Decoupling of unphysical states in the minimal pure spinor formalism I

TL;DR

This paper investigates the decoupling of unphysical states in the minimal pure spinor formalism, revealing issues with picture changing operators and proposing a Lorentz invariant prescription that ensures unphysical states decouple at all orders.

Contribution

It demonstrates that integrating over constant tensors restores decoupling of unphysical states despite non-BRST-closed picture changing operators.

Findings

Picture changing operators are not BRST closed inside correlators.

Gauge fixing in the minimal formalism is singular, causing Lorentz violation.

Including integrations over constant tensors restores decoupling of unphysical states.

Abstract

This is the first of a series of two papers where decoupling of unphysical states in the minimal pure spinor formalism is investigated. The multi-loop amplitude prescription for the minimal pure spinor superstring formulated in hep-th/0406055 involves the insertion of picture changing operators in the path integral. These operators are BRST closed in a distributional sense and depend on a number of constant tensors. One can trace the origin of these insertions to gauge fixing, so the amplitudes are formally independent of the constant tensors. We show however by explicit tree-level and one-loop computations that the picture changing operators are not BRST closed inside correlators and the amplitudes do depend on these constant tensors. This is due to the fact that the gauge fixing condition implicit in the existing minimal amplitude prescription is singular and this can lead to Lorentz violation and non-decoupling of BRST exact states. As discussed in hep-th/0406055, a manifestly Lorentz invariant prescription can be obtained by integrating over the constant tensors and in the sequel to this paper, it is shown that when one includes these integrations unphysical states do decouple to all orders despite the fact that the PCO's are not BRST closed inside correlators.