Cross sections for the $\gamma p\to K^{*+}\Lambda$ and $\gamma p\to K^{*+}\Sigma^{0}$ reactions measured at CLAS

Because the $K^{*+}$ is an unstable particle, it will quickly decay to $K\pi$ (see Table 1) by the strong interaction. By applying energy and momentum conservation, the $K^{*+}$ momentum is reconstructed from its decay particles, $K^{0}\pi^{+}$. The $K^{0}$ is a mixture of 50% $K_{S}$ and 50% $K_{L}$, but only the $K_{S}$ decay is detected by the CLAS detector. The $K^{0}$ is reconstructed from the $K_{S}$ decay to $\pi^{+}\pi^{-}$ with a decay branching fraction of 69.2%. The same branching fractions are reproduced by the Monte Carlo detector simulations (see section 3.6), and hence are implicit in the detector acceptance values.

To summarize, we report on the differential and total cross sections of the photoproduction channel(s):

followed by

and

The $K^{*+}$ and $K_{S}$ are reconstructed directly from their decay products, while the $\Lambda$ and $\Sigma^{0}$ are reconstructed using the missing mass technique.

The Time of Flight (TOF) difference method was used to identify events with three pions (two positive and one negative charge) in the final state. Explicitly,

where $tof_{mea}$ is the measured TOF of the particle and $tof_{cal}$ is the calculated TOF with the measured momentum $p$ and the mass of a pion. In more detail,

where $t_{tof}$ is the time when the particle hits the TOF scintillators and $t_{st}$ is the time when the photon hits the target. This information is determined by the CLAS Start Counter. In comparison, $tof_{cal}$ is given by:

where

Thus

where $L$ is the path length from the target to the TOF scintillators, $c$ is the speed of light, $p$ is the particle’s momentum, and $m$ is the mass of a pion. The pion candidates are required to have $|\Delta tof|<1.0$ ns. Fig. 1 shows the TOF difference spectrum. The solid lines define the region of the cut, the small peaks on the both side of the cuts are due to photons coming from other beam bunches, showing evidence of the $\sim$2 ns beam bunch structure of CEBAF.

After applying the $|$$\Delta tof$$|$ $<$ 1.0 ns cut, particles that came from different RF beam buckets were removed naturally. Of the photons measured by the photon tagger, we want those that come within 1.0 ns of the particle vertex time, which are called “good” photons. However, there might still be more than one “good” photon in each event. To select the correct photon, all “good” photons were scanned to find the one that gave the three-pion missing mass closest to the known mass of the $\Lambda$($\Sigma^{0}$), where

is the missing mass summed over all three pions in the event, while $E_{\gamma}$ and $\vec{p}_{\gamma}$ are the energy and momentum vector of the photon. The two-pion missing mass is similarly defined.

Several cuts were applied to the data to reduce the background and to remove events below threshold for the reaction of interest. In general, the strategy is to use geometric and kinematic constraints to eliminate backgrounds while ensuring that the signal remains robust. The efficiency of various cuts was tested with Monte Carlo simulations (see section III.7).

The geometric and kinematic constraints used here are listed below:

• •

  Fiducial cuts were applied to remove events that were detected in regions of the CLAS detector where the calibration of the detector is not well understood.

• •

  A cut on the vertex position along the beam axis (the $z$-axis) to be within the target position was applied. All pions were required to be generated from the same vertex position within the experimental position uncertainty.

• •

  The missing mass from the $K^{0}$ was required to satisfy the relation $MM(\pi^{+}\pi^{-})>1.0$ GeV to include all hyperon mass peaks, for pion pairs with an invariant mass inside the $K^{0}$ mass window (see next section). Similarly, the missing mass from the $K^{*+}$ was required to be greater than the nucleon mass, $MM(\pi^{+}\pi^{-}\pi^{+})>1.0$ GeV.

