Coordinate Space Representation Research Articles

Visualizing the zero order basis of the spectroscopic Hamiltonian

Recent works have shown that a generalization of the spectroscopic effective Hamiltonian can describe spectra in surprising regions, such as isomerization barriers. In this work, we seek to explain why the effective Hamiltonian is successful where there was reason to doubt that it would work at all. All spectroscopic Hamiltonians have an underlying abstract zero-order basis (ZOB) which is the "ideal" basis for a given form and parameterization of the Hamiltonian. Without a physical model there is no way to transform this abstract basis into a coordinate representation. To this end, we present a method of obtaining the coordinate space representation of the abstract ZOB of a spectroscopic effective Hamiltonian. This method works equally well for generalized effective Hamiltonians that encompass above-barrier multiwell behavior, and standard effective Hamiltonians for the vicinity of a single potential minimum. Our approach relies on a set of converged eigenfunctions obtained from a variational calculation on a potential surface. By making a one-to-one correspondence between the energy eigenstates of the effective Hamiltonian and those of the coordinate space Hamiltonian, a physical representation of the abstract ZOB is calculated. We find that the ZOB basis naturally adjusts its complexity depending on the underlying nature of phase space, which allows spectroscopic Hamiltonians to succeed for systems sampling multiple stationary points.

Anomalous pairing vibration in neutron-rich Sn isotopes beyond theN=82magic number

Two-neutron transfer associated with the pair correlation in superfluid neutron-rich nuclei is studied with focus on low-lying ${0}^{+}$ states in Sn isotopes beyond the $N=82$ magic number. We describe microscopically the two-neutron addition and removal transitions by means of the Skyrme-Hartree-Fock-Bogoliubov mean-field model and the continuum quasiparticle random phase approximation formulated in the coordinate space representation. It is found that the pair transfer strength for the transitions between the ground states becomes significantly large for the isotopes with $A\ensuremath{\geqslant}140$, reflecting very small neutron separation energy and long tails of the weakly bound $3p$ orbits. In ${}^{132--140}$Sn, a peculiar feature of the pair transfer is seen in transitions to low-lying excited ${0}^{+}$ states. They can be regarded as a kind of pair vibrational mode which is characterized by an anomalously long tail of the transition density extending to far outside of the nuclear surface, and a large strength comparable to that of the ground-state transitions. The presence of the weakly bound neutron orbits plays a central role for these anomalous behaviors.

Iterative solution of a Dirac equation with an inverse Hamiltonian method

We solve a singe-particle Dirac equation with Woods-Saxon potentials using an iterative method in the coordinate space representation. By maximizing the expectation value of the inverse of the Dirac Hamiltonian, this method avoids the variational collapse, in which an iterative solution dives into the Dirac sea. We demonstrate that this method works efficiently, reproducing the exact solutions of the Dirac equation.

Evolution equation for theB-meson distribution amplitude in the heavy-quark effective theory in coordinate space

The $B$-meson distribution amplitude (DA) is defined as the matrix element of a quark-antiquark bilocal light-cone operator in the heavy-quark effective theory, corresponding to a long-distance component in the factorization formula for exclusive $B$-meson decays. The evolution equation for the $B$-meson DA is governed by the cusp anomalous dimension as well as the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi-type anomalous dimension, and these anomalous dimensions give the ``quasilocal'' kernel in the coordinate-space representation. We show that this evolution equation can be solved analytically in the coordinate space, accomplishing the relevant Sudakov resummation at the next-to-leading logarithmic accuracy. The quasilocal nature leads to a quite simple form of our solution which determines the $B$-meson DA with a quark-antiquark light-cone separation $t$ in terms of the DA at a lower renormalization scale $\ensuremath{\mu}$ with smaller interquark separations $zt$ ($z\ensuremath{\le}1$). This formula allows us to present rigorous calculation of the $B$-meson DA at the factorization scale $\ensuremath{\sim}\sqrt{{m}_{b}{\ensuremath{\Lambda}}_{\mathrm{QCD}}}$ for $t$ less than $\ensuremath{\sim}1\text{ }\text{ }{\mathrm{GeV}}^{\ensuremath{-}1}$, using the recently obtained operator product expansion of the DA as the input at $\ensuremath{\mu}\ensuremath{\sim}1\text{ }\text{ }\mathrm{GeV}$. We also derive the master formula, which reexpresses the integrals of the DA at $\ensuremath{\mu}\ensuremath{\sim}\sqrt{{m}_{b}{\ensuremath{\Lambda}}_{\mathrm{QCD}}}$ for the factorization formula by the compact integrals of the DA at $\ensuremath{\mu}\ensuremath{\sim}1\text{ }\text{ }\mathrm{GeV}$.

