# Holographic Non-Abelian Flavour Symmetry Breaking

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## Abstract

**:**

## 1. Introduction

## 2. Summary of the Non-Abelian DBI Action in the D3/Probe D7 System

## 3. A Bottom-Up Non-Abelian Model of the $\mathcal{N}=\mathbf{2}$ Theory

#### 3.1. Example 1—Equal, Real Masses

#### 3.2. Example 2—${N}_{f}=2$ Split, Real Masses

#### 3.2.1. Diagonal States

#### 3.2.2. Off-Diagonal States

## 4. A Bottom-Up Non-Abelian Dynamic AdS/QCD Model

#### 4.1. Kinetic Terms

#### 4.2. Potential

#### 4.3. The Higgs Mechanism for the Vector Gauge Field

#### 4.4. Scenario 1—${N}_{f}$ Equal Masses

#### 4.5. Scenario 2—Two Equal-Mass Quarks and $1/N$ Effects

#### 4.6. Scenario 3—${N}_{f}=2$ Split Masses

#### 4.7. Scenario 4—${N}_{f}=3$ Split Masses, ${m}_{u}={m}_{d}\ll {m}_{s}$

## 5. Other Symmetry Breaking Patterns

#### 5.1. SU$\left(2N\right)\to $ Sp$\left(2N\right)$

#### 5.2. SU$\left(2N\right)\to $ SO$\left(2N\right)$

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Equations of Motions, Nf = 2 Split Masses

**Scalars**

**Pseudo-Scalars**

**Vectors**

**Axial-Vectors**

## Appendix B. Equations of Motions, Nf = 3

## References

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**Figure 1.**Here, we show a numerical method to the solution of the mixed equations, Equations (31) and (32). The diagonal basis is known and the solutions in each of the u and d sectors are ${M}^{2}=4(n+1)(n+2){m}_{q}$. Here, with fixed $\sigma \left(0\right)=1,\prime {\sigma}^{\prime}\left(0\right)=0,\prime {\tau}^{\prime}\left(0\right)=0$, we vary the mass squared and the value of $\tau \left(0\right)$. We plot the quantity $1/\left(\right|\sigma (\infty )|+|\tau (\infty )\left|\right)$, which diverges when both the fields vanish asymptotically. In this case, we have set one quark mass to unity and the other to 0.5 there are clear peaks at $\tau \left(0\right)=-1$ and ${M}^{2}=2,\prime 6$, which are the zeroth and first excited states of the d quark state. The peak at $\tau \left(0\right)=1$ and ${M}^{2}=8$ is the ground state of the u quark.

**Figure 2.**Left plot: different embeddings for various quark masses. We input the boundary quark mass values (2.3, 4.6, 95.5) MeV for ${L}_{i}$, $i=u,d,s$, respectively, in units where ${m}_{\rho}=775$ MeV. The quark condensates can be calculated numerically using Equation (51)—we find ($(65.7$ ${\mathrm{MeV})}^{3}$, $(67.3$ ${\mathrm{MeV})}^{3}$, $(180.2$ ${\mathrm{MeV})}^{3}$) for the three flavours. Right plot: the dashed line shows the effect of a double trace term with $\kappa =-0.3$ (dashed line) compared to $\kappa =0$ (full line). (

**a**) Embedding vs. quark masses. (

**b**) Embedding with $\kappa $.

**Figure 3.**Dependence of the scalar masses on $\kappa $ when quark mass ${m}_{q}=2$ MeV. $\kappa $ is the coupling of the double trace term introduced in Equation (76), which gives a small splitting in the singlet and triplet scalar masses ${m}_{\sigma}$ and ${m}_{{\xi}_{3}}$.

**Figure 4.**Left plot: The ${\pi}^{\pm}$ mass increases with the growing quark mass splitting at fixed ${m}_{u}$ and $\kappa =0$. Right plot: The scalar masses in the presence of $\kappa \rho \mathrm{Tr}\phantom{\rule{0.166667em}{0ex}}{\left({X}^{\u2020}X\right)}^{2}$ in the split mass case. The shaded blue area marks the valid data range of the ${f}_{0}\left(980\right)$, and the yellow area marks that of the ${a}_{0}\left(980\right)$. This indicates a range $\kappa \in [-0.4,0]$. (

**a**) ${\pi}^{\pm}$ vs. $\Delta {m}_{q}$. (

**b**) The diagonal scalar fields vary with $\kappa $.

**Table 1.**Meson masses of the lowest lying states in the U(N${}_{f}$) model with equal quark masses. The $\rho $ meson mass is fixed to 775 MeV (therefore marked with an asterisk) and we have set ${m}_{q}={m}_{u}=2.3$ MeV and $\kappa =-0.3$ (the asterisk next to 994 means when ${m}_{{f}_{0}\left(980\right)}$ = 994 MeV we get $\kappa =-0.3$). The QCD masses are taken from the PDG [53]. The mass difference in ${f}_{0}\left(980\right)$ and ${a}_{0}\left(980\right)$ is introduced by the double trace term with the coupling $\kappa $. The ${\pi}^{0}$ mass is very sensitive to the quark mass, this explains the large deviation.

