Level truncation and the tachyon in open bosonic string field theory

TL;DR

This work applies the level-truncation program to open bosonic string field theory, extending scalar fields to level $10$ and interactions to level $20$ to study tachyon condensation. By summing planar-tree diagrams with a new recursive scheme, it computes the first $60$ coefficients $c_n$ of the tachyon effective potential and reveals a finite radius of convergence that shrinks with level, while the nonperturbative stable vacuum lies well outside this radius. Nonperturbative analysis shows the stable vacuum energy approaches the D25-brane tension $T_{25} = 1/(2\pi^2 g^2)$ with a $99.91\%$ energy gap at level $(10,20)$, supporting Sen’s conjecture. Branch structure near the singular point suggests a possible global nonnegativity of the effective potential and hints at a deeper, perhaps periodic, structure of the tachyon potential in the full theory. The results demonstrate that level truncation can capture detailed nonperturbative string dynamics and point to future, more ambitious calculations in bosonic and superstring field theories.

Abstract

The tachyonic instability of the open bosonic string is analyzed using the level truncation approach to string field theory. We have calculated all terms in the cubic action of the string field theory describing zero-momentum interactions of up to level 20 between scalars of level 10 or less. These results are used to study the tachyon effective potential and the nonperturbative stable vacuum. We find that the energy gap between the unstable and stable vacua converges much more quickly than the coefficients of the effective tachyon potential. By including fields up to level 10, 99.91% of the energy from the bosonic D-brane tension is cancelled in the nonperturbative stable vacuum. It appears that the perturbative expansion of the effective tachyon potential around the unstable vacuum has a small but finite radius of convergence. We find evidence for a critical point in the tachyon effective potential at a small negative value of the tachyon field corresponding to this radius of convergence. We study the branch structure of the effective potential in the vicinity of this point and speculate that the tachyon effective potential is globally nonnegative.

Figures (4)

• Figure 1: $(\ln | c_n |)/n$ for coefficients $c_n$ in effective tachyon potential in different level truncations
• Figure 2: Expectation values of fields $u = -\psi^3$ and $v = \psi^2/\sqrt{52}$ for $g = 2$ in different level truncations
• Figure 3: Branch 1 of the effective tachyon potential at different truncation levels
• Figure 4: Structure of branches 1, 2, and 3 at different levels