The measurement of the pion form factor and, more generally, of the cross section for electron-positron annihilation into hadrons through the radiative return has become an important task for high luminosity colliders such as the Phi- or B-meson factories. This quantity is crucial for predictions of the hadronic contributions to the anomalous magnetic moment of the muon, and to the running of the electromagnetic coupling. But the radiative return opens the possibility of many other physical applications. The physics potential of this method at high luminosity meson factories is discussed, the last upgraded version of the event generator PHOKHARA is presented, and future developments are highlighted.

We sum up the next-to-leading logarithmic corrections to the heavy-quarkonium hyperfine splitting using the nonrelativistic renormalization group. On the basis of this result, we predict the mass of the $\eta_b$ meson to be $M(\eta_b)=9421 \pm 10\,{(\rm th)} \,{}^{+9}_{-8}\,(\delta\alpha_s)~{\rm MeV}$. The experimental measurement of $M(\eta_b)$ with a few MeV error would be sufficient to determine $\alpha_s(M_Z)$ with an accuracy of $\pm 0.003$. The use of the nonrelativistic renormalization group is mandatory to reproduce the experimental value of the hyperfine splitting in charmonium.

A review of the calculation of the two loop bosonic corrections to $\Delta r$ is presented. Factorization and matching onto the Fermi model are discussed. An approximate formula, describing the quantity over the interesting range of Higgs boson mass values from 100~GeV to 1~TeV is given.

The last missing correction to the muon lifetime in the Standard Model at ${\cal O}(\alpha^2)$ coming from gauge and Higgs boson loops is presented. The associated contribution to the parameter $\Delta r$ in the on-shell scheme ranges from $6 \times 10^{-5}$ to $-4 \times 10^{-5}$ for Higgs boson masses from 100~GeV to 1~TeV. This result translates into a shift of the $W$ boson mass which does not exceed $\pm 1$~MeV in the same range and amounts in particular to approximately $- 0.8$~MeV for a 115~GeV Higgs boson.

The 4-loop sunrise graph with two massless lines, two lines of equal mass M and a line of mass m, for external invariant timelike and equal to m^2 is considered. We write differential equations in x=m/M for the Master Integrals of the problem, which we Laurent-expand in the regularizing continuous dimension $d$ around $d=4$, and then solve exactly in x up to order (d-4)^3 included; the result is expressed in terms of Harmonic PolyLogarithms of argument x and maximum weight 7. As a by product, we obtain the x=1 value, expected to be relevant in QED 4-loop static quantities like the electron (g-2). The analytic results were checked by an independent precise numerical calculation

We present a brief (mainly bibliographical) report on recently performed calculations of terms of order O(\alpha_s^4 n_f^2) and O(\alpha_s^4 n_f^2 m_q^2) for hadronic Z and \tau decay rates. A few details about the analytical evaluation of the masters integrals appearing in the course of calculations are presented.

We present the Master Integrals needed for the calculation of the two-loop QCD corrections to the forward-backward asymmetry of a quark-antiquark pair produced in electron-positron annihilation events. The abelian diagrams entering in the evaluation of the vector form factors were calculated in a previous paper. We consider here the non-abelian diagrams and the diagrams entering in the computation of the axial form factors, for arbitrary space-like momentum transfer $Q^2$ and finite heavy quark mass $m$. Both the UV and IR divergences are regularized in the continuous $D$-dimensional scheme. The Master Integrals are Laurent-expanded around $D=4$ and evaluated by the differential equation method; the coefficients of the expansions are expressed as 1-dimensional harmonic polylogarithms of maximum weight 4.

We review the status of theoretical predictions for the production of neutral Higgs bosons at the LHC. Special emphasis is put on the role of bottom quarks in the gluon fusion process and in the associated production of Higgs bosons with bb-bar pairs.

The impact of final-state radiation (FSR) on the radiative

return method for the extraction of the e+e- hadronic cross section
is discussed in detail. Possible experimental tests of the model
dependence of FSR are proposed for the pi+pi- hadronic final state.
The importance of the (pi+pi-photon) final state contribution
to the muon anomalous magnetic moment is investigated,
and a method based on the radiative return is proposed 
to extract these contributions from data. 

pi+pi-e+e-: a potential background for sigma(e+e- →pi+pi-) measurement via radiative return method > A Monte Carlo generator (EKHARA) has been constructed to simulate the reaction e+e- → pi+pi-e+e- based on initial and final state emission of a e+e- pair from e+e- → pi+pi- production diagram. A detailed study of the process, as a potential background for sigma(e+e- → pi+pi-) measurement via radiative return method, is presented for phi- and B- factory energies.

