Modern Physics Letters A

We present an approach to solve the three-nucleon Faddeev equation which avoids the complicated singularity pattern going with the moving logarithmic singularities of the standard approach. Very good agreement of the new and old approaches in the application to nucleon-deuteron elastic scattering and the breakup reaction is found.

The four-nucleon bound state and scattering below three-body breakup threshold are described. The description is based on a realistic coupled-channel potential which allows the excitation of a single nucleon to a Δ isobar. In the four-nucleon system the Δ isobar mediates effective two-, three- and four-nucleon forces. The effect of the four-nucleon force is found to be much smaller than the effect of the three-nucleon force.

The method of screening and renormalization is used to include the Coulomb interaction between the charged particles in the description of few-body nuclear reactions. Calculations are done in the framework of Faddeev-type equations in momentum-space. The reliability of the method is demonstrated. The Coulomb effect on observables is discussed.

Proposed a new method which is suitable for systematic calculations in few-body systems, we call it the Jacobi-coordinate-basis antisymmetrized molecular dynamics (JAMD). We show good abilities of the method by demonstrating some examples in atoms, molecules, nuclei, and baryons systems.

The phase equivalent relativistic NN potential, which is related by a nonlinear equation to the original nonrelativistic potential, is used to construct the mass operator (rest Hamiltonian) of the 3-nucleon system. Employing the CD Bonn NN potential, the solution of the relativistic 3N Faddeev equation for ³H shows slightly less binding energy than the corresponding nonrelativistic result. The effect of the Wigner spin rotation on the binding is very small.

Observables in proton-deuteron scattering are sensitive probes of the nucleon-nucleon interaction and three-nucleon force effects. Several facilities, including the KVI, allow a detailed study of few-nucleon interactions below the pion-production threshold exploiting polarized proton and deuteron beams. In this contribution, some recent results are discussed and interpreted exploiting rigorous Faddeev calculations. Furthermore, experimental inconsistencies between two measurements of the cross section in elastic proton-deuteron scattering at 135 MeV/nucleon are reviewed.

After a brief discussion about the necessity of using the 3D approach, we present the non partial wave (PW) formalism for 3N bound state with the inclusion of 3N force (3NF). As an example the evaluation of 3NF matrix elements, which appear in the obtained coupled three-dimensional integral equations, for 2π-exchange Tucson–Melbourne 3NF show how would be this formalism efficient and less cumbersome in comparison with the PW formalism.

Cross section and analyzing power A_y of reaction at were measured and large disagreements with 3N calculations were found especially at forward angles. Effects of 2π3NF are too small to explain the discrepancy. To microscopically investigate the discrepancy, a ²H(p, pp)n experiment is planned.

We consider a pionless effective theory with dibaryon fields for the description of the weak process involving two nucleons. We construct leading order Lagrangians that contain nucleon-dibaryon weak coupling constants. We calculate the physical observable in the photodisintegration of the deuteron at threshold and obtain the result in terms of the nucleon-dibaryon weak coupling constants. Relation to existing calculations is discussed.

We made again an experiment of pd radiative capture at , and obtained A_xx, A_yy and A_zz (= -A_xx - A_yy) data using a newly-developed data-analysis method of high detection efficiency. Re-analysis of previous data on A_xx and A_yy was also made using a new method. All the new and previous data at 197 MeV agree to each other, and A_xx anomaly has been confirmed.

Few-nucleon systems can be used as fundamental laboratories for studying details of the nuclear force effects. We performed a series of deuteron-deuteron scattering experiments at intermediate energies. The experiments exploited BINA and BBS experimental setups and polarized deuteron beams with kinetic energies of 65 and 90 MeV/nucleon. These experiments aim to measure differential cross sections, vector and tensor analyzing powers of all available reaction channels in deuteron-deuteron scattering. With these data we will provide a systematic database, which will be used to test present theoretical approximations and upcoming ab-initio calculations in four-nucleon system.

High-precision measurements of the proton-deuteron elastic and break-up reaction have been performed in the past at KVI and elsewhere with the aim to study three-nucleon force (3NF) effects. In the present work, we explored 3NF effects in the break-up scattering process by performing a measurement of vector analyzing powers and differential cross sections using a 135 MeV polarized-proton beam impinging on a liquid-deuterium target. For this study, we used a new experimental setup, Big Instrument for Nuclear-polarization Analysis, BINA, which covers almost the entire kinematical phase space of the break-up reaction. The results are interpreted with the help of state-of-the-art Faddeev calculations.

