Photoproduction of $K^+K^-$ meson pairs on the proton

The exclusive reaction $\gamma p \to p K^+ K^-$ was studied in the photon energy range $3.0 - 3.8 \mbox{ GeV}$ and momentum transfer range $0.6<-t<1.3 \mbox{ GeV}^2$. Data were collected with the CLAS detector at the Thomas Jefferson National Accelerator Facility. In this kinematic range the integrated luminosity was approximately 20 pb$^{-1}$. The reaction was isolated by detecting the $K^+$ and the proton in CLAS, and reconstructing the $K^-$ via the missing-mass technique. Moments of the di-kaon decay angular distributions were extracted from the experimental data. Besides the dominant contribution of the $\phi$ meson in the $P$-wave, evidence for $S-P$ interference was found. The differential production cross sections $d\sigma/dt$ for individual waves in the mass range of the $\phi$ resonance were extracted and compared to predictions of a Regge-inspired model. This is the first time the $t$-dependent cross section of the $S$-wave contribution to the elastic $K^+K^-$ photoproduction has been measured.


I. INTRODUCTION
Data on light quark mesons comes mainly from hadron induced reactions, e.g. by using K, π, p orp beams and, more recently, from decays of heavy mesons. Up to now, only a few studies of the light meson spectrum were attempted with electromagnetic probes and, in particular, with real photons. The main reason for this is the relatively small production cross sections compared to hadronic reactions. However, this situation is changing, thanks to the recent advances in producing high-intensity and high-quality tagged, polarized photon beams. At lower energies, e.g. near single meson production thresholds, high quality data have been accumulated by the CB-ELSA [1] and CB-MAMI [2] experiments, while at higher energies, photoproduction data have come from the CLAS [3] experiment at Jefferson Lab. Moreover, two new programs, GLUEX [4] and MesonEx [5] have just been launched in the same laboratory. A typical meson photoproduction data set from past experiments in the energy range below 20 GeV, typical for meson spectroscopy, has tens of thousands of events, and only a few topologies have been studied [6][7][8]. For comparison, the data samples from the g11 run at CLAS used here, exceed the existing sets in many channels by at least an order of magnitude, and several reconstructed topologies are available for a comprehensive study [9].
Specifically, two-pseudoscalar meson photoproduction (two-pion and two-kaon) offers the possibility of investi-gating various aspects of the light meson resonance spectrum. Two-pion is the main decay mode of the lowest isoscalar-tensor, the f 2 (1270) resonance, and it is the only known hadronic decay mode of the lowest isovectorvector resonance, the ρ(770). The two-kaon channel is the main decay mode of the isoscalar-vector φ(1020) and a possible sub-threshold decay of the isoscalar-scalar f 0 (980) and the isovector-scalar a 0 (980). Both the two pion and two kaon decay modes couple to the isoscalarscalar channel, which contains the f 0 (500) and f 0 (980) resonances [10] and a few more resonances with masses above 1 GeV that are not yet well understood. For example, the f 0 (500) meson, which is now well established [11][12][13], but does not fit the naive quark model classification. The f 0 (980) is similarly difficult to classify and its composition is affected by proximity to the KK threshold. These states have been the subject of extensive investigations [14,15] since their observation in photon induced reactions can provide insights into their internal structure. In this paper we present results of the analysis of K + K − photoproduction in the photon energy range 3.0 − 3.8 GeV and momentum transfer squared −t between 0.6 GeV 2 and 1.3 GeV 2 , where the di-kaon effective mass M K + K − varies from 0.990 to 1.075 GeV. We have focused on this mass region because it is dominated by the production of the φ(1020) resonance that decays to the two kaons in the P -wave, and thus a partial wave analysis based on the lower (S and P ) waves efficiently describes it. To describe the higher mass region would require a higher number of partial waves, and is not included in this study.
Angular distributions of photoproduced mesons and related observables, such as the spherical harmonic moments and the spin density matrix elements, are the most effective tools for studying individual partial waves. For example, interference between the S-wave and the dominant P -wave was first discovered in the moment analysis of K + K − photoproduction on hydrogen in the experiments performed at DESY [16] and Daresbury [17]. In this work we applied the same methodology used in the analysis of two pion photoproduction to the same data set [18,19] and we refer the reader to those works for a detailed description of the analysis procedure.
This paper is organized as follows. In the next section we give a summary of the experimental setup and data analysis. Extraction of the angular moments of the two-kaon system is described in Section III. The fit of a phenomenological model to the extracted moments is described in Section IV, where we also present results of the partial wave analysis, including the extracted differential cross sections for each partial wave, and a physics interpretation. A summary of the results is given in Section V.

