Lectures on Multiloop Calculations

Grozin, Andrey

doi:10.1142/s0217751x04016775

Cited by 44 publications

(37 citation statements)

References 42 publications

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“…The remaining one-loop bubble integrals are also straightforward to evaluate; for arbitrary exponents n 1 and n 2 of the two propagators, they are given by [75] …”

Section: (Vmentioning

confidence: 99%

Manifest ultraviolet behavior for the three-loop four-point amplitude ofN=8supergravity

et al. 2008

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Using the method of maximal cuts, we obtain a form of the three-loop four-point scattering amplitude of N = 8 supergravity in which all ultraviolet cancellations are made manifest. The Feynman loop integrals that appear have a graphical representation with only cubic vertices, and numerator factors that are quadratic in the loop momenta, rather than quartic as in the previous form. This quadratic behavior reflects cancellations beyond those required for finiteness, and matches the quadratic behavior of the three-loop four-point scattering amplitude in N = 4 superYang-Mills theory. By direct integration we confirm that no additional cancellations remain in the N = 8 supergravity amplitude, thus demonstrating that the critical dimension in which the first ultraviolet divergence occurs at three loops is D c = 6. We also give the values of the threeloop divergences in D = 7, 9, 11. In addition, we present the explicitly color-dressed three-loop four-point amplitude of N = 4 super-Yang-Mills theory.

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“…The remaining one-loop bubble integrals are also straightforward to evaluate; for arbitrary exponents n 1 and n 2 of the two propagators, they are given by [75] …”

Section: (Vmentioning

confidence: 99%

Manifest ultraviolet behavior for the three-loop four-point amplitude ofN=8supergravity

et al. 2008

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“…This denominator does not correspond to any line, and hence the resulting integral is not a Feynman integral at all; in this case, the discussed relation becomes rather useless (though formally correct). The inversion relations [11] were used, e.g., in [12][13][14]). …”

Section: Figmentioning

confidence: 99%

“…The vector form factors F V 1,2 (13) can be written in the form (27), (28); from (29), (13) we obtain…”

Section: Appendix B: Expansion Of the Hypergeometric Function Fmentioning

confidence: 99%

Heavy-quark form factors in the large $$\beta _0$$ β 0 limit

Grozin

2017

Eur. Phys. J. C

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Heavy-quark form factors are calculated at β 0 α s ∼ 1 to all orders in α s at the first order in 1/β 0 . Using the inversion relation generalized to vertex functions, we reduce the massive on-shell Feynman integral to the HQET one. This HQET vertex integral can be expressed via a 2 F 1 function; the nth term of its ε expansion is explicitly known. We confirm existing results for n L−1 l α L s terms in the form factors (up to L = 3), and we present results for higher L.

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“…Master integrals are those f (k, n) which are independent modulo all the consequences R (see Sect.2) of (20). Thereby, the master integrals are easily determined via the leading terms of the Janet basis.…”

Section: Reduction Of Feynman Integralsmentioning

confidence: 99%

Gröbner bases applied to systems of linear difference equations

Gerdt

2008

Phys. Part. Nuclei Lett.

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In this paper we consider systems of partial (multidimensional) linear difference equations. Specifically, such systems arise in scientific computing under discretization of linear partial differential equations and in computational high energy physics as recurrence relations for multiloop Feynman integrals. The most universal algorithmic tool for investigation of linear difference systems is based on their transformation into an equivalent Gröbner basis form. We present an algorithm for this transformation implemented in Maple. The algorithm and its implementation can be applied to automatic generation of difference schemes for linear partial differential equations and to reduction of Feynman integrals. Some illustrative examples are given.

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Lectures on Multiloop Calculations

Cited by 44 publications

References 42 publications

Manifest ultraviolet behavior for the three-loop four-point amplitude ofN=8supergravity

Manifest ultraviolet behavior for the three-loop four-point amplitude ofN=8supergravity

Heavy-quark form factors in the large $$\beta _0$$ β 0 limit

Gröbner bases applied to systems of linear difference equations

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