Journal of Modern Physics, 2013, 4, 151-156
http://dx.doi.org/10.4236/jmp.2013.44A014 Published Online April 2013 (http://www.scirp.org/journal/jmp)
Discussion for the Solutions of Dyson-Schwinger
Equations at m 0 in QED3
Huixia Zhu1,2, Hongtao Feng3,4, Weimin Sun1,4,5, Hongshi Zong1,4
1Department of Physics, Nanjing University, Nanjing, China
2The College of Physics and Electronic Information, Anhui Normal University, Wuhu, China
3Department of Physics, Southeast University, Nanjing, China
4State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, CAS, Beijing, China
5Joint Center for Particle, Nuclear Physics and Cosmology, Nanjing, China
Email: zonghs@chenwang.nju.edu.cn
Received February 10, 2013; revised March 14, 2013; accepted March 25, 2013
Copyright © 2013 Huixia Zhu et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In the case of nonzero fermion mass, within a range of Ansätze for the full fermion-boson vertex, we show that
Dyson-Schwinger equation for the fermion propagator in QED3 has two qualitatively distinct dynamical chiral symme-
try breaking solutions. As the fermion mass increases and reaches to a critical value mc, one solution disappears, and the
dependence of mc on the number of fermion flavors is also given.
Keywords: QED3; Fermion Propagator; Solutions of DSEs
1. Introduction
Nowadays, it is widely accepted that Quantum Chro-
modynamics (QCD) in 3 + 1 dimensions is the funda-
mental theory for strong interaction. Dynamical chiral
symmetry breaking (DCSB) is of fundamental impor-
tance for strong interaction physics. DCSB can be ex-
plored via the gap equation, viz., the Dyson-Schwinger
equation (DSE) for the dressed-fermion self-energy. As
is well known, the gap equation has two solutions in the
chiral limit, i.e. the Nambu-Goldstone (NG) solution
which is characterized by DCSB, and the Wigner (WN)
solution in which chiral symmetry is not dynamically
broken. However, when the current quark mass m is
nonzero, the quark gap equation has only one solution
which corresponds to the NG phase and the solution cor-
responding to the WN phase does not exist [1,2]. This
conclusion is hard to understand and one will naturally
ask why the Wigner solution of the quark gap equation
only exists in the chiral limit and does not exist at finite
current quark mass. The authors of Ref. [3] first discussed
this problem and asked whether the quark gap equation
has a Wigner solution in the case of nonzero current
quark mass. Subsequently, the authors of Refs. [4-7] fur-
ther investigated the problem of possible multi-solutions
of the quark gap equation. As far as we know, partly due
to the complexity of the non-Abelian character of QCD,
this problem has not been solved satisfactorily in the lit-
erature. In the present paper we try to propose a new ap-
proach to investigate this problem in the framework of a
relatively simple Abelian toy model of QCD, namely,
quantum electrodynamics in 2 + 1 dimensions (QED3).
As a field-theoretical model, QED3 has been exten-
sively studied in recent years. It has many features simi-
lar to QCD in 3 + 1 dimensions. This is because QED3 is
known to have a phase where the chiral symmetry of the
theory is spontaneously broken and the fermions are con-
fined in this phase [8]. Moreover, QED3 is supernor-
malizable, so it is not plagued with the ultraviolet diver-
gences which are present in QED4. These are the basic
reasons why QED3 is regarded as an interesting toy
model: studying QED3 it might be possible to investigate
confinement [8-10] and dynamical chiral symmetry
breaking (DCSB) [11-16] within a theory which is struc-
turally much simpler than QCD while sharing the same
basic nonperturbative phenomena. Herein we try to use
the DSEs for the fermion and photon propagators in
QED3 to describe novel aspects of the interplay between
explicit and dynamical chiral symmetry breaking.
2. Dyson-Schwinger Equation for Fermion
Propagator
The Lagrangian of QED3 with N flavors of fermions in a
C
opyright © 2013 SciRes. JMP
H. X. ZHU ET AL.
152
general covariant gauge in Euclidean space, ignoring the
issues discuss in Ref. [15], can be written as ifications
that anticipate your paper as one part of the entire jour-
nals, and not as an independent document. Please do not
revise any of the current designations.


