QCD and Collider Physics III: Jets and Hadronization evolution equations ± how to solve ? gluon density in DLL jet evolution: particle multiplicities fragmentation function heavy quarks hadronisation models color flows in proton ± proton interactions Literature: Ellis, Stirling, Webber: QCD and Collider Physics Dissertori, Knowles, Schmelling: QCD - High Energy Exp and Theory R. Field: Applications of perturbative QCD http://www-h1.desy.de/~jung/qcd_collider_physics_wise_2006 H. Jung, QCD & Collider Physics III: LHC, Lecture 7 WS 06/07 1 From LO to NLO ... NLO for F : O(α ) 2 s NLO for dijets: O(α 2) s NLO for 3-jets: O(α 3) s NOTE: NLO for 3-jets is NOT NNLO for dijets H. Jung, QCD & Collider Physics III: LHC, Lecture 7 WS 06/07 2 Hadronization - Fragmentation Hadronization From Wikipedia, the free encyclopedia In particle physics, hadronization is the process of the formation of hadrons out of quarks and gluons. This occurs after high-energy collisions in a particle collider in which free quarks or gluons are created. Due to colour confinement, these cannot exist individually. In the independent model they combine with quarks and antiquarks spontaneously created from the vacuum to form hadrons. The details of this process are not yet fully understood. Another model is the Lund string model. The tight cone of particles created by the hadronization of a single quark is called a jet. Jets are observed in particle detectors, rather than quarks, whose existence must be inferred. Fragmentation .... (but we use it differently !!!!) From Wikipedia, the free encyclopedia Fragmentation is a term that occurs in several fields and describes a process of something breaking or being divided into pieces (fragments). See also divergence. 1Biology,2Computing,3Networking,4Economics,5Music,6Literature,7Urban sociology, 8Weaponry,9Mass spectrometry,10Waste management BUT where is HEP ???? H. Jung, QCD & Collider Physics III: LHC, Lecture 7 WS 06/07 3 Splitting functions in lowest order 4 1 + z2 P = qq 3 1 z µ ¡ ¶ 2 4 1 + (1 z) similarity to photon Pgq = ¡ 3 z radiation from electron µ ¶ 1 P = z2 + (1 + z)2 qg 2 ¡ ¢ 1 z z P = 6 ¡ + + z(1 z) gg z 1 z ¡ µ ¡ ¶ H. Jung, QCD & Collider Physics III: LHC, Lecture 7 WS 06/07 4 Solving integral equations b Integral equation of Fredholm type: Á(x) = f(x) + ¸ K(x; y)Á(y)dy solve it by iteration (Neumann series): Za Á0(x) = f(x) b Á1(x) = f(x) + ¸ K(x; y)f(y)dy a Z b b b 2 Á2(x) = f(x) + ¸ K(x; y1)f(y1)dy1 + ¸ K(x; y1)K(y1; y2)f(y2)dy2dy1 a a a n Z Z Z i Án(x) = ¸ ui(x) i=0 X u0(x) = f(x) b u1(x) = K(x; y)f(y)dy a Z b b un(x) = K(x;y1)K(y1; y2) K(yn 1; yn)f(yn)dy2 dyn ¢ ¢ ¢ ¡ ¢ ¢ ¢ Za Za n i with the solution: Á(x) = lim qn(x) = lim ¸ ui(x) n n !1 !1 i=0 X H. Jung, QCD & Collider Physics III: LHC, Lecture 7 WS 06/07 5 Approximation at small x ? For x 0 only gluon splitting function matters: ! 1 z z 1 1 P = 6 ¡ + + z(1 z) = 6 2 + z(1 z) + gg z 1 z ¡ z ¡ ¡ 1 z µ ¡ ¶ µ ¡ ¶ 1 P 6 for z 0 gg » z ! evolution equation is then: dg(x; ¹2) ® 1 d» x = s g(»; ¹2)P d log ¹2 2¼ » gg » Zx µ ¶ t 1 3®s d» 2 xg(x; t) = xg(x; t ) + d log t0 »g(»; t0) with t = ¹ 0 ¼ » Zt0 Zx H. Jung, QCD & Collider Physics III: LHC, Lecture 7 WS 06/07 6 Estimates at small x: DLL A.D. Martin,in Lectures at XXI International Meeting on Fundamental Physics, Miraflores de la Sierra, Madrid, 1993 t 1 3®s d» 2 xg(x; t) = xg(x; t ) + d log t0 »g(»; t0) with t = ¹ 0 ¼ » Zt0 Zx use constant starting distribution at small t: u0(x) = C 3®s t 1 u1(x; t) = log log C ¼ t0 x 3® 1 t 1 1 2 u (x; t) = s log log C 2 ¼ 2 t 2 x µ 0 ¶ . 1 1 3® t 1 n u (x; t) = s log log C n n! n! ¼ t x µ 0 ¶ 1 1 3® t 1 n xg(x; t) = s log log C n! n! ¼ t x n 0 X µ ¶ 3®s t 1 xg(x; t) C exp 2 log log double leading log » ¼ t x µ r 0 ¶ approximation (DLL) H. Jung, QCD & Collider Physics III: LHC, Lecture 7 WS 06/07 7 Results from DLL approximation DLL arise from taking small x limit of 3® t 1 xg(x;t) C exp 2 s log log splitting fct: » ¼ t x µ r 0 ¶ l o g 1 = x from small x limit of splitting fct l o g t = t 0 from t integration strong ordering in x from small x limit strong ordering in t from small t limit of ME... DLL gives rapid increase of gluon density from a flat starting distribution consequences: ● rise continues forever ??? ● what happens when too high gluon density ? H. Jung, QCD & Collider Physics III: LHC, Lecture 7 WS 06/07 8 Quark and Gluon jet differences DELPHI Collaboration Phys.Lett.B449:383-400,1999. hep-ex/9903073 calculate average multiplicity in gluon and quark jets: consider only gluon emissions: R. Field Appl. of pQCD, p79 < Ng > CA 3 < N > » C » 2 q r F comsider ANY type of emission: Elllis,Stirling,Webber QCD & Collider physics, p219 < Ng > CA 9 < Nq > » CF » 4 including higher order corrections NNLO: Dissertori, Knowles, Schmelling QCD, p376 < N > C n T n T C g = A 1 1 + 2 f F 4 f T F < N > C ¡ C ¡ C2 q F · µ A A ¶ ® C 25 3n T n T C ® C s A + f F 2 f F F s A £ 18¼ 8 ¡ 2C ¡ C2 18¼ Ãr µ A A ¶ !# 1:7 BUT still a »difference to the measurement H. Jung, QCD & Collider Physics III: LHC, Lecture 7 WS 06/07 9 Summary PDF evolution Jet evolution: resummation to all orders evolution is suitable for itearative procedure in some cases analytical calcs can be performed H. Jung, QCD & Collider Physics III: LHC, Lecture 7 WS 06/07 10.