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Proceedings of ISMD08, Comment: Edited by: J. Bartels, K. Borras, G. Gustafson, H. Jung, K. Kutak, S. Levonian, and J. Mnich

We discuss the problem of incorporating recoil effects into the probabilistic QCD evolution scheme based on the picture of colour dipoles as done in recent Monte Carlo programs. Such a scheme correctly describes subleading soft contributions to multiplicity distributions. However we find that a simple receipt for incorporating recoil effects into the dipole multiplication picture conflicts the collinear factorization and does not lead to the correct DGLAP equation., Comment: accepted for publication by JHEP on 21.02.2009; to appear in March 2009

We consider the bosonic sl(2) sector of the maximally supersymmetric N=4 SYM model and show that anomalous dimension of the twist-3 single-trace composite operators built of scalar fields, recently calculated up to the four-loop order, can be generated by a compact reciprocity respecting evolution kernel., Comment: 18 pages, 1 eps figure

We construct universal parton evolution equation that produces space- and time-like anomalous dimensions for the maximally super-symmetric N=4 Yang--Mills field theory model, and find that its kernel satisfies the Gribov--Lipatov reciprocity relation in three loops. Given a simple structure of the evolution kernel, this should help to generate the major part of multi-loop contributions to QCD anomalous dimensions, due to classical soft gluon radiation effects., Comment: 19 pages, 1 figure

This remark is part of an ongoing project to simplify the structure of the multi-loop anomalous dimensions for parton distributions and fragmentation functions. It answers the call for a "structural explanation" of a "very suggestive" relation found by Moch, Vermaseren and Vogt in the context of the x->1 behaviour of three-loop DIS anomalous dimensions. It also highlights further structure that remains to be fully explained., Comment: 6 pages, v2 corrects misprints and contains an additional reference

A general discussion is presented of the single logarithmic soft factor that appears in two scale QCD observables in processes involving four partons. We treat it as the ``fifth form factor'', accompanying the four collinear singular Sudakov form factors attached to colliding and outgoing hard partons. The fifth form factor is expressed in terms of the Casimir operators (squared colour charges) of irreducible representations in the crossing $t$- and $u$-channels. As an application we revisit the problem of large angle radiation in $gg\to gg$ and give a relatively simple solution and interpretation of the results. We found an unexpected symmetry of the soft anomalous dimension under exchange of internal and external variables of the problem whose existence calls for explanation., Comment: 29 pages, pictures generated with feynmf graphic package

Logarithmically enhanced effects due to radiation of soft gluons at large angles in $2\to 2$ QCD scattering processes are treated in terms of the "fifth form factor" that accompanies the four collinear singular Sudakov form factors attached to incoming and outgoing hard partons. Unexpected symmetry under exchange of internal and external variables of the problem is pointed out for the anomalous dimension that governs soft gluon effects in hard gluon--gluon scattering., Comment: 8 pages

It has been recently found that the heavy quark-antiquark QQbar pair multiplicity, in certain phase space region (QQbar at short distance, soft and with small velocity), satisfies an evolution equation formally similar to the BFKL equation for the high energy scattering amplitude. We find the exact solution of the QQbar-equation and discuss the differences with the BFKL scattering amplitude., Comment: 18 pages, 3 figures, using JHEP3.cls Corrected typos

We calculate $e^+e^- \to Q\bar{Q}(k) + anything in a certain momentum, k, region for heavy quark-antiquark (Q\bar{Q})production. In our chosen region we find that the number of heavy quark pairs produced is determined by BFKL dynamics and the energy dependence of the number of pairs is given by $\alpha p-1$ the hard pomeron intercept., Comment: 14 pages, Latex, 4 figures

