By Vladimir A. Smirnov
The publication offers asymptotic expansions of Feynman integrals in a variety of limits of momenta and lots more and plenty, and their functions to difficulties of actual curiosity. the matter of enlargement is systematically solved by means of formulating common prescriptions that categorical phrases of the growth utilizing the unique Feynman essential with its integrand extended right into a Taylor sequence in acceptable momenta and much. wisdom of the constitution of the asymptotic enlargement on the diagrammatic point is vital in knowing tips on how to practice expansions on the operator point. commonest examples of those expansions are provided: the operator product enlargement, the large-mass growth, Heavy Quark powerful idea, and Non-Relativistic QCD.
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Extra resources for Applied Asymptotic Expansions in Momenta and Masses (Springer Tracts in Modern Physics)
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. 3) for 0 < x < ε. The above formulae are modiﬁed just a little when there are more than two parameters: we multiply each small parameter by a dimensionless variable x, and deal with the resulting function of x in the limit x → 0. Another ‘experimental’ fact is the existence of a non-zero radius of convergence of the asymptotic expansion of any Feynman integral in any limit.
E. does not have divergent subgraphs, the operation is of the form RFΓ = [1 + ∆(Γ )] FΓ . 84) denotes the Feynman integral that diﬀers from FΓ/γ by insertion of the polynomial Pγ in the external momenta and internal masses of γ into the vertex vγ to which the subgraph γ was reduced. The degree of each Pγ equals the degree of divergence ω(γ). 84) because these quantities are UV divergent. The coeﬃcients of the polynomial Pγ are connected in a straightforward manner with the counterterms of the Lagrangian.
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. 53), we have 1 FΓ (q; d) = −iπ d/2 Γ (1 + ε) 0 [m2 − dξ ξ −ε . 57) In fact, the initial Feynman integral is ﬁnite. 57) and easily obtain an explicit result, FΓ (q) = iπ 2 ln 1 − q 2 /m2 . 58) Suppose we now want to perform evaluation for general d. There could be various reasons for doing this. First, when using the method of integration by parts (see below), one obtains expressions with coeﬃcients involving negative powers of ε.
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