Internal Wave Functions Research Articles

Bilocal Field Theory for Composite Scalar Bosons.

We give a bilocal field theory description of a composite scalar with an extended binding potential that reduces to the Nambu-Jona-Lasinio (NJL) model in the pointlike limit. This provides a description of the internal dynamics of the bound state and features a static internal wave function, ϕ(r→), in the center-of-mass frame that satisfies a Schrödinger-Klein-Gordon equation with eigenvalues m2. We analyze the "coloron" model (single perturbative massive gluon exchange) which yields a UV completion of the NJL model. This has a BCS-like enhancement of its interaction, ∝Nc the number of colors, and is classically critical with gcritical remarkably close to the NJL quantum critical coupling. Negative eigenvalues for m2 lead to spontaneous symmetry breaking, and the Yukawa coupling of the bound state to constituent fermions is emergent.

Wave function of the sexaquark or compact H-dibaryon

We derive and explicitly display the internal wave function of a color-, flavor- and spin- singlet dibaryon composed of uuddss quarks in a spatially symmetric state, in the approximation of exact SU(3) flavor symmetry. This wavefunction shows that the often-used superposition of ΛΛ, NΞ and ΣΣ baryons, relevant for di-baryon molecules, accounts for only 1/5 of a spatially symmetric six-quark color-flavor-spin-singlet state, with the remaining 4/5 consisting of products of color-octet baryons. Using the correct wavefunction has important implications for calculations of the mass of the lowest-lying flavor-singlet dibaryon, as we illustrate using the Cornell potential. We also provide a compact representation of the state in terms of creation operators, and comment on the impact of not using the optimal operator in lattice studies of this system.

Harmonic functions and gravity localization

In models with extra dimensions, matter particles can be easily localized to a ‘brane world’, but gravitational attraction tends to spread out in the extra dimensions unless they are small. Strong warping gradients can help localize gravity closer to the brane. In this note we give a mathematically rigorous proof that the internal wave-function of the massless graviton is constant as an eigenfunction of the weighted Laplacian, and hence is a power of the warping as a bound state in an analogue Schrödinger potential. This holds even in presence of singularities induced by thin branes.We also reassess the status of AdS vacuum solutions where the graviton is massive. We prove a bound on scale separation for such models, as an application of our recent results on KK masses. We also use them to estimate the scale at which gravity is localized, without having to compute the spectrum explicitly. For example, we point out that localization can be obtained at least up to the cosmological scale in string/M-theory solutions with infinite-volume Riemann surfaces; and in a known class of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 models, when the number of NS5- and D5-branes is roughly equal.

Semiclassical propagator approach for emission processes from deformed nuclei

We describe a more rigorous formulation of the 3D semiclassical approach, corresponding to deformed potentials, which leads to the exact results and we also compare them with the much simpler expressions given by the angular Wentzel–Krames–Brillouin (WKB) and linearized WKB with its approximation, known as Fröman WKB method. We define the semiclassical fundamental matrix of solutions, describing the effect of the deformed field. Its inverse, proportional to the so-called propagator matrix, connects the solution in two points. A comparison between the coupled channels fundamental matrix of solutions to various semiclassical counterparts evidences close results in the barrier region. We express the scattering amplitudes in terms of the propagator matrix multiplied by the ratio between internal wave function components and irregular Coulomb waves. The relative influence of nuclear and Coulomb action terms together with nuclear and Coulomb centrifugal barriers upon the penetration process reveals the dominance of the Coulombian parts. An analysis of alpha-emission processes between ground states and to excited states of even–even nuclei in terms of the propagator matrix evidenced the important role played by deformation of the Coulomb field and the shape of the nuclear interaction.

Universal proton emission systematics

We investigate proton systematics in terms of the barrier penetration and formation probability. To this purpose we approximate the realistic proton-core interaction by a parabolic dependence describing the nuclear part matched to a pure Coulomb potential. In this way we can use analytic semiclassical solutions to describe the proton dynamics inside the barrier. We then carefully investigate effects induced by the centrifugal barrier and quadrupole deformation. It turns out that the nuclear centrifugal and deformation effects are much smaller than their Coulomb counterparts. On the other hand, the Coulomb action defining the semiclassical penetrability is larger by one order of magnitude than its nuclear counterpart. We evidence a dependence of the Coulomb action upon the Coulomb parameter in terms of two parallel lines separated by $A=145$. Such a dependence was previously discovered for the logarithm of the experimental decay width corrected by the Coulomb centrifugal barrier. We also evidence an additional but smaller residual linear decrease of the proton formation probability versus ${A}^{1/3}$ along two parallel lines separated by $A=145$, induced by the internal proton wave function. Therefore, the dependence between the experimental decay width corrected by the centrifugal barrier and the Coulomb penetrability multiplied by the proton formation probability in the doubly logarithmic scale can be fitted within a factor of two by a universal straight line.

