Measurement of the $\pi^0$ electromagnetic transition form factor slope

The NA62 experiment collected a large sample of charged kaon decays in 2007 with a highly efficient trigger for decays into electrons. A measurement of the $\pi^0$ electromagnetic transition form factor slope parameter from $1.11\times10^{6}$ fully reconstructed $K^\pm \to \pi^\pm \pi^0_D, \pi^0_D \to e^+ e^-\gamma$ events is reported. The measured value $a = (3.68 \pm 0.57)\times10^{-2}$ is in good agreement with theoretical expectations and previous measurements, and represents the most precise experimental determination of the slope in the time-like momentum transfer region.


Introduction
The Dalitz decay π 0 D → e + e − γ with a branching fraction of B = (1.174±0.035)% [1] proceeds through a π 0 → γγ * process with an off-shell photon converting into an e + e − pair. The π 0 electromagnetic transition form factor (TFF) describes the deviation of this transition from a point-like interaction. It is an input to the computation of the π 0 → e + e − decay rate [2], as well as the hadronic light-by-light scattering contribution to the muon anomalous magnetic moment (g − 2) µ which at present contributes the second largest uncertainty on its Standard Model value [3]. The commonly used kinematic variables are defined in terms of the e ± and π 0 four-momenta (p e ± , p π 0 ) as , with the allowed kinematic region defined as where m e and m π 0 are the corresponding PDG [1] masses, and M ee is the invariant mass of the e + e − pair. The differential decay width reads [4] where Γ(π 0 2γ ) is the π 0 → γγ decay width, the function δ(x, y) describes the radiative corrections and F(x) is the electromagnetic transition form factor of the π 0 to a real and virtual photon. The function F(x) is expected to vary slowly in the kinematic region of the π 0 D decay and is usually approximated by a linear expansion F(x) = 1 + ax, where a is the slope parameter. The vector meson dominance (VMD) model [5,6] predicts a π 0 TFF slope value of a ≈ 0.03, in agreement with further theoretical estimates [7][8][9][10].
The TFF slope has been determined in the time-like momentum transfer region by measuring the π 0 D decay rate [11][12][13][14][15], all including radiative corrections. The TFF has been measured in the space-like momentum transfer region in the reaction e + e − → e + e − π 0 , where the π 0 is produced by the fusion of two photons radiated by the incoming beams and decays to two detected photons [16]. The current world average a = 0.032±0.004 [1] is obtained from time-like measurements [12][13][14] and the extrapolation of space-like data [16] using a VMD model.
The NA62 experiment at the CERN SPS collected in 2007 a large sample of charged kaons decaying in flight in vacuum with a minimum-bias trigger configuration [17]. The K ± decays represent a source of tagged neutral pions; the K ± → π ± π 0 (K 2π ) decay channel accounts for 63% of π 0 production. The mean free path of the neutral pion in the NA62 experimental conditions is negligible (few µm). This letter reports a model-independent measurement of the π 0 TFF slope parameter from an analysis of 1.11×10 6 K 2π decays followed by the prompt π 0 D decay (denoted K 2πD ) using the full NA62 2007 data set.