Jet shape observables which involve measurements restricted to a part of phase space are sensitive to multiplication of soft gluon with large relative angles and give rise to specific single logarithmically enhanced (SL) terms (non-global logs). We consider associated distributions in two variables which combine measurement of a jet shape V in the whole phase space (global) and that of the transverse energy flow away from the jet direction, Eout (non-global). We show that associated distributions factorize into the global distribution in V and a factor that takes into account SL contributions from multi-gluon ``hedgehog'' configurations in all orders. The latter is the same that describes the single-variable Eout distribution, but evaluated at a rescaled energy VQ., Comment: 16 pages

A new release of the Monte Carlo program HERWIG (version 6.5) is now available. The main new features are: support for the Les Houches interface to matrix element generators; additional SM and MSSM Higgs processes in lepton collisions; additional matrix elements for the spin correlation algorithm; a new version of the ISAWIG interface; interface to the MC@NLO program for heavy quark, Higgs and vector boson production in hadron collisions. boson pair production in hadron collisions. This is planned to be the last major release of Fortran HERWIG. Future developments will be implemented in a new C++ event generator, HERWIG++., Comment: Updated to version 6.510, released Oct 31, 2005. Program available at http://hepwww.rl.ac.uk/theory/seymour/herwig/ or http://home.cern.ch/seymour/herwig/

We introduce a new angular correlation in DIS process and study its differential distribution in the region in which the observable is small. We perform a perturbative resummation at single logarithmic accuracy and estimate leading non-perturbative power corrections., Comment: 4 pages, 2 figures. Talk presented by Giuseppe Marchesini at the XXXVIIth Rencontres de Moriond `QCD and high energy hadronic interactions', Les Arcs, France, and by Andrea Banfi at the X International Workshop on Deep Inelastic Scattering (DIS2002), Krakow, Poland

We consider interjet observables in hard QCD processes given by the energy flow E_out in a region away from all hard jets. Here the QCD radiation is depleted (E_out << Q), and therefore these observables provide ideal means to study non-perturbative effects. We derive an evolution equation (in the large N_c limit) which resums, for large Q/E_out, all leading terms arising from large angle soft emission (double logarithms are absent). We discuss the analytical features of the result and identify universal and geometry-dependent contributions. Our analysis confirms features found using numerical methods by Dasgupta and Salam., Comment: 29 pages, 2 figures, JHEP class included. Paper version

A new release of the Monte Carlo program HERWIG (version 6.4) is now available. The main new features are: spin correlations between the production and decay of heavy fermions, i.e. top quarks, tau leptons and SUSY particles; polarization effects in SUSY production processes in lepton-lepton collisions; an interface to TAUOLA for tau decays; MSSM Higgs processes in lepton-lepton collisions., Comment: 3 pages, program available at http://hepwww.rl.ac.uk/theory/seymour/herwig/

I recall some points on the present status of QCD results in the short distance regions and illustrate the case of event shape in QCD radiation, Comment: 11 pages, talk given at XXXI International Symposium on Multiparticle Dynamics, Sept 1-7, 2001, Datong China. URL http://ismd31.ccnu.edu.cn/

We present a QCD analysis of the cumulative out-of-event-plane momentum distribution in DIS process with emission of high p_t jets. We derive the all-order resummed result to next-to-leading accuracy and estimate the leading power correction. We aim at the same level of accuracy which, in e+e- annihilation, seems to be sufficient for making predictions. As is typical of multi-jet observables, the distribution depends on the geometry of the event and the underlying colour structure. This result should provide a powerful method to study QCD dynamics, in particular to constrain the parton distribution functions, to measure the running coupling and to search for genuine non-perturbative effects., Comment: 40 pages, 4 figures, JHEP class included. Version 2 corrects some formulae in appendix B and F and updates the figures and one reference

A new release of the Monte Carlo program HERWIG (version 6.3) is now available. The main new features are new (MRST) built-in parton distribution functions, more SM gauge boson production processes, 2 to 3 MSSM Higgs production processes, an option to handle negative event weights, and an interface to the CIRCE beamstrahlung program., Comment: 10 pages, program available at http://hepwww.rl.ac.uk/theory/seymour/herwig/ Corrected PDF nomenclature

