The electromagnetic interaction of ultrarelativistic heavy ions

The validity of a delta-function approximation for the electromagnetic interaction of relativistic heavy ions is investigated. The production of e+e− pairs, with electron capture, is used as a test of the approximation. The production of ee in peripheral collisions of relativistic heavy ions has attracted a great amount of theoretical interest due to its non-perturbative character. The calculations are hard to perform and it is common to find substantial differences between the cross sections calculated within several approaches [1]. A good simplification of the problem has been found by Baltz and collaborators [2]. They have shown that if one makes a gauge transformation in the wavefunction of the form ψ = exp {−iχ (r, t)}ψ′, where χ (r, t) = Zα v ln [ γ (z − vt) + √ b2 + γ2 (z − vt) ] (1) the interaction induced by the electromagnetic field of an ultrarelativistic particle is gauge transformed to (in our units h̄ = c = me = 1) V (ρ, z, t) = φ ( ρ, z, t) (1− vα̂z)− φ (ρ = 0, z, t) (1− α̂z/v) , (2) where φ (ρ, z, t) is the Lienard-Wiechert potential at a point r = (ρ, z), generated by a relativistic particle with velocity v = vẑ and impact parameter b, 1 φ (ρ, z, t) = γZα [ (b− ρ) + γ (z − vt) ]−1/2 . (3) In these expressions γ = (1− v2) is the Lorentz contraction factor, and α̂z is the third of the Dirac matrices. The second part of eq. (2) acts as a regularization term of the modified potential. It removes the divergence at b = 0. This new potential is very useful since the Lorentz contraction yields a delta-function in the longitudinal variables when γ ≫ 1, and b is not too large. Evidently, this is a great simplification since delta-function interactions always lead to a considerable decrease of integration steps in perturbative as well as in non-perturbative calculations. In [2] the formal derivation of the delta-function interaction was obtained by expanding the gauge transformed potential (2) into multipoles. Further manipulation of the multipole expansion and comparison with numerical calculations have shown that (2) can be expressed as V (ρ, z, t) = δ (z − t)Zα (1− α̂z) ln (b − ρ) b2 . (4) We will show that this expression can be obtained in a simpler way. The derivation is useful to study the validity of the delta-function approximation. In particular we will test the approximation in a solvable problem, namely the production of ee pairs in which the electron is captured in an orbit around one of the nuclei (bound-free pairs). Using the Bethe-integral [3], the potential (3) can be written in the form φ (ρ, z, t) = Zα 1 2π2 ∫ dq ee q2 − v2q2 z , (5) where u = b + vt and q = (qt, qz). For relativistic particles we can replace (1− vα̂) and (1− α̂/v) by (1− α̂) in the interaction (2). As shown in ref. [3] this amounts to neglect a very small (∼ O(1/γ2)) piece of the longitudinal part of the interaction. However, it is important to keep the other v factors in their respective places, as they give rise to