$K^0\Lambda$ photoproduction off the neutron with nucleon resonances

We investigate kaon photoproduction off the neutron target, i.e., $\gamma n \to K^0 \Lambda$, focusing on the role of nucleon resonances given in the Review of Particle Data Group in the range of $\sqrt{s} \approx 1600 - 2200$ MeV. We employ an effective Lagrangian method and a Regge approach. The strong couplings of nucleon resonances with $K\Lambda$ vertices are constrained by quark model predictions. The numerical results of the total and differential cross sections are found to be in qualitative agreement with the recent CLAS and FOREST experimental data. We discuss the effects of the narrow nucleon resonance $N(1685,1/2^+)$ on both the total and differential cross sections near the threshold energy. In addition, we present the results of the beam asymmetry as a prediction.

1. Kuznetsov et al. reported the measurement of the cross sections for η photoproduction off the neutron, which shows a narrow bump structure near the center-of-mass (CM) energy W = 1.68 GeV [1]. In the γp → ηp reaction, there is only a small dip structure at the same energy. The CB-ELSA and TAPS Collaborations in Bonn [2] and the A2 Collaboration in Mainz [3][4][5] have confirmed this feature of η photoproduction off the neutron. This phenomena is often called the neutron anomaly in η photoproduction. However, there is no consensus in the interpretations on the narrow enhancement at W = 1.68 GeV. In fact, the narrow nucleon resonance around 1.68 GeV was predicted by the chiral quark-soliton model [6][7][8] in which the neutron anomaly was explained in terms of the different values of the N (1685) → N γ transition magnetic moments. The A2 measurement of the helicity-dependent γn → ηn cross sections favors the existence of a narrow P 11 resonance [3]. On the other hand, Ref. [9] disputed that such the narrow enhancement arises from the interference between N (1535)S 11 and N (1650)S 11 , based on the Bonn-Gatchina multichannel partial-wave analysis. However, Ref. [10] refuted it in favor of the narrow P 11 nucleon resonance. In this situation it is of great importance to scrutinize the narrow structure around 1.68 GeV and the related neutron anomaly in other processes such as K 0 Λ photoproduction.
In the present Letter, we investigate the K 0 Λ photoproduction off the neutron, focussing on the effects of the narrow resonance structure around 1.68 GeV, which appeared in the γn → ηn reaction. Since the threshold energy of the γn → K 0 Λ is 1.61 GeV, K 0 Λ photoproduction can provide a possible clue in understanding the nature of the narrow nucleon resonance N (1685, 1/2 + ). In this regard, the investigation on K 0 Λ photoproduction will shed light on the neutron anomaly yet from the different facet. The FOREST Collaboration at the Research Center for Electron Photon Science, Tohoku University [11,12] and the CLAS Collaboration at the Thomas Jefferson National Accelerator Facility [13] have announced the experimental data on the total and differential cross sections of K 0 Λ photoproduction off the neutron 1 . Thus, it is of great interest to examine theoretically the role of the narrow nucleon resonance N (1685, 1/2 + ) also in this γn → K 0 Λ reaction. We will employ an effective Lagrangian approach in which we can consider directly the nucleon resonances in the s channel. We will introduce sixteen different nucleon resonances up to 2.2 GeV. In addition, we take into account the narrow nucleon resonance N (1685, 1/2 + ) corresponding to the narrow enhancement found in η photoproduction off the neutron. We also include the K * Reggeon exchange in the t channel, since it explains properly the high-energy behavior of the total cross section.

2.
In an effective Lagrangian approach, the γn → K 0 Λ reaction can be represented by the tree-level Feynman diagram illustrated in Fig. 1. The notations of the four momenta of the incoming and outgoing particles are given in Fig. 1(a) in which the t-channel K * -Reggeon exchange is depicted. Other exchanges such as K, K 1 , and higher strange mesons are excluded in the present process because of their small photocouplings to the K 0 meson, e.g., Br(K * (1410) → K 0 γ) < 2.2 × 10 −4 , Br(K * 2 (1430) → K 0 γ) < 9 × 10 −4 [14]. The s-channel diagrams shown in Fig. 1(b) include contributions from the neutron and their resonances, generically. We will consider the sixteen different nucleon resonances taken from the Particle Data Group (PDG) data [14]. On top of them, we include the narrow N (1685, 1/2 + ) resonance, which corresponds to the narrow enhancement found in η photoproduction [1][2][3][4][5]. Λ and Σ exchanges are included in the u-channel diagrams drawn in Fig. 1 The general expressions of the electromagnetic (EM) interaction Lagrangians can be written as where A µ , K, K * , and N designate the fields for the photon, pseudoscalar kaon, vector kaon, and nucleon, respectively. Λ and Σ denote respectively the fields for the ground-state hyperons. M N and e N stand respectively for the mass and electric charge of the nucleon, whereas e denotes the unit electric charge. Since the neutron is involved in the present work, we need only the magnetic term in the γN N vertex.
