![]() Perturbative QCD has been applied to many physical processes with the help of operator product expansion (OPE) and renormalization group equations, such as electron–nucleon scattering, e + e − annihilation, heavy quarkonia ( J/ψ’s and Ʊ’s) decays, photon–photon scattering as a subprocess of e + e − → e + e − X ( X represents unobserved hadrons), production of jets from quarks and gluons, the Drell–Yan process NN → l + l − X, inclusive e + e − annihilation e + e − → hadrons + X, large- p T hadron reactions, and multijets in e + e − annihilations. This approach in QCD is usually referred to as perturbative QCD. The QCD has the property of the asymptotic freedom at short distances, while it has the possibility of quark confinement at long distances.Īccording to the property of asymptotic freedom of QCD, one may safely use perturbation theory to discuss short-distance processes. When one considers massless gluons, serious infrared divergences exist in QCD and may be the origin of quark confinement. This property of gluons is an essential ingredient for having asymptotic freedom. Gluons carry color charges and hence interact with each other even in the absence of quarks. The non-Abelian gauge field in QCD mediates color interactions between quarks. In 1970s the idea of the extra quantum number, color, for quarks was established and the theory of quark dynamics was found to be a non-Abelian gauge theory with SU(3) symmetry that was named quantum chromodynamics (QCD). These facts strongly support the idea that quarks should have an extra quantum number, color. (5) is much smaller than experimental data. If there is no color degree of freedom, the prediction of Eq. (5) with the data strongly support N c = 3. Where s is the center-of-mass total energy squared of the e + e − system, Q i is the charge of the ith quark, and N f is the number of flavors of quarks which may contribute to the process. (5) σ ( e + e − → hadrons ) = 4 π α 2 3 s N c ∑ i = 1 N f Q i 2 , It contains one spinor conjugacy class in addition to the adjoint conjugacy class. ![]() It turns out that the particular Lie group that is appropriate is Spin (32)/ Z 2. There are several different Lie groups that have the same Lie algebra. The term SO (32) is used here somewhat imprecisely. For any other choice there are fatal anomalies. Now we see that at the quantum level, the only choice that is consistent is SO (32). (This result was obtained by Green and Schwarz, 1984a.)Īs we mentioned earlier, at the classical level one can define type I superstring theory for any orthogonal or symplectic gauge group. For these two choices, all the anomalies cancel. This theory has both gauge and gravitational anomalies for every choice of Yang–Mills gauge group except SO (32) and E 8 × E 8. Type I supergravity coupled to super Yang–Mills. (This result was obtained by Alvarez–Gaumé and Witten, 1983.) However, when their contributions are combined, the anomalies all cancel. ![]() This theory has three chiral fields each of which contributes to several kinds of gravitational anomalies. This theory is nonchiral, and therefore it is trivially anomaly-free. This theory has anomalies for every choice of gauge group. ![]() There are several possible cases in ten dimensions: The anomalies can be attributed to the massless fields, and therefore they can be analyzed in the low-energy effective field theory. In the case of ten-dimensional chiral gauge theories, the potentially anomalous Feynman diagrams are hexagons, with six external gauge fields. ![]()
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