Beam and detector
The NA62 experimental setup used in 2007 was composed of the NA48 detector [18] and a modified beam line [19] of the earlier NA48/2 experiment.
The beam line was designed to provide simultaneously K + and K − beams. The primary 400 GeV/c proton beam delivered by the SPS impinged on a beryllium target of 40 cm length and 0.2 cm diameter. The secondary beam momenta were selected by magnets in a four dipole achromat and a momentum-defining slit incorporated into a beam dump. This 3.2 m thick copper/iron block provided the possibility to block either of the K + or K − beams. The selected particles had a central momentum of 74 GeV/c with a spread of ±1.4 GeV/c (rms). The beams were focused and collimated before entering a 114 m long cylindrical vacuum tank containing the fiducial decay volume. The beams were mostly composed of π ± , with a K ± fraction of approximately 6%. Since the muon halo sweeping system was optimised for the positive beam in 2007, most of the data were recorded with the single K + beam to reduce the halo background. The K + and K − beams were deflected horizontally by a steering magnet at the entrance of the fiducial decay volume at angles of ± (0.23 to 0.30) mrad with respect to the detector axis, to compensate for the opposite ∓ 3.58 mrad deflection by the downstream spectrometer magnet. The polarities of those magnetic fields were regularly simultaneously reversed to reduce the effects caused by an asymmetry in the detector acceptance.
The momenta of charged particles were measured by a spectrometer composed of four drift chambers (DCH) and a dipole magnet placed between the second and third chamber providing a horizontal transverse momentum kick of 265 MeV/c to singly-charged particles. The measured momentum resolution was σ p /p = 0.48% ⊕ 0.009% · p, where the momentum p is expressed in GeV/c. The spectrometer was housed in a tank filled with helium at nearly atmospheric pressure, separated from the decay volume by a thin (3×10 −3 X 0 ) Kevlar TM window.
The photons were detected and measured by a liquid krypton (LKr) electromagnetic calorimeter, which is a quasi-homogeneous ionisation chamber with an active volume of 6.7 m 3 of octagonal cross-section and a thickness of 127 cm, corresponding to 27 X 0 . The LKr volume is divided into 13,248 cells of about 2 × 2 cm 2 cross section without longitudinal segmentation. The measured energy resolution was σ E /E = 3.2%/ √ E ⊕ 9%/E ⊕ 0.42%, and the spatial resolution for the transverse coordinates x and y was 0.42 cm/ √ E ⊕ 0.06 cm, where the energy is given in GeV in both cases.
A scintillator hodoscope (HOD) was located between the spectrometer and the LKr calorimeter. It consists of a set of scintillators arranged into a plane of 64 vertical counters followed by a plane of 64 horizontal counters. Each plane was divided into four quadrants of 16 counters providing a fast trigger signal for charged particles.

Data sample and trigger logic
The analysis is based on the full data set collected during 4 months in 2007, corresponding to about 2×10 10 K ± decays in the vacuum tank. A total of 65% (8%) of the K + (K − ) flux was collected in single-beam mode while the remaining 27% were collected with simultaneous K ± beams with a K + /K − flux ratio of 2.0. During part of the data taking (55% of the K ± flux), a 9.2 X 0 thick transverse horizontal electron absorber lead (Pb) bar was installed between the two HOD planes, approximately 1.2 m in front of the LKr calorimeter, to study muon-induced electromagnetic showers [17]. A total of 11 rows of LKr calorimeter cells were shadowed by the bar, corresponding to about 10% of the total number of cells.
The 100 kHz kaon decay rate in the vacuum volume during the spill enabled the use of a minimum-bias trigger configuration with a highly efficient trigger chain optimised to select events with at least one electron (e ± ) track.
The low level hardware trigger required a coincidence of hits in at least one hodoscope quadrant in both planes (the Q 1 condition), upper and lower cuts on the hit multiplicity in the drift chambers (the 1-track condition), and a minimum total energy deposit of 10 GeV in the LKr calorimeter (the E LKr condition). The high level software trigger (HLT) condition required at least one track with 5 GeV/c < p < 90 GeV/c and E/p > 0.6, where E is the energy reconstructed in the calorimeter and p is the momentum reconstructed in the spectrometer. Downscaled minimum bias trigger streams were collected to evaluate the trigger efficiencies.
3 Simulated samples and π 0 D decay simulation Monte Carlo (MC) simulations of the K 2πD decay chain and two other K ± decay chains producing π 0 Dalitz decays, K ± → π 0 D e ± ν and K ± → π 0 D µ ± ν (denoted K e3D and K µ3D , respectively), were performed with a π 0 TFF slope a MC = 3.2×10 −2 . Separate simulated samples, proportionally to the number of kaon decays recorded, were produced for each data taking condition. The total simulated sample amounts to 386 M K 2πD , 105 M K µ3D and 103 M K e3D events within the 97 m long fiducial decay region. All these modes contribute to the π 0 D sample, although the selection is optimized for K 2πD .
The radiative corrections to the total [4] and differential [20][21][22] π 0 D decay widths have been studied extensively. They have to be considered for the TFF measurement since their effect on the x spectrum is comparable to the effect of the TFF. The calculation of the radiative corrections [22] implemented in the MC simulation of the π 0 D decay for the present analysis includes real photon emission from the π 0 D decay vertex. It also includes the one-photon irreducible contribution, neglected in earlier studies, which has an effect of |∆a| ≈ 0.5×10 −2 on the slope of the x spectrum. Higher order correction terms not included in the simulation contribute to the slope by |∆a| < 0.01×10 −2 , which is considered as a systematic uncertainty (Table 1).