Transitional description of mass spectra for baryons in the U(7) model

In this study, we obtained mass spectra for baryons. The mass spectra for baryons depend on the spatial and internal symmetry Hamiltonian and wave functions. The algebraic solution for mass spectra of baryons is introduced in the spatial part. Results are obtained by using the Bethe Ansatz within an infinite-dimensional Lie algebra. The mass spectra for baryons are compared with the experimental data. The results show that the theory reproduces the experimental data quite well.

Collision clock shift of two Fermi atoms in harmonic potentials

Atomic clocks provide the most accurate definition for time. The precision of atomic clock has been improved by many orders of magnitude since the first atomic clock was built. However, the interatomic interaction usually suppress the precision of atomic clock. As a result, it is especially meaningful to study the interaction effect in atomic clock, which is considered to be helpful in improving the precision and accuracy of atomic clock. In order to characterize the collision effect induced clock shift, we theoretically study the collision clock shift in the Rabi spectrum, caused by the short-range interaction between two Fermi atoms in harmonic potential. Given that the short-range interatomic interaction is generally weak, and that the parameters of external lattice laser field are in the Lamb-Dicke regime, we make an approximation that the spatial wave-function of the Fermi atoms does not change, and then derive the motion equation for the internal wave-function under the external Rabi driving field. We solve the equation of motion by the perturbative method, and obtain the solution to first order, and thus derive the expression of the collision clock shift of the Rabi spectrum in terms of the interatomic interaction and the external Rabi driving laser field parameters for specific spatial wave-functions of atoms. Finally, we use the exact expression of the Green’s function in harmonic potential to obtain the averaged clock shift of collision at finite temperatures. Our results relate the atomic interaction with atomic clock shift, and provide a unified description of all partial waves of atomic interaction induced clock shift. Therefore, it becomes much more convenient to study the contributions of different partial waves to atomic clock shift. On the other hand, our results indicate that through precisely measuring the clock shift, the information about the interatomic interactions can also be obtained. In addition, our results for two interacting atoms can inspire the future study of real many-body interacting system which will be the next research topic.

Use of Microscopic Theoretical Methods for The Construction of The Nuclear Wavefunction of 7Li

Microscopic theoretical studies of scattering and reaction problems for light nuclei have been extensively carried out using resonating group method. In this paper we have used the nuclear cluster model, the resonating group method, the generator coordinate method and complex generator coordinate technique for the construction of microscopic antisymmetrized nuclear wavefunction of 7Li nucleus. This wavefunction can be further used to calculate the structural properties of the nucleus. The 7Li nucleus in ground state is considered as a nuclear system consisting of three clusters namely an alpha cluster, a deuteron cluster and a neutron cluster. We have chosen spatial, spin and isospin function of cluster internal functions. The arguments of internal wavefunction include the parameter coordinates. These parameters can be adjusted to some extent to obtain predictions close to experimental results. The wavefunction is written using shell model with definite parity and angular momentum. The complex generator coordinate technique allows this wavefunction to write it as an antisymmetrized product of seven single particle functions after inclusion of the wavefunction for the center- of -mass motion

Revealing all states of dewetting of a thin gold layer on a silicon surface by nanosecond laser conditioning

Dewetting is a ubiquitous phenomenon which can be applied to the laser synthesis of nanoparticles. A classical spinodal dewetting process takes place in four successive states, which differ from each other in their morphology. In this study all states are revealed by interaction of pulsed nanosecond UV laser light with thin gold layers with thicknesses between 1 nm and 10 nm on (100) silicon wafers. The specific morphologies of the dewetting states are discussed with particular emphasis on the state boundaries. The main parameter determining which state is formed is not the duration for which the gold remains liquid, but rather the input energy provided by the laser. This shows that each state transition has a separate measurable activation energy. The temperature during the nanosecond pulses and the duration during which the gold remains liquid was determined by simulation using the COMSOL Multiphysics® software package. Using these calculations, an accurate local temperature profile and its development over time was simulated. An analytical study of the morphologies and formed structures was performed using Minkowski measures. With aid of this tool, the laser induced structures were compared with thermally annealed samples, with perfectly ordered structures and with perfectly random structures. The results show that both, structures of the laser induced and the annealed samples, strongly resemble the perfectly ordered structures. This reveals a close relationship between these structures and suggests that the phenomenon under investigation is indeed a spinodal dewetting generated by an internal material wave function. The purposeful generation of these structures and the elucidation of the underlying mechanism of dewetting by short pulse lasers may assist the realisation of various technical elements such as nanowires in science and industry.