II. EXPERIMENTAL PROCEDURES AND DATA ANALYSIS
A. The photon beam and the target The measurement was performed with the CLAS detector [20] in Hall B at Jefferson Lab with a bremsstrahlung photon beam produced by a continuous 60 nA electron beam of energy E 0 = 4.02 GeV impinging on a gold foil of thickness 8 × 10 −5 radiation lengths. A bremsstrahlung tagging system [21] with a photon energy resolution of 0.1% E 0 was used to tag photons in the energy range from 1.6 GeV to a maximum energy of 3.8 GeV. In this analysis only the high-energy part of the photon spectrum, ranging from 3.0 to 3.8 GeV, was used. The e + e − pairs produced by interactions of the photon beam on an additional thin gold foil were used to continuously monitor the photon flux during the experiment. Absolute normalization was obtained by comparing the e + e − pair rate with the photon flux measured by a total absorption lead-glass counter in dedicated low-intensity runs. The energy calibration of the Hall-B tagger system was performed both by a direct measurement of the e + e − pairs produced by the incoming photons and by applying an over-constrained kinematic fit to the reaction γp → pπ + π − , where all particles in the final state were detected in CLAS [22]. The quality of the calibrations was checked by looking at the mass of known particles, as well as their dependence on other kinematic variables (photon energy, detected particle momenta and angles).
The target cell, a Mylar cylinder 4 cm in diameter and 40-cm long, was filled by liquid hydrogen at 20.4 K. The luminosity was obtained as the product of the target density, target length and the incoming photon flux corrected for data-acquisition dead time. The overall systematic uncertainty on the run luminosity was estimated to be approximately 10%, dominated by the uncertainty of the photon flux normalisation [23].

B. The CLAS detector
Outgoing hadrons were detected in the CLAS spectrometer. Momentum information for charged particles was obtained via tracking through three regions of multiwire drift chambers [24] within a toroidal magnetic field (∼ 1.25 T) generated by six superconducting coils. The polarity of the field was set to bend the positive particles away from the beam line into the acceptance of the detector. Time-of-flight scintillators (TOF) were used for charged hadron identification [25]. The interaction time between the incoming photon and the target was measured by the start counter (ST) [26]. This was made of 24 strips of 2.2 mm thick plastic scintillator surrounding the hydrogen cell with a single-ended PMT-based readout. The average time resolution of the ST strips was ∼300 ps.
The CLAS momentum resolution, σ p /p, ranged from 0.5 to 1.0%, depending on the kinematics. The detector geometrical acceptance for each positive particle in the relevant kinematic region was about 40%. It was somewhat less for low-energy negative hadrons, which could be lost at forward angles because their paths were bent toward the beam line and out of the acceptance by the toroidal field. Coincidences between the photon tagger and the CLAS detector triggered the recording of the events. The trigger in CLAS required a coincidence between the TOF and the ST in at least two sectors, in order to select reactions with at least two charged particles in the final state. A total integrated luminosity of 70 pb −1 (∼ 20 pb −1 in the range 3.0< E γ <3.8 GeV) was accumulated in 50 days of data taking in 2004.

