Difermion condensates in vacuum in 2-4D four-fermion interaction models
Bang-Rong Zhou†
College of Physical Sciences, Graduate School of
the Chinese Academy of Sciences, Beijing 100049, China
In any four fermion (denoted by q) interaction models, the couplings of (qq)2-form can
always coexist with the ones of (q̄q)2-form via the Fierz transformations. Hence, even in
vacuum, there could be interplay between the condensates 〈q̄q〉 and 〈qq〉. Theoretical anal-
ysis of this problem is generally made by relativistic effective potentials in the mean field
approximation in 2D, 3D and 4D models with two flavor and Nc color massless fermions.
It is found that in ground states of these models, interplay between the two condensates
mainly depend on the ratio GS/HS for 2D and 4D case or GS/HP for 3D case, where GS ,
HS and HP are respectively the coupling constants in a scalar (q̄q), a scalar (qq) and a
pseudoscalar (qq) channel.
In ground states of all the models, only pure 〈q̄q〉 condensates could exist if GS/HS or
GS/HP is bigger than the critical value 2/Nc, the ratio of the color numbers of the fermions
entering into the condensates 〈qq〉 and 〈q̄q〉. Below it, differences of the models will manifest
themselves.
In the 4D Nambu-Jona-Lasinio (NJL) model, as GS/HS decreases to the region below
2/Nc, one will first have a coexistence phase of the two condensates then a pure 〈qq〉 con-
densate phase. Similar results come from a renormalized effective potential in the 2D Gross-
Neveu model, except that the pure 〈qq〉 condensates could exist only if GS/HS = 0. In a
3D Gross-Neveu model, when GS/HP < 2/Nc, the phase transition similar to the 4D case
can arise only if Nc > 4, and for smaller Nc, only a pure 〈qq〉 condensate phase exists but
no coexistence phase of the two condensates happens. The GS −HS (or GS − HP ) phase
diagrams in these models are given.
The results deepen our understanding of dynamical phase structure of four-fermion inter-
action models in vacuum. In addition, in view of absence of difermion condensates in vacuum
of QCD, they will also imply a real restriction to any given two-flavor QCD-analogous NJL
model, i.e. in the model, the derived smallest ratio GS/HS via the Fierz transformations in
the Hartree approximation must be bigger than 2/3.
The project supported by the National Natural Science Foundation of China under Grant No.10475113.
Electronic mailing address: zhoubr@163bj.com
http://arxiv.org/abs/0704.0841v3
I. MAIN RESULTS
We have researched interplay between the fermion(q)-antifermion (q̄) condensates 〈q̄q〉 and
the difermion condensates 〈qq〉 in vacuum in 2D, 3D and 4D four-fermion interaction models
with two flavor and Nc color massless fermions. It is found that the ground states of the systems
could be in different phases shown in the following GS − HS and GS − HP phase diagrams
[Fig.(a)–Fig.(d)], where
GS — coupling constant of scalar (q̄q)
2 channel
HS — coupling constant of scalar color
Nc(Nc − 1)
−plet (qq)2 channel (4D, 2D case)
HP — coupling constant of pseudoscalar color
Nc(Nc − 1)
−plet (qq)2 channel (3D case)
Λ — Euclidean Momentum cutoff of loop integrals (4D,3D case)
(σ1, 0) — pure 〈q̄q〉 phase
(0,∆1) — pure 〈qq〉 phase
(σ2,∆2) — mixed phase with both 〈q̄q〉 and 〈qq〉
Fig.(a)-Fig.(d) (pages 3-6)
                                                                    3 
4D NJL Model 
y=GSΛ
2/π2 
               
 y =2x/ Nc
           (σ1, 0) 
(σ2,Δ2) 
      
1/Nc                    (0 ,Δ1)                R 
0        1/2 
x =HSΛ
R:  y=x/[1+(Nc-2)x] 
Fig. (a) 
                                                          4           
2D GN Model   
y=GS /π 
y=2x /Nc  
         
(σ1, 0)  
(σ2 ,Δ2) 
x=HS /π  (0,Δ1) 
Fig. (b)      
                                            
3D GN Model,  Nc≤4  
y=GSΛ/π
y=2x/ Nc
           (σ1, 0) 
1/4Nc
                                 (0,Δ1) 
        0         1/8 
x=HPΛ/π
Fig. (c)  
3D GN Model,  Nc≥5  
y=GSΛ/π
          y =2x/ Nc
(σ1, 0) 
1/4Nc                         
            (σ2,Δ2) 
(0,Δ1) 
0 1/8 
x =HPΛ/π
Fig. (d) 
Main Conclusions
1. In all the models, pure 〈q̄q〉 phase happens if
) > 2
(also GS must be large
enough in 3D and 4D model).
