Unitarity and the Perturbative QCD Dipole Picture

1. Unitarity and the Perturbative QCD Dipole Picture Ted C. Rogers The Pennsylvania State University • TCR, Guzey, Strikman, Zu (2004) • TCR, Strikman (2006) • TCR, Stasto, Strikman (2008) 4th Electron-Ion Collider Workshop, Hampton University, May 19-23 2008

2. Outline: • Review and Summary of Dipole Description of DIS. • Impact parameter picture and s-channel unitarity. • Extraction of impact parameter dependence. • Black disk limit (BDL) and extrapolation to very high energies. • ep vs. eA scattering. • Connection to pp scattering at UHE.

3. Dipole Picture at small-x: target rest frame Increasing gluon density • Write total cross section in terms of qq-proton/nucleus cross section. • At small-x photon fluctuates far ahead of interaction. • At large Q2 coupling is small, but gluon distribution becomes large at small-x. • Convenient picture for discussing transition between perturbative and non-perturbative physics.

4. Feynman Graphs: + others.

5. Complete Dipole Cross-Section • LO pQCD expression grows too rapidly with decreasing x. • Expect taming from gluon “saturation” effects. • Rapid growth for large sizes (soft physics). • Strategy: • Model physics for large size behavios. • Smoothly extrapolate to LO pQCD expression in DGLAP region. • Question: Where does enforcing unitarity become an issue?

6. Complete Dipole Cross-Section • Independently model small and large size configurations. • Small sizes: Use LO pQCD result: • For d ~ dπ, use pion-like behavior: • Match Bjorken-x, gluon momentum fraction: (MFGS model -McDermott, et al. (2000))

7. matching region hard regime soft regime Interpolate between small and large sizes.

8. Comparison with data No fitting: H1

9. Unitarity and the small-x limit • The MFGS model allows rapid growth for small size configs. at small-x. • Fast approach to unitarity limit driven by increasing gluon density. • Approach to unitarity limit requires new dynamical phenomena to tame growth. • Possible effects: • ln(1/x) enhancement of gluon radiation. • Long range non-perturbative effects. • Where are taming effects necessary to avoid unitarity violations? • Question is independent of specific dynamics needed for taming. • s-channel unitarity requires knowledge of t-dependence.

10. Impact Parameter Picture

11. “Black Disk Limit”: • Assume dominantly imaginary amplitude. Unitarity Constraint Black Disk Limit (Central black region growing with decrease in x.) (figure from C. Weiss.)

12. Model the Basic Amplitude small d: 2-gluon form factor unity hard diffusion large d: nucleon form factor pion form factor soft diffusion (TCR et al., (2004)) for large d for small d ( α΄ possibly smaller.)

13. Nucleon Profile Functions: Q2 = 2 GeV2 (Note: gluon dipoles come with an extra factor of 9/4.)

14. Nuclear Targets: (See also V. Guzey talk) • Glauber-Gribov shadowing formalism for relating diffraction from a nucleon to total nuclear cross sections. • Collins factorization theorem for leading twist diffraction. • Use leading twist pQCD Q2 evolution. Note that nuclear shadowing is a leading twist effect. + (Collins (2002) ) Leading twist nuclear shadowing. (Frankfurt et al. (2002))

15. 2 0 8 b P x from bottom to top: x=10-2,10-3,10-4,10-5

16. 2 0 8 b P Heavy nuclei are ideal to study the approach to the black disk.

17. Impose unitarity by “Brute-Force” Very high energy limit and Q2 0: (TCR, Strikman (2006)) With the usual definition of the profile function: Except, if this yields, Unitarity enforced. In which case, Conservative Upper Limit

18. Growth of the cross-section soft + hard Pomeron (DL) Standard parameterization grossly violates unitarity Nearly factor of 10 increase in cross section.

19. Complication with light Nuclear Targets • Use leading twist nuclear shadowing: (Frankfurt, et al. (2002)) • Small sizes: • Large sizes: • Interpolate and impose “brute force” unitarity as in proton case.

20. Yields definite upper bound: But nonsensical result!

21. Black disk limit: nucleon vs. nucleus • Nuclear PDFs smear out matter in impact parameter space. • All regions of disk approach the BDL at same rapid rate with increasing energy. • But, partons are localized in nucleons! - Gaps between nucleons. • - Large diffractive cross section implies large nuclear shadowing. • Heavy nuclei, many nucleons, partons basically uniformly spread out in transverse space.

