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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 Presentation
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Nonperturbative Effects from Soft-Collinear Effective Theory
Christopher LeeInstitute for Nuclear Theory,
University of Washington12 January 2006
12-Jan-06 Nonperturbative Effects from SCET 2
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
12-Jan-06 Nonperturbative Effects from SCET 4
Light-Cone Coordinates
Use light-cone coordinates: Define light-like directions:
e.g.
Coordinates:
where
12-Jan-06 Nonperturbative Effects from SCET 5
Degrees of Freedom
Mode Scaling
Collinear Quarks Gluons
(Ultra)soft Gluons
12-Jan-06 Nonperturbative Effects from SCET 6
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 SCET 7
SCET Lagrangian
Leading order in :
Feynman rules:
Bauer, Fleming, Pirjol, Stewart (2001)
12-Jan-06 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:
12-Jan-06 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)
Event Shapes in SCET
12-Jan-06 Nonperturbative Effects from SCET 11
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 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…
12-Jan-06 Nonperturbative Effects from SCET 13
Factorization
Differential rate becomes:
At LO in perturbation theory, first factor is just:
12-Jan-06 Nonperturbative Effects from SCET 14
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 SCET 15
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 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.
12-Jan-06 Nonperturbative Effects from SCET 17
Jet Mass Distribution
For jet masses,
So:
where
Thrust and jet mass sum receive same nonperturbative corrections!
12-Jan-06 Nonperturbative Effects from SCET 18
Other Variables?
Jet Broadening:
C Parameter:
No obvious relation…
12-Jan-06 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.
12-Jan-06 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)
12-Jan-06 Nonperturbative Effects from SCET 21
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 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)
12-Jan-06 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.