Quadrupole Form Factors Research Articles

Hamiltonian sensitivity in calculating the electromagnetic moments and electroexcitation form factor for the sd nuclei using shell model with Skyrme–Hartree–Fock

In this work, the nuclear electromagnetic moments for the ground and low-lying excited states for sd shell nuclei have been calculated, resulting in a revised database with 56 magnetic dipole moments and 41 electric quadrupole moments. The shell model calculations are performed for each sd isotope chain, considering the sensitivity of changing the sd two-body effective interactions USDA, USDE, CWH and HBMUSD in the calculation of the one-body transition density matrix elements. The calculations incorporate the single-particle wave functions of the Skyrme interaction to generate a one-body potential in Hartree–Fock theory to calculate the single-particle matrix elements. For most sd shell nuclei, the experimental data are well reproduced, except for those spans near the island of inversion. In order to interpret the structure of low-lying excited states, the electric quadrupole and magnetic dipole transition form factors and the corresponding reduced transition probabilities in the sd shell nuclei have also been calculated, for which the experimental data are available. The present results demonstrate the nuclear electromagnetic moments’ sensitivity to many forms of the understanding of nucleon–nucleon interactions and provide a crucial baseline for future improvements in conceptual calculations.

Deuteron electromagnetic form factors and tensor polarization observables in the framework of the hard-wall AdS/QCD model

We study the electromagnetic form factors and tensor polarization observables of the deuteron in the framework of the hard-wall AdS/QCD model. We find a profile function for the bulk twist vector field, which describes the deuteron on the boundary and fix the infrared boundary cut-off of AdS space in accordance with the ground state mass of the deuteron. We obtain the deuteron charge monopole, quadrupole, and magnetic dipole form factors and tensor polarization observables from the bulk Lagrangians for the deuteron and photon field interactions. We plot the momentum transfer dependence of the form factors and tensor polarization observables and compare our numerical results with those in the soft-wall model and experimental data.

Perturbative calculations of deuteron form factors

We calculate the deuteron charge, electric quadrupole, and magnetic form factors up to next-to-next-to-leading order in chiral effective field theory, treating subleading corrections, especially that of chiral nuclear forces, in perturbation theory. We examine the power counting based on naive dimensional analysis by investigating the ultraviolet cutoff variation of these form factors. We find that the N${}^2$LO magnetic form factor shows significant cutoff dependence, suggesting the contact current operator responsible be enhanced. After promotion to N${}^2$LO, it indeed renormalizes the magnetic form factor. This is in agreement with a previous work based on renormalization-group analysis. For the charge and quadrupole moments, perturbative calculation allows us to study how they scale with multiple low-energy parameters such as the pion mass, deuteron binding momentum, and momentum transfer.

Electromagnetic transitions of the singly charmed baryons with spin 3/2

We investigate the electromagnetic transitions of the singly charmed baryons with spin 3/2, based on a pion mean-field approach, also known as the chiral quark-soliton model, taking into account the rotational $1/N_c$ corrections and the effects of flavor SU(3) symmetry breaking. We examine the valence- and sea-quark contributions to the electromagnetic transition form factors and find that the quadrupole form factors of the sea-quark contributions dominate over those of the valence-quark ones in the smaller $Q^2$ region, whereas the sea quarks only provide marginal contributions to the magnetic dipole transition form factors of the baryon sextet with spin 3/2. The effects of the flavor SU(3) symmetry breaking are in general very small except for the forbidden transition $\Xi_c^0\gamma\to \Xi_c^{*0}$ by $U$-spin symmetry. We also discuss the widths of the radiative decays for the baryon sextet with spin 3/2, comparing the present results with those from other works.

Electric quadrupole form factors of singly heavy baryons with spin 3/2

Abstract We study the electromagnetic form factors of the lowest-lying singly heavy baryons in a pion mean-field approach, which is also known as the SU(3) chiral quark-soliton model. In the limit of the heavy-quark mass, the dynamics inside a singly heavy baryon is governed by the $N_c-1$ valence quarks, while the heavy quark remains as a static one. In this framework, a singly heavy baryon is described by combining the $N_c-1$ soliton with the singly heavy quark. In the infinitely heavy-quark mass limit, we can compute the electric quadrupole form factors of the baryon sextet with spin 3/2, with the rotational $1/N_c$ and linear corrections of the explicit flavor SU(3) symmetry breaking taken into account. We find that the sea-quark contributions or the Dirac-sea level contributions dominate over the valence-quark contributions in the lower $Q^2$ region. We examined the effects of explicit flavor SU(3) symmetry breaking in detail. The numerical results are also compared with the recent data from the lattice calculation with the unphysical value of the pion mass considered, which was used in the lattice calculation.

Calculation magnetic dipole moments, electric quadrupole moments and form factors for some Ti isotopes

This paper presents calculations of the magnetic moments, the quadrupole moments, and the form factors of a number of titanium isotopes as determined using the shell model space. The calculations of the shell model have adopted the model space (MS) with core-polarization effects, which were included for two types of effective g factors and different effective charges. The wave functions are generated using the harmonic oscillator potential (HO) with size parameters b, which for these isotopes are calculated. The one body density matrices (OBDM) for these isotopes were obtained using the NuShellX@MSU code, which used an adopted fpd6 interaction and fp model space. Reasonable results are obtained that show the collapse of the magicity property at neutron number N = 28.

Investigation of the Quadrupole Moment and Form Factors of Some Ca Isotopes

Nuclear shell model is adopted to calculate the electric quadrupole moments for some Calcium isotopes 20Ca (N = 21, 23, 25, and 27) in the fp shell. The wave function is generated using a two body effective interaction fpd6 and fp space model. The one body density matrix elements (OBDM) are calculated for these isotopes using the NuShellX@MSU code. The effect of the core-polarizations was taken through the theory microscopic by taking the set of the effective charges. The results for the quadrupole moments by using Bohr-Mottelson (B-M) effective charges are the best. The behavior of the form factors of some Calcium isotopes was studied by using Bohr-Mottelson (B-M) effective charges.

Transition form factors: γ*+p→Δ(1232) , Δ(1600)

Electroproduction form factors describing the ${\ensuremath{\gamma}}^{*}p\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Delta}}}^{+}(1232)$, ${\mathrm{\ensuremath{\Delta}}}^{+}(1600)$ transitions are computed using a fully dynamical diquark-quark approximation to the Poincar\'e-covariant three-body bound-state problem in relativistic quantum field theory. In this approach, the $\mathrm{\ensuremath{\Delta}}(1600)$ is an analogue of the Roper resonance in the nucleon sector, appearing as the simplest radial excitation of the $\mathrm{\ensuremath{\Delta}}(1232)$. Precise measurements of the ${\ensuremath{\gamma}}^{*}p\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Delta}}}^{+}(1232)$ transition already exist on $0\ensuremath{\le}{Q}^{2}\ensuremath{\lesssim}8\text{ }\text{ }{\mathrm{GeV}}^{2}$, and the calculated results compare favorably with the data outside the meson-cloud domain. The predictions for the ${\ensuremath{\gamma}}^{*}p\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Delta}}}^{+}(1600)$ magnetic dipole and electric quadrupole transition form factors are consistent with the empirical values at the real photon point, and extend to ${Q}^{2}\ensuremath{\approx}6{m}_{p}^{2}$, enabling a meaningful direct comparison with experiment once analysis of existing data is completed. In both cases, the electric quadrupole form factor is particularly sensitive to deformation of the $\mathrm{\ensuremath{\Delta}}$-baryons. Interestingly, while the ${\ensuremath{\gamma}}^{*}p\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Delta}}}^{+}(1232)$ transition form factors are larger in magnitude than those for ${\ensuremath{\gamma}}^{*}p\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Delta}}}^{+}(1600)$ in some neighborhood of the real photon point, this ordering is reversed on ${Q}^{2}\ensuremath{\gtrsim}2{m}_{p}^{2}$, suggesting that the ${\ensuremath{\gamma}}^{*}p\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Delta}}}^{+}(1600)$ transition is more localized in configuration space.

Combined parametrization of GEn and $\gamma^{\ast} N \rightarrow \Delta (1232)$γ*N→Δ(1232) quadrupole form factors

Models based on $SU(6)$ symmetry breaking and large $N_c$ limit provide relations between the pion cloud contributions to the $\gamma^\ast N \to \Delta(1232)$ quadrupole form factors, electric ($G_E$) and Coulomb ($G_C$), and the neutron electric form factor $G_{En}$, suggesting that those form factors are dominated by the same physical processes. Those relations are improved in order to satisfy a fundamental constraint between the electric and Coulomb quadrupole form factors in the long wavelength limit, when the photon three-momentum vanishes (Siegert's theorem). Inspired by those relations, we study alternative parametrizations for the neutron electric form factor. The parameters of the new form are then determined by a combined fit to the $G_{En}$ and the $\gamma^\ast N \to \Delta(1232)$ quadrupole form factor data. We obtain a very good description of the $G_E$ and $G_C$ data when we combine the pion cloud contributions with small valence quark contributions to the $\gamma^\ast N \to \Delta(1232)$ quadrupole form factors. The best description of the data is obtained when the second momentum of $G_{En}$ is $r_n^4 \simeq -0.4$ fm$^4$. We conclude that the square radii associated with $G_E$ and $G_C$, $r_E^2$ and $r_C^2$, respectively, are large, revealing the long extension of the pion cloud. We conclude also that those square radii are related by $r_E^2 - r_C^2 = 0.6 \pm 0.2$ fm$^2$. The last result is mainly the consequence of the pion cloud effects and Siegert's theorem.

