Renormalization

A lattice Boltzmann model which simulates and predicts, on a massively parallel computer, wave propagation in urban environments is presented. This technique takes into account complicated boundary conditions. Two-dimensional simulations are performed starting from a city map and a renormalisation scheme is proposed to extend the results to three dimensions and adjust the wave length. The method, which is simple and easy to implement, provides good path loss predictions when compared with in situ measurements.

In this paradigm shift, we embrace an updated, earlier default paradigm, which enables us to avoid narrow disciplinary confines, and to examine the entire natural order of increasing complexity from a central perspective of sentience. We replace the assumed and fundamentally flawed insentient natural world myth  with another, better myth meeting empirical demands. A single axiomatic change, reflecting this view, enables us to restructure common science, and to utilize our own experience as a first principled basis for broad empirical investigation.

Even the uninitiated will know that Quantum Field Theory cannot be introduced systematically in just four lectures. I try to give a reasonably connected outline of part of it, from second quantization to the path-integral technique in Euclidean space, where there is an immediate connection with the rules for Feynman diagrams and the partition function of Statistical Mechanics.

We calculate one-loop R-parity-violating couplings corrections to the processes H − → τν τ and H − → bt. We find that the corrections to the H − → τν τ decay mode are generally about 0.1%, and can be negligible. But the corrections to the H − → bt decay mode can reach a few percent for the favored parameters.

This manuscript stands at the interface between combinatorial Hopf algebra theory and renormalization theory. Its plan is as follows: Section 1 is the introduction, and contains as well an elementary invitation to the subject. The rest of part I, comprising Sections 2-6, is devoted to the basics of Hopf algebra theory and examples, in ascending level of complexity. Part II turns around the all-important Faà di Bruno Hopf algebra. Section 7 contains a first, direct approach to it. Section 8 gives applications of the Faà di Bruno algebra to quantum field theory and Lagrange reversion. Section 9 rederives the related Connes-Moscovici algebras. In Part III we turn to the Connes-Kreimer Hopf algebras of Feynman graphs and, more generally, to incidence bialgebras. In Section 10 we describe the first. Then in Section 11 we give a simple derivation of (the properly combinatorial part of) Zimmermann's cancellation-free method, in its original diagrammatic form. In Section 12 general incidence algebras are introduced, and the Faà di Bruno bialgebras are described as incidence bialgebras. In Section 13, deeper lore on Rota's incidence algebras allows us to reinterpret Connes-Kreimer algebras in terms of distributive lattices. Next, the general algebraic-combinatorial proof of the cancellation-free formula for antipodes is ascertained; this is the heart of the paper. The structure results for commutative Hopf algebras are found in Sections 14 and 15. An outlook section very briefly reviews the coalgebraic aspects of quantization and the Rota-Baxter map in renormalization.

Using {\it weighted traces} which are linear functionals of the type $$A\to tr^Q(A):=(tr(A Q^{-z})-z^{-1} tr(A Q^{-z}))_{z=0}$$ defined on the whole algebra of (classical) pseudo-differential operators (P.D.O.s) and where $Q$ is some positive invertible elliptic operator, we investigate the geometry of loop groups in the light of the cohomology of pseudo-differential operators. We set up a geometric framework to study a class of infinite dimensional manifolds in which we recover some results on the geometry of loop groups, using again weighted traces. Along the way, we investigate properties of extensions of the Radul and Schwinger cocycles defined with the help of weighted traces.

