Measurement of e+e- ->phi ->K+K- cross section with the CMD-2 detector at VEPP-2M Collider

The process e+ e- ->phi ->K+ K- has been studied with the CMD-2 detector using about 542 000 events detected in the center-of-mass energy range from 1.01 to 1.034 GeV. The systematic error of the cross section is estimated to be 2.2%. The phi(1020) meson parameters in the phi ->K+K- decay channel have been measured: sigma_0(phi->K+K-) = 2016 +- 8 +- 44 nb, m(phi) = 1019.441 +- 0.008 +- 0.080 MeV/c2, Gamma(phi) = 4.24 +- 0.02 +- 0.03 MeV, B(e+e-)B(K+K-) = (14.27 +- 0.05 +- 0.31)*10(-5).


Introduction
A study of the process e + e − → K + K − is of interest for a number of physical problems.Since the K + K − final state is the main φ(1020) meson decay chan-nel, the resonance parameters can be obtained by measuring the cross section of the process in the energy range around the φ(1020) meson mass [1,2].The isovector part of the e + e − → K K cross section (both K + K − and K 0 L K 0 S final states should be considered) can be related to the τ − → K − K 0 ν τ decay by using conservation of vector current (CVC) [3].Finally, the process under study is used in the calculation of the hadronic contribution to the muon anomalous magnetic moment [4].In view of the increasing experimental accuracy in the measurement of this quantity [5], any significant contribution like that from the process e + e − → K + K − should be measured with adequate precision.
At the energy around the φ(1020) meson mass low momenta kaons from the process e + e − → K + K − have large probabilities for a nuclear interaction, decays in flight and kaon stop in a thin layer of the detector material.That introduces large uncertainties in the detection efficiency and increases systematic errors in the cross section.Earlier measurement of the cross section performed by the CMD-2 collaboration [1] at the VEPP-2M collider [6], was based on a relatively small data sample and had a systematic accuracy about 4%.The SND collaboration [2] used significantly larger statistics to study the reaction e + e − → K + K − .The experiment was based on the integrated luminosity of 8.5 pb −1 , but the accuracy of the cross section was limited by systematic errors estimated to be 7.1%.
In this work we report a measurement of the e + e − → K + K − cross section based on 1.0 pb −1 of data collected with the CMD-2 Detector [7] at the VEPP-2M collider from 1.01 to 1.034 GeV center-of-mass (E c.m. = √ s) energy.A special procedure to extract the detection efficiency from data is developed and the systematic uncertainty on the cross section is estimated to be 2.2%.

Detector and experiment
The CMD-2 detector has been described in detail elsewhere [7].The detector tracking system consists of the cylindrical drift chamber (DC) [8] surrounding the interaction point, and proportional Z-chamber (ZC) [9] for a precise measurement of polar angles, both also used as a charged trigger.Both chambers are inside a thin (0.38 X 0 ) superconducting solenoid [10] with a field of 1 T. The barrel electromagnetic calorimeter [11] is placed outside the solenoid and consists of 892 CsI crystals.The muon-range system [12] of the detector, also located outside the solenoid, is based on streamer tubes.The end-cap electromagnetic calorimeter [13] based on the 680 BGO crystals makes the detector almost hermetic for photons.In this experiment we require a charged-trigger signal from at least one charged track and any (>20 MeV) energy deposition in the barrel electromagnetic calorimeter.
The data sample used in the analysis was collected in two scans of the centerof-mass energy range 1.01 -1.034 GeV.In the scans the beam energy was increased from 505 MeV to 517 MeV with a 0.5 MeV step.To determine the detection efficiency, we simulated 50000 events [14] of the process e + e − → K + K − (γ) at each energy point.

