Search for the dark photon in π 0 decays

A sample of 1.69×107 fully reconstructed π → γe+e− in the kinematic range mee > 10MeV/c 2 with a negligible background contamination collected by the NA48/2 experiment at CERN in 2003-04 is analysed to search for the dark photon (A′) via the decay chain π → γA′, A′ → ee. No signal is observed, and preliminary exclusion limits on space of dark photon mass mA′ and mixing parameter 2 are reported.


Introduction
High intensity kaon experiments provide opportunities for precision studies of π 0 decay physics due to the fact that kaons represent a source of tagged neutral pion decays, mainly via their K ± → π ± π 0 , K ± → e ± π 0 ν and K ± → μ ± π 0 ν decays.One of them is the NA48/2 experiment at the CERN SPS, which collected a large sample of charged kaon (K ± ) decays in flight in 2003-04 corresponding to ∼ 2 × 10 11 K ± decays in the fiducial decay volume [1].This large sample of π 0 mesons produced and decaying in vacuum collected by NA48/2 allows for a high sensitivity search for the dark photon (A ), a hypothetical gauge boson appearing in hidden sector new physics models with an extra U(1) gauge symmetry.In these models the interaction of the dark photon with the visible sector proceed through kinetic mixing with the Standard Model (SM) hypercharge [2].Such scenarios with GeV-scale dark matter provide possible explanations to the observed rise in the cosmic-ray positron fraction with energy and the muon gyromagnetic ratio (g − 2 ) measurement [3].From the experimental point of view, the dark photon is characterized by two a priori unknown parameters, the mixing parameter and the mass m A .The A production and the subsequent decay can be obtained from the following decay chain: with three charged particles and a photon in the final state.The expected branching fraction of the π 0 decay is [4] which is kinematically suppressed as m A approaches m π 0 .In the mass range 2m e m A < m π 0 accessible in this analysis, the dark photon is below threshold for all decays into SM fermions except A → e + e − , while the allowed loop-induced decays (A → 3γ, A → νν) are strongly suppressed.Therefore, assuming that the dark photon decay only in SM particles, B(A → e + e − ) 1.The expected total dark photon decay width is [5] It follows that, for 2m e m A < m π 0 , the dark photon mean proper lifetime τ A satisfies the relation The NA48/2 analysis is performed assuming that the dark photon decays at the production point (prompt decay), which is valid for sufficiently large values of the mass (m A > 10MeV/c 2 ) and the mixing parameter ( 2 > 5 × 10 −7 ).In this condition the dark photon signature is identical to the Dalitz decay π 0 → γe + e − , which therefore represents an irreducible background and determines the sensitivity.The largest π 0 D sample is obtained through the reconstruction of K ± → π ± π 0 and K ± → μ ± π 0 ν decays (denoted K 2π and K μ3 ).In Addition, the K ± → π ± π 0 π 0 decay (denoted K 3π ) is considered as a background in the K μ3 sample.

The NA48/experiment
The NA48/2 beam line has been designed to deliver simultaneous narrow momentum band K + and K − beams following a common beam axis derived from the primary 400 GeV/c protons extracted from the CERN SPS.Secondary beams with central momenta of (60 ± 3) GeV/c (r.m.s.) were used.The beam kaons decayed in a fiducial decay volume contained in a 114 m long cylindrical vacuum tank.The momenta of charged decay products were measured in a magnetic spectrometer, housed in a tank filled with helium placed after the decay volume.The spectrometer comprised four drift chambers (DCHs), two upstream and two downstream of a dipole magnet which provided a horizontal transverse momentum kick of 120 MeV/c to singly-charged particles.Each DCH was composed of eight planes of sense wires.A plastic scintillator hodoscope (HOD) producing fast trigger signals and providing precise time measurements of charged particles was placed after the spectrometer.Further downstream was a liquid krypton electromagnetic calorimeter (LKr), an almost homogeneous ionization chamber with an active volume of 7 m 3 of liquid krypton, 27X 0 deep, segmented transversally into 13248 projective 2 × 2 cm 2 cells and with no longitudinal segmentation.The LKr information is used for photon measurements and charged particle identification.An iron/scintillator hadronic calorimeter and muon detectors were located further downstream.A dedicated two-level trigger was in operation to collect three-track decays with an efficiency of about 98%.A detailed description of the detector can be found in Ref. [6].

