The Eikonal Approach to Calculation of the Multiphoton Exchange Contributions to the Total Cross Sections of Ππ Atom Interaction with Ordinary Atoms

The total cross sections of interaction of relativistic ππ atom with ordinary atoms are obtained in the eikonal approach, which takes into consideration all multiphoton exchange processes. Contribution of these processes strongly depends on the atom nucleus charge Z and varies from 1.5% for Titanium (Z = 22) to 14% for Tantalum (Z = 73) An accurate measurement of the ππ atom (dimesoatom) lifetime in the experiment DIRAC[1] will allow to check a high precision prediction of the Chiral Perturbation Theory. Interaction of ππ atoms with ordinary atoms is an essential part of this experiment, as atom observation bases on the their breakup (ionization) while passing through the target where they are produced. Cross sections of ππ atoms with ordinary atoms usually were calculated in the first Born approximation. In the paper[2] the cross sections for few low states of dimesoatom were calculated in Coulomb-modified Glauber approximation. It was shown that multiphoton exchange processes play significant role in the interaction of ππ atoms with atoms. In the Glauber approximation the total cross sections of the coherent interaction of ππ atom with ordinary atoms could be written[2] as: σ nlm = 2Re ∫ db dr|ψnlm(~r)| [ 1− exp ( iχ(~b− ~s/2)− iχ(~b+ ~s/2) )] . (1) Here ~s = ~r⊥ is the projection of the vector ~r on the plane perpendicular to the impact parameter~b, ψnlm(~r) is the wave function of π π atom in the state with principal, orbital and magnetic quantum numbers n, l and m respectively. The phase shift χ(~b) is expressed via the screened Coulomb potential of the target atom: χ(~b) = 1 v ∞ ∫ −∞ U( √ b + z ) dz . (2) Here v is the velocity of dimesoatom in the lab frame. 1 Let us write (1) in other form taking into account following relations: exp (iχ(~b)) = 1− γ(b) , (3) γ(b) = 1 2πi ∫ f(q) exp (−iqb) dq , (4) f(b) = 1 2πi ∫ [ 1− exp (iχ(~b)) ] exp (iqb) dq . (5) Here f(q) is the amplitude of the elastic Coulomb πA-scattering, which normalized by the relations: σ πA = 4π Im f(0) , (6) d σ πA d~q = |f(q)| . (7) Then it is easy to get the total cross section in the form: σ nlm = 2 ∫ |f(q)| [1− Snlm(~q)] dq , (8) Snlm(~q) = ∫ |ψnlm(~r)| exp (i~q~r) dr . (9) Here Snlm(~q) is the elastic form factor of the π π atom in the state with the quantum numbers n, l and m. The expressions (8) and (9) the together with results obtained in the paper[3] for the transition form factors of the hydrogen-like atoms allow to calculate the total cross sections for any state of dimesoatom. However in this paper we consider only the cross sections averaged over the magnet quantum number: σ nl = 1 2l + 1 ∑ m σ nlm . (10) The wave function of ππ can be written as a product of the radial and angular parts: ψnlm(~r) = Rnl(r) Ylm(Θ, φ) . (11) Taking into account the normalization 1 2l + 1 ∑ m |Ylm(Θ, φ)| = 1 , (12) we have Snl(q) = 1 2l + 1 ∑