$π$ and $σ$ mesons at finite temperature and density in the NJL model with dimensional regularization

0.29 MB·English

by  T. Inagaki

HUPD-0704
π and σ mesons at finite temperature and density in the NJL
model with dimensional regularization
T. Inagaki
Information Media Center, Hiroshima University, Higashi-Hiroshima, Japan
D. Kimura
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0 Department of Physical Science, Hiroshima University, Higashi-Hiroshima, Japan
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2

n A. Kvinikhidze
a
J A. Razmadze Mathematical Institute of Georgian Academy of Sciences, Tbilisi, Georgia

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2
(Dated: February 2, 2008)

]
h Abstract
p
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p Dynamical Symmetry breakingand meson masses are studied in theNambu-Jona-Lasinio (NJL)
e
h model at finite temperature and chemical potential using the dimensional regularization. Since
[

the model is not renormalizable in four space-time dimensions, physical results and parameters
2
v
6 depend on the regularization method. Following the imaginary time formalism, we introduce the
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3 temperature,T andthechemicalpotential, µ. Theparametersofthemodelarefixedbycalculating
1
. the pion mass and decay constant in the dimensional regularization at T = µ = 0.
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1
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0 PACS numbers: 11.10.Kk,11.30.Qc,12.39.-x
:
v
i
X
r
a
1
I. INTRODUCTION
QCD is a fundamental theory of quarks and gluons whose coupling constant is large
at low energy scale. Therefore one cannot adopt the perturbative expansion in powers of
the coupling constant at low energy. To study physics in a hadronic phase we can not
avoid considering the non-perturbative effect of QCD. One of the possible ways to evaluate
the phenomena in the hadronic phase is to use a low energy effective theory. Some phe-
nomenological parameters are introduced to construct the low energy effective theory which
is simpler than QCD to deal with. NJL model is one of the low energy effective theories of
QCD [1], for a review see Refs.[2, 3, 4, 5]. The model has the same chiral symmetry as QCD
and the symmetry is broken down dynamically. Thus it is often used to study symmetry
properties of QCD in the hadronic phase.
The NJL model contains four-fermion interactions. Since a four-fermion interaction is a
dimension six operator, the NJL model is not renormalizable in four space-time dimensions.
To obtain finite expressions we must regularize the theory. In such a non-renormalizable
model most of the physical quantities depend on the regularization method and a parameter
introduced to regularize thetheory. Forexample, a cut-off scale is introduced as a parameter
in a cut-off regularization. The coupling constant and the cut-off scale are determined
phenomenologically.
The regularization in general may break some of the symmetries of the theory. For
example a naive three dimensional cut-off breaks Lorentz and gauge invariance [6]. In the
present paper we employ thedimensional regularization. Itpreserves most ofthesymmetries
of the theory, including the general covariance. We regard the space-time dimensions as one
of the parameters in the effective theory. Thus the dimensions should be determined in some
low energy phenomena which have some relation with chiral symmetry breaking.
There are some works investigating NJL model in the dimensional regularization. The
general properties of the renormalization and the renormalization group is studied in arbi-
trary dimensions 2 < D < 4 [7, 8, 9]. In Refs.[10, 11, 12, 13, 14, 15, 16] the NJL model
is considered as a prototype model of composite Higgs. The phase structure of dynamical
symmetry breaking is analyzed at high temperature, density, electro-magnetic field and cur-
vature. But we have no established fundamental theory of composite Higgs models at high
energy scale. Thus the physical scale of the theory has not been fixed and the contribution
2
of the current fermion mass has not been analyzed.
In Ref.[17] the dimensional regularization is modified to keep four dimensional properties
of the non-renormalizable theory as much as possible. To achieve this goal the dimensional
regularization is applied to only the radial part in loop integrals. It is one of the analytic
regularization. Themesonloopcontributiontothechiralsymmetrybreakingisalsoanalyzed
in the NJL model with the modified dimensional regularization [18]. The procedure is easily
extended tothefinitetemperatureandchemical potential. However, thephenomenologically
consistent dimensions are less than two in this approach. Because of the infrared divergence
it seems to be difficult to obtain a finite result at finite temperature.
