Renormalization

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.

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.

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.

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.

The structure of infrared singularities in gauge theories is intrinsically linked to
the ultraviolet divergences in the correlator of Wilson lines, upon taking the eikonal
approximation. To study these singularities, we consider Feynman diagrams with soft
gluon exchanges between these Wilson lines. Webs are closed sets of such Feynman
diagrams, with elements of the set related to each other by permutation of gluon
emission/absorption order. Webs form the exponent of the correlators of Wilson
lines, and provide the basis to start computing the soft anomalous dimension. Using
the Non-Abelian Exponentiation theorem, we compute the web mixing matrix for the
4-loop, two Wilson line web, and we calculate the kernels of the kinematic integrals
that contribute to this web. This project has two parts. The rst focuses on reviewing
how Wilson lines and their correlators arise in perturbative QCD, and what purpose
they have in calculating the soft anomalous dimension.
In the second part, we discuss the diagrams involved in the 4-loop, two Wilson line
web, the replica trick, and Non-Abelian Exponentiation theorem. With these tools,
we compute the kinematic integrals of the Feynman diagrams in the 4-loop, two
Wilson line web, and we analyze their structure. We then proceed to consider the
colour structures that arise when we use the eective vertex basis.

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.

In this article, we define and study a geometry on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an integer parameter $N$. Then we emulate the theory of random matrices in a combinatorial framework: for any parameter $N$, we introduce a family of linear forms on the partition algebras which allows us to define a notion of weak convergence similar to the convergence in moments in random matrices theory.

A renormalization of the partition algebras allows us to consider the weak convergence as a simple convergence in a fixed space. This leads us to the definition of a deformed partition algebra for any integer parameter $N$ and to the definition of two transforms: the cumulants transform and the exclusive moments transform. Using an improved triangular inequality for the distance defined on partitions, we prove that the deformed partition algebras, endowed with a deformation of the linear forms converge as $N$ go to infinity. This result allows us to prove combinatorial properties about geodesics and a convergence theorem for semi-groups of functions on partitions.

At the end we study a sub-algebra of functions on infinite partitions with finite support : a new addition operation and a notion of $\mathcal{R}$-transform are defined. We introduce the set of multiplicative functions which becomes a Lie group for the new addition and multiplication operations. For each of them, the Lie algebra is studied.

The appropriate tools are developed in order to understand the algebraic fluctuations of the moments and cumulants for converging sequences. This allows us to extend all the results we got for the zero order of fluctuations to any order.

Abstract This paper discusses and applies a basis for modeling elementary forces and particles. We show that models based on isotropic quantum harmonic oscillators describe aspects of the four traditional fundamental physics forces and point to some known and possible elementary particles. We summarize results from models based on solutions to equations featuring isotropic pairs of isotropic quantum harmonic oscillators. Results include predictions for new elementary particles and possible descriptions of dark matter and dark energy.

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...

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.

We analyze the properties of a class of improved lattice topological charge density operators, constructed by a smearing-like procedure. By optimizing the choice of the parameters introduced in their definition, we find operators having (i) a better statistical behavior as estimators of the topological charge density on the lattice, i.e. less noisy; (ii) a multiplicative renormalization much closer to one; (iii) a large suppression of the perturbative tail (and other unphysical mixings) in the corresponding lattice topological susceptibility.

We construct a Cantor set of limit-periodic Jacobi operators having the spectrum on the Julia setJ of the quadratic mapz↦z 2+E for large negative real numbersE. The density of states of each of these operators is equal to the unique equilibrium measure μ onJ. The, Jacobi operators in are defined over the von Neumann-Kakutani system, a group translation on the compact topological group of dyadic integers. The Cantor set is an attractor of the iterated function system built up by the two renormalisation maps Φ±:L=ψ(D ± 2 +E) ↦D ±. To prove the contraction property, we use an explicit interpolation of the Bäcklund transformations by Toda flows. We show that the attractor is identical to the hull of the fixed pointL + of Φ±.

We prove that the invariant horseshoe for the renormalization of critical circle maps is globally uniformly hyperbolic with one-dimensional unstable direction. This extends our earlier result on hyperbolicity of periodic orbits of the renormalization, and completes the proof of Lanford's Program.

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.

We show that the non-commutative Yang Mills field forms an irreducible representation of the (undeformed) Lie algebra of rigid translations, rotations and dilatations. The non-commutative Yang Mills action is invariant under combined conformal transformations of the Yang Mills field and of the non-commutativity parameter theta. The Seiberg Witten differential equation results from a covariant splitting of the combined conformal transformations and can be computed as the missing piece to complete a covariant conformal transformation to an invariance of the action.