Semiclassical electron correlation operator

The concept of the correlation operator, introduced 10 years ago as a possible method to model the electron correlation effects with single determinant wave functions [Rassolov, J. Chem. Phys. 110, 3672 (1999)], is revisited. We derive a semiclassical limit of the correlation operator in weakly correlated systems and give its coordinate space representation. Application of this operator to the atomic systems, such as computations of energies of the neutral atoms, energies of the cations, and spin states energy gaps, demonstrates capabilities and limitations of this concept.

A simple and efficient numerical scheme to integrate non-local potentials

As nuclear wave functions have to obey the Pauli principle, potentials issued from reaction theory or Hartree-Fock formalism using finite-range interactions contain a non-local part. Written in coordinate space representation, the Schrodinger equation becomes integro-differential, which is difficult to solve, contrary to the case of local potentials, where it is an ordinary differential equation. A simple and powerful method has been proposed several years ago, with the trivially equivalent potential method, where non-local potential is replaced by an equivalent local potential, which is state-dependent and has to be determined iteratively. Its main disadvantage, however, is the appearance of divergences in potentials if the wave functions have nodes, which is generally the case. We will show that divergences can be removed by a slight modification of the trivially equivalent potential method, leading to a very simple, stable and precise numerical technique to deal with non-local potentials. Examples will be provided with the calculation of the Hartree-Fock potential and associated wave functions of 16O using the finite-range N3LO realistic interaction.

Subleading contributions to the chiral three-nucleon force: Long-range terms

We derive the long-range contributions to the tree-nucleon force at next-to-next-to-next-to-leading order in the chiral expansion. We give both momentum- and coordinate-space representations.

Exact boundary integral equations for scattering of scalar waves from perfectly reflecting infinite rough surfaces

The derivation of integral equations for scattering from an infinite rough surface is not a straightforward procedure. The usual formalism involves a single plane wave incident on the surface and the free-space Green’s function. Problems arise due to waves propagating horizontal to the surface and to the inability to evaluate in a simple way the Helmholtz–Kirchhoff or Green’s surface integral on the upper hemisphere. One approach is to change the usual Sommerfeld radiation condition to exclude horizontal waves. In this paper we take a different approach. Instead of the single incident plane wave as the Born term we choose the flat surface result of incident plus reflected field. The additional field scattered from the rough surface satisfies the Sommerfeld condition. The full Green’s function is chosen as a combination of the free-space Green’s function and its image. The main result is that the Helmholtz–Kirchhoff integral using this new Born term and the image Green’s function can be evaluated exactly (and simply) on the hemisphere. This leads directly to integral equations on the total field (flat surface plus scattered fields) for the Dirichlet and Neumann problems. Simple coordinate-space representations for the kernels of these equations are also presented. Since we use image functions we have standing waves in the z-direction, so that the result is not strictly categorized as a diffraction result. Nevertheless, the integral relations induced by their choice yield an exact and simple result for the solution of an otherwise difficult problem.

Superradiance and subradiance in the electromagnetic radiative decay ofX(3872)

It is pointed out that if the molecular interpretation of the recently observed resonance $X(3872)$ is valid, then nature may have prepared a good laboratory for us to examine the phenomenon of superradiance and subradiance of Dicke. The superradiance and supradiance factors are evaluated and the effects on the electromagnetic radiative decay of $X(3872)$ are discussed. Our results using coordinate space representation is similar to the momentum space results of Voloshin.