C. Data analysis and reaction identification
The raw data were passed through the standard CLAS reconstruction software to determine the four-momenta of the detected particles. In this phase of the analysis, corrections were applied to account for the energy loss of charged particles in the target and surrounding materials, misalignments of the drift chamber positions, and uncertainties in the value of the toroidal magnetic field.
The reaction γp → pK + K − was isolated by detecting the proton and the K + in the CLAS spectrometer, while the K − was reconstructed from the four-momenta of the detected particles by using the missing-mass technique. A combination of drift chambers and TOF information allowed for the identification of the kaon band in the β vs. p plane for positive charged particles. More details, as well as the resulting K + missing mass spectrum for the reaction γp → K + X can be found in Ref. [23]. The exclusivity of the reaction was ensured by retaining events within 3σ around the missing K − peak (492 MeV ± 30 MeV). This cut kept the contamination from pion misidentification and multi-kaon background to a minimum (∼7%) for events in the di-kaon mass range of interest for this analysis (0.990 GeV < M K + K − < 1.075 GeV). Figure 1 shows the K − missing mass. The background below the kaon peak appears as a smooth contribution to the K + K − invariant mass that can be accounted for by fitting and subtracting a polynomial function. Since the focus of the paper is about the interference of the narrow P -wave (the φ meson) with the S-wave, the experimental background, as well as the projection of high mass hyperons populating the pK + mass spectrum, enters in the K + K − mass as a smooth incoherent contribution that does not affect the results.
To cut out edge regions in the detector acceptance, only events within a fiducial phase space volume were retained in this analysis. In the laboratory reference system, cuts were defined for the minimum hadron momentum (p p > 0.32 GeV/c and p K + > 0.125 GeV/c), and the minimum angles (θ p > 10 • and θ K + > 5 • ). The fiducial cuts were defined comparing in detail the experimental data distributions with the results of the detector simulation. The minimum momentum cuts were tuned for different hadrons to take into account the energy loss as the particles pass through the target and the detector. After all cuts, 0.2M events were identified as produced in the exclusive reaction γp → pK + (K − ). The other event topologies that required the K − to be detected were not used since, in the kinematics of interest for this analysis (−t < 1.3 GeV 2 ), the collected data were about one order of magnitude less due to the reduced detector acceptance for the inbending K − . Figure 2 shows the invariant mass spectra of pK − and K + K − using the reconstructed K − four-momentum. The φ(1020) dominates the K + K − spectrum and the Λ(1520) peak is visible in the mass spectrum of the pK − invariant mass. No overlap between the Λ(1520) peak and the K + K − spectrum occurs for M K + K − < 1.25 GeV. Nevertheless, a sharp cut for M pK − < 1.6 GeV was applied to avoid any contamination in the meson spectrum from the Λ(1520). A hint of excited Λ states is visible in the bi-dimensional distribution but their contribution to the K + K − spectrum is very small and tends to be smooth when all hyperon states are integrated over.

III. MOMENTS OF THE DI-KAON ANGULAR DISTRIBUTIONS
In this section we consider the analysis of spherical harmonic moments, Y lM = Y lM (E γ , t, M K + K − ), of the di-kaon angular distribution defined as, where dσ is the four-fold differential cross section at fixed photon energy E γ . Here t is the momentum transfer squared between the target and the recoil proton, M K + K − is the di-kaon invariant mass and Y lM are spherical harmonics. The spherical angle Ω K = (θ K , φ K ) corresponds to the direction of flight of the K + in the K + K − helicity rest frame. This is the rest frame of the K + K − pair, with the y-axis perpendicular to the production plane and the z-axis pointing in the opposite direction of the recoil nucleon momentum. In equation (1) the normalization has been chosen such that the Y 00 moment is equal to the di-kaon production differential cross section dσ/dt dM K + K − . There are several advantages in using moments of the angular distribution compared to a direct partial wave analysis. Moments can be expressed as bi-linear in terms of the partial waves and, depending on the particular combination of L and M , show specific sensitivity to a particular subset of them. In addition, they can be directly and unambiguously derived from the data, allowing for a quantitative comparison to the same observables calculated in specific theoretical models. Since partial wave analysis has either intrinsic mathematical ambiguities or is model dependent, it is important to extract physical observables like moments before proceeding with a model dependent analysis [27].
The moments were extracted using two separate methods, both expanding in a model-independent set of basis functions, which were compared to the data by maximizing a likelihood function. The first of these two methods (M1) parametrized the angular distributions in terms of moments directly, while the second method (M2) used spherical harmonic partial wave amplitudes. The approximations in these two methods are dependent on the basis and on their truncation. As a check of systematics we also applied two further methods: we first binned the data and Monte Carlo simulations in all kinematical variables and divided the data by acceptance to obtain the expected angular distributions; the second used linear algebra techniques to set up an over-determined system of equations for the moments. They provided consistent results but were not as stable or reliable as the maximum likelihood methods M1 and M2 and were not included in the final determination of the experimental moments. Detailed systematic studies using both Monte Carlo and data were performed to test the stability of the results for the different methods. A summary of these studies is reported in Appendix A. Full details regarding the procedure adopted for the moment extractions are reported in [19,28].

