# Cancellation of the Sigma Mode in the Thermal Pion Gas by Quark Pauli Blocking

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## Abstract

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## 1. Introduction

## 2. The Interacting Pion Gas

#### 2.1. Beth–Uhlenbeck Equation and Cancellation of the σ Meson

#### 2.2. Analytic Justification for the Cancellation

## 3. Mesons at a Finite Temperature in the Nambu–Jona-Lasinio Model

## 4. Results and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**Left panel**): $\pi \pi $-scattering phase shifts as functions of the total center-of-mass-momentum. The blue dashed line corresponds to the phase shift ${\delta}_{0}^{2}$ calculated using Equation (12) [4]. (

**Right panel**): Interaction contribution to the pressure according to the Beth–Uhlenbeck formula with the medium-independent phase shifts of the left panel (without the contribution from the $\rho $-meson channel). The almost perfect cancellation of the $\sigma $-meson (green dashed line) by the quark Pauli blocking (red dashed-dotted line) is demonstrated by the blue solid line.

**Figure 2.**(

**Left panel**): the temperature dependence ${M}_{i}\left(T\right)$ of mass spectra for mesons ($i=\pi ,\sigma ,\rho $) and doubled quark mass $m\left(T\right)$). (

**Right panel**): scaled temperature dependence of total decay widths ${\Gamma}_{i}\left(T\right)/{\Gamma}_{i}\left(0\right)$ for mesons ($i=\sigma ,\rho $) calculated within the NJL model.

**Figure 3.**The phase shifts ${\delta}_{0}^{0}$ (

**left panel**) and ${\delta}_{0}^{2}$ (

**right panel**) for different temperatures $T=0$, 0.15, 0.185 GeV. The solid blue line in the right panel corresponds to the scattering length approximation ${\delta}_{0}^{2}=-0.12q/{M}_{\pi}$. Experimental data are taken from refs. [25,26,27,28] for ${\delta}_{0}^{0}$ and from ref. [29] for ${\delta}_{0}^{2}$.

**Figure 4.**(

**Left panel**): Contributions to the pressure of the pion gas at second order of the virial expansion ($\pi \pi $ scattering) using the Beth–Uhlenbeck equation with medium dependent phase shifts. The quark Pauli blocking term ${P}^{02}$ (red dot-dashed line) is modified to take into account that it vanishes when the pion bound state is dissociated. (

**Right panel**): Pressure of the ideal pion gas (red dashed line), compared to the total pressure with all three interaction channels (black solid line) and with just the $\rho $-meson channel (blue dotted line). The $\rho $-meson contribution is also shown separately by the blue dotted line.

T = 0 | T = 0.14 | T = 0.16 | T = 0.19 | T = 0.2 | |
---|---|---|---|---|---|

${a}_{0}^{0}$ | 0.148 | 0.155 | 0.164 | 0.202 | 0.237 |

${a}_{2}^{0}$ | −0.036 | −0.037 | −0.037 | −0.041 | −0.043 |

${a}_{0}^{0}/\left(5{a}_{2}^{0}\right)$ | −0.826 | −0.85 | −0.88 | −0.996 | −1.11 |

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**MDPI and ACS Style**

Blaschke, D.; Friesen, A.; Kalinovsky, Y. Cancellation of the Sigma Mode in the Thermal Pion Gas by Quark Pauli Blocking. *Symmetry* **2023**, *15*, 711.
https://doi.org/10.3390/sym15030711

**AMA Style**

Blaschke D, Friesen A, Kalinovsky Y. Cancellation of the Sigma Mode in the Thermal Pion Gas by Quark Pauli Blocking. *Symmetry*. 2023; 15(3):711.
https://doi.org/10.3390/sym15030711

**Chicago/Turabian Style**

Blaschke, David, Alexandra Friesen, and Yuri Kalinovsky. 2023. "Cancellation of the Sigma Mode in the Thermal Pion Gas by Quark Pauli Blocking" *Symmetry* 15, no. 3: 711.
https://doi.org/10.3390/sym15030711