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Center for Exploration of Energy and Matter


   a research center of the Office of the Vice Provost for Research at IU Bloomington

Effective Field Theory

Lorentz-covariant, meson--baryon effective field theories (often called ``quantum hadrodynamics'' or QHD) provide the basis for relativistic mean-field (RMF) calculations of nuclei. When applied within the framework of modern effective field theory (EFT) and density functional theory (DFT), they justify and generalize earlier mean-field approaches. We recently constructed an effective hadronic lagrangian consistent with the symmetries of QCD and intended for applications to finite-density nuclear systems. The degrees of freedom are (valence) nucleons, pions, and low-lying non-Goldstone bosons, which account for the intermediate-range nucleon--nucleon (NN) interactions and conveniently describe the non-vanishing expectation values of bilinear products of nucleon fields. Chiral symmetry is realized nonlinearly, with a light scalar meson included as a chiral singlet to provide the mid-range NN attraction. The low-energy electromagnetic structure of the nucleon and pion is included using vector-meson dominance, so that ad hoc form factors are not needed. We verified that for normal nuclear systems, the effective lagrangian can be expanded systematically in powers of the meson fields (and their derivatives) and can be truncated reliably after the first few orders.

Electro magnetic Interactions in the Nuclear Many-Body Problem

We have also studied more thoroughly the electromagnetic (EM) interactions in our QHD/EFT of the nuclear many-body problem. The complete EM lagrangian arising from minimal substitution was derived and shown to possess the residual chiral symmetry of massless, two-flavor QCD with EM interactions. The EM current was exhibited to all orders in the pion field, the uniqueness of the minimal EM current was proved, and the properties of isovector vector and axial-vector currents were discussed, generalizing earlier work. The residual chiral symmetry was maintained in additional (non-minimal) EM couplings expressed as a derivative expansion and in implementing vector-meson dominance.

Initial studies of nuclear currents in our QHD/EFT framework focused on axial-vector currents, including meson-exchange currents. We now propose to investigate the structure of the one- and two-body EM currents, based on the gauge-invariant lagrangian discussed earlier. Although the low-energy structure of the nucleon is described using vector-meson dominance (VMD), so that additional, external form factors are not needed, we must still augment the VMD contributions with a derivative expansion to describe the form factors accurately at momentum transfers where exchange currents should be visible. Both electron scattering and pion photoproduction will be studied; the first step is to derive explicit expressions for Lorentz-covariant, one- and two-body matrix elements with an appropriately truncated lagrangian to verify that they are consistent with EM current conservation. Although exchange currents in chiral theories have been discussed, earlier treatments are based on heavy-baryon chiral perturbation theory and do not have a unified description of the currents and the nuclear structure, unlike our covariant approach that is based on a single lagrangian. The ultimate goal of this work is to use the two-body EM amplitudes to derive meson-exchange current operators that can be used with RMF nuclear wave functions and then to compute exchange-current corrections to observables within our unified approach.

Loop Corrections and Power Counting

Although one-loop (RMF) calculations provide a realistic description of bulk and single-particle nuclear properties, it is necessary to examine multi-loop corrections to develop a systematic finite-density power-counting scheme for the nuclear many-body problem. Recent work on covariant chiral perturbation theory has shown that a consistent power counting can be achieved even in theories with baryons and heavy mesons, at least at zero density. We calculated two- and three-loop corrections to our chiral QHD/EFT lagrangian in nuclear matter; with the techniques of Infrared Regularization, it is possible to separate out the short-range contributions and to write them as local products of fields that are already present in the lagrangian. (The field variables must be re-defined at each order in loops.) The corresponding lagrangian parameters implicitly include short-range effects to all orders in the interaction, so these effects need not be calculated explicitly. The remaining (long-range) contributions that must be calculated are nonlocal and resemble those in conventional nuclear-structure calculations. Nonlinear isoscalar scalar and vector meson interactions were included, which incorporate many-nucleon forces and nucleon substructure. We verified that the coupling parameters remain of natural size when fitted to the empirical properties of equilibrium nuclear matter.

