a r X i v :n u c l -t h /9911076v 2 25 N o v 1999The role of S 11resonance in πN Scattering
1Yohanes Surya,2Kam Chan Hin,2T.C Au Yeung,2W.Z.Shangguan,3Wirawan Purwanto,4Eddy Yusuf ,5Darmadi Kusno,2Oki Gunawan 1Universitas Pelita Harapan,Indonesia 2Nanyang Technological University,Singapore 3Physics Dept.College of William and Mary 4Physics Dept University of Indonesia 5Physics Dept.Florida State University Abstract We analyze Pion Nucleon Scattering up to 700MeV using a simple,rela-tivistic,unitary model.1The kernel of the integral equation includes nucleon,roper,delta,D 13as well as S 11poles with their corresponding crossed pole terms approximated by contact interactions.The s -and p -wave phase shifts are calculated from the model and shown to agree very well with the val-ues derived from πN scattering data.2All parameters which involve S 11are
presented.
I.Introduction
Even though pion-nucleon scattering has been extensively studied for many years,there still remains a number of interesting problems to be ex-plored,especially with the construction of powerful new facilities such as TJNAF(Thomas Je?erson National Lab)in Virginia,USA.In this brief paper we analyze pion nucleon scattering up to700MeV laboratory kinetic energy of pion using a simple,covariant and unitary model ofπN scattering.1 In this work,theπN scattering amplitude is obtained as a solution of a relativistic wave equation in which the pion and eta is restricted to their mass shell in all intermediate states.The rationale for this approach has been dis-cussed by Gross and Surya.1The kernel of the relativistic integral equation includes undressed delta(?),roper(N?),D13and S11poles in addition to the undressed nucleon(N)pole.The kernel also includes contributions derived from crossed N,?,N?,D13,and S11diagrams,as well as fromσ-andρ-like exchange terms.To keep the model simple the crossed terms are approxi-mated by contact interactions as described in reference[1].
TheπNN coupling is taken to be the superposition of both pseudoscalar and pseudovector coupling and theπNS11coupling is taken as a superposi-tion of both scalar and vector coupling.However for simplicity we take only scalar coupling for theηNS11coupling(the results obtained also show this tendency).The Feynman rule for theπNN andπNS11vertex are given as follows,
πNN vertex:Γ0N=gπNN λ+1?λ
γμkμ τi
m
ηNS11vertex:Γ0η=igητ0
k is the pion momentum,λis a mixing parameter that is determined by the requirement that the nucleon mass be unshifted by the interaction,1g1,g2 and gηare the coupling constants adjusted to?t data and m is the nucleon mass.TheπNN coupling gπNN is chosen to be equal to13.5the same as in the reference[1].
II.The model
The model has been described in detail in reference[1]except for the S11 resonance.Figure(1)shows additional kernels that one needed to describe
Figure1:
Figure2:
the S11resonance.Consistent with the previous model,the crossed diagram is approximated as a contact interaction.
Figure(2)shows the self energy of S11.M c is the in?nite sum of iter-ated the contact diagrams.The Mππc,Mπηc,Mηπc and Mηηc are treated as a coupled channel.
III.Results
Due to the contact interactions approximation,the model allows us to
?t the spin1
2phase shifts separately.Our?t to S11,S31,P11and
P31phase shifts are given in Figures3and4.The?t is good.The S11 phase shifts are particularly interesting.Figure5shows how the total is
built up from individual contributions,the curves in the?gures show the result when the kernel(i)includes only the direct nucleon pole term and the contact term derived from crossed nucleon exchange plus the combined σ-andρ-like contact terms(the dot line),(ii)the terms in(i)plus the additionalρ-likeππNN contact term(the dot-dashed line),(iii)the terms in(ii)plus roper driving terms(dashed line)and?nally(iv)the total result, which includes the terms in(iii)plus S11driving terms(solid line).
From our calculation the contribution from S11(includingηN channel) plays an important role to pion nucleon interaction at energies above600 MeV as shown in Figure5.
Table1shows parameters that are used to?t the data(column2).The table includes e?ective masses(m)and widths(Γ)of the resonances(col-umn3).In the table,Λrepresents the baryon cut o?mass,g′is the inelastic coupling in the roper channel.We found the S11width is smaller compared to Particle data group(～100to250MeV).3This maybe is related
to our smaller value of gηNS
11=1.19as compared to e?ective models results3
or QCD sum rules using interpolating?eld results4which is around2.
parameter value e?ective value
Λ1205.8
Λ(N?)1961.3
gπNN? 6.924
m N?1458.01463.5
ΓN?244.3
Cρ0.911
g′0.793
g2 1.910
g11.222
Λs112666.3
m s111540.61567.0
Γs1157.4
gηNS
11
1.19
IV.Conclusions
We have successfully extended the relativistic,simple and unitary model of pion nucleon scattering1to analyze the pion nucleon scattering up to700 MeV which includes the S11resonance.We?nd that the model works well despite of our meson on-shell approximation in all intermediate states.We also?nd that the inclusion ofηN channel tends to give the correct high energy
behavior.The coupling of pion and eta to the nucleon and S11is smaller than expected.This would be clari?ed by some quark model calculations.
V.Acknowledgment
YS would like to thank School of Electrical&Electronic Engineering Nanyang Technological University for their support and for kind hospitality during the visit.
Figure3:S11phase-shifts
Figure4:P-wave phase shifts
Figure5:S-wave phase shifts
References
1Franz Gross and Yohanes Surya,Phys.Rev.C47,703(1993). 2SAID data base(1999),http:\\https://www.doczj.com/doc/1e18080059.html,\analysis\pin