By Vladimir A. Smirnov
The ebook offers asymptotic expansions of Feynman integrals in a number of limits of momenta and lots more and plenty, and their purposes to difficulties of actual curiosity. the matter of growth is systematically solved by means of formulating common prescriptions that categorical phrases of the growth utilizing the unique Feynman essential with its integrand elevated right into a Taylor sequence in applicable momenta and much. wisdom of the constitution of the asymptotic enlargement on the diagrammatic point is vital in realizing the right way to practice expansions on the operator point. most common examples of those expansions are offered: the operator product growth, the large-mass growth, Heavy Quark potent idea, and Non-Relativistic QCD.
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Extra resources for Applied Asymptotic Expansions in Momenta and Masses (Springer Tracts in Modern Physics)
E. FΓ (q; λ1 , λ2 , d) = iπ d/2 Γ (λ1 + λ2 + ε − 2) 1 × 0 dξ ξ λ1 (1 − ξ)λ2 λ1 +λ2 +ε−2 [m21 ξ + m22 (1 − ξ) − q 2 ξ(1 − ξ) − i0] . 53) Suppose that the masses are zero. 54) where G(λ1 , λ2 ) = Γ (λ1 + λ2 + ε − 2)Γ (2 − ε − λ1 )Γ (2 − ε − λ2 ) . 5 How They Are Evaluated Γ (ε)Γ (1 − ε)2 dd k = iπ d/2 . g. 56). Let us present another example of evaluation of Feynman diagrams by means of alpha parameters: consider Fig. 1 with m1 = 0, m2 = m, a1 = 1, a2 = 2. We shall use this example in the next chapter to illustrate prescriptions for asymptotic expansions.
A crucial step – an appropriate modiﬁcation of the integrand before diﬀerentiation, with an application at the two-loop level (to massless propagator diagrams) — was taken in  and, in a coordinate-space approach, in . The case of three-loop massless propagators has been treated in . 5 How They Are Evaluated 39 ¾ ½ ¿ Fig. 5. 66) can be used as a recurrence relation for the given family of integrals because, on the right-hand side, we encounter integrals where the sum a3 + a4 + a5 is less by one than that on the left-hand side.
1) n=n0 j=0 where x = m2 /Q2 and ω is the degree of divergence of the graph Γ . The sum over n runs from some minimal value. The index n can generally take, in some limits, not only integer but also half-integer values. The second index, j, is bounded, for any n, by twice the number of loops. It should be stressed that these expectations are, in general, a matter of experience rather than a corollary of a mathematical theorem. However we shall see that, in some simpler limits, these statements can indeed be justiﬁed.