From the mass term for the transformed quark fields, we obtain effective
contact interactions of the NJL type. The parameters of the model that maps a
system of non-interacting transformed fields into quarks interacting via NJL
contact terms are discussed

Lima, C. L.

Timoteo, V. S.

English

arXiv

V. S. Timóteo

C. L. Lima

Crossref

The NJL interaction from q-deformed inspired transformations

From the mass term for q-deformed quark fields, we obtain effective contact interactions of the NJL type. The parameters of the model that maps a system of non-interacting deformed fields into quarks interacting via NJL contact terms is discussed.717

Timóteo, V. S.

Repositorio da Producao Cientifica e Intelectual da Unicamp

Brazilian Journal of Physics, vol. 37, no. 1, March, 2007 71
The NJL Interaction from q-Deformed Inspired Transformations ∗
V. S. Timo´teo1 and C. L. Lima2
1Centro Superior de Educac¸a˜o Tecnolo´gica, Universidade Estadual de Campinas, 13484-370, Limeira, SP, Brazil
2Instituto de Fı´sica, Universidade de Sa˜o Paulo, CP 66318, 05315-970, Sa˜o Paulo, SP, Brazil
Received on 29 September, 2006
From the mass term for q-deformed quark fields, we obtain effective contact interactions of the NJL type.
The parameters of the model that maps a system of non-interacting deformed fields into quarks interacting via
NJL contact terms is discussed.
Keywords: Quantum Algebras; Hadronic Physics; Effective Models
In hadron physics, the NJL model [1, 2] is a very simple
effective model for strong interactions that describes impor-
tant features like the dynamical mass generation, spontaneous
chiral symmetry breaking, and chiral symmetry restoration at
finite temperature.
q-Deformed algebras provide a nice framework to incorpo-
rate, in an effective way, interactions not originally contained
in the Lagrangian of a particular system [3–5].
In recent works [6–9], we have been investigating possible
applications of quantum algebras in hadronic physics. In gen-
eral, we observed that when we deform the underlying alge-
bra, the system is affected with correlations between its con-
stituents. We have studied in detail the NJL model under the
influence of a quantum su(2) algebra.
In this work we approach the following question: is it pos-
sible to obtain an art of ”canonical transformation” connecting
the NJL model to a simpler non-interacting system? We veri-
fied that we can indeed obtain the same dynamics of the NJL
interaction with non-interacting q-deformed quark fields.
We start by writing a mass term for the q-deformed quark
fields
Lmassq = −M ΨΨ
= −M ( Ψ1Ψ1+Ψ2Ψ2 )
= −M (UU +DD ) (1)
where Ψ1,2 =U,D are the components of Ψ.
The q-deformed quark fields can be written in terms of the
standard fields as
Ψ1 = ψ1+(q−1−1) ψ1ψ2γ0ψ2 , (2)
Ψ2 = ψ2+(q−1−1) ψ2ψ1γ0ψ1 , (3)
or
U = u+(q−1−1) udγ0d , (4)
D = d+(q−1−1) duγ0u , (5)
and ψ1,2 = u,d. Here both components are modified in the
same way, so that the above expressions are different from he
∗Talk given at RETINHA 18, IFUSP, Sa˜o Paulo, May 2006.
Dedicated to Prof. Yogiro Hama in honor of his 70th birth.
ones used in [4, 5], where only one component is affected.
Extending the deformation to the two components is required
to obtain a set of terms that will form an interaction of the NJL
type. This implies that the anti-commutation relations for the
deformed fields Ψ will also be different from the ones in [4, 5].
