B-B-bar mixing involves three physical parameters: the magnitudes of the off-diagonal elements of the mass and decay matrices and their relative phase. They are related to the mass and width differences between the mass eigenstates and to the CP asymmetry in flavour-specific decays, a_fs. Introducing a new operator basis I present new, more precise theory predictions for the width differences in the B_s and B_d systems: in the Standard Model one finds Delta Gamma_s = 0.088 +/- 0.017 ps^{-1} and Delta Gamma_d = (26.7 +5.8/-6.5) * 10^{-4} ps^{-1}. Updates of the mass differences Delta M_d and Delta M_s and of a_{fs}^d and a_{fs}^s are also presented. Then I discuss how various present and future measurements can be combined to constrain new physics. The extraction of a new CP phase phi_s^Delta from data on a_{fs}^s also profits from our new operator basis. Confronting our new formulae with D0 data we find that sin phi_s^Delta deviates from zero by 2 sigma.

We describe the computer algebra program ParFORM

   which is the parallel version of the symbolic manipulation program
   FORM.

We update the theory predictions for the mass difference Delta m_s, the width difference Delta Gamma_s and the CP asymmetry in flavour-specific decays, a_fs^s, for the Bs-Bs-bar mixing system. In particular we present a new expression for the element Gamma_{12}^s of the decay matrix, which enters the predictions of Delta Gamma_s and a_fs^s. To this end we introduce a new operator basis, which reduces the troublesome sizes of the 1/m_b and alpha_s corrections and diminishes the hadronic uncertainty in Delta Gamma_s/Delta m_s considerably. Logarithms of the charm quark mass are summed to all orders. We find Delta m_s = (49.7 =/- 9.4 ) * 10^{-4} and Delta Gamma_s =(f_{B_s}/240 MeV)^2 [(0.105 +/- 0.016) B + (0.024 +/- 0.004) B_S~' - 0.027 +/- 0.015] ps^{-1} in terms of the bag parameters B, B_S~' in the NDR scheme and the decay constant f_{B_s}. The improved result for Gamma_{12}^s also permits the extraction of the CP-violating Bs-Bs-bar mixing phase from a_fs^s with better accuracy. We show how the measurements of Delta M_s, Delta Gamma_s, a_fs^s, A_CP^mix(B_s\to J/\psi \phi) and other observables can be efficiently combined to constrain new physics. Applying our new formulae to data from the D0 experiment, we find a 2-sigma deviation of the Bs-Bs-bar mixing phase from its Standard Model value. We also briefly update the theory predictions for the Bd-Bd-bar mixing system and find Delta Gamma_d/Delta m_d = (52.6+11.5/-12.8) * 10^{-4} and a_fs^d =(-4.8 +1.0/-1.2}) *10^{-4} in the Standard Model.

We have constructed an ε-finite basis of master integrals for all new types of one-scale tadpoles which appear in the calculation of the four-loop QCD corrections to the electroweak ρ-parameter. Using transformation rules from the ε-finite basis to the standard minimal-number-of-lines basis, we obtain as a by-product analytical expressions for few leading terms of the ε-expansion of all members of the standard basis. The new master integrals have been computed with the help of the Pade method and by use of difference equations independently.

Relativistic differential equations for bound states of $n$ Dirac particles ($n=1,2,3$) are put into forms that depend only on the square of the total cms energy. Instead of coupling $4^n$ Dirac components, they decompose into two equivalent equations for $4^n/2$ components $\psi$ and $\chi$ of total chiralities $+1$ and $-1$, respectively. Their time-dependent versions are of the Klein-Gordon type; they reproduce the relativistic kinematics for the emission of photons or pions. Although the equations are presently extended to mesons and baryons within perturbative QCD only, the necessity of free Klein-Gordon equations for closed systems implies $E^2$-spectra not only for mesons, but also for baryons. For binaries, there exists an intricate transformation which turns the equation for $\psi$ into an effective one-body Dirac equation. The corresponding transformation for $\chi$ is derived here.

Top-quark physics plays an important r\^ole at hadron colliders

   such as the Tevatron at Fermilab or the upcoming 
   Large Hadron Collider (LHC) at CERN. 
   Given the planned experimental 
   precision, detailed theoretical predictions are 
   mandatory. In this article we present analytic results for the
   complete weak
   corrections to gluon-induced top-quark pair production
   --- neglecting purely photonic corrections, completing
   our earlier results for the quark-induced reaction. As an
   application we discuss top-quark pair production at the Tevatron and
   LHC. In particular we show that, although they are small for inclusive 
   quantities,
   weak corrections can be sizeable for differential distributions.