A new technique has been developed to calculate scattering of and spin-0 particles. The so called momentum-helicity basis states are constructed from the helicity and the momentum states, which are not expanded in the angular momentum states. Thus, all angular momentum states are taken into account. Compared with the partial-wave approach this technique will then give more benefit especially in calculations for higher energies. Taking as input a simple spin-orbit potential, the Lippman-Schwinger equations for the T-matrix elements are solved and some observables are calculated.

We propose a new theoretical approach to calculate nuclear structure using bare nuclear interaction, in which the tensor and short-range correlations are described by using the tensor optimized shell model (TOSM) and the unitary correlation operator method (UCOM), respectively. We compare the obtained results using TOSM+UCOM for ⁴He with the rigorous calculation.

The exact treatment of the two-body problem in momentum space in the presence of Coulomb forces is discussed. Convergence of the traditionally used renormalization approximation is investigated with respect to increasing ranges of the screened Coulomb potentials. The permissible energy range for this approximation is obtained for the first time in this work.

A modified version of the Faddeev three-body equation to accommodate the long-range Coulomb interaction is applied to the proton-deuteron scattering problem at energies above the three-body breakup threshold. Numerical results for elastic and breakup cross sections in proton-deuteron scattering with effects of three-nucleon forces are compared with experimental data.

At E_d = 26 MeV, H(d,pp)n cross section was measured at 4 angle pairs corresponding to off-plane star and compared with recent pd calculations. A rotary foil target was developed for this experiment. The present data and previous data revealed dependence of the star anomaly on the star-inclination angle α. The angular dependence at E_p = 13 MeV differs much with that at 9.5 MeV.

We measured D(p, pp)n cross section at E_p = 9.5 MeV and 13 MeV at off-plane star configurations, including the space star where the star is perpendicular to the beam axis. The experimental results were compared with recent pd breakup calculations using screened Coulomb force, and disagreement was found to smoothly vary with an inclination angle of the off-plane star, α. Further systematic experiments are necessary to investigate the star anomaly.

Gamow–Teller states, possibly with narrow widths, have been searched for by the ⁴He(³He, t) reaction with an energy resolution of 100 keV. The search is motivated by theoretical discussions on the role of pions in nuclear structure and simulation of supernovae explosion. In the preliminary analysis of the spectrum at around 0°, no significant narrow structure is observed.

We solved the three-nucleon Faddeev equation in a Poincaré invariant model of the three-nucleon system. Two-body interactions are generated so that when they are added to the two-nucleon invariant mass operator (rest energy) the two-nucleon S matrix is identical to the non-relativistic S matrix with a CD Bonn interaction. Cluster properties of the three-nucleon S-matrix determine how these two-nucleon interactions are embedded in the three-nucleon mass operator. Differences in the predictions of the relativistic and corresponding non-relativistic models for elastic and breakup processes are investigated. Of special interest are the lowering of the A_y maximum in elastic nucleon-deuteron (Nd) scattering below ≈ 25 MeV caused by the Wigner spin rotations and the significant changes of the breakup cross sections in certain regions of the phase-space.

At the MIT-Bates Linear Accelerator Center, the nucleon form factors have been measured by scattering polarized electrons from vector-polarized hydrogen and deuterium. The experiment used the longitudinally polarized electron beam stored in the MIT-Bates South Hall Ring along with an isotopically pure, highly vector-polarized internal atomic hydrogen and deuterium target provided by an atomic beam source. The measurements were carried out with the symmetric Bates Large Acceptance Spectrometer Toroid (BLAST). Results are presented for the proton form factor ratio, , and for the charge form factor of the neutron, . Both results are more precise than previous data in the corresponding Q² ranges.

In the viewpoint of QCD, the nucleon is made up of constituent quarks, sea quarks and gluons. Concerning the quark sea, also strange quarks can contribute to the nucleon properties. Parity violating electron scattering offers a tool to investigate the strange quark contribution to the nucleon form factors. The measurements of different experiments are discussed and the recent results from the A4 collaboration at MAMI is presented. Altogether the existing data allow to give constraints on the strangeness contribution.

In this presentation, we briefly review recent investigations on pion and kaon structures from the instanton vacuum. Starting from the low-energy QCD partition function, we have computed the Gasser-Leutwyler low-energy constants, electromagnetic form factors of the pion and kaon, and semileptonic decay form factors of the kaon. The results are in good agreement with the experimental and empirical data.

