Selected Topics in the Theory of Heavy Ion Collisions Lecture 3 Urs Achim Wiedemann CERN Physics Department TH Division Varenna, 20 July 2010 Based on http://cdsweb.cern.ch/record/1143387/files/p277.pdf and updated material
Hard Probes at the LHC Hard probes = hard processes embedded in dense nuclear matter (and sensitive to its properties ) These are produced abundantly at the LHC.
Bjorken s original estimate and its correction Bjorken 1982: consider jet in p+p collision, hard parton interacts with underlying event collisional energy loss de coll dl 10GeV fm (error in estimate!) Bjorken conjectured monojet phenomenon in proton-proton But: radiative energy loss expected to dominate ΔE rad α sˆ q L 2 Baier Dokshitzer Mueller Peigne Schiff 1995 p+p: L 0.5 fm, ΔE rad 100 MeV A+A: L 5 fm, ΔE rad 10 GeV Negligible! Monojet phenomenon! Observed at RHIC Today: we want to understand how these estimates arise and how energetic partons lose energy in dense matter.
IV.1 Jet Quenching at RHIC So far, jet quenching is mainly tested by suppressed leading hadron production: Nuclear modification factor characterizes medium-effects: R AA ( p T ) = dn AA dp T n coll dn NN dp T R AA ( p T ) =1.0 R AA ( p T ) = 0.2 no suppression factor 5 suppression
IV.2. Suppression persists to highest p T Centrality dependence: 0-5% 70-90% L large L small
STAR azimuthal correlation function shows ~ complete absence of awayside jet IV.3. The Matter is Opaque ΔΦ = π GONE ΔΦ ΔΦ=0 ΔΦ = 0 Partner in hard scatter is completely absorbed in the dense medium
IV.4. Issues Purpose of this lectures: sketch current state of the art of the theory of jet quenching and its testable consequences. Warning: This theory is far from complete! Our presentation is simplified.
A a + IV.6. Eikonal formalism ( ) = P exp i dz T a A + a ( x i,z ) W x i T a [ ] Here, is target gauge field, is SU(3) generator in representation of the parton α i, x i, z is light cone coordinate. Interpretation of Ψ out = ˆ S Ψ in = { α i,x i } r ( ) Π i W i αi β i ψ α i,x i ( x i ) ( ) β i, x i High-energy scattering Rotation in color space Components of Ψ in decohere Inelastic particle production Measure of decoherence: (4.4) δψ = 1 Ψ in Ψ in [ ] Ψ out δψ δψ E.g. probability of inelastic scattering of projectile given by
IV.7. Example: gluon production in q+a Consider high energy quark centered at x=0 and projectile rapidity y=0. The wave packet to zeroth order in coupling is (4.5) α(0,0) = α α x = 0 But to 1st order in coupling, the quark is not any more a bare quark, it has a gluon in its wavefunction (4.6) Ψ in q = α(0,0) + dx dξ f ( x)t b αβ β(0,0),b(x,ξ) + O(g 2 ) + = α α α T αβ b b(x) This distribution of gluons is flat in rapidity. In transverse space, it follows a Coulomb-type Weizsäcker-Williams field (4.7) f (x) g x x 2 β
SUPPLEMENT (4.8) Ψ in q IV.8. Example: gluon production in q+a Note that the very past i.e. U Ψ in q results from unitary free time evolution of bare quark from t = to the present = U α = exp dx t = 0 f (x) 2 + i dx dξ creates the cloud of gluons around the bare quark. f (x)( a d (x,ξ) + a + d (x,ξ))t d α (4.9) Now comes the scattering q Ψ out = ˆ S U α = W αγ F (0) γ + dx f x ( )T αβ b W βγ F (0)W bc A (x) γ(0),c(x) (4.10) Gluons are produced in those components of, which lie in the subspace orthogonal to the incoming state with arbitrary color orientation γ α δψ α = U + Ψ out U γ γu + U + α + Ψ out γ q Ψ out To calculate this, use γu + U + α + Ψ out = γ W αδ F (0)δ + O( f 2 ) = W αγ F (0)
