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 ππ and πK 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 [4–6] and with the results from other experiments [7]. Particularly interesting is the fact that one may deter- × Nearly ﬁfty years ago, Deser et al. [1] derived the mine in this manner the nature of the SU(2) SU(2) formulae for the decay width and strong energy shift spontaneous chiral symmetry breaking experimentally of pionic hydrogen at leading order in isospin symme- [8]. New experiments are proposed for CERN PS and try breaking. Similar relations also hold for π+π− [2] J-PARC in Japan [9]. In order to determine the scat- and π−K+ atoms, which decay predominantly into tering lengths from such experiments, the theoretical 2π0 and π0K0, respectively. These Deser-type rela- expressions for the decay width and the strong en- tions allow to extract the scattering lengths from mea- ergy shift must be known to an accuracy that matches surements of the decay width and the strong energy the experimental precision. For this reason, the ground shift. The DIRAC Collaboration [3] at CERN intends state decay width of pionium has been evaluated at to measure the lifetime of pionium in its ground state next-to-leading order [10–15] in the isospin symme- at the 10% level, which will allow to extract the scat- try breaking parameter δ, where both the ﬁne-structure 2 | 0 − 2| constant α and (mu − md ) count as O(δ).Theaim tering length difference a0 a0 at 5% accuracy. The experimental result can then be compared with the- of the present Letter is to provide the corresponding oretical predictions for the S-wave scattering lengths formulae for the S-wave decay widths and strong energy shifts of pionium and the π±K∓ atom at next- to-leading order in isospin symmetry breaking. A de- E-mail address: [email protected] (J. Schweizer). tailed derivation of the results will be provided else-

0370-2693/$ – see front matter  2004 Published by Elsevier B.V. doi:10.1016/j.physletb.2004.03.007 34 J. Schweizer / Physics Letters B 587 (2004) 33–40 where [16]. The strong energy shift of the π±K∓ atom δ9/2 and δ4, respectively, is proportional to the sum of the isospin even and + − † † † † + L = C π K π−K+ + C π K π K + h.c. odd S-wave πK scattering lengths a0 a0 . This sum 2 1 − + 2 − + 0 0 [18–22] is sensitive to the combination of low-energy + † † +··· r + r C3π0 K0 π0K0 . (2) constants 2L6 L8 [23]. The consequences of this ob- servation for the SU(3) × SU(3) quark condensate [24] The ellipsis stands for higher order terms.1 We work in remain to be worked out. the center of mass system and thus omit terms proportional to the total 3-momentum. The total and reduced masses read 2. Non-relativistic framework = + Σi Mπi MKi , The non-relativistic effective Lagrangian frame- Mπi MKi work has proven to be a very efficient method to inves- µi = ,i=+, 0. (3) M i + M i tigate bound state characteristics [12,15,25,26]. The π K non-relativistic Lagrangian is exclusively determined The coupling constant C1 contains contributions com- by symmetries, which are rotational invariance, par- ing from the electromagnetic form factors of the pion ity and time reversal. It provides a systematic expan- and kaon, sion in powers of the isospin breaking parameter δ. − + What concerns the π K atom, we count both α and = − 2 = 1 2 + 2 C1 C1 e λ, λ rπ+ rK+ , (4) mu − md as order δ. The different power counting for 6 + − − + the π π and π K atoms are due to the fact that in 2 2 where r + and r + denote the charge radii of QCD, the chiral