A review is given on the theory of vortex-glass phases in impure type-II superconductors in an external field. We begin with a brief discussion of the effects of thermal fluctuations on the spontaneously broken U (1) and translation symmetries, on the global phase diagram and on the critical behaviour. Introducing disorder we restrict ourselves to the experimentally most relevant case of weak uncorrelated randomness which is known to destroy the long-ranged translational order of the Abrikosov lattice in three dimensions. Elucidating possible residual glassy ordered phases, we distinguish between positional and phase-coherent vortex glasses. The study of the behaviour of isolated vortex lines and their generalization -directed elastic manifolds -in a random potential introduces further important concepts for the characterization of glasses. The discussion of elastic vortex glasses, i.e., topologically ordered dislocation-free positional glasses in two and three dimensions occupy the main part of our review. In particular, in three dimensions there exists an elastic vortex-glass phase which still shows quasi-long-range translational order: the 'Bragg glass'. It is shown that this phase is stable with respect to the formation of dislocations for intermediate fields. Preliminary results suggest that the Bragg-glass phase may not show phase-coherent vortex-glass order. The latter is expected to occur in systems with weak disorder only in higher dimensions (or for strong disorder, as the example of unscreened gauge glasses shows). We further demonstrate that the linear resistivity vanishes in the vortex-glass phase. The vortex-glass transition is studied in detail for a superconducting film in a parallel field. Finally, we review some recent developments concerning driven vortex-line lattices moving in a random environment.

In a framework of the renormalizable theory of weak interaction, problems of CP-violation are studied. It is concluded that no realistic models of CP-violation exist in the quartet scheme without introducing any other new fields. Some possible models of CP-violation are also discussed.

The Cornwall-Jackiw-Tomboulis (CJT) effective action for composite operators at finite temperature is used to investigate the chiral phase transition within the framework of the linear sigma model as the low-energy effective model of quantum chromodynamics (QCD). A new renormalization prescription for the CJT effective action in the Hartree-Fock (HF) approximation is proposed. A numerical study, which incorporates both thermal and quantum effect, shows that in this approximation the phase transition is of first order. However, taking into account the higher-loop diagrams contribution the order of phase transition is unchanged.

Second order perturbative corrections to electron wavefunction are calculated here at generalized temperature, for the first time. This calculation is important to prove the renormalizeability of QED through order by order cancellation of singularities at higher order. This renormalized wavefunction could be used to calculate the particle processes in the extremely hot systems such as the very early universe and the stellar cores. We have to re-write the second order thermal correction to electron mass in a convenient way to be able to calculate the wavefunction renormalization constant. A procedure for integrations of hot loop momenta before the cold loop momenta integration is maintained throughout to be able to remove hot singularities in an appropriate way. Our results, not only includes the intermediate temperatures T ∼ m (where m is the electron mass), the limits of high temperature T >> m and low temperature T << m are also retrievable. A comparison is also done with the existing results.

Recent evidence suggests that quantum coherence enhances excitation energy transfer (EET) through individual photosynthetic light-harvesting protein complexes (LHCs). Its role in vivo is unclear however, where transfer to chemical reaction centres (RCs) spans larger, multi-LHC/RC networks. Here we predict maximum coherence lengths possible in fully connected chromophore networks with the generic structural and energetic features of multi-LHC/RC networks. A renormalization analysis reveals the dependence of EET dynamics on multiscale, hierarchical network structure. Surprisingly, thermal decoherence rate declines at larger length scales for physiological parameters and coherence length is instead limited by localization due to static disorder. Physiological parameters support coherence lengths up to ∼5 nm, which is consistent with observations of solvated LHCs and invites experimental tests for intercomplex coherences in multi-LHC/RC networks. Results further suggest that a semiconductor quantum dot network engineered with hierarchically clustered structure and small static disorder may support coherent EET over larger length scales, at ambient temperatures.

Dynamics of the three-dimensional flow in a cyclone with tangential inlet and tangential exit were studied using particle tracking velocimetry (PTV) and a three-dimensional computational model. The PTV technique is described in this paper and appears to be well suited for the current flow situation. The flow was helical in nature and a secondary recirculating flow was observed and well predicted by computations using the RNG k-e turbulence model. The secondary flow was characterized by a single vortex which circulated around the axis and occupied a large fraction of the cylinder diameter. The locus of the vortex center meandered around the cylinder axis, making one complete revolution for a cylinder aspect ratio of 2. Tangential velocities from both experiments and computations were compared and found to be in good agreement. The general structure of the flow does not vary significantly as the Reynolds number is increased. However, slight changes in all components of velocity and pressure were seen as the inlet velocity is increased. By increasing the inlet aspect ratio it was observed that the vortex meandering changed significantly.

