Current Commutation Relations Research Articles

Chiral dynamics of a1(1260) → 3π

We calculate the sequential decays a1 → πσ → 3π and a1 → πρ → 3π using chiral SU(2) ⊗ SU(2) current commutation relations. Proper vertices a1πσ, σππ, a1πρ, ρππ are derived from Ward identities and yield energy-dependent decay widths Γρ→ππ and Γσ→ππ necessary for the total Γa1→3π decay width. Both sequential decays (via πσ and πρ) are necessary to reproduce Γtota1. We find evidence for a substantial quenching of the σ → ππ decay width in a1 → πσ → 3π.

The axial charge sum rule for complex nuclei

By generalizing the Adler-Weisberger sum rule to complex nuclei we show that the enhancement of the effective axial charge formfactor of the nucleon in J + a3 J −, Δ T = 1 weak transitions is a direct consequence of the chiral symmetry constraints imposed by the current algebra commutation relation for axial charges.

Dirac-bracket quantization of bosonized chiral two-dimensional QCD.

We systematically derive the bosonized form of the chiral two-dimensional QCD Lagrangian exhibiting explicitly the anomalous breaking of gauge invariance, and quantize it using Dirac's algorithm for constrained systems. As a side product we also discuss the Hamiltonian formalism for the principal \ensuremath{\sigma} model and derive the commutation relations of the chiral currents in both models.

Current algebra and quark masses from a Monte Carlo simulation with Wilson fermions

We compute with a Monte Carlo simulation the normalization constants of the vector and axial vector currents. The validity of vector current commutation relations is verified. Knowledge of the currents allows to determine the decay constants F π and (ƒ p, ø) −1 . Quark masses are obtained from the pseudoscalar meson spectrum, and an evaluation of 〈 ΨΨ〉 0 is given.

Cohomology, cocyles and the current algebra for the nonlinear σ-model

Using the idea of cohomology defined for the Lie algebra of gauge transformations, we examine the extension of the current algebra for the system of the gauged nonlinear σ-model. An anomalous term in the current commutation relation is constructed and shown to be equivalent to that arising in the gauged nonlinear σ-model with the Wess-Zumino term. The relation with the anomalous Schwinger term given by Faddeev is also discussed.

On a relation between massive Yang-Mills theories and dual string models

The relations between mass terms in Yang-Mills theories, projective representations of the group of gauge transformations, boundary conditions on vector potentials and Schwinger terms in local charge algebra commutation relations are discussed. The commutation relations (with Schwinger terms) are similar to the current algebra commutation relations of the SU(N) extended dual string model.

Sum rules for the real part of a current-hadron scattering amplitude and light-cone commutators

Sum rules are derived relating the real part of a current-proton scattering amplitude to the light-cone commutator of two currents. The contribution of the different, connected and disconnected, intermediate states is examined in detail and the relative importance in the infinite (or quasi-infinite) momentum limit is exhibited. The sum rules whose dominant contribution in such a limit comes from direct and/or two-particle disconnected (Z-diagram) intermediate states are related to the parameters appearing in the quark model light-cone current commutation relations and consistent results are then obtained.

Quantum Field Theories in the Infinite-Momentum Frame III. Quantization of Coupled Spin-One Fields

Light-front quantization of spin-one fields coupled to a conserved or nonconserved current constructed from a Dirac field is studied. It is shown that an operator phase transformation must be performed on the Dirac field in order to maintain simple canonical commutation relations and a simple Hamiltonian. In this formulation quantum electrodynamics emerges as the zero-mass limit of the massive gluon model. Lorentz invariance of the vector-gluon model is explicitly verified. Vacuum expectation values of operator products and Green's functions are studied and spectral sum rules are derived. The general structure of the current commutators on a light front is formally not altered by the interactions. Feynman's parton model for deep-inelastic electron scattering is derived from canonical light-front current commutation relations. The structure function in the Bjorken scaling limit is related to the ${p}^{+}$ distribution of the constituents of the hadron target in any frame of reference.

