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

12-Jan-06 Nonperturbative Effects from SCET2 Outline Soft-Collinear Effective Theory (SCET) Event Shapes Factorization Nonperturbative Corrections Angularities Bauer, CL, Manohar, Wise, PRD 70, (2004). CL, in preparation.

Soft-Collinear Effective Theory

12-Jan-06 Nonperturbative Effects from SCET4 Light-Cone Coordinates Use light-cone coordinates: Define light-like directions: e.g. Coordinates: where

12-Jan-06 Nonperturbative Effects from SCET5 Degrees of Freedom ModeScaling Collinear Quarks Gluons (Ultra)soft Gluons

12-Jan-06 Nonperturbative Effects from SCET6 QCD  SCET 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 Bauer, Fleming, Luke (2001) Bauer, Fleming, Pirjol, Stewart (2001)

12-Jan-06 Nonperturbative Effects from SCET7 SCET Lagrangian Leading order in : Feynman rules: Bauer, Fleming, Pirjol, Stewart (2001)

12-Jan-06 Nonperturbative Effects from SCET8 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:

12-Jan-06 Nonperturbative Effects from SCET9 Decoupling of Ultrasoft Gluons Field redefinition: Removes collinear-ultrasoft couplings in leading-order SCET I 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)

Event Shapes in SCET

12-Jan-06 Nonperturbative Effects from SCET11 Match QCD current onto SCET Expansion in Typical off-shellness: Collinear Treat in perturbation theory Ultrasoft Nonperturbative physics lives here

12-Jan-06 Nonperturbative Effects from SCET12 Differential Two-Jet Rate Differential rate for Z decay to 2 Jets: Insert and integrate over phase space. Collinear and ultrasoft matrix elements factorize…

12-Jan-06 Nonperturbative Effects from SCET13 Factorization Differential rate becomes: At LO in perturbation theory, first factor is just:

12-Jan-06 Nonperturbative Effects from SCET14 Nonperturbative Effects from Ultrasoft Particles Example: Jet Energy Delta function defining reduces to: Smear over region Then we can expand in powers of

12-Jan-06 Nonperturbative Effects from SCET15 Correction to Jet Energy Distribution Jet Energy distribution becomes where the NP matrix element is defined: This can be expressed as the matrix element of an operator cf. Korchemsky, Sterman (1995)

12-Jan-06 Nonperturbative Effects from SCET16 Thrust Distribution Considerrecalling 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.

12-Jan-06 Nonperturbative Effects from SCET17 Jet Mass Distribution For jet masses, So: where Thrust and jet mass sum receive same nonperturbative corrections!

12-Jan-06 Nonperturbative Effects from SCET18 Other Variables? Jet Broadening: C Parameter: No obvious relation…

12-Jan-06 Nonperturbative Effects from SCET19 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.

12-Jan-06 Nonperturbative Effects from SCET20 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)

12-Jan-06 Nonperturbative Effects from SCET21 Angularities in SCET Apply same SCET formalism to angularities: Leading power correction: CL (in preparation) Berger, Kucs, Sterman (2003)

12-Jan-06 Nonperturbative Effects from SCET22 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)

12-Jan-06 Nonperturbative Effects from SCET23 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.