A collocation method is adopted as a numerical framework to develop approximate inertial manifolds (AIMs) in the case of partial differential problems (e.g. reaction/diffusion models) containing non-polynomial nonlinearities. The spatial discretization, based on the collocation approach, is the starting point for the alternative construction of AIMs by means of a renormalization/ decimation approach naturally derived from the incremental unknown method developed by Temam in a finite difference framework.

SPheno is a program that accurately calculates the supersymmetric particle spectrum within a high scale theory, such as minimal supergravity, gauge mediated supersymmetry breaking, anomaly mediated supersymmetry breaking, or string effective field theories. An interface exists for an easy implementation of other high scale models. The program solves the renormalization group equations numerically to two-loop order with user-specified boundary conditions. The complete one-loop formulas for the masses are used which are supplemented by two-loop contributions in case of the neutral Higgs bosons and the µ parameter. The obtained masses and mixing matrices are used to calculate decay widths and branching ratios of supersymmetric particles as well as of Higgs bosons, b → sγ, ∆ρ and (g − 2) µ . Moreover, the production cross sections of all supersymmetric particle as well as Higgs bosons at e + e − colliders can be calculated including initial state radiation and longitudinal polarization of the incoming electrons/positrons. The program is structured such that it can easily be extend to include non-minimal models and/or complex parameters.

Using weighted traces which are linear functionals of the type A → tr Q (A) := tr(AQ −z ) − z −1 tr(AQ −z ) z=0 defined on the whole algebra of (classical) pseudo-differential operators (P.D.O.s) and where Q is some positive invertible elliptic operator, we investigate the geometry of loop groups in the light of the cohomology of pseudo-differential operators. We set up a geometric framework to study a class of infinite dimensional manifolds in which we recover some results on the geometry of loop groups, using again weighted traces. Along the way, we investigate properties of extensions of the Radul and Schwinger cocycles defined with the help of weighted traces.

Supersymmetry or ad hoc methods such as renormalization are often used to tame infinities that result from divergent functions in quantum physics. Although SUSY particles have yet to be discovered and may be too massive to fulfill their purpose, and, renormalization seems to lack mathematical rigor. Here we offer an alternative method that employs the least-action and Heisenberg uncertainty principles.

Renormalization of composite fields is employed to suppress the statistical noise in lattice gauge calculations. We propose a new action which differs from the standard Wilson action by "irrelevant" operators, but suppresses the fluctuations of the plaquette. We numerically study the Creutz ratios and find a scaling window. The SU(2) mass gap is estimated. We prove that the contributions of the "irrelevant" operators to the screening mass decrease towards the continuum limit. The results obtained from the action with noise suppression are compared with those of the standard Wilson action.

We review the present status of QCD corrections to weak decays beyond the leading logarithmic approximation including particle-antiparticle mixing and rare and CP violating decays. After presenting the basic formalism for these calculations we discuss in detail the effective hamiltonians for all decays for which the next-to-leading corrections are known. Subsequently, we present the phenomenological implications of these calculations. In particular we update the values of various parameters and we incorporate new information on mt in view of the recent top quark discovery. One of the central issues in our review are the theoretical uncertainties related to renormalization scale ambiguities which are substantially reduced by including next-to-leading order corrections. The impact of this theoretical improvement on the determination of the Cabibbo-Kobayashi-Maskawa matrix is then illustrated in

Efficient methods for determining the lower and upper bounds on the probabilities of source-to-terminal communication in a multistage interconnection network are developed. A novel lower bounding strategy (shifting) and a novel upper bounding strategy (renormalization) are presented; both can be computed in polynomial time. These strategies can be combined with existing methods based on coherence, and on consecutive cuts, to obtain an improvement on previously known efficiently computable bounds. A second efficient upper bound (averaging) is developed. An empirical evaluation of the bounds is discussed. Finally, the value of these bounding strategies in assessing the reliability of interconnection networks is examined.

