Chiral Algebra Research Articles

Nonleptonic decays with exact field algebra

Nonleptonic decays due to a current-current interaction are considered in a nonlinear chiral Lagrangian model with a field-current identity. The apparently attractive possibility of preserving the complete chiral field algebra exactly even after the weak interactions (as well as symmetry-breaking effects for all particles) are included is shown to fail: 3-point vertices for parity-violating baryon B→B′π and K→2π decays vanish.

A representation ofSU 3⊗SU 3 chiral algebra for baryons

A representation ofSU 3⊗SU 3 chiral algebra for baryons

Nonlinear realizations of chiral algebras and phenomenological Lagrangians

Complete and general treatments are presented for the nonlinear realizations of SU(n) x SU(n) and the associated invariant Lagrangians. All transformation functions and covariant derivatives are given in strikingly simple closed forms. The physically important cases of n = 2 and n = 3 are treated in extra detail.

Chiral field algebra and K ℓ4 decays

The method of maintaining the minimality of the number of derivatives in a Lagrangian with spin-zero and spin-one mesons, introduced for chiral SU(2) × SU(2) by Ogievetsky and Zupnik, is generalized as far as necessary to SU(3) × SU(3). Complete field algebra is also preserved. The formalism is applied to the calculation of K e4 axial-vector form factors and decay rates and achieves good agreement with experiment. Various soft-pion limits are also considered, to investigate the conditions under which the Callan-Treiman-Weinberg relations may be obtained.

Harmonic-oscillator pattern arising from an algebraic approach to chiral symmetry

In a previous work we proposed a saturation scheme for theSU3⊗SU3 chiral algebra within an infinite set ofSU6⊗O3 states. A mixing operator was introduced which transforms the axial charges of theSU6 solution into the physical ones,Qi5. Here we saturate the Weinberg equation for the (mass)2 operator [Q5+, [Q5+,m2]]=0, between meson states, in a perturbative approach. The generatorZ of the mixing operators, for which formerly only the transformation properties underSU6×O3 were given, is now completely established asZ=(W×M)z, whereW is theW-spin operator andM is the co-ordinate of the three-dimensional harmonic oscillator. In a perturbative expansion of the (mass)2 operator, the lowest term consists of two pieces, the harmonic-oscillator energy and a spin-orbit coupling of the form (−1)L+1(L·S+1/2). The resulting (mass)2 consists of families of equispaced linearly rising trajectories.

Chiral algebra of fields and semi-leptonic weak interactions

The field algebra model with symmetry breaking which has been introduced in a previous paper is generalized to include axial-vector mesons and pseudoscalar mesons forming a non-linear realization of chiral SU(3) × SU(3). It is used in the Cabibbo theory of weak processes to obtain the form factors for K ℓ3 and baryon semi-leptonic decays. Sum rules for the latter in the presence of symmetry breaking are verified.

General Chiral Algebra andππScattering Lengths

Weinberg's and Khuri's current-algebra calculations for the low-energy $\ensuremath{\pi}\ensuremath{\pi}$ amplitude are generalized by allowing an isospin $I=2$ component in the $\ensuremath{\sigma}$ commutator (as opposed to a pure $I=0$ component given in the $\ensuremath{\sigma}$ model used by both authors). A consistency condition for the $\ensuremath{\pi}\ensuremath{\pi}$ amplitude is derived when one of the pions has zero four-momentum ${q}_{1}=0$, and the others restricted so that $0<{{q}_{2}}^{2}$, ${{q}_{3}}^{2}$, ${{q}_{4}}^{2}<{\ensuremath{\mu}}^{2}$. Using this consistency condition, the power-series expansion of the amplitude in the variables $s$, $t$, and $u$ is calculated up to fourth order in the momenta. It is found that the coefficients of the fourth-order terms are much smaller than the second-order ones in all models. The scattering lengths are calculated in several models without resorting to an expansion of the $\ensuremath{\sigma}$ commutator into a series of the pion field. It is found in each model that the current-algebra calculation of the scattering lengths agrees with the effective Lagrangian calculation in the tree-diagram approximation. The reason for the agreement can be traced to the negligible contributions of the fourth- and higher-order terms in the pion field when the $\ensuremath{\sigma}$ commutator is expanded in a series.

