Instanton induced green functions in the superfield formalism

We discuss in detail the supersymmetric instanton calculus of NSVZ and extend it to chiral matter fields in the, adjoint representation. The constant Green functions induced by the instanton of supersymmetricSU (2) gauge theories are calculated systematically for the cases with and without scalar vev's bigger than the scale of the gauge theory and for nonvanishing small masses of chiral fields. Oneinstanton contributions to the Green functions containing four fields without large vevs would disturb clustering; but they are argued to vanish; two-instanton effects then lead to a pattern which quantitatively agrees with factorization and the anomaly relation.     

Some issues in a gauge model of unparticles

We address in a recent gauge model of unparticles the issues that are important for consistency of a gauge theory, i.e., unitarity and the Ward identity of the physical amplitudes. We find that non-integrable singularities arise in physical quantities like the cross section and the decay rate from the gauge interactions of unparticles. We also show that the Ward identity is violated due to the lack of a dispersion relation for charged unparticles although the Ward–Takahashi identity for general Green functions is incorporated in the model. A previous observation that the contribution of the unparticle (with scaling dimension d) to the gauge boson self-energy is a factor (2−d) of the particle’s self-energy has been extended to the Green function of triple gauge bosons. This (2−d) rule may be generally true for Green functions for any number of points of the gauge bosons. This implies that the model would be trivial even as one that mimics certain dynamical effects on gauge bosons in which unparticles serve as an interpolating field.     

Path-integral formulation of chiral invariant fermion models in two dimensions

We study the Thirring and chiral-invariant Gross-Neveu (CGN) models using the functional integral method. By introducing an auxiliary vector field we disclose a relation with two-dimensional gauge theories coupled to fermions and then extend a technique based on a chiral change in the functional variables to study purely fermionic models.We obtain the exact Klaiber solution for the massless Thirring model (for spin $\text{1}\text{2}$) in a very simple way and we then extend our technique to investigate the CGN model. We show the factorization of a free fermionic part at the level of Green functions on very general grounds. We then impose certain restrictions on the behavior of the fields — which render our treatment exact only in the zero winding number sector, but allow the computation of the U(1) part of the CGN Green functions exactly, showing, in particular, its complete decoupling from the color part and the almost long-range order behaviour in the infrared region.In our approach, the non-triviality of the jacobian arising from the chiral transformation — directly related to the topological density and the axial anomaly — appears to be crucial for the functional integral treatment of these models.     

Complete control of gauge parameter dependence in the Abelian Higgs model

We examine the dependence on all gauge parameters in the example of the Abelian Higgs model by applying a general algebraic method which roots in an extension of the usual Slavnov-Taylor identity. This method automatically yields all information about the gauge parameter dependence of Green functions and therefore especially allows to control the range of “good” normalization conditions. In this context we show that the physical on-shell normalization conditions are in complete agreement with the restrictions dictated by the enlarged Slavnov-Taylor identity and that the coupling can be fixed in an easily handleable way on the Ward identity of local gauge invariance. As an application of the general method we also study the Callan-Symanzik equation and the renormalization group equation of the Abelian Higgs model. Received: 18 July 1997 / Revised version: 20 October 1997 / Published online: 23 February 1998     

The off-shell infrared problem in N = 1 supersymmetric yang-mills theories (II). General massless models

We prove that in a general massless N = I SYM theory off-shell Green functions exist such that Green functions of gauge invariant operators are supersymmetrically covariant. The off-shell infrared problem present in the superfield treatment of these theories is thus shown to remain a gauge artefact. The N = 2, 4 pure SYM theories are covered by this result and thus exist as N = 1 SYM theories.     

General form of the full electromagnetic Green function in materials physics

In this article, we present the general form of the full electromagnetic Green function which is suitable for the application in bulk materials physics. In particular, we show how the seven adjustable parameter functions of the free Green function translate into seven corresponding parameter functions of the full Green function. Furthermore, for both the fundamental response tensor and the electromagnetic Green function, we discuss the reduction of the Dyson equation on the four-dimensional Minkowski space to an equivalent, three-dimensional Cartesian Dyson equation.     

The Nielsen identities for the two-point functions of QED and QCD

We consider the Nielsen identities for the twopoint functions of full QCD and QED in the class of Lorentz gauges. For pedagogical reasons the identities are first derived in QED to demonstrate the gauge independence of the photon self-energy, and of the electron mass shell. In QCD we derive the general identity and hence the identities for the quark, gluon and ghost propagators. The explicit contributions to the gluon and ghost identities are calculated to one-loop order, and then we show that the quark identity requires that in on-shell schemes the quark mass renormalisation must be gauge independent. Furthermore, we obtain formal solutions for the gluon selfenergy and ghost propagator in terms of the gauge dependence of other, independent Green functions.     

