Effective interactions from q-deformed
inspired transformations
V. S. Timóteo a C. L. Lima b
aCentro Superior de Educação Tecnológica, Universidade Estadual de Campinas,
13484-370, Limeira, SP, Brasil
bInstituto de F́ısica, Universidade de São Paulo, CP 66318, 05315-970, São
Paulo, SP, Brazil
Abstract
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.
It is very common in physics to use transformations that make one particular
system mathematically simpler, yet describing the same phenomena. A clear
example is the use of canonical transformations in classical mechanics.
q-Deformed algebras provide a nice framework to incorporate, in an effective
way, interactions not originally contained in the Lagrangian of a particular
system.
In hadron physics, the NJL model is a very simple effective model for strong
interactions that describes important features like the dynamical mass genera-
tion, spontaneous chiral symmetry breaking, and chiral symmetry restoration
at finite temperature.
In recent works, we have been investigating possible applications of quantum
algebras in hadronic physics. In general, we observed that when we deform
the underlying algebra, the system is affected with correlations between its
constituents. We have studied in detail the NJL model under the influence of
a quantum su(2) algebra.
The question we approach in this letter is: is it possible to obtain a transfor-
mation connecting the NJL model to a simpler non-interacting system? We
verified that we can indeed obtain the same dynamics of the NJL interaction
Preprint submitted to Elsevier 4 November 2018
http://arxiv.org/abs/0704.0584v1
with a simple transformation of the quark fields, inspired in the q-deformed
quark fields of previous works [2], [3], [6].
Mass term
We start by writing a mass term for the transformed quark fields
Lmassq =−M ΨΨ
Ψ1Ψ1 +Ψ2Ψ2
UU +DD
where
 . (2)
The transformed quark fields can be written in terms of the standard fields as
Ψ1=ψ1 + (q
−1 − 1) ψ1ψ2γ0ψ2 , (3)
Ψ2=ψ2 + (q
−1 − 1) ψ2ψ1γ0ψ1 , (4)
U = u+ (q−1 − 1) udγ0d , (5)
D= d+ (q−1 − 1) duγ0u , (6)
 . (7)
Here both components are modified in the same way, so that the above ex-
pressions are different from the ones used in [2,3], where only one component
is affected. Extending the transformation to both components is required to
obtain a set of terms that will form an interaction of the NJL type. This im-
plies that the anti-commutation relations for the deformed fields Ψ will also
be different from the ones in [2,3]. Since obtaining the new anti-commuation
relations is not in the scope of this work, we focus on the effective interactions
contained in the non-interacting Lagrangian.
Using Eqs. (5) and (6), we can re-write the Lagragian Eq.(1) in terms of the
standard quark fields
UU = uu+Q uud†d+Q d†duu+Q2 dduudd , (8)
DD= dd+Q ddu†u+Q u†udd+Q2 uudduu , (9)
where Q = (q−1 − 1).
We can re-write the above equations as follows
1 + 2Q d†d
dduudd+ dduudd
, (10)
1 + 2Q u†u
uudduu+ uudduu
, (11)
so that we identify the contact interactions between the quarks contained in
the non-interacting deformed fields Lagrangian. Figure 1 shows the six-point
contact interactions contained in the mass term for the q-deformed quark
fields.
We can reduce the six-point interactions to four-point contact terms in a mean
field approach [5], so that we have
UU +DD
1 + 2Q
uu+ dd
dduu+ dddd+ uudd+ uuuu
, (12)
where 〈ψ†ψ〉 = 〈u†u〉 = 〈d†d〉 = ρv,
= 〈uu〉 =
= ρs, and A =
A(T ; q) has the same dimension of the condensate and will be determined
later in this letter. The reduction of the six-point to four-point contact terms
by closing one fermion line is also shown in Figure 1.
