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The role of the transverse gauge links in soft collinear effective theory

The role of the transverse gauge links in soft collinear effective theory. Ignazio Scimemi Universidad Complutense de Madrid (UCM) In collaboration with A. Idilbi , M. García Echevarría A.I., I.S. Phys. Lett . B695 (2011) 463 ,

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The role of the transverse gauge links in soft collinear effective theory

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  1. The role of thetransverse gauge links in softcollineareffectivetheory IgnazioScimemi Universidad Complutense de Madrid (UCM) In collaborationwith A. Idilbi , M. García Echevarría A.I., I.S. Phys. Lett. B695 (2011) 463, M.G.E., A.I., I.S. arXive:1104.0686[hep-ph] and work in progress

  2. Outline • SCET and itsbuilding blocks • Gauge invarianceforcovariant gauges • Gauge invariancefor singular gauges (Light-cone gauge) • A new Wilson line in SCET: T • Theorigin of T-Wilson lines in SCET Lagrangian: gauge conditionsfordifferentsectors • Phenomenology • Conclusions

  3. SCET, aneffectivetheory of QCD • SCET (softcollineareffectivetheory) isaneffectivetheory of QCD • SCET describes interactionsbetweenlowenergy ,”soft” partonicfields and collinearfields (veryenergetic in one light-conedirection) • SCET and QCD havethesameinfraredstructure: matchingispossible • SCET helps in theproof of factorizationtheorems and identification of relevantscales

  4. SCET: Kinematics Bauer, Fleming, Pirjol, Stewart, ‘00 U-soft Light-cone coordinates Softmodes do notinteractwith (anti) collinearor u-soft In covariant gauge Integratedoutwith EOM

  5. SCET Bauer, Fleming, Pirjol, Stewart, ‘00 Light-cone coordinates Leading order Lagrangian (n-collinear) The new fields do not interactanymorewith u-softfields

  6. SCET

  7. SCET building blocks The SCET Lagrangianisformedby gauge invariantbuilding blocks. Gauge Transformations: Is gauge invariant

  8. Factorization Theorem For DIS • PDF In Full QCD • Factorization In SCET [Neubert et.al, Manohar] • PDF In SCET: [Stewart et.al] is gauge invariant because each building block is gauge Invariant

  9. Factorization Theorem For SIDIS in QCD: Covariant gauge • In Full QCD And At Low Transverse Momentum: Ji, Ma,Yuan ‘04 • “Naïve” Transverse Momentum Dependent PDF (TMDPDF): Analogous to the W in SCET Thisresultis true only in “regular” gauges: Hereallfieldsvanish at infinity

  10. Transverse Gauge Link in QCD Ji, Ma, Yuan Ji, Yuan Belitsky, Ji, Yuan Cherednikov, Stefanis • For gauges not vanishing at infinity [Singular Gauges] like • the Light-Cone gauge (LC) one needs to introduce an additional • Gauge Link which connects with to make it • Gauge Invariant • In LC Gauge This Gauge Link Is Built From The Transverse Component • Of The Gluon Field:

  11. Gauge Invariant TMDPDF In SCET? Are TMDPDF fundamental matrixelements in SCET? Are SCET matrixelements gauge invariant? Where are transverse gauge link in SCET?

  12. Gauge invariance of SCET building blocks We calculate at one-loop in Feynman Gauge and In LC gauge In Feynamn Gauge

  13. Gauge invariance of SCET building blocks We calculate at one-loop in Feynman Gauge and In LC gauge In LC Gauge [Bassetto, Lazzizzera, Soldati] Canonical quantization imposes ML prescription

  14. Gauge invariance of SCET building blocks We calculate at one-loop in Feynman Gauge and In LC gauge In LC Gauge

  15. Gauge invariance of SCET building blocks We calculate at one-loop in Feynman Gauge and In LC gauge In LC Gauge The gauge invarianceisensuredwhen • Theresult of thisisindependent of • and has gotonly a single pole. Zero-bin • subtractionisnul in ML.

  16. Gauge invariance in SCET The SCET matrixelementisnot gauge invariant . Using LC gauge theresult of theone-loopcorrectiondependsontheusedprescription.

  17. Gauge invariance in SCET In ordertorestore gauge invariancewehave to introduce a new Wilson line, T, in SCET matrixelements And the new gauge invariantmatrixelementis

  18. The T-Wilson Line In covariant gauges , so werecoverthe SCET results In LC gauge

  19. The T-Wilson Line All prescription dependence cancels out and gauge invariance is restored no matter what prescription is used Covariant Gauges In All Gauges

  20. The T-Wilson Line in other prescriptions Letusconsiderthe pole part of theinteresting integral with orthe PV prescription. Theresultis And in PV theresultdoesnothaveanyimaginarypart. The gauge invarianceisrestoredeitherwiththe T with a prescriptiondependent factor OR withzero-bin Subtraction!! Thevalues of thisconstantdepend alsoontheconventionforinner/outermoments

  21. Where does the T-Wilson Line come from? Isthere a waytounderstandthe T-Wilson lines fromthe SCET Lagrangian? Anexample, the quark form factor: from QCD to SCET in LCG The T-Wilson line isbornnaturally in Oneloopmatching.

