Pion-nucleon Form Factor Research Articles

Nucleon axial-vector and pseudoscalar form factors and PCAC relations

We use a continuum quark+diquark approach to the nucleon bound-state problem in relativistic quantum field theory to deliver parameter-free predictions for the nucleon axial and induced pseudoscalar form factors, $G_A$ and $G_P$, and unify them with the pseudoscalar form factor $G_5$ or, equivalently, the pion-nucleon form factor $G_{\pi NN}$. We explain how partial conservation of the axial-vector current and the associated Goldberger-Treiman relation are satisfied once all necessary couplings of the external current to the building blocks of the nucleon are constructed consistently; in particular, we fully resolve the seagull couplings to the diquark-quark vertices associated with the axial-vector and pseudoscalar currents. Among the results we describe, the following are worth highlighting. A dipole form factor defined by an axial charge $g_A=G_A(0)=1.25(3)$ and a mass-scale $M_A = 1.23(3) m_N$, where $m_N$ is the nucleon mass, can accurately describe the pointwise behaviour of $G_A$. Concerning $G_P$, we obtain the pseudoscalar charge $g_p^\ast = 8.80(23)$, and find that the pion pole dominance approach delivers a reliable estimate of the directly computed result. Our computed value of the pion-nucleon coupling constant, $g_{\pi NN}/m_N =14.02(33)/{\rm GeV}$ is consistent with recent precision determinations.

Meson-Cloud Effects in the Electromagnetic Nucleon Structure

We study how the electromagnetic structure of the nucleon is influenced by a pion cloud. To this aim we make use of a constituent-quark model with instantaneous confinement and a pion that couples directly to the quarks. To derive the invariant 1- photon-exchange electron-nucleon scattering amplitude we employ a Poincar\'e- invariant coupled-channel formulation which is based on the point-form of relativistic quantum mechanics. We argue that the electromagnetic nucleon current extracted from this amplitude can be reexpressed in terms of pure hadronic degrees of freedom with the quark substructure of the pion and the nucleon being encoded in electromagnetic and strong vertex form factors. These are form factors of bare particles, i.e. eigenstates of the pure confinement problem. First numerical results for (bare) photon-nucleon and pion-nucleon form factors, which are the basic ingredients of the further calculation, are given for a simple 3-quark wave function of the nucleon.

The effective chiral Lagrangian from dimension-six parity and time-reversal violation

We classify the parity- and time-reversal-violating operators involving quark and gluon fields that have effective dimension six: the quark electric dipole moment, the quark and gluon chromo-electric dipole moments, and four four-quark operators. We construct the effective chiral Lagrangian with hadronic and electromagnetic interactions that originate from them, which serves as the basis for calculations of low-energy observables. The form of the effective interactions depends on the chiral properties of these operators. We develop a power-counting scheme and calculate within this scheme, as an example, the parity- and time-reversal-violating pion–nucleon form factor. We also discuss the electric dipole moments of the nucleon and light nuclei.

The effective chiral Lagrangian from the theta term

We construct the effective chiral Lagrangian involving hadronic and electromagnetic interactions originating from the QCD θ ¯ term. We impose vacuum alignment at both quark and hadronic levels, including field redefinitions to eliminate pion tadpoles. We show that leading time-reversal-violating (TV) hadronic interactions are related to isospin-violating interactions that can in principle be determined from charge-symmetry-breaking experiments. We discuss the complications that arise from TV electromagnetic interactions. Some implications of the expected sizes of various pion–nucleon TV interactions are presented, and the pion–nucleon form factor is used as an example.

Erratum: Axial nucleon and nucleon toΔform factors and the Goldberger-Treiman relations from lattice QCD [Phys. Rev. D76, 094511 (2007)

We evaluate the nucleon axial form factor, $G_A(q^2)$, and induced pseudoscalar form factor, $G_p(q^2)$, as well as the pion-nucleon form factor, $G_{\pi N N}(q^2)$, in lattice QCD. We also evaluate the corresponding nucleon to $\Delta$ transition form factors, $C_5^A(q^2)$ and $C_6^A(q^2)$, and the pion-nucleon-$\Delta$ form factor $G_{\pi N\Delta}(q^2)$. The nucleon form factors are evaluated in the quenched theory and with two degenerate flavors of dynamical Wilson fermions. The nucleon to $\Delta$ form factors, besides Wilson fermions, are evaluated using domain wall valence fermions with staggered sea quark configurations for pion masses as low as about 350 MeV. Using these form factors, together with an evaluation of the renormalized quark mass, we investigate the validity of the diagonal and non-diagonal Goldberger-Treiman relations. The ratios $G_{\pi N\Delta}(q^2)/G_{\pi NN}(q^2)$ and $2C_5^A(q^2)/G_A(q^2)$ are constant as a function of the momentum transfer squared and show almost no dependence on the quark mass. We confirm equality of these two ratios consistent with the Goldberger-Treiman relations extracting a mean value of $1.61(2)$.

