A toolbox to solve coupled systems of differential and difference equations
| Autor: | Abilio De Freitas, Jakob Ablinger, Carsten Schneider, Johannes Bluemlein |
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| Rok vydání: | 2016 |
| Předmět: |
FOS: Computer and information sciences
Computer Science - Symbolic Computation High Energy Physics - Theory Power series Pure mathematics Holonomic Laurent series FOS: Physical sciences Order (ring theory) Harmonic (mathematics) Mathematical Physics (math-ph) Symbolic Computation (cs.SC) 16. Peace & justice Hypergeometric distribution High Energy Physics - Phenomenology High Energy Physics - Phenomenology (hep-ph) High Energy Physics - Theory (hep-th) Linear differential equation Closure (mathematics) ddc:530 Mathematical Physics Mathematics |
| Zdroj: | Proceedings of Science (2016). doi:10.22323/1.235.0060 Scopus-Elsevier Proceedings of 12th International Symposium on Radiative Corrections (Radcor 2015) and LoopFest XIV (Radiative Corrections for the LHC and Future Colliders) — PoS(RADCOR2015)-Sissa Medialab Trieste, Italy, 2016.-ISBN-doi:10.22323/1.235.0060 Proceedings of 12th International Symposium on Radiative Corrections (Radcor 2015) and LoopFest XIV (Radiative Corrections for the LHC and Future Colliders) — PoS(RADCOR2015)-Sissa Medialab Trieste, Italy, 2016.-ISBN-doi:10.22323/1.235.006012th International Symposium on Radiative Corrections and LoopFest XIV (Radiative Corrections for the LHC and Future Colliders), Radcor 2015, Los Angeles, USA, 2015-06-15-2015-06-19 |
| DOI: | 10.22323/1.235.0060 |
| Popis: | We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases. Here we assume that the desired solution has a power series representation and we seek for the coefficients in closed form. In particular, if the coefficients depend on a small parameter $\ep$ (the dimensional parameter), we assume that the coefficients themselves can be expanded in formal Laurent series w.r.t.\ $\ep$ and we try to compute the first terms in closed form. More precisely, we have a decision algorithm which solves the following problem: if the terms can be represented by an indefinite nested hypergeometric sum expression (covering as special cases the harmonic sums, cyclotomic sums, generalized harmonic sums or nested binomial sums), then we can calculate them. If the algorithm fails, we obtain a proof that the terms cannot be represented by the class of indefinite nested hypergeometric sum expressions. Internally, this problem is reduced by holonomic closure properties to solving a coupled system of linear difference equations. The underlying method in this setting relies on decoupling algorithms, difference ring algorithms and recurrence solving. We demonstrate by a concrete example how this algorithm can be applied with the new Mathematica package \texttt{SolveCoupledSystem} which is based on the packages \texttt{Sigma}, \texttt{HarmonicSums} and \texttt{OreSys}. In all applications the representation in $x$-space is obtained as an iterated integral representation over general alphabets, generalizing Poincar\'{e} iterated integrals. |
| Databáze: | OpenAIRE |
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