Decay widths and energy shifts of ππ and πK atoms

Physics Letters B 587 (2004) 33–40 www.elsevier.com/locate/physletb

Decay widths and energy shifts of π π and π K atoms J. Schweizer Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland Received 20 January 2004; accepted 2 March 2004 Editor: W.-D. Schlatter

Abstract We calculate the S-wave decay widths and energy shifts for π + π − and π ± K ∓ atoms in the framework of QCD + QED. The evaluation—valid at next-to-leading order in isospin symmetry breaking—is performed within a non-relativistic effective field theory. The results are of interest for future hadronic atom experiments.  2004 Published by Elsevier B.V. PACS: 03.65.Ge; 03.65.Nk; 11.10.St; 12.39.Fe; 13.40.Ks Keywords: Hadronic atoms; Chiral perturbation theory; Non-relativistic effective Lagrangians; Isospin symmetry breaking; Electromagnetic corrections

1. Introduction Nearly fifty years ago, Deser et al. [1] derived the formulae for the decay width and strong energy shift of pionic hydrogen at leading order in isospin symmetry breaking. Similar relations also hold for π + π − [2] and π − K + atoms, which decay predominantly into 2π 0 and π 0 K 0 , respectively. These Deser-type relations allow to extract the scattering lengths from measurements of the decay width and the strong energy shift. The DIRAC Collaboration [3] at CERN intends to measure the lifetime of pionium in its ground state at the 10% level, which will allow to extract the scattering length difference |a00 − a02 | at 5% accuracy. The experimental result can then be compared with theoretical predictions for the S-wave scattering lengths E-mail address: [email protected] (J. Schweizer). 0370-2693/$ – see front matter  2004 Published by Elsevier B.V. doi:10.1016/j.physletb.2004.03.007

[4–6] and with the results from other experiments [7]. Particularly interesting is the fact that one may determine in this manner the nature of the SU(2) × SU(2) spontaneous chiral symmetry breaking experimentally [8]. New experiments are proposed for CERN PS and J-PARC in Japan [9]. In order to determine the scattering lengths from such experiments, the theoretical expressions for the decay width and the strong energy shift must be known to an accuracy that matches the experimental precision. For this reason, the ground state decay width of pionium has been evaluated at next-to-leading order [10–15] in the isospin symmetry breaking parameter δ, where both the fine-structure constant α and (mu − md )2 count as O(δ). The aim of the present Letter is to provide the corresponding formulae for the S-wave decay widths and strong energy shifts of pionium and the π ± K ∓ atom at nextto-leading order in isospin symmetry breaking. A detailed derivation of the results will be provided else-

34

J. Schweizer / Physics Letters B 587 (2004) 33–40

where [16]. The strong energy shift of the π ± K ∓ atom is proportional to the sum of the isospin even and odd S-wave πK scattering lengths a0+ + a0− . This sum [18–22] is sensitive to the combination of low-energy constants 2Lr6 + Lr8 [23]. The consequences of this observation for the SU(3) × SU(3) quark condensate [24] remain to be worked out. 2. Non-relativistic framework The non-relativistic effective Lagrangian framework has proven to be a very efficient method to investigate bound state characteristics [12,15,25,26]. The non-relativistic Lagrangian is exclusively determined by symmetries, which are rotational invariance, parity and time reversal. It provides a systematic expansion in powers of the isospin breaking parameter δ. What concerns the π − K + atom, we count both α and mu − md as order δ. The different power counting for the π + π − and π − K + atoms are due to the fact that in QCD, the chiral expansion of the pion mass difference ∆π = Mπ2 + − Mπ2 0 is of second order in mu − md , 2 − M2 while the kaon mass difference ∆K = MK + K0 starts at first order in mu − md . In the sector with one or two mesons, the non-relativistic πK Lagrangian is LNR = L1 + L2 . The first term contains the one-pion and one-kaon sectors,   1
2 ∆ † 2 L1 = E − B + h0 i∂t − Mh0 + 2 2Mh0   † ∆2 + + · · · h0 + h± iDt − Mh+ 8Mh30 ±  D2 D4 + + + · · · h± , 2Mh+ 8Mh3+

(1)

˙ B = ∇ × A and the quanwhere E = −∇A0 − A, tity h = π, K stands for the non-relativistic pion and kaon fields. We work in the Coulomb gauge and eliminate the A0 component of the photon field by the use of the equations of motion. The covariant derivatives are given by Dt h± = ∂t h± ∓ ieA0h± and Dh± = ∇h± ± ieAh± , where e denotes the electromagnetic coupling. What concerns the one-pion–one-kaon sector, we only list the terms needed to evaluate the decay width and the energy shift of the π − K + atom at order

δ 9/2 and δ 4 , respectively,
 † † π− K+ + C2 π−† K+ π0 K0 + h.c. L2 = C1 π−† K+ + C3 π0† K0† π0 K0 + · · · .

