Bled Workshops in Physics Vol. 13, No. 1 p. 66 Bled, Slovenia, July 1 - 8, 2012 Meson electro-production in the region of the Delta(1700) D33 resonance B. Colli Faculty of Education, University of Ljubljana, 1000 Ljubljana, Slovenia and JoZef Stefan Institute, 1000 Ljubljana, Slovenia Abstract. We apply a coupled channel formalism incorporating quasi-bound quark-model states to calculate the D13, D33 and D15 scattering and electro-production amplitudes. The meson-baryon vertices for nN, nA (s- and d-waves), pN, nN(1440), nN(1535), nA(1600) and aA(1600) channels are determined in the Cloudy Bag Model. We use the same values for the model parameters as in the case of the P11, P33 and S11 partial waves except for the strength of the coupling of the d-wave mesons to quarks which has to be increased in order to reproduce the width of the observed D-wave resonances. The electro-production amplitudes exhibit a consistent behavior in all channels but are too weak in the resonance region. 1 Introduction This work is a continuation of a joint project on the description of baryon resonances performed by the Coimbra group (Manuel Fiolhais and Pedro Alberto) and the Ljubljana group (Simon Sirca and B. C.) [1-9]. In our previous works [5-7] we have successfully applied our method which incorporates excited baryons represented as quasi-bound quark-model states into a coupled channel formalism using the K-matrix approach [5] to calculate the scattering and the electro-production amplitudes in the P11, P33 and S11 partial waves. In the present work we extend of the approach to low lying negative parity D-wave resonances. In the next section we give a short review of the method and in the following sections we discuss in more detail scattering and electro-production in the D13 and D33 and D15 partial waves. 2 The method We limit ourselves to a class of chiral quark models in which mesons couple linearly to the quark core. In such cases the elements of the K matrix in the basis with good total angular momentum J and isospin T can be cast in the form [5]: Kjt km 'R ' M 'B' MR (1) Here and kM are the energy and momentum of the incoming (outgoing) meson, |¥B} is properly normalized baryon state and EB is its energy, W is the invariant energy of the meson-baryon system, and |^MB} is the principal value state |VjJB> = Nmb [at(kM)|^B)]JT + ^ cmb|®r) M 'B ' The first term represents the free meson (n, n, P, K, ...) and the baryon (N, A, A,...) and defines the channel, the next term is the sum over bare tree-quark states involving different excitation of the quark core, the third term introduces meson clouds around different isobars, E(k) is the energy of the recoiled baryon. We assume that the two pion decay proceeds either through an unstable meson (P-meson, a-meson,...) or through a baryon resonance (A(1232), N* (1440) ...). The meson amplitudes xM B MB (k, kM) are proportional to the (half) off-shell matrix elements of the K-matrix and are determine by solving a Lippmann-Schwinger type of equation. The resulting matrix elements of the K-matrix take the form V iv v 1 — V" ^BK^mI^rM i v-bkg fv v ^ KM'B'mbI^M)-~2_ ZR(W)(W-Wr) +KM'B'MBlk.l V) VBR which means that the T matrix for elektro-production can be split into the resonant part and the background part; the latter is the solution of the Heitler equation with the "background" K-matrix defined as i/bkg _ _i/bkg ^NR _ V- ^BR'^NR'__, Rbkg KMByN- KMBMB VM__ Zr , (W _ Wr ,) + D MB yN • Note that V^ (ky) is proportional to the helicity amplitudes while the strong amplitude V^ (kM) to VTmb and to C the sign of the phase of the meson decay. 