Resummation of Large Endpoint Corrections to Color-Octet J/

Photoproduction

Adam LeibovichUniversity 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?

γp → J /ψ + X

Since multi-scale process, can useEffective Field Theory

An Effective Field Theory is an approximation to the trueunderlying 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

W

μ

eνμ

νe

Four-Fermi Interaction

g2eΓανe

1q2 −MW

2 νμΓαμ

μ

eνμ

νe

−g2

MW2

eΓαν e ν μ Γα μ + Oq2

MW4

⎛

⎝⎜⎞

⎠⎟

q2 = MW

2

W integrated out.Effects contained in coefficient function

Expansion in small parameter

q2

MW2

GF = 2g2

MW2

NRQCD Review

• Effective field theory relevant for heavy quark pairs (v typical velocity of quark)

– Scales:

– Expansion in αs and v 1

• Production rate written as

Calculable in perturbation series

Scale as some vn

Can be octet

Color singletmodel

New octetcontribution

Removed.o.f. ~ m, mv

J/ photoproduction• Lowest order color-singlet contribution

• Lowest order color-octet contribution

Scales like αs2 v3

Scales like αs v7 (αs ~ v2)

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 correctionsNeed 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 αs and ~ Q

• Couple NRQCD and SCET for photoproduction

– Only color-octet so far

1−z

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 modesdo not interact withcollinear modes

⇒ Rate factors!

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 Convolutedwith pdf

Includes small

Octet contribution at endpoint

Shape function onlyLogs 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