(2)

The ellipsis stands for higher order terms.1 We work in the center of mass system and thus omit terms proportional to the total 3-momentum. The total and reduced masses read Σi = Mπ i + MK i , Mπ i MK i , µi = Mπ i + MK i

i = +, 0.

(3)

The coupling constant C1 contains contributions coming from the electromagnetic form factors of the pion and kaon, C1 = C1 − e2 λ,

λ=

1
 2   2  r + + rK + , 6 π

(4)

2  denote the charge radii of where rπ2 +  and rK + the charged pion and kaon, respectively. The low energy constants C1 , . . . , C3 may be determined through matching the πK amplitude at threshold for various channels, see Section 3. To evaluate the energy shift and decay width of the π − K + atom at next-to-leading order in isospin symmetry breaking, we make use of resolvents. For a detailed discussion of the technique, we refer to Ref. [15]. Here, we simply list the results. We use dimensional regularization, to treat both ultraviolet and infrared singularities. Up to and including order δ 9/2 , the decay into π 0 K 0 is the only decay channel contributing, and we get for the total S-wave decay width

 µ2 k 2 C 2 αµ2+ C1 ξn α 3 µ3+ 2 Γn = 3 2 µ0 k0 C2 1 − 0 0 2 3 − π n π 4π 3 3 2
 5µ0 k0 Mπ 0 + MK 0 + O δ5 , + 3 3 8 Mπ 0 MK 0

(5)

where k0 = [2µ0(Σ+ − Σ0 − α 2 µ+ /(2n2 ))]1/2 is of order δ 1/2 . The function ξn develops an ultraviolet sin1 The basis of operators containing two space derivatives can be

chosen such that none of them contributes to the energy shift and decay width at next-to-leading order in isospin symmetry breaking [16].

J. Schweizer / Physics Letters B 587 (2004) 33–40

35

±;± Fig. 1. Non-relativistic π − K + → π − K + scattering amplitude. The blob describes the vector form factor of the pion and kaon. T¯NR denotes the truncated amplitude.

gularity as d → 3,



ξn = Λ(µ) − 1 + 2 ln

2µ+ α + ln n µ

1 , + ψ(n) − ψ(1) − n  1 − ln 4π − Γ  (1) , Λ(µ) = µ2(d−3) d −3

(6)

with ψ(n) = Γ  (n)/Γ (n) and the running scale µ. At order δ 4 , the total energy shift may be split into a strong part and an electromagnetic part, according to En = Enh + Enem .

(7)

where the S-wave scattering lengths2 a0+ = 1/3(a0 + 3/2 1/2 3/2 2a0 ) and a0− = 1/3(a0 − a0 ) are defined in QCD, . . at mu = md and Mπ = Mπ + , MK = MK + . By substituting these relations into the expression for the decay width (5) and the strong energy shift (8), one obtains the Deser-type formulae [1,2]. We demonstrate the matching at next-to-leading order in δ by means of the π − K + → π − K + amplitude. In the presence of virtual photons, we first have to subtract the onephoton exchange diagram from the full amplitude, as displayed in Fig. 1. The coupling constant C1 is deter±;± , which contains an mined by the truncated part T¯NR infrared singular Coulomb phase θc as d → 3, 1/2

For the discussion of the electromagnetic energy shift, we refer to Section 4. The strong S-wave energy shift reads  µ20 k02 2 αµ2+ 2 α 3 µ3+ C − ξ − C C C Enh = − 1 n 3 πn3 2π 1 4π 2 2
5 +O δ . (8) The results for the decay width (5) and energy shift (8) are valid at next-to-leading order in isospin symmetry breaking.

