Quark-Antiquark and Diquark Condensates in Vacuum in a 3D Two-Flavor
  Gross-Neveu Model
Quark-Antiquark and Diquark Condensates in Vacuum in a 3D Two-Flavor
Gross-Neveu Model∗
Zhou Bang-Rong
College of Physical Sciences, Graduate School of the Chinese Academy of Sciences, Beijing 100049, China and
CCAST (World Laboratory), P.O.Box 8730, Beijing 100080, China
(Dated:)
The effective potential analysis indicates that, in a 3D two-flavor Gross-Neveu model in vacuum,
depending on less or bigger than the critical value 2/3 of GS/HP , where GS and HP are respectively
the coupling constants of scalar quark-antiquark channel and pseudoscalar diquark channel, the
system will have the ground state with pure diquark condensates or with pure quark-antiquark
condensates, but no the one with coexistence of the two forms of condensates. The similarities and
differences in the interplay between the quark-antiquark and the diquark condensates in vacuum in
the 2D, 3D and 4D two-flavor four-fermion interaction models are summarized.
PACS numbers: 12.38Aw; 12.38.Lg; 12.10.Dm; 11.15.Pg
Keywords: 3D Gross-Neveu model, quark-antiquark and diquark condensates, effective potential
I. INTRODUCTION
It has been shown by effective potential approach that
in a two-flavor 4D Nambu-Jona-Lasinio (NJL) model [1],
even when temperature T = 0 and quark chemical poten-
tial µ = 0, i.e. in vacuum, there could exist mutual com-
petition between the quark-antiquark condensates and
the diquark condensates [2]. Similar situation has also
emerged from a 2D two-flavor Gross-Neveu (GN) model
[3] except some difference in the details of the results [4].
An interesting question is that if such mutual competi-
tion between the two forms of condensates is a general
characteristic of this kind of two-flavor four-fermion in-
teraction models? For answer to this question, on the
basis of research on the 4D NJL model and the 2D GN
model, we will continue to examine a 3D two-flavor GN
model in similar way. The results will certainly deepen
our understanding of the feature of the four-fermion in-
teraction models.
We will use the effective potential in the mean field
approximation which is equivalent to the leading order
of 1/N expansion. It is indicated that a 3D GN model is
renormalizable in 1/N expansion [5].
II. MODEL AND ITS SYMMETRIES
The Lagrangian of the model will be expressed by
L = q̄iγµ∂µq +GS [(q̄q)2 + (q̄~τq)2]
A=2,5,7
(q̄τ2λAq
C)(q̄Cτ2λAq). (1)
All the denotations used in Eq.(1) are the same as the
ones in the 2D GN model given in Ref.[4], except that
∗The project supported by the National Natural Science Founda-
tion of China under Grant No.10475113.
the dimension of space-time is changed from 2 to 3 and
the coupling constant HS of scalar diquark interaction
channel is replaced by the coupling constant HP of pseu-
doscalar diquark interaction channel. Now the matrices
γµ(µ = 0, 1, 2) and the charge conjugate matrix C are
taken to be 2× 2 ones and have the explicit forms
, γ1 =
, γ2 =
It is emphasized that, in 3D case, no ”γ5” matrix can
be defined, hence the third term in the right-handed side
of Eq.(1) will be the only possible color-anti-triplet di-
quark interaction channel which could lead to Lorentz-
invariant diquark condensates, where we note that the
matrix Cτ2λA is antisymmetric. Without ”γ5”, the La-
grangian (1) will have no chiral symmetry. Except this, it
is not difficult to verify that the symmetries of L include:
1. continuous flavor and color symmetries SUf (2) ⊗
SUc(3)⊗ Uf (1);
2. discrete symmetry R: q → −q;
3. parity P : q(t, ~x) → γ0q(t,−~x) and qC(t, ~x) →
−γ0qC(t,−~x);
4. time reversal T : q(t, ~x) → γ2q(−t, ~x) and
qC(t, ~x) → −γ2qC(−t, ~x);
5. charge conjugate C: q ↔ qC ;
6. special parity P1: q(t, x1, x2) → γ1q(t,−x1, x2) and
qC(t, x1, x2) → −γ1q(t,−x1, x2);
7. special parity P2: q(t, x1, x2) → γ2q(t, x1,−x2) and
qC(t, x1, x2) → −γ2qC(t, x1,−x2).
