Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Lattice NRQCD at non-zero temperature Seyong Kim Sejong University in collaboration with G. Aarts(Swansea), C. Allton(Swansea), S. Hands(Swansea), M.P . Lombardo(LNF), M.B. Oktay(Utah), S.M. Ryan(Trinity), D.K. Sinclair(ANL), J.I. Skullerud (NUIM) based on PRL106, (2011) 061602 arXiv:1010.3752, JHEP1111, (2011) 103 arXiv:1109.4496, and arXiv:1210.2903 and PLB 711, (2012) 199 arXiv:1202.4353 1 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Outline 1 Introduction 2 Lattice NRQCD Lattice NRQCD at T � = 0 3 4 Conclusion 2 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Lattice Gauge Theory • lattice gauge theory aims to calculate low energy non-perturbative quantities in QCD reliably and quantitatively. • despite the systematic errors from finite lattice spacing, finite spacetime volume, finite quark mass • cf. K.G. Wilson, PRD10 (1974) 2445 3 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Lattice Gauge Theory 2012 PDG summary on QCD 4 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Lattice NRQCD • NRQCD is an effective field theory • expansion in v , the heavy quark velocity in heavy quarkonium • Ma ∼ 1 • quarkonium spectrum is one of “gold plated” result from lattice QCD (PRL 92 (2004) 022001 ) 5 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Lattice NRQCD • Non-relativistic QCD G ( x , t = 0 ) S ( x ) � = (1) � n � n � � � D 2 D 2 � 1 + 1 1 + 1 G ( x , t = 1 ) U † x , t ) G ( x , 0 ) � � � = 4 ( (2) 2 n 2 m 0 2 n 2 m 0 b b � n � n � � � D 2 D 2 � 1 + 1 1 + 1 U † [ 1 − δ H ] G ( G ( x , t + 1 ) x , t ) x , t ) � � � = 4 ( 2 n 2 n 2 m 0 2 m 0 b b (3) 6 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Lattice NRQCD where S ( x ) is the source and − ( � D ( 2 ) ) 2 ig δ H b ) 2 ( � D · � E − � E · � D ) = b ) 3 + 8 ( m 0 8 ( m 0 g g σ · ( � D × � E − � E × � D ) − σ · � B − b ) 2 � � 8 ( m 0 2 m 0 b a 2 � D ( 4 ) − a ( � D ( 2 ) ) 2 + (1) 24 m 0 16 n ( m 0 b ) 2 b • calculate NRQCD propagator using gluon field which has light quark vacuum polarization effect 7 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Non-zero T on lattice • N 3 s × N τ lattice, T = 1 N τ a • boundary condition for quantum field on the time direction • high temperature means N t << N s → spectrum in finite temperature environment is not feasible 8 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Lattice study of quarkonium in non-zero T T quark-gluon • plasma T o • deconfined, χ -symmetric � � � � � � hadron gas confined, χ -SB color �� �� �� �� superconductor µ µ few times nuclear o matter density 9 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Lattice study of quarkonium in non-zero T • Recall Schroedinger eq. i ∂ψ ∂ t = H ψ (2) with H = 2 M − ∇ 2 2 M + V ( r ) • T = 0, e.g., Cornel potential; V ( r ) = − α r + σ r (3) • T � = 0, Debye screening; µ ( T )( 1 − e − µ ( T ) r ) − α σ r e − µ ( T ) r V ( r , T ) = (4) where µ ( T ) = 1 / r D ( T ) 10 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Lattice study of quarkonium in non-zero T • F. Karsch, M.T. Mehr, and H. Satz, Z.Phys.C37 (1988) 617. 11 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Lattice study of quarkonium in non-zero T • separation of scale: M (heavy quark mass) and Mv (bound state momentum) • decay rate = the probability for heavy quark and heavy anti-quark to meet × partonic cross-section for quark–anti-quark annihilation • similar to positronium 12 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Lattice study of quarkonium in non-zero T • obtain finite temperature heavy quark potential by lattice calculation → solve Schroedinger equation • obtain spectral function of heavy meson correlator by lattice calculation → observe temperature modification of spectrum • “derive” potential from Wilson loop • study heavy quarkonium correlator in finite temperature by lattice NRQCD on anisotropic lattice 13 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Anisotropic lattice study of quarkoinum in non-zero T • choose the time direction lattice spacing a τ different from