Pionium lifetime and ππ scattering lengths in generalized chiral perturbation theory H. Sazdjian

To cite this version:

H. Sazdjian. Pionium lifetime and ππ scattering lengths in generalized chiral perturbation theory. International Euroconference in QCD ’99, Jul 1999, Montpellier, France. pp.271-274. ￿in2p3-00004723￿

HAL Id: in2p3-00004723 http://hal.in2p3.fr/in2p3-00004723 Submitted on 22 Jun 2000

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Pionium lifetime and ππ scattering lengths in generalized chiral perturbation theory H. Sazdjiana aGroupe de Physique Th´eorique,Institut de Physique Nucl´eaire, Universit´e Paris XI, F-91406 Orsay Cedex, France E-mail: [email protected]

The relationship between the pionium lifetime and the ππ scattering lengths is established, including the sizable electromagnetic corrections. The formalism that is used is that of constraint theory which provides a covariant three-dimensional reduction of the Bethe–Salpeter equation. The framework of generalized chiral perturbation theory allows then an analysis of the lifetime value as a function of the ππ scattering lengths, the latter being dependent on the condensate value.

The possible measurement of the pionium ory and χP T , three different methods of evalu- + (π π− ) lifetime with a 10% precision in the ation have led to the same estimate, of the or- DIRAC experiment at CERN [1] is expected to al- der of 6%, of these corrections [6–9]. The first 0 2 low a determination of the combination (a0 a0) method uses a three-dimensionally reduced form of the ππ scattering lengths with 5% accuracy;− of the Bethe–Salpeter equation (constraint theory I here, a0 is the strong interaction (dimensionless) approach) and deals with an off-mass shell formal- S-wave scattering length in the isospin I chan- ism [6,7]. The second method uses the Bethe– 0 nel. The strong interaction scattering lengths a0 Salpeter equation with the Coulomb gauge [8]. 2 and a0 have been evaluated in the literature in the The third one uses the approach of nonrelativis- framework of chiral perturbation theory (χP T )to tic effective theory [9]. two-loop order of the chiral effective lagrangian The pionium lifetime, with the sizable O(α) [2–4]. Therefore, the pionium lifetime measure- corrections included in, can be represented as: ment provides a high precision experimental test 1 1 2 of chiral perturbation theory predictions. =Γ = 2 e 00,+ (1 + γ) τ 64πm + R M − The nonrelativistic formula of the pionium life- π  time in lowest order of electromagnetic interac- f2∆m ∆m (0) 2 π (1 π ) tions was first evaluated by Deser [5]. It ψ+ et al. ×| − | s mπ+ − 2mπ+ reads: ∆m ∆Γ Γ 1 π 1+ (2) 1 16π 2∆m (a0 a2)2 0 , π 0 0 2 ≡ s − 2mπ+ Γ0 =Γ0 = −2 ψ+ (0) , (1) + −    τ0 9 s mπ mπ+ | | where e 00,+ is the real part of the on- R M − where ∆mπ = mπ+ mπ0 and ψ+ (0) is the wave mass shell scattering amplitude of the process − − + 0 0 function of the pionium at the origin (in x-space). π π− πfπ , calculated at threshold, in the A precise comparison of the theoretical values presence→ of electromagnetic interactions and from of the strong interaction scattering lengths with which singularities of the infra-red have experimental data necessitates, however, an eval- been appropriately subtracted [10]; the factor γ uation of the corrections of order O(α)(αbeing represents contributions at second-order of per- the fine structure constant) to the above formula. turbation theory with respect to the nonrela- Such an evaluation was recently done by several tivistic zeroth-order Coulomb hamiltonian of the authors. In the frameworks of quantum field the- bound state formalism. The explicit expressions of e 00,+ and of γ may differ from one ap- A0 is expressible in terms of two-point functions proachR M to the− other, but their total contribution of scalar and pseudoscalar quark densities. In the shouldf be the same. standard χP T case, this term is relegated to the Our aim is to extend the previous analysis to next-to-leading order. the case of generalized χP T [11,4]. The latter is At the tree level of the chiral effective la- based on the observation that the fundamental