3. Matching the low-energy constants The coupling constants Ci can be determined through matching the non-relativistic and the relativistic amplitudes at threshold. The coupling C3 is needed at order δ 0 only. However, we have to determine both C1 and C2 at next-to-leading order in isospin symmetry breaking. The relativistic amplitudes are related to the non-relativistic ones through TRlm;ik (q; p) 1/2 lm;ik

TNR (q; p), = 4 ωi (p)ωk (p)ωl (q)ωm (q)

with ωi (p) = (Mi2 + p2 )1/2 . The 3-momentum p denotes the center of mass momentum of the incoming particles, q the one of the outgoing particles. The effective Lagrangian in Eqs. (1) and (2), allows us to evaluate the non-relativistic π − K + → π 0 K 0 and π − K + → π − K + scattering amplitudes at threshold at order δ. In the isospin symmetry limit, the effective couplings C1 , C2 and C3 are √ 2π
+ 2 2π − − C1 = a + a0 , C2 = − a , µ+ 0 µ+ 0 2π + a , C3 = (10) µ+ 0

(9)

±;± ±;± T¯NR (p; p) = e2iαθc TˆNR (p; p),  1

µ+ d−3 1 µ − ln 4π + Γ  (1) θc = |p| d −3 2

2|p| . + ln µ

(11)

±;± At order δ, the remainder TˆNR is free of infrared sin±;± is given gularities at threshold. The real part of TˆNR 2 a + and a − are normalized as in Ref. [18]. 0 0

36

J. Schweizer / Physics Letters B 587 (2004) 33–40

by

where

±;± (p; p) Re TˆNR

Kn =

=

B1 |p| 1 + Re A±;± + B2 ln thr |p| µ+ 4Mπ + MK + + O(p),

(12)

with B1 = C1 απµ+ + o(δ), B2 = −C12 αµ2+ /π + o(δ) and

C22 C3 µ30 (13) (Σ+ − Σ0 ) + o(δ). 2π 2 Here, the ultraviolet pole term Λ(µ) is removed by renormalizing the coupling C1 . The renormalization of C1 eliminates at the same time the ultraviolet divergence contained in the expression for the energy shift (8). The calculation of the relativistic π − K + → π − K + scattering amplitude was performed at O(p4 , e2 p2 ) in Refs. [20,21]. Both the Coulomb phase and the logarithmic singularity in Eq. (12) are absent in the real part of the relativistic amplitude at this order of accuracy, they first occur at order e2 p4 . The quantity Re A±;± thr denotes the constant term occurring in the real part of the truncated relativistic threshold amplitude. The coupling constant C2 may be determined analogously by matching the nonrelativistic π − K + → π 0 K 0 amplitude to the relativistic one at order δ. −

4. Results for the π − K + atom The result for the decay width and strong energy shift are valid at next-to-leading order in isospin symmetry breaking, and to all orders in the chiral expansion. We get for the decay width at order δ 9/2 , in terms of the relativistic π − K + → π 0 K 0 threshold amplitude, 8 3 2 ∗ 2 α µ+ pn A (1 + Kn ), n3 1 1 Re A00;± A=− √ thr + o(δ), 8 2π Σ+

The outgoing relative 3-momentum pn∗ =

1 Re A±;± thr 4Mπ + MK +  C1 αµ2+ 4µ2 = C1 1 + 1 − Λ(µ) − ln 2+ 2π µ

Γn =

(14)

Mπ + ∆K + MK + ∆π
+ 2 a0 Mπ + + MK + 
 1 α − 4αµ+ a0+ + a0− ψ(n) − ψ(1) − + ln n n + o(δ). (15)  1
2 2 2 1/2 λ En , Mπ 0 , MK , 0 2En

(16)

with λ(x, y, z) = x 2 + y 2 + z2 − 2xy − 2xz − 2yz, is chosen such that the total final state energy corresponds to En = Σ+ − α 2 µ+ /(2n2 ). The quantity is calculated as follows. One evaluates the Re A00;± thr relativistic π − K + → π 0 K 0 amplitude near threshold and removes the divergent Coulomb phase. The real part contains singularities ∼ 1/|p| and ∼ ln |p|/µ+ . The constant term in this expansion corresponds to Re A00;± thr . The normalization is chosen such that A = a0− + .

