1. Guangxi Normal University
Wei-Hong Liang
Guangxi Normal University
In collaboration with:
C.W. Xiao, and E. Oset
Wei-Hong Liang – Chiral 13, Beijing, October 27, 2013

2. Wei-Hong Liang – Chiral 13, Beijing, October 27, 2013
CONTENTS
Introduction and motivation
Formalism
Results and discussions
Summary
Wei-Hong Liang – Chiral 13, Beijing, October 27, 2013

3. Wei-Hong Liang – Chiral 13, Beijing, October 27, 2013
Introduction and motivation
Taking the Fixed Center Approximation (FCA) to the Faddeev equations is an effective tool to deal with the three-body hadronic interactions.
This method is technically simple and accurate when dealing with bound states.
M. Bayar, J. Yamagata, E. Oset, PRC84(2011)015209;
E. Oset, D. Jido, et al., NPA881(2012)127;
solved the puzzle
J.J. Xie, A. Martinez Torres, E. Oset, PRC83(2011)055204;
L. Raca, C.W. Xiao, M. Bayar, J.J. Xie, A. Martinez Torres, E. Oset, PRC84(2011)015209; PRD84(2011)034037; NPA833(2012)57; PRD82(2010)054013; PRD82(2010)094017;
Wei-Hong Liang – Chiral 13, Beijing, October 27, 2013

4. BESSIII, PRL107(2011)182001;
BESSIII, PRL106(2011)072002;

5. Weihong Liang – Chiral 13, Beijing, October 27, 2013
Formalism
the basic idea of the FCA to the Faddeev Eqs.
-- Assumes a pair of particles (1 and 2) forming a cluster;
-- Particle 3 interacts with the components of the cluster,
undergoing all possible multiple scattering with the components.
Weihong Liang – Chiral 13, Beijing, October 27, 2013

6. Weihong Liang – Chiral 13, Beijing, October 27, 2013
the Faddeev Equations under the FCA
Weihong Liang – Chiral 13, Beijing, October 27, 2013

8. a factor coming from the normalization of the fields.

9. Three coupled channels:1 2 3
The Bethe-Salpeter Equation in coupled channels:

10. (divergent)
Following the dimensional regularization scheme, there are two regularization parameters:
The change causing by a change of μ can be cancelled by a change
of the subtraction constant a(μ).
(the only free parameter in our calculation)

11. F.K. Guo, R.G. Ping, P.N. Shen, et al., NPA773(2006)78;
Bingham, NPB41(1972)1;
Mercer, NPB32(1971)381;
Estabrooks, NPB133(1978)490;

12. Results and discussions
the three-body scattering amplitues
There is a peak at around 1490 MeV.
The same result was obtained in
[M. Albaladejo, J.A. Oller, L. Roca, PRD82(2010)094019].

13. (cusp effect)
A peak at around 1940 MeV, close to the threshold (1942MeV).
This peak should be an enhancement effect of the threshold.

14. Wei-Hong Liang – Chiral 13, Beijing, October 27, 2013

15. Wei-Hong Liang – Chiral 13, Beijing, October 27, 2013
the uncertainties coming from the change in parameters
The results that we get are rather solid.
Wei-Hong Liang – Chiral 13, Beijing, October 27, 2013

16. discussions and comments
How far are we from creating a resonance structure in the
system?
The peak move by about 6 MeV below the threshold.
The cusp character of the amplitude is quite a stable result and we cannot associate a physical state to it.
comment on the results of Ref. [80]
M. Albaladejo, J.A. Oller, L. Roca, PRD82(2010)094019;
Ref. [80]:

17. Wei-Hong Liang – Chiral 13, Beijing, October 27, 2013
comment on Ref. [96]
A. Martinez Torres, et al., PRD84(2011)074027;
Ref. [96]:
unexpected, since qualitative agreement between the FCA and the Faddeev calculations for bound states was shown in many three-body systems, like
We call for further calculations of the system using different approaches in order to clarify the situation.
Wei-Hong Liang – Chiral 13, Beijing, October 27, 2013

18. Wei-Hong Liang – Chiral 13, Beijing, October 27, 2013
Summary
We investigate the three-body systems of and by taking the fixed center approximation to Faddeev equations, and find:
— a stable resonance structure around 1490 MeV in ,
associated to
— only an enhancement effect of the threshold in
Wei-Hong Liang – Chiral 13, Beijing, October 27, 2013

19. Wei-Hong Liang – Chiral 13, Beijing, October 27, 2013
Acknowledgements
We would like to thank J. A. Oller, M. Albaladejo, F. K. Guo and A. Martinez Torres for useful discussions.
Thank you for your attention!
Wei-Hong Liang – Chiral 13, Beijing, October 27, 2013