Data analysis 4.1 Event reconstruction and selection
Hits and drift times in the DCH and a detailed map of the magnetic field are used to reconstruct track directions and momenta. Three-track vertices are reconstructed by a Kalman filter algorithm extrapolating track segments from the upstream part of the spectrometer into the decay volume, taking into account multiple scattering in the helium and the Kevlar window, the Earth's magnetic field and residual vacuum tank magnetization. The reconstructed K ± → π ± π + π − invariant mass and the missing mass in the K ± → µ ± ν decay are monitored and used for fine calibration of the spectrometer momentum scale and DCH alignment. Clusters of energy deposition in the LKr calorimeter are found by locating maxima in space and time in the digitized pulses from individual cells. Reconstructed energies are corrected for energy outside the cluster boundaries, energy lost in isolated inactive cells (0.8% of the total number), sharing of energy between clusters, and non-linearity for clusters with energy below 11 GeV. Electrons produced in K ± → π 0 e ± ν decays are used to calibrate the energy response.
The main K 2πD selection criteria are the following.
• The event should contain exactly one reconstructed 3-track vertex, which should be located within the fiducial decay region and be geometrically compatible with a beam kaon decay. The vertex charge q vtx , defined as the sum of the track charges, should match the beam charge in the single-beam mode. Otherwise it should satisfy a relaxed condition |q vtx | = 1.
The track with the charge opposite to q vtx is necessarily an e ± candidate, while the samesign tracks can be either π ± or e ± candidates.
• The tracks are required to be in time (within 25 ns of the trigger time and 15 ns of each other), and within the geometrical acceptance of the drift chambers. The allowed track momentum range is (2-74) GeV/c, excluding the low momentum range where a 2% deficit of data with respect to MC simulation is seen. Events with a photon converting into an e + e − pair in the material in or in front of DCH1 (Kevlar window, helium) are suppressed by requiring a minimum distance of 2 cm between the impact points of every track pair in the first drift chamber, as verified by simulation of K 2π decays.
• Reconstructed clusters of energy deposition in the LKr calorimeter are used to identify photon candidates. A photon candidate cluster should be geometrically isolated from the track impact points in the LKr calorimeter (d t > 20 cm from the same-sign tracks and d t > 10 cm from the remaining track), within 10 ns of each track and with more than 2 GeV of energy. The photon 4-momentum is reconstructed assuming that the photon originates from the same vertex as the tracks. If more than one photon candidate is found, the event is rejected.
• The total reconstructed momentum should be compatible with the beam momentum, in the range (70-78) GeV/c, and there should be no missing transverse momentum with respect to the beam axis within the resolution: p 2 t < 10 −5 (GeV/c) 2 . Particle identification using an E/p ratio is not required thanks to the low background in the sample, reducing the systematics associated to the misidentification and increasing the K 2πD acceptance by more than a factor of two. The π/e ambiguity for the two same-sign tracks is resolved by testing the two possible mass assignments. For each hypothesis, the reconstructed kinematic variables should be |x|, |y| < 1, and the reconstructed e + e − γ and π ± π 0 masses should be close to the nominal ones: M eeγ in the range (115-145) MeV/c 2 and M ππ in the range (460-520) MeV/c 2 . Only events with a single valid hypothesis are selected. The probability of correct (incorrect) mass assignment evaluated with the K 2πD MC sample is 99.62% (0.02%). The remaining 0.36% of events have either zero or two valid hypotheses and are rejected.
• The trigger conditions described in Section 2 are reproduced on simulated samples. To eliminate edge effects due to different calibration and resolution between the trigger and the offline analysis, tighter variants of the trigger criteria are applied to both data and MC samples. The offline E LKr condition requires a minimum of 14 GeV of electromagnetic energy in the LKr calorimeter summed over the reconstructed photon and e ± clusters. The offline condition corresponding to the HLT requires at least one track whose impact point on the LKr calorimeter front plane is within its acceptance and not behind the Pb bar, p > 5.5 GeV/c and E/p > 0.8, effectively requesting that at least one e ± track is detected in the calorimeter.
• A 1% deficit in the data/MC ratio is seen for events with x < 0.01 due to the steeply falling acceptance. For this reason the signal region is defined as x > 0.01, equivalent to M ee > 13.5 MeV/c 2 .
The selected K 2πD sample amounts to 1.11×10 6 events. The overall acceptances of the selection evaluated with MC simulations are 1.90% for K 2πD decays, 0.02% for K µ3D decays and 0.01% for K e3D decays. The K 2πD acceptances for periods with and without the Pb bar installed are 1.64% and 2.23%, respectively.
The reconstructed π ± π 0 and e + e − γ invariant mass spectra are shown in Fig. 1; the mass resolutions obtained from a Gaussian fit are 3.7 MeV/c 2 and 1.5 MeV/c 2 (rms), respectively. The reconstructed spectrum of the x variable and the acceptances for the decay channels considered are shown in Fig. 2. The e + e − mass resolution determined from the K 2πD MC sample can be approximated by σ ee = 0.9% · M ee , which translates into the resolution on the x variable as σ x = 1.8% · x.

