Role of the high-spin nucleon and delta resonances in the $K\Lambda$ and $K\Sigma$ photoproduction off the nucleon

We have investigated the effect of nucleon and delta resonances with spins 11/2, 13/2, and 15/2 in the kaon photoproduction process $\gamma + N \to K + Y$ by using two covariant isobar models. The formalism for high-spin propagators and interaction Lagrangians were adopted from the works of Pascalutsa and Vrancx et al. The unknown parameters in the amplitudes, i.e., the coupling constants and hadronic form factor cutoffs, were obtained by fitting the calculated observables to experimental data. In the $K\Lambda$ channels the inclusion of $N(2600)I_{1,11}$ and $N(2700)K_{1,13}$ resonances improves the agreement between model calculations and experimental data significantly and reduces the dominance of resonances in the model by increasing the hadronic form factor cutoff of the Born terms. Furthermore, the inclusion of these resonances reduces the number of resonance structures in cross sections, including the structure in the $K^0\Lambda$ differential cross section at $W\approx 1650$ MeV, which could become a hint of the narrow resonance. In the $K\Sigma$ channels the inclusion of $N(2600)I_{1,11}$, $N(2700)K_{1,13}$, $\Delta(2420)H_{3,11}$, $\Delta(2750)I_{3,13}$, and $\Delta(2950)K_{3,15}$ states also significantly improves the model and increases the hadronic form factor cutoff of the Born terms. However, different from the $K\Lambda$ channels, the inclusion of these high-spin resonances leads to more resonance structures in the $K^+\Sigma^0$ differential cross section. This investigation reveals that the second and third peaks in the $K^+\Sigma^0$ differential cross section originate from the $\Delta(2000)F_{35}$ and $N(2290)G_{19}$ resonances, respectively. We have also evaluated the resonance properties at the pole positions and using the Breit-Wigner method.


I. INTRODUCTION
Recently, the effect of spins-7/2 and -9/2 nucleon resonances on kaon photoproduction processes has been phenomenologically investigated by using a covariant isobar model [1], in which the scattering amplitude was calculated by using the appropriate Feynman diagrams depicted in Fig. 1. The analytical calculation performed in this study made use of the consistent interaction Lagrangians proposed by Pascalutsa [2]. The calculated observables were fitted to nearly 7400 experimental data points. The result of the fitting process showed that the inclusion of spins-7/2 and 9/2 nucleon resonances could improve the agreement between the model calculation and the experimental data. The model was later extended to describe both γp → K + Λ and γn → K 0 Λ processes, simultaneously [3]. In the latter, the model was fitted to nearly 9400 data points, including recent data from the CLAS and MAMI collaborations. The extended model yielded a nice agreement between the calculated observables and experimental data in both isospin channels.
Despite the success of the model, it has not yet consider the resonances with spins-11/2, 13/2 and 15/2, which theoretic model. Presumably, this is due to the complicated formulations of propagator and vertex factors along with the problem of lower-spin background that plagued the formulation of high-spin (J > 1/2) resonance propagator. Therefore, the first purpose of this paper is to set forth the formulation of higher-spin resonances amplitude. After that we can study their effect on the six isospin channels of kaon photoproduction (see Table II) by means of a covariant isobar model.
We have organized this paper as follows. In Sec. II we present the formalism used in our study. In Sec. III we present the numerical result and discuss the comparison between model calculations and experimental data. Finally, in Sec. IV we summarize and conclude our work. The extracted form functions used to calculate the observables for the fitting process are given in Appendix A.

II. FORMALISM
As mentioned above we adopt the formalism of the nucleon propagators and the interaction Lagrangian developed by Pascalutsa [2] and Vrancx et al. [7]. In our previous work, we explained this formalism in details and constructed the reaction amplitude for the nucleon resonances with spins 7/2 and 9/2 [1]. To facilitate the reader, in this section we briefly discuss this formalism and derive the construction of the spins-11/2, -13/2, and -15/2 resonance propagators along with their interaction Lagrangians. A preliminary result for the analytical form of the production amplitudes involving nucleon resonances with spins up to 13/2 has been reported in a conference [8]. The notation of the four-momentum of photon, nucleon, kaon, and hyperon used in the following discussion is given in the caption of Fig. 1, with p R = p + k = p Y + q.

