Tutorial for Multiloop Calculations: Master Integrals & Automated Solutions
Learn about multiloop calculations in theoretical physics with a tutorial by Marvin Schnubel. Discover the step-by-step process of reducing to master integrals, creating diagrams, IBP reduction, and solving integrals with automated tools. Find out how to generate expressions and handle complex computations efficiently.
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Presentation Transcript
BNL Lunch Talk 06.12.2024 Tutorial for multiloop Tutorial for multiloop calculations calculations Marvin Schnubel
Motivation Motivation Multiloop calculations are necessary for precise predictions for many observables M Schnubel - Tutorial for Multiloop Calculations Expressions beyond 1 loop are often too long and/or complicated to be computed by hand and no automated procedure exists till now (compare 1 loop computation by Package X etc.) Many tools exist to help with various steps within the process Idea is here to give a rough overview together with exemplary Mathematica notebook 2 BNL Lunch Talk - 06.12.2024
Step by step guideline Step by step guideline Reduce to Master Integrals Get Solve Master Integrals Create diagrams Done! expressions M Schnubel - Tutorial for Multiloop Calculations Could be difficult, in practice easy automated Varying difficulty, Not automated Easy, Easy, automated automated 3 BNL Lunch Talk - 06.12.2024
1) Create diagrams & 2) Get expressions 1) Create diagrams & 2) Get expressions Generation of diagrams is in principle a combinatorial task with set rules, automated in various programs (FeynArts, Qgraf, etc.) M Schnubel - Tutorial for Multiloop Calculations Getting the expressions from the diagrams follows simple Feynman rules can be automated straight forwardly e.g. with FeynCalc Reduction programs can only solve for scalar integrals project out correct Dirac and Lorentz structure 4 BNL Lunch Talk - 06.12.2024
3) IBP Reduction 3) IBP Reduction Integration by Parts relation: M Schnubel - Tutorial for Multiloop Calculations v: combination of propagators and ext. momenta With cut propagators: inverse unitarity Large and sparse system, can be reduced to a set of Master Integrals Can be solved by algorithms (AIR, FIRE, Kira, Reduze, LiteRed, Blade ) 5 BNL Lunch Talk - 06.12.2024
4) Solve Master Integrals 4) Solve Master Integrals Numerically automated (DiffExp, AMFlow ) M Schnubel - Tutorial for Multiloop Calculations Often analytically possible! If computation with Feynman parameters doesn t work, can try using differential equations E.g. This can be again reduced to MIs system of coupled DEQs 6 BNL Lunch Talk - 06.12.2024
4) Solve Master Integrals 4) Solve Master Integrals If solving MIs fully is too complicated, often the - expansion is sufficient M Schnubel - Tutorial for Multiloop Calculations Especially straight-forward if DEQ is in canonical form (CANONICA, Fuchsia ) Solution is then given by Chen iterated integrals 7 BNL Lunch Talk - 06.12.2024
4) Solve Master Integrals 4) Solve Master Integrals Solution can be expressed in easy functions (Goncharov polylogs) if M Schnubel - Tutorial for Multiloop Calculations Finding correct boundary conditions can be further challenging, for massive integrals find often the result for the massless case in the literature. 8 BNL Lunch Talk - 06.12.2024
Example Example Gluon-gluon to Higgs fusion via light quarks, hard matching coefficient, leading mbbehavior M Schnubel - Tutorial for Multiloop Calculations 9 BNL Lunch Talk - 06.12.2024
