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Factorization and event generators

Factorization and event generators. John Collins (PennState) and Xiaomin Zu. Concentrate on Initial state shower algorithm. A nonleading order EG for DIS (non-gauge theories). Questions: How well do EGs implement QCD? Systematic improvements?. Generate hard scattering.

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Factorization and event generators

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  1. Factorization and event generators John Collins (PennState) and Xiaomin Zu Concentrate on Initial state shower algorithm. A nonleading order EG for DIS (non-gauge theories) • Questions: • How well do EGs implement QCD? • Systematic improvements?

  2. Generate hard scattering. Branch an external parton. Repeat branching recursively. Hadronize. Structure of a MCEG (DIS) O(1) thinking per step;O(N) cost;“natural” recursive structure. Looks like a Feynman diagram, but not the same. Justification based on std. factorization and DGLAP.

  3. q k+q k P Differences between EGs and Inclusive Xsection • Event Generators • Details of ‘final’ state. • Exact parton kinematics required. • Conditional probabilities given kμ, differential in final states. • Normal functions only, no +distributions. • Inclusive Xsection • Sum over ‘final’ state. • Approx. parton kinematics used. • Combined use of f(x) and DGLAP, f(x) — integrated over k- and kT. • Generalized functions OK. Photon+hadron CM frame Effect of Approx. kinematics

  4. MCEG: Reformulate from beginning • Aim: • Factorization of the fully-differential Xsections. • Not just inclusive ones. • Exact treatment of parton kinematics. • No reassignment after event generation. • Precise definition of conditional probabilities. • Can be ‘differential’ in final states. • Maintain all the good stuff of existing EGs. • O(1)-thinking per step. • O(N)-cost for event generation with N final state particles. • Recursive structure. • recursive factorization.

  5. W(f) is an arbitrary smooth function of the final state. • Any exclusive components can be obtained from σ[w] by • functional differentiation with respect to W. • j(y) is the current under consideration ~ φ2. • K is a standard leptonic factor. MC estimation of the Xsection L-luminosity MCEG: Fully-differential Xsection Ingnore all QCD complications

  6. Need of un-integrated parton correlation function See also the work by Watt, Martin and Ryskin. Can NOT sum over final state. Exact parton kinematics integration over ‘k’ NOT allowed. un-integrated pCf Φ(kμ, P) which depends on all components of k. New kind of factorizations in terms of pCf’s. • Exact parton kinematics are generated according to the probabilities • given by pCf (or target) and FS (or vacuum) jet correlation factors. • Both FS jet factor and pCf can be differential in their own final state.

  7. k k H H Φ(k, P) = ~ P Factorization of pCf and showering Kinematically just like DIS • Similar to σ[W], we can define Φ[W]. • Factorization of σ[W] applies to Φ[W] with minor modification — recursive factorization. • Differential in final states → replaces parton shower. • DGLAP evolutions are implicit. • FS factors — already worked out by J.C. Collins.

  8. How about QCD? Gluon exchange: • Gauge invariance issue. • Correct proof of factorization gives: • Wilson line(s) in operators. • Direction of Wilson lines (universality, Collins-Metz). • Collins-Soper eq. (or similar) for direction-dep. • ‘New’ operators corresponding to non-cancellation of soft gluon effects. • Need to do this precisely, completely. • Separation of final-state kinematics?

  9. Conclusions “…The mechanic, who wishes to do his work well, must first sharpen his tools …” —Chapter15, “The Analects” attributed to Confucius, translated by James Legge. • Need to formulate precise logic: QCD→MCEG •  Exact parton kinematics. •  Use of unintegrated correlation functions (vacuum, target) • Several factorization theorems • Inclusive: value of σ • value of pCf • Differential in f.s.: for both σand pCf • Showering and event generation. • Basis for MCEGs • Complete formulation in non-gauge theory models. • Techniques exist for extension to QCD.

  10. References • “Initial state parton shower beyond LO”, J.C. Collins and X. Zu, JHEP 0503:059, 2005,hep-ph/0411332. • “Monte-Carlo event generators at NLO”, J.C. Collins, Phys.Rev.D65, 094016, hep-ph/0110113. • “Universality of soft and collinear factors in hard-scattering factorization”, J.C. Collins and A. Metz, Phys.Rev.Lett.93:252001, hep-ph/0408249. • “Un-integrated parton distributions and inclusive jet production at HERA”, G. Watt, A.D. Martin and M.G. Ryskin, Eur.Phys.J.C31:73-89, hep-ph/0306169. • “Back-to-back jets in QCD”, J.C. Collins and D.E. Soper, Nucl.Phys.B193:381,1981, Erratum-ibid.B213:545,1983. • “Sudakov form-factors”, J.C. Collins, Adv.Ser.Direct.High Energy Phys.5:573-614,1989, hep-ph/0312336.

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