We present the QCD analysis of the cumulative out-of-event-plane momentum distribution in the process proton-antiproton into Z0 and a hard jet (event plane defined by the proton, antiproton and Z0 momenta). Particular attention is placed on the near-to-planar events for which we derive the all-order resummed result to next-to-leading accuracy. We consider also the leading power correction originating from the fact that, even in hard processes, the resummed QCD coupling runs into the infrared region. We aim at the same level of accuracy which, in e+e- annihilation, seems to be sufficient for making predictions. Contributions from a ``soft underlying event'' due to beam remnant interactions are discussed. Experimental data (not yet available) are needed to cast light on the predictive level of standard QCD analysis in hard hadron-hadron collisions. We plot examples of the predicted distribution at Tevatron energies. The techniques here developed can be extended to other hard hadron-hadron and hadron-lepton processes., Comment: 36 pages, 3 figures, JHEP class included

We present the QCD analysis of D-parameter distribution in near-to-planar 3-jet e+e- annihilation events. We derive the all-order resummed perturbative prediction and the leading power suppressed non-perturbative corrections both to the mean value and the distribution. Here non-perturbative corrections are larger than in 2-jet shape observables, so that higher order non-perturbative effects could be relevant. Experimental data (not yet available) are needed in order to cast light on this important point. The technique we develop aims at improving the accuracy of the theoretical description of multi-jet ensembles, in particular in hadron-hadron collisions., Comment: 26 pages, 6 figures, JHEP class included

We present the analysis of the 1/Q-suppressed non-perturbative (NP) contribution to the cumulative out-of-event-plane momentum distribution in e+e- annihilation in the near-to-planar three-jet region. It complements our previous study of the perturbative (PT) distributions resummed to single logarithmic accuracy. Due to inter-jet soft gluon radiation, the NP contributions (as well as the PT distributions) are sensitive to the geometry (the angles between jets) and the colour structure of the underlying hard process (topology of the three-prong parton antenna). The results and techniques presented here could be extended to other multi-jet processes and, in particular, to hadron-hadron collisions., Comment: 53 pages, 4 figures, JHEP class included

HERWIG is a general-purpose Monte Carlo event generator, which includes the simulation of hard lepton-lepton, lepton-hadron and hadron-hadron scattering and soft hadron-hadron collisions in one package. It uses the parton-shower approach for initial- and final-state QCD radiation, including colour coherence effects and azimuthal correlations both within and between jets. This article updates the description of HERWIG published in 1992, emphasising the new features incorporated since then. These include, in particular, the matching of first-order matrix elements with parton showers, a more correct treatment of heavy quark decays, and a wide range of new processes, including many predicted by the Minimal Supersymmetric Standard Model, with the option of R-parity violation. At the same time we offer a brief review of the physics underlying HERWIG together with details of the input and control parameters and the output data, to provide a self-contained guide for prospective users of the program. This version of the manual (version 3) is updated to HERWIG version 6.5, which is expected to be the last major release of Fortran HERWIG. Future developments will be implemented in a new C++ event generator, HERWIG++., Comment: 112 pages, updated to version 6.5; program available at http://hepwww.rl.ac.uk/theory/seymour/herwig/ and http://home.cern.ch/seymour/herwig/

We present the results of QCD analysis of out-of-event-plane momentum distribution in 3-jet e+e- annihilation events. We consider the all-order resummed perturbative prediction and the leading power suppressed non-perturbative corrections to the mean value and the distribution and explain their non-trivial colour structure. The technique we develop aims at improving the accuracy of the theoretical description of multi-jet ensembles, in particular in hadron-hadron collisions., Comment: 11 pages, 2 figures

Perturbative QCD analysis is presented of the cumulative out-of-plane momentum distribution in the near-to-planar e+e- annihilation events, K_out << Q. In this kinematical region multiple gluon radiation effects become essential. They are resummed with the single-logarithmic accuracy, which programme includes the 2-loop treatment of the basic radiation and matching with the exact O(alpha_s^2) result. Dedicated experimental analyses of 3-jet event characteristics are of special interest for the study of the non-perturbative confinement effects., Comment: 32 pages, 1 figure, JHEP class included

