Anomalous quark spectra around QCD critical points --- effects of soft modes and van Hove singularity ---

1
Anomalous quark spectra
around QCD critical points ---
effects of soft modes and van Hove singularity
---

• T. Kunihiro

Based on works done in collaboration
with M.Kitazawa, Y. Nemoto, T. Koide,
K.Mitsutani.
NFQCD2013 Nov. 27, 2013, YTP
2
Thoretical (half-conjectured) Phase Diagram of QCD
T
2Tc
QGP
chiral sym. restoreddeconfinement
LHC/RHIC
Cross over
Tc
GSI,J-PARC
CEP
?
Hadron
chiral sym. broken (antiquark-quark
condensate)confinement
(Color SuperConducting phase)CSC
quark-quark condensate
m
compact stars
3
QCD _at_ Tgt0
T
Tc
4
QCD _at_ Tgt0
T
Tc
What is the physical picture of the
elementary Excitations in the crossover
region? How do quarks and gluons disappear?
5
Quarks at Extremely High T
Klimov 82, Weldon 83 Braaten, Pisarski 89

• 2 collective excitations having thermal mass
  gT
• width g2T
• Minimum of the plasmino mode at nonzero p

w / mT
plasmino
p / mT
6
Thoretical (half-conjectured) Phase Diagram of QCD
large fluctuations owing to strong coupling
around the critical points. There may exist also
other low-lying (hadronic) excitations in the
QGP phase.
Phase transitions
T
Here we explore how they affect quark
quasi-particle picture!
2Tc
QGP
chiral sym. restoreddeconfinement
LHC/RHIC
chiral fluctuations
Tc
density fluctuations
GSI,J-PARC
CEP
Hadron
diquark fluctuations
chiral sym. broken (antiquark-quark
condensate)confinement
(Color SuperConducting phase)CSC
quark-quark condensate
m
compact stars
7
Contents

• 1.Introduction
• 2. diquark fluctuations and pseudogap in quark
  specral function in hot and dense quark matter
• --- a lesson from condensed matter physics
  ---
• 3.Soft modes of chiral transition and anomalous
  quasi-quark spectrum
• 4. Effects of para-pion with non-hyperbolic
  dispersion rel. and van Hove singularity
• 4.Summary and concluding remarks

8
2. Diquark fluctuations and pseudogap in quark
spectral function in hot and dense quark matter
9
Lesson from condensed matter physics on strong
correlations
Phase diagram of cuprates to be high-Tc
superconductor
Renner et al.(96)
(hole doping)
Fermi energy
A typical non-Fermi liq. behavior!
Anomalous depression of the density of states
near the Fermi surface in the normal phase.
Pseudogap
The mechanism of the pseudogap in High-TcSC is
still controversial, but see, Y. Yanase et al,
Phys. Rep. 387 (2003),1, where the essential role
of pair fluc. is shown.
10
Possible pseudogap formation in heated quark
matter
M. Kitazawa, T. Koide, T. K. and Y. Nemoto
Phys. Rev. D70, 956003(2004) Prog. Theor.
Phys. 114, 205(2005),
CSC
N(w)/104
m 400 MeV
Pseudogap is formed above Tc of CSC in
heated quark matter!
pseudogap !
How?
w
Fermi energy
11
Mechanism of the pseudogap formation
1. Development of precursory diquark fluctuations
above Tc
Dynamical Structure Factorof diquark fluctuations
for m 400 MeV T 1.05Tc
sharp peak at the origin ( diffusive
over-damping mode)
2.Coupling of quarks with the diquark
fluctuations
quark self-energy
400 MeV, 0.01
Cherenkov-like emission of diquark mode around
Fermi energy
Depression of the quark spectral Function around
the Fermi energy
12
Level repulsion or Pseudo gap due to resonant
scattering
M. Kitazawa, T.K. and Y. Nemoto, Phys. Lett.B
631(2005),157
GC4.67GeV-2
Janko, Maly, Levin, PRB56,R11407 (1995)
Mixing between particles and holes owing to the
Landau damp. by the collective diquark mode.
level repulsion of energy level
particle 1
particle -1
particle
hole
particle
hole
13
3. Soft modes of chiral transition and
anomalous quasi-quark spectrum
14
Chiral Transition and the collective modes
The low mass sigma in vacuum is now established
pi-pi scattering Colangero, Gasser,
Leutwyler(06) and many others
Full lattice QCD SCALAR collaboration
(03) and others.
q-qbar, tetra quark, glue balls, or their mixed
sts?
c.f. The sigma as the Higgs particle in QCD
Higgs field
Higgs particle
15
DigressionThe poles of the S matrix in the
complex mass plane for the sigma meson channel
complied in Z. Xiao and H.Z. Zheng (2001)
Softening ?
See also, I. Caprini, G. Colangero and H.
Leutwyler, PRL(2006) H. Leutwyler,
hep-ph/0608218 M_sigma441 i 272 MeV
16
Fluctuations of chiral order parameter
around Tc in Lattice QCD
T
T
m
?
?
the softening of the ? with increasing T
17
The spectral function of the degenerate
para-pion and the para-sigma at TgtTc for
the chiral transition Tc164 MeV
T. Hatsuda and T.K. (1985)
(NJL model cal.)
18
Para-pion and para-sigma modes are still seen in
PNJL model
K. Fukushima (2004)
PPolyakov-loop coupled
C. Ratti, M. Thaler, W. Weise (2006)
PNJL model
S. Rößner, T. Hell, C. Ratti, W.
Weise.arXiv0712.3152 hep-ph
W. Weise, talk at NFQCD2008 at YITP, Kyoto.
19
Quark Spectral Function

