Pion Pole Research Articles

Analysis of KL → π+π−γ in chiral perturbation theory

We study the long-distance dominated K L → π + π − γ decay at O(p 6) in chiral perturbation theory. The complete calculation of the O(p 6) loop magnetic amplitude is carried out. At this chiral order the model-dependent part of the vector meson exchange contribution to the magnetic amplitude is evaluated in the two models: FM (Factorization Model) and FMV (Factorization Model in the Vector couplings). We predict, in an almost model-independent way, a slope of K L → π + π − γ in the range c ∼- −1.6, −1.8, consistently with the experimental value. We find that the experimental result for the width of K L → π + π − γ is compatible with a bigger breaking of the nonet symmetry in the weak vertices than previously stated. Thus we conclude that our analysis does not exclude an opposite sign to the one given by the dominance of the pion pole in the poorly known K L → γγ amplitude. A complete analysis of the O(p 3) weak vector-pseudoscalar-pseudoscalar ( VPP) vertex is also performed.

How to extract theP33(1232)resonance contributions from the amplitudesM1+3/2,E1+3/2,S1+3/2of pion electroproduction on nucleons

Within the dispersion relation approach, solutions of integral equations for the multipoles M_{1+}^{3/2},E_{1+}^{3/2},S_{1+}^{3/2} are found at 0 < Q^2 < 3 GeV^2. These solutions should be used as input for the resonance and nonresonance contributions in the analyses of pion electroproduction data in the P_{33}(1232) resonance region. It is shown that the traditional identification of the amplitude M_{1+}^{3/2} (as well of the amplitudes E_{1+}^{3/2},S_{1+}^{3/2}) with the P_{33}(1232) resonance contribution is not right; there is a contribution in these amplitudes which has a nonresonance nature and is produced by rescattering effects in the diagrams corresponding to the nucleon and pion poles. This contribution is reproduced by the dispersion relations. Taking into account nonresonance contributions in the amplitudes M_{1+}^{3/2},E_{1+}^{3/2}, the helicity amplitudes A_p^{1/2},~A_p^{3/2} and the ratio E2/M1 for the gamma N -> P_{33}(1232) transition are extracted from experiment at Q^2=0. They are in good agreement with quark model predictions.

Optimal renormalization scale and scheme for exclusive processes

We use the Brodsky-Lepage-Mackenzie (BLM) method to fix the renormalization scale of the QCD coupling in exclusive hadronic amplitudes such as the pion form factor and the photon-to-pion transition form factor at large momentum transfer. Renormalization-scheme-independent commensurate scale relations are established which connect the hard scattering subprocess amplitudes that control exclusive processes to other QCD observables such as the heavy quark potential and the electron-positron annihilation cross section. The commensurate scale relation connecting the heavy quark potential, as determined from lattice gauge theory, to the photon-to-pion transition form factor is in excellent agreement with $\ensuremath{\gamma}\stackrel{\ensuremath{\rightarrow}}{e}{\ensuremath{\pi}}^{0}e$ data assuming that the pion distribution amplitude is close to its asymptotic form $\sqrt{3}{f}_{\ensuremath{\pi}}x(1\ensuremath{-}x)$. We also reproduce the scaling and normalization of the $\ensuremath{\gamma}\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ data at large momentum transfer. Because the renormalization scale is small, we argue that the effective coupling is nearly constant, thus accounting for the nominal scaling behavior of the data. However, the normalization of the space-like pion form factor ${F}_{\ensuremath{\pi}}{(Q}^{2})$ obtained from electroproduction experiments is somewhat higher than that predicted by the corresponding commensurate scale relation. This discrepancy may be due to systematic errors introduced by the extrapolation of the ${\ensuremath{\gamma}}^{*}\stackrel{\ensuremath{\rightarrow}}{p}{\ensuremath{\pi}}^{+}n$ electroproduction data to the pion pole.

A new measurement of the [formula omitted]p → [formula omitted]n charge-exchange differential cross-section at LEAR

We have measured the p p → n n charge-exchange differential cross-section at two incoming p momenta (601.5 and 1202 MeV/ c) at the CERN Low Energy Antiproton Ring. After reviewing the experimental method, the data analysis is fully detailed, and the results on the whole statistics available in the centre of mass forward hemisphere are given. A precision in the absolute normalisation of 2% and 4% at 601.5 and 1202 MeV/ c respectively is achieved, while over most of the angular range the point-to-point error is lower than 1%. The forward peak associated with the nearby pion pole and a dip-bump structure are accurately measured. The “Chew” extrapolation to the pion-pole suggests a “low” value for the πNN coupling constant g c 2.