Within the framework of QCD we compute renormalization constants for the strong coupling and the quark masses to four-loop order. We apply the DR-bar scheme and put special emphasis on the additional couplings which have to be taken into account. This concerns the epsilon-scalar–quark Yukawa coupling as well as the vertex containing four epsilon-scalars. For a supersymmetric Yang Mills theory, we find, in contrast to a previous claim, that the evanescent Yukawa coupling equals the strong coupling constant through three loops as required by supersymmetry.

Xs gamma at O(alpha_s^2) > Combining our results for various ${\cal O}(\alpha_s^2)$ corrections

 to
 the weak radiative $B$-meson decay, we are able to present the first
 estimate of the branching ratio at the next-to-next-to-leading order
 in QCD.
 We find ${\cal B}({\bar B}\to X_s\gamma) = (3.15 \pm 0.23) \times
 10^{-4}$
 for $E_{\gamma} > 1.6\;$GeV in the ${\bar B}$-meson rest frame. The
 four
 types of uncertainties: non-perturbative (5\%), parametric (3\%),
 higher-order (3\%) and $m_c$-interpolation ambiguity (3\%) have been
 added in quadrature to obtain the total error.

Decoupling of c-quark loops in b-quark HQET is considered. The decoupling coefficients for the HQET heavy-quark field and the heavy-light quark current are calculated with the three-loop accuracy. The last result can be used to improve the accuracy of extracting f_B from HQET lattice simulations (without c-quark loops). The decoupling coefficient for the flavour-nonsinglet QCD current with n antisymmetrized \gamma-matrices is also obtained at three loops; the result for the tensor current (n=2) is new.

One of the most troublesome contributions to the NNLO QCD corrections to ${\bar B}\to X_s\gamma$ originates from three-loop matrix elements of four-quark operators. A part of this contribution that is proportional to the QCD beta-function coefficient $\beta_0$ was found in 2003 as an expansion in $m_c/m_b$. In the present paper, we evaluate the asymptotic behaviour of the complete contribution for $m_c \gg m_b/2$. The asymptotic form of the $\beta_0$-part matches the small-$m_c$ expansion very well at the threshold $m_c = m_b/2$. For the remaining part, we perform an interpolation down to the measured value of $m_c$, assuming that the $\beta_0$-part is a good approximation at $m_c=0$. Combining our results with other contributions to the NNLO QCD corrections, we find ${\cal B}({\bar B}\to X_s\gamma) = (3.15 \pm 0.23) \times 10^{-4}$ for $E_{\gamma} > 1.6\;$GeV in the ${\bar B}$-meson rest frame. The indicated error has been obtained by adding in quadrature the following uncertainties: non-perturbative (5\%), parametric (3\%), higher-order %(${\cal O}(\alpha_s^3)$) perturbative (3\%), and the interpolation ambiguity (3\%).

The experimental measurable R-ratio can be used to perform a precise determination of the charm- and bottom-quark mass with the help of theoretical computable expansion coefficients of the vacuum polarization function. The four-loop contributions in perturbative QCD to the first two expansion coefficients of the vacuum polarization functions are presented.

In this paper we consider the matching coefficient of the vector current between Quantum Chromodynamics (QCD) and Non-Relativistic QCD (NRQCD) to three-loop order in perturbation theory. We evaluate the fermionic corrections containing a closed massless fermion loop. The results are building blocks both for the bottom and top quark system at threshold. We explain in detail the methods used for the evaluation of the Feynman diagrams, classify the occurring master integrals and provide results for the latter. The numerical effects are significant. They have the tendency to improve the behaviour of the perturbative series — both for the bottom and top quark system.

We describe new developments in the symbolic manipulation system FORM. Particular attention is given to the implementation of parallel processing of formulas.

In this contribution we review the status of the decoupling coefficients for $\alpha_s$ both in the Standard Model (SM) and its minimal supersymmetric extension, the so-called MSSM. We stress the importance of a consistent treatment, in particular within the framework of the MSSM.

pi e^+ e^- and K_L → pi mu^+ mu^- : a binary star on the stage of flavor physics > A systematic analysis of New Physics impacts on the rare decays KL to pi0 l+ l- is performed. Thanks to their different sensitivities to flavor-changing local effective interactions, these two modes could provide valuable information on the nature of the possible New Physics at play. In particular, a combined measurement of both modes could disentangle scalar/pseudoscalar from vector or axial-vector contributions. For the latter, model-independent bounds are derived. Finally, the KL to pi0 mu+ mu- forward-backward CP-asymmetry is considered, and shown to give interesting complementary information.

QCD penguins are responsible for about 2/3 of the Delta I=1/2 rule in K to pi pi decays, as inferred from a combined analysis of K to pi pi and KL to gamma gamma. Further tests based on the decays KS to pi0 gamma gamma and K+ to pi+ gamma gamma are proposed. New insights into the treatment of pi0, eta, eta' pole amplitudes are also reported.

