Pion-pole Dominance Research Articles

Partially Conserved Axial-Vector Current and theπ0→2γDecay Rate

The difference between applying partial conservation of axial-vector current (PCAC) to the amplitude for ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ decay, which involves products of local currents, and to amplitudes involving composite hadrons, where it has enjoyed its notable successes-viz., the Goldberger-Treiman relation, the Adler-Weisberger sum rule, and the Adler consistency condition-is analyzed. Using the Bell-Jackiw-Adler theory of the anomaly, we show that this difference provides a mechanism for removing the factor-of-10 discrepancy that is usually claimed to exist between the observed decay rate and the one calculated on the basis of the original Gell-Mann-Zweig quark model with one triplet of fractionally charged quarks. An essential dynamical assumption is that pion-pole dominance is valid only for those matrix elements of the divergence of the axial-vector current taken with composite hadronic states; this is akin to features in the weak PCAC of Brandt and Preparata. A specific model of the hadron as a Bethe-Salpeter bound state of two pointlike constituents is used to illustrate the underlying dynamical mechanism. It follows from this that there should be a sizable enhancement above the prediction by Adler for forward-angle high-energy very inelastic neutrino-hadron cross sections. Verification of this prediction will be a crucial test of our theory.

Theory of Deviations from Pion Pole Dominance

It is shown how mass dispersion relations can be effectively used to quantitatively determine the effects of extrapolating an amplitude off the mass shell. Light-cone operator-product expansions abstracted from the canonical gluon model are used to determine the asymptotic behavior in the mass variable. This determination is made quantitative via the principles of precocious asymptopia and smooth threshold behavior. These techniques are used to calculate some corrections to pion pole dominance for vertex functions and scattering amplitudes in the weak PCAC (partially conserved axial-vector current) framework. Corrections to kaon pole dominance and to SU (3) symmetry are determined in the course of the analysis. Put differently, some measurable consequences of broken SU (3) \ifmmode\times\else\texttimes\fi{} SU (3) symmetry are determined. The canonical equal-time and light-cone operator algebraic structures are presumed to be exact, but the physical states (and hence on-shell) $S$-matrix elements) may exhibit considerable, but now controllable, deviations from the underlying SU (3) \ifmmode\times\else\texttimes\fi{} SU (3) symmetry. As explicit examples of the methods, the ${K}_{l3}$ form factors ${f}_{\ifmmode\pm\else\textpm\fi{}}({q}^{2})$ and the symmetric and antisymmetric $\ensuremath{\pi}N$ scattering amplitudes are discussed in detail. The numerical results are: ${f}_{+}(0)\ensuremath{\cong}1.0$, $\ensuremath{\xi}(0)\ensuremath{\equiv}\frac{{f}_{\ensuremath{-}}(0)}{{f}_{+}(0)}\ensuremath{\cong}\ensuremath{-}0.7$, ${f}_{+}({{m}_{K}}^{2})+{f}_{\ensuremath{-}}({{m}_{K}}^{2})\ensuremath{\cong}(0.7)\ifmmode\times\else\texttimes\fi{}\frac{{f}_{K}}{{f}_{\ensuremath{\pi}}}$, ${\ensuremath{\alpha}}_{8}=[\mathrm{SU}(3)\ensuremath{-}\mathrm{b}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{g}\phantom{\rule{0ex}{0ex}}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}]\ensuremath{\cong}\ensuremath{-}140$ MeV, ${\ensuremath{\sigma}}_{N}=(\ensuremath{\pi}N \ensuremath{\sigma} \mathrm{term})\ensuremath{\sim}450$ MeV, ${g}_{A}\ensuremath{\sim}1.23$. The results for $\ensuremath{\pi}N$ scattering confirm and extend the previous weak PCAC analysis, based on Regge theory, but some of the ${K}_{l3}$ results are now different. The good result $\ensuremath{\xi}(0)\ensuremath{\sim}\ensuremath{-}0.7$ is, however, maintained and is now obtained without any assumption about the ${q}^{2}$ dependence of ${f}_{\ifmmode\pm\else\textpm\fi{}}$. None of the present results involved assumptions of smooth off-shell behaviors of Green's functions. In all cases the mass dispersion relations were used to compute the effects of off-shell extrapolations.

A modified algebra of currents in broken chiral symmetry

A method of relaxing the usual hypothesis that the algebra of currents is unaffected by chiral symmetry breaking is proposed and investigated. While the equal-time SU(3) ⊗ SU(3) charge algebra is retained, the space components of the currents are allowed to transform generally among themselves under commutation with the nonconserved charges, resulting in a modified current algebra containing five real parameters of the symmetry breaking. It is found that manifest covariance in low-energy theorems is destroyed, necessitating introduction of new techniques for sharpening the role of pion pole dominance and SU(3) symmetry. These techniques lead to the conclusion that the predictions of current algebra for pion scattering lengths G A G V and hyperon decays are unchanged, whereas those for π 13 decay and the connections among K 12, K 13 and K 14 decays are altered in definite ways, as is Weinberg's first-sum rule. All of the predictions for leptonic K-decays are found to involve only one of the five symmetry breaking parameters; kaon experiments therefore provide both measurements of one of the new parameters and tests of the original hypothesis of a generalized current algebra. Present experimental data are, however, not sufficiently accurate for this purpose.

