Quarkonium Suppresion and Entropic Force

9.11.14

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

0.8 B(x)= |P(x)| at x = µr = 1: rQQ¯ = rD entropic force 0.6 reaches maximum and 0.4 overtakes binding force, 0.2 bound state dissociates K(x)

0.5 1.0 1.5 2.0

7 Apply to quarkonium radii

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.