The four-loop QCD corrections to the electroweak $\rho$-parameter arising from top and bottom quark loops are computed. Specifically we evaluate the missing ``non-singlet'' piece. Using algebraic methods the amplitude is reduced to a set of around 50 new master integrals which are calculated with various analytical and numerical methods. Inclusion of the newly completed term halves the final value of the four-loop correction for the minimally renormalized top-quark mass. The predictions for the shift of the weak mixing angle and the W-boson mass is thus stabilized.

mu^+ mu^- at Next-to-Next-to-Leading Order > We calculate the charm quark contribution to the decay K_L → mu^+ mu^- in next-to-next-to-leading order of QCD. This new contribution reduces the theoretical uncertainty in the relevant parameter P_c from +/-22% down to +/-7%, corresponding to scale uncertainties of +/-3% and +/-6% in the short-distance part of the branching ratio and the determination of the Wolfenstein parameter rhobar from K_L → mu^+ mu^-. The error in P_c = 0.115 +/- 0.018 is now in equal shares due to the combined scale uncertainties and the current uncertainty in the charm quark mass. We find BR (K_L → mu^+ mu^-)_SD = (0.79 +/- 0.12) x 10^-9, with the present uncertainty in the Cabibbo-Kobayashi-Maskawa matrix element V_td being the dominant individual source in the quoted error.

New results at four-loop order in perturbative QCD for the first two Taylor coefficients of the heavy quark vacuum polarization function are presented. They can be used to perform a precise determination of the charm- and bottom-quark mass. Implications for the value of the quark masses are briefly discussed.

We present in analytic form the O(alpha_s^5) correction to the H→gg partial width of the standard-model Higgs boson with intermediate mass M_H<2M_t. Its knowledge is useful because the O(alpha_s^4) correction is sizeable (around 20%). For M_H=120 GeV, the resulting QCD correction factor reads 1 + (215/12) alpha_s^(5)(M_H)/pi + 152.5 (alpha_s^(5)(M_H)/pi)^2 + 381.5 (alpha_s^(5)(M_H)/pi)^3 \approx 1+0.65+0.20+0.02. The new four-loop correction increases the total Higgs-boson hadronic width by a small amount of order 1 promille and stabilizes significantly the residual scale dependence.

We present an extensive analysis of rare K decays, in particular of the two neutrino modes K+→pi+ nu nu-bar and KL→pi0 nu nu-bar, in the Minimal Supersymmetric extension of the Standard Model. We analyse the expectations for the branching ratios of these modes, both within the restrictive framework of the minimal flavour violation hypothesis and within a more general framework with new sources of flavour-symmetry breaking. In both scenarios, the information that can be extracted from precise measurements of the two neutrino modes turn out to be very useful in restricting the parameter space of the model, even after taking into account the possible information on the mass spectrum derived from high-energy colliders, and the constraints from B-physics experiments. In the presence of new sources of flavour-symmetry breaking, additional significant constraints on the model can be derived also from the two KL→pi0 l+l- modes.

We describe the implementation of facilities for the communication with external resources in the Symbolic Manipulation System FORM. This is done according to the POSIX standards defined for the UNIX operating system. We present a number of examples that illustrate the increased power due to these new capabilities.

In this paper we consider the matching coefficients up to two loops between Quantum Chromodynamics (QCD) and Non-Relativistic QCD (NRQCD) for the vector, axial-vector, scalar and pseudo-scalar currents. The structure of the effective theory is discussed and analytical results are presented. Particular emphasis is put on the singlet diagrams.

We explore the possibilities to determine the spin/CP properties of the Higgs boson at the LHC. To cover the mass region below the ZZ threshold we make use of the properties of the production in Weak Boson Fusion (WBF) and the decay chain H $\rightarrow$ W$^+$W$^-$ $\rightarrow$ $\ell^+\nu \ell^-\nu$. In particular, we study the angular correlations of the forward jets and the distribution of the invariant mass of the lepton pair for different hypothetical Higgs like particles.

In this paper we consider the production of top quark pairs close to threshold and compute the dependence on the Higgs boson mass to order $\alpha\alpha_s$. This requires the evaluation of the matching coefficient of the vector current to two-loop level and the inclusion of Higgs mass dependent operators in the non-relativistic effective theory. For Higgs boson masses below 200~GeV moderate contributions to the top quark mass and the peak cross section are observed. We also provide additional information to the on-shell wave function renormalization which is relevant for the matching coefficient.

