научная статья по теме ON THE CRITICAL BEHAVIOR OF ( )-DIMENSIONAL QED Физика

Текст научной статьи на тему «ON THE CRITICAL BEHAVIOR OF ( )-DIMENSIONAL QED»

ЯДЕРНАЯ ФИЗИКА, 2012, том 75, № 7, с. 945-947

ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ

ON THE CRITICAL BEHAVIOR OF (2 + 1)-DIMENSIONAL QED

©2012 A. V. Kotikov*

Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Russia Received March 31, 2011

It is shown the analysis [1] for QED in (2+ 1) dimensions with N four-component fermions in the leading and next-to-leading orders of the 1/N expansion. As it was demonstrated in [1], the range of the admissible values N, where the dynamical fermion mass exists, decreases strongly with the increasing of the gauge charge. So, in Landau gauge the dynamical chiral symmetry breaking appears for N < 3.78, that is very close to the results of the leading order and in Feynman gauge dynamical mass is completely absent.

Quantum Electrodynamics in (2 + 1) dimensions (QED3) has acquired increasing attention [1—7] because of its similarities to (3 + 1)-dimensional QCD. Moreover, last years a new strong interest comes to QED3 in the relation with graphene properties (see [8] and discussions and references therein). Graphene, a one-atom-thick layer of graphite, is a remarkable system with many unusual properties that was fabricated for the first five years ago [9]. Theoretically it was shown long ago [10] that quasiparticle excitations in graphene are described by the massless Dirac equation in (2 + 1) dimension. This explains why the bilayer graphene in external fields is a subject of intensive recent study [11].

A number of investigations have been performed for the study of dynamical chiral symmetry breaking in QED3 and very different results have been obtained. Using the leading order (LO) in the 1/N expansion of the Schwinger—Dyson (SD) equation, Appelquist et al. [2] showed that the theory exhibits a critical behavior as the number N of fermion flavor approaches Nc = 32/n2; that is, a fermion mass is dynamically generated only for N < Nc. On the contrary, Pennington and collaborators [3], adopting a more general non-perturbative approach to the SD equations, found that the dynamically generated fermion mass decreases exponentially with N, vanishing only as N ^<x>. This conclusion was supported also by Pisarski [4] by the use of the other methods. On the other hand, an alternative non-perturbative study by Atkinson et al. [5] suggested that chiral symmetry is unbroken at sufficiently large N. The theory has also been simulated on the lattice [6, 7]. Remarkably, the conclusions of [6] are in agreement with the existence of a critical N as predicted in the analysis of [2], while the second paper [7] contains the opposite results.

Because the critical value Nc is not large, the contribution of the higher orders in the 1/N expansion can be essential and may lead to better understanding of the problem. The purpose of this work is to consider the 1/N correction [1, 12] to LO result [2].

1. The Lagrangian of massless QED3 with N flavors is

1

I1 '

L = 4>(id -eA)4> - -F2

where ^ is taken to be a four-component complex spinor. In massless case, which we are considering, the model contains infrared divergences, which can be canceled when the model is analyzed in a 1/N expansion [13]. Since the theory is massless, the mass scale is the dimensional coupling constant a = = Ne2/8 which is kept fixed as N ^<x>.

Following [2] we study the solution of the SD equation. The inverse fermion propagator has the form

5-1(p) = -[1 + A(p)][p + X(p) ],

where A(p) is the wave-function renormalizable coefficient and £(p) is a dynamical, parity-conserving mass taken to be the same for all the fermions.

The SD equation is

^D^v(p - k)[1 + A(k)] {k + YJ(k))Tv(p, k)

[1 + A(k)]2 (k2 + £2(k))

where1)

DßV (p) =

QPßPu/p2 p2 [1 + П(р)]

E-mail: kotikov@theor.jinr.ru

^Following [12] we introduce a nonlocal gauge-fixing term. The detailed analysis of this possibility has been given in [14].

8 ЯДЕРНАЯ ФИЗИКА том 75 № 7 2012

946

KOTIKOV

is the photon propagator and rv (p,k) is the vertex function.

