By Vladimir A. Smirnov
The target of this ebook is to explain the main strong equipment for comparing multiloop Feynman integrals which are presently utilized in practice. This publication supersedes the author’s past Springer ebook “Evaluating Feynman Integrals” and its textbook model “Feynman critical Calculus.” because the ebook of those books, robust new tools have arisen and traditional tools were more desirable on in crucial methods. one more qualitative swap is the truth that many of the tools and the corresponding algorithms have now been applied in desktop codes that are usually public.
In comparability to the 2 earlier books, 3 new chapters were added: One is on area decomposition, whereas the second one describes a brand new procedure by way of Lee. The 3rd new bankruptcy issues the asymptotic expansions of Feynman integrals in momenta and lots more and plenty, which have been defined intimately in one other Springer publication, “Applied Asymptotic Expansions in Momenta and Masses,” by means of the writer. This bankruptcy describes, at the foundation of papers that seemed after the book of stated ebook, easy methods to algorithmically notice the areas appropriate to a given restrict in the technique of growth by way of areas. furthermore, the chapters at the approach to Mellin-Barnes illustration and at the approach to integration by way of elements were considerably rewritten, with an emphasis at the corresponding algorithms and laptop codes.
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Extra info for Analytic Tools for Feynman Integrals (Springer Tracts in Modern Physics)
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30) The singularity at s − (s + t)ξ = 0 is absent because the rest of the integrand is zero at this point. To calculate this integral in expansion in ε one needs, however, to separate the two terms in the square brackets. 3 One-Loop Examples 45 to the denominator one can perform an auxiliary subtraction at s − (s + t)ξ = 0. 31) where 1 f (s, t; ε) = (−t)−1−ε 0 dξ ξ −1−ε − s − (s + t)ξ s s+t −1−ε . 32) To expand the function f in a Laurent series in ε one needs to perform another subtraction, at ξ = 0, which we make by the replacement (s + t)ξ 1 1 → + .
Let us consider, for example, the massless tadpole diagram, which can be reduced by means of alpha parameters to a scaleless one-dimensional integral: dd k = −iε π d/2 k2 ∞ dα α ε−2 . 42) are convergent in different domains of the regularization parameter ε, namely, Re ε > −1 and Re ε < −1, with no intersection. A massless Feynman integral with a zero external momentum can appear either in the beginning when using Feynman rules, or after some manipulations: after using partial fractions, integration by parts, etc.