Adler's Self-consistency Condition Research Articles

Strong decays and Adler selfconsistency condition in two-dimensional QCD

Strong mesonic decays are studied in the framework of the 't Hooft model for the two-dimensional QCD in the axial gauge. Special attention is payed to the processes with pions in the final state involved and the low energy theorems have been checked including the “Adler zero”. It is demonstrated explicitly that in the chiral limit any three-meson decay amplitude with at least one pion vertex vanishes identically for any values of the external momenta. Ward identities for the dressed vector and axial-vector currents vertices are identified and their relationship to the pionic vertex are discussed.

Partially conserved axial-vector current inS-matrix theory

Mandelstam’s argument that PCAC follows from assigning Lorentz quantum numberM=1 to the massless pion is examined in the context of multiparticle dual resonance model. We construct a factorisable dual model for pions which is formulated operatorially on the harmonic oscillator Fock space along the lines of Neveu-Schwarz model. The model has bothm π andm ϱ as arbitrary parameters unconstrained by the duality requirement. Adler self-consistency condition is satisfied if and only if the conditionmϱ2−mπ2=1/2 is imposed, in which case the model reduces to the chiral dual pion model of Neveu and Thorn, and Schwarz. The Lorentz quantum number of the pion in the dual model is shown to beM=0.

A calculation of the [1,1] padé approximant for pion-nucleon phase-shifts

Low-energy pion-nucleon phase-shifts have been computed using, (i) once subtracted fixed hyperbola dispersion relations and (ii) elastic unitarity, in the [1, 1] Padé approximation. A t-channel s-wave pole (σ-meson) has been introduced, the residue of which is determined from Adler's self-consistency condition. It is found that although the inclusion of the Adler condition gives results which are considerably closer to experimental phase-shifts as compared to earlier calculations of this kind with fixed momentum transfer dispersion relations, the agreement with data is still far from satisfactory.

Smoothness of pion amplitudes and PCAC

We point out the essentially obvious result that if the massless pion amplitude is the smooth finite limit of the massive pion world, the Adler self-consistency condition must be true.

Chirality in Dual-Resonance Models

We present a dual-resonance model for $N$ pions which satisfies the Adler self-consistency condition and provides a generalization of the Lovelace-Shapiro formula for four pions. Our model has difficulty with SU(3) since there is no candidate for the $\ensuremath{\omega}$ degenerate with the $\ensuremath{\rho}$. We also discuss briefly the problem of extending the model to include fermions.

Dispersive approach to off-mass-shell ππ amplitudes

It is shown that ππ amplitudes with one pion off its mass shell are determined by a set of linear integral equations in terms of physical amplitudes once off-shell adsorptive parts are known in the inelastic region. We show how this can be used in practice to obtain off-shell amplitudes in the low energy region. The arbitrariness of off-shell extensions is discussed; it is suppressed in the soft pion limit owing to Adler's self-consistency condition. Applications of our method to Adler-Weisberger type relations, and to off-shell effects in Chew-Low extrapolation experiments are discussed.

Low-Energy Zeros and Regge Poles from an Extended Amati-Bertocchi-Fubini-Stanghellini-Tonin Multiperipheral Model

By inserting the elastic absorptive part from a multiperipheral model into a fixed-momentum-transfer dispersion relation, it is possible to obtain the low-energy amplitude in the crossed channel. We use the ABFST (Amati-Bertocchi-Fubini-Stanghellini-Tonin) version of this model so that the amplitude satisfies a BS (Bethe-Salpeter) equation. The Veneziano model for the $\ensuremath{\pi}\ensuremath{\pi}$ amplitude is used to calculate the ABFST input kernel, for which off-shell effects are neglected. We go beyond the usual pion-exchange dominance of the ABFST model by including an "inelasticity" factor in our BS equation. The strength of this factor is fixed by requiring that the Chew-Mandelstam $\ensuremath{\pi}\ensuremath{\pi}$ symmetry-point crossing condition is satisfied. It is then found that zeros arise in the $S$-wave amplitudes, which can presumably be identified with the zeros which are predicted from the Adler self-consistency condition. As a by-product, we also calculate the leading Regge trajectories in our model. These have no difficulty in rising to fairly high angular momenta and lead to relatively narrow reduced widths for the $\ensuremath{\rho}$ and ${f}^{0}$ resonances, as well as a high-energy total cross section in approximate agreement with experiment.