SUPPLEMENT (4.10) IV.9. Example: gluon production in q+a α δψ α = U + Ψ out U γ γu + U + α + Ψ out γ W F αβ (0) γ + dx f (x)t b αβ W F βγ (0) W A bc (x) γ,c(x) = U + W F αβ (0) γ dx f (x)w F b αγ (0)T γβ β,b(x) = dx f (x) T b αβ W F βγ (0)W A bc (x) T c βγ W F αβ (0) γ,c(x) [ ] The number spectrum of produced gluons reads then Use (4.6), (4.9) (4.11) (4.12) N qa ( k T ) = 1 δψ α a + d k T N c α ik. ( x y) = dx d y e To calculate this expression, use a d (x)δψ α ( )a d ( k T )δψ α 1 N c = f (x) T b W F (0) α δψ α a + d ( y)a d ( x)δψ α [( ) W A αγ bd (x) ( W F (0)T d ) ] γ αγ
SUPPLEMENT (4.12) Again: IV.10. Example: gluon production in q+a a d (x)δψ α [( ) W A αγ bd (x) ( W F (0)T d ) ] γ αγ = f (x) T b W F (0) δψ α a + A d (y) = γ W + d b (y) W F + (0)T b ( ) γα T d W F + (0) ( ) γα f (y) To calculate from this δψ α a + d ( y)a d ( x)δψ α, we use (4.13) Tr[ T a T b ] = δ ab /2 W A ab (x) = 2Tr[ T a W F (x)t b W F + (x)] f (x). f (y) = α s 2π x.y x 2 y 2 (4.14) N qa ( k T ) = α sc F 2π dx dy [ ] [ ] + Tr[ W A + (y)w A (x)] A x.y 1 Tr W + (0)W A (x) x 2 y 2 eik.(x y ) Tr W A + (y)w A (0)
V.9. BDMPS gluon radiation spectrum R. Baier et al. (BDMPS), NPB484:265, 1997 Wiedemann, NPB 588 (2000) 303 di = α C s R 2Re dy dy d lnω dk T (2π) 2 2 ω u. K(s = 0, y;u, y ω) s 0 y due ik T u e 1 4 dξ q ˆ ξ y ( )u2 Radiation off produced parton Target average includes Brownian motion: ( ) = Drexp dξ K s,y;u, y ω ω u= r(y ) s= r(y ) exp 1 q 4 ˆ L long r 2 y y iω 2 r 2 1 q 4 ˆ (ξ) r 2 BDMPS transport coefficient Tr[ W A + (0)W A (r)] Expectation value of light-like Wilson line
VI.3. Parton energy loss - what to expect? Medium characterized by BDMPS transport coefficient: q ˆ µ2 λ (6.6) (6.7) (6.8) (6.9) How much energy is lost? Phase accumulated in medium: Number of coherent scatterings: Gluon energy distribution: Average energy loss k T 2 Δz 2ω N coh t coh λ q ˆ L 2 2ω = ω c ω, where ω di med dω dz 1 N coh ω di 1 dω dz α s ΔE = L 0 dz ω c 0 dω Characteristic gluon energy t coh 2ω k T 2 ω ˆ q ˆ q ω k T 2 ˆ q t coh ω di med dω dz ~ α sω c ~ α sˆ q L 2 Quadratic increase with L!
VI.4. Medium-induced gluon energy distribution Consistent with estimate (3.6), spectrum is indeed determined by ω c = ˆ q L 2 2 Salgado,Wiedemann PRD68:014008 (2003)
VI.5. Kt-distribution of medium-induced gluons Follows transverse Brownian motion, consitent with (3.6). ω c ω = 32 ω c ω =10 ˆ q L ω c ω = 3.2 ω c ω =1 κ 2 k 2 T q ˆ L
VI.17. Estimating Time scales for parton E-loss in vacuum h h in QGP h L hadr = const E Q 2 2 ΔE E α sˆ q L therm 100 fm Dynamics of the bulk Dynamics of hadronization Partonic equilibration processes Jet Jet modification absorption L medium L therm ~ E /q 1 fm soft intermediate 2 L hadr ~ E /Q hadr hard E 1GeV 10 GeV 100 GeV
VI.11.Recall: High p T Hadron Spectra R AA ( p T,η) = dn AA dp T dη n coll dn NN dp T dη Centrality dependence = dependence on inmedium path-length L L large 0-5% 70-90% L small
VI.11. Jet Quenching: Au+Au vs. d+au Final state suppression Initial state enhancement partonic energy loss
VI.13.Trigger Bias in high-pt hadron production Triggering on high-pt hadrons, one selects a biased parton fragmentation pattern: A bias favoring hard fragmentation, determined by steepness of spectrum: z n 1 (p hadron T ) D (med ) n h / q (z,q 2 ) A bias favoring small in-medium path length (surface emission) A bias favoring initial-state pt-broadening in the direction of the trigger
VI.14. Determining the quenching parameter Eskola, Honkanen, Salgado, Wiedemann Nucl Phys A747 (2005) 511 L +ξ 0 ( ) q ˆ = 2 ξ ξ L 2 0 ξ 0 q ˆ (ξ) dξ = 5 15 GeV 2 fm
VII.1. Testing the dynamical origin of high-pt hadron suppression How does this parton thermalize? Where does this associated radiation go to? What is the dependence on parton identity? How do we detect recoil in case of collissional energy loss? ΔE gluon > ΔE quark, m= 0 > ΔE quark, m>0