Bled Workshops in Physics Vol. 11, No. 1 p. 56 S and D-wave resonances in chiral quark models: coupled channel approach 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 S11 and D13 scattering amplitudes. The meson-baryon verteces for nN, r|N, nA, pN and KA channels are determined in the Cloudy Bag Model. Using the same values for the model parameters as in the case of the P11 and P33 amplitudes the elastic as well as most of the inelastic amplitudes are reasonably well reproduced. 1 Introduction This work is a continuation of a joint project on the description of baryon resonances performed by the Coimbra group (Manuel Fiolhais, Luis Alvarez Ruso, Pedro Alberto) and the Ljubljana group (Simon Sirca and B. G.) We have developed a general method to incorporate excited baryons represented as quasi-bound quark-model states into a coupled channel formalism using the K-matrix approach [1]. In our method, the meson-baryon and the photonbaryon verteces are therefore determined by the underlying quark model rather than fitted to the experimental data as is the case in phenomenological approaches. The method can be applied to meson scattering as well as to electro and weak-production of mesons. In the previous work we have investigated the P33 and P11 amplitude dominated by the low lying positive parity resonances A (1232), A (1600) and N (1440) [1,2]. We have found a good agreement between the model prediction and experiment for the scattering as well as the electro-production amplitudes. We have shown that the pion and the a-meson considerably contribute in particular to the scattering amplitudes in the energy region just above the two pion threshold and to the electro-excitation amplitudes in the region of low Q2 transfer. In the present work we investigate the extension of the approach to low lying negative parity resonances. This implies the inclusion of new channels involving the s-wave and the d-wave pions, the n and the p mesons, and the K A channel. In the next section we give a short review of the method and in the following sections we discuss in more detail scattering in the S11 and D13 partial waves. B. Golli 2 A short overview of the K-matrix approach We consider a class of chiral quark models in which mesons couple linearly to the quark core: Hn dk^ aimt(k)almt (k) + [vlmt(k)almt(k] + Y^k) a^k)] } , Imt (1) where a{mt(k) is the creation operator for a meson with angular momentum I its third components m and isospin t (absent in the case of s-waves and isoscalar mesons). Here Vlmt(k) is a general form of the meson source involving the quark operators and is model dependent. In the following section we give a few examples for Vlmt(k) in the Cloudy Bag Model. We have shown [1] that in such models the elements of the K matrix in the basis with good total angular momentum J and isospin T take the form: K^,B'MB=-^M'B'(^B||VM,(k)||¥B'), A/mb = - (2) where and kM are the energy and momentum of the incoming (outgoing) meson, EB is the baryon energy and W is the invariant energy of the mesonbaryon system. In addition, the channels are specified by the relative angular momentum of the meson-baryon system and parity. Here |^MB} is the principal value state and assumes the form: } = Nmb { [at(kM )I^b}]it + X cMbI®r} I R M 'B' -Li^»-« • (3. 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 excitations of the quark core, the third term introduces meson clouds around different isobars, E(k) is the energy of the recoiled baryon. In our approach we assume the commonly used picture in which the two pion decay proceeds either through an unstable meson (p-meson, a-meson,...) or through a baryon resonance (A(1232), N*(1440) ...). In such a case the state ¥MB depends on the invariant mass of the subsystem (either nn or nN) and the sum over M'B' in (3) implies also integration over the invariant mass. The state ¥B is the asymptotic state of the incoming (outgoing) baryon; in the case it corresponds to an unstable baryon it depends on the invariant mass of the nN subsystem, MB, and is normalized as (¥B (MB)| VB (MB)} = 5(MB — MB), where MB is the