Heavy baryon wave functions, Bakamjian-Thomas approach to form factors, and observables in Λb→Λc(12±)ℓν¯ transitions

Motivated by the calculation of observables in the decays $\Lambda_b \to \Lambda_c\left({1 \over 2}^\pm \right) \ell \overline{\nu}$, as tests of Lepton Flavor Universality, we present a calculation of form factors in the quark model. Our scheme combines a spectroscopic model, providing the internal wave functions, and the Bakamjian-Thomas relativistic formalism to deduce the wave functions in motion and current matrix elements, that amount in the heavy quark limit to the Isgur-Wise (IW) function. For baryons we meet difficulties using standard spectroscopic models, leading us to propose a simple phenomenological model : a Q-pointlike-diquark model, non-relativistic with harmonic oscillator forces, with a reasonable low-lying spectrum and good slope of the IW function. We extract this slope from Lattice data and find $\rho_\Lambda^2 \sim 2$. We are not able to reproduce the right $\rho_\Lambda^2$ when using standard linear + Coulomb potential models, both with three quarks $Qqq$ or in a Q-pointlike-diquark picture. These difficulties seem to derive from the high sensitivity of $\rho_\Lambda^2$ to the structure of the light quark subsystem. After fixing the parameters of our interim model to yield correct spectrum and $\rho_\Lambda^2$, we compute observables. Bjorken sum rule shows that the inelastic IW function is large, and therefore $\Lambda_b \to \Lambda_c \left({1 \over 2}^-, {3 \over 2}^- \right) \ell \overline{\nu}$ could be studied at LHCb. Some observables in the $\tau$ case present zeroes for specific values of $q^2$ that could be tests of the Standard Model.

Microscopic study of 7Li-nucleus potential * * Supported by National Natural Science Foundation of China (11705009, 11575291) and Science Challenge Project (TZ2018005)

A microscopic approach is employed to study the optical potential for the 7Li-nucleus interaction system without any free parameters. It is obtained by folding the microscopic optical potentials of the constituent nucleons of 7Li over their density distributions. We employ an isospin-dependent nucleon microscopic optical potential, which is based on the Skyrme nucleon-nucleon effective interaction and derived using the Green's function method, as the nucleon optical potential. The harmonic oscillator shell model is used to describe the internal wave function of 7Li and obtain the nucleon density distribution. The 7Li microscopic optical potential is used to predict the reaction cross-sections and elastic scattering angular distributions for the target range from 27Al to 208Pb and energy range below 450 MeV. Generally, the results can reproduce the measured data reasonably well. In addition, the microscopic optical potential is comparable to a global phenomenological optical potential by fitting the presently existing measured data.

Microscopic optical potential for 7Li

A microscopic optical potential (MOP) for 7Li without free parameter is obtained by folding the MOPs of its internal nucleons with their density distributions. An isospin-dependent nucleon MOP based on the Skyrme nucleon–nucleon effective interaction is used as the nucleon optical potential. Cluster model is employed to construct the internal wave function of 7Li and derive the density distributions of the internal nucleons. The 7Li MOP is used to calculate the elastic-scattering angular distributions and reaction cross sections for target nuclei from 27Al to 208Pb at incident energies up to 450 MeV. The results are compared with experimental data and the calculated results by a global phenomenological optical model potential. Generally the MOP can reproduce the experimental data reasonably well, and in many cases it is comparable to the global phenomenological optical potential.

Microscopic optical potentials for Li isotopes

The microscopic optical potentials for Li isotopes (A=6,7) without free parameter are obtained by folding the microscopic optical potentials of their internal nucleons with density distributions generated from corresponding internal wave functions of Li isotopes. An isospin-dependent nucleon microscopic optical potential based on the Skyrme nucleon-nucleon effective interaction is used as the nucleon optical potential. Shell model is employed to construct the internal wave functions of Li isotopes and derive their density distributions of internal nucleons. The Li microscopic optical potentials are used to calculate the elastic-scattering angular distributions and reaction cross sections. The results reproduce experimental data well and are comparable to those calculated by phenomenological optical model potentials in many cases.