A new version of the Monte Carlo program HERWIG (version 6.1) is now available. This note gives brief details of the new features relative to the previous version (5.9). A fully updated manual will follow shortly. The main new features are: supersymmetric processes (both R-parity conserving and R-parity violating) in hadron-hadron collisions; the new process e+e- to 4 jets; matrix element corrections to top decay and Drell-Yan processes; new soft underlying event options; updates to particle data tables; new LaTeX and html printout options., Comment: 11 pages, latex figures, program and further details available at http://hepwww.rl.ac.uk/theory/seymour/herwig/

The fully resummed next-to-leading-order perturbative calculation of the energy-energy correlation in $e^+e^-$ annihilation is extended to include the leading non-perturbative power-behaved contributions computed using the ``dispersive method'' applied earlier to event shape variables. The correlation between a leading (anti)quark and a gluon produces a non-perturbative 1/Q contribution, while non-perturbative effects in the quark-antiquark correlation give rise to a smaller contribution $\ln Q^2/Q^2$. In the back-to-back region, the power-suppressed contributions actually decrease much more slowly, as small non-integer powers of 1/Q, as a result of the interplay with perturbative effects. The hypothesis of a universal low-energy form for the strong coupling relates the coefficients of these contributions to those measured for other observables., Comment: 41 pages, LaTeX, 4 figures, uses JHEP.cls

We show that taking into account the interplay between perturbative and non-perturbative effects, the power-suppressed shift to the broadening distributions becomes B dependent, and the non-perturbative contribution to the mean values becomes proportional to 1/(Q\sqrt{\as(Q)}). The new theoretical treatment greatly improves the consistency of the phenomenology with the notion of the universality of confinement effects in jet shapes., Comment: 44 pages, 8 .eps figures; uses amsmath. This updated version includes the revised value for the Milan factor, which follows from the correction of a mistake in hep-ph/9707532

This article presents the results of a quantitative study of the small-x data at HERA, using the CCFM equation. The first step consists of choosing the version of the CCFM equation to be used, corresponding to selecting a particular subset of next-to-leading-logarithmic corrections --- the choice is constrained by requiring a phenomenologically reasonable small-x growth. For the time being, the parts of the splitting functions that are finite at z=0 have been left out. We then examine results for F_2^c, R, the transverse energy flow, the charged-particle transverse-momentum spectrum and the forward-jet cross section and compare to data. While some of the data is reproduced better than with DGLAP-based calculations, the agreement is not entirely satisfactory, suggesting that the approach developed here is not yet suitable for detailed phenomenology. We discuss why, and suggest directions for future work., Comment: 33 pages, 10 figures, uses cite.sty and JHEP.cls (both included). Version 2 includes additional and updated references

First attempts in the application of Numerical Stochastic Perturbation Theory (NSPT) to the problem of pushing one loop further the computation of SU(3) (SU(2)) pertubative beta function (in different schemes) are reviewed and the relevance of such a computation is discussed. Other issues include the proposal of a different strategy for gauge-fixed NSPT computations in lattice QCD., Comment: 3 pages, Latex, LATTICE98(algorithms)

We describe a new method in lattice field theory to compute observables at various values of the parameters lambda_i in the action S[phi,lambda_i]. Firstly one performs a single simulation of a ``reference action'' S[phi^r, lambda_i^r] with fixed lambda_i^r. Then the phi^r-configurations are transformed into those of a field phi distributed according to S[phi,lambda_i], apart from a ``remainder action'' which enters as a \break weight. In this way we measure the observables at values of lambda_i different from lambda_i^r. We study the performance of the algorithm in the case of the simplest renormalizable model, namely the phi^4 scalar theory on a four dimensional lattice and compare the method with the ``histogram'' technique of which it is a generalization., Comment: Latex, 23 pgs, 8 eps-figures included