3 The D-wave resonances in the Cloudy Bag Model In the quark model, the negative parity D-wave resonances are described by a single quark I = 1 orbital excitation. The two D13 (flavor octet, J = §) resonances are the superposition of the S = \ and S = | configurations, the D33 resonance (flavour decouplet) has S = j, while the D15 resonance (octet, J = f) has S = |. We use the j-j coupling scheme in which the resonances take the following forms: N(1520)D13 = - sin^d|48a/2) + cos£dl%/2) = 4l(1s)21p3/2)MS + cA|(1s)21p3/2 )mA + 41 (1s)2 1p V2 ) , (8) N(1700)D13 = cos $d|483/2) + sin $d|283/2) = cSl(1s)21p3/2)MS + cAl(1s)21p3/2 ) MA + 41 (1s)2 1p V2 ) , (9) rF t A(1700)D33 = |2103/2> = s)2lp3/2) - -1(1 s)2lp1/2), (10) N(1675)D15 = | 48s/2) = l(1s)21p3/2). (11) Here MS and MA denote the mixed symmetric and the mixed antisymmetric representation, and i 2 . , /T , i V2 , , V5 . , V2 , cj sm9d+W— cos9d, c^ =—— cos$d, Cp =—— sin9d+— cos$d 3 V lo 2 3 3 (12) The I = 2 pions couple only to j = 3/2 quarks; the corresponding interaction in the Cloudy Bag Model takes the form where r 1 3 1 *— 1 m - Z CIm;imls^)(p3/2mj|) cus =2.043, u;p3/2 = 3.204. msmj In the case of P11, P33 and S11 waves we have used the bag radius R = 0.83 fm which determines the range of quark-pion interaction corresponding to the cut-off A - 550 MeV/c, and the value for fn = 76 MeV which reproduces the experimental value of the nNN coupling constant. For the d-wave pions it turns out that the range predicted by (13) is too large while the resulting coupling strength is too weak. We have therefore modified the interaction in such a way as to correspond to A - 550 MeV/c, while the coupling strength has been increased by a factor 1.7 - 2.75 (depending on the considered resonance). 4 Scattering amplitudes The effect of the form factor and the strength of quark-meson coupling discussed in the previous section is most clearly seen in the case of the D15 where the background effects as well as the influence of other resonances are almost negligible. Using our standard value for the cut-off parameter we have to increase the quark model coupling constant by a factor of 2.75 in order to obtain an almost perfect fit to the data in the region of the resonance. Fig. 1. The form factor for the D-wave pions (left panel), and the real and the imaginary part of the D15 scattering amplitude (right). The data points are from [10]. The data for elastic scattering in the D13 partial wave show almost no sign of the second resonance N(1700). Since the I = 2 pions most strongly couple to the |(1s)21p3/2)ma configuration, the absence of the second resonance can be most easily explained by the vanishing of the c\ coefficient in (9), c\ = — sin6d/\fl. This suggests 6 d = 0. In our model the resonances are mixed through the pion interaction which changes slightly the above conclusion leading to the choice 6d ~ 10° for the optimal mixing. At this energy range the effect of the cut-off is less pronounced; the quark-model prediction for the nNR coupling constant has to be increased by a factor of 1.7, while that to the A decreased by a factor of one half. 0.6 0.4 ■ D13 0.2 0 Im T - -0.2 exp ^ Re T - -0.4 exp 1.2 1.3 0.2 0.1 0 -0.1 -0.2 ImT D33 , , •'• >• , • i - Im T - exp 1—•—1 Re T..... 1$>-.......- - „ m" „ exp 1—0—1 ReT W [GeV] 1.3 1.4 1.5 1.6 1.7 1.8 1.9 W [GeV] Fig. 2. The real and the imaginary part of the D13 wave scattering amplitude (left), and for the D33 wave (right). The data points are from [10]. In the vicinity of the D33 resonance the elastic amplitude is dominated by the coupling of the elastic channel to the nA(1232) channel. The d-wave pion coupling to the nucleon is increased by a factor of 2.5 with respect to the quark model value, while the model value for s-wave coupling to the A(1232) is not modified. Increasing the latter coupling brings the real part of the amplitude closer to the data, however the behavior of the photo-production amplitudes, presented in the next section, is deteriorated. 5 Electro-production The electro-production amplitudes are obtained by evaluating the EM current consisting of the quark and the pion part between the nucleon ground state and the resonant state. The corresponding helicity amplitude V^ in (7) reads e V^rJkY) = -=, V where the resonant state stemming from the second and the third term in (2) consists of the bare-quark part and the meson cloud ^^{l^-Ll^ai«'.