Experimental study of the γ p → K 0 Σ + , γ n → K 0 Λ, and γ n → K 0 Σ 0 reactions at the Mainz Microtron

. This work measured d σ/ d Ω for neutral kaon photoproduction reactions from threshold up to a c.m. energy of 1855 MeV, focussing speciﬁcally on the γp → K 0 Σ + , γn → K 0 Λ , and γn → K 0 Σ 0 reactions. Our results for γn → K 0 Σ 0 are the ﬁrst-ever measurements for that reaction. These data will provide insight into the properties of N ∗ resonances and, in particular, will lead to an improved knowledge about those states that couple only weakly to the πN channel. Integrated cross sections were extracted by ﬁtting the diﬀerential cross sections for each reaction as a series of Legendre polynomials and our results are compared with prior experimental results and theoretical predictions.


Introduction
Most of our early knowledge of N * resonances came from experiments involving the πN channel in the initial or a e-mail: venky.avss@gmail.com final state, e.g., pion nucleon elastic or inelastic scattering [1] or single-pion photoproduction. Lattice QCD and quark models both predict more nucleon resonances in the mass range below 2000 MeV than have been observed experimentally. This is known as the "missing resonances" problem in baryon spectroscopy. For that reason, there has been a concerted effort at electromagnetic facilities, including JLab, Mainz, and Bonn, to measure N * formation reactions that do not include the πN channel at all. The data analyzed in this work bear directly on that problem. The photoproduction of a kaon on a nucleon target can provide new information on nucleon resonances. Out of six elementary kaon photoproduction reactions (γp → K 0 Σ + , γn → K 0 Λ, γn → K 0 Σ 0 , γp → K + Λ, γp → K + Σ 0 , γn → K + Σ − ), a significant amount of experimental research [2][3][4][5][6] has been done on the charged kaon reactions.
The main focus of the current work was to measure the differential cross section from threshold to c.m. energy W = 1855 MeV for the reactions γp → K 0 Σ + , γn → K 0 Λ,a n dγn → K 0 Σ 0 on a liquid deuterium target, where W was calculated from the incident beam energy assuming quasifree kinematics. Fermi-motion corrections were not possible in the present analysis due to the large combinatoric background; however, in an analysis of the quasifree production of η mesons off deuterium where Fermi-motion corrections were possible, the resolution in W rises nearly linearly from ΔW (FWHM) ∼ 30 MeV at 1680 MeV to ΔW ∼ 40 MeV at 1820 MeV [12]. We used W energy bins of width 30 MeV for measurements below W = 1765 MeV and bins of width 20 MeV for energies above 1765 MeV.
The measurements were performed at MAMI-C, the Mainz Microtron located in Mainz, Germany. We analyzed these reactions via the K 0 → π 0 π 0 decay. Further details are provided in sect. 3.
The cross-section data can be used to help determine N * resonance properties using partial-wave analyses or to test phenomenological models of kaon photoproduction. This paper reports the world's first results on differential and total cross sections for the reaction γn → K 0 Σ 0 . This paper is divided into six sections: sect. 2 describes the experimental setup, sect. 3 describes the data analysis, sect. 4 describes the calculation of uncertainties, sect. 5 describes the results and discussion for all three reactions, and sect. 6 gives the summary and conclusions. Our measured cross sections are tabulated in the appendix.

Experimental setup
Data for the photoproduction of neutral kaon reactions on a liquid deuterium target were measured using the Crystal Ball (CB) [13][14][15][16][17], particle identification detector (PID) [18] and TAPS [15][16][17] detectors. All these detectors were set up at the Mainz Microtron [19] bremsstrahlungtagged photon beam facility in Germany. At the time the measurements were performed, MAMI-C could deliver electrons with energies up to a maximum energy of 1508 MeV. The mono-energetic electron beam was used to produce photons via bremsstrahlung in a 10 μmc o pper radiator. The bremsstrahlung photons are tagged by the Glasgow photon tagger [20]. The tagged photons are then passed through a lead collimator to produce a photon beam. The hole in the lead collimator was 4 mm in diameter for this experiment. This collimation gave a photon beam spot on target with a diameter of about 1.3 cm. The photon beam was incident on a 125 μm Kapton target cylinder of length 4.72 cm and diameter 4 cm filled with liquid deuterium. Further details on the target system can be found in ref. [21].
