Field renormalization constant for unstable particles

A recently proposed normalization condition for the imaginary part of the self-energy of an unstable particle is shown to lead to closed, exact expressions for the renormalized self-energy and propagator, which are consequently free of ultraviolet divergences to all orders in perturbation theory. In turn, the corresponding closed expressions for the mass and field renormalization counterterms are necessary in some important cases to avoid power-like infrared divergences in high orders of perturbation theory. In the same examples, the width plays the rôle of an infrared cutoff, and, consequently, the field renormalization counterterm is not an analytic function of the coupling constant.     

Quantum fields and motives

This is a survey of our results on the relation between perturbative renormalization and motivic Galois theory. The main result is that all quantum field theories share a common universal symmetry realized as a motivic Galois group, whose action is dictated by the divergences and generalizes that of the renormalization group. The existence of such a group was conjectured by P. Cartier based on number theoretic evidence and on the Connes-Kreimer theory of perturbative renormalization. The group provides a universal formula for counterterms and is obtained via a Riemann-Hilbert correspondence classifying equivalence classes of flat equisingular bundles, where the equisingularity condition corresponds to the independence of the counterterms on the mass scale.     

One-loop-dimensional reduction of the linear σ model

We perform the dimensional reduction of the linear σ model at one-loop level. The effective potential of the reduced theory obtained from the integration over the nonzero Matsubara frequencies is exhibited. Thermal mass and coupling constant renormalization constants are given, as well as the thermal renormalization group equation which controls the dependence of the counterterms on the temperature. We also recover, for the reduced theory, the vacuum unstability of the model for large N.     

Taming perturbative divergences in asymptotically safe gravity

We use functional renormalization group methods to study gravity minimally coupled to a free scalar field. This setup provides the prototype of a gravitational theory which is perturbatively non-renormalizable at one-loop level, but may possess a non-trivial renormalization group fixed point controlling its UV behavior. We show that such a fixed point indeed exists within the truncations considered, lending strong support to the conjectured asymptotic safety of the theory. In particular, we demonstrate that the counterterms responsible for its perturbative non-renormalizability have no qualitative effect on this feature.     

精确计算n-n重正化链图传播下n+n→２π0反应截面

采用中子n-反中子n与中性介子π⁰强相互作用的Lorentz不变耦合模型,对n-n重正化链图传播子作了有关物理分析及其严格解析计算,获得精确理论计算结果. 进而将此结果用于n+n→2π⁰反应的物理过程分析及其截面的计算研究中,并精确计算出n-n重正化链图传播下n+n→2π⁰反应微分截面. 还将此计算结果与n-n树图和重正化单圈链图传播下n+n→2π⁰反应微分截面作了对比分析,获得了有关辐射修正的重要信息. 此结果对于深 关键词： Lorentz不变耦合模型 重正化链图传播子 微分截面 辐射修正     

The renormalization group of the model ofA 4-coupling in the abstract approach of quantum field theory

For the model ofA ⁴-interaction the postulates of the renormalization group are stated within the abstract approach of quantum field theory. In the massive case these postulates follow if an on-shell formulation of the model is assumed to exist. For the massless model the postulates of the renormalization group imply that the propagator has a pole at momentum zero. Consequently there is no dynamic mass generation and the propagator is normalizable on the mass shell. It is shown that theS-matrix elements scale with canonical dimensions. A general method of rescaling parameter values is developed which takes into account the possibility of propagator zeros and stationary points of the effective coupling.     

The symmetry group of Feynman diagrams and consistency of the BPHZ renormalization scheme

We study the relation between the symmetry group of a Feynman diagram and its reduced diagrams.We then prove that the counterterms in the BPHZ renormalization scheme are consistent with adding counterterms to the interaction Hamiltonian in all cases,including that of Feynman diagrams with symmetry factors.     

On the Summation of Non-leading Logarithms in QCD

Exploiting the relations between the expansion coefficients of Green's functions and those of β and γ functions in massless QFT, the summation up to the n-th non-leading logarithms is reduced to the solution of a system of linear differential equations, which in general differ from renormalization group equations. Applying this procedure to the gluon propagator the leading log approximation is modified near the mass shell by a constant factor only. Furthermore for the gluon propagator exponentiation in terms of the running coupling constant is argued to be restricted to leading logarithms.     

Renormalization of non-abelian gauge theories in curved space-time

We use indirect, renormalization group arguments to calculate the gravitational counterterms needed to renormalize an interacting non-abelian gauge theory in curved space-time. This method makes it straightforward to calculate terms in the trace anomaly which first appear at high order in the coupling constant, some of which would need a 4-loop calculation to find directly. The role of gauge invariance in the theory is considered, and we discuss briefly the effect of using coordinate-dependent gauge-fixing terms. We conclude by suggesting possible applications of this work to models of the very early universe.     

Cornwall-Jackiw-Tomboulis Effective Potential for Quark Propagator in Real-Time Thermal Field Theory and Landau Gauge

We complete the derivation of the Cornwall-Jackiw-Tomboulis effective potential for quark propagator at finite temperature and finite quark chemical potential in the real-time formalism of thermal field theory and in Landau gauge. In the approximation that the function A(p2) in inverse quark propagator is replaced by unity, by means of the running gauge coupling and the quark mass function invariant under the renormalization group in zero temperature Quantum Chromadynamics (QCD), we obtain a calculable expression for the thermal effective potential, which will be a useful means to research chiral phase transition in QCD in the real-time formalism.     

