Nonperturbative Effects from Soft-Collinear Effective Theory

Christopher LeeInstitute for Nuclear Theory,

University of Washington12 January 2006

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Outline

Soft-Collinear Effective Theory (SCET)

Event Shapes Factorization Nonperturbative Corrections

Angularities

Bauer, CL, Manohar, Wise, PRD 70, 034014 (2004).

CL, in preparation.

Soft-Collinear Effective Theory

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Light-Cone Coordinates

Use light-cone coordinates: Define light-like directions:

e.g.

Coordinates:

where

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Degrees of Freedom

Mode Scaling

Collinear Quarks Gluons

(Ultra)soft Gluons

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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)

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SCET Lagrangian

Leading order in :

Feynman rules:

Bauer, Fleming, Pirjol, Stewart (2001)

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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:

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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)

Event Shapes in SCET

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Match QCD current onto SCET

Expansion in

Typical off-shellness: Collinear

Treat in perturbation theory

Ultrasoft

Nonperturbative physics lives here

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Differential Two-Jet Rate

Differential rate for Z decay to 2 Jets:

Insert and integrate over phase space.

Collinear and ultrasoft matrix elements factorize…

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Factorization

Differential rate becomes:

At LO in perturbation theory, first factor is just:

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Nonperturbative Effects from Ultrasoft Particles

Example: Jet Energy

Delta function defining reduces to:

Smear over region

Then we can expand in powers of

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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)

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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.

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Jet Mass Distribution

For jet masses,

So:

where

Thrust and jet mass sum receive same nonperturbative corrections!

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Other Variables?

Jet Broadening:

C Parameter:

No obvious relation…

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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.

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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)

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Angularities in SCET

Apply same SCET formalism to angularities:

Leading power correction:

CL (in preparation)

Berger, Kucs, Sterman (2003)

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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)

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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.