Bled Workshops in Physics Vol. 8, No. 1 p. 61 Proceedings of the Mini-Workshop Hadron Structure and Lattice QCD Bled, Slovenia, July 9-16, 2007 Dynamics of P11 and P33 resonances in quark models with chiral mesons* B. Gollia,b and S. Sircab>c a Faculty of Education, University of Ljubljana, 1000 Ljubljana, Slovenia b Jozef Stefan Institute, 1000 Ljubljana, Slovenia c Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia Abstract. We apply the coupled channel formalism for the K-matrix to the calculation of the P11 and P33 scattering amplitudes in the region the N(1440) and A(1600) resonances. 1 Introduction In this work [1] we extend the coupled channel formalism for the K matrix derived in [2] using the static approximation to take into account the correct rela-tivistic kinematics of the meson-baryon system. We apply this method that has been used in [2] to explain the peculiar behaviour of the scattering amplitudes in the energy range of the Roper resonance to the calculation of the A(1600) resonance, the Roper's counterpart in the P33 partial wave. In quark models these two resonances are assumed to have a similar spatial structure, this similarity is however not reflected in the scattering amplitudes. While in the P11 partial wave the phase shift reaches 90 degrees around W ~ 1500 MeV the phase shift in the P33 case shows no sign of resonanant behaviour in the energy range of W ~ 1600 and above, which is a strong signal of the important role of inelastic channels. This is further supported by the unusual behaviour of the inelasticity which in the P11 case rapidly rises from zero to unity and remains close to this value in a broad energy region, while in the P33 case it rises rather slowly and reaches the unitarity limit only at much higher energies. We show that these features can be explained by assuming that in this energy range the inelastic channels are dominated by the two and three-pion decays proceeding mainly through two channels: (i) in the nA channel the resonance first decays into the pion and the A isobar of invariant mass M, MN +mn < M < W—mn, and (ii) the ct channel in which the resonance first decays into the cr-meson mimicking two pions in the relative s-state and either the nucleon (in the P11 case) or the A isobar (in the P33 case). * Talk delivered by B. Golli 2 Coupled-channel K-matrix formalism We consider a class of chiral quark models in which mesons (the pion and the sigma meson in our case) couple linearly to the quark core: Hn dk^ almt(k)almt(k) + [vlmt(k)almt(k] + V^M a^k)] } , Imt where a{mt(k) is the creation operator for a meson with angular momentum I and the third components of spin m and isospin t. In the case of the pion, we include only I = 1 pions, and 3 Vmt(k) =-v(k)^ « (1) m i=1 is the general form of the pion source, with the quark operator, v(k), depending on the model. It includes also the possibility that the quarks change their radial function which is specified by the reduced matrix elements Veb' = (B||V(k)11B'}, where B are the bare baryon states (e.g. the bare nucleon, A, Roper,...) In the case of the d mesons we assume only I = 0 mesons, coupled to the quark core with ~ V 1 -F Here = k2 + and wCT(^) is the mass distribution function modeling the resonant decay into two pions. In this work we take the values consistent with the recent analysis of Leutwyler [5], mCT = 450 MeV and TCT = 550 MeV. The strength parameter GCT in (2) is a free parameter of the model. Chew and Low [4] have shown that in such models it is possible to find the exact expression for the T matrix (and consequently for the K matrix) without explicitly specifying the form of asymptotic states. In the basis with good total angular momentum J and isospin T, in which the K and T matrices are diagonal, it is possible to express the K matrix for the elastic channel in the form: JN„N(k,W) = —nNN (J(W)||V(k)||¥N} , where W is the invariant mass of the meson-baryon system. In the inelastic channels we find JAnN(k,W,M) = —nNA( J(W)||V(k)||¥A(M)} , KnTA«A(k,W,M',M) = — nNA 3 3 - KJJ (k, W, M, n', M') = -7tAAffA 3 (W, M')|V^(k)|¥A(M)) , where A/b = \JcuEB/kW, A/"ctb = xA^V^b/^VV, cu is the energy of the scattering pion, k = ^ix>2 — m^, and cvn is the energy of the scattering cr-meson of invariant mass \x, kM = \]uj1 — \i2. Here is the principal value state for which we use the following ansatz that takes into account the proper relativistic kinematics and replaces the similar expression in [2] derived in the static (no-recoil) approximation: l/h (Wmh)\ _ N. J ( b J (W,mH)> = nh ch(W,mH)|$B) + [at(kH)^H>]JT + dM + t(k) - W " - --xíH(k. W, u.mH) . - 1 (3) + + En (k) - W xfTH(k, W, M,mH) , ~ JT XYemV'"'w^iMir + du XrH(k, W. u, mH) , Here H stands for either the nN, nA, ctN or the ctA channel, mH is the invariant mass of the corresponding intermediate hadron in the inelastic channels, E(k) is the energy of the recoiled baryon (nucleon or A). The first term consists of the sum over bare tree-quark states ®B, involving different excitations