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

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

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

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

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 Corresponds to short wavelengths Keep light (long wavelength) modes

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 Want to describe (low energies) Don’t know this (high energies) EFT gets here down Extra particles whose propagation not relevant for low energies

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

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 Calculable in perturbation series Scale as some v n Can be octet Color singlet model New octet contribution Remove d.o.f. ~ m, mv

J/  photoproduction Lowest order color-singlet contribution Lowest order color-octet contribution Scales like  s 2 v 3 Scales like  s v 7 (  s ~ v 2 ) Peaked at endpoint

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

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

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

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

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

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

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

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

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

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

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

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