invariant mass Mb of the Nn subsystem. The meson amplitudes xM B MB (k, kM) are proportional to the (half) off-shell matrix elements of the K-matrix Km 'b ' MB(k,kM) = n NM 'b 'Nmb XM 'B' MB(k,kM) (4) and obey a Lippmann-Schwinger type of equation: XM'B' MB(k,kM)=-X cMB VMr(k)+ KM'B' MB(k,kM) K + L M ''B ' f a^j M"B"(k, k/)xM"B" MB(k/,kM) dk <+Eb»(1c')-W ' (5) where K (k'k)-^7fBB'cuk + < + EB,(k)-W' (6) fAB = VUIa + 1 ) (2JB + 1 ) (2Ta + 1 ) (2TB + 1 ) W(1 IAIB 1 ; le, I)W(1 TATB1 ; Tc, T). The coefficients cMB obey the equation MB MB(kkM)VMK(k) dk cuk + EB'(k) — W ' (7) (W- MR0))cMB = VBR(kM) + X M 'B Here VM(k) are the matrix elements of the quark-meson interaction between the baryon state B and the bare 3-quark state and M^' is the energy of the bare state. Solving the coupled system of equations (5) and (7) using a separable approximation [1] for the kernels (6), the resulting amplitudes take the form XM'B' MB(k,kM)=-X ^MB VBMK(k)+ Dm'b' MB(k,kM) , (8) R where the first term represents the contribution of various resonances while Dm b mb (k) originates in the non-resonant background processes. Here yM -MB = _V_WR__(q) n ZR(W)(W-MR) ' K > yBM is the dressed matrix element of the quark-meson interaction between the resonant state and the baryon state in the channel MB, and ZR is the wave-function normalization. The physical resonant state R is a superposition of the dressed states built around the bare 3-quark states '. The T matrix is finally obtained by solving the Heitler's equation T = K + iTK. (10) In this work we concentrate on the negative parity partial waves S11 and D13; in both cases the amplitudes are dominated by two rather closely lying resonances, either the N(1535) St t and N(1650) Sn, or the N(1520) D^ and N(1700) D13 .We have performed the calculation of the scattering amplitudes in the same model, the Cloudy Bag Model (CBM), with the same choice of model parameters as in the case of positive parity resonances. For the bag radius we use R = 0.83 fm and fn = 76 MeV for the parameter determining the interaction strength, while the energies of the bare states are taken as free parameters. S and D-wave resonances in chiral quark models ... 59 3 The meson coupling to negative parity states in the CBM We have used the bag model description for the resonances assuming that one of the three quarks is excited from the 1 s state to the 1 p, state with the total angular momentum j either 1 /2 or 3/2. The relevant quark bispinors in the j m, basis are Ns ( -ijo(wsr/R) \ * fr) =_il»_ f -MoicUsT/R) \ Ws[ ' v^ioicus) U-iii(^sr/R)y'Xmi' NPl/2 i iji(cuPl/2r/R)CT-f 0^Pl/2 ) V j0(WPl/ NP3/2 ( -1 jl(^Ps/2 r/R) V*Hio(œp,/2){ io(^/2r/R) J Xmi ' Û (r)-_^^_ -imu,P3/2,/*j \ y f c4mj Herexm is the spinor for spin R is the bag radius, cus = 2.043, cuPl/2 = 3.811, ^P3/2 = 3.204, and 2 = CUS n2 = CUPi/2 n2 = 9cUP3/2 s 2R2(cus — 1) ' 2R2(cuPl/2 +1) ' 4R2(cuP3/2-2)' The wave function of the negative parity states in the j-j coupling scheme are taken from [5]. For the quark pion coupling we use the usual CBM form yielding v„ m 1 "WUs 1 k2 0(kR ^ vi=o,t (k = Tn 7-TTT7-77 1--F= i p Z_ Tt 1 Psp 1 ' 2f„ V (cup + l)(cus - 1) In ^œ^ kR 1 ws Ilk2 j! (kR) 3 Vlmt(k) - 2^737^ h X (ffm(i)+rp1/2s5i(i)+rp3/2s5ï(i)) , v„ 1 I tUp3/2^s V7! k2 j2(kR) y- [+4],.. v2mt (k = Tn 7-tû-TT 1--F= 1 P > Tt(r I2m' M 2f„ V (cuP3 -2 (cus - 1) In ^w^ kR ^ zm 3 Here VsV = Y_ lsmi)(Pl/2UXj| , = Y_ C|mpmlPl/2mi)(Pl/2TTLjl ' mj mj m' 4t?] = Z C|mi2mlsms)(P3/2mj , S^ = L cf ^ >3^) (pa^l , msmj mjm' and ^p (^S - 1) 2^P3/2 (^S - 1) -r — _;__r = ___ cus(cuPl/2 +1) ' 5CUs(CUP3/2 — 2) The p meson coupling to quarks is similar to the EM coupling. For the coupling of negative parity states to pN, the dominant contribution is expected to arise from the transverse p-mesons with the total J = 1 and the orbital angular momentum of the pN system equal to either 0 or 2. In the spirit of the CBM model we assume that the p meson couples to the quarks only on the bag surface [6]. Assuming further a pure vector coupling yHpH we find (note that m in (1) and below refers to the total angular momentum rather