We perform the two-loop analysis of the 1/Q power corrections to jet-shape variables. This step is necessary for producing reliable theoretical predictions for the relative magnitudes of genuine confinement effects. We show that the rescaling factor recently derived for the thrust case (the Milan factor) remains the same for the class of observables which includes the C-parameter, invariant jet masses, jet broadening and the energy-energy correlation measure. We list the expressions which should be used for extracting 1/Q power effects in jet shapes. We also envisage large non-perturbative effects, characterised by fractional powers of Q, in certain semi-inclusive observables such as the height of the spectrum of Drell-Yan lepton pairs with invariant mass Q and transverse momentum Q_t=0, and back-to-back energy flows in e^+e^- and DIS., Comment: 30 pages, uses JHEP.cls (included). This updated version includes the revised value for the Milan factor, which follows from the correction of a mistake in hep-ph/9707532

The perturbative all-order analysis of the jet-broadening B-distribution in the small-B region is carried out with single-logarithmic accuracy, which requires the control of both the sum of the moduli and the modulus of the sum of the transverse momenta of soft gluons. We confirm the master equation for the B-distribution derived by Catani, Turnock and Webber (CTW). Proper treatment of quark recoil is necessary at this accuracy. This effect was neglected in the CTW solution. We show that the answer can be expressed in terms of the CTW result but evaluated at a properly rescaled B value., Comment: 20 pages, 4 eps figures, includes JHEP.cls; corrected typos in equations 4.29 and 5.7

From the analysis of the perturbative expansion of the lattice regularized gluon condensate, toghether with MC data, we present evidence of OPE-unexpected dim-2 power corrections in the scaling behaviour of the Wilson loop. These can be interpreted as an indication that in lattice gauge theories the running coupling at large momentum contains contributions of order Q^2., Comment: 3 pages, 2 figures. Talk given at the Lattice97 conference, Edinburgh, UK

A review of new developments in numerical stochastic perturbation theory (NSPT) is presented. In particular, the status of the extension of the method to gauge fixed lattice QCD is reviewed and a first application to compact (scalar) QED is presented. Lacking still a general proof of the convergence of the underlying stochastic processes, a self-consistent method for testing the results is discussed., Comment: 3 pages, 1 figure. Poster presented at the Lattice97 conference, Edinburgh, UK

We address the problem of potential non-universality of the leading 1/Q power corrections to jet shapes emerging from the non-inclusive character of these observables. We consider the thrust distribution as an example and analyse the non-inclusive contributions which emerge at the two-loop level. Although formally subleading in $\as$, they modify the existing na{\"\i}ve one-loop result for the expected magnitude of the power term by a factor of order unity. Such a promotion of a subleading correction into a numerical factor is natural since the non-perturbative power terms are explicitly proportional to powers of the QCD scale $\Lambda$ which can be fixed precisely only at the two-loop level. The ``jet-shape scaling factor'' depends on the observable but remains perturbatively calculable. Therefore it does not undermine the universal nature of 1/Q power corrections, which remain expressible in terms of the universal running coupling and universal soft-gluon emission., Comment: 21 pages, no figures, LaTeX. This revised version corrects a mistake in the calculation of the two-loop correction factor. The conclusions remain unchanged

We present some evidence that in Lattice Gauge Theory the condensate contains a non-perturbative contribution proportional to $\Lambda^2$, the square of the physical scale. This result is based on an analysis of the Wilson loop plaquette expectation from Monte Carlo simulations and its perturbative expansion computed to eight loops. The analysis is not fully conclusive since the calculations are done on a finite lattice and one needs an extrapolation to infinite lattice. It has been recently suggested that in the gluon condensate a $\Lambda^2$-contribution could be present coming from the large momentum behaviour of the running coupling and not connected to operator product expansion., Comment: 9 pages, Latex, epsfig