  After this step, the $K^{*+}\Lambda$ and $K^{*+}\Sigma^{0}$ reaction channels were treated differently, since different backgrounds are present for each final state. For instance, the large background from

  $$\gamma p\to K^{0}\Sigma{}^{\ast{}+}(1385)$$

  (10)

  present for the $K^{*+}\Lambda$ reaction channel makes the the extraction of $K^{*+}\Lambda$ yields by simply fitting the $\Lambda$ peak impossible. On the other hand, there are only very small portions of the $\Sigma^{\ast{}+}$(1385) that contribute to the $K^{*+}\Sigma^{0}$ background, which can be easily removed based on Monte Carlo studies (see following section). Thus we could fit directly the $\Sigma^{0}$ peak in the three-pion missing mass for $K^{*+}\Sigma^{0}$ channel, whereas a different approach (given below) is necessary to extract the $K^{*+}\Lambda$ yield separately from background due to $K^{0}\Sigma^{\ast{}+}$(1385) production. The following lists the extra cuts applied for each reaction channel.

• •

  For the $K^{*+}\Lambda$ analysis, a cut was placed on the $\Lambda$ peak in the three-pion missing mass: 1.08 GeV $<MM(\pi^{+}\pi^{-}\pi^{+})<$ 1.15 GeV. This ensures that a $\Lambda$ was present in the final state.

• •

  For the $K^{*+}\Sigma^{0}$ analysis, a cut was placed on the $K^{*+}$ peak of the three-pion invariant mass: 0.812 GeV $<M(\pi^{+}\pi^{-}\pi^{+})<$ 0.972 GeV. This ensures that a $K^{*+}$ was produced.

Because reactions other than $K^{*}$ photoproduction are present, background is still mixed in with the channels of interest. Fig. 2 shows the two-pion invariant mass plot after the first three cuts in the previous section, integrated over all photon energies. A clear peak centered near 0.497 GeV sits on top of a smooth background. The invariant mass is calculated using the momentum vector of one $\pi^{+}$ in the event, along with the $\pi^{-}$ momentum. Since there are two $\pi^{+}$s, both $\pi^{+}\pi^{-}$ pairs are tested, but typically only one combination will satisfy all kinematic constraints. To avoid double-counting, in rare cases where both $\pi^{+}$ satisfy all constraints, this combinatoric background is removed, for both data analysis and Monte Carlo acceptances.

To reduce the background, a Sideband Subtraction Method (SSM) was applied. The concept of the SSM is to assume that the background in the signal region can be approximated by a combination of the left and the right regions, which are adjacent to the signal region. In our analysis, the two-pion mass of the $K_{S}$ is used as the criteria to select the signal and sideband regions. Fig. 2 shows the regions used in our analysis. The middle band is the signal region, centered at the mass of $K^{0}$ with a width of 0.03 GeV. The other two bands, with the same band sizes, are the combinatorial background.

Fig. 3 shows the sideband subtraction applied to the reconstructed three-pion invariant mass and to the three-pion missing mass. The SSM reduces the background, giving cleaner signal peaks.

After applying the SSM to each $M(\pi^{+}\pi^{+}\pi^{-})$ invariant mass plot, corresponding to different incident photon energy and different $K^{*+}$ production angle ranges, the $K^{*+}$ peak becomes clearer, but it is still not free of background. The main contribution to the background comes from the reaction channel $\gamma p\to K^{0}\Sigma^{*+}$(1385), which passed through all the cuts. In addition, the 3-body phase space reaction $\gamma p\to K^{0}\pi^{+}\Lambda$ is also present, and will contribute to the background as well.

In order to extract the correct $K^{*+}$ peak yield, instead of fitting the $K^{*+}$ peaks directly with a Breit-Wigner plus background functions, we applied a template fit. The precondition for this template fitting is that we assume there is negligible interference between the $K^{*+}\Lambda$ and $K^{0}\Sigma^{*+}$(1385) channels; in other words, we assume that the $K^{*+}\Lambda$ and $K^{0}\Sigma^{*+}$(1385) add incoherently. If we remove all other sources of background, then the $K^{*+}$ mass plot should have background only from the $\Sigma^{*+}$(1385) peak. Similarly, the background in the $\Sigma^{*+}$(1385) plot comes only from events in the $K^{*+}$ peak. Because the three-body $K^{0}$$\pi^{+}$$\Lambda$ channel is also a possible background, we assume it will add incoherently as well in both mass projections.

To justify these assumptions, we explored the effect of various levels of interference between these two final states in the simulations. The result is that the template fits correctly reproduced the generated events to within a 5% uncertainty for assumptions of maximal constructive or destructive interference.

