# On the Partition Temperature of Massless Particles in High-Energy Collisions

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## Abstract

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## 1. Introduction

## 2. Phase-Space Integration of the One-Particle Momentum Distribution

## 3. The Contour Integration

## 4. The Feasibility of Partition Temperature

## 5. Further Discussions and Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. The Derivations of the Formuale for One-Particle Distribution Function

## References

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**Figure 1.**Left: The numerical fits shown in Figure 1 but presented in the logarithmic scale. From the top to bottom, the curves correspond to the multiplicities $n=5,10,50$, and 100, respectively. Right: The extracted partition temperature $1/{T}_{p}$ as a function of the multiplicity n, where the red dashed curve is obtained by a third-order spline fit.

**Figure 2.**The numerical fits shown in Figure 1 but presented in the logarithmic scale (

**left**) and the extracted partition temperature $1/{T}_{p}$ as a function of the multiplicity n (

**right**), where the red dashed curve is obtained using a third-order spline fit.

**Table 1.**The extracted partition temperatures, the corresponding standard errors, and p-values for different multiplicities n. The quality of the fit improves as the multiplicity increases.

n | 5 | 10 | 30 | 40 | 50 | 100 |

$1/{T}_{p}$ | 0.0523 | 0.0794 | 0.0936 | 0.0952 | 0.0962 | 0.0981 |

standard error | 0.0006 | 0.0005 | 0.0002 | 0.0001 | 0.0001 | 0.00005 |

p-value | $1.2\times {10}^{-51}$ | $9.5\times {10}^{-67}$ | $3.0\times {10}^{-91}$ | $1.9\times {10}^{-97}$ | $3.2\times {10}^{-102}$ | $6.5\times {10}^{-117}$ |

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**MDPI and ACS Style**

Qian, W.-L.; Lin, K.; Yue, R.-H.; Hama, Y.; Kodama, T.
On the Partition Temperature of Massless Particles in High-Energy Collisions. *Symmetry* **2023**, *15*, 2035.
https://doi.org/10.3390/sym15112035

**AMA Style**

Qian W-L, Lin K, Yue R-H, Hama Y, Kodama T.
On the Partition Temperature of Massless Particles in High-Energy Collisions. *Symmetry*. 2023; 15(11):2035.
https://doi.org/10.3390/sym15112035

**Chicago/Turabian Style**

Qian, Wei-Liang, Kai Lin, Rui-Hong Yue, Yogiro Hama, and Takeshi Kodama.
2023. "On the Partition Temperature of Massless Particles in High-Energy Collisions" *Symmetry* 15, no. 11: 2035.
https://doi.org/10.3390/sym15112035