Resummation of Large Endpoint Corrections to Color-Octet J/ Photoproduction
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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
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Outline
• Overview of problem
• Effective field theory review
• NRQCD– What is missing
• SCET
• Results for color-octet J/ photoproduction
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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
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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
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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
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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
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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
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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
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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
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Comparison to data
Cacciari and Kramer, PRL 76, 4128 (1996)
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Comparison to data
Butterworth and Wing, Rept. Prog. Phys. 68, 2773 (2005)
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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)
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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
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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
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Gauge invariance restrictions
Factorization
Only coupling to ultrasoft sector
}Introduce usoft Wilson line:
Field redef:
Ultrasoft decouples:
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In Pictures
Heavy/soft modesdo not interact withcollinear modes
⇒ Rate factors!
W
W†
Y †
Y
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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
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Results of Calculation• Factorization theorem:
• With logs summed:
Nonperturbative shape function Gluon pdf
Logs are here Convolutedwith pdf
Includes small
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Octet contribution at endpoint
Shape function onlyLogs summed
only
Logs summed and shape function
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Differential cross section
Color singlet
Color octet
Singlet + octet
Rise due to small
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Outlook
• Unlike previous calculations:– Include small , need this data– Include diffractive part
• In sum over states
• Work in progress: Color-singlet contribution