πNN coupling determined beyond the chiral limit

Within the conventional QCD sum rules, we calculate the πNN coupling constant, g _πN, beyond the chiral limit using two-point correlation function with a pion. For this purpose, we consider the Dirac structure, iγ₅, at m _π ² order in the expansion of the correlator in terms of the pion momentum. For a consistent treatment of the sum rule, we include the linear terms in quark mass as they constitute the same chiral order as m _π ². In this sum rule, we obtain g _πN= 13.3 ± 1.2, which is very close to the empirical πNN coupling. This demonstrates that going beyond the chiral limit is crucial in determining the coupling. Received: 8 July 1999 / Revised version: 20 August 1999     

Self-consistent treatment of the quark condensate and hadrons in nuclear matter

We calculate the contribution of pions to the $\bar qq$-expectation value κ(ρ) =<M|ˉq q|M> in symmetric nuclear matter. We employ exact pion propagator renormalized by nucleon-hole and isobar-hole excitations. Conventional straightforward calculation leads to the “pion condensation” at unrealistically small values of densities, causing even earlier restoration of chiral symmetry. This requires a self-consistent approach, consisting in using the models, which include direct dependence of in-medium mass values on κ(ρ), e.g. the Nambu–Jona-Lasinio–model. We show, that in the self-consistent approach the ρ-dependence of the condensate is described by a smooth curve. The “pion condensate” point is removed to much higher values of density. The chiral restoration does not take place at least while ρ < 2.8ρ₀ with ρ₀ being the saturation value. Validity of our approach is limited by possible accumulation of heavier baryons (delta isobars) in the ground state of nuclear matter. For the value of effective nucleon mass at the saturation density we found m ^*(ρ₀) = 0.6m, consistent with nowadays results of other authors. Received: 8 October 1998     

Medium effects in the production and π0γ decay of ω-mesons in pA collisions in the GeV region

The ω resonance production and its π⁰γ decay in pA reactions close to threshold is considered within the Intranuclear Cascade (INC) model. The π⁰γ invariant-mass distribution shows two components which correspond to the ω decay “inside” and “outside” the nucleus, respectively. The “inside” component is distorted by medium effects, which introduce a mass shift as well as collisional broadening for the ω-meson and its decaying pion. The relative contribution of the “inside” component is analyzed in detail for different kinematical conditions and nuclear targets. It is demonstrated that a measurement of the correlation in azimuthal angle between the π⁰ and γ momenta allows to separate events related to the “inside”ω decay from different sources of background when uncorrelated π⁰'s and γ's are produced. Received: 2 April 2001 / Accepted: 5 June 2001     

Cross-sections for low-energy πgamma; reactions

We review the cross-sections for low-energy π^-γ reactions in the framework of chiral perturbation theory. Charged-pion Compton scattering, π^-γ → π^-γ , is considered up to one-loop order where the pion's internal structure enters through the difference of the electric and magnetic pion polarizability, α_π - β_π . The ongoing COMPASS experiment aims at measuring this important structure constant with high statistics using the Primakoff effect. In the same way, the reaction π^-γ → π^-π⁰ serves as a test of the QCD chiral anomaly (via the γ3π -coupling constant F _γ3π . Furthermore, we calculate the total cross-sections for neutral and charged pion-pair production, π^-γ → π^-π⁰π⁰ and π^-γ → π^-π⁺π^- , which are governed by the chiral ππ -interaction. Finally, we investigate the radiative (correction) process π^-γ → π^-γγ and calculate the corresponding two-photon mass spectrum. This information will be useful for analyzing the π^- Primakoff scattering events with photons in the final state.     