A. Detector efficiency
The CLAS detection efficiency for the reaction γp → pK + K − was obtained by means of detailed Monte Carlo simulations, which included knowledge of the full detector geometry and a realistic response to traversing particles. Events were generated according to three-particle phase space with a bremsstrahlung photon energy spectrum. A total of 96 M events were generated in the energy range 3.0 GeV < E γ < 3.8 GeV and covered the allowed kinematic range in −t and M K + K − . About 19 M events were reconstructed in the M K + K − and −t ranges of interest (0.990 GeV < M K < 1.365 GeV, 0.6 GeV 2 < −t < 1.3 GeV 2 ). This corresponds to more than 400 times the statistics collected in the experiment, thereby introducing a negligible statistical uncertainty with respect to the statistical fluctuations of the data.

B. Extraction of the moments via likelihood fit of experimental data
The extraction of the moments, Y LM , was performed using the extended maximum likelihood method.
As stated above, the expected theoretical yield was parametrized in terms of appropriate functions, amplitudes in one case and moments in the other. The theoretical expectation, after correction for acceptance, was compared to the experimental yield. The likelihood is then given by, Here a represents a data event, n is the number of data events in a given (E γ , t, M K + K − ) bin (i.e. the fit is done independently in each bin), τ a represents the set of kinematical variables of the a th event (here the two kaon decay angles), η(τ a ) is the corresponding acceptance derived by Monte Carlo simulations and I(τ a ) is the theoretical function representing the expected event distribution. The measure dτ includes the phase space factor and the likelihood function is normalized to the expected number of events in the bin This normalization integral was performed by Monte-Carlo integration over the reconstructed simulated events. The parameters were extracted by minimizing a function of the form, The advantage of this approach lies in avoiding binning the data and the large uncertainties related to the corrections in regions of CLAS with vanishing efficiencies.
Comparison of the results of the two different extraction methods allows one to estimate the systematic uncertainty related to the procedure. A detailed description of the two approaches is reported in Ref. [19].

C. Method comparisons and final results
Moments derived by the different procedures agreed qualitatively. The two methods were consistent in the range of interest from 0.990 GeV< M K + K − < 1.075 GeV (and 0.6 < −t < 1.3 GeV 2 ). We do not use the region M K + K − > 1.075 GeV to extract amplitude information because the choice of amplitude parametrization (see Sec. IV A) is only valid in proximity to the φ(1020) meson mass. The difference between the fit results of M1 and M2 was used to evaluate the systematic uncertainty associated with the moment extractions. The final results are given as the average of M1 (parametrization with moments) and M2 (parametrization with amplitudes),             where Y stands for Y lM (E γ , t, M K + K − ). The total uncertainty δY f inal in the final moments was evaluated by adding in quadrature the statistical uncertainty, δY MIN UIT as given by MINUIT, and two systematic uncertainty contributions: δY syst f it related to the moment extraction procedure, and δY syst norm , the systematic uncertainty associated with the photon flux normalization (see Sec. II). (6) with: As a check of the analysis procedure, the differential cross section dσ/dt for the γp → pφ(1020) meson was extracted by integrating the Y 00 moment in each t bin in the range 1.005 GeV < M K + K − <1.035 GeV after subtracting a first-order polynomial background fitted to the data (excluding the region 1.005 GeV < M K + K − < 1.035 GeV as Ỹ 00 is not linear due to the φ peak). The results are shown in Fig. 6. Despite the different energy binning of the various studies, the reasonable agreement within the quoted uncertainties with previous measurements [17,31] gives us confidence in the accuracy of the analysis method.