In the present paper we regard the NJL model as a low energy effective theory of QCD
and apply the dimensional regularization not only to the radial part but also to the angular
parts of internal momenta in loop integrals. We include the effect of the current quark mass
and fix the scale through the observed properties of pseudo-scalar mesons, the pion mass
and the decay constant at zero density and temperature. The constituent quark mass and
the meson properties are evaluated in thermal equilibrium.
The paper is organized in the following way. First we apply the dimensional regular-
ization to the NJL model with two flavors of quarks. As is well-known, the small mass of
quarks explicitly breaks the chiral symmetry. Thus the chiral symmetry is only approxi-
mate. In such a model we evaluate the phase structure of the theory. We need to perform
renormalization as well to define a positive coupling constant. In Sec.II we calculate the
mass of scalar and pseudo-scalar mesons. We take the massless quarks limit and show that
the dimensional regularization keeps the Nambu-Goldstone modes massless. A relationship
between the space-time dimensions and the physical mass scale is also discussed. In Sec.IV
we introduce the thermal effect in the imaginary time formalism and evaluate the effective
potential. In Sec.V we calculate the mass of scalar and pseudo-scalar mesons at finite T
and µ. The three massless poles of pseudo-scalar mesons survive at higher temperature and
chemical potential. At the end concluding remarks are given.
II. DYNAMICAL SYMMETRY BREAKING
Thechiralsymmetry isbroken whenthecomposite operatorofaquarkandananti-quark,
¯
ψψ, develops a non-vanishing expectation value. It is caused by the QCD dynamics and is
3
called dynamical symmetry breaking.
The NJL model is one of the simplest models of dynamical chiral symmetry breaking.
The model is originally introduced to evaluate the low energy properties of hadrons as a
bound state of some primary fermion fields [1]. The Lagrangian of the two-flavor NJL model
is defined by
= ψ¯(i∂ m)ψ +g0 (ψ¯ψ)2 +(ψ¯iγ τaψ)2 , (1)
L 6 − π 5
where g0 is an effective coupling constant, (cid:8)τa represents the iso(cid:9)spin Pauli matrices, and
π
m = diag(m ,m ) is the mass matrix of up and down quarks. In the Lagrangian (1) we
u d
omit the color and the flavor indeces.
In the limit of massless quarks this Lagrangian is invariant under the global flavor trans-
formation
ψ eiθaτaψ, (2)
→
and the chiral transformation
ψ eiθaτaγ5ψ. (3)
→
The quark mass term violates these symmetries explicitly. When the composite operator
¯
ψψ develops a non-vanishing expectation value, the quark acquires a mass and the chiral
symmetry is broken dynamically.
¯
The vacuum expectation value of the composite operator ψψ can be found by solving the
gap equation. The gap equation of the NJL model in D dimensions is
dDk
σ = 2ig0 trS(k), (4)
h i π (2π)D
Z
where S(k) is the fermion propagator and ”tr” denotes trace with respect to flavor, color
and spinor indices,
1
S(k) . (5)
≡ k m σ +iǫ
6 − −h i
Integration over k takes the gap equation to the form
2N g0 D
σ = c π Γ 1 (m + σ )[(m + σ )2](D/2−1)
j j
h i (2π)D/2 − 2 h i h i
(cid:18) (cid:19)j∈{u,d}
X
g0A(D) (m + σ )[(m + σ )2](D/2−1), (6)
≡ π j h i j h i
j∈{u,d}
X
where N is the number of colors and A(D) is defined by
c
2N D
c
A(D) Γ 1 . (7)
≡ (2π)D/2 − 2
(cid:18) (cid:19)
4
6x104
4x104
)
DV 2x104
e
M
0
(
)
sV(-2x104
-4x104
-6x104
-100 -50 0 50 100
s (MeV)
FIG. 1: Typical behavior of the effective potential (D = 2.8, m = 3MeV, m = 5MeV and
u d
g0 = 0.01MeV2−D).
π −
Thegapequation(6)hasthreesolutions. Tofindthestablesolutionweevaluatetheeffective
potential for the scalar channel, σ, (see, for example Ref.[10])
σ2 A(D)
V(σ) = [(m +σ)2]D/2. (8)
4g0 − 2D j
π
j∈{u,d}
X
Extrema of the effective potential satisfy the gap equation (6). The most stable solution is
determined by the minimum of the effective potential. As is shown in the Fig.1, we find the
minimum at a negative value of σ for a negative coupling constant, g0, with a finite current
π
1
¯
quark mass. We note that the expectation value, ψψ σ , has a negative value as
h i ∼ −2g0h i
π
usual.
At the massless limit, m 0, the gap equation (6) is simplified and the solutions are:
→
σ = 0, (9)
h i
1
( σ 2)D/2−1 = . (10)
h i 2g0A(D)
π
The non-vanishing solution (10) is stabler, see Fig.1. It shows that the composite operator
¯
ψψ develops a non-vanishing expectation value and the chiral symmetry is broken dynami-
cally. It should be noted that
A(D) < 0, for 2 < D < 4. (11)
The left hand side of the Eq.(10) is real and positive. To find a real solution of Eq.(10) we
must set a negative value for the coupling constant g0. It is one of characteristic features of
π
the dimensional regularization.
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A. Renormalization
The Lagrangian (1) is not renormalizable for D = 4. However, the renormalization is