expansion of the pion mass difference π K 2 2 the charged pion and kaon, respectively. The low en- ∆π = M + − M is of second order in mu − md , π π0 ergy constants C1,...,C3 may be determined through 2 2 while the kaon mass difference ∆ = M + − M K K K0 matching the πK amplitude at threshold for various starts at first order in mu − md . In the sector with one channels, see Section 3. or two mesons, the non-relativistic πK Lagrangian is To evaluate the energy shift and decay width of − + LNR = L1 + L2. The first term contains the one-pion the π K atom at next-to-leading order in isospin and one-kaon sectors, symmetry breaking, we make use of resolvents. For a detailed discussion of the technique, we refer to L = 1 2 − 2 + † − + ∆ Ref. [15]. Here, we simply list the results. We use di- 1 E B h0 i∂t Mh0 2 2Mh0 mensional regularization, to treat both ultraviolet and 9/2 2 infrared singularities. Up to and including order δ , ∆ † 0 0 + +··· h0 + h± iDt − Mh+ the decay into π K is the only decay channel con- M3 8 h0 ± tributing, and we get for the total S-wave decay width 2 4 D D α3µ3 µ2k2C2 αµ2 C ξ + + +··· h±, (1) = + 2 − 0 0 3 − + 1 n 2M + 3 Γn µ0k0C2 1 h 8Mh+ n3π2 4π2 π

2 3 + 3 =−∇ − ˙ =∇× 5µ0k M 0 M 0 where E A0 A, B A and the quan- + 0 π K + O δ5 , (5) tity h = π,K stands for the non-relativistic pion and 8 M3 M3 π0 K0 kaon ﬁelds. We work in the Coulomb gauge and elim- 0 2 2 1/2 inate the A component of the photon ﬁeld by the where k0 =[2µ0(Σ+ − Σ0 − α µ+/(2n ))] is of 1/2 use of the equations of motion. The covariant deriva- order δ . The function ξn develops an ultraviolet sin- tives are given by Dt h± = ∂t h± ∓ieA0h± and Dh± = ∇h± ± ieAh±,wheree denotes the electromagnetic 1 The basis of operators containing two space derivatives can be coupling. What concerns the one-pion–one-kaon sec- chosen such that none of them contributes to the energy shift and tor, we only list the terms needed to evaluate the decay decay width at next-to-leading order in isospin symmetry breaking − + width and the energy shift of the π K atom at order [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. TNR denotes the truncated amplitude.

→ = 2 + 2 1/2 gularity as d 3, with ωi (p) (Mi p ) . The 3-momentum p denotes the center of mass momentum of the incom- α 2µ+ ξn = Λ(µ) − 1 + 2 ln + ln ing particles, q the one of the outgoing particles. The n µ effective Lagrangian in Eqs. (1) and (2), allows us 1 − + → 0 0 + ψ(n)− ψ(1) − , to evaluate the non-relativistic π K π K and − + → − + n π K π K scattering amplitudes at threshold − 1 at order δ. In the isospin symmetry limit, the effective Λ(µ) = µ2(d 3) − ln 4π − Γ (1) , (6) − couplings C1, C2 and C3 are d 3 √ with ψ(n) = Γ (n)/Γ (n) and the running scale µ. 2π + − 2 2π − C1 = a + a ,C2 =− a , At order δ4, the total energy shift may be split into µ+ 0 0 µ+ 0 a strong part and an electromagnetic part, according to = 2π + C3 a0 , (10) = h + em µ+ En En En . (7) 2 + = 1/2 + For the discussion of the electromagnetic energy shift, where the S-wave scattering lengths a0 1/3(a0 − 2a3/2) and a = 1/3(a1/2 −a3/2) are deﬁned in QCD, we refer to Section 4. The strong S-wave energy shift 0 0 . 