Bled Workshops in Physics Vol. 9, No. 1 p. 87 Proceedings of the Mini-Workshop Few-Quark States and the Continuum Bled, Slovenia, September 15-22, 2008 Pion electro-production in a dynamical model including quasi-bound three-quark states* B. Gollia,c and S. Sircab>c a Faculty of Education, University of Ljubljana, 1000 Ljubljana, Slovenia b Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia c Jozef Stefan Institute, 1000 Ljubljana, Slovenia Abstract. We present a method to calculate the pion electro-production amplitude in a framework of a coupled channel formalism incorporating quasi-bound quark-model states. 1 Introduction In our previous work ([1] and [2]) we have developed a general method to incorporate excited baryons represented as quasi-bound quark-model states into a coupled channel calculation using the K matrix. The method has been applied to calculate pion scattering amplitudes in the energy region of low-lying P11 and P33 resonances. In addition to the elastic channel we have included the nA and aN (aA) channels where the a-meson models the correlated two-pion decay. We have been able to explain a rather intriguing behaviour of the scattering amplitudes in these two partial waves in the range of invariant energies from the threshold up to W ~ 1700 MeV. In this work we show how the formalism can be extended to the calculation of electro-production amplitudes. 2 Incorporating quark-model states into multi-channel 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^ a{mt(k)aimt (k) + [vlmt(k)almt(k] + V^M a{mt(k)] } , Imt where a{mt (k) is the creation operator for a meson with angular momentum 1 and the third components of spin m and isospin t. In the case of the pion, we include only 1 = 1 pions, and Vmt(k) = —v(k) Y.t=i °mTt is the general form of the pion source, with the quark operator, v(k), depending on the model. It includes also * Talk delivered by B. Golli the possibility that the quarks change their radial function which is specified by the reduced matrix elements Vbb ' (k) = (B||V (k)||B'), where B are the bare baryon states (e.g. the bare nucleon, A, Roper,...) We have shown that in such models it is possible to find an exact expression for the K matrix without explicitly specifying the form of the asymptotic states. In the basis with good total angular momentum J and isospin T, the elements of the K matrix take the form: K{7,h = -7t^<^T||V(k)||¥B,> , A/H = , (1) wEB H V kw where w and k are the energy and momentum of the meson. Here is the principal value state corresponding to channel H specified by the meson (n, CT, ...) and the baryon B (N, A,...): i^t> = a/h j^cgl®*) + [at(k)|^b)]jt+x}^+xe7k7k)w • (2) The first term is the sum over bare tree-quark states ;)), Ny = ^ • (7) Here = (W2 - M2 - Q2)/2W, k* = + Q2 , EN = W - , with Q2 measuring the photon virtuality. We perform the spin-isospin decomposition of the outgoing state ¥' + >(ms,mt;k0,t))= £ i^^j^'fMj, MT;k0)l)m)t))C^slmC™;it. ImJT (8) Commuting a* in (5) to the left and using the expansion (8), we can write the T matrix in the JT basis as tJNYN =-nNY (J(MJM T;ko,l)|VY(^,ky)|¥N(mSmt)). (9) The electro-production amplitude is proportional to (9) through T = Y^k^TSTt M, hence =--^=(W^)(MJMT;lco)l)|VY(^lcY)|VN(mX)>- (10) V koky The amplitudes proportional to the elements of the K matrix are obtained by replacing the state ¥J+ by the corresponding principal value state: =--(¥^T(MjMT;ko,l)|Vy(ia,ky)|¥N(m^m;)). (11) ko kY The procedure to calculate the electro-production amplitudes in our formalism is the following: we first evaluate (11) using (2) as obtained in pion scattering, and then compute (10) using M = MK + iTMK. (This equation trivially follows from the Heitler's equation T = K + iTK since the proportionality factor between T and M is the same as between K and MK.) In principle, this equation involves also the matrix elements corresponding to Compton scattering. They can be neglected since they are orders of magnitude smaller than those containing strong interaction. In the P11 case we have M nN (W) = mKN (W) + i TtiNtiN (W) + T^n nA (W)M7tA{W) K +T„NaN (W)MffN(W) (12) We have further simplified the equation by using averaged values for amplitudes involving the nA and the cN channels and thus avoiding integration over the corresponding invariant masses. In the P33 case we have also added the nN (1440) channel, while the cN channel has been replaced by the cA channel. 