# Several Topics on Transverse Momentum-Dependent Fragmentation Functions

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^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Factorization and Evolution

#### 2.1. Evolution Equations for TMD FFs

#### 2.2. TMD Factorization at the Higher Twist

## 3. Spin-Dependent TMD FFs

#### 3.1. The Intuitive Definition of TMD FFs

#### 3.2. The Definition of TMD FFs from the Parton Correlators

#### 3.3. The Spin Dependence

#### 3.4. Decomposition Result for Spin-Dependent TMD FFs

#### 3.5. TMD FFs of Antiquarks and Gluons

## 4. Experiment and Phenomenology

#### 4.1. $\mathsf{\Lambda}$ Hyperons

#### 4.2. Vector Mesons

## 5. Model Calculation

## 6. Summary

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Reproduction of the longitudinal polarization of ${\mathsf{\Lambda}}^{0}$ in ${e}^{+}{e}^{-}$ annihilation at $\sqrt{s}=91.2$ GeV measured by the ALEPH [115] and OPAL [116] collaborations at LEP. We have combined the statistical and systematic errors. Neglecting the mass of $\mathsf{\Lambda}$ hyperons in the high-energy limit, the definitions of z in these two experiments are the same as those of the momentum fraction in the light-cone coordinate currently used in the QCD factorizations.

**Figure 3.**Spin transfer coefficient ${\mathcal{D}}_{LL}$ as a function of z and ${x}_{B}$ in polarized SIDIS measured by the HERMES [136] and COMPASS [138] collaborations. Only the statistic errors are shown in both plots. Axes 1 and 2 refer to two different definitions of the longitudinal direction of $\mathsf{\Lambda}$ in the experiment. They are approximately the same at the high-energy limit. However, at the HERMES energy, they are not parallel to each other.

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**MDPI and ACS Style**

Chen, K.-B.; Liu, T.; Song, Y.-K.; Wei, S.-Y.
Several Topics on Transverse Momentum-Dependent Fragmentation Functions. *Particles* **2023**, *6*, 515-545.
https://doi.org/10.3390/particles6020029

**AMA Style**

Chen K-B, Liu T, Song Y-K, Wei S-Y.
Several Topics on Transverse Momentum-Dependent Fragmentation Functions. *Particles*. 2023; 6(2):515-545.
https://doi.org/10.3390/particles6020029

**Chicago/Turabian Style**

Chen, Kai-Bao, Tianbo Liu, Yu-Kun Song, and Shu-Yi Wei.
2023. "Several Topics on Transverse Momentum-Dependent Fragmentation Functions" *Particles* 6, no. 2: 515-545.
https://doi.org/10.3390/particles6020029