IV. PARTIAL WAVE ANALYSIS
In the previous section we discussed how moments of the angular distributions of the K + K − system, Y LM , were extracted from the data in each bin in photon energy, momentum transfer and di-kaon mass. In this section we describe how partial waves were parametrized and extracted by fitting the experimental moments.
The production amplitudes can be written as where λ γ , λ, λ ′ are the helicities of the photon, target and recoil nucleon, respectively, and M K + K − is the invariant mass of the K + K − system. In terms of the helicity amplitudes the cross section is given by, FIG. 6: Differential cross section for 3.0 < Eγ < 3.8 GeV derived from the Y00 moment analysis compared with other results. The differential cross section was calculated from the P -wave extracted using the partial waves analysis described in Sec. IV D. The uncertainties include fit parameter uncertainties added in quadrature with a 10% systematic uncertainty from the photon flux normalization. Results of this work are compared to CLAS published results from [31] in the energy range Eγ= 3.300 GeV ± 0.015 GeV and Daresbury data [17] in the range 2.8 < Eγ < 3.8 GeV.
with the phase space factor Φ given by where the factor of 1/4 comes from averaging over the initial photon and target polarizations and all dimensional quantities enter in units of GeV. The helicity amplitudes are decomposed into partial waves f LM {λ} in the KK channel, (12) so that the moments, defined in (1), are given by, (13) with the c's proportional to a product of Clebsch-Gordan coefficients. Note that we are using the spherical basis for the spin projection M and not the so-called reflectivity basis. Equation (13) is a bilinear relation between the moments derived from the data and the partial wave amplitudes. The fit minimized the difference of the right and the left side of Eq. (13) with respect to free parameters in the amplitude parametrization. In this way, a set of moments was used to determine the amplitudes.
A. Parametrization of the partial waves For a given L and M , there are eight independent amplitudes, f LM λγ ,λ,λ ′ (M K + K − ), in each energy and momentum transfer bin corresponding to each combination of photon and initial and final nucleon helicity. We have only one energy bin in this analysis, so the fitted amplitudes do not depend on E γ . Since the L ≥ 2 amplitudes (D-and F -waves) are expected to be small in the K + K − invariant mass range, we only include S-and P -partial-waves. The reaction γp → pK + K − was then characterized by 32 amplitudes. There were 8 amplitudes required to describe the S−wave depending on the two spin projections of the photon (λ γ = ±1), the target proton (λ = ±1/2), and the recoil proton (λ ′ = ±1/2). In addition, there were 24 P −wave amplitudes depending also on 3 spin projections of the φ. However, the photon helicity was restricted to λ γ = +1 since the other amplitudes are related by parity conservation, resulting in 16 unconstrained amplitudes. In addition, some approximations in the parametrization of the partial waves were adopted to reduce the number of free parameters in the fit as discussed below. In general, it is expected that the dominant amplitudes require minimal photon helicity flip, i.e.
corresponding to photon helicity flip by zero and one, respectively. In the s-channel helicity frame, we assume the P -wave production (L = 1) is dominated by helicity non-flip amplitudes, i.e. the non-vanishing independent amplitudes are: where ± refer to helicities of the photon and the protons, e.g. +, +, + corresponds to λ γ = +1, λ = +1/2 and λ ′ = +1/2. We introduced two additional amplitudes per each orbital angular momentum, to describe unit photon helicity flip, and In the approximations described above, the dependence of moments on the S and P amplitudes is given by, with Y 22 vanishing under our assumptions. Here we see the Y 10 and Y 11 moments contain information about the presence of the S-wave interference with the dominant P -wave. Thus a nonzero Y 10 or Y 11 moment is an indication of a non-vanishing S-wave amplitude. In order for the Y 22 moment to be non-zero, there must be two-unit photon helicity flip amplitudes. Given that there is no significant structure in any Y 22 moments of this analysis, it is justified to neglect two-unit photon helicity flip amplitudes. So far we have introduced only the nucleon helicity non-flip amplitudes. Indeed P -wave nucleon helicity flip amplitudes are expected to be small (cf. Appendix B and Ref. [33]).
Without polarization information, it is difficult to separate out amplitudes differing only by the helicity of the nucleon. We did attempt to fit the data using various configurations of nucleon helicity amplitudes and found in particular that the S − P interference signal in the Y 11 moment cannot be described solely by interference between nucleon flip amplitudes. We comment on this further in Sec. IV C. We find, however, that the moments can be well described by interference between the dominant, nucleon helicity non-flip P -and S-wave amplitudes. Details of the amplitude parametrization are given in Ap-pendix B.

