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NRQCD Factorization in Heavy Quarkonium Production · 24M

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October 17, 2007 Jianwei Qiu, ISU 1 NRQCD Factorization in Heavy Quarkonium Production Jianwei Qiu Iowa State University International workshop on Heavy Quarkonium 2007 17-20 October 2007, DESY Hamburg, Germany Based on work with Z.-B. Kang, G.C. Nayak, and G. Sterman
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Page 1: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU1

NRQCD Factorizationin

Heavy Quarkonium Production

Jianwei QiuIowa State University

International workshop on Heavy Quarkonium 2007 17-20 October 2007, DESY Hamburg, Germany

Based on work with Z.-B. Kang, G.C. Nayak, and G. Sterman

Page 2: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU2

Perturbative QCD vs NRQCD

Connect pQCD factorization to NRQCD factorization

Outline

Summary and conclusions

Heavy quarkonium production

Color transfer in associated production of heavyquarkonium

Page 3: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU3

Perturbative QCDQCD Lagrangian:

QCDnF=1

6

Asymptotic freedom and perturbative QCD:

( ) ( ) ( )Pert ,

2

ns

nn

Q QCα

σπμ

μ⎡ ⎤

= ⎢ ⎥⎣ ⎦

Infrared Safe

μRenormalization scale:

PQCD factorization for hadronic observables:

( ) ( ) ( )Hadroni Pc ert F Fˆ 1 f, m1 fm 1 m, f,Q QQ

Oμσ σ ϕ μ⎛ ⎞

≈ ⊗ + ⎜ ⎟⎝ ⎠

Infrared safe Universal – Predictive power

F :μ Factorization scale

Power series of ( )sα μ

Page 4: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU4

Non-Relativistic QCD (NRQCD)NRQCD Lagrangian:

nF=1

3 or 4

Limitation:Powerful for a process with available kinetic energy: Mv2 << Mc2

Formalism is ideal for heavy quarkonium decayAdditional complications for production with s >> (2M)2

Caswell, Lepage, Phys. Lett. B, 1986Bodwin, Braaten, Lepage, PRD, 1995

Pauli spinor for antiquark

Pauli spinor for heavy quark

Bodwin, Braaten, Lepage, PRD, 1995

Page 5: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU5

Heavy quarkonium production in NRQCDNRQCD factorization for heavy quarkonium production:

( ) [ ] ( )( )NRQCD ,ˆ, , , nn sQQn

s M s M M v Oσ σ α=∑

Infrared Safe Universal – Predictive power

[ ] ( )( ),ˆ , ,n sQQ s M vMσ α : power series of both and ( )s Mα v

( )† † ' † † † ', ,n n n n nN

H HO H H a aN Nχ κ ψ ψ κ χ χ κ ψ ψ κ χ= =∑

Bodwin, Braaten, Lepage, PRD, 1995Brambilla, et al. hep-ph/0412158Lansberg, hep-ph/0602091, …

vwith heavy quark velocity:This factorization formalism has not been proved

Expectations:

It does not work when additional heavy quark velocity is involved

It works for inclusive heavy quarkonium production when, unless for exclusive production2 4M s M< <

with a proper insertion of gauge link, the formalism works to two-loop

such as, the associated production

Nayak, Qiu, Sterman, 2005, 2006

Nayak, Qiu, Sterman, PRL 2007

Page 6: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU6

Inclusive J/ψ + charm production:

Belle:NRQCD-LO: 0.07 pb∼

Kiselev, et al 1994,Cho, Leibovich, 1996Yuan, Qiao, Chao, 1997…Zhang, Chao, 2007 (NLO)

Ratio to light flavors:

Belle:

Production rate of is larger thane e J ccψ+ − →

Message:

,e e J ggψ+ − → , ...e e J qqψ+ − →combined ?

all these channels:

Heavy quarkonium associated production

Page 7: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU7

Kinematically preferred configuration:

Production rate of a singlet charm quark pair is dominated by the phase space where s3=(P1+P2+P3)2 or s4=(P1+P2+P4)2 near its minimum

Color transfer enhances associated heavy quarkonium production

A heavy quark as a color source to enhance the transition rate for an octet pair to become a singlet pair

Associated production at B-factory

NRQCD formalism does not apply when there are more thanone heavy quark velocity involved

Singlet

( )24P p+

( )23P p+

Nayak, Qiu, Sterman, PRL 2007

Page 8: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU8

Soft gluon enhancement – color transferSoft gluons between heavy quarks: active pair: P1, P2; spectators: P3, P4

There are three heavy quark velocities:

Soft gluon between a heavy quark pair:

1 s

ij

iεαβ

NRQCD approach is not well defined in this region

Page 9: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU9

NLO correction to the amplitude:

Estimate the enhancement factor from NNLO in NRQCD approach:

2 2

8 41

22 2

2s

S in g le t O cN

ts

N Le

O L OtA AvC α π

βε→

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠∼

Associated production is enhanced

Does not contribute to NLO production rate in NRQCD Zhang, Chao, PRL 2007

Velocity expansion:

1, 1SS

vββ

< <

Velocity-ordered region:

the enhancement factor:

All other two-loop diagrams give a single pole ! Nayak, Qiu, Sterman, 2007

Page 10: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU10

LO hard parts with color factor:

OctetSinglet

( )24P p+

( )23P p+

Matrix elements:2 2 2

82 s v Oπ αε

Two terms are equally important if 0.3sβ ∼Same feature for heavy quark fragmentation

