Current Algebra Constraints Research Articles

ScalarB→πandD→πform factors in QCD

QCD sum rules on the light cone are derived for the sum ${f}^{+}{+f}^{\ensuremath{-}}$ of the $\stackrel{\ensuremath{\rightarrow}}{B}\ensuremath{\pi}$ and $\stackrel{\ensuremath{\rightarrow}}{D}\ensuremath{\pi}$ form factors taking into account contributions up to twist four. Combining the results with the corresponding ${f}^{+}$ form factors calculated previously by the same method, we obtain the scalar form factors ${f}^{0}.$ Our sum rule predictions are compared with lattice results, current-algebra constraints, and quark-model calculations. Furthermore, we calculate decay distributions and the integrated width for the semileptonic decay $\stackrel{\ensuremath{\rightarrow}}{B}\ensuremath{\pi}\overline{\ensuremath{\tau}}{\ensuremath{\nu}}_{\ensuremath{\tau}}$ which is sensitive to ${f}^{0}.$ Finally, the dependence of the sum rules on the heavy quark mass and the asymptotic scaling laws are discussed.

Effective lagrangians with U(1) axial anomaly (II)

Properties of the effective lagrangian obeying anomalous U(3) × U(3) current algebra constraints are discussed. Analytic formulae are deduced for the current interpolating constants F ab , the topological constant C (governing the strength of coupling of the topological charge to the physical meson spectrum) and the wave-function renormalization matrix (√ Z) ab . Numerical evaluations of F ab , C and the gluon content of the η and η′ are obtained.

Current algebra and the low-energyπNscattering amplitudes

We discuss low-energy pion-nucleon scattering using the current-algebra constraints on the scattering amplitudes. In order to estimate the on-mass-shell amplitudes, the "remainder terms" are calculated as a sum over the contributions from $\ensuremath{\Delta}(1236)$, ${N}^{*}(1525)$, and ${N}^{*}(1470)$ with particular emphasis on the correct evaluation of the off-mass-shell effects of spin-3/2 resonances. The invariant amplitudes are expanded in powers of $\ensuremath{\nu}$ and ${\ensuremath{\nu}}_{B}$ for small values of these variables. The expansion coefficients are calculated explicitly and compared with experiment.

Isoscalarππresonances and current-algebra constraints inψ′→ψ+2π

The difficulty of classifying three $I=0$, low-lying scalar resonances [in particular $\ensuremath{\epsilon}(700)$] within an SU(4) framework is stressed. Current-algebra constraints are worked out for the decay ${\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{\psi}+2\ensuremath{\pi}$; these lead to a matrix element disfavoring small pion momenta. A companion discussion is given of the decays ${\ensuremath{\eta}}^{\ensuremath{'}}=\ensuremath{\eta}\ensuremath{\pi}\ensuremath{\pi}$ and ${\ensuremath{\omega}}^{\ensuremath{'}}=\ensuremath{\omega}\ensuremath{\pi}\ensuremath{\pi}$.

Fixed mass and scaling sum rules: Generalized truncation and current algebra constraints on Regge residues

Using the correspondence principle (continuity in dynamics), we extend the approach of Keppel-Jones-Ward-Taha to fixed mass and scaling current algebraic sum rules so as to consider explicitly the contributions of all classes of intermediate states. A natural, generalized formulation of the truncation ideas of Cornwall, Corrigan, and Norton is introduced as a by-product of this extension. The formalism is illustrated in the familiar case of the spin independent Schwinger term sum rule. New sum rules are derived which relate the Regge residue functions of the respective structure functions to their fixed hadronic mass limits for q 2 → ∞.

Off-shell amplitudes and currents in the dual-resonance model

We consider the possibility of constructing off-shell amplitudes and currents in the conventional dual model. We determine the necessary form for a current which incorporates the current algebra constraints in addition to the constraints arising from duality and factorization. The current consists of two parts—an off-shell amplitude and a part that exhibits the current algebra properties. We prove that a dual, factorizable off-shell amplitude which has the same spectrum as the conventional on-shell dual-resonance model does not exist.