Continuum effects on the pairing in neutron drip-line nuclei studied with the canonical-basis HFB method

The canonical-basis HFB method provides an efficient way to describe pairing correlations involving the continuum part of the single-particle spectrum in coordinate-space representations. It can be applied to super-conducting deformed drip-line nuclei as easily as to stable or spherical nuclei. This method is applied to a simulation of the approach to the neutron drip line. It turns out that the HFB solution has a stronger pairing and a smaller deformation as the Fermi level is raised. However, such changes are smooth and finite. No divergences or discontinuities of the radius or other quantities are found in the limit of zero Fermi energy. The nuclear density continues to be localized even a little beyond the drip line.

Novel coordinate space representation for deformed loosely-bound nuclei(Poster presentation,YITP Workshop on New Developments in Nuclear Self-Consistent Mean-Field Theories (MF05))

Novel coordinate space representation for deformed loosely-bound nuclei(Poster presentation,YITP Workshop on New Developments in Nuclear Self-Consistent Mean-Field Theories (MF05))

Canonical-basis solution of the Hartree-Fock-Bogoliubov equation on a three-dimensional Cartesian mesh

A method is presented to obtain the canonical-form solutions of the HFB equation for atomic nuclei with zero-range interactions like the Skyrme force. It is appropriate to describe pairing correlations in the continuum in coordinate-space representations. An improved gradient method is used for faster convergences under constraint of orthogonality between orbitals. To prevent high-lying orbitals to shrink into a spatial point, a repulsive momentum dependent force is introduced, which turns out to unveil the nature of high-lying canonical-basis orbitals. The asymptotic properties at large radius and the relation with quasiparticle states are discussed for the obtained canonical basis.

Stringy profiles of gauge field tadpoles near orbifold singularities: II. field theoretical implications

Recent work has provided a direct string calculation of the internal coordinate dependence of gauge field tadpoles on the orbifold C^3/Z_3. We investigate the structure of these profiles in momentum and coordinate space representations using both analytical and numerical methods. The twisted states are determined to be localized within a few string lengths of the singularity. This provides an example of how delta-like singularities, which are typical for field theories on orbifolds, are avoided in string theory. A systematic expansion of the full string result allows us to speculate on how the leading stringy effects can be incorporated in a field theoretical description. In one heterotic model, we find that even though tachyons are not present in the physical spectrum, they totally dominate the tadpole profile in the vicinity of the fixed point.

Stringy profiles of gauge field tadpoles near orbifold singularities: I. heterotic string calculations

Closed string theories on orbifolds contain both untwisted and twisted states. The latter are normally assumed to live exactly at the orbifold fixed points. We perform a calculation of a gauge field tadpole amplitude and show that off-shell both the twisted and untwisted states give rise to non-trivial momentum profiles over the orbifold C^3/Z_3. These profiles take the form of Gaussian distributions integrated over the fundamental domain of the modular parameter of the torus. The propagators of the internal coordinate fields on the torus world sheet determine the width of the Gaussian profiles. These propagators are determined up to a single normal ordering constant which must be bounded below to allow the existence of the coordinate space representation of these Gaussians. Apart from the expected massless states, massive and even tachyonic string excitations contribute to the profiles in some anomalous U(1) models. However, when a tadpole is integrated over the internal dimensions, these tachyonic contributions cancel in a non-trivial manner.

