Atomic Nuclear Physics

Download Applied asymptotic expansions in momenta and masses by Vladimir A. Smirnov PDF

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 by way of formulating common prescriptions that categorical phrases of the growth utilizing the unique Feynman necessary with its integrand elevated 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 find out how to practice expansions on the operator point. commonest examples of those expansions are offered: the operator product growth, the large-mass enlargement, Heavy Quark potent conception, and Non-Relativistic QCD.

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After expanding the gamma functions as Laurent series in ε, we arrive at the well-known result [198, 53] J(1, 1, 1, 1, 1) = 1 1 G(1, 1) [G(2, 1) − G(2, 1 + ε)] ε (−q 2 )1+2ε 2 6ζ(3) = iπ 2 + O(ε) . 68) In this simple example, it was sufficient to use only one IBP relation, which, in fact, follows from an IBP identity for the triangle massless diagram of Fig. 6 with general indices. This ‘triangle’ rule takes the form 1= 1 a1 1+ 3− − p21 + a2 2+ 3− − p22 d − a1 − a2 − 2a3 . 69) When applied to the left triangle in Fig.

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. But one needs additional prescriptions for removing UV divergences from diagrams that involve both of the vertices connected with the composite operators. 98) ik where Zik,i k is a renormalization matrix of products of the composite operators.

By definition, the empty forest is also implied in the sum, with the corresponding term equal to unity. For example, for the one-loop graph Γ of Fig. 1, there are two forests, ∅ and Γ , so that the R-operation takes the form R = 1 − M . For the two-loop scalar graph shown in Fig. 8, we have the following forests: {∅}, {γ}, {Γ } and {γ, Γ }, where γ is the one-loop subgraph. The corresponding R-operation takes the form R = 1 − Mγ − MΓ + MΓ Mγ ≡ (1 − MΓ ) (1 − Mγ ) . 88) Moreover, for the two-loop QED graph shown in Fig.

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