By Robert T Beyer

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The matter 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 and summarize them. comparing Feynman Integrals characterizes the main strong tools, particularly these used for contemporary, really refined calculations, after which illustrates them with a number of examples, ranging from extremely simple ones and progressing to nontrivial examples.

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The three-loop RG (3-functions of the general NLSM with non-vanishing torsion are very complicated, while they depend upon the renormalization 40 2. Classical Structure and Renormalization prescription used in actual calculations [92]. The generic structure of the three-loop contribution is /3~;) '" RR2 + (\7 R)2 + R 2\7 H + R(\7 R)H + (\72 R) (\7 H) + RRH2 + R(\7 H)2 + (\7 R)(\7 H)H + RH(\72 H) + (\72 R)H2 + RH2 (\7 H) + (\7 R)H3 + (\7 H)3 + (\72 H) (\7 H)H + (\73 H)H2 + RH4 + (\7 H)2 H2 + H4 (\7 H) .

For simplicity, we only consider the case G = SU(2) with the structure constants be = c abe , a, b, ... = 1,2,3. 19) with respect to A~ yields the algebraic equations of motion for the vector fields, r 20/l-B ac/l- A + (1] A/l-8 ab + CA/l- Becabe)A~ = 0 . 21), in front of At. 26) of the NLSM with torsion, whose metric and the torsion potential are given by 13 h (B) = -cabeBe ab 1 + B2 . 19) with respect to Ba first, we arrive at the constraint 13 The (up and down) positions of the flat SU(2) indices ofthe B-field are irrelevant, while it matters for the 'world' indices of the NLSM metric, torsion and curvature tensors.

78) where, as regards the 19th and 20th terms, one may take any two linearly independent combinations of H abc H apqD p Hbms D q H cms , ct H abc H amsD k Hbmt DkHs ' H abc H pqtDaHbckD t H pqk, H abc H pqtDaHpqk D t Hbck . 78) agrees with the three-loop RG ,B-function [92] in the RVB-prescription, if the coefficients {ai} are given by [104, 105, 106] = -2, al a2 = -8/3, a3 = -2/3, a4 = 3/2, a5 = 9/2, a6 = 5/3 , = 9/8, as = 7/4, ag = -4/3, alO = 2/9, all = -3, al2 = 2 , al3 = 880/81, al4 = -281/27, al5 = 1045/81, al6 = -2626/27 , al7 = -2383/27, alS = a21 = 3/2, al9 = a20 = 0 .

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