The Crystal Ball (CB) is a multiphoton spherical spectrometer [13]. The CB geometry is based on an icosahedron, a polyhedron having 20 triangle-shaped sides. Each of the 20 major triangles is divided into four minor triangles. Each minor triangle consists of nine crystals, so for a complete sphere, there would be 720 crystals. However, for the entrance and exit tunnels, 48 crystals were not installed, resulting in 672 crystals for the Crystal Ball. The chemical composition of each crystal is thalliumdoped sodium iodide, NaI(Tl), which is a hygroscopic material so it is important to protect the crystals from moisture [12,14]. The Crystal Ball covers the polar angle range from 20 • to 160 • and the azimuthal angle range from 0 • to 360 • .
The forward moving particles are detected by TAPS [14,22], which was configured as a photon calorimeter consisting of 384 BaF 2 crystals located downstream of the Crystal Ball. These BaF 2 crystals were arranged in a honeycomb pattern to form a hexagonal wall covering the polar angle range from 4 • to 20 • .
The PID (Particle Identification Detector) [18] is a cylindrical detector oriented concentric with the target inside the Crystal Ball. The PID consists of 24 4 mm thick plastic scintillators having a length of 50 cm. It was designed to work along with the CB to provide information on charged particles. The PID effectively distinguishes between different types of charged particles and neutral particles based on the energy deposited (dE/dx) in the PID elements versus total energy measured in a CB cluster. The PID provided a clean separation of pions and protons as shown, for example, in ref. [23]. For further details about these detectors, such as their energy and angle resolutions or their calibrations, see [12,[15][16][17][24][25][26][27]. The CB and TAPS detectors are very efficient at detecting the final-state photons. A cylindrical MWPC (MultiWire Proportional Chamber) may be used to improve the angular resolution (tracking) of charged particles. During this The PID is placed inside the CB for charged particle detection. In this experiment, the PbWO 4 crystals were not installed in TAPS. experiment, the MWPC was not used. Figure 1 shows a schematic diagram of the CB and TAPS detector setup.

Data analysis
After all the detectors had been calibrated, the event selection and analysis was carried out. Detailed Monte Carlo (MC) studies were performed using 3 × 10 6 events generated according to phase space for each of the three K 0 photoproduction reactions, as well as for γp → ηp and γn → ηn, which are the leading backround reactions due to η → 3π 0 → 6γ decays.
In each reaction the K 0 was identified through its decay K 0 → π 0 π 0 → 4γ.T h eΣ + was identified through its decay Σ + → π 0 p, Λ through its decay Λ → π 0 n,a n dΣ 0 through its decay Σ 0 → γΛ → γπ 0 n. Therefore, the detection of three π 0 s in the final state was required in all cases, giving rise to six final-state photons via π 0 → γγ. Data for γp → K 0 Σ + , γn → K 0 Λ,a n dγn → K 0 Σ 0 reactions were sorted into various cases (nc), where n represents the detected number of final-state neutral particles and c represents the detected number of final-state charged particles. The spectator nucleon from the deuterium was not detected.
When a photon hits the CB or TAPS, it deposits its energy via an electromagnetic shower. This shower spreads over a group of neighboring crystals. The cluster with the highest energy deposition is called the central crystal. Typical CB clusters have 12 neighbors and typical TAPS clusters have six neighbors. The cluster energy was calculated as the weighted sum of each crystal location and the deposited energy in that particular crystal. We used a standard algorithm for identifying clusters that has been successfully used in most prior A2 experiments at MAMI. In the case of clusters produced by neutrons or protons, only the cluster location was used in the data analysis since nucleons do not deposit their full kinetic energy in the CB or TAPS. If only six neutral clusters are detected, the event is case (60). To be a viable event for γp → K 0 Σ + or γn → K 0 Λ, further analysis was needed to establish these six neutral clusters as photons produced from π 0 decays. The data analysis for case (60) starts by first selecting events that have six and only six neutral clusters. If the final proton in Σ + → π 0 p is detected then there will be six neutral clusters and one charged cluster in the final state, which defines case (61). If the neutron in Λ → π 0 n is detected then there will be seven neutral clusters and no charged cluster, which defines case (70).