Concerning the values of the coupling constants, g 0 γKK * is determined by the experimental data for the decay width Γ(K * → Kγ), resulting in −0.388 GeV −1 [14]. The sign of the coupling is fixed from the quark model. The anomalous and transition magnetic moments of the baryons are given by the PDG [14] κ N = −1.91, κ Λ = −0.61, µ ΣΛ = 1.61. ( The effective Lagrangians for the meson-nucleon-hyperon interactions are given by where Y represents generically the fields for the hyperons (Λ or Σ 0 ). The strong coupling constants are taken from the average values of the Nijmegen soft-core potential (NSC97) [15] g Note that although we use the pseudovector coupling for the latter one in Eq. (3), the numerical results of the present work almost do not change when the pseudoscholar coupling is employed, since the effects of nucleon and hyperon exchanges turn out to be tiny. In general, the invariant amplitude for photoproduction can be written by where ǫ µ represents the polarization vector of the incident photon. u N and u Λ denote the Dirac spinors for the incoming nucleon and the outgoing Λ, respectively. The isospin factors are given by I K * = I N = I Λ = 1 and I Σ = −1 in the present process. The effective Lagrangians of Eqs. (1) and (3) being considered, the individual amplitudes for the Born term are obtained as follows: where q t,s,u designate the four momenta of the exchanged particles, i.e., q t = k 2 − k 1 , q s = k 1 + p 1 , and q u = p 2 − k 1 . Considering the finite sizes of hadrons, we need to introduce a form factor at each vertex. We use the following form where q 2 is the squared momentum of q s,u and M ex the mass of the corresponding exchanged particle, respectively. Although we are mainly interested in the vicinity of the threshold energy for K 0 Λ photoproduction, future experiments are exppected to cover higher energy regions. Thus, we employ the t-channel Regge trajectory for the K * -meson exchange and follow Refs. [16,17]. This can be done by replacing the Feynman propagator in Eq. (6) with the Regge one as where either a constant phase (1) or a rotating one (e −iπα(t) ) can be considered for the Regge phase. The K * Regge trajectory reads [17] and α ′ ≡ ∂α(t)/∂t denotes the slope parameter. The energy-scale parameter is chosen to be s 0 = 1 GeV 2 for simplicity. Consequently, the entire Born amplitude is written as Unlike the charged kaon production, all the terms are manifestly gauge-invariant, so we do not need to introduce any prescription for gauge invariance.
It is worthwhile to compare these coupling constants extracted from the prediction of the quark model [21] with those calculated from the experimental data on the branching ratios [14], although the signs of the couplings can be fixed only in the quark models. In Table II, we summarize both values for the seventeen different nucleon resonances under  consideration. Only five resonances provide both of them. Comparing these two values, we find that they are close to each other. Since the experimental data exist for the N (1880, 1/2 + ) and N (1900, 3/2 + ) resonances, we determine the strong coupling constants for them by using the PDG data. Their signs are determined phenomenologically. The only exception is the N (1720, 3/2 + ). We reduce its strong coupling constant 40% smaller than that from the quark model such that we are able to reproduce the experimental data. Though we could reproduce the experimental data better by fitting the coupling constants, we have not performed it, because the main concern of the present work lies in understanding the role of each nucleon resonance and we want to avoid additional uncertainties arising from the valuse of the strong coupling constants.   [21] and the branching ratios of N * s to the KΛ state are taken from Ref. [14]. We can construct the individual amplitudes for the nucleon-resonance exchange using Eqs. (11) and (13) in the form of M = I N * ū Λ M N * u N as in Eq. (5): where Γ N * designates the full decay width of N * . The spin-3/2, -5/2, and -7/2 projection operators, given by ∆ ρ µ , ∆ ρσ µν , and ∆ ρσδ µνα , respectively, are represented in the Rarita-Schwinger formalism [24][25][26][27] as in Refs. [18,19,28,29]. The phase factors of the invariant amplitudes for the nucleon resonances cannot be determined by symmetries only, so we regard them as free parameters. These amplitudes are thus written by where the gaussian form factor is employed [30,31] (20)  , and solid (black) curves correspond to contribution from K * -Reggeon exchange, that from the sum of N * exchanges, and the total contribution, respectively. The data are taken from the CLAS experiment [13]. Right: Each contribution to the γn → K 0 Λ reaction for various nucleon resonances.
