Cutkosky rules for superstring field theory

Abstract

Superstring field theory expresses the perturbative S-matrix of superstring theory as a sum of Feynman diagrams each of which is manifestly free from ultraviolet divergences. The interaction vertices fall off exponentially for large space-like external momenta making the ultraviolet finiteness property manifest, but blow up exponentially for large time-like external momenta making it impossible to take the integration contours for loop energies to lie along the real axis. This forces us to carry out the integrals over the loop energies by choosing appropriate contours in the complex plane whose ends go to infinity along the imaginary axis but which take complicated form in the interior navigating around the various poles of the propagators. We consider the general class of quantum field theories with this property and prove Cutkosky rules for the amplitudes to all orders in perturbation theory. Besides having applications to string field theory, these results also give an alternative derivation of Cutkosky rules in ordinary quantum field theories.

One loop mass renormalization of unstable particles in superstring theory

“…However in the conventional formulation of string field theory, the vertices grow exponentially for large time-like momenta. Due to this property, while computing Feynman amplitudes by integrating over internal momenta, we cannot take the integral over internal energies along the real axis -the ends of the integration contour have to be tied to ±i∞ [23]. However in the interior of the complex plane the contour has to be deformed appropriately away from the imaginary axis following the algorithm described in [23].…”

“…Due to this property, while computing Feynman amplitudes by integrating over internal momenta, we cannot take the integral over internal energies along the real axis -the ends of the integration contour have to be tied to ±i∞ [23]. However in the interior of the complex plane the contour has to be deformed appropriately away from the imaginary axis following the algorithm described in [23]. With this prescription we get finite results for all loop corrections except where there are physical infrared divergences involving one or more divergent propagators -e.g.…”

“…Therefore this automatically gives the analytically continued result that we would have gotten from the usual world-sheet approach. Another bonus of this approach is that the amplitudes defined this way automatically satisfies the Cutkosky cutting rules [23]. While for general amplitudes one still needs few more steps to prove unitarity from the cutting rules by showing that the contribution to the cut diagrams from unphysical intermediate states cancel, for diagrams involving one loop mass renormalization this can be shown…”

“…If we try to express this as integration over Schwinger parameters, we get back the expression that we have in the world-sheet description, and it is divergent. But we can directly evaluate this Feynman diagram by performing integration over loop momenta following the prescription of [23] and this yields a finite answer. The difference between the two can be traced to the fact that the Schwinger parameter representation of the internal propagators breaks down for certain range of momentum integration.…”

“…for some positive constant A that makes the diagram ultraviolet (UV) finite [23]. In that case the contribution of this diagram to mass 2 of the heavy particle can be expressed as…”