Gauge invariant quark Green’s functions with polygonal Wilson lines

Properties of gauge invariant two-point quark Green’s functions, defined with polygonal Wilson lines, are studied. The Green’s functions can be classified according to the number of straight line segments their polygonal lines contain. Functional relations are established between the Green’s functions with different numbers of segments on the polygonal lines. An integrodifferential equation is obtained for the Green’s function with one straight line segment, in which the kernels are represented by a series of Wilson loop vacuum averages along polygonal contours with an increasing number of segments and functional derivatives on them. The equation is exactly solved in the case of two-dimensional QCD in the large-N _c limit. The spectral properties of the Green’s function are displayed.     

1+1维非线性σ模型的BRST变换的等价性和Ward恒等式

在依据Dirac约束规范理论和作推广后的条件下,导出了规范生成元,推导出了1+1维O(3)非线性σ模型的一般条件(β≠0)下的BRST变换,给出了其BRST变换与Dirac规范变换的等价关系,得到了鬼场的新的一般对易关系,且其一般参数β为零时就回到通常的鬼场的对易关系.并由规范生成元导出了BRST荷,进而完成了此模型的一种BRST量子化.还在此基础上进一步导出了此系统的Green函数生成泛函、连通Green函数生成泛函和正规顶角生成泛函,获得了3种不同的Ward恒等式     

Green functions in a super self-dual Yang-Mills background

In euclidean supersymmetric theories of chiral superfields and vector superfields coupled to a super-self-dual Yang-Mills background, we define Green functions for the Laplace-type differential operators which are obtained from the quadratic part of the action. These Green functions are expressed in terms of the Green function on the space of right chiral superfields, and an explicit expression for the right chiral Green function in the fundamental representation of an SU(n) gauge group is presented using the supersymmetric version of the ADHM formalism. The superfield kernels associated with the Laplace-type operators are used to obtain the one-loop quantum corrections to the super-self-dual Yang-Mills action, and also to provide a superfield version of the super-index theorems for the components of chiral superfields in a self-dual background.     

Green functions of vortex operators

We study the euclidean Green functions of the 't Hooft vortex operator, primarily for abelian gauge theories. The operator is written in terms of elementary fields, with emphasis on a form in which it appears as the exponential of a surface integral. We explore the requirement that the Green functions depend only on the boundary of this surface. The Dirac veto problem appears in a new guise. We present a two-dimensional “solvable model” of a Dirac string, which suggests a new solution of the veto problem. The renormalization of the Green functions of the abelian Wilson loop and abelian vortex operator is studied with the aid of the operator product expansion. In each case, an overall multiplication of the operator makes all Green functions finite; a surprising cancellation of divergences occurs with the vortex operator. We present a brief discussion of the relation between the nature of the vacuum and the cluster properties of the Green functions of the Wilson and vortex operators, for a general gauge theory. The surface-like cluster property of the vortex operator in an abelian Higgs theory is explored in more detail.     

On the cancellation of hard anomalies in gauge field models: a regularization independent proof

In this paper we prove Bardeen's conjecture that the anomaly of the Adler-Bardeen-Bell-Jackiw-Schwinger type in gauge models are definitely absent if they are cancelled at the first order of the perturbation expansion. Our analysis develops within the regularization independent B.P.H.Z. renormalization scheme. We discuss the possible appearance of anomalies in an enlarged class of gauge models admitting soft violations of the Slavnov-Taylor identities which prescribe the gauge transformation properties of the Green functions. By a repeated use of the Callan-Symanzik equation we conclude that the lowest non vanishing contributions to the anomalies must necessarily correspond to the first order in the perturbation expansion, hence if they are cancelled at this order the theory will be definitely anomaly free.     

Gauge invariance of relativistic two-particle bound-state energies (I)

The gauge invariance of relativistic two-particle energy levels in quantum electrodynamics is demonstrated for both covariant and non-covariant gauges, by considering infinitesimal gauge transformations from a fixed gauge. The proof is carried out by expressing the bound-state energies in terms of ratios of contour integrals involving four-point Green functions in the neighborhood of their poles in the energy plane. The generalized Ward-Takahashi identities are then used to help reduce the Green functions which appear into forms which make the singularity structure of the four-point function apparent. The formalism is illustrated for the special cases of one- and two-photon exchange in perturbation theory.     

Non-abelian gauge theories in a background field

Green functions of on-mass-shell transverse gauge fields in a background gauge are proved to be obtained from Green functions in the perturbative vacuum by a mere translation.     

