1. Unintegrated parton distributions and final states in DIS Anna Stasto Penn State University Work in collaboration with John Collins and Ted Rogers `

2. Outline Integrated parton distributions Collinear factorization Kinematic approximations Problems at Monte Carlo Event Generators Unintegrated parton distributions/fragmentation and correlation functions Wilson lines in non-lightlike directions New factorization framework

3. QCD cross section A general QCD cross section contains long and short distance contributions. s Even when s is very large, there is a range of scales from small scales up to s. It is impossible (or at least very hard) to compute cross section using only perturbative QCD.

4. DIS on a proton Wide angle scattering of electron off a hadron(proton): Large momentum transfer: Interactions between partons on scales much larger than the electron scattering. Legitimate approximation to treat partons as non- interacting: essentially free objects parton model

5. Factorization Process tractable in pQCD once we have an additional hard scale Q in the process. Collinear factorization: separation of short and long distance contributions. Short distance: partonic cross sections, perturbatively calculable. Long distance: parton distribution and fragmentation functions, non- perturbative, universal quantities. Various processes: DIS, Drell-Yan, e+e-

6. Partonic xsection Parton distribution Parton model Kinematic approximations k q l F J Massless, collinear, on-shell approximation:

7. Integrated PDFs Operator definition of the integrated quark distribution function: Wilson line gauge link k P Light-like vector: y 0 -

8. Collinear framework Hard scattering coefficient. On-shell matrix element Integrated parton distribution: Renormalization group equations: Factorization for structure function: Expansion for anomalous dimensions(splitting functions)

9. Exact kinematics and MCEGs Standard factorization theorems involve sums over the unobserved final states. Kinematic approximations fine for the inclusive cross sections. Standard framework derived for on-shell partons. How about the detailed final states? Ex. Monte Carlo generators resolve full final state. Approximations on the kinematics become invalid in the exclusive case where the partons are off-shell: spacelike or timelike. Violation of the exact energy-momentum conservation when implementing collinear factorization in the MCEG.

10. Breakdown of the collinear approximation In parton model one makes the replacement However if we allow for the parton to have invariant mass (for hadronization) Mismatch between parton model and the real kinematics invariant mass of the outgoing parton l k q

11. Monte Carlo Event Generator l k q l k q On-shell matrix element hadronization final state showering initial state showering

12. Kinematics and MCEG Collins, Jung The exact kinematics is very important for the detailed distributions of exclusive processes. Huge differences between standard parton model, unintegrated partons and exact kinematics. Conventional formalism with integrated parton densities is not suitable for the analysis of the final states.

13. Conventional integrated parton distributions and fragmentation functions; longitudinal momentum: Unintegrated parton distributions (and fragmentation functions); longitudinal and transverse momentum: Fully unintegrated parton correlation functions (PCFS); longitudinal, transverse momenta and virtuality: Known from small x

14. At small x, the unintegrated gluon density naturally appears. It is related with the integrated density: Would like to define these objects also for large x. Issues with positivity. At high energy there are kinematical approximations too (strong rapidity ordering) Small x see also Kimber, Martin, Ryskin; Watt,Martin,Ryskin;Kimber,Kwiecinski,Martin,Stasto Dokshitzer, Diakonov, Troyan

15. Modified DIS Additional transverse momentum dependence Soft gluon-ex. color recombination Detector Color singlet hadron

16. Exact kinematics in initial and final states. Explicit factors (bubbles) representing final states. Retain on-shell matrix elements. Define projections from exact to approximate momenta. Construct definitions of the gauge invariant PCFs and Soft factor. Use eikonal lines and Ward identities to show general factorization. Use non-light-like eikonal Wilson lines to regularize light-cone divergences in the unintegrated parton distributions ( this introduces a cutoff in rapidity, similarly to what is done at small x ). General strategy Collins, Rogers, Stasto

17. Keep the intermediate bubbles everywhere to maintain exact kinematics. Most general graph Need to disentangle soft and collinear gluons. Define the approximators for each region(soft, collinear, hard). Modification with respect to the lowest order parton model level.

18. General factorized form Wilson lines + Ward identities to disentangle gluons.

19. Factorization of one soft gluon emission Generalization to many emissions: Soft operator with Wilson lines

20. Directions of Wilson lines When using lightlike directions gluons can have infinite negative rapidity. This is NOT a problem in conventional integrated parton distributions since this divergence cancels between real and virtual terms. Use non-light-like Wilson lines to regulate. Collins, Soper,Sterman; Balitsky Obtain evolution equations with respect to the rapidity cutoff: Introduce an extra parameter. Renormalization group in rapidity.

21. Definitions: soft factor Soft factor: Direction of Wilson lines: non-light-like vectors Non-light-like Wilson lines are appropriate for a finite energy process. They provide cutoffs on rapidity divergences. I(u’), I(n’) are required for gauge invariance (closed Wilson loop).

22. Definitions: unintegrated pdfs Recall the integrated case (in momentum space): Integrated pdfs in coordinate space: y- 0 Parton correlation function (completely unintegrated) in coordinate space:

23. Definitions: correlation functions Need subtractions for the soft region to avoid double counting. Approximation of the soft region: Approximation of the target collinear region: A general graph: Parton correlation function (completely unintegrated with soft subtractions) in coordinate space: Soft factor Soft subtraction

24. Factorization for one jet production in DIS Two main directions of Wilson lines: one for quark current jet and one for target jet. In principle one could generalize this procedure to more than one jet. More directions for the Wilson lines. More complicated soft factor: many Wilson lines.

25. Relation to small x Balitsky Factorization in rapidity y. Rapidity divide y between slow and fast fields Renormalization group equation with respect to the slope of the Wilson lines. Recover BFKL limit. Note: this is not a perturbative factorization. All components (impact factor and evolution kernel) have both large and small transverse momenta, so strong coupling can take arbitrary values. Regge limit (or rapidity ordering) is non-perturbative.

26. Summary Standard kinematic approximations are inappropriate when looking at specific final states. Conventional collinear factorization formalism is not sufficient. Incompatible with Monte Carlo event generators. General formalism which does not rely on kinematic approximations. Definition of parton correlation functions and soft factors in a gauge invariant way using non-light-like Wilson lines. Factorization works with these objects. To do: evolution equations, check consistency with unintegrated gluon distributions in the appropriate limits (BFKL/CCFM and Watt,Martin,Ryskin approach ). Relation to Balitsky framework. More to do: relation with the Wilson loop amplitudes in N=4 SYM.

27. Examples One cannot still compute reliably total proton-proton cross section from QCD. Multiplicity of charged particles at mid-rapidity in AA collisions. Factors of magnitude difference between models. These quantities depend crucially on soft (or long ) distance physics.

28. Definitions: jet factor Jet factor (with subtractions): Proceed in an analogous way for the jet factor

29. Example: one soft gluon Most general description (using ‘bubbles’): Soft=small rapidity

30. Disentangle the soft gluons Momenta contracted with the bubbles Soft approximatorExact momenta For the soft gluon (momentum ): Factorized soft contribution

31. One-gluon contribution to the soft factor Eikonal denominators coming from expansion of the Wilson lines in u and n directions Wilson line expansion: