337. GUTs: The Unification of ForcesLearning Objectives
Present quests to show that the four basic forces are different manifestations of a single unified force follow a long tradition. In the 19th century, the distinct electric and magnetic forces were shown to be intimately connected and are now collectively called the electromagnetic force. More recently, the weak nuclear force has been shown to be connected to the electromagnetic force in a manner suggesting that a theory may be constructed in which all four forces are unified. Certainly, there are similarities in how forces are transmitted by the exchange of carrier particles, and the carrier particles themselves (the gauge bosons in Table: Selected Particle Characteristics) are also similar in important ways. The analogy to the unification of electric and magnetic forces is quite good—the four forces are distinct under normal circumstances, but there are hints of connections even on the atomic scale, and there may be conditions under which the forces are intimately related and even indistinguishable. The search for a correct theory linking the forces, called the Figure 1 is a Feynman diagram showing how the weak nuclear force is transmitted by the carrier particle ${Z}^{0}$, similar to the diagrams in Section 333 Figure 2 and Section 333 Figure 3 for the electromagnetic and strong nuclear forces. In the 1960s, a gauge theory, called Although the weak nuclear force is very short ranged ( ${\mathrm{<\; 10}}^{\u201318}\text{m}$, as indicated in Section 333 Table 1), its effects on atomic levels can be measured given the extreme precision of modern techniques. Since electrons spend some time in the nucleus, their energies are affected, and spectra can even indicate new aspects of the weak force, such as the possibility of other carrier particles. So systems many orders of magnitude larger than the range of the weak force supply evidence of electroweak unification in addition to evidence found at the particle scale.
The strong force is complicated, since observable particles that feel the strong force (hadrons) contain multiple quarks. Figure 3 shows the quark and gluon details of pion exchange between a proton and a neutron as illustrated earlier in Section 332 Figure 1 and Section 333 Figure 3. The quarks within the proton and neutron move along together exchanging gluons, until the proton and neutron get close together. As the $u$ quark leaves the proton, a gluon creates a pair of virtual particles, a $d$ quark and a $\stackrel{}{d}$ antiquark. The $d$ quark stays behind and the proton turns into a neutron, while the $u$ and $\stackrel{}{d}$ move together as a ${\pi}^{+}$ (Section 336 Table 2 confirms the $u\stackrel{}{d}$ composition for the ${\pi}^{+}$.) The $\stackrel{}{d}$ annihilates a $d$ quark in the neutron, the $u$ joins the neutron, and the neutron becomes a proton. A pion is exchanged and a force is transmitted. It is beyond the scope of this text to go into more detail on the types of quark and gluon interactions that underlie the observable particles, but the theory ( Making Connections: Unification of ForcesGrand Unified Theory (GUT) is successful in describing the four forces as distinct under normal circumstances, but connected in fundamental ways. Experiments have verified that the weak and electromagnetic force become identical at very small distances and provide the GUT description of the carrier particles for the forces. GUT predicts that the other forces become identical under conditions so extreme that they cannot be tested in the laboratory, although there may be lingering evidence of them in the evolution of the universe. GUT is also successful in describing a system of carrier particles for all four forces, but there is much to be done, particularly in the realm of gravity. How can forces be unified? They are definitely distinct under most circumstances, for example, being carried by different particles and having greatly different strengths. But experiments show that at extremely small distances, the strengths of the forces begin to become more similar. In fact, electroweak theory’s prediction of the ${W}^{+}$,
${W}^{}$, and ${Z}^{0}$ carrier particles was based on the strengths of the two forces being identical at extremely small distances as seen in
Figure 4. As discussed in case of the creation of virtual particles for extremely short times, the small distances or short ranges correspond to the large masses of the carrier particles and the correspondingly large energies needed to create them. Thus, the energy scale on the horizontal axis of
