A covariant model for the $\gamma^\ast N \to N^\ast(1520)$ reaction

We apply the covariant spectator quark model to the study of the electromagnetic structure of the $N^\ast(1520)$ state ($J^{P}= \frac{3}{2}^-$), an important resonance from the second resonance region in both spacelike and timelike regimes. The contributions from the valence quark effects are calculated for the $\gamma^\ast N \to N^\ast(1520)$ helicity amplitudes. The results are used to parametrize the meson cloud dominant at low $Q^2$.

The electromagnetic structure of the nucleon (N ) and the nucleon excitations (N * ) can be probed through the γ * N → N * reactions, with squared momentum transfer q 2 < 0 (spacelike region). Experimental facilities such as Thomas Jefferson Laboratory (Jlab), MIT-Bates and Mainz provide nowadays important information about those reactions for low and high Q 2 , where Q 2 = −q 2 > 0 [1][2][3].
In order to interpret the data one has to rely on theoretical models based either on the fundamental QCD degrees of freedom, quarks and gluons, or effective ones. Although the microscopic dynamics refers to quarks and gluons, those degrees of freedom can be observed only at very high Q 2 . At low and intermediate Q 2 more phenomenological descriptions with baryon and mesons or constituent quarks can be justified [1][2][3][4].
The state N * (1520) (J P = 3 2 − ) is an important state, as it is the N * (1535) (J P = 1 2 − ), from the second resonance region, and plays also an important role in the timelike region (Q 2 < 0). The extension of the CSQM to the timelike region was already made for the ∆(1232) [14]. The available data for the γ * N → N * (1520) transition, combined with model estimations suggest that valence quark effects dominate at high Q 2 , while effects of the meson cloud dressing of the baryons can be significant at low Q 2 [1,2,15,16].
In the CSQM the wave function of the baryon B, Ψ B , is determined by the baryon properties (flavor, spin, orbital angular momentum, etc.), and their symmetries, and the radial part represented by a scalar function ψ B adjusted phenomenologically to the experimental electromagnetic form factor data, and lattice QCD data for some baryon systems [5,7,8]. According to the specta-tor theory, two of the quarks can be considered on-shell in the intermediate states and the third quark is free to interact with the electromagnetic probe. Integrating in the degrees of freedom of the quark-pair we can reduce the baryon to a quark-diquark system where the diquark is on-shell with effective mass m D [6,7,17]. The quark electromagnetic current is described using vector meson dominance (VMD). The quark electromagnetic form factors are parametrized by some vector meson masses [6][7][8].
The vectors meson poles parametrize the spatial extension of the constituent quarks. VMD is very useful for the extrapolation of the model to other regimes such as the lattice regime [18] and the timelike regime [14]. Finally the electromagnetic current between two baryon states is obtained by taking the impulse approximation and summing in the individual quark currents [5][6][7]17].
For the transition between the nucleon and the N * (1520) state, here labeled as R (for resonance), we need to construct the nucleon and the R wave functions. For the nucleon, we use the wave function derived in a work where the nucleon is described as a S-wave quarkdiquark system [6]. For the R wave function we took a covariant generalization of the non relativistic form [4,19]. The non relativistic form couples angular momentum states Y 1m (m = 0, ±1) in the diquark momentum k and in the relative quark-pair momentum r to threespin states with different symmetries (mixed-symmetric, mixed-antisymmetric and totally symmetric). In principle the N * (1520) wave function is a combination of states with core spin (sum of the quark spins) 1/2 and 3/2, but hadronic decays suggests that the spin-3/2 admixture is small (sin θ D ≈ 0.1 ≪ 1) [4].
In the transition between a J P = 1 2 + (nucleon) and a J P = 3 2 − , like the N * (1520) state, one defines three independent electromagnetic form factors. In particular one can consider G M , the magnetic dipole form factor, the function G ′ 4 = −(G M + G E ), where G E is the electric quadrupole form factor, and G C , the Coulomb quadrupole form factor. Alternatively, one can use the classic helicity amplitudes, A 1/2 , A 3/2 and S 1/2 , in the R rest frame. One has then [19] The form factors and the helicity amplitudes are functions of Q 2 only. In the CSQM the transition form factors are written as a combination of quark electromagnetic form factors and the radial wave functions. The dependence on the radial wave functions is codified in a covariant function I z (Q 2 ), which is an overlap integral. In the R rest frame one has where P R (P N ) are the R (nucleon) momentum and k the diquark momentum at the R rest frame. The integration symbol is a short notation for the covariant integration in k. A covariant generalization of the equation (2) can be derived in an arbitrary frame [19]. The orthogonality between the nucleon and the R state is expressed by the condition I z (0) = 0. Based on this condition it is possible to construct a function ψ R (P R , k) with one adjustable parameter that can be fitted to the data. The calculations using the spin-1/2 component of the R wave function lead to According to equation (1) these results are a consequence of G ′ 4 ≡ 0. The results for the valence quark contributions to the helicity amplitudes are presented in the figure 1 (dash-lines). Our fit of the function ψ R was made for Q 2 > 1.5 GeV 2 , a regime where we expect small meson cloud effects.
Although in the CSQM the quarks have structure, there are processes such as meson exchange between two different quarks inside the baryon, that cannot be interpreted just as quark dressing and have to be classified at hadronic level as meson cloud [8,12]. Assuming then that the CSQM gives a good description of the valence quark effects only, we used the model to extract the meson cloud contributions from the data. Since the pion is the dominant decay, we assumed a parametrization regulated by the pion with some effective momentum ranges (cutoffs) adjusted to the global data. The results of the fit are presented also in the figure 1 (dot-dashed lines). The solid line gives the final result (total).
From figure 1, one concludes that valence quark effects dominate the high Q 2 regime (Q 2 > 1.5 GeV 2 ) for the amplitudes A 1/2 , S 1/2 , although meson cloud effects are significant at low Q 2 . As for A 3/2 , only the meson cloud contributes. This result differs from other quark model estimates, but it is consistent with the large meson cloud estimate from the EBAC group [16].  [15]. Not included are the results from the MAID analysis [20]. Finally we look at the results for the multipole form factors at very high Q 2 , extending the model to the region of the Jlab 12-GeV upgrade. In the figure 2, we plot G M , −G E and G M + G E , normalized by the dipole form factor G D , for a better observation of the falloff of those form factors. Note the scaling of G M , −G E , and also the very slow falloff of G M + G E , that is negligible for very high Q 2 , according to pQCD [21] and also to the CSQM. Those results suggest that G M + G E ≃ 0 (or G E ≃ −G M ), equivalent to A 3/2 ≃ 0, happens only for very large Q 2 .
The present parametrization of the γ * N → N * (1520) transition form factors in the spacelike regime can be extended to the timelike regime [22].