Analytic Form Factors Research Articles

Heavy quark diffusion coefficients in magnetized quark-gluon plasma

We evaluate the heavy quark momentum diffusion coefficients in a hot magnetized medium for the most general scenario of any arbitrary values of the external magnetic field. We choose to work with the systematic way of incorporating the effect of the magnetic field, by using the effective gluon and quark propagators, generalized for a hot and magnetized medium. To get gauge independent analytic form factors valid through all Landau levels, we apply the hard thermal loop technique for the resummed effective gluon propagator. The derived effective hard thermal loop gluon propagator and the generalized version of Schwinger quark propagator subsequently allow us to analytically evaluate the longitudinal and transverse momentum diffusion coefficients for charm and bottom quarks beyond the static limit. Within the static limit we also explore another way of incorporating the effect of the magnetic field, i.e. through the magnetized medium modified Debye mass and compare the results to justify the need for structural changes. Published by the American Physical Society 2024

Form factor of any polyhedron and its singularities derived from a projection method

An analytical and general form factor for any polyhedron is derived on the basis of a projection method, in terms of the vertex coordinates and topology of the polyhedron. An integral over the polyhedron equals the sum of the signed integrals over a set of dissected tetrahedra by defining a sign function, and a general tetrahedral form factor is established by defining a projection method. All possible singularities present in the formula are discussed in detail. Using a MATLAB implementation, illustrative examples are discussed to verify the accuracy and generality of the method. The use of the scalar product operation and the sign function in this work allows a general and neat formula to be obtained for any polyhedron, including convex and concave polyhedra. The formulas and discussions presented here will be useful for the characterization of nanoparticles using small-angle scattering techniques.

The quantum sine-Gordon model with quantum circuits

Analog quantum simulation has the potential to be an indispensable technique in the investigation of complex quantum systems. In this work, we numerically investigate a one-dimensional, faithful, analog, quantum electronic circuit simulator built out of Josephson junctions for one of the paradigmatic models of an integrable quantum field theory: the quantum sine-Gordon (qSG) model in 1+1 space-time dimensions. We analyze the lattice model using the density matrix renormalization group technique and benchmark our numerical results with existing Bethe ansatz computations. Furthermore, we perform analytical form-factor calculations for the two-point correlation function of vertex operators, which closely agree with our numerical computations. Finally, we compute the entanglement spectrum of the qSG model. We compare our results with those obtained using the integrable lattice-regularization based on the quantum XYZ chain and show that the quantum circuit model is less susceptible to corrections to scaling compared to the XYZ chain. We provide numerical evidence that the parameters required to realize the qSG model are accessible with modern-day superconducting circuit technology, thus providing additional credence towards the viability of the latter platform for simulating strongly interacting quantum field theories.

Rearrangement term in the folding model of the nucleon optical potential

Based on the mean-field determination of the single-particle energy in nuclear matter that contains naturally a rearrangement term (RT) implied by the Hugenholtz–van Hove theorem, the folding model of the nucleon optical potential (OP) is extended to take into account the RT using the effective, density dependent CDM3Yn interaction. With the exchange part of the nucleon folded OP treated exactly in the Hartree–Fock manner, a compact nonlocal version of the folding model is suggested in the present work to determine explicitly the isospin-dependent, nonlocal central term of the nucleon OP. To solve the optical model (OM) equation with a complex nonlocal OP, the calculable R-matrix method is used to analyse the elastic neutron and proton scattering on 40,48Ca, 90Zr, and 208Pb targets at low energies. The inclusion of the RT into the folding model calculation of the nonlocal nucleon OP was shown to be essential for the overall good OM description of elastic nucleon scattering. To validate the nonlocal version of the folding model, the OM results given by the nonlocal folded nucleon OP are also compared with those given by the global parametrization of the nonlocal OP using the analytical nonlocal form factor suggested by Perey and Buck.