Figure 4 corresponds to smaller and smaller distances, with 100 GeV corresponding to approximately, ${10}^{18}\text{m}$ for example. At that distance, the strengths of the EM and weak forces are the same. To test physics at that distance, energies of about 100 GeV must be put into the system, and that is sufficient to create and release the ${W}^{+}$, ${W}^{}$, and ${Z}^{0}$ carrier particles. At those and higher energies, the masses of the carrier particles becomes less and less relevant, and the ${Z}^{0}$ in particular resembles the massless, chargeless, spin 1 photon. In fact, there is enough energy when things are pushed to even smaller distances to transform the, and ${Z}^{0}$ into massless carrier particles more similar to photons and gluons. These have not been observed experimentally, but there is a prediction of an associated particle called the The small distances and high energies at which the electroweak force becomes identical with the strong nuclear force are not reachable with any conceivable humanbuilt accelerator. At energies of about ${\text{10}}^{\text{14}}\phantom{\rule{0.25em}{0ex}}\text{GeV}$ (16,000 J per particle), distances of about ${\text{10}}^{\text{30}}\phantom{\rule{0.25em}{0ex}}\text{m}$ can be probed. Such energies are needed to test theory directly, but these are about ${\text{10}}^{\text{10}}$ higher than the proposed giant SSC would have had, and the distances are about ${\text{10}}^{\text{12}}$ smaller than any structure we have direct knowledge of. This would be the realm of various GUTs, of which there are many since there is no constraining evidence at these energies and distances. Past experience has shown that any time you probe so many orders of magnitude further (here, about ${\text{10}}^{\text{12}}$), you find the unexpected. Even more extreme are the energies and distances at which gravity is thought to unify with the other forces in a TOE. Most speculative and least constrained by experiment are TOEs, one of which is called At the energy of GUTs, the carrier particles of the weak force would become massless and identical to gluons. If that happens, then both lepton and baryon conservation would be violated. We do not see such violations, because we do not encounter such energies. However, there is a tiny probability that, at ordinary energies, the virtual particles that violate the conservation of baryon number may exist for extremely small amounts of time (corresponding to very small ranges). All GUTs thus predict that the proton should be unstable, but would decay with an extremely long lifetime of about ${\text{10}}^{\text{31}}\phantom{\rule{0.25em}{0ex}}\text{y}$. The predicted decay mode is $$p\to {\pi}^{0}+{e}^{+}\text{, (proposed proton decay)}$$ which violates both conservation of baryon number and electron family number. Although ${\text{10}}^{\text{31}}\phantom{\rule{0.25em}{0ex}}\text{y}$ is an extremely long time (about ${\text{10}}^{\text{21}}$ times the age of the universe), there are a lot of protons, and detectors have been constructed to look for the proposed decay mode as seen in Figure 5. It is somewhat comforting that proton decay has not been detected, and its experimental lifetime is now greater than $5\times {\text{10}}^{\text{32}}\phantom{\rule{0.25em}{0ex}}\text{y}$. This does not prove GUTs wrong, but it does place greater constraints on the theories, benefiting theorists in many ways. From looking increasingly inward at smaller details for direct evidence of electroweak theory and GUTs, we turn around and look to the universe for evidence of the unification of forces. In the 1920s, the expansion of the universe was discovered. Thinking backward in time, the universe must once have been very small, dense, and extremely hot. At a tiny fraction of a second after the fabled Big Bang, forces would have been unified and may have left their fingerprint on the existing universe. This, one of the most exciting forefronts of physics, is the subject of Frontiers of Physics. Summary
Conceptual QuestionsExercise 1If a GUT is proven, and the four forces are unified, it will still be correct to say that the orbit of the moon is determined by the gravitational force. Explain why. Exercise 2If the Higgs boson is discovered and found to have mass, will it be considered the ultimate carrier of the weak force? Explain your response. Exercise 3Gluons and the photon are massless. Does this imply that the ${W}^{+}$, ${W}^{}$, and ${Z}^{0}$ are the ultimate carriers of the weak force? Problems & ExercisesExercise 1Integrated Concepts The intensity of cosmic ray radiation decreases rapidly with increasing energy, but there are occasionally extremely energetic cosmic rays that create a shower of radiation from all the particles they create by striking a nucleus in the atmosphere as seen in the figure given below. Suppose a cosmic ray particle having an energy of ${\text{10}}^{\text{10}}\phantom{\rule{0.25em}{0ex}}\text{GeV}$ converts its energy into particles with masses averaging $\text{200}\phantom{\rule{0.25em}{0ex}}\text{MeV/}{c}^{2}$. (a) How many particles are created? (b) If the particles rain down on a $1\text{.