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Gauge Invariant Factorization Proofs With Transverse Momentum Dependence

Gauge Invariant Factorization Proofs With Transverse Momentum Dependence. Ahmad Idilbi Uni. Regensburg. TUM November 30, 2011. M. G. Echevarria , A. Idilbi , I. Scimemi , arXive : :1111.4996 [ hep -ph] . M. G. Echevarria , A. Idilbi , I. Scimemi , Phys. Rev.D.

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Gauge Invariant Factorization Proofs With Transverse Momentum Dependence

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  1. Gauge Invariant Factorization Proofs With Transverse Momentum Dependence Ahmad Idilbi Uni. Regensburg TUM November 30, 2011 M. G. Echevarria, A. Idilbi, I. Scimemi, arXive: :1111.4996[hep-ph] M. G. Echevarria, A. Idilbi, I. Scimemi, Phys. Rev.D 84:011502,2011 A. Idilbi, I. Scimemi, Phys. Lett. B695 (2011) 463-468

  2. Outline • Factorization Theorems In QCD • Basics Of Soft-Collinear Effective Theory (SCET) And Limitations There-off • A Modified SCET Lagrangian And Its Applications To LHC And RHIC • TMD PDFs And Drell-Yan At Low-q_T • Conclusions And Outlook

  3. Text-Book Example: Inclusive DIS The Hard Function (Wilson Coefficient) Is Perturbative (Known To Third Order!) Non-Perturbative QCD Matrix Element: Feynman PDF Gauge Invariance And Universality

  4. Inclusive Higgs Boson Production [Nuebert et.al. 2010] Factorization Theorems Are Essential For Phenomenology

  5. SCET: Tool To Derive Factorization Theorems • Describes 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)

  6. 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

  7. Decoupling Of Soft Gluons Regular Derivative Soft Gluons Are Decoupled From The Lagrangian. However They Appear In The operators. This Decoupling Is Important For Manifest Factorization Of Soft And Collinear Sectors

  8. Drell-Yan At Low q_T In SCET The SCET Building Block:

  9. The Soft Function The NP Matrix Element: Quark PDF Is Not Gauge Invariant : No Transverse Gauge Link [Becher And Neubert, 10]

  10. Reminder From QCD • TMDPDF In QCD [Analogous To Collinear Wilson line In SCET] [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]

  11. Regular vs. Singular Gauges Notice: The Fundamental Observation: The Above Relation is Non-Trivial For Singular Gauges.

  12. SCET Revisited In LCG: SCET Lagrangian Valid For Both Classes Of Gauges: [M.I, A.I and I.S,11]

  13. Drell-Yan At Low-q_T Introduce Gauge Invariant Quark Jet: The TMDPDF Is Indeed Gauge Invariant.

  14. 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.

  15. Jet Broadening Jet Broadening All Gluons Are Glauber Gluons: They Carry Mainly Momentum In The Transverse Direction Construct An EFT Of Collinear Parton+Glauber Gluons [A.I and AbhijitMajumder 09]

  16. Lagrangian Valid In Regular And Singular Gauges [A.I and A. M 09] In Covariant Gauge

  17. The Forward Scattering Amplitude [K. Rajagopal et.al 10] Jet Quenching Parameter:

  18. In LCG It Vanishes. The Above Results Are Not Gauge Invariant. To Get Gauge Invariant Results: Impose

  19. Drell-Yan At Low q_T: Correct TMDPDF? [M.I, A.I, I.S, 2011] Partonically: How Do We Calculate The Collinear Matrix Elements?

  20. DY Continued: Double Counting Naive Collinear Soft Function The Soft Contribution Are Included In The Collinear!!

  21. Pictorially The ``Jets’’ Are The TMDPDFs [Collins, 2011]

  22. Issues With Collins` Definition First: 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

  23. 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!

  24. Universality: TMDPDF IN DIS And DY Collinear: Soft: Not Equal But Related Through The Cusp Anomalous Dimension!

  25. Conclusions And Outlook • SCET 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.

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