Lorentz Invariant Amplitudes Research Articles

Generalized parton distributions from lattice QCD with asymmetric momentum transfer: Axial-vector case

Recently, we made significant advancements in improving the computational efficiency of lattice QCD calculations for generalized parton distributions (GPDs). This progress was achieved by adopting calculations of matrix elements in asymmetric frames, deviating from the computationally-expensive symmetric frame typically used, and allowing freedom in the choice for the distribution of the momentum transfer between the initial and final states. A crucial aspect of this approach involves the adoption of a Lorentz covariant parametrization for the matrix elements, introducing Lorentz-invariant amplitudes. This approach also allows us to propose an alternative definition of quasi-GPDs, ensuring frame independence and potentially reduce power corrections in matching to light cone GPDs. In our previous work, we presented lattice QCD results for twist-2 unpolarized GPDs (H and E) of quarks obtained from calculations performed in asymmetric frames at zero skewness. Building upon this work, we now introduce a novel Lorentz covariant parametrization for the axial-vector matrix elements. We employ this parametrization to compute the axial-vector GPD H˜ at zero skewness, using an Nf=2+1+1 ensemble of twisted mass fermions with clover improvement. The light-quark masses employed in our calculations correspond to a pion mass of approximately 260 MeV. Published by the American Physical Society 2024

Generalized parton distributions from lattice QCD with asymmetric momentum transfer: Unpolarized quarks

Traditionally, lattice QCD computations of generalized parton distributions (GPDs) have been carried out in a symmetric frame, where the transferred momentum is symmetrically distributed between the incoming and outgoing hadrons. However, such frames are inconvenient since they require a separate calculation for each value of the momentum transfer, increasing significantly the computational cost. In this work, by focusing on the quasi-distribution approach, we lay the foundation for faster and more effective lattice QCD calculations of GPDs exploiting asymmetric frames, with freedom in the transferred momentum distribution. An important ingredient of our approach is the Lorentz covariant parameterization of the matrix elements in terms of Lorentz-invariant amplitudes, which allows one to relate matrix elements in different frames. We also use this amplitude approach to propose a new definition of quasi-GPDs that is frame-independent and, more importantly, may lead to smaller power corrections in the matching relations to the light-cone GPDs. We demonstrate the efficacy of the formalism through numerical calculations using one ensemble of $N_f$=2+1+1 twisted mass fermions with a clover improvement. The value of the light-quark masses lead to a pion mass of about 260 MeV. Concentrating on the proton, and limiting ourselves to a vanishing longitudinal momentum transfer to the target, we extract the invariant amplitudes from matrix element calculations in both the symmetric and asymmetric frame, and obtain results for the twist-2 light-cone GPDs for unpolarized quarks, that is, $H$ and $E$.

The relativistic Dirac–Brueckner approach to asymmetric nuclear matter

The properties of asymmetric nuclear matter have been investigated in a relativistic Dirac–Brueckner–Hartree–Fock framework using the Bonn A potential. The components of the self-energies are extracted by projecting on Lorentz invariant amplitudes. Furthermore, the optimal representation scheme for the T-matrix, the subtracted T-matrix representation, is applied and the results are compared to those of other representation schemes. Of course, in the limit of symmetric nuclear matter our results agree with those found in literature. The binding energy E b fulfills the quadratic dependence on the asymmetry parameter and the symmetry energy is 34 MeV at saturation density. Furthermore, a neutron–proton effective mass splitting of m n * < m p * is found. In addition, results are given for the mean-field effective coupling constants.