0 0 . reads at mu = md and Mπ = Mπ+ , MK = MK+ . By substi- 3 3 2 2 2 tuting these relations into the expression for the de- h α µ+ αµ+ 2 µ0k0 2 E =− C − C ξn − C C cay width (5) and the strong energy shift (8), one ob- n πn3 1 2π 1 4π2 2 3 tains the Deser-type formulae [1,2]. We demonstrate + O δ5 . (8) the matching at next-to-leading order in δ by means of the π−K+ → π−K+ amplitude. In the presence The results for the decay width (5) and energy shift (8) of virtual photons, we ﬁrst have to subtract the one- are valid at next-to-leading order in isospin symmetry photon exchange diagram from the full amplitude, as breaking. displayed in Fig. 1. The coupling constant C1 is deter- ¯ ±;± mined by the truncated part TNR , which contains an infrared singular Coulomb phase θ as d → 3, 3. Matching the low-energy constants c ±;± ±;± T¯ (p; p) = e2iαθc Tˆ (p; p), NR NR The coupling constants Ci can be determined µ+ d−3 1 1 through matching the non-relativistic and the relativis- θc = µ − ln 4π + Γ (1) |p| d − 3 2 tic amplitudes at threshold. The coupling C3 is needed 0 2|p| at order δ only. However, we have to determine both + ln . (11) C1 and C2 at next-to-leading order in isospin symme- µ try breaking. The relativistic amplitudes are related to ˆ ±;± At order δ, the remainder TNR is free of infrared sin- the non-relativistic ones through ˆ ±;± gularities at threshold. The real part of TNR is given lm;ik ; TR (q p) ; = 1/2 lm ik ; 2 + − 4 ωi (p)ωk(p)ωl (q)ωm(q) TNR (q p), (9) a0 and a0 are normalized as in Ref. [18]. 36 J. Schweizer / Physics Letters B 587 (2004) 33–40 by where

+ + + ˆ ±;± Mπ ∆K MK ∆π + 2 Re T (p; p) Kn = a NR + + + 0 Mπ MK B |p| 1 ±;± = 1 + B ln + Re A + − 1 α | | 2 + + thr − 4αµ+ a + a ψ(n)− ψ(1) − + ln p µ+ 4Mπ MK 0 0 n n + O (p), (12) + o(δ). (15) = + =− 2 2 + with B1 C1απµ+ o(δ), B2 C1 αµ+/π o(δ) The outgoing relative 3-momentum and ∗ = 1 2 2 2 1/2 1 ±;± pn λ En,Mπ0 ,MK0 , (16) Re Athr 2En 4Mπ+ MK+ = 2 + 2 + 2 − − − 2 2 with λ(x,y,z) x y z 2xy 2xz 2yz, C1αµ+ 4µ+ = C1 1 + 1 − Λ(µ) − ln is chosen such that the total final state energy cor- 2π µ2 2 2 responds to En = Σ+ − α µ+/(2n ). The quantity 2 3 00;± C2 C3µ0 Re Athr is calculated as follows. One evaluates the − (Σ+ − Σ0) + o(δ). (13) − + 2π2 relativistic π K → π0K0 amplitude near threshold Here, the ultraviolet pole term Λ(µ) is removed and removes the divergent Coulomb phase. The real part contains singularities ∼ 1/|p| and ∼ ln |p|/µ+. by renormalizing the coupling C1. The renormaliza- tion of C eliminates at the same time the ultravi- The constant term in this expansion corresponds to 1 00;± olet divergence contained in the expression for the Re Athr . The normalization is chosen such that energy shift (8). The calculation of the relativistic − A = a + . (17) π−K+ → π−K+ scattering amplitude was performed 0 at O(p4,e2p2) in Refs. [20,21]. Both the Coulomb The isospin breaking corrections have been evalu- phase and the logarithmic singularity in Eq. (12) are ated at O(p4,e2p2) in Refs. [21,27]. See also the com- absent in the real part of the relativistic amplitude mentsinSection6. at this order of accuracy, they first occur at order We now discuss the various energy shift contribu- ±;± e2p4. The quantity Re A denotes the constant term tions. According to Eq. (7), the energy shift at order thr 4 em occurring in the real part of the truncated relativis- δ is split into an electromagnetic part En and the h tic threshold amplitude. The coupling constant C2 strong part En in Eq. (8). The electromagnetic en- may be determined analogously by matching the non- ergy shift contains both pure QED corrections as well relativistic π−K+ → π0K0 amplitude to the relativis- as finite size effects due to the charge radii of the pion tic one at order δ. and kaon, contained in λ. The pure electromagnetic corrections have been evaluated in Ref. [28] for arbi- trary angular momentum l. We checked3 that the elec- 4 4. Results for the π−K+ atom tromagnetic energy shift at order α indeed amounts to 4 em = α µ+ − 3µ+ 3 − 1 The result for the decay width and strong energy Enl 1 n3 Σ+ 8n 2l + 1 shift are valid at next-to-leading order in isospin sym- 4α4µ3 λ metry breaking, and to all orders in the chiral expan- + + 9/2 3 δl0 sion. We get for the decay width at order δ ,interms n − + 0 0 4 2 of the relativistic π K → π K threshold ampli- α µ+ 1 1 3 + δl0 + − tude, Σ+ n3 n4 n3(2l + 1) 5 8 3 2 ∗ 2 + O α ln α . (18) Γn = α µ+p A (1 + Kn), n3 n 1 1 A =− √ 00;± + 3 Re Athr o(δ), (14) We thank A. Rusetsky for a very useful communication 8 2π Σ+ concerning technical aspects of the calculation. J. Schweizer / Physics Letters B 587 (2004) 33–40 37

Here, the first term is generated by the mass insertions, and 2P states is given by the second contains the finite size effects and the last stems from the one-photon exchange contribution. The E2s−2p strong S-wave energy shift reads at order δ4, = h + em − em + vac − vac E2 E20 E21 E20 E21 3 2 2α µ+ Eh =− A (1 + K ), =−1.4 ± 0.1eV. (22) n n3 n 1 ±;± h A = Re A + o(δ), (19) The uncertainty displayed is the one in E2 only. For 8πΣ+ thr the numerical values of the various energy shift con- with tributions, see Table 2 in Section 6. + − 1 α K =−2αµ+ a + a ψ(n)− ψ(1) − + ln n 0 0 n n 5. Results for pionium + o(δ). (20) The decay rate and strong energy shift of pio- In the isospin limit, the normalized relativistic ampli- nium can be obtained from the formulae in Eqs. (5) tude and (8) through the following substitutions of the + → + → A = + + − + masses MK Mπ , MK0 √Mπ0 and the cou- a0 a0 , (21) pling constants C1 → c1, C2 → 2(c2 − 2c4∆π ) and reduces to the sum of the isospin even and odd scatter- C3 → 2c3 [16]. The ci are the low-energy constants ing lengths. The corrections have been obtained at defined in Ref. [15]. The S-wave decay width of the O(p4,e2p2) in Refs. [20,21]. See also the comments π+π− atom reads at order δ9/2, in terms of the rela- h + − → 0 0 in Section 6. The result for E1 in Eq. (19) agrees tivistic π π π π threshold amplitude, with the one obtained for the strong energy shift of the ground state in pionic hydrogen [26], if we replace µ+ 2 3 ∗ 2 ±;± = A + − Γπ,n 3 α pπ,n π (1 Kπ,n), with the reduced mass of the π p atom and Re Athr 9n with the constant term in the threshold expansion for 0 2 Aπ = a − a + π , the real part of the truncated π−p → π−p amplitude. 0 0 κ What remains to be added are the vacuum polar- = 0 + 2 2 Kπ,n a0 2a0 ization contributions [14,29], which are formally of 9 higher order in α, however numerically not negligible. − 2α 0 + 2 − − 1 + α 2a a0 ψ(n) ψ(1) ln The vacuum polarization leads to an energy level shift 3 0 n n Evac as well as to a change in the Coulomb wave + o(δ), nl − + function of the π K atom at the origin δψK,n(0). vac 2 1/2 For the first two energy levels, E [14,29] is given ∗ α 2 nl p = ∆π − M + , (23) numerically in Table 2, Section 6. Formally of or- π,n 4n2 π der α2l+5, this contribution is enhanced due to its 2 2 2l+2 where κ = M + /M − 1. The quantity A is defined large coefficient containing (µ+/me) . The mod- π π0 π ified Coulomb wave function affects both, the decay as in Refs. [13,15]. The isospin symmetry breaking O 4 2 2 width and the strong energy shift, see Section 6. corrections π have been evaluated at (p ,p e ) in As discussed in Section 6, the electromagnetic con- Refs. [13,15,30]. For the decay width of the ground 9/2 tributions (18) are known to a high precision. Further, state at order δ , we reproduce the result obtained the