Factorization and Fully Unintegrated Factorization Ted C. Rogers Vrije Universiteit Amsterdam trogers@few.vu.nl Factorization and unintegrated PDFs: open problems Fully unintegrated factorization INT 09 - September, 2009
Perturbative QCD Asymptotic freedom: Strong coupling becomes small over short time/distance scales! α s (Q 2 )<<1; Q>>Λ QCD Hard scale Use standard Feynman perturbation theory to make accurate first principles calculations. 2
pqcd and Factorization The Real World: Always involves both long and short time/distance scales. Factorization Theorem: Systematic separation of long and short distance scales in QFT. Short distance part: Well-defined perturbation series in small coupling. Long distance parts: Universal correlation functions. Access quark/gluon degrees of freedom. 3
DIS: Standard Approximations LO DIS: Before approximations: q k P l J(l) Φ(k, P) P = q= ( P +, M2 p 2P +,0 t ) ( xp +, ) Q 2 2xP,0 + t Parton model kinematics: k=(k +,k,k ) k + =xp + + M2 J +k2 t 2(q +k ) l=k+q=(k + xp +,q +k,k ) l 2 0 k + xp + x B P + 4
Standard Approximations (cont.) Unapproximated structure tensor: W µν (q,p)= e2 j d 4 k 4π (2π) 4Tr[γµ J(k+q)γ ν Φ(k,P)] l 5
Standard Approximations (cont.) Unapproximated structure tensor: W µν (q,p)= e2 j d 4 k 4π (2π) 4Tr[γµ J(k+q)γ ν Φ(k,P)] l Approximate momentum inside subgraphs: k (xp +,k,k ); k+q (l +,q,0 ) Inside target subgraph Inside jet subgraph 6
Standard Approximations (cont.) Unapproximated structure tensor: W µν (q,p)= e2 j d 4 k 4π (2π) 4Tr[γµ J(k+q)γ ν Φ(k,P)] l Approximate momentum inside subgraphs: k (xp +,k,k ); k+q (l +,q,0 ) Inside target subgraph Inside jet subgraph Use parton model approximation inside hard part. k ˆk=(xP +,0,0 ); l ˆl=(0,q,0 ) 7
Standard Approximations (cont.) Unapproximated structure tensor: W µν (q,p)= e2 j d 4 k 4π (2π) 4Tr[γµ J(k+q)γ ν Φ(k,P)] l Approximate momentum inside subgraphs: k (xp +,k,k ); k+q (l +,q,0 ) Inside target subgraph Inside jet subgraph Use parton model approximation inside hard part. k ˆk=(xP +,0,0 ); l ˆl=(0,q,0 ) Project out largest Dirac components. 8
Factorized Graph 2 = Parton Distribution??? LO partonic structure functions. Note shift in kinematics!! 9
Varieties of parton correlation functions Standard Integrated PCFs: All small momentum components are integrated in definitions. 10
Factorized Graph 2 = Parton Distribution??? LO partonic structure functions. Note shift in kinematics!! 11
Varieties of parton correlation functions Standard Integrated PCFs: All small momentum components are integrated in definitions. Transverse momentum dependent (TMD) PCFs. Only the minus component (smallest) is integrated in definition. Sivers, etc 12
Factorized Graph 2 = Parton Distribution??? LO partonic structure functions. Note shift in kinematics!! 13
Varieties of parton correlation functions Standard Integrated PCFs: All small momentum components are integrated in definitions. Transverse momentum dependent (TMD) PCFs. Only the minus component (smallest) is integrated in definition. Sivers, etc Fully unintegrated PCFs. Explicit dependence on exact parton momentum. 14
Varieties of parton correlation functions Standard Integrated PCFs: All small momentum components are integrated in definitions. Transverse momentum dependent (TMD) PCFs. Only the minus component (smallest) is integrated in definition. Sivers, etc Fully unintegrated PCFs. Explicit dependence on exact parton momentum. How to set up factorization that treats kinematics more accurately? 15
Hadron Kinematics Explicit sample: cc photoproduction Try: Parton model kinematics. Keeping k T dependence, but approximating minus component. (TMD PDFs) Exact kinematics. (Fully unintegrated PDFs) 16
Errors in final state kinematics Cascade Compare: Parton model kinematics TMD kinematics (From Collins and Jung, 2005) 17
Errors in final state kinematics Cascade Compare: TMD kinematics Fully Unintegrated kinematics (From Collins and Jung, 2005) 18
Errors in final state kinematics Cascade Compare: TMD kinematics Fully Unintegrated kinematics Fully Unintegrated TMD (From Collins and Jung, 2005) 19
Large-x (SI)DIS But k T runs to order Q in the def. of the integrated PDF! Definition of standard PDF becomes inconsistent. Sensitive to remnant mass. Fully unintegrated PDFs needed. (Accardi, Qiu, JHEP 0807:090,2008)
