Strategies for baryon resonance analysis

CPC(HEP & NP), 2009, 33(12): 1127- -1131Chinese Physics CVol. 33, No.12，Dec., 2009Strategies for baryon resonance analysisM. Doring' C. Hanhart1,2 HUANG Fei(黄飞)* s. Krewaldl.2:1) U.-G. MeiBner12.41 (Institut fir Kernphysik and Jilich Center for Hadron Physics, Forschungszentrum Jilich, D-52425 Jilich, Germany)2 (Institute for Advanced Simulation, Forschungszentrum Jilich, D-52425 Jilich, Germany)3 (Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602, USA)4 (Helmholtz-Institut fir Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical Physics,Universitat Bonn, NuBallee 14-16, D-53115 Bonn, Germany)Abstract The analytic properties of scattering amplitudes provide a meeting point for experimental andtheoretical investigations of baryon resonances. Pole positions and residues allow for a parameterization ofresonances in a well- defined way which relates different reactions. The recent progress made within the Jilichmodel is summarized.Key words pion-nucleon scattering, meson exchange model, analytic structure of the scattering amplitudePacs 14.20.Gk, 13.75.Gx, 11.80.Gw1 Introductionmodel is a coupled channel approach which includeseffective two-pion nucleon channels in addition to theCurrently, there is rapid progress in experimen-pion- nucleon and the etar nucleon channels. The ana-tal investigations of the baryon resonance spectrumlytical structure of the model allows continuation intoup to center of mass energies of 3 GeV. In this en-the complex plane.ergy region, resonances which couple to many decaychannels, in particular multi-pion final states, mayResonanceanalysis within thehave large widths, and overlap. For photon-inducedJilich modelreactions, complete experimental studies of the po-larization degrees of freedom are being performed inAt a first glance, a theoretical model appearsorder to resolve ambiguities in the partial wave anal-to have no difficulty in distinguishing a backgroundysis. Unfortunately, no systematic investigations ofcontribution from a resonance contribution. Start-the polarization dependence of the pion-nucleon reac-ing from a field-theoretical Lagrangian, there is ation are available in this energy range, which makesunique way to classify diagrams into one-line re-an improvement of the classical partial wave analy-ducible and irreducible ones. The bare interactionses by Cutkosky, Hohler and Arndt and collaboratorscan be split into so-called pole diagrams and non-dificut'-司.pole diagrams. The iteraton of the non-pole diagramsThe extraction of resonance parameters from ain the Lippmann Schwinger equation results in whatgiven partial wave analysis has to provide a separa-commonly is called the non-pole T matrix. Given thetion of the partial wave armplitude into a resonantnon-pole T-matrix, one proceeds to compute dressedand non-resonant part. Here we want to study thatvertices and self energies. Ultimately, one achieves toseparation within the framework of a theoretical ap-split the exact T-matrix into a non-pole and a poleproach based on effective Lagrangians, such asos .part:The present contribution focusses on recent resultsobtained within the Jilich model9- -13]. The JilichT= rNP +TP .Received 7 August 2009* Supported by DFG (Deutsche Forschungagemneinschaft, Gz: DO 1302中国煤化工gh funds provided tothe virtual institute "'Spin and Strong QCD”(VH-V1-231), EU-Researchy "Study of StronglyInteracting Matter”(HadronPhysics2, grant n. 227431 ) under the SeventhYHCNMHGFG(TR16),COSYFFE grant No. 41445282 (COSY-58)1)E- mail:s.krewald@fz-juelich.de@2009 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Instituteof Modern Pbysics of the Chinese Academy of Sciences and IOP Publishing Ltd1128Chinese Physics C (HEP & NP)Vol. 33The construction of TP is sunmarized in Fig. 1.The pole T-matrix contributes:a_1a_12o= (品(TrE")+二2》). (5)==Adding the constant defined by Eq. (4) to the actual借”{”G TNPpole contribution, or in other words, using Eq. (2),ne obtains a reasonable appproximation to the fullE=====●=calculation (dashed red line). As one moves awayfrom the pole energy, the higher powers of the poly-r6”G rdnomial expansion become increasingly important, ofcourse. A partial cancellation in ao of the contri-butions of the non-pole T-matrix TNP and the poleSs== +一-0=T-matrix TP is observed not only for the P33, butalso for most of the other energetically low-lying res-SbSESaonances. Fig. 2 demonstrates that a separation ofa global smooth background, such as the one givenby the non- pole T-matrix(blue solid line), from theTP= .experimental data may be misleading.TdSa ryFT0.6Fig. 1. Theoretical T-matrix: Pole and Non-0.4pole contributions.