Event selection
A candidate to a e + e − → K + K − event is an event with two low-momentum tracks and high ionization losses, originating from the interaction region.There is a number of effects leading to the loss of a charged-kaon track: decays in flight, nuclear interactions, track reconstruction inefficiency etc.If one track is not reconstructed, the event can still be identified using a second detected track.Using single-track events to study detection efficiency we can significantly reduce various systematic errors.In our analysis we select events with one or two "good kaons" found, where a "good kaon" is defined according to the following criteria: Figure 1 shows a scatter plot of the track ionization losses vs. track total momentum for all two-track events.Lines show the boundaries of applied selections which allow to separate events with charged kaon(s) from other reactions.The distribution of the track impact parameter in the R − ϕ plane is shown in Fig. 2 for the remaining events.The vertical arrow shows the applied selection.
The number of events with one or two "good" kaons found is determined from the distribution of a Z-coordinate of the point closest to the interaction region along the beam axis.Figure 3 demonstrates the Z-coordinate distribution for events with one "good kaon".A background from the beam-gas and beam-pipe interactions producing low-momentum protons or pions is clearly seen.This background contributes about 15% to a sample of one "good" kaon events and is significantly smaller (0.4%) if both tracks are identified as "good kaons".
To extract the number of signal events, the distribution is fitted to a sum of a Gaussian function describing the interaction region and a smooth function describing the background.The shape of the background distribution was derived from the analysis of events collected at the energy point below the threshold of charged kaon pair production.The Z-coordinate distribution obtained at the center-of-mass energy of 0.984 GeV is shown in Fig. 4 and is fitted to a sum of three Gaussian functions.The obtained values of the fitting parameters but the number of background events are then used for the background description at each energy point.The background is relatively small, so that variations of these parameters or the alternative description of the background with the "flat" distribution do not change the final number of signal events by more than 0.4%.
After background subtraction we select 178932±432 events with one "good kaon" and 363490±604 events with two "good kaons".The number of selected events for each energy point is presented in Table 1.By varying the selection criteria we estimate the systematic error on these numbers as 1.4%.

Cross section
At each energy point the e + e − → K + K − cross section is calculated according to the formula: where N 1 and N 2 are the numbers of events with one or two "good" kaons, ε is the detection efficiency obtained from the MC simulation [14] with some corrections from data, L is the integrated luminosity calculated with a 1% accuracy using events of large angle Bhabha scattering [15] and (1 + δ rad ) is the correction for initial-state radiation determined with a 0.5% accuracy according to Ref. [16].Z 0 , cm Fig. 3.The distribution of the Z-coordinate of the point closest to the interaction region for events with one "good" kaon.The curve shows the result of the fit described in the text.The detection efficiency is determined from the following formula: where the acceptance ε geom is calculated as the ratio of the number of events passing the selection criteria to the initial number of MC simulated events, ε TF • ε CsI is the product of the charged-trigger efficiency and a probability to have energy deposition in the CsI calorimeter.
The number of events with one "good" kaon is about 50% of that with two "good" kaons.Therefore, using the sum of events with one and two "good" kaons we increase the detection efficiency and decrease the uncertainty due to an incorrect description of the track losses in the MC simulation.In Eq. 2 we introduce ∆ EXP and ∆ SIM as the corrections describing the losses of both charged kaons for experimental and MC simulated events, respectively, taking into account a different probability of nuclear interaction as well as the different number of hits for tracks of positive and negative kaons.We found no statistically significant difference in the losses of positive and negative kaons due to the effects mentioned above.For example, from the N 1 and N 2 values in Table 1 at √ s = 1020.1 MeV and assuming no correlations, we estimate a probability to lose both kaons to be ∆ EXP = 0.035 in good agreement with the MC simulation.The difference in ∆ EXP and ∆ SIM at all energy points does not exceed 0.7% and this value is taken as an estimate of the systematic error in the acceptance calculation.
The charged-trigger efficiency (ε TF ) was estimated to be 0.920 ± 0.003 using a special computer code simulating CMD-2 trigger [9].The difference between the trigger efficiency for experimental and Monte Carlo events is 1% and is used as an estimate of the corresponding systematic error.
The positive trigger decision also requires the presence of at least one cluster in the CsI calorimeter with the energy deposition greater than 20 MeV.The efficiency ε CsI is calculated in a similar way and is 0.970 ± 0.001.
The total calculated efficiency for each energy point is listed in Table 1.The beam energy at each point was determined using a procedure described in detail in [15].
The total systematic error on the cross section is estimated to be equal to 2.2% obtained by adding in quadrature contributions from various sources listed in Table 2.
) is the propagator of the vector meson V , g V γ is a constant describing the coupling of the meson V with a photon and g V KK -coupling constant of the meson V with a K + K − pair, ψ V is the phase.The coupling constants g V γ g V KK are related to the product of the branching fractions B(V → e + e − )B(V → K + K − ) according to: Since the ρ(770) and ω(782) mesons are below the K + K − pair production threshold, we calculate B(ρ, ω → K + K − ) using the corresponding relation from simple quark model (see, for example, Ref. [17]): The phases ψ φ and ψ ω are equal to π according to SU(3).If the phase ψ ρ is a free parameter of the fit, its obtained value is consistent with π in agreement with simple quark model [17].We fixed ψ ρ at π while determining the φ meson parameters.A K + K − is a constant complex amplitude describing possible contributions from excited vector states.The energy dependence of the total width for a meson V is chosen as in Ref. [18].The function Z(s) given by the relation s describes the Coulombian interaction of charged kaons in the final state [19].
The product B(φ → e + e − )B(φ → K + K − ) is related to the peak cross section σ(φ → K + K − ) according to the formula: and this parameter along with the φ meson mass and total width is determined from the fit: σ(φ → K + K − ) = 2016 ± 8 ± 44 nb, m φ = 1019.441± 0.008 ± 0.080 MeV/c 2 , Γ φ = 4.24 ± 0.02 ± 0.03 MeV.And from the other fit, where instead of the peak cross section we have a product of the branching fractions as a free parameter, we obtain: where the first error is statistical and the second is systematic.If we keep A K + K − as a free parameter, it is consistent with zero and we fixed it at this value while determining the φ meson parameters.
The systematic error in φ meson mass and total width is dominated by the accuracy of the beam energy determination described in Ref. [15].
The values of m φ and Γ φ obtained in this work are strongly correlated with the corresponding values obtained in our analysis of the neutral kaon pair production in Ref. [21] because they are based on almost the same data sample and therefore should be not averaged together.The values of both φ meson mass and width obtained in this analysis agree with the world average values and that for the width is more precise.
The parameter B e + e − • B K + K − is in good agreement with the world average value [20] (14.60 ± 0.33) × 10 −5 and has the same accuracy.
In Fig. 5 we show the energy dependence of the cross section obtained in this work as well as the results of the most precise previous experiments [1,2].The results of all experiments are in good agreement.

