Selected Topics in the Theory of Heavy Ion Collisions Lecture 1

Transcription

1 Selected Topics in the Theory of Heavy Ion Collisions Lecture 1 Urs chim Wiedemann CERN Physics Department TH Division Varenna, 19 July 2010 Based on and updated material

2 Preface Starting point: Quantum Chromodynamics, QCD, the theory of strong interactions, is a mature theory with a precision frontier. - background in search for new physics - TH laboratory for non-abelian gauge theories Open fundamental question: How do collective phenomena and macroscopic properties of matter emerge from the interactions of elementary particle physics? Heavy Ion Physics: addresses this question in the regime of the highest temperatures and densities accessible in laboratories. How? 1. Benchmark: establish baseline, in which collective phenomenon is absent. 2. Establish collectivity: by characterizing deviations from baseline 3. Seek dynamical explanation, ultimately in terms of QCD. These lectures give examples of this How?

3 I.1. The very first measurement at an Heavy Ion Collider Signal proportional to multiplicity What is the benchmark for multiplicity distributions? Multiplicity in inelastic + collisions is incoherent superposition of inelastic p+p collisions. (i.e. extrapolate p+p -> p+ -> + without collective effects) Glauber theory

4 I.2. Glauber Theory ssumption: inelastic collisions of two nuclei (-B) can be described by incoherent superposition of the collision of an equivalent number of nucleon-nucleon collisions. How many? Spectator nucleons Establish counting based on Participating nucleons To calculate N part or N coll, take σ = inelastic n-n cross section priori, no reason for this choice other than that it gives a useful parameterization. N part = 7 N coll. = 10 N quarks +gluons =? N inelastic = 1

5 We want to calculate: I.3. Glauber theory for n+ N part = number of participants = number of wounded nucleons, which undergo at least one collision N coll = number of n+n collisions, taking place in an n+ or +B collision We know the single nucleon probability distribution within a nucleus, the so-called nuclear density (1.1) ρ(b,z) dz db ρ(b,z) =1 b (1.2) Normally, we are only interested in the transverse density, the nuclear profile function T (b) = dz ρ(b,z)

6 I.4. Glauber theory for n+ The probability that no interaction occurs at impact parameter b: (1.3) P 0 (b) = Π 1 dsi T ( s i i=1 )σ b s i [ ( )] dsσ(s) inel = σ nn (1.4) (1.5) If nucleon much smaller than nucleus σ(b s) σ inel nn δ(b s) P 0 (b) = 1 T ( b)σ nn [ inel] b s i Transverse position of i-th nucleon in nucleus (1.6) (1.7) The resulting nucleon-nucleon cross section is: σ inel n = [ [ inel ] ] db( 1 P 0 (b)) = db 1 1 T ( b)σ nn >>n db 1 exp T ( b)σ nn [ [ inel] ] = db T ( b)σ inel nn 1 2 T ( b)σ nn Optical limit ( inel) 2 + Double counting correction

7 I.5. Glauber theory for n+ To calculate number of collisions: probability of interacting with i-th nucleon in is (1.8) p(b,s i ) = ds i T ( s i )σ ( b s ) inel i = T (b)σ nn (1.9) (1.10) Probability that projectile nucleon undergoes n collisions = prob that n nucleons collide and -n do not P(b,n) = 1 p n ( ) n p n verage number of nucleon-nucleon collisions in n+ n N coll (b) = n P(b,n) = n= 0 = T ( b)σ nn inel n= 0 b n 1 p n verage number of nucleon-nucleon collisions in n+ s i Transverse position of i-th nucleon in nucleus ( ) n p n = p (1.11) n N part n (b) =1+ N coll(b)

8 (1.12) I.6. Glauber theory for +B collisions We define the nuclear overlap function T B ( b ) = d s T ( s )T B ( b s ) The average number of collisions of nucleon at s B with nucleons in is s B b B (1.13) n N coll (b s B ) = T (b s B inel )σ nn The number of nucleon-nucleon collisions in an -B collision at impact parameter b is b B spectators participants (1.14) B N coll (b) = B ds B T B (s B ) n N coll (b s B ) = B inel dst B (s) T B (b s)σ nn inel = BT B (b)σ nn determined in terms of nuclear overlap only