Fit procedure
A χ 2 fit with free MC normalisation in equally populated bins comparing the data and MC reconstructed x distributions is performed to extract the TFF slope. A number of slope hypotheses a h are tested by reweighting a single set of MC events simulated with a slope a MC = 3.2×10 −2 using the weights where x MC is the true x value for each event. The minimization of the χ 2 test statistics yields the following result: where the uncertainties are statistical due to the limited data and MC sample sizes. The fit gives χ 2 /ndf = 54.8/49, which has a p-value of 26.4%. The fit result is illustrated in Fig. 3.

Calibration, resolution and beam simulation
The spectrometer momentum scale modifies proportionally the x variable. The corrections applied to the momentum calibration have a typical relative size of the order of 10 −3 . The sensitivity of the fit to a residual miscalibration has been assessed conservatively by turning the corrections off, leading to a shift of the fit result of ∆a = −0.16×10 −2 considered as the  systematic uncertainty on the spectrometer calibration. A similar procedure is applied for the chamber misalignment correction with no significant effect on the fit result.
The spectrometer mass resolution has been evaluated separately for individual data-taking periods using samples of K ± → π ± π + π − decays. The maximum relative difference observed on the resolution of the reconstructed squared 3-pions mass between data and MC is 2%. Scaling the MC resolution of the x variable by 0.98 results in a shift of ∆a = 0.05×10 −2 , which is considered as a systematic uncertainty.
The corrections applied to the energies measured in the LKr calorimeter affect the TFF slope result indirectly through the photon selection acceptance. A correction for the non-linearity in the energy response in the data sample with an alternative function is used to evaluate the sensitivity to the correction function, resulting in a shift of ∆a = 0.03×10 −2 . A global photon energy scaling factor of 1.001, which is the typical size of the energy corrections, applied only in the MC sample causes a shift of ∆a = 0.02×10 −2 . The overall systematic uncertainty due to the LKr energy calibration is assigned as the sum of these two effects in quadrature: The beam momentum is simulated according to the central value measured separately for different data taking periods from fully reconstructed K ± → π ± π + π − decays. A remaining discrepancy between data and MC in the tails of the beam momentum spectrum affects the TFF slope measurement through the K ± momentum dependence of the acceptance. After applying a correction to improve the spectrum data/MC agreement, the measured slope shifts by ∆a = 0.03×10 −2 , which is considered as a systematic uncertainty.