2. The phases with condensates 〈qq〉, including pure 〈qq〉 phase and mixed phase with 〈q̄q〉
and 〈qq〉, arise only if
) < 2
3. In 3D Gross-Neveu model, no mixed phase with 〈q̄q〉 and 〈qq〉 exists for Nc ≤ 4.
II. Motive and general approach
• In any four-fermion interaction model [1, 2], the couplings of (qq)2-form and (q̄q)2-form
can always coexist via the Fierz transformations, hence there must be interplay between
the condensates 〈q̄q〉 and 〈qq〉 in ground state of the system.
• In the vacuum, despite of absence of net fermions, based on a relativistic quantum field
theory, it is possible that the condensates 〈qq〉 and 〈q̄q̄〉 are generated simultaneously.
• The mean field approximation has been taken. In this case, we have used the Fierz
transformed four-fermion couplings in the Hartree approximation to avoid double
counting [3].
• In selecting the couplings of (qq)2-form, we always simulate SU(Nc) gauge interaction,
where two fermions are attractive in the antisymmetric
Nc(Nc − 1)
-plet.
• Euclidean momentum cutoffs in 3D and 4D models have been used so as to maintain
Lorentz invariance of effective potentials in the vacuum.
• In massless fermion limit, all the discussions can be made analytically.
• The coupling constants GS and HS (or HP ) are viewed as independent parameters.
III. 4D Nambu-Jona-Lasinio model
With 2 flavors and Nc color massless fermions, the Lagrangian
L = q̄iγµ∂µq +GS [(q̄q)
2 + (q̄iγ5~τq)
2] +HS
(q̄iγ5τ2λAq
C)(q̄C iγ5τ2λAq), (1)
where the fermion fields q are in the doublet of SUf (2) and the Nc-plet of SUc(Nc), i.e.
i = 1, · · · , Nc, (2)
http://arxiv.org/abs/0704.0841v3
qC is the charge conjugate of q and ~τ = (τ1, τ2, τ3) are the Pauli matrices acting in two-flavor
space. The matrices λA run over all the antisymmetric generators of SUc(Nc).
Assume that the four-fermion interactions can lead to the scalar condensates
〈q̄q〉 = φ (3)
with all the Nc color fermion entering them, and the scalar color
Nc(Nc − 1)
2 -plet difermion
and di-antifermion condensates (after a global SUc(Nc) transformation)
〈q̄Ciγ5τ2λ2q〉 = δ, 〈q̄iγ5τ2λ2q
C〉 = δ∗, (4)
with only two color fermions enter them. The corresponding symmetry breaking is that
SUfL(2) ⊗ SUfR(2) → SUf (2), SUc(Nc) → SUc(2), and a ”rotated” electric charge UQ̃(1)
and a ”rotated” quark number U ′q(1) leave unbroken. It should be indicated that in the case
of vacuum, the Goldstone bosons induced by spontaneous breaking of SUc(Nc) could be some
combinations of difermions and di-antifermions.
Define that
σ = −2GSφ, ∆ = −2HSδ, ∆
∗ = −2HSδ
∗. (5)
With standard technique and a 4D Euclidean momentum cutoff Λ [4], we obtain the relativistic
effective potential
V4(σ, |∆|) =
2 + 2|∆|2)Λ2 − (Nc − 2)
−(σ2 + |∆|2)2
σ2 + |∆|2
. (6)
The ground states of the system, i.e. the minimum points of V4(σ, |∆|), will be at
(σ, |∆|) =
(0, ∆1)
(σ2, ∆2)
(σ1, 0)
, 0 ≤
1 + (Nc − 2)
1 + (Nc − 2)
, (7)
Eq.(7) gives the phase diagram Fig.(a) of the 4D NJL model.
