Advanced information on the Nobel Prize in Physics, 5 October 2004 Information Department, P.O. Box 50005, SE-104 05 Stockholm, Sweden Phone: +46 8 673 95 00, Fax: +46 8 15 56 70, E-mail: [email protected], Website: www.kva.se Asymptotic Freedom and Quantum ChromoDynamics: the Key to the Understanding of the Strong Nuclear Forces The Basic Forces in Nature We know of two fundamental forces on the macroscopic scale that we experience in daily life: the gravitational force that binds our solar system together and keeps us on earth, and the electromagnetic force between electrically charged objects. Both are mediated over a distance and the force is proportional to the inverse square of the distance between the objects. Isaac Newton described the gravitational force in his Principia in 1687, and in 1915 Albert Einstein (Nobel Prize, 1921 for the photoelectric effect) presented his General Theory of Relativity for the gravitational force, which generalized Newton’s theory. Einstein’s theory is perhaps the greatest achievement in the history of science and the most celebrated one. The laws for the electromagnetic force were formulated by James Clark Maxwell in 1873, also a great leap forward in human endeavour. With the advent of quantum mechanics in the first decades of the 20th century it was realized that the electromagnetic field, including light, is quantized and can be seen as a stream of particles, photons. In this picture, the electromagnetic force can be thought of as a bombardment of photons, as when one object is thrown to another to transmit a force. In a similar way the gravitational force is believed to be transmitted by particles called gravitons, but since the gravitational force is some 1040 times weaker than the electromagnetic force, they have not yet been detected. The electromagnetic force holds the atom together since the nucleus and the electrons carry electric charges. The composition of the nucleus was, however, not understood in the early days of the quantum era, but there was a common belief that it consisted of protons and electrons. In 1932 James Chadwick (Nobel Prize, 1935) discovered electrically neutral radiation from the nucleus and could establish that it consisted of a new type of elementary particle called the neutron. Two years later, Eugene Wigner (Nobel Prize, 1963) showed that there must be two distinct nuclear forces at play within the nucleus, a weak force that is responsible for the radioactivity and a strong one that binds the protons and the neutrons together. Both of them act only over a very short range, of the size of the nucleus, hence they have no macroscopic analogue. Only a year after Wigner’s work, the young Japanese student Hideki Yukawa (Nobel Prize, 1949) proposed that the strong nuclear force is mediated by a new particle in analogy with the electromagnetic force. However, the electromagnetic force has a long range while the strong force has a short range. Yukawa realized that, while the electromagnetic force is mediated by massless particles, photons, the strong interactions must be mediated by massive particles. The mass then gives a natural scale for the range of the force. (In quantum physics length is conjugated to mass or energy, in the sense that the fundamental parameters, Planck’s constant η and the velocity of light c, can be used to convert a mass to a length scale.) Knowing the approximate range of the strong force, 10-15 m, 1 Yukawa estimated the mass of the particle, which was to be named the pion, as 100–200 MeV/c2. Only one year after Yukawa´s work a particle in that mass range was discovered in cosmic rays by Carl Anderson (Nobel Prize, 1936), but later it was understood that this particle had too weak an interaction to be the pion. Instead, it was a heavy version of the electron, the muon. The pion was not discovered until after the Second World War, in 1947, by Cecil Powell (Nobel Prize, 1950). Quantum Electrodynamics During the Second World War and the years thereafter, Quantum Electrodynamics (QED) was developed. The key physicists here were Richard Feynman, Julian Schwinger, Sin-itiro Tomonaga (Nobel Prize, 1965) and Freeman Dyson. They showed that a (seemingly) consistent quantum theory could be formulated for the electromagnetic force. In particular they showed that a systematic perturbation expansion could be defined. This means that the amplitude for scattering between electrically charged particles can be written as an expansion in powers of the fine structure constant, αem, which is a measure of the strength of the electric force. When computing these terms, most of them are found to be infinite. There is, however, a unique way to absorb all the infinities. This is done by interpreting them as contributions to the electron mass, its charge and to the norm of its wave function. By letting these parameters be free, it is possible to define finite values to each order in the expansion that can be compared with the experiments. The parameters are said to be renormalized. A quantum field theory in which only a finite number of parameters need be renormalized to define a finite perturbation expansion is called a renormalizable theory. Richard Feynman introduced a very useful diagrammatical formulation that was to be used routinely in all perturbative calculations, the Feynman diagrams. To each particle he introduced a propagator describing the free propagation of the particle. The specific theory then defines the interaction vertices, which are combined with propagators to build a specific diagram. In QED there are two propagators, one for the electron and one for the photon There is one type of vertex. 2 Feynman also introduced the rule that a particle going backwards in time corresponds to an antiparticle going forward in time (hence the arrow on the electron line). The theory also describes the scattering of positrons. For every propagator and vertex there is an analytic expression and there are definite rules for constructing a diagram. The momenta are conserved in a vertex and if there is a loop in a diagram the momentum has to be integrated. This is the root of divergences. With these rules it is possible to construct an infinite series of ever more complicated diagrams for a scattering amplitude. Consider the scattering of one electron off another electron. This is given by the following diagram: These divergences had been known since the early 1930s . It was argued at the time that they were effects from other interactions and that they would disappear when all interactions were taken into account. It was a scientific sensation when it was shown that QED by itself could handle the divergences. The perturbation calculations were also shown to give extremely exact results. One helpful factor is the smallness of the expansion parameter, α, the so-called fine structure constant. The Lamb shift in hydrogen could, for, example be computed to a very high degree of accuracy to find precise agreement with the experiment (Willis Lamb, Nobel Prize, 1955). Relativistic quantum electrodynamics is described by a four-vector potential field in which the time component has a negative norm relative to the space components. In 1929 Hermann Weyl constructed a gauge invariant formulation of the theory by introducing a local symmetry into the theory. This symmetry is the local change of the phase of the electron wave function, which cannot be gauged (measured). It is called abelian since this symmetry group is commutative, abelian. The symmetry leads to redundancy of the time and the longitudinal components of the electromagnetic field, and the physical degrees of freedom are carried only by the transverse components. The key to proving the renormalizability of QED was then to prove that the gauge invariance is still preserved by all the renormalized quantum corrections. John Ward introduced the so-called Ward identities that have to be satisfied for this to be true. This in turn provided a compact way of proving renormalizability. Both gauge invariance and the Ward identities were to be key ingredients when the non-abelian gauge theories were shown to be renormalizable some twenty-five years later. The Yukawa Theory Since QED had paved the way for describing relativistic many-body theories, it was natural to attempt a similar approach for the Yukawa theory. In this case, the symmetry, isospin symmetry, is global, i.e. the symmetry transformations are the same for all space-time points, 3 and there is no need for Ward identities. Again propagators can be introduced for the nucleons, the protons and the neutrons, and for the pions that were known to exist with the electric charge –1, 0 or 1. A natural three-point coupling was also at hand. The Feynman rules: nucleon propagator pion propagator three-point vertex the simple scattering term describe the rough behaviour of the strong force, being the first term in a series expansion. This theory can also be shown to be renormalizable. However, when compared with the experimental results, it was found that the effective coupling strength is larger than 1 (about 14). Such a perturbation theory is meaningless, since every new order is larger than the previous one. The number of diagrams also increases rapidly with every higher order. This means that no mathematical results on asymptotic series can be used. The Proliferation of Elementary Particles The failure of the Yukawa field theory to describe the strong nuclear forces satisfactorily led the physics community to doubt the relevance of relativistic quantum field theories. Perhaps the success of QED was an accident; in order to describe the other forces some alternative formalism would perhaps be needed.