Transition Radiation in Quantum Regime as a Diffractive Phenomenon

We demonstrate that the transition photon radiation and pair creation can be interpreted as a diffractive phenomenon in terms of the light-cone wave functions in a way similar to the Good-Walker approach [6] to the diffraction dissociation. Our results agree with those obtained by Baier and Katkov [5] within the quasiclassical operator method, and disagree with the analysis of Garibyan [4]. 1. The transition radiation [1]) is usually discussed within the classical electrodynamics (for reviews, see [2, 3]). For relativistic particles the classical approach applies when the momentum of the radiated photon is small as compared to the momentum of the charged particle. The emission of hard photons with momentum comparable to the charged particle momentum requires a quantum treatment. The transition radiation in quantum regime and transition pair creation was addressed many years ago by Garibyan [4]. Recently the problem was analyzed by Baier and Katkov [5] within the quasiclassical operator method. In the present note we would like to demonstrate that the transition radiation (and pair creation) in the quantum regime can be interpreted as a diffractive effect in terms of the light-cone wave functions in the spirit of the Good-Walker approach to the diffraction dissociation [6]. 2. Let us consider for definiteness the photon emission from a relativistic electron with energy Ee ≫ me (we use the units h̄ = c = 1) which moves normally to the boundary from medium 1 to medium 2. We choose the z axis along the electron momentum and assume that the boundary is located at z = 0. We assume that the photon momentum is sufficiently large and the medium effect on the photon quantum field may be treated in the plasma approximation [7, 3]. In this approximation the in-medium photon is described as a quasiparticle with nonzero mass mγ = ωp, where ωp = √ 4πnα/me is the plasma frequency [7] (hereafter α = 1/137). To leading order in the coupling constant the physical electron in the media 1 and 2 symbolically can be written as the Fock state superpositions |e i 〉 = |e〉+ ∑ x,k Ψi(x,k)|eγ〉 , (1) where i = 1, 2, Ψi(x,k) is the light-cone wave functions of the |e′γ〉 Fock states in the momentum representation (hereafter k is the transverse momentum of the photon, and x = pγ,z/pe,z is its fractional longitudinal momentum). The discontinuity of the mγ on the boundary between two media leads to a jump in the Fock states decomposition of the physical electron. Evidently, the Fock component |e′γ〉 after passing through the boundary from medium 1 to medium 2 has the same wave function as in medium 1, i.e. Ψ1(x,k), which, however, does not match with the wave function of the |e′γ〉 state in medium 2. Thus, in terms of the physical states the projectile state after passing through the boundary, |E1〉, can be written as |E1〉 = |e 2 〉+ ∑ x,k [Ψ1(x,k)−Ψ2(x,k)]|eph 2 γ 2 〉 . (2) Evidently, the second term on the right-hand side of (2) describes the photon emission. The corresponding spectrum reads dN dxdk = 1 (2π) |Ψ1(x,k)−Ψ2(x,k)|2 . (3) The situation with the transition radiation is very analogous to the Good-Walker picture of the diffraction dissociation [6]. In that case the projectile wave function after the target has the form |Ψf〉 = Ŝ|Ψin〉, where Ŝ is the S-matrix, and |Ψin〉 is the wave function of the incident particle. In the diffraction dissociation the jump in the Fock state decomposition of the projectile stems from the different scattering amplitudes for different Fock states. In the case of the transition radiation there is no scattering at all. Nonetheless, similarly to the diffraction dissociation there is a jump in the Fock state decomposition of the projectile state, which also leads to the inelastic process. The light-cone wave function can be easily obtained evaluating the wave function of the physical electron using the time-ordered perturbation theory |eph〉 = exp [−i ∫ 0 −∞ dtĤint(t)]|e〉 (Ĥint is the interaction term in the Hamiltonian). For the no electron spin flip transition (below λ denotes particle helicity) it gives Ψi(x,k) = −i 1 2π √ α 2x [λγ(2− x) + 2λex] kx − iλγky k + κ2i (4) with κ2i = m 2 ex 2 + m2γ,i(1 − x). For the electron spin flip transition the only nonzero component is that with λγ = 2λe Ψi(x,k) = − i 2π √ 2αx me k + κ2i . (5)