Pionic contribution to the muon magnetic moment

A procedure is developed to derive an optimal lower bounds for the pionic contribution to the muon magnetic moment from analyticity of the pion form factor F(t), its normalization F(0)=1 and from experimental information from both the processes e^?p → e^?π⁺n and e⁺e^? → π⁺π^?. It represents essentially the solution of a certain kind of optimization problem in Hilbert space. Numerical results are presented and compared to the recent data for the muon magnetic moment; we find a_μ(π⁺π^?) ? 42 × 10^?9.     

Scattering lengths of pionic 3He and 4He

Scattering lengths of pionic ³He and ⁴He and the charge exchange contribution to the 1s width of pionic ³He are calculated within the fixed scatterer approximation of the multiple scattering formalism. Particular attention is focussed on the nuclear physics part and on πN p-wave contributions. For the first time triple scattering and double-spin-flip contributions have been included. We find significant deviations from previous estimates and calculations. Good agreement is achieved with the experimental π^?3He scattering length, whereas in the case of ⁴He a repulsive dispersion contribution is clearly needed. We propose to use the measured 1s level shift of pionic ³He as a constraint to deduce a precise value of the isoscalar πN scattering length. Furthermore, we find that multiple scattering reduces the impulse approximation value for Γ_1s(π^?3He → π^{0 3}H) by more thsn 20 %. This result casts some doubt on impulse approximation calculations of radiative pion capture as well.     

Binding effects in the optical potential of pionic atoms

We calculate the contribution from nucleon binding to the coefficient Re B₀ in the optical potential of pionic atoms. The relevant Feynman graphs are deduced for nuclear matter. They depend on off-shell values for pion-nucleon scattering lengths a_πN and for the nucleon-binding potential V_NA. Off-shell effects are found to be very important: A pole model for a_πN, extrapolated off-shell, increases binding effects by a factor of three over the use of on-shell values. Two simple models are proposed for the off-shell continuation of V_NA. One leads to attractive, the other to repulsive contributions to Re B₀.     

Exact solution of a meiman problem with an application to the pionic contribution to the muon magnetic moment

An exact solution is obtained tp the follpwing problem: given a form factor F(t) analytic in the cut t plane, to find a lower bpund to a weighted ontegral over |F(t)|² over the cut in the presence of space-like data and also time-like data on both the modulus and the phase of F(t). The solution is used to obtain a lower bound on the pionic contribution to the muon magnetic moment (a_μ(ππ)). We obtain a_μ(ππ) ≥ 41.8 × 10^?9.     

Low-energy pions in nuclear matter and ππ photoproduction within a BUU transport model

In the present paper we investigate a method to describe low-energy scattering events of pions and nuclei within a Boltzmann-Uehling-Uhlenbeck (BUU) transport model. Implementing different scenarios of medium modifications, we studied the mean-free path of pions in nuclear matter at low momenta and compared pion absorption simulations to data. Pursuing these studies we have shown, that also in a regime of a long pionic wave length the semi-classical BUU model still generates reasonable results. We present results on π-induced events in the regime of 10 MeV≤T _kin ^π ≤130 MeV and photo-induced ππ production at incident beam energies of 400–460 MeV.     

The pionic hydrogen experiment at PSI

A new high-precision determination of the strong-interaction shift (∈_1s) and broadening (Γ_1s) of the ground state in pionic hydrogen (πH) has been started at PSI. This constitutes a direct measurement of the πN scattering lengths and is an important test of the methods of chiral perturbation theory.The experiment will be conducted in two steps. In the first step we aim for a precision of 0.2% for ∈_1s and 3% for Γ_1s. A new method of energy calibration is used: by measuring πH simultaneously with pionic oxygen (πO), systematic errors are suppressed. The goal for the second step is an accuracy of 1% for Γ_1s.     

First Precision Spectroscopy of Pionic Atoms at RI Beam Factory

We conducted an inclusive missing-mass spectroscopy of ¹²²Sn (d, ³He) reaction near the π¯ emission threshold at an incident energy of T _d = 250 MeV/nucleon. The experiment sets its goals to the research and development of the high precision spectroscopy of pionic atoms at the RI beam factory (RIBF) of RIKEN, which precedes a new project, pionic atom factory project (piAF), to observe pionic atoms systematically. Here, we report the preliminary results of the pilot experiment.     

Pions in the light-cone nucleon and electromagnetic form factors

The light-cone Fock-state decomposition of the nucleon involves pion-quark core (π′N′ andπ′Δ′) components. The size of the quark core is at present of special importance for the analysis of the onset of colour transparency in the A(e,e′p) reactions. With the light-coneNπ′N′ andNπ′Δ′ vertex functions deduced from experimental data on high-energy proton fragmentation into nucleons and isobars, we evaluate the effect of the pionic cloud on the electromagnetic form factors of nucleons. We find the quark core radius to be about 10% smaller than the nucleon's radius. Analysis of the renormalization of the total nucleon-nucleon cross section for the presence of the pionic cloud suggests a similar core radius.     