Neutron charge density from simple pion cloud models

The physical nucleon is modeled as a bare nucleon surrounded by a pion cloud using a psuedoscalar pion-nucleon coupling to examine its implications for the neutron form factor, F{sub 1}(Q{sup 2}), and the corresponding transverse charge density, {rho}(b), in the infinite momentum frame. Two versions, one with a tunable Pauli-Villars parameter and another with light-front cloudy-bag pion-nucleon form factors, are examined. A qualitative agreement with the experimental form factor (F{sub 1}<0 for all Q{sup 2}) and transverse charge density is achieved when the nucleon is treated as having a finite extent. The bare nucleon must have a finite extent if this model is to account for a negative definite F{sub 1} and consequently a negative transverse charge density of the neutron at its center.

Axial nucleon and nucleon toΔform factors and the Goldberger-Treiman relations from lattice QCD

We evaluate the nucleon axial form factor, G{sub A}(q{sup 2}), and induced pseudoscalar form factor, G{sub p}(q{sup 2}), as well as the pion-nucleon form factor, G{sub {pi}}{sub NN}(q{sup 2}), in lattice QCD. We also evaluate the corresponding nucleon to {delta} transition form factors, C{sub 5}{sup A}(q{sup 2}) and C{sub 6}{sup A}(q{sup 2}), and the pion-nucleon-{delta} form factor G{sub {pi}}{sub N{delta}}(q{sup 2}). The nucleon form factors are evaluated in the quenched theory and with two degenerate flavors of dynamical Wilson fermions. The nucleon to {delta} form factors, besides Wilson fermions, are evaluated using domain wall valence fermions with staggered sea quark configurations for pion masses as low as about 350 MeV. Using these form factors, together with an evaluation of the renormalized quark mass, we investigate the validity of the diagonal and nondiagonal Goldberger-Treiman relations. The ratios G{sub {pi}}{sub N{delta}}(q{sup 2})/G{sub {pi}}{sub NN}(q{sup 2}) and 2C{sub 5}{sup A}(q{sup 2})/G{sub A}(q{sup 2}) are constant as a function of the momentum transfer squared and show almost no dependence on the quark mass. We confirm equality of these two ratios consistent with the Goldberger-Treiman relations extracting a mean value of 1.61(2)

Axial, induced pseudoscalar, and pion-nucleon form factors in manifestly Lorentz-invariant chiral perturbation theory

We calculate the nucleon form factors ${G}_{A}$ and ${G}_{P}$ of the isovector axial-vector current and the pion-nucleon form factor ${G}_{\ensuremath{\pi}N}$ in manifestly Lorentz-invariant baryon chiral perturbation theory up to and including order $\mathcal{O}({p}^{4})$. In addition to the standard treatment including the nucleon and pions, we also consider the axial-vector meson ${a}_{1}$ as an explicit degree of freedom. This is achieved by using the reformulated infrared renormalization scheme. We find that the inclusion of the axial-vector meson effectively results in one additional low-energy coupling constant that we determine by a fit to the data for ${G}_{A}$. The inclusion of the axial-vector meson results in an improved description of the experimental data for ${G}_{A}$, while the contribution to ${G}_{P}$ is small.

Scales in nuclear matter: Chiral dynamics with pion nucleon form factors⋆

A systematic calculation of nuclear matter is performed which includes the long-range correlations between nucleons arising from one- and two-pion exchange. Three-body effects from 2π exchange with excitations of virtual Δ(1232)-isobars are also taken into account in our diagrammatic calculation of the energy per particle ¯(k f). In order to eliminate possible high-momentum components from the interactions we introduce at each pion-baryon vertex a form factor of monopole type. The empirical nuclear matter saturation point, ρ0 ≃ 0.16fm^-3, ¯0 ≃ - 16MeV, is well reproduced with a monopole mass of Λ ≃ 4πf π ≃ 1.16GeV. As in the recent approach based on the universal low-momentum NN potential V low-k, the inclusion of three-body effects is crucial in order to achieve saturation of nuclear matter. We demonstrate that the dependence of the pion exchange contributions to ¯(k f) on the “resolution” scale Λ can be compensated over a wide range of Λ by counterterms with two “running” contact couplings. As a further application we study the in-medium chiral condensate 〈¯q〉(ρ) beyond the linear density approximation. For ρ ⩽ 1.5ρ0 we find small corrections from the derivative d¯(k f)/dm π, which are stable against variations of the monopole regulator mass Λ.

