Nuclear Mean Field from Chirally Symmetric Effective Theory*

The mean-field theory of the nuclear many-body problem proposed recently by Furnstahl, Serot, and Tang (FST) is discussed. The FST chiral Lagrangian is derived in terms of an effective field theory. This new approach allows one to construct in a controlled manner the universal nuclear Lagrangian consistent with symmetries of QCD. The FST Lagrangian is constructed by using power counting, i.e., the expansion in powers of the lowest lying hadronic fields and their derivatives. Terms in the Lagrangian are organized by applying Georgi’s naive dimensional analysis and “naturalness” condition. The relevant degrees of freedom are nucleons, pions, an isoscalar-vector field (ω meson), an isoscalar-scalar field (σ meson), and an isovector-vector field (ρ meson). The chiral symmetry is realized nonlinearly using a standard WCCWZ procedure. c © 2003 MAIK “Nauka/Interperiodica”. 1. EFFECTIVE FIELD THEORIES (INSTEAD OF INTRODUCTION) The basic idea underlying the effective field theory approach [1] is relevant to the appearance of disparate characteristic energy scales, E E0, in quantum field theories. Suppose that we are interested in the physics at lower scaleE; then, we can choose a cutoff scale Λ at or slightly below E0 and divide the generic fields φ into lowand high-energy parts: φL and φH (φ = φL + φH); accordingly, their momenta are smaller or greater than Λ. The effective Lagrangian is obtained by path integration over the high-energy part φH in the generating functional Z: Z = ∫ [dφL][dφH ]e ∫ dxL(φL,φH) (1) = ∫ [dφL]e ∫ d4xLeff(φL), where ∫ dxLeff(φL) = −i ln ∫ [dφH ]e ∫ d4xL(φL,φH). (2) The next step is to write Leff in terms of local operators Leff(φL) = ∑∞ i gi(Λ)Oi(φL), where gi(Λ) are the coupling constants absorbing the contribution of the integrated-out high-energy degrees of freedom. The effective Lagrangian is represented by a tower of interactions; to make this procedure useful, we need some dimensional analysis. In units = c = 1, the action Seff = ∫ dxLeff(φL) is dimensionless. If an ∗This article was submitted by the author in English. e-mail: [email protected] 1063-7788/03/6608-1574$24.00 c © operator Oi has dimension [Oi] = [m]δi ≡ δi, in mass units, then its coupling constant has dimension [gi] = 4− δi. We can also define dimensionless coefficients ci = Λδi−4gi, which are additionally assumed to be “natural,” i.e., of orderO(1). For a process at scale E Λ, we can estimate dimensionally the magnitude of the ith operator in the action as ∫ dxOi ∼ Eδi−4, so that the ith term is of order ∫ dx ci Λδi−4 Oi ∼ ci ( E Λ )δi−4 . (3) Now, we can see that, at energies below Λ, the behavior of the different operators is determined by their dimension. If δi < 4, the operator is more and more important when E → 0 and is termed relevant. Similarly, if δi > 4, the operator is less and less important, and is termed irrelevant. An operator with δi = 4 is equally important at all scales and is calledmarginal. The general form of the effective Lagrangian is obtained by setting the operators Oi according to their dimensions,