IV. 2D Gross-Neveu model
The Lagrangian is expressed by
L = q̄iγµ∂µq +GS [(q̄q)
2 + (q̄iγ5~τq)
2] +HS(q̄iγ5τSλAq
C)(q̄Ciγ5τSλAq), (8)
All the denotations are the same as ones in 4D NJL model, except that in 2D space-time
, γ1 =
= −C, γ5 = γ
and τS = (τ0 ≡ 1, τ1, τ3) are flavor-triplet symmetric matrices. It is indicated that the product
matrix Cγ5τSλA is antisymmetric.
Assume that the four-fermion interactions could lead to the scalar quark-antiquark conden-
sates
〈q̄q〉 = φ, (9)
which will break the discrete symmetries
χD : q(t, x)
→ γ5q(t, x),
P1 : q(t, x)
→ γ1q(t,−x),
and that the coupling with HS can lead to the scalar color
Nc(Nc − 1)
-plet difermion con-
densates and the scalar color anti-
Nc(Nc − 1)
-plet di-antifermion condensates (after a global
transformation in flavor and color space)
〈q̄C iγ51fλ2q〉 = δ, 〈q̄iγ51fλ2q
C〉 = δ∗ (10)
which will break discrete symmetries Zc
(center of SUc(3)) and Z
(center of SUf (2)), besides
χD and P1. Noting that in a 2D model, no breaking of continuous symmetry needs to be
considered on the basis of Mermin-Wagner-Coleman theorem [5].
The model is renormalizable. In the space-time dimension regularization approach, we can
write down the renormalized L in D = 2− 2ε dimension space-time by the replacements
GS → GSM
2−DZG, HS → HSM
2−DZH ,
with the scale parameter M , the renormalization constants ZG and ZH . In addition, the γ
L will become 2D/2 × 2D/2 matrices.
Define the order parameters
σ = −2GSM
2−DZGφ, ∆ = −2HSM
2−DZHδ, (11)
which will be finite if ZG and ZH are selected so as to cancel the UV divergences in φ and δ. In
the minimal substraction scheme,
ZG = 1−
2NcGS
, ZH = 1−
. (12)
By similar derivation to the one made in Ref.[6], the corresponding renormalized effective po-
tential in the mean field approximation up to one-loop order becomes
V2(σ, |∆|) =
σ2 + |∆|2
+ (Nc − 2) ln
σ2 + |∆|2
, M̄2 = 2πe−γM2, (13)
where γ is the Euler constant. The ground states of the system i.e. the minimal points of
V2(σ, |∆|) will be at
(σ, |∆|) =
(0, ∆1)
(σ2, ∆2)
(σ1, 0)
GS/HS = 0
0 < GS/HS < 2/Nc
GS/HS > 2/Nc
Eq.(14) gives the phase diagram Fig.(b) of 2D GN model.
In 2D case, the GS -HS phase structure has the following feature:
1. The pure 〈qq〉 phase (0,∆1) could appear only if GS/HS = 0;
2. Formations of the condensates do not call for that the coupling constant GS and HS have
some lower bounds.
V. 3D Gross-Neveu model
The Lagrangian is expressed by
L = q̄iγµ∂µq +GS [(q̄q)
2 + (q̄~τq)2] +HP
(q̄τ2λAq
C)(q̄Cτ2λAq), (15)
where γµ(µ = 0, 1, 2) are taken to be 2× 2 matrices
, γ1 =
, γ2 =
It is noted that the product matrix Cτ2λA is antisymmetric, and since without the ”γ5” matrix,
the only possible color
Nc(Nc − 1)
-plet difermion interaction channel is pseudoscalar one. The
condensates 〈q̄q〉 will break
time reversal symmetry T : q(t, ~x) → γ2q(−t, ~x),
special parity P1 : q(t, x
1, x2) → γ1q(t,−x1, x2),
special parity P2 : q(t, x
1, x2) → γ2q(t, x1,−x2).