Topological challenges to the geometry of globalization and polarization
Space-time crystals as fundamental to comprehension of global order
Via sphere, torus and hyperbola to space-time crystals of governance?
Comprehension of requisite complexity through game-ball design?
Contrasting forms of coherence framed by positive and negative curvature
Space-time crystals as space-time polyhedra fundamental to appropriate organization
Imagining complementary images of the shape of civilization
Logic of renormalization as enabling flying metaphorically understood?

Using normal coordinates in a Poincaré-Birkhoff-Witt basis for the Hopf algebra of renormalization in perturbative quantum field theory, we investigate the relation between the twisted antipode axiom in that formalism, the Birkhoff algebraic decomposition and the universal formula of Kontsevich for quantum deformation.

This analysis shows that divergence issues in QFT can be resolved without mass and charge renormalization by postulating a two-component theory of the vacuum whose net energy and charge are zero. For the free fields, the quantum vacuum is uniformly homogeneous and isotropic, but interacting fields break the uniformity as vacuum energy is redistributed. For an irreducible self-interaction amplitude Ω, infinite field actions divide the vacuum into positive and negative self-energy components such that the net vacuum energy correction to a scattering process is finite in the presence of external fields but vanishes for free particles. For each particle mass in a loop, two mass states dressed with vacuum energy, are constructed for fermion and boson self-energy processes. For electroweak interactions, the stabilized amplitude Ω^ = Ω - <Ω> includes a correction for a vacuum energy deficit within an infinitesimal near-field (INF) region, where <Ω> is given by an average of Ω over dressed mass levels. For QCD, strong interactions redistribute vacuum energy so that there is a surplus in the INF with a corresponding deficit in the surrounding far-field resulting in a sign reversal of Ω^ relative to QED and asymptotic freedom. Stabilized amplitudes are shown to agree with renormalization for radiative corrections in Abelian and non-Abelian gauge theories.

We compute the graviton one loop contribution to a classical energy in a traversable wormhole background. Such a contribution is evaluated by means of a variational approach with Gaussian trial wave functionals. A zeta function regularization is involved to handle with divergences. A renormalization procedure is introduced and the finite one loop energy is considered as a selfconsistent source for the traversable wormhole.

The effects of quantum fluctuations in fields confined by background configurations may be simply and transparently computed using the Green's function approach pioneered by Schwinger. Not only can total energies and surface forces be computed in this way, but local energy densities, and in general, all components of the vacuum expectation value of the energy-momentum tensor may be calculated. For simple geometries this approach may be carried out exactly, which yields insight into what happens in less tractable situations. In this talk I will concentrate on the example of a scalar field in a circular cylindrical delta-function background. This situation is quite similar to that of a spherical delta-function background. The local energy density in these cases diverges as the surface of the background is approached, but these divergences are integrable. The total energy is finite in strong coupling, but in weak coupling a divergence occurs in third order. This universal feature is shown to reflect a divergence in the energy associated with the surface, the integrated local energy density within the shell itself, which surface energy should be removable by a process of renormalization.

Further arguments are offered in defence of the position that the variant of quantum field theory (QFT) that should be subject to interpretation and foundational analysis is axiomatic quantum field theory. I argue that the successful application of renormalization group (RG) methods ...

