The "expansion by regions" is a method of asymptotic expansion developed by Beneke and Smirnov in 1997. It expands the integrand according to the scaling prescriptions of a set of regions and integrates all expanded terms over the whole integration domain. This method has been applied successfully to many complicated loop integrals, but a general proof for its correctness has still been missing. This paper shows how the expansion by regions manages to reproduce the exact result correctly in an expanded form and clarifies the conditions on the choice and completeness of the considered regions. A generalized expression for the full result is presented that involves additional overlap contributions. These extra pieces normally yield scaleless integrals which are consistently set to zero, but they may be needed depending on the choice of the regularization scheme. While the main proofs and formulae are presented in a general and concise form, a large portion of the paper is filled with simple, pedagogical one-loop examples which illustrate the peculiarities of the expansion by regions, explain its application and show how to evaluate contributions within this method.Comment: 84 pages; v2: comment on scaleless integrals added to conclusions, version published in JHE

When performing asymptotic expansions using the strategy of expansion by regions, it is a non-trivial task to find the relevant regions. The recently published Mathematica code asy.m automates this task, but it has not been able to detect potential regions in threshold expansions or Glauber regions. In this work we present an algorithm and its implementation in the update asy2.m which also reveals potential and Glauber regions automatically.

We analyse subleading electroweak effects in the top anti-top resonance production region in e + e − collisions which arise due to the decay of the top and anti-top quarks into the W + W − bb final state. These are NLO corrections adopting the nonrelativistic power counting v ∼ α s ∼ √ α EW . In contrast to the QCD corrections which have been calculated (almost) up to NNNLO, the parametrically larger NLO electroweak contributions have not been completely known so far, but are mandatory for the required accuracy at a future linear collider. The missing parts of these NLO contributions arise from matching coefficients of non-resonant productiondecay operators in unstable-particle effective theory which correspond to off-shell top production and decay and other non-resonant irreducible background processes to tt production. We consider the total cross section of the e + e − → W + W − bb process and additionally implement cuts on the invariant masses of the W + b and W −b pairs.

We consider two-loop leading and next-to-leading logarithmic virtual corrections to arbitrary processes with external massless fermions in the electroweak Standard Model at energies well above the electroweak scale. Using the sector-decomposition method and alternatively the strategy of regions we calculate the mass singularities that arise as logarithms of Q 2 /M 2 W , where Q is the energy scale of the considered process, and 1/ε poles in D = 4−2ε dimensions, to one-and two-loop next-to-leading logarithmic accuracy. The derivations are performed within the complete electroweak theory with spontaneous symmetry breaking. Our results indicate a close analogy between the form of two-loop electroweak logarithmic corrections and the singular structure of scattering amplitudes in massless QCD. We find agreement with the resummation prescriptions that have been proposed in the literature based on a symmetric SU(2) × U(1) theory matched with QED at the electroweak scale and provide new next-to-leading contributions proportional to ln(M 2 Z /M 2 W ).August 2006 † Ansgar.Denner@psi.ch ‡ physics@bernd-jantzen.de §

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