of the quark core, the next term corresponds to the free meson (pion or CT-meson) and the baryon (N or A) and defines the channel, the next two terms represent the pion cloud around the nucleon and the A isobar, respectively, and the last term the CT-meson cloud around the nucleon (for JT = or the A (for JT = here b* is the creation operator for the CT-meson. The on-shell meson amplitudes XjT H, describing the corresponding meson clouds around the nucleon and the A are proportional to corresponding matrix elements of the on-shell K matrix kh'h = nnh'nh xH'H(kH' ,kH). From the variational principle for the K matrix it is possible to derive the integral equation for the amplitudes which is equivalent to the Lippmann-Schwinger equation for the K matrix. Using a simplified ansatz for the principal value states in which the terms involving the integrals are neglected amounts to taking only the non-homogeneous part of the corresponding Lippmann-Schwinger equation. Such an approximation is widely used in phenomenological analysis of scattering amplitudes and is known as the Born approximation for the K matrix. The T matrix is calculated from the K matrix through the Heitler equation: T = —K + iKT. 3 Results for the scattering amplitudes in the Cloudy Bag Model We illustrate the method by calculating scattering amplitudes for the P11 and the P33 partial waves. Though the expressions derived in the previous sections are general and can be applied to any model in which mesons linearly couple to the quark core, we choose here the Cloudy Bag Model, primarily because of its simplicity. In this model, the matrix element of the pion source (1) between the model 3-quark states can be written as where and <®B ' ||V (lc)||®B> = rq v(k) (Jb ' ,Tb ' = Jb 'llY_ , i=1 1 k2 cumit ii(kRhag) v(k) = 2f \J 11n1 cui< <^MiT - 1 kRh 1 2 3 "-'mit'^mit-1 ) 1) "mtt' a,mit 1/2 + r2 i 1 a for B = B' = (1s)3 configuration 0.457 for B = (1s)3 ,B' = (1s)2 (2s)1 for B = B' = (1s)2 (2s)1 In this work we use Rbag = 0.9 fm, f = 76 MeV yielding the correct value for the nNN coupling constant. Similar results are obtained for 0.85 fm < Rbag < 1.0 fm. In addition, the energies of the 3-quark states in different excited states are also taken as free parameters. r q nN nN J P 1300 1400 1500 1600 1700 1800 W [GeV] nN —> nN Hf y nA — -> nA / 1 100 1200 1300 1400 1500 1600 1700 1800 W [GeV] Fig. 1. The P11 (left panel) and P33 (right panel) phase shifts. The corresponding thin line in the P33 case represent the phase shift for the pion scattering on the A. The data points in this and subsequent figures are from the SAID nN —> nN partial-wave analysis [6] The model parameters are Mr = 1510 MeV, Ma = 1232 MeV, Ma* = 1700 MeV 200 50 The results in the simplest approximation, the Born approximation for the K matrix without background, are displayed in Figure 1. This approximation is equivalent to keeping only the the first term in the ansatz (3). By increasing the nNA interaction strength by 60 % and the nNRoper by 80 % with respect to the above model values, the widths of the N(1440) and A(1232) are reproduced. This simplified approach explains why the resonant behaviour of the phase shift is not observed for the A(1600) (= A*) in the elastic channel: in this energy region, the matrix element nAA* becomes stronger than nNA* in which case the resonance disappears in the nN —> nN but appears is the non-observable nA —> nA channel. Including the background in the the Born approximation for the K matrix through the term [at(kH)|¥H(M)}]JT in (3) we obtain almost perfect agreement of the calculated scattering amplitudes compared to the amplitudes extracted in the partial-wave analysis but still at the expense of considerably larger nNA and nNRoper interaction strength compared to those predicted by the quark model. This inconsistency is resolved when solving the Lippmann-Schwinger equation for the pion amplitudes; it turns out that for our particular choice of the bag radius we are able to reproduce the experimental scattering amplitudes starting from the bare values as predicted by the Cloudy Bag Model (see Fig. 2). W[GeV] W[GeV] Fig. 2. The real and the imaginary parts of the T matrix for the P11 (left panel) and P33 (right panel) partial waves. The dashed/full curves in the left panel show the effect of omitting/including the N (1710) state in the sum over B in (3) References 1. B. Golli and S. Sirca, arXiv:0708.3759v1 [hep-ph] 2. B. Golli and S. Sirca in: B. Golli, M. Rosina, S. Sirca (eds.), Proceedings of the MiniWorkshop "Progress in Quark Models", July 10-17, 2006, Bled, Slovenia, p. 82. 3. P. Alberto, L. Amoreira, M. Fiolhais, B. Golli, and S. Sirca, Eur. Phys. J. A 26 (2005) 99. 4. G. F. Chew and F. E. Low, Phys. Rev. 101,1570 (1956). 5. H. Leutwyler, Int. J. Mod. Phys. A 22 (2007) 257. 6. R. A. Arndt, W. J. Briscoe, R. L. Workman, I. I. Strakovsky, SAID Partial-Wave Analysis, http://gwdac.phys.gwu.edu/. 7. P. Alberto, M. Fiolhais, B. Golli, and J. Marques, Phys. Lett. B 523 (2001) 273.