Generalized dipole polarizabilities and the spatial structure of hadrons

We present a phenomenological discussion of spin-independent, generalized dipole polarizabilities of hadrons entering the virtual Compton scattering process ${\ensuremath{\gamma}}^{*}\stackrel{\ensuremath{\rightarrow}}{h}\ensuremath{\gamma}h.$ We introduce a new method of obtaining a tensor basis with appropriate Lorentz-invariant amplitudes which are free from kinematical singularities and constraints. The result is summarized in terms of a compact effective Lagrangian. We then motivate a gauge-invariant separation into a generalized Born term containing ground-state properties only and a residual contribution describing the model-dependent internal structure. The generalized dipole polarizabilities are defined in terms of Lorentz-invariant residual amplitudes. Particular emphasis is laid on a physical interpretation of these quantities as characterizing the spatial distributions of the induced electric polarization and magnetization of hadrons. It is argued that three dipole polarizabilities---namely, the longitudinal electric ${\ensuremath{\alpha}}_{L}{(q}^{2}),$ the transverse electric ${\ensuremath{\alpha}}_{T}{(q}^{2}),$ and the magnetic $\ensuremath{\beta}{(q}^{2})$ ones---are required in order to fully reconstruct local polarizations induced by soft external fields in a hadron. One of these polarizabilities, ${\ensuremath{\alpha}}_{T}{(q}^{2}),$ describes an effect of higher order in the soft final-photon momentum ${q}^{\ensuremath{'}}.$ We argue that the associated spatial distributions obtained via Fourier transforms in the Breit frame are meaningful even for such a light particle as the pion. The spatial distributions are determined at large distances $r\ensuremath{\sim}{1/m}_{\ensuremath{\pi}}$ for pions, kaons, and octet baryons by the use of chiral perturbation theory.

Elastic unitarity and the K-matrix in a Lorentz covariant representation of the NN interaction

A Lorentz covariant representation of the NN t-matrix has been obtaained over the energy range from 150 to 500 MeV by solving the integral equation that connects the t-matrix with the K-matrix. The K-matrix is expanded in a complete set of on-shell Lorentz invariant amplitudes represented phenomenologically by isoscalar and isovector “meson” exchanges. The real part of the K-matrix is fit over the energy range from 150 to 500 MeV using coupling strengths that are allowed to vary quadratically with energy. Above the pion production threshold at T lab = 280 MeV, the real K-matrix is supplemented by an imaginary part with linear energy dependence. The K-matrix parameters are fit to thesmost recent (January 1999) Arndt amplitudes [R.A. Arndt, D. Roper, VPI and SU Scattering Analysis Interactive Dial-in Program and Data Base]. Direct and exchange contributions to the K-matrix are handled explicitly in the formalism. The resulting t-matrix satisfies elastic unitary below the pion production threshold and contains non-local terms that are not present in direct Love-Franey parameterizations of the t-matrix. Results are given for the NN amplitudes and compared with both the Arndt amplitudes and amplitudes obtained from a direct fit of the t-matrix [O.V. Maxwell, Nucl. Phys. A 600 (1996) 509]. Results are also given for a selected set of np and pp observables.

Covariant representations of the relativistic Brueckner T-matrix and the nuclear matter problem

We investigate nuclear matter properties in the relativistic Brueckner approach. The in-medium on-shell T-matrix is represented covariantly by five Lorentz invariant amplitudes from which we deduce directly the nucleon self-energy. We discuss the ambiguities of this approach and the failure of previously used covariant representations in reproducing the nucleon self-energies on the Hartree-Fock level. To enforce correct Hartree-Fock results we develop a subtraction scheme which treats the bare nucleon-nucleon potential exactly in accordance with the different types of meson exchanges. For the remaining ladder kernel, which contains the higher order correlations, we employ two different covariant representations in order to study the uncertainty inherent in the approach. The nuclear matter bulk properties are only slightly sensitive to the explicit representation used for the kernel. However, we obtain new Coester lines for the various Bonn potentials which are shifted towards the empirical region of saturation. In addition the nuclear equation-of-state turns out to be significantly softer in the new approach.

Dirac structure of the nucleus-nucleus potential in heavy ion collisions

We investigate nuclear matter properties in the relativistic Brueckner approach. The in-medium on-shell T-matrix is represented covariantly by five Lorentz invariant amplitudes from which we deduce directly the nucleon self-energy. To enforce correct Hartree-Fock results we develop a subtraction scheme which treats the bare nucleon-nucleon potential exactly in accordance to the different types of meson exchanges. For the higher order correlations we employ two different covariant representations in order to study the uncertainty inherent in the approach. The nuclear matter bulk properties are only slightly sensitive on the explicit representation used. However, we obtain new Coester lines for the various Bonn potentials which are shifted towards the empirical region of saturation.