Our recent work on the loop expansion in QHD has taught us that power counting in loops is not sufficient; one must also compare the loop terms with the mean-field terms. Moreover, in going from two loops to three loops, the contributions to the nuclear binding energy are reduced by roughly a factor of two. We emphasize that these analyses are based on Infrared Regularization (not heavy-baryon chiral perturbation theory), which allows the short-distance physics to be separated out and parameterized covariantly. Nevertheless, we still do not have an explicit power counting scheme at finite density, nor have we reached a conclusion on the practicality of the loop expansion. To investigate these issues, we will attempt two further calculations, both of which involve a sum of loops to all orders. The first will be a ladder sum in a Dirac--Brueckner--Hartree--Fock calculation, including the implicit many-body forces and short-range structure contained in the nonlinear meson interactions. The second will be an RPA calculation to sum the loops, again including the important meson nonlinearities. By comparing these results to the lowest-order (three-loop) ring and ladder contributions, we should get some quantitative insight into the behavior of the loop expansion.

Pion Interactions in Nuclear Matter

Our QHD/EFT lagrangian contains pions interacting in a manner consistent with the spontaneously broken chiral symmetry of QCD and also provides a realistic description of nuclei and nuclear matter at the RMF level. It thus allows for an investigation of pion propagation in nuclear matter and finite nuclei that is consistent with relativity, nonlinear chiral symmetry, and observed nuclear properties. Calculations containing these three crucial dynamical aspects have never been performed, and we plan to initiate a study of the relativistic pion response function in nuclear matter. To achieve realistic results for the isovector response, we will first extend our covariant lagrangian to include the first excited state of the nucleon, namely, the Delta(1232). While lagrangians with chirally invariant pion-nucleon--Delta and pion--Delta--Delta couplings have been exhibited previously in the literature, none has been applied seriously to the propagation of pions in nuclear matter. This is essentially a relativistic isobar-hole approach with covariant isoscalar and isovector dynamics. We also propose to study the role of the Delta in axial-vector and EM exchange currents. Introducing EM gauge invariance is not trivial, due to the constraints on the Rarita- Schwinger (Delta) field that must be satisfied so that it describes the correct number of degrees of freedom. Some fraction of the work involving the Delta will make up the Ph.D. thesis of X. Zhang.

Relativistic Mean-field Energy Functionals

In our earlier work on relativistic, mean-field energy functionals, we showed that five isoscalar, non-gradient parameters, one gradient parameter, and one isovector parameter are well determined by the usual bulk nuclearobservables. We also emphasized the importance of using appropriate linear combinations of the lagrangian parameters for the parameter counting and for determining if the fitted parameters are of natural size. We have performed parameter fits at several different levels of truncation of the QHD lagrangian, keeping all (non-redundant) terms consistent with the symmetries; these fits contained as many as thirteen free parameters that were determined from empirical nuclear properties. We propose to re-examine these fits by constructing the appropriate linear combinations of parameters to reduce the correlations between terms, a procedure that is not commonly used by other practitioners. Our goal is to achieve fits that are quantitatively better than those found in the literature and to produce some new ``standard'' parameter sets that can be used by other investigators.

Selected Publications

Self-consistent Hartree Description of Finite Nuclei in a Relativistic Quantum Field Theory, C. J. Horowitz and B. D. Serot, Nucl. Phys. A368 (1981) 503.

Dirac-Equation Impulse Approximation for Intermediate-Energy Nucleon--Nucleus Scattering, B. C. Clark, S. Hama, R. L. Mercer, L. Ray, and B. D. Serot, Phys. Rev. Lett. 50 (1983) 1644.

The Relativistic Nuclear Many-Body Problem, B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16 (1986) 1.

Quantum Hadrodynamics, B. D. Serot, Rep. Prog. Phys. 55 (1992) 1855.

Phase Transitions in Warm, Asymmetric Nuclear Matter, H. Müller and B. D. Serot, Phys. Rev. C 52 (1995) 539.

A Chiral Effective Lagrangian for Nuclei, R. J. Furnstahl, B. D. Serot, and H.-B. Tang, Nucl. Phys. A615 (1997) 441; A640 (1998) 505 (E).

Recent Progress in Quantum Hadrodynamics, B. D. Serot and J. D. Walecka, Int. J. Mod. Phys. E 6 (1997) 515.

Axial-vector Current in Nuclear Many-Body Physics, S. M. Ananyan, B. D. Serot, and J. D. Walecka, Phys. Rev. C 66 (2002) 055502.

 

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