3. Before we present the numerical results, we need to mention how the model parameters are fixed. The cutoff masses are fixed to be Λ B(N,Λ,Σ),N * = 0.9 GeV for simplicity. We do not fit the values of the cutoff masses to avoid additional uncertainties arising from them. We find that at high energies above the CM energy W = 2.2 GeV, where K * -Reggeon exchange comes into a dominant play, the rotating Regge phase (e −iπα K * (t) ) and the phase angle ψ N * = π turn out to be the best choice.
In the left panel of Fig. 3, the total cross section for the γn → K 0 Λ reaction is drawn as a function of the CM energy. The N * contributions are dominant in the lower-energy region (W 2.2 GeV). K * Reggeon exchange in the t channel being included, the result is in agreement with the CLAS data [13]. As W increases, the K * Reggeon takes over N * contributions. Because of K * Reggeon exchange, the total cross section behaves asymptotically as σ ∼ s α K * (0)−1 and describes the experimental data well. As shown in the left panel of Fig. 3, the result is slightly underestimated in the vicinity of the threshold energy, compared to the CLAS data. Each contribution of various nucleon resonances is drawn in the right panel of Fig. 3. The well-known N (1650, 1/2 − ) and N (1720, 3/2 + ) resonances are the most dominant ones. While the N (1675, 5/2 − ), N (1710, 1/2 + ), and N (1900, 3/2 + ) resonances have sizable effects on the total cross section, all other resonances almost do not affect it, so we show only the contributions of the nine N * resonances in the figure. Moreover, the N (1685, 1/2 + ) resonance has only a marginal effect on the total cross section. Thus, as far as the results of the total cross section are concerned, the present ones are more or less similar to those of Ref. [9] where the Bonn-Gatchina coupled-channel partial-wave analysis was used. In Ref. [9], it was shown that the partial waves J P = 1/2 ± and 3/2 + contribute dominantly to the total cross section. . Differential cross section for the γn → K 0 Λ reaction as a function of cos θ K 0 CM for each beam energy. The dashed (blue), dot-dashed (magenta), and solid (black) curves correspond to the contribution from K * -Reggeon exchange, that from the sum of N * exchanges, and the total contribution, respectively. The dotted (green) one indicates the total contribution without the effect of the narrow N (1685, 1/2 + ) resonance. The data are taken from the FOREST experiment [12]. Figure 4 draws the differential cross sections for the γn → K 0 Λ reaction as a function of cos θ K 0 CM , being compared with the FOREST experimental data [12]. The photon energy is varied from E γ = 937.5 MeV to E γ = 1137.5 MeV. The dashed curve is drawn for the contribution of K * Reggeon exchange. As expected, its effect is rather small in the range of the photon energy given in Fig. 4. Here, main interest lies in the effect of the narrow N (1685, 1/2 + ) resonance. While the dotted curve is depicted without N (1685, 1/2 + ) taken into account, the solid one includes it. Though the effect of the N (1685, 1/2 + ) is very small at smaller values of E γ , it comes into play as E γ increases. In particular, the experimental data of the differential cross section at E γ = 1037.5 MeV and E γ = 1062.5 MeV can be explained only by including the narrow N (1685, 1/2 + ) resonance. Otherwise, the results would be overestimated in the forward direction and would be underestimated in the backward direction. Although the N (1685, 1/2 + ) does not give any significant contribution to the total cross section, it is essential to consider it to explain the differential cross section data in the range of the photon energies 1037 MeV ≤ E γ ≤ 1062 MeV.
