1. Recent study on γγ*  ηc transition form factor and χcJ  γγ
桑 文 龙
SouthWest University, ChongQing
In collaboration with Feng Feng, Yu Jia, Shuang-Ran Liang
QCD Study Group 2016 workshop
上海交大 (SJTU)

2. Contents Brief introduction to NRQCD factorization
NNLO QCD correction to γγ*  ηc form factor and confront the BaBar data
NNLO QCD correction to χcJ  γγ and confront the BESIII data
Summaries

3. Nonrelativistic QCD (NRQCD): the modern effective field theory to tackle heavy quarkonium
Caswell, Lepage (1986); Bodwin, Braaten, Lepage (1995)
This scale separation is
usually referred to as
NRQCD factorization.
The NRQCD short-dist.
coefficients can be computed in perturbation theory, order by order

4. production at hadron colliders ……
NRQCD gains popularity in quarkonium physics (see Brambilla et al. EPJC 2011 for a recent review)
Nowadays, NRQCD becomes an important approach to tackle various quarkonium production and decay processes
charmonium:
bottomonium:
Exemplified by
at B factories
production at hadron colliders ……

5. The underlying problems
Most of the NRQCD successes based on the NLO QCD predictions. However, the NLO QCD corrections are often large: Zhang et.al. Gong et.al. Campbell et.al. Mackenzie et.al. … …

6. Question: How about the convergence of the QCD correction?
So calculating the higher order QCD correction is very necessary to test NRQCD predictions!

7. Investigation on γγ*  ηc form factor Experiment
BaBar Collaboration: Phys.Rev. D81 (2010)
Babar measures the transition form factor in the momentum transfer range from 2 to 50 GeV2.

8. Investigation on γγ*  ηc form factor Experiment
BaBar Collaboration: Phys.Rev. D81 (2010)

9. Investigation on γγ*  ηc form factor Experiment
BaBar Collaboration: Phys.Rev. D81 (2010)
The solid curve is from a simple monopole fit:
The dotted curve is from pQCD prediction
Feldmann and Kroll, Phys. Lett. B 413, 410 (1997)

10. Investigation on γγ*  ηc form factor Previous investigation
factorization: Feldmann et.al., Cao and Huang
Lattice QCD: Dudek et.al.,
J/ψ -pole-dominance: Lees et.al.,
QCD sum rules: Lucha et.al.,
light-front quark model: Geng et.al.,
All yield predictions compatible with the data, at least in the small Q2 range.
So far, so good. Unlike γγ*  π0 , there is no open puzzle here

11. Investigation on γγ*  ηc form factor Our motivations
Model-independent method is always welcome.
In the normalized form factor, nonperturbative NRQCD matrix element cancels. Therefore, our predictions are free from any freely adjustable parameters!
Is LO NRQCD prediction enough?
The momentum transferred is not large enough, we are not bothered by the large logarithmic resummation Jia and Yang

12. Investigation on γγ*  ηc form factor Theoretical descriptions
Definition for form factor:
NRQCD factorization defines:
Factorization scale
Short-distance coefficient
We are going to compute it!

13. Investigation on γγ*  ηc form factor Theoretical descriptions
Upon general consideration, the SDC can be written as
IR pole matches anomalous dimension of NRQCD pseudo-scalar density
RGE invariance

14. Investigation on γγ*  ηc form factor Theoretical calculation
Tree-level SDC
NLO QCD correction

15. Investigation on γγ*  ηc form factor Feynman diagrams
Numer of diagrams 2 8
108 12

16. Investigation on γγ*  ηc form factor NNLO corrections
At , the value of is compatible with asymptotic behavior
in Jia, Yang, NPB 2009
Light-by-light
UV/IR finite
regular
Reproduce
known NNLO
corr. to ηc->γγ
Czarnecki et al. 2001

17. Investigation on γγ*  ηc form factor NNLO corrections
Contribution from lbl is not always negligible!

18. γγ*  ηc : NNLO prediction seriously fails to describe data!
Investigation on γγ*  ηc form factor
Theory vs Experiment
Our Prediction is free of any nonperturbative parameters!
γγ*  ηc : NNLO prediction seriously fails to describe data!

19. Investigation on γγ*  ηc form factor Prediction to γγ*  ηb
Convergence of perturbation series is rather good.
Wait for CEPC/ILC to test our prediction?

20. Investigation on c0,2 2γ Experiment
 BESIII gives updated measurements:
(BESIII, PRD, 2012)
Decay width of c0  2 γ
Ratio of decay width of c2  2 γ to that of c0  2 γ
Ratio of the two-photon widths for helicity λ= 0 and
helicity λ= 2 components in c2  2 γ

21. Investigation on c0,2 2γ Various theoretical predictions
 Review of previous theoretical works
Numerous work based on phenomenological models
R. Barbieri, R. Gatto, R. Kogerler (PLB, 1976)
L. Bergstrom, G. Hulth, H. Snellman (Z. Phys. C, 1983)
Z. P. Li, F. E. Close, T. Barnes (PRD, 1991)
H. W. Huang, C. F. Qiao, K. T. Chao (PRD, 1996)
S. N. Gupta, J. M. Johnson, W. W. Repko (PRD, 1996)
G. L. Wang (PLB, 2007)
C. W. Hwang, R. S. Guo (PRD, 2010)
… … …
Lattice QCD prediction for c0 2 γ by J. Dudek and R. Edwards
(PRL 2006), obtains partial width of ,
based on the algorithm proposed by X. D. Ji and C. W. Jung (PRL
2001; PRD, 2001)

22. Investigation on c0,2 2γ NRQCD formula
According to the NRQCD factorization
The short-distance coefficients can be systematically expanded

23. Investigation on c0,2 2γ NRQCD formula

24. Investigation on c0,2 2γ NRQCD calculation

25. Investigation on c0,2 2γ NRQCD calculation

26. Investigation on c0,2 2γ Theory vs Experiment
(BESIII, PRD, 2012)
By assuming HQSS, we predict

27. Investigation on c0,2 2γ Theory vs Experiment

28. Investigation on c0,2 2γ Theory vs Experiment
Giving up HQSS

29. Summaries
The NNLO NRQCD predictions to γγ*  ηc form factor and c0,2 2γ fail to explain the experiment!
The convergence of QCD expansions for the short-distance coefficients is poor!
NRQCD predictions should be tested at high order!

30. Thanks for your attention!