# New Aspect of Chiral SU(2) and U(1) Axial Breaking in QCD

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## Abstract

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## 1. Introduction

## 2. Coincidence between Chiral and Axial Symmetry Breaking

#### 2.1. Chiral and Axial Symmetry in Meson Susceptibility Functions

#### 2.2. Trivial Coincidence between Chiral and Axial Symmetry Breaking

#### 2.3. Flavor-Singlet Nature of Topological Susceptibility

#### 2.4. Correlation between Susceptibility Functions

#### 2.5. Nontrivial Coincidence between Chiral and Axial Symmetry Breaking

## 3. Chiral and Axial Symmetry Breaking in Low-Energy QCD Description

#### 3.1. Nambu–Jona–Lasinio Model

#### 3.2. Mean-Field Approximation and Vacuum of NJL Model

#### 3.3. Scalar- and Pseudoscalar-Meson Susceptibility in NJL Model

#### 3.4. Trivial and Nontrivial Coincidence of Chiral and Axial Breaking in a View of the NJL Description

## 4. Quark Mass Dependence on QCD Vacuum Structure

#### 4.1. QCD Vacuum Structure with Physical Quark Masses

#### 4.2. Crossover Domain

#### 4.3. First-Order Domain

#### 4.4. Chiral and Axial Symmetry Restorations in View of Chiral–Axial Phase Diagram

## 5. Summary and Discussion

- The predicted chiral–axial phase diagram in Figure 10 is a new guideline for exploring the influence of the $U{\left(1\right)}_{A}$ anomaly on the chiral phase transition, which is sort of giving a new interpretation of the conventional Columbia plot. Further studies are desired in lattice QCD simulations to draw definitely conclusive benchmarks on the chiral–axial phase diagram.
- The existence of the nontrivial coincidence implies that the $U{\left(1\right)}_{A}$ anomaly can be controlled by the current mass of the strange quark. The controllable anomaly can give a new understanding of the ${\eta}^{\prime}$ meson mass originated from the $U{\left(1\right)}_{A}$ anomaly. The investigation for the ${m}_{s}$-dependence on the pseudoscalar meson masses will thus be a valuable study.
- The fate of the $U{\left(1\right)}_{A}$ anomaly in the nuclear/quark matter is a longstanding problem and has attracted a lot of people so far. The nontrivial coincidence should also be realized in the finite dense matter involving the massless strange quark. The nontrivial coincidence at finite density may shed light on a novel insight for the partial $U{\left(1\right)}_{A}$ restoration in the medium with physical quark masses.
- The contribution of the $U{\left(1\right)}_{A}$ anomaly to the color confinement is an open question. It is worth including the Polyakov loop terms in the NJL model to address the correlation between the nontrivial coincidence of the chiral and axial breaking and the deconfinement–confinement phase transition.
- Though the NJL model produces the qualitative results consistent with lattice observations, it is a rough analysis due to the mean-field approximation. However, the existence of the nontrivial coincidence is robust because it is based on the anomalous Ward identity, This should thus be seen even beyond the mean-field approximation that the present NJL study has assumed or even more rigorous nonperturbative analyses, such as those based on the lattice NJL-model and the functional renormalization group method.
- The nontrivial chiral–axial coincidence is a generic phenomenon, which can also be seen in a generic class of QCD-like theories with “1 (${m}_{s}=0$) + 2 (${m}_{l}$) flavors”, involving models beyond the standard model. In particular, the coincidence in the first-order phase-transition case may impact cosmological implications of QCD-like scenarios with axionlike particles associated with the axial breaking, including the gravitational wave probes. Investigation along also this line may be interesting.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Conventional Columbia plot. The strength of chiral symmetry breaking coincides with the axial symmetry breaking strength in the meson susceptibility functions with ${m}_{s}=0$ because of the vanishing topological susceptibility. Thus, the simultaneous symmetry restoration between the chiral $SU\left(2\right)$ and $U{\left(1\right)}_{A}$ is realized on the ${m}_{u,d}$ axis independently of the order of the chiral phase transition.