sometimes useful to connect the results in the different regularization schemes. Here we
renormalize parameters in our model.
In the leading order of the 1/N expansion the radiative correction of the four-fermion
coupling for the scalar channel is given by the summation of all bubble type diagrams,
G (p2, σ ) +
s
h i ≡
+
+
Π (p2) 4(g0)2
= 2g0 1+ s = π , (12)
π 2g0 Π (p2) 2g0 Π (p2)
(cid:18) π − s (cid:19) π − s
where the self-energy Π (p2) is
s
dDk
Π (p2) = = 4i(g0)2 tr[S(k)S(k p)]
s π (2π)D −
Z
1
= 2(g0)2(D 1)A(D) dxLD/2−1, (13)
π − j
j∈{u,d}Z0
X
with L = (m + σ )2 x(1 x)p2.
j j
h i − −
We renormalize the coupling constant g0 by imposing the following renormalization con-
π
dition
G (p2 = 0, σ = M ) Z (M )Gr(p2 = 0, σ = M )
s h i 0 ≡ g 0 s h i 0
4Z (M )g0
= g 0 π
[(m +M )2]D/2−1
j 0
j∈{u,d}
X
4gr
= π , (14)
[(m +M )2]D/2−1
j 0
j∈{u,d}
X
where we introduce the renormalization constant, Z (M ). The superscript r stands for the
g 0
renormalized quantities and M is a renormalization scale.
0
6
At the limit p2 = 0 the self-energy Π (p2) simplifies to
s
Π (p2 = 0) = 2(g0)2(D 1)A(D) [(m +M )2]D/2−1. (15)
s |hσi=M0 π − j 0
j∈{u,d}
X
Thus the renormalization constant is given by
1 j∈{u,d}[(mj +M0)2]D/2−1
Z (M ) = . (16)
g 0 21 g0(D 1)A(D) [(m +M )2]D/2−1
− π −P j∈{u,d} j 0
Therefore the renormalized coupling gr is foundPto be
π
1 1 1
= +(D 1)A(D). (17)
gπ0 [(mj +M0)2]D/2−1 2gπr −
j∈{u,d}
X
We rewrite the gap equation (6) in terms of the renormalized coupling constant gr as
π
(m + σ )[(m + σ )2]D/2−1
j j
h i h i
1
j∈{u,d}
X = σ +D 1 . (18)
[(mj +M0)2]D/2−1 h i(cid:18)2gπrA(D) − (cid:19)
j∈{u,d}
X
In the massless limit m 0, the non-trivial solution of the gap equation is given by
→
1
( σ 2)(D/2−1) = +D 1 (M2)(D/2−1). (19)
h i 2grA(D) − 0
(cid:18) π (cid:19)
A real solution for σ exists for 1/gr < 1/gcr,
h i π π
1
gcr = > 0 for 2 < D < 4. (20)
π 2(1 D)A(D)
−
The criticalcoupling isrealandpositive. It shouldbenoted thatthebroken phase isrealized
for gr > gcr > 0 or gr < 0.
π π π
It may be useful to rewrite the Eqs. (17) and (18) in terms of the critical coupling,
1 1 1 1
= , (21)
gπ0 [(mj +M0)2]D/2−1 2gπr − 2gπcr
j∈{u,d}
X
and
(m + σ )[(m + σ )2]1−D/2
j j
h i h i
2 σ 1 1
j∈{u,d}
X = h i . (22)
[(mj +M0)2]1−D/2 A(D) (cid:18)gπr − gπcr(cid:19)
j∈{u,d}
X
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From Eq.(21) it is clear that the bare coupling g0 should be negative in the broken phase,
π
1/gr < 1/gcr, in the dimensional regularization.
π π
The renormalization group β function is defined by [8, 9],
∂gr
β(gr) M π . (23)
π ≡ 0 ∂M
0(cid:12)g0
π
(cid:12)
(cid:12)
Using Eq.(22) in Eq.(23) we obtain (cid:12)
M [(m +M )2](D−3)/2
0 j 0
gr(gcr gr)
β(gr) = 2(D 2) j∈X{u,d} π π − π . (24)
π − [(mj +M0)2]D/2−1 gπcr
j∈{u,d}
X
The ultraviolet stable fixed point appears at the critical coupling, gr = gcr.
π π
III. MESON MASSES
To describe a bound state in the hadronic phase non-perturbative effects of QCD should
be considered. Here we study meson masses in the NJL model at the leading order of the
1/N expansion.
A. Pseudo-scalar mesons
First we calculate the pion mass in the two-flavor NJL model. It corresponds to the
Nambu-Goldstonemodesatthemasslessquarklimit. Sincethecurrentquarkmassexplicitly
breaks the chiral symmetry, the pion mass should be proportional to the current quark mass
in the broken phase.
The pion mass is defined by the pole of the propagator for a pseudo-scalar channel. The
leading order of the 1/N expansion is described by the summation of infinite number of the
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bubble diagrams,
Gab(p2, σ ) = p a p b + p a p b
5 h i
p a p b
+
p a p b
+
4(g0)2 ZaM4−D
= π δab π 0 δab, (25)
2g0 Πa(p2) ≡ m2 p2 +O(p4)
π − 5 πa −
D/2−2
where we insert the renormalization scale M to impose the correct mass dimensions
0
on Z . Πa(p2) is the self-energy for the pseudo-scalar channel which is given by
π 5
Πa(p2) = iγ5τa iγ5τa
5
dDk
= 4i(g0)2 tr[iγ5τaS(k)iγ5τaS(k p)]. (26)
π (2π)D −
Z
Integrating over the D-dimensional momentum k, we obtain
1 D
Π1,2(p2) = 4(g0)2A(D) dx (1 D)x(1 x)p2 + x(m + σ )2
5 π − − 2 u h i
Z0 (cid:20)
D D
+ (1 x)(m + σ )2 + 1 (m + σ )(m + σ )
d u d
2 − h i − 2 h i h i
(cid:18) (cid:19) (cid:21)
x(m + σ )2 +(1 x)(m + σ )2 x(1 x)p2 D/2−2,
u d
× h i − h i − −
(cid:2) (cid:3) (27)
and
1
Π3(p2) = 2(g0)2A(D) dx (1 D)x(1 x)p2 +(m + σ )2
5 π − − j h i
Z0 j∈{u,d}
X (cid:2) (cid:3)
(m + σ )2 x(1 x)p2 D/2−2. (28)
j
× h i − −
(cid:2) (cid:3)
We can analytically calculate the momentum integral at finite p2, but it is enough to
evaluate the self-energy near p2 = 0 to observe the massless pole of the Nambu-Goldstone
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mode. Straightforward calculations lead to
[(m + σ )2](D−1)/2
Π1,2(p2 0) = 4(g0)2A(D) j∈{u,d} j h i f(D)p2
5 ∼ π [(m + σ )2]1/2 −
"P j∈{u,d} j h i #
+O(p4), P (29)
and
Π3(p2 0) = (g0)2A(D) 2 [(m + σ )2]D/2−1 (30)
5 ∼ π  j h i
j∈{u,d}
X

