Empirical Phase Shifts Research Articles

Peripheral nucleon-nucleon scattering at next-to-next-to-leading order in SU(3) heavy baryon chiral perturbation theory

We calculate the complete $T$ matrices of the elastic nucleon-nucleon scattering up to third order in SU(3) heavy baryon chiral perturbation theory. The phase shifts with orbital angular momentum $L\geq 2$ and the mixing angles with $J\geq 2$ are evaluated by using low-energy constants that were extracted from the meson-baryon analysis. It turns out that our prediction is consistent with the empirical phase shifts and mixing angles data, and also the results from SU(2) heavy baryon chiral perturbation theory. The errors from the low-energy constants are analyzed in detail. Our calculation provides reliable evidence that the nucleon-nucleon interaction calculated in SU(3) heavy baryon chiral perturbation theory leads to reasonable predictions.

Perturbative NN scattering in chiral effective field theory

Within the framework of chiral effective field theory, perturbative calculation for $NN$ scattering is carried out in partial waves with orbital angular momentum $L \geqslant 1$. The primary goal is to identify the lowest angular momenta at which perturbative treatment of chiral forces can apply. Results up to the order where the subleading two-pion exchange appears are shown. It is concluded that perturbation theory applies to all partial waves but ${}^1S_0$, ${}^3S_1 - {}^3D_1$, and ${}^3P_0$. Where it is applicable, perturbation theory with the delta-less chiral forces produces good agreement with the empirical phase shifts up to $k_\text{c.m.} \simeq 300$ MeV.

Two-nucleon S01 amplitude zero in chiral effective field theory

We present a new rearrangement of short-range interactions in the $^{1}S_{0}$ nucleon-nucleon channel within chiral effective field theory. This is intended to address the slow convergence of Weinberg's scheme, which we attribute to its failure to reproduce the amplitude zero (scattering momentum $\ensuremath{\simeq}\phantom{\rule{0.16em}{0ex}}340$ MeV) at leading order. After the power counting scheme is modified to accommodate the zero at leading order, it includes subleading corrections perturbatively in a way that is consistent with renormalization-group invariance. Systematic improvement is shown at next-to-leading order, and we obtain results that fit empirical phase shifts remarkably well all the way up to the pion-production threshold. An approach in which pions have been integrated out is included, which allows us to derive analytic results that also fit phenomenology surprisingly well.

Dynamic modeling of the behavior of permalloy for magnetic shielding

The minimization of the remanent magnetization of ferromagnetic materials is a prerequisite for a reproducible low magnetic field inside shields. To realistically describe this so-called magnetic equilibration procedure, this paper proposes two approaches for the calculation of time- and space-dependent fields in the presence of ferromagnetic materials like permalloy. The first method is based on the Jiles-Atherton model and also takes into account frequency dependent effects. The second method is the newly developed empirical phase shift model, tailored specially for the simulation of the equilibration procedure. Both approaches are compared to experimental tests and show good quantitative agreement.

Dominant contributions to the nucleon-nucleon interaction at sixth order of chiral perturbation theory

We present the dominant two- and three-pion-exchange contributions to the nucleon-nucleon interaction at sixth order (next-to-next-to-next-to-next-to-next-to-leading order, N5LO) of chiral perturbation theory. Phase shifts with orbital angular momentum L>=4 are given parameter free at this order and allow for a systematic investigation of the convergence of the chiral expansion. The N5LO contribution is prevailingly repulsive and considerably smaller than the N4LO one, thus, establishing the desired trend towards convergence. Using low-energy constants that were extracted from an analysis of pi-N scattering at fourth order, the predictions at N5LO are in excellent agreement with the empirical phase shifts of peripheral partial waves.

Dispersion relations in application to chiral two-nucleon dynamics

A novel scheme based on the chiral Lagrangian is applied to the nucleon-nucleon interaction close to threshold. Subthreshold partial-wave amplitudes are calculated in chiral perturbation theory and analytically extrapolated above threshold. The constraints imposed by analyticity and unitarity are used to stabilize the extrapolation. A reasonable description of the empirical phase shifts up to laboratory energies T lab ≃ 300 MeV is obtained in terms of the parameters relevant at next-to-next-to-next-to-leading order. The convergence properties of the method and the comparison with the conventional potential approach are discussed.