9 (1.15) (1.16) I.7. Glauber theory for +B collisions Probability that nucleon at in B is wounded by in configuration s i [ ] p(s B,{ s i }) =1 1 σ(s B s i ) P(w b,b) = i=1 Probability of finding nucleons in nucleus B: w B B w B s B i=1...p(s B wb { } wounded b B s B spectators participants B ds i ds B j T (s i )T B (s B j b) p(s B 1,{ s i })... j=1,{ s i })[ B 1 p(s wb +1, { s i })]...[ 1 p(s B B,{ s i })] (1.17) Nuclear overlap function defines inelastic +B cross section. inel σ B = dbσ B (b) = dbp(w B = 0,b) B = db 1 ds i ds B j T (s i )T B (s B j b) Π B j=1 i=1 j=1 db [ 1 [ 1 T B (b)σ inel NN ] B ] 1 p(s B j,{ s i }) [ ]

10 I.8. Glauber theory for +B collisions It can be shown Problem 1: derive the expressions (1.17), (1.19) Use e.g.. Bialas et al., Nucl. Phys. B111 (1976) 461 (1.18) (1.19) Number of collisions: Number of participants: N B inel coll (b) = BT B (b)σ NN N part inel (b) B (b) = σ B inel (b) + Bσ inel (b) N B coll (b) +1 σ B σ B inel (b) (1.20) (1.21) 1. There is a difference between analytical and Monte Carlo Glauber theory: For MC Glauber, a random probability distribution is picked from T. 2. The nuclear density is commonly taken to follow a Wood-Saxon parametrization (e.g. for > 16) 3. The inelastic Cross section is energy dependent, typically But ( ) = ρ 0 1+ exp[ (r R) c] ρ r ( ); R / 3 fm,c = fm. C.W. de Jager, H.DeVries, C.DeVries, tom. Nucl. Data Table 14 (1974) 479 inel σ nn inel σ nn 40 mb at s nn =100GeV. is sometimes used as fit parameter.

11 I.9 Event Multiplicity in wounded nucleon model (1.22) Model assumption: If n nn is the average multiplicity in an n-n collision, then n B (b) = 1 x 2 N B part(b) + x N B coll (b) n NN is average multiplicity in +B collision (x=0 defines the wounded nucleon model). The probability of having w b wounded nucleons fluctuates around the mean,, so does the multiplicity n per event (the dispersion d is a fit parameter, say d~1) (1.23) P(n,b) = 1 2π d n B (b) exp n n B (b) 2d n B (b) [ ] 2 How many events dn events have event multiplicity dn? (1.24) dn events dn = db [ ( ) B ] P(n,b) 1 1 σ NN T B (b) 1 P 0 (b )

12 I.10 Wounded nucleon model vs. multiplicity = db Compare data to multiplicity distribution (1.24): dn events dn dn events dn P(n,b) [ 1 P 0 (b)] determined by geometry only insensitive to details of particle production [there is almost no dependence on parameter x in (1.22)] insensitive to collective effects Sensitivity to geometry but insensitivity to model-dependent dynamics makes dn events dn well-suited centrality measure (i.e. a measure of the impact parameter b) b B

13 I.11. Multiplicity as a Centrality Measure The connection between centrality and event multiplicity can be expressed in terms of (1.25) u+u N part n>n 0 = n0 dn n0 db P(n,b) [ 1 P 0 (b)] N part (b) dn db P(n,b) [ 1 P 0 (b)] b B Centrality class = percentage of the minimum bias cross section s NN = 200 GeV Centrality class specifies range of impact parameters u u N part u u N coll % % b in fm

14 I.12. Cross-Checking Centrality Measurements The interpretation of min. bias multiplicity distributions in terms of centrality Measurements can be checked in multiple ways, e.g. 1. Energy E F of spectators is deposited in Zero Degree Calorimeter (ZDC) ( ) s /2 E F = N part (b) /2 2. Testing Glauber in d+u and in p+u(+ n forward)

15 I.13. Final remarks on event multiplicity in +B There is no 1st principle QCD calculation of event multiplicity, neither in p+p nor in +B Clear deviations from multiplicity of wounded nucleon model Total charged event multiplicity: models failed at RHIC gnostic estimates for HI at LHC: LHC dn ch dη η= 0 =

16 I.14. Final remarks on event multiplicity Multiplicity distribution is not only used as centrality measure but: Multiplicity (or transverse energy) thought to determine properties of produced matter Bjorken estimate ε(τ 0 ) = 1 1 de T π R 2 τ 0 dy de T dy dn dy E T This estimate is based on geometry, thermalization is not assumed, numerically: ε SPS (τ 0 1fm /c) = 3 4 GeV / fm 3

17 II.1. zimuthal nisotropies of Particle Production We know how to associate an impact parameter range b b min,b max to an event class in +, namely by selecting a multiplicity class. s NN = 200 GeV [ ] What can we learn by characterizing not only the modulus b, but also the orientation b?