strong shift in the nP state is very much suppressed in Refs. [11,13,15]. The electromagnetic energy shift em (order α5). A future measurement of the energy split- Eπ,nl is obtained from Eq. (18) through the above → 2 ting between the nSandnP states will therefore allow mass substitutions and λ 1/3 rπ+ . Finally, the to extract the strong S-wave energy shift in Eq. (19), S-wave energy shift of the π+π− atom reads at order + + − 4 and to determine the combination a0 a0 of the πK δ , in terms of the relativistic one-particle irreducible scattering lengths. The energy splitting between the 2S 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 δh,1 δh,1 δh,2 + − − − − π π atom (5.8 ± 1.2) × 10 2 (6.2 ± 1.2) × 10 2 (6.1 ± 1.2) × 10 2 ± ∓ − − − π K atom (4.0 ± 2.2) × 10 2 (1.7 ± 2.2) × 10 2 (1.5 ± 2.2) × 10 2

3 α M + Eh =− π A (1 + K ), within the uncertainties quoted in [21]. In the follow- π,n 3 π π,n −2 −1 n ing, we use [21] = (0.1 ± 0.1) × 10 M + and = π 1 −2 −1 A = 0 + 2 + (0.1 ± 0.3) × 10 M + . For the charge radii of the π 2a0 a0 π , π 6 2 2 pion and kaon, we take r + =(0.452 ± 0.013) fm α π =− 0 + 2 2 2 Kπ,n 2a0 a0 and r + =(0.363 ± 0.072) fm [33]. 3 K 1 α We obtain for the decay width of the ground state, × ψ(n)− ψ(1) − + ln + o(δ), (24) = 3 2 ∗ − 2 + n n Γ1 8α µ+p1 (a0 ) (1 δK,1), A A 3 where π is defined analogously to the quantity 2α ∗ 0 2 2 Γπ, = p a − a (1 + δπ, ), (26) discussed in Section 4. The isospin symmetry breaking 1 9 π,1 0 0 1 O 2 2 contributions π have been calculated at (e p ) where the corrections δh,1, h = π,K are given in in Refs. [31,32]. For pionium the energy splitting Table 1. The strong energy shift reads between the 2S and 2P states reads 2α3µ2 h =− + + + − + = h + em − em En a0 a0 (1 δK,n), Eπ,2s−2p Eπ,2 Eπ,20 Eπ,21 n3 vac vac 3 + E − E α M + π,20 π,21 Eh =− π 2a0 + a2 (1 + δ ). (27) π,n 3 0 0 π,n =−0.59 ± 0.01 eV. (25) 6n h For the first two energy levels, the corrections δh,n are Again the uncertainty displayed is the one in Eπ,2 specified in Table 1. As mentioned in Section 4, these only. The numerical values for the various energy corrections to the Deser-type formulae are modified by shifts are listed in Table 3, Section 6. vacuum polarization, δ → δ + δvac,δ→ δ + δvac, (28) 6. Numerical analysis h,n h,n h,n h,n h,n h,n where For the S-wave ππ scattering lengths, we use 2δψh,n(0) 0 2 vac = the chiral predictions a = 0.220 ± 0.005 and a = δh,n . (29) 0 0 ψh,n(0) −0.0444 ± 0.0010 [5,6]. The correlation matrix for a0 0 vac 2 2 Formally, the contribution δh,n is of order α ,but and a0 is given in Ref. [6]. For the isospin symmetry breaking corrections to the ππ threshold amplitudes enhanced because of the large coefficient containing −2 µ+/m . For the ground state, the corrections [14] (23) and (24), we use π = (0.61 ± 0.16) × 10 and e − vac = × −2 vac = × −2 = (0.37 ± 0.08) × 10 2 as given in Ref. [15,32], yield δK,1 0.45 10 and δπ,1 0.31 10 . π vac respectively. For the πK scattering lengths, we use The changes in δπ,1 and δπ,1 due to δπ,1 are about the values from the recent analysis of data and Roy– 5%. For δK,1 however, the correction amounts to 27%. + −1 vac Steiner equations [22], a = (0.045±0.012)M + and Here, we omit the contributions from δh,n, because the 0 π − −1 vac a = (0.090 ± 0.005)M +. The correlation parameter uncertainties in δh,n and δh,n are much larger than δh,n. 0 π . − + − The numerical values for the lifetime τ = Γ 1, for a0 and a0 is given in Ref. [22]. The isospin break- 1 1 =. −1 ing corrections to the πK threshold amplitudes (17) (τπ,1 