In Fig. 5, we compare the present results of the differential cross section with the CLAS data [13]. The CLAS experiment covers a much wider range of the photon energies (0.97 GeV ≤ E γ ≤ 2.45 GeV) than the FOREST experiment. The first three figures in the first row of Fig. 5 can be compared to the FOREST data given in Fig. 4. Though there are some discrepancies between these two experimental data, general tendency of the data is similar each other. The present results are also in qualitative agreement with the CLAS data. In particular, the narrow N (1685, 1/2 + ) resonance pulls down the differential cross section at E γ = 1.05 GeV in the forward direction. On the other hand, the N (1685, 1/2 + ) makes it enhanced in the backward direction, as we already discussed in Fig. 4. As a result, the inclusion of the N (1685, 1/2 + ) provides noticeably better agreement with the data. As E γ increases, i.e E γ ≥ 1.8 GeV (or W ≥ 2.05 GeV), the results are in good agreement with the CLAS data. This can be understood by K * Reggeon exchange which governs the γn → K 0 Λ process in the higher energy region. We want to mention that we fit the value of the KΛN (1685) coupling constant to be g KΛN (1685,1/2 + ) = −(0.8 − 1.1), which implies the branching ratio Br(N (1685, 1/2 + ) → KΛ) = (0.5 − 1.0) % with Γ N (1685,1/2 + ) = 30 MeV. We hope that future experiments may  FIG. 5. Differential cross section for the γn → K 0 Λ reaction as a function of cos θ K 0 CM for each beam energy. The notations are the same as in Fig. 4. The data are taken from the CLAS experiment [13]. clarify this prediction. Figure 6 draws the differential cross sections for the γn → K 0 Λ reaction as functions of the CM energy W with cos θ K 0 CM fixed. As in the case of Fig. 5, including the narrow N (1685, 1/2 + ) resonance improves the description of the CLAS data near the threshold region [13]. In particular, the N (1685, 1/2 + ) resonance enhances the differential cross section in the backward angle, i.e. in the range of −0.7 < cos θ K 0 CM < 0.0 around E γ = 1.68 GeV. On the other hand, the N (1685, 1/2 + ) makes it reduced in the forward direction as clearly seen in the last row of Fig. 6.

4.
In the present work, we investigated K 0 Λ photoproduction, aiming at understanding the nature of the narrow resonance structure around 1.68 GeV. We employed an effective Lagrangian method combined with a Regge approach. We inlcuded seventeen different N * resonances in the s channel together with nucleon exchange as a background. In addition, we considered Λ and Σ exchanges the u channel. In the t channel, we included K * Reggeon exchange which governs the behavior of the γn → K 0 Λ amplitude in higher energy regions. Since charged kaon photoproduction has been widely investigated in the literature, it is of great interest to compare the role of each diagram in both the charged and neutral productions. Even though the photocouplings are different, the important contributions in the s channel are similar to each other. For example, in Refs. [30][31][32][33][34], the N (1650, 1/2 − ) and N (1720, 3/2 + ) resonances are the most significant ones and the N (1900, 3/2 + ) is also required for the description of the cross secton data.
In the present work, we have taken into account the N * resonances which appeared only in the PDG data. We were able to reproduce the recent CLAS and FOREST data reasonably well without any complicated fitting procedure, even though the nucleon resonances from the PDG data are considered only. The sign ambiguities of the strong coupling constants for the nucleon resonances were resolved by relating them to informaiton on the partial-wave decay amplitudes predicted by a quark model.
We found that the narrow nucleon resonance N (1685, 1/2 + ) has a certain contribution to the differential cross sections near the threshold energy. It enhances them in the backward direction while it makes them decreased in the forward direction, whereas it is difficult to see the effect of the N (1685, 1/2 + ) resonance on the total cross section. It   6. Differential cross section for γn → K 0 Λ as a function of W for each cos θ K 0 CM . The notations are the same as Fig. 4. The data are taken from the CLAS experiment [13].
implies that it is of great importance to examine the beam asymmetries and other polarization observables of K 0 Λ photoproduction both experimentally and theoretically, to clarify the role of the narrow N (1685, 1/2 + ) resonance more clearly. The relevant work is under investigation.