Pion Degrees Of Freedom Research Articles

On the axial-vector form factor of the nucleon and chiral symmetry

We consider the chiral Lagrangian with nucleon, isobar, and pion degrees of freedom. The baryon masses and the axial-vector form factor of the nucleon are derived at the one-loop level. We explore the impact of using on-shell baryon masses in the loop expressions. As compared to results from conventional chiral perturbation theory we find significant differences. An application to QCD lattice data is presented. We perform a global fit to the available lattice data sets for the baryon masses and the nucleon axial-vector form factor, and determine the low-energy constants relevant at hbox {N}^3LO for the baryon masses and at hbox {N}^2LO for the form factor. Partial finite-volume effects are considered. We point out that the use of on-shell masses in the loops results in non-analytic behavior of the baryon masses and the form factor as function of the pion mass, which becomes prominent for larger lattice volumes than presently used.

Implications of Efimov physics for the description of three and four nucleons in chiral effective field theory

In chiral effective field theory the leading order (LO) nucleon-nucleon potential includes two contact terms, in the two spin channels $S=0,1$, and the one-pion-exchange potential. When the pion degrees of freedom are integrated out, as in the pionless effective field theory, the LO potential includes two contact terms only. In the three-nucleon system, the pionless theory includes a three-nucleon contact term interaction at LO whereas the chiral effective theory does not. Accordingly arbitrary differences could be observed in the LO description of three- and four-nucleon binding energies. We analyze the two theories at LO and conclude that a three-nucleon contact term is necessary at this order in both theories. In turn this implies that subleading three-nucleon contact terms should be promoted to lower orders. Furthermore this analysis shows that one single low energy constant might be sufficient to explain the large values of the singlet and triplet scattering lengths.

Role of diquark correlations and the pion cloud in nucleon elastic form factors

Electromagnetic form factors of the nucleon in the space-like region are investigated within the framework of a covariant and confining Nambu-Jona-Lasinio model. The bound state amplitude of the nucleon is obtained as the solution of a relativistic Faddeev equation, where diquark correlations appear naturally as a consequence of the strong coupling in the colour $\bar{3}$ $qq$ channel. Pion degrees of freedom are included as a perturbation to the "quark-core" contribution obtained using the Poincar\'e covariant Faddeev amplitude. While no model parameters are fit to form factor data, excellent agreement is obtained with the empirical nucleon form factors (including the magnetic moments and radii) where pion loop corrections play a critical role for $Q^2 \lesssim 1\,$GeV$^2$. Using charge symmetry, the nucleon form factors can be expressed as proton quark sector form factors. The latter are studied in detail, leading, for example, to the conclusion that the $d$-quark sector of the Dirac form factor is much softer than the $u$-quark sector, a consequence of the dominance of scalar diquark correlations in the proton wave function. On the other hand, for the proton quark sector Pauli form factors we find that the effect of the pion cloud and axialvector diquark correlations overcomes the effect of scalar diquark dominance, leading to a larger $d$-quark anomalous magnetic moment and a form factor in the $u$-quark sector that is slightly softer than in the $d$-quark sector.

Pion Degrees of Freedom and EMC Effect

A short review of the phenomena influenced by the pionic degrees of freedom in nuclei is given. The pionic mechanism of the EMC effect is discussed.

Photo- and electro-excitation of the Δ-resonance at MAMI

Over the last decade accurate experiments at MAMI played an essential role to improve our understanding of the nucleon to Δ(1232) transition. Originally to a large extent motivated through intraquark hyperfine interactions anticipated in QCD-inspired quark models they showed that pionic degrees of freedom are essential. The meson cloud is mainly responsible for the observed quadrupole excitation strength and affects the magnetic dipole transition strength as well

Two-flavor condensates in chiral dynamics: Temperature and isospin density effects

Isospin density and thermal corrections for several condensates are discussed, at the one-loop level, in the frame of chiral dynamics with pionic degrees of freedom. The evolution of such objects gives an additional insight into the condensed-pion phase transition that occurs basically when $|{\ensuremath{\mu}}_{\mathrm{I}}|>{m}_{\ensuremath{\pi}}$, $|{\ensuremath{\mu}}_{\mathrm{I}}|$ being the isospin chemical potential. Calculations are done in both phases, showing a good agreement with lattice results for such condensates.

Pionic degrees of freedom in atomic nuclei and quasielastic knockout of pions by high-energy electrons

Two kinds of virtual pions in the nucleus are discussed: delocalized pions forming a pionic condensate and pions localized in the pionic clouds of individual nucleons in the nucleus. The pionic degrees of freedom in nuclei may be efficiently verified in pion quasielastic knockout experiments A(e, e’π±)A*. First, the momentum distribution (MD) of collective pions has a bright maximum at K ≃ 0.3 GeV/c. Second, the excitation spectrum of the recoil nucleus is concentrated at low energies Erec, A ≃ K2/2MA≤1 MeV. The results for the pion knockout from mesonic clouds of individual nucleons in nuclei are distinctly different.