Event selection
The full NA48/2 data sample is used for the analysis.The K 2π and K μ3 with the following π 0 → γe + e − event selection requires a three-track vertex reconstructed in the fiducial decay region formed of a pion or muon candidate track and two opposite-sign electron (e ± ) candidate tracks.Charged particle identification is based on the ratio of energy deposition in the LKr calorimeter (E) to the momentum measured by the spectrometer (p).Pions and muons are kinematically constrained to the momentum range above 5 GeV/c, while the momentum spectra of electrons originating from π 0 decays are soft,  peaking at 3 GeV/c.Therefore,p > 5 GeV/c and E/p < 0.85 (E/p < 0.4) are required for the pion (muon) candidate, while p > 2.75 GeV/c and (E/p) min < E/p < 1.15, where (E/p) min = 0.80 for p < 5 GeV/c and (E/p) min = 0.85 otherwise, are required for the electron candidates.Furthermore, a single insolated LKr energy deposition cluster is required and considered as the photon candidate.The reconstructed invariant mass of the e + e − γ system is required to be compatible with the π 0 mass, |m e + e − γ − m π 0 | < 8 Mev/c 2 (this interval corresponds to ±5 times the resolution on m e + e − γ ).For the K 2π selection, the reconstructed invariant mass of the π ± e + e − γ (figure 1) system should be compatible with the K ± mass, 474 MeV/c 2 < m πe + e − γ < 514 MeV/c 2 .For the K μ3 selection, the squared missing mass m 2 miss = (P K − P μ − P π 0 ) 2 (figure 1) should be compatible with tne neutrino mass (|m 2  miss | < 0.01 GeV 2 /c 4 ), where P μ and P π 0 are the reconstructed μ ± and π 0 four-momenta, and P K is the nominal kaon four-momentum.At the end of the two selections a sample of 1.38 × 10 7 (0.31 × 10 7 ) fully reconstructed π 0 D decay candidates comining from K 2π (K μ3 ) with a negligible background is selected.Correcting the observed number of candidates for acceptance and trigger efficiency, the total number of K ± decays in the 98 m long fiducial decay region for the analysed data sample is found to be N K = (1.57± 0.05) × 10 7 , where the quoted error is dominated by the external uncertainty on the π 0 D decay branching fraction.The reconstructed e + e − invariant mass spectra are displayed in figure 2. In addition to the above individual K 2π and K μ3 selections, a joint dark photon selection is also considered: an event passes the joint selection if it passes either the K 2π or the K μ3 selection.The acceptance of the joint selection for any process is equal to the sum of acceptances of the two mutually exclusive individual selections.

The π 0 Dalitz simulation
The π 0 D decay is simulated using the following lowest-order differential decay rate with x and y, two kinematic variables, defined as r = 2m e /m π 0 , q 1 , q 2 and p are the four-momenta of the electrons (e ± ) and the neutral pion, Γ 0 is the rate of the π 0 → γγ decay, and F(x) is the pion transition form factor (TFF).
Radiative corrections to the differential rate are implemented following the classical approach of Mikaelian and Smith [7]: the differential decay rate is modified using the following formula which does not account for the emission of inner bremsstrahlung photons.
The TFF is conventionally parameterized as F(x) = 1 + ax.The transition form factor slope parameter a is expected from vector meson dominance models to be a 0.03, and detailed theoretical calculations based on dispersion theory yield a = 0.0307 ± 0.0006 [8].Experimentally, the PDG average a = 0.032 ± 0.004 [9] is determined mainly from a e + e − → e + e − π 0 rate measurement in the spacelike region by the CELLO experiment [10].The precision on the used radiative corrections to the π 0 D decay is limited: in particular, the missing correction to the measured TFF slope due to two-photon exchange is estimated to be Δa = +0.005[11].Therefore the background description cannot benefit  from the precise inputs on the TFF slope [8,9], and an "effective" TFF slope obtained from a fit to the data m ee spectrum itself is used to obtain a satisfactory background description (as quantified by a χ 2 test) over the range m ee > 8 MeV/c 2 used for the dark photon search.The low m ee region is not considered for the search as the acceptance computation is less robust due to the steeply falling geometrical acceptance at low m ee and decreasing electron identification efficiency at low momentum.