If the quark-antiquark condensates 〈q̄q〉 could be formed,
then the time reversal T , the special parities P1 and P2
will be spontaneously broken [6]. If the diquark conden-
sates 〈q̄Cτ2λ2q〉 could be formed, then the color symme-
try SUc(3) will be spontaneously broken down to SUc(2)
http://arxiv.org/abs/0704.0829v2
and the flavor number Uf (1) will be spontaneously bro-
ken but a ”rotated” electric charge U
(1) and a ”rotated”
quark number U ′q(1) leave unbroken [7]. In addition, the
parity P will be spontaneously broken, though all the
other discrete symmetries survive. This implies that the
diquark condensates 〈q̄Cτ2λ2q〉 will be a pseudoscalar.
In this paper we will neglect discussions of the Gold-
stone bosons induced by breakdown of the continuous
symmetries and pay our main attention to the problem
of interplay between the above two forms of condensates.
III. EFFECTIVE POTENTIAL IN MEAN FIELD
APPROXIMATION
Define the order parameters in the 3D GN model by
σ = −2GS〈q̄q〉 and ∆ = −2HP 〈q̄Cτ2λ2q〉, (3)
then in the mean field approximation, the Lagrangian (1)
can be rewritten by
L = Ψ̄(x)S−1(x)Ψ(x) −
, (4)
where
Ψ(x) =
qC(x)
and Ψ̄(x) =
q̄(x) q̄C(x)
are the expressions of the quark fields in the Nambu-
Gorkov basis [8]. In the momentum space, the inverse
propagator S−1(x) for the quark fields may be expressed
S−1(p) =
6p− σ −τ2λ2∆
−τ2λ2∆∗ 6p− σ
, 6p = γµpµ. (5)
The effective potential corresponding to L given by
Eq.(4) becomes
V (σ, |∆|) = σ
(2π)3
Tr lnS−1(p)S0(p).
Similar to the case of the 2D NG model [4], the calcu-
lations of Tr for (red, green) and blue color degrees of
freedom can be made separately thus Eq.(6) will be re-
duced to
V (σ, |∆|) = σ
(2π)3
p2 − (σ − |∆|)2 + iε
p2 + iε
p2 − (σ + |∆|)2 + iε
p2 + iε
p2 − σ2 + iε
p2 + iε
After the Wick rotation, we may define and calculate in
3D Euclidean momentum space
I(a2) =
(2π)3
p̄2 + a2
a3 arctan
a2Λ − π
, if Λ ≫ |a|, (8)
where Λ is the 3D Euclidean momentum cut-off. Assume
that Λ ≫ |σ − |∆||, Λ ≫ σ + |∆| and Λ ≫ σ, then by
means of Eq.(8) we will obtain the final expression of the
effective potential in the 3D GN model
V (σ, |∆|) = σ
(3σ2 + 2|∆|2)Λ
6σ2|∆|+ 2|∆|3 + σ3
+2θ(σ − |∆|)(σ − |∆|)3
. (9)
IV. GROUND STATES
Equation (9) provide the possibility to discuss
the ground states of the model analytically. The
extreme value conditions ∂V (σ, |∆|)/∂σ = 0 and
∂V (σ, |∆|)/∂|∆| = 0 will lead to the equations
θ(σ − |∆|)(σ − |∆|)2 = 0, (10)
[σ2 − θ(σ − |∆|)(σ − |∆|)2] = 0. (11)
Define the expressions
= AC −B2,
where A, B and C represent the second order derivatives
of V (σ, |∆|) with the explicit expressions
θ(σ − |∆|)(σ − |∆|),
∂σ∂|∆|
∂|∆|∂σ
[σ − θ(σ − |∆|)(σ − |∆|)],
∂|∆|2
θ(σ − |∆|)(σ − |∆|). (12)
Equations (10) and (11) have the four different solutions
which will be discussed in proper order as follows.