Currents, stress tensor and generalized unitarity in conformal invariant quantum field theory

Matrix elements of internal symmetry currents and energy momentum density tensor are constructed in Migdal Polyakov conformal invariant bootstrap field theory. Their 3-point functions satisfy Bethe Salpeter equations which determine any free coefficients that may still occur in the conformal invariant Ansatz. Ward identities are verified for alln-point functions. They imply correct equal time current commutation relations. A proof of generalized unitarity is also given. Various equivalent forms of the propagator bootstrap are discussed. Our algebraic techniques also yield an eigenvalue equation for first order correction to the exactly conformal invariant theory, assuming the latter is Gell-Mann Low large momentum asymptote of a renormalizable finite mass theory.

Consistency of Current-Algebra Results in the Presence of Electromagnetism

We demonstrate in this paper that despite the appearance of dynamical terms in the current-commutation relations as a result of the electromagnetic interaction, one can still use the algebra of currents to derive low-energy results.

Some remarks on the algebra of currents in the presence of electromagnetism

It is shown that in order to avoid a contradiction, either the equal-time commutation relations of charged vector or axial vector currents should be modified by a term of order e in the presence of electromagnetism if V 3 λ which enters on the right side of these commutation relations is to be regarded as the source of the isovector part of the electromagnetic potential or, if these equal-time commutation relations are exact and valid in the usual form in the presence of electromagnetism, V 3 λ is not the source of the isovector part of the electromagnetic potential but differs from the latter by a term of order e.

Deep inelastic neutrino-nucleon interactions

Deep inelastic neutrino-nucleon interactions are discussed under the assumptions of conserved axial vector current and equal-time commutation relations of currents. It is shown that the structure functionsW1 andW2 for vector and axial vector currents are equal in the average sense with respect to the variablev and are equal in the limitv → ∞. The resulting expressions are then used to predict relations connecting deep inelastic neutrino and electron scattering.

Factorisable representations of current groups and the Araki-Woods imbedding theorem

In order to study the current commutation relations of quantum field theory, Araki and Woods [2] and Araki [1] introduced the notion of current groups and factorisable representations of such groups. Araki and Woods [2] and Streater [5] estabhshed that such representations admit a natural imbedding in a symmetric Fock space exp H over a Itflbert space H. If G is a locally compact group, then a suitable space of Borel functions on a Borel space with values in G is made into a group under pointwise multiphcation. This is called a current group of G. Araki [1] established that the factorisable representations of the current group are based on certain coeycle valued measures. In this paper we show the existence of a measure on the Borel space over which the current group is constructed, relative to which a coeycle valued density exists. This yields a certain natural topology for the current group under which the faetorisable representation is continuous. In order to take into account all the coeycles of first order in the construction of fae, torisable representations it turns out that projective representations should also be considered. Finally, the Araki-Woods imbedding is expheitly constructed in terms of the cocycles. At this stage it may be worth remarking that our methods differ very much from that of Araki and Woods. We rely more on measure theory and not at all on lattice theory.

Local one particle approximation of the current commutators

The spectral functions of the current commutators matrix elements between vacuum and one particle boson states are calculated in a local one particle approximation. In the equal time limit, the restrictions imposed by the current algebra commutation relations without operator Schwinger terms are systematically discussed. It is found, that in this approximation, and when one of the currents is conserved, the PCAC and vector meson dominance models cannot be considered simultaneously. The comparison of the results with experiment gives preference to the first of the two models.

Stress-Tensor—Current Commutation Relations, the Pion-Mass Form Factors, and Tensor- and Scalar-Meson Dominance

The low-energy behavior of the pion-mass form factors is derived using stress-tensor---current commutation relations. The results indicate significant deviations from tensor-meson and scalar-meson pole dominance of the stress tensor. The stress-tensor---$\ensuremath{\epsilon}$-meson coupling and the scale dimension of the pion field are obtained in terms of ${g}_{{A}_{1}\ensuremath{\epsilon}\ensuremath{\pi}}$. Numerical estimates are given.