**Figure 2.**The temperature dependence of the susceptibility functions at the physical point (${m}_{l}=5.5$ MeV and ${m}_{s}=138$ MeV) for (

**a**) $T/{T}_{\mathrm{pc}}=0-2$ and (

**b**) $T/{T}_{\mathrm{pc}}=1-2$. The pseusocritical temperature for the chiral crossover is observed to be ${T}_{\mathrm{pc}}\simeq 189$ MeV. The susceptibility functions are normalized by square of the pion decay constant ($\simeq 93$ MeV), and the temperature axis is also normalized by ${T}_{\mathrm{pc}}$, so all quantities are dimensionless to reduce the systematic uncertainty (approximately about 30%) associated with the present NJL model description of QCD. See also the text.

**Figure 3.**The plot showing the nontrivial coincidence between the chiral and axial indicators (of two types) in the crossover domain with the massless strange quark (${m}_{l}=5.5$ MeV and ${m}_{s}=0$; ${T}_{\mathrm{pc}}\simeq 144$ MeV). The topological susceptibility is exactly zero for all temperatures because the flavor-singlet nature associated with the massless strange quark. Scaling factors are applied on both horizontal and vertical axes in the same way as in Figure 2.

**Figure 4.**The plot showing the finiteness of the topological susceptibility along with the temperature dependence of the chiral and axial indicators in the crossover domain with a small strange quark mass (${m}_{l}=5.5$ MeV and ${m}_{s}={10}^{-3}{m}_{l}$; ${T}_{\mathrm{pc}}\simeq 144$ MeV). The same scaling for two axes has been made as in Figure 2.

**Figure 5.**The plots clarifying the significant interference of the topological susceptibility to make the sizable discrepancy between the chiral and axial indicators in the crossover domain with the large strange quark mass (${m}_{l}=5.5$ MeV and ${m}_{s}$ = 10 ${m}_{l}$; ${T}_{\mathrm{pc}}\simeq 174$ MeV) for (

**a**) $T/{T}_{\mathrm{pc}}=0-2$ and (

**b**) $T/{T}_{\mathrm{pc}}=1-2$. The manner of scaling axes is the same as in Figure 2.

**Figure 6.**The plots clarifying the trend of the nontrivial simultaneous restoration for the chiral and axial symmetries even in the chiral first-order phase-transition domain with ${m}_{l}=0.1$ MeV and ${m}_{s}=0$. Panel (

**a**) shows a jump in the mesons susceptibility functions at around ${T}_{\mathrm{c}}\simeq 119$ MeV as a consequence of the first-order phase transition. The panel (

**b**) closes up the temperature dependence for the chiral and axial indicators after the chiral phase transition. The manner of scaling axes is the same as in Figure 2. The trend induced by the interference of ${\chi}_{\mathrm{top}}$ is similar to the one observed in the crossover domain in Figure 3.

**Figure 7.**The plots showing the still almost coincidence of the chiral and axial indicators for all temperature ranges, even in the first-order phase-transition domain with ${m}_{l}=0.1$ MeV and ${m}_{s}={10}^{-4}{m}_{l}$; ${T}_{\mathrm{c}}\simeq 119$ MeV. The two displayed axes are scaled in the same way as explained in the caption of Figure 2. The trend induced by the interference of ${\chi}_{\mathrm{top}}$ is similar to the one observed in the crossover domain in Figure 4.

**Figure 8.**The plots clarifying the significant interference of ${\chi}_{\mathrm{top}}$ with the chiral and axial indicators in the first-order phase-transition domain with ${m}_{l}=0.1$ MeV and ${m}_{s}$ = 10 ${m}_{l}$; ${T}_{\mathrm{c}}\simeq 126$ MeV. The two displayed axes are scaled in the same way as explained in the caption of Figure 2. The trend induced by the interference of ${\chi}_{\mathrm{top}}$ is similar to the one observed in the crossover domain in Figure 5.

**Figure 9.**The strange quark mass dependence on the difference between the axial indicator $|{\chi}_{\pi -\delta}|$ ($|{\chi}_{\eta -\sigma}|$) and the chiral indicator $|{\chi}_{\eta -\delta}|$ ($|{\chi}_{\pi -\sigma}|$) in (

**a**) the crossover domain and (

**b**) the first-order phase-transition domain. In the crossover domain with ${m}_{s}/{m}_{l}=0\phantom{\rule{0.166667em}{0ex}}\left(50\right)$, the pseudocritical temperature is evaluated as ${T}_{\mathrm{pc}}\simeq 144\phantom{\rule{0.166667em}{0ex}}\left(200\right)$ MeV, and the temperatures $T=300-400$ MeV displayed as in panel (

**a**) correspond to $T\simeq (1.5-2.8)\phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{pc}}$. On the other hand, in the first-order phase-transition domain, the temperatures $T=240-300$ MeV as fixed in panel (

**b**) correspond to $T\simeq (1.8-2.5)\phantom{\rule{0.166667em}{0ex}}{T}_{\mathrm{c}}$, where ${T}_{\mathrm{c}}\simeq 119\phantom{\rule{0.166667em}{0ex}}\left(130\right)$ MeV for ${m}_{s}/{m}_{l}=0\phantom{\rule{0.166667em}{0ex}}\left(20\right)$.

