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UDC 530.1; 539.1
SCHWINGER-DYSON ANALYSIS OF 1+2 DIMENSIONAL QED
T. Inagaki
information Media Center, Hiroshima University, 1-7-1 Kagamivama, Higashi-Hiroshima, 739-8521, Japan.
E-mail: inagaki@hiroshima-u.ac.jp
The QED ground state is studied for a four component Dirac fermion with the topological Chern-Simons term in 1+2 dimension. The Lagrangian for a massless Dirac fermion has the parity and the chiral symmetries. Spontaneous breaking of these symmetries can be found by evaluating the vacuum expectation values for fermion and anti-fermion pairs. We introduce approximate vertex functions and calculate the expectation values by solving the Schwinger-Dyson equation. We observe a smaller gauge dependence for the Ball-Chiu vertex function.
Keywords: l-f-2 dimensional QED, Schwinger-Dvson equation, Chern-Simons term.
1 Introduction
for fermion and anti-fermion pairs.
Spontaneous symmetry breaking is one of the fundamental concepts in elementary particle physics. Non-perturbative effect is essential for spontaneous symmetry breaking caused by gauge interactions. The Schwinger-Dyson (SD) equation is often used to evaluate it. A solution of the SD equation for a propagator gives a full self-energy. However, a full-vertex function is necessary to solve the SD equation. We usually set some approximate vertex functions in order to define a closed form for the SD equation. In the ladder approximation we replace the full-vertex function by the tree level one.
The Ward-Takahashi (WT) identity is preserved in the ladder approximation with the Landau gauge for 1+3 dimensional QED. It should be noted that the physical observable in the solution of the SD equation depends on the gauge parameter because of the approximate vertex function. The ladder approximation with the Landau gauge can not preserve the WT identity in low dimensional systems or thermal environment. The validity of the ladder approximation can not be guaranteed in such systems. Much works have been done to find a better vertex function [1-5] and a gauge parameter [6-8].
The 1+2 dimensional QED with the Chern-Simons (CS) term is one of the simple models to develop the procedure to analyze the SD equation beyond the ladder approximation. In this paper we employ-some vertex functions and solve the SD equation and show the gauge dependence of the result. In Sec. 2 we introduce the 1+2 dimensional QED with the CS term and review the chiral symmetry for a four component Dirac fermion. In Sec. 3 we give an explicit form for the SD equation and adopt some vertex functions. Evaluating the SD equation numerically, we show the gauge dependence for the vacuum expectation values
2 Chiral symmetry in 1+2 dimensional QED
The QED Lagrangian is uniquely fixed by the gauge symmetry, the Lorentz covariance and the renormalizability. We start form the Lagrangian density with a massless Dirac fermion,
L = -1 - + Ap
4
2a
(1)
The first term in the right hand side is ordinary QED Lagrangian density written by the field strength, = dMAv — dvAM. The second term is the gauge fixing part in the Lorentz gauge with the gauge parameter, a. The third term is called the CS term. This term is topologically determined and can be written as a total divergence. The coefficient p introduces a topological mass for the gauge field, AM. = +ieAM represents the covariant derivative for the Dirac fermion, —. ym denotes the 4 x 4 Dirac matrices. Since we consider the 1+2 dimensional space-time, the indices, p, v run from 0 to 2.
The Clifford algebra in 1+2 dimensions is defined by the anti-commutator for the Dirac matrices,
|7M,7V} = , P, v G{0,1, 2}. (2)
There are two Casimir operators, y5,
{Y3,Ym} = 0, {Y5,Ym} = 0. (3)
Thus we can define two-kinds of the chiral transformations,
- ^ eiY'03- ^ e^5(4) where i^d 05 are real parameters.
T. Inagaki. Schwinger-Dyson analysis of 1+2 dimensional QED
The Lagrangian (1) is invariant under the chiral transformations (4). If the fermion acquire a non-vanishing mass, the mass term, mW breaks the chiral symmetries. We can introduce a parity odd mass term,
mT t =2 {Y3,Y5}.
/p - £(p) + ie'
with the fermion self-energy,
i£(p)
d3k
(p)
p2 —
PßPv
__1 p _ • PmPV
2 2 2 eMvpp ia 4 ,
p2 — p2
and
rM(p, k) = 7M.
¿(p — k)Mr^(p, k) = S-1(k) — S-1(p).
Here we introduce the parity projection operators
1 ± T
X± =
and assume a decomposed form for the fermion propagator,
S (p)
A+/p — B-
-X+ +
A_/p — B.
-X-
(12)
(5)
This term is invariant under the chiral transformations. Therefore the spontaneous breaking for the chiral and the parity symmetries is found by observing the vacuum expectation values for the composite operators, (VV) and (V>t^), respectively [6] .