A. Consistent Interaction Theory
A consistent interaction Lagrangian is required to eliminate the appearance of the lower-spin background amplitude, which is known as an intrinsic problem in the TABLE I: Nucleon and delta resonances with spins from 11/2 to 15/2 used in our study and tabulated by the PDG in their particle listing [4].

B. Interaction Lagrangians
The basic Lagrangian for the kaon-hyperon-nucleon interaction is Following this basic Lagrangian, the standard interaction Lagrangian for the interaction with spin-1/2 resonances is written as According to Pascalutsa the hadronic interaction Lagrangian for spin-3/2 resonance with mass m R reads [9] whereΨ Y is the spinor field of the hyperon, φ is the pseudoscalar field of the kaon, and ψ ν is the massive R-S field of the nucleon or resonance. However, the above Lagrangian is inconsistent for higher-spin resonances. With the substitution proposed by Vrancx, ψ µ →Ψ µ /m R , the Lagrangian for a resonance with spin 3/2 can be written as (13) whereas the Lagrangian for hadronic interaction with spin-(n + 1/2) resonances has the form of In addition to the hadronic Lagrangian, Pascalutsa has also constructed the Lagrangian for the electromagnetic interaction, which is written as [10] where F µν is the conventional electromagnetic field strength tensor. To correctly model the interaction in discussion, a consistent interaction for the electromagnetic interaction Lagrangian is needed. The electromagnetic interaction Lagrangian for the resonance particles with spin-(n + 1/2) can be written as The consistency of the interaction is guaranteed by the operator interaction which fulfills: with i = 1, 2, ..., n and p λi R is the four-momenta of the resonance particles with spin-(n + 1/2).
The electromagnetic and hadronic vertices, which are required to calculate the scattering amplitude, can be obtained from the interaction Lagrangians constructed in the previous discussion. As the result the hadronic vertex can be written as and the electromagnetic vertex reads In the present work we use the propagator whereP n+1/2 µ1...µn;ν1...νn (p R ) is the on-shell projection operator. The complicated form of the projection operator will be discussed later.
The production amplitude is obtained by sandwiching the propagator between the two vertices, i.e., The above formulation is simplified by considering the orthogonalities of the projection operator, and p R µi P µ1..µn,ν1..νn with i = 1, 2, ..n.
Thus, the scattering amplitude can be written as The above equation shows how a consistent interaction structure constructed intuitively.

D. Hadronic and Electromagnetic Vertices
By using the above prescription we obtain that the hadronic and electromagnetic vertices for spin-11/2 reads as and respectively, whereas for the spin-13/2 resonance and respectively. The above result indicates that the number of momentum dependence increases with the number of spin. This conclusion was previously made by Ref. [11]. For the spin-15/2 resonance the hadronic and electromagnetic vertices are given by and Γ νν1ν2ν3ν4ν5ν6± respectively. The above formalism is valid for both positive and negative parities, for which the parity factors are denoted by Γ + = iγ 5 and Γ − = 1, respectively.

E. Production Amplitudes
As stated above the production amplitudes for spin-11/2, -13/2, and -15/2 resonances are obtained by sandwiching the propagators given by Eqs. (32), (33), and (34) between the corresponding hadronic vertex factors given by Eqs. (35), (37), and (39), and electromagnetic vertex factors given by Eqs. (36), (38), and (40), respectively. As a result we obtain the amplitude for spin-11/2 with: The production amplitude for spin-13/2 particle is given by with Finally, the amplitude for spin-15/2 resonance reads with Note that in the fitting process only the product of the hadronic and electromagnetic couplings, i.e., g KY R g i with i = 1, 2, 3, are extracted from the data. Furthermore, in the production amplitudes given by Eqs. (41), (45), and (49) we have used the following definitions,

F. Calculation of the observables
The production amplitudes given by Eqs. (41), (45), and (49) can be decomposed into six gauge and Lorentz invariant matrices M i through where the gauge and Lorentz invariant matrices M i are given by [12,13]: where P = 1 2 (p + p Λ ) and ε µνρσ is the Levi-Civita antisymmetric tensor. All observables required for fitting the experimental data can be calculated from the form function A i extracted from Eq. (54), after adding the contributions from all involved intermediate states. The form functions A i for baryon resonances with spins up to 9/2 are given in the previous works [1,14], whereas those with spins 11/2, 13/2, and 15/2 considered in the present work are given in Appendix A.

