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The charged pion multiplicity ratio in intermediate energy heavy-ioncollisions, a probe of the density dependence of symmetry energy above thesaturation point, has been proven in a previous study to be extremely sensitiveto the strength of the isovector $\Delta$(1232) potential in nuclear matter. Asthere is no knowledge, either from theory or experiment, about the magnitude ofthis quantity, the extraction of constraints on the slope of the symmetryenergy at saturation by using exclusively the mentioned observable is hinderedat present. It is shown that, by including the ratio of average $p_T$ ofcharged pions $\langle p_T^{(\pi^+)}\rangle/\langle p_T^{(\pi^-)}\rangle$ inthe list of fitted observables, the noted problem can be circumvented. Arealistic description of this observable requires accounting for theinteraction of pions with the dense nuclear matter environment by theincorporation of the so called $S$-wave and $P$-wave pion optical potentials.This is performed within the framework of a quantum molecular dynamicstransport model that enforces the conservation of the total energy of thesystem. It is shown that constraints on the slope of the symmetry energy atsaturation density and the strength of the $\Delta$(1232) potential can besimultaneously extracted. A symmetry energy with a value of the slope parameter$L>$ 50 MeV is favored, at 1$\sigma$ confidence level, from a comparison withpublished FOPI experimental data. A precise constraint will requireexperimental data more accurate than presently available, particularly for thecharged pion multiplicity ratio, and better knowledge of the density andmomentum dependence of the pion potential for the whole range of these twovariables probed in intermediate energy heavy-ion collisions.

Constraining the density dependence of the symmetry energy using the multiplicity and average $p_T$ ratios of charged pions