On the low-energy πN scattering amplitudes

The pion-nucleon scattering amplitudes with current algebra constraints involving “remainder terms” are calculated by saturating them with the N ∗(1236) contribution having off-mass shell effects and compared with experiments.

A field-theoretical approach to nucleon-nucleon dynamics

The usual approach to nucleon-nucleon dynamics is based on the potentials derived from the one-boson exchange approximation. Our aim is to investigate the dynamical content of simple. Lagrangians, depending at most on one free parameter, and where the pion is the only source of nuclear forces. Due to the pseudoscalar nature of the pion it is essential to use a fully relativistic formalism: indeed the nucleon negative-parity states, which are taken into account, appear to play a fundamental role especially at zero energy. The inadequacy of the Born series is over-come by applying to the Green's function an approximation (Pade approximation) derived from variational principles; this approximation enjoys unitarity, analyticity and bound-antibound-state poles. The results obtained for the standard Yukawa Lagrangian and the nonlinear σ-model. which fulfils the current algebra constraints, prove that the one- and two-pion forces are sulficient to reproduce quite well all the phase shifts up to 50 MeV, while from 50 MeV to 400 MeV some of the lowest waves still require further improvement. The inversion in theS-waves (the3P0 changes sign) is still lacking but the introduction of the ω-meson in the Lagrangian and inclusion of all possible couplings between physical and unphysical states (some of them were neglected) should provide it.

Unitarity and current algebra constraints in a low-energy Kπ model

A crossing-symmetric model for low-energy πK scattering in investigated. The model satisfies elastic unitarity to a good approximation, on and off the mass shell. With some minimal additional input from current algebra,s-wave phase shifts andp-wave scattering lengths are determined. A comparison is made with further current algebra theorems including those arising from the (3*, 3)+(3, 3*) symmetry-breaking scheme.

Questions in theoretical determinations of ππ scattering lengths

Using crossing relations on the physical region between the P- and I = 2 S-waves, we estimate the range of ππ scattering lengths allowed by ϱ meson existence. We discuss the difference of our result with some model predictions. We emphasize in particular the role of virtual bound states poles in parametrizations based on current algebra and unitarity constraints. The values of the S-wave scattering lengths can be restricted when some experimental information about the S-waves is added, and improved phenomenological lower bounds may then be obtained.

Exact representation of meson-meson scattering amplitudes consistent with current-algebra constraints

All four-point Green functions connected to the vector and axial-vector currents by Ward identities are constructed, consistent with the constraints of an arbitrary current algebra. The structure of the four-point functions is separated into one-particle reducible and irreducible parts, without approximations. That is, no single-particle approximation is made, nor is it assumed that the strangeness-changing vector currents are conserved. The result is a representation of the meson-meson scattering amplitude that explicitly satisfies all the constraints of the current algebra. In particular, it satisfies: (i) threshold theorems, (ii) crossing symmetry and (iii) reduction to the usual tree-approximation in the single-particle approximation. Application is made to kaon-pion scattering, obtaining a representation of the scattering amplitude that is suitable for the investigation of the constraints of elastic unitarity.

Implications for theΔNπInteraction andσTerm from Low-EnergyπNScattering

A model for low-energy $\ensuremath{\pi}N$ scattering consisting of $N$, $\ensuremath{\rho}$, $\ensuremath{\Delta}$, and $\ensuremath{\sigma}$ exchanges together with current-algebra constraints is presented. We derive the $\ensuremath{\Delta}$ contributions to the scattering amplitude using a general $\ensuremath{\Delta}$ propagator and $\ensuremath{\pi}N\ensuremath{\Delta}$ interaction. We show that the data are consistent with the simplest structure for both the $\ensuremath{\Delta}$ propagator and the interaction, A $\ensuremath{\sigma}$ term of 42 \ifmmode\pm\else\textpm\fi{} 10 MeV at the Cheng-Dashen point is found, and predictions of the $\ensuremath{\sigma}$ term at $t=0$, the $\ensuremath{\pi}\mathrm{NN}$ coupling constant, and the isospin-even $s$-wave scattering length are made.