Now we can write the mass term for the transformed quark fields
Lmassq = −MΨΨ = −M
1 + 2
ψψ − M
Γ2 ψψψψ , (13)
with Γ = Q/A
Kinetic energy term
Accordingly, the kinetic energy term for the transformed fields, Ψγµ∂µΨ, can
be written in terms of the standard ones as
Ψγµ∂µΨ=Uγ
µ∂µU +Dγ
= uγµ∂µu+Q
dγ0duγ
µ∂µu+ uγ
µ∂µudγ0d
+ dγµ∂µd+Q
uγ0udγ
µ∂µd+ dγ
µ∂µduγ0u
dγ0duγ
µ∂µudγ0d
uγ0udγ
µ∂µduγ0u
By using an extreme mean field approximation, namely, substituting every-
where in the kinetic energy contribution 〈ψ†ψ〉 = 〈u†u〉 = 〈d†d〉 → ρv, and
= 〈uu〉 =
→ ρs, we obtain
Ψγµ∂µΨ= uγ
µ∂µu (1 + 2Γρv)
+ dγµ∂µd (1 + 2Γρv)
+ (uγµ∂µu) Γ
2ρv +
dγµ∂µd
uγµ∂µu+ dγ
(1 + Γρv)
This corresponds to a usual kinetic energy with a shifted momentum p →
p (1 + Γρv)
The full Lagrangian
The treatment of the density dependence of the kinetic energy 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 + Γρv)
Lq = ψγµ∂µψ −M
1 + 2
(1 + Γρv)
(1 + Γρv)
ψψψψ (16)
This representative of the full Lagrangian Lq = Ψγµ∂µΨ + Lmassq , when writ-
ten in terms of the standard quark fields, can be identified with the NJL
Lagrangian
LNJL = ψγµ∂µψ −m0 ψψ +G ψψψψ . (17)
The conditions for both Lagrangians, LNJL and L′q, to be equivalent for any
values of T and q are
(1 + Γρv)
(1 + 2Γρv)
m0 , (18)
G = −M
(1 + Γρv)
. (19)
Inserting Eq. (18) in Eq. (19), we obtain an equation for Γ
Γ2 − 2αρv Γ− α = 0 , (20)
where
α = − 2G
> 0. (21)
This equation has two solutions
Γ± = αρv
. (22)
The mass of the transformed fermion fields, M , has to be positive, 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.
The scalar (ρs) and vector (ρv) densities were calculated from the NJL model
at finite temperature:
[1− n− n] , (23)
dpp2 [n− n] , (24)
where
n(p, T, µ) =
1 + exp [β (E − µ)]
, (25)
n(p, T, µ) =
1 + exp [β (E + µ)]
, (26)
are the fermions and anti-fermions distribution functions respectively with
p2 +m2.
First we solve the set of coupled gap equations for m, µ, and ρv (Eqs. 27
and 24, respectively) in the NJL model at finite temperature and chemical
potential
m = m0 − 2Gρs ,
µ = µ0 − GNcρv .
The next step is to calculate the scalar and vector densities entering in the
equation for Γ for a given value of the transformation 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 2 and 3, where we show the quantity A and the mass
M as a function of both temperature and transformation parameter in the
q > 1 and q < 1 regimes. It is worth to note that the mass of the transformed
fermion fields does not depend on the transformation parameter.
The well known results of the NJL model are mapped through A(T ; q) from the
non-interacting transformed fermion fields Lagrangian. It is worth to note that
the mass of the q-deformed fermion fields does not depend on the deformation
of the algebra.
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
transformations, when we write the non-interacting Lagrangian in terms of
the standard quark fields. These correlations, in a mean field approximation,
are effectively represented by contact interactions of the NJL type. It is also
important to mention that it inherits the phase transition. When the con-
densate and the dynamical mass vanishes with increasing T , the quantity A
also experiences the phase transition. This is an expected behavior, since it
depends on the dynamical mass. For a given temperature, T , and transfor-
mation parameter, q, there is a value of the mapping function, A(T ; q), that
makes the Lagrangians Eq.(16) and Eq.(17) 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 inspired transformations and a mapping
function.