18 (2.1) (2.2) (2.3) II.2. Particle production w.r.t. reaction plane Consider single inclusive particle momentum spectrum f ( p ) dn d p p x = p T cosφ p = p y = p T sinφ p z = p 2 T + m 2 sinhy b φ Particle with momentum p To characterize azimuthal asymmetry, measure n-th harmonic moment of (2.1) in some detector acceptance D [phase space window in (p T,Y)-plane]. v n ( D) e i n φ = dp e i n φ D D dp D f ( p ) f ( p ) Problem: Eq. (2.3) cannot be used for data analysis, since the orientation of the reaction plane is not known a priori. n-th order flow

19 II.3. Why is the study of v n interesting? Single 2->2 process Maximal asymmetry NOT correlated to the reaction plane Many 2->2 or 2-> n processes Reduced asymmetry ~ 1 N NOT correlated to the reaction plane final state interactions asymmetry caused not only by multiplicity fluctuations collective component is correlated to the reaction plane The azimuthal asymmetry of particle production has a collective and a random component. Disentangling the two requires a statistical analysis of finite multiplicity fluctuations.

20 (2.4) (2.5) II.4. Cumulant Method If reaction plane is unknown, consider particle correlations e i n (φ 1 φ 2 ) D 1 D 2 = two-particle distribution has an uncorrelated and a correlated part (2.6) Short hand D 1 D 2 d p 1 d p 2 e i n (φ 1 φ 2 ) D 1 D 2 f ( p 1, p 2 ) d p 1 d p 2 f ( p 1, p 2 ) f ( p 1, p 2 ) = f ( p 1 ) f ( p 2 ) + f c ( p 1, p 2 ) (1,2) = (1)(2) + (1,2) c Correlated part ssumption: Event multiplicity N>>1 correlated part is O(1/N)-correction to f ( p 1 ) f ( p 2 ) (2.7) (2.8) e i n ( φ 1 φ 2 ) If v n = v n D 1 D 1 D 2 ( ) v n ( D 2 ) + e i n φ 1 φ 2 ( D) >> 1 N ( ) O( 1 N) corr D 1 D 2,then non-flow corrections are negligible. What, if this is not the case? Non-flow effects

21 (2.9) II.5. 4-th order Cumulants v n >>1 2nd order cumulants allow to characterize v n, if. Consider now 4-th order cumulants: (1,2,3,4) = (1)(2)(3)(4) + (1,2) c (3)(4) + + (1,2) c (3,4) c + (1,3) c (2,4) c + (1,4) c (2,3) c + (1,2,3) c (4) + + (1,2,3,4) c If the system is isotropic, i.e. v n (D)=0, then k-particle correlations are unchanged by rotation for all i, and only labeled terms survive. This defines φ i φ i + φ (2.9) e i n ( φ 1 +φ 2 φ 3 φ 4 ) N e i n ( φ 1 +φ 2 φ 3 φ 4 ) e i n ( φ 1 φ 3 ) e i n ( φ 2 φ 4 ) e i n ( φ 1 φ 4 ) e i n ( φ 2 φ 3 ) (2.10) For small, non-vanishing v n, one finds e i n ( φ 1 +φ 2 φ 3 φ 4 ) = v n 4 + O 1 Improvement: signal can be separated from fluctuating background, if (, v 2 ) 2n N 3 N 2 v N >> 1 N 3 / 4 Borghini, Dinh, Ollitrault, PRC (2001)

22 (2.11) II.6. RHIC Data on Elliptic Flow: v 2 dn ( p T )cos( 2(φ ψ reaction plane )) E dn d 3 p = 1 2π p T dp T dη 1+ 2v 2 [ ] Momentum space: Reaction plane Signal v implies 2-1 asymmetry of particles production w.r.t. reaction plane. Non-flow effect for 2nd order cumulants (2.12) N ~ N ~ O(v 2 ) (2.13) 2nd order cumulants do not characterize solely collectivity. 1 N 3 4 ~ 0.03 << v 2 Non-flow effects should disappear if we go from 2nd to 4th order cumulants.

23 II.7. Pt-integrated elliptic flow: v 2 STR Coll, Phys. Rev. C66 (2002) Elliptic flow signal is stable if reconstructed from higher order cumulants. We have established a strong collective effect, which cannot be mimicked by multiplicity fluctuations in the reaction plane.