Selected Topics in the Theory of Heavy Ion Collisions Lecture 1 Urs chim Wiedemann CERN Physics Department TH Division Skeikampen, 4 January 2012
Heavy Ion Collisions - Experiments lternating Gradient Synchrotron (GS) at Brookhaven BNL - variety of beams, since mid 1980 s GS s NN u+u CERN SPS fixed target experiments SPS - variety of beams, Pb-beams since 1994 s NN Pb+Pb 2 5GeV 17GeV Relativistic Heavy Ion Collider RHIC at BNL - since 2000, p+p, d+u, u+u, Cu-Cu, RHIC s NN u+u 200GeV Large Hadron Collider LHC - since 2000, so far p+p, Pb+Pb, LHC s NN Pb+Pb = 2.75TeV - total cross section: - maximal luminosity: σ Pb+Pb total 8barn = 8 10 24 cm 2 Pb+Pb L max,lhc 10 27 cm 2 s 1 8000 collisions per second!
What is measured when at the LHC? σ Pb+Pb total 8barn = 8 10 24 cm 2 Pb+Pb L max,lhc 10 27 cm 2 s 1 1month 10 6 s 1LHCyr Pb + Pb When? 15 min~10 3 s (ideal) 1 st month, 2010 1 month, 2011 2-3 yrs How much data? L Pb+Pb int 1µb 1 1st L year int 7 8µb 1 L Pb+Pb int 150µb 1 p+pb What? Event multiplicity Low-p T hadron spectra bundant high-p T processes such as jets Rare and leptonic processes Strategy for these lectures: - explain basic theory for data accessible at the LHC (and say where it is incomplete) - explain theory in the order in which data will become accessible - give motivation for measurement by explaining measurement (not before)
last introductory remark 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 - for Quantum Chromodynamics (QCD, i.e. non-abelian QFT) - in the regime of the highest temperatures and densities accessible in the laboratory 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?
I.1. The very first measurement at a Heavy Ion Collider PHOBOS, RHIC, 2000 LICE, PRL 105 (2010) 252301, arxiv:1011.3916 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
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
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)
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
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)
(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
(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 }) [ ] 0 2 12
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 1.07 1/ 3 fm,c = 0.545 fm. C.W. de Jager, H.DeVries, C.DeVries, tom. Nucl. Data Table 14 (1974) 479 inel σ nn inel σ nn 40 (65) mb at s nn =100 (2700) GeV. is sometimes used as fit parameter.
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 )
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)] dominated by geometry only weakly sensitive to details of particle production [e.g. weak dependence on x in (1.22)] insensitive to collective effects Kharzeev, Nardi, PLB 507 (2001) 121 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
I.11. Multiplicity as a Centrality Measure The connection between centrality and event multiplicity can be expressed in terms of (1.25) + N part n>n 0 = n0 dn n0 dbp(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 LICE, 2010
I.12. Centrality Class fixes Impact Parameter The connection between centrality and event multiplicity can be expressed in terms of (1.25) + N part n>n 0 = n0 1200 1000 800 600 400 dn n0 dbp(n,b) [ 1 P 0 (b)] N part (b) dn u u N part db P(n,b) [ 1 P 0 (b)] Centrality class specifies range of impact parameters u u N coll 200 0-5 % 10-30 % b in fm 0 0 2 4 6 8 10 12 14 b B
I.13. 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) E F = ( N part (b) /2) s /2 LICE, PRL 105 (2010) 252301, arxiv:1011.3916
I.14. Cross-Checking Centrality Measurements The interpretation of min. bias multiplicity distributions in terms of centrality measurements can be checked in multiple ways, e.g. 2. Testing Glauber in d+u and in p+u(+ n forward) STR Coll.
I.15. Final remarks on event multiplicity in +B There is no 1st principle QCD calculation of event multiplicity, neither in p+p nor in +B Total charged event multiplicity: models failed to predict RHIC and failed to predict LHC
I.16. 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 - dependence of event multiplicity not understood in pp and LICE Coll., PRL 106, 032301 (2001) arxiv:1012.1657 s
I.17. Final remarks on event multiplicity Multiplicity distribution is not only used as centrality measure but: Multiplicity (or transverse energy) constrains density 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(τ 0 1fm / c) = 3 4GeV / fm 3
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. LICE, 2010 What can we learn by characterizing not only the modulus b, but also the orientation b?
(2.1) (2.2) II.2. Particle production w.r.t. reaction plane Consider single inclusive particle momentum spectrum f ( p) dn E 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]. (2.3) v n e i n φ = dpe i n φ f ( p) D dp f ( p) event D average n-th order flow Problem: Eq. (2.3) cannot be used for data analysis, since the orientation of the reaction plane is not known a priori.
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.
(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 { 2} >> 1 N = v n { 2} v n 2 corr { }+ e i n ( φ 1 φ 2 ) ( ) O 1 N,then non-flow corrections are negligible. Flow via 2 nd order cumulants Non-flow effects What, if this is not the case?
II.5. 4-th order Cumulants v n >>1 2nd order cumulants allow to characterize v n, if. Consider now 4-th order cumulants: (2.9) (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 N (2.9) cn { 4} 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 4 c n { 4} = v n ( ) { 4}+O 1, v 2 2 n N 3 N 2 Borghini, Dinh, Ollitrault, PRC (2001) Improvement: signal can be separated from fluctuating background, if v n { 4} >> 1 N 3/4
(2.11) II.6. LHC and RHIC Data on Elliptic Flow: v 2 dn ( p T )cos( 2(φ ψ reaction plane )) Momentum space: E dn d 3 p = 1 2π p T dp T dη 1+ 2v 2 [ ] Reaction plane Signal v 2 0.2 implies 2-1 asymmetry of particles production w.r.t. reaction plane. Non-flow effect for 2nd order cumulants (2.12) N ~ 100 1 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.
II.7. Establishing collectivity in v 2 pt-integrated v2 stabilizes at 4 th order cumulants STR Coll, Phys. Rev. C66 (2002) 034904 pt-differential v2 from 2 nd and 4 th order cumulants LICE Coll., arxiv:1011.3914 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.