Complete form factors in Yang-Mills from unitarity and spinor helicity in six dimensions

We present a systematic procedure to compute complete, analytic form factors of gauge-invariant operators at loop level in pure Yang-Mills. We consider applications to operators of the form $\mathrm{Tr}\, F^n$ where $F$ is the gluon field strength. Our approach is based on an extension to form factors of the dimensional reconstruction technique, in conjunction with the six-dimensional spinor-helicity formalism and generalised unitarity. For form factors this technique requires the introduction of additional scalar operators, for which we provide a systematic prescription. We also discuss a generalisation of dimensional reconstruction to any number of loops, both for amplitudes and form factors. Several novel results for one-loop minimal and non-minimal form factors of $\mathrm{Tr}\, F^n$ with $n>2$ are presented. Finally, we describe the \texttt{Mathematica} package \texttt{SpinorHelicity6D}, which is tailored to handle six-dimensional quantities written in the spinor-helicity formalism.

Analytic pion form factor

The pion electromagnetic form factor and two-pion production in electron-positron collisions are simultaneously fitted by a vector dominance model evolving to perturbative QCD at large momentum transfer. This model was previously successful in simultaneously fitting the nucleon electromagnetic form factors (spacelike region) and the electromagnetic production of nucleon-antinucleon pairs (timelike region). For this pion case dispersion relations are used to produce the analytic connection of the spacelike and timelike regions. The fit to all the data is good, especially for the newer sets of time-like data. The description of high-$q^2$ data, in the time-like region, requires one more meson with $\rho$ quantum numbers than listed in the 2014 Particle Data Group review.

Scattering from phase-separated vesicles. I. An analytical form factor for multiple static domains

This is the first in a series of papers considering elastic scattering from laterally heterogeneous lipid vesicles containing multiple domains. Unique among biophysical tools, small-angle neutron scattering can in principle give detailed information about the size, shape and spatial arrangement of domains. A general theory for scattering from laterally heterogeneous vesicles is presented, and the analytical form factor for static domains with arbitrary spatial configuration is derived, including a simplification for uniformly sized round domains. The validity of the model, including series truncation effects, is assessed by comparison with simulated data obtained from a Monte Carlo method. Several aspects of the analytical solution for scattering intensity are discussed in the context of small-angle neutron scattering data, including the effect of varying domain size and number, as well as solvent contrast. The analysis indicates that effects of domain formation are most pronounced when the vesicle's average scattering length density matches that of the surrounding solvent.

Scattering from Laterally Heterogeneous Vesicles: An Analytical form Factor for Multiple Domains

It is widely accepted that lateral heterogeneity in the plasma membrane (PM) plays a role in signaling, transport, and the pathogenesis of some viruses and bacteria. In many cases, the size and connectivity of “raft” domains is crucial for understanding their genesis and mode of action, yet detailed knowledge is still lacking. For example, the observation of ∼ 10 nm diameter rafts in the PM of unstimulated cells, as well as in multicomponent lipid-only model membranes, has led to several theories explaining nanodomain stability, including microemulsions, critical fluctuations, the presence of line active molecules, or competing interactions (e.g., line tension and bending energy). Unique among biophysical tools, small-angle neutron scattering can in principle give detailed information about the size, shape and spatial arrangement of domains. We present a general theory for scattering from laterally heterogeneous vesicles, including form factors for vesicles containing an arbitrary number of domains. These form factors are then applied to the analysis of neutron scattering data from phase-separated vesicles, with an emphasis on distinguishing the size and spatial arrangement of domains. We discuss important experimental considerations, including the use of contrast matching to highlight different structural features.

Modeling of Fibrin Gels Based on Confocal Microscopy and Light-Scattering Data

Fibrin gels are biological networks that play a fundamental role in blood coagulation and other patho/physiological processes, such as thrombosis and cancer. Electron and confocal microscopies show a collection of fibers that are relatively monodisperse in diameter, not uniformly distributed, and connected at nodal points with a branching order of ∼3–4. Although in the confocal images the hydrated fibers appear to be quite straight (mass fractal dimension Dm = 1), for the overall system 1<Dm<2. Based on the confocal images, we developed a method to generate three-dimensional (3D) in silico gels made of cylindrical sticks of diameter d, density ρ, and average length 〈L〉, joined at randomly distributed nodal points. The resulting 3D network strikingly resembles real fibrin gels and can be sketched as an assembly of densely packed fractal blobs, i.e., regions of size ξ, where the fiber concentration is higher than average. The blobs are placed at a distance ξ0 between their centers of mass so that they are overlapped by a factor η = ξ/ξ0 and have Dm ∼1.2–1.6. The in silico gels’ structure is quantitatively analyzed by its 3D spatial correlation function g3D(r) and corresponding power spectrum I(q) = FFT3D[g3D(r)], from which ρ, d, Dm, η, and ξ0 can be extracted. In particular, ξ0 provides an excellent estimate of the gel mesh size. The in silico gels’ I(q) compares quite well with real gels’ elastic light-scattering measurements. We then derived an analytical form factor for accurately fitting the scattering data, which allowed us to directly recover the gels’ structural parameters.