than to the orbital one): %P=0mtM — ^ If Ws 1 k2 jo(kR) p v (cus - 1) 2n v/cu£ A/8 I cu ~3~ kR P 1/2 WPl/2 + 1 I lm + ^1 W P 3/2 Wp ■2 I 11%) VP=2mt(k) 2fn P 3/2 ' P3/2 1 1 k2 j2(kR) (w- P 3/2 2)(cus — 1) In 3 v/cu£ kR *— r1 31 i=1 Here r[}l I1m r 1 3 i I1m and fp is the p-meson decay constant with the experimental value 208 MeV. For the coupling of the p meson to the nucleon we obtain a similar expression as for the coupling of the p-wave pions, with fn replaced by f p, yielding gnNN/9pnn = fp/fn which is experimentally well fulfilled. The choice of the above form with f p « 200 MeV is therefore not insensible. For the s-wave n and K mesons we assume the SU(3) symmetry yielding V11 (k) = -L 1 k2 jo(kR) PsP(i) , V (tuPl/2 + 1)(cus - 1) In y/aJ^ kR ^ vKfvi 1 I "W"* 1 k2 ioW f Vt k = TT\ 7-TTT7-77 i--F= i P / Vt 1 + Ut M /^sp M , 2f„ y (cuPl/2 + 1)(cus - 1) In y/voi kR with t = ±l,and V±t = (A4 ± iAsj/v^ and U±t = (A6 ± iA/j/v7! The peculiar oscillating shape of the CBM form factor has little influence in the case of the p and d-wave pions but leads to the unphysical behaviour of the s-wave scattering amplitude since it crosses zero already at W ~ 1950 MeV. We have cured this problem by replacing j0(kR) by an exponential tail for k > 1.6/R in such a way as not to alter the value of the self energy integral. x ms m msm WP1/2 Ws 4 S11 resonances For the S11 partial wave we have included the nN, nA, nN(1440), pN and KA channels and the N(1535) and N(1650) resonances. We have used the quarkmodel wave-functions for the negative-parity states using the j-j coupling scheme [5]: = cR|(1s)2(1p3/211) + cR|(1s)2(1pi/2)1)l + cR'|(1s)2(1pi/2)1)2 where the mixing coefficients cR, cR, and cR can be expressed in terms of the mixing angle between the spin-1 /2 and spin-3/2 3-quark configurations. The mixing is a consequence of the gluon and the meson interaction; since the quark-gluon interaction is not included in the model, the mixing angle due to gluons is taken as a free parameter independent of W. In the energy region of the N (1535) and N (1650) resonances we obtain the best results using = -34° in agreement with the phenomenological analysis [7]. Resonance Pot [MeV] P/Pot 7tN r|N 7TA KA PiN 7tN (1440) N(1535) 165 0.29 0.69 0.01 - 0.01 0 PDG 125-175 0.35-0.55 0.53 0.01 - 0.02 0 N(1650) 188 0.59 0 0.19 0.13 0.04 0.04 N(1650)mod 156 0.72 0 0.08 0.10 0.05 0.04 PDG 150 -180 0.60-0.95 0.02 0.02 0.03 0.01 0.03 Table 1. The total and the partial widths for the N(1535) and the N(1650) resonance at the K-matrix pole (1545 MeV and 1695 MeV, respectively) using the unmodified and the modified (mod) quark-model values for the quark-meson couplings. The PDG values are from [3]. Fig. 1. The real and the imaginary part of the scattering amplitudes for the S11 wave. Dotted lines are for the elastic channel with unmodified quark-model verteces, full lines are those with the modified values, dashed lines correspond to the nN channel. The data points for the elastic channel are from the SAID nN —> nN partial-wave analysis [4], those for the inelastic one are taken from [9]. In the vicinity of the lower resonance, just above the n meson threshold, the elastic and inelastic amplitudes are dominated by the s-wave nN channel. In the energy region of the upper resonance, additional channels open or become more important. We have considered the following additional channels: the nA(1232) channel with I = 2, the KA( 1116) channel with I = 0, two channels involving the p meson with I = 0 (piN) and I = 2 (p3N), and the nN*(1440) channel with I = 0. 0.2 0.1 0 -0.1 -0.2 Fig. 2. The real and the imaginary part of the inelastic amplitudes for S11 partial wave. The experimental points are from [8]. Using the quark-model values for the quark-meson coupling as introduced in the previous section we obtain a good agreement between the model prediction and the experimental analysis for the lower resonance; for the upper resonance the agreement is worse. Though the extraction of the experimental data is less reliable and considerably differs between different authors, it clearly indicates that the strengths of the nA (d-wave) vertex is overestimated in our model; the same is probably true also for the KA (s-wave) channel (Table 1). Multiplying the strength of the nA and the KA vertex by 0.6 and 0.8, respectively, yields a better agreement in particular for the imaginary part of the T matrix (Figs. 1 and 2). 