格子QCDシミュレーションによる QGP媒質中のクォーク間ポテンシャルの研究 WHOT-QCD Collaboration 前沢祐 (理研) in collaboration with 青木慎也, 金谷和至, 大野浩史 (筑波大物理) 浮田尚哉 (筑波大計算セ) 初田哲男,石井理修 (東京大学) 江尻信司 (BNL) 梅田貴士 (広島大学) WHOT-QCD Coll. arXive:0907.4203 WHOT-QCD Coll. PoS LAT2007 (2007) 207 WHOT-QCD Coll. Phys. Rev. D75 (2007) 074501 原子核・ハドロン物理 @ KEK, 2009年8月11—13日

Introduction Study of Quark-Gluon Plasma (QGP) Early universe after Big Bang Relativistic heavy-ion collision Theoretical study based on first principle (QCD) Lattice QCD simulation at finite (T, mq) Bulk properties of QGP (p, e, Tc,…) Internal properties of QGP are well investigated. are still uncertain. Properties of quarks and gluons in QGP Heavy-quark potential: free energy btw. static charged quarks in QGP Relation of inter-quark interaction between zero and finite T Temperature dependence of various color-channels Heavy-quark potential at finite density (mq) in Taylor expansion method 2

Heavy-quark potential at finite T interaction btw. static quark (Q) and antiquark (Q) in QCD matter T = 0, T > 0, string tension (s ) decreases, string breaking at rc At T > Tc, screening effect in QGP Properties of heavy-quark potential in quark-gluon plasma Heavy-quark bound state (J/y, Υ) in QGP Screening effect in QGP Inter-quark interaction btw. QQ and QQ Lattice QCD simulations 3 3

Operator similar to the Wilson-loop Heavy-quark free energy To characterize an interaction b/w static quarks, T = 0 Heavy-quark potential Brown and Weisberger (1979) Wilson-loop operator T > 0 Heavy-quark free energy Nadkarni (1986) Polyakov-line : static quark at x Correlation b/w Polyakov-lines projected to a color singlet channel in the Coulomb gauge Operator similar to the Wilson-loop at finite T Heavy-quark free energy may behave in QGP as: short r: F 1(r,T ) ~ V(r) (no effect from thermal medium) mid r : screened by plasma long r : two single-quark free energies w/o interactions 4

Approaches of lattice simulations at finite T To vary temperature on the lattice, a: lattice spacing Nt: lattice size in E-time direction Fixed Nt approach Fixed scale approach WHOT-QCD, PRD 79 (2009) 051501. Changing a (controlled by the gauge coupling (b = 6/g2)), fixing Nt Advantages: T can be varied arbitrarily, because b is an input parameter. Changing Nt , fixing a Advantages: investigation of T dependence w/o changing spatial volume possible. renormalization factor dose not change when T changes.

Results of numerical simulations Heavy-quark potential at T = 0 and T > 0 Heavy-quark potential for various color-channels Heavy-quark potential at finite density (mq) 6

Simulation details Nf = 2+1 full QCD simulations Lattice size: Ns3 x Nt = 323 x 12, 10, 8, 6, 4 Temperature: T ~ 200-700 MeV (5 points) Lattice spacing: a = 0.07 fm Light quark mass*: mp/mr = 0.6337(38) Strange quark mass*: mK/mK* = 0.7377(28) Scale setting: Sommer scale, r0 = 0.5 fm *CP-PACS & JLQCD Coll., PRD78 (2008) 011502. 7

Heavy-quark potential at T = 0 Data calculated by CP-PACS & JLQCD Coll. on a 283 x 58 lattice Phenomenological potential Our fit results = Coulomb term at short r + Linear term at long r + const. term 8

Heavy-quark free energy at T > 0 At short distance F 1(r,T ) at any T converges to V(r) = F 1(r,T=0 ). Short distance physics is insensitive to T. Note: In the fixed Nt approach, this property is used to adjust the constant term of F 1(r,T ). In our fixed scale approach, because the renormalization is common to all T. no further adjustment of the constant term is necessary. We can confirm the expected insensitivity!

