Calculation of a scalar-isoscalar hadronic correlator

Abstract

In an earlier work we calculated the continuum contribution to the hadronic current correlator for pseudoscalar and scalar-isovector currents. In this work we extend our considerations to the study of a scalar-isoscalar correlator of scalar operators. In this formalism we explore the distribution of $q\overline{q}$ strength as a function of energy. We perform a covariant random-phase approximation calculation of the scalar-isoscalar states using parameters that have been determined in earlier studies of pseudoscalar and vector mesons. The state of lowest energy that we find can be identified with the ${f}_{0}(980).$ We find very little $q\overline{q}$ strength for energies less than the energy of the ${f}_{0}(980).$ That suggests that the nonlinear sigma model is the model of choice, since there is no low-lying scalar state of $q\overline{q}$ character in our model. [The ${f}_{0}(400--1200),$ which now appears in the data tables, may be understood as a ``dynamically generated'' state that appears when one studies \ensuremath{\pi}\ensuremath{\pi} scattering. In our analysis it is not a $q\overline{q}$ state. In the Nambu--Jona-Lasinio model, chiral symmetry requires the same interaction strength ${G}_{S}$ for scalar and pseudoscalar states. If we wished to obtain a scalar state at about 600 MeV, we would require a major violation of chiral symmetry, such that ${G}_{S}$ for scalar states would be about 50% larger than the value determined from our study of the pseudoscalar mesons.] We conclude that, except for certain limited applications that we have described in earlier work, the linear sigma model, which describes a low-energy $q\overline{q}$ state of mass of about 500--600 MeV, does not provide a correct description of the physical spectrum.