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Investigation of + - , K + - and + K - atoms for pion-pion and pion-kaon scattering length measurements
Valeriy Yazkov1 on behalf of the DIRAC collaboration D. V. Skobeltsyn Institute of Nuclear Physics, M. V. Lomonosov Moscow State University Leninskie Gory 1, 119991 Moscow, Russia

Theory, using Low Energy QCD, calculated with high precision the and K scattering length. Lifetime of +
-

atoms are related with a difference of pion-pion s-wave scattering lengths with isospin 0 and 2: | a0 - a2 |.
-

Energy splitting between p-states (long-lived atoms) and s-states of + lengths with isospin 1/2 and 3/2: | a

atoms allows to investigate another

combination: 2a0 + a2 . Lifetime of K atoms gives information about difference of pion-kaon s-wave scattering
1/2

-a

3/2

|. Experimental measurements of pion-kaon lifetimes and observa-

tion of long-lived + - atoms at experiment DIRAC are presented.

1

Introduction

Chiral Perturbation Theory (ChPT) describes QCD processes at low energies. ChPT in 2-loop approximation and Roy equation predicts s-wave scattering lengths with isosprin 0 and 2 to be [1]: a0 = 0.220 ± 2.3%, a2 = -0.0444 ± 2.3%, a0 - a2 = 0.265 ± 1.5% . (1)

Also there are predictions for s-wave K scattering with isosprin 1/2 and 3/2, which have done by ChPT in 1-loop approximation [2, 3]: a
1/2

= 0.19 ± 0.02, a

3/2

= -0.05 ± 0.02, a

1/2

-a

3/2

= 0.23 ± 0.01 .

(2)

ChPT with L(2) , L(4) , L(6) in 2-loop approximation predicts s-wave scattering length difference to be [4]: a1/2 - a3/2 = 0.267. Another prediction for scattering length difference have been obtained, using RoySteiner equations [5]: a
1/2

-a

3/2

= 0.269 ± 0.015 .

(3)

The K scattering has also been studied extensively in the framework of lattice QCD. Predictions for K - scattering length a1/2 = 0.183 ± 0.039, a3/2 = -0.0602 ± 0.0040 [6] and their combination a0 have been obtained [7]:
- a0 =
1

1 (a 3

1/2

-a

3/2

) = 0.0811 ± 0.0143 .

(4)

valeri.yazkov@cern.ch

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XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

The measurement of the s-wave scattering lengths would test our understanding of the chiral SU (2) L â SU (2) R symmetry breaking (u,d quarks) and measurement of the s-wave K scattering lengths would test our understanding of the chiral SU (3) L â SU (3) R symmetry breaking of QCD (u, d and s quarks). Experimental data on the K low-energy phases are absent.

2

Method of + - and K atom observation and investigation

The + - -atom ( A2 ) is a hydrogen-like atom consisting of + and - mesons. The A2 -atom lifetime (ground state 1S), = 1/ is dominated by the annihilation process into 0 0 . There is a relation [8, 9] between the width of A2 decay 1s,20 and scattering lengths for isospin 0 and 2: 1s,20 = R · | a0 - a2 |2 ( R 1.2%). Taking into account (Eq. 1), lifetime is predicted to be: = (2.9 ± 0.1) â 10-15 s. Therefore a measurement of A2 lifetime allows to measure a value for difference of s-wave scattering lengths: | a0 - a2 | [10]. In order to get values of a0 and a2 separately from + - data [10], one may exploit the fact that the energy splitting between the levels ns and n p, Ens-n p = Ens - En p , depends on another combination of the scattering lengths: 2a0 + a2 [11]. Detailed analysis of strong and electromagnetic interactions on the A2 s energy structure has been performed in [12]. Term Entlr takes into account effects from strong interaction and contributes up to 80% (-0.47 ± 0.01 eV) of the full energy shift. Other contributions from finite-size effect, self-energy corrections, vacuum polarization and relativistic insertions have been calculated with high precision. This fact provides a high sensitivity of a Ens-n p measurement to the value of the term: 2a0 + a2 . Thus it allows to obtain a value for the new combination of s-wave scattering lengths. The K-atom ( A K ) is a hydrogen-like atom consisting of K + (K - ) and - ( + ) mesons. Lifetime of this atom is dominated by the annihilation process into 0 K0 and is related with a difference of s-wave K - scattering lengths for isospin 1/2 and 3/2 [12]: 1S,0 K0 = 83 µ2 p ( a0 )2 (1 + K ). Here is the fine structure constant, µ is the reduced mass of the ± K system, p is the outgoing 0 momentum in the K atom system, and K accounts for corrections, due to isospin breaking, at order and quark mass difference (mu - md ). With prediction of scattering length difference from Eq. (3), lifetime of A K in ground state is estimated to be: = (3.5 ± 0.4) â 10-15 . A method of investigation for + - , K and other atoms, consisting from two oppositely charged mesons, has been proposed in [10]. Pairs of + - or K + (K - ) and - ( + ) mesons are producing in proton-target interactions. Pairs, which are generated from fragmentation and strong decay ("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 ( NA ) is proportional to a number of "Coulomb pairs" ( NC ) with low relative momentum Q in a pair C.M. system: NA = K · NC . The coefficient K is calculated with an accuracy better than 1%. If at least one meson is generated from long-lived sources (electromagnetically or weakly decaying mesons 0 or baryons: , , Ks , . . .), then such pairs are not affected by interaction in the final states ("non-Coulomb pairs"). After production, A2 and A K travel through the target and could to annihilate into 0 0 ( 0 K0 ), 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: Pbr ( ) = n A / NA = n A /(K · NC ) [13, 14]. In Fig. 1 dependence of A K breakup probability is shown for two nickel target are

