First measurement of a long-lived $\pi^+ \pi^-$ atom lifetime

The adapted DIRAC experiment at the CERN PS accelerator observed for the first time long-lived hydrogen-like $\pi^+\pi^-$ atoms, produced by protons hitting a beryllium target. A part of these atoms crossed the gap of 96~mm and got broken up in the 2.1~\textmu{}m thick platinum foil. Analysing the observed number of atomic pairs, $n_A^L= \left.436^{+157}_{-61}\right|_\mathrm{tot}$, the lifetime of the 2$p$ state is found to be ${\tau_{2p}=(\left.0.45^{+1.08}_{-0.30}\right|_\mathrm{tot}) \cdot10^{-11}}$s, not contradicting the corresponding QED $2p$ state lifetime ${\tau_{2p}^\mathrm{QED}=1.17 \cdot 10^{-11}}$s. This lifetime value is three orders of magnitude larger than our previously measured value of the $\pi^+\pi^-$ atom ground state lifetime $\tau=(\left.3.15^{+0.28}_{-0.26}\right|_\mathrm{tot})\cdot 10^{-15}$s. Further studies of long-lived $\pi^+\pi^-$ atoms will allow to measure energy differences between $p$ and $s$ atomic states and so to determine $\pi\pi$ scattering lengths with the aim to check QCD predictions.


Introduction
The DIRAC collaboration aims to check low-energy QCD predictions using double-exotic π + π − and π ± K ∓ atoms, which have been observed and studied [1][2][3][4][5][6][7].In strong inclusive pro- The expression ψ nl (0) is the pure Coulomb atomic wave function at the origin with principal quantum number n and orbital quantum number l.The precision of the ratio R is 1.2% [13], and the scattering lengths and their combination have been calculated in low-energy QCD [14].The DIRAC experiment has measured these lifetimes and so derived combined ππ [1][2][3] and for the first time πK [5,7] scattering lengths.
After investigation of ππ and πK atoms with "short" lifetimes of order 10 −15 s, DIRAC continues to study ππ atomic states (A L 2π ) of "long" lifetimes of order 10 −11 s as considered in Ref. [8].Moving inside the target, relativistic short-lived A 2π interact with the electric field of the target atoms, resulting in a possible change of the A 2π orbital momentum l by one or more units.Some (N L A ) of them (N A ) will leave the target with l > 0. For such states, |ψ nl (0)| 2 = 0 and thus all decays are suppressed in accordance with (1).Therefore, the decay mechanism of such excited states is the radiative de-excitation to an ns state, annihilating subsequently with the "short" lifetime τ • n 3 into two π 0 .The shortest A L 2π lifetime is the 2p state lifetime τ 2p ≈ τ QED 2p = 1.17 • 10 −11 s, which is three orders of magnitude larger than the minimum A 2π lifetime.For an average momentum of detected A 2π p A =4.44 GeV/c (γ 15.9), the decay lengths are 5.6 cm (2p), 18.4 cm (3p), 43 cm (4p), 83 cm (5p) and 143 cm (6p).
These large atom decay lengths in the laboratory (lab) system open a possibility to measure the energy splitting between ns and np levels, which depends on an other combination of ππ scattering lengths, 2a 0 + a 2 [15,16].
In this paper, the DIRAC collaboration presents the first lifetime measurement of long-lived π + π − atoms, observed by DIRAC as described in [17].