Calculations of the deuteron form factors

On the received coefficients of the analytical forms for deuteron wave function in coordinate representation in form r^(l+1)*exp(-A*r) for the modern realistic nucleon-nucleon potentials NijmI, NijmII, Nijm93, Reid93 and Argonne v18 are calculated charge GC(p), quadrupole GQ(p) and magnetic GM(p) deuteron form factors. An original dipole fit for proton and neutron isoscalar electric and magnetic form factors is used for calculations. Theoretical calculations of values of the deuteron form factor are compared with their experimental data of world collaborations (Bates, BLAST, Bonn, JLab, Mainz, Naval Research Lab, NIKHEF, Orsay, Saclay, SLAC, Stanford, VEPP3 and VEPP4) and reviews (Abbott, Boden, Garcon and Karpius). The change of the sign of form factors is in the pulse regions at 4.7-4.9 fm-1 for GC; at 12.8-14.7 fm-1 for GQ; at 6.3-8.1 and 11.4-12.2 fm-1 for GM. The theoretical values for the form factors GQ(0) and GM(0) were calculated at the boundary condition for the momentum at p2=0. Calculated positions of the zero for deuteron form factors are compared with values for other potential models. The peculiarities of parametrizations of deuteron form factors for theoretical approaches and fits of theoretical calculations to experimental values, which are described in the cited literature, are analyzed. Formulas for spherical and quadrupole form factors are writed, which are expressed in terms of the coefficients for the indicated deuteron wave function. At large values of momentum the asymptotics of the deuteron form factors are determined by the coefficients of the analytical forms for deuteron wave function, nucleon isoscalar form factors and the order of momentum in the denominator. The further use of deuteron form factors for obtaining polarization observables in processes involving deuteron as a projectile is discussed.

$$N^*$$ N ∗ form Factors Based on a Covariant Quark Model

We discuss the results of the covariant spectator quark model for the several $\gamma^\ast N \to N^\ast$ transitions, where $N$ is the nucleon and $N^\ast$ a nucleon excitation. More specifically, we present predictions for the form factors and transition amplitudes associated with the resonances $N(1440)1/2^+$, $N(1535)1/2^-$, $N(1520)3/2^-$, $\Delta(1620)1/2^-$ and $N(1650)1/2^-$. The estimates based on valence quark degrees of freedom are compared with the available data, particularly with the recent Jefferson Lab data at low and large momentum transfer ($Q^2$). In general, the estimates are in good agreement with the empirical data for $Q^2 >2$ GeV$^2$, with a few exceptions. The results are discussed in terms of the role of the valence quarks and the meson cloud excitations for the different resonances $N^\ast$. We also review our results for the resonance $\Delta(1232)3/2^+$ and discuss the relevance of the pion cloud component for the magnetic dipole form factor and the electric and Coulomb quadrupole form factors.

New low-Q2 measurements of the $\gamma^{\ast}N\rightarrow\Delta (1232)$ γ * N → Δ ( 1232 ) Coulomb quadrupole form factor, pion cloud parametrizations and Siegert’s theorem

The novel measurements of the $\gamma^\ast N \to \Delta(1232)$ Coulomb quadrupole form factor in the range $Q^2=0.04$--0.13 GeV$^2$ changed the trend of the previous data. With the new data the electric and Coulomb form factors are both in remarkable agreement with estimates of the pion cloud contributions to the quadrupole form factors at low $Q^2$. The pion cloud contributions to the electric and Coulomb form factors can be parametrized by the relations $G_E \propto G_{En}/ \left(1 + \frac{Q^2}{2M_\Delta(M_\Delta-M)}\right)$ and $G_C \propto G_{En}$, where $G_{En}$ is the neutron electric form factor, and $M$, $M_\Delta$ are the nucleon and $\Delta$ masses, respectively. Those parametrizations are in full agreement with Siegert's theorem, which states that $G_E= \frac{M_\Delta-M}{2M_\Delta} G_C$ at the pseudothreshold, when $Q^2=-(M_\Delta -M)^2$, and improve previous parametrizations. Also a small valence quark component estimated by a covariant quark model contributes to this agreement. The combination of the new data with the new parametrization for $G_E$ concludes an intense period of studying the $\gamma^\ast N \to \Delta(1232)$ quadrupole form factors at low $Q^2$, with the agreement between theory and data.

A method for measuring D* electromagnetic form factors in e+ e− annihilation**Supported by National Natural Science Foundation of China (11675166)

We describe a method for measuring the electromagnetic form factors of the D* meson at time-like momentum transfer in e+ e− annihilation. This is to study the joint angular distribution of the e+e−→γ*→D*+D*−, D*+→D0π+, and D*−→D̅0 π− processes. The magnitudes and relative phases of the charge, magnetic and quadrupole form factors can be determined. The method can also be applied to other vector particles.