G. Pole Position
Besides the Breit-Wigner parameters, such as mass, width, and branching ratios, the Particle Data Group has recently listed new information on the resonance properties, i.e., the pole position. It is obvious that the Breit-Wigner parameters extracted in each model depend highly on the background terms of the model. Therefore, all resonance properties obtained by using such parameterization is difficult to compare with those obtained from other models. This problem does not appear in the case of pole position. Currently, the pole position has been extensively used in the realm of hadronic physics. In the Particle Data Book 2018 the pole positions of resonance are listed before the Breit-Wigner parameters. The placement shows that the pole positions is currently considered as the important properties of a resonance.
In principle, the pole position can be calculated by setting the denominator of the scattering amplitude to zero. Approaching the pole position the scattering amplitude of a resonance increases dramatically. Since the resonance scattering amplitude becomes extremely larger than contributions from other intermediate states, the resonance property calculated at the pole position is insensitive to the contribution of background terms. As a result, the evaluation of resonance properties at the pole position is practically model independent.
In the present work, the pole position properties of a resonance are the resonance mass and width. They are defined via As previously stated, this is obtained by setting the denominator of scattering amplitude to zero, i.e., Notice that the above equation cannot be directly calculated, since in the present study we use Γ(s) that depends on the total c.m. energy. Therefore, the solution of Eq. (62) must be obtained numerically.

III. RESULTS AND DISCUSSION
In the present work, the isobar model used to analyze the effect of spin-11/2 and -13/2 nucleon resonances in the KΛ channels is based on our previous model developed to describe all available data in these channels [3]. Furthermore, in this study we also investigate the effect of spin-11/2, -13/2, and -15/2 ∆ resonances in the KΣ reaction channels. Along with the nucleon resonances used in the KΛ channels, these ∆ resonances are listed in Table I. In total, there are 23 nucleon resonances included in our analysis for the KΛ and KΣ channels and, in addition, 17 ∆ resonances in the KΣ channels with spins up to spin-15/2. The result obtained in all channels will be discussed in the following subsections.

A. KΛ Channel
In Table III we present the leading coupling constants and other background parameters obtained from the previous work (Model B) [3] and current analysis (Model A). Note that in the present work we have omitted the  [3]. Error bars were not reported in Model B. See Ref. [3] for the explanation of the parameter notation.
Parameter K 0 Λ photoproduction data obtained from MAMI collaboration [15] due to the problem of data discrepancy as discussed in Ref. [3]. Furthermore, it was shown that by excluding these data from the database leads to a better model that can nicely reproduce the γn → K 0 Λ helicity asymmetry E [3]. It is important to note that Model B was also obtained from fitting without these data.
From Table III we can conclude that there is no dramatic changes in the background parameters after including the spin-11/2 and -13/2 nucleon resonances in the model. Nevertheless, the increase of g KΛN coupling and the Born hadronic cutoff Λ B shows that the inclusion of the two resonances increases the contribution of the background terms. We note that in the case of Kaon-Maid, the Born cutoff is very soft, i.e., Λ B = 0.637 GeV [17]. Clearly, the Kaon-Maid model is dominated by the resonance terms, whereas the Born terms are strongly suppressed. Such situation is completely different from the case of pion or eta photoproduction and could raise a question, whether Kaon-Maid is a realistic phenomenological model.
The listed χ 2 values indicate that the agreement between model calculation and experimental data is significantly improved after including the two nucleon resonances, as clearly expected. Since the calculation includes two isospin channels, i.e., γp → K + Λ and γn → K 0 Λ, in the followings we present comparison between model calculations and experimental data in details.   [3] and present works, compared with the experimental data from the CLAS collaboration (solid squares [16]).