• Quark Self-energy at finite T

• Quark Spectral function

0
(chiral limit)
0
particle
antiparticle
at zero density
c.f for a free quark
20
Quarks coupled to chiral soft modes near Tc
We incorporate the fluctuation mode into a single
particle Green function of a quark through a
self-energy.
Non self-consistent T-approximation (1-loop of
the fluctuation mode)
N.B. This is a complicated multiple integral
owing to the compositeness of the
para-sigma and para-pion modes.
21
Spectral Function of Quark
Kitazawa, Nemoto and T.K., Phys. Lett. B633, 269
(2006),
e 0.05
k MeV
k MeV

• Three-peak structure emerges.
• The peak around the origin is
• the sharpest.

Quasi-dispersion relation for eye-guide
22
Digression Quarks at very high T (TgtgtTc)
---- physical origin of plasmino and thermal
mass ---

• 1-loop (gltlt1) HTL approx.

Klimov, Weldon(82), Pisarski(89), A.Schaefer,
Thoma (99)
w
quark
anti-quark
Thermally excited anti-quarks
quasi-q

p
anti-plasmino
23
Mechanism of the 3-peak fromation
The level crossing is shifted by the massof the
fluctuation modes.
quark part
r-( ,p)
r( ,p)
w MeV
quark
anti-q hole
quark
p MeV
anti-quark part
p MeV
anti-q
quark hole
anti-q
the HTL result only with the normal quark
and plasmino.
24
Fluctuations of the chiral codensate
Spectrum of the fluctuations
sharp peak in time-like region
T 1.1Tcm0
CSC
s , p-mode
para
p
propagating mode
m
c.f. diffusion-like mode in diquark
fluctuations
T
Tc
25
Quark Spectrum in Yukawa models
Kitazawa, Nemoto and T.K., Prog. Theor. Phys.117,
103(2007),
Near the critical point , the soft-modes may be
represented by an elementary boson.
quark massive boson.
Yukawa model!
(at one-loop)
g1 , T/m1.5
Massie scalar/pseudoscalar boson
Massie vector/axial vector boson
The 3-peak structure emerges irrespective of the
type of the boson at .
Remark Bosonic excitations in QGP may include s,
p, r, J/y, / glue balls and even
the W/Z bosons in the electroweak theory.
26
Neutrino spectral density at electroweak-scale
temperature
K. Miura, Y. Hidaka, D. Satow, TK, PRD 88 (2013),
065024
See talk presented by K. Miura at this workshop,
Nov.20,2013.
27
The complex quasi-quark pole
Pole position
Mitsutani, Kitazawa, Nemoto, T.K. Phys. Rev. D77,
045034 (2008)
There are three poles corresponding to the three
peaks in the spectral function the pole
distribution is symmetric with respect to the
imaginary axis because mf m 0
T?
T-dependence of the residues
The sum of the three residues approximately
satisfy the sum rule
The three residues comparable at T mb which
support the 3-peak structure
28
Finite quark mass effects
Mitsutani, Kitazawa, Nemoto, T.K. Phys. Rev. D77,
045034 (2008)
mf / mb 0.1

• There still exist three poles.
• The pole at T0 (red) moves toward the origin as
  T is raised.
• The pole in the w lt 0-region has a larger
  imaginary part than that in the positive-w
  region for the same T.

