Precise measurements of the and the neutral kaon meson masses with the KLOE detector.
Abstract
We present precise measurements of the and masses using the processes , and , . The mass measurement, MeV, is in acceptable agreement with the previous measurements but is more accurate. We find MeV. Our value is the most accurate to date and is in agreement with two recent measurements based on decays, but is inconsistent, by about 10, with a measurement of comparable precision based on production at threshold.
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1 Introduction
The meson mass has changed four times in the past 40 years while the measuring accuracy never was better than 0.15 MeV till 2002. In 2005 the GEM experiment using the reaction at threshold found MeV [1], while in 2002 the NA48 collaboration using the decay found MeV [2]. The two results above differ by about eight standard deviations. Preliminary KLOE results [3] confirm the disagreement. Recently the CLEOc collaboration found MeV [4] using decays and combining different decay modes of the .
For the mass there is good agreement between the Novosibirsk (1995) and CERN(2002) measurements that have a precision of keV [5, 6]. Our measurement, similar to that of Novorsibisk with rather increased statistics, is based on the knowledge of the meson mass which is known to 20 ppm from the Novosibirsk measurement employing the g2 depolarizing resonance method. The mass is also the basis for the mass measurement which relies on a precise determination of the collision center of mass energy, in the following. is determined run by run using events ( for each run), while the absolute momentum scale is obtained from the cross section as a function of .
2 The KLOE experiment.
KLOE operates at DAΦNE, the –factory collider running at a center of mass equal to the meson mass. Positrons and electrons collide at an angle of rad. The KLOE detector consists of a 4 m diameter, 3.2m length drift chamber, DC [7], surrounded by a lead/scintillatingfiber sampling calorimeter, EMC [8], both immersed in a axial magnetic field of 0.52 T with the axis parallel to the bisectrix of the two beam lines. The transverse momentum resolution for charged particles is . The EMC consists of a barrel and two end caps. Energy deposits in the EMC are reconstructed in the calorimeter with energy and time resolutions = 0.057/, = 54 ps / in quadrature with 140 ps. The centroid of showers is measured with resolution = 1 cm/ in the coordinate parallel to the fibers and 1 cm in the transverse coordinate.
For a photon coming from the IP the angular resolution is . Closeby energy deposits are combined into “clusters”. A prompt photon is defined as a cluster with ( is the arrival time measured at the EMC, is the distance from the interaction point and is the velocity of light) not associated to a charged particle. For this latter, we require the distance between the centroid of the cluster and the extrapolation of any track reaching the calorimeter to be larger than three times the cluster position resolution.
Only calorimeter signals are used to trigger [9] events for these analyses. We require at least two energy deposits above threshold ( MeV in the barrel and MeV in the endcap). The trigger has a large time jitter with respect to the event time but is synchronized with the collider radio frequency with an accuracy of 50 ps. The time of the bunch crossing producing an event is determined offline during event reconstruction.
The large cross section for , 3 b and for elastic scattering, Bhabha scattering, allows KLOE to collect large number of events, some 450,000 per hour. KLOE takes advantage of these events to maintain a running calibration of time and energy scales of its calorimeter, of the momentum and position resolution of the drift chamber, of the machine energy and beams crossing angle, and therefore of the center of mass motion, of the mean position of the interaction point and of the detector alignment.
3 Calibration of c.m. energy.
The centerofmass energy, , has been measured for each run by fitting the invariantmass distribution for Bhabha events to a Monte Carlo generated function, including radiative effects.
Initial state radiation (ISR), where one or both initial colliding particles radiate a photon before interacting, affects the collision centerofmass energy and therefore the final state invariant mass. ISR, which is mostly collinear to the beam, is in general not detected. MC Bhabha events were generated using the BABAYAGA event generator [10], which accounts for both final and initial state radiation.
An example of this fit is shown in Fig. 1.
In a typical run, an integrated luminosity of 50 is collected and is measured with a statistical accuracy of keV. The stability of the momentum calibration has been studied measuring the twopion invariant mass in decay. It is found to be stable to within 10 keV in the analyzed runs.
