By Vladimir A. Smirnov
The booklet offers asymptotic expansions of Feynman integrals in numerous limits of momenta and much, and their purposes to difficulties of actual curiosity. the matter of enlargement is systematically solved through formulating common prescriptions that specific phrases of the growth utilizing the unique Feynman essential with its integrand increased right into a Taylor sequence in applicable momenta and much. wisdom of the constitution of the asymptotic growth on the diagrammatic point is essential in realizing tips on how to practice expansions on the operator point. commonest examples of those expansions are provided: the operator product growth, the large-mass growth, Heavy Quark powerful concept, and Non-Relativistic QCD.
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Extra resources for Applied Asymptotic Expansions in Momenta and Masses (Springer Tracts in Modern Physics)
KNk + . . )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.
98) ik where Zik,i k is a renormalization matrix of products of the composite operators. Although the monomials of the ﬁelds present in the Lagrangian and the composite operators were traditionally5 supposed, in the early days of quan5 Even the very technique of deﬁnition of the composite operators and analysing their properties was called the ‘Normal Product Algorithm’ . So, now, ‘Abnormal Product Algorithm’ is in use. 50 2 Feynman Integrals: a Brief Review tum ﬁeld theory, to be normal-ordered, it is very important to consider them as operators without normal order.
One-loop self-energy graph. The dashed line denotes a massless propagator beginning, we have chosen the second power of the lower propagator, which is indicated by a dot on the line. The corresponding dimensionally regularized Feynman integral takes the form FΓ (q 2 , m2 ; d) = (k 2 dd k . 5) Regularization has been consciously introduced for a ﬁnite initial quantity because we shall obtain various divergences in individual terms of our result for the expansion. This appearance of divergences is characteristic (well, almost) of all expansions in momenta and masses and is a negligible price paid in order to obtain explicit formulae for individual terms that are as simple as possible in the general order of the expansion.
Applied Asymptotic Expansions in Momenta and Masses (Springer Tracts in Modern Physics) by Vladimir A. Smirnov