A two-pion exchange three-nucleon force based on a realistic πN interaction

The contribution of a ππ-exchange three-body force to the three-nucleon binding energy is calculated in terms of a πN amplitude. The latter is based on a meson-theoretical model of πN interaction developed by the Jülich group. Similar to a previous study based on simple phenomenological πN potentials a very small effect of the resulting three-body force is found. Possible origins of the two-orders-of-magnitude discrepancy between the present result and the values obtained for the Tucson-Melbourne three-body force are investigated. Evidence is provided that this discrepancy is most likely due to strikingly different off-shell properties of the πN amplitudes underlying the two approaches. Received: 9 February 2000     

Reworking the Tucson-Melbourne Three-Nucleon Potential

 We introduce new values of the strength constants (i.e., a, b, c, and d coefficients) of the Tucson-Melbourne (TM) 2π-exchange three-nucleon potential. The new values come from contemporary dispersion-relation analyses of meson-factory πN-scattering data. We make variational Monte-Carlo calculations of the triton with the original and updated three-body forces to study the effects of this update. We remove a short-range–π-range part of the potential due to the c coefficient and discuss the effect on the triton binding energy. Received September 11, 1999; revised November 2, 1999; accepted February 23,     

Dynamical Study of Tensor Observables in the Energy Range of 80 keV to 95 MeV: Tests of Effective Two-Body Models

 Realistic interactions are used to study tensor observables in the energy range of 80 keV to 95 MeV deuteron laboratory energy, as well as the differential cross section for the two-body photodisintegration of . The Siegert form of the E1 multipole operator in the long-wavelength limit is taken as the sole component of the electromagnetic interaction. The three-body Faddeev equations for the bound-state and continuum wave functions are solved using the Paris, Argonne V14, Bonn-A, and Bonn-B potentials. The corresponding nucleon-nucleon t-matrices are represented in a separable form using the Ernst-Shakin-Thaler representation. The Coulomb force between protons is neglected and no three-nucleon force is included. The contribution of nucleon-nucleon P-wave components to the observables is carefully studied, not only in the angular distribution of the observables, but also as a function of the deuteron laboratory energy for fixed centre-of-mass angle. Comparison with data is shown wherever it exists. Results with simple Yamaguchi-type interactions with variable %D-state in the deuteron are compared with realistic interactions and one of these model potentials is used to study the results in terms of contributions from specific wave-function components or terms in the electromagnetic operator. Effective two-body models are examined by means of a derivation that is consistent with the underlying three-body calculation and that leads to an effective two-body t-matrix for neutron-deuteron elastic scattering carrying the same on-shell amplitudes as the original three-body equations. Received September 21, 1999; revised December 23, 1999; accepted February 9, 2000     

On the Uniqueness of the Solution to the Three-Body Problem with Zero-Range Interactions

We consider the uniqueness of the solution to a three-body problem with zero-range Skyrme interactions in configuration space. With the lowest, k⁰, two-body term alone the problem is known to have no unique solution as the system collapses – the variational estimate of the energy tends towards negative infinity, the size of the system towards zero. We argue that the next, k², two-body term removes the collapse and the three-body system acquires finite ground-state energy and size. The three-body interaction term is thus not necessary to provide a unique solution to the problem.     

np → ηd near threshold and the s -wave ηN amplitude

We calculate the process np → ηd near threshold using a separable potential model of the coupled ηN - πN - ππN subystems, and a relativistic three-body calculation for the ηd scattering amplitude. The ππN channels are represented by an effective σN channel, and we compare the case where the σ and π masses are related by m _σ = 2m _π and no width is considered, to another where the mass and width of the σ -meson are taken from ππ scattering data. The np → ηd cross-section can be well described up to about 60MeV by models where the real part of the ηN scattering length lies between 0.4≤Re(a _ηN)≤0.6 fm which allows us to determine the s -wave ηN scattering amplitude for -60≤E≤60 MeV.     

