A proof of the axial anomaly

The local form of the axial anomaly with both left and right-handed gauge fields and a metric present is given and proved using the families index theoremResearch supported by an NSF Mathematical Sciences Postdoctoral Fellowship     

Renormalization of the axial anomaly operators

An expressionE(d ³ σ/dp ³)=Kp _T ^?n exp(?E/τ, wheren, K and τ are parameters to be fitted, has been used to represent the large-p _T hadron invariant single-particle inclusive cross sections. In most cases the fitting is within the quoted experimental uncertainties. An interpretation of the expression in terms of fireballs is given. Scaling is seen to be violated in large-p _T phenomena.     

Stochastic perturbative derivation of the axial anomaly

The axial anomaly is calculated as the infinite Langevin time limit of stochastic triangle diagrams. Their regularization is insured with the help of an analytic stochastic regulator. The usual axial anomaly is recovered only when the Langevin equations used to generate the perturbative expansion are gauge covariant.     

Ultraviolet and infrared aspects of the axial anomaly II

This is the second part of a brief review of some perturbative aspects of the Adler-Bell-Jackiw axial anomaly. To begin with, we discuss here the derivation of the anomaly based on the imaginary part of the VVA triangle graph and dispersion relations. Within such an approach the infrared properties of the VVA amplitude are substantial. Next we summarize the main results obtained by the present author and co-authors concerning some particular ultraviolet and infrared aspects of the axial anomaly. These results cover a wide variety of topics, ranging from dispersion relations to a systematic analysis of gauge invariant ultraviolet regularizations of the VVA triangle graph. In the latter context, one of the highlights is a reinterpretation of dimensional regularization of the VVA diagram as a continuous superposition of Pauli-Villars cut-offs.     

Ultraviolet and infrared aspects of the axial anomaly I

This is the first part of a brief review of some perturbative aspects of the Adler-Bell-Jackiw axial anomaly, described in terms of ultraviolet and infrared behaviour of the famous VVA triangle graph. Apart from a general overview of the diversified role played by the anomaly in quantum field theory and particle physics, we present here an elementary introduction to the subject of the anomaly, which hopefully should be comprehensible to an uninitiated reader with only a basic background in quantum field theory. In this article we stress the ultraviolet aspects of the anomaly and the topics covered here are the following: Vector and axial-vector Ward identities for the VVA triangle graph, the anomaly and several ways to derive it, namely the symmetric momentum cut-off and shifting the integration variables in linearly divergent integrals, the Adler-Rosenberg argument, the Pauli-Villars method and dimensional regularization.     

Quark-hadron duality,axial anomaly and mixing

Interplay between axial anomaly and quark-hadron duality in the presence of strong mixing is considered. The anomaly sum rule for meson transition form factors based on the dispersive representation of axial anomaly and quark-hadron duality in octet channel is analyzed. The comparison of this sum rule to the experimental data on the transition form factors of the η and η′ mesons shows that the interval of duality in this channel is rather small, contradicting the usual understanding of quark-hadron duality. The same values of interval of duality are supported by considering the two-point correlator in the local duality limit. This contradiction may be resolved by introducing of some nonperturbative non-one-pion exchange correction to the relevant spectral density. The form and value of this correction are discussed.     

The Schwinger model and its axial anomaly

The Schwinger model (quantum electrodynamics with massless fermions in one spatial dimension) is solved, supposing that space is a circle. This clarifies aspects of the usual version of the model, where space is a line, without changing the physics. The Hamiltonian formalism is used. On a circle, an abelian gauge field has one physical degree of freedom, and the gauge covariant Dirac operator, which couples the fermions to this degree of freedom, exhibits spectral flow. The relationship between the spectral flow and the axial anomaly is explained. Some variants of the Schwinger model are also discussed.     

The axial anomaly and anti-symmetric tensor fields

Spin-$\text{1}\text{2}$ fermions are coupled to external axial vector and rank-2 anti-symmetic tensor fields. The chiral U(1) Ward identity is shown to have anomalous structure given by $\text{F}{}_{\mathrm{\mu \nu }}\text{F}\text{?}{}^{\mathrm{\mu \nu }}$ only with F_μν the axial vector field strength tensor. No Additional axial anomalied are introduced due to the presence of the anti-symmetric tensor fields.     

The axial anomaly for spin 32 fields

The axial anomaly for a Majorana spin $\text{3}\text{2}$ loop with external gravitons in supergravity is ?_μJ^μ₅ = (192π²)^??^αβ?σR?σ^μνRμναβ. This is twice the value of the corresponding anomaly for a Dirac spin $\text{1}\text{2}$ loop.     

On the axial anomaly at finite temperature in the Schwinger model

The axial anomaly in 2 dimensional QED at finite temperature is carefully investigated in the real time formalism in the limit of vanishing fermion mass. We follow the dispersion approach of Dolgov and Zakharov. The temperature independence of the anomaly is recovered.     

The role of the axial anomaly in measuring spin-dependent parton distributions

It is shown that forward matrix elements of j₅^μ, the flavor singlet axial vector current, do not merely measure the helicity carried by quarks and antiquarks but also include a spin-dependent gluonic component due to the anomaly. In perturbation theory the exact value of the gluonic component depends on the infrared regulator, and we argue that computing off-shell matrix elements with zero-mass quarks gives the proper method of regulation to exhibit the chiral properties of the theory. We suggest measuring two-jet production in deep inelastic scattering off polarized targets as a means of determining the spin-dependent gluonic component of j₅^μ.     

New critical point induced by the axial anomaly in dense QCD

We study the interplay between chiral and diquark condensates within the framework of the Ginzburg-Landau free energy, and classify possible phase structures of two and three-flavor massless QCD. The QCD axial anomaly acts as an external field applied to the chiral condensate in a color superconductor and leads to a crossover between the broken chiral symmetry and the color superconducting phase, and, in particular, to a new critical point in the QCD phase diagram.     

Absence of strong interaction corrections to the axial anomaly in the σ model

The absence of strong interaction corrections to the axial anomaly in the model is proved in a cut-off independent way using Zimmermann's normal product algorithm.     

Is the axial anomaly really determined in a continuous non-perturbative regularization?

In the framework of a gauge invariant continuous and non-perturbative regularization scheme based on the smearing of point-like interactions by means of cutoff functions, we show that the axial anomaly, though cutoff independent, depends on the shape of the cutoff functions. The standard value for the strength of the axial anomaly is recovered if we assume that the regularized gauge invariant axial current is in addition local. Received: 11 July 2001 / Published online: 25 January 2002     

Calculation of the color axial anomaly on the basis of the gauge-invariant spatial point-splitting regularization

The nonsinglet axial anomaly is calculated by employing Schwinger’s point-splitting regularization of the interaction between fermions and a non-Abelian gauge field. This method makes it possible to obtain a covariant expression for the anomaly directly from the effective action for the gauge field. Previously, the anomaly under study was calculated by many other methods. However, all calculations based on the point-splitting regularization (from the pioneering study of Bardeen in 1969) involve a number of intermediate steps and subtractions of specially chosen polynomials in the field.