  22. Where does the T-Wilson Line come from? In the canonical quantization of ofthe gauge field (Bassetto et al.) We define And we can show

  23. The T-Wilson Lines in SCET-I In SCET-I onlycollinear and u-softfields. Thefirststeptoobtainthe SCET Lagrangianisintegratingoutenergeticpart of spinors And thenapplyingmultipoleexpansion, Where U-softfield do notgiverisetoanytransverse gauge link!! There are no transverse u-softfields and theycannotdepend ontransversecoordinates!!

  24. The T-Wilson Lines in SCET-II Nowthedegrees of freedom are justcollinear and soft No interactionispossibleforon-shellstates Isthis true in every gauge?

  25. The T-Wilson Lines in SCET-II The gauge ghosthoweveractsonlyonsomemomentumcomponents Thusthecovariantderivativeis Thedecoupling of softfieldsrequires

  26. The T-Wilson Lines in SCET-II The new SCET-II Lagrangianis

  27. Applications TMDPDF Drell-Yan at low Pt [Becher,Neubert] Higgsproduction at low Pt [Mantry,Petriello] Beamfunctions [Jouttenus,Stewart, Tackmann,Waalewijn] Heavy Ion physics – Jet Broadening[Ovanesyan,Vitev ] …

  28. Factorization Theorem For DIS • PDF In Full QCD • Factorization In SCET [Neubert et.al] • PDF In SCET: [Stewart et.al] is gauge invariant because each building block is gauge Invariant

  29. Factorization Theorem For SIDIS in QCD: Covariant gauge • In Full QCD And At Low Transverse Momentum: Ji, Ma,Yuan ‘04  Isrenorrmalizationscale;  is a rapiditycut-off • “Naïve” Transverse Momentum Dependent PDF (TMDPDF): Analogous to the W in SCET Thisresultis true only in “regular” gauges: Hereallfieldsvanish at infinity

  30. Transverse Gauge Link in QCD Ji, Ma, Yuan Ji, Yuan Belitsky, Ji, Yuan Cherednikov, Stefanis • For gauges not vanishing at infinity [Singular Gauges] like • the Light-Cone gauge (LC) one needs to introduce an additional • Gauge Link which connects with to make it • Gauge Invariant • In LC Gauge This Gauge Link Is Built From The Transverse Component • Of The Gluon Field:

  31. TMDPDF We Can Define A Gauge Invariant TMDPDF In SCET (And Factorize SIDIS)

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

  33. Collinear Anomaly Notice That The Cross-Section Is Independent Of The Renormalization Scale (RG Invariance). Also (Up To Three Loop Calculation!) For Vanishing Soft Function, The Product Of Two TMDPDFs Has to Be Logarithmically Dependent On The Renormalization Scale. This is Impossible Unless There Is Anomaly.

  34. Origin Of Collinear Anomaly In The Absence Of Soft Interactions Different Collinear Sectors Do Not Interact So There Is No Way To Generate The Q-Dependence Classically Each Collinear Lagrangian Is Invariant Under Rescaling of Collinear Momentum. For TMDPDF Quantum Loop Effects Needs Regulation Then Classical Invariance Is Lost However The Q-Dependence Is Obtained

  35. The TMDPDF Is Ill-Defined And We need To Introduce New Set Of NP Matrix Elements Re-factorization Into The Standard PDF The Analysis Of Becher-Neubert Ignores Two Notions: Transverse Gauge Links AndSoft-Gluon Subtraction Needed To Avoid Double Counting! (Currently Investigated.)

  36. Application To Heavy-Ion Physics D´Eramo, Liu, Rajagopal In LC Gauge The Above Quantity Is Meaningless. If We Add To It The T-Wilson line Then We Get A Gauge Invariant Physical Entity.

  37. Conclusions The usual SCET building blocks havetobemodifiedintroducing a New Gauge Link, the T-Wilson line. Usingthe new formalismweget gauge invariantdefinitions of non-perturbativematrixelements. In particular the T iscompulsoryformatrixelements of fieldsseparated in thetransversedirection. Thesematrixelements are relevant in semi-inclusive crosssectionsortransversemomentumdependentones. Itispossiblethatthe use of LC gauge helps in theproofs of factorization. Theinclusion of T is so fundamental. Work in progress in thisdirection.

  38. Conclusions Itisdefinetelypossibletounderstandtheorigin of T-Wilson lines in a Lagrangianframeworkfor EFT. Every sector of the SCET can beappropriatelywritten in LCG. The LCG has peculiar propertyforloopcalculation and can avoidtheintroduction of new ad-hoc regulators Thereis a richphenomenologytobestudied… so a lot of work in progress!! THANKS!

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