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 an outstanding number of tools for comparing Feynman integrals has been built over a span of greater than fifty years, this e-book is a primary try and summarize them. Evaluating Feynman Integrals characterizes the main strong tools, specifically these used for contemporary, relatively subtle calculations, after which illustrates them with a number of examples, ranging from extremely simple ones and progressing to nontrivial examples.
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G. V. Tkachov, Nucl. Phys. B 192 (1981) 159. C. Collins, Renormalization (Cambridge University Press, Cambridge, 1984). I. A. Smirnov, Nucl. Phys. B 554 (1999) 391. 17 A. ), Higher Transcendental Functions, Vols. 1 and 2 (McGraw-Hill, New York, 1954). M. E. Shilov, Generalized Functions, Vol. 1 (Academic Press, New York, London, 1964). G. Grozin, Heavy Quark Effective Theory (Springer, Berlin, Heidelberg, 2004). 14 K. Hepp, Commun. Math. Phys. 2 (1966) 301. G. ’t Hooft and M. Veltman, Nucl. Phys.
2, with m1 = 0, m2 = m, a1 = 1, a2 = 2. 5). There is a logarithmic singularity at threshold, q 2 = m2 , so that we cannot strictly speak about the value of the integral there. Still we can certainly define the threshold Feynman integral by putting q 2 = m2 in the integrand of the integral over the loop momentum or over the alpha parameters. And this is what was really meant and will be meant by ‘on-shell’ and ‘threshold’ integrals. 13) (to be derived in Chap. 3): k 2 (k 2 Γ (ε) dd k = iπ d/2 . 40) This integral is divergent, in contrast to the original Feynman integral defined for general q 2 .
M. E. Shilov, Generalized Functions, Vol. 1 (Academic Press, New York, London, 1964). G. Grozin, Heavy Quark Effective Theory (Springer, Berlin, Heidelberg, 2004). 14 K. Hepp, Commun. Math. Phys. 2 (1966) 301. G. ’t Hooft and M. Veltman, Nucl. Phys. B 44 (1972) 189. 23 G. Leibbrandt, Rev. Mod. Phys. 47 (1975) 849. P. , Phys. Rev. D 46 (1992) 4052. V. B. Wise, Heavy Quark Physics (Cambridge University Press, Cambridge 2000). 14 N. Nakanishi, Graph Theory and Feynman Integrals (Gordon and Breach, New York, 1971).