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

Transcription

1 Introduction to the physics of hard probes in hadron collisions: lecture I Michelangelo Mangano TH Division, CERN michelangelo.mangano@cern.ch

2 Contents The structure of the proton the initial-state : parton densities their evolution nuclear modifications Some benchmark SM processes the hard probes Drell-Yan Jets Heavy quark production The structure of a hard proton-proton collision the finalstate : jet evolution hadronization

3 References: Report from the Workshop on Hard Probes in Heavy Ion Collisions at the LHC PDF s, shadowing and pa collisions: hep-ph/ Heavy quarks: hep-ph/ Photon physics: hep-ph/ Jet physics: hep-ph/ Previous lecture notes on QCD Introduction to QCD: brazil.ps.gz Introduction to heavy quark production: hep-ph/

4 QCD Feynman rules p id i j p/ m + ie a, µ p b, ν ab igµn d p 2 + ie (Feynman gauge) i j a, µ igl a i j g µ a 2, µ 2, p 2 g f a 1a 2 a 3 [g µ 1µ 2 (p 1 p 2 ) µ 3 + g µ 2µ 3 (p 2 p 3 ) µ 1 + g µ 3µ 1 (p 3 p 1 ) µ 2 ] a 1, µ 1, p 1 a 3, µ 3, p 3 a 2, µ 2 a 3, µ 3 ig 2 [ f a 1a 2 X f a 3a 4 X (g µ 1µ 3 g µ 2µ 4 g µ 1µ 4 g µ 2µ 3 ) + (2 3) + (2 4) ] a 1, µ 1 a 4, µ 4

5 Some results for the SU 3 colour algebra l a i j : SU(N c ) matrix in the fundamental representation, N c = 3, a = 1,...,Nc 2 1, i = 1,...,N c [l a,l b ] = i f abc l c ( trl a = 0) tr(l a l b ) def = T F d ab, T F = 1/2 by convention a  a  a.b  a (l a l a def ) i j = C F d i j Exercise = N2 c 1 d i j N c f abc f abd def = C A d cd Exercise = N c d cd l a i j l a kl Exercise = 1 2 (d ikd jl 1 N d i jd kl ) c i i l a b a j k j d

6 Exercises Verify the properties of the SU N algebra given in the previous pages Prove that the sums of the following sets of diagrams are gauge invariant, namely the amplitude remains invariant if we replace the polarization vector of any gluon, ε µ, with ε µ + p µ, p µ being the gluon momentum:

7 Running of the coupling constant a s de f = g2 s 4p da s d log(q 2 ) = b(a s) At 1-loop: b = b 0 a 2 s, b 0 = 33 2n f 12p 1 and a s (Q) = b 0 log(q 2 /L 2 ) At 2-loops: b = b 0 a 2 s b 1 a 3 s, and a s (Q) = [ 1 b 0 log(q 2 /L 2 1 b 1 ) b 2 0 b 1 = n f 24p 2 loglog(q 2 /L 2 ) logq 2 /L n ] Current World Average Bethke 2002 : a s (M Z ) = ± 0.003

8 Factorization Theorem ds dx = Â j,k Ú X ˆ f j (x 1,Q i ) f k (x 2,Q i ) d s ˆ jk (Q i,q f ) dx ˆ F( ˆ X Æ X;Q i,q f ) f x,q i s ˆ X ˆ F X f j (x,q) sum over all initial state histories leading, at the scale Q, to: r p j = x r P proton F( ˆ X Æ X;Q i,q f ) transition from partonic final state to the hadronic observable hadronization, fragm. function, jet definition, etc Sum over all histories with X in them

9 Universality of parton densities and factorization, a naive proof Exchange of hard gluons among quarks inside the proton is suppressed by powers of m p /Q 2 q q q>q Q q d 4 q q 6 1 Q 2 Typical time-scale of interactions binding the proton is therefore of O 1/m p in a frame in which the τ 1/m p proton has energy E, τ=γ/m p = E/m p 2 If a hard probe Q>>m p hits the proton, on a time scale =1/Q, there is no time for quarks to negotiate a coherent response

10 As a result, to study inclusive processes at large Q it is sufficient to consider the interactions between the external probe and a single parton: 1 calculable in perturbative QCD pqcd 2 do not affect f x : x before = x after q>q q Q q<q 1 x before x after affect f x! 2 for q 1 GeV not calculable in pqcd This gluon cannot be reabsorbed because the quark is gone However, since τ q 1GeV >>1/Q, the emission of low-virtuality gluons will take place long before the hard collision, and therefore cannot depend on the detailed nature of the hard probe. While it is not calculable in pqcd, f q<<q can be measured using a reference probe, and used elsewhere Universality of f(x)

11 Q dependence of parton densities Q>µ q<µ x in x= y x in x=x in µ The larger is Q, the more gluons will not have time to be reabsorbed PDF s depend on Q! f (x,q) = f (x,µ) + x dx in f (x in,µ) Q µ dq 2 dyp(y,q 2 )d(x yx in ) 0

12 f (x,q) = f (x,µ) + x dx in f (x in,µ) Q µ dq 2 0 dyp(y,q 2 )d(x yx in ) f x,q should be independent of the intermediate scale µ considered: d f (x,q) dµ 2 = 0 d f (x,µ) dµ 2 = x dy y f (y,µ)p(x/y,µ2 ) One can prove that: P(x,Q 2 ) = a s 2p 1 Q 2 P(x) calculable in pqcd and therefore Altarelli-Parisi equation : d f (x,µ) d logµ 2 = a s 2p x dy y f (y,µ)p(x/y)

