Collisional energy loss in the sqgp

Transcription

1 Parton propagation through Strongly interacting Systems ECT, Trento, September 005 Collisional energy loss in the sqgp André Peshier Institute for Theoretical Physics, Giessen University, Germany Bjorken s estimate Strongly coupled (Q)GP 3 = +

2 Why collisional energy-loss? T c SB QGP * CSC AIP Bulletin, April 0 005: Now, for the first time since starting nuclear collisions at RHIC in the year 000 and with plenty of data in hand, all four detector groups operating at the lab [BNL]... believe that the fireball is a liquid of strongly interacting quarks and gluons rather than a gas of weakly interacting quarks and gluons. A. Peshier, Collisional energy loss Bjorken s estimate p.

3 Why collisional energy-loss? paradigm: radiative energy-loss [BDMPS] dominates collisional energy-loss How realistic are commonly used (pqcd based) input parameters for e-loss estimates? observed at RHIC: strongly coupled QGP (sqgp) collective phenomena, in line with hydrodynamics fast equilibration, low viscosity large Xsections, large coupling dense liquid collisional e-loss Fokker-Planck eqn. with drag and diffusion parameters related to de coll /dx [Mustafa, Thoma] compatible quenching factors A. Peshier, Collisional energy loss Bjorken s estimate p. 3

4 Framework things are involved simplify consider energy-loss of hard jet in static infinite thermalized medium (QGP) consider mostly quenched QCD; pqcd expectation quarks and gluons differ by group factors (coupling Casimirs, d.o.f.) seen also for large coupling (lattice) rescale: Tc quench = 60 MeV 70 MeV Universality in QCD p/t 4 p SB /T 4 3 flavour + flavour flavour pure gauge T [MeV] p/p SB 3 flavour flavour + flavour pure gauge T/T c A. Peshier, Collisional energy loss Bjorken s estimate p. 4

5 Bjorken s formula [Bjorken 8] considers energy-loss per length due to elastic collisions de dx = ρ(k) Φ dt dσ E k 3 }{{} dt E flux θ small t dominate: dσ dt = πc gg t E, E k T : t = ( cos θ)k E Φ = cos θ α k t E = E E divergences cut-offs: t t dt dσ 9 dt E = 4 πα k( cos θ) ln t screening: t = µ t E < E max : t = ( cos θ)k E max jet persist: E max 0.5E A. Peshier, Collisional energy loss Bjorken s estimate p. 5

6 Bjorken s formula de dx = 9π ρ(k) 4 α k 3 k ln ( cos θ)ke µ pragmatically: ( cos θ) dk kρ(k) ln k ln k dk kρ(k) k T with ρ(k) = 6n b : de B dx = 3πα T ln 4TE µ µ m D = 4παT shortcomings: sloppy k-integral de/dx < 0?? phenomenological IR cut-off 3 relevant scale for coupling? A. Peshier, Collisional energy loss Bjorken s estimate p. 6

7 Bjorken s estimate collisional energy-loss of hard gluons(+) and quarks ( ) de B dx = ( ) 3 ± ( + 6 n ) f πα T ln 4TE µ de/dx [GeV/fm] M = m D 0.6 g in GP 0.4 q in QGP E [GeV] T = 300 MeV α = 0. µ = GeV m D = 4παT compare to cold e-loss GeV/fm radiative e-loss E rad E β, β = 0,, A. Peshier, Collisional energy loss Bjorken s estimate p. 7

8 Bjorken s formula improvements orderly t and k integrals µ t = ( cos θ)k E max t! < µ : E > 0 constrains k 3 0 screening systematically within HTL perturbation theory de dx = 3πα T ln.7te m D [Braaten, Thoma] limit ET µ : dẽ B dx = 3πα T ln 0.64TE µ µ 0.5m D NB: conform with general form of collisional energy loss, in leading-log approximation, related to cut dressed -loop diagrams [Thoma] A. Peshier, Collisional energy loss Bjorken s estimate p. 8

9 Bjorken s formula improvements running coupling α pert (Q) = 4π ln(q /Λ ) often: Q πt, Λ = T c /.4 m D [GeV] (Q), Q = [..4 ]T = 0. T 3 4 de/dx [GeV/fm] 6 E = 0 GeV T/ T c T / T c NB: even conservative α(t ): m D > T (pqcd: m D = gt T ) what is relevant scale for α? how reliable is extrapolated pqcd? A. Peshier, Collisional energy loss Bjorken s estimate p. 9