Birkhoff Normal Form for Some Nonlinear PDEs

 We consider the problem of extending to PDEs Birkhoff normal form theorem on Hamiltonian systems close to nonresonant elliptic equilibria. As a model problem we take the nonlinear wave equation

The c-Term of the Tucson-Melbourne Three-Body Force: To Be or Not to Be

 By Faddeev calculations of ³H we study the dependence of the binding energy on the three-nucleon force. We adopt the 2π-exchange Tucson-Melbourne three-nucleon force and investigate phenomenologically the dependence on the strength of the individual three-body force operators (the a-, b-, c-, and d-terms). While it is well known that the a-term is not as important as the b- and d-terms to gain the experimental binding energy, we find two solutions for the c-term, one around the value used in the original Tucson-Melbourne model and a new one close to zero. A tensor-analyzing power T ₂₀ of the pd elastic scattering using the modified Tucson-Melbourne model, which follows the recommendation by chiral perturbation theory that the short-range c-term should be dropped, describes the data well. Received April 22, 1999; revised June 6, 2000; accepted for publication June 16, 2000     

Triton Photodisintegration with Effective Field Theory

Effective field theory (EFT) has been recently used for the calculation of neutron–deuteron radiative capture at very low energies. We present here the use of EFT to calculate the two-body photodisintegration of the triton, considering the three-body force. The calculated cross section shows sharp rising from threshold to maximum about 0.88 mb at ~13 MeV and decreasing slightly to about 0.81 mb at ~19 MeV, in agreement with the experimental data. Our results are in good agreement with the experimental data and the other calculations using modern realistic two- and three-nucleon forces, like AV18/UrbanaIX potential.     

QCD corrections to QED vacuum polarization

We compute QCD corrections to QED calculations for vacuum polarization in background magnetic fields. Formally, the diagram for virtual eē loops is identical to the one for virtual qq̄ loops. However, due to confinement, or to the growth of α_s as p² decreases, a direct calculation of the diagram is not allowed. At large p² we consider the virtual qq̄ diagram, in the intermediate region we discuss the role of the contribution of quark condensates 〈qq̄〉 and at the low-energy limit we consider the π⁰, as well as charged pion π⁺π^- loops. Although these effects seem to be out of the measurement accuracy of photon–photon laboratory experiments, they may be relevant for γ-ray burst propagation. In particular, for emissions from the center of the galaxy (8.5 kpc), we show that mixing between the neutral pseudo-scalar pion π₀ and photons renders a deviation from the power-law spectrum in the TeV range. For scalar quark condensates 〈qq̄〉 and virtual qq̄ loops are relevant only for very high radiation density ∼300 MeV/fm³ and very strong magnetic fields of order ∼10¹4 T. PACS 12.20.Ds; 14.40.-n; 12.38.Aw; 14.65.Bt     

New parameterization of the trinucleon wave function and its application to the π <Superscript>3</Superscript>He scattering length

We present a new parameterization of the trinucleon wave function. As a novel feature a separable parameterization for the complete wave function is given. In this way any calculation that considers two-body currents only is largely simplified. To demonstrate this we calculate the π³He scattering length in chiral-perturbation theory. We find reasonable agreement with experimental values inferred from data on level shifts in pionic ³He bound states. The relevance of the π-triton system for an alternative determination of the πN scattering lengths is discussed. Received: 12 August 2002 / Accepted: 25 November 2002 / Published online: 25 February 2003 RID="a" ID="a"e-mail: c.hanhart@fz.juelich.de Communicated by V. Vento     

Analysis of the low-energy ηNN-dynamics within a three-body formalism

The interaction of an η-meson with two nucleons is studied within a three-body approach. The major features of the ηNN-system in the low-energy region are accounted for by using a s-wave separable ansatz for the two-body ηN and NN amplitudes. The calculation is confined to the (J ^π;T) = (0^-;1) and (1^-;0) configurations which are assumed to be the most promising candidates for virtual or resonant ηNN-states. The eigenvalue three-body equation is continued analytically into the nonphysical sheets by contour deformation. The position of the poles of the three-body scattering matrix as a function of the ηN-interaction strength is investigated. The corresponding trajectory, starting on the physical sheet, moves around the ηNN three-body threshold and continues away from the physical area giving rise to virtual ηNN-states. The search for poles on the nonphysical sheets adjacent directly to the upper rim of the real energy axis gives a negative result. Thus no low-lying s-wave ηNN-resonances were found. The possible influence of virtual poles on the low-energy ηNN-scattering is discussed. Received: 27 June 2000 / Accepted: 3 August 2000     

