# Gauge Sector Dynamics in QCD

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## Abstract

**:**

Contents | ||

1 | Introduction | 2 |

2 | Basic Concepts and General Theoretical Framework | 5 |

3 | Schwinger Mechanism in Yang–Mills Theories | 10 |

4 | Dynamical formation of Massless Poles | 13 |

5 | Generation of the Gluon Mass | 16 |

5.1 Gluon Mass from the qμqν Component | 17 | |

5.2 Gluon Mass from the gμν Component: Seagull Identity and Ward Identity Displacement | 18 | |

6 | Renormalization Group Invariant Interaction Strength | 20 |

7 | Three-Gluon Vertex and Its Planar Degeneracy | 23 |

8 | Ghost Dynamics from Schwinger–Dyson Equations | 25 |

9 | Divergent Ghost Loops and Their Impact on the QCD Green’s Functions | 30 |

10 | Ward Identity Displacement of the Three-Gluon Vertex | 34 |

11 | The Ghost-Gluon Kernel Contribution to the Ward Identity | 35 |

12 | Displacement Function from Lattice Inputs | 39 |

13 | Conclusions | 40 |

A. | Appendix A | 41 |

B. | Appendix B | 43 |

References | 43 |

## 1. Introduction

- In Section 2, we introduce some basic notations and review certain prominent features of Green’s functions within the linear gauges and the PT-BFM formalism [109,192]. We stress, in particular, the properties of the auxiliary function $G\left(q\right)$ [16,131,212,213], which relates the gluon propagators with quantum and background gluons, and is intimately connected with the definition of the process-independent and RGI interaction strength [16], to be discussed in detail in Section 6. In addition, we elucidate (with a concrete example) the important property of “block-wise” transversality, displayed by the background gluon self-energy [18,109,112].
- In Section 3, we review the general principles associated with the Schwinger mechanism [127,128] that endows gauge bosons with an effective mass, focusing on the details associated with its realization in the context of Yang–Mills theories. We place particular emphasis on the pivotal requirement that must be satisfied by the fundamental vertices of the theory, namely the appearance of massless poles in their form factors [18,93,109,111,112,113,117,159,214].
- In Section 4, we examine the dynamical formation of colored composite excitations (bound states) of vanishing masses, which provide the required structures in the vertices in order for the Schwinger mechanism to be activated [18,117,159,214]. The formation of these states out of a pair of gluons or a ghost–anti-ghost pair is controlled by a set of coupled Bethe–Salpeter equations (BSEs) [18,117,124,214,215], which are found to have nontrivial solutions for the corresponding Bethe–Salpeter (BS) amplitudes, to be denoted by $\mathbb{C}\left(r\right)$ and $\mathcal{C}\left(r\right)$, respectively.
- In Section 5, we explain in detail how the presence of the massless poles in the dressed vertices that enter the SDE of the gluon propagator give rise to a gluon mass. The demonstration is carried out separately for the ${g}_{\mu \nu}$ and ${q}_{\mu}{q}_{\nu}/{q}^{2}$ components of the gluon self-energy. The former case requires the evasion of the so-called “seagull identity” [113,166]; this becomes possible by virtue of the crucial Ward identity (WI) displacement, to be further considered in Section 10.
- In Section 6, we go over the basic notions underpinning the PT [14,96,100,193,194], and show how their application leads naturally to the definition of a dimensionful process-independent RGI interaction strength [3,16,20,79,96,129,130,131], denoted by $\widehat{d}\left(q\right)$. The genuine process independence of this quantity is concretely exemplified by demonstrating its appearance in two processes involving fundamentally different external fields. Next, $\widehat{d}\left(q\right)$ is computed by combining lattice data for the gluon propagator and SDE results for the function $G\left(q\right)$. Finally, the dimensionless quantity is derived that constitutes the physical definition of the one-gluon exchange interaction appearing in standard bound-state computations [15,16,17,216,217,218,219,220,221,222].
- In Section 7, we focus on the structure of the “transversely projected” three-gluon vertex [126,174,175,223], and discuss briefly the property of planar degeneracy [86], satisfied, at a high level of accuracy [86,87,88,174,175,223], by the vertex form factors. This special property induces a striking simplification to the structure of this vertex, captured by a particularly compact expression [86], which will be extensively used in some of the following sections.
