Complex-mass scheme and perturbative unitarity

For the description of resonances one needs to take into account the finite widths of these states. The complex-mass scheme (CMS) provides with a consistent framework of handling this problem. This approach has proven successful in various applications in the standard model and chiral effective field theory. In the framework of quantum field theory the usage of the CMS leads to complex-valued renormalized parameters. While the problems of unitarity and causality in field theories containing unstable particles have been resolved long ago, the issue of perturbative unitarity of the S-matrix in the CMS is still open. Using the CMS one does not change the bare Lagrangian, therefore unitarity is not violated in the complete theory. On the other hand, the perturbation theory is based on order-by-order approximation to the exact results. Therefore it is not obvious that the approximate expressions to the S-matrix also satisfy the unitarity condition. We derive cutting rules for loop diagrams and demonstrate at the one-loop level perturbative unitarity of the scattering amplitude within the complex-mass scheme. As an application of the CMS we consider the electromagnetic form factor of the pion in the timelike region.