Wilson lines Ward identity: Target collinear gluons decouple from hard part. Ward identity collinear lines extracted into overall factor. Wilson line becomes trivial in light-cone gauge. 21
Standard (Integrated) PDF Operator definition: f(x,µ)= dw 4π e ixp+ w p ψ(0,w,0 t )V w(u J )γ + V 0 (u J )ψ(0) p u J =(0,1,0 t ) Light-like Wilson lines enforce gauge invariance. 22
TMD PDFs: Generally accepted definition: P(x,k t,µ)= Link at infinity dw dw t 16π 3 e ixp+ w +ik t w t p ψ(0,w,w t )V w(n)i n;w,0 γ + V 0 (n)ψ(0) p n=u J =(0,1,0 t ) Fields are no longer evaluated along light-like separation. Connection at infinity needed for closed Wilson line / gauge invariance. 23
Complications with K T - Factorization: Uncanceled light-cone divergences corresponding to gluons moving with infinite rapidity in minus direction. + Use non-light-like Wilson lines. Introduces new arbitrary parameter. Predictability recovered with new evolution equations (e.g. CSS). 24
TMD PDFs: Wilson Lines Paths of Wilson lines: + + w w,w t Standard (Integrated) Unintegrated First Try _ w,w t + Same issues arise in treating fully unintegrated PCFs Unintegrated tilted Wilson line directions 25
Hadron-Hadron Scattering: Modified Universality? Unintegrated-factorization generally fails in hadroproduction of high-p T hadrons. No simple Wilson line structure in PDFs. (Bomhof et al., Phys.Lett.B596:277-286,2004) (Collins and Qiu, PRD75:114014,2007) Recall Piet Mulders talk + 26
Modified Universality? Consider one correlator at a time. Wilson line structure emerges. (Bomhof et al, Eur.Phys.J.C47:147-162, 2006, Bomhof, Mulders Nucl.Phys.B795:409-427,2008) Φ [,( )] q Φ [+] q 27
Modified Universality? (Mulders et al, Eur.Phys.J.C47:147-162, 2006) Consider one correlator at a time. Wilson line structure emerges. Φ [,( )] q Φ [+] q σ N2 N 2 1 H 2 Φ [,( )] q Φ [+][ ] g [ ] q 1 N 2 1 H 2 Φ [+] q Φ [+][ ] g [ ] (Boer et al, Phys.Lett.B660:360-368, 2008) q 28
Modified Universality? Complex Wilson Lines:?? σ C Φ [WL 1] q Φ [WL 2] g [WL 3] q + Sum over color structures - + - + 29
Modified Universality? P B 0 P A?? σ Φ [+( )] Φ [+( )] One gluon contribution gives zero! 30
Modified Universality? Open questions: Can a generalization of K T -factorization be formulated with well-defined but non-standard Wilson lines??? σ C P A F1 WL 1 F 1 P A P B F2 WL 2 F 2 P B + Or,?? σ C P A P B F1 F2 WL 1 WL 2 F 1 F 2 P A P B + After P T weighting? 31
Summary So Far Details of kinematic approximations in standard factorization. 32
Summary So Far Details of kinematic approximations in standard factorization. Motivation for more exact formalism. 33
Summary So Far Details of kinematic approximations in standard factorization. Motivation for more exact formalism. Emphasis on need for well-defined operator definition. Problems 34
Summary So Far Details of kinematic approximations in standard factorization. Motivation for more exact formalism. Emphasis on need for well-defined operator definition. Problems Basic Problem: In standard treatment, factorization works after integrations, but final states are changed! 35
Fully Unintegrated Factorization Back to Deep Inelastic Scattering Proposal: Set up a factorization formalism with exact kinematics for initial and final states. (Collins, TCR, Stasto, PRD77:085009,2008) 36
Fully Unintegrated Factorization Back to Deep Inelastic Scattering Proposal: Set up a factorization formalism with exact kinematics for initial and final states. (Collins, TCR, Stasto, PRD77:085009,2008) Factorization should work point-by-point in phase space. (correct of up to power suppressed terms.) 37
Fully Unintegrated Factorization Back to Deep Inelastic Scattering Proposal: Set up a factorization formalism with exact kinematics for initial and final states. (Collins, TCR, Stasto, PRD77:085009,2008) Factorization should work point-by-point in phase space. (correct of up to power suppressed terms.) To obtain factorization, only apply approximations to the hard part. 38
Fully Unintegrated Factorization Back to Deep Inelastic Scattering Proposal: Set up a factorization formalism with exact kinematics for initial and final states. (Collins, TCR, Stasto, PRD77:085009,2008) Factorization should work point-by-point in phase space. (correct of up to power suppressed terms.) To obtain factorization, only apply approximations to the hard part. Hard scattering should involve ordinary functions. 39
