Scattering Amplitudes from Multivariate Polynomial Division
By Pierpaolo Mastrolia, E. Mirabella, G. Ossola, Tiziano Peraro. 2012
138 citations recorded by the source corpus.
Abstract We show that the evaluation of scattering amplitudes can be formulated as a problem of multivariate polynomial division, with the components of the integration-momenta as indeterminates. We present a recurrence relation which, independently of the number of loops, leads to the multi-particle pole decomposition of the integrands of the scattering amplitudes. The recursive algorithm is based on the weak Nullstellensatz theorem and on the division modulo the Grobner basis associated to all possible multi-particle cuts. We apply it to dimensionally regulated one-loop amplitudes, recovering the well-known integrand-decomposition formula. Finally, we focus on the maximum-cut, defined as a system of on-shell conditions constraining the components of all the integration-momenta. By means of the Finiteness Theorem and of the Shape Lemma, we prove that the residue at the maximum-cut is parametrized by a number of coefficients equal to the number of solutions of the cut itself.
Subfields: Computational Mechanics, Computational Theory and Mathematics, Information Systems
View Paper: https://www.semanticscholar.org/paper/dfe9acf6c1dc0c3cb828b7fbc5bc249434f16e5c
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In Computational Mechanics
In Computational Theory and Mathematics
- Gudrun Heinrich
- Hjalte Frellesvig
- Ansgar Denner
- Stefan Kallweit
- M. Grazzini
- R. Frederix
- Pierpaolo Mastrolia
- L. Tancredi
In Information Systems
Citations in the corpus
- A brief introduction to modern amplitude methods (2013, unknown): “The latter direction has seen important recent progress [61, 62, 63, 64], but there is still a ways to go before two-loop QCD amplitudes for generic 2 → 3 processes will be available.”
- Handbook of LHC Higgs Cross Sections: 3. Higgs Properties (2013, methodology): “For the computation of the virtual corrections we use a code generated b y the program package G OSAM, which combines automated diagram generation and algebraic manip ul tion with integrand-level reduction techniques [133, 300, 301].”
- A two-loop five-gluon helicity amplitude in QCD (2013, methodology): “Furthermore, the integrand reduction method was systematically generalized to all looporders using computational algebraic geometry [51, 52].”
- GoSam-2.0: a tool for automated one-loop calculations within the Standard Model and beyond (2014, methodology): “The principle of an integrand-reduction method, which is valid at any order in perturbation theory [19–23], is the underlying multi-particle pole expansion for the integrand of any scattering amplitude, or, equivalently, a representation where the numerator of each Feynman integral is expressed as a”
- Lectures on differential equations for Feynman integrals (2014, background): “[77–80] and references therein, to our surprise [13] does not seem to be widely known in the QCD community, and we hope that these lecture notes help to disseminate these useful ideas.”
- Collider physics at the precision frontier (2020, methodology): “It employs a Gröbner basis technique [497, 438] combined with a module intersec-”
- On the maximal cut of Feynman integrals and the solution of their differential equations (2016, background): “it can be trivially expressed as a linear combination of diagrams with a reduced number of propagators [30].”