A Note on a Proposal for the Tachyon State in Vacuum String Field Theory

TL;DR

This paper provides a BCFT-based construction of the HK tachyon state in vacuum string field theory, showing that the on-shell condition for the tachyon on a D25-brane arises from the linearized equations of motion. It derives an explicit analytic form for the tachyon state and demonstrates agreement with the oscillator description at the level of current-conservation constraints, reinforcing the HK-state identification. However, the authors show that a naive computation of the D25-brane tension using the on-shell three-tachyon coupling yields ${\cal E}_c/{\cal T}_{25} = \frac{\pi^2}{3} ( {16\over 27 \ln 2} )^3 \approx 2.0558$, which indicates the sliver does not represent two branes and reveals that the HK state does not satisfy the equations of motion in a strong sense. The work emphasizes that the tension calculation is not valid and that a fully consistent tachyon fluctuation remains an open problem, with regularization ambiguities playing a crucial role in correlators of such states.

Abstract

We discuss the proposal of Hata and Kawano for the tachyon fluctuation around a solution of vacuum string field theory representing a D25 brane. We give a conformal field theory construction of their state -- a local insertion of a tachyon vertex operator on the sliver surface state, and explain why the on-shell momentum condition emerges correctly. We also show that a naive computation of the D25-brane tension using data for the three point coupling of this state gives an answer that is $(π^2/3)(16/27\ln 2)^3 \simeq 2.0558$ times the expected answer. We demonstrate that this problem arises because the HK state does not satisfy the equations of motion in a strong sense required for the computation of D-brane tension from the on-shell 3-tachyon coupling.