We compute the gluon distribution in deep inelastic scattering at small x by solving numerically the angular ordering evolution equation. The leading order contribution, obtained by neglecting angular ordering, satisfies the BFKL equation. Our aim is the analysis of the subleading corrections. Although not complete - the exact next-to-leading contribution is not yet available - these corrections are important since they come from the physical property of coherence of QCD radiation. In particular we discuss the subleading correction to the BFKL characteristic function and the gluon distribution's dependence on the maximum available angle. Conformal invariance of the BFKL equation is lost, however this is not enough to bring the small-x gluon distribution into the perturbative regime: although large momentum regions are enhanced by angular ordering, the small momentum regions are not fully suppressed. As a consequence, the gluon anomalous dimension is finite and tends to the BFKL value $\gamma=1/2$ for $\alpha_S \to 0$. The main physical differences with respect to the BFKL case are that angular ordering leads to 1) a larger gluon anomalous dimension, 2) less singular behaviour for $x \to 0$ and 3) reduced diffusion in transverse momentum., Comment: 22 pages LaTeX2e, including 7 eps figures

We analyze the perturbative expansion of a condensate in the O(N) non-linear sigma model for large N on a two dimensional finite lattice. On an infinite volume this expansion is affected by an infrared renormalon. We extrapolate this analysis to the case of the gluon condensate of Yang-Mills theory and argue that infrared renormalons can be detected by performing perturbative studies even on relatively small lattices., Comment: LaTeX file, 6 figures in postscript

We show that it is possible to formulate a gauge theory starting from a local action at the ultraviolet (UV) momentum cutoff which is BRS invariant. One has to require that fields in the UV action and the fields in the effective action are not the same but related by a local field transformation. The few relevant parameters involved in this transformation (six for the $SU(2)$ gauge theory), are perturbatively fixed by the gauge symmetry., Comment: 5 pages, Latex, no figures

HERWIG (Hadron Emission Reactions With Interfering Gluons) is a multipurpose Monte Carlo event generator for the simulation of hard collisions between hadrons, leptons and photons. This note describes the new features of version 5.9. These include adding over 100 new particles, 1800 new decay modes and supporting new decay types. A complete space-time picture, including heavy meson mixing and secondary decays, is available for all event types and includes an optional colour rearrangement model. Initial state radition of photons in e^+e^- is is now included. Many other processes and features have been upgraded., Comment: 57k of gzipped ASCII text. The source code and more information can be obtained from http://surya11.cern.ch/users/seymour/herwig/

We discuss the relation between the Gell-Mann-Low beta function and the ``flowing couplings'' of the Wilsonian action $S_\L[\phi]$ of the exact renormalization group (RG) at the scale $\L$. This relation involves the ultraviolet region of $\L$ so that the condition of renormalizability is equivalent to the Callan-Symanzik equation. As an illustration, by using the exact RG formulation, we compute the beta function in Yang-Mills theory to one loop (and to two loops for the scalar case). We also study the infrared (IR) renormalons. This formulation is particularly suited for this study since: $i$) $\L$ plays the r\^ole of a IR cutoff in Feynman diagrams and non-perturbative effects could be generated as soon as $\L$ becomes small; $ii$) by a systematical resummation of higher order corrections the Wilsonian flowing couplings enter directly into the Feynman diagrams with a scale given by the internal loop momenta; $iii$) these couplings tend to the running coupling at high frequency, they differ at low frequency and remain finite all the way down to zero frequency., Comment: 19 pages, 6 figures, LaTex, uses epsfig, rotating

We consider power-behaved contributions to hard processes in QCD arising from non-perturbative effects at low scales which can be described by introducing the notion of an infrared-finite effective coupling. Our method is based on a dispersive treatment which embodies running coupling effects in all orders. The resulting power behaviour is consistent with expectations based on the operator product expansion, but our approach is more widely applicable. The dispersively-generated power contributions to different observables are given by (log-)moment integrals of a universal low-scale effective coupling, with process-dependent powers and coefficients. We analyse a wide variety of quark-dominated processes and observables, and show how the power contributions are specified in lowest order by the behaviour of one-loop Feynman diagrams containing a gluon of small virtual mass. We discuss both collinear safe observables (such as the e+e- total cross section and \tau hadronic width, DIS sum rules, e+e- event shape variables and the Drell-Yan K-factor) and collinear divergent quantities (such as DIS structure functions, e+e- fragmentation functions and the Drell-Yan cross section)., Comment: 44 LaTeX pages including 10 figures; REVISED version with some typos fixed and one reference added