K + Λ Channel
Comparison between calculated γp → K + Λ total cross sections from Models A and B, Kaon-Maid, and experimental data is displayed in Fig. 2. Note that the experimental data shown in this figure are only for visual comparison. The data were not included in the fitting process, since differential and total cross sections data come from the same experiment.
Compared to the prediction of Kaon-Maid, both models A and B displayed in Fig. 2 show substantial improvement. However, since our main motivation in this work is to investigate the effect of spin-11/2 and 13/2 nucleon resonance, we will not compare our result with the prediction of Kaon-Maid in the following discussion, except in the case of total cross section, in which recent experimental data are in good agreement with Kaon-Maid for certain isospin channel. Both current and previous models seem to have a great agreement, with a tiny difference only at higher energy region, i.e., W ≥ 2.6 GeV. The difference originates from the use of the high spin nucleon resonances, as obviously seen from their masses. However, since there are no available data in this energy region, no conclusion can be drawn at this point. Future experiments with 12 GeV electron source at JLab could be expected to reveal more information in this energy regime.
Both peaks shown by the two models seem to agree with each other, with minuscule difference at W ≈ 1.85 GeV, where the second peak is attributed to the P 13 (1990) state, as discussed in Ref. [18].
More information can be obtained from the differential cross sections shown in Figs. 3 and 4. Figure 3 shows that the difference in the total cross sections of models A and B originate from the forward and backward regions of the differential cross section. Furthermore, as shown in Fig. 3 at backward angle (cos θ = −0.70), it is also apparent that the result of model A has a better agree-ment than model B, except in the higher energy region, W > 2.4 GeV. Another interesting result is that the second peak of Model A is slightly shorter, but wider, than that of Model B. As a result, Model A yields a more accurate explanation of experimental data, especially for the CLAS 2010 [20] and Crystal Ball [21] ones. However, in the forward region Model A yields fewer peaks than Model B. Nevertheless, Model A seems to produce more natural shape of the cross section at the very forward angle, cos θ = 0.90, where unfortunately, the available experimental data from different collaborations produce uncertainty in differential cross section up to nearly 40%.
The angular distribution of differential cross section displayed in Fig. 4 shows that both models are in good agreement with experimental data. Furthermore, experimental data in higher energy region are better reproduced. This figure again shows that the inclusion of the high-spin resonances with higher masses does not influence the the cross section behavior in the lower energy region, where experimental data exist. Figure 5 shows the single-and double-polarization observables, for which experimental data are abundantly available at present. In this case we do not see a dra- matic changes after including the spin-11/2 and 13/2 nucleon resonance in the model, except improvement in the agreement between model calculation and experimental data at forward direction and high energy region, which is expected due to the higher masses of these resonances. Nevertheless, we still see significant improvement in the beam-recoil double-polarization observables C x and C z at higher energies, where experimental data have large error bars. Note that only a small part of experimental data can be displayed in Fig. 5. More data are available in the fitting database, especially for the recoil polarization P , and are not shown in the figure due to their different kinematics.

K 0 Λ Channel
The available data for the K 0 Λ channel are significantly fewer than the K + Λ one, given that the experiment with neutron target is more difficult to perform. The number of data included in this study is less than 1000, which will affect the accuracy between model calculation and experimental data. The calculation of this channel is performed by using the isospin relation of the hadronic coupling constants, as well as information on the neutral kaon transition moment and neutron helicity photon coupling obtained from PDG [4], in the K + Λ   model as discussed in Refs. [22,23]. Thus, investigation of the K 0 Λ channel can be considered as a direct test of isospin symmetry in kaon photoproduction. Experimental data of this channel are already available from the CLAS g10 and g13 collaboration [24] and MAMI 2018 collaboration [15]. In the previous work [3], both data sets were included in the analysis. However, in the present work we exclude the data from the MAMI 2018 collaboration, since it was found that the data are more difficult to fit and have a discrepancy problem with the CLAS g10 and g13 data. Furthermore, in the present work our main motivation is to investigate the effect of the higher spin nucleon resonances, in which we need an accurate isobar model. Figure 6 shows the calculated total cross section of the γn → K 0 Λ channel. Obviously, both models A and B yield similar cross section trend, except in the lower energy region and especially near the production threshold, where the cross section obtained from model B is steeper than that of the present work. All models give the total cross sections within the experimental error bars. As discussed in Ref. [23], threshold behavior of K 0 Λ photoproduction provides important information that can shed more light on the difference between pseudovector and pseudoscalar theories in the kaon photoproduction process. The absence of K 0 exchange in this channel also reduces the number of unknown parameters in the model. As a consequence, threshold properties of the K 0 Λ can be more accurately investigated. Note also that the over prediction of Kaon-Maid model is understandable, since it is pure prediction and the model was fitted to old data.   [24] and MAMI 2018 collaboration (solid circles) [15]. Note that the data shown in this figure were not used in the fitting process and shown here only for comparison. Figure 7 shows the differential cross section of the γn → K 0 Λ channel. At a glance, both models seem to be similar, especially at W > 1.8 GeV region. However, the inclusion of the two high-spin nucleon resonances leads to different differential cross section near the threshold. FIG. 7: Energy distribution of the γn → K 0 Λ differential cross section for different values of cos θ shown in each panel.
Notation of the curves and experimental data is as in Fig. 6.
The difference is more apparent at the forward angle, i.e., cos θ = 0.75. The angular distribution of differential cross section shown in Fig. 8 corroborates this result. In Fig. 8 we can see that the difference between the two models is more obvious in the forward and backward regions.
In general, the K 0 Λ differential cross sections also show that the inclusion of the two high-spin nucleon resonances improves the model. However, there is an important phenomenon appears in the K 0 Λ channel. As in the case of the K + Λ channel, the inclusion of these resonances leads to fewer structures in differential cross section, especially at the forward region (see Fig. 3). The same phenomenon is also displayed by the K 0 Λ channel as shown in Fig. 7. In this channel the previous work displays a clear structure at W ≈ 1.65 GeV, which appears in a wide range of angular distribution, but is more apparent near forward region. The structure is eliminated by the inclusion of the two high-spin nucleon resonances. We found that this structure is very interesting because it originates from the N (1650) resonance contribution and the corresponding width is less than 50 MeV (see the dashed curves in Fig. 7, especially at cos θ = 0.75). In the previous work [23] it was concluded that the structure could be a hint of the narrow resonance, which was found to have the mass of 1650 MeV.