The centerofmass energy scale has been calibrated by obtaining the mass from a fit to the cross section measurements for the process . The cross section is measured at different values of around by counting the number of events, correcting for selection efficiency, background and the factor of the pair, and normalizing to the integrated luminosity. events are selected by requiring two tracks with opposite charge that form a vertex within a cylinder of 5 cm radius and 10 cm length centered on the interaction point. The invariant mass computed from the two pion tracks is required to be within 20 MeV of the nominal neutral kaon mass. The luminosity is measured by using verylargeangle () Bhabha events[11].
The measured cross section is fitted to a theoretical function [12] that depends on the parameters, takes into account the effect of ISR, and includes the interference with the and the mesons. The mass, total width, and peak cross section are the only free parameters of the fit, the and the parameters being fixed. The results of the fit to the data are shown in Fig. 2. The fitted mass is , to be compared with , measured by CMD2 at VEPP2M [13]. The ratio of these two values is used to fix the overall energy scale. The correction factor / is , corresponding to a shift in the value of of 150 keV.
4 Measurement of the neutral kaon mass.
The events offer a unique possibility to obtain a precise value of the neutral kaon mass. To obtain a crude estimate of the resolution and explain the method used we observe that if the meson is at rest the kaon mass can be extracted from the kaon momentum using the relation:
(1) 
Since MeV, measuring it at 1% level, well within the KLOE capability, results in a measurement of the mass better than %. 50,000 events are enough to reach a statistical accuracy of about 1 keV.
mesons are produced with a momentum along the axis, MeV at DAFNE. From the measured momenta of the two pions from , we measure the momentum. The momentum is given by , where is the average momentum measured with Bhabha events collected in the same runs. The center of mass energy of the pair () is related to the kaon mass , according to:
with
On the other hand the collision center of mass energy is computed from Bhabha events as described above.
Corrections due to ISR have to be taken into account when relating to . The correction function has been evaluated using a full detector simulation where the radiation from both beams has been implemented, and is reconstructed as in the data. The expression of the radiator function has been taken from Ref. [14], including corrections. The correction is very small below the resonance: about 40 keV. It increases up to about 100 keV when is above the mass. In this region radiative return begins to be important. The neutral kaon mass is then obtained solving the equation:
(2) 
The single event mass resolution is about 430 keV. Contributions to the mass resolution are: experimental resolution about 370 keV, beam energy spread about 220 keV, as measured by KLOE ( in agreement with machine theory ), and ISR about 100 keV. The kaon mass distribution for a single run is shown in Fig. 3 together with a gaussian fit to the distribution.
The source of systematic errors considered for this measurement are:

the momentum calibration;

the theoretical uncertainty on the radiator function ;

the absolute calibration of the beam energy.
The systematic error due to the momentum miscalibration has been evaluated by changing the momentum scale in computing the pion momenta. A momentum miscalibration, , translates to a miscalibration on = 0.06 , in agreement with the qualitative calculation made above. The momentum scale is obtained by using several processes covering a wide momentum range 50 to 500 MeV (), with a fractional accuracy below , in agreement with the estimate obtained using Bhabhas [15], resulting in a systematic error of 6 keV.
The systematic error coming from theoretical uncertainty of the radiator function has been evaluated considering the contribution from higher order terms in . The correction function has been evaluated by excluding the constant term in the . The corresponding change in is corresponding to a variation on of 7 keV. Further checks have been made by using the function given in Ref. [16]: no significant differences were observed. Additional systematics come from the dependence of the measured mass from the value: we compare the average of the measurements with data collected at MeV with data at MeV, where the value of is more than a factor two larger. The difference between the two mass values is keV, consistent with zero.