The Higgs naturalness principle served as the basis for the so far failed prediction that signatures of physics beyond the Standard Model (SM) would be discovered at the LHC. One influential formulation of the principle , which prohibits fine tuning of bare Standard Model (SM) parameters, rests on the assumption that a particular set of values for these parameters constitute the "fundamental parameters" of the theory, and serve to mathematically define the theory. On the other hand, an old argument by Wetterich suggests that fine tuning of bare parameters merely reflects an arbitrary, inconvenient choice of expansion parameters and that the choice of parameters in an EFT is therefore arbitrary. We argue that these two interpretations of Higgs fine tuning reflect distinct ways of formulating and interpreting effective field theories (EFTs) within the Wilsonian framework: the first takes an EFT to be defined by a single set of physical, fundamental bare parameters, while the second takes a Wilsonian EFT to be defined instead by a whole Wilsonian renormalization group (RG) trajectory, associated with a one-parameter class of physically equivalent parametrizations. From this latter perspective, no single parametrization constitutes the physically correct, fundamental parametrization of the theory , and the delicate cancellation between bare Higgs mass and quantum corrections appears as an eliminable artifact of the arbitrary, unphysical reference scale with respect to which the physical amplitudes of the theory are parametrized. While the notion of fundamental parameters is well motivated in the context of condensed matter field theory, we explain why it may be superfluous in the context of high energy physics.

We examine the renormalization of an effective theory description of a general initial state set in an isotropically expanding space-time, which is done to understand how to include the effects of new physics in the calculation of the cosmic microwave background power spectrum. The divergences that arise in a perturbative treatment of the theory are of two forms: those associated with the properties of a field propagating through the bulk of space-time, which are unaffected by the choice of the initial state, and those that result from summing over the short-distance structure of the initial state. We show that the former have the same renormalization and produce the same subsequent scale dependence as for the standard vacuum state, while the latter correspond to divergences that are localized at precisely the initial time hypersurface on which the state is defined. This class of divergences is therefore renormalized by adding initial-boundary counterterms, which render all of the p...

Let δ(t) denote the Dirac delta function. We show how, when the renormalization constant c > 0 in δ 2 (t) = c δ(t) is large or approaches +∞, the commutation relations for the Renormalized Powers of Quantum White Noise (RPQWN) can be truncated to yield either the Heisenberg Canonical Commutation Relations (CCR) or the Renormalized Square of White Noise (RSWN) commutation relations of , parametrized by the order of the white noise functionals. The, still open, problem of choosing a renormalization of the powers of the delta function that will lead to a Fock representation of the RPQWN commutation relations is described.

We present calculations of the magnetic-field-induced changes of the heavy quasiparticles in YbRh 2 Si 2 which are reflected in thermodynamic and transport properties. The quasiparticles are determined by means of the Renormalized Band Method. The progressive de-renormalization of the quasiparticles in the magnetic field is accounted for using field-dependent quasiparticle parameters deduced from Numerical Renormalization Group studies. Consequences for the interpretation of experimental data are discussed.

We show that it is possible to find an extension of the matter content of the standard model with a unification of gauge and Yukawa couplings reproducing their known values. The perturbative renormalizability of the model with a single coupling and the requirement to accomodate the known properties of the standard model fix the masses and couplings of the additional particles. The implications on the parameters of the standard model are discussed.

Biological thinking is structured by the notion of level of organization. We will show that this notion acquires a precise meaning in critical phenomena: they disrupt, by the appearance of infinite quantities, the mathematical (possibly equational) determination at a given level, when moving at an "higher" one. As a result, their analysis cannot be called genuinely bottom-up, even though it remains upward in a restricted sense. At the same time, criticality and related phenomena are very common in biology. Because of this, we claim that bottom-up approaches are not sufficient, in principle, to capture biological phenomena. In the second part of this paper, following (Bailly, 1991b), we discuss a strong criterium of level transition. The core idea of the criterium is to start from the breaking of the symmetries and determination at a "first" level in order to "move" at the others. If biological phenomena have multiple, sustained levels of organization in this sense, then they should be interpreted as extended critical transitions.