Acknowledgments
C. L. L. thanks Profs. D. Galetti and B. M. Pimentel for most helpful discus-
sions. This work was partially supported by FAPESP Grant No. 2002/10896-7.
V.S.T. would like to thank FAEPEX/UNICAMP for financial support.
References
[1] D. Galetti and B. M. Pimentel, An. Acad. Bras. Ci. 67 (1995) 7; S. S. Avancini,
A. Eiras, D. Galetti, B. M. Pimentel, and C. L. Lima, J. Phys. A: Math. Gen. 28
(1995) 4915; D. Galetti, J. T. Lunardi, B. M. Pimentel, and C. L. Lima, Physica
A242 (1997) 501.
[2] M. Ubriaco, Phys. Lett. A 219 (1996) 205.
[3] L. Tripodi and C. L. Lima, Phys. Lett. B 412 (1997) 7.
[4] Y. Nambu and G. Jona-Lasinio, Physical Review 122 (1961) 345.
[5] U. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27 (1991) 195.
[6] V. S. Timóteo and C. L. Lima, Phys. Lett. B 448 (1999) 1.
[7] V. S. Timóteo and C. L. Lima, Mod. Phys. Lett. A 15 (2000) 219.
[8] V. S. Timóteo and C. L. Lima, nucl-th/0509089.
http://arxiv.org/abs/nucl-th/0509089
d dd d
Gq Gq
Gq Gq
u u d d
d d u u
〈uu〉〈dd〉
〈dd〉〈uu〉
Fig. 1. Contact interactions generated by the mass term for the q-deformed fermion
fields and their reduction from six-point to four-point by closing one fernion line.
0.2 1
T (GeV)
0.2 0.40.5
qT (GeV)
Fig. 2. The quantity A, in units of the chiral condensate at zero temperature
ρs(T = 0) = −1.42 × 10−2 GeV3, as a function of temperature and q-deformation
for the q > 1 and q < 1 regimes.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
T (GeV)
Fig. 3. 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.
	References

  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.

Effective interactions from q-deformed
inspired transformations
V. S. Timóteo a C. L. Lima b
aCentro Superior de Educação Tecnológica, Universidade Estadual de Campinas,
13484-370, Limeira, SP, Brasil
bInstituto de F́ısica, Universidade de São Paulo, CP 66318, 05315-970, São
Paulo, SP, Brazil
Abstract
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.
It is very common in physics to use transformations that make one particular
system mathematically simpler, yet describing the same phenomena. A clear
example is the use of canonical transformations in classical mechanics.
q-Deformed algebras provide a nice framework to incorporate, in an effective
way, interactions not originally contained in the Lagrangian of a particular
system.
In hadron physics, the NJL model is a very simple effective model for strong
interactions that describes important features like the dynamical mass genera-
tion, spontaneous chiral symmetry breaking, and chiral symmetry restoration
at finite temperature.
In recent works, we have been investigating possible applications of quantum
algebras in hadronic physics. In general, we observed that when we deform
the underlying algebra, the system is affected with correlations between its
constituents. We have studied in detail the NJL model under the influence of
a quantum su(2) algebra.
The question we approach in this letter is: is it possible to obtain a transfor-
mation connecting the NJL model to a simpler non-interacting system? We
verified that we can indeed obtain the same dynamics of the NJL interaction
Preprint submitted to Elsevier 4 November 2018
http://arxiv.org/abs/0704.0584v1
with a simple transformation of the quark fields, inspired in the q-deformed
quark fields of previous works [2], [3], [6].
