By Vladimir A. Smirnov

The challenge of comparing Feynman integrals over loop momenta has existed from the early days of perturbative quantum box theory.

Although a very good number of tools for comparing Feynman integrals has been built over a span of greater than fifty years, this booklet is a primary try to summarize them. 'Evaluating Feynman Integrals' characterizes the main robust tools, specifically these used for fresh, rather refined calculations, after which illustrates them with a variety of examples, ranging from extremely simple ones and progressing to nontrivial examples.

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**Extra info for Evaluating Feynman Integrals**

**Example text**

16) 38 3 Evaluating by Alpha and Feynman Parameters Such a reduction of numerators can be performed for any Feynman integral. The corresponding algebraic manipulations can easily be implemented on a computer. 3 One-Loop Examples Let us present examples of evaluation of Feynman diagrams by means of alpha parameters with results which are not written in terms of gamma functions for general d. We ﬁrst turn to the example considered in the introduction. 1. 2) corresponding to Fig. 1. 1 (q 2 , m2 ; a1 , a2 , d) = iπ d/2 (−1)a1 +a2 1 × 0 Γ (a1 + a2 + ε − 2) Γ (a1 )Γ (a2 ) dξ ξ a2 −1 (1 − ξ)1−a2 −ε [m2 − q 2 ξ − i0] a1 +a2 +ε−2 .

1). A very important point here is that all the properties of dimensionally regularized integrals given above, apart from the last one, can be justiﬁed in a purely algebraic way [8, 27], through identities between functions in the alpha representation. Then, using sector decompositions described in Sect. 1, with a control over convergence at hand, one can see that all the resulting massless Feynman integrals with zero external momenta indeed vanish – see details in [8, 27]. Let us now remind ourselves of reality and observe that it is necessary to deal in practice with diagrams on a mass shell or at a threshold.

Two-loop vacuum diagram of Fig. 8 with the masses m, 0, m and general complex powers of the propagators. 1 2 3 Fig. 8. 4 (m2 ; λ1 , λ2 , λ3 , d) dd k dd l . 37) The two basic functions in the alpha representation are U = α1 α2 +α2 α3 + α3 α1 and V = 0. 4 = iπ 2 d/2 ×δ Γ (λ + 2ε − 4) Γ (λl )(m2 )λ+2ε−4 αl − 1 l ∞ 0 ∞ 0 ∞ 0 (α1 α2 + α2 α3 + α3 α1 )ε−2 . 38) Now we exploit the freedom provided by the Cheng–Wu theorem and choose the argument of the delta function as α1 + α3 − 1. The integration over α2 is 44 3 Evaluating by Alpha and Feynman Parameters performed from 0 to ∞.