The weak interaction is easily treated with the same sort of quantum field theory as we just described for electromagnetism. There are some important differences, however. The big one is that the exchanged particle is not a photon, but one of a set of three particles, called the Z0, W+, and W–. Like the photon, these three particles are bosons, but unlike the photon they have mass.
The upshot of this is that the weak interaction can’t have an infinite range like the electromagnetic interaction. The energy of a photon is determined by its frequency, as we have seen before. There is no limit to how low the frequency of a free photon can be, so the energy of a free photon can be arbitrarily low. It travels at the speed of light, of course, so a low energy photon can travel a long way in the short time permitted by the uncertainty principle. If the charged particles are separated by even more distance, all you need is a lower energy photon to mediate the force. Since there is no limit to how low the free photon’s energy can be, there is no limit on how far the electromagnetic interaction can be felt.
If an exchange boson has mass, however, the restriction on how far it can go becomes much more stringent. Mass is, after all, a form of energy, so (unlike the photon) a massive exchange boson cannot have arbitrarily low energy (i.e., this boson exchange mandates a larger ΔE than photon exchange does). This means that a spontaneously emitted W or Z boson would have to disappear sooner (i.e., have a smaller Δt than for a photon, so as not to violate the uncertainty principle).
The Z or W boson also can’t travel faster than the speed of light, which severely limits the distance over which the weak interaction can operate. And, given the sizeable mass of the W and Z bosons, two particles that are going to interact via the weak force have to be very close together, otherwise the interaction couldn’t happen at all.
Once you take this difference into account, the math of quantum field theory works out very well. You can even use the known mass of the W and Z bosons to calculate the range of the weak interaction, and it gives precisely the same answer as what experimentalists have observed.
QUANTUM LEAP
Actually, from a historic perspective, physicists predicted the mass of the W and Z bosons from the estimated range of the weak interaction before anyone had ever seen any W or Z bosons. Then by adding enough energy to the right particle collisions with a big enough accelerator, the experimentalists were able to turn virtual bosons into real Ws and Zs and observe them directly. The newly observed particles turned out to have the mass and other properties that theory had predicted, another great success for a quantum field theory, for which a Nobel Prize was duly awarded.
The weak interaction has other unique aspects that are worth mentioning here. It is the only interaction that has any appreciable effect on neutrinos. It is also the only interaction that can change the “flavor” of a quark, from down to up for example. This latter process is what happens in beta decay, including the decay of isolated neutrons.
This is the Feynman diagram for the decay of a neutron into a proton via the weak interaction.
A neutron that is off by itself does not live forever. It only stays a neutron for (on average) about 15 minutes. At some point in time, one of the down quarks in a neutron will spontaneously emit a W– boson, turning into an up quark in the process. The W– immediately decays into an electron and an electron’s antineutrino. The resulting three-quark composite then consists of two ups and one down quark, making a proton where the neutron used to be. The electron and the anti-neutrino escape with the excess energy, due to the fact that the neutron is a little more massive than a proton. Thus all of the conservation laws are obeyed.
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