Applied Asymptotic Expansions in Momenta and Masses by Vladimir A. Smirnov

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By Vladimir A. Smirnov

The e-book offers asymptotic expansions of Feynman integrals in quite a few limits of momenta and much, and their functions to difficulties of actual curiosity. the matter of enlargement is systematically solved via formulating common prescriptions that specific phrases of the growth utilizing the unique Feynman essential with its integrand increased right into a Taylor sequence in acceptable momenta and lots more and plenty. wisdom of the constitution of the asymptotic growth on the diagrammatic point is essential in figuring out find out how to practice expansions on the operator point. most common examples of those expansions are offered: the operator product enlargement, the large-mass enlargement, Heavy Quark powerful conception, and Non-Relativistic QCD.

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8) which, in the limit ε = 0, gives F (q, m, 0) = ln(q/m) . 7) following two different strategies. As with its four-dimensional prototype, we shall immediately obtain an integral that is divergent for ε = 0 if we begin to expand the integrand into a Taylor series in m: ∞ 0 k −ε−1 dk + ... 7) is finite in ε. The breakdown of this ‘naive’ expansion is explained by the fact that the Taylor series, in m, of the factor 1/(k + m) converges only for |k| < m, rather than for arbitrary values of k. Still, let us continue to be naive and evaluate the whole series that results from this expansion: Flarge ∼ ∞ dk 0 ∞ = k −ε−1 −m k+q ∞ (−1)n mn k dk 0 −n−ε−1 k −ε−2 + ...

The dimensionally regularized Feynman integral is then by definition [61] obtained as the analytic continuation of this function to the point (ε; 1, . . , 1) where the analytic regularization is switched off. And what does one do with the massless detachable subdiagrams? To provide a definition for them, let us observe that we can separate the UV and IR contributions of every propagator by dividing the integration region for each αl into UV and IR regions, 0 ≤ αl ≤ µ−2 and µ−2 ≤ αl < ∞. The αparametric integral is then divided into 2L pieces, which can be characterized by subsets of UV contributions (for example).

The breakdown of this ‘naive’ expansion is explained by the fact that the Taylor series, in m, of the factor 1/(k + m) converges only for |k| < m, rather than for arbitrary values of k. Still, let us continue to be naive and evaluate the whole series that results from this expansion: Flarge ∼ ∞ dk 0 ∞ = k −ε−1 −m k+q ∞ (−1)n mn k dk 0 −n−ε−1 k −ε−2 + ... 11) k+q 0 n=0 ∞ Λ m q m q n n . 12) We attach the label ‘large’ in the sense that the expansion of the integrand is still legitimate at large values of k, of order q.

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