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XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

Pbr

0.5

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 2 4 6 8 10 12 14 16 18 20 , fs

Figure 1: Dependence of the breakup probability Pbr on A K 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. used in experiment DIRAC for pair laboratory momentum range 4.8 Â 7.6 GeV/c. Value is averaged, using experimentally measured spectrum of atoms. To observe atoms in states with orbital momentum l > 0 this approach has been modified. In inclusive processes A2 are produced in s-states distributed over the principal quantum number n according to n-3 . When moving inside the target, relativistic A2 interact with the electric field of target atoms, and some of L L them ( NA ) leave the target with an orbital quantum number l > 0 ( A2 ). For these states the decays are suppressed, due to 0 value of wave function for small distances, where strong interaction acts. Therefore, the decay mechanism of such excited states is the radiative deexcitation to an ns state, annihilating subL sequently into two 0 . Thus, the A2 decay probability is given by the shortest radiative lifetime, the 2 p -11 s. For an average A lifetime 2 p = 1.17 · 10 16), the decay lengths are 2 momentum of 4.5 GeV/c ( 5.7 cm (2 p), 19 cm (3 p) and 43 cm (4 p). Using a 100µm thick Be target and inserting a 2µm thick Pt foil L downstream of this target [15], a large fraction of the long-lived atoms A2 , generated in Be, reaches the Pt L of atomic pairs (see Fig. 2). foil and breaks up, thus providing an extra number n A

3 DIRAC setup
DIRAC setup was created to detect + - with small relative momenta [16]. In 2004-2006 it has been modified in order to detect both + - and K pairs [17]. In 2011-2012 years Pt foil and additional permanent magnet between Be target and Pt foil (see Fig. 2) have been added for searching long-lived A2 [18].

4

Investigation of + K - and K + - atoms

Experimental distribution of + K - and K + - have been analysed and total number of K "atomic pairs" +- +- is found to be [17]: nK + n K = 178 ± 49. Effect-to-error ratio (3.6) is not sufficient for observation, A A 3


XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

Be target 103µm

Magne6c field Pt-
foil 2.1µm 100 mm

+
QY = 12.9MeV/c

Coulomb, non-Coulomb & atomic pairs

p

Magne6c B Field =0.25T
X

-
Breakup

Excitation: 1s, 2s ... Þ 2p, 3p, 4p ...

+
QY = 2.3MeV/c

p
Be

Long-lived states

Atomic pairs: n

l A

Pt
l N A 0.06 N A

-

l N A 0.03 N

A

Figure 2: Method to observe long-lived A but these statistic allow to make estimation of A = (2.5

L 2

by means of a breakup foil (Pt).

K

atom lifetime:

+3.0 +0.3 -1.8 |st at -0.1 |syst

) f s = (2.5

+3.0 -1.8 |to t

) fs.

(5)

Lifetime estimation (Eq. 5) provides the first model-independent estimation of s-wave K scattering length combination (Eq. 4):
- | a0 | M = 0.107 +0.093 -0.035

= 0.11

+0.09 -0.04

(6)

5

Observation of long-lived + - atoms

The event reconstruction has been performed by means of the DIRAC + - analysis software already used for the analysis of the 2001-2003 data [19] and 2008-2010 [17]. One modification of procedure has been induced by presence of additional permanent magnet between Be target and Pt foil. Horizontal field of this magnet shifts QY projection of relative momentum Q by 12.9 MeV/c for pairs generated in Be target and by 2.3 MeV/c for pairs generated in Pt foil (Fig. 2). Therefore new definition of transverse component of relative momentum has been used: Q T = Q2 + ( QY - 2.3 MeV /c)2 . To improve background conditions the X procedure selects events with small values Q T . One-dimensional (over | Q L |) and 2-dimensional (| Q L |, Q T ) distributions of experimental data have been analysed, using simulated distributions of "atomic pairs"

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XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

L from ionisation of A2 in Pt foil, "Coulomb" and "non-Coulomb" pairs, generated in Be target. Simulation takes into account initial distributions of pairs over pair laboratory momentum and relative momentum Q and its projections, multiples scattering in a target, Pt foil, the setup detectors and partitions, detector resolution and tracking efficiency.