grangian, the ππ scattering amplitude A(s t, u) order parameter of spontaneous chiral symmetry has the expression A(s t, u)=(s 2ˆmB )|/F 2, | − 0 breaking is Fπ, the decay coupling constant of the which displays explicit dependence on the quark , which is related to the two-point function of condensate parameter. It is useful to introduce left- and right-handed currents in the chiral limit. two parameters, α and β, that allow one to ex- The other order parameters, such as the quark press the amplitude A in terms of the physical condensate in the chiral limit, < 0 qq 0 >0,have constants Fπ and mπ [11,4]: values depending on the details of the| | mechanism β 4 m2 of chiral symmetry breaking and require indepen- ( )= ( 2)+ π (5) A s t, u 2 s mπ α 2. dent experimental tests. Standard χP T is based | Fπ −3 3Fπ on the assumption that the value of the Gell- At leading order, Fπ = F and β =1.The Mann–Oakes–Renner (GOR) parameter [12], de- GOR parameter (3) is then related to the pa- fined as rameter α by the relation: xGOR =(4 α)/3. For α = 1, one recovers the standard χP− T case, 2ˆm<0qq 0 >0 xGOR = | | , (3) where x =1.Whenαincreases, x de- − F 2m2 GOR GOR π π creases, corresponding to decreasing values of the where 2m ˆ = mu +md and mπ is the pion physical quark condensate. For α = 4, one reaches the ex- mass, is close to one. Stated differently, the quark treme case of generalized χP T ,wherexGOR and condensate parameter the quark condensate vanish. At higher orders, the relationship between α and x becomes <0qq 0 > GOR 0 (4) more complicated, involving the parameter β and B0 | 2| , ≡− F other low energy constants, but the main quali- where F is F in the chiral SU(2) SU(2) limit, tative feature found above remains: values of α π × is of the order of the hadronic mass scale ΛH 1 close to 1 correspond to the case of standard χP T , GeV. This assumption fixes the way standard∼ while large values of α, above 1.5-2, say, cover the χP T is expanded: the quark condensate parame- complementary domain of generalized χP T . ter B0 is assigned dimension zero in the infra-red At two-loop order, the ππ scattering amplitude external momenta of the Goldstone , while is described by six parameters, α, β, λ1, λ2, λ3 the quark masses are assigned dimension two [2]. and λ4. The five parameters other than α are Generalized χP T relaxes the previous assump- weakly dependent on the quark condensate and tion and treats the order of magnitude of the the values of the λ’s are fixed by a detailed anal- quark condensate parameter B0 as an a priori ysis of sum rules using available data on ππ scat- unknown quantity (awaiting a precise experimen- tering at medium energies. The parameter β is tal information about it) leaving to it the possibil- essentially sensitive to the deviation of Fπ from ity of reaching small or vanishing values. To this F and remains in general close to 1. One ends aim, B0 is assigned dimension one in the infra- up, in generalized χP T , with a complete deter- red momenta of the external Goldstone bosons mination of the threshold parameters of the ππ and accordingly quark masses are also assigned scattering amplitude in terms of the possible nu- dimension one. Due to this rule, at each order merical values of the parameter α together with of the perturbative expansion, generalized χP T a best value of the other parameters. The expres- contains more terms than standard χP T . For in- sions of the scattering lengths to two-loop order, stance, the pion mass formula becomes, at leading as well as the values of the parameters λ,canbe 2 2 order, mπ =2ˆmB0 +4ˆm A0, where the constant found in Ref. [4]. 