Resummation of Large Endpoint Corrections to Color-Octet J/  Photoproduction

1. Resummation of Large Endpoint Corrections to Color-Octet J/Photoproduction Using Effective Field Theories Near Corners of Phase Space Adam Leibovich University of Pittsburgh 10/17/07 With Sean Fleming and Thomas Mehen International Workshop on Heavy Quarkonium 2007, DESY

2. Outline • Overview of problem • Effective field theory review • NRQCD • What is missing • SCET • Results for color-octet J/ photoproduction

3. Overview of problem • Looking at • Interesting because… • Want to compare theory to experiment • How do we calculate this theoretically? Since multi-scale process, can use Effective Field Theory

4. An Effective Field Theory is an approximation to the true underlying theory, with enough in it to describe the physics of interest. i.e., true theory = Standard Model? EFT = depends on question Particularly useful when there are multiple well-separated scales

5. Effective Field Theory Review • When there are well-separated scales, can find small dimensionless numbers • Goal is to try to expand in one of these small numbers • Equivalently, shrink large energies (small distances) to a point • Think multipole expansion • In Field Theory, remove heavy d.o.f. • Effects turn into coefficients Keep light (long wavelength) modes Corresponds to short wavelengths

6. What is needed? • The “light” degrees of freedom • If EFT contains correct d.o.f., get the IR correct • If missing d.o.f., get problems Don’t know this (high energies) Extra particles whose propagation not relevant for low energies EFT gets here down Want to describe (low energies)

7. W integrated out. Effects contained in coefficient function Expansion in small parameter Quick EFT Example Four-Fermi Interaction

8. Remove d.o.f. ~ m, mv NRQCD Review • Effective field theory relevant for heavy quark pairs (v typical velocity of quark) • Scales: • Expansion in s and v 1 • Production rate written as Color singlet model New octet contribution Calculable in perturbation series Scale as some vn Can be octet

9. Peaked at endpoint J/ photoproduction • Lowest order color-singlet contribution • Lowest order color-octet contribution Scales like s2v3 Scales like sv7 (s ~ v2)

10. Comparison to data Cacciari and Kramer, PRL 76, 4128 (1996)

11. Comparison to data Butterworth and Wing, Rept. Prog. Phys. 68, 2773 (2005)

12. Color-Octet Growth at Endpoint • As approach endpoint • Fixed-order perturbative and nonperturbative calculation breaks down • Large perturbative corrections • Large nonperturbative corrections (Sudakov logs) Need to sum (Motion of quark pair)

13. Need for SCET • NRQCD doesn’t contain correct d.o.f. • At endpoint need both soft and collinear modes • NRQCD only has soft d.o.f • SCET couples collinear and soft d.o.f. • Was created to sum Sudakov logarithms • Expansion in s and  ~ Q • Couple NRQCD and SCET for photoproduction • Only color-octet so far

14. SCET Intro • Systematic expansion in • Degrees of freedom: • Collinear particles with • Soft particles with • Ultrasoft particles with • By using gauge invariance, operators constrained • Field redefinition allows leading order factorization theorems

15. Gauge invariance restrictions Factorization } Only coupling to ultrasoft sector Introduce usoft Wilson line: Field redef: Ultrasoft decouples:

16. In Pictures Y Y † W† W Heavy/soft modes do not interact with collinear modes ⇒ Rate factors!

17. Overview of Calculation • Match QCD onto SCET (and NRQCD) • Calculate rate (includes low , high z) • Run down to low scale (sums logs) Near z ~ 1 Up to few GeV

18. Includes small Results of Calculation • Factorization theorem: • With logs summed: Nonperturbative shape function Gluon pdf Logs are here Convoluted with pdf

19. Octet contribution at endpoint Shape function only Logs summed only Logs summed and shape function

20. Differential cross section Rise due to small Singlet + octet Color singlet Color octet

21. Outlook • Unlike previous calculations: • Include small , need this data • Include diffractive part • In sum over states • Work in progress: Color-singlet contribution