The nature of the sigma meson and

the soft modes of the QCD critical points

Teiji Kunihiro (Kyoto)

HHIQCD2015, YITP Feb.16 --- 20 March, 2015

Based on the works done in collaboration with

・ SCALAR collaboration+M. Wakayama,・ Y.Minami,・ T.Yokota and K. Morita

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

Introduction continued

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

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,

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.

1. Then what is the sigma?2. What about at hot and dense matter?

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 : “f0(600) of ” 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+ 　 . E.P. Shabalin (1988); T. Morozumi, C.S. Lim and I. Sanda (1990).

-N sigma term 40-60 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

S. Sakai and T.K., PTEP (’15) 013D03

Significance of the final-state int.

Some issues to understand the sigma in QCD

● In the constituent quark model;

　　 the mass in the 1.2 --- 1.6 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?

The Scalar mesons on the Lattice

The Scalar Collaboration:S. Muroya,A. Nakamura,C. Nonaka,M. Sekiguchi,H. Wada,T. K.

(Phys. Rev. D70, 034504(2004))

---- A full QCD calculation -----

Scalar Mesons in Lattice QCDScalar Mesons in Lattice QCD

2004

DeTar and Kogut, PRD36(1987)28281987 screening mass

Alford and Jaffe, NPB578(2000)367

quench

Lee and Weingarten, PRD61(2000)014015

SCALAR ,NPProc.Suppl.106(2002)272

McNeile and Michael, PRD63(2001)114503

Prelovsek and Orginos, NPProc.Suppl.119(2003)822

2000

2001

2002

2003

mixing with glueball

disconnected diagram

+glueball

domain wall fermion, propagators in quench

disconnected diagram SCALAR,PRD70 (2004)034504

dynamical

mesonas

• Operator (two flavor)

• Propagator

SCALAR, Phys. Rev. D70 (2004)034504

Quark modelcolor Dirac

Disconnected diagram

- Vacuum contribution

connected

disconnected

Simulation SetupSimulation Setup• 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

SCALAR, Phys. Rev. D70 (2004)034504

Disconnected DiagramsDisconnected Diagrams• Propagators

• Due to the existence of disconnected diagram, m becomes smaller.

SCALAR, Phys. Rev. D70 (2004)034504

connected

disconnected

Light Scalar MesonLight Scalar MesonSCALAR,Phys. Rev. D70 (2004)034504

• Only connected diagrams

• Disconnected diagrams

• At chiral limit

Possible tetra/molecular property of the sigma Possible tetra/molecular property of the sigma

With not only connected but also disconnected diagrams

Use of tetra and molecular operators as the interpolationOperators to see the overlap with the physical sigma fromThe signal of the propagators.

Various diagrams including disconnected ones and hence

Very time-consuming!

CautionCaution• “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.

arXiv1412.3909[hep-lat]

How does the sigma mode manifest itself or modify its properties at finite temperature, density, magnetic field and so on?

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 revealedby seeing possible change of its propertiesat varying environment.

The sigma in the hot and dense medium

Chiral Transition and the sigma mode (meson)

0

c.f. Higgs particle in WS model

; Higgs fieldHiggs particle

para sigmapara pion

the softening ofthe with increasing T

and

:Screening masses

The poles of the S matrix in the complex mass plane forthe 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), 022201. T. Hyodo, D. Jido and TK, NPA848 (2010). Possible spectral change depending on the strength of the exotic components.

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

The same universality class; Z2

Density fluctuation is the soft mode of QCD critical point!The sigma mode is a slaving mode of the density.

H. Fujii, PRD 67 (03) 094018;H. Fujii and M.Ohtani,Phys.Rev.D70(2004)Dam. T. Son and M. A. Stephanov, PRD70 (’04) 056001

T

P

SolidLiq.

Classical Liq.-Gas

Critical point

Triple.P

gas

Plausible QCD phase diagram:

What is the soft mode at CP?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?

T-dependence (=CP )

-mode

Space-like region

(the soft modes)

T>Tc

(non-soft mode)

Phonon mode in the space-like region softens at CP.

H. Fujii (2003)H. Fujii and M.Ohtani(2004)

Spectral function of the chiral condensate

It couples to hydrodynamicalmodes,

See also, D. T. Son and M. Stephanov (2004)

does not affect particlecreation in the time-like region.

leading to interesting dynamical criticalphenomena.