For γn → K 0 Σ 0 events, the detection of seven photon candidates is required, six coming from π 0 decays and one coming from Σ 0 → γΛ. If the final-state neutron is not detected, then the event corresponds to case (70); however, if the final-state neutron is detected, then the event corresponds to case (80). Table 1 tabulates the reactions and the corresponding cases for the present work. It also lists, for each case, the total number of events included in the observed π 0 π 0 invariant-mass distributions, summed over all energy and angle bins, after all cuts discussed below.
Once events had been separated according to the number of neutral and charged clusters, the next step was to identify the final three π 0 s from the neutral clusters. To identify the three π 0 s, all distinct possible combinations of two-photon candidates were constructed. There are 15, 21, and 28 possible ways to construct distinct two-γ combinations from six, seven, and eight neutral clusters, respectively. A histogram of the invariant-mass of all distinct two-γ combinations for case (60) is shown in fig. 2. Only those distinct two-γ combinations whose invariantmass m(γγ) was between 90 and 160 MeV are the actual π 0 candidates. This invariant-mass cut is represented by solid red vertical lines in fig. 2. A typical event had several combinations that satisfied this criterion. Only those events that had a minimum of three distinct π 0 candidates were kept. Major sources of background for the reactions of interest are γp → ηp and γn → ηn, where η → 3π 0 . In order to eliminate this background, only those three π 0 candidates whose combined invariant mass is greater than 600 MeV were selected for further analysis [10,28]. This cut significantly reduces the η background contribution while only slightly reducing events from the reactions of interest. If the three π 0 candidates for a given combination are labeled as π 0 1 , π 0 2 , π 0 3 , then there are three ways to construct the two π 0 s that could correspond to a K 0 decay; that is, (π 0 1 π 0 2 ), (π 0 2 π 0 3 ), or (π 0 1 π 0 3 ). A histogram of the mass of one π 0 candidate m(γγ) versus the invariant mass m(π 0 π 0 ) of the other two π 0 candidates is shown in fig. 3. This two-dimensional plot provided information on where best to impose a cut on m(π 0 π 0 ) to reduce the background further. Only combinations in which m(π 0 π 0 ) was between 435 and 482 MeV were selected for further analysis. This cut was applied before the energy correction discussed below. After this correction, the K 0 peaks in the π 0 π 0 invariant-mass distribution were very close to 498 MeV.
The energy reconstruction of the K 0 mesons was improved by applying a correction that makes use of the nominal mass of the π 0 . After the best choice for the correct three-π 0 combination had been determined, the measured photon energies E 1,2 were replaced by where E 1 and E 2 are the measured energies of the two photon clusters, m γγ is the invariant mass of the decay photons, and m π 0 = 135 MeV is the known π 0 mass. Before scaling, the invariant mass for π 0 → γ 1 γ 2 is given by where θ γγ is the measured opening angle for π 0 → γ 1 γ 2 .
