Fermionic Collective Modes in QGP near Critical Temperatures
Yukio Nemoto1,∗) , Masakiyo Kitazawa2,∗∗) , Tomoi Koide3,∗∗∗) and Teiji
Kunihiro
1 Department of Physics, Nagoya University, Nagoya 464-8602, Japan
2 RIKEN-BNL Reseach Center, Brookhaven National Laboratory, Upton, NY
11973, USA
3 Instituto de F́ısica, Universidade Federal do Rio de Janeiro, C. P. 68528,
21945-970, Rio de Janeiro, Brasil
4 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606–8502,
Japan
We investigate the quark spectrum in the quark-gluon plasma phase near color super-
conducting (CS) and chiral phase transitions. Owing to the precursory soft modes of the
phase transitions, there appear novel excitaion spectra: In the CS transition, the quark mat-
ter shows non-Fermi liquid behavior and leads to the pseudogap in the density of states of
quarks. In the chiral transition, three collective excitations appear in the quark spectrum.
§1. Introduction
Recently, the quark-gluon plasma (QGP) just above the chiral and deconfine-
ment phase transition is believed to be an unexpectedly strongly interacting system,
which is based on the facts that the created matter at RHIC behaves like a perfect
fluid and that some hadronic bound states of heavy quarks can survive above the
critical temperature (Tc) from Lattice QCD. Because the fundamental degrees of
freedom in QGP are quarks and gluons, it is also important to study their properties
in such a strongly interacting system. Here, we investigate the quark spectrum just
above Tc of the chiral transition at zero density, and the color superconducting (CS)
transition around µB = 1 GeV with µB being the baryon number density, focusing
on the precursory soft modes of these transitions. It is known that these soft modes
exist over a wide range of temperature above Tc owing to a strong coupling nature
between quarks.1), 2) In this paper, we show that they affect the quark spectrum
significantly in a region just above Tc.
§2. Precursory soft modes
To describe the quark matter near Tc, we employ the two-flavor Nambu–Jona-
Lasinio type interaction with the scalar diquark correlation included, L = ψ̄i/∂ψ +
GS [(ψ̄ψ)
2 +(ψ̄iγ5~τψ)
2] +GC(ψ̄iγ5τ2λAψ
C)(ψ̄C iγ5τ2λAψ), with ψ
C ≡ Cψ̄T and C =
∗) e-mail address: nemoto@hken.phys.nagoya-u.ac.jp
∗∗) e-mail address: kitazawa@quark.phy.bnl.gov
∗∗∗) e-mail address: koide@if.ufrj.br
†) e-mail address: kunihiro@yukawa.kyoto-u.ac.jp
http://arxiv.org/abs/0704.1188v1
2 Y. Nemoto, M. Kitazawa, T. Koide and T. Kunihiro
-30-20-10 0 10 20 30 40 50 ω[MeV]
k[MeV]
S(k,ω) [GeV2]
Fig. 1. The dynamic structure factor SC for the diquark mode for µ = 400 MeV and ε ≡ (T −
Tc)/Tc = 0.02 (left), and the spectral function ρσ ≡ ρS for the quark-antiquark mode for µ = 0
and ε = 0.1 (right). k and p denote the momentum.
iγ2γ0 being the charge conjugation operator. The matrices τ2 and λA (A = 2, 5, 7)
are the antisymmetric components of the Pauli and Gell-Mann matrices for the flavor
SU(2)f and color SU(3)c, respectively. The coupling constants, GS and GC , and
the three-momentum cutoff, Λ, are taken from Refs. 1) and 3).
The fluctuations of the diquark (chiral) condensate are described by the diquark
(quark-antiquark) Green function in the random phase approximation, DC,S(p, νn) =
−[1/2GC,S +QC,S(p, νn)]
−1, where the subscript C(S) denotes the diquark (quark-
antiquark) sector. νn = 2πnT is the Matsubara frequency for bosons and QC,S(p, νn)
is the undressed quark-antiquark (diquark) polarization function at one-loop. To
evaluate strengths of the fluctuations, we employ the spectral function, ρ, and the
dynamic structure factor, S, given by ρC,S(p, ω) = −(1/π)ImDC,S(p, νn)|iνn=iω+iη
and SC,S(p, ω) = ρC,S(p, ω)/(1−e
−ω/T ), respectively. SC(p, ω) for the diquark mode
and ρS(p, ω) for the quark-antiquark mode near TC are plotted in Fig. 1. One can
see that there appear pronounced peaks which denote the precursory soft modes.1), 2)
The peak positions of these modes are approximately expressed as ωC ≃ p
2 for the
diquark mode, and ωS ≃ ±
m∗σ(T )
2 + p2 for the quark-antiquark mode. A T -
dependent ‘mass’ m∗σ(T ) becomes smaller as T approaches Tc, which means the
softening at Tc. As will be seen in Sec.3, the difference between ωC and ωS leads to
a quite different quark spectrum.