2. The LO approximations in the 1/N expansion

are

A(p) = 0, n(p) = a/\p\, Tv (p,k) = y v,

where we neglect the fermion mass in the calculation of n(p). The gap equation is

Let us get the exact critical value Nc from Eq. (6). Supposing a = ac = —1/4 we obtain the critical values in the following form

Nc,± = (7)

8

n2

(2+e ) ± ((2+e )2 + 4/ + )/4)

1/2-

i.e.

m = + 0 x

N

(2)

Tr

d3k

E(k)

(2n)3 k2 [(p — k)2 + a\p — k\]

where we ignore the term £2(k) in the denominator of r.h.s.

Following [2], we set

£(k) = (k2)a. (3)

One can see that for large a the r.h.s. of (2) together with condition (3) (and the contributions of higher orders) can be calculated by the standard rules for massless diagrams of the perturbation theory (see, for example, [15]). Thus, we have for large a

with 3 = (-a)(a + 1/2) and L = n2N, or

Nc,+(£) = (3.31, 3.35, 3.09, 2.81), N—£) = (-0.07, 0.38,1.29,1.88)

for £ = (0.0,0.3,0.7,0.9), respectively.

Notice the intriguing fact that follows from (7). The addition of 1/N correction leads to the occurrence of the second critical point (for 0.05 < £ < 0.95) such that for N < Nc— the chiral symmetry does not break. The dynamical mass generation exists in the interval between the critical points Nc— and Nc,+. For £ > 0.95 this interval disappears and the chiral symmetry breaking is absent. For small values of gauge parameter £ (£ < 0.05) new critical point does not occur.

The solution of the Eq. (6)

=

1

2L

œ(e)

a± = (—1 ± [1 — 16(2 + e)/L]1/2)/4. (5) ± ((2 + e)2 + 4/(e) + 2(2 + e)

L

L

,

L2 J

We reproduce the solution given in [2]. That analysis yields a critical number of fermions Nc = = 16(2 + £)/n2 w 1.62(2 + £) (i.e. Lc = 16(2 + £)), such that for N > Nc £(p) = 0 and

£(0) ~ exp\-2n/(N/Nc - 1)1/2

for N < Nc. Thus, chiral symmetry breaks when a becomes complex, that is for N < Nc.

3. The next-to-leading order (NLO) approximation has been included in [1, 16] using the differential equation method [ 17]. The results have a cumbersome form [16], which is similar to results for complicated massless diagrams obtained using Gegenbauer polynomials [18]. In [1] we have analyzed a simplified form, which contains only the terms ~(-a)-k and ~(a + + 1/2)-k (k = 1,2,3) from the series given in [16]. These terms are most important in the neighborhood of the critical point Nc. Equation (4) is replaced now by

has the simple form in Landau gauge

i±(e = o) =

1/2

(8)

1

I

1/2

1 +

(i +

3

14 L ) ,

where the last term in r.h.s. of Eq. (8) is very small for L ~ Lc. Leaving it out we get the following equation for 3+

i+(e = o)

+ ( 1 + ^7/3^(0)

1

2L2

2.52

~zT

1

6.52

L

1 = ^ + [/(0+^(01^,

(6)

where / (e) = 4(1 — e)/3 — e2, p(e) = 176/9 — 4n2 — — (16/3)e + 4e2.

which has coefficients close to those from paper [12].

Resume. We reviewed the results of [1], where the 0(1/N2) terms have been included into SD equation and the strong gauge dependence of the result has been found. Hence, the addition of 1/N correction does not lead to the essential improvement in the understanding of dynamical chiral symmetry breaking.

x

x

MEPHA^ OH3HKA tom 75 № 7 2012

ON THE CRITICAL BEHAVIOR

947

However, as it was shown in [1], in the Landau gauge the inclusion of 0(1/N2) terms slightly changes only quantitative (but not qualitative) properties of the LO results. Thus, in the Landau gauge the analysis in [1] gives further evidence in favor of the solution given by Appelquist et al. [2].

The author thanks the Organizing Committee of the International Conference "Critical Point and Onset of Deconfinement (CPOD)" for invitation.

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