arXiv:nucl-th/9704037v1 17 Apr 1997 h eaieinadcl it ion p negative call positive and the ion as the negative lifetime Since the and energy binding ions. same pionium the have positive obviously and negative the as systems agtnce.I hs ecin eti ubro olm boun (( Coulomb of pions number charged certain three a reactions these In nuclei. target xeieta ENt measure to CERN at experiment cteiglntso h hadronic the of lengths scattering neet uhaoshv lal enosre tDba[]ada and [1] Dubna lifetime at the observed for been obtained clearly have atoms Such interest. 4,5,weeterfrne oteerirltrtr a efound), be can literature earlier the to references the where [4],[5], o e enivsiae.I steamo hsnt oso o h w the how show to note for this results of experimental aim and the is It investigated. been yet not nteCR experiment, CERN the In h ( The hlttehdoi nrysitadtelifetime the and shift energy hadronic the Whilst ntttfu hoeicePyi e Universit¨at Z¨urich, der Physik f¨ur Theoretische Institut π hre in)aeetmtduigterslsfrteposit the for results the using estimated are pions) charged ftepoiminaeapoiaeytesm sfrpionium. for as same the approximately a are energy ion state pionium ground the the of of shift hadronic the that out turns + h arncpoete ftepoimin(olm on sy bound (Coulomb ion pionium the of properties hadronic The π − atom ) arncEet ntePoimIon Pionium the in Effects Hadronic A π 2 π − π τ poim a eoeo osdrbeeprmna n theoretica and experimental considerable of become has (pionium) + ftegon tt 2.Alreclaoaini nae nan in engaged is collaboration large A [2]. state ground the of A π Ps 2 − − sbitdt hsc etr B) Letters Physics to (submitted π A n ( and ) − 2 H85 Z¨urich, Switzerland CH-8057 o short. for π teCuobbudsse ( system bound Coulomb (the .Rsh n .Gashi A. and Rasche G. τ ilb rdcdi neatoso 4GVpooswith protons GeV 24 of interactions in produced be will ππ iha cuayo bu 0 [3]. 10% about of accuracy an with itrcinhv entetdehutvl see.g. (see exhaustively treated been have -interaction π + Abstract π − π 1 + )wl lob rdcd erfrt these to refer We produced. be also will )) τ of A 2 itrhrrtas 190, Winterthurerstrasse π n hi oncin othe to connections their and e oimion ronium − h rprisof properties the e + dtelifetime the nd ytm ossigof consisting systems d e tmo three of stem − l nw theoretical known ell oe ii a been has limit lower a eue sa as used be can ) ) eawy ee to refer always we Ps oiminwill ion ionium − It . A 2 − π have l − − starting point to treat the binding energy and the lifetime τ of A2π. In the quantum mechanical treatment of the Coulomb binding energy and the three

− − particle wave function of A2π one is faced with the same difficulties as for Ps . Analytically the problem can not be solved and one has to resort to numerical variational methods ([6],[7],[8]). Since in each case only one mass is involved and the same charges are present, the

− − results for Ps can be taken over directly for A2π by the usual dimensional arguments. From the results of [6], [7], [8], [9] for Ps− we conclude that there exists only one bound

− − − state of A2π. For the π -affinity of A2π (i.e. the binding energy of A2π against dissociation − into A2π and a free π ) we get

m 0.3267 c eV = 89.2 eV, (1) me

mc where me is the ratio of the mass of the charged pion to the electron mass. For the binding − energy of A2π ( with respect to three free charged pions ) we therefore get

1 m α2 + 89.2 eV = 1858.1eV+89.2 eV = 1947.3 eV. (2) 4 c

From [4] we know that in addition to this Coulombic part of the energy we have to take into account the hadronic and vacuum polarisation shifts. The hadronic shift ∆EHAD of

+ − − the ground state energy of A2π is proportional to ρ(π π ), the probability density of π at

+ − the position of π . Analogously the hadronic shift ∆EHAD of the ground state energy of

− − + − − + − A2π consists of two parts. The first part is proportional to 2ρ(π π π ), where ρ(π π π ) is the probability density of one π− at the position of π+, integrated over all positions of the π+ and of the other π−. The proportionality constants are the same. It is evident from dimensional arguments that

2ρ(π−π+π−) 2ρ(e−e+e−) = . (3) ρ(π+π−) ρ(e+e−)

The second part of the hadronic shift is proportional to the probability density of the two π− being in the same position. Because of the Coulomb repulsion this is very small compared

2 to ρ(π−π+π−). A simple estimate using fig. 6 of [8] shows that the ratio is ≈ 0.017. Since the hadronic interaction in the π−π−-system is of the same order of magnitude as in the

− + − π π -system, this second part of ∆EHAD is completely unimportant numerically. Anticipating the result that the right hand side of (3) is ≈ 1, we thus have

− ≈ ∆EHAD ∆EHAD, (4)

where ∆EHAD ≈ 2.8 eV from [4], [5]. With identical arguments one finds that

τ − ≈ τ, (5)

≈ × −15 − → − 0 0 where τ 3.3 10 s from [4], [5]. (5) holds for the dominant hadronic decay A2π π π π − → − as well as for the negligible electromagnetic decay A2π π γγ. − One should note that the vacuum polarisation shift ∆EV AC of the ground state energy of

− − − + − A2π cannot be derived from the results on Ps or A2π: it is not proportional to ρ(π π π ) and it does not depend on only one mass (mc), since me also goes into a calculation of − ∆EV AC . This shows the limitations of our simple procedure. On the other hand there is little doubt that

− ≈ ∆EV AC ∆EV AC , (6)

where ∆EV AC ≈ 0.9 eV is the vacuum polarisation shift of the ground state of A2π [5]. 2ρ(e−e+e−) ≈ We have anticipated the result ρ(e+e−) 1 and will now justify it. For this purpose we use the results of [10], [11], [12]. From [12] we have

t 1 2ρ(e−e+e−) = . (7) t− 4 ρ(e+e−)

− − 1 Here t and t are the lifetimes of Ps and Ps respectively; the factor 4 takes into account the fact that the two electrons in Ps− are in a singlet state, so that the probability of finding one of the electrons in a singlet state with the positron (from which they can annihilate into

1 − × −9 γγ) is 4 . From [10] we have t =0.478 10 s, which agrees well with the experiment [11]. Using t =1.25 × 10−10s for the lifetime of Ps from the singlet state, we get

3 2ρ(e−e+e−) ≈ 1.05. (8) ρ(e+e−)

− In a second note we will estimate the formation probability of A2π in the experiment [3]. We thank L. Nemenov, G. C. Oades and W. S. Woolcock for useful discussions and suggestions.

4 REFERENCES

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[2] L. Afanasyev et al., Phys. Lett. B 338 (1994) 478.

[3] B. Adeva et al., Lifetime measurement of π+π− atoms to test low energy QCD predic-

tions: proposal to the SPSLC, CERN-SPSLC-95-1 (December 1994).

[4] U. Moor et al., Nucl. Phys. A 587 (1995) 747.

[5] A. Gashi et al., preprint nucl-th/9704017 and to be published.

[6] W. Kolos et al. Rev. Mod. Phys. 32 (1960) 178.

[7] E. A. Hylleraas Phys. Rev. 71 (1947) 491.

[8] U. Schr¨oder Zeitschr. f. Physik 173 (1963) 221.

[9] A. M. Frolov et al. J. Phys. B22 (1989) 1263.

[10] Y. K. Ho J. Phys. B16 (1983) 1503.

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[12] G. Ferrante Phys. Rev. 170 (1968) 76.

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