Compton scattering on the charged pion and the process γγ→π0π0

The Compton scattering on a charged pion and the process γγ→π⁰π⁰ are studied using the dispersion relations. Unknown parameters of the S-wave ππ interaction and a sum and a difference of the π⁰ meson polarizabilities are found from a fit to the experimental data for the γγ→π⁰π⁰ process. The found parameters of the ππ interaction are used for the calculation of the cross section of the elastic γπ^± scattering. The analysis of the obtained results shows that the experimental data for the elastic γπ^± scattering in the energy region up to 1 GeV together with the data for the γγ→π⁰π⁰ process could be used both for a determination of the pion polarizability values and for study of the S-wave ππ interaction. Received: 20 November 1998 / Revised version: 22 January 1999     

The σK coupling in the chiral unitary approach and the isoscalar ¯N , ¯A interaction

We evaluate the “σ " exchange contribution to the ˉN → ˉN scattering within a chiral unitary approach. We show that the chiral transition potentials for ππ → Kˉ in the t -channel lead to a “σ " contribution that vanishes in the ˉ forward direction and, hence, would produce a null “σ " exchange contribution to the K^- optical potential in nuclear matter in a simple impulse approximation. This is a consequence of the fact that the leading-order chiral Lagrangian gives an I = 0 ππ → Kˉ amplitude proportional to the squared momentum transfer, q². This finding poses questions on the meaning or the origin of “σ " exchange potentials used in relativistic mean-field approaches to the K^- nuclear self-energy. This elementary “σ ” exchange potential in ˉN → ˉN is compared to the Weinberg-Tomozawa term and is found to be smaller than the present theoretical uncertainties but will be relevant in the future when aiming at fitting increasingly more accurate data.     

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.     

Towards a high-precision calculation for the pion-nucleus scattering lengths

We calculate the leading isospin-conserving few-nucleon contributions to pion scattering off ²H, ³He, and ⁴He. We demonstrate that the strong contributions to the pion-nucleus scattering lengths can be controlled theoretically to an accuracy of a few percent for isoscalar nuclei and of 10% for isovector nuclei. In particular, we find the π-³He scattering length to be (62 ± 4 ± 7) × 10⁻³ m _π⁻¹ where the uncertainties are due to ambiguities in the π-N scattering lengths and few-nucleon effects, respectively. To establish this accuracy we need to identify a suitable power counting for pion-nucleus scattering. For this purpose we study the dependence of the two-nucleon contributions to the scattering length on the binding energy of ²H. Furthermore, we investigate the relative size of the leading two-, three-, and four-nucleon contributions. For the numerical evaluation of the pertinent integrals, a Monte Carlo method suitable for the momentum space is devised. We observe that, so far, no power counting is able to provide a quantitative understanding of the relative strength of N- and (N + 1)]]-nucleon operators. Empirically, we find a relative suppression by a factor of 5 compared to a factor of 50 predicted from dimensional analysis. On the other hand, the relative importance of different contributions within each class of N-nucleon operators can be understood within Weinberg counting. The relevance of our findings for the extraction of the isoscalar π-N scattering length from pionic ²H and ⁴He is outlined. We also discuss the applicability of heavy pion effective field to the π-²H system.     

Novel approach to pion and eta production in proton-proton collisions near threshold

We evaluate the threshold matrix–element for the reaction pp→ppπ⁰ in a fully relativistic Feynman diagrammatic approach. We employ a simple effective range approximation to take care of the S–wave pp final–state interaction. The experimental value for the threshold amplitude A = (2.7 −i0.3) fm⁴ can be reproduced by contributions from tree level chiral (long–range) pion exchange and short–range effects related to vector meson exchanges, with ω-exchange giving the largest individual contribution. Pion loop effects appear to be small. We stress that the commonly used heavy baryon formalism is not applicable in the NN–system above the pion production threshold due to the large external momentum, |p|≃ (Mm _π)^−1/2, with M and m _π the nucleon and the pion mass, respectively. We furthermore investigate the reaction pp→pnπ⁺ near threshold within the same approach. We extract from the data the triplet threshold amplitude as B = (2.8 −i1.5) fm⁴. Its real part can be well understood from (relativistic) tree level meson–exchange diagrams. In addition, we investigate the process pp→ppη near threshold. We use a simple factorization ansatz for the ppη final–state interaction and extract from the data the modulus of the threshold amplitude, |C|= 1.32 fm⁴. With g _ηN= 5.3, this value can be reproduced by (relativistic) tree level meson–exchange diagrams and η–rescattering, whose strength is fixed by the ηN scattering length. We also comment on the recent near threshold data for η^′–production. Received: 27 November 1998     