Reactionπ−p→ηnwith Veneziano Terms

The Veneziano amplitude for ${\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}\ensuremath{\eta}n$ scattering is constructed using suitable combinations of beta-function terms. If certain constraints for the pole structure are imposed, the amplitude shows agreement and consistency with the experimental data over the whole energy region. The shape of the high-energy ${\ensuremath{\pi}}^{\ensuremath{-}}p\ensuremath{\rightarrow}\ensuremath{\eta}n$ differential cross section is reproduced. The decay widths for the ${\frac{5}{2}}^{+}$ and ${\frac{5}{2}}^{\ensuremath{-}}$ pion-nucleon resonances are obtained, and Adler's self-consistency condition is also approximately satisfied.

Veneziano Model and the Adler-Weisberger Sum Rules forK−πScattering

It is shown that the contributions to the Adler-Weisberger sum rule (AWSR) in the simplest Veneziano model for $K\ensuremath{-}\ensuremath{\pi}$ scattering are too large by a factor of 2. A modification of the simplest Veneziano amplitude is proposed which satisfies (i) the Adler self-consistency condition, (ii) the AWSR, and (iii) the ${K}^{*}(890)$ having the experimentally observed width of 50 MeV. If this modified Veneziano amplitude is required to satisfy the Adler-Weisberger-type sum rule for zero-mass $K$ mesons scattering from physical pions, the ratio $\frac{{F}_{K}}{F\ensuremath{\pi}}$ is predicted to be 1.26, in good agreement with experiment.

Internal symmetry implications of the Veneziano model in pseudoscalar-pseudoscalar scattering

The leading order Veneziano model amplitudes for ( J P ) 0 − × 0 − → 0 − × 0 − scattering ∗∗ ∗∗ While in preparation we were informed of the presentation by Kawarabayashi et al. [4] of a paper that contains results similar to ours. constrained by the absence of “exotic” resonances, is shown to predict the coupling constant relation of SU(3) coupled with the quark-model/SU(6) ideal singlet-octet mixing in the 1 −, 2 +, nonets. The Adler self-consistency condition is found to hold (for the most part) both for pions and kaons fulfilling Lovelace's conjecture.

Pion scattering lengths, the veneziano amplitude and ρ universality

The Veneziano amplitude, Adler's self-consistency condition, and universality of the ρ-coupling are shown to determine theS-wave scattering lengths for the process πA→πA. A discussion is presented with reference to the current algebra result.

New Representation of the Scattering Amplitude

A new representation of the scattering amplitude that has good analytic properties, Regge asympotic behavior, and an arbitrary double-spectral boundary is proposed. The representation automatically yields partial waves with the correct threshold behavior for both their real and imaginary parts. The presence of the correct double-spectral boundary should be very important in decay problems, where an unstable external particle considerably modifies the analytic properties of the amplitude. This representation in its simplest version is applied to pion-pion scattering. Unitarity is enforced near threshold. By using the (degenerate) $\ensuremath{\rho}$- and $f$-meson trajectories, the $\ensuremath{\rho}$ width, and the Adler self-consistency condition, the three isotopic scattering lengths, the $f$ width, and the Regge-scale mass are predicted.

Adler's self-consistency condition for nuclei

It is shown that Adler's self-consistency condition for light nuclei ( 4He, 12C, 9Be, etc.) implies the smallness of the contribution from the unphysical region to charge symmetric π-nucleus forward dispersion relations.

