By Vladimir A. Smirnov
The e-book provides asymptotic expansions of Feynman integrals in numerous limits of momenta and lots more and plenty, and their functions to difficulties of actual curiosity. the matter of growth is systematically solved via formulating common prescriptions that categorical phrases of the growth utilizing the unique Feynman essential with its integrand increased right into a Taylor sequence in acceptable momenta and much. wisdom of the constitution of the asymptotic enlargement on the diagrammatic point is essential in realizing find out how to practice expansions on the operator point. commonest examples of those expansions are awarded: the operator product enlargement, the large-mass enlargement, Heavy Quark powerful concept, and Non-Relativistic QCD.
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Additional info for Applied Asymptotic Expansions in Momenta and Masses
This procedure of introducing dimensional regularization is easily generalized [28, 35, 61] to an arbitrary Feynman integral. 37) to the case of an arbitrary number of loop integration momenta: FΓ (q; d) = iπ d/2 Γ (ε) 0 dd k1 . . dd kh exp i qi ·ki Aij ki ·kj + 2 i,j i = eiπh(1−d/2)/2 π hd/2 (det A)−d/2 exp −i . 1 The elements of the inverse matrix involved here are rewritten in graphtheoretical language (see details in [181, 27]), and the resulting alpha representation takes the form [28, 35] FΓ (q1 , .
H . 50) l=1 This property is the basis of a powerful method of evaluation of Feynman integrals – see the next section. The next property says that • any diagram with a detachable massless subgraph is zero. Setting massless detachable subdiagrams to zero turns out to be a speciﬁc case of a more general prescription (which we shall discuss in the next chapter and use throughout the book). This property can also be shown to be a consequence of the above definition [61, 208], by use of an auxiliary analytic regularization using pieces of the α-integral considered in diﬀerent domains of the regularization parameters.
Nk ] distinct ways. 94). 95) is generated by a polynomial j(x) composed of the ﬁelds present in the theory. As will be explained shortly (and, in detail, in Chap. 4), it is very convenient to include powers of the masses of the theory in the composite operators. The canonical dimension of the operator is deﬁned in the standard way, in mass units, as Nb + Nf + Nm , where Nb Nf and Nm are the number of boson ﬁelds, the number of fermion ﬁelds and the total power of the masses involved in the product j, respectively.