Transfer reactions with the Lagrange-mesh method

We apply the $R$-matrix method in distorted wave Born approximation (DWBA) calculations. The internal wave functions are expanded over a Lagrange mesh, which provides an efficient and fast technique to compute matrix elements. We first present an outline of the theory, by emphasizing the $R$-matrix aspects. The model is applied to the $^{16}\mathrm{O}(d,p)^{17}\mathrm{O}$ and $^{12}\mathrm{C}{(}^{7}\mathrm{Li},t)^{16}\mathrm{O}$ reactions, typical of nucleon and of $\ensuremath{\alpha}$ transfer, respectively. We illustrate the sensitivity of the cross sections with respect to the $R$-matrix parameters and show that an excellent convergence can be achieved with relatively small bases. We also discuss the effects of the remnant term in DWBA calculations and address the question of the peripherality in transfer reactions. We suggest that uncertainties on spectroscopic factors could be underestimated in the literature.

Atom interferometers with weak-measurement path detectors and their quantum mechanical analysis**Project supported by the National Key Research and Development Program of China (Grant No. 2018YFA 0306200), the National Natural Science Foundation of China (Grant No. 11434017), and the Guangdong Innovative and Entrepreneurial Research Team Program, China (Grant No. 2016ZT06C594).

According to the orthodox interpretation of quantum physics, wave-particle duality (WPD) is the intrinsic property of all massive microscopic particles. All gedanken or realistic experiments based on atom interferometers (AI) have so far upheld the principle of WPD, either by the mechanism of the Heisenberg’s position-momentum uncertainty relation or by quantum entanglement. In this paper, we propose and make a systematic quantum mechanical analysis of several schemes of weak-measurement atom interferometer (WM-AI) and compare them with the historical schemes of strong-measurement atom interferometer (SM-AI), such as Einstein’s recoiling slit and Feynman’s light microscope. As the critical part of these WM-AI setups, a weak-measurement path detector (WM-PD) deliberately interacting with the atomic internal electronic quantum states is designed and used to probe the which-path information of the atom, while only inducing negligible perturbation of the atomic center-of-mass motion. Another instrument that is used to directly interact with the atomic center-of-mass while being insensitive to the internal electronic quantum states is used to monitor the atomic center-of-mass interference pattern. Two typical schemes of WM-PD are considered. The first is the micromaser-cavity path detector, which allows us to probe the spontaneously emitted microwave photon from the incoming Rydberg atom in its excited electronic state and record unanimously the which-path information of the atom. The second is the optical-lattice Bragg-grating path detector, which can split the incoming atom beam into two different directions as determined by the internal electronic state and thus encode the which-path information of the atom into the internal states of the atom. We have used standard quantum mechanics to analyze the evolution of the atomic center-of-mass and internal electronic state wave function by directly solving Schrödinger’s equation for the composite atom-electron-photon system in these WM-AIs . We have also compared our analysis with the theoretical and experimental studies that have been presented in the previous literature. The results show that the two sets of instruments can work separately, collectively, and without mutual exclusion to enable simultaneous observation of both wave and particle nature of the atoms to a much higher level than the historical SM-AIs, while avoiding degradation from Heisenberg’s uncertainty relation and quantum entanglement. We have further investigated the space–time evolution of the internal electronic quantum state, as well as the combined atom–detector system and identified the microscopic origin and role of quantum entanglement, as emphasized in numerous previous studies. Based on these physics insights and theoretical analyses, we have proposed several new WM-AI schemes that can help to elucidate the puzzling physics of the WPD of the atoms. The principle of WM-AI scheme and quantum mechanical analyses made in this work can be directly extended to examine the principle of WPD for other massive particles.

Suppression of light nuclei production in collisions of small systems at the Large Hadron Collider

We show that the recently observed suppression of the yield ratio of deuteron to proton and of helium-3 to proton in p+p collisions compared to those in p+Pb or Pb+Pb collisions by the ALICE Collaboration at the Large Hadron Collider (LHC) can be explained if light nuclei are produced from the coalescence of nucleons at the kinetic freeze-out of these collisions. This suppression is attributed to the non-negligible sizes of deuteron and helium-3 compared to the size of the nucleon emission source in collisions of small systems, which reduces the overlap of their internal wave functions with those of nucleons. The same model is also used to study the production of triton and hypertriton in heavy-ion collisions at the LHC. Compared to helium-3 in events of low charged particle multiplicity, the triton is less suppressed due to its smaller size and the hypertriton is even more suppressed as a result of its much larger size.

