Cancellation of the $\sigma$ meson in thermal models

The by now well-established scalar-isoscalar resonance $f_0\left(500\right)$ (the $\sigma$ meson) seems potentially relevant in the evaluation of thermodynamic quantities of a hadronic gas since its mass is low. However, we recall that its contribution to isospin-averaged observables is, to a surprising accuracy, canceled by the repulsion from the pion-pion scalar-isotensor channel. As a result, in practice one should not incorporate $f_0\left(500\right)$ in standard hadronic resonance-gas models for studies of isospin-averaged quantities. In our analysis we use the formalism of the virial expansion, which allows one to calculate the thermal properties of an interacting hadron gas in terms of derivatives of the scattering phase shifts, hence in a model-independent way directly from experimentally accessible quantities. A similar cancellation mechanism occurs for the scalar kaonic interactions between the $I=1/2$ channel [containing the alleged $K_0^{*}\left(800\right)$ or the $\kappa$ meson] and the $I=3/2$ channel.

Figure 1

The parametrization of the pion-pion scattering phase shifts plotted vs invariant mass for various isospin-spin $\left(IJ\right)$ channels, taken from Refs. [6, 7, 31].

Figure 2

The distributions $1/\pi f_{IJ}d{\delta }_{IJ}/dM$, corresponding to the phase shifts of Fig. 1, plotted as functions of the $\pi \pi$ invariant mass $M$. We note a nearly complete cancellation for the $S$-wave interactions between the $I=0$ and the $I=2$ channels up to $M\sim 0.85\mathrm{GeV}$.

Figure 3

Pion, $\rho$-meson, isoscalar-scalar, and isoscalar-tensor contributions to the volume density of the trace of the energy-momentum tensor divided by $T^4$, plotted as functions of $T$.

Figure 4

Relative contribution to the pion yield from $f_0\left(500\right)$, evaluated with share [10, 38] applied to $A+A$ collisions at various collision energies. The thick lines correspond to the naive Breit-Wigner implementation of $f_0\left(500\right)$, whereas the thin lines indicate the correct inclusion of combined isoscalar-scalar and isoscalar-tensor channels. The points indicate the corresponding contributions at the Large Hadron Collider (LHC).

Figure 5

The (a) pion-kaon $S$-wave phase shifts and (b) their derivatives, multiplied with the isospin-spin degeneracy factors, plotted as functions of the invariant mass.