the space direction lattice spacing a s • more lattice points along the temperature direction (or time direction) 14 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Anisotropic lattice study of quarkoinum in non-zero T • Anisotropic lattice on 12 3 × N t (ref. G. Aarts et al, PRD 76 (2007) 094513) N s N t a − 1 T/ T c T(MeV) No. of Conf. τ 12 80 7.35GeV 90 0.42 250 12 32 7.35GeV 230 1.05 1000 12 28 7.35GeV 263 1.20 1000 12 24 7.35GeV 306 1.40 500 12 20 7.35GeV 368 1.68 1000 12 18 7.35GeV 408 1.86 1000 12 16 7.35GeV 458 2.09 1000 Table: summary for the lattice data set • two-plaquette Symanzik improved gauge action, fine-Wilson, coarse-Hamber-Wu fermion action with stout-link smearing 15 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Anisotropic lattice study of quarkoinum in non-zero T • bound state → exponential decay of the propagator G ( τ ) ∼ Ae − E τ (2) • free state → power-like decay G ( τ ) ∼ A ′ τ − γ (3) 16 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Effective mass for Υ m eff ( τ ) = − log [ G ( τ ) / G ( τ − a τ )] , (4) 17 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Effective mass for Υ 0.25 3 S 1 ( Υ ) 1 T =0.42 T c T =1.05 T c 0.2 T =1.40 T c a τ m eff T =2.09 T c 0.15 0.1 0 20 40 60 80 τ/ a τ 18 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion Effective mass for χ b 1 0.45 3 P 1 ( χ b1 ) 1 T =0.42 T c T =1.05 T c 0.35 T =1.40 T c a τ m eff T =2.09 T c 0.25 0.15 0 20 40 60 80 τ / a τ 19 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion χ b propagator 10 T= 2.09 T c χ b 0 χ b 1 χ b 2 G ( τ ) 1 0.1 2 4 8 16 τ/ a τ 20 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion effective exponent for Υ γ eff ( τ ) = − τ G ′ ( τ ) G ( τ ) = − τ G ( τ + a τ ) − G ( τ − a τ ) (4) 2 a τ G ( τ ) 21 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion effective exponent for Υ 5 3 S 1 ( Υ ) 1 T=0.42T c T=1.05T c 4 T=1.40T c T=2.09T c 3 free field γ eff 2 1 0 0 8 16 24 32 τ/ a τ 22 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion effective exponent for χ b 7 3 P 1 ( χ b1 ) 1 T =0.42 T c 6 T =1.05 T c T =1.40 T c 5 T =2.09 T c 4 free field γ eff 3 2 1 0 0 8 16 24 32 τ/ a τ 23 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion S-wave bottomonium spectral function = ∑ G Γ ( τ ) � ψ ( τ ,� x )Γ ψ ( τ ,� x ) ψ ( 0 ,� 0 )Γ ψ ( 0 ,� 0 ) � (4) x � � ∞ d ω d 3 p � 2 π K ( τ , ω ) ρ Γ ( ω ,� p ) = (5) ( 2 π ) 3 0 and K ( τ , ω ) = cosh [ ω ( τ − 1 / 2 T )] . (6) sinh ( ω / 2 T ) With ω = 2 M + ω ′ and T / M << 1, � ∞ d ω ′ G ( τ ) = 2 π exp ( − ω ′ τ ) ρ ( ω ′ ) (7) − 2 M 24 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion S-wave bottomonium spectral function 20 T/T c =0.42 2 15 1 2 ρ(ω)/ M 10 0 9 10 11 12 13 14 5 3 S 1 (vector) Upsilon 0 9 10 11 12 13 14 15 ω [GeV] 25 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion S-wave bottomonium spectral function 3 T/T c =0.42 T/T c =1.05 T/T c =1.20 T/T c =1.05 T/T c =1.20 T/T c =1.40 2 1 2 ρ(ω)/ M 0 T/T c =1.40 T/T c =1.68 T/T c =1.86 T/T c =1.68 T/T c =1.86 T/T c =2.09 2 3 S 1 (vector) 1 Upsilon 0 9 10 11 12 13 14 9 10 11 12 13 14 9 10 11 12 13 14 15 ω [GeV] 26 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion S-wave bottomonium spectral function 20 T/T c =0.42 2 15 1 2 ρ(ω)/ M 10 0 9 10 11 12 13 14 5 1 S 0 (pseudoscalar) η b 0 9 10 11 12 13 14 15 ω [GeV] 27 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion S-wave bottomonium spectral function 3 T/T c =0.42 T/T c =1.05 T/T c =1.20 T/T c =1.05 T/T c =1.20 T/T c =1.40 2 1 2 ρ(ω)/ M 0 T/T c =1.40 T/T c =1.68 T/T c =1.86 T/T c =1.68 T/T c =1.86 T/T c =2.09 2 1 S 0 (pseudoscalar) η b 1 0 9 10 11 12 13 14 9 10 11 12 13 14 9 10 11 12 13 14 15 ω [GeV] 28 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion S-wave bottomonium spectral function 0.19 3 S 1 (vector) Upsilon 0.18 ∆ E / M 0.17 0.16 0.15 0 0.5 1 1.5 2 T / T c 29 / 47
Introduction Lattice NRQCD Lattice NRQCD at T � = 0 Conclusion S-wave bottomonium spectral function 2 3 S 1 (vector) Upsilon 1.5 Γ/ T 1 0.5 0 0 0.5 1 1.5 2 T/T c 30 / 47
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