The difermion condensates 〈q̄Cτ2λ2q〉 (after a global rotation in the color space) will break
SUc(Nc) → SUc(2)
and leave a ”rotated” electrical charge U
(1) and a ”rotated” fermion number U ′q(1) unbroken.
It also breaks
parity P : q(t, ~x) → γ0q(t,−~x)
and this shows pseudoscalar feature of the difermion condensates.
Define the order parameters in the 3D GN model
σ = −2GS〈q̄q〉, ∆ = −2HP 〈q̄
Cτ2λ2q〉, (16)
on bases of the same method used in Ref.[7], we find out the effective potential in the mean field
approximation
V3(σ, |∆|) =
2 + 2|∆|2)Λ
6σ2|∆|+ 2|∆|3 + (Nc − 2)σ
3 + 2θ(σ − |∆|)(σ − |∆|)3
, (17)
where Λ is a 3D Euclidean momentum cutoff. The ground states of the system correspond to
the least value points of V3(σ, |∆|) which will respectively be at
(σ, |∆|) =
(0,∆1),
(0,∆1),
(σ2,∆2),
(σ1, 0),
, for Nc ≤ 4
for Nc > 4
, for all Nc
Eq.(18) gives the GS −HP phase diagrams Fig.(c) and Fig.(d) of the 3D GN model.
VI. Summary
• Present research deepens our theoretical understanding of the four-fermion interaction
models:
1. Even in vacuum, it is possible that the difermion condensates are generated as long
as the coupling constants of the difermion channel are strong enough (bigger than
zero or some finite values).
2. Interplay between the condensates 〈q̄q〉 and 〈qq〉 mainly depends on GS/HS (or
GS/HP ), the ratio of the coupling constants of scalar fermion-antifermion channel
and scalar (or pseudoscalar ) difermion channel.
3. In all the discussed 2-flavor models, if GS/HS (GS/HP ) > 2/Nc, the ratio of the
color numbers of the fermions entering into the condensates 〈qq〉 and 〈q̄q〉, (and also
with sufficiently large GS in 4D and 3D model), then only pure 〈q̄q〉 condensates
phase may exist. Below 2/Nc, (and also with sufficiently large HS or HP in 4D or
3D model), one will always first have a mixed phase with condensates 〈q̄q〉 and 〈qq〉,
then a pure 〈qq〉 condensate phase, except that in the 3D GN model, no the mixed
phase appears when Nc ≤ 4.
• In view of absence of 〈qq〉 condensates in vacuum of QCD, the result here also implies
a real restriction to any given two-flavor QCD-analogue NJL model: in such model, the
derived smallest ratio GS/HS via the Fierz transformation in the Hartree approximation
must be bigger than 2/3 [4].
[1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345; 124 (1961) 246.
[2] D.J. Gross and A. Neveu, Phys. Rev. D 10 (1974) 3235.
[3] M. Buballa, Phys. Rep. 407 (2005) 205.
[4] Zhou Bang-Rong, Commun. Theor. Phys. 47 (2007) 95.
[5] N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133; S. Coleman, Commun. Math. Phys.
31 (1973) 259.
[6] Zhou Bang-Rong, Commun. Theor. Phys. 47 (2007) 520.
[7] Zhou Bang-Rong, Commun. Theor. Phys. 47 (2007) 695.
	Main results
	References

  Theoretical analysis of interplay between the condensates $<\bar{q}q>$ and
$<qq>$ in vacuum is generally made by relativistic effective potentials in the
mean field approximation in 2D, 3D and 4D models with two flavor and $N_c$
color massless fermions. It is found that in ground states of these models,
interplay between the two condensates mainly depend on the ratio $G_S/H_S$ for
2D and 4D case or $G_S/H_P$ for 3D case, where $G_S$, $H_S$ and $H_P$ are
respectively the coupling constants in a scalar $(\bar{q}q)$, a scalar $(qq)$
and a pseudoscalar $(qq)$ channel. In ground states of all the models, only
pure $<\bar{q}q>$ condensates could exist if $G_S/H_S$ or $G_S/H_P$ is bigger
than the critical value $2/N_c$, the ratio of the color numbers of the fermions
entering into the condensates $<qq>$ and $<\bar{q}q>$. As $G_S/H_S$ or
$G_S/H_P$ decreases to the region below $2/N_c$, differences of the models will
manifest themselves. Depending on different models, and also on $N_c$ in 3D
model, one will have or have no the coexistence phase of the two condensates,
besides the pure $<qq>$ condensate phase. The $G_S-H_S$ (or $G_S-H_P$) phase
diagrams in these models are given. The results also implicate a real
constraint on two-flavor QCD-analogous NJL model.