Fig. 4 shows an example of the template fitting, where the solid dots with error bars are from the data, while the curve is from the fit, which contains contributions from both the $K^{*+}\Lambda$, $K^{0}\Sigma^{*+}$ and $K^{0}\pi^{+}\Lambda$ channels. The $K^{*+}$ peak is seen in the left plots and the $\Sigma^{*+}$ peak is seen in the right plots. The template shape for each contribution comes from the simulation for that channel, and the magnitude of each channel is a free parameter to optimize the fit, with the result for each component of the fit shown in the bottom plots of Fig. 4. Both mass projections of Fig. 4 are fit simultaneously to minimize the overall $\chi^{2}$.

For the $K^{*+}\Sigma^{0}$ reaction, the counts from the $\Sigma^{0}$ were extracted by using a Gaussian fit, then the yields were corrected bin by bin based on a Monte Carlo study of how much $K^{0}\Sigma^{\ast{}+}$(1385) leakage there is to $K^{*+}\Sigma^{0}$ reaction channel. The correction was studied and found to be less than 0.1%, which was included in our cross section calculation. Fig. 5 shows an example of the fitting. There are two peaks in the three-pion missing mass plot, one corresponding to the $\Lambda$ and the other to the $\Sigma^{0}$. The fitting function used two Gaussians plus a second order polynomial, for the $\Lambda$ peak, $\Sigma^{0}$ peak and background, respectively.

A computational simulation package, the CLAS GEANT Simulation (GSIM), was used for the Monte Carlo modeling of the detector acceptance. GSIM is based on the CERN GEANT simulation code with the CLAS detector geometry. Thirty million $\gamma p\to K^{*+}\Lambda\ (\Sigma^{0})$ events were randomly generated, with all possible decay channels of the final state particles ($K^{*+}$, $\Lambda$, $\Sigma^{0}$…). The Monte Carlo files were generated with a Bremsstrahlung photon energy distribution and a tunable angular distribution that best fit the $K^{\ast}$ data. The energy bin size was 0.1 GeV and the total cross section was assumed constant across the bin. This assumption is reasonable based on the slowly-varying total cross sections shown below. Because the simulations have a better resolution than the real CLAS data, the output from GSIM are put through a software program to smear the particle momentum, timing, etc. to better match the real data.

An extensive study of the g11a trigger mike showed a small inefficiency for the experimental trigger. To account for the trigger inefficiency, an empirical correction was mapped into the Monte Carlo. The trigger corrections applied here is the same as used for other CLAS analyses of this same dataset mike .

The detector acceptance is calculated by:

where $\epsilon$ represents the detector acceptance, $D_{MC}$ is the number of simulated events after processing and $G_{MC}$ is the number of generated events.

The same software used for the experimental data was applied directly to the Monte Carlo data. Simulated events are extracted by fitting each reconstructed $K^{*+}$ peak for a given photon energy and $K^{*+}$ production angle. In our analysis, a non-relativistic Breit-Wigner function

was used to extract the counts of the $K^{*+}$ peaks for $K^{*+}\Lambda$ channel. Here, $\Gamma$ is the full width at half maximum of the resonance peak, $E$ is the scattering energy and $E_{R}$ is the center of the resonance.

As described in section III.4, different methods were used for the $K^{*+}\Lambda$ and $K^{*+}\Sigma^{0}$ channels due to the presence of $\Lambda^{*}$ resonance contributions in the former. For the $K^{*+}\Sigma^{0}$ channel, where there is no kinematic overlap from hyperon states, the counts under the $\Sigma^{0}$ peak were fitted directly using a Gaussian function. Fitting the three-pion missing mass of the $\Sigma^{0}$ has less uncertainty than fitting the $K^{*+}$ peak, since the $\Sigma^{0}$ peak is relatively narrow on top of a nearly flat background. This method was used for both simulated and experimental data.

Systematic uncertainties come from several sources: the applied cut parameters, the choice of fitting functions, the Monte Carlo used for the detector acceptance and so on.

