Mellin–Barnes (MB) representations have become a widely used tool for the evaluation of Feynman loop integrals appearing in perturbative calculations of quantum field theory. Some of the MB integrals may be solved analytically in closed form with the help of the two Barnes lemmas which have been known in mathematics already for one century. The original proofs of these lemmas solve the integrals by taking infinite series of residues and summing these up via hypergeometric functions. This paper presents new, elegant proofs for the Barnes lemmas which only rely on the well-known basic identity of MB representations, avoiding any series summations. They are particularly useful for presenting and proving the Barnes lemmas to students of quantum field theory without requiring knowledge on hypergeometric functions. The paper also introduces and proves an additional lemma for a MB integral ∫dz involving a phase factor exp (±iπz).

We analyse the subleading non-resonant contributions to the e+ e- -> W+ W- b bbar cross section at energies near the top-antitop threshold. These correspond to next-to-next-to-leading-order (NNLO) corrections with respect to the leading-order resonant result. We show that these corrections produce 1/epsilon endpoint singularities which precisely cancel the finite-width divergences arising in the resonant production of the W+ W- b bbar final state from on-shell decays of the top and antitop quarks at the same order. We also provide analytic results for the (m_t/Lambda)^2, (m_t/Lambda) and (m_t/Lambda)^0 log(Lambda) terms that dominate the expansion in powers of (Lambda/m_t) of the complete set of NNLO non-resonant corrections, where Lambda is a cut imposed on the invariant masses of the b W pairs that is neither too tight nor too loose (m_t Gamma_t << Lambda^2 << m_t^2).

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.

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.

We analyze 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 non-relativistic power counting v∼αs∼αEW. 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 production-decay 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.

The production of W+ W- b bbar from e+ e- collisions at energies close to the t tbar threshold is dominated by the resonant process with a nearly on-shell t tbar intermediate state. The W b pairs in the final state can also be reached through the decay of off-shell tops or through background processes containing no or only single top quarks. This non-resonant production starts to contribute at NLO to the W+ W- b bbar total cross section in the non-relativistic power-counting v ~ alpha_s ~ sqrt(alpha_EW). The NLO non-resonant corrections presented in this talk represent the non-trivial NLO electroweak corrections to the e+ e- -> W+ W- b bbar cross section in the top anti-top resonance region. 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. We consider the total cross section of the e+ e- -> W+ W- b bbar process and additionally implement cuts on the invariant masses of the W+ b and W- bbar pairs.

We study the asymptotic behaviour of two-loop electroweak corrections at energies Q >> M_W, where logarithms of the type ln(Q/M_W) become dominant. The calculation of the leading and next-to-leading logarithmic terms for massless and massive fermion-scattering processes is summarized. The derivations are performed diagrammatically within the spontaneously broken electroweak theory. We find that the soft and collinear singularities resulting from photons can be factorized into a QED-like term and that, up to logarithms of the Z-W mass ratio, the effects of symmetry breaking cancel. This result supports resummation prescriptions that are based on a symmetric SU(2)xU(1) theory matched with QED at the electroweak scale.

We derive logarithmically enhanced two-loop virtual electroweak corrections for arbitrary fermion-scattering processes at the TeV scale. This extends results previously obtained for massless fermion scattering to processes that involve also bottom and top quarks. The contributions resulting from soft, collinear, and ultraviolet singularities in the complete electroweak Standard Model are explicitly extracted from two-loop diagrams to next-to-leading-logarithmic accuracy including all effects associated with symmetry breaking and Yukawa interactions.

This report summarizes the activity on comparisons of existings tools for the standard model and on issues in jet physics by the SMHC working group during and subsequent to the Workshop "Physics at TeV Colliders", Les Houches, France, 11-29 June, 2007.

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 Q2/MW2, 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(MZ2/MW2).

Recently the two-loop next-to-leading logarithmic (NLL) virtual corrections to arbitrary processes with massless external fermions have been calculated. Within the spontaneously broken electroweak theory the one- and two-loop mass singularities have been derived to NLL accuracy and expressed as universal correction factors depending only on the quantum numbers of the external particles. This talk summarizes the results for massless fermionic processes and presents new aspects arising in the extension of the corresponding loop calculations to massive external fermions. As a preliminary result, the Abelian form factor for massive fermions is given.

We present the complete analytical result for the two-loop logarithmically enhanced contributions to the high energy asymptotic behavior of the vector form factor and the four-fermion cross section in a spontaneously broken SU(2) gauge model. On the basis of this result we derive the dominant two-loop electroweak corrections to the neutral current four-fermion processes at high energies. Previously neglected effects of the gauge boson mass difference are included through the next-to-next-to-leading logarithmic approximation.

Recently two-loop electroweak corrections to the neutral current four-fermion processes at high energies have been presented. The basic ingredient of this calculation is the evaluation of the two-loop corrections to the Abelian vector form factor in a spontaneously broken SU(2) gauge model. Whereas the final result and the derivation of the four-fermion cross sections from evolution equations have been published earlier, the calculation of the form factor from the two-loop Feynman diagrams is presented for the first time in this paper. We describe in detail the individual contributions to the form factor and their calculation with the help of the expansion by regions method and Mellin-Barnes representations.

Experiments at the future colliders LHC and ILC will, for the first time, explore electroweak reactions at energies in the TeV region, i.e. far beyond the masses of the W and Z bosons. In exclusive processes without the emission of massive gauge bosons double-logarithmic contributions, so-called Sudakov logarithms, will then become relevant. Precise theoretical predictions require the knowledge of the complete logarithmic two-loop corrections in the high energy limit, thus exceeding the approximation through the leading powers of logarithms known so far. This PhD thesis is focussed on the calculation of analytic two-loop corrections to the form factor of an Abelian vector current. It also studies with the help of evolution equations the consequences of these results for electroweak cross sections of four-fermion processes. Contributions from the Higgs boson and the mass difference between the W and the Z boson are considered as well, and the factorization of the singular contributions from the massless photon is demonstrated. The results represent theoretical two-loop predictions for four-fermion processes in the accuracy needed for the precision experiments at the ILC.

We present the complete analytical result for the two-loop logarithmically enhanced contributions to the high energy asymptotic behavior of the vector form factor and the four-fermion cross section in a spontaneously broken SU(2) gauge model. On the basis of this result we derive the dominant two-loop electroweak corrections to the neutral current four-fermion processes at high energies. Previously neglected effects of the gauge boson mass difference are included through the next-to-next-to-leading logarithmic approximation.

We present the complete analytical result for the two-loop logarithmically enhanced contributions to the high energy asymptotic behavior of the vector form factor and the four-fermion cross section in a spontaneously broken SU(2) gauge model. On the basis of this result we derive the dominant two-loop electroweak corrections to the neutral current four-fermion processes at high energies.

We present the complete analytical result for the two-loop logarithmically enhanced contributions to the high-energy asymptotic behavior of the vector form factor and the four-fermion cross section in a spontaneously broken $SU(2)$ gauge model. On the basis of this result we derive the dominant two-loop electroweak corrections to the neutral current four-fermion processes at high energies.