9.11.14 Quarkonium Suppresion and Entropic Force Helmut Satz Universit¨at Bielefeld, Germany QUARKONIUM2014 CERN,November10,2014 An entropic force is a phenomenological force resulting from the entire system’s statistical tendency to increase its entropy, rather than from a particular underlying microscopic force. Example: free expansion of an ideal gas (Joule experiment) ∂S ∂U P (V,T ) = T + ∂V T ∂V T Pressure P : in general energetic & entropic components. Here (∂U/∂V )T = 0, so that the pressure of an ideal gas ∂S P (V,T ) = T = K(V,T ) ∂V T is a purely entropic force. Windmill: entropic Watermill: energetic 2 Quarkonium in deconfined medium: two distinct effects • medium modifies QQ¯ interaction (string softens/melts) • medium interacts with individual Q (polarization clouds) in vacuum in QGP For QQ¯ separation r = 0: no clouds, formation with increasing r increasing r implies increasing entropy, ∃ entropic force separating QQ¯ pair 3 Potential model for in-medium quarkonium binding 1 2 − ∇ 2mQ + V (r, T ) Φi(r, T ) = Mi(T )Φi(r, T ). mQ in vacuum V (r, T = 0) = σr (modulo Coulombic term) in medium V (r, T ) = ? Common: screened string potential −µr 1 − e σ −µr − V (r, T ) = σr = [1 e ] µr µ temperature dependence given through screening mass µ(T ) only for µ ≤ µi, bound state i exists: µi specifies dissociation temperature Ti for state i as µ → µi, bound state radius diverges: ri → ∞ binding energy vanishes: 2mQ + σ/µ − Mi(T ) → 0 4 10 What is a quarkonium state 8 – of radius 10 fm ψ ’(2S) J/ψ (1S) −1 6 – of binding energy 10 MeV (GeV) ] Υ’(2S) Υ(1S) r[ in a medium of temperature 4 200 MeV ∼ scale 1fm? 2 0.2 0.4 0.6 0.8 1.0 1.2 1.4µ[ GeV] Propose: entropic force breaks up quarkonium states before they reach dissociation point from two-body potential treatment Consider thermodynamics of QGP with and without a static QQ¯ pair at separation r, form differences of thermodynamic potentials 5 σ free energy F (r, T ) = [1 − e−x] µ ∂F (r, T ) σ −x entropy T S(r, T ) = −T = [1−(1+x)e ] ∂T r µ internal energy U(r, T ) = T S(r, T )+ F (r, T ) [σ/µ] 2 U(r,T) F (r,T) 1 what forces correspond T S (r,T) to these potentials? 1 2 3 x= µ r 6 ∂F (r, T ) −x attractive pressure P (x) = − = −σe ∂r T ∂S(r, T ) −x repulsive entropic force K(x) = T = σ x e ∂r T 1.0 at x = µr = 1: rQQ¯ = rD 0.8 entropic force B(x)= |P(x)| reaches maximum and 0.6 overtakes binding force, 0.4 bound state dissociates K(x) 0.2 0.5 1.0 1.5 2.0 x 7 Apply to quarkonium radii 10 Schr¨odinger equation with 8 ψ ’(2S) J/ψ (1S) screened string potential −1 6 neglects effects of (GeV) ] Υ’(2S) Υ(1S) r[ entropic force 4 r µ =1 2 0.2 0.4 0.6 0.8 1.0 1.2 1.4µ[ GeV] Example: – at µ(T )= 0, rJ/ψ = 0.4 fm – at µ(T )= 00.32 GeV, rJ/ψ = 0.6 fm: entropic force takes over for larger µ(T ), i.e., for larger T , no more bound state J/ψ 8 Now turn to more realistic world: free energy −µr σ −µr e − − F (r, T ) = (1 e ) α + µ µ r µ(T) Coulombic term & more general µ(T ) T/Tc 0 1 α −µr B(r, T ) = σ + (1 + µr) e binding force r2 K(r, T ) = (Tµ′)[σr + αµ] e−µr entropic force NB: critical behavior (µ′ → ∞) affects only entropic force! 9 thermal quarkonium features in (2+1) finite T lattice QCD [Kaczmarek & Zantow 2007] 4000 1.6 TS∞[MeV] µ/σ1/2 F∞[MeV] 3500 1.4 3000 1.2 2500 1 2000 0.8 1500 0.6 1000 0.4 500 0 0.2 T/Tc T/Tc -500 0 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 F (∞, T ) & TS(∞, T ) µ(T ) 7 α (r,T) 1.4 eff T[MeV] α (r,T) 156 U 186 6 T=191 MeV 1.2 197 T=220 MeV 200 T=270 MeV 203 T=304 MeV 5 T=394 MeV 1 215 226 T=514 MeV 240 4 0.8 259 281 326 3 0.6 365 424 459 0.4 548 2 0.2 1 r[fm] r[fm] 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 r2B(r, T ) r2[K(r, T )+ B(r, T )] 10 Use this to extract binding and entropic forces, compare K(r,T=190 MeV) B(r,T=270) 9 9 6 6 B(r,T=190 MeV) 3 3 K(r,T=270) r r 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 near Tc: strong entropic variation; – consequences on charmonium? – two-body treatment inadequate – need to include entropic forces. 11.