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Springer--Applied asymptotic expansions in momenta and masses 2002
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Introduction The problem of obtaining asymptotic expansions of Feynman integrals in various limits of momenta and masses is a typical mathematical problem in elementary-particle physics. In this book, it is explained how this problem is solved. To characterize, briefly, the main steps of the solution let us consider, for example, the process of e+e− annihilation, where an incoming electron and positron produce, according to the quantum-theoretical description, a virtual photon, e+e− → γ∗, which in turn produces some particles, for ex- ample a quark–antiquark pair, γ∗ → q¯q, which may then be transformed into mesons. The process of quark production is described, in perturbative quantum field theory, by Feynman integrals corresponding to various graphs generated by the Feynman rules. One of the three external legs of such a dia- gram corresponds to a triple vertex associated with the quark vector current, and the other two external legs correspond to the external quarks. If we are interested in the total cross-section for the production of the quarks the prob- lem reduces to the evaluation of the imaginary part of diagrams contributing to the vacuum polarization and containing only two external vertices for the two vector currents. To fulfil our programme, a sketch of basic facts connected with Feynman integrals is first presented in Chap. 2. Chapter 3 then introduces the two basic strategies for expanding Feynman integrals in infinite series in powers and logarithms. Chapters 4 and 5 are devoted to typical Euclidean limits, while Chaps. 6–8 deal with typical pseudo-Euclidean on-shell and threshold limits. The threshold expansion in the case of one heavy (non-zero) mass in the threshold is studied in Chap. 6, the case of two non-zero masses is treated in Chap. 7 and limits of the Sudakov type are investigated in Chap. 8. The structure of each of Chaps. 4–8 is the same: we start with one-loop examples, then formulate prescriptions for a given limit, present two-loop examples and, finally, go up to the operator level and discuss applications to physical problems. We shall (almost) always take scalar Feynman integrals as examples, not only because this choice simplifies our discussion, but also because, in real practical calculations with non-scalar numerators, one usually performs a tensor decomposition and reduces the problem to scalar diagrams. In Chap. 9, I conclude by presenting some alternative approaches, charac- terizing the status of the methods described and conclude with some advice. In Appendix A, one can find a table of basic integrals and useful formulae, in particular the definitions and basic properties of some special functions that are used in the book. Basic notational conventions are presented below. The notation is described in more detail in the List of Symbols. In Appendix B, an analysis of the convergence of Feynman integrals and a proof of the pre- scriptions for the off-shell large-momentum limit are presented. Download link：http://www.isload.com.cn/store/6600u8p6073kk [ Last edited by zhq025 on 2008-1-16 at 10:40 ] |
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