Chiral symmetry and light resonances in hot and dense matter

We present a study of the π π scattering amplitude in the σ and ρ channels at finite temperature and nuclear density within a chiral unitary framework. Meson resonances are dynamically generated in our approach, which allows us to analyze the behavior of their associated scattering poles when the system is driven towards chiral-symmetry restoration. Medium effects are incorporated in three ways: (a) by thermal corrections of the unitarized scattering amplitudes, (b) by finite nuclear-density effects associated to a renormalization of the pion decay constant, and complementarily (c) by extending our calculation of the scalar–isoscalar channel to account for finite nuclear-density and temperature effects in a microscopic many-body implementation of pion dynamics. Our results are discussed in connection with several phenomenological aspects relevant for nuclear-matter and heavy-ion collision experiments, such as ρ mass scaling versus broadening from dilepton spectra and chiral restoration signals in the σ channel. We also elaborate on the molecular nature of π π resonances.     

The kappa in J/Ψ↦K<Superscript>+</Superscript>π

BES II data for J/Ψ↦K ^*(890)Kπ reveal a strong κ peak in the Kπ S-wave near threshold. Both magnitude and phase are determined in slices of Kπ mass by interferences with strong K ₀(1430), K ₁(1270) and K ₁(1400) signals. The phase variation with mass agrees within errors with LASS data for Kπ elastic scattering. A combined fit is presented to both BES and LASS data. The fit uses a Breit-Wigner amplitude with an s-dependent width containing an Adler zero. The κ pole is at (760±20(stat)±40(syst)) - i(420±45(stat)±60(syst)) MeV. The S-wave I = 0 scattering length a ₀ = 0.23±0.04 m _π ^-1 is close to the prediction 0.19±0.02 m _π ^-1 of Chiral Perturbation Theory at O(p ⁴). An erratum to this article can be found at .     

Scales in nuclear matter: Chiral dynamics with pion nucleon form factors<Superscript>⋆</Superscript>

A systematic calculation of nuclear matter is performed which includes the long-range correlations between nucleons arising from one- and two-pion exchange. Three-body effects from 2π exchange with excitations of virtual Δ(1232)-isobars are also taken into account in our diagrammatic calculation of the energy per particle ˉ(k _f). In order to eliminate possible high-momentum components from the interactions we introduce at each pion-baryon vertex a form factor of monopole type. The empirical nuclear matter saturation point, ρ₀ ≃ 0.16fm^-3, ˉ₀ ≃ - 16MeV, is well reproduced with a monopole mass of Λ ≃ 4πf _π ≃ 1.16GeV. As in the recent approach based on the universal low-momentum NN potential V _low-k, the inclusion of three-body effects is crucial in order to achieve saturation of nuclear matter. We demonstrate that the dependence of the pion exchange contributions to ˉ(k _f) on the “resolution” scale Λ can be compensated over a wide range of Λ by counterterms with two “running” contact couplings. As a further application we study the in-medium chiral condensate 〈ˉq〉(ρ) beyond the linear density approximation. For ρ ⩽ 1.5ρ₀ we find small corrections from the derivative dˉ(k _f)/dm _π, which are stable against variations of the monopole regulator mass Λ.     

<Emphasis Type="Italic">ππ</Emphasis> scattering in a nonlocal chiral quark model

We consider a nonlocal version of the Nambu and Jona-Lasinio model with the SU(2) × SU(2) chiral symmetry broken by the current-quark-mass term. The nonlocality is contained in the quark-antiquark bilinears of the four-quark vertices as a form factor of the Gaussian type. The model has three parameters which can be fixed in favor of the values of the pion mass mπ, the pion decay constant fπ, and the current quark mass m _c . It is shown that, in the model, the main low-energy theorems which are known for pions are fulfilled. The s-, p-, and d-wave scattering lengths in all isotopic channels and the s-wave slope parameters are calculated, and the results are in satisfactory agreement with phenomenological data. The text was submitted by the authors in English.     