In this talk, we report our recent work on the pion weak decay constant (F_π) and pion mass (m_π) using the nonlocal chiral quark model with the finite quark-number chemical potential (μ) taken into account. Considering the breakdown of Lorentz invariance at finite density, the time and space components are computed separately, and the corresponding results turn out to be: and at μ_c ≈ 320 MeV, respectively. Using the in-medium Gell-Mann-Oakes-Renner (GOR) relation, we show that the pion mass increases by about 15% at μ_c.

The three-body resonance of system is investigated by using the coupled channels Faddeev equation. The resonance energy is determined from the pole of S-matrix on the unphysical sheet. It is found that the pole positions of the predicted amplitudes are significantly modified when the three-body dynamics is approximately treated by introducing the effective two-body interaction.

Pionium and kaonium are studied in an accurate numerical approach based on Sturmian functions. It is found that the ground-state wave functions of the exotic atoms in realistic strong interactions, particularly for kaonium, are considerably different from the hydrogen-like ones at small distances. The kaon-kaon scattering length derived from the 1s energy shift of kaonium by applying the Deser-Trueman formula is strongly inconsistent with the one derived directly by solving the Schödinger equation. The theoretical results indicate that it is arguable to treat kaonium perturbatively.

We have measured (K^-,K⁺) reaction at 1.67 GeV/cK^- at the KEK proton synchrotron. A scintillating fiber active target was employed to view both secondary interactions and decays of particles produced in the (K^-,K⁺) reaction. This tracking capability for the vertex region provides us for studying the fate of Ξ^- hypernuclei, which is advantageous over the previous two experiments. We present a preliminary result on the production cross sections for , and also on the branching fractions into the decay modes with charged particles and/or Λ hyperon(s).

The formation of a deeply-bound K^-pp state by the ³He(in-flight K^-, n) reaction is investigated theoretically in a distorted-wave impulse approximation using the Green's function method. The expected inclusive and semi-exclusive spectra at p_K^- = 1.0 GeV/c and θ_n = 0° are calculated for the forthcoming J-PARC E15 experiment. We discuss these spectra with some K^--"pp" optical potentials, taking into account the energy-dependence of the imaginary part of the potentials by a phase space suppression factor.

After a brief discussion of effective field theory applied to nuclear clusters, I present the aspect of Coulomb interactions, with applications to low-energy alpha-alpha and nucleon-alpha scattering.

Using chiral effective field theory (EFT) with explicit Δ degrees of freedom we calculated nuclear forces up to next-to-next-to-leading order (N²LO). We find a much improved convergence of the chiral expansion in all peripheral partial waves. We also present a novel lattice EFT method developed for systems with larger number of nucleons. Combining Monte Carlo lattice simulations with EFT allows one to calculate the properties of light nuclei, neutron and nuclear matter. Accurate description of two-nucleon phase-shifts and ground state energy ratio of dilute neutron matter up to corrections of higher orders show that lattice EFT is a promising tool for quantitative studies of low-energy few- and many-body systems.

Nuclear electromagnetic currents derived in a chiral-effective-field-theory framework including explicit nucleons, Δ isobars, and pions up to N²LO, i.e. ignoring loop corrections, are used in a study of neutron radiative captures on protons and deuterons at thermal energies, and of A=2 and 3 nuclei magnetic moments. With the strengths of the Δ-excitation currents determined to reproduce the n-p cross section and isovector combination of the trinucleon magnetic moments, we find that the cross section and photon circular polarization parameter, measured respectively in n-d and processes, are significantly underpredicted by theory.

We outline our renormalization-group analysis for low-energy scattering two charged particles. This uses dimensional regularization and extends previous analyses by including both momentum- and energy-dependent perturbations.

Meson-baryon potentials are proposed as an extension of baryon-baryon one-boson-exchange potentials. All coupling constants satisfy the SU(3) symmetry and all mesons and baryons are assumed to have physical masses and widths and to behave as gaussian sources of meson fields. Assuming the parameters determined so as to reproduce the πN and K N scattering, our model inevitably predicts a strongly attractive S-wave interaction both in isospin 0 and 1 states.

By exploiting the chiral Ward identities based on the QCD Lagrangian for pion weak production, we derive a transition amplitude by the weak current and show analytical results of relevant multipole amplitudes without any expansion. Their physical implications and numerical results are also discussed.

