1. The nature of the sigma meson and the soft modes of the QCD critical points
Teiji Kunihiro (Kyoto)
Based on the works done in collaboration with
・SCALAR collaboration+M. Wakayama,
・Y.Minami,
・T.Yokota and K. Morita
HHIQCD2015, YITP
Feb March, 2015

2. Contents Introduction Lattice simulations of the sigma meson
The sigma mode at finite temperature
The sigma mode at QCD critical point
On-going FRG analysis
Summary

3. Introduction continued
A condensed matter physics of vacuum (Y. Nambu; 1960)

4. What about at hot and dense matter?
Chiral Transition = a phase transition of QCD　vacuum,
being the order parameter. Lattice QCD;
eg. F. Karsch, Nucl. Phys. Proc. Suppl. 83, 14 (2000).
The wisdom of many-body theory tells us:
If a phase transition is of 2nd order or weak 1st order,
9 soft modes » the fluctuations of the order parameter
For chiral transition,
Then what is the sigma?
What about at hot and dense matter?
Eg. The  meson becomes the soft mode of chiral transition at
T. Hatsuda and T. K. , Phys. Rev. Lett.; Prog. Theor. Phys (1985):
It was also shown that hadronic excitations (para pion and　sigma)
exisit even in the ``QGP” phase.

5. re-identification of the s: “f0(600) of s” in PDG2002
The significance of the  meson in low energy hadron physics and QCD
1. The pole in this mass range observed in the pi-pi S-matrix.
As a compilation of the pole positions of the  obatined in the modern
analyses: Significance of respecting chiral symmetry,unitarity and crossing
symmetry to reproduce the phase shifts both in the  (s)- and , (t)-channels
with a low mass  pole;(Igi and Hikasa(1999)).
re-identification of the s: “f0(600) of s” in PDG2002
2. Seen in decay processes from heavy particles;
E. M. Aitala et al, Phys. Rev. Lett. (86), 770 (2001)
3. Responsible for the intermediate range attraction in the nuclear force.
4. Accounts for  I=1/2 enhancement in K 　 2 compared with K+　+0.
E.P. Shabalin (1988); T. Morozumi, C.S. Lim and I. Sanda (1990).
5.-N sigma term MeV (naively » 15 MeV) enhanced by the collectiveness of the  (.T.Hatsuda and T.K.(1990))
6. Coupled to the quantum fluctuation of the chiral order parameter.
The Higgs particle in the WSG model

6. S. Sakai and T.K., PTEP (’15) 013D03
Significance of the final-state int.

7. Some issues to understand the sigma in QCD
In the constituent quark model;
　　 the mass in the GeV region.
Some mechanism needed to down the mass;
(i) Color magnetic interaction between the di-quarks?
(Jaffe; 1977, Maiani, ‘tHooft ….)
(ii) The collectiveness of the scalar mode as the ps mode; a superposition of states Chiral symmetry (NJL)
(iii) The - molecule as suggested in -  scatt.
What would the Lattice tell us about the sigma?

8. The Scalar mesons on the Lattice
---- A full QCD calculation -----
The Scalar Collaboration:
S. Muroya,A. Nakamura,C. Nonaka,M. Sekiguchi,
H. Wada,T. K.
(Phys. Rev. D70, (2004))

9. Scalar Mesons in Lattice QCDas
meson
quench
1987
screening mass
DeTar and Kogut, PRD36(1987)2828
2000
Alford and Jaffe, NPB578(2000)367
mixing with glueball
Lee and Weingarten, PRD61(2000)014015
dynamical
2001
+glueball
McNeile and Michael, PRD63(2001)114503
2002
disconnected diagram
SCALAR ,NPProc.Suppl.106(2002)272
domain wall fermion, propagators in quench
2003
Prelovsek and Orginos, NPProc.Suppl.119(2003)822
2004
disconnected diagram
SCALAR,PRD70 (2004)034504

10. Operator (two flavor) Propagator Quark model connected disconnected
SCALAR, Phys. Rev. D70 (2004)034504
Operator (two flavor)
Propagator
Quark model
color
Dirac
connected
disconnected
Disconnected diagram
- Vacuum contribution

