Applied Asymptotic Expansions in Momenta and Masses (Springer Tracts in Modern Physics, 177)

The Book Presents Asymptotic Expansions Of Feynman Integrals In Various Limits Of Momenta And Masses, And Their Applications To Problems Of Physical Interest. The Problem Of Expansion Is Systematically Solved By Formulating Universal Prescriptions That Express Terms Of The Expansion Using The Original Feynman Integral With Its Integrand Expanded Into A Taylor Series In Appropriate Momenta And Masses. Knowledge Of The Structure Of The Asymptotic Expansion At The Diagrammatic Level Is Key In Understanding How To Perform Expansions At The Operator Level. Most Typical Examples Of These Expansions Are Presented: The Operator Product Expansion, The Large-mass Expansion, Heavy Quark Effective Theory, And Non-relativistic Qcd. Introduction -- Off-shell Expansions -- On-shell Expansions -- Expansions Near Threshold -- Conclusion. Vladimir Smirnov. Includes Bibliographical References And Index. 'The sturgeon they sent was second grade fresh,'said the barman.'Really, what nonsense/''Why nonsense?'''Second grade fresh'that's what I call nonsense/ There's only one degree of freshness the first, and it's the last) (M. A. Bulgakov, The Master and Margarita) The goal of this book is to describe in detail how Feynman integrals can be expanded in suitable parameters, when various momenta or masses are small or large. In a narrow sense, this problem is connected with practical calcula tions. In a situation where a given Feynman integral depends on parameters of very different scales, a natural idea is to replace it by a sufficiently large number of terms of an expansion of it in ratios of small and large scales. It will be explained how this problem of expansion can be systematically solved, by formulating universal prescriptions that express terms of the expansion by using the original Feynman integral with its integrand expanded into a Taylor series in appropriate momenta and masses. It turns out that knowledge of the structure of the asymptotic expansion at the diagrammatic level is a key point in understanding how to perform expansions at the operator level. There are various examples of these ex pansions: the operator product expansion, the large mass expansion, Heavy Quark Effective Theory, Non Relativistic QCD, etc. Each of them serves as a realization of the factorization of contributions of different scales. Annotation The book presents asymptotic expansions of Feynman integrals in various limits of momenta and masses, and their applications to problems of physical interest. The problem of expansion is systematically solved by formulating universal prescriptions that express terms of the expansion using the original Feynman integral with its integrand expanded into a Taylor series in appropriate momenta and masses. Knowledge of the structure of the asymptotic expansion at the diagrammatic level is key in understanding how to perform expansions at the operator level. Most typical examples of these expansions are presented: the operator product expansion, the large-mass expansion, Heavy Quark Effective Theory, and Non-Relativistic QCD 1Introduction......Page 1 Index......Page 0 2Feynman Integrals a Brief Review......Page 16 3Why, Where and How to Expand......Page 50 4Off-Shell Large-Momentum Expansion......Page 65 5Large-Mass Expansion......Page 93 6Threshold Expansion. One Heavy Mass in the Threshold......Page 113 7Threshold Expansion. Two Masses in the Threshold......Page 132 8Sudakov Limits......Page 162 9Conclusion......Page 206 10Tables......Page 222 11Technical Details Convergence and Asymptotic Behaviour......Page 231 12References......Page 248 13List of Symbols......Page 256 14Index......Page 258