Transcript QCD Factorization for Semi

P? Dependence of SSA
-- from nonpert. to pert.
Feng Yuan
RBRC , Brookhaven National Laboratory
References: Ji, Ma, Yuan, Phys. Rev. D70, 074021;
Phys. Lett. B597, 299;
hep-ph/0503015;
to be published;
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Outline
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Introduction
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Low PT region: TMD parton
distributions and QCD factorization
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High PT region: quark-gluon-quark
correlation (ETQS mechanism)
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Summary
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Inclusive and
Semi-inclusive DIS
Inclusive DIS:
Partonic Distribution depending on
the longitudinal momentum fraction
Semi-inclusive DIS:
Probe additional information for parto
transverse distribution in nucleon
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Different PT Region
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Integrate out PT
-- similar to inclusive DIS, probe int. PDF
Large PT (>>QCD)
-- hard gluon radiation, can be calculated from
perturbative QCD,
Polarized ->q-g-q correlations
Low PT (~QCD)
-- nonperturbative information (TMD): new
factorization formula
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TMD Physics
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A way to measure Transversity Distribution, the
last unkown leading twist distribution
Collins 1993
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The Novel Single Spin Asymmetries
Connections with GPDs, and Quantum Phase
Space Wigner distributions
Quark Orbital Angular Momentum and
Many others …
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TMD Distribution: the definition
Gauge Invariance requires the Gauge Link
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Brodsky,Hwang,Schmidt 02’
Collins 02’
Belitsky,Ji,Yuan 02’
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This definition is also
consistent with the
QCD factorization
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Illustrate the Factorization
(one-loop order)
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Take an on-shell quark target
Calculate the TMD Dis. and F.F.
Separate the cross section into different pieces
Show the soft divergence is cancelled out
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TMD Dis. At One-loop
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No soft divergence
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Collinear divergence : ln(m2)
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Double Logarithms: ln2(2 b2)
--- Collins-Soper equation
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Factorization
Soft Factor
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The Factorization Applies to
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Semi-inclusive DIS (polarized and unpolarized)
Drell-Yan at Low transverse momentum
Di-hadron production in e+e- annihilation
(extract the Collins function)
Di-jet and/or di-hadron correlation at hadron
collider (work in progress)
Many others, …
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Large Logarithms Resummation
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At low PT, PT<< Q, we must resum the large
logarithms snln2n-1(Q2/PT2)
-- Dokshitzer, Diakonov, Troian, 1978
-- Parisi, Petronzio, 1979
These large logarithms can be resummed by
solving the energy evolution equation for the
TMD parton dis.
-- Collins-Soper 1981
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TMD: the Energy Dependence
 The TMD distributions depend on the
energy of the hadron! (or Q in DIS)
 One can write down an evolution equation
in ζ=2(P+)2v-/v+
Collins and Soper (81’)
 K and G obey an RG equation,μ independent!
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Spin-dep. Collins-Soper Eq.
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The evolution kennel is the same for all the
leading-twist TMD quark distributions
Kt-even ones: q(x, k┴), ΔqL, δqT
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Kt-odd ones: qT(x, k┴), δq, ΔqT, δqL
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Idilbi, Ji, Ma, Yuan: PRD70 (2004)
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Large Logarithms Resumm. (II)
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After resummation, large logarithms associated
with Q2 can be factorized into the Sudakov
form factors, e.g.
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And the Sudakov form factor
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Double Logarithmic (DL) Approx.
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If Q2 is not too large, DL approx. applies.
The Sudakov suppression form factor then
only depends on Q2
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The Q2 dependence of the structure
functions can be factorized out
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We can predict the PT distribution at higher
Q2 from that of lower Q2
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Phenomenogical applications of the QCD
resummation to the PT spectrum of EW bosons
production have been very successful
Yuan, Nadolsky, Ladinsky, Landry,
Qiu, Zhang, Berger, Li,
Laenen, Sterman, Vogelsang, Kulesza,
Bozzi, Catani, deFlorian,
Kulesza, Stirling, and many others, …
working even at NNLL level for some
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Our interest however, is in the region of not so
large Q2, where DLA (LLA) should apply
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Drell-Yan at Fixed Target
QT spectrum from E288, PRD23,604(81)
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Normalized to the same spectrum!
Consequence of DLA
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At very large Q2 (e.g., Z0 and W boson),
DL Approx. breaks down
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Power Counting at Large PT
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kt-even distributions have the same dependence
on kt
kt-odd distributions are suppressed at large kt
Power Counting Rule
kt-even: 1/kt2
kt-odd: 1/kt3
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SIDIS: at Large PT
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Large PT SIDIS can be calculated from
perturbative QCD (same for TMDs)
Polarized scattering depends on the quark-gluon
correlations in the nucleon
Single Spin Asymmetry (Sivers and Collins) at
large P? is not suppressed by 1/Q, but by 1/PT
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An example: SSA at Large PT
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ETQS mechanism
Efremov & Teryaev: 1982 & 1984
Qiu & Sterman: 1991 & 1999
Quark-gluon correlations
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It is suppressed by 1/PT
Ji,Ma,Yuan, work in progress
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Transition from Perturbative region
to Nonperturbative region?
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Compare different region of PT
Nonperturbative TMD
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Perturbative region
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Summary
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Q2 evolution of the PT spectrum is calculable
from perturbative QCD, by solving the
associated Collins-Soper eqs
Large PT SSA depends on the quark-gluonquark correlation of nucleon (ETQS), which is
suppressed by 1/PT, not 1/Q
Detailed study of the PT dependence will
provide us information for the perturbative to
nonperturbative transition of QCD
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