Applied Asymptotic Expansions in Momenta and Masses

1. Front Matter

   Pages I-IX

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2. Introduction

   Pages 1-15

3. Feynman Integrals: a Brief Review

   Pages 17-50

4. Why, Where and How to Expand

   Pages 51-65

5. Off-Shell Large-Momentum Expansion

   Pages 67-94

6. Large-Mass Expansion

   Pages 95-114

7. Threshold Expansion. One Heavy Mass in the Threshold

   Pages 115-133

8. Threshold Expansion. Two Masses in the Threshold

   Pages 135-164

9. Sudakov Limits

   Pages 165-208

10. Conclusion

    Pages 209-224

11. Tables

    Pages 225-233

12. Technical Details: Convergence and Asymptotic Behaviour

    Pages 235-251

13. Back Matter

    Pages 253-264

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'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.

• Feynman diagrams
• Quark
• asymptotic expansions
• behavior
• momentum
• quantenchromodynamik
• quantum chromodynamics
• structure
• threshold expansion

• Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow, Russia

  Vladimir A. Smirnov

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