Introduction to the physics of hard probes in hadron collisions: lecture II. Michelangelo Mangano TH Division, CERN

Transcription

1 Introduction to the physics of hard probes in hadron collisions: lecture II Michelangelo Mangano TH Division, CERN

2 Jet production gg gg gg qq _ qg qg _ qq gg qq qq qq qq

3 Inclusive production of jets is the largest component of high-q phenomena in hadronic collisions QCD predictions are known up to NLO accuracy Intrinsic theoretical uncertainty at NLO is approximately 10% Uncertainty due to knowledge of parton densities varies from 5-10% at low transverse momentum, p T to 100% at very high p T, corresponding to high-x gluons Jet are used as probes of the quark structure possible substructure implies departures from point-like behaviour of cross-section, or as probes of new particles peaks in the invariant mass distribution of jet pairs

4 Phase space and cross-section for LO d[ps] = d3 p 1 (2p) 2 2p 0 1 a b jet production d 3 p 2 (2p) 2 2p 0 (2p) 4 d 4 (P in P out ) dx 1 dx 2 2 d(e in E out )d(pin z Pz out)dx 1 dx 2 = d p z = dy dh p 0 d[ps] = 1 4pS p T d p T dh 1 dh 2 1 2E 2 beam d 3 s d p T dh 1 dh 2 = p T 4pS Â i, j f i (x 1 ) f j (x 2 ) 1 2ŝ Â kl M(i j kl) 2 The measurement of pt and rapidities for a dijet final state uniquely determines the parton momenta x 1 and x 2. Knowledge of the partonic cross-section allows therefore the determination of partonic densities f x

5 Prove as an exercise that where Some more kinematics y = h 1 h 2 2 x 1,2 = p T E beam cosh y e ±y b, y b = h 1 + h 2 2 We can therefore reach large values of x either by selecting large invariant mass events: p T cosh y t 1 E beam or by selecting low-mass events, but with large boosts y b large in either positive of negative directions. In this case, we probe large-x with events where possible new physics is absent, thus setting consistent constraints on the behaviour of the cross-section in the high-mass region, which could hide new phenomena.

6 Example, at the Tevatron DO jet data, and PDF fits 0<η<0.5 2<η< <η<2 1<η< <η<1 CDF data, using fits from high-η region 0<η<0.9

7 Small-angle jet production, a useful approximation for the determination of the matrix elements and of the cross-section At small scattering angle, t = (p 1 p 3 ) 2 (1 cosq) 0 and the 1/t 2 propagators associated with t-channel gluon exchange dominate the matrix elements for all processes. In this limit it is easy to evaluate the matrix elements. For example: p p k q q (l a ) i j (l a ) kl (2p µ ) 1 t (2q µ) = 2s t (la ) i j (l a ) kl where we used the fact that, for k=p-p <<p small angle scattering, ū(p )g µ u(p) ū(p)g µ u(p) = 2p µ Using our colour algebra results, we then get: Â col,spin M 2 = 1 N 2 c N 2 c 1 4 Noting that the result must be symmetric under s u exchange, and setting Nc=3, we finally obtain: s 2 + u 2 Â col,spin M 2 = 4 9 which turns out to be the exact result! t 2 4s 2 t 2

8 Quark-gluon and gluon-gluon scattering We repeat the exercise in the more complex case of qg scattering, assuming the dominance of the t-channel gluon-exchange diagram: i,q a,p j,q c,k f abc l c 1 i j 2p µ t 2q µ = 2 s t f abc l c i j b,p Using the colour algebra results, and enforcing the s u symmetry, we get: which differs by only 20% from the exact result even in the large-angle region, at 90 o In a similar way we obtain for gg scattering using the t u symmetry : compared to the exact result with a 20% difference at 90 o  col,spin  col,spin  col,spin  col,spin M 2 = s2 + u 2 t 2 M 2 = s2 + u 2 t 2 4 s 2 + u 2 9 us M(gg gg) 2 = 9 ( ) s 2 2 t 2 + s2 u 2 ( 3 ut s 2 us t 2 st ) u 2 M(gg gg) 2 = 9 2

9 Note that in the leading 1/t approximation we get the following result: ŝ gg : ŝ qg : ŝ qq = 9 4 : 1 : 4 9 and therefore ds jet = dx 1 dx 2 Â i j f i (x 1 ) f j (x 2 )dŝ i j = dx 1 dx 2 Â i j where we defined the `effective parton density F x : F(x 1 )F(x 2 )dŝ gg F(x) = g(x) Â i [q i (x) + q i (x)] As a result jet data cannot be used to extract separately gluon and quark densities. On the other hand, assuming an accurate knowledge of the quark densities say from HERA, jet data can help in the determination of the gluon density Exercise: prove that the 1/t 2 behaviour of the cross-section implies Rutherford s scattering law. Event rates in PbPb:

10 Final-state evolution Typical final state resulting from a jet includes many particles How can a calculation of a 2->2 cross-section have anything to do with reality? What are the quantities which can be properly described by such calculation? Hadronization takes place at time scales much larger than the hard 2->2 scattering 1/ΛQCD>>1/pT. What happens between the hard process and hadronization can be described by perturbative QCD. However hadronization itself is a phenomenon of catastrophic intensity, which could totally disrupt the structure of the final state, by forcing any pair of colour-connected partons which are separated by more than 1/ ΛQCD to bind together. Fortunately, QCD is kind enough to ensure that the perturbative evolution prepares a partonic state which will be left almost unperturbed by hadronization. We ll now discuss how.