At finite density, scalar-vector mixing is present.

p

p P=40 MeV

Spectral function of density fluctuations in the Landau frame

In the long-wave length limit, k→0

/p nc c t

Long. Dynamical ：： specific heatratio

： sound velocity

thermal expansion rate ：

Rel. effects appearonly in the sound mode.

Rel. effects appear only in the width of the peaks.

rate of isothermal exp.

Notice:

As approaching the critical point, the ratio of specific heats diverges!

P

The strength of the sound modes vanishes out at the critical point.

enthalpy

sound modesthermal mode

20 0

1( / 2 )

2 s Pc T w

Y.Minami and TK, Prog. Theor. Phys.122 (2010),881

Spectral function of density fluctuation at CP

( ) /c cT T T

( ) /c ct T T T

The sound mode (Brillouin) disappearsOnly an enhanced thermal mode remains. Furthermore, the Rayleigh peak is enhanced, meaning the large energy dissipation.

Suggesting interesting criticalphenomena related to sound mode/density fluctuation.

Spectral function at CP

0.4

0. 1t

The soft mode around QCD CP is thermally induced density fluctuations,but not the usual sound mode.

Y. Minami and T.K., (2009)

Mach cone?

FRG analysis of spectral function of collective excitations

Based on the pioneering work byR-A. Tripolt et al, PRD89 (2014), 034010; PRD90 (2014), 074031andT.Yokota, K. Morita andTK, in progress.

Model:

Taken from R-A. Tripolt et al, PRD89 (2014), 034010; PRD90 (2014), 074031

T.Yokota et al

T_c=10 MeV

c= 293 MeV

Collective excitations around the CP

p-h excitation due to Fermi sphere is seen in the sigma channel!

T. Yokota et al

Slightly different from the previous results, probably due to the difference in the treatment of thermally excited modes. More works are needed.

Preliminary!

Summary• 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 collectiveexcitations in the hot and dense medium(even under magnetic field?) to reveal the rich physics of such a medium as condensed matter.

BACK-UPS

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!

Scalar Mesons in the Di-quark picture

(Jaffe(1977), Alford and Jaffe (2000))

Extrapolation

5.1410±0.0747 c = 0.1945±0.0029

( CP-PACS c = 0.19286(14) )

0.8093

a = 0.207±0.009 fm

CP-PACSa = 0.197(2) fm

0.270

334.08093.0

270.0

m

m

mσ=257MeV

Propagators of Molecule

Connected diagrams

Singly disconnected diagram

• Singly disconnected diagram is dominant.• Slopes (~masses) of them are almost the same.

total

arXiv1412.3909[hep-lat]

Propagators of Tetra

totalConnected diagram

Singly disconnected diagram

• Singly disconnected diagram is dominant.• Due to the singly disconnected diagram, the mass of tetra becomes smaller.

arXiv1412.3909[hep-lat]

Experimental results for meson

● It is remarked that the with I=1/2 is reported to exist with a mass m ～ 800MeV.

● The is supposed to constitute the nonet scalar states of chiral SU(3) X SU(3) symmetry together with . Fermilab E791, E. Aitala et al,. Phys. Rev. Lett. 89 (2002) 121801.

Scalar Mesons in Lattice QCDScalar Mesons in Lattice QCD2004 Full QCDSCALAR, PRD70(2004)034504

SCALAR, PLB652(2007)250

UKQCD,PRD74(2006)114505

UKQCD,PRD74(2006)014508

BGR,PRD85(2012)034508

ETM,JHEP1304(2013)137

S.Prelovsek et al, PRD79(2009)014503

S.Prelovsek et al, PRD82(2010)094507

2006

20092010

2013

2014 SCALAR(+Wakayama), arXiv1412.3909[hep-lat]

2007

2012 a

a

a

aG

connected diagrams

connected + singly disconnected diagrams

Extrapolation ( hop_s = 0.1845 fix)

Chiral limit

527 MeV

846 MeV

1.729 GeV

SCALAR, PLB652(2007)250

Summary on

m ～ 800MeV is not reproduced but twice of it.

P.D.G. による m K/m K*

Possible disappearance or strong suppression of Mach cone at the QCD critical point

Mach cone sound mode as the density fluctuationdeveloped from

Around the CP;

However,

(i) Attenuation of the sound mode; the dynamical density fluctuation is hardly developed.(ii) 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.

scvsc

M si n sM

c

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