Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops

It is found that the number, $M_n$, of irreducible multiple zeta values (MZVs) of weight $n$, is generated by $1-x^2-x^3=\prod_n (1-x^n)^{M_n}$. For $9\ge n\ge3$, $M_n$ enumerates positive knots with $n$ crossings. Positive knots to which field theory assigns knot-numbers that are not MZVs first appear at 10 crossings. We identify all the positive knots, up to 15 crossings, that are in correspondence with irreducible MZVs, by virtue of the connection between knots and numbers realized by Feynman diagrams with up to 9 loops.Comment: 15 pages, Latex, figures using EPSF, replaced version has references and conclusions updated, Eq.(7) revised; as to appear in Phys.Lett.

The effective potential of N-vector models: a field-theoretic study to O(\epsilon^3)

We study the effective potential of three-dimensional O($N$) models. In statistical physics the effective potential represents the free-energy density as a function of the order parameter (Helmholtz free energy), and, therefore, it is related to the equation of state. In particular, we consider its small-field expansion in the symmetric (high-temperature) phase, whose coefficients are related to the zero-momentum $2j$-point renormalized coupling constants $g_{2j}$. For generic values of $N$, we calculate $g_{2j}$ to three loops in the field-theoretic approach based on the $\epsilon$-expansion. The estimates of $g_{2j}$, or equivalently of $r_{2j}\equiv g_{2j}/g_4^{j-1}$, are obtained by a constrained analysis of the series that takes into account the exact results in one and zero dimensions.Comment: 22 pages, RevTex, Nucl. Phys. B, in pres

The V-A sum rules and the Operator Product Expansion in complex q^2-plane from tau-decay data

The operator product expansion (OPE) for the difference of vector and axial current correlators is analyzed for complex values of momentum q^2. The vector and axial spectral functions, taken from hadronic tau-decay data, are treated with the help of Borel, Gaussian and spectral moments sum rules. The range of applicability, advantages and disadvantages of each type are discussed. The general features of OPE are confirmed by the data. The vacuum expectation values of dimension 6 and 8 operators are found to be O_6=-(6.8\pm 2.1)*10^{-3} GeV^6, O_8=(7\pm 4)*10^{-3} GeV^8.Comment: 1 latex + 10 eps files, 14 page

RS-invariant all-orders renormalon resummations for some QCD observables

We propose a renormalon-inspired resummation of QCD perturbation theory based on approximating the renormalization scheme (RS) invariant effective charge beta-function coefficients by the portion containing the highest power of $b$=$\frac{1}{6}(11N$--$2N_{f})$, for SU($N$) QCD with $N_{f}$ quark flavours. This can be accomplished using exact large-$N_{f}$ all-orders results. The resulting resummation is RS-invariant and the exact next-to-leading order (NLO) and next-to-NLO (NNLO) coefficients in any RS are included. This improves on a previously employed naive resummation of the leading-$b$ piece of the perturbative coefficients which is RS-dependent, making its comparison with fixed-order perturbative results ambiguous. The RS-invariant resummation is used to assess the reliability of fixed-order perturbation theory for the $e^{+}e^{-}$ $R$-ratio, the analogous $\tau$-lepton decay ratio $R_{\tau}$, and Deep Inelastic Scattering (DIS) sum rules, by comparing it with the exact NNLO results in the effective charge RS. For the $R$-ratio and $R_{\tau}$, where large-order perturbative behaviour is dominated by a leading ultra-violet renormalon singularity, the comparison indicates fixed-order perturbation theory to be very reliable. For DIS sum rules, which have a leading infra-red renormalon singularity, the performance is rather poor. In this way we estimate that at LEP/SLD energies ideal data on the $R$-ratio could determine $\alpha_{s}(M_{Z})$ to three-significant figures, and for the $R_{\tau}$ we estimate a theoretical uncertainty $\delta\alpha_{s}(m_{\tau})\simeq0.008$ corresponding to $\delta\alpha_{s}(M_{Z})\simeq0.001$. This encouragingly small uncertainty is much less than has recently been deduced from comparison with the ambiguous naive resummation.Comment: 25 pages, uses LaTeX, 12 Postscript figures, epsfig.sty 'elsart.sty' and 'elsart12.sty' are available via anonymous-ftp at ftp://ftp.tex.ac.uk/tex-archive/macros/latex/contrib/supported/elsevie

Recent progress in hadronic tau decays

The determination of alpha_s from hadronic tau decays is impeded by the fact that two choices for the renormalisation group resummation, namely fixed-order (FOPT) and contour-improved perturbation theory (CIPT), yield systematically differing results. On the basis of a model for higher-order terms in the perturbative series, which incorporates well-known structure from renormalons, it is found that FOPT smoothly approaches the Borel sum for the tau hadronic width, while CIPT is unable to account for the resummed series. An example for the behaviour of QCD spectral function moments, displaying a similar behaviour, is presented as well.Comment: 6 pages, 3 figures. Talk presented at 11th International Workshop on Tau Lepton Physics, Manchester, UK, 13-17 September 201

Hubble Space Telescope imaging of the CFRS and LDSS redshift surveys - IV. Influence of mergers in the evolution of faint field galaxies from z~1

