Bled Workshops in Physics Vol. 4, No. 1 A Proceedings of the Mini-Workshop Effective q-q Interaction (p. 83) Bled, Slovenia, July 7-14, 2003 Calculation of electroproduction amplitudes in the K-matrix formalism Bojan Golli Faculty of Education, University of Ljubljana, and J. Stefan Institute, Ljubljana, Slovenia Abstract. We present the K-matrix approach to calculate pion electroproduction amplitudes in the framework of chiral quark models. We derive the relation between the K-matrix and the experimentally measured T-matrix and show how to separate the resonant contribution to the amplitudes from the background. The work is being done in collaboration with Manuel Fiolhais (Coimbra), Pedro Alberto (Coimbra), Ze Amoreira (Covilha) and Simon Sirca (Ljubljana). 1 Motivation In our previous work [1] on electroweak excitations we have shown that different versions of chiral quark models may successfully describe the properties of the low lying nucleon resonances. In these calculations the excited states have been treated as bound states which is justified if we are interested only in the resonant part of the production amplitudes. The total amplitudes as measured in the experiment however include also non-resonant (background) processes related to the outgoing pion. Incorporating the decaying channel in the model calculation may therefore represent a stringent test for the model as well as yield interesting information on the production mechanism and in particular on the role of non-quark degrees in freedom in baryons. In several models such as the linear ff-model or the chromodielectric model, which include nonlinear effects and in which the interplay between quark and non-quark degrees of freedom is treated in a self-consistent way, the calculation is only feasible in a variational approach. While such an approach can be easily implemented in the bound state calculations its application to the description of scattering processes is much more complicated. In [2] the Kohn variational approach has been adopted to calculate the iN phase shifts and the structure of the resonant state in the Cloudy-Bag-type models. In this work we extend the approach to be able to calculate also the electroproduction amplitudes for pions. We shall limit ourself to the description of the A resonance. 84 B. Golli 2 The variational approach to the K-matrix We shall be interest in the class of models in which the pion is coupled linearly to the quark source: H dk^ a^t(k)amt(k) + Vmt(k)amt(k) + ^(k) EBcjjOB mt B (1) where 3 Vmt(k) = -V(k) £ TtW , vmt(k) = (-1)t+mV-m-t(k) . (2) i=1 Here amt (k) and amt(k) are annihilation and creation operators for the I = 1 pions with the third component of the spin m and isospin t; cB and cB are annihilation and creation operators for the "bare" baryons made up of three quarks, EB are the corresponding bare energies, V(k) is the source function determined from the quark profiles. We shall limit here only to two state, the nucleon N, and the A. In the present stage we do not include meson self-interactions. In the variational approach to the K-matrix when only a single channel is opened (such as the resonant scattering in the P33 channel below the 2 pion threshold) the resonant state is assumed in the form: IW = Ca Oa> + dkT!(k,ko) [a^Jk)!®^)]" . (3) Here |®N) and |Oa) represent the nucleon and the A bound states normalized as I®n) = 1 and (Oa|Oa) = 1 , respectively; "(k,ko) describes the scattering pion and []ST denotes the coupling of spin and isospin of the pion and the bare quark core to the quantum numbers of the A. Asymptotically the pion wave function behaves as "(r,ko)= ko ji (kor) — tan5koyi (kor) , r -> oo . (4) Note that in the K-matrix approach the standing waves rather than outgoing (and incoming) waves are used. In k-space this leads to Tl(k,ko) = ,/^5(k-ko)+ X(k'k°] , K = tan6 = \/2^x(ko,k0). (5) v 2 !k — !o ko The nucleon state |®N) in (3), modified in presence of the scattering pion, i.e. it depends on k and k0 of the pion, should asymptotically go over to the ground states, |®N) ON for k^ ko. Before introducing the variational principle which determines the parameters of the trial function of the type (3), i.e. the pion wave function "(k, ko), the parameter ca as well as structure of the states Oa and O^, we first prove an important relation which hold in this type of models: X(k0,k0) = (¥|(H-E) [a^lkollON)]1' . (6) Calculation of electroproduction amplitudes in the K-matrix formalism 85 We assume that both |¥) and |®) are exact states, i.e. H|¥) = E|¥) and H|®) = EN |®). We should keep in mind that H is not Hermitian since ¥ is not a square integrable function, hence (¥|H = (¥|E, and the above expression does not vanish. We write H = H + (H - H): <¥|(H-E) [qLOco^n)]" =(¥|(H-Ht) [atmt(Tc0)|n>] The non-Hermitian part of H is the pion kinetic energy term: H kin dr^_ (-1)' i-t(r) -A, it(r) . (7) (8) Only those terms in ¥ that asymptotically (r —> oo) behave as r 1 contribute to (7), i.e. the terms involving "(k, k0). Expression (7) yields dk"(k,ko) dr^[(k)]M74(r) A A it(r) [afmt(ko)|®n>] (9) We now commute a and a through the pion field; only the commutators produce a non vanishing contribution. We next perform the k-integration yielding the pion wave function in r-space (4). After performing the angular integration we end up with the following integral k2 2iv0 dr Í ji (kor) - (Tcor) ko d 2 d d 2 d dr dr dr dr ji (kor) (10) The first term in () vanishes. We perform an integration per partes and we are left with the Wronskian of j1 and y1, W(j(k0r),y1 (k0r)) = 1/(k0r)2, multiplied by k2r2x(k0,k0). The resulting integral is finite for r —> oo and equal to x(k0,k0) which