13 More in general, one should consider additional processes which lead to the evolution of partons at high Q t=logq 2 : [g(x)] + : 0 dx f (x)g(x) + 0 [ f (x) f (1)]g(x)dx dq(x, Q) dt = a s 2p x dy y [q(y,q)p qq ( xy ) + g(y,q)p qg( xy ) ] ( ) 1 + x 2 P qq (x) = C F 1 x + P qg (x) = 1 [ x 2 + (1 x) 2] 2 dg(x, Q) dt = a s 2p x dy y [ g(y,q)p gg ( x y ) + Â q, q q(y,q)p gq ( x y ) ] ( ) 1 + (1 x) 2 P gq (x) = C F x [ x P gg (x) = 2N c + 1 x (1 x) + x ] ( ) 11Nc 2n f + x(1 x) + d(1 x) 6

14 Examples of PDFs and their evolution Valence up Sea up Note: sea 10% glue Gluon All, at Q=1TeV Note: charm up at high Q

15 Example: charm in the proton Assuming a typical behaviour of the gluon density: we get: dc(x, Q) dt = a s 2p and therefore: x dc(x, Q) dt = a s 2p g(x,q) A/x dy y g(x/y,q)p qg(y) = a s 2p x dy y g(y,q)p qg( x y ) x dy A x 1 2 [y2 + (1 y) 2 ] = a s 6p A x c(x,q) a s 6p log(q2 ) g(x,q) m 2 c Corrections to this simple formula will arise due to the Q dependence of g x and of αs

16 PDF uncertainties up at Q=3.16 GeV Green bands represent the convolution of theoretical and experimental systematics in the determination of PDFs gluon at Q=3.16 GeV Proton PDFs known to 10-20% for 10-3 <x<0.3, with uncertainties getting smaller at larger Q

17 Nuclear modifications Interactions among various nucleons may change the parton densities shadowing, Cronin effect, etc The evolution at high-q, however, should be decoupled from the nuclear environment. Nuclear PDFs for hard processes can therefore be obtained from AP evolution of PDFs fit at low Q. Usually parameterized by R A x,q =f A x,q /f p x,q R x,q 2 =5 GeV HKM Armesto EKS Hijng Large differences between different parameterizations, approaches. Only data will provide a solid basis for extrapolation from pp to PbPb at the LHC

18 Recent NLO analysis of nuclear PDF s: DeFlorian, Sassot, hepph/

19 Examples of impact of shadowing on some hard-probe observables at RHIC and LHC

20 all y central production fwd production Examples of x ranges probed by charm production at the LHC

21 Drell-Yan processes: q q _ W ln Z l + l Very clean probe of the initial state: no interaction with the plasma! Very well understood theoretically: σ W,Z known up to NNLO 2- loops Excellent experimental monitor of energy-scale for jets, when produced at large Et: gluon jet quark jet # events /month detected by CMS: barrel barrel+ endcap

22 s(pp W) =  q,q where:  spin,col LO Cross-section calculation dx 1 dx 2 f q (x 1,Q) f q (x 2,Q) 1 2ŝ M(q q W) 2 = 1 3 d3 p W 1 4 8g2 W V qq 2 ŝ = 2 G F mw d[ps]  spin,col M(q q W) 2 V qq 2 ŝ d[ps] = (2p) 3 pw 0 (2p) 4 d 4 (P in p W ) = 2pd 4 p W d(pw 2 mw)d 2 4 (P in p W ) = 2pd(ŝ mw) 2 leading to exercise! : where: s(pp W) =  i j pa u d m 2 W pa i j m 2 W t = 6.5nb and t = m2 W S t dx x f i(x,q) f j ( t x,q) pa i j  i j mw 2 tl i j (t)

23 Some useful relations and definitions Rapidity: y = 1 2 log E W + p z W E W p z W Pseudorapidity: where: h = log(tan q 2 ) and tanq = p T p T = p z p 2 x + p 2 y Exercise: prove that for a massless particle rapidity=pseudorapidity: Exercise: using t = ŝ S = x 1x 2 and { EW = (x 1 + x 2 )E beam pw z = (x y = 1 1 x 2 )E beam 2 log x 1 x 2 x 1,2 = te ±y dy = dx 1 x 1 prove the following relations: dx 1 dx 2 = dydt dtd(ŝ m 2 W) = 1 S

24 Study the function τl τ) Assume, for example, that f (x) 1 Then: L(t) = t x, 0 < d < 1 1+d dx 1 x x (x 1+d t )1+d = 1 t log(1 1+d t ) ( ) d ( ) S S log and: s W = s 0 W m 2 W m W Therefore the W cross-section grows at least logarithmically with the hadronic CM energy. This is a typical behavior of cross-sections for production of fixedmass objects in hadronic collisions, contrary to the case of e+e- collisions, where cross-sections tend to decrease with CM energy.

25 Photon plus jet production qg initial state: _ qq initial state: As in the case of Z+jet, provides a good calibration for the absolute experimental determination of the energy of the recoil jet. Rates are larger than for Z s: _ g x >>q x, therefore the first process dominates by at least a factor 10 throughout the phase-space. Potentially a good observable to constrain g x! Affected however by large higher-order, bremstrahlung-like corrections:, therefore up-type quarks are enhanced. In particular, the fraction of charm contribution is large, and a good fraction of recoling jets is charm-like.