Corrections to the Pagels-Stokar formula for fπ

Within the composite operator formalism we derive a formula for the pion decay constant $f_{\pi}$, as defined directly from the residue at the pion pole of the meson propagator, rather than from the matrix element of the axial current. The calculation is performed under some simplifying assumptions, and we verify the complete consistency with soft-pion results, in particular with the Adler-Dashen relation. The formula one obtains for (the pole-defined) $f_{\pi}^2$ differs from the previous Pagels-Stokar expression by an additive term, and it still provides $f_{\pi}^2$ in terms of the quark self-energy. We make some numerical estimates leading to $(30 \div 40)%$ deviation for $f_{\pi}^2$ with respect to the Pagels-Stokar formula.

Pion electromagnetic form factor in the instanton vacuum

We have calculated the 3-point correlation function for the electromagnetic interaction of the pion in an ensembles of instantons and anti-instantons, modelling the QCD vacuum. The results are well described by a pion pole and the pion formfactor is extracted, nicely following a standard monopole fit. The experimental data on the formfactor are well reproduced, provided an average instanton size is fixed at ${\bar\rho}=0.35\pm 0.03\fm$, the same as found for a variety of other correlation functions and was found on the lattice.

Dispersive approach to the axial anomaly, the t'Hooft's principle and QCD sum rules

The dispersive approach to the axial anomaly is revisited. Considering the familiar VVA triangle graph, the anomalous Ward identity is proved in the case of the external momenta corresponding to one real and one virtual photon. We also comment on a recent claim that the anomaly pole in QCD fails to reproduce the pion pole. In this connection, it is emphasized that there is no need to introduce a massless axial meson in the chiral limit. In the framework of QCD sum rules method a constraint for the Borel transform of relevant form factors imposed by the anomaly is considered.

Pi NN coupling from high precision np charge exchange at 162 MeV.

Differential {ital np} cross sections for unpolarized neutrons of 162 MeV have been measured to high precision with particular attention to the absolute normalization. These data can be extrapolated precisely and model independently to the pion pole and give a {pi}{ital NN} coupling constant {ital g}{sup 2}=14.62{plus_minus}0.06{plus_minus}0.29 or {ital f}{sup 2}=0.0808{plus_minus}0.0003{plus_minus}0.0017. This is higher than recently suggested values.

Pi NN and pseudoscalar form factors from lattice QCD.

The $\pi NN$ form factor $g_{\pi NN}(q^2)$ is obtained from a quenched lattice QCD calculation of the pseudoscalar form factor $g_P(q^2)$ of the proton with pion pole dominance. We find that $g_{\pi NN}(q^2)$ fitted with the monopole form agrees well with the Goldberger-Treiman relation and is much preferred over the dipole form. The monopole mass is determined to be $0.75 \pm 0.14 {\rm GeV}$ which shows that $g_{\pi NN}(q^2)$ is rather soft. The extrapolated $\pi N$ coupling constant $g_{\pi NN} = 12.7 \pm 2.4$ is quite consistent with the phenomenological values. We also compare $g_{\pi NN}(q^2)$ with the axial form factor $g_A(q^2)$ to check the pion dominance in the induced pseudoscalar form factor $h_A(q^2)$ vis \`{a} vis chiral Ward identity.

Chiral calculations of the K→2π and K→3π weak decays

We show in many chiral schemes how to relate consistently K→2π to K to vacuum transitions. Then we discuss how the observed scale of ΔI=1/2 enhancement is set by charmed quark or charmed D-meson W-mediated loops. Lastly we demonstrate that all four K→3π decay amplitudes are linked to the above K0→2π transitions via PCAC and the dominant pion pole part. We conclude that all six K0→2π, K→3π weak decays are compatible with data.

Determination of the charged pion-nucleon coupling constant from [formula omitted]p → [formula omitted]n differential cross-section

A recent experiment at LEAR, the CERN Low Energy Antiproton Ring, has provided very precise differential cross-section data for the p p → n n charge-exchange reaction. Extrapolating the data to the pion pole we have determined the charged-pion coupling constant to be f c 2 = 0.071 ± 0.002 from the data at 601 MeV/ c. Details of the fitting procedure are given, as well as results obtained when applying the same method to recent np → pn data.

Manifestation of the pion pole in [formula omitted]p → [formula omitted]n charge-exchange scattering

A recent experiment at LEAR, the CERN Low Energy Antiproton Ring, has provided very precise differential cross-section data for the p p → n n charge-exchange reaction. When performing the extrapolation procedure to the pion pole originally proposed by G.F. Chew the data exhibit the same dynamical mechanism than the np → pn line reversal reaction. The residue at the pole is consistent with the known value of the charged πNN coupling constant, as expected from charge conjugation invariance.