}{\text{00km}}^{2}$ area, how many particles are there per square meter? Show/Hide Solution Solution(a) $5\times {\text{10}}^{\text{10}}$ (b) $5\times {\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}{\text{particles/m}}^{2}$ Exercise 2Integrated Concepts Assuming conservation of momentum, what is the energy of each $\gamma $ ray produced in the decay of a neutral at rest pion, in the reaction ${\pi}^{0}\to \gamma +\gamma $? Exercise 3Integrated Concepts What is the wavelength of a 50GeV electron, which is produced at SLAC? This provides an idea of the limit to the detail it can probe. Show/Hide Solution Solution$2.5\times {\text{10}}^{\text{17}}\phantom{\rule{0.25em}{0ex}}\text{m}$ Exercise 4Integrated Concepts (a) Calculate the relativistic quantity $\gamma =\frac{1}{\sqrt{1{v}^{2}/{c}^{2}}}$ for 1.00TeV protons produced at Fermilab. (b) If such a proton created a ${\pi}^{+}$ having the same speed, how long would its life be in the laboratory? (c) How far could it travel in this time? Exercise 5Integrated Concepts The primary decay mode for the negative pion is ${\pi}^{}\to {\mu}^{}+{\stackrel{}{\nu}}_{\mu}$. (a) What is the energy release in MeV in this decay? (b) Using conservation of momentum, how much energy does each of the decay products receive, given the ${\pi}^{}$ is at rest when it decays? You may assume the muon antineutrino is massless and has momentum $p=E/c$, just like a photon. Show/Hide Solution Solution(a) 33.9 MeV (b) Muon antineutrino 29.8 MeV, muon 4.1 MeV (kinetic energy) Exercise 6Integrated Concepts Plans for an accelerator that produces a secondary beam of Kmesons to scatter from nuclei, for the purpose of studying the strong force, call for them to have a kinetic energy of 500 MeV. (a) What would the relativistic quantity $\gamma =\frac{1}{\sqrt{1{v}^{2}/{c}^{2}}}$ be for these particles? (b) How long would their average lifetime be in the laboratory? (c) How far could they travel in this time? Exercise 7Integrated Concepts Suppose you are designing a proton decay experiment and you can detect 50 percent of the proton decays in a tank of water. (a) How many kilograms of water would you need to see one decay per month, assuming a lifetime of ${\text{10}}^{\text{31}}\phantom{\rule{0.25em}{0ex}}\text{y}$? (b) How many cubic meters of water is this? (c) If the actual lifetime is ${\text{10}}^{\text{33}}\phantom{\rule{0.25em}{0ex}}\text{y}$, how long would you have to wait on an average to see a single proton decay? Show/Hide Solution Solution(a) $7\text{.}2\times {\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{kg}$ (b) $7\text{.}2\times {\text{10}}^{2}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{3}$ (c) $\text{100 months}$ Exercise 8Integrated Concepts In supernovas, neutrinos are produced in huge amounts. They were detected from the 1987A supernova in the Magellanic Cloud, which is about 120,000 light years away from the Earth (relatively close to our Milky Way galaxy). If neutrinos have a mass, they cannot travel at the speed of light, but if their mass is small, they can get close. (a) Suppose a neutrino with a $7\text{eV/}{c}^{2}$ mass has a kinetic energy of 700 keV. Find the relativistic quantity $\gamma =\frac{1}{\sqrt{1{v}^{2}/{c}^{2}}}$ for it. (b) If the neutrino leaves the 1987A supernova at the same time as a photon and both travel to Earth, how much sooner does the photon arrive? This is not a large time difference, given that it is impossible to know which neutrino left with which photon and the poor efficiency of the neutrino detectors. Thus, the fact that neutrinos were observed within hours of the brightening of the supernova only places an upper limit on the neutrino’s mass. (Hint: You may need to use a series expansion to find v for the neutrino, since its $\gamma $ is so large.) Exercise 9Construct Your Own Problem Consider an ultrahighenergy cosmic ray entering the Earth’s atmosphere (some have energies approaching a joule). Construct a problem in which you calculate the energy of the particle based on the number of particles in an observed cosmic ray shower. Among the things to consider are the average mass of the shower particles, the average number per square meter, and the extent (number of square meters covered) of the shower. Express the energy in eV and joules. Exercise 10Construct Your Own Problem Consider a detector needed to observe the proposed, but extremely rare, decay of an electron. Construct a problem in which you calculate the amount of matter needed in the detector to be able to observe the decay, assuming that it has a signature that is clearly identifiable. Among the things to consider are the estimated half life (long for rare events), and the number of decays per unit time that you wish to observe, as well as the number of electrons in the detector substance.
