# Mean-Field Expansion, Regularization Issue, and Multi-Quark Functions in Nambu–Jona-Lasinio Model

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## Abstract

**:**

## 1. Introduction

## 2. The Mean-Field Expansion in the Bilocal-Source Formalism

#### 2.1. First-Step Equations

#### 2.2. Pion and Sigma-Meson

## 3. Regularization Issue: DAR

- (i)
- All computations are made in 4D Euclidean space;
- (ii)
- Translational invariance is supposed;
- (iii)
- The regularization procedure includes of modification of integration measure via the weight function which provides the convergence of integrals.

#### Two-Quark Amplitude and Model Parameters in Leading Approximation

## 4. Regularization Issue: Predictive Formulation of the Nambu-Jona-Lasinio Model and Ghost Problem

- (1)
- (2)
- (3)
- at $a>1$ Equation (57) possesses one negative root ${x}_{1}<0$ and two positive roots $0<{x}_{2}<1$ and ${x}_{3}>1$.

## 5. Meson Contributions in Chiral Condensate and in Quark Propagator

#### 5.1. Pion Contribution

#### 5.2. Scalar Contribution

#### 5.3. Improved Model Parameters

## 6. The Corrections to the Two-Quark Function and the Legendre Transform

## 7. The Equations for Multi-Quark Functions

#### 7.1. Four-Quark and Three-Quark Functions

#### 7.2. VERTEX $\sigma \pi \pi $

#### 7.3. NLO Two-Particle Function and Correction to Pion-Decay Constant

#### 7.4. Structure of Third Step of Iterations

## 8. The Generalization of the Method for Other Types of Multi-Quark Sources

## 9. Results and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Garibli, A.A.; Jafarov, R.G.; Rochev, V.E.
Mean-Field Expansion, Regularization Issue, and Multi-Quark Functions in Nambu–Jona-Lasinio Model. *Symmetry* **2019**, *11*, 668.
https://doi.org/10.3390/sym11050668

**AMA Style**

Garibli AA, Jafarov RG, Rochev VE.
Mean-Field Expansion, Regularization Issue, and Multi-Quark Functions in Nambu–Jona-Lasinio Model. *Symmetry*. 2019; 11(5):668.
https://doi.org/10.3390/sym11050668

**Chicago/Turabian Style**

Garibli, Aydan A., Rauf G. Jafarov, and Vladimir E. Rochev.
2019. "Mean-Field Expansion, Regularization Issue, and Multi-Quark Functions in Nambu–Jona-Lasinio Model" *Symmetry* 11, no. 5: 668.
https://doi.org/10.3390/sym11050668