nonperturbative effects from soft-collinear effective theory
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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. - PowerPoint PPT PresentationTRANSCRIPT
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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.
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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)
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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.