Angular distributions of differential cross sections from the latest CLAS data sets,⁶ for the reaction γ + p→K⁺ + Λ have been analyzed using associated Legendre polynomials. This analysis is based upon theoretical calculations in Ref. 1 where all sixteen observables in kaon photoproduction can be classified into four Legendre classes. Each observable can be described by an expansion of associated Legendre polynomial functions. One of the questions to be addressed is how many associated Legendre polynomials are required to describe the data. In this preliminary analysis, we used data models with different numbers of associated Legendre polynomials. We then compared these models by calculating posterior probabilities of the models. We found that the CLAS data set needs no more than four associated Legendre polynomials to describe the differential cross section data. In addition, we also show the extracted coefficients of the best model.

We present the neutral-current (NC) and charged-current (CC) reactions by incident neutrino (antineutrino) scattering on the nucleon, and also on the ¹²C target in quasi-elastic region within the framework of a relativistic single particle model. Effects of the strangeness are studied on the cross sections, the ratios between the NC and CC reactions, and the asymmetries. We found that there exists some cancellation of the strange quark contributions between the knocked-out protons and neutrons in the neutrino (antineutrino)-nucleus scattering. The effect of the strangeness on the asymmetry exhibits larger than that on the ratio and the cross section.

We apply QCD factorization to the quasi two-body B → (Kπ)π decays where the (Kπ)-pair effective mass is limited to 1.8 GeV. Our strong interaction phases constrained by theory and πK experimental data yield useful information for studies of CP violation.

The new experimental data of kaon photoproduction on the nucleon γp → K⁺Λ have been analyzed by means of a multipoles model. Different from the previous models, in this analysis the resonance decay widths are constrained to the values given by the Particle Data Group (PDG). The result indicates that constraining these parameters to the PDG values could dramatically change the conclusion of the important resonances in this reaction found in the previous studies.

Neutral kaon photoproduction on the deuteron has been investigated by including the final state effects and compared with the experimental data. Comparison shows that the models used in this calculation can reproduce the data in the Σ channel regions fairly well but still give over predictions in the Λ channel. It seems that the tensor target asymmetries are more suitable for studying the final state effects. The extractions of the elementary photoproduction amplitude are also demonstrated.

It is proposed to extend the nuclear physics by including the magnetic monopole as the additional component. It is indicated that the nuclear reaction of such a system is drastically different from the ordinary nuclear reaction.

A large solid angle γ-ray detector system, FOREST, has been constructed at LNS-Sendai to study the π⁰ and η photo-production reactions. The FOREST detector consists of three electro-magnetic calorimeters: pure CsI crystals, Lead/SciFi blocks and Lead Glass Cherenkov counters. It covers about 90% of the total solid angle. A new data acquisition system for the FOREST detector has been developed to handle high rate trigger events with a high efficiency. To investigate performances of FOREST and the DAQ system, a test experiment has been conducted. The π⁰ and η mesons are clearly observed. A DAQ efficiency of 76% has been obtained at a 2 kHz trigger request.

The D+D → ⁴He reaction can be one of the methods to count the magnetic monopole. In order to prove this, the mechanism of bound state of nucleons and monopole system is clarified. We calculated bound state of nucleon-monopole system by using the Jacobi-coordinate bases antisymmetrized molecular dynamics (or JAMD) method. This calculation is done to show the efficiency of JAMD as a first step of main subject. The binding energy of neutron and proton in the magnetic monopole field is calculated. Our results are compared with those of analytical solution.

The dynamics of spinor Bose–Einstein condensates has been solved via a method of few body theory in which the spin degrees of freedom are treated strictly. Our approach is appropriate to the condensates of several thousands or less atoms. The Bose gas of ⁸⁷Rd spin-1 atoms, trapped in a spin-independent potential and interacting via spin-dependent force, has been calculated as an example. Rich evolution patterns that are sensitive to the initial states have been found.

One of the methods to obtain a precise solution of a few-body equation is to expand the orbital part of the wave function into the correlated Gaussian with double global vectors. In this contribution, we report our recent studies applying this method to light nuclei, (i) excited states of ⁴He and (ii) two-nucleon momentum distribution in ⁶He and ⁶Li, especially focusing on the importance of the tensor force.

The universal character of non-relativistic large three-body bosonic systems is addressed, with focus on highly interesting situations that occur in low-energy nuclear and atomic physics. Investigations on the trajectory of the first excited Efimov state, in a renormalized zero-range three-body model for a system with two bound and one virtual two-body subsystems, are reported. The approach is applied to n-n-¹⁸C, where the n - n virtual energy and the three-body ground state are kept fixed. It is shown that the real part of the elastic s-wave phase-shift () presents a zero, or a pole in , when the system has an Efimov excited or virtual state. A brief discussion is given on the relevance of the approach for ultracold atom physics with tunable scattering lengths.