§3. Quark spectrum near the phase transitions
The effect of the soft modes on the quark spectrum is incorporated in the quark
self-energy in the non-selfconsistent way, Σ̃C,S(p, ωn) = −4T
(2π)3
DC,S(p −
q, ωn − ωm)G0(q, ωm), where G0(q, ωm) is the free quark propagator with ωn =
(2n+1)πT being the Matsubara frequency for fermions. The quasi-quark and quasi-
antiquark spectral functions, ρ±(p, ω), are obtained from the retarded self-energies,
ΣR±(p, ω) = (1/2)Tr[Σ
Rγ0Λ±], respectively, i.e. ρ± = −(1/π)Im[ω+µ∓|p|−Σ
with the analytic continuation ΣR(p, ω) = Σ̃(p, ωn)|iωn=ω+iη and the projection op-
Fermionic Collective Modes in QGP 3
Fig. 2. The quasi-quark spectral function ρ+ above the CS transition for µ = 400 MeV and ε = 0.01
(left), and ρ+ above the chiral transition for µ = 0 and ε = 0.1 (right). k and p denote the
momentum.
-400 -300 -200 -100  0  100
ω [MeV]
ε=0.20
ε=0.05
ε=0.02
ε=0.01
Fig. 3. The density of states of quarks above the CS transition for µ = 400 MeV. The thin dotted
curve represents that of free quarks.
erators Λ± = (1 ± γ
γ · p/|p|)/2. In the following, we show the quark spectral
function obtained from the self-energy Σ̃C(Σ̃S) for the CS (chiral) transition.
3.1. Color superconducting transition
The quasi-quark spectral function ρ+ for µ = 400 MeV and ε = 0.01 is plotted in
the left panel of Fig. 2. We see that the peak has a clear depression around the Fermi
energy, ω = 0.4) Owing to the softening of the soft mode near Tc, a quark near the
Fermi energy is scattered by the soft mode and create a hole, while a hole can create
a quark by absorbing the soft mode. Then, the incident quark and a quark near
the Fermi surface make a resonant scattering to form the soft mode and vice versa.
This resonant processes induce a virtual mixing between quarks and holes, which
leads to the level repulsion of the energy spectrum near the Fermi energy, making
the gap-like structure as shown in the left panel of Fig. 2. This behavior, which
is quite different from the conventional Fermi liquid, is due to the strong coupling
nature between quarks. In fact, we see that the depression is more significant as the
diquark coupling GC becomes larger.
This depression leads to a depression in the density of states (DOS) of quarks
around the Fermi energy, as shown in Fig. 3. Thus, we see a gap-like structure in
DOS even above Tc, which we call the pseudogap.
4) The non-Fermi liquid behav-
ior is essential for the formation of the pseudogap, which is analogous to high-Tc
superconductors, although the origin of the pseudogap of the latter is not known
4 Y. Nemoto, M. Kitazawa, T. Koide and T. Kunihiro
precisely.
3.2. Chiral transition
The quasi-quark spectral function ρ+ for µ = 0 MeV and ε = 0.1 is plotted in
the right panel of Fig. 2. We see a clear three-peak structure at low momentum,
which exists even at ε = 0.2.6) Although not shown in the figure, the quasi-antiquark
spectrum, ρ−, has also a three-peak structure for a relation, ρ−(p, ω) = ρ+(p,−ω).
The mechanism of the appearance of the three-peak structure in ρ+ is as follows: The
imaginary part of ΣR+(0, ω) has two peaks at nonzero values of ω, which means that
there exist two large damping modes of the quasi-quark there. From a kinematical
consideration, we see that one is a collision of a thermally excited antiquark and the
quasi-quark creating the soft mode, and the other is a collision of the quasi-quark
and the soft mode creating an on-shell quark. Both the processes are interpreted as
a Landau damping of the quasi-quark. The point is that the quasi-quark is a mixed
state between quarks and ‘antiquark-holes’ which are annihilation of thermally ex-
cited antiquarks and have the positive quark number. Then, these damping modes
cause a mixing between quarks and antiquark-holes. This mixing mechanism can be
described in terms of the resonant scattering as in the case of the color superconduc-
tivity, although a crucial difference arises owing to the different nature of the soft
modes. We can show that a coupling with the soft mode with a nonzero mass m∗σ(T )
is essential for the appearance of the three-peak structure in the quark spectrum.