Testing Production Scenarios for (Anti-)(Hyper-)nuclei with Multiplicity-dependent Measurements at the LHC

The production of light anti- and hyper-nuclei provides unique observables to characterise the system created in high energy proton-proton (pp), proton-nucleus (pA) and nucleus-nucleus (AA) collisions. In particular, nuclei and hyper-nuclei are special objects with respect to non-composite hadrons (such as pions, kaons, protons, etc.), because their size is comparable to a fraction or the whole system created in the collision. Their formation is typically described within the framework of coalescence and thermal-statistical production models. In order to distinguish between the two production scenarios, we propose to measure the coalescence parameter B$_{A}$ for different anti- and hyper-nuclei (that differ by mass, size and internal wave-function) as a function of the size of the particle emitting source. The latter can be controlled by performing systematic measurements of light (anti-)(hyper-)nuclei in different collision systems (pp, pA, AA) and as a function of the multiplicity of particles created in the collision. While it is often argued that the coalescence and the thermal model approach give very similar predictions for the production of light nuclei in heavy-ion collisions, our study shows that large differences can be expected for hyper-nuclei with extended wave-functions, as the hyper-triton. We compare the model predictions with data from the ALICE experiment and we discuss perspectives for future measurements with the upgraded detectors during the High-Luminosity LHC phase in the next decade.

Quantifying wave-function overlaps in inhomogeneous Majorana nanowires

A key property of Majorana zero modes is their protection against local perturbations. In the standard picture, this protection is guaranteed by a high degree of spatial nonlocality of the Majoranas, namely a suppressed wave-function overlap, in the topological phase. However, a careful characterisation of resilience to local noise goes beyond mere spatial separation, and must also take into account the projection of wave-function spin. By considering the susceptibility of a given zero mode to different local perturbations, we find the relevant forms of spin-resolved wave-function overlaps that measure its resilience. We quantify these overlaps and study their dependence with nanowire parameters in several classes of experimentally relevant configurations. These include nanowires with inhomogeneous depletion and induced pairing, barriers and quantum dots. Smooth inhomogeneities have been shown to produce near-zero modes, so-called pseudo-Majoranas, below the critical Zeeman field in the bulk. Surprisingly, their resilience is found to be comparable or better than that of topological Majoranas in realistic systems. We further study how accurately their overlaps can be estimated using a purely local measurement on one end of the nanowire, accessible through conventional transport experiments. In uniform nanowires this local estimator is remarkably accurate. In inhomogeneous cases it is less accurate but can still provide reasonable estimates for potential inhomogeneities of the order of the superconducting gap. We further analyse the zero mode wave-function structure, spin texture and spectral features associated with each type of inhomogeneity. All our results highlight the strong connection between internal wave-function degrees of freedom, nonlocality and protection in smoothly inhomogeneous nanowires.

Microscopic optical potential for He6

The microscopic optical potential for $\phantom{\rule{0.16em}{0ex}}^{6}\mathrm{He}$ with no free parameters is obtained by folding the microscopic optical potentials of its constituent nucleons with the internal wave function of $\phantom{\rule{0.28em}{0ex}}^{6}\mathrm{He}$. We use the isospin-dependent nucleon microscopic optical potential, which is derived by using the Green's function method through the nuclear matter approximation and the local density approximation based on the Skyrme nucleon-nucleon effective interaction. The internal wave function of $^{6}\mathrm{He}$ is described in a harmonic-oscillator form. The $^{6}\mathrm{He}$ microscopic optical potential is used to calculate the reaction cross sections and elastic-scattering angular distributions for target nuclei in the mass range $12\ensuremath{\le}\mathrm{A}\ensuremath{\le}209$ at incident energies up to 350 MeV. The results are compared with the experimental data and those calculated by a global phenomenological optical potential; in most cases, the microscopic optical potential reproduces the experimental data less well than the global potential. The sensitivity of scattering to the potentials as a function of radius has been investigated by using the notch perturbation method. The investigation shows that the scattering is sensitive to the optical potential in the nuclear surface region. It is concluded from the discussion that the microscopic optical potential can be improved by increasing the surface absorption contribution.

Evidence of enhanced threeαradius inα+12C inelastic scattering

The microscopic coupled-channels calculations (MCC), which is based on precise internal wave functions and a realistic nucleon-nucleon interaction, is performed for the α + 12 C inelastic scattering in the energy range of Eα = 80 to 400 MeV. The MCC calculations nicely reproduce the observed differential cross sections for the elastic and inelastic scattering, which goes to the 21+, 31−, 02+ states. The partial wave analysis for the differential cross sections has also performed. From the partial wave analysis, the nuclear radius of three α rotational state in 12 C with a life time of 10-21 second, which has been expected to have much more extended radius than the ground 12 C nucleus, is speculated. Present analysis predicts about 1.0 fm enhancement in the matter radius of the three α rotational state in comparison to the normal radius of the ground state, which is known to be proportional to the mass number to the one third. The spatial extension of the three α rotational state is comparable to the extended radius observed in the neutron halo phenomena. Constraint on the recent ab-initio calculation for the 3α states in 12 C is also discussed.