# Theta Expansion of First Massive Vertex Operator in Pure Spinor

## TL;DR

This paper derives covariant superspace equations for the complete theta expansion of the first massive open string vertex operator in pure spinor formalism, simplifying superfield degrees of freedom and confirming consistency with BRST conditions.

## Contribution

It provides explicit theta expansions of superfields up to third order and demonstrates their use in computing 3-point amplitudes matching RNS results.

## Key findings

- Explicit theta expansion of superfields up to O(θ^3)
- Covariant superspace equations for massive vertex operators
- 3-point amplitude calculation consistent with RNS

## Abstract

We provide the covariant superspace equations that are sufficient to determine the complete $\theta$ expansion of the vertex operator of the open string massive states with $(mass)^2=1/\alpha'$ in pure spinor formalism of superstring theory. These equations get rid of the redundant degrees of freedom in superfields and are consistent with the BRST conditions derived in [1]. Further, we give the explicit $\theta$ expansion of the superfields appearing in the unintegrated vertex to $O(\theta^3)$. Finally, we compute the contribution to a 3-point tree amplitude with the resulting vertex operator upto $O(\theta^3)$ and find its kinematic structure to be identical to the corresponding RNS computation.