(17)

The isospin breaking corrections  have been evaluated at O(p4 , e2 p2 ) in Refs. [21,27]. See also the comments in Section 6. We now discuss the various energy shift contributions. According to Eq. (7), the energy shift at order δ 4 is split into an electromagnetic part Enem and the strong part Enh in Eq. (8). The electromagnetic energy shift contains both pure QED corrections as well as finite size effects due to the charge radii of the pion and kaon, contained in λ. The pure electromagnetic corrections have been evaluated in Ref. [28] for arbitrary angular momentum l. We checked3 that the electromagnetic energy shift at order α 4 indeed amounts to   1 α 4 µ+ 3µ+ 3 em − Enl = 1 − n3 Σ+ 8n 2l + 1 4α 4 µ3+ λ δl0 n3  α 4 µ2+ 1 1 3 + δl0 + 4 − 3 Σ+ n3 n n (2l + 1)
5  + O α ln α . (18) +

3 We thank A. Rusetsky for a very useful communication

concerning technical aspects of the calculation.

J. Schweizer / Physics Letters B 587 (2004) 33–40

Here, the first term is generated by the mass insertions, the second contains the finite size effects and the last stems from the one-photon exchange contribution. The strong S-wave energy shift reads at order δ 4 , 2α 3 µ2+  Enh = − A (1 + Kn ), n3 1 Re A±;± thr + o(δ), 8πΣ+

37

and 2P states is given by E2s−2p em em vac vac = E2h + E20 − E21 + E20 − E21

= −1.4 ± 0.1 eV.

(22)

with

The uncertainty displayed is the one in E2h only. For the numerical values of the various energy shift contributions, see Table 2 in Section 6.

Kn

5. Results for pionium

A =

(19)


+ α 1 − = −2αµ+ a0 + a0 ψ(n) − ψ(1) − + ln n n + o(δ). (20)

In the isospin limit, the normalized relativistic amplitude A = a0+ + a0− +   ,

(21)

reduces to the sum of the isospin even and odd scattering lengths. The corrections   have been obtained at O(p4 , e2 p2 ) in Refs. [20,21]. See also the comments in Section 6. The result for E1h in Eq. (19) agrees with the one obtained for the strong energy shift of the ground state in pionic hydrogen [26], if we replace µ+ with the reduced mass of the π − p atom and Re A±;± thr with the constant term in the threshold expansion for the real part of the truncated π − p → π − p amplitude. What remains to be added are the vacuum polarization contributions [14,29], which are formally of higher order in α, however numerically not negligible. The vacuum polarization leads to an energy level shift vac as well as to a change in the Coulomb wave Enl function of the π − K + atom at the origin δψK,n (0). vac For the first two energy levels, Enl [14,29] is given numerically in Table 2, Section 6. Formally of order α 2l+5 , this contribution is enhanced due to its large coefficient containing (µ+ /me )2l+2 . The modified Coulomb wave function affects both, the decay width and the strong energy shift, see Section 6. As discussed in Section 6, the electromagnetic contributions (18) are known to a high precision. Further, the strong shift in the nP state is very much suppressed (order α 5 ). A future measurement of the energy splitting between the nS and nP states will therefore allow to extract the strong S-wave energy shift in Eq. (19), and to determine the combination a0+ + a0− of the πK scattering lengths. The energy splitting between the 2S

The decay rate and strong energy shift of pionium can be obtained from the formulae in Eqs. (5) and (8) through the following substitutions of the masses MK + → Mπ + , MK 0 →√Mπ 0 and the coupling constants C1 → c1 , C2 → 2(c2 − 2c4 ∆π ) and C3 → 2c3 [16]. The ci are the low-energy constants defined in Ref. [15]. The S-wave decay width of the π + π − atom reads at order δ 9/2 , in terms of the relativistic π + π − → π 0 π 0 threshold amplitude, Γπ,n =

2 3 ∗ α pπ,n A2π (1 + Kπ,n ), 9n3

Aπ = a00 − a02 + π , 2 κ
Kπ,n = a00 + 2a02 9   α 2α
0 1 2 2a0 + a0 ψ(n) − ψ(1) − + ln − 3 n n + o(δ),   α 2 2 1/2 ∗ , pπ,n = ∆π − 2 Mπ + (23) 4n where κ = Mπ2 + /Mπ2 0 − 1. The quantity Aπ is defined as in Refs. [13,15]. The isospin symmetry breaking corrections π have been evaluated at O(p4 , p2 e2 ) in Refs. [13,15,30]. For the decay width of the ground state at order δ 9/2 , we reproduce the result obtained in Refs. [11,13,15]. The electromagnetic energy shift em is obtained from Eq. (18) through the above Eπ,nl mass substitutions and λ → 1/3rπ2 + . Finally, the S-wave energy shift of the π + π − atom reads at order δ 4 , in terms of the relativistic one-particle irreducible π + π − → π + π − amplitude at threshold,