Compatibility between current algebra and unitarity constraints in a low-energy πK model

An attempt is made to describe the πK scattering amplitude, on and off mass-shell, in the low-energy region, using the requirements of analyticity and unitarity.A three variables crossing symmetric expansion at third order is considered, allowing for the unitarity cuts and allowing one particles to go off its mass-shell. Threshold unitarity is required on-shell and off-shell for one pion or one kaon extrapolated between its physical mass and the Adler's point, thus fixing all the parameters of our model in terms of the two S-wave scattering lengths a O 1 2 and a O 3 2 . Then elastic unitarity is enforced on-shell by a minimization procedure, giving the domains of best unitarity in the (a O 1 2 , a O 3 2 ) plane. Finally some current algebra constraints or a dynamical input provided by the existence of the K ∗ are used to restrict those domains. In the π-extrapolation approach, selected solutions are in agreement with the Adler's condition, give a reasonable value of the π-decay constant f π and satisfy conveniently unitarity and positivity. Moreover their scattering lengths are close to the current algebra results. The predicted phase shifts agree well with the experimental data. Conclusions from the K-extrapolation study, though less accurate are in agreement with the pion extrapolation ones.

Dual-Pion Model Satisfying Current-Algebra Constraints

Dual-Pion Model Satisfying Current-Algebra Constraints

Adler-Weisberger Theorem Reexamined

We consider the nature of the current-algebra constraints on the on-mass-shell amplitudes of pion-nucleon scattering at $s=m_{N}^{}{}_{}{}^{2}$, $t=2m_{\ensuremath{\pi}}^{}{}_{}{}^{2}$. We prove that, at this point, an even amplitude is determined up to corrections of order $m_{\ensuremath{\pi}}^{}{}_{}{}^{4}$ and that the corrections to an odd amplitude are of order $m_{\ensuremath{\pi}}^{}{}_{}{}^{2}$. Thus, the $\ensuremath{\sigma}$ term in the even amplitude, which is of order $m_{\ensuremath{\pi}}^{}{}_{}{}^{2}$, can be determined, confirming the recent work of Cheng and Dashen. However, such a determination cannot be made with a mixture of even and odd amplitudes as von Hippel and Kim have attempted to do. We estimate the actual magnitude of the corrections from what should be the dominant physical mechanisms: the large nucleon size and the proximity of the strong 3-3 resonance state. These corrections are insignificant for the even amplitude. For the odd amplitude, they are separately large, about 10%, but they cancel for the most part and give a much smaller contribution. Corrections due to other resonances are insignificant compared to the 3-3 contribution. The Adler-Weisberger theorem, with our on-shell method, relates the pion-nucleon coupling constant $f$ to the pion decay constant and an integral over the absorptive part of the physical $\ensuremath{\pi}\ensuremath{-}N$ scattering amplitude; it does not involve the axial-vector coupling constant ${g}_{A}$. We find that this relation gives ${f}^{2}=0.077$, which is to be compared to direct evaluations which range from ${f}^{2}=0.082$ to ${f}^{2}=0.076$. We conclude that these current-algebra theorems may well be satisfied to a considerable degree of accuracy. More precise low-energy pion-nucleon scattering data are needed, however, for a definitive test.