In fact, we have investigated the quark spectrum in Yukawa models with a
massive scalar (pseudo-scalar) and vector (axial-vector) boson of a nonzero mass m,
and find that there appears a three-peak structure in the quark spectral function
with a collective nature when temperature is compatible with m.7) Because the
employed Yukawa models are rather generic, the findings may represent a universal
phenomenon for fermions coupled with a massive bosonic excitation with a vanishing
or small width.
§4. Conclusions
We have investigated the quark spectrum near the CS and chiral transitions
taking into account the fluctuating soft modes. We have shown that the quark
spectrum shows a non-Fermi liquid behavior leading to the pseudogap in the density
of states above the CS transition, and a clear three-peak structure above the chiral
transition. The gap-like structure in the spectral function near both the transitions
can be uderstood in terms of the resonant scattering of each soft mode.
References
1) T. Hatsuda and T. Kunihiro, Phys. Rev. Lett. 55 (1985), 158.
2) M. Kitazawa, T. Koide, T. Kunihiro and Y. Nemoto, Phys. Rev. D 65 (2002), 091504.
3) T. M. Schwarz, S. P. Klevansky and G. Papp, Phys. Rev. C 60 (1999), 055205.
4) M. Kitazawa, T. Koide, T. Kunihiro and Y. Nemoto, Phys. Rev. D 70 (2004), 056003.
M. Kitazawa, T. Koide, T. Kunihiro and Y. Nemoto, Prog. Theor. Phys. 114 (2005), 205.
5) M. Kitazawa, T. Kunihiro and Y. Nemoto, Phys. Lett. B 631 (2005), 157.
6) M. Kitazawa, T. Kunihiro and Y. Nemoto, Phys. Lett. B 633 (2006), 269.
7) M. Kitazawa, T. Kunihiro and Y. Nemoto, Prog. Theor. Phys. 117 (2007), 103.
	Introduction
	Precursory soft modes
	Quark spectrum near the phase transitions
	Color superconducting transition
	Chiral transition
	Conclusions

  We investigate the quark spectrum in the quark-gluon plasma phase near color
superconducting (CS) and chiral phase transitions. Owing to the precursory soft
modes of the phase transitions, there appear novel excitaion spectra: In the CS
transition, the quark matter shows non-Fermi liquid behavior and leads to the
pseudogap in the density of states of quarks. In the chiral transition, three
collective excitations appear in the quark spectrum.

Introduction
Recently, the quark-gluon plasma (QGP) just above the chiral and deconfine-
ment phase transition is believed to be an unexpectedly strongly interacting system,
which is based on the facts that the created matter at RHIC behaves like a perfect
fluid and that some hadronic bound states of heavy quarks can survive above the
critical temperature (Tc) from Lattice QCD. Because the fundamental degrees of
freedom in QGP are quarks and gluons, it is also important to study their properties
in such a strongly interacting system. Here, we investigate the quark spectrum just
above Tc of the chiral transition at zero density, and the color superconducting (CS)
transition around µB = 1 GeV with µB being the baryon number density, focusing
on the precursory soft modes of these transitions. It is known that these soft modes
exist over a wide range of temperature above Tc owing to a strong coupling nature
between quarks.1), 2) In this paper, we show that they affect the quark spectrum
significantly in a region just above Tc.
