Properties of equations of the continuous Toda type

We study a modified version of an equation of the continuous Toda type in 1+1 dimensions. This equation contains a friction-like term which can be switched off by annihilating a free parameter \ep. We apply the prolongation method, the symmetry and the approximate symmetry approach. This strategy allows us to get insight into both the equations for \ep =0 and \ep \ne 0, whose properties arising in the above frameworks are mutually compared. For \ep =0, the related prolongation equations are solved by means of certain series expansions which lead to an infinite- dimensional Lie algebra. Furthermore, using a realization of the Lie algebra of the Euclidean group $E_{2}$, a connection is shown between the continuous Toda equation and a linear wave equation which resembles a special case of a three-dimensional wave equation that occurs in a generalized Gibbons-Hawking ansatz \cite{lebrun}. Nontrivial solutions to the wave equation expressed in terms of Bessel functions are determined. For \ep\,\ne\,0, we obtain a finite-dimensional Lie algebra with four elements. A matrix representation of this algebra yields solutions of the modified continuous Toda equation associated with a reduced form of a perturbative Liouville equation. This result coincides with that achieved in the context of the approximate symmetry approach. Example of exact solutions are also provided. In particular, the inverse of the exponential-integral function turns out to be defined by the reduced differential equation coming from a linear combination of the time and space translations. Finally, a Lie algebra characterizing the approximate symmetries is discussed.Comment: LaTex file, 27 page

Sixth-Order Vacuum-Polarization Contribution to the Lamb Shift of the Muonic Hydrogen

The sixth-order electron-loop vacuum-polarization contribution to the $2P_{1/2} - 2S_{1/2}$ Lamb shift of the muonic hydrogen ($\mu^{-} p^+$ bound state) has been evaluated numerically. Our result is 0.007608(1) meV. This eliminates the largest uncertainty in the theoretical calculation. Combined with the proposed precision measurement of the Lamb shift it will lead to a very precise determination of the proton charge radius.Comment: 4 pages, 5 figures the totoal LS number is change

What two models may teach us about duality violations in QCD

Though the operator product expansion is applicable in the calculation of current correlation functions in the Euclidean region, when approaching the Minkowskian domain, violations of quark-hadron duality are expected to occur, due to the presence of bound-state or resonance poles. In QCD finite-energy sum rules, contour integrals in the complex energy plane down to the Minkowskian axis have to be performed, and thus the question arises what the impact of duality violations may be. The structure and possible relevance of duality violations is investigated on the basis of two models: the Coulomb system and a model for light-quark correlators which has already been studied previously. As might yet be naively expected, duality violations are in some sense "maximal" for zero-width bound states and they become weaker for broader resonances whose poles lie further away from the physical axis. Furthermore, to a certain extent, they can be suppressed by choosing appropriate weight functions in the finite-energy sum rules. A simplified Ansatz for including effects of duality violations in phenomenological QCD sum rule analyses is discussed as well.Comment: 17 pages, 6 figures; version to appear in JHE

New perturbation theory representation of the conformal symmetry breaking effects in gauge quantum field theory models

We propose a hypothesis on the detailed structure for the representation of the conformal symmetry breaking term in the basic Crewther relation generalized in the perturbation theory framework in QCD renormalized in the ${\rm \bar{MS}}$ scheme. We establish the validity of this representation in the $O(\alpha_s^4)$ approximation. Using the variant of the generalized Crewther relation formulated here allows finding relations between specific contributions to the QCD perturbation series coefficients for the flavor nonsinglet part of the Adler function $D^{ns}_A$ for the electron-positron annihilation in hadrons and to the perturbation series coefficients for the Bjorken sum rule $S_\text{Bjp}$ for the polarized deep-inelastic lepton-nucleon scattering. We find new relations between the $\alpha_s^4$ coefficients of $D^{ns}_A$ and $S_\text{Bjp}$. Satisfaction of one of them serves as an additional theoretical verification of the recent computer analytic calculations of the terms of order $\alpha_s^4$ in the expressions for these two quantities.Comment: 12 pages, Title modified, abstract modified, improved and extended variant of the talks, presented at Int. Seminar "Quarks-2010" (6-12 June, 2010, Kolomna) and Int. Workshop Hadron Structure and QCD: From Low to High Energies (5-9 July 2010, Gatchina

The Cross Section of e^+ e^- Annihilation into Hadrons of Order alpha_s^4 n_f^2 in Perturbative QCD

We present the first genuine QCD five-loop calculation of the vacuum polarization functions: analytical terms of order alpha_s^4 n_f^2 to the absorptive parts of vector and scalar correlators. These corrections form an important gauge-invariant subset of the full O(alpha_s^4) correction to $e^+ e^- annihilation into hadrons and the Higgs decay rate into hadrons respectively. They discriminate between different widely used estimates of the full result.Comment: 4 pages, revtex4 styl