B. Fit of the moments
To account for detector resolution, the moments calculated from the amplitudes were smeared by a Gaussian function. The φ width apparent in the Y 00 moment determined the smearing needed in order for the P -wave parametrization (with fixed φ width) to match the data. This lead to a width in the Gaussian smearing of 4 MeV, which is compatible with the CLAS detector resolution measured in other reactions [23]. We fit the moments Y LM with L ≤ 2 and M ≤ 2 using up to L = 1 (P ) waves as described above. In Figs. 7 -13, we present the fit results of this analysis from 0.6 < −t < 1.3 GeV 2 . To properly take into account the uncertainty contributions (statistical and systematic) to the experimental moments described in Sec. III C, the two sets of moments from methods M1 and M2 were individually fit, and the fit results were averaged, obtaining the central value shown by the black line in the figures. The error band, shown as a grey area, was calculated following the same procedure adopted for the experimental moments (Sec. III C). The two lowest momentum transfer bins 0.4 ≤ t ≤ 0.6 GeV 2 were excluded from the analysis because the moment reconstruction procedure was found not to be reliable in this region. In addition, the Y 10 moment was not used to extract the S-wave magnitude because the procedure could not always reproduce an accurate Y 10 moment based on tests performed on pseudo-data.

C. Partial wave amplitudes
As an example, the square of the magnitude of the S-and P partial-waves derived by fit for the momentum transfer bin 0.7 < −t < 0.8 GeV 2 are shown in Fig. 14. The S-wave threshold enhancement provides a hint of the scalar f 0 (980) or a 0 (980) states, which have been parametrized by the exchange of the ω and ρ vector mesons in the t-channel. The top and the middle plots show the partial waves summed over all helicities. The two bottom plots show the amplitudes for two possible values of M = 1, 0, the helicity of the di-kaon system. Note that we use the wave with photon helicity λ γ = +1 as a reference. Thus, M = 1 corresponds to the no-helicity flip (s−channel helicity conserving) am-plitude, which, as expected, is the dominant one, and M = 0 corresponds to unit photon helicity flip. The nonvanishing Y 22 moments show the presence of a small two unit helicity flip amplitude. By neglecting the M = −1 amplitudes, we have focused on describing the dominant structure in the Y 11 and Y 20 moments and reducing the number of fit parameters. To check sensitivity to various helicity components we performed the fit in three configurations. In the first configuration we included S-and P -wave amplitudes with vanishing photon helicity flip and unit photon helicity flip. Nucleon helicity flip amplitudes were excluded. In the second configuration, we used Regge factorization to reduce the number of independent amplitudes. Specifically, the parity relation applied to the nucleon vertex [32] reduces the number of unconstrained amplitudes by a factor of two, since S + is related to S − , P + to P − , and P 0+ to P 0− . Finally in the third configuration we used the above Regge-constrained P -wave amplitudes and we added to them the nucleon helicity flip amplitudes. In this configuration we tested if the interference signal in the moments could be described by interfering nucleon flip amplitudes by attempting to extract the nucleon helicity flip amplitudes from the Y 10 and Y 11 moments. Specifically, we added two nucleon helicity flip P −wave amplitudes f 1,1 ++− , f 1,0 ++− and one nucleon flip S−wave amplitude f 0,0 ++− . It is only necessary to consider one-half of all the nucleon flip amplitudes because the others are not independent after using the Regge factorization condition. We found that the first two configurations gave similar results, and specifically, in Figs. 7-13, we show the results obtained with the second configuration described above. In the third configuration a fit was first performed using the Y 00 and Y 20 moments to extract the dominant nucleon non-flip P −wave, while setting the nucleon flip amplitudes to zero. After fixing the strength of the non-flip P -wave in this way, we introduced nucleon flip P − and S-waves and added the Y 10 and Y 11 moments to the fit. As shown in Fig. 15 , we found the nucleon flip amplitudes cannot be large enough to significantly affect the Y 11 moment. We thus conclude that the non-flip amplitudes dominate the measured moments.