Other sources of systematics are due to the uncertainties on the W calibration, i.e., the statistic and systematic error on and on obtained from our fit. The total contribution from these sources amounts to a mass uncertainty of 15 keV. Systematic uncertainties are treated as uncorrelated. The result is:
(3) 
5 The mass
The decay , for meson at rest, is a source of monochromatic mesons of 362.8 MeV momentum, recoiling against a photon of equal momentum. Detection of such a photon signals the presence of an meson. Photons from have a flat spectrum in the range MeV in the laboratory frame. In the laboratory, the opening angle of the two photons has a distribution peaked at its minimum value of 113 . KLOE measures this angle with an accuracy of 0.4. The value of the minimum angle is a function of the mass and its measurement determines the mass with a resolution of 2 MeV, without energy measurements. In fact we do measure the photon energies. The mass accuracy is however ultimately due to the accurate measurement of the photon angles. Together with the stability of the continuously calibrated detector and the very large sample of mesons collected we have been able to obtain a very accurate measurement of the mass [17].
Events are selected requiring at least three energy clusters in the barrel calorimeter with polar angle . A kinematic fit imposing energymomentum conservation is performed. The fitted photon energy resolution is vastly improved over the EMC measurement because of the good angular resolution. The kinematic fit uses the value of the total energy, the transverse momentum and the average value of the beambeam interaction point; these values are determined with good precision run by run by analyzing elastic scattering events. Fig. 4 shows the of the kinematic fit for the data and for Monte Carlo [15] simulated signal events. If more than three photons are selected, the combination with the lowest is chosen. Events with are kept for the analysis.
Fig. 5 shows the , Dalitz plot population, with the energies ordered as . The , and bands are clearly visible. We apply a cut GeV, “backgroundrejection cut” in the following, shown by the line in Fig. 5. Events below the line are retained for the analysis. The resulting distribution, for a data subsample, is shown in Fig. 6,top. The distribution in the 542.5 to 552.5 interval is fitted well with a single gaussian with MeV as shown in Fig. 6,bottom.
To estimate systematic uncertainties we have studied the effects of the detector response and alignment, event selection cuts, kinematic fit and beam energy calibration that can influence our measurement. The values of the systematic errors are summarized in Table 1.
First, to check the effect of the interaction point and the alignment of the calorimeter relative to the drift chamber, we have selected a high purity sample of events [18]. The average position of the interaction point, determined run by run with events, has been compared with the average position of reconstructed vertex . The difference between the two values was computed run by run and the rms of these points () was used to evaluate the systematic error introduced in the kinematic fit by varying the IP position by . To check for misalignments between the calorimeter and the drift chamber, each pion track was extrapolated to the calorimeter and compared with the centroid of the cluster. A small correction of 1.1 mm along the vertical coordinate, , and of 2.0 mm along the longitudinal coordinate, , was applied. The rms () of the mean difference between the coordinates of the extrapolated point and the cluster centroid was used to compute the systematic uncertainty on the mass by shifting the photon point of arrival at the EMC by .
The energy scale of the calorimeter response and its linearity are checked using two different samples of and events. The energy of the photon, determined from the track momenta and the average value of , was compared with the calorimeter cluster energy. The calorimeter energy scale was calibrated to better than 1% and the response is linear to better than 2% in the range of interest. The systematic effect on the twophoton invariant mass is 4 keV from the energy scale miscalibration and 4 keV from the nonlinearity. The values above confirm that the mass measurement has little sensitivity to the calorimeter energy response.
It is however important to check the correctness of the position measurement in the 24 calorimeter modules of the barrel. We compute the twophoton invariant mass for different orientations of the plane. The rms width of the mass distribution was assumed as systematic error: 10 keV and 15 keV respectively for variations of the polar and azimuth angle of the normal to the plane.
Systematic effects from event selection criteria were studied by varying the cut (Fig. 4) and the backgroundrejection cut of Fig. 5. The first has no influence ( keV) on the result; the second introduces a systematic error of 17 keV determined by translating the solid line in Fig. 5 and rotating as shown by the dashed line in the same figure. The mass value is very sensitive to the center of mass energy of the system used in the kinematic fit. Due to initial state radiation emission (ISR) the available center of mass energy is a bit lower than the value computed from the nominal energy of the beams. The effect has been studied with a detailed Monte Carlo simulation of the events in the detector, and a shift of 100 keV was found for the mass measurement. Since this correction is relatively large, we have checked the MC correction due to ISR emission also for runs taken at different values of . The data were divided in eight energy bins; moreover, two offpeak energy bins, centered at = 1017 and 1022 MeV, were also analyzed in the same way as the peak data. Fig. 7, top shows the shift of the mass evaluated by MC as a function of together with the measured shift of the mass respect to the value obtained at MeV. Fig. 7, bottom shows the value of the mass corrected for the ISR effect. The rms of these points is used as systematic error (8 keV).