, the scaled invariant mass m(γγ) was exactly the π 0 mass, 135 MeV. This correction has been used effectively in several prior works [12,22,24,26,[29][30][31][32]. The scaled 4-momenta of the π 0 s were used to calculate m(π 0 π 0 )andm(π 0 N ), where N represents the nucleon. All three π 0 π 0 combinations were considered for further analysis. Figure 4 shows a histogram of the invariant mass m(π 0 n) plotted versus the invariant mass m(π 0 π 0 ). The quantity m(π 0 n) was actually calculated as the missing mass of the same π 0 π 0 combination, since the two quantities should be equal. This plot provided information on where best to impose a cut on the invariant mass m(π 0 n). Only combinations in which m(π 0 n) was between 1000 and 1300 MeV were selected for further analysis. After the energy correction, the peaks in the m(π 0 n) distributions were very close to the Λ mass (1116 MeV) for the MC simulated γn → K 0 Λ events. Monte Carlo studies on the polar angle of the undetected nucleon showed that most of the undetected nucleons go forward at our kinematics. A cut was therefore imposed that the cosine of the polar angle of the final-state nucleon, whether measured or calculated, must be greater than or equal to 0.7. All these cuts were used to reduce the number of incorrect three π 0 combinations. Even after all these cuts, there were still a number of events with more than one candidate for the correct three-π 0 combination. Monte Carlo studies were made of the opening angle between two photons for π 0 → γγ decays. While the distribution is broad, it is more likely for our kinematics that the opening angle is less than 90 • than greater than 90 • . The average opening angle for each remaining three π 0 combination was therefore calculated and the combination with the minimum average opening angle was selected as the best choice for the correct 3π 0 combination. Although several methods for reconstructing the 3π 0 combination were investigated using Monte Carlo simulations, this method produced the largest K 0 yields.
For case (61), events with six neutral clusters and one charged cluster were selected. The PID was used to select the proton candidate. Similar analysis steps were used to select the best choice for the correct three-π 0 combination as for case (60).
For case (70), there were seven neutral clusters. Similar analysis steps were followed as for case (60) to identify the best choice for the correct three-π 0 combination. Here for each three-π 0 combination there was one unpaired particle.
For case (80), there were eight neutral clusters. Again, similar analysis steps were followed as for case (60) to identify the best choice for the correct three π 0 combination. Here for each three-π 0 combination there were two unpaired particles; i.e., the seventh and eighth particles (a photon and a neutron). The missing mass of the seven photons should equal the mass of the neutron. Therefore a cut was imposed that the missing mass of the three π 0 s and the seventh particle (a photon) be greater than 800 MeV and a cut that cosine of the polar angle of the eighth particle (a neutron) should be greater than or equal to 0.7. These cuts were used to reduce the number of incorrect three-π 0 combinations for case (80), and helped to distinguish which of the other neutral particles was a neutron.
In MC simulations for each γN → K 0 Y event, there are two incorrect π 0 π 0 combinations for every correct combination corresponding to K 0 → π 0 π 0 . In real data, there can be additional contributions to background in the m(π 0 π 0 ) distributions.
The π 0 π 0 invariant-mass distributions were fitted using a binned likelihood method with the parametrization where α, β, δ, μ, σ B ,andσ K were fitting parameters. The first factor ensured that the distribution goes to zero when x =2m π 0 = 270 MeV. The exponent α is a small number (0 <α<1) determined by fitting the m(π 0 π 0 ) distribution for given energy bins. The parameter β measures the yield of the background contribution. The background was represented by a scaled Gaussian distribution with centroid μ and standard deviation σ B . The parameter δ measures the yield of the kaon signal. The kaon signal distribution was represented by a scaled Gaussian with centroid 498 MeV (the K 0 mass) and standard deviation σ K . High-statistics Monte Carlo simulations showed that this simple function provided a very good description of the simulated π 0 π 0 invariant-mass distributions for all energy and angle bins in the kinematic range of the measurements. These Monte Carlo simulations showed that there was very little angular variation of the α and σ K parameter values for given energy bins. Therefore, the observed m(π 0 π 0 ) distributions for each energy bin, summed over all angle bins, were fitted to determine α and σ K parameter values for each energy bin. Next the observed m(π 0 π 0 ) distributions for each angle bin, for a particular energy bin, were fitted with the values of α and σ K held fixed at their fitted values for that particular energy bin. The fitting parameters β, δ, μ,andσ B were allowed to vary freely in each