Numerical enhancement from NNLO

Kang, et al. 2007

Nayak, Qiu, Sterman, 2007

Page 11: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU11

NRQCD factorization has not been proved theoretically

NRQCD Factorization fails for low pT:

NRQCD

Quantuminteference

NRQCD Factorization might work for large pT

Spectator interactions are suppressed by (1/pT)n

Factorization is necessary for the predictive power

Hadronic heavy quarkonium production

Low pT requires pQCD kT-factorization

Collins, Qiu, PRD 2007

Page 12: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU12

Combination of pQCD and NRQCDHeavy quarkonium production when PT >> 2M

( )2 211

2

T MP×∝

2

2 1

TP⎛ ⎞⎜ ⎟⎝ ⎠

When , fragmentation contribution dominates the production( )2 22TP M

Combination of pQCD and NRQCD factorization (not yet proved):pQCD factorization to isolate heavy quarkonium physics intothe universal fragmentation function

2 22

NRQCD factorization to isolate non-perturbative physics ofthe fragmentation function into NRQCD matrix elements

2

2

+ T

MP

Also see LansbergIn the workshop

Page 13: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU13

Step 1: Fragmentation factorizes from the rest

fragmentation

Still has long-distance physicsdue to incoming hadrons

Reduced diagram

Fragmentationfunction

PQCD collinear factorization – 2 stepsNayak, Qiu, Stermen, 2005

Page 14: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU14

Step 2: Cancellation of remaining IR final state:Note: Uncut loops are short distance

Remaining soft-interaction absorbed into the Wilson lines of PDFsH is IR safe!

Page 15: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU15

PQCD Factorization at large pT:

Universal fragmentation function – single parton to a hadron H:

Fragmentation function

e.g. a gluon to a hadron H (such as J/ψ):

For the fragmentation function, all that is left is the gluon source: 2 Gauge link:

A necessary condition for the universality:

The fragmentation function is independent of the nμ

Page 16: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU16

Proposed NRQCD factorization:

Proved pQCD factorization for single hadron production:

Prove NRQCD Factorization

with IR safe

gauge invariant and universal

independent of the direction of the Wilson lines

Connection to NRQCD factorization

To prove:at 0 2 cmμ ∼

Page 17: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU17

Conventional operator definition

Products of color and spin matrices, covariant derivatives

Fields at is not truly local

Operator-valued gauge transformations (as to gauge) do not commute with

Only color-singlet give gauge invariant ,the color-octet operators are not gauge invariant

, are heavy quark, antiquark fields

Need to modify NRQCD matrix elements

Nayak, Qiu, Stermen, 2005

Page 18: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU18

Resolution: supplement fields by Wilson lines – gauge links:

Our new, gauge invariant operators:

Are the “coefficient” functions IR safe?

Our NNLO answer is no The gauge links are necessary

Do the gauge links absorb all IR divergences?

Can’t tell yet for sure. OK at NNLO

Two remaining questions for NRQCD factorization:

An all-order proof of NRQCD factorization at high PTis still lacking, and urgently needed

Page 19: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU19

Summary and conclusionsNRQCD formalism is ideal for heavy quarkonium decay

NRQCD formalism does not apply for associated production

due to multiple heavy quark “velocities”

Associated production of heavy quarkonium could be strongly

enhanced by the soft color transfer

Combination of PQCD and NRQCD factorization might work

for hadronic heavy quark production at high PT

NRQCD factorization for the fragmentation function is valid

to the NNLO order in αs at a finite v

An all-order proof of NRQCD factorization at high PT is still lacking,

and urgently needed

Thank you!

Page 20: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU20

Backup slices

Page 21: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU21

LO:

NLO:

Color neutralization is IR divergent – nonperturbative!

IR divergences cancel between real and virtual diagrams

Eikonal line

Short-distance

n=[QQ]singlet

Topologially-factorized the matrix element

Factorization works to NLO at v2

Velocity expansion:

Page 22: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU22

Diagrams:

Factorization fails at NNLO

All IR divergences cancel between real and virtual diagrams, except

2k

Page 23: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU23

2k

Explicit calculation at NNLO at v2 – I

The infrared divergent expression to order q2 ~ v2:

The result is:

Page 24: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU24

This non-topological IR divergence Cannot beabsorbed into the conventional NRQCD matrix elements

Explicit calculation at NNLO at v2 - II

In heavy quark pair’s rest frame:

IR poles would appear in coefficient function at NNLOunless we have eikonal interactions to absorb them

Can be absorbed into the modified matrix elements

Page 25: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU25

Factorization at a finite v?

Velocity expansion is not efficient for charmonium

Large phase space available for gluon radiation:

Large possible velocity in production:

Very different from decay:

2 2 2 2 24 4 4 6 GeVC D CQ M M M− ⇒ − ≈

Still no solution for the polarization data even if NRQCD factorization is valid

High order terms are very important

Page 26: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU26

Factorization at NNLO and all order in v2

with

Calculation with a finite v

Reproduce the v2 result when expanded

Page 27: NRQCD Factorization in Heavy Quarkonium Production · 24M

October 17, 2007 Jianwei Qiu, ISU27

Significance?

The IR poles at all orders of v-expansion at NNLO are independentof the direction of the Wilson line and universal – consistent with factorization

Although limited to NNLO, our result suggests that the decouplingof light parton dynamics from heavy quark pair production isrobust in perturbation theory at the level of infrared divergence– high orders?

Although the eikonal approximation do not cover many terms in general NRQCD velocity expansion, in particular, those dealing with spin, it should cover all perturbative infrared divergences


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