§2. Precursory soft modes
To describe the quark matter near Tc, we employ the two-flavor Nambu–Jona-
Lasinio type interaction with the scalar diquark correlation included, L = ψ̄i/∂ψ +
GS [(ψ̄ψ)
2 +(ψ̄iγ5~τψ)
2] +GC(ψ̄iγ5τ2λAψ
C)(ψ̄C iγ5τ2λAψ), with ψ
C ≡ Cψ̄T and C =
∗) e-mail address: nemoto@hken.phys.nagoya-u.ac.jp
∗∗) e-mail address: kitazawa@quark.phy.bnl.gov
∗∗∗) e-mail address: koide@if.ufrj.br
†) e-mail address: kunihiro@yukawa.kyoto-u.ac.jp
http://arxiv.org/abs/0704.1188v1
2 Y. Nemoto, M. Kitazawa, T. Koide and T. Kunihiro
-30-20-10 0 10 20 30 40 50 ω[MeV]
k[MeV]
S(k,ω) [GeV2]
Fig. 1. The dynamic structure factor SC for the diquark mode for µ = 400 MeV and ε ≡ (T −
Tc)/Tc = 0.02 (left), and the spectral function ρσ ≡ ρS for the quark-antiquark mode for µ = 0
and ε = 0.1 (right). k and p denote the momentum.
iγ2γ0 being the charge conjugation operator. The matrices τ2 and λA (A = 2, 5, 7)
are the antisymmetric components of the Pauli and Gell-Mann matrices for the flavor
SU(2)f and color SU(3)c, respectively. The coupling constants, GS and GC , and
the three-momentum cutoff, Λ, are taken from Refs. 1) and 3).
The fluctuations of the diquark (chiral) condensate are described by the diquark
(quark-antiquark) Green function in the random phase approximation, DC,S(p, νn) =
−[1/2GC,S +QC,S(p, νn)]
−1, where the subscript C(S) denotes the diquark (quark-
antiquark) sector. νn = 2πnT is the Matsubara frequency for bosons and QC,S(p, νn)
is the undressed quark-antiquark (diquark) polarization function at one-loop. To
evaluate strengths of the fluctuations, we employ the spectral function, ρ, and the
dynamic structure factor, S, given by ρC,S(p, ω) = −(1/π)ImDC,S(p, νn)|iνn=iω+iη
and SC,S(p, ω) = ρC,S(p, ω)/(1−e
−ω/T ), respectively. SC(p, ω) for the diquark mode
and ρS(p, ω) for the quark-antiquark mode near TC are plotted in Fig. 1. One can
see that there appear pronounced peaks which denote the precursory soft modes.1), 2)
The peak positions of these modes are approximately expressed as ωC ≃ p
2 for the
diquark mode, and ωS ≃ ±
m∗σ(T )
2 + p2 for the quark-antiquark mode. A T -
dependent ‘mass’ m∗σ(T ) becomes smaller as T approaches Tc, which means the
softening at Tc. As will be seen in Sec.3, the difference between ωC and ωS leads to
a quite different quark spectrum.
§3. Quark spectrum near the phase transitions
The effect of the soft modes on the quark spectrum is incorporated in the quark
self-energy in the non-selfconsistent way, Σ̃C,S(p, ωn) = −4T
(2π)3
DC,S(p −
q, ωn − ωm)G0(q, ωm), where G0(q, ωm) is the free quark propagator with ωn =
(2n+1)πT being the Matsubara frequency for fermions. The quasi-quark and quasi-
antiquark spectral functions, ρ±(p, ω), are obtained from the retarded self-energies,
ΣR±(p, ω) = (1/2)Tr[Σ
Rγ0Λ±], respectively, i.e. ρ± = −(1/π)Im[ω+µ∓|p|−Σ
with the analytic continuation ΣR(p, ω) = Σ̃(p, ωn)|iωn=ω+iη and the projection op-
Fermionic Collective Modes in QGP 3
Fig. 2. The quasi-quark spectral function ρ+ above the CS transition for µ = 400 MeV and ε = 0.01
(left), and ρ+ above the chiral transition for µ = 0 and ε = 0.1 (right). k and p denote the
momentum.
-400 -300 -200 -100  0  100
ω [MeV]
ε=0.20
ε=0.05
ε=0.02
ε=0.01
Fig. 3. The density of states of quarks above the CS transition for µ = 400 MeV. The thin dotted
curve represents that of free quarks.
erators Λ± = (1 ± γ
γ · p/|p|)/2. In the following, we show the quark spectral
function obtained from the self-energy Σ̃C(Σ̃S) for the CS (chiral) transition.
3.1. Color superconducting transition
The quasi-quark spectral function ρ+ for µ = 400 MeV and ε = 0.01 is plotted in
the left panel of Fig. 2. We see that the peak has a clear depression around the Fermi
energy, ω = 0.4) Owing to the softening of the soft mode near Tc, a quark near the
Fermi energy is scattered by the soft mode and create a hole, while a hole can create
a quark by absorbing the soft mode. Then, the incident quark and a quark near
the Fermi surface make a resonant scattering to form the soft mode and vice versa.