D
+ 1 p2 [(m + σ )2]D/2−2 +O(p4),
j
− 2 h i 
(cid:18) (cid:19) j∈{u,d}
X

where f(D) is given by
f(D)
1
=
[(m + σ )2 (m + σ )2]3
d u
h i − h i
2
1 (m + σ )D+2 (m + σ )D+2
d u
×{ D − h i − h i
(cid:18) (cid:19)
2 (cid:2) (cid:3)
+ +1 (m + σ )D(m + σ )2 (m + σ )2(m + σ )D
d u d u
D h i h i − h i h i
(cid:18) (cid:19)
4 (cid:2) (cid:3)
1 (m + σ )D+1(m + σ ) (m + σ )(m + σ )D+1
d u d u
− D − h i h i − h i h i
(cid:18) (cid:19)
(m + σ )(cid:2)D−1(m + σ )3 +(m + σ )D−1(m + σ )3 . (cid:3) (31)
d u d u
− h i h i h i h i }
In the isospin symmetric case, m = m , Π1,2(p2 0) coincides with Π3(p2 0).
u d 5 ∼ 5 ∼
Further we obtain the pion wave function renormalization constants,
Z−1 = A(D)f(D)M4−D, (32)
π1,2 − 0
and
A(D) D
Z−1 = 1 M4−D [(m + σ )2]D/2−2. (33)
π3 4 − 2 0 j h i
(cid:18) (cid:19) j∈{u,d}
X
Neglecting O(p4) terms, the pion masses are found to be
1 ( σ +m )D−1 +( σ +m )D−1
Z−1 MD−4m2 = A(D) h i u h i d , (34)
π1,2 0 π1,2 2g0 − 2 σ +m +m
π h i u d
and
1 1
Z−1MD−4m2 = A(D) [(m + σ )2]D/2−1. (35)
π3 0 π3 2g0 − 2 j h i
π
j∈{u,d}
X
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