Analytic two-loop form factors in $ \mathcal{N} = 4 $ SYM

We derive a compact expression for the three-point MHV form factors of half-BPS operators in N=4 super Yang-Mills at two loops. The main tools of our calculation are generalised unitarity applied at the form factor level, and the compact expressions for supersymmetric tree-level form factors and amplitudes entering the cuts. We confirm that infrared divergences exponentiate as expected, and that collinear factorisation is entirely captured by an ABDK/BDS ansatz. Next, we construct the two-loop remainder function obtained by subtracting this ansatz from the full two-loop form factor and compute it numerically. Using symbology, combined with various physical constraints and symmetries, we find a unique solution for its symbol. With this input we construct a remarkably compact analytic expression for the remainder function, which contains only classical polylogarithms, and compare it to our numerical results. Furthermore, we make the surprising observation that our remainder is equal to the maximally transcendental piece of a closely related two-loop amplitude in QCD.

Universal Analytical Scattering Form Factor for Shell–, Core–Shell, or Homogeneous Particles with Continuously Variable Density Profile Shape

A novel analytical and continuous density distribution function with a widely variable shape is reported and used to derive an analytical scattering form factor that allows us to universally describe the scattering from particles with the radial density profile of homogeneous spheres, shells, or core-shell particles. Composed by the sum of two Fermi-Dirac distribution functions, the shape of the density profile can be altered continuously from step-like via Gaussian-like or parabolic to asymptotically hyperbolic by varying a single "shape parameter", d. Using this density profile, the scattering form factor can be calculated numerically. An analytical form factor can be derived using an approximate expression for the original Fermi-Dirac distribution function. This approximation is accurate for sufficiently small rescaled shape parameters, d/R (R being the particle radius), up to values of d/R ≈ 0.1, and thus captures step-like, Gaussian-like, and parabolic as well as asymptotically hyperbolic profile shapes. It is expected that this form factor is particularly useful in a model-dependent analysis of small-angle scattering data since the applied continuous and analytical function for the particle density profile can be compared directly with the density profile extracted from the data by model-free approaches like the generalized inverse Fourier transform method.

Small-angle scattering from phospholipid nanodiscs: derivation and refinement of a molecular constrained analytical model form factor

Nanodiscs™ consist of small phospholipid bilayer discs surrounded and stabilized by amphiphilic protein belts. Nanodiscs and their confinement and stabilization of nanometer sized pieces of phospholipid bilayer are highly interesting from a membrane physics point of view. We demonstrate how the detailed structure of Di-Lauroyl-Phosphatidyl Choline (DLPC) nanodiscs may be determined by simultaneous fitting of a structural model to small-angle scattering data from the nanodiscs as investigated in three different contrast situations, respectively two SANS contrasts and one SAXS contrast. The article gives a detailed account of the underlying structural model for the nanodiscs and describe how additional chemical and biophysical information can be incorporated in the model in terms of molecular constraints. We discuss and quantify the contribution from the different elements of the structural model and provide very strong experimental support for the nanodiscs as having an elliptical cross-section and with poly-histidine tags protruding out from the rim of the protein belt. The analysis also provides unprecedented information about the structural conformation of the phospholipids when these are localized in the nanodiscs. The model paves the first part of the way in order to reach our long term goal of using the nanodiscs as a platform for small-angle scattering based structural investigations of membrane proteins in solution.