Axial form factor of the nucleon in the perturbative chiral quark model

We apply the perturbative chiral quark model (PCQM) at one loop to analyse the axial form factor of the nucleon. This chiral quark model is based on an effective Lagrangian, where baryons are described by relativistic valence quarks and a perturbative cloud of Goldstone bosons as dictated by chiral symmetry. We apply the formalism to obtain analytical expressions for the axial form factor of the nucleon, which is given in terms of fundamental parameters of low-energy pion–nucleon physics (weak pion decay constant, strong pion–nucleon form factor) and of only one model parameter (radius of the nucleonic three-quark core).

Electromagnetic form factors of the baryon octet in the perturbative chiral quark model

We apply the perturbative chiral quark model at one loop to analyze the electromagnetic form factors of the baryon octet. The analytic expressions for baryon form factors, which are given in terms of fundamental parameters of low-energy pion-nucleon physics (weak pion decay constant, axial nucleon coupling, strong pion-nucleon form factor), and the numerical results for baryon magnetic moments, charge and magnetic radii are presented. Our results are in good agreement with experimental data.

Nucleon mass and pion loops: Renormalization

Using Dyson-Schwinger equations, the nucleon propagator is analyzed nonperturbatively in a field-theoretical model for the pion-nucleon interaction. Infinities are circumvented by using pion-nucleon form factors which define the physical scale. It is shown that the correct, finite, on-shell nucleon renormalization is important for the value of the mass shift and the propagator. For physically acceptable forms of the pion-nucleon form factor the rainbow approximation together with renormalization is inconsistent. Going beyond the rainbow approximation, the full pion-nucleon vertex is modeled by its bare part plus a one-loop correction including an effective $\ensuremath{\Delta}.$ It is found that a consistent value for the nucleon mass shift can be obtained as a consequence of a subtle interplay between wave function and vertex renormalization. Furthermore, the bare and renormalized pion-nucleon coupling constants are approximately equal, consistent with results from the cloudy bag model.

Electromagnetic structure of the nucleon in the perturbative chiral quark model

We apply the perturbative chiral quark model (PCQM) at one loop to analyze the electromagnetic structure of nucleons. This model is based on an effective Lagrangian, where baryons are described by relativistic valence quarks and a perturbative cloud of Goldstone bosons. Including the electromagnetic interaction we first develop the formalism up to one loop in the Goldstone boson fluctuation relying on renormalization by use of counterterms. Local gauge invariance is satisfied both on the Lagrangian level and also for the relevant baryon matrix elements in the Breit frame. We apply the formalism to obtain analytical expressions for the nucleon charge and magnetic form factors, which are expressed in terms of fundamental parameters of low-energy pion-nucleon physics (weak pion decay constant, axial nucleon coupling, strong pion-nucleon form factor) and of only one model parameter (radius of the nucleonic three-quark core). A detailed numerical analysis of the nucleon magnetic moments, charge and magnetic radii, and also the momentum dependence of form factors is presented.

Mesonic cloud contribution to the nucleon andΔmasses

Pion-nucleon elastic scattering in the dominant ${P}_{33}$ channel is examined in the model in which the interaction is of the form $\ensuremath{\pi}+N\ensuremath{\leftrightarrow}N, \ensuremath{\Delta}(1232).$ New expressions are found for the elastic pion-nucleon scattering amplitude which differ from existing formula both in the kinematics and in the treatment of the renormalization of the nucleon mass and coupling constant. Fitting the model to the phase shifts in the ${P}_{33}$ channel does not uniquely fix the parameters of the model. The cutoff for the pion-nucleon form factor is found to lie in the range $\ensuremath{\beta}=750\ifmmode\pm\else\textpm\fi{}350 \mathrm{MeV}/c.$ The masses of the nucleon and the $\ensuremath{\Delta}$ which would arise if there were no coupling to mesons are found to be ${m}_{{}_{N}}^{(0)}=1200\ifmmode\pm\else\textpm\fi{}200 \mathrm{MeV}$ and ${m}_{\ensuremath{\Delta}}^{(0)}=1500\ifmmode\pm\else\textpm\fi{}200 \mathrm{MeV}.$ The difference in these bare masses, a quantity which would be accounted for by a residual gluon interaction, is found to be $\ensuremath{\delta}{m}^{(0)}=350\ifmmode\pm\else\textpm\fi{}100 \mathrm{MeV}.$

Chiral corrections to baryon properties with composite pions

A calculational scheme is developed to evaluate chiral corrections to properties of composite baryons with composite pions. The composite baryons and pions are bound states derived from a microscopic chiral quark model. The model is amenable to standard many-body techniques such as the BCS and RPA formalisms. An effective chiral model involving only hadronic degrees of freedom is derived from the macroscopic quark model by projection onto hadron states. Chiral loops are calculated using the effective hadronic Hamiltonian. A simple microscopic confining interaction is used to illustrate the derivation of the pion-nucleon form factor and the calculation of pionic self-energy corrections to the nucleon and Delta(1232) masses.

