Solving Witten's string field theory using the butterfly state

TL;DR

This work develops a regulated butterfly-state expansion to solve Witten's cubic open string field theory, using a midpoint operator insertion to construct approximate solutions that satisfy the EOM with well-defined energy densities. At leading order, the solution ${|\Psi^{(0)}\rangle = (x/\sqrt{1-t})|B_t(c)\rangle}$ yields ${\mathcal E}/T_{25} \approx -0.6838$, and also satisfies the VSFT equation at this order, hinting at a deep connection between the two frameworks. Incorporating next-to-leading terms, two additional solutions emerge with energies ${\mathcal E}/T_{25} \approx -0.8826$ and ${-1.0898}$, corresponding to roughly ${88\%}$ and ${109\%}$ of the D25-brane tension, which are close to level-2 truncation results without gauge fixing. The approach generalizes to other projectors and to VSFT-like kinetic operators, offering a controlled, regular method to study tachyon condensation and D-brane energetics beyond standard level truncation. Overall, the results support Sen's conjecture within a novel, nonperturbative scheme and suggest fruitful connections to alternative string-field formulations.

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.