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

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

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

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

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

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

a factor coming from the normalization of the fields.

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

(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)

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;

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].

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

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

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

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]:

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

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

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