angle and energy bin. The fitted value of μ for a particular angle and energy bin, with α and σ K held fixed as described above, was called the nominal background centroid. The values of the nominal background centroid for each energy and angle bin were recorded for further analysis. The background contribution was obtained after the fit by setting δ equal to zero. Numerical integration was used to calculate the total number of kaons (the kaon yield, N K 0 ) by subtracting the areas under the total and background curves. The kaon yield was sensitive to the background contribution. Although high-statistics Monte Carlo studies and the observed π 0 π 0 distributions summed over all energy and angle bins showed that the combinatoric background had a simple form that was easy to model, the peak position in the background contribution for individual energy/angle bins was not always well defined. For that reason, a second fit of the observed m(π 0 π 0 ) distributions was performed with a different value of μ called the modified centroid. The modified centroid was chosen to be the average of the nominal centroid of the background and the signal centroid (498 MeV). This modified centroid was the maximum value of the background centroid that produced a good fit of the data. In these second fits, the free parameters were β, δ,a n dσ B . The use of these two background centroids is discussed further in sect. 4. Figure 5 shows the observed π 0 π 0 invariant-mass distributions for γp → K 0 Σ + , γn → K 0 Λ,a n dγn → K 0 Σ 0 summed over all energy and angle bins. The fitted total invariant-mass distributions are represented by solid red curves and the background contributions are represented by solid black curves. Examples showing typical fits of the observed π 0 π 0 invariant-mass distributions with both the nominal and modified background centroids are shown for case (60) at W = 1815 MeV in figs. 6 and 7. As these figures show, the quality of the fits was quite good for all the angle bins. The complete set of fits with both the nominal and background centroids for all energy-angle bins and for all cases can be found in ref. [33]. For calculating the differential cross sections, eight angle bins were used to cover the range from cos θ cm = −1.0 to +1.0. The c.m. energy range W = 1615 to 1765 MeV was divided into five bins of width 30 MeV, and the c.m. energy range W = 1765 to 1865 MeV was divided into five bins of width 20 MeV. After subtracting the background, the differential cross section for a specified energyangle bin was calculated using where N K 0 = N K 0 (E γ ,θ cm ) is the kaon yield for a given energy-angle bin, N γ = N γ (E γ ) is the photon flux for a given energy bin, ǫ = ǫ(E γ ,θ cm ) is the acceptance for a specified energy-angle bin calculated from Monte Carlo simulations, N t is the number of target nucleons per cm 2 , B is a product of branching ratios for the particular reaction, and Δ cos θ cm is the bin width for cos θ cm . The differential cross section for γn → K 0 Σ 0 for case (80) was calculated using dσ dΩ  where N 80 K 0 is the measured K 0 yield for case (80) and For case (70), the measured K 0 yield has contributions from both γn → K 0 Σ 0 and γn → K 0 Λ: N 70 where B Λ = B Σ 0 =0.05301±0.00074. Values of B Λ , B Σ 0 , and B Σ + were calculated using branching ratios taken from the Review of Particle Physics [34]. For details, see ref. [33]. The γn → K 0 Λ differential cross section for case (70) is then dσ dΩ The measured γn → K 0 Σ 0 cross sections for case (80) and the measured K 0 yields for case (70) were used to calculate the γn → K 0 Λ cross sections for case (70). Similarly, the differential cross section for γp → K 0 Σ + for case (61) was calculated using dσ dΩ where N 61 K 0 is the measured K 0 yield for case (61) and B Σ + =0.07637 ± 0.00046.  For case (60), the measured K 0 yield has contributions from both γn → K 0 Λ and γp → K 0 Σ + : N 60 Thus, dσ dΩ The measured K 0 yields for case (60) and the results of a 15-parameter global fit of dσ/dΩ for γp → K 0 Σ + ,d i scussed in sect. 5.A, were used to calculate the γn → K 0 Λ cross sections for case (60). It was not possible to determine meaningful values of (dσ/dΩ) 60 γp→K 0 Σ + due to the large subtractions required.
The final task was to determine the γn → K 0 Σ 0 cross section for case (70). For this case, recall that N 70 Thus, dσ dΩ The average of the differential cross sections for the cases with and without detection of the final-state neutron, weighted according to the statistical uncertainties, was calculated for γn → K 0 Λ and γn → K 0 Σ 0 and then integrated cross sections were obtained by fitting these values with two-parameter expansions in Legendre polynomials. The Legendre fits include P 0 and P 1 terms for the γn → K 0 Λ and γn → K 0 Σ 0 results but just a P 0 term for the γp → K 0 Σ + results. We used only our case (61) results for γp → K 0 Σ + . Given the simple linear behavior observed for our γn → K 0 Λ differential cross-section measurements and their relatively large statistical uncertainties, it was deemed that more complicated parametrizations were unnecessary.