Model of Low-Energy Pion-Pion Scattering

A model of the low-energy pion-pion scattering amplitudes is constructed which is consistent with the usual partial-wave analyticity, elastic unitarity, and crossing symmetry in the $s$- and $p$- wave channels. The crossing relations are expressed as five Roskies conditions on the $s$- and $p$- wave amplitudes. The subtraction constants are chosen to agree with currentalgebra constraints. The partial-wave amplitudes are constructed by the $\frac{N}{D}$ method after approximating the left-hand singularities by a few poles. The pole structure is then determined self-consistently by requiring the Roskies conditions and the current-algebra conditions on the derivatives of the amplitudes at subtraction points to be satisfied. A solution is found which contains the $\ensuremath{\rho}$ and $\ensuremath{\sigma}$ mesons as dynamical bound states of the two-pion system. The resulting amplitudes, which should be a good approximation below 1 GeV, are in qualitative agreement with available experimental data. The $I=0$, $s$-wave amplitude has a broad resonance at about 700 MeV. The location of the $\ensuremath{\rho}$ meson is in agreement with experiment, but the width is somewhat larger than the experimental value. The $I=2$, $s$-wave phase shifts are repulsive and small, in agreement with experiment.

Effective-dipole form-factors for the K [formula omitted]3 -decay

To describe the latest experimental situation on the K l 3 decays (which shows large λ +≈0.07 along with a ξ≈−0.9), effective-dipole form-factors, satisfying exact results of first order breaking of the chiral SU 3 ⊗ SU 3 symmetry down to SU 2, are proposed. To the same order in the symmetry breaking the relation, F +(t) + tF −(t) (m 2 K − m 2 π) = F +(0) , holds for a wide enough range of t, so that soft-meson current-algebra constraints are automatically satisfied to that order.

Veneziano Formula,KAMeson Decays, and Weinberg Limits ofKl4Form Factors

$s$- and $d$-wave ratios of ${g}_{{K}_{A}{K}_{\ensuremath{\rho}}}$ and ${g}_{{K}_{A}{K}^{*}\ensuremath{\pi}}$ couplings have been determined using Veneziano representations for ${K}_{A}\ensuremath{\pi}\ensuremath{\rightarrow}K\ensuremath{\pi}$ amplitudes. The results are in agreement with those obtained by the hard-meson techniques of current algebra. Assuming that the axial-vector current is dominated by ${K}_{A}$- and $K$-meson poles only, we obtain the ${K}_{l4}$ form factors from $K\ensuremath{\pi}\ensuremath{\rightarrow}{K}_{A}\ensuremath{\pi}$ and $K\ensuremath{\pi}\ensuremath{\rightarrow}K\ensuremath{\pi}$ scattering amplitudes when the ${K}_{A}$ and one of the $K$ mesons are taken off shell. Current-algebra constraints due to Weinberg, when both pions are soft, are then used to fix the arbitrary parameters of the theory. We find that consistency of our results with current-algebra predictions requires the presence of nonleading Veneziano amplitudes.

Pion-Pion Dynamics in theσModel

We propose an unambiguous way of constructing amplitudes which satisfy both unitarity and the current-algebra constraints. This consists in working out higher-order corrections on a Lagrangian which produces the correct soft-pion limit in the tree approximation. We consider $\ensuremath{\pi}\ensuremath{\pi}$ scattering in the $\ensuremath{\sigma}$ model, and we compute the perturbation series up to second order. The renormalization procedure preserves the partially conserved axial-vector current condition and the current-algebra constraints at each order. In order to sum the strong-coupling perturbation series, we use the Pad\'e-approximation technique. Thereby, our partial-wave amplitudes satisfy unitarity. The $\ensuremath{\rho}$ and ${f}_{0}$ resonances are generated, although they were not present in the Lagrangian. Our unitary amplitudes satisfy crossing symmetry to a very good accuracy, showing the consistency of the results. Our results are in agreement with the "up-down" solution of the $I=0$, $s$-wave $\ensuremath{\pi}\ensuremath{\pi}$ phase shift, with a very broad $\ensuremath{\sigma}$ resonance; the $I=2$ $s$-wave phase shift is repulsive, and agrees very well with experiment.

Local current commutators

The predictive power of current algebra relations is discussed using models for the spectral functions of current commutator matrix elements. The functions constructed fulfill the requirements of covariance, locality and spectrum conditions. Thus, the current algebra constraints can easily be imposed. In these models the formfactors are described by superpositions of poles. For simplicity only spin zero states are taken into account.