The generalized two-center cluster model (GTCM), which can treat various single particle configurations in general two center systems, is applied to the light neutron-rich system, ¹²Be= α+α+4N. We discuss the change of the neutrons' configuration around two α-cores as a variation of an excitation energy. We show that the covalent, ionic and atomic configurations appear in the unbound region above the α+⁸He_g.s. particle-decay threshold, and they coexist within a quite small energy interval.

We developed an m-scheme approach of the cluster-orbital shell model formalism. The radial wave function is treated as the super position of the Gaussian functions with different width parameters. Energies and r.m.s. radii of oxygen isotopes are studied.

To describe excited nuclear states, we developed the Brueckner-AMD combining the Brueckner theory with Antisymmetrized Molecular Dynamics (AMD), which has been recently proposed as a new framework to study nuclear structure of light nuclei based on the realistic nuclear force. In the present framework, we formulate a multi-configuration calculation with Generator Coordinate Method (GCM) in the Brueckner-AMD. An application is shown for the excited states with positive and negative parities in ⁴He, and the successful results and reliability of the present framework are discussed.

The K⁰ photoproduction process is investigated in the threshold region, E_γ = 0.8-1.1 GeV at LNS-Tohoku. K⁰ events are measured via π⁺π^- decay channel. Obtained momentum spectra for the deuteron target are compared with those calculated in the spectator model using a realistic deuteron wave function. Recent isobar models and a simple phenomenological model are used as elementary amplitudes to investigate the effect of the new data on the angular behavior of the elementary cross section. The data favor a backward-peaked angular distribution of the elementary n(γ, K⁰)Λ process in the center-of-mass (c.m.) system, which provides additional constraints on current models of kaon photoproduction. The present study shows that the n(γ, K⁰)Λ reaction can provide key information on the mechanism of the photoproduction of strangeness.

The long standing Γ_n/Γ_p puzzle of NMWD has finally been solved recently. It appears that the reason behind the puzzle was the universal quenching of nucleon yields from the prediction of one-nucleon induced (1N-) NMWD. We have shown that the quenching of nucleon yields can be explained only when we include a fairly large contribution of the three-body process (Γ_2N) in Γ_nm.

We present a Faddeev-type coupled integrodifferential equations describing unequal mass A-particle systems. The equations are obtained by assuming an expansion of the A-particle wave function in Faddeev amplitudes which, in turn, are expanded in terms of Potential Harmonics. By projecting the resulting system on r_ij-space we obtained a system of coupled equations which can be used to study multi-strange hypernuclei.

We reexamine the spin-orbit splitting of excited states in terms of the SU₆ quark-model baryon-baryon interaction. The previous folding procedure to generate the Λα spin-orbit potential from the quark-model ΛN LS resonating-group kernel predicted three to five times larger values for ΔE_ℓs = E_x(3/2⁺) - E_x(5/2⁺) in the model FSS and fss2. This time, we calculate Λα LS Born kernel, starting from the LS components of the nuclear-matter G-matrix for the Λ hyperon. This framework makes it possible to take full account of an important P-wave ΛN-ΣN coupling through the antisymmetric LS^(-) force involved in the Fermi-Breit interaction. We find that the experimental value, , is reproduced by the quark-model G-matrix LS interaction with a Fermi-momentum around k_F = 1.0 fm^-1, when the model FSS is used in the energy-independent renormalized RGM formalism. On the other hand, the model fss2 gives too large splitting of almost 200 keV, owing to the uncanceled contribution of the scalar-meson exchange LS components.

Gaussian nonlocal potentials for the quark-model baryon–baryon interactions are derived by using the Gauss-Legendre quadrature for the special functions. The reliability of the approximation is examined with respect to the phase shifts and the deuteron binding energy. The potential is accurate enough if one uses seven-point Gauss-Legendre quadrature.

We present the effects of Fermi motion and different off-shell assumptions on the differential cross sections of photo- and electroproduction of the hypertriton. We found that Fermi motion plays an important role in these processes. Different off-shell assumptions yield different cross section magnitudes. The few available electroproduction data favor the assumption that the initial nucleon is off-shell and the final hyperon is on-shell. Measurement of the hypertriton photoproduction is urgently required to further support our findings.