4 The behaviour of the amplitudes close to a resonance From (3) and (4) it follows that close to a resonance, denoted by R, the K matrix element between the elastic channel and the nB channel can be cast in the form „ EnEb Bd n 1 , „background IWN = -nyj kokBW2 CRVNR(Tco) + K„B7t8N . After some rearrangements, the principal value states (2) take the form = —K^BtiN \/ ^^ + |¥Hnon-reS) n2^otN Vnr with VNR(k)[at(k)|¥N)]JT v- f VBR(k)[at(k)|¥B)]JT cuk + EN (k) - M 2— dk- W + EB(k)- M (the inclusion of the cN channel in the P11 case is straightforward). We can now split the K-matrix type amplitudes (11) into the resonant part containing the pole and the "non-resonant" part: "s = w /sf knh<*T(W)I W - A*'»' ■ (13, We see that the resonant matrix elements depend on a particular channel (H) only through the K matrix element referring to that channel. Next we plug (13) into (12) and take into account the relation between the T and the K matrix (T = K + iTK). The resonant part of the electro-production amplitudes then reads (res) , /<^YEN I k0W V^" ^(res) nAn ,a7 m, >T n .. MN =V^W V^W^TW ( K (W)|Vy|¥n) T^N^N , (14) while the non-resonant part satisfies MNnon) = mN (non) + i t ./(non) x -r-rK (non) = -j-jK (non) l«NnNMN + I nNnAMA + I nNsNM, Let us note that (¥Res) (W)|VY|¥n) is the electro-excitation amplitude for the resonance R. For a sufficiently weak meson field the state ¥ is dominated by the bare-three quark core surrounded by a cloud of pions, which is a familiar form of a baryon state in chiral-quark models. The relation (14) can be rewritten in a more familiar form by noting that the elastic part of the K matrix can be written as CUQEN ^nk _ I^ei KtiN 7tN — 71 - (15) koW Zr(Mr) Mr - W' where rel is the elastic width of the resonance. Expressing VNR from (15) we get where we have taken into account that at the resonance TnNnN = irel/rtot. (16) 5 Multipole decomposition for the P11 and P33 wave Expanding (6) into multipoles, we have in the P33 case: M (3/2) 1 + Y-N(V + )||yM1||¥N)) (3/2) 1+ 30koW JT 11 Y and in the Pll (I = T = i) case (17) M (1/2) ^Y^N ad( + )|Ia7M1 ii-vi; \ A/t(0) _ . I ^Y^N ai,( + ) IIA7M1 6k 0W related to 7t° production amplitude on the proton as = M^ + j , and ^ M\m). Here IV and IS denote the isovector ,(1/2) on the neutron as Mn_ M (0) 1- 3 and the isoscalar part of the interaction, respectively. The same formulas apply to the Mk amplitudes. (Similar relations can be derived for the scalar amplitudes.) The transverse electro-excitation amplitudes are defined in terms of the he-licity amplitudes Amj . In the P33 case we separate them into the magnetic dipole and the electric quadrupole part: Ml E2 1 2a/3 \/3 Ai +Ai As - \/3 Ai = -^(^eS)IIVy(Ml)||¥N) (^llVy^)^). (18) (19) Taking into account (17) and (16) we reproduce the familiar relation f = m1+/2) = if Ml El3+/2) = if E2 , 3CUYEY rd I 8nkoWr20t ■ In the P11 case only one transverse helicity amplitude appears and we find A^1 7 <^es)||VM1 {IS)\\WN) ± 4= (^es)||VM1 (IV)||¥N) (the reduced matrix elements appear only in the angular momentum, the third component of the isospin are MT = m^ = 6 Preliminary results in the N(1440) sector The P33 wave amplitudes in the region of the A(1232) have been extensively investigated in our previous works (see e.g. [3] and [4]). Since the electro-production amplitudes are dominated by the resonant contribution, they follow the shape of the elastic T matrix accordingly to (14). This is not the case in the P11 wave. In Fig. 1 we show some preliminary results (without including the nA and the cN channels) for the electro-production amplitude in the region of the N(1440) resonance showing the important role of the background processes. These are dominated by the nucleon pole contribution, the contribution from the second term in (2) (t-channel), and by a u-channel-type process with the A in the intermediate state. Below the resonance, the contribution of the resonant term is almost negligible. The resonant contribution itself is dominated by the pion cloud and the admixture of the nucleon component which considerably reduces the contribution. This point is still under investigation; we expect that inclusion of higher resonances may cure this deficiency. 1 1 exp i—i—i MAID fit - x resonant ................ - y \ nucl. pole ........ s \ background ------- / \ total - - \ \ ! 1300 1400 W [MeV] 1300 1400 W [MeV] Fig.1. The real (left panel) and the imaginary (right panel) parts of the electro-production amplitudes for the P11 partial waves. The MAID result is taken from [5]; the experimental points from [6]. The thin dashed curve in the right panel shows the effect of omitting the nucleon component in the resonant state. References 1. B. Golli and S. Sirca, to be published in Eur. Phys. J. A, arXiv:0708.3759 [hep-ph] 2. B. Golli and S. Sirca in: B. Golli, M. Rosina, S. Sirca (eds.), Proceedings of the MiniWorkshop "Hadron Structure and Lattice QCD", July 9-16, 2007, Bled, Slovenia, p. 61. 3. M. Fiolhais, B. Golli, S. Sirca, Phys. Lett. B 373, 229 (1996) 4. P. Alberto, L. Amoreira, M. Fiolhais, B. Golli, and S. Sirca, Eur. Phys. J. A 26 (2005) 99. 5. D. Drechsel, S. S. Kamalov, L. Tiator, Eur. Phys. J. A 34 (2007) 69. 6. R. Arndt et al., Phys. Rev. C 52 (1995) 2120; R. Arndt et al., Phys. Rev. C 69 (2004) 035213. 0 1100 1200 1500 1600 1700 1100 1200 1500 1600 1700