This resonant processes induce a virtual mixing between quarks and holes, which
leads to the level repulsion of the energy spectrum near the Fermi energy, making
the gap-like structure as shown in the left panel of Fig. 2. This behavior, which
is quite different from the conventional Fermi liquid, is due to the strong coupling
nature between quarks. In fact, we see that the depression is more significant as the
diquark coupling GC becomes larger.
This depression leads to a depression in the density of states (DOS) of quarks
around the Fermi energy, as shown in Fig. 3. Thus, we see a gap-like structure in
DOS even above Tc, which we call the pseudogap.
4) The non-Fermi liquid behav-
ior is essential for the formation of the pseudogap, which is analogous to high-Tc
superconductors, although the origin of the pseudogap of the latter is not known
4 Y. Nemoto, M. Kitazawa, T. Koide and T. Kunihiro
precisely.
3.2. Chiral transition
The quasi-quark spectral function ρ+ for µ = 0 MeV and ε = 0.1 is plotted in
the right panel of Fig. 2. We see a clear three-peak structure at low momentum,
which exists even at ε = 0.2.6) Although not shown in the figure, the quasi-antiquark
spectrum, ρ−, has also a three-peak structure for a relation, ρ−(p, ω) = ρ+(p,−ω).
The mechanism of the appearance of the three-peak structure in ρ+ is as follows: The
imaginary part of ΣR+(0, ω) has two peaks at nonzero values of ω, which means that
there exist two large damping modes of the quasi-quark there. From a kinematical
consideration, we see that one is a collision of a thermally excited antiquark and the
quasi-quark creating the soft mode, and the other is a collision of the quasi-quark
and the soft mode creating an on-shell quark. Both the processes are interpreted as
a Landau damping of the quasi-quark. The point is that the quasi-quark is a mixed
state between quarks and ‘antiquark-holes’ which are annihilation of thermally ex-
cited antiquarks and have the positive quark number. Then, these damping modes
cause a mixing between quarks and antiquark-holes. This mixing mechanism can be
described in terms of the resonant scattering as in the case of the color superconduc-
tivity, although a crucial difference arises owing to the different nature of the soft
modes. We can show that a coupling with the soft mode with a nonzero mass m∗σ(T )
is essential for the appearance of the three-peak structure in the quark spectrum.
In fact, we have investigated the quark spectrum in Yukawa models with a
massive scalar (pseudo-scalar) and vector (axial-vector) boson of a nonzero mass m,
and find that there appears a three-peak structure in the quark spectral function
with a collective nature when temperature is compatible with m.7) Because the
employed Yukawa models are rather generic, the findings may represent a universal
phenomenon for fermions coupled with a massive bosonic excitation with a vanishing
or small width.
§4. Conclusions
We have investigated the quark spectrum near the CS and chiral transitions
taking into account the fluctuating soft modes. We have shown that the quark
spectrum shows a non-Fermi liquid behavior leading to the pseudogap in the density
of states above the CS transition, and a clear three-peak structure above the chiral
transition. The gap-like structure in the spectral function near both the transitions
can be uderstood in terms of the resonant scattering of each soft mode.
References
1) T. Hatsuda and T. Kunihiro, Phys. Rev. Lett. 55 (1985), 158.
2) M. Kitazawa, T. Koide, T. Kunihiro and Y. Nemoto, Phys. Rev. D 65 (2002), 091504.
3) T. M. Schwarz, S. P. Klevansky and G. Papp, Phys. Rev. C 60 (1999), 055205.
4) M. Kitazawa, T. Koide, T. Kunihiro and Y. Nemoto, Phys. Rev. D 70 (2004), 056003.
M. Kitazawa, T. Koide, T. Kunihiro and Y. Nemoto, Prog. Theor. Phys. 114 (2005), 205.
5) M. Kitazawa, T. Kunihiro and Y. Nemoto, Phys. Lett. B 631 (2005), 157.
6) M. Kitazawa, T. Kunihiro and Y. Nemoto, Phys. Lett. B 633 (2006), 269.
7) M. Kitazawa, T. Kunihiro and Y. Nemoto, Prog. Theor. Phys. 117 (2007), 103.
	Introduction
	Precursory soft modes
	Quark spectrum near the phase transitions
	Color superconducting transition
	Chiral transition
	Conclusions