V−ATheory for Nucleon Resonances

The conserved-axial-vector-current hypothesis and current commutation relations are used to predict that the weak-interaction coupling of nucleon resonances to nucleons is described by a $V\ensuremath{-}A$ theory.

Unitarity at Threshold and Veneziano-Type Form Factors for Mesons

A simple form of Veneziano type form factors is investigated and applied to Kwdecay form factors. The p-wave part is determined by using the unitarity condition at thresh­ old and the Veneziano model for scattering amplitudes. The PCAC condition and the broken chiral symmetry with current commutation relations are used for determining the s­ wave part. The results arefKif~f+ (0) =1.24 and~= -0.15. Suura's method is also applied and shown to give fKif~f+ (0) = 1.41.

Current algebras and universal divergent radiative corrections

We examine, using current algebras, the ultraviolet divergences occurring in the calculation of electromagnetic radiative corrections to any lowest order weak process at arbitrary momentum transfer. We consider all orders in perturbation theory in the fine structure constant α. The divergent parts of the radiative corrections are expressed in terms of matrix elements of equal-time current commutators by using the Bjorken expansion of time-ordered products of currents at large momenta. We assume the validity of this expansion and of the use of “naive” current commutation relations in discussing various current algebra models. We impose the condition that the divergences contribute only to an unobservable, universal weak coupling constant renormalization. It is shown that, in models with operator Schwinger terms in the current commutators, this condition cannot be satisfied for nonzero momentum transfer. Also, it is not satisfied for a weak interaction theory mediated by a vector boson. Two current algebra models are exhibited which are satisfactory if the weak Hamiltonian has a local curren-current form. For these models, the weak and electromagnetic currents of both the hadrons and the leptons obey the same commutation relations, and the Schwinger terms are c numbers. One, a quark model of hadrons with integrally charged quarks together with the conventional lepton currents, gives finite radiative corrections. The second, the algebra of fields model for the total electromagnetic and weak currents, including leptons, contains only a universal divergent factor. These two results are shown to hold to all orders in α. In obtaining these results, divergent contributions to electromagnetic mass shifts and to electromagnetic renormalization effects in strong interaction processes are isolated and removed by adding a counter term to the interaction Hamiltonian. These divergences may thus be treated as a separate problem, which we do not discuss in detail.

High-Energy Inelastic Neutrino-Nucleon Interactions

We discuss high-energy inelastic neutrino-nucleon processes in the light of recent theoretical and experimental developments for the corresponding electroproduction processes. We review the kinematics for the process in a form especially convenient for experimental analysis. We discuss sum rules and results related to current commutation relations. Consequences of the parton model and diffractive models are considered. Other results are: (1) The vector and axial-vector contributions to the total cross section are equal, provided the only symmetry-breaking term in the energy density transforms like a quark-mass term under $U(6)\ensuremath{\bigotimes}U(6)$. (2) Scale invariance of one of the three form factors ($\ensuremath{\nu}\ensuremath{\beta}$ or $\ensuremath{\nu}{\mathrm{W}}_{2}$) describing the process implies a neutrino total cross section which rises linearly with laboratory energy, provided the lepton current is local and there is no $W$ boson. The effect of a $W$ boson on this result is studied. (3) The relation of existing neutrino data and electroproduction data given by the conserved-vector-current hypothesis is studied and found compatible with experiment.

Current Algebra on the Mass-shell

Pion-nucleon scattering on the mass-shell is studied in a field-theoretical framework where the weak currents together with their divergences are taken as interpolating fields for mesons and where the currents obey the Gell-Mann current-algebra (CA) scheme. On the mass-shell the PCAC assumption has a clear physical meaning and the condition for not having any polynomial high-energy behaviour in the πN scattering amplitude is an on mass-shell form of the Adler-Weisberger sum-rule. With this sum-rule we find gA(mπ2)=1.25±0.02 which is consistent with gA(0)=1.23±0.01, the experimental value. The CA and PCAC πN threshold formula is discussed and finite mass corrections to the soft pion limit are considered. Finally we express the condition for the absence of polynomial high-energy behaviour in field-theoretic language as a new current commutation relation.