Setup for investigation of long-lived π + π − atoms
To investigate long-lived A L 2π atoms, the primary 24 GeV/c CERN PS proton beam with intensity of 3.0 •10 11 protons per spill [18] hits a 103 µm thick Be target (Figures 1 and 2).On their way to the 2.1 µm thick Pt foil at a distance of 96 mm behind the Be target, a part of the produced long-lived atoms A L 2π , depending on their lifetimes, decays, whereas the other part enters the Pt foil.By interacting with Pt atoms, the atoms A L 2π break up (get ionized) and generate atomic π + π − pairs with relative momenta Q < 3 MeV/c in their pair centre-of-mass (c.m.) system.The foil is introduced at 7.5 mm above the primary proton beam to avoid interaction of the beam halo with Pt.The beam position in the vertical plane was permanently monitored.Between target and breakup foil, there was installed a permanent retractable magnet [19] with a pole distance of 60 mm (insertion of Figure 2) and a maximum horizontal field strength of 0.25 T (bending power 0.02 Tm).The magnet enlarges the value of the vertical component Q Y of π + π − pairs, generated in Be, on Q Y = 13.15MeV/c (see Figure 1).For simulated  events, this value is Qy = 13.12MeV/c.For all pairs generated in Pt foil, the fringing magnetic field modifies the vertical component Q Y only by 2.3 MeV/c.This large difference in the Q Y shift for pairs from Be and Pt allows to suppress the background by a factor of about 6 in detecting atomic pairs from long-lived A L 2π .For e + e − pairs generated in Be, the peak is shifted to Q Y = 12.9 MeV/c and its position used to measure the magnetic field stability during the 6 month data taking [17].
The DIRAC setup [20], sketched in Figure 2, identifies pions, kaons, protons, electrons and muons by means of Cherenkov, preshower and muon detectors and time-of-flight measurement.The achieved resolution in the particle momentum is ∆p/p 3 • 10 −3 and the precision of the Q components σ QX ≈ σ QY ≈ 0.44 MeV/c and σ QL ≈ 0.50 MeV/c.This high resolution enables the extraction of an atom signal in form of small Q atomic pairs.The secondary channel with the solid angle Ω = 1.2 • 10 −3 sr is vertically inclined relative to the proton beam by 5.7 • upward.
The events are reconstructed by means of the DIRAC ππ analysis software described in [17], exploiting information from DC, SFD, IH and the proton beam position on the target.The setup is aligned using properties of Λ ( Λ) decays [21].
In complete analogy, the production of π + π − Coulomb pairs is expressed via the inclusive production cross section of π + π − pairs without FSI, modified by the Coulomb enhancement function A C (q), which depends on the pair relative momentum q (∼ 1/q for small q).This function, the well-known Gamov-Sommerfeld-Sakharov factor [22][23][24], is used for simulation of Coulomb pairs.
The relative yield between atoms and Coulomb pairs [25] is expressed by the ratio of the above cross sections.Thus, the total number N A of produced A 2π is determined via the number of Coulomb pairs N C by the model-independent relation: where N C (q ≤ q 0 ) is the number of Coulomb pairs with q ≤ q 0 and K(q 0 ) a known function of q 0 [25].
In the above consideration, the production of π + π − pairs and their FSI are assumed to be pointlike.Contributions from medium-lived resonances [26,27] are taken into account as small corrections to the production cross sections.

Production of long-lived π + π − atoms
The short-lived π + π − atoms, which propagate in the Be target after their production in ns states, are either decaying or interacting with Be.This interaction with the electric field of target atoms will exite/de-excite or break up π + π − atoms.In the excitation/de-excitation processes, only transitions with Zparity (P Z = (−1) l+m ) conservation are permitted [28], changing the orbital momentum by one unit or more and so forming long-lived atoms.The quantization axis is along the lab atom momentum.The lifetime of π + π − atoms in the lab system depends on the quantum numbers n,l and the atom momentum.The populations of all atomic states with quantum numbers n, l, m, as a multi-level quantum system, are described by an infinite set of transport equations in terms of probabilities [29,30].These equations account for excitation/de-excitation and annihilation processes and describe the atomic state populations from the production point up to the target exit.For the calculations below, our measured value of the A 2π ground state lifetime τ = 3.15 • 10 −15 s [3] and the spectrum of the atom lab momentum as extracted from the experimental distribution of the Coulomb pairs (Figure 3) are used.

