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.D

A. Idilbi, I. Scimemi, Phys. Lett. B695 (2011) 463-468

84:011502,2011

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

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

Factorization Theorems Are Essential For Phenomenology

Inclusive Higgs Boson Production

[Nuebert et.al. 2010]

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)

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

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

Drell-Yan At Low q_T In SCET

The SCET Building Block:

The Soft Function

The NP Matrix Element: Quark PDF

Is Not Gauge Invariant : No Transverse Gauge Link

[Becher And Neubert, 10]

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]

Regular vs. Singular Gauges

Notice:

The Fundamental Observation:

The Above Relation is Non-Trivial For Singular Gauges.

SCET Revisited

In LCG:

SCET Lagrangian Valid For Both Classes Of Gauges:

[M.I, A.I and I.S,11]

Drell-Yan At Low-q_T

Introduce Gauge Invariant Quark Jet:

The TMDPDF Is Indeed Gauge Invariant.

Application To Heavy-Ion Physics

Collide 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.

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

Lagrangian Valid In Regular And Singular Gauges

[A.I and A. M 09]

In Covariant Gauge

The Forward Scattering Amplitude

[K. Rajagopal et.al 10]

Jet Quenching Parameter:

In LCG It Vanishes. The Above Results Are Not Gauge Invariant.

To Get Gauge Invariant Results:

Impose

Drell-Yan At Low q_T: Correct TMDPDF?

Partonically:

[M.I, A.I, I.S, 2011]

How Do We Calculate The Collinear Matrix Elements?

DY Continued: Double Counting

Naive Collinear

Soft Function

The Soft Contribution Are Included In The Collinear!!

The ``Jets’’ Are The TMDPDFs [Collins, 2011]

Pictorially

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

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!

Universality: TMDPDF IN DIS And DY

Collinear:

Soft:

Not Equal But Related Through The Cusp Anomalous Dimension!

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.