The value of the mass was measured with the same method fitting the low mass region of Fig. 6 and the ratio was also determined. All systematic effects discussed above were also evaluated for the mass of the and for the ratio ; the corresponding values are listed in Table 1.
Systematic effect  (keV)  (keV)  R () 

Vertex position  4  6  19 
Calorimeter energy scale  4  1  6 
Calorimeter nonlinearity  4  11  31 
angular uniformity  10  44  120 
angular uniformity  15  12  37 
cut  1  4  13 
Backgroundrejection cut  17  4  18 
ISR emission  8  9  28 
Total  27  49  136 
Finally, the stability of the results as function of running conditions was checked by dividing the data set in eight different periods and determining the values of , and for each period. The results are shown in Fig. 8 and in Table 2 together with their statistical significance. Fit to a common value are good.
Result of the fit  /d.o.f.  C.L. %  

547,791 7 (keV)  6.9/7  44 %  
134,886 12 (keV)  7.7/7  36 %  
4.0610 0.0004  8.9/7  26 % 
The constraint on the total momentum of photons in the kinematic fit is very effective in reducing the error on the twophoton invariant mass. The absolute scale of is determined using the CMD2 value as in the mass measurement section.
Rescaling the values of Table 2 for the ratio we obtain
(4) 
(5) 
the mass value is in agreement with the world average [19] within 1.4.
As a check of this result, we can use the measured ratio
(6) 
and the world average value of the mass, MeV [19] to derive MeV, consistent with the results quoted above although affected by a larger systematic error due to a worse cluster position reconstruction of the two photons from decays which have a lower energy.
6 Conclusions
Our mass measurement is in acceptable agreement with the previous measurements shown in Table 3, but more accurate. Averaging [5, 6] and our result we obtain MeV.
Experiment  Method  (MeV)  events  

CMD  [5]  497.661 0.033  3713  
NA48  [6]  497.625 0.001 0.031  665 k  
KLOE  497.583 0.005 0.020  35 k 
Our measurement of the (eq. 5) mass is the most accurate result today. It is in good agreement with the recent measurements based on decays listed in Table 4. Averaging the mass values from [2, 4] and our result we obtain MeV, a value different by 10 from the average of the measurements done studying the production of the meson at threshold in nuclear reactions.
7 Acknowledgements
We thank the DAFNE team for their efforts in maintaining low background running conditions and their collaboration during all datataking. We want to thank our technical staff: G.F. Fortugno and F. Sborzacchi for their dedicated work to ensure an efficient operation of the KLOE Computing Center; M. Anelli for his continuous support to the gas system and the safety of the detector; A. Balla, M. Gatta, G. Corradi and G. Papalino for the maintenance of the electronics; M. Santoni, G. Paoluzzi and R. Rosellini for the general support to the detector; C. Piscitelli for his help during major maintenance periods. This work was supported in part by EURODAPHNE, contract FMRXCT980169; by the German Federal Ministry of Education and Research (BMBF) contract 06KA957; by Graduiertenkolleg ‘H.E. Phys. and Part. Astrophys.’ of Deutsche Forschungsgemeinschaft, Contract No. GK 742; by INTAS, contracts 96624, 9937; and by the EU Integrated Infrastructure Initiative HadronPhysics Project under contract number RII3CT2004506078.
Footnotes
 Corresponding author: B. Di Micco, email dimicco@fis.uniroma3.it, M. Dreucci, email marco.dreucci@lnf.infn.it
 footnotemark:
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