1 The evaluation of the corrections contained in ious sets of inputs for a1 and for the λ’s within the decay width formula (2) can be done in much the intervals of their possible values. Generally, 0 2 the same way as in the standard χP T case, with foragiven(a0 a0), the lifetime τ varies little for the difference that whenever scattering lengths such changes of− the inputs. The variations of the or the quark condensate parameter appear, these λ’s induce at most a variation of 0.8% on τ, while 1 should be expressed through the generalized χP T the variations of a1 induce at most a variation of formulas or parametrizations. We have followed 1.1%. Assuming these variations as being uncor- the same method of approach as in Refs. [6,7] related, one may consider that 2% is an upper and skip here the details of the calculations. We bound for the possible variations of τ. Consider- only note that we made the approximation of tak- ing the latter as un uncertainty and adding it to ing the same values for the electromagnetic low the 2% uncertainty obtained in the course of cal- energy constants ki in generalized and standard culation of ∆τ/τ0, one obtains a total uncertainty χP T [13,14,10]. Actually, the effective lagrangian of 4% around the value of τ calculated with the 1 with electromagnetism is not yet available in gen- central values of a1 and of the λ’s. eralized χP T , where the number of such con- The above analyzes are graphically summa- stants should be larger. However, the contri- rized in Fig. 1. The full line represents the butions of the constants ki not being dominant lifetime τ as a function of the the combination 0 2 in the total decay width correction, such an ap- (a0 a0), corresponding to the central values of 1 − proximation seems to be justified. Furthermore, a1 and of the λ’s. The band around it corresponds a 100% uncertainty has been assigned to their to the estimated 4% total uncertainty. As already global contributions in the various types of cor- mentioned, the parameter β is almost insensitive rection, their values being taken from Ref. [15]. to the variations of α; for instance, on the central Also, weakly dependent terms on the parameter line, it varies between 1.10 and 1.11. For a given α have been numerically taken the same as in the experimental value of the lifetime, with a possi- standard case. Numerical estimates have been ble uncertainty, one can deduce from Fig. 1 the done with the following inputs: F =92.4MeV, corresponding value of the combination (a0 a2) π 0 − 0 mπ+ = 139.57 MeV, mπ0 = 134.97 MeV. with the related uncertainty. We have chosen to analyze the total correction The interpretation of the experimental value in terms of the scattering lengths, which are the of the pionium lifetime will depend on its or- physical quantities of interest, rather than of the der of magnitude. We assume in the follow- intermediate parameters α and β. Since the pio- ing discussion that, as the DIRAC experimen- nium lifetime concerns at leading order the com- tal project foresees, the experimental uncertainty 0 2 bination (a0 a0) of the scattering lengths, it on the lifetime is of 10%. For larger uncertain- is natural to− promote this quantity as the main ties, the discussion should accordingly be mod- variable of the problem. As a second variable, ified. Three different possibilities may be con- we choose the P -wave scattering length, which sidered. First, the central value of the lifetime 15 15 is most sensitive to the parameter β and almost is close to 3 10− s, lying above 2.9 10− insensitive to α. One then expresses the total cor- s, say. Then× standard χP T is firmly confirmed,× 0 2 1 0 2 rection in terms of (a0 a0), a1 and the λ’s. (It is since its predictions of (a0 a0) lie between 0.250 sufficient for the correction− term to use the one- and 0.258 [3]. Second, the− central value of the 15 loop expressions of the scattering lengths, from lifetime lies below 2.4 10− s. Then it is the × which λ3 and λ4 are absent.) The experimental scheme of generalized χP T which is confirmed, 1 2 value of a1mπ is 0.038 0.002 [16]. One can then since the corresponding central values of α would study the dependence± of the decay width (or of lie above 2, which would mean that the quark 0 2 the lifetime) upon the combination (a0 a0)of condensate is not as large as assumed in the stan- the scattering lengths for fixed values of the− above dard scheme. The third possibility is the most parameters. difficult to interpret. If the central value of the 15 The previous analysis can be repeated with var- lifetime lies in the interval 2.4 2.9 10− s, − × 4 Rusetsky and J. Stern for useful discussions.