10 Quasiparticle perspective of s(q)gp PI formalism: thermodynamic potential in terms of full propagator ) Ω = (ln( Tr ) + Π Φ, Φ = Π = δφ δ = truncation thermodyn. consistent resummed approximations entropy functional of dressed (=quasiparticle) propagator [Riedel,...] s[ ] = n ( ) b d i Im ln( i ) + ImΠ i Re i p 4 T i=t,l (in Fermi liquid theory: dynamical quasiparticle entropy) A. Peshier, Collisional energy loss Strongly coupled (Q)GP p. 0

11 Quasiparticle perspective of s(q)gp dressed propagator for momenta p T (d g = 6 transverse d.o.f.) Ansatz: Lorentzian spectral function corresponds to self-energy Π = m iγω γ transport properties (, ) collisional broadening mass shift quasiparticle mass and (collisional) width parameterized in form of pqcd results (gauge invariant, momentum-independent) m = παt, γ = 3 π α T ln c α to extrapolate to T T c use effective coupling α(t ) = 4π ln(λ(t T s )/T c ) A. Peshier, Collisional energy loss Strongly coupled (Q)GP p.

12 Quasiparticle perspective of s(q)gp QP model [AP] vs. lattice data [Okamoto et al.].0 (e-3p)/3p 0, s/s s/s 0 (e-3p)/3p 0 /T c, m/t c 4 m T/T c T/T c non-perturbative quasiparticles m T heavy excitations γ T short mean free path except very near T c (crit. slow down) A. Peshier, Collisional energy loss Strongly coupled (Q)GP p.

13 s(q)gp: coupling α(t ) pqcd α pert (Q) = 4π ln(q /Λ ) with Q πt, Λ T c analyze lattice data entropy within QP model for T O(T c ) IR enhancement of α lattice (F) lattice (s) & QP pqcd: Q = [..4 ]T static q q free energy F (r, T ) C r α r exp( m Dr) [Kaczmarek et al.] T / T c A. Peshier, Collisional energy loss Strongly coupled (Q)GP p. 3

14 s(q)gp: cross section pqcd & cut-off m D = 4παT σ pert = dt πα t = 9 α 8 T α(πt ) T O( mb) phenomenology [Molnar, Gyulassy] σ RHIC O(0 mb) [mb] 0 0. QP: eff pqcd 3 4 T/T c QP model, from m interaction rate d 4 N/dx 4 [AP, Cassing] [ ] [ ] λ Tr p,p n b (ω )n b (ω ) σ Θ(P )Θ(P ) = Tr p ω ω ω γ n b(ω)θ(p ) average σ and γ σ eff = γ N + I O(0 mb) A. Peshier, Collisional energy loss Strongly coupled (Q)GP p. 4

15 Interlude: quasiparticle model implications QP-model indicates an almost ideal liquid ( ) [AP, Cassing] large plasma parameter 0 Γ = N cα N /3 E kin liquid large percolation measure κ = σ eff N / gas dilute regime 3 4 T/T c T/T c ( ) ideal gas!!! very low shear viscosity η s 0. near T c A. Peshier, Collisional energy loss Strongly coupled (Q)GP p. 5

16 Interlude: quasiparticle model implications quantities relevant for radiative energy-loss mean free path λ = γ transport coefficient ˆq = m D /λ [fm] q T c = 70MeV g [GeV /fm] T c = 70MeV 0. ˆq T / T c T / T c NB: γ (p T ) as a lower estimate for mean free path of hard jet A. Peshier, Collisional energy loss Strongly coupled (Q)GP p. 6

17 Non-perturbative parameterization: α(t ) Can pqcd, by using appropriate scales, be extrapolated to near T c? α pert (Q) = pqcd: Q Λ πt T c 4π ln(q /Λ ) non-pert. parameterization (aim at T >.3T c ): Q Λ.7 T T c lattice (F) NP: Q/ =.7 T/T c lattice (s) & QP pqcd: Q = [..4 ]T T / T c lqcd data [Kaczmarek et al.] A. Peshier, Collisional energy loss Strongly coupled (Q)GP p. 7

18 Non-perturbative parameterization: Debye mass Can pqcd be consistently extrapolated to near T c? pqcd: m D = 4παT α α pert (Q πt ) m D / T 4 3 NP pqcd QP:.7 non-pert. parameterization: α α NP (T ) observation within QP model: m D.7γ 3 4 T / T c lattice ( ) lattice (F) lqcd data [Nakamura et al.], [Kaczmarek et al.] A. Peshier, Collisional energy loss Strongly coupled (Q)GP p. 8