^,]^ (14, The background term entering (7) is dominated by the pion-pole term and the u-channel process which originate from the first term in (2). In Figs. 3-6 the transverse photo-production amplitudes for the partial D13, D33 and D15 partial waves calculated in our model are compared to the data as well as to the analysis of the MAID group [11]. While our calculation correctly reproduce the behavior of the amplitudes at the energies close to the threshold where they are dominated by the pion-pole term, their strength in the resonance region is typically a factor 0.5 to 0.7 weaker compared to the value of the electric transverse amplitude as deduced from the experiment, and even weaker in the case of the magnetic amplitude. The pertinent multipoles are sensitive to the nucleon's periphery which is apparently not adequately reproduced in the bag model, as we have already noticed when analyzing the coupling of the resonance to the d-wave pions. Here the pion cloud effect are relatively weak as a consequence of cancellations of different terms, and contribute at the level of 10 % to 20 % to the amplitudes. W [GeV] W [GeV] Fig. 3. The real and the imaginary part of the proton and neutron multipoles E2- for the D13 wave in units 10-3/mn (preliminary). The data points are from [10], "maid" corresponds to the partial wave analysis from [11]. Nonetheless, we should stress that the amplitudes exhibit a consistent behavior in all considered partial waves. In particular, our model correctly predicts 1 /2 that in the D13 partial wave the nE2- multipole amplitude is weaker than the 1/2 1/2 corresponding nE2- amplitude, and that the nM2- amplitude almost vanishes. Similarly, for the D15 partial wave the quark model predicts that the quark con- 1/2 tribution to the pM2- multipole vanishes and only the pion cloud contributes to the resonant part of the amplitude. The non-zero quark contribution in the case of the neutron multipole is however too weak to reproduce the data. 6 Discussion Comparing the present results with the results for other partial waves obtained in chiral quark models we notice a general trend that the quark core alone does not provide sufficient strength to reproduced the observed resonance excitation amplitudes. The best known example is the P33 partial wave in which case the quark contribution to the electric dipole excitation of the A(1232) is estimated 1.5 1 0.5 0 -0.5 -1 2.5 2 1.5 1 0.5 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 W [GeV] 1 0.5 0 -0.5 -1 -1.5 0 -0.5 -1 -1.5 -2 - Re nM2- ii-j^i ? i exp i— 1 res--- -- total - maid - 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 W [GeV] Fig. 4. The M2- multipole, notation as in Fig. 3. 1 Im E2- - 0 -1 \ \ -2 exp i—s— Mr - -3 res total - -4 maid Î y -5 - 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 W [GeV] -0.5 -0.5 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 W [GeV] Fig. 5. E2- and M2- amplitudes for the D33 wave, notation as in Fig. 3. by only 60 % while the rest is attributed to the pion cloud [1]. In the present calculation the pion cloud effects turn out not to be that important. In fact, we have noticed a considerable cancellation of different contributions of the meson cloud, e.g. the vertex correction due to pion loops and the genuine contribution Re pM2+ exp 1—6—1 res----- total - maid----- Im pM2+ exp i—®—i res total - maid----- " 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 W [GeV] -1.5 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 W [GeV] Fig. 6. The M2+ amplitudes for the D15 wave, notation as in Fig. 3. 2 0 of the pion cloud to the EM current. It is therefore possible that a calculation in a more elaborate chiral quark model could provide a better agreement with the data. To conclude, the overall qualitative agreement with the multipole analysis in the D13, D33 and D15 partial waves prove that the quark-model explanation of the D-wave resonance as the p-wave excitation of the quark core supplemented by the meson cloud is sensible and that no further degrees of freedom are needed. References 1. M. Fiolhais, B. Golli, S. Sirca, Phys. Lett. B 373, 229 (1996) 2. P. Alberto, M. Fiolhais, B. Golli, and J. Marques, Phys. Lett. B 523, 273 (2001). 3. B. Golli, S. Sirca, L. Amoreira, M. Fiolhais Phys.Lett. B553 (2003) 51-60 4. P. Alberto, L. Amoreira, M. Fiolhais, B. Golli, and S. Sirca, Eur. Phys. J. A 26, 99 (2005). 5. B. Golli and S. Sirca, Eur. Phys. J. A 38, (2008) 271. 