Instanton effects in supersymmetric gauge theories

We investigate how in supersymmetric gauge theories non-perturbative effects are able to generate non-trivial vacuum properties otherwise forbidden by perturbative non-renormalization theorems. This conclusion can be reliably drawn since the constancy of certain Green functions — due to supersymmetry (SUSY) — allows one to connect vacuum-dominated large distances with short-distance behaviour which is reliably computed by instanton methods. In all the cases we discuss (without matter, with massive or massless matter in real representations and, finally, with matter in complex representations) instanton calculations imply the occurrence of a variety of condensates. For the pure SUSY gauge theory, a gluino condensate induces the spontaneous breaking of Z_2N. For massive super-quantum chromodynamics (SQCD) we find a peculiar mass dependence of matter condensates whose origin is traced to mass singularities of non-zero mode instanton contributions. These contributions force the massless limit of SQCD to differ from the strictly massless case, in which the spontaneous breaking of chiral symmetries is induced. Inconsistency with an anomaly equation forces either infinite matter condensates or spontaneous SUSY breaking in the massless cases. For non-constant Green functions, instantons are shown to provide new calculable short-distance singularities of an obvious non-perturbative nature.     

Anomaly poles as common signatures of chiral and conformal anomalies

One feature of the chiral anomaly, analyzed in a perturbative framework, is the appearance of massless poles which account for it. They are identified by a spectral analysis of the anomaly graph and are usually interpreted as being of an infrared origin. Recent investigations show that their presence is not just confined in the infrared, but that they appear in the effective action under the most general kinematical conditions, even if they decouple in the infrared. Further studies reveal that they are responsible for the non-unitary behaviour of these theories in the ultraviolet (UV) region. We extend this analysis to the case of the conformal anomaly, showing that the effective action describing the interaction of gauge fields with gravity is characterized by anomaly poles that give the entire anomaly and are decoupled in the infrared (IR), in complete analogy with the chiral case. This complements a related analysis by Giannotti and Mottola on the trace anomaly in gravity, in which an anomaly pole has been identified in the corresponding correlator using dispersion theory in the IR. Our extension is based on an exact computation of the off-shell correlation function involving an energy–momentum tensor and two vector currents (the gauge–gauge–graviton vertex) which is responsible for the appearance of the anomaly.     

Gauge parameter dependence in the background field gauge and the construction of an invariant charge

By using the enlarged BRS transformations we control the gauge parameter dependence of Green functions in the background field gauge. We show that it is unavoidable — also if we consider the local Ward identity — to introduce the normalization value ξ₀ of the gauge parameter ξ. The dependence of Green functions on ξ₀ is governed by a partial differential equation in a similar manner as the dependence on the normalization point κ is governed by the RG equation. By modifying the Ward identity we are able to construct in 1-loop order a gauge parameter independent combination of 2-point vector and background vector functions. By explicit construction of the next orders we show that this combination can be used to construct a gauge parameter independent RG-invariant charge. However, it is seen that this RG-invariant charge does not satisfy the differential equation of the normalization value ξ₀ of the gauge parameter, and, hence, is not ξ₀-independent as required.     

Nonperturbative Time-Independent Green Function of Matrix Schrödinger Equation. General Formalism and Quasiclassical Representation

A Green function of time-independent multichannel Schrödinger equation is considered in matrix representation beyond a perturbation theory. Nonperturbative Green functions are obtained through the regular in zero and at infinity solutions of the multichannel Schrödinger equation for different cases of symmetry of the full Hamiltonian. The spectral expansions for the nonperturbative Green functions are obtained in simple form through multichannel wave functions. The developed approach is applied to obtain simple analytic equations for the Green functions and transition matrix elements for compound multipotential system within quasiclassical approximation. The limits of strong and weak interchannel interactions are studied.Alexander I. Pegarkov:On leave from Physics Faculty     

A Gauge Invariant Description for the General Conic Constrained Particle from the FJBW Iteration Algorithm

We consider a second-degree algebraic curve describing a general conic constraint imposed on the motion of a massive spinless particle. The problem is trivial at classical level but becomes involved and interesting concerning its quantum counterpart with subtleties in its symplectic structure and symmetries. We start with a second-class version of the general conic constrained particle, which encompasses previous versions of circular and elliptical paths discussed in the literature. By applying the symplectic FJBW iteration program, we proceed on to show how a gauge invariant version for the model can be achieved from the originally second-class system. We pursue the complete constraint analysis in phase space and perform the Faddeev-Jackiw symplectic quantization following the Barcelos-Wotzasek iteration program to unravel the essential aspects of the constraint structure. While in the standard Dirac-Bergmann approach there are four second-class constraints, in the FJBW they reduce to two. By using the symplectic potential obtained in the last step of the FJBW iteration process, we construct a gauge invariant model exhibiting explicitly its BRST symmetry. We obtain the quantum BRST charge and write the Green functions generator for the gauge invariant version. Our results reproduce and neatly generalize the known BRST symmetry of the rigid rotor, clearly showing that this last one constitutes a particular case of a broader class of theories.     

Greens functions of 2-dimensional Yang-Mills theories on nonorientable surfaces

By using the path integral method, we calculate the Green functions of the field strength of Yang-Mills theories on arbitrary nonorientable surfaces in Schwinger-Fock gauge.We show that the non-gauge invariant correlators consist of a free part and an almost x-independent part. We also show that the gauge invariant n-point functions are those corresponding to the free part, as in the case of orientable surfaces.