B. KΣ Channel
As in the KΛ channels, the four available channels of KΣ photoproduction can be also simultaneously analyzed by exploiting the isospin symmetry and some information on the resonance properties from PDG [4]. Recently, we have studied these channels by using a partial wave approach for the resonance part, whereas the background part was still constructed from the covariant Feynman diagram technique [28]. The model was fitted to nearly 8000 experimental data points available from all four channels, but dominantly from the K + Σ 0 one.
In addition, a fully covariant model to describe photoproduction of KΣ has been also constructed by including nucleon resonances with spins up to 9/2 and the result has been submitted for publication [29]. In the present work we add the nucleon and delta resonances that are not available in this covariant model. They include the nucleon resonances with spins 11/2 and 13/2, as well as the delta resonances with spins 11/2, 13/2, and 15/2, listed in Table I. The experimental data used in this study were obtained from the CLAS, Crystal Ball, GRAAL, SAPHIR, LEPS, and SPring8 collaborations. Thus, to observe the effect of including these resonances, we will compare the result of our present work to that of the covariant model reported in Ref. [29]. Table IV lists the leading coupling constants and other background parameters extracted from the present analysis. For the sake of discussion, the present model and the model reported in Ref. [29] will be referred to as Model C and Model D, respectively. As seen from the values of χ 2 /N dof in Table IV, the agreement between model IV: Extracted coupling constants and other background parameters in the KΣ channels obtained from the present work (Model C) and the previous one (Model D) [29]. Note that error bars were not reported in Model D.

Parameter
Model Among all possible KΣ isospin channels, the γp → K + Σ 0 channel has the most abundant experimental data. This is understandable since, as the production of K + Λ, the production of K + Σ 0 is relatively easier to measure due to stable proton target and relatively simpler technique to measure the decay of Λ or Σ 0 hyperon in the final states. The experimental data mentioned here include those obtained from the Crystal Ball (at MAMI) [21], CLAS [16,30,31], GRAAL [32], LEPS, SAPHIR and SPring8 collaborations. Figure 9 shows the comparison between the calculated total cross sections before and after the inclusion of the high-spin resonances, where the prediction of Kaon-Maid is also displayed to show the improvement made by the current models. It is seen that the prediction of Model C (solid red curve) is practically similar to that of Model D (dashed black curve). The difference between both models is very subtle and can be seen only at W ≈ 2.0 GeV and W ≈ 2.25 GeV. Overall, both models fit nicely the experimental data, with an exception at W 2.15 GeV, where we observe that there is a discrepancy problem in the existing experimental data of total and differential cross sections. This problem will be clarified later when we discuss the result for differential cross section.
Comparison between differential cross sections obtained from the two models is shown in Figs. 10 and 11. The energy distribution of differential cross section shown in Fig. 10 reveals that the difference between the two models is most obvious at cos θ = 0.95. At this for-   ward angle we can see that the inclusion of the high-spin resonances yields at least two structures in differential cross section at the W ≥ 2.6 GeV, which can be traced back to the resonance masses. Furthermore, the second peak in differential cross section becomes more apparent and much closer to experimental data after the inclusion of these resonances. This peak still clearly appears at cos θ = 0.80 and 0.65, and quickly disappears as we move to larger kaon angles. We have investigated the origin of this second peak and found that it is due to the ∆(2000)F 35 resonance.
In addition, we also observe a third peak at W ≈ 2.3 GeV, which originates from the N (2290)G 19 state with spin 9/2, has positive parity, and earns a status of fourstar in the Particle Data Book [4]. Interestingly, this state appears in the K + Σ 0 channel after the inclusion of higher nucleon resonances. This peak also quickly disappears as we increase the kaon angle, but appears again in the backward angles. In the latter, both models are in agreement with each other.
The angular distributions of differential cross section shown in Fig. 11 support this finding. In general, the agreement with experimental data are similar for the two models. In the case when the experimental data from different collaborations are scattered, the models try to reproduce their average.