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

We study the action of diffeomorphisms on spin foam models. We prove that in 3 dimensions, there is a residual action of the diffeomorphisms that explains the naive divergences of state sum models. We present the gauge fixing of this symmetry and show that it explains the original renormalization of Ponzano-Regge model. We discuss the implication this action of diffeomorphisms has on higher dimensional spin foam models and especially the finite ones. * Electronic address: [email protected] † Electronic address: [email protected] ‡ UMR 5672 du CNRS

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 $m_s \vev{\bar \psi \psi}$.

The use of power series to model natural phenomena is a common practice. Sometimes, however, when the practical situation at hand imposes us to acknowledge our own limits, facing infinity, then, one is driven to approximations. This is when the convenience to resort to a polynomial model may appear as a natural, pragmatic way to go ahead. This is also the reason why, around 1997, trying to reach for a deeper understanding of turbulence, and having thus become interested in studying the dynamics of those formal power series associated with some elliptic curves over Q, I turned my attention to certain ninth degree polynomials, which I found to be canonically deducible from them.

In this paper we extend M. Lyubich's recent results on the global hyperbolicity of renormalization of quadratic-like germs to the space of C^r unimodal maps with quadratic critical point. We show that in this space the bounded-type limit sets of the renormalization operator have an invariant hyperbolic structure provided r ge 2+ alpha with alpha close to one. As an intermediate step between Lyubich's results and ours, we prove that the renormalization operator is hyperbolic in a Banach space of real analytic maps. We construct the local stable manifolds and prove that they form a continuous lamination whose leaves are C^1 codimension one, Banach submanifolds of the ambient space, and whose holonomy is C^{1+\beta} for some beta >0. We also prove that the global stable sets are C^1 immersed (codimension one) submanifolds as well, provided r ge 3+ alpha with alpha close to one. As a corollary, we deduce that in generic, one-parameter families of C^r unimodal maps, the set of parameters corresponding to infinitely renormalizable maps of bounded combinatorial type is a Cantor set with Hausdorff dimension less than one.

In this paper we extend M. Lyubich's recent results on the global hyperbolicity of renormalization of quadratic-like germs to the space of C^r unimodal maps with quadratic critical point. We show that in this space the bounded-type limit sets of the renormalization operator have an invariant hyperbolic structure provided r ge 2+ alpha with alpha close to one. As an intermediate step between Lyubich's results and ours, we prove that the renormalization operator is hyperbolic in a Banach space of real analytic maps. We construct the local stable manifolds and prove that they form a continuous lamination whose leaves are C^1 codimension one, Banach submanifolds of the ambient space, and whose holonomy is C^{1+\beta} for some beta >0. We also prove that the global stable sets are C^1 immersed (codimension one) submanifolds as well, provided r ge 3+ alpha with alpha close to one. As a corollary, we deduce that in generic, one-parameter families of C^r unimodal maps, the set of parameters corresponding to infinitely renormalizable maps of bounded combinatorial type is a Cantor set with Hausdorff dimension less than one.

I consider a simple mechanical problem where the ``particle acceleration'' due to an external force creates sound waves. Theoretical description of this phenomenon should provide the total energy conservation. In order to better couple seemingly insufficiently coupled ``mechanical'' and ``wave'' equations at disposal, I advance an ``interaction Lagrangian'' similar to that of the Classical Electrodynamics (a self-action ansatz). New coupled ``mechanical'' and ``wave'' equations manifest unexpectedly wrong dynamics. I show that renormalization of the fundamental constants in the wrong equations works: the original ``inertial'' properties of solutions are restored. The corresponding exactly renormalized equations contain only physical fundamental constants, describe the experimental data, and reveal a deeper physics -- that of permanently coupled constituents of the ``particle". The perturbation theory is then just a routine calculation giving small and physical corrections. Thus, I show that with advancing a self-action Lagrangian we make a mathematical and physical error, that renormalization is just illegitimately discarding harmful corrections compensating this error, that the exactly renormalized equations may sometimes accidentally coincide with the correct equations, and that the right theoretical formulation of permanently coupled constituents can be fulfilled directly, if realized.

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.

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 show that the noncommutative Yang-Mills field forms an irreducible representation of the (undeformed) Lie algebra of rigid translations, rotations and dilatations. The noncommutative Yang-Mills action is invariant under combined conformal transformations of the Yang-Mills field and of the noncommutativity parameter θ. The Seiberg-Witten differential equation results from a covariant splitting of the combined conformal transformations and can be computed as the missing piece to complete a covariant conformal transformation to an invariance of the action.

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

We rigorously define the self-adjoint one-dimensional Salpeter Hamiltonian perturbed by an attractive δ-interaction, of strength β, centred at the origin, by explicitly providing its resolvent. Our approach is based on a 'coupling constant renormalization', a technique used first heuristically in quantum field theory and implemented in the rigorous mathematical construction of the selfadjoint operator representing the negative Laplacian perturbed by the δ-interaction in two and three dimensions. We show that the spectrum of the self-adjoint operator consists of the absolutely continuous spectrum of the free Salpeter Hamiltonian and an eigenvalue given by a smooth function of the parameter π β / . The method is extended to the model with two twin attractive deltas symmetrically situated with respect to the origin in order to show that the discrete spectrum of the related self-adjoint Hamiltonian consists of two eigenvalues, namely the ground state energy and that of the excited antisymmetric state. We investigate in detail the dependence of these two eigenvalues on the two parameters of the model, that is to say both the aforementioned strength β and the separation distance. With regard to the

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.