Discussion
Significant improvement of the systematic accuracy of the cross section (from 4% to 2.2%) is achieved due to additional analysis of events with only one charged kaon.It allows to take into account a possible difference of nuclear interactions, decays in flight and reconstruction efficiency of the charged kaons in data and MC simulation.The trigger efficiency is also extracted directly from the data and is in good agreement with the MC simulation.
Using this precise measurement we recalculate the contribution of the reaction e + e − → φ → K + K − to the hadronic part of the theoretical prediction for the anomalous magnetic moment of muon.It can be calculated via the dispersion integral [22]: where K(s) is the QED kernel [23], R(s) denotes the ratio of the "bare" crosssection for e + e − annihilation into hadrons to the muon pair cross section.Using data on the e + e − → φ → K + K − cross section from [1,2]

Conclusions
Using a data sample of 5.42×10 5 reconstructed events with one or two reconstructed charged kaons collected at CMD-2, the most precise measurement of the cross section of the reaction e + e − → φ → K + K − has been performed.The estimated systematic error in the cross section is 2.2%.The following φ meson parameters have been determined: The obtained results agree with and are more precise than the results of other measurements.

Fig. 4 .
Fig. 4. The distribution of the Z-coordinate of the point closest to the interaction region for background events.The curve shows the results of the fit described in the text.

Fig. 5 .
Fig. 5. (Top) The deviations of the measured cross section from the fitting curve.(Bottom)The experimental cross section of the reaction e + e − → φ → K + K − obtained in the present analysis (squares), earlier CMD-2 experiment[1] (circles) and SND experiment[2] (triangles) one obtains the following average K + K − contribution to a had,LO µ in the c.m. energy range √ s = 1.011 -1.055 GeV: (15.28±0.16±0.78)•10−10 .From the results of the present work the new value of the K + K − contribution to a had,LO µ in the same energy range is (15.53±0.15±0.33)•10−10 .It agrees with the previous one and is more precise.

Table 1
The number of events, integrated luminosity, detection efficiency, radiative correction, cross section of the e + e − → φ → K + K − process.

Table 2
Contributions to the systematic error of the e + e − → K + K − cross section