The Re-Analysis of the 1700 MeV Structure of the P 11 Partial Wave Using the πN → KΛ Production Data

We have used the Breit-Wigner resonance model with S ₁₁, P ₁₁, and P ₁₃ resonances in the s-channel to reanalyze the old πN → KΛ data with the aim to establish the origin of the prominent structure in the total cross section in the vicinity of 1700 MeV. We have found a new set of resonance parameters enforcing the experimentally observed structure of the total cross-section data simultaneously with the linear dependence of the differential cross sections with cos θ in the energy range 1650 MeV < W < 1800 MeV. Owing to the differential cross-section linearity, the P ₁₃ partial wave has been strongly attenuated in this model, and the total cross-section structure is attributed to the resonant behavior of the P ₁₁ partial wave. In this paper we show that, at least in the Breit-Wigner resonance model, it is not possible to achieve a detailed reproduction of the narrow 1700 MeV total cross-section peak using the standard partial widths. To understand the phenomenon, a much narrower width of a resonant state, the N(1710) P ₁₁ in our case, is required (Γ ≈ 68 MeV), but then the agreement of the model predictions with the total cross-section data at higher energies is lost. One way out is to allow for the existence of another P ₁₁ resonance in that energy range. The same feature is shown by the polarization data: The introduction of a much narrower resonance spoils the level of agreement which the Breit-Wigner resonance model is able to achieve with experiment, but the consistency is restored when another resonance is introduced. Analyzing the qqq or qqqq nature of the recommended narrow P ₁₁ structure in the neighborhood of 1700 MeV we reopen (remind of) the possibility that another P ₁₁ resonant state exists in addition to the standard N(1710) P ₁₁ PDG-resonance, and that one of the two states can be identified with the yet undiscovered cryptoexotic pentaquark state. To clarify the situation, we strongly recommend a remeasurement of the πN → KΛ process in the energy range 1650 MeV < W < 1800 MeV.     

Critical Temperature of Lequid-gas Phase Transition for Hot Nuclear Matter and Three-body Force Effect

The finite temperature Brueckner-Hartree-Fock (FTBHF) approach is extended by introducing a microscopic three-body force. Within the extended approach, the three-body force effects on the equation of state of hot nuclear matter and its temperature dependence have been investigated. The critical properties of the liquid-gas phase transition of hot nuclear matter have been calculated. It is shown that the three-body force provides a repulsive contribution to the equation of state of hot nuclear matter. The repulsive effect of the three-body force becomes more pronounced as the density and temperature increase and consequently inclusion of the three-body force contribution in the calculation reduces the predicted critical temperature from about 16MeV to about 13MeV. By separating the contribution originated from the 2σ-exchange process coupled to the virtual excitation of a nucleon-antinucleon pair from the full three-body force, the connection between the three-body force effect and the relativistic correction from the Dirac-Brueckner-Hartree-Fock has been explored. It turns out that the contribution of the 2σ-NN part is more repulsive than that of the full three-body force and the calculated critical temperature is about 11MeV if only the 2σ-NN component of the three-body force is included which is lower than the value obtained in the case of including the full three-body force and is close to the value predicted by the Dirac-Brueckner-Hartree-Fock (DBHF) approach. Our result provides a reasonable explanation for the discrepancy between the values of critical temperature predicted from the FTBHF approach including the three-body force and the DBHF approach.     