2
11
A
42
F
 

1, ,jN
 
22
Ap Bp


1,Smp

1,Smp
2
1
N
jj
j
LieAm



(1)
where the 4 × 1 spinor j is the fermion field with
being the flavor indices.
Based on Lorentz structure analysis, the inverse fer-
mion propagator in the chiral limit can be written as

1
Spip
 (2)
One assumes that dressed fermion propagator at finite
m is analytic in the neighborhood of m = 0,
so the can be written as
 

 
1
11
0
22
22
,
,
mSmp
Smp S pm
ipApBp ipCmp
ipEp Fp



 
 

2 2
d
,,
m
Dmp
 
 
2
2
,,
,.
Cmp
Dmp
(3)
where
 
 
22
22
Ep Ap
Fp Bp


Setting e2 = 1, the DSE for the fermion propagator can
be written as
 
 
11
0
3
3
,,
d,;,
2π
Smp Smp
kSmkmpkD


 
 
,mpk


1
0,Smp

;,mpk
(4)
where is the bare inverse fermion propagator
and is the full fermion-photon vertex. Sub-
stituting Equation (3) into Equation (4), one can obtain



3
2
23
1d
142π
Tr,; ,
k
Ep p
ipSmkm


 

,pkDmq

(5)
and



3
2
3
1d
42π
Tr,; ,
k
Fp m
Smk mp



,kD mq

(6)
where q = p – k. The full photon propagator can be writ-
ten as


2
4
22
q qq
q
q
,1,
qq
Dmq
qm






2
,mq
(7)
with the vacuum polarization defined by
 
22 2
,,mqq qqmq
 



(8)
The DSE satisfied by the photon vacuum polarization
tensor reads
 
3
2
3
d
,2π
Tr,; ,,
k
mq N
Smp mpkSmk




2
,mq
(9)
The boson polarization has an ultraviolet
divergence which is present only in the longitudinal part.
By applying the projection operator
2
3qq
Pq
 
 (10)
one can remove this divergence and obtain a finite vac-
2
,mq [15]. uum polarization
The DSEs for the photon and fermion propagators
form a set of coupled integral equations for the three
 
22 2
,and, Ep Fpmq

;,mpk
once scalar functions
the full fermion-photon-vertex is known.
Unfortunately, although several works attempts to re-
solve the problem, none of them are completely satisfac-
tory [17-23]. Thus, in phenomenological applications,
one often proceed by adopting reasonable approximation
for
;,mpk


such that Equations (5), (6) and (9) are
reduced to a closed system of equations which may be
solved directly. In this letter, following Ref. [15], we
choose the following AnsÄatzefor the full fermion-pho-
ton vertex
22
;, ,mpkfE pEk (11)

and the form of function f is: 1) 1; 2)
 
22
1
2Ep Ek
 
22
Ep Ek.
; 3)
The first one is the bare vertex. This structure plays the
most dominant role in the full fermion-photon vertex in
high energy region and the full fermion-photon vertex
reduces to it in large momentum limit. The second form
is inspired by the BC-vertex [18]. Previous works [15,24]
show that the numerical results of DSEs employing this
Ansatz is as good as that employing BC and CP vertex
[19]. Since the numerical results obtained using the last
Ansatze coincide very well with earlier investigation
[15,25], we choose this one as a reasonable Ansatze to be
used in this work. Using those AnsÄatzefor the full fer-
mion-photon vertex, the coupled DSEs for the fermion
propagator and photon vacuum polarization reduce to the
following form
Copyright © 2013 SciRes. JMP
H. X. ZHU ET AL.
Copyright © 2013 SciRes. JMP
153

 
22
2
1
f q
q
3
2
23
2222 22
2d
1(2π)
Ekpq kq
k
Ep pqEkk Fk

 

 
(12)


 
2
3
2
32 22222
d
22π
Fk f
k
Fp m
qEkk Fk
 2
1q
 

 
(13)



 
22
22
222
3
2
23
222 22222
23
4d
2π
E
kE pkkq
Nk
qqEkkFkE p

 

kqq f
pF p

 

 
 
22
Ep Ap
 
22
(14)
where the Landau gauge has been chosen. In the chira
limit, and
F
pBp
2

2
Bp
2

2
Bp

20Bp
. From
Equations (12)-(14), it is not difficult to find that the
above coupled equations have one Wigner solution
and two dynamical symmetry breaking solu-

20Bp
Bp
tions: and . As was point-


2
0Bp  
ed out in Ref. [4], if is a solution of the gap
equation in the chiral limit, then so is . While
these two solutions are distinct, the chiral symmetry en-
tails that each yields the same pressure. In the chiral limit,
the two dynamical symmetry breaking solutions are
symmetric about the Wigner solution
Bp
.
However, just as will be shown below, this might be
changed when the fermion mass is not zero.
3. Numerical Results
Our next task is to solve for the two scalar functions
and