The measurement of the pion form factor and, more generally, of the cross section for electron–positron annihilation into hadrons through the radiative return has become an important task for high luminosity colliders such as the $\Phi$- or $B$-meson factories. For detailed understanding and analysis of this reaction, the construction of a Monte Carlo program, PHOKHARA, has been undertaken. Version 2.0 was based on a next-to-leading order (NLO) treatment of the corrections from initial-state radiation (ISR) and leading order final-state radiation (FSR). The most recent extension of PHOKHARA (version 3.0), which incorporates NLO corrections to FSR and the impact of combined ISR and FSR on selected distributions, is presented.

In this paper we present the master integrals necessary for the analytic calculation of the box diagrams with one electron loop ($N_{F}=1$) entering in the 2-loop ($\alpha^3$) QED virtual corrections to the Bhabha scattering amplitude of the electron. We consider on-shell electrons and positrons of finite mass $m$, arbitrary squared c.m. energy $s$, and momentum transfer $t$; both UV and soft IR divergences are regulated within the continuous $D$-dimensional regularization scheme. After a brief overview of the method employed in the calculation, we give the results, for $s$ and $t$ in the Euclidean region, in terms of 1- and 2-dimensional harmonic polylogarithms, of maximum weight 3. The corresponding results in the physical region can be recovered by analytical continuation. For completeness, we also provide the analytic expression of the 1-loop scalar box diagram including the first order in $(D-4)$.

The large-\beta_0 limit of QCD is discussed, with the emphasize on simple technical methods of calculating various quantities at the order 1/\beta_0. Many examples, mainly from heavy quark physics, are considered. Some QCD results based on renormalization group (and not restricted to the large-\beta_0 limit) are also discussed.

The differential equations for the 2-loop sunrise graph, at equal masses but arbitrary momentum transfer, are used for the analytic evaluation of the coefficients of its Laurent-expansion in the continuous dimension $d$.

Extensions of the Standard Model of elementary particle physics to noncommutative geometries have been proposed as a low energy limit of string models. Independent of this motivation, one may consider such a model as an effective field theory with higher-dimensional operators containing an antisymmetric rank-two background field. We study the signals of such a Noncommutative Standard Model (NCSM) and analyze the discovery potential of a future photon collider, considering angular distributions in fermion pair production.

The mechanism of electroweak symmetry breaking in Little Higgs Models is analyzed in an effective field theory approach. This enables us to identify observable effects irrespective of the specific structure and content of the heavy degrees of freedom. We parameterize these effects in a common operator basis and present the complete set of anomalous contributions to gauge-boson, Higgs, and fermion couplings. If the hypercharge assignments of the model retain their standard form, electroweak precision data are affected only via the S and T parameters and by contact interactions. As a proof of principle, we apply this formalism to the minimal model and consider the current constraints on the parameter space. Finally, we show how the interplay of measurements at LHC and a Linear Collider could reveal the structure of these models.

J/Psi K_S > We study the corrections to the determination of sin(2 beta) from the time dependent CP asymmetry of B → J/Psi K_S which arise in the standard model. Although a precise prediction of these corrections is not possible we find that they are indeed extremely small, of the order of less than a per mil of the observed value. This means in turn that any deviation visible at the B factories will be a clear signal for new physics.

The measurement of the pion form factor and, more generally, of the cross section for electron–positron annihilation into hadrons through the radiative return has become an important task for high luminosity colliders such as the $\Phi$- or $B$-meson factories. For detailed understanding and analysis of this reaction, the construction of a Monte Carlo program, PHOKHARA, has been undertaken. Version 2.0 was based on a next-to-leading order (NLO) treatment of the corrections from initial-state radiation (ISR). In the present paper a further extension of PHOKHARA (version 3.0) is described, which incorporates NLO corrections to final-state radiation (FSR). The impact of combined ISR and FSR on various distributions is investigated and methods are presented, which will allow the extraction of the form factor, and even give access to inclusive photon emission due to FSR. The dependence of the results on the model for FSR is discussed and the impact of this contribution on the anomalous magnetic moment of the muon is evaluated.

