Collisional energy loss in the sqgp

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 = +

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.

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

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 5 4 3 0.0 0.8 0.6 0.4 0. 0.0 Universality in QCD p/t 4 p SB /T 4 3 flavour + flavour flavour pure gauge T [MeV] 00 00 300 400 500 600 p/p SB 3 flavour flavour + flavour pure gauge T/T c.0.5.0.5 3.0 3.5 4.0 A. Peshier, Collisional energy loss Bjorken s estimate p. 4

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

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

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].0 0.8 M = m D 0.6 g in GP 0.4 q in QGP 0. 0 5 0 5 0 E [GeV] T = 300 MeV α = 0. µ = 0.5... 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

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

Bjorken s formula improvements running coupling α pert (Q) = 4π ln(q /Λ ) often: Q πt, Λ = T c /.4 m D [GeV].0.5.0 0.5 (Q), Q = [..4 ]T = 0. T 3 4 de/dx [GeV/fm] 6 E = 0 GeV 4 3 4 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

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

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.

Quasiparticle perspective of s(q)gp QP model [AP] vs. lattice data [Okamoto et al.].0 (e-3p)/3p 0, s/s 0 0.8 0.6 0.4 0. s/s 0 (e-3p)/3p 0 /T c, m/t c 4 m 0.0 3 4 0 3 4 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.

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.0.5.0 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.] 0.5 3 4 T / T c A. Peshier, Collisional energy loss Strongly coupled (Q)GP p. 3

s(q)gp: cross section pqcd & cut-off m D = 4παT σ pert = 9 8 9 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

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 /3.5.0.5 0 - gas 3 4.0 0.5 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

Interlude: quasiparticle model implications quantities relevant for radiative energy-loss mean free path λ = γ transport coefficient ˆq = m D /λ [fm] 5 0.5 q T c = 70MeV g [GeV /fm] 3.0.5.0.5.0 T c = 70MeV 0. ˆq 0.5 0. 3 4 3 4 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

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.0.5.0 lattice (F) NP: Q/ =.7 T/T c lattice (s) & QP pqcd: Q = [..4 ]T 0.5 3 4 T / T c lqcd data [Kaczmarek et al.] A. Peshier, Collisional energy loss Strongly coupled (Q)GP p. 7

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

Non-perturbative parameterization: cross section Can pqcd be consistently extrapolated to near T c? σ = µ dt pqcd: σ = 9 8 9 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

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

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.

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.

Collisional e-loss in s(q)gp numerical results non-perturbative parameterization (T >.3T c ) de/dx [GeV/fm] 0 5 5 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 300 400 500 600 T [MeV] A. Peshier, Collisional energy loss = + p. 3

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] 0 5 5 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 300 400 500 600 T [MeV] A. Peshier, Collisional energy loss = + p. 4

Collisional e-loss in s(q)gp numerical results unquenching: energy-loss of a quark in the sqgp de/dx [GeV/fm] 0 5 5 quenched E = 0GeV E = 50GeV 00 300 400 500 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

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 ) 0.3 0. 0. dr dn(p T + p T ) d p T 5 0 5 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

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] 0 5 5 3 4 T / T c lattice ( ) lattice (F) 00 300 400 500 T [MeV] quenched E = 0GeV E = 50GeV A. Peshier, Collisional energy loss = + p. 7