Calculation of uncertainties
There are two types of uncertainty involved in calculating the differential cross section. One is the statistical uncertainty and the other is the systematic uncertainty. The statistical uncertainty describes our imprecise knowledge of the kaon signal yield. The systematic uncertainty is the combination of uncertainties from the photon flux, acceptance, and branching ratios. The kaon signal yield in real data was correlated with the centroid of the background. As mentioned earlier, the π 0 π 0 invariant-mass distributions were fitted with a sum of scaled Gaussians, with background and signal parts. First the invariant-mass histogram was fitted, the background centroid was noted and the kaon yield was calculated; this is called the nominal case. Next a centroid for the background was chosen, which is an average of nominal case background centroid and kaon signal centroid (498 MeV), and the m(π 0 π 0 )distribution was refitted and the kaon yield was recalculated. This is called the modified case. The statistical uncertainty was conservatively calculated using Here, Poisson error = √ N K 0 + 1, where N K 0 is the average number of K 0 s determined by fitting the m(π 0 π 0 ) distributions using the nominal and modified values for the background centroid. The model error was taken as half the difference in the number of K 0 s determined using the two different background centroids. Explicit examples of the extracted number of K 0 s are provided in figs. 6 and 7 for case (60) at W = 1815 MeV. As these examples show, the dominant contribution to the statistical uncertainty is from the model error, which, for this case and energy bin the fractional uncertainty varied from a maximum of 34% at cos θ cm =+ 0 .625 to a minimum of 14% at cos θ cm = −0.625. Similar trends were observed for the other cases and energy bins.
The statistical uncertainty in dσ/dΩ is given by 14) and the systematic uncertainty is given by where the contribution from the uncertainty in the photon flux varied from 1.1% to 2.4% and the contribution from the acceptance varied from about 2% to about 4% for γn → K 0 Λ and γn → K 0 Σ 0 . The contribution from the product of branching ratios was 1.4% for γn → K 0 Λ and γn → K 0 Σ 0 and was 0.6% for γp → K 0 Σ + . 5 Results and discussion 5.1 γp → K 0 Σ + Figure 8 shows the differential cross section for γp → K 0 Σ + for the seven energy bins from 1720 to 1855 MeV. Our results are shown as solid black circles. Our results in the bins at cos θ cm = ±0.875 and −0.675 and in the energy bin at 1690 MeV had very low statistics and are not shown in fig. 8. Prior results from Lawall et al. [7], measured with the SAPHIR detector at ELSA in Bonn, are shown as solid magenta squares. Prior results from Castelijns et al. [8], measured with the Crystal Barrel and TAPS spectrometers at ELSA, are shown as solid blue triangles. The most precise prior results are from Aguar-Bartolomé et al. [10], measured on a liquid hydrogen target with the Crystal Ball and TAPS spectrometers at MAMI, and shown as solid red circles. The solid blue curves in fig. 8 are from a 15-parameter global fit to all the data, which is described below. The solid red curves are from a three-parameter global fit in which the angular distributions were approximated as being isotropic in each energy bin. The measurements in fig. 8 are compared with isobar-model predictions by Mart [35], which are shown as dashed green curves. In general, these predictions do not agree well with the measured angular distributions.