Difermion condensates in vacuum in 2-4D four-fermion interaction models
Bang-Rong Zhou†
College of Physical Sciences, Graduate School of
the Chinese Academy of Sciences, Beijing 100049, China
In any four fermion (denoted by q) interaction models, the couplings of (qq)2-form can
always coexist with the ones of (q̄q)2-form via the Fierz transformations. Hence, even in
vacuum, there could be interplay between the condensates 〈q̄q〉 and 〈qq〉. Theoretical anal-
ysis of this problem is generally made by relativistic effective potentials in the mean field
approximation in 2D, 3D and 4D models with two flavor and Nc color massless fermions.
It is found that in ground states of these models, interplay between the two condensates
mainly depend on the ratio GS/HS for 2D and 4D case or GS/HP for 3D case, where GS ,
HS and HP are respectively the coupling constants in a scalar (q̄q), a scalar (qq) and a
pseudoscalar (qq) channel.
In ground states of all the models, only pure 〈q̄q〉 condensates could exist if GS/HS or
GS/HP is bigger than the critical value 2/Nc, the ratio of the color numbers of the fermions
entering into the condensates 〈qq〉 and 〈q̄q〉. Below it, differences of the models will manifest
themselves.
In the 4D Nambu-Jona-Lasinio (NJL) model, as GS/HS decreases to the region below
2/Nc, one will first have a coexistence phase of the two condensates then a pure 〈qq〉 con-
densate phase. Similar results come from a renormalized effective potential in the 2D Gross-
Neveu model, except that the pure 〈qq〉 condensates could exist only if GS/HS = 0. In a
3D Gross-Neveu model, when GS/HP < 2/Nc, the phase transition similar to the 4D case
can arise only if Nc > 4, and for smaller Nc, only a pure 〈qq〉 condensate phase exists but
no coexistence phase of the two condensates happens. The GS −HS (or GS − HP ) phase
diagrams in these models are given.
The results deepen our understanding of dynamical phase structure of four-fermion inter-
action models in vacuum. In addition, in view of absence of difermion condensates in vacuum
of QCD, they will also imply a real restriction to any given two-flavor QCD-analogous NJL
model, i.e. in the model, the derived smallest ratio GS/HS via the Fierz transformations in
the Hartree approximation must be bigger than 2/3.
The project supported by the National Natural Science Foundation of China under Grant No.10475113.