Cornwall-Jackiw-Tomboulis Effective Potential for Quark Propagator in Real-Time Thermal Field Theory and Landau Gauge

We complete the derivation of the Cornwall-Jackiw-Tomboulis effective potentiM for quark propagator at finite temperature and finite quark chemical potential in the real-time formalism of thermal field theory and in Landau gauge. In the approximation that the function A（p^2） in inverse quark propagator is replaced by unity, by means of the running gauge coupling and the quark mass function invariant under the renormalization group in zero temperature Quantum Chromadynamics （QCD）, we obtain a calculable expression for the thermal effective potential, which will be a useful means to research chiral phase transition in QCD in the real-time formalism.     

Ultraviolet renormalons in abelian gauge theories

We analyze the large-order behaviour in perturbation theory of classes of diagrams with an arbitrary number of chains (i.e. photon lines, dressed by vacuum polarization insertions). We derive explicit formulae for the leading and subleading divergence as n, the order in perturbation theory, tends to infinity, and a complete result for the vacuum polarization at the next-to-leading order in an expansion in l / N f, where N f is the number of fermion species. In general, diagrams with more chains yield stronger divergence. We define an analogue of the familiar diagrammatic R-operation, which extracts ultraviolet renormalon counterterms as insertions of higher-dimension operators. We then use renormalization group equations to sum the leading (in n/ N f ) k corrections to all orders in l INf and find the asymptotic behaviour in n up to a constant that must be calculated explicitly order by order in 1/N_f.     

d-Mott phases in one and two dimensions

We use exact diagonalization to determine the spectrum of reduced Hamiltonians based on renormalization group flows to strong coupling. For the half-filled two-leg Hubbard ladder we reproduce the known insulating d-Mott ground state with spin and charge gaps. For the saddle point regions of the two-dimensional Hubbard model near half filling we find a crossover to a similar strong coupling state, which truncates the Fermi surface near the saddle points. At lower scales d-wave superconductivity appears on the remaining Fermi surface.     

Adler–Bardeen theorem and manifest anomaly cancellation to all orders in gauge theories

We reconsider the Adler–Bardeen theorem for the cancellation of gauge anomalies to all orders, when they vanish at one loop. Using the Batalin–Vilkovisky formalism and combining the dimensional-regularization technique with the higher-derivative gauge invariant regularization, we prove the theorem in the most general perturbatively unitary renormalizable gauge theories coupled to matter in four dimensions, and we identify the subtraction scheme where anomaly cancellation to all orders is manifest, namely no subtractions of finite local counterterms are required from two loops onwards. Our approach is based on an order-by-order analysis of renormalization, and, differently from most derivations existing in the literature, does not make use of arguments based on the properties of the renormalization group. As a consequence, the proof we give also applies to conformal field theories and finite theories.     

TheO(ɛ2) scaling law fordσ/dt in the Reggeon Field Theory

We calculate the two loop contributions within the ε-expansion to the Reggeon Field Theory scaling law fordσ/dt, derived using the renormalization group and a general renormalization point for the Pomeron propagator. This generalizes theO(ε) work of Abarbanel, Bartels, Bronzan, and Sidhu. The invariance of the results under certain coupling constant rescalings is demonstrated. We also make some qualitative comments regarding phenomenological applications. Our amplitude in a certain limit approximates the form of the low energy diffractive amplitude advocated by Kane.     

The background field method and the ultraviolet structure of the supersymmetric nonlinear σ-model

Calculations of the ultraviolet counterterms of the bosonic and supersymmetric nonlinear σ-models in two space-time dimensions are undertaken in order to verify conclusions of a recent argument based on differential geometry in the supersymmetric case. The background field method and the normal coordinate expansion are discussed in detail, and the generalized renormalization group pole equations applicable to the nonlinear σ-model are derived. Both component and superfield calculations of the counterterms are presented.     

Renormalization Proof for Massive $$\phi_4^4$$ Theory on Riemannian Manifolds

In this paper we present an inductive renormalizability proof for massive theory on Riemannian manifolds, based on the Wegner-Wilson flow equations of the Wilson renormalization group, adapted to perturbation theory. The proof goes in hand with bounds on the perturbative Schwinger functions which imply tree decay between their position arguments. An essential prerequisite is precise bounds on the short and long distance behaviour of the heat kernel on the manifold. With the aid of a regularity assumption (often taken for granted) we also show that for suitable renormalization conditions the bare action takes the minimal form, that is to say, there appear the same counterterms as in flat space, apart from a logarithmically divergent one which is proportional to the scalar curvature.     

Can renormalization group flow end in a Big Mess?

The field theoretical renormalization group equations have many common features with the equations of dynamical systems. In particular, the manner how Callan–Symanzik equation ensures the independence of a theory from its subtraction point is reminiscent of self-similarity in autonomous flows towards attractors. Motivated by such analogies we propose that besides isolated fixed points, the couplings in a renormalizable field theory may also flow towards more general, even fractal attractors. This could lead to Big Mess scenarios in applications to multiphase systems, from spin-glasses and neural networks to fundamental string (M?) theory. We consider various general aspects of such chaotic flows. We argue that they pose no obvious contradictions with the known properties of effective actions, the existence of dissipative Lyapunov functions, and even the strong version of the c-theorem. We also explain the difficulties encountered when constructing effective actions with chaotic renormalization group flows and observe that they have many common virtues with realistic field theory effective actions. We conclude that if chaotic renormalization group flows are to be excluded, conceptually novel no-go theorems must be developed.