Heavy-quark free energy at T > 0 At short distance F 1(r,T ) at any T converges to V(r) = F 1(r,T=0 ). Short distance physics is insensitive to T. T ~ 200 MeV: F 1 ~ V up to 0.3--0.4 fm T ~ 700 MeV: F 1 = V even at 0.1 fm Range of thermal effect is T-dependent Debye screening effect WHOT-QCD Coll. arXive:0907.4203 WHOT-QCD Coll. Phys. Rev. D75 (2007) 074501

Heavy-quark free energy at T > 0 2FQ calculated from Polyakov-line operator Long distance F 1(r,T ) becomes flat and has no increase linealy. Confinement is destroyed due to a thermal medium effect. At large r, correlation b/w Polyakov-lines will disappear, F 1 converges to 2x(single-quark free energy) at long distance

Results of numerical simulations Heavy-quark potential at T = 0 and T > 0 Heavy-quark potential for various color-channels Heavy-quark potential at finite density mq 12 12

Heavy-quark potential for various color-channels Interaction for various color-channels are induced in QGP Separation to each color channel Projection of Polyakov-line correlators Nadkarni (1986) Normalized free energy QQ correlator: 13

Heavy-quark potential for various color-channels QQ potential QQ potential Temperature increases becomes weak 1c, 3*c: attractive force, 8c, 6c: repulsive force Single gluon exchange ansatz Consistent with Casimir scaling e.g.) Casimir factor 14

Results of numerical simulations Heavy-quark potential at T = 0 and T > 0 Heavy-quark potential for various color-channels Heavy-quark potential at finite density (mq) 15 15

Heavy-quark potential at finite mq Motivation Taylor expansion method in terms of mq /T Properties of QGP at 0 < mq /T << 1 Expectation values at finite mq At mq /T << 1, Taylor expansion of quark determinant detD(mq) where Early universe after Big Bang Relativistic heavy-ion collisions Low density e.g. mq /Tc ~ 0.1 at RHIC c.f.) sign problem detD becomes complex when mq≠0 As one of the reweighting method, we use the Taylor expansion method with respect to the chemical potential. Let me explain the Taylor expansion method. The expectation value of observable O at finite \mu is written like this, where the “Delta” is the quark determinant. The Taylor expansion of the quark determinant can be written like this. The expansion coefficients can be expressed here, using the inverse of the quark determinant. If “O” is independent of \mu, the expectation value can be expanded with respect to \mu, like this, and we can calculate the expansion coefficients from the simulation at zero chemical potential. 16

Heavy-quark potential at finite mq Heavy-quark potential up to 2nd order with expansion coefficients: QQ potential (1c, 8c) QQ potential (3*c, 6c) Odd term of QQ potential vanishes because of symmetry under mq→ -mq

QQ potential 1c channel: attractive force 8c channel: repulsive force becomes weak at 0 < mq /T << 1 1c channel: attractive force 8c channel: repulsive force

QQ potential 3*c channel: attractive force 6c channel: repulsive force becomes strong at 0 < mq /T << 1 3*c channel: attractive force 6c channel: repulsive force

Heavy-quark potential at finite mq QQ potential (1c, 8c) QQ potential (3*c, 6c) Attractive channel in heavy-quark potential Heavy-meson correlation (1c) weak at 0 < mq /T << 1 Diquark correlation (3*c) strong

Summary Heavy-quark potential at T = 0 and T > 0 Properties of quark-gluon plasma Heavy-quark potential (F 1(r,T )) defined by free energy btw. static charged quarks Heavy-quark potential at T = 0 and T > 0 Heavy-quark potential for various color-channels Heavy-quark potential at finite density (mq) At short r: F 1 at any T converges to heavy-quark potential at T = 0 Short distance physics is insensitive to T. At mid r : screening effect appears At long r: F 1 becomes flat and has no linear behavior. Correlation b/w Polyakov-lines disappears and F 1 converges to 2FQ 1c, 3*c: attractive force, 8c, 6c: repulsive force Consistent with Casimir scaling based on single gluon exchange ansatz Taylor expansion in terms of mq /T Heavy-meson correlation (1c) weak Diquark correlation (3*c) strong at 0 < mq /T << 1