Pure Spinor Vertex Operators in Siegel Gauge and Loop Amplitude Regularization

TL;DR

This work addresses the challenge of enforcing Siegel gauge in the pure spinor superstring, where the $b$-ghost is composite and depends on non-minimal variables. It shows how to construct Siegel-gauge vertex operators from antifields via $V_S=b_0 V^*$ and $U_S=b_{-1} b_0 V^*$, and demonstrates that these operators necessitate BRST-invariant regularization. The authors introduce a new 1-loop prescription that computes $n$-point amplitudes using only integrated Siegel-gauge vertices, and verify it explicitly by evaluating the 4-point massless amplitude, including a detailed treatment of the $(\overline{\lambda}\lambda)\sim0$ regularization. The results advance the practical use of the pure spinor formalism in loop computations and offer a framework for gauge-fixing open superstring field theories in a covariant setting.

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.