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

The charged pion multiplicity ratio in intermediate energy heavy-ion collisions, a probe of the density dependence of symmetry energy above the saturation point, has been proven in a previous study to be extremely sensitive to the strength of the isovector $\Delta$(1232) potential in nuclear matter. As there is no knowledge, either from theory or experiment, about the magnitude of this quantity, the extraction of constraints on the slope of the symmetry energy at saturation by using exclusively the mentioned observable is hindered at present. It is shown that, by including the ratio of average $p_T$ of charged pions $\langle p_T^{(\pi^+)}\rangle/\langle p_T^{(\pi^-)}\rangle$ in the list of fitted observables, the noted problem can be circumvented. A realistic description of this observable requires accounting for the interaction of pions with the dense nuclear matter environment by the incorporation of the so called $S$-wave and $P$-wave pion optical potentials. This is performed within the framework of a quantum molecular dynamics transport model that enforces the conservation of the total energy of the system. It is shown that constraints on the slope of the symmetry energy at saturation density and the strength of the $\Delta$(1232) potential can be simultaneously extracted. A symmetry energy with a value of the slope parameter $L>$ 50 MeV is favored, at 1$\sigma$ confidence level, from a comparison with published FOPI experimental data. A precise constraint will require experimental data more accurate than presently available, particularly for the charged pion multiplicity ratio, and better knowledge of the density and momentum dependence of the pion potential for the whole range of these two variables probed in intermediate energy heavy-ion collisions.