6. B. Golli, S. Sirca, and M. Fiolhais, Eur. Phys. J. A 42,185 (2009) 7. B. Golli, S. Sirca, Eur. Phys. J. A 47 (2011) 61. 8. B. Golli, talk given at the Sixth International Workshop on Pion-Nucleon Partial-Wave Analysis and the Interpretation of Baryon Resonances, 23-27 May, 2011, Washington, DC, U.S.A., http: //gwdac . phys . gwu. edu/pwa2 011/Thursday/b_golli . pdf 9. Simon Sirca, Bojan Golli, Manuel Fiolhais and Pedro Alberto, in Proceedings of the XIV International Conference on Hadron Spectroscopy (hadron2011), Munich, 2011, edited by B. Grube, S. Paul, and N. Brambilla, eConf C110613 (2011) [http://arxiv.org/abs/110 9.0163]. 10. R. A. Arndt, W. J. Briscoe, I. I. Strakovsky, R. L. Workman, Phys. Rev. C 74 (2006) 045205. 11. D. Drechsel, S.S. Kamalov, L. Tiator, Eur. Phys. J. A 34, 69 (2007). bled workshops A Proceedings of the Mini-Workshop in physics l"l Hadronic Resonances VOL. 13, No. 1 Bled, Slovenia, July 1 - 8,2012 p. 74 Scattering phase shifts and resonances from lattice QCD S. Prelovseka'b a Jozef Stefan Institute, Jamova 39,1000 Ljubljana, Slovenia b Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19,1000 Ljubljana, Slovenia Most of hadrons are hadronic resonances - they decay quickly via the strong interactions. Among all the resonances, only the p meson has been properly simulated as a resonance within lattice QCD up to know. This involved the simulation of the nn scattering in p-wave, extraction of the scattering phase shift and determination of mR and r via the Breit-Wigner like fit of the phase shift. In the past year, we performed first exploratory simulations of Dn, D*n and Kn scattering in the resonant scattering channels [1,2]. Our simulations are done in lattice QCD with two-dynamical light quarks at a mass corresponding to mn ~ 266 MeV and the lattice spacing a = 0.124 fm. > Dw thresholds. 0 1+ + 2+ + 2- + 2 - - 3- - 3 We also simulated Kn scattering in s-wave and p-wave for both isospins I = 1 /2, 3/2 using quark-antiquark and meson-meson interpolating fields [2]. Fig. 3 shows the resulting energy levels of Kn in a box. In all four channels we observe the expected K(n)n(-n) scattering states, which are shifted due to the interaction. In both attractive I = 1/2 channels we observe additional states that are related s-wave, I=1/2 s-wave, I=3/2 1.2 0.8 0.6 0.4 T~ fï ;xp rit -L 2 exp fit^-L_L. _ T~ ........... ~T "T" -------- -l-^Jli ...... _L K(2)n(-2) K(1)n(-1) K(0)n(0) 0 5 10 15 0 5 10 15 t t p-wave, I=1/2 p-wave, 1=3/2 K(2)n(-2) K(1)n(-1) 1 Fig. 3. The energy levels E(t)a of the Kn in the box for all four channels (multiply by a.-1 = 1.59 GeV to get the result in GeV). The horizontal broken lines show the energies E = Ek + En of the non-interacting scattering states K(n)n(-n) as measured on our lattice; K(n)7t(—n) corresponds to the scattering state with p* = Note that there is no K(0)n(0) scattering state for p-wave. Black and green circles correspond to the shifted scattering states, while the red stars and pink crosses correspond to additional states related with resonances. to resonances; we attribute them to K0(143O) in s-wave and K*(892), K*(1410) and K*(1680) in p-wave. We extract the elastic phase shifts 6 at several values of the Kn relative momenta. The resulting phases exhibit qualitative agreement with the experimental phases in all four channels, as shown in Fig. 4. In addition to the values of the phase shifts shown in Fig. 4, we also extract the values of the phase shift close to the threshold, which are expressed in terms of the scattering lengths in [2]. s-wave, I=1/2 s-wave, I=3/2 180 150 IB 120 a 90 to 60 30 180 150 Is 120 tS> S 90 (O 60 30 lat: present work exp: Estabrooks (elastic) exp: Aston (elastic) exp: Aston (almost elastic) 1 1.2 1.4 sqrt(s) [GeV] p-wave, I=1/2 f It - ff-20 - -o EIP • lat: present work + exp: Estabrooks (elastic) III 1 1.2 1.4 sqrt(s) [GeV] p-wave, I=3/2 • lat: present work + exp: Estabrooks (elastic) * exp: Aston (elastic) x exp: Aston (almost elastic) i - S? 0 - -o 1 1.2 1.4 sqrt(s) [GeV] 1 1.2 1.4 sqrt(s) [GeV] 0 Fig.4. The extracted Kn scattering phase shifts in all four channels I = 0,1 and I = 1 /2, 3/2. The phase shifts are shown as a function of the Krc invariant mass