The investigation of K + π − , π + K − and π + π − atoms

Theory, using Low Energy QCD, predicts with high precision the pion-pion and pion-kaon scattering lengths. There is accurate relation between K+π− and π+K− atoms lifetime and pion-kaon S-wave scattering lengths with isospin 1/2 and 3/2. Experiment DIRAC at CERN PS detects 345 ± 61 pairs from K+π− and π+K− atoms breakup. It allows to achieve the first observation of exotic atoms consisted of pion and kaon. Measured values of πK atom lifetime and corresponding pion-kaon scattering length difference are presented. It is shown, that experimental accuracy for pion-kaon scattering length difference could be significantly improved with an experiment at SPS energy.

Experimental data on the πK low-energy phases are absent.The only experimental pion-kaon scattering length measurement has been done with estimation of πK atom lifetime [8]: 2 Method of πK atom observation and investigation πK-atom (A πK ) is a hydrogen-like atom consisting of K + (K − ) and π − (π + ) mesons.The πK-atom lifetime (ground state 1S), τ = 1 Γ is dominated by the annihilation process into π 0 K 0 .There is a relation between the width of A πK decay and S-wave πK scattering lengths for isospin 1/2 and 3/2 [9]: Here α is the fine structure constant, μ is the reduced mass of the π ± K ∓ system, p * is the outgoing π 0 momentum in the πK atom system, and δ K accounts for corrections, due to isospin breaking, at order α and quark mass difference (m u − m d ).
Prediction of scattering length difference in Eq. ( 3) provides an estimation of lifetime of A πK in ground state to be: τ = (3.5 ± 0.4) × 10 −15 .
A method of investigation for π + π − , πK and other atoms, consisted of two oppositely charged mesons, has been proposed in [10].Pairs of K + (K − ) and π − (π + ) mesons are producing in protontarget interactions.Pairs, which are generated from fragmentation and strong decays ("short-lived" sources), are affected by Coulomb interaction in the final state.Some of them form Coulomb bound states -atoms, other are generated as free pairs ("Coulomb pairs").Number of produced atoms (N A ) is proportional to a number of "Coulomb pairs" (N C ) with low relative momentum Q in a pair C.M. system: N A = K • N C .The coefficient K is calculated with an accuracy better than 1% [11].
If at least one meson is generated from long-lived sources (electromagnetically or weakly decaying mesons or baryons: η, η , K 0 s , . ..), then such pairs ("non-Coulomb pairs") are not affected by interaction in the final states.
After production, A πK travel through the target and could to annihilate into π 0 K 0 , or to be ionised due to interaction with the target matter, producing specific "atomic pairs".These pairs have small relative momentum (Q < 3 MeV/c) and a number of such pairs n A could be measured experimentally.Ratio of "atomic pair" number to a number of atom produced is a breakup probability: [12,13].In Fig 1 dependence of A πK breakup probability is shown for two nickel target are used in experiment DIRAC for pair laboratory momentum range 5.1 ÷ 8.5 GeV/c.Value is averaged, using experimentally measured spectrum of atoms.

DIRAC setup
DIRAC setup was created to detect π + π − with small relative momenta [14].In 2004-2006 it has been modified in order to detect both π + π − and πK pairs.New detectors for particle identification have been added: Cherenkov counters with heavy gas and aerogel for identification of K-mesons among background of pions and protons, correspondingly.Taking into account kinematic of πK "atomic pairs", new detectors cover only internal parts of each arm (see Fig. 2).Compared to the previous investigation [15], the Pt data was analysed including the upstream detectors.The consequence is a decrease of the statistics, but on the other hand an increase of the Q T resolution.This better resolution improves the quality of data.Concerning the Ni target, the increase of n A , compared to [8], is caused by optimizing the time-of-flight criteria, which decreases atomic pair losses for the same fraction of background in the final distributions.
The evaluation of the atomic pair number n A is affected by several sources of systematic errors [16,17].These uncertainties lead to differences in the shapes of experimental and MC distributions for "atomic", "Coulomb" and to a much lesser extent for "non-Coulomb" pairs.The shape differences induce a bias in the value of the fit parameter n A , corresponding to a systematic error of the atomic pair number.Sources of systematic error and estimation of error values are listed in Table 2  Taking into account both statistical and systematic errors, the one-dimensional π ∓ K ± analysis in Q yields n A = 349 ± 61(stat) ± 9(syst) = 349 ± 62(tot) atomic pairs (5.6 σ) for both combinations of charge and two-dimensional analysis in (|Q L |, Q T ) yields n A = 314±59(stat)±10(syst) = 314±60(tot) atomic pairs (5.2 σ).This is the first statistically significant observation of the strange dimesonic πK atom.