In order to ensure a smooth variation with energy and that the cross section vanishes at threshold, a 15parameter global fit of our results and prior differential cross-section data was performed. This fit used the parametrization where W T = 1687 MeV is the threshold energy for γp → K 0 Σ + and P ℓ (cos θ cm ) is a Legendre polynomial. The a nℓ coefficients were constant fitting parameters. Uncertainties in the fitted cross sections were conservatively calculated as twice the difference between results of the 15parameter global fit and a separate three-parameter global fit in which the angular distributions were approximated as being isotropic in each energy bin (only the a n0 coefficients were varied). Our measured integrated cross sections for γp → K 0 Σ + were obtained by making one-parameter Legendre fits of our measured differential cross sections. They are shown in fig. 9 as solid black circles. Prior results from Lawall et al. [7], Castelijns et al. [8], and Aguar-Bartolomé et al. [10] are shown as solid magenta squares, solid blue triangles, and solid red circles, respectively. The results of our 15-parameter global fit are shown as solid cyan circles. The experimental results are compared with Mart's isobar-model predictions [35] shown as a dashed green curve.

γn → K 0 Λ
Since the measured γp → K 0 Σ + cross sections for case (61) were imprecise due to low statistics and the low acceptance at backward and forward angles, the fitted world values of (dσ/dΩ) γp→K 0 Σ + and the measured K 0 yields for case (60) were used to calculate γn → K 0 Λ cross sections for case (60). Because the associated uncertainties in the fitted world values were relatively large at cos θ cm = ±0.875, those angle bins were excluded for all three K 0 photoproduction reactions. The c.m. energy range W = 1615 to 1765 MeV was divided into five bins Fig. 9. Integrated cross section for γp → K 0 Σ + . Our results, shown as solid black circles, were obtained by fitting our measured differential cross sections with a one-parameter Legendre expansion. The solid magenta squares show prior results from Lawall et al. [7], the solid blue triangles show prior results from Castelijns et al. [8], the solid red circles show prior results from Aguar-Bartolomé et al. [10], and the dashed green curve represents an isobar-model prediction by Mart [35]. The solid cyan circles were obtained from a 15-parameter global fit of our results combined with prior differential cross-section data. (See text for details.).  Fig. 10. Differential cross section for γn → K 0 Λ. The solid black circles represent the weighted average of our results for cases (60) and (70). The solid magenta triangles and solid blue triangles respectively show g10 and g13 results from Compton et al. [11]. The solid green curves are a prediction [36,37] based upon a partial-wave analysis. The solid red curves show the results of two-parameter Legendre polynomial fits to our measurements.
of width 30 MeV, and the c.m. energy range W = 1765 to 1865 MeV was divided into five bins of width 20 MeV. The first two c.m. energy bins W = 1630 and 1660 MeV were below γp → K 0 Σ + threshold 1687 MeV. Therefore only γn → K 0 Λ events can contribute to these bins. Figure 10 shows the differential cross section for γn → K 0 Λ for these ten energy bins. Solid black circles show our results (weighted average of cases (60) and (70)). The solid magenta triangles and solid blue triangles respectively show the g10 and g13 results from Compton et al. [11] measured at JLab. Our results agree, within uncertainties, with the g10 results (at W = 1720 and 1835 MeV) but agree with the g13 results only at W = 1855 MeV. At lower energies, results of g13 are systematically higher. It should be noted that the g10 and g13 results, where they overlap, are consistent for c.m. energies above about 1800 MeV, but the g13 results below that energy are all larger (especially at forward angles) than the g10 result that falls into our energy bin at 1690 MeV. The solid red curves in fig. 10 show results of two-parameter Legendre polynomial fits to our measurements. The solid green curves show predictions based upon a partial-wave analysis [36,37]. We have checked various factors that might affect the normalizations of our results (e.g., the photon flux N γ and detector acceptance) and have been unable to find any problems that would explain the differences between our results and the low-energy g13 results. Our results for all energy bins were handled in exactly the same manner as each other. Figure 11 shows the differential cross section for γn → K 0 Λ as a function of c.m. energy W for individual angle bins. The results in this plot show a generally smooth energy variation, which implies we do not have normalization inconsistencies in individual energy bins.
Measured integrated cross sections for γn → K 0 Λ are shown in fig. 12. Solid black circles show our results, which were obtained by making two-parameter Legendre fits of the weighted average of our measured γn → K 0 Λ differential cross sections for cases (60) and (70). The solid magenta triangles and solid blue triangles, respectively, show the g10 and g13 results from Compton et al. [11] measured at JLab. The solid green curve shows a prediction based upon a partial-wave analysis [36,37].