Infinite-Cutoff Renormalization of the Chiral Nucleon–Nucleon Interaction up to N3LO

Naively, the "best" method of renormalization is the one where a momentum cutoff is taken to infinity while maintaining stable results due to a cutoff-dependent adjustment of counterterms. We have applied this renormalization method in the non-perturbative calculation of phase-shifts for nucleon-nucleon (NN) scattering using chiral NN potentials up to next-to-next-to-next-to-leading order (N3LO). For lower partial waves, we find that there is either no convergence with increasing order or, if convergence occurs, the results do not always converge to the empirical values. For higher partial waves, we always observe convergence to the empirical phase shifts (except for the 3G5 state). Furthermore, no matter what the order is, one can use only one or no counterterm per partial wave, creating a rather erratic scheme of power counting that does not allow for a systematic order-by-order improvement of the predictions. The conclusion is that infinite-cutoff renormalization is inappropriate for chiral NN interactions, which should not come as a surprise, since the chiral effective field theory, these interactions are based upon, is designed for momenta below the chiral-symmetry breaking scale of about 1 GeV. Therefore, this value for the hard scale should also be perceived as the appropriate upper limit for the momentum cutoff.

Effective short-range interaction for spin-singletP-wave nucleon-nucleon scattering

Distorted-wave methods are used to remove the effects of one- and two-pion exchange up to order Q^3 from the empirical 1P1 phase shift. The one divergence that arises can be renormalised using an order-Q^2 counterterm which is provided by the (Weinberg) power counting appropriate to the effective field theory for this channel. The residual interaction is used to estimate the scale of the underlying physics.

Deconstructing 1S0 nucleon-nucleon scattering

A distorted-wave method is used to analyse nucleon-nucleon scattering in the 1S0 channel. Effects of one-pion exchange are removed from the empirical phase shift to all orders by using a modified effective-range expansion. Two-pion exchange is then subtracted in the distorted-wave Born approximation, with matrix elements taken between scattering waves for the one-pion exchange potential. The residual short-range interaction shows a very rapid energy dependence for kinetic energies above about 100 MeV, suggesting that the breakdown scale of the corresponding effective theory is only 270MeV. This may signal the need to include the Delta resonance as an explicit degree of freedom in order to describe scattering at these energies. An alternative strategy of keeping the cutoff finite to reduce large, but finite, contributions from the long-range forces is also discussed.

A consistent 3 α and 2 αΛ Faddeev calculation using the 2 α RGM kernel

A Consistent description of the 3α and 2αΛ systems is attempted in the three-cluster Faddeev Formalism, which explicitly employs the 2α resonating-group-method (RGM) kernel reproducing the empirical αα phase shifts. The 3-range Minnesota (MN) force for the effective NN interaction gives a larger 3α binding energy than the 2-range Volkov No.1 and No.2 forces. The 2αΛ ground-state and 2+ excited-state energies are reasonably reproduced for the 3-range MN force, if an appropriate ΛN force is used for the Λα folding potential. Here, the effective ΛN force is spin-singlet and triplet two-range Gaussian central forces, constructed from the phase-shift behavior of the quark-model hyperon-nucleon (YN) interaction, fss2, by using an inversion method based on supersymmetric quantum mechanics. © 2004 Published by Elsevier B.V.

Family of Hermitian low-momentum nucleon interactions with phase shift equivalence

Using a Schmidt orthogonalization transformation, a family of Hermitian low-momentum NN interactions is derived from the non-Hermitian Lee-Suzuki (LS) low-momentum NN interaction. As special cases, our transformation reproduces the Hermitian interactions for Okubo and Andreozzi. Aside from their common preservation of the deuteron binding energy, these Hermitian interactions are shown to be phase shift equivalent, all preserving the empirical phase shifts up to decimation scale Lambda. Employing a solvable matrix model, the Hermitian interactions given by different orthogonalization transformations are studied; the interactions can be very different from each other particularly when there is a strong intruder state influence. However, because the parent LS low-momentum NN interaction is only slightly non-Hermitian, the Hermitian low-momentum nucleon interactions given by our transformations, including the Okubo and Andreozzi ones, are all rather similar to each other. Shell model matrix elements given by the LS and several Hermitian low-momentum interactions are compared.