Lorentz covariant representation of the NN interaction at energies above 500 MeV

Using the formalism developed in the paper by Maxwell [Nucl. Phys. A 600 (1996) 509] a relativistic Love-Franey representation of the NN t-matrix has been obtained over the laboratory energy range from 500 to 800 MeV. The real and imaginary parts of this t-matrix are expanded separately in complete sets of on-shell Lorentz invariant amplitudes, which are represented by isoscalar and isovector “meson” exchanges. The masses associated with these exchanges are assumed to be independent of energy, while the coupling strengths and form factor masses are allowed to vary quadratically and linearly with energy, respectively. Direct and exchange contributions to the interaction are handled explicitly in the model. The model parameters are fit to the Summer 1997 Arndt amplitudes [Arndt and Roper, VPI and SU scattering analysis interactive dial-in program and data base (SAID)] and results given for various np and pp observables. The spin-averaged differential cross section and the vector asymmetry are generally reproduced quite well. The results for the other observables considered are not quite as good, especially at higher energy.

Asymmetric nuclear matter in the relativistic Brueckner-Hartree-Fock approach

We present a calculation of asymmetric nuclear matter properties at zero temperature in a relativistic Brueckner-Hartree-Fock framework. Following other calculations the components of the self-energies are extracted by projecting on Lorentz-invariant amplitudes. It is shown that for asymmetric nuclear matter one needs a sixth invariant. We present a set of invariants which in the limit of symmetric nuclear matter reduces to the conventional set. We argue that the existence of such a properly behaving set is also crucial for the application of the projection method in symmetric nuclear matter. Results for the equation of state and other observables are presented. Special attention is paid to an analysis in terms of mean-field effective coupling constants. Apart from the usual ones we also find significant strength in the isovector scalar channel, which can be interpreted as an effective $\ensuremath{\delta}$ meson.

Energy-dependent Lorentz covariant representation of the NN interaction

A phenomenological Lorentz covariant representation of the NN t -matrix has been obtained with parameters that are explicitly energy dependent. The real and imaginary parts of the t -matrix are separately expanded in a complete set of on-shell Lorentz invariant amplitudes, each of which is represented by the exchange of isoscalar and isovector “mesons” modified by form factors. Direct and exchange contributions are handled explicitly in the model. The model parameters are fit to the summer 1994 Arndt amplitudes over the laboratory kinetic energy range from 200 to 500 MeV. In contrast to the earlier work of Horowitz, a single χ 2 minimization is carried out over the full energy range considered. Two separate fits are obtained, one with energy dependence confined to the coupling strengths, the other with energy dependence in the form factor masses as well. Although the second fit is somewhat better, both fits reproduce the empirical amplitudes reasonably well. Results are also given for various np and pp observables and compared with the data.

Elastic scattering of protons by 16O, 40Ca, and 208Pb at 200, 500, and 800 MeV: Relativistic and nonrelativistic analyses based on the impulse approximation.

Systematics of Dirac impulse approximation predictions for cross sections and spin observables in elastic proton scattering by $^{16}\mathrm{O}$, $^{40}\mathrm{Ca}$, and $^{208}\mathrm{Pb}$ at energies of 200, 500, and 800 MeV are presented. The analysis is based on an optical potential constructed from complete sets of Lorentz-invariant NN amplitudes. The NN amplitudes are determined from a relativistic meson exchange model of the nuclear force. Comparisons are made with the original form of the Dirac impulse approximation, which is based on five Fermi terms to represent the NN interaction, and with the Schr\"odinger form of the impulse approximation. For the Dirac analyses, there is implicit coupling to virtual nucleon-antinucleon states. In order to illustrate these contributions, the Dirac equation is recast in the form of the Schr\"odinger equation and the resulting Schr\"odinger potentials are separated into ``no-pair'' and virtual-pair parts. The resulting virtual-pair part of the central potential is typically 25 MeV at the center of the nucleus for 200--800 MeV protons. A reasonable description of the experimental data is obtained over a broad energy range and over a wide variation of nuclear size when the analysis is based on complete sets of Lorentz-invariant amplitudes.