Fully Unintegrated Factorization Back to Deep Inelastic Scattering Proposal: Set up a factorization formalism with exact kinematics for initial and final states. (Collins, TCR, Stasto, PRD77:085009,2008) Factorization should work point-by-point in phase space. (correct of up to power suppressed terms.) To obtain factorization, only apply approximations to the hard part. Hard scattering should involve ordinary functions. Need well-defined fully unintegrated parton correlation functions. 40
Factorization Strategy Consider general unapproximated Feynman diagrams. Classify leading regions. 41
Factorization Strategy Consider general unapproximated Feynman diagrams. Classify leading regions. Apply approximations appropriate for each region starting with the smallest (but never changing final state momentum!). 42
Factorization Strategy Consider general unapproximated Feynman diagrams. Classify leading regions. Apply approximations appropriate for each region starting with the smallest (but never changing final state momentum!). Obtain contribution from larger regions by subtracting smaller regions. 43
Factorization Strategy Consider general unapproximated Feynman diagrams. Classify leading regions. Apply approximations appropriate for each region starting with the smallest (but never changing final state momentum!). Obtain contribution from larger regions by subtracting smaller regions. Use Ward identities to disentangle soft and collinear gluons from hard part in sum over graphs. 44
Factorization Strategy Consider general unapproximated Feynman diagrams. Classify leading regions. Apply approximations appropriate for each region starting with the smallest (but never changing final state momentum!). Obtain contribution from larger regions by subtracting smaller regions. Use Ward identities to disentangle soft and collinear gluons from hard part in sum over graphs. Identify well-defined operator definitions for the PCFs. 45
General Graphical Structure Should start with: Must disentangle soft and collinear gluons to get factorization 46
Graphical Example: Single extra gluon: real emission Jet Collinear Soft Target Collinear y J y Q y p 47
Soft Region: Factorized Structure 48
Graphical Example (Cont.): Subtractions Consider target-collinear region After applying Ward identities. The contribution to the PDF (target PCF) requires double counting subtractions. 49
Recall the starting point: Factorization After steps outlined above, this becomes (Ward identities + subtractions) 50
Topological Factorization: A formula of this type is the goal. Also need double counting subtractions. 51
Full Factorization 52
Next-to-Leading Order (TCR, PRD78:074018,2008) Wide angle jets: Non-trivial hard scattering matrix element. + Cross terms Relies on details of LO treatment. 53
Next-to-Leading Order (Cont.) NLO hard scattering overlaps with LO. Factorization requires double counting subtractions. Standard integrated factorization: Hard scattering involves generalized functions (distributions)! Exactly analogous subtractive formalism used in fully unintegrated case. (Collins, Zu, JHEP 03 (2005) 059) Double counting subtractions give factorized expression with power suppressed errors point-by-point in phase. 54
Fully Unintegrated Approach: Full factorization for scalar theory. Collins, Zu, JHEP 03 (2005) 059 Abelian Gauge Theory: Detailed treatment of factorization for case of a single outgoing jet in DIS (SIDIS). (Collins, TCR, Stasto, PRD77:085009,2008) Suggestive of complete all-orders treatment in QCD Simplest NLO wide-angle hard scattering calculation in DIS. LO + NLO consistent with factorization point-by-point in phase space (TCR, PRD78:074018,2008) 55
Outlook Practical Issues: Fully Unintegrated PDFs: Need to finish calculation of NLO hard scattering. Simplifications? Implementation in MCEGs. Open problems in unintegrated formalism: Hadron-hadron scattering? Definitions of TMD/Fully Unintegrated PDFs. Do we have consistent definitions? Other UV divergences? (Cherednikov et al., arxiv:0904.2727) Evolution equations (CSS formalism??). Relationship to other approaches? Full extension to non-abelian case. 56