Γπ,1 ) and the energy shifts at next-to-leading and (21) have been worked out in [20,21,27]. Whereas order in isospin symmetry breaking are given in Ta- the analytic expressions for and obtained in [20, bles 2 and 3. The energy shifts due to vacuum po- vac 21,27] are not identical, the numerical values agree larization Enl are taken from Ref. [14,29]. In the 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 ± ∓ em vac h π K atom Enl [eV] Enl [eV] En [eV] τn [s] − n = 1, l = 0 −0.095 −2.56 −9.0 ± 1.1 (3.7 ± 0.4) × 10 15 n = 2, l = 0 −0.019 −0.29 −1.1 ± 0.1 n = 2, l = 1 −0.006 −0.02

Table 3 + − Numerical values for the energy shift and the lifetime of the π π atom + − em vac h π π atom Eπ,nl [eV] Eπ,nl [eV] Eπ,n [eV] τπ,n [s] − n = 1, l = 0 −0.065 −0.942 −3.8 ± 0.1 (2.9 ± 0.1) × 10 15 n = 2, l = 0 −0.012 −0.111 −0.47 ± 0.01 n = 2, l = 1 −0.004 −0.004 evaluation of the uncertainties, the correlations be- R. Kaiser, A. Rusetsky, H. Sazdjian and J. Schacher tween the S-wave scattering lengths have been taken for useful discussions as well as for helpful comments into account. For the decay width and the strong en- on the manuscript. This work was supported in part by ergy shift of the π±K∓ atom, the dominant source of the Swiss National Science Foundation and by RTN, uncertainty is due to the uncertainties in the scatter- BBW-Contract No. 01.0357 and EC-Contract HPRN- + − ing lengths a0 and a0 . We do not display the error CT2002-00311 (EURIDICE). bars for the electromagnetic energy shifts, which stem 4 2 2 at order α from the uncertainties in rπ+ and rK+ References em only. For pionium, the uncertainties of Eπ,10 at order α4 amount to about 0.7%, while for the π±K∓ atom [1] S. Deser, M.L. Goldberger, K. Baumann, W. Thirring, Phys. em Rev. 96 (1954) 774; E10 is known at the 5% level. To estimate the or- T.L. Trueman, Nucl. Phys. 26 (1961) 57. der of magnitude of the electromagnetic corrections at [2] T.R. Palfrey, J.L. Uretsky, Phys. Rev. 121 (1961) 1798; higher order, we may compare with positronium. Here, S.M. Bilenky, V.H. Nguyen, L.L. Nemenov, F.G. Tkebuchava, the α5 and α5 ln α corrections [34] amount to about Yad. Fiz. 10 (1969) 812. 2% with respect to the α4 contributions. [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, 7. Summary and conclusions 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; We provided the formulae for the energy shifts and J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, M.E. Sainio, decay widths of the π+π− and π±K∓ atoms at next- Nucl. Phys. B 517 (1998) 639, Erratum, hep-ph/9707291; to-leading order in isospin symmetry breaking. To J. Nieves, E. Ruiz Arriola, Eur. Phys. J. A 8 (2000) 377, hep- ph/9906437; confront these predictions with data presents a chal- G. Amoros, J. Bijnens, P. Talavera, Nucl. Phys. B 585 (2000) lenge for future hadronic atom experiments. Should it 293; turn out that these predictions are in conﬂict with ex- G. Amoros, J. Bijnens, P. Talavera, Nucl. Phys. B 598 (2001) periment, one would have to revise our present under- 665, Erratum, hep-ph/0003258. standing of the low-energy structure of QCD. [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. Acknowledgements [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; It is a great pleasure to thank J. Gasser for many in- A.A. Bolokhov, M.V. Polyakov, S.G. Sherman, Eur. Phys. J. teresting discussions and suggestions. Further, I thank A 1 (1998) 317, hep-ph/9707406; 40 J. Schweizer / Physics Letters B 587 (2004) 33–40

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