Factorization, Kinematic Singularities, and Conspiracies

Factorization of residues of poles of the $S$ matrix is derived from the requirements of unitarity for partialwave helicity amplitudes. Careful attention is given to questions of spin and the kinematic singularities of the relevant amplitudes, especially at $t=0$. Residues $\ensuremath{\beta}$ of a pole in the full partial-wave amplitude satisfy factorization in the simple form ${{\ensuremath{\beta}}_{\mathrm{ba}}}^{2}={\ensuremath{\beta}}_{\mathrm{aa}}{\ensuremath{\beta}}_{\mathrm{bb}}$. In general, $\ensuremath{\beta}$ can be written as $\ensuremath{\beta}=K\ensuremath{\gamma}$, where $K$ contains the standard kinematic singularities of the Hara-Wang type, plus threshold behavior, and $\ensuremath{\gamma}$ is a reduced residue. The $K'\mathrm{s}$ for various mass classes are exhibited in a compact and consistent form, and the corresponding factorization statements for the reduced residues are derived. These factorization relations are of the form ${t}^{x}{{\ensuremath{\gamma}}_{\mathrm{ba}}}^{2}={\ensuremath{\gamma}}_{\mathrm{aa}}{\ensuremath{\gamma}}_{\mathrm{bb}}$, where $x$ is an integer. The reduced residues are analytic in the neighborhood of thresholds and pseudothresholds, but may, in the case of conspiracies, contain poles at $t=0$. Various examples are presented to illustrate the use of factorization. These include LeBellac's argument on the behavior of the pion residue at $t=0$ and its circumvention with a type-II conspiracy. mandelstam's treatment of Adler's self-consistency condition and the hypothesis of partially conserved axial-vector current using an $M=1$ pion is discussed from the viewpoint of factorization. It is shown that factorization for an $M=1$ pion seems to imply smallness of both soft-pion and hard-pion amplitudes. The smallness of the latter casts some doubt on the $M=1$ assignment for the pion. The nature of the relations between amplitudes and the behavior of the reduced residues at $t=0$ for conspiracies with unequal masses is also considered.

A novel application of Regge trajectories

We suggest that Dalitz plots for 3π decays and annihilations are “measurements” of ππ scattering in the double spectral region. Veneziano's formula for crossing-symmetric Regge trajectories both justifies the external mass extrapolation and correctly predicts the experimental results. It also fulfills Adler's self-consistency condition if αϱ(m π 2) = 1 2 .

Method of Pole Dominance and Applications

A method for calculating scattering amplitudes at low to moderate energies in terms of their bound-state and resonant poles is presented. By comparing expressions obtained for $s$-wave scattering lengths with corresponding predictions of current algebra, the widths of the $\ensuremath{\rho}$, ${K}^{*}$, $\ensuremath{\Delta}$, and $\ensuremath{\omega}$ resonances are calculated in close agreement with experiment. Furthermore, without reference to current algebra, all the detailed features of $\ensuremath{\pi}N$ elastic scattering (such as the behavior of the ${P}_{11}$ phase shift) up to an energy of 400 MeV are obtained within an error of about 10 to 20%. The principal advantage of the present method over previous pole-dominance models is that the calculation of amplitudes has been reduced to the evaluation of a number of rapidly convergent integrals. In some cases, one or two undetermined subtraction constants must be introduced, but in the examples of $\ensuremath{\pi}K$ and $\ensuremath{\pi}N$ elastic scattering only one unknown constant arises, and it can be evaluated by means of the Adler self-consistency condition or an $M=1 O(4)$ assignment for the pion.

Low-energy theorem for pions

Under the assumption that incoming and outgoing pions have zero mass a gauge principle for theS-matrix is developed, leading to the generalized Adler self-consistency condition that all pion amplitudes vanish at zero pion four-momentum. In the second part the theory of composite particles is discussed from a more fundamental point of view, in connection with р-meson and low-energy\(\pi \mathcal{N}\)-scattering. A result equivalent to that of Adler and Weisberger is derived and comparison with other models discussed. The generalization to forward scattering of low-energy zero-mass pions off any target is also indicated.

Relations between Partially Conserved Axial-Vector Current, Axial-Vector Charge Commutation Relations, and Conspiracy Theory

The Adler self-consistency condition is shown to be true without reference to currents, provided that the pion trajectory is a member of a conspiracy with $|M|\ensuremath{\geqq}1$. It is then shown from the Adler self-consistency condition that the antisymmetric part of the amplitude for forward scattering of low-energy low-mass pions off any target is equal to a universal constant multiplied by the isotopic spin of the target. Thus all restrictions which current algebra imposes on hadron scattering amplitudes can be derived from conspiracy theory; production amplitudes are not examined here. We next investigate the question whether a system which contains massless pions satisfying the Adler self-consistency condition possesses a conserved axial-vector current. Assuming that the usual Omnes-type equation can be solved, we find that it does. Finally, the commutation relations between total axial-vector charges are shown to follow from the conservation equation under very general assumptions.