(i) (σ, |∆|)=(0,0). It is a maximum point of V (σ, |∆|),
since in this case we have
< 0 and K = A
assuming Eqs. (10) and (11) have solutions of non-zero
σ and |∆|.
(ii) (σ, |∆|)=(σ1 ,0), where the non-zero σ1 satisfies the
equation
= 0. (13)
When Eq. (13) is used, we obtain
K = = A
> 0, if
Hence (σ1,0) will be a minimum point of V (σ, |∆|) when
GS/HP > 2/3.
(iii) (σ, |∆|)= (0, ∆1), where non-zero ∆1 obeys the
equation
= 0. (14)
By using Eq.(14) we may get
, K = A
Obviously, (0,∆1) will be a minimum point of V (σ, |∆|)
when GS/HP < 2/3.
(iv) (σ, |∆|)=(σ2,∆2). In view of existence of the func-
tion θ(σ− |∆|) in Eqs.(10) and (11), we have to consider
the case of σ2 > ∆2 and σ2 < ∆2 respectively.
(a) σ2 > ∆2. In this case, Eqs.(10) and (11) will be-
+ 2∆2
) = 0,
From them we can get
− 2∆2
> 0, K = −
Thus it is turned out that (σ2, ∆2) will be neither a
maximum nor a minimum point of V (σ, |∆|) if σ2 > ∆2.
(b) σ2 < ∆2. Now Eqs. (10) and (11) are changed into
4∆2 + σ2
= 0, (15)
= 0. (16)
Hence we will have the results that
, K =
− 8σ2∆2),
from which it may be deduced that only if
σ2 < (
17− 4)∆2, (17)
(σ2,∆2) is just a minimum point of V (σ, |∆|). On the
other hand, from Eqs. (15) and (16) obeyed by σ2 and
∆2 we may get
13− 1
∆2 + σ2
13 + 1
∆2 − σ2
. (18)
Equation (18) indicates that for the minimum point
(σ2,∆2) satisfying Eq.(17) one will certainly have
GS/HP > 2/3. Taking this and the result obtained in
case (ii) into account we see that if GS/HP > 2/3 the
effective potential V (σ, |∆|) will have two possible min-
imum points (σ1, 0) and (σ2,∆2). To determine which
one of the two minimum points is the least value point
of V , we must make a comparison between V (σ1, 0) and
V (σ2,∆2) with the constraint given by Eq.(17). In fact,
it is easy to find out that when Eq.(13) is used,
V (σ1, 0) = −
, (19)
and that when Eqs. (15) and (16) are used,
V (σ2,∆2) = −
+ 3σ2
. (20)
By comparing Eq.(13) with Eq.(15) we may obtain the
relation
3σ1 = σ2 + 4∆2. (21)
By means of Eqs.(19)-(21) it is easy to verify that
V (σ1, 0)− V (σ2,∆2) = −
(23∆3
− 4σ3
(8∆2 − 7σ2)
when Eq.(17) is satisfied. This result indicates that
when GS/HP > 2/3, the least value point of V (σ, |∆|)
will be (σ1, 0) but not (σ2,∆2).