Mass term
We start by writing a mass term for the transformed quark fields
Lmassq =−M ΨΨ
Ψ1Ψ1 +Ψ2Ψ2
UU +DD
where
 . (2)
The transformed quark fields can be written in terms of the standard fields as
Ψ1=ψ1 + (q
−1 − 1) ψ1ψ2γ0ψ2 , (3)
Ψ2=ψ2 + (q
−1 − 1) ψ2ψ1γ0ψ1 , (4)
U = u+ (q−1 − 1) udγ0d , (5)
D= d+ (q−1 − 1) duγ0u , (6)
 . (7)
Here both components are modified in the same way, so that the above ex-
pressions are different from the ones used in [2,3], where only one component
is affected. Extending the transformation to both components is required to
obtain a set of terms that will form an interaction of the NJL type. This im-
plies that the anti-commutation relations for the deformed fields Ψ will also
be different from the ones in [2,3]. Since obtaining the new anti-commuation
relations is not in the scope of this work, we focus on the effective interactions
contained in the non-interacting Lagrangian.
Using Eqs. (5) and (6), we can re-write the Lagragian Eq.(1) in terms of the
standard quark fields
UU = uu+Q uud†d+Q d†duu+Q2 dduudd , (8)
DD= dd+Q ddu†u+Q u†udd+Q2 uudduu , (9)
where Q = (q−1 − 1).
We can re-write the above equations as follows
1 + 2Q d†d
dduudd+ dduudd
, (10)
1 + 2Q u†u
uudduu+ uudduu
, (11)
so that we identify the contact interactions between the quarks contained in
the non-interacting deformed fields Lagrangian. Figure 1 shows the six-point
contact interactions contained in the mass term for the q-deformed quark
fields.
We can reduce the six-point interactions to four-point contact terms in a mean
field approach [5], so that we have
UU +DD
1 + 2Q
uu+ dd
dduu+ dddd+ uudd+ uuuu
, (12)
where 〈ψ†ψ〉 = 〈u†u〉 = 〈d†d〉 = ρv,
= 〈uu〉 =
= ρs, and A =
A(T ; q) has the same dimension of the condensate and will be determined
later in this letter. The reduction of the six-point to four-point contact terms
by closing one fermion line is also shown in Figure 1.
Now we can write the mass term for the transformed quark fields
Lmassq = −MΨΨ = −M
1 + 2
ψψ − M
Γ2 ψψψψ , (13)
with Γ = Q/A
Kinetic energy term
Accordingly, the kinetic energy term for the transformed fields, Ψγµ∂µΨ, can
be written in terms of the standard ones as
Ψγµ∂µΨ=Uγ
µ∂µU +Dγ
= uγµ∂µu+Q
dγ0duγ
µ∂µu+ uγ
µ∂µudγ0d
+ dγµ∂µd+Q
uγ0udγ
µ∂µd+ dγ
µ∂µduγ0u
dγ0duγ
µ∂µudγ0d
uγ0udγ
µ∂µduγ0u
By using an extreme mean field approximation, namely, substituting every-
where in the kinetic energy contribution 〈ψ†ψ〉 = 〈u†u〉 = 〈d†d〉 → ρv, and
= 〈uu〉 =
→ ρs, we obtain
Ψγµ∂µΨ= uγ
µ∂µu (1 + 2Γρv)
+ dγµ∂µd (1 + 2Γρv)
+ (uγµ∂µu) Γ
2ρv +
dγµ∂µd
uγµ∂µu+ dγ
(1 + Γρv)
This corresponds to a usual kinetic energy with a shifted momentum p →
p (1 + Γρv)
The full Lagrangian
The treatment of the density dependence of the kinetic energy 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 + Γρv)
Lq = ψγµ∂µψ −M
1 + 2
(1 + Γρv)
(1 + Γρv)
ψψψψ (16)
This representative of the full Lagrangian Lq = Ψγµ∂µΨ + Lmassq , when writ-
ten in terms of the standard quark fields, can be identified with the NJL
Lagrangian
LNJL = ψγµ∂µψ −m0 ψψ +G ψψψψ . (17)
The conditions for both Lagrangians, LNJL and L′q, to be equivalent for any
values of T and q are
(1 + Γρv)
(1 + 2Γρv)
m0 , (18)
G = −M
(1 + Γρv)
. (19)
Inserting Eq. (18) in Eq. (19), we obtain an equation for Γ
Γ2 − 2αρv Γ− α = 0 , (20)
where
α = − 2G
> 0. (21)
This equation has two solutions
Γ± = αρv
. (22)
The mass of the transformed fermion fields, M , has to be positive, 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.
