By Vladimir A. Smirnov
The booklet offers asymptotic expansions of Feynman integrals in numerous limits of momenta and lots more and plenty, and their purposes to difficulties of actual curiosity. the matter of enlargement is systematically solved by means of formulating common prescriptions that specific phrases of the growth utilizing the unique Feynman imperative with its integrand multiplied right into a Taylor sequence in acceptable momenta and lots more and plenty. 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. commonest examples of those expansions are provided: the operator product growth, the large-mass enlargement, Heavy Quark powerful conception, and Non-Relativistic QCD.
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Additional resources for Applied asymptotic expansions in momenta and masses
Usually, one chooses a basis of composite operators Oi generated by some polynomials oi . 97) k where Zik is a renormalization matrix. One says that the operators are mixed by the renormalization. An example of mixing of operators (in QCD) can be found in Sect. 5. In diagrams that contribute to bare products of two renormalized composite operators [T Oi (x)Ok (0)]B , all the UV divergences present in purely Smatrix 1PI (sub)diagrams and diagrams involving only one vertex corresponding to a composite operator are removed.
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. 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.
Z) dependence (let us call them IR poles) are to the left of the contour and the poles with a Γ (. . − z) dependence (UV poles) are to the right of it. Such factors can be chosen as functions in parametric integrals. But the simplest way to apply this representation is to write down a massive propagator in terms of massless ones: 1 1 1 = 2 2 λ (m − k ) Γ (λ) 2πi +i∞ dz −i∞ (m2 )z Γ (λ + z)Γ (−z) . 75) We consider once again Fig. 54) and obtain the following result: Γ (1 + ε) FΓ (q; d) = −iπ d/2 (−q 2 )1+ε × 1 2πi +i∞ dz −i∞ m2 −q 2 z Γ (1 + ε + z)Γ (−ε − z)Γ (−z) .
Applied asymptotic expansions in momenta and masses by Vladimir A. Smirnov