22. Approach to black disk limit based onnuclear parton distribution picture:

23. ¼ black disk limit for individual nucleons: transition energy,where each nucleon is black

24. Behavior in transition region: • Use Glauber-Gribov formula for nuclear shadowing with diffraction. • Impose unitarity constraints on x-section for configuration to scatter from the nucleon. • Define: • We automatically get:

25. Effective Nuclear Profile Function: b (fm)

26. Nearly factor of 7 increase as compared with 10 for nucleon (effect of shadowing.)

28. Modelling pp x-sections at very high energies • Assume dominantly imaginary amplitude (real Γ) - Unitarity constraint on profile: • Extrapolation to very high energies: • Use extrapolation of elastic differential cross section. = (e.g., Islam, Luddy, Prokudin (2003))

29. Modelling pp x-sections at very high energies (TCR, Stasto, Strikman (2008)) • Extrapolation to very high energies. • Important component of cosmic ray event generators. • Both hard and soft contributions: • Typically modelled in an eikonal/parton picture: • Hard part modeled by pQCD dijet formula. • Impact parameter dependence modeled / obtained from fits. • Soft part modelled with Regge theory; fits from elastic and total cross sections. hard soft (DPMJET, QGSJET, SIBYLL…) (Review: Engel (2003))

30. Hard Scattering • At least one (semi-)hard jet pair. distribution function P Hard scattering:use pQCD + … P

31. Hard Scattering

32. i n c ¾ j 2 t e s Hard Scattering 2.5 GeV 3.5 GeV 1.5 GeV

33. Information from DIS: GPDs • The generalized gluon PDF and deep inelastic J/ψ production. • Impact parameter space gluon distribution function. Describes transverse distribution of hard partons

34. Information from DIS: GPDs • Frankfurt, Strikman, Weiss (FSW(2004)): fit to 2-gluon form factor from J/ψ production: • In impact parameter space:

35. Information from DIS • pp 2jet + X cross section in impact parameter space.

36. Comparison with extrapolation:UHE • Identical partons, • CTEQ6M gluon PDF ptc = 3.5 GeV ptc = 2.5 GeV Islam et al.

37. Information from DIS

38. Inelastic 2-jet profile function exclusive probability

39. Inelastic 2-jet profile function

40. Comparison with extrapolation • Large contribution from large impact parameters. • Identical partons, • CTEQ6M gluon PDF extrapolated profile function using elastic cross section Small effect from correlations

41. Source of low-pt taming: • Very rapid growth of gluon distribution at small-x. • Below some value of x, non-linear effects come into play, growth is tamed. • In pp 2jets + X jets x-section, what is the role “saturation” of the gluon distribution? • Look at jet rapidity distribution. • Compare with Golec-Biernat-Wustoff model in DIS to estimate “saturation” scale.

42. Source of taming: Region where gluon PDF would lead to saturation in dipole-proton scattering • Apparently small effect from gluon saturation. (solid line for octet dipoles) pt = 2.5 GeV pt = 1.5 GeV • Dominant effect from multiple parton scattering.

43. Conclusion • Can we observe the transition between: • light nuclei (unitarity driven by black disk limit of the nucleon) and, • heavy nuclei (black disk limit of nucleus as a whole)? • What is the energy dependence of DIS cross sections in the black disk limit? • What can we say about extrapolations to very high energies? And how does this relate to pp / pA collisions in LHC and/or UHE cosmic rays? • Need detailed information about transverse structure in nuclei. (EIC)

44. Back up slides

45. Taming small-x growth: • Impose exponential cut-off at small-x or large sizes: • Small size x-section has hard Pomeron-type behavior: • Does taming occur near unitarity limit? (GBW model K. Golec-Biernat and M. Wüsthoff Phys.Rev.D59:014017,1999) ; (Q2 independent.)

46. Approach to unitarity limit: (using same t-dependence)

47. Comparison with extrapolation • Sensitivity to large-b at -t = .02 GeV2. • Integrand of Fourier transform to t-space. • Integrate to b = 1.5 fm. Percent of integral; • ptc = 1.5 GeV: 65 % • ptc = 2.5 GeV: 80 % • ptc = 3.5 GeV: 87 %

48. Comparison with extrapolation • Sensitivity to large-b at -t = .1 GeV2. • Integrand of Fourier transform to t-space. • Oscillations at large b.

49. Comparison with extrapolation • Transition from small to large b, affects cross section at t = 0. • Fraction of differential inelastic cross section from dijet production to expectation from Islam,Luddy,Prokudin fit. 1.5 GeV 2.5 GeV 3.5 GeV

50. Source of low-pt taming: • Very rapid growth of gluon distribution at small-x. • Below some value of x, non-linear effects come into play, growth is tamed. • In pp 2jets + X jets x-section, what is the role “saturation” of the gluon distribution? • Look at jet rapidity distribution. • Compare with Golec-Biernat-Wustoff model in DIS to estimate “saturation” scale.