Determination of the hadronic width of the ground state in pionic hydrogen

In a new high-precision experiment at the Paul Scherrer Institut (PSI), the hadronic shift (ε _1s) and width (Γ_1s) of the ground state of pionic hydrogen, which are directly connected to the pion-nucleon isospin scattering lengths, were remeasured (PSI-Experiment R-98-01 Gotta et al. 1998). The results can be confronted with recent work of effective field theories defined in the low-energy limit of quantum chromodynamics (QCD), such as, e. g., chiral perturbation theory (ChPT). In addition, Γ_1s is connected to the pion-nucleon coupling constant f _πN. A precisely known value for f _πN allows an accurate determination of the Goldberger-Treiman discrepancy, which constitutes a measure of chiral symmetry breaking. This contribution is mainly devoted to describe a method for an accurate extraction of the hadronic width from the π ^??H data.     

Taming the pion cloud of the nucleon

We present a light-front determination of the pionic contribution to the nucleon self-energy, Σ(π), to second order in pion-baryon coupling constants that allows the pion-nucleon vertex function to be treated in a model-independent manner constrained by experiment. The pion mass μ dependence of Σ(π) is consistent with chiral perturbation theory results for small values of μ and is also linearly dependent on μ for larger values, in accord with the results of lattice QCD calculations. The derivative of Σ(π) with respect to μ(2) yields the dominant contribution to the pion content, which is consistent with the d[over ˉ]-u[over ˉ] difference observed experimentally in the violation of the Gottfried sum rule.     

The strong interaction shift and width of the ground state of pionic hydrogen

The 3p-1s transition in pionic hydrogen was investigated with a high-resolution crystal spectrometer system. From the precisely measured transition energy, together with the (calculated) electromagnetic energy, the strong interaction shift of the 1s state was obtained as ϵ_1s = −7.127 ± 0.028(stat.)± 0.036(syst.) eV (attractive). From the natural line width, measured for the first time, we determine the decaywidth of the 1s state: Γ_1s^(decay) = 0.97 ± 0.10(stat.)± 0.05(syst.) eV. With the recently calculated electromagnetic corrections the s-wave scattering lengths of an isospin symmetric strong interaction are deduced. The scattering length for elastic scattering of a negative pion on a proton is a_{π⁻p→π⁻p}^h = 0.0885±0.00003(stat.)±0.0006(syst.)m_π⁻¹. The scattering lengthe for single charge exchange is found to be a_{π⁻p→π⁰n}^h = −0.136 ± 0.007(stat.) ± 0.003(syst.)m_π⁻¹.The experiment was performed at the Paul Scherrer Institute (PSI) in Switzerland. A focussing crystal spectrometer with an array of bent crystals, the cyclotron trap (a magnetic system designed to increase the particle stop density) and a CCD (charge-coupled device) detector system were employed. The results from the pionic hydrogen experiment — together with those from the pionic deuterium experiment — were used to test the isospin symmetry of the strong interaction. The present data are still consistent with isospin sysmmetry.     

Relativistic chiral mean field model for finite nuclei

We present a method for proper treatment of pion-exchange interaction in the nuclear many-body problem in a relativistic chiral mean field (RCMF) model. The pionic correlation is expressed in 2-particle 2-hole (2p–2h) states in addition to the standard mean field state, to describe the full strength of pionic correlations. The effect of the short-range repulsion is included by way of the unitary correlation operator method (UCOM) for the central part of the pion-exchange interaction. We apply the RCMF model to ¹²C and ¹⁶O. The convergence of pionic energy contributions is realized with pionic quantum number Jπ$J^{\pi }$ up to 10⁻ for ¹²C and 11⁺ for ¹⁶O. The pion-exchange interaction gives the dominant contribution to the binding energy. The pion plays an important role in the formation of the jj  -magic shell effect by way of the Pauli-blocking mechanism of the pion-exchange interaction. The lower pionic quantum numbers Jπ?³⁺$J^{\pi }?3^{+}$ are the dominant components of the nuclear surface structure.     

Empirical formula for energy level shifts of pionic atoms

An empirical formula for energy level shifts of pionic atoms is proposed. Numerical results show remarkably good agreement with experimental data. We predict energy level shifts and widths for pionic deuterium, tritium and helium. With these values, scattering lengths of π-d, π-t and π-³He are also calculated. The results are (in fm) ?0.083 + 0.024 i, ?0.287 + 0.028 i and 0.067 + 0.035 i, respectively.     

Critical scattering and pionic instabilities in heavy ion collisions

We investigate the effect of collective instabilities in heavy ion collisions. The focus is on critical scattering phenomena associated with pionic instabilities. The decay rate Γ of excited many-body systems is calculated in RPA. Γ is shown to give the rate of spontaneous “phonon” pair production. We express Γ as a sum of a collective, Γ_col, and a scattering, Γ_scat, rate. Γ_col is the pair production rate of phonons in unstable states. In the case of pionic instabilities, Γ_col is the condensation rate of π⁺π^? and π⁰π⁰ pairs into unstable states. Γ_scat is the pair production rate of phonons in the particle-hole excitation region and gives the two-body scattering rate in the medium. An effective (density-dependent) two-body cross section is obtained. The difference between critical scattering of external particles in a system near equilibrium and that of constituents of systems far from equilibrium is investigated. A model calculation suggests the existence of pionic instabilities in heavy ion collisions. Growth rates of unstable modes and the effective cross sections displaying critical scattering are calculated. Finally, we estimate Γ_scat and Γ_col.     

Electromagnetic corrections to the s-wave scattering lengths in pionic hydrogen

Motivated by the recently performed X-ray precision experiments on pionic hydrogen (preceding paper), we reconsider the problem of electromagnetic corrections to the π - N scattering lengths. Based on a relativistic two-channel approach, we find corrections to the π⁻p elastic and single-charge-exchange (SCE) scatering lengths due to the point-Coulomb interaction, the finite-size Coulomb interaction (including the pion size), the first-order vacuum polarization (Uehling potential) and the (π⁻p) - (π⁰n) mass difference (mass difference effect). We also estimate the contribution due to the (γ,n) decay channel. The total corrections to the elastic and the SCE scattering lengths are found to be δ_ε = −(2.1 ± 0.5) × 10⁻² and δ_Γ = −(1.3 ± 0.5) × 10⁻². Previously published values for the corrections are critically compared with our results.     

Optimal bounds for the pion form factor from analyticity and experimental data

Optimal bounds for the pion electromagnetic form factor F(t) below threshold and on the pion mean-square charge radius 〈r_π²〉 = 6F'(0) are derived. Use is made of analyticity arguments and of experimental data on F(t) from e⁺e^? → π⁺π^? as well as e^?p → e^?nπ⁺. The method accounts in an approximate way for the statistical errors of the experimental information. Numerical results for F(t) are calculated for the CEA as well as the DESY electroproduction data.     

Excitation of ΔL = 0 spin-flip transitions in the N = 28 isotones by 201 MeV proton inelastic scattering

The effect of the three-nucleon forces on the triton binding energy (E_T) is investigated. Two different types of three-nucleon potentials are considered, one involving 2π-exchange (V_ππF) and the other involving 2σ-exchange (V_SSF) which is an approximation to 3π exchange. Both V_ππF and V_SSF are constructed using the πN scattering amplitude extrapolated to off (pion) mass shell subject to PCAC. The contribution of V_ππF and V_SSF to e_t are calculated in the first-order perturbation using the triton wave function obtained by solving the Faddeev equations with the Reid soft-core potential. The Fujita-Miyazawa force is also investigated as a special case. We find that the effect of V_ππF on E_T is sensitive to the assumed pionic form factor and is small due to cancellation between the πN s-d wave and p-wave contributions. The effect of V_SSF is also found to be small.     

关于核子的同位旋矢量电磁结构

本文根据Sugawara和Kanazawa所证明的一个定理和Omnes关于渐近行为的研究,讨论了G_i(t)和Г_i(t)色散关系的減出问题。比较仔细地考虑了Г_i(t)和F_π(t)中远区奇异性的贡献,所得的结果可以同时很好地符合电形式因子和磁形式因子。其中Г_i(t)并满足Vick从πN实验分析中所提出的要求。计算结果还表明,核子的矢量总电荷和总磁矩绝大部分是由2π中间态贡献的,核子的核心点电荷和点磁矩     

Isospin violation in τ → 3πντ

Isospin violating signals in the τ^? → (3π)^?ν_τ decay mode are discussed. For the τ^? → π^?π^?π⁺ν_τ decay mode, isospin violation arises from the vector current contribution in the τ^? → ωπ^?ν_τ decay with the subsequent isospin violating ω decay into π⁺π^?. We demonstrate that such effects may be observed in presently available data through the measurement of the interference effects of these vector current contributions with the dominating axial vector current,i. e. through a measurement of the structure functionsW _F, WG, WH andW _I. In the case of the τ^? → π⁰π⁰π^?ν_τ decay mode, a vector current contribution is generated by ηπ⁰ mixing in the decay chain τ^? → η?^?ν_τ → π⁰π⁰π^?ν_τ. We find that this effect is rather small, the magnitude of the associated interference terms being too low for present statistics.     

Faddeev πNN dynamics with ρ-exchange contribution for the 1D2 and 3F3 diproton anomalies

The ¹D₂ and ³F₃ diproton anomalies are studied in a framework of Faddeev πNN dynamics with ρ-exchange contribution. Two-body interactions in πN-P₃₃, πN-P₁₁, NN-³P₂; and NN-³Si₁ states are input. The pp-pp, pp-πd and πd-πd processes are treated in a unified way and the results are compared with partial wave analysis results of each process. The ¹D₂ and ³F₃ diproton anomalies are qualitatively explained by the present models. The ρ-contributions are remarkable in the amplitudes of pp-pp and pp-πd processes from the pp-¹D₂ state.