Dimensional Reduction is applied to \qcd{} in order to compute various

 renormalization constants in the \drbar{} scheme at higher orders in
 perturbation theory. In particular, the $\beta$ function and the
 anomalous dimension of the quark masses are derived to three-loop
 order. Special emphasis is put on the proper
 treatment of the so-called $\varepsilon$-scalars and the additional
 couplings which have to be considered.

pi^+ nu anti-nu at Next-to-Next-to-Leading Order > the charm contribution of the rare decay K+ → pi+ nu nu-bar. We encounter several new features, which were absent in lower orders. We discuss them in detail and present the results for the 2-loop matching conditions of the Wilson coefficients, the 3-loop anomalous dimensions, and the 2-loop matrix elements of the relevant operators that enter the NNLO renormalization group analysis of the Z-penguin and the electroweak box contribution. The inclusion of the NNLO QCD corrections leads to a significant reduction of the theoretical uncertainty from 9.8% down to 2.4% in the relevant parameter Pc, implying the leftover scale uncertainties in BR(K+ → pi+ nu nu-bar) and in the determination of |V_td|, sin(2 beta), and gamma from the K → pi nu nu system to be 1.3%, 1.0%, 0.006, and 1.2 degrees, respectively. For the charm quark MSbar mass mc=(1.30+-0.05) GeV and |V_us|= 0.2248 the NLO value Pc=0.37+-0.06 is modified to Pc=0.38+-0.04 at the NNLO level with the latter error fully dominated by the uncertainty in mc. We present tables for Pc as a function of mc and alphas(MZ) and a very accurate analytic formula that summarizes these two dependences as well as the dominant theoretical uncertainties. Adding the recently calculated long-distance contributions we find BR(K+ → pi+ nu nu-bar)=(8.0+-1.1)*10^-11 with the present uncertainties in mc and the Cabibbo-Kobayashi-Maskawa elements being the dominant individual sources in the quoted error. We also emphasize that improved calculations of the long-distance contributions to K+ → pi+ nu nu-bar and of the isospin breaking corrections in the evaluation of the weak current matrix elements from K+ → pi0 e+ nu would be valuable in order to increase the potential of the two golden K → pi nu nu decays in the search for new physics.

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.

We investigate the electroweak corrections to the Z-boson and direct photon production at large transverse momentum at the LHC. Analytical results for the high energy next-to-next-to-leading logarithmic approximation of the one-loop weak corrections and the two-loop electroweak corrections in the next-to-leading logarithmic approximation are presented. Numerical results showing the relevance of these corrections are discussed.

We discuss recent developments in multiloop calculations aiming eventually in computing the total cross section for e^+e^- annihilation into hadrons in order alpha_s^4.

It is shown that for every problem within dimensional regularization, using the Integration-By-Parts method, one is able to construct a set of master integrals such that each corresponding coefficient function is finite in the limit of dimension equal to four. We argue that the use of such a basis simplifies and stabilizes the numerical evaluation of the master integrals. As an example we explicitly construct the ep-finite basis for the set of all QED-like four-loop massive tadpoles. Using a semi-numerical approach based on Pade approximations we evaluate analytically the divergent and numerically the finite part of this set of master integrals.

The calculations confirm the recent results of Schroder and Vuorinen. All the contributions found there by fitting the high precision numerical results have been confirmed by direct analytical calculation without using any numerical input.

The $\tau$-decay library TAUOLA has gained popularity over the last decade. However, with the continuously increasing precision of the data, some of its functionality has become insufficient. One of the requirements is the implementation of decays into five mesons plus a neutrino with a realistic decay amplitude. This note describes a step into this direction. For the $2\pi^-\pi^+2\pi^0$ mode the three decay chains $\tau^- \to a_1^- \nu \to \rho^-(\to \pi^-\pi^0) \omega (\to \pi^-\pi^+\pi^0) \nu$, $\tau^- \to a_1^- \nu \to a_1^-(\to 2\pi^-\pi^+) f_0 (\to 2\pi^0) \nu$, and $\tau^- \to a_1^- \nu \to a_1^-(\to \pi^-2\pi^0) f_0 (\to \pi^+\pi-) \nu$ are introduced with simple assumptions about the couplings and propagators of the various resonances. Similar amplitudes (without the $\rho\omega$ contributions) are adopted for the $\pi^-4\pi^0$ and $3\pi^-2\pi^+$ modes.

The five-pion amplitude is thus based on a simple model, which, however, can be considered as a first realistic example. Phase-space generation includes the possibility of presampling the $\omega$ and $a_1$ resonances, in one channel only, however. This is probably sufficient for the time being, both for physics applications and for tests.

The technical test of the new part of the generator is performed by comparing Monte Carlo and analytical results. To this end a non-realistic, but easy to calculate, purely scalar amplitude for the decay into five massless pions was used.