Pion-nucleon scattering in chiral perturbation theory (I): Isospin-symmetric case

We construct the complete effective chiral pion-nucleon Lagrangian to third order in small momenta based on relativistic chiral perturbation theory. We then perform the so-called heavy baryon limit and construct all terms up to and including order 1 m 2 with fixed and free coefficients. As an application, we discuss in detail pion-nucleon scattering. In particular, we show that for this process andto third order, the 1 m expansion of the Born graphs calculated relativistically can be recovered exactly in the heavy baryon approach without any additional momentum-dependent wave function renormalization. We fit various empirical phase shifts for pion laboratory momenta between 50 and 100 MeV. This leads to a satisfactory description of the phase shifts up to momenta of about 200 MeV. We also predict the threshold parameters, which turn out to be in good agreement with the dispersive analysis. In particular, we can sharpen the prediction for the isovector S-wave scattering length, 0.083 M π −1 ⩽ a 0+ − ⩽ 0.093 M π −1. We also consider the subthreshold parameters and give a short comparison to other calculations of πN scattering in chiral perturbation theory or modifications thereof.

Peripheral NN-scattering: role of delta-excitation, correlated two-pion and vector meson exchange

We evaluate, within one-loop chiral perturbation theory, the two-pion exchange diagrams with single and double delta-isobar excitation contributing to elastic NN-scattering. We find that virtual Δ-excitation processes (in the static limit) produce the correct amount of isoscalar central attraction as needed in the peripheral partial waves with L ⩾ 3. Furthermore we compute the two-loop diagrams involving the ππ-interaction (so-called correlated 2π-exchange). Contrary to common believe these processes lead to negligibly small and repulsive corrections to the NN-potential. The exchange of vector mesons (ϱ, ω) turns out to be important for the F-wave phase shifts above T lab = 150 Mev. Without adjustable parameters we are able to reproduce the empirical NN phase shifts up to 350 MeV for L ⩾ 3 and up to about (50–80) MeV for the D-waves. This is therefore the characteristic window in which the NN-interaction is basically governed by chiral symmetry. Not surprisingly, the lower partial waves require non-perturbative methods and additional short-distance parametrizations of the NN-dynamics.

Peripheral nucleon-nucleon phase shifts and chiral symmetry

Within the one-loop approximation of baryon chiral perturbation theory we calculate all one-pion and two-pion exchange contributions to the nucleon-nucleon interaction. In fact we construct the elastic NN-scattering amplitude up to and including third order in small momenta. The phase shifts with orbital angular momentum L ≥2 and the mixing angles with J ≥ 2 are given parameter free and thus allow for a detailed test of chiral symmetry in the two-nucleon system. We find that for the D-waves the 2π-exchange corrections are too large as compared with empirical phase shifts, signaling the increasing importance of shorter range effects in lower partial waves. For higher partial waves, especially for G-waves, the model-independent 2π-exchange corrections bring the chiral prediction close to empirical NN phase shifts. We propose to use the chiral NN phase shifts with L ≥ 3 as input in a future phase-shift analysis. Furthermore, we compute the irreducible two-pion exchange NN potentials in coordinate space. They turn out to be of van der Waals type, with exponential screening of two-pion mass range.

SeparableNNpotentials from inverse scattering for nuclear matter studies

Low-rank separable potentials greatly simplify perturbation-theory based many-body computations and are especially useful in finite temperature and nonequilibrium nuclear matter studies. With local potentials such calculations become very lengthy. In this paper, we present a first version of a separable potential constructed directly from available empirical nucleon-nucleon phase shifts (Elab<1.6 GeV), and established deuteron properties, via nonrelativistic inverse scattering. In our approach, the on-shell and off-shell properties of the potential can be independently varied: the on-shell scattering data are exactly fitted by construction while physically motivated off-shell features can be systematically included. This advantage is illustrated by the application of the procedure in the 3S1−3D1 channel, where the deuteron wave function serves as off-shell input at the binding energy. The simplest potential thus constructed in this channel has rank 4. The deuteron wave function is nevertheless empirically undetermined at high momenta, prompting us to adopt as well as construct several model wave functions that all fit the low momentum deuteron data while allowing variations at high momenta. The effects of these off-shell variations on predicted nuclear matter properties are discussed. No off-shell information is included in the other channels, leading to potentials of rank either 1 or 2. With this simple model potential we perform standard Brueckner nuclear matter ground state calculations and compare the results with Machleidt’s using Bonn OBEP. The agreement is good in the S channels and in the singlet D2 channel. Other channels show larger discrepancies, the most significant of which coming from the 3P1 and 3D1 channels. These results are explained by the off-shell behavior of our model potential as compared to the Bonn OBEP. Further off-shell input, empirical and/or theoretical, will be explored in future improved versions of the model.