38

J. Schweizer / Physics Letters B 587 (2004) 33–40

Table 1 Next-to-leading order corrections to the Deser-type formulae π + π − atom π ± K ∓ atom

δh,1

 δh,1

 δh,2

(5.8 ± 1.2) × 10−2 (4.0 ± 2.2) × 10−2

(6.2 ± 1.2) × 10−2 (1.7 ± 2.2) × 10−2

(6.1 ± 1.2) × 10−2 (1.5 ± 2.2) × 10−2

α 3 Mπ +   Aπ (1 + Kπ,n ), n3  1
Aπ = 2a00 + a02 + π , 6  α
 Kπ,n = − 2a00 + a02 3  1 α × ψ(n) − ψ(1) − + ln + o(δ), n n

within the uncertainties quoted in [21]. In the follow ing, we use [21]  = (0.1 ± 0.1) × 10−2 Mπ−1 + and  = −1 −2 (0.1 ± 0.3) × 10 Mπ + . For the charge radii of the pion and kaon, we take rπ2 +  = (0.452 ± 0.013) fm2 2  = (0.363 ± 0.072) fm2 [33]. and rK + We obtain for the decay width of the ground state,

h Eπ,n =−

(24)

where Aπ is defined analogously to the quantity A discussed in Section 4. The isospin symmetry breaking contributions π have been calculated at O(e2 p2 ) in Refs. [31,32]. For pionium the energy splitting between the 2S and 2P states reads

2 2α 3 ∗
0 (26) p a − a02 (1 + δπ,1 ), 9 π,1 0 where the corrections δh,1 , h = π, K are given in Table 1. The strong energy shift reads Γπ,1 =

 2α 3 µ2+
+  ), a0 + a0− (1 + δK,n 3 n  α 3 Mπ +
0 h  Eπ,n (27) =− ). 2a0 + a02 (1 + δπ,n 3 6n  are For the first two energy levels, the corrections δh,n specified in Table 1. As mentioned in Section 4, these corrections to the Deser-type formulae are modified by vacuum polarization,

Enh = −

h Eπ,2s−2p = Eπ,2

em em + Eπ,20 − Eπ,21 vac vac + Eπ,20 − Eπ,21

= −0.59 ± 0.01 eV.

Γ1 = 8α 3 µ2+ p1∗ (a0− )2 (1 + δK,1 ),

(25)

h Again the uncertainty displayed is the one in Eπ,2 only. The numerical values for the various energy shifts are listed in Table 3, Section 6.

6. Numerical analysis

vac , δh,n → δh,n + δh,n

  vac δh,n → δh,n + δh,n ,

(28)

where For the S-wave ππ scattering lengths, we use the chiral predictions a00 = 0.220 ± 0.005 and a02 = −0.0444 ± 0.0010 [5,6]. The correlation matrix for a00 and a02 is given in Ref. [6]. For the isospin symmetry breaking corrections to the ππ threshold amplitudes (23) and (24), we use π = (0.61 ± 0.16) × 10−2 and π = (0.37 ± 0.08) × 10−2 as given in Ref. [15,32], respectively. For the πK scattering lengths, we use the values from the recent analysis of data and Roy– Steiner equations [22], a0+ = (0.045 ± 0.012)Mπ−1 + and a0− = (0.090 ± 0.005)Mπ−1 . The correlation parameter + for a0+ and a0− is given in Ref. [22]. The isospin breaking corrections to the πK threshold amplitudes (17) and (21) have been worked out in [20,21,27]. Whereas the analytic expressions for  and   obtained in [20, 21,27] are not identical, the numerical values agree

vac δh,n =

2δψh,n (0) . ψh,n (0)