Electronic mailing address: zhoubr@163bj.com
http://arxiv.org/abs/0704.0841v3
I. MAIN RESULTS
We have researched interplay between the fermion(q)-antifermion (q̄) condensates 〈q̄q〉 and
the difermion condensates 〈qq〉 in vacuum in 2D, 3D and 4D four-fermion interaction models
with two flavor and Nc color massless fermions. It is found that the ground states of the systems
could be in different phases shown in the following GS − HS and GS − HP phase diagrams
[Fig.(a)–Fig.(d)], where
GS — coupling constant of scalar (q̄q)
2 channel
HS — coupling constant of scalar color
Nc(Nc − 1)
−plet (qq)2 channel (4D, 2D case)
HP — coupling constant of pseudoscalar color
Nc(Nc − 1)
−plet (qq)2 channel (3D case)
Λ — Euclidean Momentum cutoff of loop integrals (4D,3D case)
(σ1, 0) — pure 〈q̄q〉 phase
(0,∆1) — pure 〈qq〉 phase
(σ2,∆2) — mixed phase with both 〈q̄q〉 and 〈qq〉
Fig.(a)-Fig.(d) (pages 3-6)
                                                                    3 
4D NJL Model 
y=GSΛ
2/π2 
               
 y =2x/ Nc
           (σ1, 0) 
(σ2,Δ2) 
      
1/Nc                    (0 ,Δ1)                R 
0        1/2 
x =HSΛ
R:  y=x/[1+(Nc-2)x] 
Fig. (a) 
                                                          4           
2D GN Model   
y=GS /π 
y=2x /Nc  
         
(σ1, 0)  
(σ2 ,Δ2) 
x=HS /π  (0,Δ1) 
Fig. (b)      
                                            
3D GN Model,  Nc≤4  
y=GSΛ/π
y=2x/ Nc
           (σ1, 0) 
1/4Nc
                                 (0,Δ1) 
        0         1/8 
x=HPΛ/π
Fig. (c)  
3D GN Model,  Nc≥5  
y=GSΛ/π
          y =2x/ Nc
(σ1, 0) 
1/4Nc                         
            (σ2,Δ2) 
(0,Δ1) 
0 1/8 
x =HPΛ/π
Fig. (d) 
Main Conclusions
1. In all the models, pure 〈q̄q〉 phase happens if
) > 2
(also GS must be large
enough in 3D and 4D model).
2. The phases with condensates 〈qq〉, including pure 〈qq〉 phase and mixed phase with 〈q̄q〉
and 〈qq〉, arise only if
) < 2
3. In 3D Gross-Neveu model, no mixed phase with 〈q̄q〉 and 〈qq〉 exists for Nc ≤ 4.
II. Motive and general approach
• In any four-fermion interaction model [1, 2], the couplings of (qq)2-form and (q̄q)2-form
can always coexist via the Fierz transformations, hence there must be interplay between
the condensates 〈q̄q〉 and 〈qq〉 in ground state of the system.
• In the vacuum, despite of absence of net fermions, based on a relativistic quantum field
theory, it is possible that the condensates 〈qq〉 and 〈q̄q̄〉 are generated simultaneously.
• The mean field approximation has been taken. In this case, we have used the Fierz
transformed four-fermion couplings in the Hartree approximation to avoid double
counting [3].
• In selecting the couplings of (qq)2-form, we always simulate SU(Nc) gauge interaction,
where two fermions are attractive in the antisymmetric
Nc(Nc − 1)
-plet.
• Euclidean momentum cutoffs in 3D and 4D models have been used so as to maintain
Lorentz invariance of effective potentials in the vacuum.
• In massless fermion limit, all the discussions can be made analytically.
• The coupling constants GS and HS (or HP ) are viewed as independent parameters.
III. 4D Nambu-Jona-Lasinio model
With 2 flavors and Nc color massless fermions, the Lagrangian
L = q̄iγµ∂µq +GS [(q̄q)
2 + (q̄iγ5~τq)
2] +HS
(q̄iγ5τ2λAq
C)(q̄C iγ5τ2λAq), (1)
where the fermion fields q are in the doublet of SUf (2) and the Nc-plet of SUc(Nc), i.e.
i = 1, · · · , Nc, (2)
http://arxiv.org/abs/0704.0841v3
qC is the charge conjugate of q and ~τ = (τ1, τ2, τ3) are the Pauli matrices acting in two-flavor
space. The matrices λA run over all the antisymmetric generators of SUc(Nc).
Assume that the four-fermion interactions can lead to the scalar condensates
〈q̄q〉 = φ (3)
with all the Nc color fermion entering them, and the scalar color
Nc(Nc − 1)
2 -plet difermion
and di-antifermion condensates (after a global SUc(Nc) transformation)
〈q̄Ciγ5τ2λ2q〉 = δ, 〈q̄iγ5τ2λ2q
C〉 = δ∗, (4)
with only two color fermions enter them. The corresponding symmetry breaking is that
SUfL(2) ⊗ SUfR(2) → SUf (2), SUc(Nc) → SUc(2), and a ”rotated” electric charge UQ̃(1)
and a ”rotated” quark number U ′q(1) leave unbroken. It should be indicated that in the case
of vacuum, the Goldstone bosons induced by spontaneous breaking of SUc(Nc) could be some
combinations of difermions and di-antifermions.