Possibilities of investigating pionic degrees of freedom in nuclei by using quasielastic pion knockout by high-energy electrons

Possibilities of investigating pionic degrees of freedom in nuclei by using quasielastic pion knockout by high-energy electrons

Correlated charged pions from chiral dynamics in high-energy nuclear collisions

It is expected that chiral symmetry is approximately restored in the very excited matter produced in a high-energy nuclear collision and that the chiral order parameter will thereafter quickly relax towards its aligned vacuum value. This non-equilibrium evolution of the environment endows the pionic degrees of freedom with a time-dependent effective mass and this agency, in turn, may create correlated soft pions that could provide specific observable signatures of the chiral dynamics that are based solely on charged pions.

Practical approximation scheme for the pion dynamics in the three-nucleon system

We discuss a working approximation scheme to a recently developed formulation of the coupled piNNN-NNN problem. The approximation scheme is based on the physical assumption that, at low energies, the 2N-subsystem dynamics in the elastic channel is conveniently described by the usual 2N-potential approach, while the explicit pion dynamics describes small, correction-type effects. Using the standard separable-expansion method, we obtain a dynamical equation of the Alt-Grassberger-Sandhas (AGS) type. This is an important result, because the computational techniques used for solving the normal AGS equation can also be used to describe the pion dynamics in the 3N system once the matrix dimension is increased by one component. We have also shown that this approximation scheme treats the conventional 3N problem once the pion degrees of freedom are projected out. Then the 3N system is described with an extended AGS-type equation where the spin-off of the pion dynamics (beyond the 2N potential) is taken into account in additional contributions to the driving term. These new terms are shown to reproduce the diagrams leading to modern 3N-force models. We also recover two sets of irreducible diagrams that are commonly neglected in 3N-force discussions, and conclude that these sets should be further investigated, because a claimed cancellation is questionable.

Axially symmetricB=2solution in the chiral quark soliton model

The baryon-number-two $(B=2)$ solution based on the SU(2) chiral quark soliton model (\ensuremath{\chi}QSM) is solved numerically, including fully the sea quark degrees of freedom. We confirm that the axially symmetric meson configurations yield the energy minimum for the $B=2$ state in the \ensuremath{\chi}QSM when taking into account quark dynamics. Due to the axially symmetric meson fields, six valence quarks occupy the lowest energy level, consistent with the Pauli exclusion principle. The minimal-energy of the entire system is obtained within the framework of a symmetric ansatz. The fermion determinant with axially symmetric meson fields is calculated by diagonalizing the corresponding Dirac Hamiltonian in a nonperturbative way, using a cylindrical Dirac basis. The baryon number density, calculated with quark fields corresponding to a soliton, is toroidal in shape. We also calculate the mean radius of the toroid from the quark fields. These results are closely related to Skyrme model calculations based upon pion degrees of freedom. Our model calculations clarify the underlying dynamical structure of the baryons at the quark level.

Chiral restoration and the extended photoabsorption sum rule in nuclei

The Bethe-Levinger sum rule is extended beyond the potential model. The pion degrees of freedom are taken into account and the modifications of the potential theory are analysed within a dispersion relation on the Compton amplitude. Our aim is to extract from the photon data experimental information on the expectation value of the square of the pion field, a quantity which enters also in the restoration of chiral symmetry in nuclei. We are led to incorporate in the description the Δ resonance, which is strongly excited by the pion degrees of freedom.

Construction of gauge invariant effective nucleonic theories: Functional approach

Starting from relativistic quantum field theories, describing interacting nucleons and pions coupled to the dynamical electromagnetic field, the pion degrees of freedom are eliminated by means of functional integration. Apart from taking into account some operators perturbatively in e, e.g. the vacuum polarization, this procedure is exact, giving effective theories for nucleons and photons. The subsequent nonrelativistic reduction yields the corresponding nonrelativistic quantum field.theory. The latter is unique, irrespective of the precise form of the original nucleon-pion interaction. Nucleonic potentials and electromagnetic interactions are mutually consitent. Local gauge invariance is satisfied at any stage of the formal development.

Algebraic Method for Large-Nc QCD

We discuss an algebraic method for large-Nc baryon structure. In addition to the widely used quark representation for the large-Nc algebra, we consider several other representations that reflect important dynamical effects. The axial vector coupling constants both in the isoscalar (flavour-singlet) and isovector sectors are discussed in detail. Importance of the pion degrees of freedom is pointed out.