We carry out a systematic investigation of all the 2-loop integrals occurring in the electron vertex in QED in the continuous $D$-dimensional regularization scheme, for on-shell electrons, momentum transfer $t=-Q^2$ and finite squared electron mass $m_e^2=a$. We identify all the Master Integrals (MI's) of the problem and write the differential equations in $Q^2$ which they satisfy. The equations are expanded in powers of $\epsilon = (4-D)/2$ and solved by the Euler's method of the variation of the constants. As a result, we obtain the coefficients of the Laurent expansion in $\epsilon$ of the MI's up to zeroth order expressed in close analytic form in terms of Harmonic Polylogarithms

We present the closed analytic expression of the form factors of the two-loop QED vertex amplitude for on-shell electrons of finite mass $m$ and arbitrary momentum transfer $S=-Q^2$. The calculation is carried out within the continuous $D$-dimensional regularization scheme, with a single continuous parameter $D$, the dimension of the space-time, which regularizes at the same time UltraViolet (UV) and InfraRed (IR) divergences. The results are expressed in terms of 1-dimensional harmonic polylogarithms of maximum weight 4.

I discuss methods of calculation of propagator diagrams (massless, those of Heavy Quark Effective Theory, and massive on-shell diagrams) up to 3 loops. Integration-by-parts recurrence relations are used to reduce them to linear combinations of basis integrals. Non-trivial basis integrals have to be calculated by some other method, e.g., using Gegenbauer polynomial technique. Many of them are expressed via hypergeometric functions; in the massless and HQET cases, their indices tend to integers at $\varepsilon\to0$. I discuss the algorithm of their expansion in $\varepsilon$, in terms of multiple $\zeta$ values. These lectures were given at Calc-03 school, Dubna, 14–20 June 2003.

QCD sum rules are used to estimate the flavour SU(3)-symmetry violation in two-body B decays to pions and kaons. In the factorizable amplitudes the SU(3)-violation manifests itself in the ratio of the decay constants f_K/f_pi and in the differences between the B→K, B_s→K and B→pi form factors. These effects are calculated from the QCD two-point and light-cone sum rules, respectively, in terms of the strange quark mass and the ratio of the strange and nonstrange quark-condensate densities. Importantly, QCD sum rules predict that SU(3) breaking in the heavy-to-light form factors can be substantial and does not vanish in the heavy-quark mass limit. Furthermore, we investigate the strange-quark mass dependence of nonfactorizable effects in the B→K pi decay amplitudes. Taking into account these effects we estimate the accuracy of several SU(3)-symmetry relations between charmless B-decay amplitudes.

The current status of the QCD sum rule predictions for charmed mesons is overviewed.

We recalculate the next-to-leading order Altarelli-Parisi kernel using a method which relates it to the splitting amplitudes describing the collinear factorization properties of scattering amplitudes. The method breaks up the calculation of the kernel into individual pieces which have an independent physical interpretation.

We consider a special class of dimension-six operators which are assumed to be induced by some new physics with typical scales of order \Lambda. This special class are the operators with two quarks which can mediate transitions between quark flavours. We show that under quite general assumptions the effect of these operators can be parametrised in terms of six parameters, leading to a modification of the (at tree level flavour diagonal) neutral currents and to an ffective CKM matrix'' for the charged currents, which is not necessarily unitary any more. The effects of these operators on charged and neutral currents are studied.

We discuss the spin properties of top quark pairs produced at hadron colliders at next-to-leading order in the coupling constant alpha_s of the strong interaction. Specifically we present, for some decay channels, results for differential angular distributions that are sensitive to t tbar spin correlations.

tau tau decay > We show how the transverse tau tau spin correlations can be used to measure the parity of the Higgs boson and hence to distinguish a CP-even H boson from CP-odd A in the future high energy accelerator experiments. We investigate the subsequent decays of the tau into pi nu and rho nu. The prospects for the measurement of the Higgs boson parity with a mass of 120 GeV are quantified for the case of e+ e- collisions of 500 GeV center of mass energy. The Standard Model Higgsstrahlung production process is used as an example.

Employing the QCD light-cone sum rule approach we calculate the B → pi pi hadronic matrix element of the current-current operator with c quarks in the penguin topology (``charming penguin''). The

      dominant contribution to the sum rule is due to the c-quark loop at short distances and is of O(alpha_s) with respect to the factorizable B -> pi pi amplitude. The effects of soft gluons are suppressed at least
      by O(alpha_s m_b^{-2}). Our result indicates that sizable nonperturbative effects generated by charming penguins at finite m_b are absent. The same is valid for the penguin contractions of the
      current-current operators with light quarks. 