Population of long-lived states
In this paper, the transport equations are solved numerically for all states with principal quantum numbers n ≤ 10.The populations of states with higher n are taken into account by an extrapolation procedure.Only total and excitation/de-excitation cross sections are considered: the cross sections are calculated in the first Born approximation (one-photon exchange), using Molière parametrization of the Thomas-Fermi formfactors for Be and Pt atoms.
Figure 4 shows the atomic state population P L n versus n, summed over l and m, at the exit of the Be target and before the Pt foil.The almost pure exponential behaviour at high n allows to extrapolate to n > 10 and to evaluate the population of the non-negligible infinite "tail".Thus, the population of all n of all long-lived atomic states at the exit of the Be target as a function of n used for "tail" estimation.For each n, the two upper curves show the sum of state populations for the given n plus different "tail" estimations calculated from populations for n and n − 1 (exponential "tail" -•, hyperbolic "tail" -).The lower curve ( ) presents the sum of the population for the given n plus the population for n + 1 instead of the "tail".discrete states can be estimated.The population at n = 10 is excluded from the extrapolation procedure, as this population is underestimated because of cutting the infinite set of transport equations, excluding de-excitations to this state from higher n states.The cut does almost not affect the populations with smaller n, as transitions between nearest n levels dominate.A fitting procedure over all points is badly suited for the "tail" estimation, as the n dependence is mainly dictated by two different dependences: production rates as n −3 and total cross sections roughly as n 2 .The following two n dependences are chosen for extrapolation: exponential, a exp (−bn), and hyperbolic, cn −3 + dn −5 .The free parameters a, b and c, d have been calculated for n, using the populations P L n and P L n−1 .Figure 5 illustrates the "tail" estimations.The numerical solution of the transport equations are obtained for n ≤ 10.Then, the populations at n = 9 and 8 are used for the "tail" estimation.The two upper points at n = 9 in Figure 5 show the sum of the populations for n ≤ 9 plus the different "tail" estimations (exponential -•, hyperbolic -).The lower point ( ) presents the sum of the populations with n ≤ 9 plus the population for n = 10 instead of the "tail".The curves are obtained by applying the same procedure to the points with lower n.The exponential "tail" estimation is chosen as the mean value for the population of long-lived states, as it has the smallest slope and is nearly flat for n ≥ 7. The two other curves are used as error band.The illustrative numerical values are calculated for the average atom momentum p A = 4.44 GeV/c (Figure 3): the sum of the long-lived state population at the Be exit is 6.04% for n ≤ 10 (5.99% for n ≤ 9) of the total number N A of produced A 2π ; the exponential "tail" n ≥ 10 is 0.14%, the hyperbolic "tail" 0.25%, and the population of all long-lived states at the Be exit is given as N L,Be A = (6.13+0.11  −0.09 ) • 10 −2 × N A .Moreover, about 17% of A 2π are produced in short-lived states (ns) and annihilate just behind the Be target.About 4.6% of the atoms break up in Be producing atomic pairs.Let us underline that accuracy of all above numbers accounts for the accuracy of the extrapolation procedure.
As mentioned above, the cut of the infinite set of transport equations at n ≤ 10 leads to a small underestimation of all lowlying states because of neglecting the de-excitation from the truncated "tail" with n > 10.A possible influence of this cut has been estimated by comparing the above solution with a solution, cut at n ≤ 9, where the de-excitation from states with n = 10 are also truncated.Estimated in this way, the population of long-lived states at the Be exit is with a very conservatively error band (6.29 +0.33 −0.13 ) • 10 −2 [31].An additional correction to this value may arise from the accuracy of atom-atom cross sections.The most accurate cross sections are obtained in the Glauber approach accounting for multi-photon exchange and the Hartree-Fock-Dirac form factors of the target atoms [32][33][34].As shown in [35], the accuracy of the used cross sections, compared to the most accurate ones, depends on the target material.In the case of the Be target, an estimation shows that the summed population of long-lived states for n ≤ 8 at the Be exit, calculated with the most accurate cross sections, is 2.8% less than the current value.Thus, this value can be used as the scale factor accounting for uncertainties of the used atom-atom cross sections.Further, the accuracy of the most accurate cross sections is estimated to be about 1% [32][33][34], which can be converted roughly to a 1% error in the populations.Let us conservatively add this error linearly to the latter error band.Finally, the population of long-lived states at the Be exit is (6.12 +0.39 −0.19 ) • 10 −2 , compared to (6.13 +0.11 −0.09 ) • 10 −2 in the above calculations.Thus, this crude accuracy estimation leads mainly to an increase of the error band for the calculated probabilities.Nevertheless, it is significantly smaller than the experimental error.Hence, its influence is applied only to the final results as described below.

Decay in gap and breakup in Pt for long-lived atoms
In the 96 mm gap between the Be target and the Pt foil, the populations of the atomic states alter depending on their lifetime [36], calculated in QED.The decay length of atoms with fixed n is minimal for l = 1 and strongly increasing with l. Figure 4 indicates that mainly states with n = 2 − 4 are decaying in the gap.The summed population of long-lived states at the entry of the Pt foil is 0.037 ± 0.001.Note that all cascade radiation transitions between atomic states as well as annihilation of ns states are considered, using the technique of transport equations.The cascade accounting increases the summed population by about 10% compared to a pure exponential decay.
Table 1 shows the summed populations of long-lived states at the Be target exit and at the Pt foil entry as a function of n.Illustrative values of np state breakup probabilities in the Pt foil are shown as well.The pionium breakup in Pt is treated in the same way as the breakup in the target.The breakup probability of A L 2π entering in Pt, averaged over populations, is 0.944.Finally, the expected total probability P tot br (Pt), that pionium produced in Be will break up in Pt, is calculated via the difference between the total populations at the entry and exit of the Pt foil.For the average atom momentum (see above), this probability is found to be:

Number of atoms produced in the Be target
The analysis of π + π − experimental data, produced in the Be target, is similar to the π + π − analysis as presented in [3].The main difference is a shift in Q Y by 13.1 MeV/c, due to the magnetic field of the inserted permanent magnet.For the analysis in the current section, the found Q Y value of each experimental and MC event is lessened by the above 13.1 MeV/c.
Events with transverse Q T < 4 MeV/c and longitudinal |Q L | < 15 MeV/c are selected to be analysed.The experimental Q-distribution N(Q i ) and the distributions of its projections are fitted by simulated distributions of atomic . Using the difference of the particle production times, the admixture of accidental pairs is subtracted from the experimental distributions.The distributions of simulated events, , are normalized to 1.In the experimental distributions, the numbers of atomic (n A ), Coulomb (N C ) and non-Coulomb (N nC ) pairs are free fit parameters in the minimizing expression: The sum of these parameters is equal to the number of analysed events.The fitting procedure takes into account the statistical errors of the experimental distributions.
Figure 6 shows the experimental and simulated Q L distributions of π + π − pairs for the data obtained from the Be target.One observes a very good agreement of the sum of simulated Coulomb and non-Coulomb pairs with experimental data in the region of Q L > 1.5 MeV/c, where no atomic pairs are expected.The main result of this fit is the number of Coulomb pairs: using (2), this number allows to obtain the full number of π + π − atoms produced in the Be target.
The Coulomb pairs from this distribution are used to reconstruct the lab momentum of produced atoms, shown in Figure 3.The average momentum is p A = 4.44 GeV/c.This spectrum is used in the atom simulation.
Table 2 summarizes the number of Coulomb pairs N C from the 1-dimensional Q and |Q L | analyses and from the 2-dimensional (|Q L |,Q T ) analysis.The efficiency of Coulomb pair recording is evaluated from the simulated data as ratio of the MC Coulomb pair number N MC C , passing the corresponding cutsin each of the above analyses -to the full number of generated  for the beryllium (Be) target fitted by the sum of simulated distributions of "atomic", "Coulomb" and "non-Coulomb" pairs.The summed distribution of free ("Coulomb" and "non-Coulomb") pairs is shown as black line.
Coulomb pairs . The full number of produced Coulomb pairs in the Be target is then given by N C /ε C .This number allows to calculate the number N A of atoms produced in the Be target: using the theoretical ratio K = N A /N C (2), calculated for q ≤ 2 MeV/c, and the simulated probability To recover the losses due to the coplanarity cut in the trigger system (Sec.2), the number N A is additionally corrected (MC).The corrected values N tot A are shown in the last column of Table 2.

Number of long-lived atoms broken up in Pt
Atomic pairs from the A L 2π atom breakup in the Pt foil are observed [17] above background π + π − pairs produced mainly in the Be target.For the analysis, the same approach is applied as in Section 5, but this time Q Y is shifted by 2.3 MeV/c for all pairs instead of 13.1 MeV/c (Figure 1).
In A is mainly due to different signal shapes in the experiment and simulation and is taken as an estimate for the systematic error (only with positive sign) σ syst shape = 144.The systematic error of the number of atoms produced in Be (N tot A ) is calculated as a maximum difference between the value of the analysis with the current variables and other analyses, σ syst A = 260.In summary, the analysis of the 2-dimensional (|Q L |, Q T ) distributions yields the following value for the total probability that a π + π − atom produced in Be breaks up in the Pt foil: In the second more sophisticated analysis, the populations of the pionium states are described in terms of transport equations for the whole path from the production point in the Be target to the exit of the Pt foil.For the gap between Be and Pt, radiation transition rates and annihilation of short-lived states are included in the transport equations.The variation of the different A L 2π lifetimes τ i from their QED values τ QED i [36] is done via one common factor α = τ i /τ QED i .All the probabilities involved in these calculations are averaged over the spectrum of the observed atoms (Figure 3).
The solid line curve in Figure 8 presents the breakup probability P tot br (Pt) (see (3)) as a function of the factor α. The error band (dashed curves) accounts for the different extrapolations of the state population for n ≥ 10 (Section 4).The horizontal lines are the measured value P tot br (Pt) together with the total errors (4).This value corresponds to α = 0.376 +0.804  −0.183 .As α = 1 is included in the error band, one concludes that the measured lifetime does not contradict the QED calculations.The lifetime of the 2p state, which is the shortest-lived of all long-lived states, is found to be τ 2p = 0.44 +0.94 −0.21 tot • 10 −11 s.This value is in agreement with the calculation in QED, τ 2p = 1.17 The magnetic field between the Be target and the Pt foil transforms into an electric field in the atom c.m. system.This field mixes the wave functions of long-lived np states and shortlived ns states resulting in shortening the observed lifetime of np states.The influence of the magnetic field on the lifetime for 2p and 2s states has been evaluated in [15].The magnetic field in the gap shortens the 2p lifetime by 1.002 and even more for higher n.To estimate a maximum reduction, all np states with n > 3, around 8.6% of all long-lived states at the Be exit, have been considered as decayed in the gap.In this extreme case, the value of the 2p lifetime is τ 2p = ( 0.60 +1.34 −0.30 tot ) • 10 −11 s, not contradicting the result in (5), evaluated without taking into account the influence of the magnetic field.