5 D13 resonances In the D13 partial wave the model yields a consistent picture for the upper resonance but fails to reproduce the behaviour of the scattering amplitudes for the lower resonance. In the latter case, the quark-model values for the d-wave nN vertex and the s-wave nA vertex are of comparable strength and relatively weak. Dressing the verteces and introducing the mixing of the two (bare) resonances considerably enhances the verteces. However, the enhancement is stronger in the case of the nA channel and, as a consequence, the resonance disappears in the elastic channel. A reasonable agreement is obtained if the quark-model strength of the nA is multiplied by 0.3 (Table 2 and Fig. 3). Resonance Hot [MeV] Fi/Ftot 7tN 7tA (s-wave) 7tA (d-wave) pN (s-wave) N(1520) 64 0.56 0.40 0.00 0.03 PDG 100 -125 0.55-0.65 0.15 0.11 0.09 N(1700) 55 0.02 0.10 0.70 0.18 PDG 50 -150 0.05-0.15 0.11 0.79 0.07 Table 2. The total and the partial widths for the N(1520) and the N(1700) resonance at the K-matrix pole (1515 MeV and 1700 MeV, respectively) using the unmodified quark-model values for the quark-meson couplings except for the s-wave nA coupling which is taken with only 30 % of the model strength. The PDG values are from [3]. n N ^ n A (D-wave) n N ^ K A (S-wave) n N ^ p N (S-wave) n N ^ n N(1440) (S-wave) W [MeV] W [MeV] W [MeV] W [MeV] 0.6 0.4 0.2 0 -0.2 -0.4 - n N ^ n N - n N ^ n(l=0) A --------- n N ^ n(l=2) A ---------- "n N ^ p(l=0) N .................... SAID i—^ ^ * /A " + + / \ rX -N-* + + —..........- "V.Í i^E------- 3È - Re T \k * 0.6 0.4 0.2 0 -0.2 -0.4 1100 1200 1300 1400 1500 1600 1700 1800 1100 1200 1300 1400 1500 1600 1700 1800 W [MeV] W [MeV] Fig. 3. The real and the imaginary part of the elastic and the dominant inelastic scattering amplitudes for the D13 wave. The data points for the elastic channel are from [4] For the upper resonance (N(1700)), the model predicts the dominance of the d-wave nA channel in agreement with the phenomenological analysis. 6 Concluding remarks The model reasonably well reproduces the behaviour of the amplitude in the S11 partial wave. The bare quark values for the meson verteces are generally too weak but are considerably enhanced through the meson cloud effects and the mixing of the bare quark resonances. At higher energies around and above N (1650), the model amplitudes are too small; here the contribution of higher resonances not included in our model becomes important as suggested by the phenomenological analysis [9]. The background contribution is in our model generated by the u-channel processes. We have not considered the Weinberg-Tomazawa term which would enhances the behaviour of the amplitudes near the pion threshold. The situation is less favourable for the D13 partial wave. The model fails to reproduce a rather intriguing behaviour of the s and d-wave nA amplitudes in the energy region of the N(1520) resonance. Here the phenomenological analysis suggests that they are comparable in strength which is difficult to explain in a model calculation where at relatively low pion momenta the 1 = 0-wave coupling is strongly favoured over the 1 = 2 one. References 1. B. Golli and S. Sirca, Eur. Phys. J. A 38, (2008) 271. 2. B. Golli, S. Sirca, and M. Fiolhais, Eur. Phys. J. A 42,185 (2009) 3. C. Amsler et al. (Particle Data Group), Phys. Lett. B 667,1 (2008). 4. R. A. Arndt, W. J. Briscoe, I. I. Strakovsky, R. L. Workman, Phys. Rev. C 74 (2006) 045205. 5. F. Myhrer and J. Wroldsen, Z. Phys. C 25, 281 (1984). 6. R. F. Alvarez-Estrada and A. W. Thomas, J. Phys. G: Nucl. Phys. 9,161 (1983). 7. A. J. Hey et al. Nucl. Phys. A362, 317 (1981). 8. D. M. Manley, E. M. Saleski, Phys. Rev. D 45 (1992) 4002 and private communication. 9. A. Kiswandhi and S. Capstick, Phys. Rev. C 69 (2004) 025205