Form Factor of the Relativistic Scalar Bound State Calculated in Minkowski Space

where p and k are relative four-momenta of the constituent pair in final and intermediate states, and the free two-particle propagator G0(k, P ) = g(k + P/2,m1)g(k − P/2,m2) is a product of renormalized one body propagators g. The kernel V is a sum of all Bethe-Salpeter irreducible diagrams . For solution, the BSE is usually analytically continued (with the help of the Wick rotation) into the Euclidean space. This avoids the singularities in the kernel and propagators and standard numerical techniques can be employed. An interesting alternative has been developed by Kusaka and Williams . It is based on the Perturbation Theory Integral Representation (PTIR) , i.e., on the fact that any n-point Green function can be expressed as a unique integral over spectral variables. Then, the known structure of the singularities can be factorized and the BSE is cast into the real finite integral equation for the real vertex weight function. This approach is, in principle, applicable also for complicated kernels and/or for the case when the propagators are arbitrarily dressed, which may make the Wick rotation difficult or impossible. However, the derivation of the spectral decomposition is nontrivial and has been so far performed only for kernels induced by cubic type derivative free interaction of scalars.