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 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.

This paper contrasts two alternative approaches to statistical quantum field theory in curved spacetimes, namely (1) a canonical Hamiltonian approach, in which the basic object is a density matrix ϱ characterizing the noncovariant, but globally defined, modes of the field; and (2) a Wigner function approach, in which the basic object is a Wigner function f defined quasilocally from the Hadamard, or correlation, function G1( x1, x2). The key object is to isolate on the conceptual biases underlying each of these approaches and then to assess their utility and limitations in effecting concrete calculations. The following questions are therefore addressed and largely answered. What sorts of spacetimes (e.g., de Sitter or Friedmann-Robertson-Walker) are comparatively easy to consider? What sorts of objects (e.g., average fields or renormalized stress energies) are easy to compute approximately? What, if anything, can be computed exactly? What approximations are intrinsic to each approach or convenient as computational tools? What sorts of "field entropies" are natural to define?

The analysis of fractal structures by conventional methods gets harder and harder when the number of length scales rises. In this study, we try to combine the renormalization method and the surface impedance model to help rigorous studying of electromagnetic diffraction of fractal structures at their final (infinitum) iteration. As an application, we applied the method to a couple of fractal structures: Cantor Iris 1D and 2D.

We explore the possibility that, while the Higgs mechanism provides masses to the weak-gauge bosons at the electroweak scale as in the standard model, fermion masses are generated by an unknown mechanism at a higher energy scale. At low energies, the standard model can then be regarded as an effective field theory, where fermion masses explicitly break the electroweak SU(2) L × U(1) Y gauge symmetry. If Λ is the renormalization scale where the renormalized Yukawa couplings vanish, then at energies lower than Λ, effective Yukawa couplings will be radiatively induced by nonzero fermion masses. In this scenario, Higgs-boson decays into photons and weak gauge-bosons pairs are in general quite enhanced for a light Higgs. However, depending on Λ, a substantial decay rate into bb can arise, that can be of the same order as, or larger than, the enhanced H → γγ rate. A new framework for Higgs searches at hadron colliders is outlined, vectorboson fusion becoming the dominant production mechanism at the CERN LHC, with an important role also played by the W H/ZH associated production. A detailed analysis of the Higgs branching fractions and their implications in Higgs searches is provided, versus the energy scale Λ. * This is not true in the case of the so-called flavor-changing Higgs penguin diagrams, where a Higgs-boson is exchanged between a flavor-conserving current and a flavor-changing one induced at one-loop. However, this contribution to low-energy FCNC processes is tiny, being suppressed by the Yukawa couplings to the external light quarks.

A model is presented for a system of N two-level excitons interacting with each other via optical near fields represented as localized photons. In a low exciton density limit, quantum dynamics of the dipole moments or quantum coherence between any two energy levels is linear. As the exciton density becomes higher, the dynamics becomes nonlinear, and the system has several kinds of quasi-steady states of the dipole distribution depending on the system parameters. These quasi-steady states are classified with the help of the effective Hamiltonian that is derived from the renormalization of degrees of freedom of localized photons with a unitary transformation. Among them there exist a “ferromagnetic” state (dipole-ordered state), in which all electric dipoles are aligned in the same direction, and an “anti-ferromagnetic” state, where all dipoles alternatingly change the direction. In addition, we show that an arbitrary state can be transformed into a dipole-ordered state by manipulating initial values of the population differences appropriately. For example, if we initially prepare a dipole-forbidden state, which is similar to the “anti-ferromagnetic” state and cannot be coupled with propagating far fields, and if we manipulate the distribution of the population differences properly, the initial state evolves into a dipole-ordered state. The radiation property of such dipole-ordered states is examined in detail. Neglecting energy dissipation by radiation, we find that some of the ordered states show strong radiation equivalent to Dicke’s superradiance. Then by introducing a radiation reservoir, the dissipative master equation is derived. Solving the equation with and without quantum correlations, we numerically show that multiple peaks in the radiation profile can survive in both cases. The mechanism of this phenomenon is discussed, and a brief comment on an application to photonic devices on a nanometer scale is given.

We prove that the non-commutative Gross–Neveu model on the two-dimensional Moyal plane is renormalizable to all orders. Despite a remaining UV/IR mixing, renormalizability can be achieved. However, in the massive case, this forces us to introduce an additional counterterm of the form $\bar{\psi}\imath\gamma^{0}\gamma^{1}\psi$ . The massless case is renormalizable without such an addition.