Single top quark production in electron photon interactions provides a clean environment for the measurement of the Cabbibo-Kobayashi-Maskawa matrix element Vtb. Aiming an experimental precision at the percent level the knowledge of radiative corrections is important. In this paper we present results for the radiative corrections in quantum chromodynamics.

The anomalous dimension of the heavy-light quark current in HQET is calculated with three-loop accuracy, as well as the renormalized heavy-quark propagator. The NNL perturbative correction to f_B/f_D is obtained.

We present results for the three-loop top quark contributions to the rho parameter in the limit of large top quark mass. The simultaneous dependence on the mass of the Higgs boson M_H and the mass of the top quark m_t is obtained from expansions in the range of M_H around m_t and in the limit M_H » m_t. In combination with the previous result for M_H = 0 the dependence of the rho parameter on the leading Yukawa contributions, i.e. on m_t and M_H, is well under control for all mass values of practical importance. The effects lead to a shift in the W mass in the order of 5 MeV and are relevant for precision measurements at TESLA.

Two-loop corrections for the form factor in a massive Abelian theory are evaluated, which result from the insertion of massless fermion or scalar loops into the massive gauge boson propagator. The result is valid for arbitrary energies and gauge boson mass. Power-suppressed terms vanish rapidly in the high energy region where the result is well approximated by a polynomial of third order in ln(s/M^2). The relative importance of subleading logarithms is emphasised.

We evaluate the on shell form factors of the electron for arbitrary momentum transfer and finite electron mass, at two loops in QED, by integrating the corresponding dispersion relations, which involve the imaginary parts known since a long time. The infrared divergences are parameterized in terms of a fictitious small photon mass. The result is expressed in terms of Harmonic Polylogarithms of maximum weight 4. The expansions for small and large momentum transfer are also given

We have calculated the fermion contributions to the shift of the position of the poles of the massive gauge boson propagators at two–loop order in the Standard Model. Together with the bosonic contributions calculated previously the full two–loop corrections are available. This allows us to investigate the full correction in the relationship between MS and pole masses of the vector bosons $Z$ and $W$. Two–loop renormalization and the corresponding renormalization group equations are discussed. Analytical results for the two-loop propagator type diagrams are presented. A new approach of summing multiple binomial sums has been used.

We present results on the two-loop leading and angular-dependent next-to-leading logarithmic virtual corrections to arbitrary processes at energies above the electroweak scale. In the `t Hooft-Feynman gauge the relevant Feynman diagrams involving soft and collinear gauge bosons \gamma, Z, W^\pm coupling to external legs are evaluated in the eikonal approximation in the region where all kinematical invariants are much larger than the electroweak scale. The logarithmic mass singularities are extracted from massive multi-scale loop integrals using the Sudakov method and alternatively the sector-decomposition method in the Feynman-parameter representation. The derivations are performed within the spontaneously broken phase of the electroweak theory, and the two-loop results are in agreement with the exponentiation prescriptions that have been proposed in the literature based on a symmetric SU(2) x U(1) theory matched with QED at the electroweak scale.

Contributions from light-by-light scattering to $(g_\mu-2)/2$, the anomalous magnetic moment of the muon, are mediated by the exchange of charged fermions or scalar bosons. Assuming large masses $M$ for the virtual particles and employing the technique of large mass expansion, analytical results are obtained for virtual fermions and scalars in the form of a series in $(m_\mu /M)^2$. This series is well convergent even for the case $M=m_\mu$. For virtual fermions, the expansion confirms published analytical formulae. For virtual scalars, the result can be used to evaluate the contribution from charged pions. In this case our result confirms already available numerical evaluations, however, it is significantly more precise.

rho nu decay > We demonstrate that a measurement of the impact parameter in one-prong tau decay can be useful for the determination of the Higgs boson parity in the H/A –> tau tau; tau –> rho nu decay chain. We have estimated that for a detection set-up such as TESLA, use of the information from the tau impact parameter can improve the significance of the measurement of the parity of the Standard Model 120 GeV Higgs boson to 4.5 sigma, and in general by factor of about 1.5 with respect to the method where this information is not used.

We also show that the variation in the assumption on the precision of the measurement of the impact parameter and/or pi's momenta does not affect the sensitivity of the method. This is because the method remains limited by the type of twofold ambiguity in reconstructing the tau momentum.