胃0.2On the other hand, the S-matrix is uniquely char-acterized by its analytical structure. In order to ex--0.2tract the pole residue, we expand the amplitude T(2)-0.4on the second sheet in a Laurent series around thepole position,a7T(2)←→1=+ +a5→+O(z- zo), .(2)。只0.6where i and j denote the various channels consid-目0.4Fered. In the following, we compare the theoretical de-composition into pole and non-pole diagrams, Eq. (1)0.2with the one based on Eq. (2). Fig. 2 shows the par-tial wave amplitude for the P33 partial wave. The fullsolution of the Jilich model (solid red line) agrees11001200 1300 1400 1500with the Arndt solution!l. Adding the pole contri-Z [MeV]butiona-1- to the non-pole T-matrix produces aFig. 2. The P33 partial wave amplitude, dez- Zocomposed into real(upper figure) and imagi-drastic deviation from the amplitude of the full solu-nary(lower figure) parts. The Jilich modeltion (dashed- dotted black line). The reason for thisis given by the solid(red) line. The dashed-behavior can be traced to the fact that the pole T.dotted and dasbed lines refer to two approxi-matrix is an analytic function by itself which has amations discussed in the text.non-trivial polynomial contribution in addition to thepole term. Explicitly, the residue readsThere is another observation to be made in con-a_1=rr{t)(3)necti中国煤化工y generated reso-1-i8nancqCNMH(_。ow theory, the uchanr.rtial wave for pionwhereas the polynomial part starts with the constant:nucleon scattering generates attraction. After itera-tion, the orresponding T-matrix may have a pole onao=TNP+aF .(4)the second Riemann sheet, see Fig. 3, blue surface.No. 12M. Doring et al: Strategies for baryon resonance analysis1129The residue of that pole is canceled, once a bare poleIn Fig. 4, the partial wave amplitude of the Suis added, and the pole of the full amplitude is at thepartial wave is plotted. The full solution of thephysical position (red surface) while the pole visibleJilich model is indicated with the solid red lines.in TNP has moved far away into the complex plane.It describes well the SES solution of Ref. [5] up toThis observation shows that conclusions concerningz~ 1.9 GeV. The gray dotted lines indicate TNP. Wethe nature of resonances should be made only after acan also plot the pole approximation from Eq. (2)quantitative fit to the data has been achieved.For simplicity, we set ao = 0. On the physical axis, thepole approximations of the N(1535) and N(1650)appear as resonant like structures indicated with theDela(1232)black dashed-dotted lines.At first sight, the shapes of the real parts of thepartial wave amplitudes of the two resonances areP33|quite different: While the pole approximation of theN*(1535) shows a familiar shape with a maximumand a minimum in ReS1, the N*(1650) looks quitediferent. The reason is that a_1 is a complex num-229ReZber that mixes real and imaginary parts of a classicalBreit-Wigner shape. In other words, the phase of theImZ45~1074resonances is responsible for this twisting of resonanceshapes and can have a very large effect.Fig. 3. The second Riemann sheet of the P33The individual contributions from the two reso-partial wave amplitude. The red surface showsnances (black dashed-dotted lines) are quite differentthe magnitude of the amplitude evaluated infrom the full solution. However, the sumthe full Jilich model. The blue surface repre-,N*(1535)N (1650)sents the amplitude generated by the non-poleT(?)(z)=-"+(13 +167，(6)T-matrix.indicated as the purple dashed lines in Fig. 5, fts3 Resonance interference in S11the full solution quite well over the entire resonanceregion.The S1 partial wave is of particular interest be-cause there are two resonances in this partial wave.0.20.4-0.2-工&-0.4-0.2if+++++++0.81-+++++++: 0.6目0.4-二0.60.2 t.4 F120014001600 1800中国煤化工_shows the sum16001800HCNMHGddtedlines)，z (MeV]from two different sheets that are connectedFig. 4. Amplitude in the S11 partial wave. Seeto the physical axis below and above the ηNtext for legend.thresbold, respectively.1130Chinese Physica C (HEP & NP)Vol. 33Thus, the two resonances cannot be treated sepa- 4 Quark mass dependencerately but must be treated together; the residue fromthe N*(1535) provides a strongly energy dependentbackground in the N*(1650) region and viceversa.Lattice calculations of the mucleon octet are relarNote that the real part of the partial wave ampli-tively staightforward to performn because the lttietude shows a strong energy dependence of the tailapproach emphasizes the ground state of a given sys-of the N*(1535) in the region of the N*(1650). Thetem, but suppresses the excitation modes. Despiteresonances