K 0 Σ + Channel
The K 0 Σ + channel is the last channel measured for the proton target. Although proton is stable, detection of the neutral kaon and positively charged Σ hyperon in the final state is more challenging. Figure 12 shows the K 0 Σ + total cross sections obtained from both models, as well as Kaon-Maid for comparison. Obviously, the three models display different shapes of total cross section, which originate from different nucleon and delta resonances used in the models. We note that experimental data for this channel are relatively scattered, especially in the energy range 1.77 W 1.90 GeV, where the new SAPHIR data are almost 50% smaller than the older SAPHIR data. Fortunately, near the production threshold all data are in agreement with each other and, interestingly, closer to the prediction of Kaon-Maid. Nevertheless, at this point we still observe that the inclusion of higher-spin resonances improves the model prediction. Furthermore, as shown in Fig. 12, very close to the threshold the predictions of the three models are very different. While the predicted cross section of Kaon-Maid is slowly increasing with energy, the calculated cross section of the present model (Model C) is rising steeply and reveals the contributions of resonances with masses near 1.7 GeV. We note that there are four nucleon resonances and one delta resonance to this end. As stated in Ref. [38], experimental data near the threshold region are very crucial to understand the production mechanism and related phenomenological applications [39].
Despite the fact that there are limited data available for this reaction, the agreement between model calculation and experimental data increases significantly with the inclusion of high-spin resonances. As in the case of previous channels, the inclusion of high-spin resonances leads also to a number of structures in the total cross sections, as clearly shown in Fig. 12.
The limited number of experimental data for the K 0 Σ + photoproduction clearly impose a strong constraint on the range of validity of our present model. Figure 13 obviously shows that the available data are relatively scattered, with apparent peak at W ≈ 1.95 GeV. This peak is more obvious in the forward regions. Furthermore, in this figure we can also observe that the inclusion of high-spin resonances eliminates the small structure at W ≈ 1.85 GeV and emphasizes the contribution of resonances with m ≈ 1.7 GeV as in the case of total cross section. More experimental data are strongly required, especially at these energy points, to clarify the effects of the inclusion of high-spin resonances in the present model. The angular distributions of differential cross section shown in Fig. 14 clear up the difference between Models C and D, which is visible in the whole angles covered by experimental data. The improvement of the model after including the high-spin resonances is relatively unclear due to the scattered experimental data. In general, the inclusion of the high-spin resonances slightly improves the agreement between model calculation and experimental data. Both Figs. 13 and 14 indicate that the present model prefers the new MAMI A2 2019 data set [15] in the energy range 1.8 W 1.9 GeV, where experimental data from different collaborations are significantly scattered.

