Chew-Low Model Research Articles

Asymptotic Helicity Conservation and Pion-Nucleon Dynamics

Motivated by the hypothesis of the asymptotic conservation of helicity, we exploit the hypothesis that the $P$ and ${P}^{\ensuremath{'}}$ Regge trajectories decouple from the ${A}^{+}$ amplitude for pion-nucleon scattering. From the vantage point of recent theoretical developments, viz., the concept of duality and the Freund-Harari conjecture, a set of dispersion relations is selected, by means of which we have shown, in good agreement with what is known, the correlation between low-energy resonances in the direct channel and non-Pomeranchukon $t$-channel exchanges. Furthermore, we have shown that there exists a dynamical limit in which the $t$-channel ($\ensuremath{\rho}$ and $\ensuremath{\sigma}$) exchanges decouple, thus providing us with new insight into the relationship between our present approach and the Chew-Low model, the reciprocal-bootstrap, and the strong-coupling theories. Finally, some remarks of a speculative nature are given.

Weak axial-vector coupling constant in the static Chew-Low model

By properly defining weak currents so as to include the mesonic contributions and thereby have them satisfy the chiralSU2×SU2 algebra, it is shown that the axial-vector coupling constant is renormalized by a factor |GA/Gv|=1.13 due to static pion-nucleon interaction, ℋ = (f0/μ)τασ ·▽ϱα. It is also shown that 1≦(GA/Gv)2≦3/2 in this model.

Unified Fermi-Yang and Chew-Low Models

We consider a unified bootstrap theory which incorporates the features of the Chew-Low and Fermi-Yang models. The baryons are treated as being much heavier than the mesons. In this approximation the self-consistent baryon-meson vertices are shown to incorporate simultaneously the $\mathrm{SU}{(6)}_{\mathrm{JJ}}$ and $\mathrm{SU}{(6)}_{W}$ relations. It is pointed out that if the baryons are taken as very heavy, a nearly degenerate sequence of excited rotational states must exist in the crossed channel (baryon-antibaryon); the Chew-Low model is not really self-consistent unless these mesons are all included. The rotational sequence includes exactly the same states as are obtained in the quark model. However, it may be possible to distinguish the two theories through some differences in detail (for example, in the predicted mass splittings).

Chew-Low Model for Regge-Pole Couplings

The couplings of a meson trajectory α(t) to the baryon octet B and the decimet Δ are studied in the Chew-Low model. The model predicts ratios, though not absolute magnitudes, for SU(3)-symmetric couplings of the 0- octet Π, 1- octet V, and 2+ octet T trajectories at the small t of either sign for which static kinematics is applicable. For non-spin-flip, the V and T trajectories are predicted to couple to B B like F+1 / 4D, independent of t. For magnetic dipole terms, the Π, V, and T trajectories are all predicted to couple to B B like D+2 / 3F, and to Δ B with the same relative strength as the 0- octet, independent of t. The electric quadrupole couplings of the Π, V, and T trajectories are predicted to be small, independent of t. These results generally agree with existing data, improve Sawyer's explanation of the Johnson-Treiman relations, provide a partial justification of the recent suggestion that V and T couplings are similar, predict that T exchange produces large spin flips, and predict certain ratios such as dσ / dt(π-p→π0n) / (dσ / dt)(π+p→π0N*++).

Consequences of analyticity and unitarity for partial-wave amplitudes

Asymptotic and integral relationships between the functions associated with the unphysical and physical spectral functions of partial-wave amplitudes with l ≧ 1 for the scattering of spinless particles are developed. The results are used to show that: (a) The function associated with the unphysical singularities must have a closely limited asymptotic behavior if the dispersion relation is to be consistent. (b) The unphysical singularities determine precisely the leading asymptotic term in the function associated with the physical cut and in the absence of unphysical singularities, there can be no scattering. (c) In many cases, the unphysical singularities determine the asymptotic limits of the transmission factor and the real part of the phase shift. (d) If the transmission factor goes to zero (maximal inelasticity), the rate at which it approaches this limit is sometimes bounded by a known function. In these cases, the unphysical singularities and the transmission factor together determine the leading asymptotic term of the phase shift. (e) In some situations, a sign inconsistency in the asymptotic constraints implies the existence of poles in any N D solution with residues whose signs prevent their interpretation as bound states. These poles are then “ghosts.” (f) In an exact partial-wave dispersion relation, at least when l ≧ 2, there is no simple way by which to determine, from the sign of the unphysical function at threshold or at infinity, whether the corresponding “force” is attractive or repulsive. This sign is fixed at these energies by general arguments. As a result of these constraints, one can conclude that the usual approximations of the unphysical cut in terms of the singularities of pole diagrams have a chance of satisfying consistency requirements only for the s- and p-waves with spin zero and one particle exchanges. For p-waves, the exchange of a spin one particle is required. Certain rapidly convergent integral identities are developed which can be used to test for the presence of ghosts or of bound states in N D solutions or to test the accuracy of solutions. Finally, the relativistic Chew-Low model is used to demonstrate numerically the importance of some of the restrictions obtained.