Precision calculation of γd↦πnn within chiral perturbation theory

The reaction γd↦π⁺nn is calculated up to order χ^5/2 in chiral perturbation theory, where χ denotes the ratio of the pion to the nucleon mass. Special emphasis is put on the role of nucleon recoil corrections that are the source of contributions with fractional power in χ. Using the known near-threshold production amplitude for γp↦π⁺n as the only input, the total cross-section for γd↦π⁺nn is described very well. A conservative estimate suggests that the theoretical uncertainty for the transition operator amounts to 3% for the computed amplitude near threshold.     

pp → ppπnear threshold in pionless effective theory

The total cross-section of the pp → ppπ⁰ reaction near threshold is calculated in pionless effective field theory with a di-baryon and external pions. The amplitudes for a leading one-body and subleading contact neutral pion production vertex are obtained including the initial- and final-state interactions. After estimating a low-energy constant in the contact vertex, we compare our results for the total cross-section with the experimental data.     

Photoproduction of neutral pion pairs from the proton

Double neutral pion photoproduction from the proton has been measured at MAMI for photon energies between threshold and 820 MeV. The reaction was identified by an invariant mass and missing mass analysis. From threshold up to 370 MeV the total cross-section does not exceed 30 nb. For higher energies it shows a smooth rise until it reaches a maximum of about 10 μb at E _γ = 740 MeV. Dalitz plots of m ²(π⁰π⁰) versus m ²(p,π⁰) for seven bins of incident photon energy have been analysed. For E _γ > 610 MeV, a strong contribution of a sequential decay is observed with the Δ(1232)-resonance as intermediate state. A comparison to model calculations shows that these sequential decays presumably originate from the D ₁₃(1520) and also the P ₁₁(1440)-resonance. Received: 6 June 2000 / Accepted: 18 August 2000     

On parity-violating three-nucleon interactions and the predictive power of few-nucleon EFT at very low energies

We address the typical strengths of hadronic parity-violating three-nucleon interactions in “pion-less” Effective Field Theory (EFT) in the nucleon-deuteron (iso-doublet) system. By analysing the superficial degree of divergence of loop diagrams, we conclude that no such interactions are needed at leading order, O(eQ^-1)\ensuremath {O}(\epsilon Q^{-1}) . The only two distinct parity-violating three-nucleon structures with one derivative mix ²S_\frac12\ensuremath ^2S_{\frac{1}{2}} and ²P_\frac12\ensuremath ^2P_{\frac{1}{2}} waves with iso-spin transitions D \Delta I = 0 or 1. Due to their structure, they cannot absorb any divergence ostensibly appearing at next-to-leading order, O(eQ⁰)\ensuremath {O}(\epsilon Q^0) . This observation is based on the approximate realisation of Wigner’s combined SU(4) spin-isospin symmetry in the two-nucleon system, even when effective-range corrections are included. Parity-violating three-nucleon interactions thus only appear beyond next-to-leading order. This guarantees renormalisability of the theory to that order without introducing new, unknown coupling constants and allows the direct extraction of parity-violating two-nucleon interactions from three-nucleon experiments.     

Electromagnetic corrections in <Emphasis Type="Italic">η</Emphasis>→3<Emphasis Type="Italic">π</Emphasis> decays

We re-evaluate the electromagnetic corrections to η→3π decays at next-to-leading order in the chiral expansion, arguing that effects of order e ²(m _u −m _d ) disregarded so far are not negligible compared to other contributions of order e ² times a light-quark mass. Despite the appearance of the Coulomb pole in η→π ⁺ π ⁻ π ⁰ and cusps in η→3π ⁰, the overall corrections remain small.