

## Resonances

Scattering is enhanced if it occurs through a particle nearly on mass shell. This is called a resonance. Suppose an electron and a positron collide, and they happen to have a combined kinetic energy equal to the mass of the Z boson times c2. Then they will annihilate and produce a real Z, as in Figure 3. The Z is unstable and decays very quickly, here into a quark-antiquark pair. Z bosons can be studied by measuring the increase in electron-positron inelastic scat-tering at the resonance energy.

FIGURE 2

FIGURE 3

## Long-range Forces

Quantum field theory aims to explain all particle interactions. Therefore, although it excels at predicting short-distance scattering, it should apply to long-range forces as well. However, an explanation in terms of virtual particles is tricky. The discussion here will be limited to trying to understand heuristically how virtual carriers of force operate with attractive and repulsive forces.

As previously discussed, virtual particles transfer momentum between scattering particles, so it is easy to see how this can result in a repulsive force. Imagine two players, Alice on the left and Bob on the right, who are exchanging a virtual ball. As Alice throws the ball to Bob, she recoils to the left, away from him. When Bob catches it, he is pushed to the right, away from Alice. Hence, they are pushed apart. How then do attractive forces arise?

Imagine that Alice throws the ball to the left. She does get pushed toward Bob, but now the ball has momentum away from Bob. However, it is a virtual ball, and where and when it can go defy classical intuition. In virtual processes, both forwards and backwards-in-time exchanges occur, as explained in Feynman's famous QED published in1985. The ball can thus have momentum to the left but arrive at Bob on the right by going backward through time. (The change in position is just the momentum divided by the mass times the change in time, so if the change in time is negative, the change in position is to the right. ) When Bob catches the ball, he receives an impulse toward Alice, since it has momentum to the left. In a naive classical view, they can exchange a virtual ball that pulls them closer together because the catch can precede the throw!

## Decays

Most elementary particles are unstable and decay in a fraction of a nanosecond. This is also due to virtual particles. For example, in Figure 4, a muon decays via a virtual W boson. The muon weighs about0.1 proton mass, whereas the W boson weighs about 80 proton masses. This has two effects. First, the virtual W is very far off mass shell and appears for only a trillionth of a trillionth of a second. Second, the overall process is very suppressed by the ratio of these masses, so that the muon lives for about two millionths of a second, which is a long time by unstable particle standards. So, the W appears for only a tiny fraction of the muon's lifetime, roughly the same ratio as a centimeter to a light-year!

## Virtual Loops

As previously discussed, virtual particles can spawn other virtual particles in loops, as in Figure 1(b). Virtual particles in such a loop can have any energy, and although the contribution from each energy is small, the sum over the infinite range of energies is infinite. An infinity signals a breakdown of a theory. In this case, it shows that the theory is valid only up to some finite energy. This problem can be fixed by a procedure called renormalization. For example, one can cut off the sum at some finite high energy, and then the contributions from loops become finite. Using renormalization group equations, it can be shown that the loops make the strength of

FIGURE 4

the forces of nature scale with the energy of the process. At an energy of about 10 quadrillion times the proton mass times c2, the strength of the electromagnetic, strong, and weak forces all reach the same value, leading to the speculation that they can be unified into one theory at that scale.

## Bibliography

Cheng, T. P., and Li, L. F. Gauge Theory of Elementary Particle Physics (Oxford Science Publications, Oxford, UK, 1988).

Feynman, R. P. "Space-Time Approach to Quantum Electro-dynamics.Physical Review76 , 769–789 (1949).

Feynman, R. P. QED: The Strange Theory of Light and Matter (Princeton Science Library, Princeton, NJ, 1985).

Halzen, F., and Martin, A. D. Quarks and Leptons: An Introductory Course in Modern Particle Physics (Wiley, New York,1984).

Robert Garisto

Rashmi Ray