interfere which each other.these dificulties, first lattice investigations of the ex-For a theoretical description one needs a unitarycited states of the nucleon have become available bothcoupled channel model like the present one, whichfor the positive and negative parity excitations. Ralso alws for resonance interference. Otherwise, ifcently, new techniques have been proposed how toone tries to extract resonance parameters individu-extract even the second and third excited states ofally for each resonance, one needs a substantial phe-a given multipolarityl17. Lattice calculations do notnomenological background. Then, the parameters de-use quark masses consistent with the physical ma8spend very strongly on that particular background andof the pion. For technical reasons, much larger quarkresults are not reliable.masses have to be used. The results obtained for aSo far, only the pole structure above the ηN cusp .set of large quark masses are extrapolated to the ex-has been discused. At energies above z= mn +mn,perimental pion mass. It turns out that the extrapo-the physical axis is directly connected to the Riemannlation deviates from a straight line. Chiral perturba-sheet, where the physical poles have been analysed.tion theory offers a possiblity to study the quark massThis sheet is the unphysical one with respect to thedependence of the nucleon mass within a controlledπN and the nηN channel, called sheet 22 in Fig. 6.theoretical framework, see for example Ref. [18]. Ap-Sheet 21 is the second sheet to the nN channel, butplications of this formalism to the masses of the Ropera physical sheet with respect to the ηN channel.resonancel and the 03320) are available.The physical N*(1535) and N*(1655) are on sheetWithin the present approach, a link to the quark22, but on sheet 21, there are hidden poles. As amasss dependence of the resonance masses is not pos-result a prominent cusp in the amplitude becomessible. Nevertheless we can vary the pion mass in thevisible.pion-nucleon propagator and study the sensitivity ofthe pole positions of the resonances on the pion mass.N"(1535), sheet 22For suficiently large pion masses, the multi-pion finalN"(1535), shee(21channels close. A variation of the pion mass thereforecorresponds to a modifcation of the final state inter-action. In Fig. 7, we show the real part of the poleposition of resonances on the pion mass. In the caseof the Pss resonance, the present approach introducesa relatively strong modification due to final state in-teractions, a8 shown in Fig.3. The non-pole T-matrixby itself is able to generate a pole, although not at thecorrect physical energy. Indeed, one finds an increasemzof the mass of the Ps3 of about 50 MeV, when the pionRezMevnmass is increased from 140 MeV to 280 MeV. Manyauthors have claimed that the N*(1535) is generatedFig 6. Impact of poles on diferent sheets onby Kaon Lambda and Kaon Sigma dynamics,see e.g.the physical amplitude.Ref. (21]. Fig.7 shows a strong sensitivity of theN*(1535) to meson-nucleon dynamics already whentaking into account only the ηN and the effective two-A full discussion of the poles and residues obtainedpion nucleon channels. An extension of the presentin the Jilich model will be presented in the talk bylode中国煤化Irange particles isM. Doring"4. In the limit of high excitation ener-n prfYHCNMH G. the N~(1650)rgies, cro8s sections become forward peaked and showmainhanges of the piona smooth energy dependence. The implications of themass. The width of the N(1650) strongly decreasesphysics above 3 GeV for the resonance analysis[15] are .with increasing pion mass, however, see Fig. 8. Thediscussed in the talk by F. Huangl@.present investigation shows that the N*(1650) mightNo. 12M. Doring et al: Strategies for baryon resonance analysis1131be particularly interesting for lattice studies, as its100mass appears to be stable against modifcations ofthe final state interaction due to changes of the pionN*(1535)N*(1650)80mass. Eventually, a matching of the low-energy limitof the present approach to chiral perturbation theorym。2.3m%may make possible quantitative comparisons with lat-tice results.2.3m,1700-N*(16501500 1550 1600 1650 1700 1750Re z [MeV]1600_△* (1620)Fig. 8. Trajectories of the N*(1535) andr N*(1535)N*(1650) poles in the complex z plane as a兰1500N(1520)function of mn.& 14005 Conclusions1300l 0* (1232)The analytical structure of the meson-nucleon T-matrix offers a meeting point for theoretical and ex-150200250300perimental baryon resonance analysis. We have ex-m, [MeV]tracted the poles and tbe zeroes of the T-matrix inFig.7. 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