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Nonperturbative Effects from Soft-Collinear Effective Theory

Nonperturbative Effects from Soft-Collinear Effective Theory. Christopher Lee Institute for Nuclear Theory, University of Washington 12 January 2006. Outline. Soft-Collinear Effective Theory (SCET) Event Shapes Factorization Nonperturbative Corrections Angularities.

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Nonperturbative Effects from Soft-Collinear Effective Theory

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  1. Nonperturbative Effects from Soft-Collinear Effective Theory Christopher Lee Institute for Nuclear Theory,University of Washington 12 January 2006

  2. Outline • Soft-Collinear Effective Theory (SCET) • Event Shapes • Factorization • Nonperturbative Corrections • Angularities Bauer, CL, Manohar, Wise, PRD 70, 034014 (2004). CL, in preparation. Nonperturbative Effects from SCET

  3. Soft-Collinear Effective Theory

  4. Light-Cone Coordinates • Use light-cone coordinates: • Define light-like directions:e.g. • Coordinates:where Nonperturbative Effects from SCET

  5. Degrees of Freedom Nonperturbative Effects from SCET

  6. QCD  SCET • Bauer, Fleming, Luke (2001) • Bauer, Fleming, Pirjol, Stewart (2001) • Full QCD: • Divide momenta into “label” and “residual” components: • Factor out label momenta from collinear fields: • Project out large & small components of quark spinors: cf. HQET Nonperturbative Effects from SCET

  7. SCETLagrangian Bauer, Fleming, Pirjol, Stewart (2001) • Leading order in : • Feynman rules: Nonperturbative Effects from SCET

  8. Wilson Lines in SCET • Collinear Wilson Lines: Arise from integrating out off-shell propagators between emitted collinear gluons: • Ultrasoft Wilson Lines:Arise from summing up emission of ultrasoft gluons: Nonperturbative Effects from SCET

  9. Decoupling of Ultrasoft Gluons • Field redefinition: • Removes collinear-ultrasoft couplings in leading-order SCETI Lagrangian, for example:Now, satisfies:so the above term in reduces to: • At leading order in , becomes free of couplings to ultrasoft gluons, but Wilson lines must appear in operators containing collinear fields. Bauer, Pirjol, Stewart (2001) Nonperturbative Effects from SCET

  10. Event Shapes in SCET

  11. Match QCD current onto SCET • Expansion in • Typical off-shellness: • CollinearTreat in perturbation theory • UltrasoftNonperturbative physics lives here Nonperturbative Effects from SCET

  12. Differential Two-Jet Rate • Differential rate for Z decay to 2 Jets: • Insert and integrate over phase space. • Collinear and ultrasoft matrix elements factorize… Nonperturbative Effects from SCET

  13. Factorization • Differential rate becomes: • At LO in perturbation theory, first factor is just: Nonperturbative Effects from SCET

  14. Nonperturbative Effects from Ultrasoft Particles • Example: Jet EnergyDelta function defining reduces to: • Smear over region Then we can expand in powers of Nonperturbative Effects from SCET

  15. Correction to Jet Energy Distribution • Jet Energy distribution becomeswhere the NP matrix element is defined: • This can be expressed as the matrix element of an operator cf. Korchemsky, Sterman (1995) Nonperturbative Effects from SCET

  16. Thrust Distribution • Consider recalling • This time insert into the differential rate: • The thrust distribution can be written:where (a) and (b) denote the hemispheres containing the quark and antiquark jets. Nonperturbative Effects from SCET

  17. Jet Mass Distribution • For jet masses, • So: • where • Thrust and jet mass sum receive same nonperturbative corrections! Nonperturbative Effects from SCET

  18. Other Variables? • Jet Broadening: • C Parameter: • No obvious relation… Nonperturbative Effects from SCET

  19. Universality of NP corrections • Dokshitzer-Webber (1995, 1997):NP corrections shift observable f in the perturbatively-calculated distributions: where for thrust and jet mass, for C • At the level of NP matrix elements, we reproduce the relation between T and , but not with C. Nonperturbative Effects from SCET

  20. Single Gluon Emission • Relation between Thrust and C parameter can be obtained in the approximation of single gluon emission at fixed transverse momentum: cf. Catani, Webber (1998) Nonperturbative Effects from SCET

  21. Angularities in SCET CL (in preparation) • Apply same SCET formalism to angularities: • Leading power correction: Berger, Kucs, Sterman (2003) Nonperturbative Effects from SCET

  22. Scaling Rule for Angularities • NP power corrections to angularities shown to obey a universal scaling rule: • Obtain from SCET with single gluon emission approximation: Berger, Sterman (2003)Berger, Magnea (2004) Nonperturbative Effects from SCET

  23. Summary: Jets in SCET • Decoupling of ultrasoft from collinear modes in leading-order SCET Lagrangian facilitates proof of factorization theorem. • Weighted matrix elements of ultrasoft Wilson lines give power corrections to event shape distributions. • Universality of power corrections limited in SCET alone. • Relations between T and C, and between angularities recovered in single gluon approximation. Nonperturbative Effects from SCET

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