(29)

vac Formally, the contribution δh,n is of order α 2 , but enhanced because of the large coefficient containing µ+ /me . For the ground state, the corrections [14] vac vac yield δK,1 = 0.45 × 10−2 and δπ,1 = 0.31 × 10−2 . vac  The changes in δπ,1 and δπ,1 due to δπ,1 are about  5%. For δK,1 however, the correction amounts to 27%. vac , because the Here, we omit the contributions from δh,n  are much larger than δ vac . uncertainties in δh,n and δh,n h,n . The numerical values for the lifetime τ1 = Γ1−1 , . −1 (τπ,1 = Γπ,1 ) and the energy shifts at next-to-leading order in isospin symmetry breaking are given in Tables 2 and 3. The energy shifts due to vacuum povac are taken from Ref. [14,29]. In the larization Enl

J. Schweizer / Physics Letters B 587 (2004) 33–40

39

Table 2 Numerical values for the energy shift and the lifetime of the π ± K ∓ atom π ± K ∓ atom

em [eV] Enl

vac [eV] Enl

Enh [eV]

τn [s]

n = 1, l = 0 n = 2, l = 0 n = 2, l = 1

−0.095 −0.019 −0.006

−2.56 −0.29 −0.02

−9.0 ± 1.1 −1.1 ± 0.1

(3.7 ± 0.4) × 10−15

Table 3 Numerical values for the energy shift and the lifetime of the π + π − atom π + π − atom

em [eV] Eπ,nl

vac [eV] Eπ,nl

h [eV] Eπ,n

τπ,n [s]

n = 1, l = 0 n = 2, l = 0 n = 2, l = 1

−0.065 −0.012 −0.004

−0.942 −0.111 −0.004

−3.8 ± 0.1 −0.47 ± 0.01

(2.9 ± 0.1) × 10−15

evaluation of the uncertainties, the correlations between the S-wave scattering lengths have been taken into account. For the decay width and the strong energy shift of the π ± K ∓ atom, the dominant source of uncertainty is due to the uncertainties in the scattering lengths a0+ and a0− . We do not display the error bars for the electromagnetic energy shifts, which stem 2  at order α 4 from the uncertainties in rπ2 +  and rK + em only. For pionium, the uncertainties of Eπ,10 at order α 4 amount to about 0.7%, while for the π ± K ∓ atom em is known at the 5% level. To estimate the or E10 der of magnitude of the electromagnetic corrections at higher order, we may compare with positronium. Here, the α 5 and α 5 ln α corrections [34] amount to about 2% with respect to the α 4 contributions.

7. Summary and conclusions We provided the formulae for the energy shifts and decay widths of the π + π − and π ± K ∓ atoms at nextto-leading order in isospin symmetry breaking. To confront these predictions with data presents a challenge for future hadronic atom experiments. Should it turn out that these predictions are in conflict with experiment, one would have to revise our present understanding of the low-energy structure of QCD.

Acknowledgements It is a great pleasure to thank J. Gasser for many interesting discussions and suggestions. Further, I thank

R. Kaiser, A. Rusetsky, H. Sazdjian and J. Schacher for useful discussions as well as for helpful comments on the manuscript. This work was supported in part by the Swiss National Science Foundation and by RTN, BBW-Contract No. 01.0357 and EC-Contract HPRNCT2002-00311 (EURIDICE). References [1] S. Deser, M.L. Goldberger, K. Baumann, W. Thirring, Phys. Rev. 96 (1954) 774; T.L. Trueman, Nucl. Phys. 26 (1961) 57. [2] T.R. Palfrey, J.L. Uretsky, Phys. Rev. 121 (1961) 1798; S.M. Bilenky, V.H. Nguyen, L.L. Nemenov, F.G. Tkebuchava, Yad. Fiz. 10 (1969) 812. [3] B. Adeva, et al., CERN-SPSLC-95-1. [4] S. Weinberg, Phys. Rev. Lett. 17 (1966) 616; J. Gasser, H. Leutwyler, Phys. Lett. B 125 (1983) 321; J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, M.E. Sainio, Phys. Lett. B 374 (1996) 210, hep-ph/9511397; J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, M.E. Sainio, Nucl. Phys. B 508 (1997) 263; J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, M.E. Sainio, Nucl. Phys. B 517 (1998) 639, Erratum, hep-ph/9707291; J. Nieves, E. Ruiz Arriola, Eur. Phys. J. A 8 (2000) 377, hepph/9906437; G. Amoros, J. Bijnens, P. Talavera, Nucl. Phys. B 585 (2000) 293; G. Amoros, J. Bijnens, P. Talavera, Nucl. Phys. B 598 (2001) 665, Erratum, hep-ph/0003258. [5] G. Colangelo, J. Gasser, H. Leutwyler, Phys. Lett. B 488 (2000) 261, hep-ph/0007112. [6] G. Colangelo, J. Gasser, H. Leutwyler, Nucl. Phys. B 603 (2001) 125, hep-ph/0103088. [7] L. Rosselet, et al., Phys. Rev. D 15 (1977) 574; C.D. Froggatt, J.L. Petersen, Nucl. Phys. B 129 (1977) 89; M.M. Nagels, et al., Nucl. Phys. B 147 (1979) 189; A.A. Bolokhov, M.V. Polyakov, S.G. Sherman, Eur. Phys. J. A 1 (1998) 317, hep-ph/9707406;

