Karlsruhe 2011 – wissenschaftliches Programm

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T: Fachverband Teilchenphysik

T 12: QCD (Theorie) / Quantenfeldtheorie

T 12.7: Vortrag

Freitag, 1. April 2011, 15:35–15:50, 30.23: 10-1

Understanding and proving the expansion by regions — •Bernd Jantzen — Institut für Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University, 52056 Aachen

When loop integrals involve many different scales from masses and kinematical parameters, it can be hard or even impossible to evaluate them exactly. The integrand may be simplified before integration by exploiting hierarchies of parameters and expanding in powers of small parameter ratios. Naive expansions of the integrand usually fail and generate new singularities, but there are sophisticated methods of asymptotic expansions to solve this problem. One of them is the so-called “strategy of regions” or “expansion by regions” developed by M. Beneke and V.A. Smirnov. It considers a set of regions in the integration domain, expands the integrand with respect to the small parameters in each region and integrates all expanded terms over the whole integration domain. The sum of these contributions yields the full, expanded result.

This method has successfully been applied to many complicated loop integrals, but a general proof for its correctness is still missing. This talk will show how the expansion by regions manages to correctly reproduce the exact result in an expanded form and clarify what are the conditions on the choice and completeness of the considered regions. A more general expression for the full result will be presented that involves additional overlap contributions. These extra pieces normally yield scaleless integrals which are consistently set to zero, but they may be needed when special regularization schemes are used.