Search for the dark photon
A scan for a dark photon signal in the mass range 9 MeV/c 2 ≤ m A < 120 MeV/c 2 is performed.The lower extent of the considered mass range is determined by the limited precision of MC simulation of background at low mass, while at the upper limit of the mass range the signal acceptance drops to zero.The mass step of the scan and the width of the dark photon signal mass window around the assumed mass are determined by the resolution δm ee on the e + e − invariant mass.The resolution on m ee as a function of m ee evaluated with MC simulation is parameterized as σ m (m ee ) = 0.067 MeV/c 2 + 0.0105 • m ee , and varies from 0.16 MeV/c 2 to 1.33MeV/c 2 over the mass range of the scan.The mass step of the scan is set to be σ m /2, while the signal region mass window for each dark photon mass hypothesis is defined as ±1.5σ m around the assumed mass.The mass window width has been optimised with MC simulations to obtain the highest sensitivity to the dark photon signal, determined by a trade-off between π 0 D background fluctuation and signal acceptance.For each considered dark photon mass value, the number of observed data events N obs passing the joint selection is compared to the expected number of background events N exp .The latter is evaluated from MC simulations, corrected for the trigger efficiency measured from control data samples passing the joint selection.The numbers of observed and expected events for each mass value and their estimated uncertainties δN obs and δN exp are shown in figure 3. The uncertainty δN obs = √ N obs is statistical, while the uncertainty δN exp has contributions from the limited size of the generated MC samples and Confidence intervals at 90% CL for the number of A → e + e − decay candidates (N DP ) in each mass hypothesis are computed from N obs , N exp and δN exp using the Rolke-López method [12] assuming Poissonian (Gaussian) errors on the numbers of observed (expected) events.
Upper limits at 90% CL on the branching fraction B(π 0 → A γ) for each dark photon mass value with the assumption B(A → e + e − ) = 1 (which is a good approximation for m A < 2m μ if A decays to SM fermions only) are computed using the relation where A DP (K 2π ), A DP (K μ3 ) and A DP (K 3π ) are the acceptances of the joint dark photon selection for K 2π , K μ3 and K 3π decays, respectively, followed by the prompt π 0 → A γ, A → e + e − decay chain.The trigger efficiencies e 1 ane e 2 are taken into account neglecting their variations over the m ee invariant mass, variations measured to be at the level of a few permille.Event distributions in the angle between e + momentum in the e + e − rest frame and the e + e − momentum in the π 0 rest frame are identical for the decay chain involving the dark photon (π 0 → A γ, A → e + e − ) and the π 0 D decay, up to a negligible effect of the radiative corrections that should not be applied in the former case (found to influence the acceptance at the level below 1% in relative terms).Therefore dark photon acceptances are evaluated using the MC samples produced for background description, and no dedicated MC productions are required.The resulting upper limits on B(π 0 → A γ) and signal acceptances are shown in figure 4. The upper limits are O(10 −6 ) and do not exhibit a strong dependence on the A mass.Upper limits at 90% CL on the mixing parameter 2 for each dark photon mass value calculated from the B(π 0 → A γ) upper limits using equation 1 are shown in figure 5, together with the constraints from the SLAC E141 and FNAL E774 [13], KLOE [14], WASA [15], HADES [16], A1 [17], APEX [18] and BaBar [19] experiments.Also shown is the band in the (m A , 2 ) plane where the discrepancy between the measured and calculated muon (g − 2) values falls into the ±2σ range due to the dark photon contribution, as well as the region excluded by the electron (g − 2) measurement [3,20].The obtained upper limits on 2 represent an improvement over the existing data in the dark photon mass range 9 − 70 MeV/c 2 , and exclude the whole favoured region by muon g − 2 [21].The most stringent limits (2 × 10 −7 ) are achieved at m A ∼ 10 MeV/c 2 .The sensitivity of the prompt A decay search is limited by the irreducible π 0 D background.The achievable upper limit on 2 scales as the inverse square root of the integrated beam flux, which means that the possible improvements to be made with this technique using larger future K ± samples are modest.

Figure 1 .
Figure 1.Data and Monte Carlo reconstructed π ± π 0 D invariant mass for the event passing the K 2π selection (left).Data and Monte carlo reconstructed invariant mass for the event passing the K μ3 selection (right).The arrows represent the selection cuts.

Figure 2 .
Figure 2. Data and Monte Carlo reconstructed m ee invariant mass for the event passing the K 2π selection (left).Data and Monte carlo reconstructed m ee invariant mass for the event passing the K μ3 selection (right).

Figure 3 .
Figure 3. Numbers of observed data events and expected π 0 D background events passing the joint selection (indistinguishable in a logarithmic scale), estimated uncertainties and obtained upper limits at 90% CL on the numbers of dark photon candidates (N DP ) for each mass value m A (left).Estimated local significance of the dark photon signal for each A mass value (right).

Figure 4 .
Figure 4. Acceptances of the joint dark photon selection for K 2π , K μ3 and K 3π depending on the assumed DP mass, evaluated with MC simulations.The K 3π acceptance is scaled by a factor of 10 for visibility (left).Obtained upper limits on B(π 0 → A γ) at 90% CL for each A mass value (right).

Figure 5 .
Figure 5.The NA48/2 preliminary upper limits at 90% CL on the mixing parameter 2 versus the A mass, compared to the other published exclusion limits from meson decay, beam dump and e + e − collider experiments.Also shown are the band for the muon (g − 2) and the region excluded by the electron g − 2 measurement.