The scalar (ρs) and vector (ρv) densities were calculated from the NJL model
at finite temperature:
[1− n− n] , (23)
dpp2 [n− n] , (24)
where
n(p, T, µ) =
1 + exp [β (E − µ)]
, (25)
n(p, T, µ) =
1 + exp [β (E + µ)]
, (26)
are the fermions and anti-fermions distribution functions respectively with
p2 +m2.
First we solve the set of coupled gap equations for m, µ, and ρv (Eqs. 27
and 24, respectively) in the NJL model at finite temperature and chemical
potential
m = m0 − 2Gρs ,
µ = µ0 − GNcρv .
The next step is to calculate the scalar and vector densities entering in the
equation for Γ for a given value of the transformation 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 2 and 3, where we show the quantity A and the mass
M as a function of both temperature and transformation parameter in the
q > 1 and q < 1 regimes. It is worth to note that the mass of the transformed
fermion fields does not depend on the transformation parameter.
The well known results of the NJL model are mapped through A(T ; q) from the
non-interacting transformed fermion fields Lagrangian. It is worth to note that
the mass of the q-deformed fermion fields does not depend on the deformation
of the algebra.
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
transformations, when we write the non-interacting Lagrangian in terms of
the standard quark fields. These correlations, in a mean field approximation,
are effectively represented by contact interactions of the NJL type. It is also
important to mention that it inherits the phase transition. When the con-
densate and the dynamical mass vanishes with increasing T , the quantity A
also experiences the phase transition. This is an expected behavior, since it
depends on the dynamical mass. For a given temperature, T , and transfor-
mation parameter, q, there is a value of the mapping function, A(T ; q), that
makes the Lagrangians Eq.(16) and Eq.(17) 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 inspired transformations and a mapping
function.
Acknowledgments
C. L. L. thanks Profs. D. Galetti and B. M. Pimentel for most helpful discus-
sions. This work was partially supported by FAPESP Grant No. 2002/10896-7.
V.S.T. would like to thank FAEPEX/UNICAMP for financial support.
References
[1] D. Galetti and B. M. Pimentel, An. Acad. Bras. Ci. 67 (1995) 7; S. S. Avancini,
A. Eiras, D. Galetti, B. M. Pimentel, and C. L. Lima, J. Phys. A: Math. Gen. 28
(1995) 4915; D. Galetti, J. T. Lunardi, B. M. Pimentel, and C. L. Lima, Physica
A242 (1997) 501.
[2] M. Ubriaco, Phys. Lett. A 219 (1996) 205.
[3] L. Tripodi and C. L. Lima, Phys. Lett. B 412 (1997) 7.
[4] Y. Nambu and G. Jona-Lasinio, Physical Review 122 (1961) 345.
[5] U. Vogl and W. Weise, Prog. Part. Nucl. Phys. 27 (1991) 195.
[6] V. S. Timóteo and C. L. Lima, Phys. Lett. B 448 (1999) 1.
[7] V. S. Timóteo and C. L. Lima, Mod. Phys. Lett. A 15 (2000) 219.
[8] V. S. Timóteo and C. L. Lima, nucl-th/0509089.
http://arxiv.org/abs/nucl-th/0509089
d dd d
Gq Gq
Gq Gq
u u d d
d d u u
〈uu〉〈dd〉
〈dd〉〈uu〉
Fig. 1. Contact interactions generated by the mass term for the q-deformed fermion
fields and their reduction from six-point to four-point by closing one fernion line.
0.2 1
T (GeV)
0.2 0.40.5
qT (GeV)
Fig. 2. The quantity A, in units of the chiral condensate at zero temperature
ρs(T = 0) = −1.42 × 10−2 GeV3, as a function of temperature and q-deformation
for the q > 1 and q < 1 regimes.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
T (GeV)
Fig. 3. 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.
	References