What Are theπ−πScattering Lengths?

Weinberg has calculated the $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering lengths assuming current algebra, (3, $\overline{3}$)-symmetry breaking, an expansion to first order in the squared energies $s$, $t$, $u$ and mass variables ${{q}_{i}}^{2}$, and a strong form of pion-pole dominance. We show that essentially the same scattering lengths are obtained if the strong pion-pole dominance and expansion in ${{q}_{i}}^{2}$ assumptions are replaced by a much weaker form of pion-pole dominance. Canonical lightcone and Regge-pole assumptions and techniques are used in the calculation.

Pion amplitudes in a nonchiral world

The popular strong PCAC (i.e., pion pole dominance of all Green's functions involving the divergence D of the axial vector current) and good chiral SU(2) × SU(2) symmetry framework has led to many successful predictions. It also, however, leads to some difficulties since, for example, it predicts the approximate vanishing of the K l 3 form factor ratio ξ = f − f + , of the π 0 → γγ decay amplitude F π (except in models with anamolous PCAC), and of the η → 3π decay amplitude A η . We therefore propose an alternate framework based on weak PCAC (i.e., pion pole dominance of physical matrix elements of D) and good SU(3). We show that a precise formulation of these older ideas can reproduce all of the good results of the chiral scheme and none of the bad ones. We assume the usual Gell-Mann current algebra and the (3, 3) + ( 3, 3) symmetry breaking Hamiltonian H′ = α 0 S 0 + α 8 S 8. We make, and motivate with reference to a composite picture of the hadrons, the following four assumptions: (1) approximate SU(3), (2) weak PCAC, (3) minimal subtractions, (4) Regge theory. In this framework, the dominance of a given Green's function T by the pion pole depends on the nature of the despersion relations it satisfies in variables for which the absorptive part is expressible in terms of physical matrix elements of D. An unsubtracted dispersion relation leads to pole dominance but a subtracted one introduces a constant (or worse) difference between T and the corresponding on-shell pion amplitude. From assumptions (2)–(4), we derive the following successful results of the chiral framework: the Adler consistency condition, the Adler-Weisberger relation, the Weinberg-Tomozawa pion-target scattering lengths, the Weinberg π-π scattering lengths, the Fubini one-pion production relations, the Weinberg-Callan-Treiman K l 4 form factors, and the K 3 π – K 2 π relations. The extra parameters (subtraction constants) encountered in applying weak PCAC, as determined by Regge theory, turn out not to contribute to these relations. Adding assumption (1), we obtain the further results: (α 0/ 2 α ∞)≃−0.17 , ξ ≅ −1, and an excellent simultaneous fit to the S-wave and P-wave hyperon decays. Here our extra parameters turn out to be present precisely in the places where they are required to insure the consistency of our approach. (For example, the σ terms in our scheme need not be small). None of these results are common to the chiral framework. We show further that F π and A η need not approximately vanish in our framework. We therefore conclude that our theory seems to provide a better description of the present empirical situation than does the chiral theory.

Some Predictions of the Amati-Bertocchi-Fubini-Stanghellini-Tonin Multiperipheral Model

The predictions of the Amati-Bertocchi-Fubini-Stanghellini-Tonin multiperipheral model based on pion pole dominance are calculated quantitatively and compared with experimental data, by using the Veneziano representation for the $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ amplitude, with a cutoff at the mass of the $g$ resonance for the input kernel, and neglecting off-shell effects. We find ${\ensuremath{\alpha}}_{P}(0)=0.30$, ${\ensuremath{\alpha}}_{\ensuremath{\rho}}(0)=0.16$, an average multiplicity growing as 0.74 $\mathrm{ln}s$, an elasticity factor of 0.69, an average pion-pair transverse momentum equal to 0.43 $\frac{\mathrm{GeV}}{c}$, and an average invariant mass squared equal to 0.84 Ge${\mathrm{V}}^{2}$. The only serious discrepancies with experiment are the trajectory heights, which correspond to a kernel strength $2\frac{1}{2}\ensuremath{-}5$ times too weak. The inclusion of the high-subenergy contribution and the inclusion of off-shell effects by several methods of off-shell continuation are considered, but none is found to be satisfactory. The importance of interference terms, due to different ways of arranging the final-state particles along the multiperipheral chain, is discussed.

SU 3 orSU 2⊗SU 2: Kol3 form factors as a symmetry probe

We studyKl3 form factors assumingSU3⊗SU3 current commutation relations and pion-pole dominance of the axial-current divergence for small momentum transfer. We derive a correction to the Callan-Treiman relation which is small in the limit of nearSU2⊗SU2 invariance and large and negative in the limit of nearSU3 invariance. In the latter limit, the corrected result conforms to the requirements of small «leakage» forSU3 generators and predicts a negative value for the parameter ξ.