We compute the one loop fermion self-energy for massless Dirac + Einstein in the presence of a locally de Sitter background. We employ dimensional regularization and obtain a fully renormalized result by absorbing all divergences with BPHZ counterterms. An interesting technical aspect of this computation is the need for a noninvariant counterterm owing to the breaking of de Sitter invariance by our gauge condition. Our result can be used in the quantum-corrected Dirac equation to search for inflation-enhanced quantum effects from gravitons, analogous to those which have been found for massless, minimally coupled scalars.

The one-loop radiative corrections to W ^{+}W^{-} to W^ {+}W^{-}, W^0 W^0 to W^{+}W^{-} and W^0 W^0 to W^0 W^0 scatterings are calculated in the limit where m_{higgs } gg E_{c.m.} gg M_ {W}. Leading terms that grow like E _sp{c.m.}{4} and E _sp{c.m.}{4}log(E _sp{c.m.}{2}) are given. The results are shown to be independent of the experiments chosen to fix the free parameters of the theory.

We present estimates of the masses of light quarks using lattice data. Our main results are based on a global analysis of all the published data for Wilson, Sheikholeslami-Wohlert (clover), and staggered fermions, both in the quenched approximation and with $n_f=2$ dynamical flavors. We find that the values of masses with the various formulations agree after extrapolation to the continuum limit for the $n_f=0$ theory. Our best estimates, in the MSbar scheme at $\mu=2 GeV$, are $\mbar=3.4 +- 0.4 +- 0.3 MeV$ and $m_s = 100 +- 21 +- 10 MeV$ in the quenched approximation. The $n_f=2$ results, $\mbar = 2.7 +- 0.3 +- 0.3 MeV$ and $m_s = 68 +- 12 +- 7 MeV$, are preliminary. (A linear extrapolation in $n_f$ would further reduce these estimates for the physical case of three dynamical flavors.) These estimates are smaller than phenomenological estimates based on sum rules, but maintain the ratios predicted by chiral perturbation theory. The new results have a significant impact on the extraction of $\epsilon'/\epsilon$ from the Standard Model. Using the same lattice data we estimate the quark condensate using the Gell-Mann-Oakes-Renner relation. Again the three formulations give consistent results after extrapolation to $a=0$, and the value turns out to be correspondingly larger, roughly preserving $m_s \vev{\bar \psi \psi}$.