D. Differential cross sections
Differential cross section (dσ/dt) L for individual waves can be obtained by integrating the corresponding ampli- tude obtained from fits to the moments. The rCrossesults are shown in Figs. 16 and 17. All cross sections are found by integrating the mass region 1.0195 ± 0.0225 GeV. It is worth noting that the magnitudes of the S and P 0 waves found in this analysis (see Table I) are consistent with predictions (summarized in Table II) of a model constrained on a somewhat higher photon energy data [8,16,17]. The discrepancy can be explained by the different −t integration range.

E. Uncertainty evaluation
The final uncertainty was computed as the sum in quadrature of the statistical uncertainty of the fit, and two systematic uncertainty contributions: the first related to the moment extraction procedure, evaluated as the variance of the two fit results, and the second associated with the photon flux normalization estimated to be 10%. The central values and uncertainties for all of the observables of interest discussed in the next sections were derived from the fit results with the same procedure.

V. SUMMARY
In summary, we performed a partial wave analysis of the reaction γp → pK + K − in the photon energy range 3.0-3.8 GeV and momentum transfer range −t = 0.6−1.3 GeV 2 . Peripheral photoproduction of meson resonances is an important reaction to study their structure. On one side, photons have a point-like coupling to quarks, which enhances production of compact states. On the other, pion exchange amplitudes in photoproduction on the nucleon can be used to determine rate of resonance production through final state interactions. Theoretical analysis of these process are currently underway [35]. Moments of the di-kaon angular distributions, defined as bi-linear functions of the partial wave amplitudes, were fitted to the experimental data by means of an un-binned likelihood procedure. Different parametrisation bases were used and detailed systematic checks were performed to ensure the reliability of the analysis procedure. We extracted moments Y LM with L ≤ 4 and M ≤ 2 by using amplitudes with L ≤ 2 (up to P -waves). The production amplitudes have been parametrized using a Regge-theory inspired model. The P −wave, dominated by the φ(1020)meson, was parametrized by Pomeron exchange, while the f 0 (980) meson in the S-wave was described by the exchange of the ω and ρ vector mesons in the t-channel. This model also accounts for the final state interaction (FSI) of the emitted kaons. The moment Y 00 is dominated by the φ(1020) meson contribution in the P -wave, while the moments Y 10 and Y 11 show contributions of the S-wave through interference with the P -wave. The cross sections of S-and P -waves in the mass range of the φ(1020), were computed. This is the first time the t-dependent cross section of the S-wave contribution to the elastic K + K − photoproduction has been measured.

VI. ACKNOWLEDGMENTS
We would like to acknowledge the outstanding efforts of the staff of the Accelerator and the Physics Divisions at Jefferson Lab that made this experiment possible. This work was supported in part by the Chilean Comisión Nacional de Investigación Científica y Tecnológica (CONI-CYT), the Italian Istituto Nazionale di Fisica Nucleare, the French Centre National de la Recherche Scientifique, the French Commissariatà l'Energie Atomique, the U.S. Department of Energy, the National Science Founda- Two energy bin configurations were studied: a single bin with 3.0 < E γ < 3.8 GeV and two bins 3.0 < E γ < 3.4 GeV and 3.4 < E γ < 3.8 GeV. The moments were more stable for the single energy bin configuration due to larger statistics. However, the kaon-nucleon mass distributions were better reproduced using the smaller bin size. The angular moments obtained from both configurations are shown to be in good agreement in Fig. 18.