Observables | QCD [MeV] | $\mathit{U}({\mathit{N}}_{\mathit{f}}=2)$ [MeV] | Deviation |
---|---|---|---|

${M}_{\rho \left(770\right)}$ | $775.26\pm 0.23$ | 775 * | fitted |

${M}_{{a}_{1}\left(1260\right)}$ | $1230\pm 40$ | 1194 | 3% |

${M}_{{f}_{0}\left(980\right)}$ | $990\pm 20$ | 994 * | <1% |

${M}_{{a}_{0}\left(980\right)}$ | $980\pm 20$ | 997 | 2% |

${\pi}^{0}$ | $134.9768\pm 0.0005$ | 117 | 14% |

**Table 2.**Meson masses of the lowest lying states in the U(N${}_{f}$) model with unequal quark masses, ${m}_{u}=2.3$ MeV and ${m}_{d}=4.6$ MeV. The $\rho $ meson mass is fixed to 775 MeV (therefore marked with an asterisk). The QCD masses are taken from the PDG [53]. Notice the deviation of pion from the initial ${\pi}^{0}$ state to the ${\pi}^{\pm}$ state.

Observables | QCD [MeV] | $\mathit{U}({\mathit{N}}_{\mathit{f}}=2)$ [MeV] | Deviation |
---|---|---|---|

${M}_{\rho \left(770\right)}$ | $775.26\pm 0.23$ | 775 * | fitted |

${M}_{{a}_{1}\left(1260\right)}$ | $1230\pm 40$ | 1196 | 3% |

${M}_{{a}_{0}\left(980\right)}$ | $980\pm 20$ | 998 | 2% |

${\pi}^{\pm}$ | $139.57039\pm 0.00017$ | 146 | 2% |

**Table 3.**Meson masses in the three flavour case compared with the experimental data [53]. We have fixed the masses for the vector bosons $\rho $ and $\omega $ and calculated the masses for the axial vectors (therefore the asterisk next to 775), the scalars and the pNGBs. The quark masses used are ${m}_{u}={m}_{d}=3.1$ MeV and ${m}_{s}=95.7$ MeV.

Observables | QCD [MeV] | ${\mathit{N}}_{\mathit{f}}=3$—Split Masses [MeV] | Deviation |
---|---|---|---|

${M}_{\rho \left(770\right),\omega \left(782\right)}$ | $775.26\pm 0.23$ | 775 * | fitted |

${M}_{{K}^{*}\left(892\right)}$ | $891.67\pm 0.26$ | 1009 | 12% |

${M}_{\varphi \left(1020\right)}$ | $1019.461\pm 0.016$ | 1048 | 3% |

${M}_{{a}_{1}\left(1260\right)}$, ${M}_{{f}_{1}\left(1285\right)}$ | $1230\pm 40$ | 1104 | 11% |

${M}_{{K}_{1}\left(1400\right)}$ | $1403\pm 7$ | 1377 | 2% |

${M}_{{f}_{1}\left(1420\right)}$ | $1426.3\pm 0.9$ | 1713 | 18% |

${M}_{{a}_{0}\left(980\right)}$, ${M}_{{f}_{0}\left(980\right)}$ | $980\pm 20$ | 929 | 5% |

${M}_{{K}_{0}^{*}\left(700\right)}$ | $845\pm 17$ | 876 | 4% |

${M}_{{f}_{0}\left(1370\right)}$ | 1370 | 970 | 34% |

${M}_{\pi}$ | $139.57039\pm 0.00017$ | 139 | 1% |

${M}_{K}$ | $497.611\pm 0.013$ | 584 | 16% |

${M}_{{\eta}^{\prime}}$ | $957.78\pm 0.06$ | 791 | 19% |

${M}_{\pi \left(1300\right)}$ | $1300\pm 100$ | 1438 | 10% |

${M}_{K\left(1460\right)}$ | 1460 | 1807 | 21% |

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**MDPI and ACS Style**

Erdmenger, J.; Evans, N.; Liu, Y.; Porod, W.
Holographic Non-Abelian Flavour Symmetry Breaking. *Universe* **2023**, *9*, 289.
https://doi.org/10.3390/universe9060289

**AMA Style**

Erdmenger J, Evans N, Liu Y, Porod W.
Holographic Non-Abelian Flavour Symmetry Breaking. *Universe*. 2023; 9(6):289.
https://doi.org/10.3390/universe9060289

**Chicago/Turabian Style**

Erdmenger, Johanna, Nick Evans, Yang Liu, and Werner Porod.
2023. "Holographic Non-Abelian Flavour Symmetry Breaking" *Universe* 9, no. 6: 289.
https://doi.org/10.3390/universe9060289