A mixing operator for theSU 3×SU 3 chiral algebra

A saturation scheme within an infinite set of states is proposed for the chiral algebra. Mesons are classified in the (35, anyL) representations ofSU 6×O 3, while the (56,L=even) and (70,L=odd) representations are chosen for baryons. A mixing operator exp [−iθZ] is proposed which transforms the axial charges of theSU 6 solution into the physical ones. The specific form we choose forZ gives rise to many predictions. All the axial couplings of the lower positive-parity meson states to π, ϱ and ω are obtained in terms of only one parameter and the results are in fair agreement with experiment. In particular, the B-meson is predicted to decay transversely, whileg 1/g0 ratio for the A1 is found to be −1/2. For the 1/2+ baryon octet, we getD/F=3/2 andG *=4/5G A (which impliesΓ Δ→Nπ=125 MeV) up to second order in the mixing angle θ. For eachSU 6×O 3 multiplet all the decays intoN+π are given in terms of only one parameter (at lowest possible order in the mixing angle) in general agreement with experiment. Finally, we are able to obtain strong restrictions on the chiral content of the 1/2+ octet, which are perfectly compatible with nature.

Veneziano Amplitude, Current Algebra, and Vertex Functions

The possibility is examined of extending the $\ensuremath{\pi}\ensuremath{\pi}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ and $\ensuremath{\pi}{A}_{1}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ Veneziano amplitudes to the case of one or two mesons off the mass shell in a fashion consistent with chiral current algebra, partial conservation of axial-vector current, and conserved vector current. Soft-pion formulas for the pion-vector form factor and the $\ensuremath{\sigma}$-commutator vertex are deduced. The hard-meson analysis shows that if one wishes to maintain Veneziano form with two mesons off the mass shell, some of the $\ensuremath{\pi}{A}_{1}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ amplitudes must develop poles in one of the off-shell momenta at ${p}^{2}=0$. The soft-pion vertex function equations are seen to be a consequence of removing the leading part of this singularity at the soft-pion point ${p}^{\ensuremath{\mu}}=0$. However, the resultant amplitudes are still singular at ${p}^{2}=0$, ${p}^{\ensuremath{\mu}}\ensuremath{\ne}0$ for a continuum of $s$, $t$, and $u$. The fixed poles implied by the presence of form factors (when two mesons are off shell) are seen to occur only in those amplitudes orthogonal to the ${A}_{1}$ polarization vector and do not affect the amplitudes that make nonzero contributions for $\ensuremath{\pi}{A}_{1}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ scattering on the mass shell.

Partially conserved dilatation and special Liouville currents combined with chiral current algebras

Combining recent results from an analysis of partially broken dilatation and special Liouville (conformal) symmmetries with the concepts of chiral SU 2 × SU 2 and SU 3 × SU 3 algebras of currents yields predictions for the following quantities: The width of the σ(700), s-wave pion-pion scattering lengths, the σ-contribution to the pion-pion Adler-Weisberger sum rule, the partial widths of the η o +(1070) and - in the framework of SU 3 - the partial widths for the other members of the nonet of scalar mesons. Almost all predictions are in good agreement with available data.

Covariant one-particle approximation to chiral algebra and the π ϱ A 1 system

A direct saturation of the commutation relations of axial charges with isospin currents is shown to lead in the covariant one particle approximation to frame independent relations among weak form factors. Assuming the absence of I = 1 operator Schwinger terms we obtain values for the weak decay constants f A and f ϱ and the width Γ Aϱπ of the A 1 meson in reasonable agreement with experiment.

Inconsistency of AsymptoticSU(2)×SU(2)for the Proper Amplitudes with Local Chiral Algebra

Inconsistency of Asymptotic<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>SU</mml:mi><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo><mml:mn /><mml:mo>×</mml:mo><mml:mi>SU</mml:mi><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo><mml:mn /></mml:math>for the Proper Amplitudes with Local Chiral Algebra

Algebraic Realization of Chiral Symmetry and Veneziano Model

It is shown that the algebraic realization of chiral symmetry in which one assigns the mesons to a representation of the chiral algebra is not consistent with the Veneziano model.Received 6 June 1969DOI:https://doi.org/10.1103/PhysRevLett.23.103©1969 American Physical Society

Application of the chiral algebra to the pion-nucleus elastic scattering

Commutation relations of the chiral algebra and P.C.A.C. are used to derive sum rules for the pion-nucleus elastic scattering at the threshold. Saturation properties of the sum rules are discuused.

