# Study of Dynamical Chiral Symmetry Breaking in (2 + 1) Dimensional Abelian Higgs Model

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## Abstract

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**PACS**11.10.Kk; 11.15.Tk; 11.30.Qc

## 1. Introduction

_{3}) has been extensively studied for over twenty years. It has many features similar to quantum chromodynamics (QCD), such as spontaneous chiral symmetry breaking in the massless fermion limit and confinement [1,2,3,4,5,6,7,8,9,10,11,12,13]. Moreover, it is super-renormalizable, so it does not suffer from the ultraviolet divergence which are present in QED

_{4}. Due to these reasons it can serve as a toy model of QCD. In parallel with its relevance as a tool to gain insight into the aspects of QCD, QED

_{3}is also found to be equivalent to the low-energy effective theories of strongly correlated electronic systems. Recently, QED

_{3}has been studied in high T

_{c}cuprate superconductors [14,15,16,17,18,19,20,21,22,23], quantum Heinsenberg antiferromagnets [24], and fractional quantum Hall effect [25]. In particular, some authors have made progress in studying the property of graphene based on QED

_{3}[26,27,28].

_{3}with N fermion flavors by solving the DSE for fermion self-energy in the lowest-order of $1/N$ expansion and found DCSB occurs when N is less than a critical number ${N}_{c}$. Later Nash showed that the critical number of fermion flavor still exists by considering higher order corrections and he obtained ${N}_{c}=\frac{128}{3{\pi}^{2}}$ [5]. In 1995, Maris solved the coupled DSEs with a set of simplified vertex functions and obtained the critical number of fermion flavor ${N}_{c}$ = 3.3 [7,8]. Soon after that, Fisher et al. [30] self-consistently solved a set of coupled DSE and obtained ${N}_{c}^{crit}\approx 4$ by using more sophisticated vertex ansatz which satisfies the Ward–Takahashi identity. Here it should be noted that all the above results hold under the condition that the gauge boson is massless. Once a finite gauge boson mass ${m}_{a}$ is generated by Anderson–Higgs mechanism, it weakens the strength of interaction and affects DCSB. QED

_{3}with Abelian Higgs model has been widely studied as the effective theory of the high T

_{c}superconductors [17,18]. Recently, Liu et al. [31,32,33] studied the DSEs for the fermion self-energy in Landau gauge in QED

_{3}with Abelian Higgs model and found that DCSB occurs only when the gauge boson mass ${m}_{a}$ is smaller than a critical value. However, we note that in [31,32,33], the authors used the so-called nonlocal gauge function approach to solve the nonlinear DSE where the wave function renormalization and the vertex correction are simply absent (in connection to the use of the nonlocal gauge in QED

_{3}, one can see, e.g., [34]). Because of its importance, this problem deserves further study. In this paper, instead of adopting the approximations in [31,32,33], we numerically solve the coupled DSE for the fermion and gauge boson propagators of QED

_{3}with Abelian Higgs model using a specific truncation for the fermion-photon vertex ansatz for a range of finite gauge boson mass.

## 2. Results and Discussions

_{3}with N massless fermion flavors and N scalar boson flavors is $\mathcal{L}$ = ${\mathcal{L}}_{F}$ + ${\mathcal{L}}_{B}$ [7,33], where

_{3}with Abelian Higgs model the gauge field couples to both the fermion field and the scalar boson field. ${\Pi}_{\mu \nu}\left(q\right)={\Pi}_{\mu \nu}^{F}\left(q\right)+{\Pi}_{\mu \nu}^{B}\left(q\right)$ is the total vacuum polarization tensor and the full inverse gauge boson propagator is

_{1}vertex). This choice has the advantage that the equations are simplified significantly and it already contains all qualitative features of the solution employing the CP/BC vertex in the infrared region, as was demonstrated by the numerical calculations given in [30]. Based on the above discussion, we obtain the coupled DSEs with gauge boson mass ${m}_{a}$ and Higgs mass ${m}_{h}$ [47,48]:

_{1}vertex ansatz. It can be seen that for both the bare vertex and the BC

_{1}vertex ansatz, the mass function almost remains constant for small ${p}^{2}$, and decreases monotonously with ${p}^{2}$ increasing after ${p}^{2}$ reaches a certain value. In the whole range of ${p}^{2}$, the mass function obtained using the BC

_{1}vertex is much larger than that obtained using the bare vertex. This shows that the dressing effect of the fermion-photon vertex is very important in the study of dynamical mass generation in the Abelian Higgs Model in $2+1$ dimensions. In addition, due to the fact that increasing gauge boson mass weakens the attractive force between a pair of fermion and antifermion, the larger infrared value of the mass function for the case of BC

_{1}ansatz implies that the critical gauge boson mass for the case of BC

_{1}ansatz should be larger that the one for the case of bare vertex.

_{3}theory (and its generalization to the Abelian Higgs model) with $N=2$ is employed as an effective continuum theory for the 2D quantum antiferromagnetic (Neel) ordering corresponding to dynamical fermion mass generation [49,50,51,52]. So a reliable value of the critical number of fermion flavors is important for the study of dynamical fermion mass generation and chiral symmetry breaking.

## 3. Conclusions

_{3}with Abelian Higgs model is studied. We numerically solve the coupled DSEs by means of iteration method using the BC

_{1}vertex ansatz and compare our result with that obtained using the bare vertex. It is found that the mass function obtained using the BC

_{1}vertex ansatz is much larger than the one obtained using the bare vertex. This shows that the dressing effect of the fermion-photon vertex is very important in the study of dynamical mass generation in the Abelian Higgs model in $2+1$ dimensions. It is also found that the gauge boson mass ${m}_{a}$ suppresses the critical number of fermion flavor ${N}_{c}$ for a fixed ratio $r=\frac{{m}_{h}}{{m}_{a}}$. When ${m}_{a}$ exceeds a critical value ${m}_{a}^{crit}$, DCSB will be completely suppressed. One the other hand, the gauge boson mass is reduced rapidly as r increases. These results imply that for a fixed value of ${m}_{a}$, the smaller is the ratio r, the easier is it to generate a finite fermion mass by the mechanism of DCSB. The above results qualitatively accord with the conclusion of Liu et al. [33]. Finally, we note that the BC

_{1}vertex ansatz employed in our calculation does not satisfy the Ward–Takahashi identity when the dynamical mass function $B\left({p}^{2}\right)$ is present in the fermion propagator. In a more reliable calculation the CP [39] vertex should be employed. This work will be done in the future.

## Acknowledgments

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**Figure 1.**The mass function $M\left({p}^{2}\right)$ versus ${p}^{2}$ at $r=\frac{{m}_{h}}{{m}_{a}}=20$ and ${m}_{h}=0.001$ for $N=1$ case calculated using both the bare vertex and the BC

_{1}vertex ansatz.

**Figure 2.**Dependence of the number of fermion flavor ${N}_{c}$ on ${m}_{a}$ for several values of the ratio $r=\frac{{m}_{h}}{{m}_{a}}$.

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**MDPI and ACS Style**

Li, J.-F.; Huang, S.-S.; Feng, H.-T.; Sun, W.-M.; Zong, H.-S.
Study of Dynamical Chiral Symmetry Breaking in (2 + 1) Dimensional Abelian Higgs Model. *Symmetry* **2010**, *2*, 907-915.
https://doi.org/10.3390/sym2020907

**AMA Style**

Li J-F, Huang S-S, Feng H-T, Sun W-M, Zong H-S.
Study of Dynamical Chiral Symmetry Breaking in (2 + 1) Dimensional Abelian Higgs Model. *Symmetry*. 2010; 2(2):907-915.
https://doi.org/10.3390/sym2020907

**Chicago/Turabian Style**

Li, Jian-Feng, Shi-Song Huang, Hong-Tao Feng, Wei-Min Sun, and Hong-Shi Zong.
2010. "Study of Dynamical Chiral Symmetry Breaking in (2 + 1) Dimensional Abelian Higgs Model" *Symmetry* 2, no. 2: 907-915.
https://doi.org/10.3390/sym2020907