Since obtaining the new anti-commutation relations is not in
the scope of this work, we focus on the effective interactions
contained in the non-interacting Lagrangian.
Using Eqs. (4) and (5), we can re-write the Lagragian
Eq.(1) in terms of the non-deformed quark fields
UU = uu+Q uud†d+Q d†duu+Q2 dduudd , (6)
DD = dd+Q ddu†u+Q u†udd+Q2 uudduu , (7)
where Q = (q−1−1).
We can re-write the above equations as follows
UU =
(
1+2Qd†d
)
uu+
Q2
2
(
dduudd+dduudd
)
,
DD =
(
1+2Qu†u
)
dd+
Q2
2
(
uudduu+uudduu
)
,
so that we identify the contact interactions between the quarks
contained in the non-interacting deformed fields Lagrangian.
The six-point contact terms are depicted in Fig. 1.
d d
d d
u u
u u
u u
d dd d
u u
d d
d d
u u
u u
Gq Gq
Gq Gq
FIG. 1: Contact interactions generated by the mass term for the q-
deformed fermion fields.
We can reduce the six-point interactions of Fig. 1 to four-
point contact terms in a mean field approach [2], so that we
72 V. S. Timo´teo and C. L. Lima
have
(
UU +DD
)
=
(
1+2Q
〈
ψ†ψ
〉
A
)(
uu+dd
)
+
Q2
2
〈ψψ〉
A2
× (dduu+dddd+uudd+uuuu) , (8)
where 〈ψ†ψ〉 = 〈u†u〉 = 〈d†d〉 = ρv, 〈ψψ〉 = 〈uu〉 =
〈
dd
〉
=
ρs, and A = A(T ;q) has the same dimension of the conden-
sate and will be determined later in this work. This procedure
corresponds to have one fermion line closed and can be repre-
sented by the diagrams of Fig. 2.
u u d d
d d u u
〈uu〉〈dd〉
u u
u u
d d
d d
〈dd〉〈uu〉
Gq
Gq
Gq
Gq
FIG. 2: Reduction of the six-point interactions of Fig. 1 to four-point
interactions by closing one fernion line.
Now we can write the mass term for the q-deformed quark
fields
Lmassq = −MΨΨ
= −M
(
1+2
〈
ψ†ψ
〉
Γ
)
ψψ
− M
2
〈ψψ〉Γ2 ψψψψ , (9)
with Γ = Q/A
Accordingly, the kinetic energy term for the deformed
fields, Ψγµ∂µΨ, can be written in terms of the non-deformed
ones as
Ψγµ∂µΨ = Uγµ∂µU +Dγµ∂µD
= uγµ∂µu+Q
(
dγ0duγµ∂µu+uγµ∂µudγ0d
)
+ dγµ∂µd+Q
(
uγ0udγµ∂µd+dγµ∂µduγ0u
)
+ Q2
(
dγ0duγµ∂µudγ0d
)
+ Q2
(
uγ0udγµ∂µduγ0u
)
(10)
By using an extreme mean field approximation, namely,
substituting everywhere in the kinetic energy contribution
〈ψ†ψ〉= 〈u†u〉= 〈d†d〉→ ρv, and 〈ψψ〉= 〈uu〉=
〈
dd
〉→ ρs,
we obtain
Ψγµ∂µΨ = uγµ∂µu(1+2Γρv)
+ dγµ∂µd (1+2Γρv)
+ (uγµ∂µu)Γ2ρv+
(
dγµ∂µd
)
Γ2ρv
=
(
uγµ∂µu+dγµ∂µd
)
(1+Γρv)2 (11)
This corresponds to a usual kinetic energy with a shifted mo-
mentum p→ p(1+Γρv)2.
The treatment of the density dependence of the kinetic en-
ergy term is rather cumbersome and will be postponed to a
further contribution. We will consider the influence of this
momentum dependent kinetic energy term in an effective way.
Therefore, we will study a class of Lagrangians of the type
L ′q =
1
(1+Γρv)2
Lq = ψγµ∂µψ
− M
(
1+2
〈
ψ†ψ
〉
Γ
) 1
(1+Γρv)2
ψψ
− M
2
〈ψψ〉Γ2 1
(1+Γρv)2
ψψψψ (12)
This representative of the full Lagrangian Lq = Ψγµ∂µΨ+
Lmassq , when written in terms of the standard quark fields, can
be identified with the NJL Lagrangian
LNJL = ψγµ∂µψ−m0 ψψ+G ψψψψ . (13)
The conditions for both Lagrangians, LNJL and L ′q, to be
equivalent for any values of T and q are
M =
(1+Γρv)2
(1+2Γρv)
m0 , (14)
and
G =−M
2
ρsΓ2
(1+Γρv)2
. (15)
If we insert Eq. (14) in Eq. (15), we obtain an equation for Γ
Γ2−2αρv Γ−α = 0 , (16)
where
α =− 2G
m0ρs
> 0. (17)
This equation has two solutions
Γ± = αρv
(
1±
√
1+
1
αρ2v
)
. (18)
The mass of the q-deformed fermion fields, M, has to be posi-
tive, so we associate the two solutions Γ− and Γ+ with the two
regimes q < 1 and q > 1, respectively. The quantity A will be
negative in both cases.