γn → K 0 Σ 0
Our measured γn → K 0 Λ differential cross sections for case (60) and our measured K 0 yields for case (70) were Integrated cross section for γn → K 0 Λ. The solid black circles represent our results. The solid magenta triangles and solid blue triangles respectively show the g10 and g13 results from Compton et al. [11]. The solid green curve shows a prediction [36,37] based upon a partial-wave analysis.
used to calculate the γn → K 0 Σ 0 differential cross sections for case (70). The c.m. energy range W = 1675 to 1765 MeV was divided into three bins of width 30 MeV, and the c.m. energy range W = 1765 to 1865 MeV was divided into five bins of width 20 MeV. Figure 13 shows the differential cross section for γn → K 0 Σ 0 (weighted average of cases (70) and (80)) for these eight c.m. energy bins. Our results are compared with isobar-model predic-tions (dashed blue curves) by Mart [35] and the solid red curves show results of two-parameter Legendre polynomial fits to our measurements. Our differential cross section results are in reasonable agreement within error bars with Mart's predictions except at the highest energy bin, W = 1855 MeV. Figure 14 shows the differential cross section for γn → K 0 Σ 0 as a function of c.m. energy W for individual angle bins. As for γn → K 0 Λ, these results show a generally smooth energy variation, which supports the fact that the normalizations were determined consistently for the different energy bins. Our measured integrated cross section values for γn → K 0 Σ 0 are shown in fig. 15 as solid black circles. Our integrated cross sections were obtained by calculating the weighted average of our differential cross sections for cases (70) and (80) and then making two-parameter Legendre fits. Our experimental results are compared with an isobar-model prediction (solid blue curve) by Mart [35]. Our results are in reasonable agreement with Mart's predictions except at the highest energy. These are the first experimental results for γn → K 0 Σ 0 . As in the case of the differential cross sections, our results are in good agreement with Mart's predictions except at the highest energy bin.

Summary and conclusions
Our results for γp → K 0 Σ + in the four energy bins above 1775 MeV tend to be smaller in magnitude than prior MAMI measurements [10], which were measured using a liquid hydrogen target and have better statistical precision. For this reason, we used γp → K 0 Σ + world data to extract the γn → K 0 Λ cross section for case (60). An  isobar-model prediction by Mart [35] generally disagrees with all the measured differential cross sections.
Only one published set of prior measurements for γn → K 0 Λ was available for comparing with our results. These prior results were measured with the CLAS spectrometer at JLab [11] in two separate datasets. In the seven energy bins where our results can be compared, our results agree within uncertainties with the g10 results but our results have a somewhat similar shape, but smaller magnitude, compared with the g13 results below W = 1800 MeV. The results presented in ref. [11] show that the g10 and g13 results, where they overlap, are gen-erally consistent above about W = 1800 MeV but not at lower energies. Our results for γn → K 0 Λ provide new measurements in the c.m. energy range from threshold (1614 MeV) to 1855 MeV.
Our results for γn → K 0 Σ 0 are the first experimental results for that reaction and span the c.m. energy range from the threshold (1691 MeV) to 1855 MeV. Our differential cross sections for γn → K 0 Σ 0 are in reasonable agreement within error bars with isobar-model predictions by Mart [35] except at the highest energy bin. Our two independent measurements for cases (70) and (80) are consistent within error bars. In summary, our new cross-section measurements for γn → K 0 Λ and γn → K 0 Σ 0 will provide valuable data for future partial-wave analyses and may help better de-  Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors' comment: All data generated during this study are contained in this published article.] Publisher's Note The EPJ Publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A. Tabulation of results
In this appendix, we provide our measured differential and integrated cross sections for γp → K 0 Σ + in tables 2 and 3, our measured differential and integrated cross sections for γn → K 0 Λ in tables 4 and 5, and our measured differential and integrated cross sections for γn → K 0 Σ 0 in tables 6 and 7. Open Access This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.