Pore Structure and Fluid Sorption in Ordered Mesoporous Silica. I. Experimental Study by in situ Small-Angle X-ray Scattering

Ordered and disordered pores in SBA-15 silica and the gradual filling of these pores by an adsorbed fluid (dibromomethane, CH2Br2) are investigated by in situ small-angle X-ray scattering. Adsorption into the microporous corona and film formation at the corrugated surface of the ordered cylindrical pores is described by two different geometrical models. The analytical form factor resulting from these models is used to fit the integrated intensities of up to 10 Bragg diffraction peaks from the mesopore lattice. Model fits for the evacuated sample yield the porosity caused by the ordered pores. From these results and the total porosity obtained by nitrogen sorption, we determine the contribution of the disordered porosity, which is nearly 20% for the present sample. The model fits also provide new insight into the adsorbate structure in the ordered pores at different stages of pore filling, while the analysis of diffuse scattering provides information about fluid adsorption in the disordered pores in the wa...

Model-independent form factors for spin-independent neutralino–nucleon scattering fromelastic electron scattering data

Theoretical calculations of neutralino–nucleon interaction rates with various nuclei areof great interest to direct dark matter searches such as CDMS, EDELWEISS,ZEPLIN and other experiments since they are used to establish upper bounds on theWIMP–proton cross section. These interaction rates and cross sections are generallycomputed with standard, one- or two-parameter model-dependent nuclear form factors,which may not exactly mirror the actual form factor for the particular nucleus inquestion. As is well known, elastic electron scattering can allow for very precisedeterminations of nuclear form factors and hence nuclear charge densities for spherical ornear-spherical nuclei. We use charge densities derived from elastic electron scattering datato calculate model-independent, analytic form factors for various target nucleiimportant in dark matter searches, such as Si, Ge, S, Ca and others. We havefound that for nuclear recoils in the range of 1–100 keV significant differences incross sections and rates exist when the model-independent form factors are used:at 30 keV nuclear recoil the form factors squared differ by a factor of 1.06 for28Si, 1.11 for40Ca, 1.27for 70Ge and1.92 for 129Xe. We show the effect of different form factors on the upper limiton the WIMP–proton cross section obtained with a hypothetical70Ge detector during a 50 kg-d effective exposure. Helm form factors with various parameterchoices differ at most by 10–20% from the best (Fourier–Bessel) form factor, and canapproach it to better than 1% if the parameters are chosen to mimic the actual nucleardensity.

Reverse Monte Carlo analysis of small-angle scattering data on colloids and nanoparticles

Applications of Reverse Monte Carlo-type simulations are summarized for understanding the inter-particle structure of colloids and nanoparticles. Results are shown for simple spherical polymer latex, for micelles and for a catalyst containing nanoparticles. Different methodological extensions of the traditional Reverse Monte Carlo simulations are developed in order to get reliable interpretation of small-angle scattering data and they are compared: the traditional Reverse Monte Carlo technique, a method for polydisperse spheres, a combination of direct minimization and Reverse Monte Carlo procedures, and a method for particles with arbitrary shape and without analytical particle form factor.

Method of separated form factors for polydisperse vesicles

Use of the Schulz or Gamma distribution in the description of particle sizes facilitates calculation of analytic polydisperse form factors using Laplace transforms, {\cal L}[f(u)]. Here, the Laplace transform approach is combined with the separated form factor (SFF) approximation [Kiselevet al.(2002).Appl. Phys. A,74, S1654–S1656] to obtain expressions for form factors,P(q), for polydisperse spherical vesicles with various forms of membrane scattering length density (SLD) profile. The SFF approximation is tested against exact form factors that have been numerically integrated over the size distribution, and is shown to represent the vesicle form factor accurately for typical vesicle sizes and membrane thicknesses. Finally, various model SLD profiles are used with the SFF approximation to fit experimental small-angle neutron scattering (SANS) curves from extruded unilamellar vesicles.

