Gauge Invariant Factorization Proofs With Transverse Momentum Dependence
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Gauge Invariant Factorization Proofs With Transverse Momentum
DependenceAhmad Idilbi
Uni. Regensburg
TUMNovember 30, 2011
M. G. Echevarria, A. Idilbi, I. Scimemi, arXive: :1111.4996[hep-ph] M. G. Echevarria, A. Idilbi, I. Scimemi, Phys. Rev.DA. Idilbi, I. Scimemi, Phys. Lett. B695 (2011)
463-468
84:011502,2011
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OutlineFactorization Theorems In QCDBasics Of Soft-Collinear Effective Theory
(SCET) And Limitations There-offA Modified SCET Lagrangian And Its Applications To LHC And RHICTMD PDFs And Drell-Yan At Low-q_T Conclusions And Outlook
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Text-Book Example: Inclusive DIS
Non-Perturbative QCD Matrix Element: Feynman PDF
The Hard Function (Wilson Coefficient) Is Perturbative (Known To Third Order!)
Gauge Invariance And Universality
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Factorization Theorems Are Essential For Phenomenology
Inclusive Higgs Boson Production
[Nuebert et.al. 2010]
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SCET: Tool To Derive Factorization TheoremsDescribes Interactions between Soft (low energetic)
partonic fields and Collinear (highly energetic in one light-cone direction) ones
In Perturbation Theory: Soft and Collinear modes capture the infra-red behavior of QCD. Matching of the two theories: QCD And SCET [Sterman 95]
In A Process With Several Scales: A series Of Matching Procedures Is Applicable. Every Matching Coefficient Corresponds To A Term In The Factorization Theorem.
(Notice: In Any Factorization Theorem: Different Contributions Live At Different Scales)
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SCET Lagrangian
Interaction with Soft Gluons:
Collinear Wilson Line:
Collinear ``Gauge Invarint’’Quark Jet:
Essential To Obtain Gauge Invariant Operators And NP Matrix Elements In SCET
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Decoupling Of Soft Gluons
Soft Gluons Are Decoupled From The Lagrangian. However They Appear In The operators. This Decoupling Is Important For Manifest Factorization OfSoft And Collinear Sectors
Regular Derivative
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Drell-Yan At Low q_T In SCET
The SCET Building Block:
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The Soft Function
The NP Matrix Element: Quark PDF
Is Not Gauge Invariant : No Transverse Gauge Link
[Becher And Neubert, 10]
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Reminder From QCDTMDPDF In QCD
[Collins and Soper, 81]
In Covariant Gauges The Gluon Fields Vanish At Infinity: No Additional Links Required
In Light-Cone Gauge:
[Ji and Yuan,02]
)0,0(
)0,(
),( b
),( b
[Analogous To Collinear Wilson line In SCET]
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Regular vs. Singular Gauges
Notice:
The Fundamental Observation:
The Above Relation is Non-Trivial For Singular Gauges.
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SCET Revisited
In LCG:
SCET Lagrangian Valid For Both Classes Of Gauges:
[M.I, A.I and I.S,11]
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Drell-Yan At Low-q_T
Introduce Gauge Invariant Quark Jet:
The TMDPDF Is Indeed Gauge Invariant.
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Application To Heavy-Ion PhysicsCollide Highly-Energetic Beams (LH, HH,
HN, NN) And Create Hot and Dense QCD Medium (Quark-Gluon Plasma.)
Study The Properties Of The Emerging , Medium-Modified Jet .
Compare It With A Vacuum-Propagating Jet.
Extract The Properties Of The Produced Medium.
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Jet Broadening
All Gluons Are Glauber Gluons: They Carry Mainly Momentum In The Transverse Direction
[A.I and Abhijit Majumder 09]Construct An EFT Of Collinear Parton+Glauber Gluons
Jet Broadening
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Lagrangian Valid In Regular And Singular Gauges
[A.I and A. M 09]
In Covariant Gauge
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The Forward Scattering Amplitude
[K. Rajagopal et.al 10]
Jet Quenching Parameter:
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In LCG It Vanishes. The Above Results Are Not Gauge Invariant.
To Get Gauge Invariant Results:
Impose
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Drell-Yan At Low q_T: Correct TMDPDF?
Partonically:
[M.I, A.I, I.S, 2011]
How Do We Calculate The Collinear Matrix Elements?
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DY Continued: Double Counting
Naive Collinear
Soft Function
The Soft Contribution Are Included In The Collinear!!
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The ``Jets’’ Are The TMDPDFs [Collins, 2011]
Pictorially
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Issues With Collins` DefinitionFirst: the Soft Function Is Process-Dependent
Is The Universality Of TMDPDF Lost?
Second: Collins Goes Off-The-Light-Cone
Reason: To Regulate Light-Cone Singularities
Light-Cone Limit Becomes Singular: We Cannot Go Back!Physical Properties Of TMDPDF Becomes Dependent on y_n
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Anomalous Dimension
The Evolution Of The TMDPDF Depends ``Off-Lightconess’’!(Collins-Soper Evolution Equation)
Different Regulator While Staying On-The-Light-Cone:
Take:
The TMDPDF Evolves With The Physical Hard Probe!
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Universality: TMDPDF IN DIS And DY
Collinear:
Soft:
Not Equal But Related Through The Cusp Anomalous Dimension!
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Conclusions And OutlookSCET Is Modified To Allow Gauge-Invariant
Factorization Theorems For q_T-Dependent Processes.We Applied The New Formalism For Drell-Yan Process.The Results Could Be Easily Extended To Higgs Boson
Production.New Definition Of TMDPDF And Its Properties.Ideas For The Future:Collins-Soper On-The-Light-Cone.Applications To Spin-Dependent And q_T-Dependent
Hard Reactions.