11. Simulation Setup Full QCD, Hybrid Monte Carlo
SCALAR, Phys. Rev. D70 (2004)034504
Full QCD, Hybrid Monte Carlo
Plaquette gauge action, Wilson Fermion
Lattice size
Disconnected diagrams
Z2 noise method (number of noise: 1000)
CP-PACS, Phys. Rev. D60 (1999)114508
CP-PACS
our results

12. Disconnected Diagrams
Propagators
SCALAR, Phys. Rev. D70 (2004)034504
disconnected
Due to the existence
of disconnected
diagram,
ms becomes smaller.
connected

13. Light Scalar Meson Only connected diagrams Disconnected diagrams
SCALAR,Phys. Rev. D70 (2004)034504
Only connected
diagrams
Disconnected diagrams
At chiral limit

14. Possible tetra/molecular property of the sigma
Use of tetra and molecular operators as the interpolation
Operators to see the overlap with the physical sigma from
The signal of the propagators.
Various diagrams including disconnected ones and hence
Very time-consuming!
With not only connected but also
disconnected diagrams

15. Caution
“Molecule” contains mixing with tetra and two quark state
“Tetra” contains mixing with molecule and two quark state
Application of the variational method for the possible
interpolators is needed.
Results:
Molecular op: Singly disconnected diagram is dominant.
Slopes (~masses) of them are almost the same.
Due to the singly disconnected diagram,
the mass of tetra becomes smaller.
arXiv [hep-lat]

16. The sigma in the hot and dense medium
How does the sigma mode manifest itself or modify its properties at finite temperature, density, magnetic field and so on?
The nature of the sigma may be revealed
by seeing possible change of its properties
at varying environment.

17. Chiral Transition and the sigma mode (meson)
para sigma
para pion
c.f. Higgs particle in WS model
; Higgs field
Higgs particle

18. :Screening masses  
the softening of
the  with increasing T
and

19. G.Colangero, J. Gasser and Leutwyler (2001)
The poles of the S matrix in the complex mass plane for
the sigma meson channel:
complied in Z. Xiao and H.Z. Zheng (2001)
G.Colangero, J. Gasser and Leutwyler (2001)
Softening !
T.Hatsuda and TK, Prog. Theor. Phys. 74 (1985) 765; PRL55 (1985) 158.
cf. K. Yokokawa et al, PRC 66 (2002),
T. Hyodo, D. Jido and TK, NPA848 (2010).
Possible spectral change depending on the strength of the exotic components.

20. Finite T and with finite quark mass
--- soft modes at QCD critical point ----

21. Density fluctuation is the soft mode of QCD critical point!
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) ;H. Fujii and M.Ohtani,Phys.Rev.D70(2004)
Dam. T. Son and M. A. Stephanov, PRD70 (’04)
Density fluctuation is the soft mode of QCD critical point!
The sigma mode is a slaving mode of the density.

22. 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 (m=mCP )
Phonon mode in the space-like region softens at CP.
T>Tc
H. Fujii (2003) H. Fujii and M.Ohtani(2004)
P=40 MeV
See also, D. T. Son and
M. Stephanov (2004)
does not affect particle
creation in the time-like region.
s-mode
It couples to hydrodynamical
modes,
(non-soft mode)
leading to interesting dynamical critical
phenomena.
Space-like region
(the soft modes)

23. Spectral function of density fluctuations in the Landau frame
Y.Minami and TK, Prog. Theor. Phys.122 (2010),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：
：sound velocity
：specific heat
ratio
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.

24. Spectral function of density fluctuation at CP
Y. Minami and T.K., (2009)
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
The soft mode around QCD CP
is thermally induced density fluctuations,
but not the usual sound mode.
Suggesting interesting critical
phenomena related to sound
mode/density fluctuation.
Mach cone?

25. FRG analysis of spectral function of collective excitations
Based on the pioneering work by
R-A. Tripolt et al, PRD89 (2014), ; PRD90 (2014),
and
T.Yokota, K. Morita andTK,
in progress.