Brazilian Journal of Physics, vol. 37, no. 1, March, 2007 73
The scalar (ρs) and vector (ρv) densities were calculated
from the NJL model at finite temperature:
ρs = −NcN fpi2
∫ Λ
0
d pp2
m
E
[1−n−n] , (19)
ρv =
NcN f
pi2
∫ Λ
0
d pp2 [n−n] , (20)
where
n(p,T,µ) =
1
1+ exp [β(E−µ)] , (21)
and
n(p,T,µ) =
1
1+ exp [β(E +µ)]
, (22)
are the fermions and anti-fermions distribution functions re-
spectively with E =
√
p2+m2.
First solve the set of coupled gap equations for m, µ, and
ρv (Eqs. 23 and 20, respectively) in the NJL model at finite
temperature and chemical potential{
m = m0−2Gρs ,
µ = µ0− GNc ρv .
(23)
The next step is to calculate the scalar and vector densities
entering in the equation for Γ for a given value of the defor-
mation parameter q. In this way we obtain A(T ;q), which in
turn is used to obtain M. The numerical results are displayed
in Figures 3 and 4, where we show the quantity A, in units
of the condensate at zero temperature, as a function of both
temperature and deformation for the q> 1 and q< 1 regimes.
0 0.05 0.1 0.15 0.2 1
1.1
1.2
1.3
1.4
1.50
0.01
0.02
0.03
0.04
0.05
q
T (GeV)
A
/ρ
0
FIG. 3: The quantity A, in units of the chiral condensate at zero tem-
perature ρ0 = ρs(T = 0) = −1.42× 10−2 GeV3, as a function of
temperature and q-deformation for the q > 1 regime.
It is worth to note that the mass of the q-deformed fermion
fields, M, does not depend on the deformation of the algebra
while its temperature dependence is shown in 5.
The quantity A(T ;q) maps the simple non-interacting
model into the NJL model. It represents, in an effective way,
the correlations introduced by the quantum algebra, when we
write the non-interacting Lagrangian in terms of the standard
0
0.05
0.1
0.15
0.2 0.40.50.60.70.80.91
0
0.2
0.4
0.6
0.8
qT (GeV)
A
/ρ
0
FIG. 4: The quantity A, in units of the chiral condensate at zero tem-
perature ρ0 = ρs(T = 0) = −1.42× 10−2 GeV3, as a function of
temperature and q-deformation for the q < 1 regime.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
0
20
40
60
80
100
120
140
T (GeV)
M
 / 
m
0
FIG. 5: The mass of the q-deformed quark fields, in units of the
current quark mass m0 = 5 MeV, as a function of T for both q > 1
and q < 1 regimes. For small temperatures, M = m0.
quark fields. These correlations, in a mean field approxima-
tion, are effectively represented by contact interactions of the
NJL type. It is also important to mention that it inherits the
phase transition. When the condensate and the dynamical
mass vanishes with increasing T , the quantity A also experi-
ences the phase transition. This is an expected behavior, since
it depends on the dynamical mass. For a given temperature,
T , and deformation, q, there is a value of the mapping func-
tion, A(T ;q), that makes the Lagrangians Eq.(12) and Eq.(13)
equivalent.
Summarizing, we have shown that it is possible to describe
the dynamics of an interacting system of the NJL type with
a simple non-interacting system by using a set of quantum
algebra transformations and a mapping function.
Acknowledgments
This work was partially supported by FAPESP Grant No.
2002/10896-7. V.S.T. would like to thank FAPESP for finan-
cial support under grant number 2005/01911-0.
74 V. S. Timo´teo and C. L. Lima
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Effective interactions from q-deformed inspired transformations

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