proves (6). Relation (6) holds quite generally since the only assumption in the derivation was on the asymptotic form of the wave function For the class of models (2) we can derive another useful relation by commuting a in (6) through (2): (H-E) a^t (k0) = ! a^t (k0) + Vmt (k0) + a^t (^0 )(H—E). Since (H — E) |®) = — !0|®) the last term cancels the first one and we are left only with the matrix element of (2): K™ = K(k0,k0) = o.Tco) = ko ko fffi||®)V(ko). (11) This is an exact relation since in the derivation we have not made any assumption about the structure of the resonant state or the ground state; we have only referred to the form of the pion field far away from the source. The result is similar to the expression derived by Chew and Low for the T-matrix [3]. We now sketch a more general derivation of the Kohn variational principle for the K-matrix in the case of pion scattering compared to the derivation given in [2]. It is valid for a more general class of Hamiltonians than (2); we only require that the exact solution (as well as the trial state) has the form of (3) where the pion field asymptotically behaves as (4). 3 3 1 7 3 3 1 7 86 B. Golli We start by observing that the matrix element between the exact wave function We and a trial wave function satisfying the boundary condition (4) with an approximate phase shift 5t, can be written in the form (¥e|H-E|¥t> (tan6e-tan6t) . (12) The proof goes similarly as from (7) to (10), the only difference is that in (10) the expression ^/2/7t(ji (k0r) — tan6tyi (k0r) appears instead of the second }i (k0r). Introducing = ¥t — We and taking into account (H — E) |¥e) = 0 we can write tan 6e = tan 6t - ^ ((¥t|H - E|Yt) - (S^IH - E|6¥>) (13) ko which means that the functional =tan6t-^{¥t|H-E|¥t> (14) ko is stationary with respect to variations of ¥t. 3 The K-matrix for the pion production We may now extend the model by introducing the coupling to the photon, H H + H,, where H, has the usual form of the EM Hamiltonian with the minimal coupling to the hadronic EM current. Then the resonant state will include the emission and absorption of the M1 and E2 photons (denoted by the common index M) which will modify our ansatz (3) in the following way: = Oa I®a) + [afmt(ko )|0 n; + dk Xti7i(Tc,Tco) !k - !0 [amt(k) I ®N>] M dq ym (q,ky) [ak(q)|®n>]' (15) We make the standard assumption that the EM coupling is much weaker compared to the strong coupling and does not modify the structure of the resonant state. The functions xM1 and xE2 are related to the corresponding K-matrices for the pion electroproduction. The proof of these relations is analogous to the derivation of the K-matrix for the elastic pion scattering. We find: k, K M .,1(ky, ky) r i11 -(¥|(H-E) [a^milON)]22 (16) leading to K M k, {VI I [ H- I ®>. (17) 3 3 1 7 3 3 7 "z Calculation of electroproduction amplitudes in the K-matrix formalism 87 4 Splitting the amplitudes in the resonant and the non resonant part The K-matrix calculated in a model as discussed above exhibits a typical resonant behavior (provided of course the bare A-N splitting is sufficiently large). The energy at which the phase goes through 90° corresponds to the energy of the physical A, Ea. It can be parametrized in the form suggested by Davidson et al.[5]: C knn ee tan 6 = --- + D = tan + tan 6b . (18) Ea — E Here 5 is the full phase shift, is called the resonant phase shift and 5b the background phase shift (BPS), which - as shown below - is identical to the BPS in the T-matrix approach. Note that both, 5 and go through 90° at the same E, (e.g. E = 1232 MeV for A). The width of the resonance is simply related to the parameter C by FK = 2C. The T-matrix which is directly related to experimentally measured amplitudes is related to the K-matrix as T = ——— =_C+(EA-E)D_ 1-iK (EA-E)-i(C+(EA-E)D) ' In contradistinction to the K-matrix which is a real quantity the T-matrix is complex. The position of the pole in the complex plain can be easily determined if we assume that the coefficients C and D do not depend on the energy. Then the above expression can be brought in the familiar form: F T /2 T,„IE) =e2li> Ma_ £_irT/2 +8inSne'i'' CO) with CD 2C Ma =EA + T—^ =EA + lrTtan6b, rj = —^ = F* cos2 6b . (21) From the experimental phase shift in the P33 channel the following values are extracted MA = 1210 MeV, Fj = 100 MeV and 6b ss -23.5°. Turning to the T-matrix matrix for the electroproduction, T,„, we note that the effect of the EM coupling to the photon has a negligible effect on the structure of the state (15). The position and the width of the pole is not changed with respect to the pure iN channel; the phase shift of the amplitudes is that of the iN scattering. This is the so called Watson theorem which in our case can be expressed in the form [5] K T-yn = KY7t(1 +iT7t7t) = --—— . (22) I — iK„„ Using the popular parametrization A KY7t = --- + B (23) EA — E 88 B. Golli we can express (22) in the form: A + B(EA - E) (EA-E)-i[c + d(EA-E)]' (24) Assuming again that the parameters A, B, C and D do not depend on the energy, we can relate our result to the parametrization of T,n used in parameterizing the experimental data [4]: V = + ««'"■. TP5) We obtain: A B D2 (A B 1 - D2\ , B te'=c-2dttd^ + ld(c + dttd^j and d sm ' (26) We have to comment on the above assumption of constant parameters. If we identify the parameters in (18) and (23) with what comes out from a model calculation it turns out that the parameters exhibit a strong dependence on the pion momenta k0 and therefore also on the energy. This can be seen already from the lowest order expression for the scattering matrix which takes the form: k7t7t=7T^v(ko)2 ko (A|| X crr||N)2 4