High-precision measurement of the [formula omitted] charge-exchange differential cross-section

The differential p p → n n charge-exchange cross section has been measured at the CERN Low Energy Antiproton Ring (LEAR), at two incident p momenta, 601 and 1202 MeV/c. features of the differential cross-section near the forward direction, i.e. a sharp peak at 0° scattering angle followed by an energy dependent dip-bump structure, are confirmed and measured with good precision and high statistical accuracy. The data show very clearly that the shape of the cross-section is a manifestation of the pion-exchange amplitude, and a simple extrapolation to the pion pole already indicates that the pion-nucleon coupling constant f c 2 can be determined with good precision.

Antiproton-proton partial-wave analysis below 925 MeV/c.

A partial-wave analysis of all antiproton-proton scattering data below 925 MeV/c antiproton laboratory momentum is presented. The method used is adapted from the Nijmegen phase-shift analyses of pp and np scattering data. The Nijmegen 1993 antiproton-proton database, consisting of 3646 antiproton-proton scattering data, is presented and discussed. The best fit to this database results in chi^2_min/Ndata = 1.043. The pseudovector coupling constant of the charged pion to nucleons is determined to be (f_c)^2 = 0.0732(11) at the pion pole, where the error is statistical.

Compton scattering on the pion and radiative pion photoproduction from the proton

The contribution of the pion polarizabilities to radiative pion photoproduction has been investigated. It has been shown how an extrapolation of future experimental data on radiative pion photoproduction to the pion pole can give information on the polarizabilities of pion.

Meson poles and screening masses near the chiral transition

Evidence against the proposed existence of the pion pole above the chiral phase transition, based on recent lattice QCD results, is discussed. Preliminary results concerning the effect of temperature on the pion and rho below, but close to, the chiral transition are also discussed.

Axial and pseudoscalar nucleon form factors from low energy pion electroproduction.

The cross section for exclusive ${\mathrm{\ensuremath{\pi}}}^{+}$ electroproduction on the proton has been measured near threshold for the first time at two different values of the virtual photon polarization (\ensuremath{\varepsilon}\ensuremath{\sim}0.2 and \ensuremath{\varepsilon}\ensuremath{\sim}0.7). Using the low energy theorem for this reaction we deduce the axial and pseudoscalar weak form factors ${\mathit{G}}_{\mathit{A}}$ and ${\mathit{G}}_{\mathit{P}}$ at \ensuremath{\Vert}t\ensuremath{\Vert}=0.073, 0.139, and 0.179 (GeV/c${)}^{2}$. The slope of ${\mathit{G}}_{\mathit{A}}$ agrees with the value obtained in neutrino experiments. ${\mathit{G}}_{\mathit{P}}$ satisfies the pion pole dominance hypothesis, which is thus verified for the first time in this range of transfer.

Pion photoproduction in the Skyrme model and low-energy theorem

We investigate pion photoproduction on the nucleon in the Skyrme model. We employ the formulation, which was recently developed by Hayashi et al., that the full pion field is treated as an interpolating field between asymptotic in and out fields. It is shown that the amplitude of the pion photoproduction is correctly given by the direct and the crossed baryon-pole terms, and the equal-time commutator terms between the axial-vector current and the electromagnetic current and between the pion field and the latter. We show that the lowest-order Kroll-Ruderman and the pion pole terms are reproduced, and that the seagull terms inherent to the Skyrme model are present. Further, the threshold behavior of the amplitude is discussed.

Pion-photoproduction pole terms in the Skyrme model

We derive the isolated-pole terms for pion photoproduction on a nucleon in the Skyrme model. We employ a quantization method in which the Born terms of pion-nucleon elastic scattering are described by a pion-baryon linear interaction of O(N C − 1 2 ) and a pion-baryon quadratic interaction of O( N C 0). The baryon and pion pole terms in the photoproduction amplitude are generated by the pion-baryon linear interaction and the interactions with the external electromagnetic field. The identical result is also obtained by the use of a unitary-transformation method which reproduces the Yukawa coupling missing in the quantized hamiltonian. We also present a method which enables us to extract relations between the meson-baryon interactions and the Yukawa coupling diagrammatically.

Meson exchange currents in pion double charge exchange.

We examine the influence of meson exchange currents (MEC) on pion double charge exchange (DCX) using the Weinberg Lagrangian. We calculate the corresponding amplitude for DCX and examine its interplay with more conventional mechanisms of DCX for $^{14}\mathrm{C}$ at incident energies of 10 to 110 MeV and for the Ca isotopes at 35 MeV. The MEC effect includes the pion pole term and the contact term. In comparison to previous estimates, we find smaller but nevertheless non-negligible contributions of these effects to DCX. We stress the fact that the MEC contribution to DCX is strongly A dependent, with its effect in light nuclei comparatively larger than in heavy ones.