Define that
σ = −2GSφ, ∆ = −2HSδ, ∆
∗ = −2HSδ
∗. (5)
With standard technique and a 4D Euclidean momentum cutoff Λ [4], we obtain the relativistic
effective potential
V4(σ, |∆|) =
2 + 2|∆|2)Λ2 − (Nc − 2)
−(σ2 + |∆|2)2
σ2 + |∆|2
. (6)
The ground states of the system, i.e. the minimum points of V4(σ, |∆|), will be at
(σ, |∆|) =
(0, ∆1)
(σ2, ∆2)
(σ1, 0)
, 0 ≤
1 + (Nc − 2)
1 + (Nc − 2)
, (7)
Eq.(7) gives the phase diagram Fig.(a) of the 4D NJL model.
IV. 2D Gross-Neveu model
The Lagrangian is expressed by
L = q̄iγµ∂µq +GS [(q̄q)
2 + (q̄iγ5~τq)
2] +HS(q̄iγ5τSλAq
C)(q̄Ciγ5τSλAq), (8)
All the denotations are the same as ones in 4D NJL model, except that in 2D space-time
, γ1 =
= −C, γ5 = γ
and τS = (τ0 ≡ 1, τ1, τ3) are flavor-triplet symmetric matrices. It is indicated that the product
matrix Cγ5τSλA is antisymmetric.
Assume that the four-fermion interactions could lead to the scalar quark-antiquark conden-
sates
〈q̄q〉 = φ, (9)
which will break the discrete symmetries
χD : q(t, x)
→ γ5q(t, x),
P1 : q(t, x)
→ γ1q(t,−x),
and that the coupling with HS can lead to the scalar color
Nc(Nc − 1)
-plet difermion con-
densates and the scalar color anti-
Nc(Nc − 1)
-plet di-antifermion condensates (after a global
transformation in flavor and color space)
〈q̄C iγ51fλ2q〉 = δ, 〈q̄iγ51fλ2q
C〉 = δ∗ (10)
which will break discrete symmetries Zc
(center of SUc(3)) and Z
(center of SUf (2)), besides
χD and P1. Noting that in a 2D model, no breaking of continuous symmetry needs to be
considered on the basis of Mermin-Wagner-Coleman theorem [5].
The model is renormalizable. In the space-time dimension regularization approach, we can
write down the renormalized L in D = 2− 2ε dimension space-time by the replacements
GS → GSM
2−DZG, HS → HSM
2−DZH ,
with the scale parameter M , the renormalization constants ZG and ZH . In addition, the γ
L will become 2D/2 × 2D/2 matrices.
Define the order parameters
σ = −2GSM
2−DZGφ, ∆ = −2HSM
2−DZHδ, (11)
which will be finite if ZG and ZH are selected so as to cancel the UV divergences in φ and δ. In
the minimal substraction scheme,
ZG = 1−
2NcGS
, ZH = 1−
. (12)
By similar derivation to the one made in Ref.[6], the corresponding renormalized effective po-
tential in the mean field approximation up to one-loop order becomes
V2(σ, |∆|) =
σ2 + |∆|2
+ (Nc − 2) ln
σ2 + |∆|2
, M̄2 = 2πe−γM2, (13)
where γ is the Euler constant. The ground states of the system i.e. the minimal points of
V2(σ, |∆|) will be at
(σ, |∆|) =
(0, ∆1)
(σ2, ∆2)
(σ1, 0)
GS/HS = 0
0 < GS/HS < 2/Nc
GS/HS > 2/Nc
Eq.(14) gives the phase diagram Fig.(b) of 2D GN model.
In 2D case, the GS -HS phase structure has the following feature:
1. The pure 〈qq〉 phase (0,∆1) could appear only if GS/HS = 0;
2. Formations of the condensates do not call for that the coupling constant GS and HS have
some lower bounds.