40

[8]

[9]

[10]

[11]

[12]

[13]

J. Schweizer / Physics Letters B 587 (2004) 33–40

D. Pocanic, hep-ph/9809455; J.B. Lange, et al., CHAOS Collaboration, Phys. Rev. Lett. 80 (1998) 1597; J.B. Lange, et al., Phys. Rev. C 61 (2000) 025201; NA48 Collaboration, R. Batley, et al., CERN-SPSC-2000-003; S. Pislak, et al., BNL-E865 Collaboration, Phys. Rev. Lett. 87 (2001) 221801, hep-ex/0106071; S. Descotes-Genon, N.H. Fuchs, L. Girlanda, J. Stern, Eur. Phys. J. C 24 (2002) 469, hep-ph/0112088; S. Pislak, et al., Phys. Rev. D 67 (2003) 072004, hep-ex/ 0301040; V.N. Maiorov, O.O. Patarakin, hep-ph/0308162. M. Knecht, B. Moussallam, J. Stern, N.H. Fuchs, Nucl. Phys. B 457 (1995) 513, hep-ph/9507319; M. Knecht, B. Moussallam, J. Stern, N.H. Fuchs, Nucl. Phys. B 471 (1996) 445, hep-ph/9512404; L. Girlanda, M. Knecht, B. Moussallam, J. Stern, Phys. Lett. B 409 (1997) 461, hep-ph/9703448. B. Adeva, et al., CERN-SPSC-2000-032; L. Afanasyev, et al., Lifetime measurement of π + π − and π ± K ∓ atoms to test low energy QCD (2002), letter of intent of the experiment at Japan Proton Accelerator Research Complex, available at http://dirac.web.cern.ch/DIRAC/future.html. Z.K. Silagadze, JETP Lett. 60 (1994) 689, hep-ph/9411382; A. Gashi, G. Rasche, G.C. Oades, W.S. Woolcock, Nucl. Phys. A 628 (1998) 101, nucl-th/9704017; Z.K. Silagadze, hep-ph/9803307; M.A. Ivanov, V.E. Lyubovitskij, E.Z. Lipartia, A.G. Rusetsky, Phys. Rev. D 58 (1998) 094024, hep-ph/9805356; P. Minkowski, Proceedings of the International Workshop Hadronic Atoms and Positronium in the Standard Model, Dubna, 26–31 May, 1998; M.A. Ivanov, et al., (Eds.), Dubna, 1998, p. 74, hep-ph/ 9808387; E.A. Kuraev, Phys. At. Nucl. 61 (1998) 239; U. Jentschura, G. Soff, V. Ivanov, S.G. Karshenboim, Phys. Lett. A 241 (1998) 351; H.W. Hammer, J.N. Ng, Eur. Phys. J. A 6 (1999) 115, hepph/9902284. H. Jallouli, H. Sazdjian, Phys. Rev. D 58 (1998) 014011; H. Jallouli, H. Sazdjian, Phys. Rev. D 58 (1998) 099901, Erratum, hep-ph/9706450; H. Sazdjian, Phys. Lett. B 490 (2000) 203, hep-ph/0004226. P. Labelle, K. Buckley, hep-ph/9804201; X. Kong, F. Ravndal, Phys. Rev. D 59 (1999) 014031; X. Kong, F. Ravndal, Phys. Rev. D 61 (2000) 077506, hepph/9905539; B.R. Holstein, Phys. Rev. D 60 (1999) 114030, nuclth/9901041; A. Gall, J. Gasser, V.E. Lyubovitskij, A. Rusetsky, Phys. Lett. B 462 (1999) 335, hep-ph/9905309; D. Eiras, J. Soto, Phys. Rev. D 61 (2000) 114027, hep-ph/ 9905543; V. Antonelli, A. Gall, J. Gasser, A. Rusetsky, Ann. Phys. 286 (2001) 108, hep-ph/0003118. J. Gasser, V.E. Lyubovitskij, A. Rusetsky, Phys. Lett. B 471 (1999) 244, hep-ph/9910438.