We use a numerical method to obtain the weak coupling perturbative coefficients of local operators with lattice regularization. Such a method allows us to extend the perturbative expansions obtained so far by analytical Feynman diagrams calculations. In SU(3) lattice gauge theory in four dimensions we compute the first eight coefficients of the expectation value of the Wilson loop on the elementary plaquette which is related to the gluon condensate. The computed eight coefficients grow with the order much faster than predicted by the presence of the infrared renormalon associated to the dimension of the gluon condensate. However the renormalon behaviour for large order is quite well reproduced if one considers the expansion coefficients in a new coupling related to the lattice coupling by large perturbative corrections. This is expected since the lattice and continuum Lambda scales differ by almost two orders of magnitude., Comment: 11 pages, LaTeX

By using the exact renormalization group formulation we prove perturbatively the Slavnov-Taylor (ST) identities in SU(2) Yang-Mills theory. This results from two properties: {\it locality}, i.e. the ST identities are valid if their local part is valid; {\it solvability}, i.e. the local part of ST identities is valid if the couplings of the effective action with non-negative dimensions are properly chosen., Comment: 9 pages, LaTex, to be published in Phys. Lett. B

In the exact renormalization group (RG) flow in the infrared cutoff $\Lambda$ one needs boundary conditions. In a previous paper on $SU(2)$ Yang-Mills theory we proposed to use the nine physical relevant couplings of the effective action as boundary conditions at the physical point $\Lambda=0$ (these couplings are defined at some non-vanishing subtraction point $\mu \ne 0$). In this paper we show perturbatively that it is possible to appropriately fix these couplings in such a way that the full set of Slavnov-Taylor (ST) identities are satisfied. Three couplings are given by the vector and ghost wave function normalization and the three vector coupling at the subtraction point; three of the remaining six are vanishing (\eg the vector mass) and the others are expressed by irrelevant vertices evaluated at the subtraction point. We follow the method used by Becchi to prove ST identities in the RG framework. There the boundary conditions are given at a non-physical point $\Lambda=\Lambda' \ne 0$, so that one avoids the need of a non-vanishing subtraction point., Comment: 22 pages, LaTeX style, University of Parma preprint UPRF 94-412

We describe a stochastic technique which allows one to compute numerically the coefficients of the weak coupling perturbative expansion of any observable in Lattice Gauge Theory. The idea is to insert the exponential representation of the link variables $U_\mu(x) \to \exp\{A_\mu(x)/\sqrt\beta\}$ into the Langevin algorithm and the observables and to perform the expansion in \beta^{-1/2}. The Langevin algorithm is converted into an infinite hierarchy of maps which can be exactly truncated at any order. We give the result for the simple plaquette of SU(3) up to fourth loop order (\beta^{-4}) which extends by one loop the previously known series., Comment: 9 pages. + 5 figures (postscript) appended at the end, (University of Parma, Dept.of Physics, report uprf-397-1994)

The global chiral symmetry of a $SU(2)$ gauge theory is studied in the framework of renormalization group (RG). The theory is defined by the RG flow equations in the infrared cutoff $\L$ and the boundary conditions for the relevant couplings. The physical theory is obtained at $\L=0$. In our approach the symmetry is implemented by choosing the boundary conditions for the relevant couplings not at the ultraviolet point $\L=\L_0\to\infty$ but at the physical value $\L=0$. As an illustration, we compute the triangle axial anomalies., Comment: 11 pages + 1 appended EPS figure, LaTeX, UPRF 94-392