BRS-invariant vertex operators in the bosonic membrane theory

BRS-invariant vertex operators are constructed to compute Lorentz-invariant amplitudes in the bosonic membrane theory. A brief discussion is given on whether the amplitudes satisfy duality.

Vacuum polarization effects in elastic scattering of protons by nuclei

Sensitivity of spin observables in elastic proton scattering by nuclei to the vacuum polarization correction δϱ VAC is explored. The Dirac optical potential is calculated from a complete set of Lorentz invariant amplitudes, Dirac-Hartree densities and the existing estimate of δϱ VAC based on quantum hadrodynamics.

Meson theoretical basis for Dirac impulse approximation.

Basic features of proton-nucleus optical potentials for use in the Dirac equation are discussed. The original form of the Dirac impulse approximation follows from using Fermi covariants (scalar, vector, tensor, pseudoscalar, and axial vector) to extend physical NN amplitudes into operators in the full Dirac space. Overly large scalar and vector optical potentials are shown to follow at low energy in this case due to forcing pion exchange contributions to be pseudoscalar. The pair contributions to proton-nucleus scattering are much too large at low energy. A variant of the impulse approximation is developed by replacing pseudoscalar covariants by pseudovector ones. Much reduced scalar and vector strengths are obtained at low energy in the pseudovector case. The pair contributions are similarly reduced to reasonable values. However, the large differences between optical potentials based on pseudoscalar and pseudovector covariants are not controlled by physical NN scattering data. These differences represent a basic ambiguity in NN amplitudes when the only constraint is positive energy scattering data. Using a complete and unambiguous set of NN Lorentz invariant amplitudes obtained from a relativistic one-meson-exchange model, the scalar and vector optical potential strengths are found to be reasonably constant over the range of 50 to 1000 MeV of proton energy. The meson theory results for nuclear matter are found to be comparable to those obtained when the pseudoscalar covariant is replaced by the pseudovector covariant in the original impulse approximation.

Asymptotic behavior of the proton electromagnetic form factor with pion exchanges in the ladder approximation

The asymptotic behavior of the proton electromagnetic form factor due to virtual pion exchanges is studied by restricting to a class of ladder diagrams. We set up an off-mass-shell integral equation for the modified proton-proton-photon vertex due to virtual pion exchanges by summing up an infinite number of ladder diagrams. This integral equation is converted into a set of coupled differential equations involving Lorentz-invariant amplitudes. From the solutions of the differential equations it is found that the usual form factor is modified by a factor $\mathrm{exp}{[\ensuremath{-}\frac{1}{2}\ifmmode\pm\else\textpm\fi{}(\frac{1}{2}){(1\ensuremath{-}\frac{{g}^{2}}{{\ensuremath{\pi}}^{2}})}^{\frac{1}{2}}]\mathrm{ln}t}$ at large momentum transfers, where $g$ is the pion-nucleon coupling constant.

Analytic invariant amplitudes for processes involving photons and particles of arbitrary spin and displaying discrete symmetries

This paper is devoted to the construction of a complete and minimal set of Lorentz-invariant amplitudes which describe respectively the electromagnetic interaction of a photon and of a hadron of spins, the electromagnetic decay of a hadron of spinr into a photon and a hadron of spins, the inelastic photoproduction of a pion on a target of spinr and the Compton effect on a hadronic target of spins; these amplitudes are free of kinematical zeros and singularities and transform in a trivial fashion under discrete symmetries; finally the expressions of the transition amplitudes for these processes and interactions in terms of the invariant amplitudes satisfy transversality and current conservation.