Re Z

• The residue at the pole in the negative w region
  is suppressed at T mb , corresponding to the
  suppression of the peak in the negative-energy
  region.
• The sum of the residues approximately satisfy the
  sum rule also in this case.

29
Structure change of the pole behavior
mf / mb 0.3
mf / mb 0.2
Level crossing in the complex energy plane
The pole at T0 moves toward the origin as T is
raised. This behavior is qualitatively the same
as in the case of lower masses.
The pole at T0 moves toward the large-w region
as T is raised. This behavior is qualitatively
different from that in the smaller mass cases..
The physics contents of the three poles change
at a critical mass . We find
30
Beyond one-loop
Schwinger-Dyson approach for lin. sigma model
Harada-Nemoto(08)
Harada, Nemoto,0803.3257(hep-ph)
The three peak structure in the quark spectral
function is still there for small
momenta, although the central peak gets to have a
width owing to multiple scattering. See also, S.
x Qin et al, PRD 84, (2011), 014017 H.
Nakkagawa et al, PRD 85 (2012) 031902.
Possible confirmation in Lattice QCD
Unquenched lattice simulation with hopefully
chiral fermion action on a large lattice
is necessary for accommodate the possible chiral
fluctuations with energy comparable to
MeV. Nevertheless, see F. Karsch and M.
Kitazawa, PLB 685 (2007), 45 PRD 80 (2009),
056001 O. Kaczmarek, Karsch and Kitazawa, PRD86
(2012) 036006.
31
Introducing Nonzero m0
M.Kitazawa, Y.Nemoto and TK, in prep.
2-flavor NJL model
Whats NEW!

• Effect of m0

• Phase transition becomes crossover.
• Constituent quarks stay massive.
• Soft modes do not become massless.
• The critical point appears, where the phonon mode
  (density fluctuation) becomes the soft mode.

32
Stable p modes above TPC
p modes can be stable even above TPC
for mplt2M
33
Pion Dispersion Relation
T206MeV
Continuum threshold
Dispersion for a rela. free particle (hyperbolic
curve)

• Dispersion relation of stable pion deviates from
  Lorentz form.
• Pionic modes become unstable at nonzero p.

34
Anomalous pion dispersion relation in hot and
dense hadronic matter
A.B. Migdal(78), T.Ericson and F. Myhre (78)
,C.Gale and J. Kapusta(87), G.Bertsch et al
(88), G. E/ Brown, E. Oset , M. Vicente Vacas,
and W. Weise (89), E.Shuryak (90), R.Pisarski
and M.Tytgat (96) and many others.
C. Gale and J. Kapusta, PRC35 (1987), 2107
G. Bertsch et al, NPA 490 (1988), 745.
Density of states/ (inverse) group velocity
G.E. Brown et al, NPA 505 (1989), 823
35
Quark self-energy with composite bosons at finte T
(Joint) Density of states (DOS)
difference of the group velocities
Hilbert tr. of the imaginary part
A singularity in Im. Part affect the real part.
36
van Hove Singularity
w
Relative group velocity of quarks and pions
Zeros of group velocity
Divergences of joint DoS
van Hove singularity
37
Quark Spectrum
T206 MeV, p0
M
sharp peak at
broad peak at
w MeV

• Strong modification of the quark spectrum
• Appearance of sharp peak at low energy

38
T206 MeV, p0
Real part
w MeV

• There exist divergences in Im S(w)!

39
T Dependence of Quark Spectrum
40
5. Summary and concluding remarks
If a QCD phase transition is of a second order or
close to that, there should exist specific soft
modes, which may be easily thermally excited.
In the fermion-boson system with mFltltmB, the
fermion spectral function has a 3-peak structure
at 1-loop approximation at T mB.
If the chiral transition is close to a second
order, quarks may have a 3-peak structure in the
QGP phase near Tc.
The physical origin of the 3-peak structure is
the Landau damping of quarks and anti-quarks
owing to the thermally excited massive
boson, which induces a mixing between quarks and
anti-quark hole,
The boson may be vector-type or glueballs. The
logic to produce the three-peak structure is
rather robust/universal. Thus it would be
interesting to explore the fermion spectrum
coupled with bosonic excitations at nonzero
temperature, e.g. electro-weak theory and
condensed matter.
41
The quark spectrum near but above TPC coupled
with the would-be soft modes off the chiral limit.
Effects of nonzero m0

• nonvanishing masses of
• constituent quarks
• would-be soft modes

• pionic modes
• stable even above TPC
• non-hyperbolic dispersion

• Quark spectrum is significantly modified by the
  van Hove singularity induced by the scattering of
  quarks and para-pions.
• Quark spectrum near TPC can have a sharp peak
  with a far small energy.

