Solving Witten's string field theory using the butterfly state

TL;DR

This paper develops a series expansion method using the regulated butterfly state to solve Witten's cubic open string field theory, achieving solutions that approximate D-brane tension and support the vacuum string field theory conjecture.

Contribution

It introduces a novel series expansion approach with the butterfly state to solve string field theory equations beyond level truncation, including next-to-leading order improvements.

Findings

Solution energy density remains finite in the singular limit.

Leading order solution approximates 68% of D25-brane tension.

Next-to-leading order solutions approximate 88% and 109% of the tension.

Abstract

We solve the equation of motion of Witten's cubic open string field theory in a series expansion using the regulated butterfly state. The expansion parameter is given by the regularization parameter of the butterfly state, which can be taken to be arbitrarily small. Unlike the case of level truncation, the equation of motion can be solved for an arbitrary component of the Fock space up to a positive power of the expansion parameter. The energy density of the solution is well-defined and remains finite even in the singular butterfly limit, and it gives approximately 68% of the D25-brane tension for the solution at the leading order. Moreover, it simultaneously solves the equation of motion of vacuum string field theory, providing support for the conjecture at this order. We further improve our ansatz by taking into account next-to-leading terms, and find two numerical solutions which give approximately 88% and 109%, respectively, of the D25-brane tension for the energy density. These values are interestingly close to those by level truncation at level 2 without gauge fixing studied by Rastelli and Zwiebach and by Ellwood and Taylor.