K + Σ − Channel
The K + Σ − photoproduction channel has nearly 300 experimental data points in the form of differential cross section [34,40] and photon asymmetry [34]. These data were included in the fitting process. Thus, the total cross sections shown in Fig. 15 are pure prediction and cannot be compared with experimental measurement. Nevertheless, we still can see the small effect of the higher- spin resonances near the K + Σ − threshold, as in the case of K 0 Σ + channel, and at high energy W 2.1 GeV where no experimental data are available to constrain the model. Otherwise, both models C and D show similar trend.
The calculated K + Σ − differential cross sections obtained from both models are compared with experimental data in Fig. 16. This figure reveals that the origin of the structures shown in the total cross section. Near the threshold the peak appears in the whole angular distribution, whereas the difference between the two models at high energy originates from the backward angles. It is interesting to note that at high energy the inclusion of the high-spin resonances slightly increases the cross section, but in the backward region this situation dramatically changes (see the panel with cos θ = −0.65).
For completeness, we have also checked the angular distribution of K + Σ − differential cross section. Figure 17 shows the comparison between both models and experimental data. Similar to Fig. 16, we can see that the difference between models C and D is also small, except  [15] shown in this figure were not included in the fitting process of the present model. tion by measuring photoproduction of neutral kaon on a deuteron target. Although the number is very limited, the existence of experimental data in this channel significantly helps to constrain the prediction of the present model. Figure 18 shows the comparison between calculated total cross sections obtained from previous and present models and experimental data. It can be seen that the inclusion of the high-spin resonances improves the model, although the cross section trend is relatively well reproduced by the three models. The importance of the nucleon and delta resonances with m ≈ 1.7 GeV is slightly shown by Model D near the production threshold.
As shown in Fig. 18 the total cross sections increase monotonically with increasing energy. In fact, the predicted total cross section of the present model is more than 3 µb at W ≈ 2 GeV, which seems to be unrealistic if we compare it with those of the neutral kaon productions shown in Figs. 6 and 12. Thus, total cross section data up to 2.5 GeV are very important to this end. Figures 19 and 20 compare the calculated differential cross sections obtained from the previous and present models with experimental data. Figure 19 shows that models C and D start to differ at W ≈ 1.86 GeV, where no experimental data are available to constrain them. The inclusion of the high-spin resonances in this channel improves the cross section divergence, which is urgently required in forward regions. Interestingly, the two models predict a resonance structure above this energy point, albeit with different positions. Certainly, experimental data in the energy range 1.8 W 2.2 GeV are very important to determine which resonance is responsible for this structure.
The angular distributions of differential cross section shown in Fig. 20 reveals that the cross section of the K 0 Σ 0 photoproduction has backward-peaking behavior. This indicates the dominance of u-channel in this process, which is easily understood from the fact that this process does not have a t-channel in the Born terms since a neutral kaon cannot interact with real photon.  Fig. 19, but for angular distribution.

C. Extracted Resonance Properties
Having investigated the effect of high-spin resonances on our models, we are ready to discuss the resonance properties, i.e., their masses and widths, at their pole positions, before and after including the high-spin resonances. Note that during the fitting process we allowed the resonance masses and widths to vary within the estimated error bars of PDG. In Table V we show the resonance masses and widths evaluated at their pole positions extracted from the present and previous works, compared with those listed by PDG [4]. Note that for the N (2600)I 1,11 , N (2700)K 1,13 , ∆(2750)I 3,13 and ∆(2950)K 3,15 resonances, the PDG does not have any information yet. Therefore, in this case, the result shown in Table V provides the first estimate for these resonances.
In the KΛ photoproduction the agreement between the extracted properties of nucleon resonances and those of PDG is in general from fair to good. The same situation is also seen in the KΣ one. Good agreement with the V: Masses and widths of nucleon and ∆ resonances evaluated at the pole position in MeV, obtained from the present work (Models A and C), previous works (Models B [3] and D [29]) and PDG [4]. The status of resonances is due to the PDG [4].  whose masses are less than 2000 MeV, as well as for the N (2220)H 19 and N (2250)G 19 states. Furthermore, we can also observe that Model D seems to produce higher extracted masses than Model C. On the other hand, the calculated widths obtained from Model C are found to be closer to the PDG values, except for the N (2190)G 17 , N (2220)H 19 and N (2250)G 19 resonances.