Consistent Dynamical Formulation of Current-Current Theories and the Validity of Current Algebra

It is shown that the usual formulations of current-current theories are not dynamically consistent, because of the occurrence of Schwinger terms. On the basis of the Poincar\'e-group algebra, an alternative and consistent formulation is constructed and generalized to allow for parity and isospin violation. It is found that, with the appropriate normalization of currents and the corresponding definitions of coupling constants, the chiral algebra of currents for time components is always satisfied in four-dimensional theories of the current-current type. In the two-dimensional case, this statement needs modification, but it is essentially valid there as well. The algebra involving spatial components is highly sensitive to the dynamics, and consequently is not identical to the free-quark-field case. In certain cases of parity violation, the usual currents which couple in the field equations do not transform as four-vectors even though the theory itself is covariant. This is somewhat analogous to radiation-gauge electrodynamics. It is found, however, that it is possible to construct new covariant currents as linear combinations of the components of the old currents which have the property that they couple covariantly into the field equation. Because of these complications, the spatial components of the covariant currents do not have a definite parity and do not transform as vectors in isospin.

Four-Point Function andϱ′→2πγDecay

Radiative decay of the $\ensuremath{\rho}$ meson has been studied within the framework of the four-point function of currents obeying $\mathrm{SU}(2)\ensuremath{\bigotimes}\mathrm{SU}(2)$ chiral algebra. A hard-pion calculation of the decay amplitude by means of the four-point function shows the correct gauge-invariant electromagnetic structure of the radiative process, while the same amplitude evaluated in the off-shell limits ${q}^{2}\ensuremath{\rightarrow}0$, ${p}^{2}\ensuremath{\rightarrow}0$ for the pions fails to account for the dominant dynamical features, such as the internal bremsstrahlung mechanism and gauge invariance. A comparative study of both on-shell and off-shell current-algebra predictions for decay rates and the photon energy spectrum have been presented, along with the results of the vector-meson pole-dominance model.

Pion-Nucleon Scattering Lengths

Pion-nucleon scattering lengths have been calculated using a model in which $N$, ${N}^{*}$, $\ensuremath{\rho}$, and $\ensuremath{\epsilon}$ exchanges are assumed to dominate the low-energy pion-nucleon interaction. This simple model yields remarkably good agreement with the experimental $s$- and $p$-wave scattering lengths. We find that the coupling of the $\ensuremath{\epsilon}$ meson to the nucleon is consistent with estimates made using chiral symmetry and current algebra.

Electromagnetic Mass Differences and Sum Rules in a Lagrangian Model

We find the canonical commutation rules of the Brown-Munczek Lagrangian and evaluate other equal-time commutators of the theory. The fields do not obey a chiral $\mathrm{SU}(2)$ algebra exactly; it is in the time components of the axial-vector field where the algebra fails. We relate these commutators to the pion electromagnetic mass difference and to sum-rule calculations. Using the Bjorken criteria, we conclude that the mass-difference calculation diverges in general, but in the tree-diagram approximation the coefficient of the logarithmic divergence vanishes. This result is analyzed in detail and compared with other models. In particular, even in this model in the tree approximation, the $\ensuremath{\rho}$-mass calculation diverges. The spectral sum rules of the algebra are seen to be equivalent to those due to Weinberg and lead to the same successful predictions if the partially conserved axial-vector current and Kawarabayashi-Suzuki-Fayyazuddin-Riazuddin relations are included in the model. It is also possible to construct an axial-vector current which obeys the chiral algebra of charges.

Lower hadron states classification and SU(3) ⊗ SU(3) chiral algebra approximate saturation

A classification scheme for hadrons is proposed, which fits well with the experimental π-N resonances. Under the hypothesis of SU(3) ⊗ SU(3) chiral algebra approximate saturation, G A and α = D/ D + F are evaluated in terms of the squares of the axial coupling constants; their value is in good agreement with experiment. Further model dependent predictions are obtained for the mixing angles of the baryon wave function and for the branching ratios of the predicted strange resonances. The values of the mixing angles are in agreement with what we expect from the e.m. structure of the nucleons.

Chiral dynamics, SU (3) × SU (3) field algebra and symmetry breaking

Chiral field algebra Lagrangians are constructed in SU (3) × SU (3) consistent with the assumption that chiral symmetry breaking transforms as the (3,3 ∗) ⊕ (3 ∗,3) representation of SU(3) × SU(3). Alternative models are constructed based on stated assumptions and consequences are obtained for two point functions (in the K ∗ and ϱ directions). The magnitude of symmetry breaking is related to the status of K A(1 +) and ϰ(0 +) mesons. The techniques are useful for the study of (3,3 ∗) ⊕ (3 ∗,3) symmetry breaking in higher point functions within the framework of chiral dynamics.