Electromagnetic properties of one- and two-baryon systems in the quark potential model

These lectures review calculations of the electromagnetic properties of one- and two-baryon systems using the nonrelativistic quark model. We start with a short discussion of spontaneous chiral symmetry breaking, which is essential in understanding the transition from QCD to the constituent quark model. We then discuss the chiral version of the nonrelativistic quark model which simulates the symmetries and dynamical content of the underlying field theory in terms of gluon, pion, and sigma exchange between constituent quarks. In the second lecture, the electromagnetic properties of the one-baryon system are investigated in this framework. In particular, we focus on the effect of gluon and pion degrees of freedom in the electromagnetic current operators. These must be included in order for the electromagnetic current to be conserved. We also study the effect of scalar exchange currents connected with the confinement and sigma exchange potentials. By including these two-body exchange currents we go beyond the single-quark impulse approximation which has mainly been used up to now. The consistent treatment of gluon-, pion-, and scalar-exchange currents in the quark potential model is the new point of the present work. We show that exchange currents have a large effect on various electromagnetic properties, such as magnetic moments, charge and magnetic radii of the nucleon and Δ(1232). We point out that one cannot extract quantitative information about the quark-quark potential from electromagnetic observables without taking two-body exchange currents into account. In the third lecture, we use the quark potential model to study the electromagnetic properties of the deuteron. We emphasize that a quark-model description leads to short-range modifications of the conventional single-nucleon and two-nucleon currents, due to the quark-exchange currents. These appear as a consequence of the Pauli principle at the quark level.

Microscopic nuclear structure and nuclear reactions at intermediate energies

A hamiltonian for the microscopic description of nuclear phenomena at low and intermediate energies is discussed. It employs nucleon, Δ-isobar and pion degrees of freedom. It is tested in the two-nucleon system below and above pion production threshold in hadronic and electromagnetic processes. It can be applied in nuclear structure and nuclear reactions.

Inclusive electron scattering and pion degrees of freedom in light nuclei.

We demonstrate that realistic models of nuclear interactions and currents quantitatively reproduce the [alpha]-particle longitudinal and transverse responses as measured in inclusive electron scattering at intermediate energies. Pion degrees of freedom in both the nuclear interactions and currents play a crucial role in reproducing the experimental data. The charge-exchange character of the pion-exchange interaction leads to shifts of both longitudinal and transverse strength to higher excitation energies, thus producing a significant quenching of the response in the region of the quasielastic peak. However, in the transverse channel this mechanism is more than offset by the two-body pion-exchange currents required by gauge invariance, and hence the response is enhanced over the entire quasielastic spectrum. We relate these results to responses to idealized single-nucleon couplings and to quasielastic data from hadronic reactions on nuclei.

THEORETICAL ASPECTS OF THE POLARIZABILITY OF THE NUCLEON

Two aspects of the nucleon electromagnetic polarizabilities, αN and βN, are discussed. The first one is the interrelation between static and Compton polarizabilities. Relativistically consistent treatment of the polarization effects based on effective Lagrangians results in the conclusion that the Compton polarizabilities, rather than the static ones, drive the energy shift of the nucleon in external fields. The cluster substructure of the static polarizabilities is shown to be crucial for consistent calculations of the polarizabilities. The second aspect concerns the dominant role of pion degrees of freedom in αN as seen through dispersion relations, chiral perturbation theory, and constituent models. The reported calculations of αN within soliton models are shown to be inconsistent with gauge invariance. The difficulties in explaining the observed large diamagnetic component of βN are emphasized.

Three-nucleon force and the Delta mechanism for pion production and pion absorption.

The description of the three-nucleon system in terms of nucleon and \ensuremath{\Delta} degrees of freedom is extended to allow for explicit pion production (absorption) from single dynamic \ensuremath{\Delta} deexcitation (excitation) processes. This mechanism yields an energy dependent effective three-body Hamiltonian. The Faddeev equations for the trinucleon bound state are solved with a force model that has already been tested in the two-nucleon system above pion-production threshold. The binding energy and other bound-state properties are calculated. The contribution to the effective three-nucleon force arising from the pionic degrees of freedom is evaluated. The validity of previous coupled-channel calculations with explicit but stable \ensuremath{\Delta} isobar components in the wave function is studied.

Absorption mechanisms in photon induced two-body knockout

Calculations have been performed for the 16O(γ, pn) and the 16O(γ, pp) reaction in the photon-energy range E γ =60–300 MeV. Besides the contribution from the more common photoabsorption on the pionic degrees of freedom, we have investigated the influence of heavier meson exchange ( ϱ, σ, ω) and intermediate Δ creation with π and ϱ exchange. Whereas the π meson is found to set the main trends, the ϱ meson is found not to be discardable in a theoretical description of the ( γ, pn) reaction. The incorporation of an energy dependence and a decay width in the Δ propagator is observed to be essential in order to arrive at a more realistic description of ( γ, N N) reactions at higher photon energies.