Results of fit procedure for 1- and 2-dimensional distributions are collected in Table 1. It is shown that for different criteria on Q T amount of background events differs up to 17 times, but a total number of "atomic pairs", obtained as detected number of "atomic pairs" n L divided by efficiency of selection criterion, coinA cide within statistics. It demonstrates stability of results. Table 1: Analysis of data collected in 2012 for different Q T cuts. The detected numbers n L of atomic pairs A and the corresponding total numbers n L, tot (via selection efficiency) are presented together with the backA ground contribution (Coulomb, non-Coulomb and accidental pairs) and the fit quality 2 /n (n = degrees of freedom). Errors are only statistical. Q T cut (MeV/c) n
L A

n

L, t o t A T

Background

2 /n

Fit over | Q L |, Q 2.0 436 ± 57 488 ± 64 Fit over | Q L | 467 ± 88 489 ± 75 454 ± 91 495 ± 117

16790

138/140

0.5 1.0 1.5 2.0

152 ± 29 349 ± 53 386 ± 78 442 ± 105

971 3692 9302 16774

29/27 19/27 22/27 22/27

Systematic effects have been analysed and systematic error is found to be 22 [20]. From the 2-dimensional analysis the evaluated number of atomic pairs is n deviations, taking into account statistical as well as systematic errors.
L A

= 436 ± 61 or 7.1 standard

6

Summary

The analysis of K pairs statistic, collected from 2008 to 2010, allows to evaluate the number of atomic K pairs (178 ± 49) as well as the number of produced K atoms (653 ± 42) and thus the breakup (ionisation) probability. Value of K atom lifetime has been extracted to be = (2.5+3.0 ) f s. It provides a measurement of the -1.8 - - S-wave isospin-odd K scattering length: | a0 | = (0.11+0.09 ) · M 1 . -0.04 Analysis of data collected in 2012 with Be-Pt target allows to make observation of "atomic pairs" from + - atoms in long-lived states: n L = 436 ± 61. It provides possibility to plan experiments for measureA ment of "Lamb shift like" effect in + - system. Acknowledgments

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XXIInd International Workshop "High-Energy Physics and Quantum Field Theory", June 24 ­ July 1, 2015, Samara, Russia

We are grateful to CERN for continuous support and the PS team for the excellent performance of the accelerator. This work was funded by CERN, INFN (Italy), INCITE and MICINN (Spain), IFIN-HH (Romania), the Ministry of Education and Science and RFBR grant 01-02-17756-a (Russia), the Grant-in-Aid from JSPS and Sentanken-grant from Kyoto Sangyo University (Japan).

References
[1] G. Colangelo et al., scattering, Nucl. Phys. B 603 (2001) 125. [2] V. Bernard, N. Kaiser and U. Meissner, Threshold Parameters of K Scattering in QCD, Phys. Rev. D43 (1991) 2757. [3] A. Roessl, Pion-Kaon Scattering near the Threshold in Chiral SU(2) Perturbation Theory, Nucl. Phys B555 (1999) 507. [4] J. Bijnens et al., K scattering in three flavour ChPT, JHEP 0405 (2004) 036. [5] P. Buttiker, S. Descotes-Genon and B. Moussallam, A new analysis of piK scattering from Roy and Steiner type equations, Eur. Phys. J. C33 (2004) 409. [6] C.B. Lang et al., K scattering for isospin 1/2 and 3/2 in lattice QCD Phys. Rev., D86 (2012) 054508. [7] K. Sasaki et al., Scattering lengths for two pseudoscalar meson systems, Phys. Rev. D89 (2014) 054502. [8] J. Uretsky and J. Palfrey, Photoproduction And Detection Of The Two Meson Bound State, Phys. Rev., 121 (1961) 1798. [9] S.M. Bilenky et al., Sov. J. Nucl. Phys. 10 (1969) 469. [10] L. Nemenov, Elementary Relativistic Atoms, Sov. J. Nucl. Phys. 41 (1985) 629. [11] G.V. Efimov, M.A. Ivanov and V.E. Lyubovitskij, Sov. J. Nucl. Phys. 44 (1986) 296. [12] J. Schweizer, Decay widths and energy shifts of and K atoms, Phys. Lett. B 587 (2004) 33. [13] L. Afanasyev and A. Tarasov, Phys. At. Nucl. 59 (1996) 2130. [14] M. Zhabitsky, Direct calculation of the probability of pionium ionization in the target, Phys. At. Nucl. 71 (2008) 1040 [hep-ph/0710.4416]. [15] B. Adeva et al., Search for Long-Lived States of + - Atoms : Addendum to the DIRAC Proposal, CERNSPSC-2011-001 (2011), cds.cern.ch/record/1319290. [16] B. Adeva et al., DIRAC: A high resolution spectrometer for pionium detection, Nucl. Instrum. Methods A 515 (2003) 467. [17] B. Adeva et al., First K atom lifetime and K scattering length measurements, Phys. Lett. B 735 (2014) 288. [18] B. Adeva et al., First observation of long-lived + - atoms, Phys. Lett. B 751 (2015) 12. [19] B. Adeva et al., Determination of scattering lengths from measurement of + - atom lifetime, Phys. Lett. B 704 (2011) 24. [20] V. Yazkov, Investigation of systematic errors of metastable "atomic pair" number, DN-2015-02 (2015), cds.cern.ch/record/2012230.

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