Conclusion
The evaluated lifetime τ 2p = ( 0.45 +1.08 −0.30 tot ) • 10 −11 s is three orders of magnitude larger than our previously measured value of the A 2π ground state lifetime τ = ( 3.15 +0.28 −0.26 tot ) • 10 −15 s [3].This outcome opens a possibility to measure the Lamb shift of this atom and, herewith, a new ππ scattering length combination 2a 0 + a 2 .These scattering lengths have been calculated in the framework of Chiral Perturbation Theory and are nowadays investigated in Lattice QCD.

Figure 1 :
Figure 1: Method to investigate long-lived A L 2π by means of a breakup foil (side view).N A is the number of the totally produced A 2π and N L A the number of excited A L 2π with l > 0.

Figure 3 :LFigure 4 :
Figure 3: Lab momentum spectrum of A 2π produced in Be, as used in the simulation.

Figure 5 :
Figure 5: Summed populations P Ln of all long-lived atomic states at the exit of the Be target as a function of n used for "tail" estimation.For each n, the two upper curves show the sum of state populations for the given n plus different "tail" estimations calculated from populations for n and n − 1 (exponential "tail" -•, hyperbolic "tail" -).The lower curve ( ) presents the sum of the population for the given n plus the population for n + 1 instead of the "tail".

Figure 6 :
Figure 6: Experimental Q L distribution of π + π − pairs (+ with error bar)for the beryllium (Be) target fitted by the sum of simulated distributions of "atomic", "Coulomb" and "non-Coulomb" pairs.The summed distribution of free ("Coulomb" and "non-Coulomb") pairs is shown as black line.

LFigure 7 :
Figure 7: |Q L | distribution of π + π − pairs for Q T < 2.0 MeV/c.Plot a) presents the experimental distribution (+ with statistical error bar) and the simulated background (solid line).Plot b) shows the experimental distribution after background subtraction (+ with statistical error bar) and the simulated distribution of atomic pairs from A L 2π broken up in Pt (dotted-dashed line).The fit procedure has been applied to the 2-dimensional (|Q L |, Q T ) distribution.

Figure 8 : 4 ) 7 .
Figure 8: Probability P tot br (Pt) calculated as a function of α (see text).The horizontal lines correspond to the measured value P tot br (Pt) = 0.0257 +0.0097 −0.0036 tot (4) together with the total errors.The value α = 1, which corresponds to pure QED calculations, is within the error band of the measurement.

Table 1 :
Summed (over l and m) populations of long-lived atomic states versus n.The populations at the Be target exit and at the Pt foil entry are given in % of the total number of produced A 2π .P Pt br (np) is the breakup probability of the A 2π np states in the 2.1 µm thick Pt foil.The values are calculated for the average atom momentum 4.44 GeV/c and the ground state lifetime τ = 3.15 • 10 −15 s.

Table 2 :
Number N C of Coulomb pairs from different analyses and fit quality as χ 2 /n (n is degree of freedom).N A is the corresponding number of atoms produced in the Be target and N tot A the total number of atoms after correction for the coplanarity cut.The errors are statistical ones.