Chiral corrections to the isovector double scattering term for the pion-deuteron scattering length

The empirical value of the real part of the pion-deuteron scattering length can be well understood in terms of the dominant isovector $\ensuremath{\pi}N$ double scattering contribution. We calculate in chiral perturbation theory all one-pion loop corrections to this double scattering term which in the case of $\ensuremath{\pi}N$ scattering close the gap between the current-algebra prediction and the empirical value of the isovector threshold T-matrix ${T}_{\ensuremath{\pi}N}^{\ensuremath{-}}.$ In addition to closing this gap, there is in the $\ensuremath{\pi}d$ system a loop-induced off-shell correction for the exchanged virtual pion. Its coordinate space representation reveals that it is equivalent to $2\ensuremath{\pi}$ exchange in the deuteron. We evaluate the chirally corrected double scattering term and the off-shell contribution with various realistic deuteron wave functions. We find that the off-shell correction contributes at most $\ensuremath{-}8%$ and that the isovector double scattering term explains at least 90% of the empirical value of the real part of the $\ensuremath{\pi}d$-scattering length.

Continuum linear response in coordinate space Hartree–Fock–Bogoliubov formalism for collective excitations in drip-line nuclei

We formulate a continuum linear response theory on the basis of the Hartree–Fock–Bogoliubov formalism in the coordinate space representation in order to describe low-lying and high-lying collective excitations which couple to one-particle and two-particle continuum states. Numerical analysis is done for the neutron drip-line nucleus 24O. A low-lying collective mode that emerges above the continuum threshold with large neutron strength is analyzed. The collective state is sensitive to the density-dependence of the pairing. The present theory satisfies accurately the energy weighted sum rule. This is guaranteed by treating the pairing self-consistently both in the static HFB and in the dynamical linear response equation.

Continuum QRPA in the coordinate space representation

We formulate a quasiparticle random phase approximation (QRPA) in the coordinate space representation. This model is a natural extension of the RPA model of Shlomo and Bertsch to open-shell nuclei in order to take into account pairing correlations together with the coupling to the continuum. We apply it to the 120Sn nucleus and show that low-lying excitation modes are significantly influenced by the pairing effects although the effects are marginal in the giant resonance region. The dependence of the pairing effect on the parity of low-lying collective mode is also discussed.

Nuclear isotope shifts within the local energy-density functional approach

The foundation of the local energy-density functional method to describe the nuclear ground-state properties is given. The method is used to investigate differential observables such as the odd–even mass differences and odd–even effects in charge radii. For a few isotope chains of spherical nuclei, the calculations are performed with an exact treatment of the Gor'kov equations in the coordinate-space representation. A zero-range cutoff density-dependent pairing interaction with a density-gradient term is used. The evolution of charge radii and nucleon separation energies is reproduced reasonably well including kinks at magic neutron numbers and sizes of staggering. It is shown that the density-dependent pairing may also induce sizeable staggering and kinks in the evolution of the mean energies of multipole excitations. The results are compared with the conventional mean field Skyrme–HFB and relativistic Hartree–BCS calculations. With the formulated approach, an extrapolation from the pairing properties of finite nuclei to pairing in infinite matter is considered, and the dilute limit near the critical point, at which the regime changes from weak to strong pairing, is discussed.

Scale dependence of QCD string operators

We have obtained a general solution of evolution equations for QCD twist-2 string operators in form of expansion over complete set of orthogonal eigenfunctions of evolution kernels in coordinate-space representation. In the leading logarithmic approximation the eigenfunctions can be determined using constraints imposed by conformal symmetry. Explicit formulae for the LO scale-dependence of quark and gluon twist-2 string operators are given.

Four-point Green functions in the Schwinger model

The evaluation of the 4-point Green functions in the 1+1 Schwinger model is presented both in momentum and coordinate space representations. The crucial role in our calculations play two Ward identities: i) the standard one, and ii) the chiral one. We demonstrate how the infinite set of Dyson-Schwinger equations is simplified, and is so reduced, that a given n-point Green function is expressed only through itself and lower ones. For the 4-point Green function, with two bosonic and two fermionic external `legs', a compact solution is given both in momentum and coordinate space representations. For the 4-fermion Green function a selfconsistent equation is written down in the momentum representation and a concrete solution is given in the coordinate space. This exact solution is further analyzed and we show that it contains a pole corresponding to the Schwinger boson. All detailed considerations given for various 4-point Green functions are easily generizable to higher functions.