Sources of systematic errors
Experimentally measured numbers of "atomic pairs" n A and produced atoms N A allow (see section 2) to obtain new estimation of A πK lifetime in ground state, based on two-dimensional analysis in which corresponds to isospin-odd πK scattering length estimation to be: This result has been used for estimation of time which is needed for measurement a − 0 with accuracy 5% [18].It provides possibility to plan experiments for measurement of "Lamb shift like" effect in π + π − system and new combination of S-wave pion-pion scattering lengths: 2 • a 0 0 + a 2 0 .

Summary
In the DIRAC experiment at CERN, the dimesonic Coulomb bound states involving strangeness, the π − K + and π + K − atoms, were observed for the first time with reliable statistics.The one-dimensional π ∓ K ± analysis in Q yields 349 ± 62(tot) atomic pairs (5.6 σ) for both combinations of charge.Analogously, a two-dimensional analysis in (|Q L |, Q T ) was performed with the result of 314 ± 60(tot) atomic pairs (5.2 σ).DIRAC-like experiment at SPS energy provides possibility to check prediction of ChPT and LQCD for pion-kaon scattering lengths with accuracy at the level 5%.
Observation of "atomic pairs" from π + π − atoms in long-lived states (n L A = 436 ± 61(tot)) provides possibility to plan experiments for measurement of "Lamb shift like" effect in π + π − system and new combination of S-wave pion-pion scattering lengths: 2 • a 0 0 + a 2 0 .

Figure 1 .Figure 2 .
Figure1.Dependence of the breakup probability P br on A πK lifetime for 108μm (solid blue line) and 98μm (dashed red line) nickel targets, and an example how lifetime could be obtained from experimentally measured breakup probability

Table 1 .
Figure 3. Distribution of π + K − and K + π − pairs over Q (upper pictures), Atomic pair numbers n |Q L | and two-dimensional (|Q L |, Q T ) distribution are presented in Table 1 (Ni and Pt target together).The best statistical accuracy are achieved by analysis of Q and (|Q L |, Q T ) distributions.Signal to error ratio is more than 5.The 1-dimensional |Q L | analysis for all πK data yields n A = 230 ± 92, which does not contradict the values, obtained in the other two statistically more precise analyses.
A by analysing the 1-dimensional Q and |Q L | distributions and the 2-dimensional (|Q L |, Q T ) distribution.Only statistical errors are given.+ K − and K + π − "atomic pairs" obtained with analysis of one-dimensional distributions over Q,

Table 2 .
. Estimations of systematic errors, which are induced by different sources, for analysis of data distribution over relative momentum Q, its longitudinal projection |Q L | and two dimensional distribution over (|Q L |, Q T ).

Table 3 .
Estimation of time needed for measurement a − 0 with statistical accuracy δ a − 0 for present DIRAC setup and beam condition, and for versions Mod1 and Mod2, modified for proton beam energy E p = 450 GeV and intensity I B .It is assumed, that at 450 GeV beam the setup would obtain 3000 spills (4.5s) per day.
[19]e 3presents expected beam time and run time are needed for achievement of this accuracy for present (Nickel target only) and modified DIRAC setup.It is seen that experiment at SPS energy allows to measure S-wave pion-kaon scattering length difference with sufficient accuracy for checking theoretical predictions made by ChPT and LQCD.Investigation of π + π − atoms DIRAC experiment collected data with Be-Pt target in 2012.These data allow to make observation of A 2π in states with non-zero orbital momentum (long-lived atoms): n L A = 436 ± 61[19].Lifetime of long-lived atoms (τ ≥ 1.17 • 10 −11 s) is sensitive to energy splitting between ns and np atomic states.