Sigma-term physics in the perturbative chiral quark model

We apply the perturbative chiral quark model at one loop to analyze meson-baryon sigma terms. Analytic expressions for these quantities are obtained in terms of fundamental parameters of low-energy pion-nucleon physics (weak pion decay constant, axial nucleon coupling, strong pion-nucleon form factor) and of only one model parameter (radius of the nucleonic three-quark core). Our result for the $\ensuremath{\pi}N$ sigma term ${\ensuremath{\sigma}}_{\ensuremath{\pi}N}\ensuremath{\approx}45 \mathrm{MeV}$ is in good agreement with the value deduced by Gasser, Leutwyler, and Sainio using dispersion-relation techniques and exploiting the chiral symmetry constraints.

A chiral bag model with a soft surface : structure and solutions of the Fuzzy and the modified Fuzzy Bag Model

In the present work we propose a new bag model for hadrons, called the modified fuzzy bag model (MFBM). The distinguishing feature of this model is the suppression of the pion field, as it enters the bag, by means of a scalar potential for the pions, while still preserving chiral symmetry. The mechanism of pion suppression in the MFBM is similar to the mechanism of quark suppression in the fuzzy bag model (FBM). The standard chiral transformation for the pion field suffers a natural alteration in the MFBM, and as a result the model is chiral invariant. We present also a discussion of the FBM and study, in the quark sector, the implications of the soft surface of the bag on the expectation value of the mass operator. In the pion-quark sector, we study the effects of the suppression of the pion field on the form factor for the pion-nucleon interaction, on the pion-nucleon coupling constant $g_{ \pi{\scriptscriptstyle{N}}{\scriptscriptstyle{N}} }$ and on the nucleon axial charge $g_A$ . Calculations of the pion-nucleon form factor exhibit, in particular, an improvement over previous results. The pionic axial current induces, in the MFBM, a nonvanishing and orientation dependent contribution to axial charge. An analysis of the asymptotic behaviour of the axial charge shows that the role of the surface is to increase the difference of the contributions associated to different orientations.

Baryons as non-topological chiral solitons

The present review gives a survey of recent developments and applications of the Nambu-Jona-Lasinio model with N f = 2 and N f = 3 quark flavors for the structure of baryons. The model is an effective chiral quark theory which incorporates the SU(N f) L ⊗SU(N f) R ⊗U(1) V approximate symmetry of Quantum chromodynamics. The approach describes the spontaneous chiral symmetry breaking and dynamical quark mass generation. Mesons appear as quark-antiquark excitations and baryons arise as non-topological solitons with three valence quarks and a polarized Dirac sea. For the evaluation of the baryon properties the present review concentrates on the non-linear Nambu-Jona-Lasinio model with quark and Goldstone degrees of freedom which is identical to the Chiral quark soliton model obtained from the instanton liquid model of the QCD vacuum. In this non-linear model, a wide variety of observables of baryons of the octet and decuplet is considered. These include, in particular, electromagnetic, axial, pseudoscalar and pion nucleon form factors and the related static properties like magnetic moments, radii and coupling constants of the nucleon as well as the mass splittings and electromagnetic form factors of hyperons. Predictions are given for the strange form factors, the scalar form factor and the tensor charge of the nucleon.

The extended Nambu-Jona-Lasinio model with separable interaction: Low energy pion physics and pion-nucleon form factor

A Lagrangian formulation of the Nambu-Jona-Lasinio model with separable interaction is given. The electromagnetic interaction is introduced in a nonminimal way to the nonlocal quark current. Various choices of the vertex form factors characterizing the composite structure of mesons and baryon are investigated. We find that the physical observables depend very weakly on form factor shapes. We calculate the πNN form factor considering nucleon as a three-quark system.

Structure function of off-mass-shell pions and the calculation of the Sullivan process.

We construct a model for the pion (valence) structure function that fits the experimental data obtained in the study of the Drell-Yan process. The model may also be used to calculate the structure function of [ital off[minus]mass[minus]shell] pions. We apply our model in the study of deep-inelastic scattering from off-mass-shell pions found in the nucleon and are thus able to resolve a problem encountered in the standard analysis of such processes. The usual analysis is made using the structure function of [ital on[minus]mass[minus]shell] pions and requires the use of a soft [pi][ital NN] form factor that is inconsistent with standard nuclear physics phenomenology. The use of our off-mass-shell structure functions allows for a fit to the data for nonperturbative aspects of the nucleon sea'' with a pion-nucleon form factor of the standard form.