Pure Spinor Vertex Operators in Siegel Gauge and Loop Amplitude Regularization

TL;DR

This paper constructs Siegel gauge vertex operators in the pure spinor formalism, addressing the challenge posed by the composite b ghost, and demonstrates a regularized one-loop amplitude calculation without unintegrated operators.

Contribution

It introduces a method to construct Siegel gauge vertex operators using antifield operators and applies a regularization scheme to compute one-loop amplitudes in this gauge.

Findings

Successfully constructed Siegel gauge vertex operators in pure spinor formalism.

Performed the first one-loop amplitude calculation without unintegrated vertex operators.

Validated the regularization scheme for scattering amplitudes involving non-minimal variables.

Abstract

Since the b ghost in the pure spinor formalism is a composite operator depending on non-minimal variables, it is not trivial to impose the Siegel gauge condition b_0 V=0 on BRST-invariant vertex operators. Using the antifield vertex operator V* of ghost-number +2, we show that Siegel gauge unintegrated vertex operators can be constructed as b_0 V* and Siegel gauge integrated vertex operators as \int dz b_{-1} b_0 V*. These Siegel gauge vertex operators depend on the non-minimal variables, so scattering amplitudes involving these operators need to be regularized using the prescription developed previously with Nekrasov. As an example of this regularization prescription, we compute the four-point one-loop amplitude with four Siegel gauge integrated vertex operators. This is the first one-loop computation in the pure spinor formalism that does not require unintegrated vertex operators.