A direct test of T symmetry in the neutral K meson system with Ks->pi l nu and Kl->3pi0 at KLOE-2

Quantum entanglement of K and B mesons allows for a direct experimental test of time-reversal symmetry independent of CP violation. The T symmetry can be probed by exchange of initial and final states in the reversible transitions between flavour and CP-definite states of the mesons which are only connected by the T conjugation. While such a test was successfully performed by the BaBar experiment with neutral B mesons, the KLOE-2 detector can probe T-violation in the neutral kaon system by investigating the process with Ks->pi l nu and Kl->3pi0 decays. Analysis of the latter is facilitated by a novel reconstruction method for the vertex of Kl->3pi0 decay which only involves neutral particles. Details of this new vertex reconstruction technique are presented as well as prospects for conducting the direct T symmetry test at the KLOE-2 experiment.


Introduction
A direct test of the time-reversal symmetry in a single experiment is of great interest among possible ways to probe the T symmetry violation 1) . For particles with spin 0 such as pseudo-scalar mesons, a direct test may be obtained by observation of an asymmetry between a reaction from state i to state f and a reversed reaction f → i. While the CPLEAR experiment measured a nonzero value of the Kabir asymmetry in neutral kaon oscillations 2) , a controversy was raised as to whether this result was independent of CP violation as the K 0 →K 0 andK 0 → K 0 transitions are connected by both the T and CP symmetries. Therefore, an idea was proposed to exploit the quantum correlations of neutral B and K meson pairs to observe reversible transitions between flavour and CP-definite states of the mesons 3, 4) . Such a T symmetry test was successfully performed by the BaBar experiment with the entangled neutral B meson system 5) . In turn, the KLOE-2 detector at the DAΦNE φ-factory is capable of performing a statistically significant direct observation of T symmetry violation with neutral kaons independently of CP violation 4) .

Transitions between flavour and CP-definite neutral kaon states
Neutral kaon states may be described in a number of bases including flavourdefinite states: as well as the states with definite CP parity: State of the kaon can be identified at the moment of decay through observation of the decay final state. With the assumption of ∆S = ∆Q rule 1 , semileptonic kaon decays with positively and negatively charged leptons (later denoted as + , − ) unambiguously identify the decaying state as K 0 andK 0 respectively.
Similarly, the CP-definite states K + and K − are implied by decays to hadronic 1 Althought an assumption, the ∆S = ∆Q rule is well tested in semileptonic kaon decays 6) final states with respectively two and three pions (denoted ππ, 3π). In order to observe a transition between the {K 0 ,K 0 } and {K + , K − } states, both the in and out states must be identified in the respective basis. This is uniquely possible in the entangled system of neutral K mesons produced at a φ-factory. Due to conservation of φ(1 −− ) quantum numbers, the φ → K 0K0 decay yields an anti-symmetric non-strange final state of the form: which exhibits quantum entanglement between the two kaons in the EPR sense 7) . Thus, at the moment of decay of first of the K mesons (and, consequently, identification of its state) state of the partner kaon is immediately known to be orthogonal. This property allows for identification of state of the still-living kaon only by observing the decay of its partner. Its state can be then measured at the moment of decay after time ∆t, possibly leading to observation of a transition between strangeness and CP-definite states. A list of all possible transitions is presented in Table 1. It is immediately visible that time-reversal conjugates of these transitions are not identical with neither their CP-nor CPT-conjugates which is crucial for independence of the test. Transition These quantities can be measured experimentally through numbers of events with certain pairs of decays occurring in time difference ∆t. A deviation of these ratios from 1 would be an indication of T symmetry violation. Bernabeu et al. have simulated the behaviour of these ratios expected at KLOE-2 for 10f b −1 of data 4) (Figure 1). At KLOE-2 the asymptotic region of R 2 and R 4 can be observed where their theoretical behaviour may be expressed as: where = ( S + L )/2 is a T-violating parameter 4) .

Reconstruction of events for the test
The T symmetry test requires reconstruction of the processes with K S → ππ, K L → π ± ∓ ν and K S → π ± ∓ ν, K L → 3π 0 pairs of decays. While for K S → ππ the π + π − final state can be chosen to take advantage of good vertex and momentum reconstruction from charged pion tracks in the KLOE drift chamber, the K L → 3π 0 → 6γ decay reconstruction is a challenging task.
This process only involves neutral particles resulting in the calorimeter clusters from six γ hits being the only recorded information. Moreover, this decay has to be reconstructed in cases where the partner K S decays semileptonically and the missing neutrino prevents the use of kinematic constraints to aid K L → 3π 0 reconstruction. Therefore, this process requires independent reconstruction.
The aim of the new reconstruction method is to obtain the spatial coordinates and time of the K L decay point by only using information on electromagnetic calorimeter clusters created by γ hits from K L → 3π 0 → 6γ. Information available for i-th cluster includes its spatial location and recording time The problem of localizing the vertex is then in its principle similar to GPS positioning and can be solved in a similar manner. For each cluster a set of possible origin points of the incident γ is a sphere centered at the cluster with radius parametrized by an unknown γ origin time t (Figure 2, left). Then, definition of such sets for all available clusters yields a system of up to six equations: with the unknowns x,y,z and t. It is then easily noticed that the K L → 3π 0 → 6γ vertex is a common origin point of all photons which lies on an intersection of the spheres found as a solution of the above system (Figure 2, right). At least 4 clusters are required to obtain an analytic solution although additional two may be exploited to obtain a more accurate vertex numerically. It is worth noting that this vertex reconstruction method directly yields kaon decay time in addition to spatial location which is useful for time-dependent interferometric studies such as the T symmetry test.