The Hyperspherical coordinates method (HCM), allows one to obtain the spectrum of the system and develop the classification of excited states. We have solved the radial Schrödinger wave equation exactly in three dimensions for an N-body problem. Using the obtained results, we have studied the N family baryon masses.

We describe a problem associated with the role the ω meson at short distance in applying the Skyrme model to the dense matter: the ω coupling to the baryon density prevents the scale-anomaly dilaton field from developing a vanishing vacuum expectation value at the chiral restoration, which is at variance with standard scenario of QCD at the phase transition. We suggest a possible phenomenological solution to this puzzle and discuss its physical consequences.

Neutrinos are produced copiously in the early universe. Neutrinos and antineutrinos ceased to be in equilibrium with radiation when the weak interaction rate becomes slower than the rate expansion of the universe. The ratio of the temperature of the photon to the temperature of the neutrino at this stage is T_γ/T_ν = (11/4)^1/3. We investigate the neutrino energy loss due to the oscillation of the electron neutrino into a different flavor in the charged-current interaction of ν_e-e^- based on the work of Sulaksono and Simanjuntak. The energy loss from the neutrinos ΔE_ν during the decoupling of the neutrinos with the rest of the matter would be a gain in the energy of the electrons and can be obtained from the integration of stopping power equation ΔE_ν = (dE_ν/dT^-1)dT^-1 where E_ν and T are the energy of the neutrinos and the temperature respectively. When the universe expands and matter-radiation decouples, an extra energy will be transferred to the photons via the annihilation of the electron-positron pairs, e⁺+e^-→γ+γ. This consequently will increase the temperature of the photons. The net effect to the lowest order is an increase in the ratio of the photon temperature to the neutrino temperature. The magnitude of energy loss of the neutrino is ∼10^-4-10^-5 MeV for the probability of conversion of ν_e → ν_i (i = μ,τ) between 0 to 1.0.

The phase transition at temperature variation is washed out in the finite system. Quantum or statistical fluctuations may give rise to this situation. It is still uncertain whether these fluctuations are caused by approximation error or finiteness of the system. Here we use another approximation the functional Schrödinger picture formalism founded on the variational principle and study the few-body nucleon system by using this formalism.

The role of the Coulomb and isovector effects in dense stellar matter with neutrino trapping in the region of perturbative momentum q ≤ 0.6 fm^-1 has been investigated. The result might have an impact on the neutrino transport in stellar matter.

The study of the various collective flows of nuclear matter is one of the main subjects in heavy ion physics. The interest in collective nuclear motion under extreme conditions, like high density and/or high temperature originates from the equation of state (EOS) of nuclear matter. We investigate the collective transverse flow in heavy ion collisions at incident energies of 0.25AGeV to 1.15AGeV for Au + Au system within the quantum molecular dynamical (QMD) model. Some preliminary results are given and discussed.

Effects of the ω meson self coupling (OMSC) on the thermal properties of asymmetric nuclear matter (ANM) are studied within the framework of relativistic mean field (RMF) model that includes contributions of all possible mixed interactions among meson fields involved up to quartic order. In particular, we study the mechanical and chemical instabilities (spinodal), as well as the liquid-gas phase transition (binodal) at finite temperature. It is found that the onset of spinodal instabilities and the binodal curve are only marginally affected by variation of the OMSC parameter, whereas the binodal curve shows a strong correlation to the symmetry energy. Comparison with other ERMF parameter sets is also performed.

Gamow peak describes the most effective energy E₀ for a nonresonant nuclear reaction to occur. At astrophysical low energies much lower than the Coulomb barrier, the ability of the interacting nuclei to tunnel through the barrier depends on the primitive probability which is proportional to the Gamow factor. The reaction rate will then be determined by the primitive tunneling probability and the astrophysical S-factor. In the literature, tables on the thermonuclear reaction rates are compiled for many reactions of interest in astrophysics by using this low energy approximation. In this paper, we describe a method to obtain E₀ by using the exact tunneling probability that is valid for higher energy. We illustrate the method by using three major reactions in the proton-proton chain, ³He(³He, 2p)⁴He, ³He(⁴He, γ)⁷Be and ⁷Be(γ, p)⁸B. In all cases, E₀ starts to divert to lower values than the low energy approximation limit at around T = 10⁹K and hence generate lower reaction rates. This result might be significantly important for the thermonuclear reactions in the advanced stages of the evolution of massive stars.

A personal perspective is given on topics which appears to be most promising during the fourth Asia Pacific Conference on Few-Body Problem in Physics 2008.