3 SD euqation in 1+2 dimensional QED
The vacuum expectation values, (VV) and (VtV) are calculated by taking the trace of the fermion propagator. These values vanish for the massless fermion without considering the non-perturbative contribution of the gauge interaction. Here we introduce the non-perturbative effect by solving the SD equation. The SD equation for the fermion propagator, S(p), is given by
where A± and B± are functions of p. Then the WT identity (10) fix the longitudinal part of the vertex function [91,
rL(p,k) =rf-CMX+ +rB
-^BCß
X_
(13)
with
rBCß 1 L±
2 [A± (p) + A±(k)]7 M
1 + k^ + -[A±(p) — A± (k)] (/p + /k)
— [B±(p) — B± (k)]
pM + k^ p2 - k2
(14)
(6)
(p — k)S(k)rv (p,k), (7)
where (p — k) and TM(p, k) are the full gauge boson propagator and the full vertex function, respectively. The self-energy £(p) h^ an 4 x 4 matrix form and can contain a parity odd and a chirality odd mass terms.
If we insert some functional forms for (p — k) and rM(p, k), we can solve the SD equation (6) and find the fermion self-energy, £(p). In the ladder approximation we employ a tree-level gauge boson propagator and a vertex function,
The transverse part of the vertex function can not be fixed by the WT identity. We simply drop the transfers part below.
The Ball-Chiu (BC) vertex function (13) satisfies the WT identity by definition and may be a better assumption for the vertex function on the SD equation for the 1+2 dimensional QED with the CS term. Employing the Ball-Chiu vertex function and a tree-level gauge boson propagator, we numerically solve the SD equation and calculate the vacuum expectation values for fermion and anti-fermion pairs. It is found that the obtained expectation value (VtV) depends on the gauge parameter, a. Between a = 0 and 1 we fit the expectation value 3B cl linear function of the gauge a
(V>T^) — (V'T^I^q = aa + O(a2),
(15)
(8)
(9)
It is known that the ladder approximation is not consistent with the WT identity,
(10)
(11)
and find a = —(0.495 ± 0.005) x 10-3. In the ladder approximation we obtain a = (1.79 ± 0.06) x 10-3. A smaller gauge dependence is observed for the BC vertex function compared with the ladder approximation. It should be noted that both results approach near the a=0
seems to be a good assumption with the Landau gauge. Details of the numerical algorithm and the results are given in Ref. [10].
4 Concluding remarks
We have performed the SD analysis of 1+2 dimensional QED with the CS term. The often used ladder approximation is extended to avoid a large gauge dependence. For the BC vertex function a non-trivial vertex correction is introduced through the WT identity. The solution of the SD equation shows a smaller gauge dependence for the BC vertex function. We have also pointed out that the ladder approximation is not so bad with the Landau gauge.
2
e
2
p
Here we have applied some assumptions to solve the SD equation. The tree level propagator is adopted for the gauge boson. It is interesting to introduce the SD equation for the gauge boson propagator and simultaneously solve it with the SD equation (6). We have simply droped the transverse part of the vertex function. It is generated even at the one-loop level [11]. It is also interesting to study the gauge dependence for some possible assumptions of the transverse part of the vertex function. We hope to report any progress
for these problems in future.
Acknowledgement
The main part of this paper is based on the work in collaboration with Y. Mizutani and Y. Hoshino. The authors thank Wataru Sakamoto for a preliminary-contribution at the early stage of this work. TI is supported by JSPS KAKENHI Grant Number 26400250.
References
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[2] Bashir A. and Raya A. 2007 Few Body Syst. 41, 185.
[3] Bashir A., Raya A. and Sanchez-Madrigal S. 2011 Phys. Rev. D84 036013.
[4] Madrigal S. S., Hofmann C. P. and Raya A. 2011 J. Phys. Conf. Ser. 287 012028.
[5] Zhu H. x., Feng H. t„ Sun W. m. and Zong H. s. 2013 J. Mod. Phys. 4, 151.
[6] Ebihara Т., lizuka Т., Kondo К. I. and Tanaka E. 1995 Nucl. Phys. B434 85.
[7] Kondo К. I. and Maris P. 1995 Phys. Rev. Lett. 74 18.
[8] Kondo К. I. and Maris P. 1995 Phys. Rev. D52 1212.
[9] Ball J. S. and Chili T. W. 1980 Phys. Rev. D22 2542.
[10] Hoshino Y„ Inagaki T. and Mizutani Y. 2014 arXiv:1409.3629 [hep-ph],
[11] Concha-Sanchez Y„ Raya A. and Tejeda-Yeomans M. E. 2013 Phys. Rev. D87 035001.
Received, 08.11.2014
Т. Инагаки
ФОРМАЛИЗМ ШВИНГЕРА-ДАЙСОНА В 1+2 РАЗМЕРНОЙ КЭД
Исследуется основное состояние четырехкомионентного фермнона Дирака в КЭД с топологическим членом Черна-Саймона в 1+2 размерностях. Лагранжиан безмассового дираковского фермиона обладает киральной и Р-симметриями. Спонтанное нарушение этих симметрий может быть найдено с помощью расчета значения вакуумного ожидания для фермион-антифермионных пар. Мы вводим приближенную вершинную функцию и вычисляем значения вакуумного ожидания решая уравнение Швингера^Дайсона. Мы наблюдаем слабую калибровочную зависимость вершинной функции Болла^Чыо.
Ключевые слова: 1+2 размерная КЭД, уравнение Швингера-Дайеона, член Черна - Саймона.
Инагаки Т., доктор, профессор. Хиросимский университет.
1-7-1 Kagamiyama, Higashi-Hiroshima, 739-8521, Япония. E-mail: inagaki@hiroshima-u.ac.jp