2
,Emp

2
,
F
mp . These two functions can be
obtained by numerically solving the three coupled inte-
gral Equations (12)-(14). Starting from E = 1, F = 1 and
П = 1, we iterate the three coupled integral equations
until all the three functions converge to a stable solution
which is plotted in Figure 1 (solid line).
From Figure 1, it is easy to find that all the three sca-
lar functions in the DCSB phase (N = 1) are constant in
the infrared region, while in the ultraviolet region the
vector function behaves as
21Ap

and the photon
vacuum polarization behaves as 21qq . Never-
theless, in contrast to the case of massless QED3 [15], in
the large momentum region, the fermion self-energy re-
duces to the bare mass m in Equation (6). Since all the
three functions are positive in the whole range of p2, we
define them as the “+” solution.
If we do interation starting from F = 1, E = 1 and П =
1, we can obtain another stable solution. The typical be-
haviors of the three functions in the DCSB phase for a
fixed mass and number of fermion flavors are also plot-
ted in Figure 1 (the dotted line). From Figure 1, we see
that the DSEs for the fermion propagator has two distinct
nonzero solutions. Especially, the infrared value of the
fermion self-energy is negative, so we define it as the “
solution. In the low energy region, each of the three
functions in the second solution is also almost constant,
but it is different from the corresponding one in the “+”
solution. As p2 or q2 increases, each function of the “
solution approach to the corresponding one of the “+”
solution.
To reveal the difference between these two solutions,
we consider m as a continuous parameter in the DSEs.
We plot the infrared value of E, F, П in Figure 2. When
m = 0, from DSEs one obtains one E and П, but two F
which are symmetric about F = 0 in Figure 2 for each
vertex ansatze. For the “+” solution of DSEs, as m in-
creases, E(0) and F(0) increases while П(0) decreases.
However, the three infrared values in the “” solution
show a different trend as m increases. When m reaches its
critical value, we obtain only one solution for DSEs. In
addition, from Figure 2, it can be seen that the critical
mass exists for any truncated scheme of DSEs used in
this work.
Furthermore, we investigate the influence of the num-
ber of fermion flavors on the critical mass. By employing
ansatze 2, we can obtain the relation between the critical
mass and the number of fermion flavors and it is plotted
in Figure 3. We observe that the critical mass decreases
as N increases and it vanishes at N = Nc, which is similar
to the critical number of fermion flavors for DCSB in the
chiral limit [16].
4. Conclusion
To summarize, in this paper, working in the framework
of Dyson-Schwinger equations and employing a range of
ansatze for the full fermion-photon vertex of QED3,
westudy the interplay between explicit and dynamical
chiral symmetry breaking in QED3. In the case of non-
zero fermion mass, it is found that, besides the ordinary
solution, the fermions gap equation has another solution
which has not been reported in the previous work of
QED3. In the low energy region, one observes that these
two solutions are apparently different, but in the high
energy region they coincide with each other. In addition,
it is found that this solution exists only when the mass is
H. X. ZHU ET AL.
154
0.92
0.90
0.88
0.86
0.84
0.82
0.0000 0.0005 0.0010
1.00
0.96
0.92
0.88
0.84
0.80
0.0015
m
0.0000 0.0005 0.0010 0.0015
m
0.06
0.04
0.02
0.00
-0.02
-0.04
F (m,0)
0.0000 0.0005 0.0010 0.0015
m
3.2
3.0
2.8
2.6
2.4
2.2
2.0
1.8
1.6
1.4
Π(m.q
2
)
E (m,0)
s > 0 for Ansatze 1
s < 0 for Ansatze 1
s > 0 for Ansatze 2
s < 0 for Ansatze 2
s > 0 for Ansatze 3
s < 0 for Ansatze
3
E(m, p
2
)
1E-5 1E-4 1E-3 0.01 0.1 1 10 100
p
2
0.06
0.04
0.02
0.00
-0.02
F(m, p
2
)
1E-5 1E-4 1E-3 0.01 0.1 1 10 100
p
2
Figure 1. The typical behaviors of the two solutions of DSEs
for the fermion propagator at 3 for Ansatze 2. ,
Nm110
1E-5 1E-4 1E-3 0.01 0.1 1 10 100
1
0.1
0.01
Π(m, q
2
)
“+” Solution
“-” Solution
q
2
Figure 2. The infrared value of E; F; П at N = 1 as a func-
tion of m (S > 0 represents the “+” solution and S < 0 repre-
sents the “” solution).
Copyright © 2013 SciRes. JMP
H. X. ZHU ET AL. 155
0.01
1E-3
1E-4
1E-5
1E-6
N
c
m
0.0 0.5 1.0 1.5 2.0 2.5
N
Figure 3. The relation between the critical mass and the
number of fermion flavors.
smaller than a critical value. The critical mass decreases
apparently with the rise of the number of fermion flavors
and vanishes at a critical value Nc, which corresponds to
the critical number of fermion flavors of QED3 in the
chiral limit. It is an interesting phenomena which deserve
further investigations.
5. Acknowledgements
This work was supported by the National Natural Sci-
ence Foundation of China (Grant No. 11047005,
11105029, 10935001, and 11075075) and the Research
Fund for the Doctoral Program of Higher Education (un-
der Grant No 2012009111002).
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