Resonances
The difference between the mass of N (2700)K 1,13 resonance extracted from the KΛ and KΣ channels seems to be large, i.e., nearly 100 MeV. Furthermore, the extracted masses are smaller than the Breit-Wigner ones, as can be seen in Table V. Nevertheless, the masses and widths of the two resonances extracted in this work provide the first prediction of their pole properties.
In contrast to the N (2600)I 1,11 and N (2700)K 1,13 resonances, the extracted properties of the ∆(2420)H 3,11 resonance are in a good agreement with the PDG values. For the other two high-spin ∆ resonances, i.e., ∆(2750)I 3,13 and ∆(2950)K 3,15 states, the PDG does not have data for comparison. For both ∆(2750)I 3,13 and ∆(2950)K 3,15 resonances the extracted masses at the pole position are much smaller than those of the Breit-Wigner. Since the resonances affect the cross section and other polarization observables only at high energies, i.e., W 2.7 GeV, a study devoted for high energy photoproduction would be very relevant to this end. This would be also in line with the 12 GeV JLab experiments that are currently in progress.
In Tables V and VI we have also included the error bars both from PDG and the present work. Note that the previous works did not report the uncertainties in the extracted resonance masses and widths. The quoted error bars of the present work originate from the CERN-MINUIT output used for the fitting process, where we employed the MIGRAD minimizer that produces both estimated error bars of the fitted parameters and error matrix [41].
The error bars obtained from the MINUIT indicate the flexibility of the model to the variation of the resonance masses and widths to reproduce the data, i.e., the larger the error bars the more flexible the model. As a consequence, in the energy region where precise experimental data are abundantly available the error bars are forced to be small. Thus, we might expect that the error bars are large below and near the threshold region and for W 2.4 GeV (see Fig. 21). This is proven in Table V, where we can see that relatively larger error bars are obtained in Model A (KΛ photoproduction with W thr. ≈ 1610 MeV) for the N (1440)P 11 , N (1520)D 13 , N (1535)S 11 resonances. In Model C (KΣ photoproduction with W thr. ≈ 1690 MeV) three more resonances, i.e., the N (1650)S 11 , N (1675)D 15 , and N (1680)F 15 states, also exhibit this phenomenon. Only the ∆(1232)P 33 resonance has very small error bars, since PDG has estimated this resonance with very precise mass and width, whereas during the fitting process we allowed these parameters to vary within the PDG uncertainties. In the higher energy region we observe that all baryon resonances with m R 2400 MeV show the relatively larger error bars.
In conclusion, we have observed that the addition of the high-spin resonances, with spins from 11/2 up to 15/2, in our isobar models improves the agreement between their pole properties extracted in this work and those listed by PDG.

IV. SUMMARY AND CONCLUSION
We have derived the spin-11/2, -13/2, and -15/2 resonance amplitudes for kaon photoproduction off the nucleon by using the covariant Feynman diagrammatic technique. For this purpose we made use of the consistent interaction Lagrangians proposed by Pascalutsa and Vrancx et al., as well as the formulation of spin-(n+1/2) resonance propagator put forwarded by Vrancx et al. We have studied the effect of high-spin resonances in the kaon photoproduction processes by including the N (2600)I 1,11 and N (2700)K 1,13 states in our previous model for KΛ photoproduction and the N (2600)I 1,11 , N (2700)K 1,13 , ∆(2420)H 3,11 , ∆(2750)I 3,13 , and ∆(2950)K 3,15 states in our previous model developed for KΣ photoproduction. In general, the inclusion of these high-spin resonances improves the agreement between model calculations and experimental data, which is indicated by the smaller values of χ 2 in all isospin channels. The inclusion of the high-spin resonances also helps to overcome the problem of resonance-dominated model, since the inclusion increases the hadronic form factor cutoff of the Born terms and, therefore, increases the role of the Born terms in both KΛ and KΣ models. Specifically, in the K + Λ channel the inclusion leads to fewer resonance structures in the cross sections and polarization observables. The effect is significant in the high energy differential cross section, near the resonance masses and at forward direction. Different from the K + Λ channel, the effect in the K 0 Λ channel is more obvious and can be observed in both low and high energy regions. Furthermore, in this channel the effect is found to be large in both forward and backward angles. In the K + Σ 0 channel the effect is only significant in the forward region, where a number of resonance structures appear after including the high-spin resonances. Two of them are important to note here, i.e., the ∆(2000)F 35 and N (2290)G 19 resonances, which are responsible for the second and third peaks in the K + Σ 0 differential cross section. In contrast to the K + Σ 0 channel, the K 0 Σ + channel is found to be sensitive to these high-spin resonances. In this case, the effect can be ob-served in the whole energy range covered by experimental data and the whole angular distribution. The effect is, however, not observed in the K + Σ − and K 0 Σ 0 channels, at least in the whole kinematics where experimental data are available. Finally, we found that the addition of the high-spin resonances leads to a better agreement between the extracted resonance properties and those listed by PDG.