42
Future problems

• Full self-consistent calculation
• Confirmatin in the lattice QCD
• experimental observables eg. Lepton-pair
  production (PHENIX?)
• 
  transport coefficients
• Soft mode (density-fluctuations) at the CP(s)
  and quark spectrum
• M. Kitazawa,
  Y. Nemoto and TK, in preparation.
• Incorporation of CSC with/without inhomogenious
  condensates
• Density(phonon)/entropy fluctuations
• Implications to condensed matter physics
• eg. Graphen at finite temperature, in particular
  searching for possible
• van Hove singularity.

43
Vector and Anomaly terms
Z. Zhang and T.K., Phys.Rev. D83 (2011) 114003
1. Effects of mismatched Fermi sphere by
charge-neutrality
2. Then effect of G_V comes in to make
ph. tr. at low T cross over.
44
S.Carignano, D. Nickel and M. Buballa,
arXiv1007.1397
Interplay between G_V and Polyakov loop is not
incorporated
see also P. Buescher, Mater thesis submitted by
Darmstadt University,where Ginzburg-Levanyuk
analysys shows also an existence of Lifschitz
point at finite G_V.
Spatial dependence of Polyakov loop should be
considered explicitly.
45
Plausible QCD phase diagram
Classical Liq.-Gas
P
Critical point
Liq.
Solid
gas
Triple.P
T
The same universality class Z2
H. Fujii, PRD 67 (03) 094018H. Fujii and
M.Ohtani,Phys.Rev.D70(2004) Dam. T. Son and M. A.
Stephanov, PRD70 (04) 056001
Fluctuations of conserved quantities such as the
number and energy are the soft mode of QCD
critical point! The sigma mode is a slaving mode
of the density.
46
What is the soft mode at CP?
Sigma meson has still a non-zero mass at CP.
This is because the chiral symmetry is
explicitly broken.
What is the soft mode at CP?
At finite density, scalar-vector mixing is
present.
Spectral function of the chiral condensate
T-dependence (mmCP )
Phonon mode in the space-like region softens at
CP.
TgtTc
H. Fujii (2003)H. Fujii and M.Ohtani(2004)
P40 MeV
See also, D. T. Son and M. Stephanov (2004)
does not affect particle creation in the
time-like region.
s-mode
(non-soft mode)
It couples to hydrodynamical modes,
leading to interesting dynamical
critical phenomena.
Space-like region
(the soft modes)
47
Spectral function of density fluctuations in the
Landau frame
Y. Minami and T.K., Prog. Theor. Phys.122 (2009),
881.
In the long-wave length limit, k?0
sound modes
thermal mode
Rel. effects appear only in the width of the
peaks.
Rel. effects appear only in the sound mode.
rate of isothermal exp.
thermal expansion rate
Long. Dynamical
enthalpy
Notice
As approaching the critical point, the ratio of
specific heats diverges!
The strength of the sound modes vanishes out at
the critical point.
48
Spectral function of density fluctuation at CP
Y. Minami,T.K., Prog. Theor. Phys.122 (2009), 881.
0.4
The sound mode (Brillouin) disappears Only an
enhanced thermal mode remains. Furthermore,
the Rayleigh peak is enhanced, meaning
the large energy dissipation.
Spectral function at CP
Suggesting interesting critical phenomena
related to sound mode.
The soft mode around QCD CP is thermally induced
density fluctuations, but not the usual sound
mode.
49
Back Ups
50
Quarks at Extremely High T
Klimov 82, Weldon 83 Braaten, Pisarski 89

• 2 collective excitations having thermal mass
  gT
• width g2T
• Minimum of the plasmino mode at nonzero p

normal
w / mT
plasmino
51
Quark Self-Energy at p0
Im S(w,0)

• Two peaks develop at low energy.

T 1 T 1.5 T 2
T
w / mB
Re S(w,0)

• Two oscillating behavior grows as T is raised.

w / mB
52
Landau Damping
Im S(w,0)
T 1 T 1.5 T 2
T
w / mB
quark
boson
w
w
Landau damping
53
Im S
T206 MeV, p0
w MeV

• There exist divergences in Im S(w)!