11 Soft gluon emission p k = p0 k0 1-cosq singularities for collinear cosqfi1) or soft k0fi0) emission Collinear emission does not alter the global structure of the final state, since its preserves its pencil-like-ness. Soft emission at large angle, however, could spoil the structure, and leads to strong interferences between emissions from different legs. So soft emission needs to be studied in more detail. In the soft k0fi0) limit the amplitude simpifies and factorizes as follows: ( p e A so ft = gl a i j p k pe ) A Born p k Factorization: it is the expression of the independence of long-wavelength soft emission on the nature of the hard short-distance process.

12 Similar, but more structured, result in the case of more complex colour configurations: [ Qe A so ft = g (l a l b ) i j Qk pe ] pk The two terms correspond to the two possible ways colour can flow in these diagrams: [ pe + g (l b l a ) i j pk Qe ] Qk The interference between the two colour structures is suppressed by 1/N c 2  a,b,i, j  a,b,i, j (l a l b ) i j 2 = Âtr ( l a l b l b l a) = N2 1 C F = O(N 3 ) a,b 2 (l a l b ) i j [(l b l a ) i j ] =  a,b tr(l a l b l a l b ) = N2 1 2 (C F C A 2 ) }{{} 1 2N = O(N) As a result, the emission of a soft gluon can be described, to the leading order in 1/N c 2, as the incoherent sum of the emission from the two colour currents

13 Angular ordering in soft-gluon emission You can easily prove that: 1 cosq i j (1 cosq ik )(1 cosq jk ) = 1 2 ds g = Â A so ft 2 [ = ds 0 a s C F p d 3 k (2p) 3 2k 0 Â A 0 2 2pµ p n (pk)( pk) g2 Âe µ e d 3 k n (2p) 3 2k 0 dk 0 k 0 df 2p 1 cosq i j (1 cosq ik )(1 cosq jk ) d cosq ] cosq jk cosq i j (1 cosq ik )(1 cosq jk ) cosq ik 2 [i j] W (i) +W ( j) where: W (i) f inite i f k j (cosq jk 1) W ( j) f inite i f k i (cosq ik 1) The probabilistic interpretation of W i and W j is a priori spoiled by their nonpositivity. However, you can prove that after azimuthal averaging: 2 j j 1 2 = + j 2 2 Q(j-j 1 ) Q(j-j 2 ) df 2p W (i) = df 2p W ( j) = 1 1 cosq ik if q ik < q i j, 0 otherwise 1 1 cosq jk if q jk < q i j, 0 otherwise Further branchings will obey angular ordering relative to the new angles. As a result emission angles get smaller and smaller, squeezing the jet

14 Total colour charge of the system is equal to the quark colour charge. Treating the system as the incoherent superposition of N gluons would lead to artificial growth of gluon multiplicity. Angular ordering enforces coherence, and leads to the proper evolution with energy of particle multiplicities.

15 The structure of the perturbative evolution therefore leads naturally to the clustering in phase-space of colour-singlet parton pairs preconfinement. Long-range correlations are strongly suppressed. Hadronization will only act locally, on low-mass colour-singlet clusters.

16 Hadronization At the end of the perturbative evolution, the final state consists of quarks and gluons, forming, as a result of angular-ordering, lowmass clusters of colour-singlet pairs: p p p p p p N p p p p p p p p p N p p p p p Thanks to the cluster pre-confinement, hadronization is local and independent of the nature of the primary hard process, as well as of the details of how hadronization acts on different clusters. Among other things, one therefore expects: N(pions) = C N(gluons), C=constant 2

17 Heavy quark production for pedagogical introduction, see: hep-ph/ Singularity structure of t-channel propagator: (p 1 Q) 2 m 2 = 2p 1 Q = ŝ (1 bcosq) 2 2p 1 Q ŝ 2 (1 b) = ŝ 2(1 + b) (1 b2 ) = 2m2 1 + b m2 Total HQ cross-sections are finite, and calculable in pqcd

18 Matrix elements: Phase-space: Total cross-section: Differential distributions: Very strong rapidity correlation: y<1

19 I-VI: different input parameters Total cross-sections

20

21 Evolution of a heavy quark mq 0 A so ft = gl a i j ( p e p k pe ) p k A Born same as for mq=0. However now: pk = p 0 k βcosθ, where β is the quark velocity, β<1. Therefore radiation in the very forward region collinear radiation is suppressed. A massive quark therefore looses less energy during the evolution, and its quenching profile will be different than the one of a light quark or gluon.

22

23 Some interesting questions * Plot dn/dφ vs d/d. Possible? Trigger jet, J1 d D Quenched jet, J2 * Particle pt in J2 is degraded; this means that more particles must share the same amount of momentum -> increased multiplicity. Can it be measured? * What do we learn about the QGP form the study of quenching? What is the quantitative relation between the parameters of the QGP Eq of State and the fragmentation properties of J2? * What are the PDG-like parameters that we can extract from these measurements?