Investigation of the Nuclear Structure for Some p-Shell Nuclei by Harmonic Oscillator and Woods-Saxon Potentials

The single particle radial wave functions of harmonic oscillator potential and Woods-Saxon potential have been used to calculate the charge density distributions, charge form factor and the matter and charge root mean square (rms) radii for some odd-A 1p shell nuclei, namely 9Be, 11B, 13C and 15N nuclei. The calculated results are compared with the measured data. Calculations show that both the Wood-Saxson (WS) and harmonic oscillator (HO) potentials are capable of providing theoretical predictions on the structure of p-shell nuclei and be in a satisfactory description with those of experimental data. The quadrupole form factors contribution for 9Be and 11B nuclei are carried out using undeformed p-shell model and give a well accordance with the measured results.

Parametrizations of the γ*N→Δ(1232) quadrupole form factors and Siegert’s theorem

The large $N_c$ limit provides relations that can be used to calculate the $\gamma^\ast N \to \Delta(1232)$ quadrupole form factors at low and intermediate $Q^2$ under the assumption of the pion cloud dominance. There are two limitations in those parametrizations. First, the parametrization of the Coulomb quadrupole form factor underestimate the low $Q^2$ data. Second, when extrapolated for the timelike region, the form factors violate Siegert's theorem by terms of the order $1/N_c^2$. We propose here corrections to the parametrization of the electric quadrupole form factor, which violate Siegert's theorem only by terms of the order $1/N_c^4$. Combining the improved large $N_c$ pion cloud parametrizations with the valence quark contributions based on a covariant quark model for the quadrupole transition form factors, we obtain an extrapolation to the timelike region consistent with Siegert's theorem, and accomplish also a very good description of the data.

Look inside charmed-strange baryons from lattice QCD

The electromagnetic form factors of the spin-3/2 $\Omega$ baryons, namely $\Omega$, $\Omega_c^\ast$, $\Omega_{cc}^\ast$ and $\Omega_{ccc}$, are calculated in full QCD on $32^3\times 64$ PACS-CS lattices with a pion mass of 156(9) MeV. The electric charge radii and magnetic moments from the $E0$ and $M1$ multipole form factors are extracted. Results for the electric quadrupole form factors, $E2$, are also given. Quark sector contributions are computed individually for each observable and then combined to obtain the baryon properties. We find that the charm quark contributions are systematically smaller than the strange-quark contributions in the case of the charge radii and magnetic moments. $E2$ moments of the $\Omega_{cc}^\ast$ and $\Omega_{ccc}$ provide a statistically significant data to conclude that their electric charge distributions are deformed to an oblate shape. Properties of the spin-1/2 $\Omega_c$ and $\Omega_{cc}$ baryons are also computed and a thorough comparison is given. This complete study gives valuable hints about the heavy-quark dynamics in charmed hadrons.

Deuteron form factors in a phenomenological approach**Supported by NSFC (11475192, 10975146, 11035006, 10775148)

The electromagnetic form factors of the deuteron, particularly its quadrupole form factor, are studied with the help of a phenomenological Lagrangian approach where the vertex of the deuteron–proton–neutron with D-state contribution is explicitly taken into account. The results show the importance of this contribution to the deuteron quadrupole form factor in the approach.

Nucleon momentum distributions and elastic electron scattering from 19F, 25Mg, 27Al, and 29Si nuclei

The nucleon momentum distributions and elastic electron scattering form factors of the ground state for some odd 2s–1d shell nuclei, such as 19F, 25Mg, 27Al, and 29Si, have been investigated using the coherent density fluctuation model and expressed in terms of the fluctuation function (weight function) |f(x)|2. The fluctuation function has been related to the nucleon density distribution of the nuclei and determined from the theory. The property of the long-tail manner at high-momentum region of the nucleon momentum distribution has been obtained by theoretical fluctuation function. The calculated form factors F(q) of all nuclei under study are in very good agreement with those of experimental data throughout all values of momentum transfer q. It is concluded that the contributions of the quadrupole form factor F C2(q) in 25Mg and 27Al nuclei, which are characterized by the undeformed 2s–1d shell model, are necessary for getting a remarkable agreement between the theoretical and experimental form factors.

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We study the electromagnetic Ωcγ→Ωc⁎ transition in 2+1 flavor lattice QCD, which gives access to the dominant decay mode of Ωc⁎ baryon. The magnetic dipole and the electric quadrupole transition form factors are computed. The magnetic dipole form factor is found to be mainly determined by the strange quark and the electric quadrupole form factor to be negligibly small, in consistency with the quark model. We also evaluate the helicity amplitudes and the decay rate.

Nucleon and Δ electromagnetic form factors

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