Trigger efficiency
The efficiencies of individual components of the signal trigger chain have been measured using control data samples collected via alternative trigger chains. Since no inefficient events have been found, upper limits on the inefficiencies at 90% CL have been evaluated for each trigger conditions: 0.06% (Q 1 ), 0.10% (1-track), 0.03% (E LKr ) and 0.03% (HLT).
Possible systematic effects caused by each trigger condition have been investigated separately by removing potentially inefficient events either from the data or the MC sample. The Q 1 efficiency is modeled by introducing fully inefficient gaps between the HOD quadrants with 0.2 mm width tuned using data/MC comparison in other decay channels. This leads to a Q 1 inefficiency of 0.02% for K 2πD events. Energetic photons may initiate showers by interacting with the beam pipe material, causing the DCH hit multiplicities to exceed the limits allowed by the 1-track trigger condition. The sensitivity to this effect is tested by removing from the MC K 2πD sample 0.10% of events with a radiative photon with an energy above 0.5 GeV traversing the beam pipe. For the E LKr and HLT triggers, events closest to failing a trigger condition are removed from the data sample. Those are events with the lowest reconstructed energy in the LKr calorimeter for the E LKr condition, and events with the lowest maximum track E/p ratio for the HLT condition. In both cases the fraction of removed events is equal to the upper limits on inefficiencies quoted above. The only sizeable change in the TFF slope result has been observed by testing the E LKr trigger condition, resulting in a systematic uncertainty estimate of ∆a = 0.06×10 −2 .

Backgrounds
The effect of accidental background is investigated by releasing independently the timing cuts and constraints on the numbers of tracks and vertices in the selection. The number of additional events included into the data sample for each variation of the selection is less than 6×10 3 . The total systematic uncertainty due to accidentals is evaluated to be ∆a = 0.15×10 −2 .
The misidentification of charged particles is studied by a modification of the selection criteria. The pion mass is assigned to the track with the charge opposite to q vtx in the kinematic event identification to select K ± → e ± e ± π ∓ γ candidates. Since this process violates lepton number conservation, all events passing this "LNV selection" are considered to be events with misidentified tracks. A total of 188 events from the full data set pass the LNV selection. Using the same selection on the MC samples, it is estimated that most of those events are genuine K 2πD decays with misidentified π ± and e ∓ tracks, while 42±18 data events are not accounted for. The x distribution of these events is added to the reconstructed K 2πD MC one. The TFF slope shifts by ∆a = 0.06×10 −2 , which is considered as a systematic uncertainty.
Removing the K µ3D and K e3D MC samples from the fit procedure results in a shift of the slope of ∆a = 0.01×10 −2 . This is considered as an estimate of the systematic uncertainty on the TFF slope due to the neglected π 0 sources as the other neglected kaon decay modes producing neutral pions account for less than 4% of π 0 production. The acceptance of the K 2πD selection for the K 2π decay followed by π 0 → γγ is estimated with MC simulations to be smaller than 10 −7 , confirming that the minimal distance requirement between tracks in the first DCH efficiently removes the events with photon conversion. The reduction of detector acceptance by the Pb bar (Section 2) does not lead to any systematic uncertainties since events with a particle within the lead bar acceptance are discarded.

Conclusions
The slope of the electromagnetic transition form factor of the π 0 is measured from a sample of 1.11 × 10 6 π 0 Dalitz decays. The result a = (3.68 ± 0.57) × 10 −2 represents the most precise measurement of the form factor slope in the time-like momentum region. The 15% relative uncertainty represents an improvement by a factor of 2 with respect to the previous best measurement [15].