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.16 --- 20 March, 2015

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)

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.

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 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 ---- A full QCD calculation ----- The Scalar Collaboration: S. Muroya,A. Nakamura,C. Nonaka,M. Sekiguchi, H. Wada,T. K. (Phys. Rev. D70, 034504(2004))

Scalar Mesons in Lattice QCD as 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

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

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

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

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

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

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. arXiv1412.3909[hep-lat]

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.

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

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

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), 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 ----

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) 094018;H. Fujii and M.Ohtani,Phys.Rev.D70(2004) Dam. T. Son and M. A. Stephanov, PRD70 (’04) 056001 Density fluctuation is the soft mode of QCD critical point! The sigma mode is a slaving mode of the density.

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)

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.

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?

FRG analysis of spectral function of collective excitations Based on the pioneering work by R-A. Tripolt et al, PRD89 (2014), 034010; PRD90 (2014), 074031 and T.Yokota, K. Morita andTK, in progress.

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

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

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.

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.

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 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 = 0.1945±0.0029 ( CP-PACS k c = 0.19286(14) ) mσ=257MeV

Propagators of Molecule arXiv1412.3909[hep-lat] total Connected diagrams Singly disconnected diagram Singly disconnected diagram is dominant. Slopes (~masses) of them are almost the same.

Propagators of Tetra total Connected diagram arXiv1412.3909[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.

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) 121801.

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), arXiv1412.3909[hep-lat] connected + singly disconnected diagrams

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

Summary on k mk ～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 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.