Cut on
The Λ(1520) peak in the K − p mass distribution cannot be reproduced with λ max < 4 with any of the four methods. Fig. 19 shows the fit results before cutting out the region containing the Λ(1520). This region is not a main focus of this study, so the kinematical region with M K − p < 1.6 GeV was removed from this analysis. Dalitz plots of the whole pK + K − data set before and after this cut, show that the number of events in the M K + K − region near the φ mass were not affected by this cut. Therefore, the systematic effect of this cut on the determined cross sections is negligible.
3. Sensitivity to λmax and effect of truncation to λmax = 4 Fig. 20 shows results from method M1 in which the intensity was parametrized by moments and the likelihood was maximized in one energy and t bin (3.0 ≤ E ≤ 3.8 GeV, 0.6 ≤ −t ≤ 0.7 GeV 2 ). λ max was varied from 2 up to λ max = 6. The fits became unstable as the number of free parameters increases to λ max = 6.
The λ max = 4 fit reproduced the main features of the data in the region of interest (M K + K − ≤ 1.1 GeV). We compare the helicity angles and invariant masses in Fig. 22 and Fig.  21 between data and reconstruction from the fit results (plotting the average of methods M1 and M2) for three different M K + K − intervals (M K + K − = 0.995 ± .01 GeV M K + K − = 1.0275 ± .01 GeV, M K + K − = 1.0575 ± .01 GeV). The rationale for this choice of mass regions is as follows. The first region lies to the left of the φ peak, the second is directly on the peak where the signal is dominated by the φ, and the third region is to the right of the φ peak. In the first mass region shown on the top of the figures, a large momentum transfer range (0.4 ≤ −t ≤ 1.0 GeV 2 ) was integrated over to obtain an appreciable number of events. In general, it was found as expected, that the reconstructed distributions from smaller bin sizes in t and E better reproduce the data.
The helicity angle distributions reproduced from the fits are in good agreement with the data. There is a similarity in the φ K helicity angular distributions between events in the second (M K + K − = 1.0275 ± .01 GeV) and third (M K + K − = 1.0575 ± .01 GeV) mass range. This is counterintuitive because the angular distribution for M K + K − = 1.0275 GeV resembles a P -wave signal as expected, but the angular distribution in Fig. 21, which is away from the φ peak (M K + K − = 1.0575 GeV), looks similar. We found this can be attributed to the CLAS detector acceptance and not to the presence of a large P -wave in the third mass interval. The accepted Monte Carlo events, with primary events generated from a flat phase-space distribution, also takes the same form as the data in this region due to the detector acceptance. The shape of the φ K angular distribution from the data outside of the φ meson mass region can therefore be explained by the angular dependence of the detector acceptance.
The invariant mass distributions of the data are also described well by the fit. The two regions away from the φ are shown in the top and bottom plots of Fig. 22. The kaon-nucleon mass distributions directly on the φ peak (middle plots) are consistent within one sigma, except for just a few bins.  and taking the high energy limit, s ≫ t and s ≫ M 2 K + K − , the amplitudes derived in [33] result in the following helicity partial waves, (B4) Before comparing with data we multiplied each of these amplitudes by a slowly varying function of M K + K − , with w(z) conformally mapping the complex M 2 K + K − plane cut at M 2 KK = 0 and M 2 KK = 4m 2 K onto a unit circle. coefficients a, b, and c are allowed to vary independently for each helicity amplitude.

b. S−wave
The S−wave component of the K + K − amplitude is parametrized by the double t−channel exchange of the ρ and ω vector mesons as described in [34]. In the upper meson vertex, a simple meson exchange is used, allowing  for an interaction of two produced mesons in the final state. The normal propagator (t − m 2 e ) −1 , where m e is the mass of the exchanged vector meson, was used at the nucleon vertex. Both the π + π − and K + K − channels were included in the final state interactions. The S−wave in the mass region considered is dominated by the f 0 (980) and a 0 (980) resonances. Each partial wave helicity S-wave amplitude was multiplied by the function f (M K + K − ) given in Eq. (B5), which contains three independent fit parameters.