By Vladimir A. Smirnov
The challenge of comparing Feynman integrals over loop momenta has existed from the early days of perturbative quantum box thought. The objective of the publication is to summarize these equipment for comparing Feynman integrals which were built over a span of greater than fifty years. `Feynman vital Calculus' characterizes the main robust tools, specifically these used for fresh, relatively refined calculations, after which illustrates them with quite a few examples, ranging from extremely simple ones and progressing to nontrivial examples This booklet characterizes tools of overview of Feynman integrals in a scientific approach. It concentrates at the equipment which were hired lately for many subtle calculations. Feynman quintessential Calculus explains how the matter of assessment has turn into ever extra vital given that what may be simply evaluated has already been evaluated years in the past. It demonstrates and explains tips on how to practice the most recent very important calculations, whereas exhibiting the best way to decide upon sufficient equipment and mix assessment equipment in a non-trivial approach. the main strong tools of calculation of Feynman integrals are characterised after which illustrated via various examples, ranging from extremely simple ones and culminating in quite nontrivial examples.
This is a textbook model of the former publication (Evaluating Feynman integrals, STMP 211) of the writer. difficulties and strategies were integrated, Appendix G has been further, extra information were awarded, contemporary courses on comparing Feynman integrals were taken under consideration and the bibliography has been updated.
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Additional resources for Feynman Integral Calculus
5 (q 2 ; 1, 1, 1, 1, 1, 4) = (iπ 2 )2 q2 ∞ ∞ dα1 . . 0 dα5 δ( 0 αl − 1) . 46) We exploit the Cheng–Wu theorem by choosing the delta function δ (α5 − 1), with the integration over the rest of the four variables from zero to inﬁnity. 47) where ζ(z) is the Riemann zeta function. 4 This result was ﬁrst obtained in  by means of expansion in Chebyshev polynomials in momentum space. In , it was reproduced using Gegenbauer polynomials in coordinate space. 5 Two-Loop Examples 47 1 2 3 Fig. 13.
Qn ; d) = (−1) ∞ × ∞ dα1 . . 0 iπ d/2 h Γ (a − hd/2) l Γ (al ) αl − 1 dαL δ 0 U a−(h+1)d/2 (−V + U l αlal −1 m2l αl ) a−hd/2 . 37) l∈ν where ν is an arbitrary subset of the lines 1, . . e. for l∈ν, is extended to the integration from zero to inﬁnity. Observe that the integration over αl for l ∈ ν is bounded at least by 1 from above, as in the case where all the α-variables are involved in the sum in the argument of the delta function. One can prove this theorem straightforwardly by changing variables and calculating the corresponding Jacobian.
6): dk [−(k − ξ1 p1 − ξ2 p2 − k)2 ]a4 +a6 G(a3 + a5 , a4 + a6 ) = iπ d/2 . 61) where a = a1 + . . + a6 and A(ξ1 , ξ2 , ξ3 , ξ4 ) = ξ3 ξ4 + (1 − ξ3 − ξ4 )[ξ2 ξ3 (1 − ξ1 ) + ξ1 ξ4 (1 − ξ2 )] . 7 (Q2 ; a1 , . . , a6 , d) = (−1)a iπ d/2 Γ (2 − ε − a35 )Γ (2 − ε − a46 ) Γ (al )Γ (4 − 2ε − a3456 ) (Q2 )a+2ε−4 1 ×Γ (a + 2ε − 4) 1 dξ1 . . 0 dξ4 ξ1a3 −1 (1 − ξ1 )a5 −1 ξ2a4 −1 (1 − ξ2 )a6 −1 0 ×ξ3a1 −1 ξ4a2 −1 (1 − ξ3 − ξ4 )a+3456 +ε−3 A(ξ1 , ξ2 , ξ3 , ξ4 )4−2ε−a . 5 Two-Loop Examples 51 We use the shorthand notation a35 = a3 + a5 , a3456 = a3 + a4 + a5 + a6 .
Feynman Integral Calculus by Vladimir A. Smirnov