**Figure 10.**The predicted chiral–axial phase diagram on the ${m}_{u,d}$-${m}_{s}$ plane, in which the discrepancy of the chiral and axial symmetry restorations at around $T\sim (1.5-2.0)\phantom{\rule{0.166667em}{0ex}}{T}_{\left(\mathrm{p}\right)\mathrm{c}}$ is drawn by the shaded area. When the strength of the axial symmetry breaking deviates from the chiral breaking strength to be large, the shaded areas become thick. The nontrivial coincidence, as in Equation (22), is associated with the vanishing ${\chi}_{\mathrm{top}}$, which is located on the ${m}_{u,d}$ axis. When the strange quark mass obtains a finite mass, the axial restoration deviates from the chiral restoration. At around ${m}_{l}=O\left(10{m}_{l}\right)$, the axial restoration is much later than the chiral restoration because of the significant interference of ${\chi}_{\mathrm{top}}$. Namely, at the physical quark masses, the topological susceptibility provides the large discrepancy between the chiral and axial restorations in the meson susceptibilities.

**Figure 11.**The split in the restorations of the chiral $SU\left(2\right)$ symmetry and the $U\left(1\right)$ axial symmetry at hot QCD. (

**a**): Ordinary way to address the symmetry restorations. The ambiguous origin of the effective $U{\left(1\right)}_{A}$ restoration is often measured by the topological susceptibility (which is normalized by the quark mass, ${\overline{\chi}}_{\mathrm{top}}=4{\chi}_{\mathrm{top}}/{m}_{l}^{2}$). (

**b**): New point of view for symmetry restorations at ${m}_{s}=0$. Because of the anomalous Ward–Takahashi identity at hot QCD, the chiral $SU\left(2\right)$ symmetry breaking exactly coincides with the $U{\left(1\right)}_{A}$ symmetry breaking, and this coincidence holds for any temperatures. As a robust consequence, when the chiral $SU\left(2\right)$ symmetry is restored at the (pseudo)critical temperature, the $U{\left(1\right)}_{A}$ symmetry is simultaneously restored. Therefore, the limit ${m}_{s}=0$ manifests the symmetry restorations on the quark mass plane: it can be unambiguously understood that the strange quark mass handles the split in the restorations of chiral $SU\left(2\right)$ symmetry and the $U{\left(1\right)}_{A}$ symmetry at hot QCD with the three quark flavors having finite masses.

Parameters | Values |
---|---|

light quark mass ${m}_{l}$ | 5.5 MeV |

strange quark mass ${m}_{s}$ | 138 MeV |

four-fermion coupling constant ${g}_{s}$ | 0.358 ${\mathrm{fm}}^{2}$ |

six-fermion coupling constant ${g}_{D}$ | −0.0275 ${\mathrm{fm}}^{5}$ |

cutoff $\Lambda $ | 631.4 MeV |

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**MDPI and ACS Style**

Cui, C.-X.; Li, J.-Y.; Matsuzaki, S.; Kawaguchi, M.; Tomiya, A.
New Aspect of Chiral *SU*(2) and *U*(1) Axial Breaking in QCD. *Particles* **2024**, *7*, 237-263.
https://doi.org/10.3390/particles7010014

**AMA Style**

Cui C-X, Li J-Y, Matsuzaki S, Kawaguchi M, Tomiya A.
New Aspect of Chiral *SU*(2) and *U*(1) Axial Breaking in QCD. *Particles*. 2024; 7(1):237-263.
https://doi.org/10.3390/particles7010014

**Chicago/Turabian Style**

Cui, Chuan-Xin, Jin-Yang Li, Shinya Matsuzaki, Mamiya Kawaguchi, and Akio Tomiya.
2024. "New Aspect of Chiral *SU*(2) and *U*(1) Axial Breaking in QCD" *Particles* 7, no. 1: 237-263.
https://doi.org/10.3390/particles7010014