# String Scattering Amplitudes and Deformed Cubic String Field Theory

## TL;DR

This paper compares string scattering amplitudes calculated via deformed cubic string field theory and the first quantized approach, demonstrating their equivalence through conformal transformations for various string states.

## Contribution

It shows that the deformed cubic string field theory reproduces the first quantized string scattering amplitudes via conformal mappings, including for non-primary states.

## Key findings

- Amplitudes for highest spin states agree in both methods.
- Conformal invariance holds for primary states.
- Non-primary states require conformal transformation for equivalence.

## Abstract

We study string scattering amplitudes by using the deformed cubic string field theory which is equivalent to the string field theory in the proper-time gauge. The four-string scattering amplitudes with three tachyons and an arbitrary string state are calculated. The string field theory yields the string scattering amplitudes evaluated on the world sheet of string scattering whereas the coventional method, based on the first quantized theory brings us the string scattering amplitudes defined on the upper half plane. For the highest spin states, generated by the primary operators, both calculations are in perfect agreement. In this case, the string scattering amplitudes are invariant under the conformal transformation, which maps the string world sheet onto the upper half plane. If the external string states are general massive states, generated by non-primary field operators, we need to take into account carefully the conformal transformation between the world sheet and the upper half plane. We show by an explicit calculation that the string scattering amplitudes calculated by using the deformed cubic string field theory transform into those of the first quantized theory on the upper half plane by the conformal transformation, generated by the Schwarz-Christoffel mapping.