- In Section 8, we take a close look at the ghost sector of the theory, and solve the coupled system of SDEs governing the ghost propagator and ghost–gluon vertex [85,224,225,226,227,228]; as is well-known, the ghost remains massless, but its dressing function saturates at the origin [21,42,47,49,51,56,62,63,73,79,85,112,178,225,227,228,229,230,231,232,233], because the infrared-finite gluon propagator used in the ghost SDE provides an effective infrared cutoff. In the SDE of the ghost–gluon vertex, we employ as central input the compact expression for the three-gluon vertex presented in the previous section. The results are in excellent agreement with the available lattice data for the ghost dressing function [73,85] and the form factor of the ghost–gluon vertex evaluated in the soft-gluon limit [42,43].
- In Section 9, we discuss two important consequences of the masslessness of the ghost propagator, which manifest themselves at the level of both the gluon propagator and the three-gluon vertex. Specifically, the diagrams comprised by a ghost loop induce “unprotected” logarithms, i.e., of the type $ln{q}^{2}$; instead, gluonic loops give rise to “protected” logarithms, of the type $ln({q}^{2}+{m}^{2})$, where m is the effective gluon mass [172,234]. As ${q}^{2}\to 0$, the unprotected contributions diverge, driving the appearance of a maximum in the gluon propagator and a divergence in its first derivative, as well as a zero-crossing and a corresponding divergence in the form factors of the three-gluon vertex. As we comment in this section, of particular phenomenological importance [234,235,236,237,238,239,240] is the relative suppression that the above features induce to the dominant vertex form factors in the intermediate range of momenta.
- In Section 10, we discuss an outstanding feature of the WI satisfied by the pole-free part of the three-gluon vertex, namely the displacement induced by the presence of the aforementioned massless poles [93,124]. In this context, we introduce the key quantity denominated “displacement function”, whose appearance serves as a smoking gun signal of the action of the Schwinger mechanism in QCD; quite interestingly, it coincides [93,124] with the BS amplitude $\mathbb{C}\left(r\right)$ for the formation of a massless scalar out of a pair of gluons, introduced in Section 4. In addition, we derive a crucial relation, which ultimately permits the indirect determination of $\mathbb{C}\left(r\right)$ from lattice QCD [93,124,126]; an important ingredient in this relation is a partial derivative [124,241], denoted by $\mathcal{W}\left(r\right)$, of the ghost–gluon kernel [228], to be determined in the next section.
- In Section 11, we set up and solve the SDE that governs the evolution of $\mathcal{W}\left(r\right)$ [124,126,241,242]; the main component of this SDE is a special projection of the three-gluon vertex, which is computed by appealing to formulas established in Section 7, and allows for the accurate determination of $\mathcal{W}\left(r\right)$ in the entire range of relevant momenta [126].
- In Section 12, we substitute into the central relation derived in Section 10 the solution for $\mathcal{W}\left(r\right)$ found in the previous section, together with the lattice data [84,85] for the gluon propagator, the ghost dressing function, and the form factor of the three-gluon vertex associated with the soft-gluon limit, in order to obtain the form of the displacement function $\mathbb{C}\left(r\right)$[124,126]. As we discuss, the results exclude—with near-absolute certainty—the null hypothesis (absence of Schwinger mechanism, $\mathbb{C}\left(r\right)=0$), and corroborate the action of the Schwinger mechanism in QCD [126]. In addition, we show that the form of $\mathbb{C}\left(r\right)$ found is statistically completely compatible with that obtained from the BSE-based analysis presented in Section 4.
- In Section 13, we present our conclusions.
- Finally, in Appendix A, we derive the BQIs related to the displacement functions of the conventional and background vertices, while in Appendix B, we provide details about the renormalization scheme employed in our computations.