Identification of different forms of carbon by extended energy loss fine structure

Using extended energy loss fine structure spectroscopy (EXELFS), we obtained the local structure around the carbon atoms in amorphous carbon, highly oriented pyrolytic graphite and a polycrystalline diamond thin film grown by the HF-CVD technique. The experiments were performed in a scanning Auger microprobe in the reflection mode and in a high resolution transmission electron microscope with a PEELS attachment. The values obtained for the local structure applying these techniques, show an error around 12% for EXELFS in the reflection mode and 0.3% for the transmission mode, with respect to their reported crystallographic values. In order not to alter the raw experimental data obtained for the different forms of carbon under the same experimental conditions, the introduction of empirical phase shifts was not applied here. This was done in order to avoid the introduction of empirical phase shifts which are normally used to fit the data to know structural parameters. One of the main objectives of this work is to test the reproducibility of the data when taken in the reflection and in the transmission modes.

Local 4He-p potentials from resonating-group method phase shifts.

Phase shifts for \ensuremath{\alpha}+nucleon scattering generated by a multichannel resonating-group method (RGM) model of the five-nucleon system are subjected to iterative-perturbative ``mixed-case'' inversion for energies below the reaction threshold. The resulting phase-equivalent potentials are compared with local potentials calculated from the inversion of empirical phase shifts. A strong similarity is revealed between the two sets of potentials, most notably in the description by a parity- and energy-dependent local potential. In particular, the RGM-derived and empirical potentials share a distinctive form of parity dependence in which the odd-parity component is of greater radial extent. Comparison with potentials representing single-channel RGM phase shifts exhibits the importance of the coupled channels in terms of local potentials. The relative wave functions derived from RGM are very different from those for the phase-equivalent local potentials. Connections are made with general nucleon-nucleus scattering.

Screening mass from chiral perturbation theory, virial expansion, and the lattice.

We calculate the electric screening mass in hot hadronic matter using two different approaches, chiral perturbation theory and the relativistic virial expansion with empirical phase shifts, and compare the results to each other and to a gas of free pions and $\rho$ mesons. We also compute the electric screening mass for noninteracting, charged bosons with mass $m$ on a lattice to study likely finite size effects in lattice gauge theory simulations of continuum QCD. For a lattice of given size, the continuum can be properly represented only for a window in the ratio $T/m$.

Thermal properties of interacting hadrons

We study hadronic matter at temperatures close to the pion mass. In the dilute-gas stage, the relativistic virial expansion is used with empirical phase shifts to investigate the thermodynamic properties of an interacting gas of pions, kaons and nucleons. The interplay between the attractive and repulsive interactions which governs the size of the interacting contributions is discussed. In the dense gas stage characterized by the presence of many massive resonances, schematic models that include the effects of repulsive interactions are analysed. Limits on the size of repulsive interactions are established by causality constraints. The causal behaviour of some of the conventional excluded-volume approaches is examined. Since our work is also aimed at studying those equations of state which may be used in dynamical simulations of heavy-ion collisions, phase diagrams are constructed including the thermodynamics of the quark-gluon phase.

Microscopic study of ? + ? bremsstrahlung with resonating-group wave functions

Bremsstrahlung double cross sections for α + α collisions are computed with the wave functions obtained by using a single-configuration resonating-group method (RGM). With no adjustable parameters, the calculated values are shown to agree well with experiment. For reliable results, it is found that the intercluster interaction must yield a good fit to the empirical phase shifts in the energy region where relatively sharp resonance states exist. A simplified procedure (A-RGM), in which the antisymmetrization of the wave function is only approximately taken into account, has also been adopted to consider the bremsstrahlung problem. Comparisons between RGM and A-RGM results yield the useful conclusion that fitting empirical phase shifts alone is not a sufficient criterion to ensure accuracy in the bremsstrahlung cross sections and a proper description of the nuclear system in the region of strong cluster overlap is also important.