Systematic uncertainties were estimated for each cut by varying the cut intervals and then recalculating the differential cross sections. The changes to cut parameters were applied to both the experimental data and the simulated output. The relative difference between the new cross sections and the original cross sections was calculated bin by bin using:

and then the resulting $\delta\sigma$ values were histogrammed. This histogram was fitted with a Gaussian function, and the width from the Gaussian fit was taken as the systematic uncertainty for each variation. The cut intervals were varied to both larger and smaller values, and we chose the larger of the systematic uncertainties calculated from each variation.

Similar estimation were done for the detector acceptance, by varying the inputs to the Monte Carlo. Also, different fitting functions and background shapes were used to determine the systematic uncertainties associated with the peak yields. The total systematical uncertainty is then given by

which assumes no correlated uncertainties.

The total systematic uncertainty from all sources, added in quadrature, is shown in Table 2, where the other sources include the target length, density and so on. For the $K^{*+}\Lambda$ final state the overall systematic uncertainty is 14% and for $K^{*+}\Sigma^{0}$ the systematic uncertainty is 12%.

The models that are currently available for $K^{*}$ photoproduction are based on effective Lagrangians, which fall into two groups: isobar models and Reggeized meson exchange models. Isobar models evaluate tree-level Feynman diagrams, which include resonant and nonresonant exchanges of baryons and mesons. The reggeized models, on the other hand, emphasize the $t$-channel meson exchange, which is expected to dominate the reaction at energies above the resonance region. The standard propagators in the Lagrangian are replaced by Regge propagators, which take into account an entire family of exchanged particles with the same quantum numbers instead of just one meson exchange. In this section, the $K^{*+}\Lambda$ cross section results will be compared with calculations from these two theoretical models.

One model we use is by Oh and Kim (O-K Model) oh2 , which is an isobar model. This model starts with Born terms, which include $t$-channel (with $K$, $K^{*}$ and $\kappa$ exchanges), $s$-channel ground state nucleon exchanges and $u$-channel $\Lambda$, $\Sigma$ and $\Sigma^{*}$ exchanges. Additional $s$-channel nucleon resonance exchanges were added to the model using the known resonances from the PDG in Ref. KNOK , referred to here as the K-N-O-K model. One attractive point of these models is the inclusion of diagrams with a light $\kappa$ meson exchange in the $t$-channel. As mentioned in the introduction, the $\kappa$ meson has not yet been firmly established, and these models allows us to study the effect of possible $\kappa$ exchange.

The other model shown here is the Ozaki, Nagahiro and Hosaka (O-N-H) Model ONH , which is a reggeized model. This model takes into account all possible hadron exchanges with the same quantum numbers (except for the spin). The coupling constants and $\kappa$ exchange parameters are the same as those used in O-K Model oh2 .

Fig. 10 shows those calculations compared with our differential cross sections, where the solid curves represent the theoretical calculations from the K-N-O-K Model and the dashed curves represent the O-N-H Model. The corresponding curves are shown in Fig. 11 for the total cross sections, where the curves are explaind in the figure caption. The O-K model includes $s$-channel diagrams with most well-established nucleon resonances below 2 GeV pdg , whereas the K-N-O-K model includes two additional $s$-channel resonances up to 2.2 GeV. Interpretation of these results are discussed in section VI.

Fig. 12 shows the total cross section ratio of the reactions $\gamma p\to K^{*0}\Sigma^{+}$ to $\gamma p\to K^{*+}\Lambda$. The $K^{*+}\Lambda$ data alone are not very sensitive to the $\kappa$ exchange due to the unknown strength of the coupling constant, $g_{\kappa N\Lambda}$. However, the coupling constants of these two reactions is related in the effective Lagrangian models, and so the ratio is sensitive to the effects of $\kappa$ exchange. The dots with error bars in Fig. 12 use the present data along with the previously published CLAS data for $K^{*0}\Sigma^{+}$ hleiqawi . We note that another data set exists for the $K^{*0}\Sigma^{+}$ reaction from CBELSAcbelsa , but we have chosen to use CLAS data in both numerator and denominator to reduce systematics. The two curves are the theoretical predictions from O-K Model I and II oh2 , where Model I includes minimal $t$-channel $\kappa$ exchange, while Model II has a significant contribution from $\kappa$ exchange.