## 2. Basic Concepts and General Theoretical Framework

- (i)
- The propagator $\langle 0\left|\phantom{\rule{0.166667em}{0ex}}T\phantom{\rule{-0.166667em}{0ex}}\left[{Q}_{\mu}^{a}\left(q\right){Q}_{\nu}^{b}(-q)\right]\phantom{\rule{-0.166667em}{0ex}}\right|0\rangle $ that connects two quantum gluons. Notice that this propagator coincides with the conventional gluon propagator of the covariant gauges, defined in Equation (5), under the assumption that the corresponding gauge-fixing parameters, $\xi $ and ${\xi}_{{\scriptscriptstyle Q}}$, are identified, i.e., $\xi ={\xi}_{{\scriptscriptstyle Q}}$.
- (ii)
- The propagator $\langle 0\left|\phantom{\rule{0.166667em}{0ex}}T\phantom{\rule{-0.166667em}{0ex}}\left[{Q}_{\mu}^{a}\left(q\right){B}_{\nu}^{b}(-q)\right]\phantom{\rule{-0.166667em}{0ex}}\right|0\rangle $ that connects a ${Q}_{\mu}^{a}\left(q\right)$ with a ${B}_{\nu}^{b}(-q)$, to be denoted by ${\tilde{\Delta}}_{\mu \nu}^{ab}\left(q\right)=-i{\delta}^{ab}{\tilde{\Delta}}_{\mu \nu}\left(q\right)$.
- (iii)
- The propagator $\langle 0\left|\phantom{\rule{0.166667em}{0ex}}T\phantom{\rule{-0.166667em}{0ex}}\left[{B}_{\mu}^{a}\left(q\right){B}_{\nu}^{b}(-q)\right]\phantom{\rule{-0.166667em}{0ex}}\right|0\rangle $ that connects a ${B}_{\mu}^{a}\left(q\right)$ with a ${B}_{\nu}^{b}(-q)$, to be denoted by ${\widehat{\Delta}}_{\mu \nu}^{ab}\left(q\right)=-i{\delta}^{ab}{\widehat{\Delta}}_{\mu \nu}\left(q\right)$. Note that its full definition requires an additional gauge-fixing term, with the associated “classical” gauge-fixing parameter, ${\xi}_{{\scriptscriptstyle C}}$ [14,202,206].

## 3. Schwinger Mechanism in Yang–Mills Theories

- 1.
- $\mathbb{C}\left(r\right)$ and $\mathcal{C}\left(r\right)$ are the BS amplitudes describing the formation of gluon–gluon and ghost–anti-ghost colored composite bound states, respectively, see Section 4.
- 2.
- The gluon mass is determined by certain integrals that involve $\mathbb{C}\left(r\right)$ and $\mathcal{C}\left(r\right)$, given explicitly in Section 5.
- 3.
- $\mathbb{C}\left(r\right)$ and $\mathcal{C}\left(r\right)$ lead to smoking-gun displacements of the WIs. In fact, the displacement induced by $\mathbb{C}\left(r\right)$, has been confirmed by lattice QCD, by combining judiciously the results of several lattice simulations, see Section 5.2.

## 4. Dynamical formation of Massless Poles

- In order to exploit Equation (38), multiply the first equation by the factor ${P}_{{\mu}^{\prime}\mu}\left(r\right){P}_{\nu}^{{\mu}^{\prime}}\left(p\right)$.
- Take the limit of the system as $q\to 0$: this activates Equation (40) and introduces the functions $\mathbb{C}\left(r\right)$ and $\mathbb{C}\left(r\right)$.
- Isolate the tensor structures proportional to ${q}^{\alpha}$, and match the terms on both sides.
- Employ the “one-particle exchange” approximation for the kernels ${\mathcal{K}}_{ij}$, to be denoted by ${\mathcal{K}}_{ij}^{0}$, shown in Figure 5.

## 5. Generation of the Gluon Mass

#### 5.1. Gluon Mass from the ${q}_{\mu}{q}_{\nu}$ Component

#### 5.2. Gluon Mass from the ${g}_{\mu \nu}$ Component: Seagull Identity and Ward Identity Displacement

## 6. Renormalization Group Invariant Interaction Strength

## 7. Three-Gluon Vertex and Its Planar Degeneracy

## 8. Ghost Dynamics from Schwinger–Dyson Equations

## 9. Divergent Ghost Loops and Their Impact on the QCD Green’s Functions

## 10. Ward Identity Displacement of the Three-Gluon Vertex

## 11. The Ghost-Gluon Kernel Contribution to the Ward Identity

## 12. Displacement Function from Lattice Inputs

## 13. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BFM | background field method |

BQI | background-quantum identity |

BRST | Becchi–Rouet–Stora–Tyutin |

BS | Bethe–Salpeter |

BSE | Bethe–Salpeter equation |

EHM | emergent hadron mass |

MOM | momentum subtraction (renormalization scheme) |

PT | pinch technique |

QCD | quantum chromodynamics |

QED | quantum electrodynamics |

RGI | renormalization group invariant |

SDE | Schwinger–Dyson equation |

STI | Slavnov–Taylor identity |

WI | Ward identity |

## Appendix A. BQIs for the BSE Amplitudes

**Figure A1.**The auxiliary functions ${K}_{\mu}(q,r,p)$ and ${K}_{\mu \nu}(q,r,p)$ in the BQI of Equation (A1).

## Appendix B. The Asymmetric MOM Scheme

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