V. 3D Gross-Neveu model
The Lagrangian is expressed by
L = q̄iγµ∂µq +GS [(q̄q)
2 + (q̄~τq)2] +HP
(q̄τ2λAq
C)(q̄Cτ2λAq), (15)
where γµ(µ = 0, 1, 2) are taken to be 2× 2 matrices
, γ1 =
, γ2 =
It is noted that the product matrix Cτ2λA is antisymmetric, and since without the ”γ5” matrix,
the only possible color
Nc(Nc − 1)
-plet difermion interaction channel is pseudoscalar one. The
condensates 〈q̄q〉 will break
time reversal symmetry T : q(t, ~x) → γ2q(−t, ~x),
special parity P1 : q(t, x
1, x2) → γ1q(t,−x1, x2),
special parity P2 : q(t, x
1, x2) → γ2q(t, x1,−x2).
The difermion condensates 〈q̄Cτ2λ2q〉 (after a global rotation in the color space) will break
SUc(Nc) → SUc(2)
and leave a ”rotated” electrical charge U
(1) and a ”rotated” fermion number U ′q(1) unbroken.
It also breaks
parity P : q(t, ~x) → γ0q(t,−~x)
and this shows pseudoscalar feature of the difermion condensates.
Define the order parameters in the 3D GN model
σ = −2GS〈q̄q〉, ∆ = −2HP 〈q̄
Cτ2λ2q〉, (16)
on bases of the same method used in Ref.[7], we find out the effective potential in the mean field
approximation
V3(σ, |∆|) =
2 + 2|∆|2)Λ
6σ2|∆|+ 2|∆|3 + (Nc − 2)σ
3 + 2θ(σ − |∆|)(σ − |∆|)3
, (17)
where Λ is a 3D Euclidean momentum cutoff. The ground states of the system correspond to
the least value points of V3(σ, |∆|) which will respectively be at
(σ, |∆|) =
(0,∆1),
(0,∆1),
(σ2,∆2),
(σ1, 0),
, for Nc ≤ 4
for Nc > 4
, for all Nc
Eq.(18) gives the GS −HP phase diagrams Fig.(c) and Fig.(d) of the 3D GN model.
VI. Summary
• Present research deepens our theoretical understanding of the four-fermion interaction
models:
1. Even in vacuum, it is possible that the difermion condensates are generated as long
as the coupling constants of the difermion channel are strong enough (bigger than
zero or some finite values).
2. Interplay between the condensates 〈q̄q〉 and 〈qq〉 mainly depends on GS/HS (or
GS/HP ), the ratio of the coupling constants of scalar fermion-antifermion channel
and scalar (or pseudoscalar ) difermion channel.
3. In all the discussed 2-flavor models, if GS/HS (GS/HP ) > 2/Nc, the ratio of the
color numbers of the fermions entering into the condensates 〈qq〉 and 〈q̄q〉, (and also
with sufficiently large GS in 4D and 3D model), then only pure 〈q̄q〉 condensates
phase may exist. Below 2/Nc, (and also with sufficiently large HS or HP in 4D or
3D model), one will always first have a mixed phase with condensates 〈q̄q〉 and 〈qq〉,
then a pure 〈qq〉 condensate phase, except that in the 3D GN model, no the mixed
phase appears when Nc ≤ 4.
• In view of absence of 〈qq〉 condensates in vacuum of QCD, the result here also implies
a real restriction to any given two-flavor QCD-analogue NJL model: in such model, the
derived smallest ratio GS/HS via the Fierz transformation in the Hartree approximation
must be bigger than 2/3 [4].
[1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345; 124 (1961) 246.
[2] D.J. Gross and A. Neveu, Phys. Rev. D 10 (1974) 3235.
[3] M. Buballa, Phys. Rep. 407 (2005) 205.
[4] Zhou Bang-Rong, Commun. Theor. Phys. 47 (2007) 95.
[5] N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133; S. Coleman, Commun. Math. Phys.
31 (1973) 259.
[6] Zhou Bang-Rong, Commun. Theor. Phys. 47 (2007) 520.
[7] Zhou Bang-Rong, Commun. Theor. Phys. 47 (2007) 695.
	Main results
	References