19 Non-perturbative parameterization: cross section Can pqcd be consistently extrapolated to near T c? σ = µ dt pqcd: σ = πα t α Q πt T running α(t) = A/ ln( t/λ ): µ [ ] 0. σ = 9π A dt t ln( t/λ ) = 9π A [Ei( Λ ln(µ /Λ )) + Λ /µ ] ln(µ /Λ ) 9π α (µ) µ [ +...] [mb] 0 pqcd 3 4 T/T c µ NP = 0.6m D Λ NP = 40 MeV NP QP: eff A. Peshier, Collisional energy loss Strongly coupled (Q)GP p. 9

20 Non-perturbative parameterization Indeed, pqcd can consistently be extrapolated to near T c. lattice results for α(t ), m D (T ), for T /T c [.3, 4], consistent with α NP (T ) = 4π ln(.7t /T c ) cross section ( 0) enhancement consistent with pert. Xsection dσ/dt α /t running coupling α(t) = 4π ln( t/λ NP ), Λ NP = 40 MeV cut-off µ = 0.6m D (compare to µ = 0.7m D ) relation between α NP (T ) and α(t), assuming t = κt, κ =.7 Λ NP k kρ(k).74 (compare to k = 3 T c k ρ(k).70t ) 3 A. Peshier, Collisional energy loss Strongly coupled (Q)GP p. 0

21 Interlude: Cut-off and running coupling α 0 P α P Π (resummation, renormalization) vacuum: medium: Π = Π ren αp ln( P /µ ) α(p ) = 4π ln( P /Λ ) Π = Π ren + Π mat Π mat αt m D IR cut-off: P > m D running important when m D gt Λ (non-pert. regime) A. Peshier, Collisional energy loss Strongly coupled (Q)GP p.

22 Collisional e-loss with running coupling Bjorken: de dx = ρ(k) Φ k 3 t-integral, with α(t) = A/ ln( t/λ ) Φ t t dt dσ 9 dt E = 4 πa k = dt dσ dt E 9 4 πa t dt t t ln ( t/λ ) [ α(µ ) α(( cos θ)ke) ] k constraint ( cos θ)k µ /E θ-integration leads to logarithmic integrals, li(x) = P x 0 dt/ ln(t) de dx = 9A 8π k [ µ /(Ek) dkkρ(k) ln(µ /Λ ) de B dx [ + O(α)] α(µ) = T F ( µ TE, µ Λ ) )] + (li Λ µ Ek Λ liek µ Λ, k = E A. Peshier, Collisional energy loss = + p.

23 Collisional e-loss in s(q)gp numerical results non-perturbative parameterization (T >.3T c ) de/dx [GeV/fm] Bjorken = m D E = 0GeV E = 50GeV de dx = T F ( µ TE, µ Λ ) parameters Λ Λ NP µ = 0.6m D m D 4παNP (T )T T [MeV] A. Peshier, Collisional energy loss = + p. 3

24 Collisional e-loss in s(q)gp numerical results lqcd/qp parameterization of Debye mass: m D.7γ small m D (T c ) ( phase transition) increased energy loss de/dx [GeV/fm] NP E = 0GeV E = 50GeV de dx = T F ( µ TE, µ Λ ) parameters Λ Λ NP µ = 0.6m D m D m D,eff T [MeV] A. Peshier, Collisional energy loss = + p. 4

25 Collisional e-loss in s(q)gp numerical results unquenching: energy-loss of a quark in the sqgp de/dx [GeV/fm] quenched E = 0GeV E = 50GeV de dx = T F ( µ TE, µ Λ ) parameters T c 70 MeV C gg C q ( + n f /6) T [MeV] A. Peshier, Collisional energy loss = + p. 5

26 Collisional e-loss in s(q)gp numerical results Is de/dx GeV/fm enough? assume constant de/dx Bjorken dynamics quenching factor dn d = Q(p T ) dn 0 p T d p T π = πr dφ 0 R 0 Q(p T ) dr dn(p T + p T ) d p T p T de/dx = 0.8GeV/fm de/dx =.0GeV/fm de/dx =.GeV/fm Mueller (BDMPS) [GeV] comparable to [Müller]: BDMPS + transv. profile + Bjorken dynamics A. Peshier, Collisional energy loss = + p. 6

27 Resumé realistic parameters for sqgp enhanced de coll dx m D / T 4 3 NP pqcd QP:.7 (quasi) critical screening T c quenching does sqgp ( ) quench too much? far/near side jets vs. geometry+delay (talk Cassing) retardation (talk Gossiaux) ( ) strongly Quenching (Q)GP de/dx [GeV/fm] T / T c lattice ( ) lattice (F) T [MeV] quenched E = 0GeV E = 50GeV A. Peshier, Collisional energy loss = + p. 7