3.5 τ REFERENCES 1. B. Adeva et al., CERN proposal 3 CERN/SPSLC 95-1 (1995). 2. J. Gasser and H. Leutwyler, Ann. Phys. 2.5 (N.Y.) 158 (1984) 142. 3. J. Bijnens, G. Colangelo, G. Ecker, J. Gasser 2 and M. Sainio, Phys. Lett. B374 (1996) 210; Nucl. Phys. B508 (1997) 263. 1.5 4. M. Knecht, B. Moussallam, J. Stern and N. H. Fuchs, Nucl. Phys. B457 (1995) 513; B471 1 (1996) 445. 0.25 0.27 0.29 0.31 0.33 0.35 0.37 5. S. Deser, M. L. Goldberger, K. Baumann and a0 a2 W. Thirring, Phys. Rev. 96 (1954) 774. 0 − 0 6. H. Jallouli and H. Sazdjian, Phys. Rev. D 58 (1998) 014011; 099901(E). 7. H. Sazdjian, in the proceedings of the In- 15 Figure 1. The pionium lifetime, in units of 10− ternational Workshop Hadronic and 0 2 in the ,M.A. s, as a function of the combination (a0 a0)of the S-wave scattering lengths. The full line− corre- Ivanov et al. (eds.), Dubna, 1998, p. 60; 1 2 preprint hep-ph/9809425. sponds to the central value a1mπ =0.038 of the P -wave scattering length. The band delineated 8. M. A. Ivanov, V. E. Lyubovitskij, E. Z. Li- by the dotted lines takes account of the uncer- partia and A. G. Rusetsky, Phys. Rev. D 58 tainties, coming from theoretical evaluations, low (1998) 094024. 1 9. A. Gall, J. Gasser, V. E. Lyubovitskij and A. energy constants and a1. G. Rusetsky, preprint hep-ph/9905309. 10. M. Knecht and R. Urech, Nucl. Phys. B519 (1998) 329. 11. N. H. Fuchs, H. Sazdjian and J. Stern, Phys. then, because of the uncertainties, ambiguities in Lett. B 269 (1991) 183; J. Stern, H. Sazd- the interpretation may arise. Let us consider as jian and N. H. Fuchs, Phys. Rev. D 47 (1993) an illustrative possibility the hypothetical result 3814. τ =(2.60 0.26) 10 15 s. Without the in- − 12. M. Gell-Mann, R. J. Oakes and B. Renner, clusion of the± uncertainties× discussed above, this Phys. Rev. 175 (1968) 2198; S. Glashow and would imply (a0 a2)=0.278 0.015 and α = 0 0 S. Weinberg, Phys. Rev. Lett. 20 (1968) 224. 1.68 0.42. Taking− into account∓ the 4% uncer- 13. R. Urech, Nucl. Phys. B433 (1995) 234. tainties∓ represented by the band, one would have: 14. Ulf-G. Meissner, G. M¨ullerand S. Steininger, (a0 a2)=0.278 0.022 and α =1.68 0.61. 0 0 Phys. Lett. B 406 (1997) 154; 407 (1997) Clearly,− the upper∓ bound of τ is in favor of∓ stan- 454(E). dard χP T , while its lower bound in favor of gen- 15. R. Baur and R. Urech, Phys. Rev. D 53 (1996) eralized χP T . Such a situation would not dis- 6552; Nucl. Phys. B499 (1997) 319. entangle the two schemes from each other. The 16. M. M. Nagels et al., Nucl. Phys. B 147 (1979) uncertainties might be reduced in this case by a 189. more refined analysis, taking into account the cor- 1 relations between variations of a1 and of the λ’s. I thank J. Gasser, L. Girlanda, M. Knecht, P. Minkowski, B. Moussallam, L. Nemenov, A.