We review the theoretical foundations and the most important physical applications of the Pinch Technique (PT). This general method allows the construction of off-shell Green's functions in non-Abelian gauge theories that are independent of the gauge-fixing parameter and satisfy ghost-free Ward identities. We first present the diagrammatic formulation of the technique in QCD, deriving at one loop the gauge independent gluon self-energy, quark-gluon vertex, and three-gluon vertex, together with their Abelian Ward identities. The generalization of the PT to theories with spontaneous symmetry breaking is carried out in detail, and the profound connection with the optical theorem and the dispersion relations are explained within the electroweak sector of the Standard Model. The equivalence between the PT and the Feynman gauge of the Background Field Method (BFM) is elaborated, and the crucial differences between the two methods are critically scrutinized. A variety of field theoretic techniques needed for the generalization of the PT to all orders are introduced, with particular emphasis on the Batalin-Vilkovisky quantization method and the general formalism of algebraic renormalization. The main conceptual and technical issues related to the extension of the technique beyond one loop are described, using the twoloop construction as a concrete example. Then the all-order generalization is thoroughly examined, making extensive use of the field theoretic machinery previously introduced; of central importance in this analysis is the demonstration that the PT-BFM correspondence persists to all orders in perturbation theory. The extension of the PT to the non-perturbative domain of the QCD Schwinger-Dyson equations is presented systematically, and the main advantages of the resulting self-consistent truncation scheme are discussed. A plethora of physical applications relying on the PT are finally reviewed, with special emphasis on the definition of gauge-independent off-shell form-factors, the construction of non-Abelian effective charges, the gauge-invariant treatment of resonant transition amplitudes and unstable particles, and finally the dynamical generation of an effective gluon mass. ρ parameter beyond one loop 128 5.4 Self-consistent resummation formalism for resonant transition amplitudes 131 5.4.1 The Breit-Wigner Ansatz and the Dyson summation 131 5.4.2 The non-Abelian setting 133 6 Beyond one loop: from two loops to all orders 143 6.1 The pinch technique at two loops 143 6.1.1 The one-particle reducible graphs 144 6.1.2 Quark-gluon vertex and gluon self-energy at two loops 146 6.1.3 The two-loop absorptive construction 151 6.2 The PT to all orders in perturbation theory 157 6.2.1 The four-point kernel AAqq and its Slavnov-Taylor identity 157 6.2.2 The fundamental all-order s-t cancellation 159 6.2.3 The PT to all orders: the quark-gluon vertex and the gluon propagator 162 7 PT in the Batalin-Vilkovisky framework 166 7.1 Green's functions: conventions 167 4 7.2 The Batalin-Vilkovisky formalism for pedestrians 168 7.3 Faddeev-Popov equation(s) 172 7.4 The (one-loop) PT algorithm in the BV language 173 7.5 The two-loop case 174 8 The PT Schwinger-Dyson Equations for QCD Green's functions 178 8.1 SDEs for non-Abelian gauge theories: difficulties with the conventional formulation 178 8.2 The PT algorithm for Schwinger-Dyson equations 180 8.2.1 Three-gluon vertex 181 8.2.2 The gluon propagator 186 8.3 The new Schwinger-Dyson series 190 8.3.1 The PT as a gauge-invariant truncation scheme: advantages over the conventional SDEs 191 8.3.2 Some important theoretical and practical issues 195 9 Applications part II: Infrared properties of QCD Green's functions and dynamically generated gluon mass 199 9.1 PT Schwinger-Dyson equations for the gluon and ghost propagators 201 9.2 Schwinger mechanism, dynamical gauge-boson mass generation, and bound-state poles 205 9.3 Results and comparison with the lattice 208 9.4 The non-perturbative effective charge of QCD 210 10 Concluding remarks 214 A SU (N ) group theoretical identities 216 B Feynman rules 217 B.1 R ξ and BFM gauges 217 B.2 Anti-fields 218 B.3 BFM sources 218 5 C Faddev-Popov equations, Slavnov-Taylor Identities and Background Quantum Identities for QCD 220 C.1 Faddeev-Popov Equations 220 C.2 Slavnov-Taylor Identities 221 C.2.1 STIs for gluon proper vertices 221 C.2.2 STIs for mixed quantum/background Green's functions 223 C.2.3 STIs for the gluon SD kernel 224 C.3 Background-Quantum Identities 226 C.3.1 BQIs for two-point functions 226 C.3.2 BQIs for three-point functions 228 C.3.3 BQI for the ghost-gluon trilinear vertex 230 References 231

This article is devoted to the perturbative renormalization of the abelian Higgs-Kibble model, within the class of renormalizable gauges which are odd under charge conjugation. The Bogoliubov Parasiuk Hepp-Zimmermann renormalization scheme is used throughout, including the renormalized action principle proved by Lowenstein and Lam. The whole study is based on the fulfillment to all orders of perturbation theory of the Slavnov identities which express the invariance of the Lagrangian under a supergauge type family of non-linear transformations involving the Faddeev-Popov ghosts. Direct combinatorial proofs are given of the gauge independence and unitarity of the physical S operator. Their simplicity relies both on a systematic use of the Slavnov identities as well as suitable normalization conditions which allow to perform all mass renormalizations, including those pertaining to the ghosts, so that the theory can be given a setting within a fixed Fock space. Some simple gauge independent local operators are constructed.

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We study the fields rp and IJ in a class of finite quantum field models on the cylindrical space-time S' x R.. We show that cp and + satisfy a system of local, nonlinear partial differential equations. These are just the equations of evolution for the critical points of the corresponding action functional.