Perturbation theory and the goldberger-treiman relation

The Goldberger-Treiman relation was originally derived (~) by postulating that the form factor c~(q), which is the decay constant for a pion of four-momentum qg, satisfies unsubtracted dispersion relations, and that these dispersion relations could be saturated by the contribution of the nueleon-antinueleon intermediate state. Under certain approximations, it was then possible to relate e~ to the pion-nucleon coupling constant g ~ , , the nucleon mass M, and the axial vector coupling constant in neutron beta decay, g~. The resulting relation, cr: =-g .~ M/gr~x.~c, holds to an accuracy of about 10%. The approximations in this derivation were quickly recognized as difficult to justify, but several other derivations were suggested (2), based on assumed properties of matrix elements of the derivative of the axial vector current, taken between single-nucleon states. Here we describe another derivation, related to but distinct from earlier derivations. Following GOLDBERGER and TREIMAN (~), we consider the decay of the pion as occurring through a nueleon-antinueleon loop; we later discuss the extent to which the result depends on this assumption. We assume isospin-invariant derivative coupling at the pion-nucleon vertex; the Lagrangian is (3) *~fi = i O~¢P~u7~5 ~ru. For the weak vertex, we assume that only first-class currents (4) contribute; there is a contribution to the axial vector current of the form g~(q2)~r~,~su, and for the induced pseudosealar term iH(q~)~ybuq~ we assume pion-pole dominance. However, instead of using dispersion relations as in ref. (~), we do simple perturbation theory. After taking a trace, doing some simple algebra and making a translation in momentum space, one obtains

Decaysπ0→2γ,η→2γand Sum Rules for Nucleon Compton Scattering

Motivated by the success of the Goldberger---Treiman calculation of charged-pion decay and the assumption of pion-pole dominance of the divergence of the axial current subsequently introduced, we study the decays of the neutral pion and eta in the same way. Examining Compton scattering of real photons from the nucleons presents a fruitful analogy to the nuclear $\ensuremath{\beta}$-decay process. Identifying the dominant nucleon and meson-pole-term contributions to Compton scattering and utilizing the content of the exact low-energy theorem on Compton scattering, a no-subtraction hypothesis, supported by the Regge-pole phenomenology, then enables us to establish exact sum rules for the lifetimes ${\ensuremath{\tau}}_{{\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}}$ and ${\ensuremath{\tau}}_{\ensuremath{\eta}\ensuremath{\rightarrow}2\ensuremath{\gamma}}$ and also the Drell---Hearn sum rules for $\ensuremath{\kappa}_{p}^{}{}_{}{}^{2}\ifmmode\pm\else\textpm\fi{}\ensuremath{\kappa}_{n}^{}{}_{}{}^{2}$. We also consider in detail the relation of our sum rule to the Goldberger---Treiman calculation of ${\ensuremath{\pi}}^{0}$ decay and forward-Compton-scattering sum rules for systems of spin=1. Neglecting continuum contributions, which are small, we find from our sum rules $\ensuremath{\kappa}_{p}^{}{}_{}{}^{2}=\ensuremath{\kappa}_{n}^{}{}_{}{}^{2}$ and $\ensuremath{\tau}_{{\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}}^{}{}_{}{}^{\ensuremath{-}1}=\frac{\ensuremath{\pi}{\ensuremath{\alpha}}^{2}m_{\ensuremath{\pi}}^{}{}_{}{}^{3}\ensuremath{\kappa}_{p}^{}{}_{}{}^{2}}{4{g}_{\ensuremath{\pi}{N}^{2}}{M}_{{N}^{2}}}\ensuremath{\simeq}3.1$ eV or ${\ensuremath{\tau}}_{{\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}}\ensuremath{\simeq}2.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}$ sec in approximate agreement with the experimental value ${\ensuremath{\tau}}_{{\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}}=(1.0\ifmmode\pm\else\textpm\fi{}0.5)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}16}$ sec. Better photopion-production data for the nonresonant multipoles would enable us to accurately estimate the continuum contributions. Including the dominant contributions to the photopion production continuum, we find that the Drell-Hearn sum rule for $\ensuremath{\kappa}_{p}^{}{}_{}{}^{2}+\ensuremath{\kappa}_{n}^{}{}_{}{}^{2}$ is well satisfied. We can estimate the $\ensuremath{\eta}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ lifetime but our result depends on a knowledge of the eta-nucleon coupling and on nonresonant background contributions to the photopion multipoles. Once better photopion production data for the nonresonant multipoles ${E}_{l\ifmmode\pm\else\textpm\fi{}}$, ${M}_{l\ifmmode\pm\else\textpm\fi{}}$, $ll~2$ for energies 1 BeV become available, these sum rules can offer reliable theoretical estimates for the decays ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}$, $\ensuremath{\eta}\ensuremath{\rightarrow}2\ensuremath{\gamma}$.