Semileptonic decays of heavy Λ baryons in a quark model

The semileptonic decays of ${\ensuremath{\Lambda}}_{c}$ and ${\ensuremath{\Lambda}}_{b}$ are treated in the framework of a constituent quark model. Both nonrelativistic and semirelativistic Hamiltonians are used to obtain the baryon wave functions from a fit to the spectra, and the wave functions are expanded in both the harmonic-oscillator and the Sturmian bases. The latter basis leads to form factors in which the kinematic dependence on ${q}^{2}$ is in the form of multipoles, and the resulting form factors fall faster as a function of ${q}^{2}$ in the available kinematic ranges. As a result, decay rates obtained in the two models with the Sturmian basis are significantly smaller than those obtained with the harmonic-oscillator basis. In the case of the ${\ensuremath{\Lambda}}_{c}$, decay rates calculated with the Sturmian basis are closer to the experimentally reported rates. However, we find a semileptonic branching fraction for the ${\ensuremath{\Lambda}}_{c}$ to decay to excited ${\ensuremath{\Lambda}}^{*}$ states of 11 to 19%, in contradiction to what is assumed in available experimental analyses. Our prediction for the ${\ensuremath{\Lambda}}_{b}$ semileptonic decays is that decays to the ground state ${\ensuremath{\Lambda}}_{c}$ provide a little less than 70% of the total semileptonic decay rate. For the decays ${\ensuremath{\Lambda}}_{b}\ensuremath{\rightarrow}{\ensuremath{\Lambda}}_{c}$, the analytic form factors we obtain satisfy the relations expected from heavy-quark effective theory at the nonrecoil point, at leading and next-to-leading orders in the heavy-quark expansion. In addition, some features of the heavy-quark limit are shown to naturally persist as the mass of the heavy quark in the daughter baryon is decreased.

Evaluation of small-angle x-ray scattering data of a Raney-type Ni catalyst with computer simulation

A reverse Monte Carlo-type simulation method was developed for the evaluation of anomalous small-angle x-ray scattering data of a Raney-type Ni catalyst. Based on other experimental data the catalytic Ni particles were modeled as small crystalline cylinders dispersed in the matrix. The average size of the Ni particles and their pair-correlation function were determined. Despite the unknown density of the catalyst, it is shown that each particle has about 2 neighbors in the first neighboring shell independent of the modeling density, and the position of the first peak of the pair-correlation function does not depend on the modeling density. A method was elaborated to get reasonable performance of the Reverse Monte Carlo-type simulation. The scattered intensity was calculated on the basis of probe scattering atoms put inside the cylinders. The effects of the omission of the real number of the atoms, the unknown density, the lack of normalization and the uncertainties in the cross sections were unified in two constants that were determined during the simulation. The method can be used for nanoparticles with other shape, where analytic form factors are complicated, and it may be powerful in the investigation of the usually neglected or simplified inter-particle structure of these systems.

Use of Bersons-Kulsh Form Factor of Short-Pulse Approximation in Rydberg-State Physics

The experiments on high Rydberg states interacting with short electromagnetic pulses were hitherto mainly explained by using numerical integration of the time-dependent Schrodinger equation in a restricted state basis. In this study we apply a different approach based on the Bersons-Kulsh analytical form factor of the short-pulse approximation. This analytical approach is shown to well reproduce the recent experimental results and those of numerical integration of the time-dependent Schrodinger equation both in the case of terahertz half-cycle pulses and optical many-cycle pulses. This fact enables a recommendation of the analytical Bersons-Kulsh form factor as an alternative and efficient method of quantum calculations of electromagnetically induced Rydberg state redistribution.

Analytical treatment of Rydberg states interacting with terahertz half-cycle pulses

The Bersons-Kulsh analytical form factor of the short-pulse approximation along with its transformation from the parabolic to spherical bases are used as a tool alternative to numerical solution of the time-dependent Schr\"odinger equation to explain the recently measured [Wetzels et al., Eur. Phys. J. D 14, 157 (2001)] dynamics of the Rydberg wave packets generated from the initial $n=40,$ 41 rubidium states by the impact of half-cycle electrical pulses of a frequency spectrum from the terahertz regime. When a comparison was possible, both theoretical approaches were found to give the same results and offer the same level of explanation of the measured dynamics but the analytical approach had the advantage of being simpler in use. In addition, this analytical approach was applied to an experimental situation that was not given a recent theoretical attention and results of qualitative agreement with experimental findings were obtained.