In summary, if the necessary conditions GSΛ > π
and HPΛ > π
2/8 for non-zero σ and ∆ are satisfied,
then the least value points of the effective potential
V (σ, |∆|) will be at
(σ, |∆|) =
(0,∆1)
(σ1, 0)
0 ≤ GS/HP < 2/3
GS/HP > 2/3
. (22)
As a result, in the ground state of the 3D two-flavor GN
model, depending on that the ratio GS/HP is either big-
ger or less than 2/3, one will have either pure quark-
antiquark condensates or pure diquark condensates, but
no coexistence of the two forms of condensates could hap-
V. CONCLUDING REMARKS
The result (22) in the 3D GN model can be compared
with the ones in the 4D NJL model and in the 2D GN
model. The minimal points of the effective potential
V (σ, |∆|) for the latter models have been obtained and
are located respectively at
(σ, |∆|) =
(0, ∆1)
(σ2, ∆2)
(σ1, 0)
0 ≤ GS/HS < 2/[3(1 + C)]
2/[3(1 + C)] < GS/HS < 2/3
GS/HS > 2/3
with C = (2HSΛ
/π2 − 1)/3 and Λ4 denoting the 4D
Euclidean momentum cutoff in the 4D two-flavor NJL
model, if the necessary conditions GSΛ
> π2/3 and
> π2/2 for non-zero σ and ∆ are satisfied [2], and
(σ, |∆|) =
(0, ∆1)
(σ2, ∆2)
(σ1, 0)
GS/HS = 0
0 < GS/HS < 2/3
GS/HS > 2/3
in the 2D two-flavor GN model [4]. In Eqs.(23) and (24),
GS and HS always represent the coupling constants in
scalar quark-antiquark channel and scalar diquark chan-
nel separately.
By a comparison among Eqs.(22)-(24) it may be found
that the three models lead to very similar results. In
all the three models, the interplay between the quark-
antiquark and the diquark condensates in vacuum de-
pends on the ratio GS/HD (D = S for the 4D and 2D
model and D = P for the 3D model). In particular, the
diquark condensates could emerge (in separate or coex-
istent pattern) only if GS/HD < 2/3. This is probably
a general characteristic of the considered two-flavor four-
fermion models, since in these models the color number
of the quarks participating in the diquark condensates
and in the quark-antiquark condensates is just 2 and 3
respectively. However, there are also some differences
in the pattern realizing the diquark condensates among
the three models, though the pure quark-antiquark con-
densates arise only if GS/HD > 2/3 in all of them. In
the 2D GN model, the pure diquark condensates emerge
only if GS/HS = 0 and this is different from the 4D NJL
model where the pure diquark condensates may arise if
GS/HS is in a finite region below 2/3. Another difference
is that in the 3D GN model, there is no coexistence of the
quark-antuquark condensates and the diquark conden-
sates but such coexistence is clearly displayed in the 4D
and 2D model. This implies that in the 3D GN model,
GS/HP = 2/3 becomes the critical value which distin-
guishes between the ground states with the pure diquark
condensates and with the pure quark-antiquark conden-
sates.
It is also indicated that if the two-flavor four-fermion
interaction models are assumed to be simulations of QCD
(of course, only the 4D NJL model is just the true
one) and the four-fermion interactions are supposed to
come from the heavy color gluon exchange interactions
−g(q̄γµλaq)2 (a = 1, · · · , 8;µ = 0, · · · , D − 1) via the
Fierz transformation [7], then one will find that in all the
three models, for the case of two flavors and three colors
the ratio GS/HD are always equal to 4/3 which is larger
than the above critical value 2/3. From this we can con-
clude that there will be only the pure quark-antiquark
condensates and no diquark condensates in the ground
states of all these models in vacuum.
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[2] Zhou Bang-Rong, Commun. Theor. Phys. 47 (2007) 95.
[3] D.J. Gross and A. Neveu, Phys. Rev. D 10 (1974) 3235.
[4] Zhou Bang-Rong, Commun. Theor. Phys., 47 (2007) 520.
[5] B. Rosenstein, B. J. Warr, and S. H. Park, Phys. Rep. 205
(1991) 59.
[6] Bang-Rong Zhou, Phys. Lett. B444 (1998) 455.
[7] M. Buballa, Phys. Rep. 407 (2005) 205.
[8] Y. Nambu, Phys. ReV. 117 (1960) 648; L. P. Gorkov,
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