[14] D. Eiras, J. Soto, Phys. Lett. B 491 (2000) 101, hepph/0005066. [15] J. Gasser, V.E. Lyubovitskij, A. Rusetsky, A. Gall, Phys. Rev. D 64 (2001) 016008, hep-ph/0103157. [16] J. Schweizer, in preparation. The calculation of the same quantities in the framework of constrained Bethe–Salpeter equation is in progress [17]. [17] H. Sazdjian, private communication. [18] V. Bernard, N. Kaiser, U.G. Meissner, Phys. Rev. D 43 (1991) 2757. [19] A. Roessl, Nucl. Phys. B 555 (1999) 507, hep-ph/9904230. [20] A. Nehme, Eur. Phys. J. C 23 (2002) 707, hep-ph/0111212; B. Kubis, U.G. Meissner, Phys. Lett. B 529 (2002) 69, hepph/0112154. [21] B. Kubis, Dissertation Bonn University, 2002, Berichte des Forschungszentrums Jülich, 4007, ISSN 0944-2952. [22] P. Buttiker, S. Descotes-Genon, B. Moussallam, hep-ph/ 0310283. [23] J. Gasser, H. Leutwyler, Nucl. Phys. B 250 (1985) 465. [24] S. Descotes-Genon, L. Girlanda, J. Stern, JHEP 0001 (2000) 041, hep-ph/9910537; B. Moussallam, JHEP 0008 (2000) 005, hep-ph/0005245; S. Descotes-Genon, L. Girlanda, J. Stern, Eur. Phys. J. C 27 (2003) 115, hep-ph/0207337; S. Descotes-Genon, N.H. Fuchs, L. Girlanda, J. Stern, hep-ph/ 0311120. [25] W.E. Caswell, G.P. Lepage, Phys. Lett. B 167 (1986) 437. [26] V.E. Lyubovitskij, A. Rusetsky, Phys. Lett. B 494 (2000) 9, hep-ph/0009206. [27] B. Kubis, U.G. Meissner, Nucl. Phys. A 699 (2002) 709, hepph/0107199; A. Nehme, P. Talavera, Phys. Rev. D 65 (2002) 054023, hepph/0107299. [28] I.T. Todorov, Phys. Rev. D 3 (1971) 2351; A. Nandy, Phys. Rev. D 5 (1972) 1531; H. Jallouli, H. Sazdjian, Ann. Phys. 253 (1997) 376, hep-ph/ 9602241. [29] G.E. Pustovalov, JETP 5 (1957) 1234; S.G. Karshenboim, Can. J. Phys. 76 (1998) 169; S.G. Karshenboim, U.D. Jentschura, V.G. Ivanov, G. Soff, Eur. Phys. J. D 2 (1998) 209. [30] M. Knecht, R. Urech, Nucl. Phys. B 519 (1998) 329, hep-ph/ 9709348. [31] U.G. Meissner, G. Muller, S. Steininger, Phys. Lett. B 406 (1997) 154; U.G. Meissner, G. Muller, S. Steininger, Phys. Lett. B 407 (1997) 454, Erratum, hep-ph/9704377. [32] M. Knecht, A. Nehme, Phys. Lett. B 532 (2002) 55, hepph/0201033. [33] J. Bijnens, P. Talavera, JHEP 0203 (2002) 046, hep-ph/ 0203049; See also, S.R. Amendolia, et al., NA7 Collaboration, Nucl. Phys. B 277 (1986) 168; G. Colangelo, hep-ph/0312017. [34] R. Karplus, A. Klein, Phys. Rev. 87 (1952) 848.