Energy Dependence of the πN Amplitude and the Three-Nucleon Interaction

By calculating the contribution of the ππ three-body force to the three-nucleon binding energy in terms of the πN amplitude using perturbation theory, we are able to determine the importance of the energy dependence and the contribution of the different partial waves of the πN amplitude to the three-nucleon force. A separable representation of the non-pole πN amplitude allows us to write the three-nucleon force in terms of the amplitude for NN → NN*, propagation of the NNN* system, and the amplitude for NN* → NN , with N* being the πN quasi-particle amplitude in a given state. The division of the πN amplitude into a pole and non-pole part gives a procedure for the determination of the πNN form factor within the model. The total contribution of the three-body force to the binding energy of the triton for the separable approximation to the Paris nucleon-nucleon potential (PEST) is found to be very small mainly as a result of the energy dependence of the πN amplitude, the cancellation between the S- and P-wave πN amplitudes, and the soft πNN form factor. Received April 12, 1994; revised November 11, 1994; accepted for publication December 1, 1994     

Phenomenological Estimate of the Binding Energy of Heavy Dimesons

 A phenomenological estimate is derived such that the binding energies of dimesons are expressed as combinations of masses of different mesons and baryons. The estimate is almost model-independent, the only major assumptions being that the wave functions of the two light quarks in Λ _c , Λ _b and in the and dimesons are very similar, and that for heavy quarks the QQ interaction is half as strong as the interaction. We get (I = 0, J = 1) bound by about 100 MeV and unbound. Received July 4, 2000; revised November 28, 2000; accepted for publication February 5, 2001     

Sensitivity of the three-body calculations to different effects

The bound state of few-body systems in light nuclei is studied as a three-body problem. The three-body problem is solved following the different approaches of the Faddeev formalism as well as the unitary pole approximation. Separable approximations are introduced to reduce the three-body problem to a set of coupled integral equations. Numerical calculations are carried out for the resulting integral equations and the separable expansion. In the present work, we calculate the ground-state binding energy of the bound three-nucleon system³H. The main interest of the present work is to investigate the sensitivity of the three-body binding energy to different effects in the problem. For this reason, we study the dependence of the three-body binding energy of different forms of local and separable two-body potentials, on the effective range of the two-body potentials, and on the percent of theD state in the deuteron wave function. Also, we test the sensitivity of the three-body binding energy to the considered number of terms from the separable expansion.     

Role of the <Emphasis Type="Italic">ππ</Emphasis> interaction in pairwise photoproduction of <Emphasis Type="Italic">π</Emphasis><Superscript>0</Superscript> mesons on a proton

Photoproduction of two neutral pions on a proton is considered. Qualitative analysis of the available experimental data demonstrates that a linear dependence of the total cross section on the energy testifies to the dominating contribution of s-waves for E _γ  < 800 MeV. Since in the tree approximation, the s-wave is almost completely absent from the π ⁰ π ⁰ channel, the effects of π ⁺ π ⁻ meson rescattering into the neutral channel are included in calculations. Owing to the high intensity of π ⁺ π ⁻ pair formation, the influence of rescattering appears considerable. Its consideration yields a better agreement with the available experimental data.     

Structure and Occurrence of Three-Body Halos in Two Dimensions

 The quantum-mechanical three-body problem is reformulated in two dimensions by use of hyperspherical coordinates and an adiabatic expansion of the Faddeev equations. The effective radial potentials are calculated and their large-distance asymptotic behavior is derived analytically for short-range two-body interactions. Energies and wave functions are computed numerically for various potentials. An infinite series of Efimov states does not exist in two dimensions. Borromean systems, i.e. bound three-body systems without bound binary subsystems, can only appear when a short-range repulsive barrier at finite distance is present in the two-body interaction. The corresponding Borromean state is never spatially extended. For a system of three weakly interacting identical bosons we find two bound states with both binding energies proportional to the two-body binding energy. In the limit of small binding these states are spatially located at the very large distances characterized by the scattering length. Their properties are universal and independent of the details of the potential. We compare throughout with the corresponding properties in three dimensions. Received September 25, 1998; accepted for publication January 30, 1999