HST images of a sample of 285 galaxies with measured z from the CFRS and Autofib-LDSS redshift surveys are analysed to derive the evolution of the merger fraction out to z~1. We have performed visual and machine-based merger identifications, as well as counts of bright pairs of galaxies with magnitude differences less than 1.5 mag. We find that the pair fraction increases with z, with up to ~20% of the galaxies being in physical pairs at z~0.75-1. We derive a merger fraction varying with z as (1+z)^{3.2 +/- 0.6}, after correction for line-of-sight contamination, in excellent agreement with the merger fraction derived from the visual classification of mergers for which m = 3.4 +/- 0.6. After correcting for seeing effects on the ground-based selection of survey galaxies, we conclude that the pair fraction evolves as (1+z)^{2.7 +/- 0.6}. This implies that an average L* galaxy will have undergone 0.8 to 1.8 merger events from z=1 to 0, with 0.5 to 1.2 merger events occuring in a 2 Gyr time span at z~0.9. This result is consistent with predictions from semi-analytical models of galaxy formation. From the simple co-addition of the observed luminosities of the galaxies in pairs, physical mergers are computed to lead to a brightening of 0.5 mag for each pair on average, and a boost in star formation rate of a factor of 2, as derived from the average [O II] equivalent widths. Mergers of galaxies are therefore contributing significantly to the evolution of both the luminosity function and luminosity density of the Universe out to z~1.Comment: 14 pages, 6 PS figures included. Accepted for publication in MNRA

alpha_s and the tau hadronic width

Different choices exist for the renormalisation group resummation in the determination of $\alpha_s$ from hadronic $\tau$ decays: namely fixed-order (FOPT) and contour-improved perturbation theory (CIPT). The two approaches lead to systematic differences in the resulting $\alpha_s$. On the basis of a model for higher-order terms in the perturbative series, which incorporates well-known structure from renormalons, it is found that while CIPT is unable to account for the fully resummed series, FOPT smoothly approaches the Borel sum. Employing the model to determine $\alpha_s$ yields $\alpha_s(M_\tau)=0.316 \pm 0.006$, which after evolution leads to $\alpha_s(M_Z) = 0.1180 \pm 0.0008$.Comment: 6 pages, 2 figures. Talk given at the 10th International Workshop on Tau Lepton Physics, Novosibirsk, Russia, Sep. 22-25, 200

Renormalon-inspired resummations for vector and scalar correlators- estimating the uncertainty in {alpha}_{s}({m}_{\tau}^{2}) and and {\alpha}({M}_{Z}^{2})

We perform an all-orders resummation of the QCD Adler D-function for the vector correlator, in which the portion of perturbative coefficients involving the leading power of b, the first beta-function coefficient, is resummed. To avoid a renormalization scale dependence when we match the resummation to the exactly known NLO and NNLO results, we employ the Complete Renormalization Group Improvement (CORGI) approach. These fixed-order and resummed CORGI results are analytically continued by numerically performing a contour integral to obtain corresponding fixed and all-orders ``contour-improved'' results for the e+e- R-ratio ands its tau decay analogue R_{\tau}. The difference between these fixed-order and all-order results is used to estimate the uncertainty in the extraction of {alpha}_{s}({M}_{Z}^{2}} from R_{\tau} measurements, and that in the QED coupling {\alpha}({M}_{Z}^{2}) due to hadronic corrections related to R. Analogous resummations for the scalar correlator are performed, and used to assess the uncertainty in the Higgs decay width to a heavy quark pair. We point out that CORGI fixed-order contour-improved results for R and the Higgs decay width, can be given explicitly in terms of the Lambert-W function and hypergeometric functions, avoiding the need for numerical integration.Comment: 33 pages, 8 figure

Reconstruction of heavy quark current correlators at O(\alpha_s^3)

We construct approximate formulas for the O(\alpha_s^3) QCD contributions to vector, axial-vector, scalar and pseudo-scalar quark current correlators, which are valid for arbitrary values of momenta and masses. The derivation is based on conformal mapping and the Pade approximation procedure and incorporates known expansions in the low energy, threshold and high energy regions. We use our results to estimate additional terms in these expansions.Comment: 24 pages, 4 figure

A wide-field spectroscopic survey of the cluster of galaxies Cl0024+1654: I. The catalogue

We present the catalogue of a wide-field CFHT/WHT spectroscopic survey of the lensing cluster Cl0024+1654 at z=0.395. This catalogue contains 618 new spectra, of which 581 have identified redshifts. Adding redshifts available from the literature, the final catalogue contains data for 687 objects with redshifts identified for 650 of them. 295 galaxies have redshifts in the range 0.37<z<0.41, i. e. are cluster members or lie in the immediate neighbourhood of the cluster. The area covered by the survey is 21x25 arcmin2 in size, corresponding to 4x4.8 h^-2 Mpc2 at the cluster redshift. The survey is 45% complete down to V=22 over the whole field covered; within 3 arcmin of the cluster centre the completeness exceeds 80% at the same magnitude. A detailed completeness analysis is presented. The catalogue gives astrometric position, redshift, V magnitude and V-I colour, as well as the equivalent widths for a number of lines. Apart from the cluster Cl0024+1654 itself, three other structures are identified in redshift space: a group of galaxies at z=0.38, just in front of Cl0024+1654 and probably interacting with it, a close pair of groups of galaxies at z~0.495 and an overdensity of galaxies at z~0.18 with no obvious centre. The spectroscopic catalogue will be used to trace the three-dimensional structure of the cluster Cl0024+1654 as well as study the physical properties of the galaxies in the cluster and in its environment.Comment: 14 pages - figures included - A&A (re)submitted versio