Looking through the QCD conformal window with perturbation theory

We study the conformal window of QCD using perturbation theory, starting from the perturbative upper edge and going down as much as we can towards the strongly coupled regime. We do so by exploiting the available five-loop computation of the $overline{{m MS}}$ $eta$-function and employing Borel resummation techniques both for the ordinary perturbative series and for the Banks-Zaks conformal expansion. Large-$n_f$ results are also used. We argue that the perturbative series for the $overline{{m MS}}$ $eta$-function is most likely asymptotic and non-Borel resummable, yet Borel resummation techniques allow to improve on ordinary perturbation theory. We find substantial evidence that QCD with $n_f=12$ flavours flows in the IR to a conformal field theory. Though the evidence is weaker, we find indications that also $n_f=11$ might sit within the conformal window. We also compute the value of the mass anomalous dimension $gamma$ at the fixed point and compare it with the available lattice results. The conformal window might extend for lower values of $n_f$, but our methods break down for n_f<11, where we expect that non-perturbative effects become important. A similar analysis is performed in the Veneziano limit

The Physics of Hadronic Tau Decays

Hadronic tau decays represent a clean laboratory for the precise study of quantum chromodynamics (QCD). Observables (sum rules) based on the spectral functions of hadronic tau decays can be related to QCD quark-level calculations to determine fundamental quantities like the strong coupling constant, parameters of the chiral Lagrangian, |V_us|, the mass of the strange quark, and to simultaneously test the concept of quark-hadron duality. Using the best available measurements and a revisited analysis of the theoretical framework, the value alpha_s(m_tau) = 0.345 +- 0.004[exp] +- 0.009[theo] is obtained. Taken together with the determination of alpha_s(m_Z) from the global electroweak fit, this result leads to the most accurate test of asymptotic freedom: the value of the logarithmic slope of 1/alpha_s(s) is found to agree with QCD at a precision of 4%. In another approach, the tau spectral functions can be used to determine hadronic quantities that, due to the nonperturbative nature of long-distance QCD, cannot be computed from first principles. An example for this is the contribution from hadronic vacuum polarization to loop-dominated processes like the anomalous magnetic moment of the muon. This article reviews the measurements of nonstrange and strange tau spectral functions and their phenomenological applications.Comment: 89 pages, 32 figures; final version accepted for publication by Reviews of Modern Physic

Five-loop renormalisation of QCD in covariant gauges

We present the complete set of vertex, wave function and charge renormalisation constants in QCD in a general simple gauge group and with the complete dependence on the covariant gauge parameter $\xi$ in the minimal subtraction scheme of conventional dimensional regularisation. Our results confirm all already known results, which were obtained in the Feynman gauge, and allow the extraction of other useful gauges such as the Landau gauge. We use these results to extract the Landau gauge five-loop anomalous dimensions of the composite operator $A^2$ as well as the Landau gauge scheme independent gluon, ghost and fermion propagators at five loops.Comment: 17 pages; FORM and Mathematica result files available with the source; corrected minor typos, added references, journal ref, 1 remark, 1 note and 1 additional result fil

High-precision QCD at hadron colliders: electroweak gauge boson rapidity distributions at NNLO

We compute the rapidity distributions of W and Z bosons produced at the Tevatron and the LHC through next-to-next-to leading order in QCD. Our results demonstrate remarkable stability with respect to variations of the factorization and renormalization scales for all values of rapidity accessible in current and future experiments. These processes are therefore ``gold-plated'': current theoretical knowledge yields QCD predictions accurate to better than one percent. These results strengthen the proposal to use W and Z production to determine parton-parton luminosities and constrain parton distribution functions at the LHC. For example, LHC data should easily be able to distinguish the central parton distribution fit obtained by MRST from that obtained by Alekhin.Comment: 47 pages, 17 figures. Minor typos, 1 reference correcte

Two-loop amplitudes with nested sums: Fermionic contributions to e+ e- --> q qbar g

We present the calculation of the nf-contributions to the two-loop amplitude for e+ e- --> q qbar g and give results for the full one-loop amplitude to order eps^2 in the dimensional regularization parameter. Our results agree with those recently obtained by Garland et al.. The calculation makes extensive use of an efficient method based on nested sums to calculate two-loop integrals with arbitrary powers of the propagators. The use of nested sums leads in a natural way to multiple polylogarithms with simple arguments, which allow a straightforward analytic continuation.Comment: 31 pages, a file "coefficients.h" with the results in FORM format is include