26. Model:
Taken from
R-A. Tripolt et al, PRD89 (2014), ; PRD90 (2014),

27. T_c=10 MeV
= 293 MeV c
T.Yokota et al

28. Collective excitations around the CP
T. Yokota et al
Preliminary!
p-h excitation due to Fermi
sphere is seen in the sigma channel!
Slightly different from the previous results, probably due to the difference in the treatment of thermally excited modes. More works are needed.

29. Summary Applicable to find unexpected collective
The sigma meson (scalar mesons) is an interesting hadron(s) reflecting the non-pert. Dynamics of QCD, such as chiral symmetry and its SSB, possible significance of tetra/molecular/diquark correlations in hadron
dynamics.
Exploring the possible change of the spectral properties may reveal the
nature of the sigma and the roles of the above-mentioned dynamics.
The soft mode around the QCD CP is the hydrodynamic modes coupled
to the sigma mode, which may be analyzed by the application of FRG
(combined with fluid dynamics/dynamical RG.)
Applicable to find unexpected collective
excitations in the hot and dense medium
(even under magnetic field?)
to reveal the rich physics of such a medium
as condensed matter.

30. BACK-UPS

31. The numbers in ( , ) are those in the naive quark model.
(T.K. and T. Hatsuda, Phys. Lett. B240 (1990) 209)
The quark content (or the scalar charge of the quarks) isenhanced by the collective  mode in the scalar channel!
C.f.
The empirical value of -N Sigma term is reproduced due to the enhancement of the scalar charge due to the -mesonic collective mode!

32. Scalar Mesons in the Di-quark picture
(Jaffe(1977), Alford and Jaffe (2000))

33. Extrapolation k c = 0.1945±0.0029 mσ=257MeV a = 0.207±0.009 fm 0.8093
CP-PACS
a = 0.197(2) fm
0.270
5.1410±0.0747
k c = ±0.0029
( CP-PACS k c = (14) )
mσ=257MeV

34. Propagators of Molecule
arXiv [hep-lat]
total
Connected diagrams
Singly disconnected diagram
Singly disconnected
diagram is dominant.
Slopes (~masses) of them
are almost the same.

35. Propagators of Tetra total Connected diagram
arXiv [hep-lat]
total
Connected diagram
Singly disconnected diagram
Singly disconnected
diagram is dominant.
Due to the singly
disconnected diagram,
the mass of tetra becomes
smaller.

36. Experimental results for k meson
● It is remarked that the k with I=1/2 is reported
　　to exist with a mass mk ～800MeV.
● The k is supposed to constitute the nonet
　　scalar states of chiral SU(3) X SU(3) symmetry
　　together with s.
Fermilab E791, E. Aitala et al,.
Phys. Rev. Lett. 89 (2002)

37. Scalar Mesons in Lattice QCD
Full QCD s:
SCALAR, PRD70(2004)034504
2004
2006
s, k, a0:
UKQCD,PRD74(2006)114505 G
UKQCD,PRD74(2006)014508
2007 k:
SCALAR, PLB652(2007)250
s, k, a0:
2009
S.Prelovsek et al, PRD79(2009)014503
connected
diagrams
2010
S.Prelovsek et al, PRD82(2010)094507
k, a0:
2012
BGR,PRD85(2012)034508
2013
k, a0:
ETM,JHEP1304(2013)137 s:
2014
SCALAR(+Wakayama), arXiv [hep-lat]
connected + singly disconnected diagrams

38. Extrapolation ( hop_s = 0.1845  fix)
SCALAR, PLB652(2007)250
1.729 GeV
846 MeV
527 MeV
Chiral limit

39. Summary on k mk ～800MeV is not reproduced but twice of it.
P.D.G.による m K/m K*

40. Possible disappearance or strong suppression of Mach cone at the QCD critical point
developed from
Mach cone
sound mode as the density fluctuation
However,
Around the CP;
Attenuation of the sound mode; the dynamical density
fluctuation is hardly developed.
The enhancement of the Rayleigh peak suggests
that the energy dissipation is so large that the possible
density fluctuation gets dissipated rapidly.
Possible disappearance or strong suppression of Mach cone
at the QCD critical point!
Thus, if the identification of the Mach cone in the RHIC
experiment is confirmed, possible disappearance or
suppression along with the variation of the incident energy
can bea signal of the existence of the critical point belonging
to the same universality class as liq.-gas transition.